Hadronic molecules

TL;DR

This paper reviews the evidence, theoretical frameworks, and methods used to identify and study hadronic molecules, which are bound states of hadrons analogous to light nuclei, especially in the heavy quarkonium sector.

Contribution

It provides a comprehensive overview of experimental evidence, effective field theories, lattice QCD results, and analysis techniques for hadronic molecules, highlighting recent advances and challenges.

Findings

Experimental candidates for hadronic molecules are identified.

Nonrelativistic effective field theories are effective for studying these structures.

Lattice QCD results support the existence and properties of hadronic molecules.

Abstract

A large number of experimental discoveries especially in the heavy quarkonium sector that did not at all fit to the expectations of the until then very successful quark model led to a renaissance of hadron spectroscopy. Among various explanations of the internal structure of these excitations, hadronic molecules, being analogues of light nuclei, play a unique role since for those predictions can be made with controlled uncertainty. We review experimental evidences of various candidates of hadronic molecules, and methods of identifying such structures. Nonrelativistic effective field theories are the suitable framework for studying hadronic molecules, and are discussed in both the continuum and finite volumes. Also pertinent lattice QCD results are presented. Further, we discuss the production mechanisms and decays of hadronic molecules, and comment on the reliability of certain assertions often made in the literature.

Figures

Tables

Table 1. Table 1: Mesons that contain at most one heavy quark that cannot be easily accommodated in the q ​ q ¯ 𝑞 ¯ 𝑞 q\bar{q} quark model. Their quantum numbers I G ​ ( J P ​ C ) superscript 𝐼 𝐺 superscript 𝐽 𝑃 𝐶 I^{G}(J^{PC}) , masses, widths, the nearby S 𝑆 S -wave thresholds, m threshold subscript 𝑚 threshold m_{\text{threshold}} , where we add in brackets M − m threshold 𝑀 subscript 𝑚 threshold M-m_{\text{threshold}} , and the observed decay modes are listed in order. The data without references are taken from the 2016 edition of the Review of Particle Physics  Patrignani  et al. ( 2016 ) .

State	$I^G\left(J^{PC}\right)$	$M\left[\mathrm{MeV}\right]$	$\Gamma \left[\mathrm{MeV}\right]$	$S$-wave threshold(s) [MeV]	Decay mode(s) [branching ratio(s)]
$f_0\left(500\right)$ Peláez (2016) 111The mass and width are derived from the pole position quoted in Ref. Peláez (2016) via $\sqrt{s_p}=M-i\Gamma /2$.	$0^{+}\left(0^{++}\right)$	${449}_{-16}^{+22}$	$550\pm 24$	$\pi \pi \left({173}_{-16}^{+22}\right)$	$\pi \pi$ [dominant]
					$\gamma \gamma$
$\kappa \left(800\right)$	$\frac{1}{2}\left(0^{+}\right)$	$682\pm 29$	$547\pm 24$	$K\pi \left(48\pm 29\right)$	$\pi K$
$f_0\left(980\right)$	$0^{+}\left(0^{++}\right)$	$990\pm 20$	$10\sim 100$	$K^{+}{K}^{-}\left(3\pm 20\right)$	$\pi \pi$ [dominant]
				$K^0{\overline{K}}^0\left(-5\pm 20\right)$	$K\overline{K}$
					$\gamma \gamma$
$a_0\left(980\right)$	$1^{-}\left(0^{++}\right)$	$980\pm 20$	$50\sim 100$	$K\overline{K}\left(-11\pm 20\right)$	$\eta \pi$ [dominant]
					$K\overline{K}$
					$\gamma \gamma$
$f_1\left(1420\right)$	$0^{+}\left(1^{++}\right)$	$1426.4\pm 0.9$	$54.9\pm 2.6$	$K{\overline{K}}^{\ast }\left(39.1\pm 0.9\right)$	$K{\overline{K}}^{\ast }$(dominant)
					$\eta \pi \pi$ [possibly seen]
					$\varphi \gamma$
$a_1\left(1420\right)$	$1^{-}\left(1^{++}\right)$	${1414}_{-13}^{+15}$	${153}_{-23}^{+8}$	$K{\overline{K}}^{\ast }({27}_{-13}^{+15}$)	$f_0\left(980\right)\pi$ [seen]
$X\left(1835\right)$	${?}^{?}\left(0^{-+}\right)$	${1835.8}_{-3.2}^{+4.0}$	$112\pm 40$	$p\overline{p}\left(-{40.7}_{-3.2}^{+4.0}\right)$	$p\overline{p}$
					${\eta }^{\prime }\pi \pi$
					$K_S^0{K}_S^0\eta$
$D_{s0}^{\ast }{\left(2317\right)}^{+}$	$0\left(0^{+}\right)$	$2317.7\pm 0.6$		$DK\left(-45.1\pm 0.6\right)$	$D_s^{+}{\pi }^0$
$D_{s1}{\left(2460\right)}^{+}$	$0\left(1^{+}\right)$	$2459.5\pm 0.6$		$D^{\ast }K\left(-44.7\pm 0.6\right)$	$D_s^{\ast +}{\pi }^0\left[\left(48\pm 11\right)\%\right]$
					$D_s^{+}\gamma \left[\left(18\pm 4\right)\%\right]$
					$D_s^{+}{\pi }^{+}{\pi }^{-}\left[\left(4\pm 1\right)\%\right]$
					$D_{s0}^{\ast }{\left(2317\right)}^{+}\gamma \left[\left(4_{-2}^{+5}\right)\%\right]$
$D_{s1}^{\ast }{\left(2860\right)}^{+}$	$0\left(1^{-}\right)$	$2859\pm 27$	$159\pm 80$	$D_1\left(2420\right)K\left(-59\pm 27\right)$	$DK$
					$D^{\ast }K$

Table 2. Table 2: Same as Table  1 but in the charmonium and bottomonium sectors. A blank in the fifth column means that there is no relevant nearby S 𝑆 S -wave threshold.

State	$I^G\left(J^{PC}\right)$	$M\left[\mathrm{MeV}\right]$	$\Gamma \left[\mathrm{MeV}\right]$	$S$-wave threshold(s) [$\mathrm{MeV}$]	Observed mode(s) (branching ratios)
$X\left(3872\right)$	$0^{+}\left(1^{++}\right)$	$3871.69\pm 0.17$		$D^{\ast +}{D}^{-}+c.c.\left(-8.15\pm 0.20\right)$	$B\to K\left[{\overline{D}}^{\ast 0}{D}^0\right]\left(>24\%\right)$
				$D^{\ast 0}{\overline{D}}^0+c.c.\left(0.00\pm 0.18\right)$	$B\to K\left[D^0{\overline{D}}^0{\pi }^0\right]\left(>32\%\right)$
					$B\to K\left[J/\psi {\pi }^{+}{\pi }^{-}\right]\left(>2.6\%\right)$
					$B\to K\left[J/\psi {\pi }^{+}{\pi }^{-}{\pi }^0\right]$
					$p\overline{p}\to \left[J/\psi {\pi }^{+}{\pi }^{-}\right]\dots$
					$pp\to \left[J/\psi {\pi }^{+}{\pi }^{-}\right]\dots$
					$B\to K\left[J/\psi \omega \right]\left(>1.9\%\right)$
					$B\to \left[J/\psi \gamma \right]\left(>6\times {10}^{-3}\right)$
					$B\to \left[\psi \left(2S\right)\gamma \right]\left(>3.0\%\right)$
$X\left(3940\right)$	${?}^{?}\left({?}^{??}\right)$	$3942.0\pm 9$	${37}_{-17}^{+27}$	$D^{\ast }{\overline{D}}^{\ast }\left(-75.1\pm 9\right)$	$e^{+}{e}^{-}\to J/\psi \left[D{\overline{D}}^{\ast }\right]$
$X\left(4160\right)$	${?}^{?}\left({?}^{??}\right)$	${4156}_{-25}^{+29}$	${139}_{-60}^{+110}$	$D^{\ast }{\overline{D}}^{\ast }\left({139}_{-25}^{+29}\right)$	$e^{+}{e}^{-}\to J/\psi \left[D^{\ast }{\overline{D}}^{\ast }\right]$
$Z_c\left(3900\right)$	$1^{+}\left(1^{+-}\right)$	$3886.6\pm 2.4$	$28.1\pm 2.6$	$D^{\ast }\overline{D}\left(10.8\pm 2.4\right)$	$e^{+}{e}^{-}\to \pi [D{\overline{D}}^{\ast }+c.c.]$
					$e^{+}{e}^{-}\to \pi \left[J/\psi \pi \right]$
$Z_c\left(4020\right)$	$1\left({?}^{?}\right)$	$4024.1\pm 1.9$	$13\pm 5$	$D^{\ast }{\overline{D}}^{\ast }\left(7.0\pm 2.4\right)$	$e^{+}{e}^{-}\to \pi \left[D^{\ast }{\overline{D}}^{\ast }\right]$
					$e^{+}{e}^{-}\to \pi \left[h_c\pi \right]$
					$e^{+}{e}^{-}\to \pi \left[{\psi }^{\prime }\pi \right]$
$Y\left(4260\right)$	${?}^{?}\left(1^{--}\right)$	$4251\pm 9$	$120\pm 12$	$D_1\overline{D}+c.c.\left(-38.2\pm 9.1\right)$	$e^{+}{e}^{-}\to J/\psi \pi \pi$
				${\chi }_{c0}\omega \left(53.6\pm 9.0\right)$	$e^{+}{e}^{-}\to \pi D{\overline{D}}^{\ast }+c.c.$
					$e^{+}{e}^{-}\to {\chi }_{c0}\omega$
					$e^{+}{e}^{-}\to X\left(3872\right)\gamma$
$Y\left(4360\right)$	${?}^{?}\left(1^{--}\right)$	$4346\pm 6$	$102\pm 10$	$D_1{\overline{D}}^{\ast }+c.c.\left(-85\pm 6\right)$	$e^{+}{e}^{-}\to \psi \left(2S\right){\pi }^{+}{\pi }^{-}$
$Y\left(4660\right)$	${?}^{?}\left(1^{--}\right)$	$4643\pm 9$	$72\pm 11$	$\psi \left(2S\right)f_0\left(980\right)\left(-33\pm 21\right)$	$e^{+}{e}^{-}\to \psi \left(2S\right){\pi }^{+}{\pi }^{-}$
				${\Lambda }_c^{+}{\Lambda }_c^{-}\left(70\pm 6\right)$
$Z_c{\left(4430\right)}^{+}$	$?\left(1^{+}\right)$	${4478}_{-18}^{+15}$	$181\pm 31$	$\psi \left(2S\right)\rho \left({17}_{-18}^{+15}\right)$	$B\to K\left[\psi \left(2S\right){\pi }^{+}\right]$
					$B\to K\left[J/\psi {\pi }^{+}\right]$
$Z_c{\left(4200\right)}^{+}$	$?\left(1^{+}\right)$	${4196}_{-32}^{+35}$	${370}_{-32}^{+100}$		${\overline{B}}^0\to K^{-}\left[J/\psi {\pi }^{+}\right]$
$Z_c{\left(4050\right)}^{+}$	$?\left({?}^{?}\right)$	${4051}_{-40}^{+24}$	${82}_{-28}^{+50}$	$D^{\ast }{\overline{D}}^{\ast }\left({34}_{-40}^{+24}\right)$	${\overline{B}}^0\to K^{-}\left[{\chi }_{c1}{\pi }^{+}\right]$
$Z_c{\left(4250\right)}^{+}$	$?\left({?}^{?}\right)$	${4248}_{-50}^{+190}$	${177}_{-70}^{+320}$	${\chi }_{c1}\rho \left(-{37}_{-50}^{+24}\right)$	${\overline{B}}^0\to K^{-}\left[{\chi }_{c1}{\pi }^{+}\right]$
$X\left(4140\right)$Aaij et al. (2017a, b)	$0^{+}\left(1^{++}\right)$	$4146.5\pm {4.5}_{-2.8}^{+4.6}$	$83\pm {21}_{-14}^{+21}$	$D_s{\overline{D}}_s^{\ast }\left(-{66.1}_{-3.2}^{+4.9}\right)$	$B^{+}\to K^{+}\left[J/\psi \varphi \right]$
$X\left(4274\right)$Aaij et al. (2017a, b)	$0^{+}\left(1^{++}\right)$	$4273.3\pm {8.3}_{-3.6}^{+17.2}$	$56\pm {11}_{-11}^{+8}$	$D_s^{\ast }{\overline{D}}_s^{\ast }\left(-{49.1}_{-9.1}^{+19.1}\right)$	$B^{+}\to K^{+}\left[J/\psi \varphi \right]$
$X\left(4500\right)$Aaij et al. (2017a, b)	$0^{+}\left(0^{++}\right)$	$4506\pm {11}_{-15}^{+12}$	$92\pm {21}_{-20}^{+21}$	$D_{s0}^{\ast }\left(2317\right){\overline{D}}_{s0}^{\ast }\left(2317\right)\left(-{129}_{-19}^{+16}\right)$	$B^{+}\to K^{+}\left[J/\psi \varphi \right]$
$X\left(4700\right)$Aaij et al. (2017a, b)	$0^{+}\left(0^{++}\right)$	$4704\pm {10}_{-24}^{+14}$	$120\pm {31}_{-33}^{+42}$	$D_{s0}^{\ast }\left(2317\right){\overline{D}}_{s0}^{\ast }\left(2317\right)\left({69}_{-26}^{+17}\right)$	$B^{+}\to K^{+}\left[J/\psi \varphi \right]$
$Z_b\left(10610\right)$	$1^{+}\left(1^{+}\right)$	$10607.2\pm 2.0$	$18.4\pm 2.4$	$B{\overline{B}}^{\ast }+c.c.\left(4.0\pm 3.2\right)$	${\rm Y}\left(10860\right)\to \pi [B{\overline{B}}^{\ast }+c.c.]$
					${\rm Y}\left(10860\right)\to \pi \left[{\rm Y}\left(1S\right)\pi \right]$
					${\rm Y}\left(10860\right)\to \pi \left[{\rm Y}\left(2S\right)\pi \right]$
					${\rm Y}\left(10860\right)\to \pi \left[{\rm Y}\left(3S\right)\pi \right]$
					${\rm Y}\left(10860\right)\to \pi \left[h_b\left(1P\right)\pi \right]$
					${\rm Y}\left(10860\right)\to \pi \left[h_b\left(2P\right)\pi \right]$
$Z_b\left(10650\right)$	$1^{+}\left(1^{+}\right)$	$10652.2\pm 1.5$	$11.5\pm 2.2$	$B^{\ast }{\overline{B}}^{\ast }\left(2.9\pm 1.5\right)$	${\rm Y}\left(10860\right)\to \pi \left[B^{\ast }{\overline{B}}^{\ast }\right]$
					${\rm Y}\left(10860\right)\to \pi \left[{\rm Y}\left(1S\right)\pi \right]$
					${\rm Y}\left(10860\right)\to \pi \left[{\rm Y}\left(2S\right)\pi \right]$
					${\rm Y}\left(10860\right)\to \pi \left[{\rm Y}\left(3S\right)\pi \right]$
					${\rm Y}\left(10860\right)\to \pi \left[h_b\left(1P\right)\pi \right]$
					${\rm Y}\left(10860\right)\to \pi \left[h_b\left(2P\right)\pi \right]$

Table 3. Table 3: The reported branching fractions of the known decay modes of Z b ​ ( 10610 ) + subscript 𝑍 𝑏 superscript 10610 Z_{b}(10610)^{+} and Z b ​ ( 10650 ) + subscript 𝑍 𝑏 superscript 10650 Z_{b}(10650)^{+}   Garmash  et al. ( 2016 ) with the statistical and systematical uncertainties in order.

channel	$ℬ$ of $Z_b\left(10610\right)$ ($\%$)	$ℬ$ of $Z_b\left(10650\right)$$(\%)$
${\rm Y}\left(1S\right){\pi }^{+}$	${0.54}_{-0.13-0.08}^{+0.16+0.11}$	${0.17}_{-0.06-0.02}^{+0.07+0.03}$
${\rm Y}\left(2S\right){\pi }^{+}$	${3.62}_{-0.59-0.53}^{+0.76+0.79}$	${1.39}_{-0.38-0.23}^{+0.48+0.34}$
${\rm Y}\left(3S\right){\pi }^{+}$	${2.15}_{-0.42-0.43}^{+0.55+0.60}$	${1.63}_{-0.42-0.28}^{+0.53+0.39}$
$h_b\left(1P\right){\pi }^{+}$	${3.45}_{-0.71-0.63}^{+0.87+0.86}$	${8.41}_{-2.12-1.06}^{+2.43+1.49}$
$h_b\left(2P\right){\pi }^{+}$	${4.67}_{-1.00-0.89}^{+1.24+1.18}$	${14.7}_{-2.8-2.3}^{+3.2+2.8}$
$B^{+}{\overline{B}}^{\ast 0}+{\overline{B}}^0{B}^{\ast +}$	${85.6}_{-2.0-2.1}^{+1.5+1.5}$	–
$B^{\ast +}{\overline{B}}^{\ast 0}$	–	${73.7}_{-4.4-3.5}^{+3.4+2.7}$

Table 4. Table 4: Same as Table  1 but in the baryon sector.

state	$I\left(J^P\right)$	$M\left[\mathrm{MeV}\right]$	$\Gamma \left[\mathrm{MeV}\right]$	$S$-wave threshold(s) [$\mathrm{MeV}$]	Observed mode(s) (branching ratios)
$\Lambda \left(1405\right)$	$0\left({\frac{1}{2}}^{-}\right)$	${1405.1}_{-1.0}^{+1.3}$	$50.5\pm 2.0$	$N\overline{K}\left(-{29.4}_{-1.0}^{+1.3}\right)$	$\Sigma \pi \left(100\%\right)$
				$\Sigma \pi \left({76.2}_{-1.0}^{+1.3}\right)$
$\Lambda \left(1520\right)$	$0\left({\frac{3}{2}}^{-}\right)$	$1519.5\pm 1.0$	$15.6\pm 1.0$	$\Sigma {\left(1385\right)}^{-}{\pi }^{+}\left(-7.3\pm 1.1\right)$	$N\overline{K}\left(45\pm 1\right)\%$
				$\Sigma {\left(1385\right)}^{+}{\pi }^{-}\left(-2.9\pm 1.1\right)$	$\Sigma \pi \left(42\pm 1\right)\%$
				$\Sigma {\left(1385\right)}^0{\pi }^0\left(0.8\pm 1.4\right)$	$\Lambda \pi \pi \left(10\pm 1\right)\%$
$\Lambda \left(1670\right)$	$0\left({\frac{1}{2}}^{-}\right)$	$\approx 1670$	$\approx 35$	$\Lambda \eta \left(4\right)$	$N\overline{K}\left(20\sim 30\right)\%$
					$\Sigma \pi \left(25\sim 55\right)\%$
					$\Lambda \eta \left(10\sim 25\right)\%$
${\Lambda }_c\left(2595\right)$	$0\left({\frac{1}{2}}^{-}\right)$	$2592.25\pm 0.28$	$2.6\pm 0.6$	${\Sigma }_c{\left(2455\right)}^{++}{\pi }^{-}\left(-1.04\pm 0.31\right)$	${\Sigma }_c{\left(2455\right)}^{++}{\pi }^{-}\left(24\pm 7\right)\%$
				${\Sigma }_c{\left(2455\right)}^0{\pi }^{+}\left(-0.82\pm 0.31\right)$	${\Sigma }_c{\left(2455\right)}^0{\pi }^{+}\left(24\pm 7\right)\%$
				${\Sigma }_c{\left(2455\right)}^{+}{\pi }^0\left(4.62\pm 0.49\right)$	${\Lambda }_c^{+}{\pi }^{+}{\pi }^{-}\text{3-body}\left(18\pm 10\right)\%$
${\Lambda }_c\left(2625\right)$	$0\left({\frac{3}{2}}^{-}\right)$	$2628.11\pm 0.19$		${\Sigma }_c\left(2455\right)\pi \left(36.53\pm 0.24\right)$	${\Lambda }_c^{+}{\pi }^{+}{\pi }^{-}\left(67\%\right)$
${\Lambda }_c\left(2880\right)$	$0\left({\frac{5}{2}}^{+}\right)$	$2881.53\pm 0.35$	$5.8\pm 1.1$	$N{D}^{\ast }\left(-65.91\pm 0.35\right)$	${\Lambda }_c^{+}{\pi }^{+}{\pi }^{-}$
					${\Sigma }_c{\left(2455\right)}^{0,++}{\pi }^{\pm }$
					${\Sigma }_c{\left(2520\right)}^{0,++}{\pi }^{\pm }$
					$p{D}^0$
${\Lambda }_c\left(2940\right)$	$0\left({\frac{3}{2}}^{-}\right)$	${2939.3}_{-1.5}^{+1.4}$	${17}_{-6}^{+8}$	$N{D}^{\ast }\left(-{8.1}_{-1.5}^{+1.4}\right)$	${\Sigma }_c{\left(2455\right)}^{0,++}{\pi }^{\pm }$
	Aaij et al. (2017d)				$p{D}^0$
${\Sigma }_c\left(2800\right)$	$1\left({?}^{?}\right)$	${2800}_{-4}^{+5}$	${70}_{-15}^{+23}$	$ND\left(-6\pm 5\right)$	${\Lambda }_c^{+}\pi$
${\Xi }_c\left(2970\right)$	$\frac{1}{2}\left({?}^{?}\right)$	$2969.4\pm 1.7$	$19.0\pm 3.9$	${\Sigma }_c\left(2455\right)K\left(20.2\pm 1.7\right)$	${\Lambda }_c^{+}\overline{K}\pi$
					${\Sigma }_c\left(2455\right)\overline{K}$
					${\Xi }_{c}2\pi$
					${\Xi }_c\left(2645\right)\pi$
${\Xi }_c\left(3055\right)$	$?\left({?}^{?}\right)$	$3055.1\pm 1.7$	$11\pm 4$	${\Sigma }_c\left(2520\right)K\left(41.1\pm 1.7\right)$
				${\Xi }_c\left(2970\right)\pi \left(-52.3\pm 2.4\right)$
${\Xi }_c\left(3080\right)$	$\frac{1}{2}\left({?}^{?}\right)$	$3078.4\pm 0.7$	$5.0\pm 1.3$	${\Sigma }_c\left(2520\right)K\left(64.4\pm 0.7\right)$	${\Lambda }_c^{+}\overline{K}\pi$
				${\Xi }_c\left(2970\right)\pi \left(-29.0\pm 1.8\right)$	${\Sigma }_c\left(2455\right)\overline{K}$
					${\Sigma }_c\left(2520\right)\overline{K}$
$P_c\left(4380\right)$	$\frac{1}{2}\left({\frac{3}{2}}^{?}/{\frac{5}{2}}^{?}\right)$	$4380\pm 8\pm 29$	$205\pm 18\pm 86$	${\Sigma }_c\left(2520\right)\overline{D}$ ($-6\pm 30$)	$J/\psi p$
Aaij et al. (2015b)				${\Sigma }_c\left(2455\right){\overline{D}}^{\ast }$ $\left(-82\pm 30\right)$
$P_c\left(4450\right)$	$\frac{1}{2}\left({\frac{3}{2}}^{?}/{\frac{5}{2}}^{?}\right)$	$4449.8\pm 1.7\pm 2.5$	$39\pm 5\pm 19$	${\chi }_{c1}p$ $\left(0.9\pm 3.0\right)$	$J/\psi p$
Aaij et al. (2015b)				${\Lambda }_c\left(2595\right)\overline{D}$ $\left(-9.9\pm 3.0\right)$
				${\Sigma }_c\left(2520\right){\overline{D}}^{\ast }$ ($-77.2\pm 3.0$)
				${\Sigma }_c\left(2520\right)\overline{D}$ ($64.2\pm 3.0$)
				${\Sigma }_c\left(2455\right){\overline{D}}^{\ast }$ ($-12.3\pm 3.0$)

Table 5. Table 5: Predictions of the partners of the X ​ ( 3872 ) 𝑋 3872 X(3872) for Λ = 0.5 Λ 0.5 \Lambda=0.5  GeV in Ref.  Guo  et al. ( 2013b ) .

$J^{PC}$	States	Thresholds (MeV)	Masses (MeV)
$1^{++}$	$\frac{1}{\sqrt{2}}\left(D{\overline{D}}^{\ast }+D^{\ast }\overline{D}\right)$	3875.87	3871.68 (input)
$2^{++}$	$D^{\ast }{\overline{D}}^{\ast }$	4017.3	${4012}_{-5}^{+4}$
$1^{++}$	$\frac{1}{\sqrt{2}}\left(B{\overline{B}}^{\ast }+B^{\ast }\overline{B}\right)$	10604.4	${10580}_{-8}^{+9}$
$2^{++}$	$B^{\ast }{\overline{B}}^{\ast }$	10650.2	${10626}_{-9}^{+8}$
$2^{+}$	$D^{\ast }{B}^{\ast }$	7333.7	${7322}_{-7}^{+6}$

Table 6. Table 6: The diagonal contact terms for the S 𝑆 S -wave interaction between a pair of j ℓ P = 1 / 2 − superscript subscript 𝑗 ℓ 𝑃 1 superscript 2 j_{\ell}^{P}=1/2^{-} and 3 / 2 + 3 superscript 2 3/2^{+} heavy and anti-heavy mesons.

$J^{PC}$	Meson pairs	Contact terms
$1^{--}$	$\frac{1}{\sqrt{2}}\left(D{\overline{D}}_1-D_1\overline{D}\right)$	$\frac{1}{8}\left(-F_{I1}^c-5F_{I2}^c+3F_{I1}^d+5F_{I2}^d\right)$
	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_1+D_1{\overline{D}}^{\ast }\right)$	$\frac{1}{16}\left(7F_{I1}^c-5F_{I2}^c+11F_{I1}^d+5F_{I2}^d\right)$
	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_2-D_2{\overline{D}}^{\ast }\right)$	$\frac{1}{16}\left(-5F_{I1}^c-F_{I2}^c+15F_{I1}^d+F_{I2}^d\right)$
$0^{--}$	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_1-D_1{\overline{D}}^{\ast }\right)$	$F_{I1}^c+F_{I1}^d$
$2^{--}$	$\frac{1}{\sqrt{2}}\left(D{\overline{D}}_2-D_2\overline{D}\right)$	$\frac{1}{8}\left(3F_{I1}^c-F_{I2}^c+3F_{I1}^d+5F_{I2}^d\right)$
	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_1-D_1{\overline{D}}^{\ast }\right)$	$\frac{1}{16}\left(F_{I1}^c-3F_{I2}^c+F_{I1}^d+15F_{I2}^d\right)$
	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_2+D_2{\overline{D}}^{\ast }\right)$	$\frac{1}{16}\left(9F_{I1}^c+5F_{I2}^c+9F_{I1}^d+7F_{I2}^d\right)$
$3^{--}$	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_2-D_2{\overline{D}}^{\ast }\right)$	$F_{I2}^d-F_{I2}^c$
$0^{-+}$	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_1+D_1{\overline{D}}^{\ast }\right)$	$F_{I1}^d-F_{I1}^c$
$1^{-+}$	$\frac{1}{\sqrt{2}}\left(D{\overline{D}}_1+D_1\overline{D}\right)$	$\frac{1}{8}\left[5\left(F_{I2}^c+F_{I2}^d\right)+F_{I1}^c+3F_{I1}^d\right]$
	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_1-D_1{\overline{D}}^{\ast }\right)$	$\frac{1}{16}\left[5\left(F_{I2}^c+F_{I2}^d\right)-7F_{I1}^c+11F_{I1}^d\right]$
	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_2+D_2{\overline{D}}^{\ast }\right)$	$\frac{1}{16}\left(5F_{I1}^c+F_{I2}^c+15F_{I1}^d+F_{I2}^d\right)$
$2^{-+}$	$\frac{1}{\sqrt{2}}\left(D{\overline{D}}_2+D_2\overline{D}\right)$	$\frac{1}{8}\left(-3F_{I1}^c+F_{I2}^c+3F_{I1}^d+5F_{I2}^d\right)$
	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_1+D_1{\overline{D}}^{\ast }\right)$	$\frac{1}{16}\left[3\left(F_{I2}^c+5F_{I2}^d\right)-F_{I1}^c+F_{I1}^d\right]$
	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_2-D_2{\overline{D}}^{\ast }\right)$	$\frac{1}{16}\left(-9F_{I1}^c-5F_{I2}^c+9F_{I1}^d+7F_{I2}^d\right)$
$3^{-+}$	$\frac{1}{\sqrt{2}}\left(D^{\ast }{\overline{D}}_2+D_2{\overline{D}}^{\ast }\right)$	$F_{I2}^c+F_{I2}^d$

Table 7. Table 7: The triangle loops [ M 1 ​ M 2 ​ M 3 ] delimited-[] subscript 𝑀 1 subscript 𝑀 2 subscript 𝑀 3 [M_{1}M_{2}M_{3}] corresponding to Fig.  11 which have shown large impact on the production of hadronic molecules and conventional hadrons in experiment are listed in the first column. The second column is the measured process with the final states in the bracket. The checkmarks in the last column indicate that the triangle singularity of the corresponding process locates in the physical region, i.e. satisfying Eq.( 49 ). Although the singularity of the process without check mark is not located in the physical region, since it is not far away, it can still enhance the corresponding production rate significantly.

$\left[M_1{M}_2{M}_3\right]$	$A\to BC\left(\to \text{final states}\right)$	Eq.(49)
$\left[K^{\ast }KK\right]$	$\eta \left(1405/1475\right)\to a_0\left(980\right)\pi \left(\to 3\pi \right)$ Wu et al. (2012, 2013)	$✓$
	$a_1\left(1420\right)\to f_0\left(980\right)\pi \left(\to \eta \pi \pi \right)$ Mikhasenko et al. (2015); Liu et al. (2016a)	$✓$
$\left[D_{1}D{D}^{\ast }\right]$	$Y\left(4260\right)\to X\left(3872\right)\gamma$  Guo et al. (2013a)
	$Y\left(4260\right)\to Z_c\left(3900\right)\pi$ Wang et al. (2013a, b); Szczepaniak (2015); Gong et al. (2016b)
$\left[\Lambda \left(1890\right){\chi }_{c1}p\right]$	${\Lambda }_b\to P_c\left(4450\right)K$ Guo et al. (2015c); Liu et al. (2016b); Bayar et al. (2016)	$✓$
$\left[D_{s3}\left(2860\right){\Lambda }_c\left(2595\right)D\right]$	${\Lambda }_b\to P_c\left(4450\right)K$ Liu et al. (2016b)

Table 8. Table 8: Integrated cross sections (in units of nb) reported in  Albaladejo  et al. ( 2017b ) for the inclusive p ​ p / p ¯ → X ​ ( 3872 ) → 𝑝 𝑝 ¯ 𝑝 𝑋 3872 pp/\bar{p}\to X(3872) processes in comparison with the CDF  Bauer ( 2005 ) and CMS  Chatrchyan  et al. ( 2013a ) data converted into cross sections  Guo  et al. ( 2014e ) . The ranges of the results cover those obtained using both Pythia and Herwig. Here, we have converted the experimental data into cross sections  Guo  et al. ( 2014e ) .

$\sigma \left(pp/\overline{p}\to X\right)$	$\Lambda =0.1$ GeV	$\Lambda =[0.5,1]$ GeV	Experiment
Tevatron	0.05-0.07	5-29	37-115
LHC 7	0.04-0.12	4-55	13-39

Table 9. Table 9: Possible spin and flavor partners of heavy-flavor hadronic molecules. For the experimentally established states, the masses and decay modes are taken from  Patrignani  et al. ( 2016 ) . The predicted partners are denoted by question marks. The predictions from  Guo  et al. ( 2013b ) are those computed with a 0.5 GeV cutoff, and the result from  Baru  et al. ( 2016 ) is taken from the results with the cutoff limited between 0.8 and 1.0 GeV.

$J^{P\left(C\right)}$	State	Main component	Mass (MeV)	(Expected) main decay mode(s)
$0^{+}$	$D_{s0}^{\ast }\left(2317\right)$	$DK$	$2317.7\pm 0.6$	$D_s^{+}{\pi }^0$
$1^{+}$	$D_{s1}\left(2460\right)$	$D^{\ast }K$	$2459.5\pm 0.6$	$D_s^{\ast +}{\pi }^0,D_s^{(\ast )+}\gamma$
$0^{+}$	$B_{s0}^{\ast }\left(?\right)$	$B\overline{K}$	$5730\pm 16$	$B_s^{\ast 0}\gamma ,B_s^0{\pi }^0$
$1^{+}$	$B_{s1}\left(?\right)$	$B^{\ast }\overline{K}$	$5776\pm 16$	$B_s^{(\ast )0}\gamma ,B_s^{\ast 0}{\pi }^0$
$1^{-}$	$D_{s1}^{\ast }\left(2860\right)$	$D_1\left(2420\right)K$	$2859\pm 27$	$DK$, $D^{\ast }K$
$2^{-}$	$D_{s2}^{\ast }\left(?\right)$	$D_2\left(2460\right)K$	$2910\pm 9$ Guo and Meißner (2011)	$D^{\ast }K,D_s^{\ast }\eta$
$1^{-}$	$B_{s1}^{\ast }\left(?\right)$	$B_1\left(5720\right)\overline{K}$	$6151\pm 33$ Guo and Meißner (2011)	$B^{(\ast )}\overline{K}$, $B_s^{(\ast )}\eta$
$2^{-}$	$B_{s2}^{\ast }\left(?\right)$	$B_2\left(5747\right)\overline{K}$	$6169\pm 33$ Guo and Meißner (2011)	$B^{\ast }\overline{K}$, $B_s^{\ast }\eta$
$1^{++}$	$X\left(3872\right)$	$D{\overline{D}}^{\ast }$	$3871.69\pm 0.17$	$D^0{\overline{D}}^0{\pi }^0$, $J/\psi \pi \pi$, $J/\psi \pi \pi \pi$
$2^{++}$	$X_2\left(?\right)$	$D^{\ast }{\overline{D}}^{\ast }$	${4012}_{-5}^{+4}$ Guo et al. (2013b)	$D{\overline{D}}^{(\ast )},J/\psi \omega$
			$3980\pm 20$ Baru et al. (2016)
$1^{++}$	$X_b\left(?\right)$	$B{\overline{B}}^{\ast }$	${10580}_{-8}^{+9}$ Guo et al. (2013b)	${\rm Y}\left(nS\right)\omega ,{\chi }_{bJ}\pi \pi$
$2^{++}$	$X_{b2}\left(?\right)$	$B^{\ast }{\overline{B}}^{\ast }$	${10626}_{-9}^{+8}$ Guo et al. (2013b)	$B{\overline{B}}^{(\ast )},{\rm Y}\left(nS\right)\omega ,{\chi }_{bJ}\pi \pi$
$2^{+}$	$X_{bc}\left(?\right)$	$D^{\ast }{B}^{\ast }$	${7322}_{-7}^{+6}$ Guo et al. (2013b)	$DB,D{B}^{\ast },D^{\ast }B$

Equations

$$M_{D^0}+M_{D^{\ast 0}}-M_{X(3872)}=(0.00\pm 0.18)\mathrm{MeV},$$

$$\mathcal{B}(X(3872)\to {\bar{D}}^0{D}^0{\pi }^0)>32\%,$$

$$R^I\equiv \frac{\mathcal{B}\left(X(3872)\to J/\psi {\pi }^{+}{\pi }^{-}{\pi }^0\right)}{\mathcal{B}\left(X(3872)\to J/\psi {\pi }^{+}{\pi }^{-}\right)}$$

$$\frac{\mathcal{B}\left(X(3872)\to {\psi }^{\prime }\gamma \right)}{\mathcal{B}\left(X(3872)\to J/\psi \gamma \right)}=2.46\pm 0.64\pm 0.29.$$

$$\frac{\mathcal{B}(B^0\to X(3872)K^0\to J/\psi {\pi }^{+}{\pi }^{-}{K}^0)}{\mathcal{B}(B^{-}\to X(3872)K^{-}\to J/\psi {\pi }^{+}{\pi }^{-}{K}^{-})}$$

$$\displaystyle B_{6}=\left(\begin{array}[]{ccc}\Sigma_{c}(2455)^{++}&\frac{1}{\sqrt{2}}\Sigma_{c}(2455)^{+}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime+}\\ \frac{1}{\sqrt{2}}\Sigma_{c}(2455)^{+}&\Sigma_{c}(2455)^{0}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime 0}\\ \frac{1}{\sqrt{2}}\Xi_{c}^{\prime+}&\frac{1}{\sqrt{2}}\Xi_{c}^{\prime 0}&\Omega_{c}^{0}\end{array}\right),$$

$$\displaystyle B_{6}^{*}=\left(\begin{array}[]{ccc}\Sigma_{c}(2520)^{++}&\frac{1}{\sqrt{2}}\Sigma_{c}(2520)^{+}&\frac{1}{\sqrt{2}}\Xi_{c}(2645)^{+}\\ \frac{1}{\sqrt{2}}\Sigma_{c}(2520)^{+}&\Sigma_{c}(2520)^{0}&\frac{1}{\sqrt{2}}\Xi_{c}(2645)^{0}\\ \frac{1}{\sqrt{2}}\Xi_{c}(2645)^{+}&\frac{1}{\sqrt{2}}\Xi_{c}(2645)^{0}&\Omega_{c}(2770)^{0}\end{array}\right),$$

$$\displaystyle B_{\bar{3}}=\left(\begin{array}[]{ccc}0&\Lambda_{c}^{+}&\Xi_{c}^{+}\\ -\Lambda_{c}^{+}&0&\Xi_{c}^{0}\\ -\Xi_{c}^{+}&-\Xi_{c}^{0}&0\end{array}\right).$$

$$\gamma =\sqrt{2\mu E_B},$$

$$E_B=m_1+m_2-M$$

$$|\Psi ⟩=\left(\genfrac{}{}{0ex}{}{\lambda |{\psi }_0⟩}{\chi (\mathbf{k})|h_1{h}_2⟩}\right),$$

$$\hat{\mathcal{H}}|\Psi\rangle=E|\Psi\rangle,~{}~{}\hat{\mathcal{H}}=\left(\begin{array}[]{cc}\hat{H}_{c}&\hat{V}\\ \hat{V}&\hat{H}_{hh}^{0}\end{array}\right).$$

$$⟨{\psi }_0|\hat{V}|h_1{h}_2⟩=f(\mathbf{k}),$$

$$\chi (\mathbf{k})=\lambda \frac{f(\mathbf{k})}{E-k^2/(2\mu )}.$$

$$1$$

$$1={\lambda }^2\left[1+\frac{{\mu }^2{g}_0^2}{2\pi \sqrt{2\mu E_B}}+\mathcal{O}\left(\frac{\gamma }{\beta }\right)\right].$$

$$g_0^2=\frac{2\pi \gamma }{{\mu }^2}\left(\frac{1}{{\lambda }^2}-1\right),$$

$$\Sigma (E)$$

$$T_{\text{NR}}(E)=\frac{g_0^2}{E-E_0+\Sigma (E)}+\text{mbox}non-poleterms,$$

$$T_{\text{NR}}(E)=\frac{g_0^2}{E+E_B+g_0^2\mu /(2\pi )(ik+\gamma )},$$

$$T_{\text{NR}}(E)=-\frac{2\pi }{\mu }\frac{1}{1/a+(r/2)k^2-ik},$$

$$a$$

$$r$$

$$a=-5.419(7)\text{mbox}fmandr=1.764(8)\text{mbox}fm,$$

$$E_B=2.22\text{mbox}MeV⟹\gamma =45.7\text{mbox}MeV=0.23\text{mbox}{fm}^{-1}.$$

$$\frac{1}{\beta }\sim \frac{1}{M_{\pi }}\simeq 1.4\text{mbox}fm.$$

$$a_{\rm mol.}=-(4.3\pm 1.4)\ \mbox{fm}\$$

$$\gamma =-\frac{1}{a}+\frac{{\gamma }^{2}r}{2}.$$

$$g_{\text{NR}}^2\equiv Z{g}_0^2=\frac{2\pi \gamma }{{\mu }^2}(1-{\lambda }^2).$$

$$\frac{g_{\mathrm{eff}}^2}{4\pi }=4M^2\left(\frac{\gamma }{\mu }\right)\left(1-{\lambda }^2\right).$$

$$\frac{g_{\mathrm{eff}}^2}{4\pi }=\frac{4M^2}{\mu }\frac{-a\gamma }{a+2/\gamma },$$

Full text

1. I Introduction

2. II Candidates of hadronic molecules — experimental evidences

3. II.1 Light mesons

4. II.1.1 Scalars below 1 GeV

5. II.1.2 Axial vectors $f_{1}(1420)$, $a_{1}(1420)$ and implications of the triangle singularity

6. II.2 Open heavy-flavor mesons

7. II.3 Heavy quarkonium-like states: $XYZ$

8. II.3.1 $X(3872)$

9. II.3.2 $Z_{b}(10610)$, $Z_{b}(10650)$ and $Z_{c}(3900)$, $Z_{c}(4020)$

10. II.3.3 $Y(4260)$ and other vector states

11. II.4 Baryon candidates for hadronic molecules

12. II.4.1 Candidates in the light baryon sector

13. II.4.2 Candidates in the charm baryon sector

14. II.4.3 Pentaquark-like structures with hidden-charm

15. III Identifying hadronic molecules

16. III.1 Properties of the $S$-matrix

17. III.2 Definition of hadronic molecules

18. III.2.1 The Weinberg compositeness criterion

19. III.2.2 The pole counting approach

20. III.2.3 Remarks about pole trajectories

21. III.2.4 Generalizations to resonances

22. III.3 Characteristic line shapes of hadronic molecules

23. III.4 Heavy quark spin symmetry

24. IV Nonrelativistic effective field theories

25. IV.1 Power counting schemes

26. IV.1.1 Analytic structure of the three-point loop integral

27. IV.1.2 NREFT${}_{\text{I}}$

28. IV.1.3 NREFT${}_{\text{II}}$ and XEFT

29. IV.1.4 From NREFT${}_{\text{I}}$ to XEFT

30. IV.2 Formation of hadronic molecules

31. IV.3 Impact of hadron loops on regular quarkonia

32. V Hadronic molecules in lattice QCD

33. V.1 Resonances in a finite volume

34. V.2 Quark mass dependence

35. V.3 Measuring compositeness on lattice

36. V.4 Lattice QCD results on the charm-strange mesons and $\bm{XYZ}$ states

37. V.5 Lattice QCD results on hadrons built from light quarks

38. VI Phenomenological manifestations of hadronic molecules

39. VI.1 Long-distance production and decay mechanisms

40. VI.1.1 Decays into the constituents and transitions between molecular states

41. VI.1.2 More on line shapes

42. VI.1.3 Enhanced isospin violations in molecular transitions

43. VI.1.4 Enhanced production of hadronic molecules and conventional hadrons due to triangle singularities

44. VI.2 Short-distance production and decay mechanisms

45. VI.3 Implications of heavy quark spin and flavor symmetries

46. VI.4 Baryon candidates for hadronic molecules

47. VII Summary and outlook

I Introduction

With the discovery of the deuterium in 1931 and the neutron in 1932, the first bound state of two hadrons, i.e., the deuteron composed of one proton and one neutron, became known. The deuteron is very shallowly bound, by a mere MeV per nucleon, i.e. it is located just below the neutron-proton continuum threshold. Furthermore, it has a sizeable spatial extension. These two features can be used for defining a hadronic molecule. A more precise definition will be given in the course of this review.

Then the first meson, the pion, as the carrier particle of the nuclear force proposed in 1935 by Yukawa was discovered in 1947, followed by the discovery of a second meson, the kaon, in the same year. Since then, many different hadrons have been observed. Naturally hadronic molecules other than the deuteron have been expected. The first identified meson-baryon molecule, i.e., the $\Lambda(1405)$ resonance composed of one kaon and one nucleon, was predicted by Dalitz and Tuan in 1959 Dalitz and Tuan (1959) and observed in the hydrogen bubble chamber at Berkeley in 1961 Alston et al. (1961) several years before the quark model was proposed. With the quark model developed in the early 1960s, it became clear that hadrons are not elementary particles, but composed of quarks and antiquarks. In the classical quark model, a baryon is composed of three quarks and a meson is composed of one quark and one antiquark. In this picture, the $\Lambda(1405)$ resonance would be an excited state of a three-quark ($uds$) system with one quark in an orbital $P$-wave excitation. Ten years later, the theory of the strong interactions, Quantum Chromodynamics (QCD), was proposed to describe the interactions between quarks as well as gluons. The gluons are the force carriers of the theory that also exhibit self-interactions due to the non-abelian nature of the underlying gauge group, SU(3)C, where $C$ denotes the color degree of freedom. In QCD the basic constituents of the hadrons are both quarks and gluons. Therefore, the structure of hadrons is more complicated than the classical quark model allows. There may be glueballs (which contain only valence gluons), hybrids (which contain valence quarks as well as gluons) and multiquark states (such as tetraquarks or pentaquarks). Note, however, that in principle the quark model also allows for certain types of multiquark states Gell-Mann (1964).

While the classical quark model is very successful in explaining properties of the spatial ground states of the flavor SU(3) vector meson nonet, baryon octet and decuplet, it fails badly even for the lowest spatial excited states in both meson and baryon sectors.

In the meson sector, the lowest spatial excited SU(3) nonet is supposed to be the lowest scalar nonet which includes the $f_{0}(500)$, the $\kappa(800)$, the $a_{0}(980)$ and the $f_{0}(980)$. In the classical constituent quark model, these scalars should be $q\bar{q}~{}(L=1)$ states, where $L$ denotes the orbital angular momentum, with the $f_{0}(500)$ as an $(u\bar{u}+d\bar{d})/\sqrt{2}$ state, the $a_{0}^{0}(980)$ as an $(u\bar{u}-d\bar{d})/\sqrt{2}$ state and the $f_{0}(980)$ as mainly an $s\bar{s}$ state. This picture, however, fails to explain why the mass of the $a_{0}(980)$ is degenerate with the $f_{0}(980)$ instead of being close to the$f_{0}(500)$, as it is the case of the $\rho$ and the $\omega$ in the vector nonet. Instead, this kind of mass pattern can be easily understood in the tetraquark picture Jaffe (1977a) or in a scenario where these states are dynamically generated from the meson-meson interaction Weinstein and Isgur (1982); Janssen et al. (1995); Oller et al. (2000), with the $f_{0}(980)$ and the $a_{0}(980)$ coupling strongly to the $\bar{K}K$ channel with isospin 0 and 1, respectively.

In the baryon sector, a similar phenomenon seems also to be happening Zou (2008). In the classical quark model, the lowest spatial excited baryon is expected to be a ($uud$) $N^{*}$ state with one quark in an orbital angular momentum $L=1$ state to have spin-parity $1/2^{-}$. However, experimentally, the lowest negative parity $N^{*}$ resonance is found to be the $N^{*}(1535)$, which is heavier than two other spatial excited baryons: the $\Lambda^{*}(1405)$ and the $N^{*}(1440)$. This is the long-standing mass reversal problem for the lowest spatial excited baryons. Furthermore, it is also difficult to understand the strange decay properties of the $N^{*}(1535)$, which seems to couple strongly to the final states with strangeness Liu and Zou (2006), as well as the strange decay pattern of another member of the $1/2^{-}$-nonet, the $\Lambda^{*}(1670)$, which has a coupling to $\Lambda\eta$ much larger than to $NK$ and $\Sigma\pi$ according to its branching ratios listed in the tables in the Review of Particle Physics by the Particle Data Group (PDG) Patrignani et al. (2016). All these difficulties can be easily understood by assuming large five-quark components in them Zou (2008); Liu and Zou (2006); Helminen and Riska (2002) or considering them to be dynamically generated meson-baryon states Oller et al. (2000); Kaiser et al. (1995); Oset and Ramos (1998); Oller and Meißner (2001); Inoue et al. (2002); Garcia-Recio et al. (2004); Hyodo et al. (2003); Magas et al. (2005); Huang et al. (2007); Bruns et al. (2011).

No matter which configurations are realized in multiquark states, such as colored diquark correlations or colorless hadronic clusters, the mass and decay patterns for the lowest meson and baryon nonets strongly suggest that one must go beyond the classical, so-called quenched, quark model. The unquenched picture has been further supported by more examples of higher excited states in the light quark sector, such as the $f_{1}(1420)$ as a $K^{*}\bar{K}$ molecule Törnqvist (1994), and by many newly observed states with heavy quarks in the first decade of the new century, such as the $D^{*}_{s0}(2317)$ as a $DK$ molecule or tetraquark state, $X(3872)$ as $D^{*}\bar{D}$ molecule or tetraquark state Chen et al. (2016a). In fact, the possible existence of hadronic molecules composed of two charmed mesons was already proposed 40 years ago by Voloshin and Okun Voloshin and Okun (1976) and supported by Törnqvist later within a one-pion exchange model Törnqvist (1994).

However, although many hadron resonances were proposed to be dynamically generated states from various hadron-hadron interactions or multiquark states, most of them cannot be clearly distinguished from classical quark model states due to tunable ingredients and possible large mixing of various configurations in these models. A nice example is the already mentioned $\Lambda(1405)$. Until 2010, i.e., 40 years after it was predicted and observed as the $\bar{K}N$ molecule, the PDG Nakamura et al. (2010) still claimed that “the clean $\Lambda_{c}$ spectrum has in fact been taken to settle the decades-long discussion about the nature of the $\Lambda(1405)$ — true 3-quark state or mere $\bar{K}N$ threshold effect? — unambiguously in favor of the first interpretation.” Only after many delicate analyses of various relevant processes, the PDG Patrignani et al. (2016) now acknowledges the two-pole structure of the $\Lambda(1405)$ Oller and Meißner (2001) and thus a dynamical generation is most probable.

One way to unambiguously identify a multiquark state (including hadronic molecular configurations) is the observation of resonances decaying into a heavy quarkonium plus a meson with nonzero isospin made of light quarks or plus a baryon made of light quarks. Since 2008, several such states have been claimed, six $Z_{c}$ states, two $Z_{b}$ states and two $P_{c}$ states — details on the experimental situation are given in the next section. Among these newly claimed states, the two $P_{c}$ states are quite close to the predicted hadronic molecular states Wu et al. (2010); Wang et al. (2011b); Yang et al. (2012b); Xiao et al. (2013a). However, many of those states are challenged by some proposed kinematic explanations, such as threshold cusp effects Bugg (2011); Swanson (2015), triangle singularity effects Chen et al. (2013); Wang et al. (2013a); Guo et al. (2015c), etc. Some of these claims were challenged in Ref. Guo et al. (2015b) where strong support is presented that at least some of the signals indeed refer to $S$-matrix poles.

Further experimental as well as theoretical studies are necessary to settle the question which of the claimed states indeed exist. Nevertheless the observation of at least some of these new states opens a new window for the study of multiquark dynamics. Together with many other newly observed states in the heavy quarkonium sector, they led to a renaissance of hadron spectroscopy. Among various explanations of the internal structure of these excitations, hadronic molecules, being analogues of the deuteron, play a unique role since for those states predictions can be made with controlled uncertainty, especially for the states with one of or both hadrons containing heavy quark(s). In fact most of these observed exotic candidates are indeed closely related to open flavor $S$-wave thresholds. To study these hadronic molecules, both nonrelativistic effective field theories and pertinent lattice QCD calculation are the suitable frameworks. Especially, Weinberg’s famous compositeness criterion Weinberg (1963a, b) (and extensions thereof), which pinned down the nature of the deuteron as a proton-neutron bound state, is applicable here. The pole location in the corresponding hadron-hadron scattering $S$-matrix could also shed light on the nature of the resonances as extended hadronic molecules or compact states.

The revival of hadron spectroscopy is also reflected in a number of review articles. A few years ago, Klempt and his collaborators have given two broad reviews on exotic mesons Klempt and Zaitsev (2007) and baryons Klempt and Richard (2010). Other more recent pertinent reviews include Brambilla et al. (2011); Olsen (2015); Oset et al. (2016); Chen et al. (2016a, 2017c); Esposito et al. (2016b); Lebed et al. (2017); Hosaka et al. (2016); Dong et al. (2017); Olsen et al. (2017). Among various theoretical models for these new hadrons, we mainly cite those focusing on hadronic molecules and refer the interesting readers to the above mentioned comprehensive reviews for more references on other models.

This paper is organized as follows: In Sec. II, we discuss the experimental evidences for states that could possibly be hadronic molecules. In Sec. III, after a short review of the basic $S$-matrix properties, we give a general definition of hadronic molecules and discuss related aspects. Then, in Sec. IV, nonrelativistic effective field theories tailored to investigate hadronic molecules are formulated, followed by a brief discussion of hadronic molecules in lattice QCD in Sec. V. Sec. VI is devoted to the discussion of phenomenological manifestations of hadronic molecules, with a particular emphasis on clarifying certain statements from the literature that have been used to dismiss certain states as possible hadronic molecules. We end with a short summary and outlook in Sec. VII. We mention that this field is very active, and thus only references that appeared before April 2017 are included.

II Candidates of hadronic molecules — experimental evidences

In this section we briefly review what is known experimentally about some of the most promising candidates for exotic states. Already the fact that those are all located close to some two-hadron continuum channels indicates that the two-hadron continuum is of relevance for their existence. We will show that many of those states are located near $S$-wave thresholds in both light and heavy hadron spectroscopy, which is not only a natural property of hadronic molecules, which are QCD bound states of two hadrons (a more proper definition will be given in Sec. III.2), but also a prerequisite for their identification as will be discussed in Sec. III. In the course of this review, we will present other arguments why many of these states should be considered as hadronic molecules and what additional experimental inputs are needed to further confirm this assignment.

In Tables 1 and  2 we present the current status for exotic candidates in the meson sector. Exotic candidates in the baryon sector are listed later in Tab. 4. Besides the standard properties we also quote for each state the nearest relevant $S$-wave threshold as well as its distance to that threshold. Note that only thresholds of narrow states are quoted since these are the only ones of relevance here Guo and Meißner (2011). Otherwise, the bound system would also be broad Filin et al. (2010). In addition, as a result of the centrifugal barrier one expects that if hadronic molecules exist, they should first of all appear in the $S$-wave which is why in this review we do not consider $P$- or higher partial waves although there is no principle reason for the non-existence of molecular states in the $P$-wave.

II.1 Light mesons

II.1.1 Scalars below 1 GeV

The lowest $S$-wave two-particle thresholds in the hadron sector are those for two pseudoscalar mesons, $\pi\pi$, $\pi K$, $\eta\pi$, and $K\bar{K}$. Those channels carry scalar quantum numbers. The pion pair is either in an isoscalar or an isotensor state, and the isovector state is necessarily in a $P$-wave. It turns out that there is neither a resonant structure in the isotensor $\pi\pi$ nor in the isospin $3/2$ $\pi K$ $S$-wave, however, there are resonances observed experimentally in all other channels. According to the conventional quark model, a scalar meson of $q\bar{q}$ with $J^{P}=0^{+}$ carries one unit of orbital angular momentum. Thus, the mass range of the lowest scalars is expected to be higher than the lowest pseudoscalars or vectors of which the orbital angular momentum is zero. However, the lightest scalars have masses below those of the lightest vectors. Moreover, the mass ordering of the lightest scalars apparently violates the pattern of other $q\bar{q}$ nonets: Instead of having the isovectors to be the lowest states, the isovector $a_{0}(980)$ states are almost degenerate with one of the isoscalar states, $f_{0}(980)$, and those are the heaviest states in the nonet. The other isoscalar, $f_{0}(500)$, also known as $\sigma$, has the lightest mass of the multiplet and an extremely large width. The strange scalar $K_{0}^{*}(800)$, also known as $\kappa$, has a large width as well. All these indicate some nontrivial substructure beyond a simple $q\bar{q}$ description.

The mass ordering of these lightest scalars is seen as a strong evidence for the tetraquark scenario proposed by Jaffe in the 1970s Jaffe (1977a, b). Meanwhile, they can also be described as dynamically generated states through meson-meson scatterings Pennington and Protopopescu (1973); Au et al. (1987); Morgan and Pennington (1993); Peláez (2016). For a theoretical understanding of the $f_{0}(500)$ pole it is crucial to recognize that as a consequence of the chiral symmetry of QCD the scalar isoscalar $\pi\pi$ interaction is proportional to $(2s-M_{\pi}^{2})/F_{\pi}^{2}$ at the leading order (LO) in the chiral expansion. Here, $M_{\pi}(F_{\pi})$ denotes the pion mass (decay constant). As a result, the LO scattering amplitude has already hit the unitarity bound for moderate energies necessitating some type of unitarization, which at the same time generates a resonance-like structure Meißner (1991). This observation, deeply nested in the symmetries of QCD, has indicated the significance of the $\pi\pi$ interaction for the light scalar mesons. The history and the modern developments regarding the $f_{0}(500)$ was recently very nicely reviewed in Peláez (2016). Similar to the isoscalar scalar $f_{0}(500)$ generated from the $\pi\pi$ scattering, the whole light scalar nonet appears naturally from properly unitarized chiral amplitudes for pseudoscalar-pseudoscalar scatterings Oller et al. (1998, 1999); Gomez Nicola and Peláez (2002). Similar conclusions also follow from more phenomenological studies Weinstein and Isgur (1990); Janssen et al. (1995). One of the most interesting observations about $a_{0}(980)$ and $f_{0}(980)$ is that their masses are almost exactly located at the $K\bar{K}$ threshold. The closeness of the $K\bar{K}$ threshold to $a_{0}(980)$ and $f_{0}(980)$ and their strong $S$-wave couplings makes both states good candidates for $\bar{K}K$ molecular states Weinstein and Isgur (1990); Baru et al. (2004).

II.1.2 Axial vectors $f_{1}(1420)$, $a_{1}(1420)$ and implications of the

triangle singularity

The $S$-wave pseudoscalar meson pair scatterings can be extended to $S$-wave pseudoscalar-vector scatterings and vector-vector scatterings where again dynamically generated states can be investigated. The $S$-wave pseudoscalar-vector scatterings can access the quantum numbers $J^{P}=1^{+}$, while the vector-vector scatterings give $J^{P}=0^{+}$, $1^{+}$ and $2^{+}$. This suggests that some of the states with those quantum numbers can be affected by the $S$-wave open thresholds if their masses are close enough to the thresholds. Or, it might be possible that such scatterings can dynamically generate states as discussed in the literature Lutz and Kolomeitsev (2004b); Roca et al. (2005); Geng and Oset (2009). Note that not all states found in these studies survive once a more sophisticated and realistic treatment as outlined in Ref. Gülmez et al. (2017) is utilized.

In addition, the quark model also predicts regular $q\bar{q}$ states in the same mass range such that it appears difficult to identify the most prominent structure of the states.

Let us focus on the lowest $1^{++}$ mesons. Despite that these states could be dynamically generated from the resummed chiral interactions Lutz and Kolomeitsev (2004b); Roca et al. (2005), there are various experimental findings consistent with a usual $q\bar{q}$ nature of the members of the lightest axial nonet, $f_{1}(1420)$, $f_{1}(1285)$, $a_{1}(1260)$, and $K_{1A}(1^{3}P_{1})$ Patrignani et al. (2016). However, two recent experimental observations expose novel features in their decay mechanisms which illustrate the relevance of their couplings to the two-meson continua. The BESIII Collaboration observed an anomalously large isospin symmetry breaking in $\eta(1405)/\eta(1475)\to 3\pi$ Ablikim et al. (2012), which could be accounted for by the so-called triangle singularity (TS) mechanism as studied in Ref. Wu et al. (2012); Aceti et al. (2012). This special threshold phenomenon arises in triangle (three-point loop) diagrams with special kinematics which will be detailed in Sect. IV.1. Physically, it emerges when all the involved vertices in the triangle diagram can be interpreted as classical processes. For it to happen, one necessary condition is that all intermediate states in the triangle diagram, $\bar{K}K^{*}(K)+c.c.$ for the example at hand, should be able to reach their on-shell condition simultaneously. As a consequence, the $f_{1}(1420)$, which is close to the $\bar{K}K^{*}$ threshold and couples to $\bar{K}K^{*}$ in an $S$-wave as well, should also have large isospin violations in $f_{1}(1420)\to 3\pi$. This contribution has not been included in the BESIII analysis Ablikim et al. (2012). However, a detailed partial wave analysis suggests the presence of the $f_{1}(1420)$ contribution via the TS mechanism Wu et al. (2013). Moreover, the TS mechanism predicts structures in different $C$-parity and isospin (or $G$ parity) channels via the $\bar{K}K^{*}(K)+c.c.$ triangle diagrams. The $f_{1}(1420)$ was speculated long time ago to be a $\bar{K}^{*}K$ molecule from a dynamical study of the $K\bar{K}\pi$ three-body system Longacre (1990).

Apart from the $I=0$, $J^{PC}=1^{++}$ state $f_{1}(1420)$, one would expect that the TS will cause enhancements in $I=1$ channels with $C=\pm$. It provides a natural explanation for the newly observed $a_{1}(1420)$ by the COMPASS Collaboration Adolph et al. (2015) in $\pi^{-}p\to\pi^{-}\pi^{-}\pi^{+}p$ and $\pi^{-}\pi^{0}\pi^{0}p$ Liu et al. (2016a); Mikhasenko et al. (2015). It should be noted that in Refs. Aceti et al. (2016); Cheng et al. (2016); Debastiani et al. (2017a) the $a_{1}(1420)$ enhancement is proposed to be caused by the $a_{1}(1260)$ together with the TS mechanism and similarly $f_{1}(1420)$ is produced by $f_{1}(1285)$. However, as shown by the convincing experimental data from MARK-III, BESII, BESIII, and the detailed partial wave analysis of Ref. Wu et al. (2013), the $f_{1}(1420)$ matches the behavior of a genuine state in the $K\bar{K}\pi$ channel that is distorted in other channels by an interference with the TS. This appears to be a more consistent picture to explain the existing data and underlying mechanisms Zhao (2017). These issues are discussed further in Sec. VI.

II.2 Open heavy-flavor mesons

Since 2003, quite a few open heavy-flavor hadrons have been observed experimentally. Some of them are consistent with the excited states predicted in the potential quark model, while the others are not (for a recent review, see Chen et al. (2017c)). Particular interest has been paid to the positive-parity charm-strange mesons $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ observed in 2003 by the BaBar Aubert et al. (2003) and CLEO Besson et al. (2003) Collaborations. The masses of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ are below the $DK$ and $D^{*}K$ thresholds, respectively, by about the same amount, only 45 MeV (see Table 1 and references therein), which makes them natural candidates for hadronic molecules Barnes et al. (2003); van Beveren and Rupp (2003); Szczepaniak (2003); Kolomeitsev and Lutz (2004); Hofmann and Lutz (2004); Guo et al. (2006, 2007); Gamermann et al. (2007); Faessler et al. (2007); Flynn and Nieves (2007); Cleven et al. (2011b); Wu and Zhao (2012); Cleven et al. (2014a); Albaladejo et al. (2016c), while also other explanations such as $P$-wave $c\bar{s}$ states and tetraquarks exist in the literature. We will come back to the properties of these states occasionally in this review. Here, we collect the features supporting the $DK/D^{*}K$ molecular hypothesis:

• •

Their masses are about $160~{}\mathrm{MeV}$ and $70~{}\mathrm{MeV}$, respectively, below the predicted $0^{+}$ and $1^{+}$ charm-strange mesons by the Godfrey–Isgur quark model Godfrey and Isgur (1985); Di Pierro and Eichten (2001), making them not easy to be accommodated by the conventional $c\bar{s}$ states.

• •

The mass difference between these two states is equal to the energy difference between the corresponding $D^{(*)}K$ thresholds. This appears to be a natural consequence in the hadronic molecular scenario, since the involved interactions is approximately heavy quark spin symmetric Guo et al. (2009a).

• •

The small width of both $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ can only be understood if the are isoscalar states 222Negative result was reported in a search for the isospin partner of the $D_{s0}^{*}(2317)$ Choi et al. (2015), for then, since both of them are below the $DK/D^{*}K$ thresholds, the only possible hadronic decay modes are the isovector channels $D_{s}^{+}\pi^{0}$ and $D_{s}^{*+}\pi^{0}$, respectively. The molecular nature together with the proximity to the $DK/D^{*}K$ thresholds leads to a prediction for the width of the states above 100 keV while other approaches give a width about 10 keV Colangelo and De Fazio (2003); Godfrey (2003). These issues are discussed in detail in Sections V.4 and VI.1.3.

• •

Their radiative decays, i.e., $D_{s0}^{*}(2317)\to D_{s}\gamma$ and $D_{s1}(2460)\to D^{(*)}_{s}\gamma$, and production in $B$ decays proceed via short-range interactions Lutz and Soyeur (2008); Chen et al. (2015a); Cleven et al. (2014a). They are therefore insensitive to the molecular component of the states.

• •

As will be discussed in Secs. III.2 and V.4, the $DK$ scattering length extracted from LQCD calculations Liu et al. (2013a) is compatible with the result extracted in the molecular scenario for $D_{s0}^{*}(2317)$ based on Weinberg’s compositeness theorem.

The $D_{sJ}(2860)$ observed by the BaBar Collaboration Aubert et al. (2006a) presents another example of an interesting charm-strange meson. It decays into both $DK$ and $D^{*}K$ with similar branching fractions Patrignani et al. (2016). One notices that the difference between the $D_{sJ}(2860)$ mass and the $D_{1}(2420)K$ threshold is similar to that between the $D_{s0}^{*}(2317)$ and $DK$. Assuming the $D_{s0}^{*}(2317)$ to be a $DK$ hadronic molecule, an $S$-wave $D_{1}(2420)K$ bound state with quantum numbers $J^{P}=1^{-}$ was predicted to have a mass $(2870\pm 9)$ MeV, consistent with that of the $D_{sJ}(2860)$, in Guo and Meißner (2011), where the ratio of its partial widths into the $DK$ and $D^{*}K$ also gets naturally explained. As a result of heavy quark spin symmetry, a $D_{2}(2460)K$ hadronic molecule with $J^{P}=2^{-}$ and a mass of around 2.91 GeV was predicted in Guo and Meißner (2011). A later analysis by the LHCb Collaboration suggests that this structure corresponds to two states: $D_{s1}^{*}(2860)$ with $J^{P}=1^{-}$ and $D_{s3}^{*}(2860)$ with $J^{P}=3^{-}$ Aaij et al. (2014b). Regular $c\bar{s}$ interpretations for these two states have been nicely summarized in Chen et al. (2017c).

The most recently reported observation of an exotic singly-heavy meson candidate is a narrow structure in the $B_{s}^{0}\pi^{\pm}$ invariant mass distribution, dubbed as $X(5568)$, by the D0 Collaboration Abazov et al. (2016). Were it a hadronic state, it would be an isovector meson containing four different flavors of valence quarks $(\bar{b}s\bar{u}d)$. However, the peak is located at only about 50 MeV above the $B_{s}\pi$ threshold. The existence of a tetraquark, whether or not being a hadronic molecule, at such a low mass is questioned from the quark model point of view in Burns and Swanson (2016), and, more generally, from chiral symmetry and heavy quark flavor symmetry in Guo et al. (2016d). Both the LHCb Aaij et al. (2016c) and CMS CMS Collaboration (2016) Collaborations quickly reported negative results on the existence of $X(5568)$ in their data sets. An alternative explanation for the $X(5568)$ observation is necessary. One possibility is provided in Ref. Yang et al. (2017b). Because of these controversial issues with the $X(5568)$, we will not discuss this structure any further.

II.3 Heavy quarkonium-like states: $XYZ$

The possibility of hadronic molecules in the charmonium mass region was suggested in Voloshin and Okun (1976); De Rujula et al. (1977) only a couple of years after the “November Revolution” due to the discovery of the $J/\psi$. Such an idea became popular after the discovery of the famous $X(3872)$ by Belle in 2003 Choi et al. (2003).

Since then, numerous other exotic candidates have been found in the heavy quarkonium sector as listed in Table 2. In fact, it is mainly due to the observation of these structures that the study of hadron spectroscopy experienced a renaissance. The naming scheme currently used in the literature for these $XYZ$ states assigns isoscalar $J^{PC}=1^{--}$ states as $Y$ and the isoscalar (isovector) states with other quantum numbers are named as $X(Z)$. Note that the charged heavy quarkonium-like states $Z_{c}(3900)^{\pm}$, $Z_{c}(4020)^{\pm}$, $Z_{b}(10610)^{\pm}$, $Z_{b}(10650)^{\pm}$ and $Z_{c}(4430)^{\pm}$ are already established as being exotic, since they should contain at least two quarks and two anti-quarks with the hidden pair of $c\bar{c}$ or $b\bar{b}$ providing the dominant parts of their masses.

In the heavy quarkonium mass region, there are quite a few $S$-wave thresholds opened by narrow heavy-meson pairs. In the charmonium mass region, the lowest-lying thresholds are $D\bar{D}$, $D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$. They are particularly interesting for understanding the $X$ and $Z$ states which can couple to them in an $S$-wave. The relevant quantum numbers are thus $J^{PC}=1^{+-}$ and $(0,1,2)^{++}$ (for more details, see Section IV.2). The $S$-wave thresholds for the $XYZ$ exotic candidates are also shown in Table 2. In addition, the exotic candidates in the charmonium sector and the $S$-wave open-charm thresholds are shown in Fig. 1. Here, the thresholds involving particles with a large width, $\gtrsim 100$ MeV, have been neglected.

Since only $S$-wave hadronic molecules with small binding energies are well-defined (Sec. III.2), in the following, we will focus on those candidates, i.e., $X(3872)$, $Z_{c}(3900)$, $Z_{c}(4020)$, $Y(4260)$ in the charmonium sector and $Z_{b}(10610)$, $Z_{b}(10650)$ in the bottomonium sector. All of them have extremely close-by $S$-wave thresholds except for the $Y(4260)$, as will be discussed below. For the experimental status and phenomenological models of other exotic candidates, we refer to several recent reviews Swanson (2006); Eichten et al. (2008); Brambilla et al. (2014); Esposito et al. (2015b); Lebed et al. (2017); Richard (2016); Chen et al. (2016a); Esposito et al. (2016b) and references therein.

II.3.1 $X(3872)$

In 2003, the Belle Collaboration reported a narrow structure $X(3872)$ in the $J/\psi\pi^{+}\pi^{-}$ invariant mass distribution in $B^{\pm}\to K^{\pm}J/\psi\pi^{+}\pi^{-}$ Choi et al. (2003) process. It was confirmed shortly after by BaBar Aubert et al. (2005c, 2008) in $e^{+}e^{-}$ collisions, and by CDF Acosta et al. (2004); Abulencia et al. (2006, 2007); Aaltonen et al. (2009) and D0 Abazov et al. (2004) in $p\bar{p}$ collisions. Very recently LHCb also confirmed its production in $pp$ collisions Aaij et al. (2012, 2013, 2014a, 2015c) and pinned down its quantum numbers to $J^{PC}=1^{++}$, which are consistent with the observations of its radiative decays Abe et al. (2005); Aubert et al. (2006b); Bhardwaj et al. (2011) and multipion transitions Abulencia et al. (2006); Abe et al. (2005); del Amo Sanchez et al. (2010). The negative result of searching for its charged partner in $B$ decays Aubert et al. (2005b) indicates that the $X(3872)$ is an isosinglet state.

The most salient feature of the $X(3872)$ is that its mass coincides exactly with the $D^{0}\bar{D}^{*0}$ threshold Patrignani et al. (2016)333Here we use the updated “OUR AVERAGE” values in PDG2016 for the masses: $M_{D^{0}}=(1864.84\pm 0.05)$ MeV, $M_{D^{*0}}=(2006.85\pm 0.05)$ MeV, and $M_{X}=(3871.69\pm 0.17)$ MeV from the $J/\psi\pi^{+}\pi^{-}$ and $J/\psi\omega$ modes Patrignani et al. (2016).

[TABLE]

which indicates the important role of the $D^{0}\bar{D}^{*0}$ in the $X(3872)$ dynamics. That this should be the case can be seen most clearly from the large branching fraction Gokhroo et al. (2006); Aushev et al. (2010) (see Table 2)

[TABLE]

although the $X(3872)$ mass is so close to the $D^{0}\bar{D}^{*0}$ and $\bar{D}^{0}D^{0}\pi^{0}$ thresholds. These experimental facts lead naturally to the interpretation of the $X(3872)$ as a $D\bar{D}^{*}$ hadronic molecule Törnqvist (2003),444 See also, e.g., Törnqvist (2004); Swanson (2004b); Close and Page (2004); Pakvasa and Suzuki (2004); Wong (2004); Voloshin (2004b); Swanson (2004a); AlFiky et al. (2006); Braaten and Lu (2007); Fleming et al. (2007); Liu et al. (2009a); Dong et al. (2009); Ding et al. (2009); Zhang and Huang (2009); Wang et al. (2010); Lee et al. (2009); Gamermann et al. (2010); Mehen and Springer (2011); Nieves and Valderrama (2011); Lee et al. (2011); Nieves and Valderrama (2012); Li et al. (2013d); Li and Zhu (2012); Sun et al. (2012b, a); Guo et al. (2013b); Hidalgo-Duque et al. (2013a); Wang and Wang (2013); Yamaguchi et al. (2013); Guo et al. (2014a); He (2014); Zhao et al. (2014); Karliner and Rosner (2015a); Baru et al. (2015a); Jansen et al. (2015); Baru et al. (2015b); Molnar et al. (2016); Yang et al. (2017a). which had been predicted by Törnqvist with the correct mass a decade earlier Törnqvist (1994). As will be discussed in Section VI, precise measurements of the partial widths of the processes $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ and $X(3872)\to D^{0}\bar{D}^{0}\gamma$ are particularly important in understanding the long-distance structure of the $X(3872)$. In the $D^{0}\bar{D}^{*0}$ hadronic molecular scenario, one gets a tremendously large $D^{0}\bar{D}^{*0}$ scattering length of $\geq 10$ fm, c.f. Eq. (21). However, a precision measurement of its mass is necessary to really distinguish a molecular $X(3872)$ from, e.g., a tetraquark scenario Maiani et al. (2005); Esposito et al. (2015b). This will be discussed further in Sec. III.2.3 and in Sec. III.3.

Other observables are also measured which could be sensitive to the internal structure of the $X(3872)$. The ratio of branching fractions

[TABLE]

was measured to be $1.0\pm 0.4\pm 0.3$ by Belle Abe et al. (2005) and $0.8\pm 0.3$ by BaBar del Amo Sanchez et al. (2010). The value about unity means a significant isospin breaking because the three and two pions are from the isoscalar $\omega$ Abe et al. (2005); del Amo Sanchez et al. (2010) and from the isovector $\rho$ Abulencia et al. (2006), respectively. Notice that there is a strong phase space suppression on the isospin conserved three-pion transition through the $J/\psi\omega$ channel. The fact that the molecular scenario of $X(3872)$ provides a natural explanation for the value of $R^{I}$ will be discussed in Sec. VI.1.3.

The experimental information available about the radiative decays of the $X(3872)$ is Aaij et al. (2014a)

[TABLE]

A value larger than 1 for this ratio was argued to favor the $\chi_{c1}(2^{3}P_{1})$ interpretation Swanson (2004a) over the $D^{0}\bar{D}^{*0}$ hadronic molecular picture. This, however, is not the case Mehen and Springer (2011); Guo et al. (2015a) as will be demonstrated in Sec. VI.

The production rates of $X(3872)$ in $B^{0}$ and $B^{-}$ decays was measured by BaBar Aubert et al. (2006c), i.e.,

[TABLE]

We show in Sec. VI that this value is also consistent with a molecular nature of the $X(3872)$.

One expects mirror images of charmonium-like states to be present in the bottomonium sector. The $Z_{c}$ and $Z_{b}$ states to be discussed in the next subsection suggest that such phenomena do exist. The analogue of the $X(3872)$ in the bottom sector, $X_{b}$, has not yet been identified. A search for the $X_{b}$ was carried out by the CMS Collaboration, but no signal was observed in the $\Upsilon\pi^{+}\pi^{-}$ channel Chatrchyan et al. (2013b). However, as pointed out in Ref. Guo et al. (2013b) before the experimental results and stressed again in Refs. Guo et al. (2014e); Karliner and Rosner (2015b) afterwards, the $X_{b}\to\Upsilon\pi^{+}\pi^{-}$ decay requires an isospin breaking which should be strongly suppressed due to the extremely small mass differences between the charged and neutral bottomed mesons and the large difference between the $B\bar{B}^{*}$ threshold and the $\Upsilon(1S)\omega$, $\Upsilon(1S)\rho$ thresholds. In contrast, other channels such as $X_{b}\to\Upsilon\pi^{+}\pi^{-}\pi^{0}$, $X_{b}\to\chi_{bJ}\pi^{+}\pi^{-}$ Guo et al. (2013b, 2014e); Karliner and Rosner (2015b) and $X_{b}\to\gamma\Upsilon(nS)$ Li and Wang (2014) should be a lot more promising for an $X_{b}$ search.

II.3.2 $Z_{b}(10610)$, $Z_{b}(10650)$ and $Z_{c}(3900)$, $Z_{c}(4020)$

From an analysis of the $\Upsilon(10860)\to\pi^{+}\pi^{-}(b\bar{b})$ processes in 2011 the Belle Collaboration reported the discovery of two charged states decaying into $\Upsilon(nS)\pi$ with $n=1,2,3$ and $h_{b}(mP)\pi$ with $m=1,2$ Bondar et al. (2012). Their line shapes in a few channels are shown in Fig. 2. A later analysis at the same experiment allowed for an amplitude analysis where the quantum numbers $I^{G}(J^{P})=1^{+}(1^{+})$ were strongly favored Garmash et al. (2015).555The existence of an isovector $b\bar{b}q\bar{q}$ state with exactly these quantum numbers was speculated long time ago for explaining the puzzling $\Upsilon(3S)\to\Upsilon(1S)\pi\pi$ transition Voloshin (1983); Anisovich et al. (1995). The $Z_{b}$ effects in dipion transitions among $\Upsilon$ states were reanalyzed using the dispersion technique recently in Chen et al. (2016f, 2017d). This, together with the fact that the $Z_{b}(10610)$ and $Z_{b}(10650)$ have masses very close to the $B\bar{B}^{*}$ and $B^{*}\bar{B}^{*}$ thresholds, respectively, makes both excellent candidates for hadronic molecules Bondar et al. (2011)666See also, e.g., Zhang et al. (2011); Yang et al. (2012a); Danilkin et al. (2012); Sun et al. (2011); Cleven et al. (2011a); Ohkoda et al. (2012b); Li et al. (2013b); Dong et al. (2013a); Wang and Huang (2014); Wang (2014b); Dias et al. (2015); Karliner and Rosner (2015a). This statement finds further support in the observation that both states also decay by far most probably into $B\bar{B}^{*}$ and $B^{*}\bar{B}^{*}$, respectively Garmash et al. (2016) (see Tab. 3).777The branching fractions were measured by assuming that these channels saturate the decay modes and using the Breit–Wigner (BW) parameterization for the $Z_{b}$ structures Garmash et al. (2016). However, there could be non-negligible modes such as the $\eta_{b}\rho$, and the branching fractions measured in this way for near-threshold states should not be used to calculate partial widths by simply multiplying with the BW width. This point is discussed in detail in Chen et al. (2016f) for the $Z_{b}$ case. The neutral partner is so far observed only for the lighter state Krokovny et al. (2013). Very recently, the Belle Collaboration reported the invariant mass distributions of $h_{b}(1P)\pi$ and $h_{b}(2P)\pi$ channels at $\Upsilon(11020)$ energy region Abdesselam et al. (2016), see Fig. 3, clearly showing a resonant enhancement in the $Z_{b}$ mass region. However, due to the limited statistics it is impossible to judge whether there are two peaks or just one.

Employing sums of BW functions for the resonance signals the experimental analyses gave masses for both $Z_{b}$ states slightly above the corresponding open flavor thresholds together with narrow widths. It seems in conflict with the hadronic molecular picture, and was claimed to be consistent with the tetraquark approach Esposito et al. (2016a). It is therefore important to note that a recent analysis based on a formalism consistent with unitarity and analyticity leads for both states to below-threshold pole positions Hanhart et al. (2015); Guo et al. (2016a).888Notice that this, however, does not exclude the possibility of above-threshold poles. In the used parameterization, the contact terms are taken to be constants. The possibility of getting an above-threshold pole is available once energy-dependence is allowed in the contact terms. Nevertheless, the analyses at least show that the below-threshold-pole scenario is consistent with the current data.

A few years after the discovery of $Z_{b}(10610)$ and $Z_{b}(10650)$ in the Belle experiment, the BESIII and Belle Collaborations almost simultaneously claimed the observation of a charged state in the charmonium mass range, $Z_{c}(3900)$ Ablikim et al. (2013a); Liu et al. (2013b). It was shortly after confirmed by a reanalysis of CLEO-c data Xiao et al. (2013b), and its neutral partner was also reported in Refs. Xiao et al. (2013b); Ablikim et al. (2015e). Soon after these observations, the BESIII Collaboration reported the discovery of another charged state $Z_{c}(4020)$ Ablikim et al. (2013b), and its neutral partner was reported in Ref. Ablikim et al. (2014c). These charmonium-like states show in many respects similar features as the heavier bottomonium-like states discussed in the previous paragraphs, although there are also some differences. On the one hand, while the $Z_{c}(3900)$ is seen in the $J/\psi\pi$ channel and $Z_{c}(4020)$ is seen in $h_{c}\pi$, there is no clear signal of $Z_{c}(4020)$ in $J/\psi\pi$ and $Z_{c}(3900)$ in $h_{c}\pi$, although in the latter case there might be some indications of $Z_{c}(3900)\to h_{c}\pi$. This pattern might reflect a strong mass dependence of the production mechanism Wang et al. (2013b). On the other hand, in analogy to $Z_{b}(10610)$ and $Z_{b}(10650)$, $Z_{c}(3900)$ and $Z_{c}(4020)$ have masses very close to the the $D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$ thresholds, respectively, and they couple most prominently to these open-flavor channels regardless of the significant phase space suppression Ablikim et al. (2014b, a, 2015d, 2015c). The two $Z_{c}$ states are also widely regarded as hadronic molecules Wang et al. (2013a); Guo et al. (2013b); Voloshin (2013); Cui et al. (2014); Wilbring et al. (2013); Li (2013); Zhang (2013); Dong et al. (2013b); Ke et al. (2013); He et al. (2013); Karliner and Rosner (2015a); Chen et al. (2015e); Gong et al. (2016a).

Analogous to the $Z_{b}$ case, the experimental analyses of the two $Z_{c}$ states based on sums of BW distributions result in masses above the continuum thresholds as well. However, this does not allow the correct extraction of the pole locations. In order to obtain reliable pole locations an analysis in the spirit of Refs. Hanhart et al. (2015); Guo et al. (2016a) is necessary for these charmonium-like states. Such an analysis was done for the $Z_{c}(3900)$ in Ref. Albaladejo et al. (2016b). By fitting to the available BESIII data in the $Y(4260)\to J/\psi\pi^{+}\pi^{-}$ Ablikim et al. (2013a) and the $Y(4260)\to J/\psi\pi^{+}\pi^{-}$ Ablikim et al. (2015a) modes, it is found that the current data are consistent with either an above-threshold resonance pole or a below-threshold virtual state pole. A comparison of the resonance pole obtained therein with various determinations in experimental papers is shown in Fig. 4.

II.3.3 $Y(4260)$ and other vector states

At present, the vector channel with $J^{PC}=1^{--}$, in both the bottomonium and the charmonium sector, is the best investigated one experimentally, since it can be accessed directly in $e^{+}e^{-}$ annihilations. Note that a pair of ground state open-flavor mesons, such as $D\bar{D}$, $D\bar{D}^{*}+c.c.$, $D^{*}\bar{D}^{*}$, $D_{s}\bar{D}_{s}$, etc., carry positive parity in the $S$ wave, and thus cannot be directly accessed in $e^{+}e^{-}$ annihilations. Accordingly, if the $S$-wave hadronic molecules exist in the vector channel, they should be formed (predominantly) by constituents different from them. In particular, it suggests that the $S$-wave molecular states in the vector channel should be heavier than those thresholds opened by a pair of ground state $D^{(*)}$ mesons.

As the first $Y$ state, the $Y(4260)$ was observed by the BaBar Collaboration in the $J/\psi\pi^{+}\pi^{-}$ channel in the initial state radiation (ISR) process $e^{+}e^{-}\to\gamma_{\text{ISR}}J/\psi\pi^{+}\pi^{-}$ Aubert et al. (2005a). The fitted mass and width are $\left(4259\pm 8^{+2}_{-6}\right)\mathrm{MeV}$ and $50\ldots 90~{}\mathrm{MeV}$, respectively. It was confirmed by CLEO-c He et al. (2006), Belle Yuan et al. (2007) and an additional analysis of BaBar Lees et al. (2012), with, however, mass values varying in different analyses. We notice that a recent combined analysis of the BESIII data in four different channels $e^{+}e^{-}\to\omega\chi_{c0}$ Ablikim et al. (2016a), $\pi^{+}\pi^{-}h_{c}$ Ablikim et al. (2017a), $\pi^{+}\pi^{-}J/\psi$ Ablikim et al. (2017c), and $D^{0}D^{*-}\pi^{+}+c.c.$ Yuan (2017) gives a mass of $(4219.6\pm 3.3\pm 5.1)$ MeV and a width of $(56.0\pm 3.6\pm 6.9)$ MeV Gao et al. (2017).

The $Y(4260)$ was early recognized as a good candidate for an exotic state since there are no quark-model states predicted around its mass. Moreover, it does not show a strong coupling to $D\bar{D}$ as generally expected for vector $c\bar{c}$ states, and it does not show up as a pronounced enhancement in the inclusive cross sections for $e^{+}e^{-}\to$ hadrons (or the famous $R$ value plot). It is still believed to be a prime candidate, e.g., for a hybrid state Close and Page (2005) (for a recent discussion see Ref. Kalashnikova and Nefediev (2016)) or a hadrocharmonium state Dubynskiy and Voloshin (2008); Li and Voloshin (2014). However, it is also suggested to be a $D_{1}(2420)\bar{D}$ molecular state Ding (2009); Wang et al. (2013a); Cleven et al. (2014b); Li et al. (2013c); Li and Liu (2013a); Wu et al. (2014b) (the hadrocharmonium picture and the molecular picture are contrasted in Ref. Wang et al. (2014)). This picture is further supported by the fact that the recent high-statistics data from BESIII Ablikim et al. (2017c) (see Fig. 5) shows an enhancement at the $D_{1}(2420)\bar{D}$ threshold in the $J/\psi\pi\pi$ channel.999In this context it is interesting to note that also the hybrid picture predicts a large coupling of $Y(4260)$ to $D_{1}(2420)\bar{D}$ Barnes et al. (1995); Close and Page (2005); Kou and Pene (2005), which could be interpreted as the necessity of considering $D_{1}\bar{D}$ as a component. The observations of $Z_{c}(3900)\pi$ (Sec. II.3.2) and $X(3872)\gamma$ Ablikim et al. (2014d) in the mass region of the $Y(4260)$ provide further support for a sizable $D_{1}(2420)\bar{D}$ component in its wave function as will be discussed in Sec. VI. The suppression of an $S$-wave production, in the heavy quark limit, of the $1^{--}$ $D_{1}(2420)\bar{D}$ pair in $e^{+}e^{-}$ collisions Eichten et al. (1978, 1980); Li and Voloshin (2013) could be the reason for the dip around the $Y(4260)$ mass in the inclusive cross section of $e^{+}e^{-}\to$ hadrons Wang et al. (2014). In addition, the data from Belle in $e^{+}e^{-}\to\bar{D}D^{*}\pi$ Pakhlova et al. (2009) and from BESIII on $e^{+}e^{-}\to h_{c}\pi\pi$ Ablikim et al. (2017a), $\chi_{c0}\omega$ Ablikim et al. (2015f) are highly nontrivial (Fig. 6) and are claimed to be consistent with the molecular picture Cleven et al. (2014b); Cleven and Zhao (2017). A combined analysis of the BESIII data in different channels is presented in Ref. Gao et al. (2017).

The absence of a signal for $Y(4260)$ in $J/\psi K\bar{K}$ He et al. (2006); Yuan et al. (2008); Shen et al. (2014) questions the tetraquark picture of $Y(4260)$ with a diquark-antidiquark $[cs][\bar{c}\bar{s}]$ configuration Esposito et al. (2015b). In addition, the ground state in the tetraquark picture Esposito et al. (2015b), $Y(4008)$, is not confirmed by the recent high-statistics data from BESIII Ablikim et al. (2017c). Meanwhile, the cross sections for $e^{+}e^{-}\to\psi^{\prime}\pi\pi$ Aubert et al. (2007); Wang et al. (2007), $\eta^{\prime}J/\psi$ Ablikim et al. (2016b) and $\eta J/\psi$ and $\pi^{0}J/\psi$ Ablikim et al. (2015b) do not show any structure around the $Y(4260)$ energy region. It remains to be seen if these findings allow for further conclusions on the nature of the $Y(4260)$.

It is interesting to observe that some properties of the $Y(4260)$, like its proximity and strong coupling to the $D_{1}\bar{D}$ threshold, are mirror imaged by the $\Upsilon(11020)$ in the bottomonium sector Bondar and Voloshin (2016). Belle II appears to be an ideal instrument to investigate this connection in more detail in the future Bondar et al. (2017).

Searching for new decay modes of the $Y(4260)$, BaBar scanned the line shapes of $e^{+}e^{-}\to\psi(2S)\pi^{+}\pi^{-}$ and found a new structure, named $Y(4360)$ with a mass of $(4324\pm 24)~{}\mathrm{MeV}$ and a width of $(172\pm 33)~{}\mathrm{MeV}$ Aubert et al. (2007). In the same year, Belle Wang et al. (2007) analyzed the same process and found two resonant structures: a lower one consistent with $Y(4360)$ and a higher one, named $Y(4660)$, with a mass of $(4664\pm 11\pm 5)~{}\mathrm{MeV}$ and a width of $(48\pm 15\pm 3)~{}\mathrm{MeV}$. A combined fit  Liu et al. (2008) to the cross sections of the process $e^{+}e^{-}\to\psi(2S)\pi^{+}\pi^{-}$ from both BaBar and Belle gives the parameters for the two resonances $M_{Y(4360)}=\left(4355^{+9}_{-10}\pm 9\right)\mathrm{MeV}$, $\Gamma_{Y(4360)}=\left(103^{+17}_{-15}\pm 11\right)\mathrm{MeV}$ and $M_{Y(4660)}=\left(4661^{+9}_{-8}\pm 6\right)\mathrm{MeV}$, $\Gamma_{Y(4660)}=\left(42^{+17}_{-12}\pm 6\right)\mathrm{MeV}$ for $Y(4360)$ and $Y(4660)$, respectively. The fit at the same time provides an upper limit for $\mathcal{B}\left(Y(4260)\to\psi(2S)\pi^{+}\pi^{-}\right)\Gamma_{e^{+}e^{-}}$ as $4.3~{}\mathrm{eV}$. Those measurements were updated in Ref.  Wang et al. (2015b). Later on the Belle Collaboration found a structure in the $\Lambda_{c}^{+}\Lambda_{c}^{-}$ channel with a peak position $30~{}\mathrm{MeV}$ lower than that of $Y(4660)$ Pakhlova et al. (2008), which might either point at an additional state, called $Y(4630)$, or, be an additional decay channel of the $Y(4660)$ Cotugno et al. (2010); Guo et al. (2010a). The latter is the view taken in the 2016 Review of Particle Physics Patrignani et al. (2016). Of particular interest to this review is the observation of Ref. Guo et al. (2010a) that within the $\psi^{\prime}f_{0}(980)$ hadronic molecular picture Guo et al. (2008b); Wang and Zhang (2010b) the line shape of $Y(4630)$ in the $\Lambda_{c}^{+}\Lambda_{c}^{-}$ channel could be understood as the signal of $Y(4660)$ with the $\Lambda_{c}^{+}\Lambda_{c}^{-}$ final state interaction. As a byproduct, Ref. Guo et al. (2009a) predicts the properties of its spin partner, an $\eta_{c}^{\prime}f_{0}(980)$ hadronic molecule, at around 4.61 GeV.

As stressed in Refs. Wang et al. (2013b); Bondar and Voloshin (2016) the production of $Z_{c}(3900)$ and $Z_{c}(4020)$ in the mass region of $Y(4260)$ and $Y(4360)$ as well as that of $Z_{b}(10610)$ and $Z_{b}(10650)$ in the mass region of $\Upsilon(10860)$ and $\Upsilon(11020)$, respectively, are sensitive to the TS mechanism. A peculiar feature of such a mechanism is that whether peaks appear in certain invariant mass distributions depends strongly on the kinematics. The recent observation of a peak in the $\psi^{\prime}\pi$ invariant mass distribution by the BESIII Collaboration Ablikim et al. (2017b) shows exactly this behavior. The correlations between the initial $S$-wave thresholds and the final $S$-wave thresholds could be a key for understanding the rich phenomena observed in this energy region Liu et al. (2016a).

II.4 Baryon candidates for hadronic molecules

We now switch to the experimental evidences for hadronic molecules in the baryon sector. In analogy to the meson sector we will focus on states which are located close to $S$-wave thresholds of narrow meson-baryon pairs 101010Note that also $P_{11}(1440)$ was proposed to have a prominent $f_{0}(500)N$ substructure Krehl et al. (2000). However, the large width of the $f_{0}(500)$ prohibits a model-independent study of this claim.. In the light baryon spectrum the $\Lambda(1405)$ has been broadly discussed as a $\bar{K}N$ molecular state. A few charm baryons discovered in recent years are close to $S$-wave thresholds, and has been suggested to be hadronic molecules in the literature. The recently observed very interesting $P_{c}(4450)$ and $P_{c}(4380)$ have also been proposed to be hadronic molecules with hidden charm.

II.4.1 Candidates in the light baryon sector

The $\Lambda(1405)$ was discovered in the $\pi\Sigma$ subsystems of $Kp\to\Sigma\pi\pi\pi$ Alston et al. (1961) (see also Ref. Kim (1965); Hemingway (1985)). Further experimental information about this state comes from old scattering data Humphrey and Ross (1962); Sakitt et al. (1965); Watson et al. (1963); Ciborowski et al. (1982) complemented by the recent $\bar{K}N$ threshold amplitude extracted from data on kaonic hydrogen Bazzi et al. (2011, 2012) as well as the older so-called threshold ratios Tovee et al. (1971); Nowak et al. (1978). There are further data on $\Sigma\pi$ distributions from $pp\to\Sigma^{\pm}\pi^{\mp}K^{+}p$ Zychor et al. (2008); Agakishiev et al. (2013), the photoproduction $\gamma p\to K^{+}\Sigma\pi$ Moriya et al. (2014) and additional reactions. It appears also feasible that high energy experiments like BaBar, Belle, BESIII, CDF, D0 and LHCb investigate the $\Lambda(1405)$, e.g. via the decays of heavy hadrons such as $\Lambda_{b}\to J/\psi\Lambda(1405)$ Roca et al. (2015a). Note that a signal of $\Lambda(1405)$ was clearly visible in an analysis of $\Lambda_{b}\to J/\psi Kp$ performed by the LHCb Collaboration Aaij et al. (2015b).

The $\Lambda(1405)$ has strangeness $S=-1$ with $I(J^{P})=0(1/2^{-})$ and a mass about 30 MeV (see Table 4) below the $\bar{K}N$ threshold. Note that a direct experimental determination of the spin-parity quantum numbers was only given recently by the CLAS Collaboration Moriya et al. (2014). Since its mass is smaller than that of the nucleon counterpart $N^{*}(1535)\,1/2^{-}$ and the mass difference from its spin-splitting partner state $\Lambda(1520)$ $I(J^{P})=0(3/2^{-})$ is larger than that between $N^{*}(1535)\,1/2^{-}$ and $N^{*}(1520)\,3/2^{-}$, it can hardly be accepted by the conventional three-quark picture of the constituent quark model Hyodo and Jido (2012). It is fair to say that the $\Lambda(1405)$ was most probably the first exotic hadron observed Alston et al. (1961). The theoretical aspects of the $\Lambda(1405)$ will be discussed in Sec. VI.4.

II.4.2 Candidates in the charm baryon sector

The two light quarks in a charm baryon can be either in the symmetric sextet or antisymmetric anti-triplet representation of SU(3). Since the color wave function is totally antisymmetric, the spin-flavor-space wave functions must be symmetric. Hence the light-quark system in the $S$-wave flavor sextet (anti-triplet) has spin $1$ ([math]). After combining with a heavy quark, the sextet and anti-triplet give the $B_{6}(1/2^{+})$, $B_{6}^{*}(3/2^{+})$ and $B_{\bar{3}}(1/2^{+})$ baryon multiplets, respectively, as Yan et al. (1992)

[TABLE]

[TABLE]

[TABLE]

All the ground state charm baryons within these three multiplets have been well established in experiments Patrignani et al. (2016). Among the other charm baryons, $\Lambda_{c}(2765)$, $\Xi_{c}(2815)$ and $\Xi_{c}(3123)$ are not well-established from the experimental analysis Patrignani et al. (2016). The $P$-wave $1/2^{-}$ and $3/2^{-}$ antitriplet states are identified as $[\Lambda_{c}(2595)^{+},\Xi_{c}(2790)^{+},\Xi_{c}(2790)^{0}]$ Cheng and Chua (2007, 2015) and $[\Lambda_{c}(2625)^{+},\Xi_{c}(2815)^{+},\Xi_{c}(2815)^{0}]$, respectively Cheng and Chua (2015). Among the remaining charm baryons, the $\Lambda_{c}(2880)^{+}$ has a definite spin of $5/2$ Patrignani et al. (2016); Aaij et al. (2017d) and the quantum numbers of the $\Lambda_{c}(2940)^{+}$ were measured to be $J^{P}=3/2^{-}$ Aaij et al. (2017d). Besides these two charmed baryons, LHCb also measured another charm baryon $\Lambda_{c}(2860)^{+}$ which is consistent with the predication of the orbital $D$-wave $\Lambda_{c}^{+}$ excitation Chen et al. (2016c, 2017b, 2017a). The only available information of other measured charm baryons are their masses and some of their decay modes. For recent reviews on the heavy baryons, we recommend Klempt and Richard (2010); Chen et al. (2017c).

Although the $\Lambda_{c}(2595)^{+}$ may be accommodated as a regular three-quark baryon in quark models Copley et al. (1979); Pirjol and Yan (1997); Tawfiq et al. (1998); Zhu (2000); Blechman et al. (2003); Migura et al. (2006); Zhong and Zhao (2008), one cannot neglect one striking feature of it Hyodo (2013b) which could provide other potential interpretations: It lies between the $\Sigma_{c}(2455)^{+}\pi^{0}$ and $\Sigma_{c}(2455)^{0}\pi^{+}$, $\Sigma_{c}(2455)^{++}\pi^{-}$ thresholds as shown in Table 4. Thus, the $\Lambda_{c}(2595)$ is proposed as a dynamically generated state of the nearby $\Sigma_{c}(2455)\pi$ coupling with other possible higher channels Lutz and Kolomeitsev (2004a); Hofmann and Lutz (2005); Mizutani and Ramos (2006); Garcia-Recio et al. (2009); Jimenez-Tejero et al. (2009); Haidenbauer et al. (2011); Romanets et al. (2012); Lu et al. (2015); Liang et al. (2015); Garcia-Recio et al. (2015); Lu et al. (2016a); Long (2016), such as $ND$, $ND^{*}$ and so on. The strong coupling between the $\Lambda_{c}(2595)$ and the $\Sigma_{c}(2455)\pi$ channels even leads to a prediction of the existence of a three-body $\Sigma_{c}\pi\pi$ resonance in Refs. Long (2016); Long et al. (2017). The analysis in Guo and Oller (2016b), however, indicates that the compositeness of $\Sigma_{c}(2455)^{+}\pi^{0}$ is smaller than $10\%$ leaving $\Lambda_{c}(2595)$ dominated by either other heavier hadronic channels (such as $ND$ and $ND^{*}$) or compact quark-gluon structures. Some other charm baryons, such as $\Lambda_{c}(2880)$ Lutz and Kolomeitsev (2004a), $\Lambda_{c}(2940)$ He et al. (2007, 2010); Ortega et al. (2013); Zhang (2014); Zhao et al. (2017) and $\Sigma_{c}(2800)$ Jimenez-Tejero et al. (2009, 2011); Zhang (2014), have also been considered as dynamically generated states from meson-baryon interactions. In particular, the $\Lambda_{c}(2940)^{+}$ is very close to the $ND^{*}$ threshold — it even overlaps with the threshold if using the recent LHCb measurement Aaij et al. (2017d), and it can couple to $ND^{*}$ in an $S$-wave. Both are favorable features for treating it as an $ND^{*}$ hadronic molecule Ortega et al. (2013); Zhao et al. (2017).

II.4.3 Pentaquark-like structures with hidden-charm

Recently, LHCb reported two pentaquark-like structures $P_{c}(4380)^{+}$ and $P_{c}(4450)^{+}$ in the $J/\psi p$ invariant mass distribution of $\Lambda_{b}\to J/\psi pK^{-}$ Aaij et al. (2015b). Their masses (widths) are $(4380\pm 8\pm 29)$ MeV ($(205\pm 18\pm 86)$ MeV) and $(4449.8\pm 1.7\pm 2.5)$ MeV ($(39\pm 5\pm 19)$ MeV), respectively. In this analysis the $\Lambda^{*}$ states that appear in the crossed channel were parametrized via BW functions. The LHCb analysis reported preference of the spin-parity combinations $(3/2^{-},5/2^{+})$, $(3/2^{+},5/2^{-})$ or $(5/2^{+},3/2^{-})$ for these two states, respectively. The branching ratio for $\Lambda_{b}\to J/\psi pK^{-}$ was also measured Aaij et al. (2016d).

The data for the Cabibbo suppressed process $\Lambda_{b}\to J/\psi p\pi^{-}$ are consistent with the existence of these two $P_{c}$ structures Aaij et al. (2016a). The same experiment also published a measurement of $\Lambda_{b}^{0}\to\psi(2S)pK^{-}$, but no signals for the $P_{c}$ states were observed due to either the low statistics or their absence in the $\psi(2S)p$ channel Aaij et al. (2016b).

The production mechanism and the decay pattern imply a five-quark content of these two states with three light quarks and a hidden heavy $c\bar{c}$ component if they are hadronic states. In fact, pentaquark-like states with hidden charm have been predicted in the right mass region as dynamically generated in meson-baryon interactions a few years before the LHCb discovery Wu et al. (2010, 2011). There are several thresholds in the mass region of the two $P_{c}$ structures, namely $\chi_{c1}p$, $\Sigma_{c}(2520)\bar{D}$, $\Sigma_{c}(2455)\bar{D}^{*}$, $\Lambda_{c}(2595)\bar{D}$, and $\Sigma_{c}(2520)\bar{D}^{*}$ (see Table 4), though not all of them couple in $S$-waves to the reported preferred quantum numbers, suggesting different interpretations of the two $P_{c}$ states, such as $\Sigma_{c}(2455)\bar{D}^{*}$, $\Sigma_{c}(2520)\bar{D}$ or $\Sigma_{c}(2520)\bar{D}^{*}$ hadronic molecules Chen et al. (2015d, b); Roca et al. (2015b); Chen et al. (2016b); He (2016); Karliner and Rosner (2015a); Ortega et al. (2017); Shimizu et al. (2016). It has been suggested that their decay patterns could be used to distinguish among various hadronic molecular options Wang et al. (2016a); Lü and Dong (2016); Shen et al. (2016); Lin et al. (2017). There are also other dynamical studies with different channel bases Xiao (2017); Yamaguchi and Santopinto (2017); Azizi et al. (2016); Geng et al. (2017).

The extreme closeness of the $P_{c}(4450)$ to the $\chi_{c1}p$ threshold and to a TS from a $\Lambda^{*}(1890)\chi_{c1}p$ triangle diagram was pointed out in Ref. Guo et al. (2015c). In Ref. Bayar et al. (2016), it is stressed that the $\chi_{c1}p$ needs to be in an $S$-wave so as to produce a narrow observable peak in the $J/\psi p$ invariant mass distribution, and correspondingly $J^{P}$ needs to be $1/2^{+}$ or $3/2^{+}$. This TS and other possible relevant TSs are also discussed in Liu et al. (2016b); Guo et al. (2016b). It is worthwhile to emphasize that the existence of TSs in the $P_{c}$ region does not exclude a possible existence of pentaquark states whether or not they are hadronic molecules. In Ref. Meißner and Oller (2015), the possibility that the $P_{c}(4450)$ could be a $\chi_{c1}p$ molecule was investigated.

In order to confirm the existence of the two $P_{c}$ states and distinguish them from pure kinematic singularities, c.f. discussions in Sec. VI.1.4, one can either search for them in other processes, such as $\Lambda_{b}\to\chi_{c1}pK^{-}$ Guo et al. (2015c), photoproduction Karliner and Rosner (2016); Kubarovsky and Voloshin (2015); Wang et al. (2015a); Kubarovsky and Voloshin (2016); Gryniuk and Vanderhaeghen (2016); Hiller Blin et al. (2016); Huang et al. (2016), heavy ion collisions Wang et al. (2016b), pion-nucleon reactions Kim et al. (2016); Liu and Oka (2016b); Lin et al. (2017), or search for their strange Feijoo et al. (2016, 2015); Ramos et al. (2016); Chen et al. (2016d); Lu et al. (2016b), neutral Lebed (2015); Lü et al. (2016) and bottomonium Xiao and Meißner (2015) partners. The $\Lambda_{b}\to\chi_{c1}pK^{-}$ decay process has been observed at the LHCb experiment Aaij et al. (2017c).

III Identifying hadronic molecules

Hadronic molecules are analogues of light nuclei, most notably the deuteron. They can be treated to a good approximation as composite systems made of two or more hadrons which are bound together via the strong interactions. In this section the general notion of a molecular state is introduced. As will be demonstrated for near-threshold bound states this picture can be put into a formal definition that even allows one to relate observables directly to the probability to find the molecular component in the bound state wave function. However, it appears necessary to work with a more general notion of hadronic molecules as also resonances can be of molecular nature in the sense formulated above. Before we proceed it appears necessary to review some general properties of the $S$-matrix. In this subsection also the terminology of a bound state, a virtual state and a resonance are discussed for those notions will be heavily used throughout this review.

III.1 Properties of the $S$-matrix

The unitary operator that connects asymptotic $in$ and $out$ states is called the $S$-matrix. It is an analytic function in the Mandelstam plane up to its branch points and poles. The $S$-matrix is the quantity that encodes all physics about a certain scattering or production reaction. In general it is assumed that the $S$-matrix is analytic up to:

• •

Branch points, which occur at each threshold. On the one hand, there are the so called right-hand cuts starting from the branch points at the thresholds for an $s$-channel kinematically allowed process (e.g. at the $\bar{K}K$ threshold in the $\pi\pi$ scattering amplitude). On the other hand, when reactions in the crossed channel become possible one gets the left-hand cuts, which are usually located in the unphysical region for the reaction studied but may still influence significantly, e.g., the energy dependence of a reaction cross section. Branch points can also be located inside the complex plane of the unphysical Riemann sheets: This is possible when the reaction goes via an intermediate state formed by one or more unstable states. It is clear that these threshold branch points/cuts are determined kinematically and happen at the loop level of Feynman diagrams.

In general, a loop Feynman diagram with more-than two intermediate particles has more complicated kinematical singularities. They are called Landau singularities Landau (1959), see, e.g., Eden et al. (1966); Chang (1965); Gribov et al. (2009). For instance, in triangle diagrams the branch points of two intermediate pairs can be very close to the physical region simultaneously, and such a situation gives rise to the so-called triangle singularity already introduced in Sec. II. We come back to those in detail in Sec. IV.1.1.111111The Landau singularity can even be a pole if the one-loop Feynman diagram has at least five intermediate particles Gribov et al. (2009). However, this case is irrelevant for us and will not be considered.

• •

Poles, which appear due to the interactions inherited in the dynamics of the underlying theory. Depending on the locations, poles can be further classified as follows:

– Bound states, which appear as poles on the physical sheet. By causality they are only allowed to occur on the real $s$-axis below the lowest threshold. The deuteron in the isospin-0 and spin-1 proton-neutron system, which can be regarded as the first established hadronic molecule, is a nice example.

– Virtual states, which appear on the real $s$-axis, however, on the unphysical Riemann sheet. A well-known example in nuclear physics is the pole in the isospin-1 and spin-0 nucleon-nucleon scattering. It is within 1 MeV from the threshold and drives the scattering length to a large value of about 24 fm.

– Resonances, which appear as poles on an unphysical Riemann sheet close to the physical one. There is no restriction for the location of poles on the unphysical sheets. Yet, Hermitian analyticity requires that, if there is a pole at some complex value of $s$, there must be another pole at its complex conjugate value, $s^{*}$. Normally, the pole with a negative imaginary part is closer to the physical axis and thus influences the observables in the vicinity of the resonance region more strongly. However, at the threshold both poles are always equally important. This is illustrated in Fig. 7.

For a discussion of the analytic structure of the $S$-matrix with focus on scattering experiments we refer to Ref. Döring et al. (2009) and references therein. Any of these singularities leads to some structure in the observables. In a partial-wave decomposed amplitude additional singularities not related to resonance physics may emerge as a result of the partial-wave projection. For a discussion see, e.g., Ref. Höhler (1983).

From the above classification, it is clear that the branch points are kinematical so that they depend completely on the masses of the involved particles in a certain physical process, while the poles are of dynamical origin so that they should appear in many processes as long as they are allowed by quantum numbers.

We will call a structure observed experimentally a state if and only if the origin of this structure is a pole in the $S$-matrix due to dynamics. On the one hand this definition is quite general as it allows us to also call the above mentioned pole in the isovector nucleon-nucleon scattering a state. From the point of view of QCD, this definition appears to be quite natural since it takes only a marginal change in the strength of the two-hadron potential (e.g. via a small change in quark masses) to switch from a shallow bound state to a near-threshold virtual state, and both leave a striking imprint in observables (we come back to this in Sec. III.3). For example, various lattice QCD groups have observed the di-neutron to become a bound state at quark masses heavier than the physical value Beane et al. (2013); Yamazaki et al. (2012, 2015); Berkowitz et al. (2017). On the other hand, if a structure in the data finds its origin purely in a kinematical singularity without a nearby pole, it would not be called a state. There is currently a heated discussion going on in the literature whether some of the $XYZ$ states are just threshold cusps or triangle singularities Bugg (2004); Chen and Liu (2011); Chen et al. (2013); Swanson (2015, 2016); Pilloni et al. (2016); Gong et al. (2016b). It should be stressed, however, that pronounced near-threshold signals in the continuum channel related to that threshold must find their origin in a nearby pole Guo et al. (2015b).

In the physical world, basically all candidates for hadronic molecules, except for nuclei, can decay strongly and thus can not be bound states in the rigorous sense of the word, since the lowest threshold is defined by the production threshold of the decay products. However, it still appears justified to e.g. call the $f_{0}(980)$ a $K\bar{K}$ bound state, or a quasi-bound state in a more rigorous sense, if the corresponding pole is located on the physical sheet for the two-kaon system, or a virtual state if it is on the unphysical sheet for the two-kaon system, although the lowest threshold is the two-pion threshold.

III.2 Definition of hadronic molecules

In order to proceed it is necessary to first of all define the notion of a molecular state. Naively one might be tempted to argue that if data can be described by a model where all interactions between continuum states come from $s$-channel pole terms the resulting states have to be interpreted as “elementary” states. However, as we will discuss below, this is in general not correct. Analogously, a model that contains only non-pole interactions can still at the end lead to a pole structure of the $S$-matrix that needs to be interpreted as non-molecular. The origin of the failure of intuition in these circumstances is the fact that a hadronic description of hadron dynamics can only be understood in the sense of an effective field theory with limited range of applicability. In particular the very short-ranged parts of the wave function as well as the interaction potential are model-dependent and can not be controlled within the hadronic prescription.

However, at least for near-threshold bound states (the term “near” will be quantified in the next subsection) there is a unique property of the wave function of a molecular state as long as it is formed by a (nearly) stable particle pair in an $S$-wave: The very fact that this particle pair can almost go on-shell leaves an imprint with observable consequences in the analytic structure of the corresponding amplitude, a feature absent to all other possible substructures. In fact, as a consequence of this feature, hadronic molecules can be very extended. To see this observe that a bound state wave function at large distances scales as $\exp(-\gamma r)/r$, where $r$ is the distance between the constituents and $\gamma$ denotes the typical momentum scale defined via

[TABLE]

where $\mu=m_{1}m_{2}/(m_{1}+m_{2})$ denotes the reduced mass of the two-hadron system and

[TABLE]

the binding energy of the state with mass $M$ (note that we chose $E_{B}$ positive so that the bound state is located at $E=-E_{B}$ with $E$ the energy relative to the threshold). Thus, the size $R$ of a molecular state is given by $R\sim 1/\gamma$. Accordingly, if $X(3872)$ with a binding energy of less than 200 keV with respect to the $D^{0}\bar{D}^{*\,0}$ threshold were a molecule, it would be at least as large as 10 fm. For a review of properties of systems with large scattering length we refer to Ref. Braaten and Hammer (2006).

All these issues will be discussed in detail in the sections below. The arguments will start in Sec. III.2.1 from the classic definition introduced by Weinberg long ago to model-independently capture the nature of the deuteron as a proton-neutron bound state. A detailed discussion of the derivation will allow us to explain at the same time the limitations of this definition. Then in Sec. III.2.2 it is demonstrated that the Weinberg criterion is actually identical to the pole counting arguments by Morgan. In Sec. III.2.3 the generalization of the arguments to resonances is prepared by a detailed discussion of pole trajectories that emerge when some strength parameter that controls the location of the the $S$-matrix poles varies. The compositeness criteria for resonances are very briefly discussed in Sec. III.2.4.

III.2.1 The Weinberg compositeness criterion

We start from the following ansatz for the physical wave function of a bound state Weinberg (1965) :

[TABLE]

where $|\psi_{0}\rangle$ denotes the compact component of the state and $|h_{1}h_{2}\rangle$ its two-hadron component. Here compact denotes an object whose size is controlled by the confinement radius $R_{\rm conf.}<1$ fm. Thus this component is assumed to be more compact than $R\sim 1/\gamma$, which denotes the characteristic size of a shallow bound state 121212Actually, we may define the notion “shallow” by the request the $R>R_{\rm conf.}$, which translates into $E_{B}<1/(2\mu R_{\rm conf.}^{2})$. In addition, $\chi(\bm{k})$ is the wave function of the two-hadron part, where $\bm{k}$ denotes the relative momentum of the two particles. In this parameterization, by definition, $\lambda$ quantifies the contribution of the compact component of the wave function to the physical wave function of the state. Accordingly $\lambda^{2}$ denotes the probability to find the compact component of the wave function in the physical state, which corresponds to the wave function renormalization constant $Z$ in quantum field theory. Thus, the goal is to relate $\lambda$ to observables.

In order to proceed one needs to define the interaction Hamiltonian. As shown by Weinberg in Ref. Weinberg (1963b) under very general conditions one may write

[TABLE]

This expression exploits the observation that it is possible by a proper field redefinition to remove all hadron-hadron interactions from the theory and to cast them into $\psi_{0}$ Weinberg (1963a, b). Then the two-hadron Hamiltonian is given simply by the kinetic term $\hat{H}_{hh}^{0}=k^{2}/(2\mu)$, where $\mu=m_{1}m_{2}/(m_{1}+m_{2})$ denotes the reduced mass of the two-hadron system and $m_{i}$ the mass of hadron $h_{i}$. Introducing the transition form factor,

[TABLE]

one finds the wave function in the momentum space as

[TABLE]

The wave function of a physical bound state needs to be normalized to have a probabilistic interpretation. We thus get

[TABLE]

As mentioned above, $\lambda^{2}$ is in fact the wave function renormalization constant $Z$, since the integral in the last line of Eq. (14) is nothing but the energy derivative of the self-energy. Because of the positivity of the integral, $\lambda^{2}$ is bound in the range between 0 and 1, and thus allows for a physical probabilistic interpretation for a bound state.

At this point a comment is necessary: in many textbooks on quantum field theory it is written that the wave function renormalization constant $Z$ is scheme dependent and is to be used to absorb the ultraviolet (UV) divergence of the vertex corrections. Clearly this is correct. However, the scheme dependence and UV divergence are only for the terms analytic in $E$. What we find here is the LO piece of $Z$ in an energy expansion around the threshold,131313More discussion on this point can be found in Sec. IV.2. and as this piece is proportional to $\sqrt{E}$ it can not be part of the Lagrangian. Thus, the Weinberg criterion as outlined is based explicitly on the presence of the two-particle cut which is responsible for the appearance of the square root, whose presence is a distinct feature of the two-hadron component.

The integral in Eq. (14) converges if $f(\bm{k})$ is a constant. The denominator contains solely model independent parameters, while the momentum dependence of the numerator is controlled by the relevant momentum range of the vertex function that may be estimated by $\beta$, the inverse range of forces. Thus, if $\beta\gg\gamma$, the integral can be evaluated model-independently for the case of $S$-wave coupling which implies the constant $g_{0}=f(0)$ as the LO piece of $f(\bm{k})$.141414Note that in some works a model for the form factor $f(\bm{k})$ is employed Faessler et al. (2007). Then one finds

[TABLE]

From this we find the desired relation, namely

[TABLE]

which provides a relation between $\lambda^{2}$, the probability of finding the compact component of the wave function inside the physical wave function, and $g_{0}$, the bare coupling coupling constant of the physical state to the continuum, or $\lambda g_{0}$, the physical coupling constant.

The quantity $g_{0}$ appears also in the physical propagator of the bound state since the self energy is given by

[TABLE]

We may therefore write for the $T$-matrix of the two continuum particles whose threshold is close to the location of the bound state,

[TABLE]

where the subscript “NR” is a reminder of the nonrelativistic normalization used in the above equation. As long as the pole is close to the threshold, the amplitude near threshold should be dominated by the pole term (the non-pole terms are again controlled quantitatively by the range of forces). Using $E_{B}=-E_{0}+\Sigma(-E_{B})-g_{0}^{2}\mu\gamma/(2\pi)$, which absorbs the (divergent) leading contribution of the real part into the bare pole energy and at the same time takes care of the fact that the analytic continuation of the momentum term also contributes at the pole,151515When $E$ takes real values, the square root on the first sheet is defined by $\sqrt{2\mu E+i\epsilon}=+i\sqrt{-2\mu E}\,\theta(-E)+\sqrt{2\mu E}\,\theta(E)$. we get

[TABLE]

where we have introduced the two-hadron relative momentum $k=\sqrt{2\mu E}$. Note that Eq. (19) is nothing but the one channel version of the well-known Flatté parametrization Flatté (1976). Thus, a measurement of near-threshold data allows one in principle to measure the composition of the bound state wave function, in line with the effective field theory analysis to be discussed later in Sec. VI, although in practice a reliable extraction of the coupling might be hindered by a scale invariance of the Flatté parametrization that appears for large couplings Baru et al. (2005). The phenomenological implications especially of Eq. (16) on Eq. (19) and generalizations thereof will be discussed in Sec. III.3.

To make the last statement explicit we may match Eq. (19) onto the effective range expansion

[TABLE]

and find

[TABLE]

Thus, for a pure molecule ($\lambda^{2}=0$) one finds that the scattering length gets maximal, $a=-1/\gamma$, and in addition $r={\mathcal{O}}(1/\beta)$, where the latter term is typically positive, while for a compact state ($\lambda^{2}=1$) one gets $a=-{\mathcal{O}}(1/\beta)$ (in the presence of a bound state the scattering length is necessarily negative within the sign convention chosen here) and $r\to-\infty$. These striking differences have severe implications on the line shapes of near-threshold states as will be discussed in detail in Sec. III.3.

It is illustrative to apply the Weinberg criterion to the deuteron, basically repeating the analysis presented already in Ref. Weinberg (1965). The scattering length and effective range extracted from proton-neutron scattering data in the deuteron channel are Klarsfeld et al. (1984)

[TABLE]

where the sign of the scattering length was adapted to the convention employed here. Furthermore, the deuteron binding energy reads Van Der Leun and Alderliesten (1982) 161616The reference quotes $E_{B}=2.224575(9)$ MeV, however, for the analysis here such a high accuracy is not necessary.

[TABLE]

On the other hand, in case of the deuteron the range of forces is provided by the pion mass — accordingly the range corrections that appear in Eqs. (21) may in this case be estimated via

[TABLE]

Thus the effective range is of the order of the range corrections (and positive!) as required by the compositness criterion for a molecular state. Using $\lambda^{2}=0$ in the expression for the scattering length we find

[TABLE]

also consistent with Eq. (22). Based on these considerations Weinberg concluded that the deuteron is indeed composite.

As mentioned previously, a location of a molecular state very near a threshold is quite natural, while a near-threshold compact state is difficult to accomplish Jaffe (2007); Hanhart et al. (2014). This can now be nicely illustrated on the basis of Eqs. (20) and (21). By construction for $k=i\gamma$ the $T$ matrix develops a pole which may be read off from Eq. (20)

[TABLE]

For a (nearly) molecular state $a\simeq-1/\gamma$ and $r\sim{\mathcal{O}}(1/\beta)$. Thus, for this case Eq. (26) is largely saturated by the scattering length term, and the range term provides only a small correction. However, for a predominantly genuine state we have $-1/a\simeq\beta\gg\gamma$ and $r\to-\infty$. Thus in this case a subtle fine tuning between the range term and the scattering length term appears necessary for the pole to be located very near threshold.

While the low-energy scattering of the hadrons that form the bound state is controlled by scattering length and effective range, production reactions are sensitive to the residue of the bound state pole, which serves as the effective coupling constant, to be called $g_{\rm eff}$, squared of the bound state to the continuum. It is simply given by the bare coupling constant $g_{0}^{2}$ introduced above multiplied by the wave function renormalization constant $Z$, which is $\lambda^{2}$ as explained above,

[TABLE]

After switching to a relativistic normalization by multiplying with $\left(\sqrt{2m_{1}}\sqrt{2m_{2}}\sqrt{2M}\right)^{2}$, and dropping terms of order $(E_{B}/M)$, we thus get

[TABLE]

What is interesting about this equation is that it is bounded from above: The effective coupling constant of a bound system to the continuum gets maximal for a pure two-hadron bound state. Since $1-\lambda^{2}$ is the probability of finding the two-hadron composite state component in the physical wave function, it is sometimes called “compositeness”. Using Eq. (21), the effective coupling can be expressed in terms of the scattering length

[TABLE]

which reduces to $-4M^{2}/(\mu a)$ in the limit of $\lambda^{2}=0$, reflecting the universality of an $S$-wave system with a large scattering length Braaten and Hammer (2006).

Before closing this section some comments are necessary.

• •

The approach allows for model-independent statements only for $S$-waves, since otherwise in the last integral of Eq. (14) there appears in the numerator of the integrand an additional factor $k^{2L}$ from the centrifugal barrier. Accordingly, the integral can no longer be evaluated model-independently without introducing additional parameters (regulator) to cope with the UV divergence.

• •

For the same reason the continuum channel needs to be a two-body channel, since otherwise the momentum dependence of the phase space calls for an additional suppression of the integrand.

• •

The binding momentum must be small compared to the inverse range of forces, since otherwise the range corrections get larger than the terms that contain the structure information.

• •

For the applicability of the formalism as outlined and an unambiguous probabilistic interpretation, the state studied must be a bound state, since otherwise the normalization condition of Eq. (14) is not applicable which is at the very heart of the derivation. However, nowadays there exist generalizations of the Weinberg approach also to resonances which will be discussed in Sec. III.2.4.

• •

The constituents that form the bound state must be narrow, since otherwise the bound system would also be broad Filin et al. (2010); Guo and Meißner (2011).

For long it seemed that the conditions are satisfied only by the deuteron and Weinberg therefore closed his paper with the phrase Weinberg (1965): “One begins to suspect that Nature is doing her best to keep us from learning whether the ‘elementary’ particles deserve that title.” However, as outlined in the introduction, there are now various near-threshold states confirmed experimentally that appear to be consistent with those criteria, like $X(3872)$, $D_{s0}^{*}(2317)$ and less rigorously $f_{0}(980)$ and others.

For illustration we would like to compare what is known about the effect of the $D_{s0}^{*}(2317)$ on $DK$ scattering to the Weinberg criterion. Clearly, $DK$ scattering can not be measured directly in experiment, however, it can be studied in lattice QCD using the so-called Lüscher method Lüscher (1991). A first study using this method for the $DK$ system is presented in Ref. Mohler et al. (2013a). The scattering length and effective range extracted in this work for the lowest pion mass ($M_{\pi}=156$ MeV) are $-(1.33\pm 0.20)$ fm and $(0.27\pm 0.17)$ fm, respectively. This number is to be compared to the Weinberg prediction for a purely molecular state of $a=-(1\pm 0.3)$ fm and $r\sim 0.3$ fm, where the inverse $\rho$-mass was assumed for the range of forces and we used that for molecular states the effective range is positive and of the order of the range of forces. Scattering lengths of the same size were also extracted from a study of the scattering of the light pseudoscalars off $D$-mesons using unitarized chiral perturbation theory Liu et al. (2013a). Thus from both chiral dynamics on the hadronic level as well as lattice QCD there are strong indications that $D_{s0}^{*}(2317)$ indeed is a $DK$ molecule. The lattice aspects will be further discussed in Sec. V. Clearly, a direct experimental confirmation of the molecular assignment for $D_{s0}^{*}(2317)$ is very desirable. A possible observable could be the hadronic width of $D_{s0}^{*}(2317)$ as discussed in Sec. VI.1.3.

So far we only focused on bound states. However, also very near-threshold poles on the second sheet not accompanied by a first sheet pole, so-called virtual states, leave a striking imprint in observables, c.f. Sec. III.1. A $T$-matrix that has its pole on the second, instead of on the first, sheet reads, in distinction to Eq. (19),

[TABLE]

where now the virtual pole is located on the second sheet at $E=-E_{v}$, with $E_{v}>0$ and we still use $\gamma$ to denote $\sqrt{2\mu E_{v}}$. Here we use that on the second sheet below threshold the momentum is $-i|\sqrt{2\mu E}|$.

III.2.2 The pole counting approach

One of the classic approaches put forward to distinguish molecular states from genuine ones is the so-called pole counting approach Morgan (1992), which may be summarized as: A bound state that is dominated by its compact component (in the language of the previous section this implies $\lambda^{2}$ close to 1) manifests itself in two near–threshold singularities (one on the first sheet, one on the second) while a predominantly molecular bound state gives rise only to a single near-threshold pole on the first sheet.

To see that these criteria actually map perfectly on the Weinberg criterion it is sufficient to observe that the poles of Eq. (20) are given by

[TABLE]

Based on the sign convention employed in this work, cf. Eq. (20), in the presence of a bound state the scattering length is negative. In addition, keeping only the leading terms for both the scattering length $a$ and the effective range $r$ as shown in Eqs. (21), one obtains

[TABLE]

Thus It is easy to see that $k_{1}$ and $k_{2}$ are positive and negative imaginary numbers, respectively, which implies that the former is a pole on the first Riemann sheet (a bound state pole), while the latter is located on the second sheet. When $\lambda$ approaches 0, which implies that the molecular component of the state becomes increasingly important, the second pole disappears towards negative imaginary infinity, which leaves $k_{1}$ as the only relevant pole. In particular one gets from this for the asymmetry of the pole locations

[TABLE]

Thus the asymmetry of the pole locations is a direct measure of the amount of molecular admixture in the bound state wave function (as defined within the Weinberg approach) in line with the findings of Ref. Morgan (1992). The close relation between the two approaches was first observed in Ref. Baru et al. (2004).

III.2.3 Remarks about pole trajectories

QCD is characterized by a small number of parameters, namely the quark masses, the number of colors ($N_{c}$) and $\Lambda_{\rm QCD}$ (the running coupling constant). Accordingly those parameters determine completely the hadron spectrum. With advanced theoretical tools it became possible recently to investigate the movement of the QCD poles as QCD parameters are varied. There exist studies for varying quark masses as well as varying numbers of colors, $N_{c}$,—both of them allowing for deeper insights into the structure of the investigated states.

Studies that vary the number of colors are available mostly for light quark systems. For a recent review, we refer to Ref. Peláez (2016). In order to connect the $N_{c}$ dependence of a given state in the spectrum to QCD in Ref. Peláez (2016) a unitarized version of chiral perturbation theory, the so-called inverse amplitude method, is employed, where the fact is exploited that the leading $N_{c}$ behavior of the low-energy constants (LECs) is known. Thus, once the unitarized amplitudes are fitted, e.g., to phase shifts it is possible to investigate the impact of a varying $N_{c}$ by proper rescaling of these LECs. This kind of study was pioneered by the work reported in Ref. Peláez (2004), where it was demonstrated that the $N_{c}$ scaling of the vector mesons $\rho$ and $K^{*}$ is in line with expectations for $\bar{q}q$ states, however, that of $f_{0}(500)$ and $K_{0}^{*}(800)$ is completely at odds with them. While the $N_{c}$ studies allow one to distinguish the quark content of different states, they do not allow one to disentangle hadronic molecules from other four-quark structures. In addition, both mentioned resonances are very broad and as such do not allow one to straightforwardly quantify their molecular component following the approach of the previous section.

Existing studies where quark masses are varied allow for a more direct contact to the discussion of the previous section. Understanding the quark mass dependence of hadrons with different composition is not only interesting on its own sake, it is also important since lattice QCD studies can be performed at arbitrary quark masses (and in fact are often performed at enlarged quark masses for practical reasons). Thus, as soon as we can relate certain pole trajectories to the structure of the hadron it becomes feasible to “measure” the nature of the state using lattice QCD. More direct methods to use the lattice to determine the nature of certain hadrons will be discussed in Sec. V.

Let us consider pole trajectories of resonances as some generic strength parameter is varied. Here we follow the presentation in Ref. Hanhart et al. (2014). In this work it is shown that in the presence of a pole the one-channel $S$-matrix can be written as (for simplicity assuming the masses of the continuum particles to be equal)

[TABLE]

where $\xi\geq 0$. The unimodular form of Eq. (34) is because of unitarity of the $S$-matrix. The above parameterization accounts for the facts that if the $S$-matrix has a pole at some complex momentum on the second sheet, $k_{p}-i\xi$, it also has to have a pole $-k_{p}-i\xi$, which is the realization of the Schwarz reflection principle in momentum space, and that any pole on the second sheet is accompanied by a zero on the first. As shown by the last equality, the momentum space expression can be straightforwardly mapped onto the $s$-plane, where $s_{0}=4(k_{p}^{2}+\xi^{2}+m^{2})$ was introduced. In the $s$-plane the Schwarz reflection principle calls for poles at complex conjugate points.

To investigate the general behavior of the pole trajectories it is sufficient to vary the parameter $k_{p}^{2}$ from some finite positive value to some finite negative value. Typical trajectories are shown in Fig. 8. The trajectories for $S$-waves are depicted by the solid lines and for higher partial waves by the dashed lines. As long as $k_{p}^{2}$ is positive ($k_{p}$ is real), Eq. (34) develops two complex conjugate poles for all partial waves. When $k_{p}^{2}$ gets decreased, the poles approach each other and eventually, for $k_{p}^{2}=0$, meet on the real axis. One of the poles switches to the first sheet at the point were $k_{p}^{2}+\xi^{2}=0$, which is the threshold and at least for $S$-waves requires a negative value of $k_{p}^{2}$ ($k_{p}$ is imaginary).

The first nontrivial observation that can be read off Eq. (34) and Fig. 8 straightforwardly is that $S$-waves and higher partial waves behave very differently: The reason for this is the centrifugal barrier that forces one to introduce a momentum dependence into $\xi$ according to

[TABLE]

This has a striking impact on the pole trajectories: For any $L>0$ the $\xi$-term is zero at $k=0$ and therefore the point where the two pole trajectories meet ($k_{p}^{2}=0$) coincides with $k=0$ which denotes the threshold. This is different for $S$-waves: in this case the poles can meet somewhere below the threshold. Then, when $k_{p}^{2}$ is decreased further to negative values, both poles move away from the meeting point, such that one approaches the threshold while the other one goes away from the threshold. This behavior can easily be interpreted via the pole counting approach: The further away from the threshold the point is located where the two trajectories meet (this point is determined by the value of $\xi$ at the point where $k_{p}^{2}=0$), the more asymmetric are the two poles once one of them has switched to the first sheet, and thus the molecular component of the state is more pronounced.

To make more explicit the above connection between the trajectories and the molecular nature of the states we may study the scattering length and the effective range that emerge from Eq. (34):

[TABLE]

If we now use that the binding momentum $\gamma=\kappa_{p}-\xi$, where we introduced $\kappa_{p}=ik_{p}$, we can read off from the above equations and Eq. (21)

[TABLE]

To see the implications of this expression we may parameterize the relevant quantities via

[TABLE]

where $\delta>0$ and $\epsilon>0$. Here it was already used that for a bound state to exist with a finite binding energy $\kappa_{p}$ must exceed $\xi$. A vanishing binding momentum ($\gamma\to 0$) can be achieved either by $\epsilon\to 0$ for finite $\delta$, which immediately implies that $\lambda^{2}\to 0$, so that the state is purely molecular, or $\delta\to 0$ for finite $\epsilon$. This case allows for a compact admixture of the very near-threshold pole, however, at the price of an extreme fine tuning of $\kappa_{p}$ and $\xi$ as both then need to go to zero simultaneously. This is the same kind of fine tuning already observed for non-molecular near-threshold states in Sec. III.2 from a different perspective.

As an example in Ref. Hanhart et al. (2014) it was demonstrated explicitly that the pole trajectories of the $f_{0}(500)$ meson as well as the $\rho$ meson that emerge when the quark masses are varied can be easily parameterized in terms of the parameters $\xi$ and $\kappa_{p}$ introduced above. In particular it was shown that while $\xi$ changes only mildly in the parameter range studied $k_{p}^{2}$ changes a lot and in particular $\xi$ is sizable at the point where $k_{p}^{2}$ is zero in the $f_{0}(500)$ channel. Accordingly the authors of Ref. Hanhart et al. (2014) conclude that, at least for unphysically large quark masses, the $f_{0}(500)$ meson behaves like a hadronic molecule.

What should be clear from the considerations above is that states born off hadron-hadron dynamics with poles above the relevant threshold are necessarily broad; after all their coupling to this continuum channel is maximal. This should also be clear from the pole trajectories illustrated in Fig. 8. An example for such a scenario is the very broad $f_{0}(500)$ most probably generated by nonperturbative $\pi\pi$ interactions. Such a property is in contrast to the tetraquark picture advocated in Ref. Esposito et al. (2016a), where the authors argue that tetraquarks that are visible in experiment must be narrow and slightly above threshold. It is therefore of utmost importance that the pole locations of exotic candidates are determined with high precision.

III.2.4 Generalizations to resonances

The first work where the Weinberg approach was generalized to resonances is Ref. Baru et al. (2004), where the spectral density was employed to supplement the parameter $\lambda^{2}$ introduced above for bound states. The subject was later elaborated in various papers Hyodo et al. (2012); Aceti and Oset (2012); Hyodo (2013b, a); Sekihara et al. (2015); Sekihara (2017); Guo and Oller (2016a); Xiao and Zhou (2016b, 2017); Kang et al. (2016); Xiao and Zhou (2016a); however, what is common to all of them is that a quantitative, probabilistic extraction of the level of compositeness is not possible rigorously as soon as one moves to resonances. The reason is that states that belong to poles on the second sheet are not normalizable and as such one looses the condition of Eq. (14) that is crucial for the probabilistic interpretation.

However, it still appears reasonable to take over the key finding of Sec. III.2, namely that the coupling of a state is larger for a larger molecular component, also to resonance states. As we will see in the following paragraphs, in certain situations this leads to quite striking observable consequences. When being translated to an effective field theory this observation implies that for molecular states loop contributions appear always already at LO. The resulting power counting will be detailed in Sec. IV.

When the state of interest is located below the production threshold of the two hadrons that possibly form the molecular state, one can still distinguish between quasi-bound states and virtual states, depending on whether the leading pole is located on the first or the second sheet with respect to the mentioned two-hadron system. The phenomenological implications of this will be discussed in some detail in Sec. III.3.

III.3 Characteristic line shapes of hadronic molecules

Besides the deuteron all other (candidates for) hadronic molecules are unstable. Then the scattering $T$-matrix needs to be modified compared to the form discussed in Sec. III.2. In particular Eq. (19) now reads

[TABLE]

where $E=k^{2}/(2\mu)$ and $\Gamma_{0}$ accounts for inelasticities not related to the channel whose threshold is nearby. Those channels will be called inelastic channels below. We also changed the parameter that controls the pole location from $E_{B}$ to $-E_{r}$ since it now refers to a resonance instead of a bound state. Following the logic of the previous sections in the near-threshold regime the dominant energy/momentum dependence for molecular states comes from the term proportional to $g^{2}$ which is very large in this case, c.f. Eq. (16). On the contrary, for compact states the $k^{2}$ term controls the momentum dependence. As a result of this in the former case the line shape of the state that appears in any of the inelastic channels is very asymmetric while in the latter it is symmetric. The two scenarios are sketched in Fig. 9. In addition, the line shape for the molecular state shows a clearly visible nonanalyticity at the two-particle threshold which would be much weaker in the other case.171717It is completely absent only if the coupling between the state with the two hadrons vanishes. However, in this case the $T$-matrix given in Eq. (39) vanishes as well. Its presence follows directly from Eq. (39), since

[TABLE]

with $\partial k/\partial E=\sqrt{\mu/(2E)}$. It is this derivative that is not continuous when the energy crosses zero, the location of the threshold. One might expect from this discussion that the coupling $g^{2}$ can be read off from the line shape directly allowing for a direct interpretation of the structure of the underlying state. However, a scale invariance of the expression for the line shape appears as soon as the $g^{2}$-term dominates, and it hinders a quantitative study in practice Baru et al. (2005). In addition, even for large values of $g^{2}$ the nonanalyticity might well not show up clearly in the line shape since its visibility depends in addition on a subtle interplay of $E_{r}$, $\Gamma_{0}$ and $\gamma$.

It was already mentioned at the end of Sec. III.2.2 that also near-threshold virtual poles leave a striking impact on observables. In the presence of inelastic channels a virtual state always leads to a peak of the line shape exactly at the threshold while a near-threshold bound state still has strength even below the threshold. For very near-threshold states like the $X(3872)$ these two scenarios might be difficult to disentangle.

As first stressed in Ref. Braaten and Lu (2007) the line shapes in the elastic channel might well be very interesting, if at least one of the constituents of the system studied is unstable so that some strength of the amplitude leaks into the region below the threshold 181818At the same time the nonanalyticity discussed above gets smeared out.. To be concrete: If a state is located near the threshold of particles $A$ and $B$ and $A$ decays to $a$ and $b$, then the spectra in the $abB$ channel need to be studied both below and above the nominal $AB$ threshold. To implement the necessary changes, in the formulae above for narrow constituents one may simply replace the momentum $k$ in Eq. (39) by Braaten and Lu (2007)

[TABLE]

where $\Gamma$ denotes the width of the unstable constituent, and $\mu$ is the reduced mass of the two-hadron system evaluated using the mass of the unstable state. In addition, the subtraction term $\gamma$ needs to be replaced by

[TABLE]

where the upper sign leads to a (quasi-)bound state while the lower one to a virtual state. Clearly, for $\Gamma\to 0$ these expressions map nicely on those used for $k$ and $\gamma$ above for $E_{r}<0$. Eqs. (41) and (42) hold as long as the line shape for the unstable constituent is well described by a BW distribution, namely for $\Gamma/2\ll m_{A}-m_{a}-m_{b}$. As soon as the energy dependence of $\Gamma$ starts to matter, more sophisticated expressions need to be used, c.f. Ref. Hanhart et al. (2010). For simplicity we here use the expressions given above. The resulting line shapes in the elastic channel are shown for various values of $\Gamma$ in Fig. 10, where for illustration we used the parameters

[TABLE]

The left panel shows the results for the (quasi-)bound state ($\gamma_{\rm eff}>0$), and the right one is for the virtual state ($\gamma_{\rm eff}<0$). Clearly, above the nominal two-hadron channel ($E=0$) the spectra in the left and the right panels look very much alike, however, for $\Gamma>0$ drastic differences appear between the two cases for negative values of $E$.

Following the Weinberg criterion, for the bound state case it is the relative height of the peak for $E<0$ and the bump for $E>0$, which are difficult to distinguish for the largest value of $\Gamma$, that is a measure of the molecular admixture of the studied state. Therefore a high resolution measurement of the line shape of $X(3872)$ would be very valuable to deduce its nature Hanhart et al. (2007a); Braaten and Lu (2007); Hanhart et al. (2010); Meng et al. (2015); Kang and Oller (2017).

In this context it is interesting to note that a line shape very similar to the one shown in the left panel of Fig. 10 was also predicted for $Y(4260)\to D^{*}\pi\bar{D}$ under the assumption that $Y(4260)$ is a $D_{1}\bar{D}$ molecular state (note that $D^{*}\pi$ is the most prominent decay channel of $D_{1}(2420)$) Wang et al. (2014), see the middle panel of Fig. 6. A similar line shape also shows up in the calculation in Ref. Debastiani et al. (2017a) for the $f_{1}(1285)$ strongly coupled to $K^{*}\bar{K}$.

Everything said so far had the implicit assumption that there are at most two near-threshold poles on the two relevant sheets. The possible line shapes change dramatically as soon as additional poles are located in the near-threshold regime as is discussed in detail in Refs. Baru et al. (2010); Artoisenet et al. (2010); Hanhart et al. (2011).

III.4 Heavy quark spin symmetry

In the limit of infinitely heavy quarks, the spin of heavy quarks decouples from the system and is conserved individually. As a result, the total angular momentum of the light degrees of freedom becomes a good quantum number as well. This gives rise to the so-called heavy quark spin symmetry (HQSS) Isgur and Wise (1989). In the real world quarks are not infinitely heavy, however, heavy quark effective field theory allows one to systematically include corrections that emerge from finite quark masses in a systematic expansion in $\Lambda_{\rm QCD}/M_{Q}$, where $M_{Q}$ denotes the heavy quark mass. For an extensive review we refer to Ref. Neubert (1994). HQSS is the origin for the near degeneracy of, e.g., $D^{*}$ and $D$ as well as $B^{*}$ and $B$. Similarly, it also predicts straightforwardly multiplets of hadronic molecules made of a heavy hadron or heavy quarkonium and light hadrons Guo et al. (2009a); Yamaguchi et al. (2015), and of hadroquarkonium as well Cleven et al. (2015). In addition, HQSS allows one to predict ratios of different transitions involving heavy hadrons in the same spin multiplet, and in particular for your interest transitions of hadronic molecules. Examples for those predictions can be found in Refs. Fleming and Mehen (2008, 2012) where the decays of the $X(3872)$ into the final states $\chi_{cJ}\pi$ and $\chi_{cJ}\pi\pi$ are discussed in the XEFT framework which will be discussed in Sec. IV.1. The ratios among various decays of the $Z_{b}(10610)$ and $Z_{b}(10650)$ into $h_{b}(mP)\pi$ and $\chi_{bJ}(mP)\gamma$ (from the neutral $Z_{b}$ states) were computed in both XEFT Mehen and Powell (2011) and NREFT${}_{\text{I}}$ Cleven et al. (2013) frameworks to be discussed in the next section. They are consistent with the result solely based on the HQSS Ohkoda et al. (2012a). For other predictions based on HQSS on the radiative and strong decays of hadronic molecules in the heavy-quarkonium sector, we refer to Ma et al. (2014, 2015).

In special cases HQSS also allows one to make predictions for bound systems of two or more heavy mesons Voloshin (2011); Mehen and Powell (2011); Nieves and Valderrama (2012); Guo et al. (2013b); Liu (2013); Baru et al. (2016) since certain potentials get linked to each other. This will be discussed in details in Sec. IV.2.

IV Nonrelativistic effective field theories

All the candidates for hadron resonances, and in particular the candidates of hadronic molecules, which are the focus of this review, were discovered via their strong decays into other hadrons. Therefore, to understand these structures requires also a study of their decays. Because of the nonperturbative nature of QCD at hadronic energy scales, a first-principle calculation of the spectrum of hadronic resonances at the level of quarks and gluons can only be done using lattice QCD. Although there has been tremendous progress in lattice QCD, a reliable calculation of the full hadronic resonance spectrum for physical quark masses is still out of reach. In addition, even if such calculations were available, the interpretation of the emerging spectra still requires additional theoretical analyses.

Only in the special case discussed in Sec. III, i.e. for shallow bound states coupling in an $S$-wave to a nearby continuum channel comprised of two stable or at least narrow hadrons, one finds a direct and physical interpretation for the leading and nonanalytic contribution of the wave function renormalization constant $Z$ as the (normalizable) probability to find the continuum contribution in the physical state. Because of the closeness of the threshold to the mass of the physical composite state, such systems are ideal objects to apply the concept of effective field theories (EFTs), which makes use of the separation of scales and which per definition include a cutoff Lepage (1989). Of particular relevance here are the nonrelativistic EFTs (NREFTs). Note that the general principles underlying any EFT are formulated in Weinberg’s paper on phenomenological Lagrangians Weinberg (1979).

As mentioned in Sec. III, hadronic molecules are located close to some strongly coupled thresholds. We denote the low-energy (low-momentum) scale characterizing such a system, given by the binding energy (binding momentum) defined in Eq. (9) (Eq. (8)), generically by $Q$. All other hadronic scales that we may collectively label as $\Lambda$ are thus regarded as hard. This enables one to construct a perturbation theory in $Q/\Lambda$, which for near-threshold states should be a small number. As will become clear below, it depends on the system which scale is appropriate for $\Lambda$. For example when investigating the $f_{0}(980)$ as a candidate for a $\bar{K}K$ molecular state, the inverse range of forces, the natural candidate for $\Lambda$, is given by the mass of the allowed lightest exchange meson, the rho meson. A phenomenologically adequate value for the binding energy is 10 MeV. It corresponds to a binding momentum of 70 MeV, and thus $Q/\Lambda\sim 1/10$ is a good expansion parameter.191919The subtle interplay of scales in molecular transitions is is discussed in detail in Ref. Hanhart et al. (2007b) on the example of decays of the $f_{0}(980)$. Furthermore, the closeness to threshold also means that the constituent hadrons can be treated nonrelativistically.

As discussed in the preceding section, the most interesting information about the structure of a near-threshold state is contained in its coupling strength to the threshold channel, which measures the probability for finding the two-body bound state component in the physical state. This is consistent with the intuition that a state is the more composite the larger its coupling to the continuum. As shown in Eq. (28), for a bound state the coupling reaches its maximal value, if the physical state is purely an $S$-wave bound state, $\lambda^{2}=0$. Hence, it is important to extract the value of the coupling constant for understanding the nature of near-threshold structures. In addition, a large coupling implies the prominence of hadronic loops not only in the formation of the state but also in transitions and decays. In this section, we will discuss the NREFT formalism which is a natural framework for studying the transitions involving hadronic molecules with a small energy release. It can also be used to compute the universal long-distance part of the production/decay processes of hadronic molecules, which will be discussed in Sec. VI.

The analytic structure of the three-point scalar loop integral (including the TS) will be discussed in Sec. IV.1.1. The power counting rules for the NREFT treating all intermediate particles on the same footing will be detailed in Sec. IV.1.2. We denote such a theory as NREFT${}_{\text{I}}$. When one of the intermediate particles is much more off-shell than the others, it can be integrated out from NREFT${}_{\text{I}}$ and one gets another effective field theory, here called NREFT${}_{\text{II}}$, which was originally introduced as XEFT to study the properties of the $X(3872)$. The XEFT and its relation to NREFT${}_{\text{I}}$ will be discussed in Secs. IV.1.3 and IV.1.4. Sec. IV.2 is devoted to a brief description of the formation of hadronic molecules.

The formation of hadronic molecules may be viewed as a result of nonperturbative hadron-hadron interactions. It is therefore natural to ask if there is also an impact of hadron loops on the properties of more regular excited hadrons. Indeed, for certain transitions the effective field theory NREFT${}_{\text{I}}$ mentioned above predicts prominent loop effects. As examples, we briefly discuss single-pion/eta transitions and hindered M1 transitions between heavy quarkonia in Sec. IV.3. It will become clear that whether the hadron-loop effects are important for properties of an excited hadron is process-dependent. In particular, the location of an excited hadron close to a threshold is a necessary but not a sufficient condition.

IV.1 Power counting schemes

As demonstrated in Sec. III, the decisive feature of molecular states as compared to more compact structures is the prominence of a two-hadron cut. In some decays the cuts induced by intermediate particles might also matter. To illustrate this point, we start this section by a discussion of the analytic structure of three-point loop functions. This will shed light on the NREFT power counting as well.

IV.1.1 Analytic structure of the three-point loop integral

If a hadronic molecule has at least one unstable constituent, it can decay directly through the decays of that unstable particle when phase space allows. It can also decay into another heavy particle with a mass of the same order by emitting light particles such as pions or photons from its constituents. The mechanism for a transition accompanied by the emission of a single light particle is depicted in Fig. 11. In the figure the two vertical dashed lines show the relevant branch cuts: They correspond to the time slices at which the intermediate particles can go onto their mass shells.

We denote the intermediate particles as $M_{1,2,3}$ with masses $m_{1,2,3}$, and the external particles as $A,B,C$ with masses $m_{A,B,C}$, as shown in Fig. 11. If all intermediate particles are nonrelativistic we can formulate a power counting based on the velocities of the intermediate particles. Let us start from the scalar triangle loop integral

[TABLE]

where $\epsilon=0^{+}$, $N_{m}=8m_{1}m_{2}m_{3}$, $T_{i}(p)=p^{2}/2m_{i}$ denotes the kinetic energy for a heavy meson with mass $m_{i}$, and $E_{C}$ the energy of the particle $C$ in the rest frame of the initial particle $A$. The second line is obtained by treating all the intermediate states nonrelativistically in the rest frame of the initial particle. Performing the contour integration over $l^{0}$, one gets a convergent integral over the three-momentum. Defining $\mu_{ij}=m_{i}m_{j}/(m_{i}+m_{j})$, $b_{12}=m_{1}+m_{2}-m_{A}$ and $b_{23}=m_{2}+m_{3}+E_{C}-m_{A}$, one has

[TABLE]

where $c_{1}=2\mu_{12}b_{12}$, and $c_{2}=2\mu_{23}b_{23}+\left(\mu_{23}/m_{3}\right){q}^{2}$ with $q\equiv|\bm{q}|$. The two terms in the denominator of the integrand contain a unitary cut each, as indicated by the vertical dashed lines in Fig. 11. The other two-body cut crossing the lines of $M_{1}$ and $M_{3}$ corresponds to the case that the particle $M_{3}$ is propagating back in time (we assume implicitly that it is $M_{1}$ and not $M_{2}$ that can decay to $M_{3}$ and $C$ in near on-shell kinematics). This is a relativistic effect which is neglected here. The intermediate particles $M_{1}$ and $M_{2}$ are on shell when $\bm{l}^{\,2}+c_{1}=0$; $M_{2}$ and $M_{3}$ (as well as $C$) are on shell for $\bm{l}^{\,2}+c_{2}-{2\mu_{23}}\bm{l}\cdot\bm{q}/m_{3}=0$. Accordingly, $\sqrt{|c_{1}|}$ and $\sqrt{|c_{2}|}$ define two different momentum scales where the corresponding intermediate states go on shell. Their values depend on all of the masses involved and may be very different from each other. For the nonrelativistic approximation to hold both must be small compared to $m_{i}(i=1,2,3)$.

The integral of Eq. (45) can be presented in closed form Guo et al. (2011); Mehen (2015)

[TABLE]

where ${\cal N}=\mu_{12}\mu_{23}/(2\pi m_{1}m_{2}m_{3})$, and $a=\left(\mu_{23}/m_{3}\right)^{2}{q}^{2}$. Especially Eq. (47) highlights the presence of a special singularity at

[TABLE]

When rewriting the inverse trigonometric functions in terms of logarithms, one finds that this is a logarithmic divergence. The solution of this equation gives the leading Landau singularity Landau (1959) (for early and recent reviews, see Eden et al. (1966); Chang (1965); Aitchison (2015)) for a triangle diagram, also called triangle singularity, evaluated in nonrelativistic kinematics Guo et al. (2014b). The singularity location is slightly shifted from that found by solving the relativistic Landau equation. A comparison for a specific example can be found in the appendix of Ref. Guo et al. (2014b).

Being nonlinear in all of the involved masses, Eq. (48) as well as the Landau equation allow for different solutions. However, a direct evaluation of the loop integral reveals that only in a very restricted kinematics, one of the solutions produces an observable effect, namely when this solution is located on the physical boundary, i.e., the upper edge of the branch cut in the first Riemann sheet or alternatively the lower edge of the branch cut in the second Schmid (1967), see Fig. 7. In this case, the TS can produce a narrow peak in the invariant mass distribution, which may even mimic a resonance. This effect was already indicated in Sec. II.1.2 and will be further illustrated in Sec. VI.1.4. We therefore discuss now under which circumstances the singularity appears on the physical boundary. This case is contained in the Coleman–Norton theorem Coleman and Norton (1965) (for triangle diagrams see Ref. Bronzan (1964)). The physical picture becomes most transparent using the simple triangle singularity equation derived in Ref. Bayar et al. (2016):

[TABLE]

where $q_{\rm on+}$ is the center-of-mass (CM) momentum of particles $M_{1}$ and $M_{2}$ when they are on shell, and $q_{a-}$ is the momentum of particle $M_{2}$ in the rest frame of A when $M_{2}$ and $M_{3}$ are on shell (being on shell is necessary but not sufficient to define $q_{a-}$ as will be discussed immediately). One finds

[TABLE]

where

[TABLE]

are the energy and the magnitude of the three-momentum of particle $M_{2}$ in the rest frame of particle B, i.e. the CM frame of the ($M_{2}$, $M_{3}$) system, respectively, $\beta=q/E_{B}$ is the magnitude of the velocity of particle B in the rest frame of A, and $\gamma=1/{\sqrt{1-\beta^{2}}}={E_{B}}/{m_{B}}$ is the Lorentz boost factor. Eq. (49) is the condition for the amplitude $I(q)$ to have a TS on the physical boundary. Note that if particle $M_{1}$ can go on shell simultaneously with $M_{3}$ and $C$, it must be unstable. Consequently, its width moves the logarithmic divergence into the complex plane and the physical amplitude becomes finite.

Let us consider the kinematical region where the momentum of particle $M_{2}$ is positive so that Eq. (49) can be satisfied: $p_{2}=q_{a-}=\gamma(\beta\,E_{2}^{*}-p_{2}^{*})>0$. Then $p_{3}=\gamma(\beta\,E_{3}^{*}+p_{2}^{*})$ (where $E_{3}^{*}$ is the energy of particle $M_{3}$ in the rest frame of particle B), the momentum of particle $M_{3}$ in the rest frame of the initial particle, is positive as well. This means that particles $M_{2}$ and $M_{3}$ move in the same direction in that frame. The corresponding velocities are given by

[TABLE]

respectively. It is easy to see that $p_{2}>0$ leads to

[TABLE]

which means that particle $M_{3}$ moves faster than $M_{2}$ and in the same direction in the rest frame of the initial particle A. This, together with the requirement that all intermediate particles are on their mass shells, gives the condition for having a TS on the physical boundary. This is in fact the Coleman–Norton theorem Coleman and Norton (1965) applied to the triangle diagram: the singularity is on the physical boundary if and only if the diagram can be interpreted as a classical process in spacetime. For other discussions about TSs using the Mandelstam variables, we refer to two recent works Szczepaniak (2015); Liu et al. (2016a) and references therein.

To finish this section, we point out again that the TS mechanism has been around for more than half a century, but only in recent years has become a viable tool in hadron physics phenomenology due to the data discussed in this review. In fact, many of the calculations outlined in which the TS plays a dominant role can be and often are done without recourse to an EFT. Still, in a broader view it can nicely be embedded in the framework outlined here. In any case, whenever the TS can play a role, it has to be included.

IV.1.2 NREFT${}_{\text{I}}$

A key component for any EFT is the power counting in terms of some dimensionless small quantity, which allows for a systematic expansion and an estimate for the uncertainty of the calculation caused by the truncation of the series at some finite order. The natural small quantity in nonrelativistic systems is the velocity $v$ (measured in units of the speed of light) which is much smaller than one by assumption.

As mentioned in Sec. IV.1.1, triangle diagrams with all three intermediate particles being nonrelativistic in fact have two momentum scales given by $\sqrt{|c_{1}|}$ and $\sqrt{|c_{2}|}$. Accordingly, one may define $v_{1}=\sqrt{|c_{1}|}/(2\mu_{12})$ and $v_{2}=\sqrt{|c_{2}-a|}/(2\mu_{23})$ for the velocities of the intermediate mesons.

From the previous analysis, three-point loop diagrams have two kinds of singularities: two-body threshold cusps and TSs. The two-body threshold singularities are encoded in the two velocities defined above. When the TS, with its location implicitly defined via Eq. (48), is not in the considered kinematic region, the loop function of Eq. (47) can be expanded in a power series as

[TABLE]

When the masses of all three intermediate particles are similar, $m_{i}\sim m$, the LO term in the above equation may be written as

[TABLE]

Thus the arithmetic mean of the two velocities characterizes the size of the triangle loop. It is therefore the relevant parameter to estimate the leading loop contribution for the transition of a heavy state into a light state and another heavy state.

The power counting in nonrelativistic velocities for a given loop diagram can be obtained by applying the following rules: The three-momentum of the intermediate nonrelativistic particles counts as $\mathcal{O}\!\left(v\right)\!\,$, the nonrelativistic energy counts as $\mathcal{O}\!\left(v^{2}\right)\!\,$, and each nonrelativistic propagator is of $\mathcal{O}\!\left(v^{-2}\right)\!\,$. Thus, $I(q)$ scales as $\mathcal{O}\!\left(v^{5}/(v^{2})^{3}\right)\!\,=\mathcal{O}\!\left(v^{-1}\right)\!\,$. Comparing with Eq. (55), one sees that the velocity in the power counting should be understood as the average of $v_{1}$ and $v_{2}$ Guo and Meißner (2012b). In addition to the parts discussed the amplitude for a given process can have factors of the external momentum $q$. To be general we do not count the external momentum $q$ in powers of $v$, but keep it explicitly. This defines the power counting as detailed in Refs. Guo et al. (2009c, 2010b, 2011). We denote this theory as NREFT${}_{\text{I}}$.

In order to demonstrate how the power counting rules work, we compare in Fig. 12 the values of $1/v$ with $v=(v_{1}+v_{2})/2$ and an explicit calculation of the loop function as given in Eq. (46). The curves are normalized at $m_{A}=4.22$ GeV. The values used for the calculation are:

[TABLE]

For the external light particle we take $m_{C}=0.140$ GeV for Fig. 12 (a) and $m_{C}=0$ for Fig. 12 (b). In addition, we take two values for $m_{B}$, 3.886 GeV and $3.872$ GeV, and the results are shown as (a) and (b), respectively. Then (a) and (b) correspond to the loop integrals in the amplitudes for the $Y(4260)\to Z_{c}(3900)\pi$ and $Y(4260)\to X(3872)\gamma$, respectively, which will be discussed later in Sec. VI.1. Using Eq. (49) or (48), we find that for $m_{B}=3.886$ GeV, there is a TS at $m_{A}=4.288$ GeV, which is the reason for the sharp peak in the dashed line in Fig. 12 (a). Note that in the plots the widths of the intermediate mesons were neglected. For $m_{B}=3.872$ GeV, which is smaller than $m_{2}+m_{3}$, and $m_{C}=0$, the TS moves to the complex plane at $m_{A}=(4.301-i\,0.018)$ GeV with a clearly visible effect on the line shape, see the dashed line in Fig. 12 (b). One sees from the figure that the simple power counting rule of Eq. (55) agrees remarkably well with the explicit calculation except for energies very close to the TS.

In addition to counting the loop integral as discussed above, one also needs to take into account the vertices in order to obtain a proper estimate for a given loop amplitude. To illustrate the method let us start from the simplest two-point self-energy diagram shown in Fig. 13. We assume that the mass of the state is close to the threshold of the internal particles that can therefore be treated nonrelativistically. If the coupling is in an $S$-wave, then the loop scales as $\mathcal{O}\!\left(v^{5}/(v^{2})^{2}\right)\!\,=\mathcal{O}\!\left(v\right)\!\,$.202020Here we only focus on the velocity scaling and neglect the geometric factor of $1/(4\pi)$. If the coupling is in a $P$-wave, each vertex contributes an additional factor of $v$ and the loop scales as $\mathcal{O}\!\left(v^{3}\right)\!\,$. Of course, the real part of the loop integral is divergent, and the resulting correction to the mass is scale-dependent. However, since the scale dependence can be formally absorbed into the bare mass of the state this discussion is not of relevance here. Thus, we find that the effect of the two-hadron continuum on the self-energy of heavy quarkonia is parametrically suppressed, if the state is close to the threshold which implies a small value of $v$, and that this suppression increases for increasing orbital angular momentum of the two-hadron state.

Next we consider the one loop diagram for the decay process $\text{A}\to\text{B\,C}$, with A and B heavy and C light, as depicted in Fig. 11. To be concrete, we assume that C couples to the intermediate states $M_{1}$ and $M_{3}$ in a $P$-wave (such as the pion couples to the ground state heavy mesons). This coupling structure leads to a factor of $q$ (in the rest frame of A). The generalization to other situations is easy. The power counting rules for a few typical cases are then as follows:

• (1)

Both A and B couple to the intermediate states in an $S$-wave. As a result the final state particles $B$ and $C$ must be in a $P$-wave. Therefore the expression of Eq. (55) needs to be multiplied by $q$. Still, the $1/v$ enhancement factor quantifies the relative importance of the triangle diagram for the transition: the closer both A and B to the corresponding thresholds, the more important the intermediate states. On top of this may come an additional enhancement driven, e.g., by large couplings characteristic for molecular states as derived in Sec. III.

• (2)

Either A or B couples to the intermediate states in an $S$-wave with the other one in a $P$-wave. In this case, because there is only one possible linearly independent external momentum for two-body decays, the internal momentum at the $P$-wave vertex must be turned into an external momentum. The amplitude scales as $\mathcal{O}\!\left(q^{2}/(m^{2}v)\right)\!\,$. Since the decay should be in an $S$-wave in this case, we have introduced a factor $1/m^{2}$ to balance the dimension of the $q^{2}$ factor Guo et al. (2011) as in this case the loop contribution needs to be compared to a constant tree-level contribution.

• (3)

Both A and B couple to the intermediate states in a $P$-wave. Each $P$-wave vertex contributes a factor of the internal momentum. In the power counting of NREFT${}_{\text{I}}$, the external momentum is kept explicitly. As a result, there are two possibilities for the scaling of the $P$-wave vertices: Each $P$-wave vertex scales either as $m\,v$ or as the external momentum $q$. More insights can be obtained if we take a closer look at the relevant tensor loop integral:

[TABLE]

In the rest frame of the initial particle, it can be decomposed into an $S$-wave part and a $D$-wave part as

[TABLE]

where

[TABLE]

are the $S$- and $D$-wave projectors, respectively, which satisfy $P_{S}^{ij}P_{S}^{ij}=1$, $P_{D}^{ij}P_{D}^{ij}=1$, and $P_{S}^{ij}P_{D}^{ij}=0$. Then in the $S$-wave part $I_{S}(q)$, the internal momentum scales as $\mathcal{O}\!\left(v\right)\!\,$, and $I_{S}(q)\sim\mathcal{O}\!\left(v\right)\!\,$. In the $D$-wave part $I_{D}(q)$, the internal momentum turns external, and one gets $I_{D}(q)\sim\mathcal{O}\!\left(q^{2}/(m^{2}v)\right)\!\,$, which would have the same scaling as $I_{S}(q)$ if $q/m\sim v$.212121Noticing that $P_{S}^{ij}l^{i}l^{j}=l^{2}/\sqrt{3}$ and $P_{D}^{ij}l^{i}l^{j}=2l^{2}P_{2}(\cos\theta)/\sqrt{6}$ with $P_{2}(\cos\theta)$ the second Legendre polynomial, it can be shown that $I_{S}(q)$ is UV divergent while $I_{D}(q)$ is UV convergent Albaladejo et al. (2015); Shen et al. (2016). The power counting of the $D$-wave part was not discussed in Ref. Guo et al. (2011). For the decay amplitude, the factor of $q$ from the vertex coupling C to intermediate states needs to be taken into account additionally.

This nonrelativistic power counting scheme was proposed in Ref. Guo et al. (2009c) to study the coupled-channel effects of charm meson loops in charmonium transitions, and studied in detail later Guo et al. (2011). Applications to transitions between two heavy quarkonium states can be found in Refs. Guo et al. (2010b, c); Guo and Meißner (2012a); Mehen and Yang (2012); Guo and Meißner (2012b); Guo et al. (2014b), and to transitions involving one or two $XYZ$ states in Refs. Cleven et al. (2011a, 2013); Guo et al. (2013a); Esposito et al. (2015c); Mehen (2015); Huo and Chen (2016); Wu et al. (2016); Abreu and Lafayette Vasconcellos (2016); Chen et al. (2017d). In particular in Ref. Cleven et al. (2013) the implications of items (1) and (2) are demonstrated. It is shown that, while the transitions of the $Z_{b}$ states to $\Upsilon(nS)\pi$ potentially suffer from large higher order corrections, the transitions to $h_{b}(mP)\pi$ and $\chi_{bJ}(mP)\pi$ should be dominated by the triangle topology. The near-threshold cross section for $e^{+}e^{-}\to D\bar{D}$ was studied in Chen and Zhao (2013) using NREFT as well.

It is clear that the power counting can only be applied to processes where the intermediate hadrons are nonrelativistic and especially close to their mass shells. Otherwise the loop diagrams receive contributions from large momenta and cannot be treated in a simple EFT including only the hadronic degrees of freedom of A, B, C, $M_{1}$, $M_{2}$ and $M_{3}$.

IV.1.3 NREFT${}_{\text{II}}$ and XEFT

Because the $X(3872)$ is arguably the most important and interesting candidate for a hadronic molecule, here we will discuss in some detail one NREFT designed specifically for studying the properties of the $X(3872)$. It is called XEFT and was proposed in Ref. Fleming et al. (2007) following the Kaplan–Savage–Wise approach to describe the nucleon-nucleon system Kaplan et al. (1998a, b). It can be regarded as a special realization of NREFT${}_{\text{II}}$. Similar effective theories can be constructed for other possible hadronic molecules which are located very close to thresholds. For instance, in the framework of a similar theory, the $Z_{b}(10610)$ and $Z_{b}(10650)$ were studied in Refs. Mehen and Powell (2011, 2013) and the $Z_{c}(3900)$ in Ref. Wilbring et al. (2013).

The XEFT assumes the $X(3872)$ to be a hadronic molecule of $D^{0}\bar{D}^{*0}+c.c.$. The tiny binding energy Patrignani et al. (2016)

[TABLE]

implies that the long-distance part of the $X(3872)$ wave function is universal and is insensitive to the binding mechanism which takes place at a much shorter distance. The long-distance degrees of freedom are $D^{0}$, $D^{*0}$, $\bar{D}^{0}$, $\bar{D}^{*0}$ and $\pi^{0}$. All of them are treated nonrelativistically. For processes dominated by the long-distance scales such as the decays $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ and $X(3872)\to D^{0}\bar{D}^{0}\gamma$ which can occur via the decay of the vector charm meson directly, the XEFT at LO can reproduce the results from the effective range theory which makes use of the universal two-body wave function of the $X(3872)$ at asymptotically long distances Voloshin (2004b, 2006)

[TABLE]

where the $X(3872)$ is assumed to be below the $D^{0}\bar{D}^{*0}$ threshold, and $\gamma_{0}=\sqrt{2\mu_{0}B_{X}}\leq 20$ MeV with $\mu_{0}$ the reduced mass of $D^{0}$ and $\bar{D}^{*0}$. Yet, it has the merit of being improvable order by order by including local operators and pion exchanges although unknown short-distance coefficients will be involved. For processes involving shorter-distance scales such as the decays of the $X(3872)$ into a charmonium and light particles, the XEFT can still be used by parameterizing the short-distance physics in terms of local operators employing factorization theorems and the operator product expansion Braaten and Kusunoki (2005b); Braaten and Lu (2006). The XEFT can also be used even if the $X(3872)$ is a virtual state with a nonnormalizable wave function Hanhart et al. (2007a) or a resonance above threshold.

The power counting and the NLO corrections to the decay $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ were studied in Ref. Fleming et al. (2007). The XEFT was also used to study the decays of the $X(3872)$ to the $\chi_{cJ}$ with one and two pions Fleming and Mehen (2008, 2012), the radiative transitions $X(3872)\to\psi(2S)\gamma$, $\psi(4040)\to X(3872)\gamma$ Mehen and Springer (2011) and $\psi(4160)\to X(3872)\gamma$ Margaryan and Springer (2013), the scattering of an ultrasoft pion Braaten et al. (2010) or $D$ and $D^{*}$ Canham et al. (2009) off the $X(3872)$, and the quark mass dependence and finite volume corrections of the $X(3872)$ binding energy Jansen et al. (2014, 2015). The relation between the XEFT and the formalism of NREFT${}_{\text{I}}$ was clarified in Ref. Mehen (2015). As an extension of the XEFT in Ref. Alhakami and Birse (2015) a modified power counting was suggested to take into account also an expansion in the ratio between the pion mass and the charm meson masses. The need for such an expansion is removed, however, as soon as Galilean invariance is imposed on the interactions Braaten (2015).

In the following, we use the decay $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ as an example to illustrate the power counting of the XEFT. The binding momentum $\gamma_{0}\leq 20$ MeV sets the long-distance momentum scale in this theory. The typical momenta for the $D^{0}$ and $D^{*0}$ are of the order $p_{D}\sim p_{D^{*}}\sim\gamma_{0}$. The pion kinetic energy is less than 7 MeV, and thus the momentum for either an internal or external pion is also counted as $p_{\pi}\sim\gamma_{0}$. Furthermore, the pion exchange introduces another small scale $\mu=\sqrt{\Delta^{2}-M_{\pi^{0}}^{2}}\simeq 44$ MeV, with $\Delta=M_{D^{*0}}-M_{D^{0}}$. Denoting all the small momentum scales by $Q$, we have

[TABLE]

Thus, the measure for one-loop integral is of $\mathcal{O}\!\left(Q^{5}\right)\!\,$, and each nonrelativistic propagator is of $\mathcal{O}\!\left(Q^{-2}\right)\!\,$. All Feynman diagrams can then be assigned a power of $Q$.

The XEFT keeps as the degrees of freedom only those modes with a very low-momentum $\sim\gamma_{0}$. The binding momentum for the $D^{+}D^{*-}+c.c.$ channel at the $X(3872)$ mass is $\gamma_{c}\simeq 126$ MeV. It is treated as a hard scale, and the charged charm mesons are integrated out from the XEFT.

Denoting the field annihilating the $D^{0},\bar{D}^{0}$, $D^{*0}$ and $\bar{D}^{*0}$ by $D,\bar{D}$, $\bm{D}$ and $\bm{{\bar{D}}}$, respectively, and taking the phase convention that $X(3872)$ is $\left(D^{0}\bar{D}^{*0}+\bar{D}D^{*0}\right)/\sqrt{2}$, the relevant Lagrangian for the calculation up to the NLO is written as Fleming et al. (2007)

[TABLE]

where $\delta=\Delta-M_{\pi^{0}}\simeq 7$ MeV and $F_{\pi}=92.2$ MeV is the pion decay constant. The first line contains the kinetic terms for the pseudoscalar and vector charm mesons as well as for the nonrelativistic pion, the second is for the axial coupling of the pion to charm mesons with $g\simeq 0.6$ determined from the $D^{*}$ width, the third line contains the LO and NLO contact interaction terms, and the fourth line contains the terms for a short-distance emission of a pion. The contact terms in the last line were not considered in Ref. Fleming et al. (2007), but, as is argued below, also contribute to $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ at NLO. In particular, the $C_{0D}$ term may have a significant impact on the line shapes as will become clear in the discussion below.

The Feynman diagrams relevant for the calculation of the $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ decay width up to NLO are shown in Fig. 14. Diagram (a) contributes at LO, (b,c) and (e,f) are the NLO diagrams calculated in Ref. Fleming et al. (2007), and (d,g) are two new diagrams from the new terms in the last line of the Lagrangian in Eq. (63). Here we only discuss the power counting for each diagram and the contributions missing in the original work Fleming et al. (2007), and refer to Ref. Fleming et al. (2007) for details of the calculation. One essential point of the XEFT is that the pion-exchange is treated perturbatively based on the observation that the two-pion exchange contribution is suppressed relative to the one-pion exchange by

[TABLE]

Then the $X(3872)$ is generated through a resummation of the $D\bar{D}^{*}$ contact terms (the charge conjugated $\bar{D}D^{*}$ channel is always implied). The pole of the $X(3872)$ is at $E=-B_{X}$, and thus at LO

[TABLE]

where

[TABLE]

is the two-point one-loop integral containing nonrelativistic $D^{0}$ and $\bar{D}^{*0}$ propagators in the power divergence subtraction (PDS) scheme Kaplan et al. (1998a, b), where $E$ is the energy defined relative to the threshold and $\Lambda_{\rm PDS}$ is the PDS scale. For Eq. (65) to be renormalization group invariant, $C_{0}$ needs to absorb the scale dependence of the loop integral:

[TABLE]

Keeping only momentum modes of order $Q$, the power counting for the loop integral is $\Sigma_{0}(E)=\mathcal{O}\!\left(Q^{5}/(Q^{2})^{2}\right)\!\,=\mathcal{O}\!\left(Q\right)\!\,$. One sees that the scale-independent part of $C_{0}$, $\bar{C}_{0}=[\Lambda_{\rm PDS}+1/C_{0}(\Lambda_{\rm PDS})]^{-1}=2\pi/(\mu_{0}\gamma_{0})$, indeed scales as $Q^{-1}$.

Now we consider the power counting of the decay amplitudes from the diagrams in Fig. 14. The decay rate can be obtained from these amplitudes taking into account properly the wave function renormalization $Z$ Fleming et al. (2007) which accounts for the insertion of the $X(3872)$ interpolating field shown as circled crosses in Fig. 14. Notice that for the calculation of the decay rate up to the NLO, one needs $Z$ up to NLO (LO) for the LO (NLO) amplitude. The amplitude from diagram (a) scales as $\mathcal{O}\!\left(Q/Q^{2}\right)\!\,=\mathcal{O}\!\left(Q^{-1}\right)\!\,$ since there is one nonrelativistic propagator and one $P$-wave vertex which gives a factor of $p_{\pi}\sim Q$. Both one-loop diagrams (b) and (c) have four nonrelativistic propagators and three $P$-wave vertices, and thus scale as $\mathcal{O}\!\left(Q^{0}\right)\!\,$, one order higher than the LO diagram (a). The coefficients $C_{2}$ and $B_{1}$ scale as $Q^{-2}$ Fleming et al. (2007). Noticing that there are two derivatives in the $C_{2}$ term and one derivative in the $B_{1}$ term in the Lagrangian, the amplitudes from diagrams (e) and (f) should be counted as $\mathcal{O}\!\left(Q^{0}\right)\!\,$ as well.

Let us discuss diagrams (d) and (g) which were missing in the original calculation in Ref. Fleming et al. (2007). The $C_{\pi}$ contact term can be matched to the chiral Lagrangian for the interaction between heavy and light mesons Burdman and Donoghue (1992); Wise (1992); Yan et al. (1992); Guo et al. (2008a). At LO of the chiral expansion the interaction between pions and pseudoscalar heavy mesons receives contributions from the Born term from the exchange of $D^{*}$, which constitutes a subdiagram to (b) and (c), and the Weinberg–Tomozawa term. It turns out that the amplitude for $D^{0}\pi^{0}\to D^{0}\pi^{0}$ vanishes at LO. At NLO of the chiral expansion, there are several operators, see Refs. Guo et al. (2008a, 2009b). In particular, it is easy to see that the $h_{0}$ and $h_{1}$ terms therein are proportional to the light quark mass or equivalently to $M_{\pi}^{2}$. The Feynman rule for the $D^{0}\pi^{0}\to D^{0}\pi^{0}$ vertex from these two terms (using relativistic normalization for all the fields) is

[TABLE]

The value of $h_{1}$ is fixed to be 0.42 from the mass splitting between the $D_{s}$ and $D$ mesons, and the $1/N_{c}$ suppressed parameter $h_{0}\simeq 0.01$ from fitting to the lattice data for the pion mass dependence of charm meson masses Liu et al. (2013a). One sees $\mathcal{A}_{h_{0},h_{1}}\simeq 0.65$. Hence, by matching to the chiral Lagrangian, $C_{\pi}$ should scale as $Q^{0}$, which leads to the scaling of $\mathcal{O}\!\left(Q^{0}\right)\!\,$ for diagram (d).

Diagram (g) involves a short-distance contact interaction between $D^{0}$ and $\bar{D}^{0}$. If the vertex $C_{0D}$ scales as $Q^{0}$, then diagram (g) $=\mathcal{O}\!\left(Q^{0}\right)\!\,$. However, the situation could be more complicated. From the HQSS analysis of the $X(3872)$ in Sec. III.4, the $X(3872)$ as a $D\bar{D}^{*}$ hadronic molecule should have three spin partners in the strict heavy quark limit. One of them has quantum numbers $J^{PC}=0^{++}$ and couples to $D\bar{D}$ and $D^{*}\bar{D}^{*}$. Therefore, there is the possibility that the $D\bar{D}$ interaction needs to be resummed to generate a near-threshold pole. In this case, $C_{0D}$ needs to be promoted to be $\mathcal{O}\!\left(Q^{-1}\right)\!\,$, analogous to $C_{0}$. Then diagram (g) appears at $\mathcal{O}\!\left(Q^{-1}\right)\!\,$ making it a LO contribution. Clearly this can cause a large correction to the $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ decay rate. This effect can be seen clearly in Fig. 15, which is the result obtained in Ref. Guo et al. (2014a) using NREFT${}_{\text{I}}$ in combination with the framework to be discussed in Sec. IV.2. The unknown parameter $C_{0a}$ in the figure parameterizes the isoscalar part of $D^{\dagger}\bar{D}^{\dagger}D\bar{D}$ contact interaction, see Eq. (81) below, playing a role similar to $C_{0D}$ introduced in Eq. (63).

Since $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ is an important process sensitive to the long-distance structure of the $X(3872)$, it would be interesting to revisit it considering the missing diagrams in XEFT. In particular, it was found that the nonanalytic corrections from the pion-exchange diagrams (b) and (c) of Fig, 14 only contribute to $\sim 1\%$ of the decay rate Fleming et al. (2007). Whether this remains true after considering diagram (d) remains to be seen.

It should be stressed that the role of nonperturbative pions on the $X(3872)$ properties is studied in various papers Baru et al. (2011, 2013, 2016) which in many cases confirm the results of XEFT. However, also in these studies diagrams of the types shown in diagrams (d) and (g) of Fig. 14 were not included.

IV.1.4 From NREFT${}_{\text{I}}$ to XEFT

From the discussions above, we see that all momentum scales much larger than $\gamma_{0}\leq 20$ MeV have been integrated out from the XEFT. This is different from NREFT${}_{\text{I}}$, where all nonrelativistic modes are kept as effective degrees of freedom including those with a momentum of the order of a few hundreds of MeV. NREFT${}_{\text{I}}$ when applied to the $X(3872)$ can be regarded as the high-energy theory for the XEFT. The short-distance operators in XEFT at the scale of a few hundreds of MeV can be matched to NREFT${}_{\text{I}}$. This is discussed in some detail in Ref. Mehen (2015) in the context of calculations of the reactions $X(3872)\to\chi_{cJ}\pi^{0}$.

To show the relation between NREFT${}_{\text{I}}$ and XEFT explicitly let us consider the case $c_{2}\gg c_{1}$. The quantities $c_{2}$ and $c_{1}$ introduced in Eq. (45) define the locations of the two-body cuts of the triangle diagram. In the low-momentum region $l\sim\sqrt{c_{1}}$, the second factor in the integrand of Eq. (45) can be expanded in powers of $l^{2}/c_{2}$ and one gets

[TABLE]

The resulting momentum integral in the first line is divergent and needs to be regularized. The natural UV cutoff of the new effective theory is set by $\Lambda<\sqrt{c_{2}}$. We denote such a theory as NREFT${}_{\text{II}}$. It reduces to the XEFT when applied to the $X(3872)$. In order to compare with the XEFT, in the second line of the above equation we evaluate the integral in the PDS scheme which is equivalent to the sharp cutoff regularization by letting $\Lambda_{\rm PDS}=2\Lambda/\pi$ and dropping the terms of $\mathcal{O}\!\left(1/\Lambda\right)\!\,$. For a detailed comparison of dimensional versus cutoff regularization we refer to Ref. Phillips et al. (1999).

For $m_{1}=M_{D^{*0}}$, $m_{2}=M_{D^{0}}$, $E_{C}=E_{\pi}$, and $m_{A}=M_{X}$, the second line of the above equation reduces to

[TABLE]

where $\Delta_{H}=M_{D^{0}}+m_{3}-M_{X}$, and the term $q^{2}/(2m_{3})$ has been neglected. Terms of the above form appear in the XEFT amplitudes for transitions between the $X(3872)$ and a charmonium with the emission of a light particle Fleming and Mehen (2008, 2012); Mehen and Springer (2011); Margaryan and Springer (2013).

The different power countings of XEFT and NREFT${}_{\text{I}}$ has various implications that we now illustrate by two examples:

Since NREFT${}_{\text{I}}$ keeps all nonrelativistic modes explicitly, the charged $D\bar{D}^{*}$ channel which has a momentum of $\gamma_{c}\simeq 126$ MeV needs to be kept as soft degrees of freedom. On the contrary, the XEFT only keeps the ultrasoft neutral charm mesons dynamically and the charged ones are integrated out. It was pointed out in Ref. Mehen (2015) that it is crucial to take into account the charged charm mesons for the calculation of the $X(3872)\to\chi_{cJ}\pi^{0}$ decay rate in NREFT${}_{\text{I}}$ because their contribution cancels to a large extent the one from the neutral charm mesons as usual in isospin violating transitions (c.f. the discussion in Sec. VI.1.3).222222The role of the charged charm mesons for certain decays of the $X(3872)$ was already stressed in Gamermann and Oset (2009). The situation for decays into an isoscalar pion pair, $X(3872)\to\chi_{cJ}\pi\pi$, is different. We expect that the charged and neutral channels are still of similar order, but add up constructively.

Furthermore, in the XEFT calculation for $X(3872)\to\chi_{cJ}\pi^{0}$, there appears a new, reaction specific short-distance operator, labeled by $C_{\chi,0}$ in Fig. 16 (c). To estimate its size it is matched onto two contributions in heavy meson chiral perturbation theory in Refs. Fleming and Mehen (2008, 2012). Those are given by the exchange of a charm meson, which is proportional to the $\chi_{cJ}H\bar{H}$ coupling constant $g_{1}$, and a contact term accompanied by a low energy constant $c_{1}$, shown as diagram (a) and (b) in Fig. 16, respectively. The final result in XEFT then depends on the unknown ratio $g_{1}/c_{1}$. In NREFT${}_{\text{I}}$, however, the two contributions appear at different orders, since the amplitude from diagram (b) is suppressed by $v^{2}$ compared with that from diagram (a).

IV.2 Formation of hadronic molecules

While so far the focus was on transitions of molecular candidates, we now turn to their formation through two-hadron scattering. For illustration we focus in this chapter on the scattering of open-flavor heavy mesons off their antiparticles in a framework of NREFT similar to the EFT for nucleon-nucleon interactions Epelbaum et al. (2009). The example of the formation of $\Lambda(1405)$ from similar dynamics is discussed in Sec. VI.4. In this section we mainly discuss the method used in Refs. Nieves and Valderrama (2011, 2012); Valderrama (2012); Hidalgo-Duque et al. (2013b); Guo et al. (2013b, c). It is based on the Lippmann–Schwinger equation (LSE) regularized using a Gaussian vertex form factor. The coupled-channel LSE reads

[TABLE]

where $\mu_{n}$ is the reduced mass in the $n$-th channel, $E$ is the energy defined relative to threshold of the first channel, and $\Delta_{n1}$ is the difference between the $n^{\rm th}$ threshold and the first one. When the potential takes a separable form $V_{ij}(\bm{k}^{\prime},\bm{k})=\xi_{i}{(\bm{k}^{\prime})}V_{ij}\varphi_{j}{(\bm{k})}$, where the $V_{ij}$ are constants, the equation can be simplified greatly. In addition, for very near-threshold states one should expect a momentum expansion for the potential to converge fast and a dominance of $S$-waves. Both the separability as well as the absence of higher partial waves will be spoiled as soon as the one-pion exchange is included on the potential level; this case will be discussed briefly later in this section.

With a UV regulator such as of the Gaussian form, see, e.g., Ref. Epelbaum et al. (2009),

[TABLE]

the LSE can be solved straightforwardly. If the $T$-matrix has a near-threshold bound state pole, the effective coupling of this composite state to the constituents can be obtained by calculating the residue of the $T$-matrix element at the pole. For simplicity, we consider a single-channel problem with the LO contact term: $V(\bm{k}^{\prime},\bm{k})=C_{0}e^{-\bm{k}^{\prime 2}/\Lambda^{2}}e^{-\bm{k}^{2}/\Lambda^{2}}$. The nonrelativistic $T$-matrix element for the scattering of the two hadrons is then given by

[TABLE]

where

[TABLE]

is the nonrelativistic two-point scalar loop function defined in Eq. (66) but evaluated with a Gaussian regulator. After renormalization by absorbing the cutoff dependence into $C_{0}$, we obtain

[TABLE]

The binding momentum $\gamma$ was defined in Eq. (8). The effective coupling is obtained by taking the residue at the pole $E=-E_{B}$:

[TABLE]

It does not depend on $C_{0}$, and is scale independent up to terms suppressed by $1/\Lambda$. Multiplying $g_{\text{NR}}^{2}$ by the factor $(8m_{1}m_{2}M)$ to get the relativistic normalization, we recover the expression for $g_{\text{eff}}$ derived in Eq. (28) for $\lambda^{2}=0$. Thus we find that a potential of the kind given in Eq. (72) generates hadronic molecules. Deviations of this result behavior can be induced, e.g., by momentum dependent interactions (or terms of order $\gamma/\Lambda$). This observation formed the basis for the generalization of the Weinberg compositeness criterion presented in Refs. Aceti and Oset (2012); Hyodo et al. (2012); Hyodo (2013a); Sekihara et al. (2015).

To proceed we first need to say a few words about the scattering of heavy mesons. For infinitely heavy quarks the spin of the heavy quark decouples, and accordingly in a reaction not only the total angular momentum is conserved but also the spin of the heavy quark and thus the total angular momentum of the light quark system as well. Therefore, a heavy-light quark system can be labeled by the total angular momentum of the light quark system $j_{\ell}$. Accordingly the ground state mesons $D$ and $D^{*}$ ($\bar{B}$ and $\bar{B}^{*}$) form a doublet with $j^{P}_{\ell}=1/2^{-}$, where we on purpose deviate from the standard notation $s_{\ell}^{P}$ to remind the reader that the light quark part can well be a lot more complicated than just a single quark. Candidates of the next doublets of excited states are $D^{*}_{0}(2400)$232323The $D\pi$ $S$-wave resonant structure is probably more complicated than a single broad resonance, as demonstrated by a two-pole structure in Ref. Albaladejo et al. (2017a). and $D_{1}(2430)$ (the corresponding $B$-mesons are still to be found), characterized by $j^{P}_{\ell}=1/2^{+}$ and a width of about 300 MeV, and $D_{1}(2420)$ and $D_{2}^{*}(2460)$ ($\bar{B}_{1}(5721)$ and $\bar{B}^{*}_{2}(5747)$) with $j^{P}_{\ell}=3/2^{+}$ and a width of about 30 MeV. Since the states with $j^{P}_{\ell}=1/2^{+}$ are too broad to support hadronic molecules Filin et al. (2010); Guo and Meißner (2011), in what follows we focus on the scattering of the ground state mesons off their anti-particles as well as on that of the $j^{P}_{\ell}=3/2^{+}$ mesons off the ground state ones with one of them containing a heavy quark and the other a heavy anti-quark.

We start with the former system. To be concrete, we take the charm mesons. In the particle basis, there are six $S$-wave meson pairs with given $J^{PC}$ Nieves and Valderrama (2012):

[TABLE]

where the individual partial waves are labelled as ${}^{2S+1}L_{J}$, with $S$, $L$, and $J$ denoting the total spin, the angular momentum, and the total momentum of the two-meson system, respectively. We define the $C$-parity eigenstates as

[TABLE]

which comply with the convention242424 Notice that a different convention for the $C$-parity operator was used in Ref. Nieves and Valderrama (2012). As a consequence, the off-diagonal transitions of $V_{\rm LO}^{(0{++})}$ in Ref. Nieves and Valderrama (2012) have a different sign as compared to Eq. (81), see also Sec. VI A in Ref. Guo et al. (2016a) for further details of our convention. for the $C$-parity transformation $\hat{C}{\cal M}=\bar{\cal M}$. Because of HQSS, the interaction at LO is independent of the heavy quark spin, and thus can be described by the matrix elements $\langle j_{1\,\ell}^{\prime},j_{2\,\ell}^{\prime},{j_{\ell}}|\hat{\mathcal{H}}_{I}|j_{1\,\ell},j_{2\,\ell},{j_{\ell}}\rangle$ where the light quark systems get coupled to a total light-quark angular momentum of the two-meson system, $j_{\ell}$. Thus, for the systems under consideration, we have two independent terms for each isospin ($I=0$ or $1$): $\langle 1/2,1/2,0|\hat{\mathcal{H}}_{I}|1/2,1/2,0\rangle$ and $\langle 1/2,1/2,1|\hat{\mathcal{H}}_{I}|1/2,1/2,1\rangle$. This simple observation leads to the conclusion that in the strict heavy quark limit the six pairs in Eq. (77) are grouped into two multiplets with $j_{\ell}=0$ and 1, respectively. In the heavy quark limit, it is convenient to use a basis of states characterized via ${j_{\ell}^{PC}}\otimes{s_{c\bar{c}}^{PC}}$, where $s_{c\bar{c}}$ refers to the total spin of the $c$ and $\bar{c}$ pair. For the case of $S$-wave interactions only, both ${j_{\ell}^{PC}}$ and ${s_{c\bar{c}}^{PC}}$ can only be in $0^{-+}$ or $1^{--}$. Therefore, the spin multiplet with ${j_{\ell}=0}$ contains two states with quantum numbers:

[TABLE]

and the spin multiplet for ${j_{\ell}=1}$ has the following four states:

[TABLE]

It becomes clear that if the ${1^{++}}$ state $X(3872)$ is a $D\bar{D}^{*}$ molecule, then it is in the multiplet with $j_{\ell}=1$ Voloshin (2004a), and has three spin partners with $J^{PC}=0^{++}$, $2^{++}$ and $1^{+-}$ in the strict heavy quark limit as pointed out in Refs. Hidalgo-Duque et al. (2013a); Baru et al. (2016). Based on an analogous reasoning it was suggested already earlier that $Z_{b}(10610)$ and $Z_{b}(10650)$ might have four more isovector partners $W_{b0}^{(\prime)}$, $W_{b1}$ and $W_{b2}$ Bondar et al. (2011); Voloshin (2011); Mehen and Powell (2011). A detailed and quantitative analysis of these $W_{bJ}$ states can be found in Ref. Baru et al. (2017).

It is worthwhile to notice that the two $1^{+-}$ states are in different multiplets with $j_{\ell}=0$ and $1$, respectively, and thus cannot be related to each other via HQSS. However, the isovector $Z_{b}(10610)$ and $Z_{b}(10650)$ are located with similar distances to the $B\bar{B}^{*}$ and $B^{*}\bar{B}^{*}$ thresholds, respectively. Such an approximate degeneracy suggests that the isovector interactions in the $j_{\ell}=0$ and $j_{\ell}=1$ sectors are approximately the same, and the off-diagonal transition strength in the isovector channel between the two meson pairs with $J^{PC}=1^{+-}$ in Eq. (77) approximately vanishes. A fit to the Belle data of the $Z_{b}$ line shapes with HQSS constraints implemented also leads to nearly vanishing channel coupling Guo et al. (2016a). This points towards an additional “light quark spin symmetry” as proposed by Voloshin very recently Voloshin (2016b). While a deeper understanding for such a phenomenon is still missing, it seems to be realized in the charm sector as well for the charged $Z_{c}(3900)$ Ablikim et al. (2013a); Liu et al. (2013b) and $Z_{c}(4020)$ Ablikim et al. (2013b) observed by the BESIII and Belle collaborations. Note that in Ref. Valderrama (2012) it is argued that channel couplings are suppressed while in Ref. Baru et al. (2016) they were claimed to be important to keep a well defined spin symmetry limit. We come back to this controversy briefly later in this section.

When the physical nondegenerate masses for the heavy mesons are used, one needs to switch to the basis in terms of physical states in Eq. (77). In this basis and for a given set of quantum numbers $\{JPC\}$, the LO EFT potentials $V^{(JPC)}_{\rm LO}$, which respect HQSS, read AlFiky et al. (2006); Nieves and Valderrama (2012); Valderrama (2012)

[TABLE]

where $C_{0a}$ and $C_{0b}$ are two independent low-energy constants. Thus, since in the spin symmetry limit $D$ and $D^{*}$ are degenerate, implying that $D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$ loops are equal, the above equality of the potentials in the $1^{++}$ and $2^{++}$ channels immediately predicts equal binding energies for the two states in this limit.

Once HQSS violation is introduced into the system by the use of the physical masses, the two-multiplet pattern gets changed, however, the close connection between the $1^{++}$ and $2^{++}$ states persists. An inclusion of the one-pion exchange necessitates an extension of the basis, since now also $D$-waves need to be included. In fact, HQSS is preserved only if all allowed $D$-waves are kept in the system, even if the masses of the open flavor states are still kept degenerate Baru et al. (2016). The probably most striking effect of the $D$-waves, once the $D^{*}$-$D$ mass difference is included, is that now transitions of the $2^{++}$ $D^{*}\bar{D}^{*}$ $S$-wave state to the $D\bar{D}$ and $D\bar{D}^{*}$ $D$-wave become possible. It allows for a width of this state of up to several tens of MeV Albaladejo et al. (2015); Baru et al. (2016), which might be accompanied by a sizeable shift in mass. In addition, spin symmetry relations might get modified via the coupling of the molecular states with regular charmonia as discussed recently in Ref. Cincioglu et al. (2016).

For near-threshold states it is natural to assume that the contact terms are independent of the heavy quark mass — phenomenologically they can be viewed as parameterizing the exchange of light meson resonances. Then one can also predict the heavy quark flavor partners of the $X(3872)$. The heavy quark spin and flavor partners of the $X(3872)$ predicted in Ref. Guo et al. (2013b) with $\Lambda=0.5$ GeV are listed in Table 5.

The $Z_{b}(10610)$ can be related to the $Z_{b}(10650)$ when the off-diagonal interaction is neglected as discussed above. Their hidden-charm partners are found to be virtual states in this formalism Guo et al. (2013b), which may correspond to the $Z_{c}(3900)$ and $Z_{c}(4020)$. In fact, it is shown in Ref. Albaladejo et al. (2016b) that the BESIII data for the $Z_{c}(3900)$ in both the $J/\psi\pi$ Ablikim et al. (2013a) and $D\bar{D}^{*}$ Ablikim et al. (2015a) modes can be well fitted with either a resonance above the $D\bar{D}^{*}$ threshold or a virtual state below.

The number of the LO contact terms is larger for the interaction between a pair of $j_{\ell}=1/2$ and $j_{\ell}=3/2$ heavy and anti-heavy mesons. For each isospin, 0 or 1, in the heavy quark limit, there are four independent interactions denoted as $\langle j_{1\,\ell},j_{2\,\ell},j_{\ell}|\hat{\mathcal{H}}_{I}|j_{1\,\ell}^{\prime},j_{2\,\ell}^{\prime},j_{\ell}\rangle$, where now $j_{\ell}$ can take values 1 or 2

[TABLE]

The relevant combinations of these constants for a given heavy meson pair can be worked out by changing the basis by means of a unitary transformation (see, e.g., Ohkoda et al. (2012a); Xiao et al. (2013a)):

[TABLE]

where $j_{1}$ and $j_{2}$ are the spins of the two heavy mesons, $J$ is the total angular momentum of the whole system, and $s_{1\,c}$ and $s_{2\,c}$ are the spins of the heavy quark.

Consider two mesons $A$ and $B$; each of them is not a $C$ parity eigenstate, but their linear combination can form $C$ parity eigenstates. With the phase convention specified below (78), the $C=\pm$ eigenstates of a flavor-neutral system consisting of a pair of mesons are given by

[TABLE]

where $J_{A}$ and $J_{B}$ are the spins of the mesons $A$ and $B$, and $J$ is the total spin of the two-body system.

Noticing that the total spin of the heavy quark and anti-quark $s_{c\bar{c}}$ is conserved in the heavy quark limit, and combining the meson pairs into eigenstates of $C$-parity, one can obtain the contact terms for the $S$-wave interaction between a pair of $j_{\ell}=\frac{1}{2}$ and $\frac{3}{2}$ heavy and anti-heavy mesons. The diagonal ones are listed in Table 6.

One sees that the linear combinations are different for all channels, and it is not as easy as in case of the $X(3872)$ to predict spin partners for the $Y(4260)$ based on the assumption that it is predominantly a $D_{1}\bar{D}$ state. The possibility of $S$-wave hadronic molecules with exotic quantum numbers $1^{-+}$ was discussed in Wang (2014a). Here we also give the off-diagonal contact terms. There are three channels in each of $1^{--}$, $2^{--}$, $1^{-+}$ and $2^{-+}$ sectors, we label them in each sector listed in Table 6 from top to bottom as 1, 2, and 3. Then the off-diagonal contact terms for the $1^{--}$ sector are

[TABLE]

The off-diagonal contact terms for the $2^{--}$ sector are

[TABLE]

The off-diagonal contact terms for the $1^{-+}$ sector are

[TABLE]

The off-diagonal contact terms for the $2^{-+}$ sector are

[TABLE]

However, one non-trivial prediction for the spectrum of molecular states in the heavy quarkonium spectrum becomes apparent immediately from the discussion above: Since the most bound states appear in $S$-waves the lightest negative parity vector state can be formed only from $j_{\ell}^{P}=1/2^{-}$ and $3/2^{+}$ heavy and anti-heavy mesons. Therefore the mass difference of $X(3872)$ as bound state of two ground state $j_{\ell}^{P}=\frac{1}{2}^{-}$ mesons ($D$ and $D^{*}$) and the lightest exotic vector state $Y(4260)$ (388 MeV) should be of the order of the mass difference of the lightest $3/2^{+}$ state and the $D^{*}$ (410 MeV). Clearly this prediction is nicely realized in nature. Note that from this reasoning it also follows that $if$ the $Y(4008)$ indeed were to exist it could not be a hadronic molecule. In this context it is interesting to note that the most resent data from BESIII on $e^{+}e^{-}\to J/\psi\pi\pi$ Ablikim et al. (2017c) does not seem to show evidence for the $Y(4008)$, c.f. Fig. 5.

IV.3 Impact of hadron loops on regular quarkonia

In the previous sections we argued that meson loops play a prominent role in both the formation and the decays of hadronic molecules. One may wonder if they also have an impact on the properties of regular charmonia. In this section we demonstrate that certain processes for regular hadrons, largely well described by the quark model, can also be influenced by significant meson loop effects, since reaction rates can receive an enhancement due to the nearly on-shell intermediate heavy mesons. The origin of this mechanism is that for most heavy quarkonium transitions $M_{Q\bar{Q}}-2M_{Q\bar{q}}\ll M_{Q\bar{q}}$, where $M_{Q\bar{Q}}$ and $M_{Q\bar{q}}$ are the masses of the heavy quarkonium and an open-flavor heavy meson, respectively. As a result, the intermediate heavy mesons are nonrelativistic with a small velocity

[TABLE]

and the meson loops in the transitions can be investigated using NREFT${}_{\text{I}}$. We will highlight this effect on two examples in what follows.252525The effects of meson loops in heavy quarkonium spectrum are investigated in, e.g., Refs. Eichten et al. (1978, 1980); Ono and Törnqvist (1984); Kalashnikova (2005); Eichten et al. (2006); Pennington and Wilson (2007); Barnes and Swanson (2008); Li et al. (2009); Danilkin and Simonov (2010b); Ortega et al. (2010); Danilkin and Simonov (2010a); Liu and Ding (2012); Bali et al. (2011); Zhou and Xiao (2014); Ferretti et al. (2013); Ferretti and Santopinto (2014); Ferretti et al. (2014); Hammer et al. (2016); Du et al. (2016); Lu et al. (2016, 2017); Zhou and Xiao (2017), and in heavy quarkonium transitions in Refs. Ono et al. (1985); Lipkin and Tuan (1988); Moxhay (1989); Zhou and Kuang (1991); Li and Zhao (2008); Meng and Chao (2008a, b); Liu et al. (2009b); Zhang et al. (2009b); Zhang and Zhao (2010); Wang et al. (2011a); Li and Zhao (2011); Wang et al. (2012); Guo et al. (2012); Li et al. (2013a); Cao et al. (2016).

We start with the hindered M1 transitions between two $P$-wave heavy quarkonia with different radial excitations, such as the $h_{c}(2P)\to\gamma\chi_{cJ}(1P)$. It was proposed in Ref. Guo and Meißner (2012a); Guo et al. (2016c) that such transitions are very sensitive to meson-loop effects, and the pertinent partial widths provide a way to extract the coupling constants between the $P$-wave heavy quarkonia and heavy open flavor mesons.

In quark models the amplitude for such a transition is proportional to the overlap of the wave functions of the initial and final heavy quarkonia, which is tiny and quite model-dependent due to the different radial excitations — this is why they are called “hindered”. This suppression is avoided in the coupled-channel mechanism of heavy-meson loops. In this mechanism, the initial and final $P$-wave heavy quarkonia couple to the ground state pseudoscalar and vector heavy mesons in an $S$-wave. A few diagrams contributing to this mechanism are shown in Fig. 17. In (a), the photon is emitted via its magnetic coupling to intermediate heavy mesons. In (b), since the $S$-wave vertices do not have any derivative at LO, the photon couples in a gauge invariant way to one of the vertices in the two-point loop diagram. (c) and (d) are two typical two-loop diagrams. From the power counting rules discussed in Sec. IV.1.2, Fig. 17 (a) provides the leading contribution, while (b) is of higher order in the velocity counting because there is one less nonrelativistic propagator. Their amplitudes scale as

[TABLE]

respectively, where $E_{\gamma}$ is the photon energy, and the dependence on the coupling constants is dropped. The $1/m_{Q}$ suppression comes from the fact that the polarization of a heavy (anti-)quark needs to be flipped in the M1 transitions. For the two-loop diagram in (c), the amplitude scales as

[TABLE]

where the factor $1/(4\pi)^{2}$ appears because there is one more loop and the hadronic scale $\Lambda_{\chi}=4\pi F_{\pi}\sim 1$ GeV was introduced as the hard scale for the chiral expansion. The factor of $M_{H}^{2}$ was introduced to match dimensions of the above equations to those of Eqs. (93). Diagram (d) has the same scaling as (c). Since the axial coupling constant $g\simeq 0.6$ for the charm case as determined from the width of $D^{*}\to D\pi$, and about 0.5 for bottom Flynn et al. (2016), one has $gM_{D}/\Lambda_{\chi}\lesssim 1$ and $gM_{B}/\Lambda_{\chi}\simeq 2$. The value for $v$ defined as $(v_{1}+v_{2})/2$ is about 0.4 for the transitions from the 2P to 1P charmonium states Guo and Meißner (2012a), and ranges from 0.3 to 0.2 for the transitions between $1P,2P$ and $3P$ bottomonia Guo et al. (2016c). Hence, the two-loop diagrams are suppressed in the charm sector, while they are of the same order as (a) for the bottom sector. Therefore, one can make predictions for the charmonium transitions by calculating the loops corresponding to (a). The results depend on a product of two unknown coupling constants of the $1P$ and $2P$ charmonia to the charm mesons. Taking model values for them, the decay width of the $\chi_{c2}(2P)\to\gamma h_{c}(1P)$ is of $\mathcal{O}(100~{}\text{keV})$, two orders of magnitude larger than the quark model prediction, 1.3 keV Barnes et al. (2005). Although quantitative predictions cannot be made for the bottomonium transitions, it is expected that once such transitions would be observed they must be due to coupled-channel effects as the partial widths were predicted to be in the range from sub-eV to eV level in a quark model calculation that does not include meson-loop effects Godfrey and Moats (2015). It is suggested in Guo and Meißner (2012a); Guo et al. (2016c) that the coupled-channel effects can be checked by comparing results from both fully dynamical and quenched lattice QCD which has and has no coupled-channel effects, respectively. Recent developments in lattice QCD calculations of radiative decays Dudek et al. (2006, 2009); Shultz et al. (2015); Briceño et al. (2016); Agadjanov et al. (2014); Feng et al. (2015); Leskovec et al. (2016); Meyer (2011); Owen et al. (2015) should be helpful in illuminating this issue.

There are other heavy quarkonium transitions driven mainly by the coupled-channel effects. A detailed study on the transitions between two charmonia ($S$- and $P$-wave) with the emission of a pion or eta can be found in Ref. Guo et al. (2011). It is found that whether the coupled-channel effects play a sizable role depends on the process. This is a result of the power counting analysis; see the itemized discussion in Sec. IV.1.2. In particular, the single-pion/eta transitions between two $S$-wave and $P$-wave charmonia receive important contribution from charm-meson loops. Therefore, the long-standing suggestion that the $\psi^{\prime}\to J/\psi\eta/\pi^{0}$ transitions can be used to extract the light quark mass ratio Ioffe (1979) needs to be reexamined. In fact, if we assume that the triangle charm meson-loop diagrams saturate the transitions, the resulting prediction of $\mathcal{B}(\psi^{\prime}\to J/\psi\pi^{0})/\mathcal{B}(\psi^{\prime}\to J/\psi\eta)$ is consistent with the experimental data. These transitions were revisited considering both the loop and tree diagrams in Ref. Mehen and Yang (2012). Again based on the same power counting rules it was argued that the transitions $\Upsilon(4S)\to h_{b}\pi^{0}/\eta$ have only a small pollution from the bottom-meson loops, and are dominated by short-distance contribution proportional to the light quark mass difference Guo et al. (2010c). They could be used for the extraction of light quark mass ratio. Furthermore, the prediction, made before the discovery of the $h_{b}(1P)$, on the branching fraction of the order of $10^{-3}$ for the decay $\Upsilon(4S)\to h_{b}\eta$ was verified by the Belle measurement, $(2.18\pm 0.11\pm 0.18)\times 10^{-3}$ Tamponi et al. (2015).

Parameter-free predictions can be made for ratios of partial widths of decays dominated by the coupled-channel effects of heavy mesons in the same spin multiplet, since all the coupling constants will get canceled in the ratios. Such predictions on the hindered M1 transitions between $P$-wave heavy quarkonia can be found in Refs. Guo and Meißner (2012a); Guo et al. (2016c).

In Ref. Guo and Meißner (2012b), it is pointed out that coupled-channel effects can even introduce sizable and nonanalytic pion mass dependence in heavy quarkonium systems which couple to open-flavor heavy meson pairs in an $S$-wave.

To summarize this subsection, we stress that whether meson-loop effects are important for the properties of quarkonia or not does not only depend on the proximity to the relevant threshold, it is also depends on the particular transition studied.

V Hadronic molecules in lattice QCD

Lattice QCD is, in principle, the tool to calculate the spectrum of QCD from first principles. There has been a remarkable progress in the last years in this field, see e.g. Dürr et al. (2008); Baron et al. (2010); Edwards et al. (2011); Liu et al. (2012); Liu (2017). Still, the extraction of the properties of resonances and, in particular, of hadronic molecules, from finite volume calculations poses severe challenges. When QCD is put on an Euclidean space-time lattice with a finite space-time volume, asymptotic states cannot be defined and right-hand cuts are replaced by poles, thus preventing a direct calculation of scattering and resonance properties. This obstacle was overcome by Lüscher a long time ago. He derived a relation between the energy shift in the finite volume and the scattering phase shift in the continuum Lüscher (1991, 1986), see also Refs. Wiese (1989); DeGrand (1991). This approach has become known and used as Lüscher’s method. More precisely, in order to determine the mass and width from the measured spectrum, one first extracts the scattering phase shift by using the Lüscher equation. In the next step, using some parameterization for the $K$-matrix (e.g., the effective range expansion), a continuation into the complex energy plane is performed. As noted in Sec. III.1, resonances correspond to poles of the scattering $T$-matrix on the second Riemann sheet, and the real and imaginary parts of the pole position define the mass and the width of a resonance. A nice example is given by the $\rho$-meson, that has been considered using Lüscher’s method, e.g., in Refs. Feng et al. (2011); Lang et al. (2011); Aoki et al. (2011); Dudek et al. (2013). In these papers it has already been shown that even for such realistic calculations of a well isolated resonance, the inclusion of hadron-hadron type interpolating operators is mandatory, it is simply not sufficient to represent the decaying resonance by properly chosen quark bilinears (for mesons). For the discussion of hadronic molecules (or most other hadron resonances), this method needs to be extended in various directions, such as considering higher partial waves and spins Bernard et al. (2008); Luu and Savage (2011); König et al. (2011, 2012); Briceño and Davoudi (2013a); Göckeler et al. (2012), moving frames Rummukainen and Gottlieb (1995); Bour et al. (2011); Davoudi and Savage (2011); Fu (2012); Leskovec and Prelovsek (2012); Göckeler et al. (2012), multi-channel scattering Liu et al. (2006); Lage et al. (2009); Bernard et al. (2011); Döring et al. (2011a); Li and Liu (2013b); Guo et al. (2013d), including the use of unitarized chiral perturbation theory (and related methods) Döring et al. (2011b); Martínez Torres et al. (2012); Döring and Meißner (2012); Albaladejo et al. (2012); Wu et al. (2014a); Hu et al. (2016), and three-particle final states Kreuzer and Hammer (2011); Polejaeva and Rusetsky (2012); Briceño and Davoudi (2013b); Hansen and Sharpe (2014); Meißner et al. (2015); Hansen and Sharpe (2015, 2016a, 2017, 2016b); Guo and Gasparian (2017).

Here, we will not attempt to review the lattice QCD approach to the hadron spectrum in any detail but just focus on the bits and pieces that are relevant for the investigation of possible hadronic molecules. In Sec. V.1 we summarize the Lüscher method and its extension to the multi-channel space, followed by a discussion of the compositeness criterion in a finite volume, see Sec. V.3. In Sec. V.2, we discuss how the quark mass dependence of certain observables can be used to differentiate hadronic molecules from more compact multi-quark states and in Sec. V.4, we briefly summarize pertinent lattice QCD calculations for the possible molecular states containing charm quarks. A short final subsection is devoted to certain states made of light quarks only.

V.1 Resonances in a finite volume

The essence of the Lüscher approach can be understood in a simple nonrelativistic model for the scattering of identical, spinless particles of mass $m$ in 1+1 dimensions. In the CM frame, the relative momentum is quantized according to $p=(2\pi/L)n$, with $L$ the spatial lattice extension and $n$ an integer. In case of no interactions between these particles, the energy of the two-particle system is simply given by $E=2m+p^{2}/m$, which means that the free energy level-$n$ scales as $n^{2}/L^{2}$ with the volume and thus levels with different $n$ do not intersect. In the presence of interactions, this behaviour is modified. Let us assume that this interaction leads to a narrow resonance at $\sqrt{s_{R}}=E_{R}-i\Gamma_{R}/2$, that is $\Gamma_{R}\ll E_{R}$. In the infinite volume limit, this interaction leads to a phase shift $\delta(p)$ in the asymptotic wave function. Furthermore, in the presence of a resonance, the phase shift will change by $\pi$ (known as Levinson’s theorem Levinson (1949)). In a finite volume, this behavior translates into the boundary condition

[TABLE]

This condition provides the link between the volume dependence of the energy levels in the interacting system and the continuum phase shift. If one follows an energy level inwards from the asymptotic region to smaller lattice sizes, in the vicinity of a resonance, this boundary condition causes a visible distortion, the so-called avoided level crossing, c.f. Fig. 18. The plateau, where the energy of the two-particle system is almost volume-independent, corresponds to the real part of the pole $E_{R}$. The imaginary part of the pole is given by the slope according to $\left.d\delta(p)/dE\right|_{E_{R}}=2/\Gamma_{R}$. Clearly, this method can only work when certain conditions are fulfilled. First, the method as described here is restricted to the elastic two-particle case. Second, one has to make sure that the interaction range of the particles is much smaller than the size of the box to make the notion of asymptotic states possible. Third, to suppress polarization effects that arise from the interactions of the lightest particles in the theory with each other around the torus, one has to choose $L$ such that $1/m\ll L$.

We now consider the extension of the Lüscher method to the multi-channel case, as most hadronic molecules are located close to a two-particle threshold or between two close-by thresholds. To achieve this extension, an appropriate tool is a particular version of an NREFT, because up to the energies where multi-particle inelastic states become significant, such a framework is completely equivalent to the relativistic field theory, provided the couplings in the nonrelativistic framework are determined from matching to the relativistic $S$-matrix elements, for details and further references, see Bernard et al. (2008); Colangelo et al. (2006); Gasser et al. (2011). For the one-channel case, it was already shown in Ref. Beane et al. (2005) that using such an NREFT, one obtains at a very simple and transparent proof of Lüscher’s formula.

To keep the presentation simple, we first consider a two-channel LSE in NREFT in the infinite volume. Let us consider antikaon-nucleon scattering in the region of the $\Lambda(1405)$ resonance, $\bar{K}N\to\bar{K}N,\Sigma\pi$. The channel number 1 refers to $\bar{K}N$ and 2 to $\Sigma\pi$ with total isospin $I=0$. The resonance $\Lambda(1405)$ is located between two thresholds, on the second Riemann sheet, close to the real axis. The two thresholds are given by $s_{t}=(m_{N}+M_{K})^{2}$ and $s^{\prime}_{t}=(m_{\Sigma}+M_{\pi})^{2}$. We work in the isospin limit and neglect the fact that there are really two poles — see Refs. Oller and Meißner (2001); Jido et al. (2003) and Sec. VI.4.262626Note that in this two-channel formulation one only has one pole corresponding to one $\Lambda(1405)$. To deal with the two-pole scenario requires the inclusion of more channels and explicit isospin breaking. For energies above the $\bar{K}N$ threshold, $s>(m_{N}+M_{K})^{2}$, the coupled-channel LSE for the $T$-matrix elements $T_{ij}(s)$ in dimensionally regularized NREFT reads (we only consider $S$-waves here)

[TABLE]

with $q_{1}=\lambda^{1/2}(s,m_{N}^{2},M_{K}^{2})/(2\sqrt{s})$, $q_{2}=\lambda^{1/2}(s,m_{\Sigma}^{2},M_{\pi}^{2})/(2\sqrt{s})$ and $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$ is the Källén function. Furthermore, the $H_{ij}(s)$ denote the driving potential in the corresponding channel, i.e. the matrix element of the interaction Hamiltonian between the free two-particle states. Continuation of the CM momentum $q_{1}$ below threshold $(m_{\Sigma}+M_{\pi})^{2}<s<(m_{N}+M_{K})^{2}$ is obtained via

[TABLE]

The resonance corresponds to a pole on the second Riemann sheet in the complex $s$-plane. Its position is given by the solution of the secular equation

[TABLE]

with $\kappa_{1}^{R}=-(-\lambda(s_{R},m_{N}^{2},M_{K}^{2}))^{1/2}/(2\sqrt{s_{R}})$ and $\kappa_{2}^{R}=(-\lambda(s_{R},m_{\Sigma}^{2},M_{\pi}^{2})^{1/2})/(2\sqrt{s_{R}})$. The energy and width of the resonance are then given by $\sqrt{s_{R}}=E_{R}-i\Gamma_{R}/2$.

Consider next the same problem in a finite volume. Only discrete values of the three-momentum are allowed, given by $\bm{k}=(2\pi/L)\bm{n}$, with $\bm{n}$ a triplet of integer numbers. Thus, we replace the three-momentum integration in the loops by a discrete sum (see Ref. Bernard et al. (2008) for more details). The rotational symmetry is broken to a cubic symmetry, so mixing of different partial waves occurs. Here, however, we only consider $S$-waves, neglecting the small mixing to higher partial waves. If necessary, the mixing can be easily included at a later stage, see e.g. Bernard et al. (2008); Döring et al. (2012). The finite-volume version of the LSE Eq. (V.1) then takes the form

[TABLE]

with

[TABLE]

Here, we have neglected the terms that vanish exponentially at large $L$. The secular equation that determines the spectrum can be brought into the form

[TABLE]

This can be rewritten as

[TABLE]

with

[TABLE]

$\delta(s,L)$ is called the pseudophase. The dependence of the pseudophase on $s$ and $L$ is very different from that of the usual scattering phase. Namely, the elastic phase extracted from the lattice data by using Lüscher’s formula is independent of the volume modulo terms that exponentially vanish at a large $L$. Further, the energies where the phase passes through $\pi/2$ lie close to the real resonance locations. In contrast with this, the pseudophase contains the function $Z_{00}(1;k_{1}^{2})$, which does not vanish exponentially at a large $L$ and a positive $k_{1}^{2}$. Moreover, the tangent of the pseudophase contains a tower of poles at the energies given by the roots of the equation $1-({2}/{\pi L})\,Z_{00}(1;k_{1}^{2})\,H_{11}=0$. On the other hand, in the infinite volume this equation reduces to $1+\kappa_{1}^{R}\,H_{11}=0$, c.f. with Eq. (98), which has only one root below threshold very close to the position of the $\Lambda(1405)$. Other roots in a finite volume stem from oscillations of $Z_{00}(1,k_{1}^{2})$ between $-\infty$ and $+\infty$ when the variable $k_{1}^{2}$ varies along the positive semi-axis. Their locations depend on $H_{11}$ and thus contain information of the infinite-volume interaction. This is an effect of discrete energy levels in the “shielded” channel, which is the channel with the lower threshold in the coupled-channel system. The pseudophase depends on the three real functions $H_{11},H_{12},H_{22}$. Based on synthetic data it was shown in Ref. Lage et al. (2009) that a measurement of the lowest two eigenvalues at energies between 1.4 and 1.5 GeV allows one to reconstruct the pseudophase and to extract in principle the pole position. It was further pointed out in that work that two-particle thresholds also lead to an avoided level crossing, so the extraction of the resonance properties from the corresponding plateaus in the energy dependence of certain levels is no longer possible. In the case of real data, taking into account the uncertainties of each measurements, one has to measure more levels on a finer energy grid. To obtain a sufficient amount of data in a given volume, twisting and asymmetric boxes can also be helpful. First such calculations have become available recently and will be discussed below.

An alternative formulation, that allows the use of fully relativistic two-particle propagators and can easily be matched to the representation of a given scattering amplitude based on unitarized chiral perturbation theory (UCHPT) was worked out in Döring et al. (2011b). The method is based on the observation that in coupled-channel UCHPT, certain resonances are dynamically generated, e.g. the light scalar mesons in the coupled $\pi\pi/\bar{K}K$ system. The basic idea is to consider this approach in a finite volume to produce the volume-dependent discrete energy spectrum. Reversing the argument, one is then able to fit the parameters of the chiral potential to the measured energy spectrum on the lattice and, in the next step, determine the resonance locations by solving the scattering equations in the infinite volume. By construction, this method fulfills the constraints from chiral symmetry such as the appearance of Adler zeros at certain unphysical points. For recent developments using a relativistic framework, we refer to Briceño and Hansen (2015); Briceño et al. (2015); Briceño and Hansen (2016).

V.2 Quark mass dependence

To reduce numerical noise as well as to speed up algorithms, lattice calculations are often performed at unphysical values of the light quark masses. While this at first sight may appear as a disadvantage, it is indeed a virtue as it enables a new handle on investigating the structure of certain states. However, in the case of multiple coupled channels, one also has to be aware that thresholds and poles can move very strongly as a function of the quark masses. This intricate interplay between $S$-wave thresholds and resonances needs to be accounted for when one tries to extract the resonance properties.

To address the first issue, we specifically consider the charm-strange mesons $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$. As shown in Cleven et al. (2011b), in the molecular picture describing these as $DK$ and $D^{*}K$ bound states, a particular pion and kaon mass dependence arises. Consider first the dependence on the light quark masses, that can be mapped onto the pion mass dependence utilizing the Gell-Mann–Oakes–Renner relation Gell-Mann et al. (1968), $M_{\pi^{\pm}}^{2}=B(m_{u}+m_{d})$, that naturally arises in QCD as the leading term in the chiral expansion of the Goldstone boson mass. Here, $B$ is related to the vacuum expectation value of the quark condensate. In fact, this relation is fulfilled to better than 94% in QCD Colangelo et al. (2001). As shown in Ref. Cleven et al. (2011b), the pion mass dependence of such a molecular state is much more pronounced than for a simple $c\bar{s}$ state. Even more telling and unique is, however, the kaon mass dependence. For that, consider the $M_{K}$ dependence of the mass of a bound state of a kaon and some other hadron. The mass of such a kaonic bound state is given by

[TABLE]

where $M_{h}$ is the mass of the other hadron, and $E_{B}$ denotes the binding energy. Although both $M_{h}$ and $E_{B}$ have some kaon mass dependence, it is expected to be a lot weaker than that of the kaon itself. Thus, the important implication of this simple formula is that the leading kaon mass dependence of a kaon-hadron bound state is linear, and the slope is unity. The only exception to this argument is if the other hadron is also a kaon or anti-kaon. In this case, the leading kaon mass dependence is still linear but with the slope being changed to two. Hence, as for the $DK$ and $D^{*}K$ bound states, one expects that their masses are linear in the kaon mass, and the slope is approximately one. This expectation is borne out by the explicit calculations performed in Cleven et al. (2011b). Early lattice QCD attempts to investigate this peculiar kaon mass dependence have led to inconclusive results McNeile (2011). Other papers that discuss methods to analyze the structure of states based on their quark mass dependence or the behavior at large number of colors are e.g. Hanhart et al. (2008); Peláez and Ríos (2010); Bernard et al. (2011); Albaladejo and Oller (2012); Guo and Oller (2011); Nebreda et al. (2011); Peláez and Ríos (2006); Guo et al. (2015e).

The second issue we want to address briefly is the intricate interplay of $S$-wave thresholds and resonance pole positions with varying quark masses, as detailed in Ref. Döring et al. (2013). In that paper, pion-nucleon scattering in the $J^{P}=1/2^{-}$ sector in the finite volume and at varying quark masses based on UCHPT was studied. In the infinite volume, both the $N(1535)$ and the $N(1650)$ are dynamically generated from the coupled channel dynamics of the isospin $I=1/2$ and strangeness $S=0$ $\pi N,\eta N,K\Lambda$ and $K\Sigma$ system. Having fixed the corresponding LECs in the infinite volume, one can straightforwardly calculate the spectrum in the finite volume provided one knows the octet Goldstone boson masses, the masses of the ground-state octet baryons and the meson decay constants. Such sets of data at different quark masses are given by ETMC and QCDSF. ETMC provides masses and decay constants for $M_{\pi}=269$ MeV and the kaon mass close to its physical value Alexandrou et al. (2009); Ottnad et al. (2012). Quite differently, the QCDSF Collaboration Bietenholz et al. (2011) obtains the baryon and meson masses from an alternative approach to tune the quark masses. Most importantly, while the lattice size and spacing are comparable to those of the ETMC, the strange quark mass differs significantly from the physical value. The latter results in a different ordering of the masses of the ground-state octet mesons and, consequently, in a different ordering of meson-baryon thresholds. For the ETMC parameters, all thresholds are moved to higher energies. The cusp at the $\eta N$ threshold has become more pronounced, but no clear resonance shapes are visible. Indeed, going to the complex plane, one finds that the poles are hidden in the Riemann sheets which are not directly connected to the physical one by crossing the cut at the energies corresponding to the real parts of the poles. Using the QCDSF parameters, the situation is very different. In contrast to the ETMC case, a clear resonance signal is visible below the $K\Lambda$ threshold, that is the first inelastic channel in this parameter setup. Indeed, one finds a pole on the corresponding Riemann sheet. Unlike in the ETMC case, it is not hidden behind a threshold. Between the $K\Lambda$ and the $K\Sigma$ threshold, there is only a hidden pole. The $K\Sigma$ and $\eta N$ thresholds are almost degenerate, and on sheets corresponding to these higher-lying thresholds one only finds hidden poles. For more details, the reader is referred to Ref. Döring et al. (2013). In that paper, strategies to overcome such type of difficulties are also discussed.

It is worthwhile to mention that the composition of a hadron in general may vary when changing the quark masses. However, as long as the quark masses are not very different from the physical values, the quark mass dependence is rather suggestive towards revealing the internal structure as different structures should result in different quark mass dependence.

V.3 Measuring compositeness on lattice

As discussed in Sec. III.2.1, the Weinberg compositeness criterion offers a possibility to disentangle compact bound states from loosely bound hadronic molecules. By measuring the low-energy scattering observables in lattice using the Lüscher formalism discussed before, one can extract the compositeness by using Eqs. (21). For related work, see, e.g., Refs. Suganuma et al. (2007); Martínez Torres et al. (2012); Ozaki and Sasaki (2013); Albaladejo et al. (2013). It is pointed out in Agadjanov et al. (2015) that the use of partially twisted boundary conditions is cheaper than studying the volume dependence in lattice for measuring the compositeness. The basic object in that work is the scattering amplitude in the finite volume, which can be obtained from the corresponding loop function $\tilde{G}_{L}^{\bm{\theta}}(s)=G(s)+\Delta G_{L}^{\bm{\theta}}(s)$ Döring et al. (2011b), where $\Delta G_{L}^{\bm{\theta}}$ can be related to the modified Lüscher function $Z_{00}^{\bm{\theta}}$ via

[TABLE]

where $\hat{k}=kL/(2\pi)$ and ellipsis denote terms that are exponentially suppressed with the lattice size $L$ Döring et al. (2011b). Here, in case of twisted boundary conditions, the momenta also depend on the twist angle $\bm{\theta}$ according to $\bm{q}_{n}=(2\pi/L)\bm{n}+(\bm{\theta}/L),~{}0\leq\theta_{i}<2\pi$. In case of a bound state with mass $M$ in the infinite volume, the scattering amplitude should have a pole at $s=M^{2}$, with the corresponding binding momentum $k_{B}\equiv i\kappa$, $\kappa>0$, given by the equation

[TABLE]

with $\psi(k^{2})$ the analytic continuation of $k\cot\delta(k)$ for arbitrary complex values of $k^{2}$. From this, it is straightforward to evaluate the pole position shift,

[TABLE]

where the prime denotes differentiation with respect to $k^{2}$. This equation gives the bound state pole position $\kappa_{L}$ (and thus the finite volume mass $M_{L}$) as a function of the infinite-volume parameters $g^{2}$ and $\kappa$. Having determined these parameters from the bound state levels $\kappa_{L}$, one is then able to determine the wave function renormalization constant $Z$ in the infinite volume. In Ref. Agadjanov et al. (2015), this procedure is scrutinized using synthetic lattice data, for a simple toy model and a molecular model for the charm scalar meson $D_{s0}^{*}(2317)$. An important finding of this work is that the extraction of $Z$ is facilitated by using twisted boundary conditions, measuring the dependence of the spectrum on the twist angle. Also, the limitations of this approach are discussed in detail. It remains to be seen how useful this method is for real lattice data. For related papers, also making use of twisted boundary conditions to explore the nature of states, see e.g. Refs. Ozaki and Sasaki (2013); Briceño et al. (2014); Körber and Luu (2016). A different approach to quantify compositeness in a finite volume has recently been given in Ref. Tsuchida and Hyodo (2017). Using this method, the $\bar{K}N$ component of the $\Lambda(1405)$ was found to be 58%, and the $\Sigma\pi$ and other components also contribute to its structure. This is interpreted as a reflection of the two-pole scenario of the $\Lambda(1405)$.

V.4 Lattice QCD results on the charm-strange mesons

and $\bm{XYZ}$ states

There have been quite a few studies of the charm-strange mesons and some of the $XYZ$ states in lattice QCD. However, there are very few conclusive results at present, so we expect that this section will be outdated most quickly.

Let us consider first the charm-strange mesons. A pioneering lattice study of the low-energy interaction between a light pseudoscalar meson and a charmed pseudoscalar meson was presented in Ref. Liu et al. (2013a). The scattering lengths of the five channels $D\bar{K}(I=0)$, $D\bar{K}(I=1)$, $D_{s}K$, $D\pi(I=3/2)$ and $D_{s}\pi$ were calculated based on four ensembles with pion masses of 301, 364, 511 and 617 MeV. These channels are free of contributions from disconnected diagrams. SU(3) UCHPT as developed in Ref. Guo et al. (2009b) was used to perform the chiral extrapolation. The LECs of the chiral Lagrangian were determined from a fit to the lattice results. With the same set of LECs and the masses of the involved mesons set to their physical values, predictions for the other channels including $DK(I=0)$, $DK(I=1)$, $D\pi(I=1/2)$ and $D_{s}\bar{K}$ were made. In particular, it was found that the attractive interaction in the $DK(I=0)$ channel is strong enough so that a pole is generated in the unitarized scattering amplitude. Within $1\sigma$ uncertainties of the parameters, the pole is at $2315^{+18}_{-28}$ MeV, and it is always below the $DK$ threshold. From calculating the wave function normalization constant, it was found that this pole is mainly an $S$-wave $DK$ bound state (the pertinent scattering length being close to $-1$ fm as predicted in Guo et al. (2009b) for such a molecular state using Eq. (21)). Further, a much sharper prediction of the isospin breaking decay width of the $D_{s0}^{*}(2317)\to D_{s}\pi$ could be given

[TABLE]

to be contrasted with the molecular prediction without lattice data, $\Gamma(D_{s0}^{*}(2317)\to D_{s}\pi)=(180\pm 110)$ keV Guo et al. (2008a), and typical quark model predictions for a $c\bar{s}$ charm scalar meson of the order of 10 keV, see e.g. Refs. Godfrey (2003); Faessler et al. (2007). For a similar study using a covariant UCHPT instead of the heavy-baryon formalism, see Ref. Altenbuchinger et al. (2014).

A systematic study of the charm scalar and axial mesons at lighter pion masses ($M_{\pi}=156$ and $266$ MeV) was performed in Refs. Mohler et al. (2013b, a); Lang et al. (2014). These data were later reanalyzed with the help of finite-volume UCHPT Martínez Torres et al. (2015). Most notably, the $DK$ scattering with $J^{P}=0^{+}$ was investigated in Mohler et al. (2013a), using $DK$ as well as $c\bar{s}$ interpolating fields. Clear evidence of a bound state below the $DK$ threshold was found and the corresponding scattering length was $a_{0}=-1.33(20)\,$fm, consistent with the molecular scenario. The analysis of Ref. Martínez Torres et al. (2015) found a 70% $DK$ ($D^{*}K$) component in the $D_{s0}^{*}(2317)\,(D_{s1}(2460))$ state.

The most systematic study in the coupled-channel $D\pi,D\eta$ and $D_{s}\bar{K}$ system with isospin $1/2$ and $3/2$ was reported in Ref. Moir et al. (2016). Using a large basis of quark-antiquark and meson-meson basis states, the finite volume energy spectrum could be calculated to high precision, allowing for the extraction of the scattering amplitudes in the $S$-, $P$- and $D$-waves. With the help of the coupled-channel Lüscher formalism and various parameterizations of the $T$-matrix, three poles were found in the complex plane: a $J^{P}=0^{+}$ near-threshold bound state, $M_{S}=(2275.9\pm 0.9)\,$MeV, with a large coupling to $D\pi$, a deeply bound $J^{P}=1^{-}$ state, $M_{P}=(2009\pm 2)\,$MeV, and evidence for a $J^{P}=2^{+}$ narrow resonance coupled predominantly to$D\pi$, $M_{D}=(2527\pm 3)\,$MeV. An interesting observation was made in Ref. Albaladejo et al. (2017a). Using UCHPT, it was shown that there are in fact two ($I=1/2,J^{P}=0^{+}$) poles in the region of the $D_{0}^{*}(2400)$ in the coupled-channel $D\pi,D\eta,D_{s}\bar{K}$ scattering amplitudes. They couple differently to the involved channels and thus should be understood as two states. Having all the parameters fixed from earlier studies in Ref. Liu et al. (2013a), the energy levels for the coupled-channel system in a finite volume were predicted. These agree remarkably well with the lattice QCD results in Moir et al. (2016). The intricate interplay of close-by thresholds and resonance poles already pointed out in Döring et al. (2013) is also found, and it is stressed that more high-statistics data are needed to better determine the higher mass pole.

We now turn to the $XYZ$ states. Consider first the $X(3872)$. There have been a number of studies using diquark-diquark or tetraquark interpolating fields over the years, but none of these has been conclusive, see e.g. Chiu and Hsieh (2007); Yang et al. (2013). Evidence for a bound state with $J^{PC}=1^{++}$ $(11\pm 7)$ MeV below the $D\bar{D}^{*}$ threshold was reported in Ref. Prelovsek and Leskovec (2013a). This establishes a candidate for the $X(3872)$ in addition to the near-by scattering states $D\bar{D}^{*}$ and $J/\psi\rho$. This computation was performed at $M_{\pi}=266\,$MeV but in a small volume $L\simeq 2\,$fm. This finding was validated using the Highly Improved Staggered Quark action Lee et al. (2014). Finally, a refined study allowing also for the mixing of tetraquark interpolators with $\bar{c}c$ components was presented in Padmanath et al. (2015). A candidate for the $X(3872)$ with $I=0$ is observed very close to the experimental state only if both $\bar{c}c$ and $D\bar{D}^{*}$ interpolators are included. However, the candidate is not found if diquark-antidiquark and $D\bar{D}^{*}$ are used in the absence of $\bar{c}c$. Note that in Refs. Jansen et al. (2014); Garzon et al. (2014); Jansen et al. (2015); Baru et al. (2015a) strategies for extracting the properties of the $X(3872)$ from finite-volume data (at unphysical quark masses) have been worked out.

Consider next the $Z_{c}(3900)$. Various lattice calculations have been performed, which, however, did not lead to conclusive results, see e.g. Refs. Prelovsek and Leskovec (2013b); Prelovsek et al. (2015); Chen et al. (2014); Ikeda et al. (2016). For example, in the most recent work Ikeda et al. (2016) it was argued that this state is most probably a threshold cusp. Also, a systematic analysis of most of these data using a finite volume version of the framework in Ref. Albaladejo et al. (2016b) did not allow for a definite conclusion on the nature of the $Z_{c}(3900)$ Albaladejo et al. (2016a).

The Chinese Lattice QCD Collaboration has also studied $D^{*}\bar{D}_{1}$ Meng et al. (2009); Chen et al. (2016e) and $D^{*}\bar{D}$ scattering Chen et al. (2015f) with the aim of investigating the structure of the $Z_{c}(4430)$ and $Z_{c}(4025)$, respectively. These studies were mostly exploratory and no definite statements could be drawn.

V.5 Lattice QCD results on hadrons built from light quarks

Here we summarize briefly some very recent results on hadrons made entirely of light $u,d,s$ quarks, more precisely, the scalar mesons $f_{0}(500)$ and $a_{0}(980)$ as well as $\Lambda(1405)$.

The first determination of the energy dependence of the isoscalar $\pi\pi$ elastic scattering phase shift and the extraction of the $f_{0}(500)$ based on dynamical QCD using the methods described above was given by the Hadron Spectrum Collaboration in Ref.  Briceño et al. (2017). From the volume dependence of the spectrum the $S$-wave phase shift up to the $K\bar{K}$ threshold could be extracted. The calculations were performed at pion masses of 236 and 391 MeV. The resulting amplitudes are described in terms of a scalar meson which evolves from a bound state below the $\pi\pi$ threshold at the heavier quark mass to a broad resonance at the lighter quark mass. This is in line with the prediction of Ref. Hanhart et al. (2008) based on UCHPT to one loop. Earlier, the same collaboration had analyzed the coupled channel $\pi\eta,K\bar{K},\pi\eta^{\prime}$ system with isospin $I=1$ and extracted properties of the $a_{0}(980)$ meson Dudek et al. (2016). The model-independent lattice data on energy levels were reanalyzed using UCHPT in Refs. Guo et al. (2017b); Döring et al. (2016). In particular, Ref. Guo et al. (2017b) pointed out some ambiguities in the $I=1$ solution.

There have been quite a few studies of the $\Lambda(1405)$ as a simple three-quark baryon state by various lattice collaborations. In view of the intricacies of the coupled channel $K^{-}p$ scattering discussed earlier, we will not further consider these as coupled-channel effects must be considered. An exception is the analysis of Ref. Hall et al. (2015) based on the PCAS-CS ensembles Aoki et al. (2009) with three-quark sources allowing for scalar and vector diquark configurations that leads to the vanishing of the strange magnetic form factor of the $\Lambda(1405)$ at the physical pion mass. It is argued that this can only happen if the $\Lambda(1405)$ is mostly an antikaon-nucleon molecule. This is further validated by applying a finite-volume Hamiltonian approach to the measured energy levels Wu et al. (2014a). This lattice QCD result appears to be at odds with the accepted two-pole scenario. However, as pointed out in the UCHPT analysis of Ref. Molina and Döring (2016), these results exhibit some shortcomings. It is argued in that work, that what is really discussed in Hall et al. (2015) is the heavier of the two poles. In particular the complete absence of the $\pi\Sigma$ threshold in these data is discussed, as this threshold would couple stronger to the lighter pole. This effect is presumably due to the neglect of the baryon-meson interpolating fields in Ref. Hall et al. (2015). The required operators are also specified in Molina and Döring (2016). It will be interesting to see lattice QCD calculations including all the relevant channels and required interpolating fields. We also point out that better methods to calculate the matrix elements of unstable states has been given in Bernard et al. (2012); Briceño and Hansen (2016).

VI Phenomenological manifestations of hadronic molecules

A large number of theoretical studies on the recently discovered exotic candidates focus on the computation of masses (for recent reviews see Refs. Brambilla et al. (2014); Chen et al. (2016a); Lebed et al. (2017); Esposito et al. (2016b)). However, from the discussions in Secs. III and IV, it is clear that the internal structure and especially the molecular nature of a physical state manifests itself predominantly in some properly identified dynamical production and decay processes. For near-threshold hadronic molecules, the pertinent observables are provided by a set of low-energy quantities: the scattering length and effective range for the constituent-hadron system or, equivalently, the effective coupling of the hadronic molecular candidate to its constituents, since these quantities are heavily intertwined as demonstrated in Sec. III. As discussed in detail there, the probability to find the two-hadron component in the physical state, $(1-\lambda^{2})$, can be extracted directly from these quantities. However, due to the presence of various energy scales driven by different physics aspects, not all production and decay processes are sensitive to the effective coupling as will be discussed with various examples mainly on the $XYZ$ states in this section. In addition, the implications of heavy quark spin and flavor symmetries on the spectrum of hadronic molecules as well as the interplay between hadronic molecules and nearby triangle singularities are presented.

VI.1 Long-distance production and decay mechanisms

As in Sec. III, we denote the wave function of a hadronic molecule candidate as $\Psi$. In order to allow for a quantitatively controlled analysis, the state must be located close to the relevant two-body threshold of $h_{1}$ and $h_{2}$. Then the long-distance momentum scale is given by $\gamma=\sqrt{2\mu E_{B}}$, c.f. Eq. (8), with $\gamma\ll\beta$, where $\beta$ is the inverse of the range of forces. We define two classes of production and decay processes, namely

• •

long-distance processes, in which the momenta of all particles in the CM frame of $h_{1}h_{2}$ are of $\mathcal{O}\!\left(\gamma\right)\!\,$;

• •

short-distance processes, which involve particles with a momentum $\gtrsim\beta$ in the CM frame of $h_{1}h_{2}$.

It is shown in this section that only the former are sensitive to the molecular component of the state investigated. The complications in the latter will be discussed in the next section.

VI.1.1 Decays into the constituents and transitions between molecular states

The long-distance processes involving hadronic molecules can be computed using the EFT machinery introduced in Sec. IV. When only the degrees of freedom with momenta of $\mathcal{O}\!\left(\gamma\right)\!\,$ are kept, the EFT is XEFT or, more generally, NREFT${}_{\text{II}}$. When all particle energies are much smaller than $\beta^{2}/(2\mu)$ in the $h_{1}h_{2}$ CM frame, the amplitudes involving pure molecular states for the pertinent processes are at LO determined by the scattering length universality Braaten and Hammer (2006). Decay amplitudes are then proportional to the effective coupling $g_{\rm eff}$ defined in Eq. (29) which can also be expressed in terms of the scattering length. Clearly such an approach cannot be simply applied to predominantly compact states since then $g_{\rm eff}$ becomes small and short-distance mechanisms become more important than hadronic loops.

For instance, as soon as the $X(3872)$ is treated as a $D\bar{D}^{*}$ molecule, the most important long-distance processes are its decays into $D^{0}\bar{D}^{*0}\to D^{0}[\bar{D}^{0}\pi^{0}/\bar{D}^{0}\gamma]$, discussed in Sec. IV.1.4. The decay rates and the momentum distributions of the final states serve as a good probe for the structure of the $X(3872)$ Voloshin (2004b).272727These decays were also discussed in Refs. Swanson (2004b); Voloshin (2006); Meng and Chao (2007); Fleming et al. (2007); Liang et al. (2010); Braaten and Stapleton (2010); Baru et al. (2011); Guo et al. (2014a); Polosa (2015). Higher order corrections can be calculated using NREFT${}_{\text{I}}$ detailed in Sec. IV.1.2. For the $Y(4260)$, the most important process for the detection of its $D_{1}\bar{D}$ component would be the decay into $D_{1}\bar{D}\to[D^{*}\pi/D^{*}\gamma]\bar{D}$ Cleven et al. (2014b); Qin et al. (2016).

It may happen that two of the particles in the final states in the above mentioned three-body decays form another hadronic molecule in the final state. The transition of a shallow bound state into a light particle and another shallow bound state receives two enhancements: large coupling constants for the vertices involving the molecular states, and the $1/v\simeq 2/(v_{1}+v_{2})$ factor as shown in Sec. IV.1.2, where $v_{1}$ and $v_{2}$ denote the relative velocities of the heavy mesons before and after the emission of the light particle (see Fig. 11 and Eq. (55)).

The possibility of a near-threshold pole in the $D\bar{D}$ final state interaction and its possible influence on the $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ transition was studied in Ref. Guo et al. (2014a), c.f. Fig. 15. Note that experimental information on this distribution does not exist yet.282828A few calculations based on phenomenological models suggested the possible existence of a $D\bar{D}$ bound state Wong (2004); Zhang et al. (2006); Fernández-Caramés et al. (2009); Zhang and Huang (2009); Liu et al. (2010); Li et al. (2012). However, the interplay of hadronic molecules in the final and the initial state might well have been observed already as detailed in the remaining parts of this section.

The $D_{1}(2420)\bar{D}$ threshold in the $J^{PC}=1^{--}$ channel is the closest $S$-wave open-charm threshold that the $Y(4260)$ can couple to. It is at the same time the lowest $S$-wave open-charm threshold with vector quantum numbers, which provides a natural explanation why the first (established) exotic vector state is significantly heavier than the $X(3872)$. Assuming that $Y(4260)$ is a $D_{1}\bar{D}$ molecule and $X(3872)$ and $Z_{c}(3900)$ are $D^{*}\bar{D}$ molecules with $J^{PC}=1^{++}$ and $1^{+-}$, respectively, the decays of $Y(4260)$ into $Z_{c}\pi$ and $X(3872)\gamma$ will occur through the mechanisms shown in Fig. 19 (a) and (b), where the type of the light particle accompanying the hadronic molecular state is controlled by the positive and negative $C$ parity of the $\pi^{0}$ and the photon, respectively.

Since $v\lesssim 0.1$ for both transitions, $1/v$ indeed provides a large factor, shown as the solid lines in Fig. 12. Therefore, a copious production of $Z_{c}(3900)$ from $Y(4260)\to Z_{c}(3900)\pi$ transitions as observed at both BESIII and Belle Ablikim et al. (2013a); Liu et al. (2013b) appears naturally within this dynamical picture Wang et al. (2013a). In addition, $if$ the above explanation is indeed correct, also the $X(3872)$ must be necessarily produced with a large rate for the production in $Y(4260)$ radiative decays Guo et al. (2013a). Indeed, assuming that the $Y(4260)$ and $X(3872)$ are pure bound states of $D_{1}\bar{D}$ and $D^{0}\bar{D}^{*0}$,292929It has been emphasized in Sec. IV.1.4 that the charged charm mesons need to be taken into account for the $X(3872)$ in the framework of NREFT${}_{\text{I}}$ since they are below the hard scale and should be treated explicitly. The reason for neglecting them here is that the rate for the $D_{1}^{0}\to D^{*0}\gamma$ is at least one order of magnitude larger than that for the $D_{1}^{+}\to D^{*+}\gamma$ from nonrelativistic quark model calculations Fayyazuddin and Mobarek (1994); Godfrey (2005); Close and Swanson (2005). respectively, one can get the coupling strengths using the relation in Eq. (27) with $\lambda^{2}=Z=0$

[TABLE]

where we have taken the $X(3872)$ binding energy to be $(90\pm 90)$ keV, the first errors are from the uncertainties of the binding energies, and the second ones are from the approximation of Eq. (27) due to neglecting terms suppressed by $\gamma/\beta$. Here, the inverse of the range of forces is conservatively estimated by $\beta\sim\sqrt{2\,\mu\,\Delta_{\text{th}}}$, with $\Delta_{\text{th}}$ the difference between the threshold of the components and the next close one, which is $M_{D^{*+}}+M_{D^{+}}-M_{D^{*0}}-M_{D^{0}}$ for the $X(3872)$ and $M_{D_{1}}+M_{D^{*}}-M_{D_{1}}-M_{D}$ for the $Y(4260)$, respectively. The partial width for the $Y(4260)\to X(3872)\gamma$ in NREFT${}_{\text{I}}$ can reach a few tens of keV depending on the exact value of the $X(3872)$ binding energy Guo et al. (2013a). The predicted copious production of the $X(3872)$ in the $Y(4260)$ radiative decays was later confirmed by BESIII Ablikim et al. (2014d).

It is worthwhile to mention that if the CM energy of the $e^{+}e^{-}$ collisions is very close to the $D_{1}(2420)\bar{D}$ threshold at around 4.29 GeV,303030The production of an $S$-wave pair of $D_{1}(2420)$ and $\bar{D}$, which are $j_{\ell}^{P}=3/2^{+}$ and $j_{\ell}=1/2^{-}$ states, respectively, breaks HQSS Li and Voloshin (2013). This point was in fact already noticed in the classical papers of the Cornell model, see Table VI in Ref. Eichten et al. (1978) and Table VIII in Ref. Eichten et al. (1980). However, in the energy region about 4.2 GeV HQSS breaking can be sizable Wang et al. (2014). the production of the $Z_{c}(3900)$ and $X(3872)$ via the mechanism under consideration gets even more enhanced since the kinematical condition for the TS discussed in Sec. IV.1.1 is then (nearly) satisfied. However, the resulting enhancement is balanced by the fact that the energy is away from the peak of the $Y(4260)$ spectral function.

It should be mentioned that some experimental observations consistent with hadronic molecules are also claimed to be consistent with other models. For instance, treating both the $X(3872)$ and $Y(4260)$ as tetraquarks also leads to a sizable width for the radiative decay $Y(4260)\to X(3872)\gamma$  Chen et al. (2015c). However, large transitions to the constituents of the hadronic molecules appear to be unique signatures for the molecular states. In addition, their importance leaves visible imprints in the line shapes of states like the$Y(4260)$ Cleven et al. (2014b); Qin et al. (2016).

VI.1.2 More on line shapes

The line shape of a hadronic molecule near its constituent threshold reflects a long-distance phenomenon and can be used as a criterion for establishing their nature. The energy dependence of a hadronic molecule production line shape generally does not appear to be trivial.

The data available at present for the line shapes of $X(3872)$ appear to be insufficient for a unique conclusion about the pole locations of the state. For example, a simultaneous fit of the line shape of the $X(3872)$ in the $J/\psi\pi\pi$ and the $D^{*0}\bar{D}^{0}$ invariant mass distributions employing a generalized Flatté parametrization Hanhart et al. (2007a) revealed that $X(3872)$ is a virtual state. However, as soon as an explicit quarkonium pole is included in the analysis, the authors of Refs. Zhang et al. (2009a); Kalashnikova and Nefediev (2009); Meng et al. (2015) find that $X(3872)$ is the $2^{3}P_{1}$ charmonium with a large coupling to the $D^{0}\bar{D}^{*0}$ channel — in the light of the discussion of Sec. III.2 one needs to conclude that also in this case the $X(3872)$ has a sizeable molecular admixture. It should also be stressed that in Ref. Hanhart et al. (2007a) the width of the $D^{*}$ was omitted, which might distort the line shapes (c.f. Sec. III.3) as was pointed out in Refs. Braaten and Stapleton (2010); Braaten and Lu (2007). According to these analyses, once this effect is included, the fit favors a bound state solution. Another study based on an improved Flatté formula Artoisenet et al. (2010) notices that the current data can accept $X(3872)$ as both a $D^{0}\bar{D}^{*0}$ hadronic molecule or the fine-tuned $2^{3}P_{1}$ charmonium coupled with the $D^{0}\bar{D}^{*0}$ channel. Also the more recent analysis of Ref. Kang and Oller (2017) finds solutions with either a bound state pole or virtual states. Thus, to further investigate the nature of $X(3872)$ a high resolution scan of its line shapes, especially within a few $\mathrm{MeV}$ of the $D^{0}\bar{D}^{*0}$ threshold, is necessary.

A study of the line shapes of the two $Z_{b}$ states in $h_{b}(1P,2P)\pi$ channels is presented in Ref. Cleven et al. (2011a) based on the NREFT${}_{\text{I}}$ framework discussed in Sec. IV. By fitting to the $h_{b}(1P,2P)\pi$ invariant mass distribution, it is found that the current data are consistent with the two $Z_{b}$ states being $B\bar{B}^{*}$ and $B^{*}\bar{B}^{*}$ bound states, respectively. Their line shapes in the elastic channels are also studied in the XEFT/NREFT${}_{\text{II}}$ (see Sec. IV.1.3) framework with the HQSS breaking operator included explicitly in Ref. Mehen and Powell (2013) where, however, no pole locations were extracted. The line shapes of the two $Z_{b}$ states were investigated in both elastic and inelastic channels in Refs. Hanhart et al. (2015); Guo et al. (2016a) based on separable interactions, see Fig. 2. An explicit calculation revealed that the line shapes get distorted very little when a non-separable interaction is included, such as the one-pion-exchange potential. Similar studies of the $Z_{c}(3900)$ line shape in both $J/\psi\pi$ and $D\bar{D}^{*}$ can be found in Refs. Albaladejo et al. (2016b); Zhou and Xiao (2015); Pilloni et al. (2016). Based on these results one needs to state that also the data currently available for the $Z_{c}(3900)$ are insufficient to pin down the pole locations.

The situation that it is not easy to extract the pole locations is partly because that the observed peaks for the $X(3872)$, $Z_{c}(3900)$ and $Z_{b}$ states are too close to the thresholds. Being tens of MeV below the $D_{1}\bar{D}$ threshold, which, however, also means a larger uncertainty, the $Y(4260)$ situation is different. One sees clearly a nontrivial structure around the $D_{1}\bar{D}$ threshold in the $Y(4260)$ line shape in the $J/\psi\pi\pi$ invariant mass distribution, c.f. Fig. 5. It indicates that the coupling to the $D_{1}\bar{D}$ plays an essential role in understanding the $Y(4260)$ in line with the analysis in Sec. III.3. Correspondingly the molecular picture predicts a highly nontrivial energy dependence for $e^{+}e^{-}\to D_{1}\bar{D}\to[D^{*}\pi]D$ Cleven et al. (2014b), c.f. the middle panel in Fig. 6, that awaits experimental confirmation at, e.g., BESIII.

VI.1.3 Enhanced isospin violations in molecular transitions

As argued above in Sec. III.2 for molecular states the coupling of the pole to the continuum channel that forms the state is large. As a consequence of this loop effects get very important that can lead to very much enhanced isospin violations if the molecular state is close to the relevant threshold. To be concrete we start with a detailed discussion of the implications of this observation for the $f_{0}$–$a_{0}$ mixing, first discussed in Ref. Achasov et al. (1979).

If both the isoscalar $f_{0}(980)$ as well as the isovector $a_{0}(980)$ were $\bar{K}K$ molecular states, the leading mixing effect of the two scalar mesons would the difference of the loop functions of charged and neutral kaons as depicted in Fig. 20 (in isospin conserving transitions the sum enters). In the near-threshold regime we may approximate the loops by their leading energy dependence provided by the respective unitarity cuts:

[TABLE]

where $p_{K}$ denotes the modulus of the relative momentum of the kaon pair. Obviously, this leading contribution model-independently provides a measure of the product of the effective couplings of $a_{0}$ and $f_{0}$ to the kaons and therefore, as discussed in Sec. III.2.1, to the molecular admixture of both states. In the isospin limit the loops cancel exactly. However, as soon as the masses of the two-hadron states are different due to isospin violations ($M_{K^{0}}-M_{K^{+}}=4$ MeV), there appears an offset in the thresholds and the mentioned cancellation is no longer complete. This results in a transition between different isospins with all its strength located in between the two thresholds. In Refs. Wu et al. (2007); Hanhart et al. (2007c) it was predicted that this very peculiar effect should show up prominently in the transition $J/\psi\to\phi\pi^{0}\eta$, if both $f_{0}(980)$ and $a_{0}(980)$ are $K\bar{K}$ molecular states, since only then the coupling of the states to the $K\bar{K}$ is sufficiently strong for the effect to be observable. A few years later the predicted very narrow signal was measured at BESIII Ablikim et al. (2011) providing strong evidence for a prominent molecular admixture in these light scalar mesons.

Another prominent example where a molecular nature of a near-threshold state leads to a natural explanation of a huge isospin violation is the equal decay rate of the $X(3872)$ to the isoscalar $\pi\pi\pi J/\psi$ and the isovector $\pi\pi J/\psi$ channels, where in both cases the few-pion systems carry vector quantum numbers and thus may be viewed as coming from the decay of an $\omega$ and a $\rho^{0}$ meson, respectively. The argument goes as follows: The mass of the $X(3872)$ is located 7 MeV below the nominal $\omega J/\psi$ threshold, but 5 MeV above the nominal $\rho^{0}J/\psi$ threshold. In addition, the width of the $\omega$ is only 8 MeV such that the decay of the $X(3872)$ into the $\rho^{0}J/\psi$ channel is strongly favored kinematically. Therefore, the $X(3872)$ cannot have a significant isovector component in its wave function since otherwise it would significantly more often decay into the $\rho^{0}J/\psi$ than into the $\omega J/\psi$ channel. However, a calculation for an isoscalar $X(3872)$ that predominantly decays via $D^{*}\bar{D}$ loops naturally gives the experimental branching ratios for the pions-$J/\psi$ channels simply because the close proximity of the mass of the $X$ to the neutral $D^{*}\bar{D}$-threshold automatically produces an enhanced isospin violation in the necessary strength, if the $X(3872)$ is a $D^{*}\bar{D}$ molecule, since only then the coupling to continuum is strong enough Gamermann and Oset (2009). An isoscalar nature of the $X(3872)$ is also required from a study of other possible decay channels Mehen (2015), and its effective couplings to the charged and neutral channels are basically the same Hidalgo-Duque et al. (2013b); Guo et al. (2014a).

As the last example in this context we would like to mention the hadronic width of the isoscalar $D_{s0}^{*}(2317)$. This state is located below the $DK$ threshold and as such can decay strongly only via isospin violation into the isovector $D_{s}\pi$ channel. The two most prominent decay mechanisms of $D_{s0}^{*}(2317)$ are an isospin conserving transition into $D_{s}\eta$, followed by the isospin violating $\pi\eta$ mixing amplitude, and the isospin violating difference between a $D^{0}K^{+}$ and a $D^{+}K^{0}$ loop (subleading contributions to this transition were studied in Ref. Guo et al. (2008a)). The former mechanism should be present regardless the nature of the state, which typically leads to widths of the order of 10 keV Colangelo and De Fazio (2003). The latter mechanism, however, is large, if indeed the $D_{s0}^{*}(2317)$ were a $DK$ molecule. In fact typical calculations for molecular states give a width of the order of 100 keV Faessler et al. (2007); Lutz and Soyeur (2008); Liu et al. (2013a) — for more details we refer to Sec. V.4. Thus, if this admittedly small width could be measured, e.g., at $\overline{\text{P}}$ANDA, its value would provide direct experimental access to the nature of $D_{s0}^{*}(2317)$.

VI.1.4 Enhanced production of hadronic molecules and conventional

hadrons due to triangle singularities

From the analysis in Sec. IV.1.1, we see that the TS on the physical boundary is always located close to the threshold of the intermediate particles. Furthermore, its effect is most pronounced if the two intermediate particles are in an $S$-wave, since otherwise the centrifugal barrier suppresses small momenta. Hadronic molecules are located naturally near-thresholds as well, and in all cases considered in this review couple in $S$-wave to its constituents. Therefore, in the course of this review there are two important aspects of TS that need to be discussed: On the one hand, a TS may lead to a pronounced structure in experimental observables that could be mistaken as a state; on the other hand, a TS can enhance the production of a hadronic molecule in a given reaction. Note that also the production of the conventional hadrons can by strongly enhanced by a TS within small energy regions. This is accompanied by a significant distortion of the line shapes since the location of the TS depends on the invariant masses of the external states. For example, the signal in the $\eta\pi\pi$ channel interpreted as $\eta(1405)$ and the signal in the $K\bar{K}\pi$ interpreted as $\eta(1475)$ could find their origin in a single pole accompanied by a TS Wu et al. (2012, 2013). We note that although $P$-wave couplings are present in the triangle loop for the $\eta(1405/1475)$ decays, the perfect satisfaction of the TS condition, Eq. (49), causes detectable effects in these decays.

Since the $K\bar{K}^{*}$ system can contribute to both $I=0$ and $I=1$ channels with $J^{PC}=1^{++}$ and $1^{+-}$, one expects that the TS may cause enhancements also in these channels. In the following we list those possible enhancements and their quantum numbers which can be searched for in experiment:

[TABLE]

where $\tilde{h}_{1}(1420)$ and $\tilde{b}_{1}(1420)$ refer to the TSs whether or not there exists resonances around. Note that the $f_{1}(1420)$ needs to be taken into account for the angular distribution in the $J/\psi\to\gamma 3\pi$ process Ablikim et al. (2012), and a detailed partial wave analysis suggests the presence of the $f_{1}(1420)$ resonance together with a TS mechanism Wu et al. (2013), while it is argued in Ref. Debastiani et al. (2017a) that the $f_{1}(1420)$ is the manifestation of the $f_{1}(1285)$ at higher energies due to the TS. The $a_{1}(1420)$ has been reported by the COMPASS Collaboration in $\pi^{-}p\to\pi^{-}\pi^{-}\pi^{+}p$ and $\pi^{-}\pi^{0}\pi^{0}p$ Adolph et al. (2015) and can be well explained by the TS mechanism Liu et al. (2016a); Mikhasenko et al. (2015). The decay channels in the square brackets are $G$-parity violating and those in the round brackets are limited by the phase space. One notices that there are states observed in the relevant mass regions, namely, $a_{1}(1260)$, $f_{1}(1285)$, $h_{1}(1170)$, and $b_{1}(1235)$ Patrignani et al. (2016). Although most of these states have masses outside of the TS favored mass region, i.e. $1.385\sim 1.442$ GeV Liu et al. (2016a), the $h_{1}(1380)$ is located at the edge of the TS kinematics and some detectable effects could be expected Guo et al. (2013a); Ablikim et al. (2014d). Notice that when there is a TS in action, the peak position for a resonance could be shifted towards its location. Some structures around thresholds of a pair of other light hadrons were also suggested to be due to a TS, for instance, the $f_{2}(1810)$ around the $K^{*}\bar{K}^{*}$ threshold Xie et al. (2017) and the $\phi(2170)$ around the $N\bar{\Delta}$ threshold Lorenz et al. (2015).

In the heavy quarkonium sector, the most famous example for an enhanced production rate via the TS is the observation of the $Z_{c}(3900)$ at the mass region of $Y(4260)$ Wang et al. (2013a, b); Liu and Li (2013); Szczepaniak (2015); Gong et al. (2016b); Pilloni et al. (2016). In addition, the sensitivity of the TS to the kinematics of the reaction might well be the reason why the $Z_{c}(4020)$ is not seen in the same decay Wang et al. (2013a, b). As discussed in Sec. VI.1, the same mechanism also enhances the transition $Y(4260)\to\gamma X(3872)$ Guo et al. (2013a) and suggests that the rate for $e^{+}e^{-}\to\gamma X_{2}$, which can be used to search for the spin-2 partner of the $X(3872)$, $X_{2}$ (see Sec. IV.2), gets most enhanced if the $e^{+}e^{-}$ collision energy is between 4.4 and 4.5 GeV Guo et al. (2015d).313131This suggestion is based on the assumption that the $X_{2}$ mass is very close to the $D^{*}\bar{D}^{*}$ threshold. If its mass is tens of MeV below the threshold as suggested in Baru et al. (2016), then the TS would be further away from the physical boundary and the production would get less enhanced. A candidate for the analogue of $Y(4260)$ in the bottomonium sector is $\Upsilon(11020)$, since it is located close to the $B_{1}\bar{B}$ threshold Wang et al. (2013b); Bondar and Voloshin (2016). Here, the TS could affect both the $Z_{b}(10610)$ and $Z_{b}(10650)$ Wang et al. (2013b), although at the mass of $\Upsilon(11020)$ the production of the lower $Z_{b}$ state is more favored since the corresponding TS is closer Bondar and Voloshin (2016). Based on the current statistics in Belle Bondar and Voloshin (2016), it is difficult to judge whether there is one peak or two peaks present in Fig. 3. In Refs. Pakhlov and Uglov (2015); Uglov and Pakhlov (2016) the structure identified as the charged exotic $Z_{c}(4430)$ observed in the $\pi\psi^{\prime}$ final states by Belle Mizuk et al. (2009); Chilikin et al. (2013) and LHCb Aaij et al. (2015a) was claimed to be not connected to a pole but to owe its existence to the presence of a TS.

Suggestions to search for resonance-like structures due to the TS in the heavy meson and heavy quarkonium mass regions can be found in Liu (2014); Liu and Oka (2016a); Liu et al. (2016a); Liu and Meißner (2017). In particular, the very recent BESIII results on the $\pi^{\pm}\psi^{\prime}$ invariant mass distributions for the $e^{+}e^{-}\to\pi^{+}\pi^{-}\psi^{\prime}$ process at different CM energies seem in line with the predictions made in Liu (2014). Some of the strongly favored triangle loops are listed in Table 7.

The $P_{c}(4450)$ structure observed by LHCb Aaij et al. (2015b) in $\Lambda_{b}$ decays, no matter what its nature is, also contains a TS contribution, as long as it strongly couples in an $S$-wave Bayar et al. (2016) to either $\chi_{c1}p$ Guo et al. (2015c); Liu et al. (2016b); Guo et al. (2016b); Meißner and Oller (2015) or $\Lambda_{c}(2595)\bar{D}$ Liu et al. (2016b). Recent discussions on the role of TSs in the light baryonic sector can be found in Refs. Wang et al. (2017); Roca and Oset (2017); Debastiani et al. (2017b); Samart et al. (2017).

It is worthwhile to emphasize that once the kinematics for a process (nearly) satisfies the TS condition given in Eq. (49), the enhancement in reaction rates due to TS contribution is always there, and can produce a narrow peak once the relevant coupling is in an $S$-wave Bayar et al. (2016). The only question is whether it is strong enough to produce the observed or an observable structure. Complications due to the interference between the TS and a tree-level contribution is discussed in Schmid (1967); Goebel et al. (1983) in the single-channel case, and in Anisovich and Anisovich (1995); Szczepaniak (2016) for coupled channels. The key to distinguishing whether a structure is solely due to a TS or originates from a genuine resonance is the sensitivity of the TS on kinematics: If in reactions with different kinematics the same structure is observed, it most likely reflects the existence of a pole (resonance).

VI.2 Short-distance production and decay mechanisms

In Sec. VI.1 we discussed both decay and production mechanisms sensitive to the long-range parts of the wave function of a state and therefore sensitive to its molecular nature. Here we demonstrate that there are also decays and production reactions that do not allow one to extract the molecular component of a given state. We start with an example of the former to then switch to the latter.

It was claimed long ago Swanson (2004a) that the ratio

[TABLE]

with the measured value given in Eq. (3), is very sensitive to the molecular component of the $X(3872)$ wave function. In particular using vector meson dominance and a quark model, in Ref. Swanson (2004a) it was predicted to be about $4\times 10^{-3}$ if the $X(3872)$ is a hadronic molecule with a dominant $D^{0}\bar{D}^{*0}$ component plus a small admixture of the $\rho J/\psi$ and $\omega J/\psi$. However, as demonstrated in Ref. Guo et al. (2015a), when radiative decays of $X(3872)$ are calculated using NREFT${}_{\text{I}}$ (see Sec. IV.1.2) field theoretic consistency calls for the inclusion of a counter term at LO. In other words, the transitions are controlled by short-distance instead of long-distance dynamics and therefore do not allow one to extract any information on the molecular component of the $X(3872)$ wave function.

There also have been many claims that production rates of multiquark states in high-energy collisions are sensitive to a molecular admixture of those states. The production of the $X(3872)$ at hadron colliders was debated in Bignamini et al. (2009); Artoisenet and Braaten (2010); Bignamini et al. (2010); Artoisenet and Braaten (2011); Butenschoen et al. (2013); Esposito et al. (2013); Meng et al. (2013b), and that of the spin and flavor partners of the $X(3872)$ was discussed in Bignamini et al. (2010); Guo et al. (2014c, e). For the production of the $X(3872)$ in $B$ decays, we refer to Braaten and Kusunoki (2005a); Braaten et al. (2004); Meng et al. (2013a); Fan et al. (2012); Meng et al. (2013b). The production of the $X(3872)$ in heavy ion collisions was discussed in Cho and Lee (2013); Martínez Torres et al. (2014) and by the ExHIC Collaboration including other hadronic molecular candidates Cho et al. (2011a, b, 2015). In Ref. Larionov et al. (2015) it was proposed that the hadronic molecular component of the $X(3872)$ could be extracted from its production in antiproton-nucleus collisions. Prompt productions of diquark-antidiquark tetraquarks at the LHC were studied in Guerrieri et al. (2014). Here, we discuss to what extent high-energy reactions can be used to disentangle the structure of a near-threshold state.

The underlying physics for the short-distance production and decay processes of a shallow hadronic molecule is characterized by vastly different scales. This allows for a derivation of factorization formulae for the corresponding amplitudes Braaten and Kusunoki (2005a); Braaten et al. (2004); Braaten and Kusunoki (2005b); Braaten and Lu (2006).

We illustrate the production of a $h_{1}h_{2}$ pair and a near-threshold state with wave function $\Psi$ in Fig. 21, where $\Gamma$ denotes a short-distance source. The pair of $h_{1}$ and $h_{2}$ can be produced directly at short distances shown as diagram (a), and through rescattering shown as diagram (b). If the rescattering strength were weak, the production would be well approximated by only the $\Gamma$ term. There would also be no drastic energy dependence in the near-threshold region, and the strength of the $S$-wave cusp exactly at the threshold would not be strong enough to produce a narrow peak Guo et al. (2015b). However, if the rescattering is strongly attractive, the amplitude $T$ possesses a pole, which we assume to be located close to the threshold with a small binding energy $E_{B}$. Then one gets for the production amplitude of a given decay channel $j$ of the state of interest:

[TABLE]

where $G_{i}$ ($\Gamma_{i}^{\Lambda}$) is the propagator (short-distance production amplitude) for the $i$-th intermediate channel and $\Lambda$ denotes the cutoff of the integral. In principle all dynamical degrees of freedom below the energy scale $\Lambda$ should be accounted for. The short-distance contribution $\Gamma_{i}^{\Lambda}$ also serves to absorb the $\Lambda$ dependence, and as a result $\mathcal{M}$ is $\Lambda$ independent.

Let us consider the kinematic situation that the CM momentum of $h_{1}$ and $h_{2}$ is very small, $\sim\gamma$, and $\Lambda$ is much larger than $\gamma^{2}/(2\mu)$ but still small enough to prevent other channels from being dynamical. In this case, the intermediate state for diagram (b) is $h_{1}h_{2}$. The LO term of the momentum expansion of $\Gamma^{\Lambda}$ is simply a constant. The nonrelativistic two-body propagator is $G(\bm{q};E)=\left[E-\bm{q}^{2}/(2\mu)+i\epsilon\right]^{-1}$, and the $T$-matrix is given by Eq. (73). Thus, one obtains

[TABLE]

If $\Gamma^{\Lambda}\propto\Lambda^{-1}$, the LO $\Lambda$ dependence will be absorbed Braaten and Kusunoki (2005a), and the factorization formula Braaten and Kusunoki (2005a); Braaten et al. (2004); Braaten and Kusunoki (2005b) for the production of the low-momentum $h_{1}h_{2}$ pair follows:

[TABLE]

where $\Gamma\equiv\Gamma^{\Lambda}\Lambda$ is the short-distance part, and the long-distance part $T_{\text{NR}}(E)=(2\pi/\mu)\left(\gamma-\sqrt{-2\mu E}\right)^{-1}$ is provided by the scattering $T$-matrix. From the derivation in Braaten and Lu (2006), it becomes clear that the short-distance part is the Wilson coefficient of the operator production expansion in the EFT. A similar factorization formula was derived in Guo et al. (2014d) with the help of chiral symmetry for high-energy productions of kaonic bound states predicted in Guo and Meißner (2011).

The amplitude for the production of the near-threshold state is obtained from Eq. (114) by replacing $T_{\text{NR}}(E)$ by the square root of its residue $g_{\text{NR}}^{2}$ given in Eq. (76) and multiplying the factor $1/\sqrt{2\mu}$ to account for the difference in normalization factors

[TABLE]

Hence, the production rate, $\propto g_{\text{NR}}^{2}\propto\sqrt{E_{B}}$, c.f. Eq. (28), seems suppressed for very loosely bound states, which is consistent with the common intuition Braaten et al. (2004); Artoisenet and Braaten (2010). In particular, one expects a suppression of a loosely bound state in high-energy reactions. The factorization explained above is the foundation for the proposal to extract the short-distance production mechanism of $X(3872)$ in $B_{c}$ semi-leptonic and hadronic decays Wang and Zhao (2016).

Note that it is a straightforward consequence of Eq. (115) that in ratios of short-distance production rates for two hadronic molecules related to each other through some symmetry the long-distance part containing the information of the structure of the states cancels, and the remaining part solely reflects the difference in the short-distance dynamics and phase space. It could be misleading if such ratios are taken as evidence in support of or against dominantly composite nature of given states.

All the derivations of the factorization formula are based on the LO NREFT${}_{\text{II}}$ so far, which allows self-consistently only the possibility that $\Psi$ is a composite system of $h_{1}$ and $h_{2}$, see the discussion below Eq. (76). If we go to higher orders, momentum-dependent terms need to be kept in the potential as well as the short-distance production amplitude. The probability of $\Psi$ to be a $h_{1}h_{2}$ composite system would be $1-\lambda^{2}<1$, and thus one would also need to introduce a contact production term for $\Psi$. These new terms parameterize the short-distance physics though a detailed dynamics which depends on the more fundamental theory that cannot be specified within the EFT. Such contributions could be interpreted as a short-distance core of the physical wave function $\Psi$. The factorization and the related renormalization at higher orders for the near-threshold production of $h_{1}h_{2}$ and $\Psi$ in short-distance processes remain to be worked out.

The long-distance contribution in Eqs. (114) and (115) is calculable in NREFT, and the short-distance contribution is subject to the more fundamental theories which are QCD and/or electroweak theory. For inclusive high-energy hadron collisions, one is not able to calculate the short-distance contribution model-independently (for estimates using Monte Carlo event generators, see Bignamini et al. (2009); Artoisenet and Braaten (2010); Bignamini et al. (2010); Artoisenet and Braaten (2011); Esposito et al. (2013); Guo et al. (2014c, d, e)), and thus only the order of magnitude of the production cross sections can be estimated. For the production of hadronic molecules in heavy meson decays such as $B$ decays, again one normally is only able to get an order-of-magnitude estimate at the best due to the nonperturbative nature of QCD which dominates the hadronic effects in the short-distance part. The production rate of the $X(3872)$ in $B$ decays was estimated in Ref. Braaten and Kusunoki (2005a); Braaten et al. (2004). In particular, Ref. Braaten and Kusunoki (2005a) predicted that the branching fraction of $B^{0}\to X(3872)K^{0}$ should be much smaller than that of $B^{+}\to X(3872)K^{+}$ assuming that the $X(3872)$ is a $D^{0}\bar{D}^{*0}$ hadronic molecule. The prediction seems to be in contradiction with the later measurements Patrignani et al. (2016) summarized in Sec. II giving a ratio of around 0.5, c.f. Eq. (4). However, as already mentioned before the $X(3872)$ is to a very good approximation an isoscalar state. Its couplings to the neutral $D^{0}\bar{D}^{*0}$ and the charged $D^{+}D^{*-}$ channels is almost the same even if the isospin breaking is taken into account Gamermann and Oset (2009); Guo et al. (2014a). Therefore Eq. (115) needs to be generalized to coupled channels. In particular the production of $\Psi$ gets modified to

[TABLE]

where the summation runs over all possible intermediate channels below the cutoff $\Lambda$. Notably, $g_{\text{NR},i}$ denotes the coupling of the $\Psi$ state to the $i$-th channel. As a result, the $X(3872)$ production rates in neutral and charged $B$ decays should be similar. This may also be understood as that the short-distance parts in Braaten and Kusunoki (2005a) should include the charged channel and were not properly estimated.

We now turn to the discussion of production rates of shallow bound states like $X(3872)$ at hadron colliders. It was claimed that the cross section for the inclusive $X(3872)$ production at high $p_{T}$ at the Tevatron in $\bar{p}p$ collisions is too large to be consistent with the interpretation of $X(3872)$ as a $D^{0}\bar{D}^{*0}$ molecule Bignamini et al. (2009). The reasoning was based on the following estimate for an upper bound of the cross section:

[TABLE]

where $\cal R$ means that the momentum integration only receives support up to some characteristic scale $\cal R$. The upper bound quoted above depends drastically on the value of $\cal R$. A value of ${\cal R}=35$ MeV$\simeq\gamma$, the binding momentum of the $X(3872)$, was chosen in Bignamini et al. (2009). The so-estimated upper bound of 0.071 nb is orders of magnitude smaller than the Tevatron result of 37 to 115 nb.

However, for the derivation in Eq. (117) to be valid $\cal R$ must be large enough that the wave function of the bound state gets largely probed for otherwise the symbol between the first and the second integral needs to be changed from $\simeq$ to $\gg$ and the whole line of reasoning gets spoiled. But this requirement calls for values of $\cal R$ much larger than the binding momentum. To demonstrate this claim we switch to the deuteron wave function. Fig. 22 shows

the averaged deuteron wave function calculated from $\bar{\Psi}_{\Lambda}({\cal R})=\int_{\cal R}d^{3}\bm{k}\,\Psi_{\Lambda}(\bm{k})$, where the subindex $\Lambda$ indicates that a regulator needs to be specified to get a well defined wave function (for more details, see Nogga and Hanhart (2006)). The right panel in the figure is a zoom in linear scale to the relevant $\cal R$ range. From the figure, it is clear that $\bar{\Psi}_{\Lambda}({\cal R})$ is far from being saturated for ${\cal R}\simeq$45 MeV which is the deuteron binding momentum. One needs to take ${\cal R}\sim 2M_{\pi}\sim$300 MeV, the order of the inverse range of forces as pointed out in Artoisenet and Braaten (2010, 2011) based on rescattering arguments, to get a reasonable estimate so that the second line in Eq. (117) can be a good approximation of the first line. With such a large support $\cal R$, the upper bound becomes consistent with the Tevatron measurement Bignamini et al. (2009); Artoisenet and Braaten (2010, 2011). In addition, as discussed in case of the $B$ decays, the charged channels need to be considered as well.

In fact, with the factorization derived above, it was suggested that one can estimate the cross section by combining the use of Monte Carlo event generators, such as Pythia Sjostrand et al. (2008) and Herwig Bahr et al. (2008), and EFT to get the short-distance and long-distance contributions, respectively Artoisenet and Braaten (2010, 2011); Guo et al. (2014e). In Table 8, we show the estimates obtained in Albaladejo et al. (2017b). Indeed, if a small cutoff is used, $\Lambda=0.1$ GeV, and only the neutral charmed mesons are considered by using Guo et al. (2014d, e)

[TABLE]

the obtained cross sections are orders of magnitude smaller than the data, in line with the observation in Bignamini et al. (2009). Here $C=d\sigma[D^{0}\bar{D}^{*0}]/dk/k^{2}$, playing the role of $|\Gamma^{\Lambda}|^{2}$ in Eq. (113), is a constant determined from fitting to the differential cross section of the direct production of the charmed-meson pair from Pythia and Herwig, $\Psi_{X}(\bm{k})$ is the momentum-space wave function of the $X(3872)$, $g_{\text{NR},X}$ is the coupling of the $X(3872)$ to $D^{0}\bar{D}^{*0}$, $\Sigma_{\text{NR}}(-E_{B})$ is the loop function in Eq. (74) but keeping the full $\Lambda$ dependence from the Gaussian regulator. However, when a larger cutoff in the range of, e.g., $[0.5,1.0]$ GeV is used, the cross sections become consistent with both the CDF and CMS measurements. One important point is that the charged $D^{+}D^{*-}+c.c.$ channel needs to be taken into account for this case as discussed above. This is because the binding momentum for the charged channel $\gamma_{\pm}\simeq 126$ MeV is well below the cutoff so that the charged charmed mesons should also play a dynamical role.

Finally, in Ref. Esposito et al. (2015a), the cross sections for the production of light (hyper-)nuclei at small $p_{T}$ at ALICE Adam et al. (2016) were extrapolated to large $p_{T}$, and it was found that they are much smaller than the $X(3872)$ production at large $p_{T}$ at CMS Chatrchyan et al. (2013a). Since light (hyper-)nuclei are loosely bound states of baryons, the authors concluded that loosely bound states are hardly produced at high $p_{T}$, and therefore disfavored the hadronic molecular interpretation of the $X(3872)$. However, although the long-distance contributions for these productions can be managed in the EFT or universality framework, the short-distance contribution for the $X(3872)$ is completely different from that for light nuclei. This leads to a different energy dependence of the cross sections for the light nuclei and the $X(3872)$, and makes such a direct comparison questionable. One essential difference is: At short distances, the $X(3872)$ can be produced through $c\bar{c}$ or $u\bar{u}(d\bar{d})$, which hadronizes into a pair of charmed mesons at larger distances, while the minimal quark number in the light nuclei is always $3N$, with $N$ the number of baryons, giving rise to a suppression. Therefore, it is natural that the $X(3872)$ production cross section at very high $p_{T}$ is orders of magnitude larger than that of light nuclei. This point also leads to the critique Guo et al. (2017a),323232For a response, see Brodsky et al. (2017). which was appreciated in Voloshin (2016a), against the use of constituent counting rules in hard exclusive processes as a way to identifying mulqituark states with a hidden-flavor $q\bar{q}$ pair Kawamura et al. (2013); Kawamura and Kumano (2014); Brodsky and Lebed (2015); Chang et al. (2016).

To summarize, the production and decay processes with a large energy release involve both long and short distance scales. Only the long-distance part is sensitive to the low-energy quantities, thus to the hadronic molecular structure, and can be dealt with in the EFT framework. On the contrary, high energy production rates depend crucially on what happens at short distances, which is often unknown though in principle could be extracted from other reactions depending on the same short-distance physics. However, despite that it is hard to calculate the integrated production rates, the differential invariant mass distributions around the near-threshold states provide a direct access to their line shapes and precise, high-resolution data on those are urgently called for. This has been discussed in general in Sec. III.3 and also in Sec. VI.1.2.

VI.3 Implications of heavy quark spin and flavor symmetries

It turns out that HQSS, and especially its breaking, is also an important diagnostic tool when it comes to understanding the structure of certain states Cleven et al. (2015). The most straightforward example to illustrate this point is the spin doublet made of $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$. On the one hand, both are not only significantly lighter than the prediction of the quark model as reported in, e.g., Ref. Di Pierro and Eichten (2001), also their spin splitting differs. On the other hand, the mass difference of the two states agrees exactly to the mass difference between the $D$ and the $D^{*}$, which would be a natural result if $D_{s0}(2317)$ and $D_{s1}^{*}(2460)$ were $DK$ and $D^{*}K$ molecular states, respectively, since at LO in the heavy quark expansion the $DK$ interaction agrees to the $D^{*}K$ interaction Kolomeitsev and Lutz (2004). In complete analogy it was argued in Ref. Guo et al. (2009a) that, if indeed the $J^{PC}=1^{--}$ state $Y(4660)$ were a bound system of $\psi^{\prime}$ and $f_{0}(980)$ as conjectured in Ref. Guo et al. (2008b), there should exist a $J^{PC}=0^{-+}$ state which is $\eta_{c}^{\prime}f_{0}(980)$ bound system. The mass difference of the latter state to the $\eta_{c}^{\prime}f_{0}(980)$ should agree to that of $Y(4660)$ to the $\psi^{\prime}f_{0}(980)$ threshold,333333This prediction receives support from a calculation using QCD sum rules Wang and Zhang (2010a). and it was even possible to estimate the decay width of the state as well as a most suitable discovery channel. For more discussion on the implications of HQSS for the spectrum of exotic states we refer to Ref. Cleven et al. (2015), where also predictions from other approaches are contrasted to those of the molecular picture.

More predictions can be made for heavy-flavor hadronic molecules by using heavy quark flavor symmetry. The LO predictions are rather straightforward if there is only a single heavy quark in the system. For instance, one would expect the $D_{s0}^{*}(2317)$ as a $DK$ bound state to have a bottom partner, a $\bar{B}K$ bound state, with almost the same binding energy. This prediction together with the one for the bottom partner of the $D_{s1}(2460)$ are given in the fourth and fifth rows in Table 9, where the error of 16 MeV accounts for the use of heavy quark flavor symmetry and is estimated as $2\left(M_{D}+M_{K}-M_{D_{s0}^{*}(2317)}\right)\times\Lambda_{\text{QCD}}(m_{c}^{-1}-m_{b}^{-1})$. Such simple predictions are in a remarkable agreement with the lattice results of the lowest-lying $0^{+}$ and $1^{+}$ bottom-strange mesons: $(5711\pm 13\pm 19)$ MeV for the $B_{s0}^{*}$ and $(5750\pm 17\pm 19)$ MeV for the $B_{s1}$ Lang et al. (2015). This agreement may be regarded as a further support of the hadronic molecular nature of the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ states. For more complicated predictions of these two states using various versions of UCHPT, we refer to Refs. Kolomeitsev and Lutz (2004); Guo et al. (2006, 2007); Cleven et al. (2011b); Altenbuchinger et al. (2014); Torres-Rincon et al. (2014); Cleven et al. (2014a).

A more detailed discussion of spin symmetry partners of hadronic molecules formed by a pair of $S$-wave heavy mesons can be found in Sec. IV.2. Some of the spectroscopic consequences of HQSS and heavy quark flavor symmetry for hadronic molecules are listed in Table 9.

VI.4 Baryon candidates for hadronic molecules

For a discussion of and references for charmed baryons and the $P_{c}$ pentaquark structures as possible hadronic molecules we refer to Sec. II. The closing part of this section will be used to describe the most recent developments about the $\Lambda(1405)$ which basically settled the debate on the nature of this famous state. What is described below is yet another example how the interplay of high quality data and systematic theoretical investigations allows one to identify the nature of certain states.

As already stressed in Sec. II.4.1, there are good reasons to classify the $\Lambda(1405)$ as an exotic particle, since it does not at all fit into the pattern of the otherwise in this mass range quite successful quark models.

Using information on $\bar{K}N$ scattering the existence of the $\Lambda(1405)$ was predicted by Dalitz and Tuan Dalitz and Tuan (1959, 1960) before its observation. Already this study highlights the importance of the $\bar{K}N$ dynamics for the $\Lambda(1405)$. In most modern investigations it appears as a dynamically generated state through coupled-channel effects among all the ten isospin channels ($K^{-}p$, $\bar{K}^{0}n$, $\pi^{0}\Lambda$, $\pi^{0}\Sigma^{0}$, $\pi^{+}\Sigma^{-}$, $\pi^{-}\Sigma^{+}$, $\eta\Lambda$, $\eta\Sigma^{0}$, $K^{+}\Xi^{-}$, $K^{0}\Xi^{0}$) or some of them, in other words, a hadronic molecule. This coupled-channel problem has been studied by solving the LSE or Bethe–Salpeter equation with interaction kernels derived from chiral perturbation theory within a given accuracy. This procedure was first proposed in Ref. Kaiser et al. (1995) and further refined in Ref. Oller and Meißner (2001) and various follow-ups. The major finding of Ref. Oller and Meißner (2001) was the fact that there are indeed two poles, one stronger coupled to the $\bar{K}N$ channel and the other to $\Sigma\pi$, which should thus be understood as two distinct states. Both poles are located at the second Riemann sheet, and have shadow poles in the third one Oller and Meißner (2001). This two-pole scenario can be understood by considering the SU(3) limit and its subsequent breaking, see Jido et al. (2003). Presently, various groups have performed calculations to the NLO accuracy Ikeda et al. (2012); Guo and Oller (2013); Mai and Meißner (2013), see also the recent comparison of all these works in Ref. Cieplý et al. (2016) and the mini-review by the PDG Patrignani et al. (2016). Including the photoproduction data on $\gamma p\to K^{+}\Sigma\pi$ from CLAS Moriya et al. (2013), one finds that the heavier of the poles is fairly well pinned down, while the lighter one still shows some sizable spread in its mass and width Mai and Meißner (2015). All this is captured nicely in Fig. 23. Fitting only the scattering and threshold ratio data with the NLO kernel, one finds two poles, but with very limited precision Borasoy et al. (2006). Adding the precise kaonic hydrogen data, the situation changes markedly, as shown by the different solutions found by various groups, see Ikeda et al. (2012); Mai and Meißner (2013); Guo and Oller (2013). Still, as first pointed out in Ref. Mai and Meißner (2015), even with these data there is a multitude of solutions with almost equal $\chi^{2}$, as in depicted by the boxes labeled $1,\ldots,8$ in Fig. 23. However, the photoproduction data severely constrain this space of solutions. From the eight solutions only two survive, the blue (solution 2) and the green (solution 4) boxes in the figure from Ref. Mai and Meißner (2015) as well as the modified LO solution depicted by the spades from Ref. Roca and Oset (2013). This again is a nice example that only through an interplay of various reactions one is able to pin down the precise structure of hadronic molecules (or other hadronic resonances). Clearly, more data on $\pi\Sigma$ mass distributions would be needed to further sharpen these conclusions, see e.g. Ohnishi et al. (2016).

VII Summary and outlook

In this review, we have discussed the experimental indications for and the theoretical approaches to hadronic molecules, which are a particular manifestation of non-conventional states in the spectrum of QCD. The observation that these multi-hadron bound states appear close to or in between two-particle thresholds allows one to write down nonrelativistic effective field theories. This gives a systematic access to the production, decay processes and other reactions involving hadronic molecules. In the last decade or so, through precise measurements of the spectrum of QCD invloving charm and bottom quarks, more and more potential hadronic molecules have been observed. We have shown how explicit calculations of various decay modes can be used to test this scenario. This is the only way to eventually disentangle hadronic molecules from other multi-quark states like, e.g., tetraquarks. More detailed and accurate measurements are therefore called for, complemented by first-principle lattice QCD calculations with parameters close to the physical point and accounting for the involved coupled-channel dynamics related. More than 60 years after Weinberg’s groundbreaking work on the question whether the deuteron is an elementary particle, we are now in the position to identify many more of such loosely bound states in the spectrum of QCD and to obtain a deeper understanding of the mechanism underlying the appearance and binding of hadronic molecules.

Acknowledgments

We are grateful to all our collaborators for sharing their insights into the topics discussed here. This work is supported in part by DFG and NSFC through funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 11621131001, DFG Grant No. TRR110), by NSFC (Grant Nos.11425525, 11521505 and 11647601), by the Thousand Talents Plan for Young Professionals, by the CAS Key Research Program of Frontier Sciences (Grant No. QYZDB-SSW-SYS013), by the CAS President’s International Fellowship Initiative (PIFI) (Grant No. 2017VMA0025), and by the National Key Basic Research Program of China under Contract No. 2015CB856700.