Scale Invariance in Heavy Hadron Molecules

TL;DR

This paper explores the potential for scale invariance and Efimov-like phenomena in heavy hadron molecules, focusing on the role of near on-shell pion exchange and long-range 1/r^2 potentials in forming exotic bound states.

Contribution

It introduces the concept of approximate scale invariance in heavy hadron molecules and suggests conditions under which Efimov-like spectra could emerge in these systems.

Findings

Long-range 1/r^2 potential arises from pion exchange near on-shell conditions.

Discrete scale invariance may manifest in certain heavy hadron molecules.

Potential candidates include specific $rac{1}{2}^{+}$ pentaquarks and baryonia involving $ar{ ext{Xi}}_b$.

Abstract

We discuss a scenario in which the $P_c(4450)^{+}$ heavy pentaquark is a $\Sigma_c \bar{D}^*$-$\Lambda_{c}(2595) \bar{D}$ molecule. The $\Lambda_{c1} \bar{D} \to \Sigma_c \bar{D}^*$ transition is mediated by the exchange of a pion almost on the mass shell that generates a long-range $1/r^2$ potential. This is analogous to the effective force that is responsible for the Efimov spectrum in three-boson systems interacting through short-range forces. The equations describing this molecule exhibit approximate scale invariance, which is anomalous and broken by the solutions. If the $1/r^2$ potential is strong enough this symmetry survives in the form of discrete scale invariance, opening the prospect of an Efimov-like geometrical spectrum in two-hadron systems. For a molecular pentaquark with quantum numbers $\frac{3}{2}^{-}$ the attraction is not enough to exhibit discrete scale invariance, but this prospect might very well be realized in a $\frac{1}{2}^{+}$ pentaquark or in other hadron molecules involving transitions between particle channels with opposite intrinsic parity and a pion near the mass shell. A very good candidate is the $\Lambda_{c}(2595) \bar{\Xi}_b - \Sigma_c \bar{\Xi}_b'$ molecule. Independently of this, the $1/r^2$ force is expected to play a very important role in the formation of this type of hadron molecule, which points to the existence of $\frac{1}{2}^{+}$ $\Sigma_c D^*$-$\Lambda_{c}(2595) D$ and $1^+$ $\Lambda_{c}(2595) {\Xi}_b - \Sigma_c {\Xi}_b'$ molecules and $0^{+}$/$1^{-}$ $\Lambda_{c}(2595) \bar{\Xi}_b - \Sigma_c \bar{\Xi}_b'$ baryonia.

Equations

$$-u^{\prime \prime }(r)+\frac{g}{r^2}u(r)=0,$$

$$u(r)=c_{+}{r}^{\frac{1}{2}+\nu }+c_{-}{r}^{\frac{1}{2}-\nu },$$

$$u(r)=c{r}^{1/2}\sin (\nu \log {\Lambda }_{2}r),$$

$$⟨{\Lambda }_{c1}\bar{D}|V_{\mathrm{OPE}}(\vec{r})$$

$$W_E(r)=\frac{g_1{h}_2{\mu }_{\pi }^2}{4\pi \sqrt{2}{f}_{\pi }^2}\frac{e^{-{\mu }_{\pi }r}}{{\mu }_{\pi }r}\left(1+\frac{1}{{\mu }_{\pi }r}\right),$$

$$⟨{\Lambda }_{c1}\bar{D}|V_{\mathrm{OPE}}(\vec{r})$$

$$-{\mathbf{u}}^{\prime \prime }+\left[2{\mu }_{P_c^{\ast }}{\mathbf{V}}_{\mathrm{OPE}}+\frac{{\mathbf{L}}^2}{r^2}\right]\mathbf{u}=0,$$

$$2{\mu }_{P_c^{\ast }}{\mathbf{V}}_{\mathrm{OPE}}+\frac{{\mathbf{L}}^2}{r^2}$$

$$-u_i^{\prime \prime }+\frac{g_i}{r^2}{u}_i=0,$$

$$g_i=\{6,2,3+\sqrt{9+3g^2},3-\sqrt{9+3g^2}\},$$

$$\mathbf{g}({\frac{1}{2}}^{+})$$

$$⟨{\Sigma }_c{\bar{\Xi }}_b^{\prime }|V_{\mathrm{OPE}}(\vec{r})$$

$$0^{+}$$

$$0^{-}$$

$$1^{-}$$

$$\mathbf{g}(0^{+})$$

$$\mathbf{g}(0^{-})$$

$$\mathbf{g}(1^{-})$$

$$R_s<r<\frac{1}{|{\mu }_{\pi }|},$$

$$V(r)=V_{\mathrm{OPE}}(r)\theta (r-R_s)+\frac{C_0(R_s)}{4\pi R_s^2}\delta (r-R_s),$$

Full text

The onset of scale invariance in two-body systems is a remarkable property. It connects a series of seemingly disparate low-energy phenomena in atomic, nuclear and particle physics under the same theoretical description Braaten and Hammer (2006). When the scattering length $a_{0}$ of a two-body system is much larger than any other scale, i.e. $a_{0}\to\infty$, the system is invariant under the scale transformation $r\to\lambda\,r$ with arbitrary $\lambda$ Pavon Valderrama and Ruiz Arriola (2008). The low energy properties of this two-body system can be fully explained independently of the underlying short-range dynamics. That is, few-body systems with a large scattering length admit a universal description. Efimov discovered that three-boson systems exhibit a characteristic three-body spectrum for $a_{0}\to\infty$, where the binding energy of the states is arranged in a geometric series Efimov (1970). The continuous scale invariance of the three-body equations is anomalous and the spectrum only shows discrete scale invariance under the transformation $r\to\lambda_{0}r$ where the value of $\lambda_{0}$ is now fixed. Conversely if $E_{n}$ is the binding energy of a three-body state there is another state with binding $E_{n+1}=E_{n}/\lambda_{0}^{2}$, a prediction that was confirmed experimentally with Cs atoms a decade ago Kraemer et al. (2006). This type of discrete geometrical spectrum also happens in three-body systems containing at least two-identical particles Helfrich et al. (2010), or when the scattering is resonant in higher partial waves Helfrich and Hammer (2011); Braaten et al. (2012). This mechanism might be responsible for the binding of the triton Bedaque et al. (1999a), ${}^{4}{\rm He}$ König et al. (2017), a series of halo nuclei Federov et al. (1994); Horiuchi and Suzuki (2006); Canham and Hammer (2008); Acharya et al. (2013); Ji et al. (2014) and the Hoyle state Hammer and Higa (2008); Higa et al. (2008).

There is a two-body system that is intimately related to the Efimov effect, which is the $1/r^{2}$ potential 111While a two-body system with infinite scattering length is scale invariant, a two-body system with a $1/r^{2}$ potential can display at most discrete scale invariance, a possibility which will depend on the strength of the potential. Continuous scale invariance for $1/r^{2}$ is broken by the existence of a fundamental state with $E\neq 0$. Equivalently, as happens in the three-boson system Bedaque et al. (1999b), the renormalization of $1/r^{2}$ is non-trivial and requires the appearance of a new energy scale and henceforth that scale invariance is broken.. At zero energy the reduced Schrödinger equation for the s-wave becomes

[TABLE]

which is obviously scale invariant (for a finite energy analysis we refer to Bawin and Coon (2003)). The connection with the three-body system is apparent when one realizes that it also contains a similar equation with an effective $1/\rho^{2}$ potential in the hyperradius $\rho$ Fedorov and Jensen (1993). For $g>-1/4$ the equation above admits power-law solutions of the type

[TABLE]

with $c_{\pm}$ constants and $\nu=\sqrt{1/4+g}$, where scale invariance is lost. For $g<-1/4$ we have instead solutions of the type

[TABLE]

with $c$ a constant, $\nu=\sqrt{-1/4-g}$ and $\Lambda_{2}$ an energy scale that depends on the short-range physics (it can be obtained from the energy of the fundamental state). $\Lambda_{2}$ is the reason why exact scale invariance is broken and its appearance resembles dimensional transmutation Camblong et al. (2001a, b). Now the solutions display discrete scale invariance with $r\to\lambda_{0}r$, where $\lambda_{0}=e^{\pi/\nu}$ Braaten and Phillips (2004); Hammer and Swingle (2006). In turn there is a geometric bound state spectrum where $E_{n+1}=E_{n}/\lambda_{0}^{2}$, with $E_{n}$ and $E_{n+1}$ the energy of two consecutive states. Incidentally this is a rare example of an anomaly in quantum mechanics Coon and Holstein (2002). Here we make the observation that the $1/r^{2}$ potential can appear in heavy hadron molecules, for instance the $P_{c}(4450)^{+}$ heavy pentaquark if it happens to be molecular (the only other known example of a $1/r^{2}$ potential is the atom-dipole interaction Camblong et al. (2001c)). There might be other two-hadron systems where the potential might be attractive enough to exhibit discrete scale invariance. The ideas presented here involve long-range physics and hence only apply to molecular hadrons (i.e. non-relativistic bound states of two-hadrons) that fulfill a series of conditions, but not to compact hadrons.

The heavy pentaquarks $P_{c}(4380)^{+}$ and $P_{c}(4450)^{+}$, $P_{c}$ and $P_{c}^{*}$ from now on, were discovered by the LHCb Aaij et al. (2015) and are a recent and interesting addition to a growing family of exotic hidden charm (and bottom) hadrons that began with the $X(3872)$ more than ten years ago Choi et al. (2003). There is still a lot of discussion regarding the nature of the $P_{c}$ and $P_{c}^{*}$, from the role of threshold effects Guo et al. (2015); Mikhasenko (2015); Liu et al. (2016); Meißner and Oller (2015), to baryocharmonia Kubarovsky and Voloshin (2015), a compact pentaquark Diakonov et al. (1997); Jaffe and Wilczek (2003); Yuan et al. (2012); Maiani et al. (2015); Lebed (2015); Li et al. (2015), a heavy baryon-antimeson molecule Chen et al. (2015a, b); Roca et al. (2015); He (2016); Xiao and Meißner (2015) or other more exotic possibilities Mironov and Morozov (2015); Scoccola et al. (2015). The $P_{c}^{*}$ is an interesting molecular candidate because its width is not particularly big $\Gamma=35\pm 5\pm 19\,{\rm MeV}$ and its closeness to the $\Sigma_{c}\bar{D}^{*}$ threshold, see Fig. (1). As a matter of fact a series of works predicted the possibility of a heavy baryon-antimeson molecule before the discovery of the $P_{c}^{*}$ Wu et al. (2010); Yang et al. (2012); Xiao et al. (2013); Karliner and Rosner (2015). The probable quantum numbers of the $P_{c}$ and $P_{c}^{*}$ are $\frac{3}{2}^{-}$ and $\frac{5}{2}^{+}$ respectively, followed by $\frac{5}{2}^{+}$ and $\frac{3}{2}^{-}$. The standard molecular explanation for the $P_{c}^{*}$ heavy pentaquark is that of a $\Sigma_{c}\bar{D}^{*}$ bound state, which prefers the quantum number ${\frac{3}{2}}^{-}$ for the $P_{c}^{*}$. Here we discuss the scenario in which the molecular $P_{c}^{*}$ also contains a $\Lambda_{c1}\bar{D}$ component in addition to $\Sigma_{c}\bar{D}^{*}$, where $\Lambda_{c1}$ denotes the $\Lambda_{c}(2595)$. Burns Burns (2015) proposed this idea on the analogy between the $D\bar{D}^{*}+D^{*}\bar{D}$ and the $Y_{c}^{*}\bar{D}+Y_{c}\bar{D}$ systems, i.e. the X(3872) and the $P_{c}^{*}$, where $Y_{c}$, $Y_{c}^{*}$ are charmed baryons. He argued that the most natural analog to the $D\bar{D}^{*}+D^{*}\bar{D}$ system is $\Sigma_{c}\bar{D}^{*}-\Lambda_{c1}\bar{D}$ on the basis that the mass difference of the $\Lambda_{c1}$ and $\Sigma_{c}$ is very close to the $D^{*}$ and $D$ splitting. Here we will explore this possibility.

The low-energy dynamics of $\Sigma_{c}\bar{D}^{*}-\Lambda_{c1}\bar{D}$ is driven by one pion exchange (OPE) and is fascinating for two reasons. First, if the $\Lambda_{c1}\bar{D}$ pair exchanges a pion to become a $\Sigma_{c}\bar{D}^{*}$ pair the pion will be almost on the mass shell, leading to a unusual long-range potential for strong interactions. Second, the intrinsic parities of $\Lambda_{c1}(\frac{1}{2}^{-})$ and $\Sigma_{c}(\frac{1}{2}^{+})$ are different while the ones for the $D$ and $D^{*}$ are the same. As a consequence OPE will switch odd (even) waves in the $\Lambda_{c1}\bar{D}$ channel to even (odd) waves in the $\Sigma_{c}\bar{D}^{*}$ one. That is, there is a vector force: the analogous to the tensor one, except that it carries orbital angular momentum $L=1$ instead of $L=2$. The tensor force behaves as $1/r^{3}$ for $m_{\pi}r<1$, while the vector force as $1/r^{2}$. This short-range property becomes long-range if the pion is near the mass shell.

We can compute the $\Sigma_{c}\bar{D}^{*}\to\Lambda_{c1}\bar{D}$ potential from the heavy baryon chiral lagrangian of Cho Cho (1994)

[TABLE]

with $\omega_{\pi}=m(\Lambda_{c1})-m(\Sigma_{c})$ the energy of the pion, $\tau$ an isospin factor such that $\tau=\sqrt{3}$ for $I=1/2$ and $\tau=0$ for $I=3/2$ and $\vec{\epsilon}$ the polarization vector of the incoming $\bar{D}^{*}$ meson. $W_{E}$ reads

[TABLE]

with $g_{1}$ the axial coupling for the heavy mesons, $h_{2}$ the coupling for the $\pi\Lambda_{c1}\Sigma_{c}$ vertex, $f_{\pi}\simeq 130\,{\rm MeV}$ the pion decay constant and $\mu_{\pi}=m_{\pi}^{2}-\omega_{\pi}^{2}$ is the effective pion mass. Besides, there is standard-range OPE in the $\Sigma_{c}\bar{D}^{*}$ channel while OPE vanishes in the $\Lambda_{c1}\bar{D}$ channel.

Actually $|\mu_{\pi}|\simeq 5-35\,{\rm MeV}\ll m_{\pi}$ depending on whether we exchange a charged or neutral pion, i.e. the $\Sigma_{c}\bar{D}^{*}\to\Lambda_{c1}\bar{D}$ transition potential dominates the long-range dynamics of the system for $1/m_{\pi}<r<1/|\mu_{\pi}|$ (which is also the region of validity of the equations we will write below). We stress that scale invariance is only approximate and broken by two interrelated factors: (i) the pion is not exactly on the mass shell, (ii) the $\Sigma_{c}\bar{D}^{*}$ and $\Lambda_{c1}\bar{D}$ thresholds are a pair of ${\rm MeV}$ away from each other. For the moment we will assume $\mu_{\pi}=0$, which implies overlapping thresholds. In principle the widths of the $\Sigma_{c}$ and $\Lambda_{c1}$ baryons are another factor to consider. Yet the widths can be ignored if the time required for the formation of the state is shorter than the life-time of its components: $\Gamma\ll m$, with $\Gamma$ the width of the component and $m$ the mass of the exchanged particle Guo and Meissner (2011). For the $\Sigma_{c}$ and $\Lambda_{c1}$ the widths are about a pair of MeVs, well below $m_{\pi}\sim 140\,{\rm MeV}$. The potential in the $I=1/2$ channel reads

[TABLE]

i.e. the $\mu_{\pi}=0$ limit of Eqs. (4) and (5). If we consider $J^{P}=\frac{3}{2}^{-}$ (the standard quantum numbers for a molecular pentaquark), the partial waves contributing are $\Sigma_{c}\bar{D}^{*}({}^{2}D_{3/2})$, $\Sigma_{c}\bar{D}^{*}({}^{4}S_{3/2})$, $\Sigma_{c}\bar{D}^{*}({}^{4}D_{3/2})$ and $\Lambda_{c1}\bar{D}({}^{2}P_{3/2})$. In this partial wave basis the reduced Schrödinger equation at zero energy reads

[TABLE]

where ${\bf u}$ is the wave function in vector notation and $\mu_{P_{c}^{*}}$ the reduced mass of the molecule (actually there is one reduced mass for each particle channel, but here we can take their geometric mean). The combination of the vector OPE potential and the centrifugal barrier reads

[TABLE]

That is, we have a four channel version of Eq. (1). We can diagonalize the matrix ${\bf g}({\frac{3}{2}}^{-})$, in which case we end up with four equations of the type

[TABLE]

where the $g_{i}$’s ($i=1,2,3,4$) are the eigenvalues of ${\bf g}({\frac{3}{2}}^{-})$. There are three positive and one negative eigenvalue

[TABLE]

where the negative one can trigger discrete scale invariance. This will happen if $|g|>5/(4\sqrt{3})\simeq 0.7217$. However the value of $g$ for the $P_{c}^{*}$ molecule is $g=0.60^{+0.10}_{-0.10}\,h_{2}$, where we have used $g_{1}=0.59\pm 0.01\pm 0.07$ from $D^{*}\to D\pi$ and $D^{*}\to D\gamma$ decays Ahmed et al. (2001); Anastassov et al. (2002). This requires $|h_{2}|>1.21^{+0.25}_{-0.19}$, which is well above $h_{2}=0.60\pm 0.07$ from CDF Aaltonen et al. (2011) or $h_{2}=0.63\pm 0.07$ from the analysis of Ref. Cheng and Chua (2015), where in both cases $h_{2}$ is extracted from $\Gamma(\Lambda_{c1}\to\Sigma_{c}\pi)$. That is, there is not enough attraction to achieve discrete scale invariance. For the $\frac{1}{2}^{-}$ molecule the matrix is different but the attractive eigenvalue is still $g_{-}(\frac{1}{2}^{-})=3-\sqrt{9+3g^{2}}$, requiring $|g|>5/(4\sqrt{3})$. The most interesting pentaquark-like molecule is the $\frac{1}{2}^{+}$, with partial waves $\Sigma_{c}\bar{D}^{*}({}^{2}P_{1/2})$, $\Sigma_{c}\bar{D}^{*}({}^{4}P_{1/2})$ and $\Lambda_{c1}\bar{D}({}^{2}S_{1/2})$, where

[TABLE]

The attractive eigenvalue is $g_{-}(\frac{1}{2}^{+})=1-\sqrt{1+3g^{2}}$, which requires $|g|>\sqrt{3}/4\simeq 0.4330$ and $|h_{2}|>0.73^{+0.11}_{-0.06}$, i.e. overlapping with current estimations of $h_{2}$. Finally the $\frac{3}{2}^{+}$, $\frac{5}{2}^{+}$ and $\frac{5}{2}^{-}$ cases require $|g|>7\sqrt{3}/4$, $|g|>7\sqrt{3}/4$ and $|g|>15\sqrt{3}/4$. That is, the strength of the vector force is in general too weak in the pentaquark-like molecules to achieve discrete scale invariance, with the notable exception of $\frac{1}{2}^{+}$ which lies on the limit.

Yet the $P_{c}^{*}$ is not the only system where this can happen. The general conditions for a $H_{1}H_{2}-H_{1}^{\prime}H_{2}^{\prime}$ hadronic molecule to have scale invariance are: (i) the hadrons are particularly long-lived, (ii) the mass difference of the hadrons in each vertex is similar to that of a pseudo-Goldstone boson $m(H_{1})^{\prime}-m(H_{1})\simeq m(H_{2})-m(H_{2})^{\prime}\simeq m_{P}$ (iii) the intrinsic parity of $H_{2}$ and $H_{2}^{\prime}$ is the same, while that of $H_{1}$ and $H_{1}^{\prime}$ is different and (iv) $H_{1}$ and $H_{1}^{\prime}$ have the same spin for the pseudo-Goldstone boson to be emitted in s-wave. This applies as well if we substitute hadrons by antihadrons in one of the vertices: the vector force will change sign but the eigenvalues of the $1/r^{2}$ potential matrix will remain the same. Notice that it is not strictly necessary to exchange a pion near the mass-shell to have a long-range $1/r^{2}$ force. A kaon near the mass-shell will also generate this type of force.

If we have the $\Lambda_{c1}$-$\Sigma_{c}$ on the one side, besides the $D$-$D^{*}$, the $\Xi_{b}$-$\Xi_{b}^{\prime}$ bottom baryon combination also fulfills the previous conditions, see Fig. (1) for the threshold location. In this regard the $\Lambda_{c1}\bar{\Xi}_{b}$-$\Sigma_{c}\bar{\Xi}_{b}^{\prime}$ system seems to be the best candidate for a scale invariant molecule in the heavy sector. The $\Lambda_{c1}\bar{\Xi}_{b}\to\Sigma_{c}\bar{\Xi}_{b}^{\prime}$ potential for $I=1/2$ reads

[TABLE]

where $g_{3}$ is the axial coupling for the $\bar{\Xi}_{b}^{\prime}\bar{\Xi}_{b}\pi$ vertex and $\sigma_{2}$ is the Pauli matrix for that vertex. If we consider states in which the $\Sigma_{c}\bar{\Xi}_{b}^{\prime}$ is in s-wave or alternatively in p-wave where the tensor force is attractive, we have

[TABLE]

In these partial wave bases the ${\bf g}$ matrices read

[TABLE]

[TABLE]

For $|g|>3/4$ the attractive eigenvalue of the matrices above will trigger discrete scale invariance. The evaluation of $g$ depends on the axial coupling $g_{3}$, which can be extracted from the $\Sigma_{c}\to\Lambda_{c}\pi$ decay, yielding $g_{3}=0.973^{+0.019}_{-0.042}$ Cheng and Chua (2015). This translates into $g=1.12^{+0.03}_{-0.05}\,h_{2}$ requiring $|h_{2}|>0.67^{+0.03}_{-0.02}$, which is within the error of $h_{2}=0.63\pm 0.07$ Cheng and Chua (2015).

The approximate scale invariance of the Schrödinger equation describing these hadronic molecules has long- and short-range consequences, where the former — the appearance of a geometric spectrum — depends on how far these systems are from $\mu_{\pi}=0$. For $\mu_{\pi}\neq 0$ scale invariance holds for

[TABLE]

with $R_{s}$ the short-range scale, $1/m_{\pi}$ in this case 222 On a related note, a purely imaginary $\mu_{\pi}$ triggers a repulsive correction at second order perturbation theory. The effect is small and is suppressed as $|\mu_{\pi}r|^{3}/3$ in the scale invariant region. (this also applies to three-boson systems after the substitution $\mu_{\pi}\to 1/a_{0}$). The existence of a geometric excited state requires the relative size of the scale invariant window to be bigger than the discrete scaling factor. For $P_{c}^{*}$-like molecules this window is $1/(R_{s}\mu_{\pi})\sim 10-20$, requiring the coupling $|g_{-}|$ to be about $1$ at least, which is considerably larger than $1/4$. That is, the observation of geometric states in hadron and atomic physics share a similar difficulty: the fine-tuning of the pion mass (hadrons) or the scattering length (atoms). For atoms near a Feschback resonance this is solved with a magnetic field Braaten and Hammer (2006). The equivalent for hadrons will be to fine-tune the pion mass in the lattice. There is also the possibility of increasing $|g_{-}|$, for instance by having a larger reduced mass (i.e. two bottom hadrons) or if the exchanged particle is a kaon. This can happen naturally in the heavy sector where there are still plenty of hadrons to be discovered, of which a few might be candidates for a long-range vector force.

Concerning the short-range consequences, even if the vector force is not enough to trigger discrete scale invariance it will still play a remarkable role in binding. This is indeed analogous to the conjectured importance of Efimov physics in light nuclei König et al. (2017) (despite the glaring absence of Efimov states). If the binding mechanism is s-wave short-range attraction, a way to see this is the following: for $r\leq R_{s}$ we will assume that OPE is not valid and model the short-range interaction with a delta-shell:

[TABLE]

where $R_{s}$ is the short-range radius. Then we calculate the relative strength of the coupling $C_{0}$ required to have a bound state at zero energy in the presence/absence of a vector force. In the one-channel problem of Eq. (1) for $g>-1/4$ and in the absence of tensor OPE, the relative strength of $C_{0}$ is $(1/2+\nu)$ of that required to bind if $g=0$ (for $\mu_{\pi}R_{s}<1$), while for $g<-1/4$ it always binds (for $\mu_{\pi}=0$). Owing to scale invariance this happens independently of $R_{s}$. Thus if $\nu\to 0$ ($g\to-1/4$) the short-range potential only has to be half the normal strength to be able to bind the system. If there is standard-range OPE or other intermediate range physics this binding enhancement will change. Taking $R_{s}=1\,{\rm fm}$, $\mu_{\pi}=0$ and $h_{2}=0.63$, the $\Sigma_{c}\bar{D}^{*}$-$\Lambda_{c1}\bar{D}$ $P_{c}^{*}(\frac{3}{2}^{-})$ requires $70\,\%$ the attraction of a standard $\Sigma_{c}\bar{D}^{*}$ $P_{c}^{*}$ to bind (for the $\Sigma_{c}\Sigma_{c}\pi$ axial coupling we use $g_{3}=-1.38$ Cheng and Chua (2015)). For the heavy baryonium the numbers are $46\,\%$ ($0^{-}$) and $53\,\%$ ($1^{-}$) respectively. The probability of binding is enhanced but dependent on unknown short-range physics.

If binding happens for distances in which the present picture is valid, short-range physics will not be necessary. The radius below which the $P_{c}^{*}(\frac{3}{2}^{-})$ binds is $0.94\,{\rm fm}$ while for the $0^{-}$ / $1^{-}$ baryonia we have $0.40\,{\rm fm}$ / $0.84\,{\rm fm}$ respectively. For the $1^{+}$ $\Sigma_{c}\Xi_{b}^{\prime}$-$\Lambda_{c1}\Xi_{b}$ molecule we have $0.87\,{\rm fm}$ instead. For $r<1/2m_{\pi}$ ($\sim 0.7\,{\rm fm}$) two-pion exchange and hadron finite-size effects dominate, setting the limits of the OPE description and providing a criterion for binding. From this we can be confident about the existence of the $1^{-}$ baryonium and the $1^{+}$ $\Sigma_{c}\Xi_{b}^{\prime}$-$\Lambda_{c1}\Xi_{b}$ molecule, while the $0^{-}$ baryonium is contingent on the unknown short-range physics. But the more interesting cases are that of the $\Sigma_{c}D^{*}$ $(\frac{1}{2}^{+})$ / $\Sigma_{c}\bar{\Xi}_{b}^{\prime}$ $(0^{+})$ which bind in p-wave. Here the vector force effectively induces the existence of a channel behaving much like an s-wave. For the $1/2^{+}$ $\Sigma_{c}D^{*}$-$\Lambda_{c}D$ system binding happens for $0.92\,{\rm fm}$ while for the $0^{+}$ baryonium we have $0.86\,{\rm fm}$, radii which point towards the existence of these states. The bottom-line is that the vector force induces a series of binding mechanisms which do not require the ratio $m_{\pi}/\mu_{\pi}$ to be particularly large (a factor of $2-3$ is probably enough) and which in a few cases lead to predictions of new molecules.

Scale invariant hadron molecules are an intriguing theoretical possibility. They are the two-body realization of a type of universality that is usually only found in three-body atomic and nuclear systems. There are clear theoretical requirements for a hadron molecule to show scale invariance at long distances, where the most natural mechanism is the exchange of a pion almost on the mass shell between initial and final two-hadron states with opposite intrinsic parities. If we consider heavy hadrons, the candidates include $\Lambda_{c1}\bar{D}$-$\Sigma_{c}\bar{D}^{*}$, i.e. the molecular interpretation of the recently discovered $P_{c}^{*}$ pentaquark state, while the most likely scale invariant molecule is probably the $\Lambda_{c1}\bar{\Xi}_{b}$-$\Sigma_{c}\bar{\Xi}_{b}^{\prime}$ baryonium. Discrete scale invariance requires that the couplings have a minimal strength, a condition that a $\frac{1}{2}^{+}$ heavy pentaquark and a $\Lambda_{c1}\bar{\Xi}_{b}$-$\Sigma_{c}\bar{\Xi}_{b}^{\prime}$ molecule can meet. The same ideas apply to the $\Lambda_{c1}D$-$\Sigma_{c}D^{*}$ and $\Lambda_{c1}{\Xi}_{b}$-$\Sigma_{c}{\Xi}_{b}^{\prime}$ molecules as the vector force attraction is independent of whether we have hadrons or antihadrons. The appearance of a geometrical spectrum actually requires the effective mass of the pion to be considerably smaller than the other hadronic scales in the molecule. This condition is not likely to be met in nature, but could very well be realized in the lattice. Even if there is no geometrical spectrum in these molecules, the long-range attraction provided by the vector force plays an important role as a binding mechanism, which cannot be ignored and in a few cases guarantees binding. The vector force is indeed a new type of long range dynamics that has not been previously considered neither in the $P_{c}^{*}$ pentaquark nor in other hadronic molecules where it is present and can be relevant. An illustrative example is the enhancement of P-wave interactions, as happens in the $\frac{1}{2}^{+}$ $\Sigma_{c}D^{*}$ system after we include the $\Lambda_{c1}D$ channel. In this type of hadronic molecule the role of scale invariance is analogous to that in the triton, ${}^{4}{\rm He}$, a few halo nuclei and a series of cold atoms systems, to name a few examples.

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China under Grants No. 11375024, No.11522539 and the Fundamental Research Funds for the Central Universities.