Tetraquarks in holographic QCD

TL;DR

This paper uses holographic QCD to derive and solve equations predicting tetraquark properties, confirming experimental findings and extending understanding to heavy quark states.

Contribution

It introduces an analytical approach within soft-wall AdS/QCD to predict tetraquark masses, including effects of finite quark masses, and confirms experimental tetraquark observations.

Findings

Predicted the $a_1(1414)$ tetraquark mass as 1.414 GeV, matching experimental data.

Derived an analytical Schrödinger-type equation for tetraquark wave functions.

Extended the model to include heavy quark effects.

Abstract

Using a soft-wall AdS/QCD approach we derive the Schr\"odinger-type equation of motion for the tetraquark wave function, which is dual to the dimension-4 AdS bulk profile. The latter coincides with the number of constituents in the leading Fock state of the tetraquark. The obtained equation of motion is solved analytically, providing predictions for both the tetraquark wave function and its mass. A low mass limit for possible tetraquark states is given by $M \ge 2 \kappa = 1$ GeV, where $\kappa = 0.5$ GeV is the typical value of the scale parameter in soft-wall AdS/QCD. We confirm results of the COMPASS Collaboration recently reported on the discovery of the $a_1(1414)$ state, interpreted as a tetraquark state composed of light quarks and having $J^{PC} = 1^{++}$. Our prediction for the mass of this state, $M_{a_1} = \sqrt{2}$ GeV $\simeq 1.414$ GeV, is in good agreement with the COMPASS result $M_{a_1} = 1.414^{+0.015}_{-0.013}$ GeV. Next we included finite quark mass effects, which are essential for the tetraquark states involving heavy quarks.

Tables

Table 1. Table 1: Masses of possible light tetraquarks (in MeV).

Tetraquark	Quark Content	Quantum Numbers	$\kappa$	Mass	Mass	Data Olive:2016xmw
		$(J^P,n,L,S)$	(MeV)	(Zero Quark Masses)	(Finite Quark Masses)
$f_0\left(980\right)$	$\left[q\overline{s}\right]\left[s\overline{q}\right]$	($0^{+}$,0,0,0)	351	702	990	990 $\pm$ 20
$f_1\left(1215\right)$	$\left[q\overline{s}\right]\left[s\overline{q}\right]$	($1^{+}$,0,1,0)	351	993	1214
$f_2\left(1400\right)$	$\left[q\overline{s}\right]\left[s\overline{q}\right]$	($2^{+}$,0,2,0)	351	1216	1402
$f_3\left(1570\right)$	$\left[q\overline{s}\right]\left[s\overline{q}\right]$	($3^{+}$,0,3,0)	351	1404	1568
$a_1\left(1420\right)$	$\left[q\overline{s}\right]\left[s\overline{q}\right]$	($1^{+}$,0,1,1)	435	1230	1414	${1414}_{-13}^{+15}$ Adolph:2015pws

Equations

$$\mathrm{Mesons}\to N_c\mathrm{Baryons}:{\bar{q}}^a{q}^a\to {ϵ}^{a{b}_2\dots b_n}{q}^a{q}^{b_2}\dots q^{b_n},$$

$$\mathrm{Mesons}\to 2(N_c-1)\mathrm{Multiquarks}:{\bar{q}}^a{q}^a\to {ϵ}^{a{a}_2\dots a_n}{ϵ}^{a{b}_2\dots b_n}{q}^{a_2}\dots q^{a_n}{\bar{q}}^{b_2}\dots {\bar{q}}^{b_n}.$$

$$\displaystyle M^{2}=4\kappa^{2}\Big{(}n+3\Big{)}\,,$$

$$\displaystyle M^{2}=4\kappa^{2}\Big{(}n+L+1+\frac{S}{2}\Big{)}$$

$$\displaystyle M^{2}=4\kappa^{2}\Big{(}n+\frac{L+J}{2}+1\Big{)}\,.$$

$$\Delta M^2=2{\kappa }^2(J-L-S).$$

$$\displaystyle S_{J}=\int d^{d}xdz\sqrt{g}e^{-\varphi(z)}\Big{[}\partial_{M}\Phi_{J}\,\partial^{M}\Phi^{J}-(\mu_{J}^{2}+\hat{V}_{J}(z))\Phi_{J}\Phi^{J}\Big{]}\,.$$

$${\hat{V}}_J(z)=e^{-2A(z)}{\hat{U}}_J(z),{\hat{U}}_J(z)={\phi }^{\prime \prime }(z)+(d-1-2J){\phi }^{\prime }(z)A^{\prime }(z).$$

$$\displaystyle S_{J}=\int d^{d}xdz\sqrt{g}\Big{[}\partial_{M}\Phi_{J}\,\partial^{M}\Phi^{J}-(\mu_{J}^{2}+V_{J}(z))\Phi_{J}\Phi^{J}\Big{]}\,,$$

$$\displaystyle V_{J}(z)=e^{-2A(z)}U_{J}(z)\,,\quad U_{J}(z)=\frac{1}{2}\Big{(}\varphi^{\prime\prime}(z)+\frac{(\varphi^{\prime}(z))^{2}}{2}+(d-1-2J)\,\varphi^{\prime}(z)\,A^{\prime}(z)\Big{)}\,.$$

$$U_J(z)={\kappa }^4{z}^2+2{\kappa }^2(J-1).$$

$${\Phi }_J(x,z)$$

$${\Phi }_{n\tau }(z)$$

$$\displaystyle\biggl{[}-\frac{d^{2}}{dz^{2}}+U_{J}(z)+W_{\tau}(z)\biggr{]}\varphi_{n\tau}(z)=M^{2}_{nJ\tau}(z)\varphi_{n\tau}(z)\,,$$

$$W_{\tau }(z)=\frac{4\Delta (\Delta -d)+d^2-1}{4z^2}=\frac{4\tau (\tau -d)+d^2-1}{4z^2}$$

$$\mathrm{Mesons}{W}_{2+L}(z)=\frac{4L^2-1}{4z^2},$$

$$\mathrm{Tetraquarks}{W}_{4+L}(z)=\frac{4(L+2{)}^2-1}{4z^2},$$

$$\mathrm{Sixquarks}{W}_{6+L}(z)=\frac{4(L+4{)}^2-1}{4z^2},$$

$${\phi }_{n\tau }(z)=\sqrt{\frac{2\Gamma (n+1)}{\Gamma (n+\tau -1)}}{\kappa }^{\tau -1}{z}^{\tau -3/2}{e}^{-{\kappa }^2{z}^2/2}{L}_n^{\tau -2}({\kappa }^2{z}^2).$$

$${\Phi }_{n\tau }(z)\sim z^{3/2}{\phi }_{n\tau }(z)\to z^{\tau }\mathrm{at}\mathrm{small}z,{\Phi }_{n\tau }(z)\to 0\mathrm{at}\mathrm{large}z$$

$$\underset{0}{\overset{\infty }{\int }}dz{\phi }_{n\tau }^2(z)=1.$$

$$\displaystyle M^{2}_{nJ\tau}=4\kappa^{2}\,\biggl{[}n+\frac{\tau+J-2}{2}\biggr{]}=4\kappa^{2}\,\biggl{[}n+\frac{L+J}{2}+1\biggr{]}\,.$$

$${\psi }_{q_1{\bar{q}}_2}(x,\zeta ,m_1,m_2)=\frac{{\varphi }_{nL}(\zeta )}{\sqrt{2\pi \zeta }}f(x,m_1,m_2)e^{im\varphi }\sqrt{x(1-x)}.$$

$$M_{\pi }^2=2\hat{m}B$$

$$4M_K^2=M_{\pi }^2+3M_{\eta }^2$$

$$M_{Q\bar{q}}$$

$$M_{Q\bar{Q}}$$

$$M_{Q\bar{q}}^V-M_{Q\bar{q}}^P\sim \frac{1}{m_Q}.$$

$$f(x,m_1,m_2)=N{x}^{{\alpha }_1}(1-x{)}^{{\alpha }_2}$$

$$1=\underset{0}{\overset{1}{\int }}dx{f}^2(x,m_1,m_2)$$

$$\displaystyle M^{2}_{2q;nJ}=4\kappa^{2}\biggl{(}n+\frac{L+J}{2}\biggr{)}+\int\limits_{0}^{1}dx\biggl{(}\frac{m_{1}^{2}}{x}+\frac{m_{2}^{2}}{1-x}\biggr{)}f^{2}(x,m_{1},m_{2})\,.$$

$$\displaystyle M^{2}_{2q;nJ}=4\kappa^{2}\biggl{(}n+\frac{L+J}{2}\biggr{)}+(1+2\alpha_{1}+2\alpha_{2})\,\biggl{(}\frac{m_{1}^{2}}{2\alpha_{1}}+\frac{m_{2}^{2}}{2\alpha_{2}}\biggr{)}\,.$$

$$(M_{4q;nJ}^{\mathrm{HM}}{)}^2$$

Full text

gauge/gravity duality, AdS/QCD, tetraquarks, mass spectrum

I Introduction

This work is addressed to the problem of constructing hadronic wave functions of tetraquark states using the AdS and light-front QCD correspondence Brodsky:2007hb . The idea for such a correspondence works very well for conventional hadrons — mesons and baryons Brodsky:2007hb ; Brodsky:2008pf ; Brodsky:2011xx ; Branz:2010ub ; Gutsche:2012ez ; Gutsche:2014oua ; Gutsche:2013zia . In particular, from the matching of matrix elements for physical processes one can relate the bulk profile of the AdS field in a holographic dimension to the transverse part of the hadronic light-front wave function (LFWF) for the case of massless quarks Brodsky:2007hb . The LFWF was generalized in Ref. Brodsky:2008pg for the case of a two-parton state by the explicit inclusion of the constituent quark masses in the LF kinetic energy $\sum_{i}({\bf k}_{\perp i}^{2}+m_{i}^{2})/x_{i}$ while the introducing of quark masses for tetraquark states was done in Ref. Brodsky:2016yod . In the LFWF the inclusion of quark masses corresponds to the introduction of the longitudinal wave function (WF), which was done in this particular case through the so-called Brodsky-Huang-Lepage (BHL) or Gaussian ansatz Brodsky:1982nx ; Huang:1994dy . In Refs. Branz:2010ub ; Gutsche:2012ez ; Gutsche:2014oua we studied the problem of the longitudinal part of the LFWF following the ideas of Ref. Brodsky:2008pg . In particular, in Ref. Gutsche:2012ez we derived the longitudinal part of the LFWF, using constraints of chiral symmetry in the sector of light quarks, and heavy quark effective theory in the sector of heavy quarks. The idea that explicit breaking of chiral symmetry is a property of the longitudinal part of the hadronic LFWF and that it can be induced via the current quark mass dependence of the longitudinal LWFW, had been proposed before in large $N_{c}$ two-dimensional QCD 'tHooft:1974hx . This mechanism was later used in the context of the two-dimensional massive Schwinger model Bergknoff:1976xr ; Ma:1987wi ; Mo:1992sv and was reexamined in Refs. Chabysheva:2012fe ; Forshaw:2012im .

In the present paper we extend the soft-wall AdS/QCD model SW to the description of tetraquarks — compact exotic states composed of two quarks and two antiquarks (for reviews see e.g. Ref. Nielsen:2009uh -Ali:2017jda ). In Refs. Ebert:2005nc ; Dubnicka:2011mm ; Gutsche:2016cml tetraquarks have been analyzed in the context of relativistic quark models. Discussions on tetraquark production can be found, e.g. in Refs. Brodsky:2014xia -Brodsky:2016uln . The formalism for tetraquark production consistent with quark counting rules has been developed in Refs. Brodsky:2014xia ; Voloshin:2016phx ; Brodsky:2016uln . In Ref. Chow:1994hg tetraquarks were studied in the large $N_{c}$ limit and it was correctly stressed that in the context of $SU(N_{c})$ color symmetry the quark $q^{a}$ and antiquark $\bar{q}^{a}$ fields can be replaced by $N_{c}-1$ antiquark and quarks, respectively, because of similar transformations concerning the $SU(N_{c})$ color group. It means that baryons and multiquark states emerge from quark-antiquark mesons under the replacements $\bar{q}^{a_{1}}\to\epsilon^{a_{1}\ldots a_{n}}q^{a_{2}}\ldots q^{a_{n}}$ and $q^{a_{1}}\to\epsilon^{a_{1}\ldots a_{n}}\bar{q}^{a_{2}}\ldots\bar{q}^{a_{n}}$ as

[TABLE]

In the case of QCD we have $N_{c}=3$ and arrive at a picture where mesons, baryons and tetraquarks appear as fundamental color singlet states. This is consistent with arguments of superconformal symmetry where mesons, baryons with $L=0$ and $L=1$, tetraquarks are classified as members of a superquadruplet Brodsky:2016yod .

First applications of AdS/QCD to the description of tetraquarks have been performed in Refs. Forkel:2010gu ; Brodsky:2016yod ; Guo:2016uaf . In particular, in Ref. Forkel:2010gu the effective action for light scalar tetraquarks was derived. Based on this action the equation of motion for the wave functions and mass spectrum $M^{2}$ of scalar tetraquarks are derived. The spectrum results in

[TABLE]

where $\kappa\sim 500$ MeV is the dilaton scale parameter in soft-wall AdS/QCD and $n$ is the radial quantum number. From Eq. (2) it follows that the lower bound for the ground-state mass of the tetraquark is set by $2\kappa\sqrt{3}$. In Ref. Brodsky:2016yod the tetraquark state was introduced as a partner of the supersymmetric quadruplet consisting of two baryon states with positive and negative chirality, meson state and tetraquark, which it is consistent with the ideas of Ref. Chow:1994hg . In this vein, it was shown in Ref. Brodsky:2016yod that the lowest-lying light-quark tetraquark $f_{0}(980)$, with isospin $I=0$ and spin-parity $J^{PC}=0^{++}$, is a partner of the $b_{1}(1235)$ with $J^{PC}=0^{+-}$ and the nucleon, with $J^{P}=\frac{1}{2}^{+}$, while possible more heavier tetraquarks, the axial state $a_{1}(1260)$ with $J^{PC}=1^{++}$, could be a partner of the $\Delta(1230)$ with $J^{P}=\frac{3}{2}^{+}$ and the $a_{2}(1320)$ with $J^{PC}=2^{++}$. The mass formula for tetraquarks with quantum numbers $n,L,S$ (radial quantum number, orbital angular momentum and internal spin) derived in Ref. Brodsky:2016yod , in the limit of massless quarks reads

[TABLE]

The formalism for the study of tetraquarks proposed in Ref. Guo:2016uaf contains a basic error, because of the use of the conformal dimension for the two-partonic states. This gives misleading results for the twist-scaling for small values of the holographic coordinate of the tetraquark wave function and results in a underestimate of the tetraquark mass spectrum. E.g. the lower bound for the tetraquark mass is incorrect. In our consideration we derive the action for the tetraquarks with adjustable quantum numbers of $n,J,L$. Then we derive the equation of motion for the tetraquark wave function and the resulting mass spectrum $M^{2}$

[TABLE]

Our equation of motion and solution for the tetraquark mass spectrum is different from the ones derived in Ref. Forkel:2010gu , because of the different choice for the dimension of the AdS fields dual to tetraquarks. In our case the conformal dimension is $\Delta=\tau=N+L$, where $\tau_{L=0}=N=4$ is the leading twist-dimension of the tetraquark Fock state (corresponding to the number of partons in the leading Fock state) and $L$ is the maximal magnitude of the orbital angular momentum in the four-quark configuration. Such choice for the conformal dimension guarantees the correct power scaling of hadronic form factors at large values of the Euclidean transfer momentum squared $Q^{2}$. In Ref. Forkel:2010gu , the conformal dimension was chosen as $\Delta=6$ for the light scalar tetraquark, which differs from our choice $\Delta=4,5$ at $N=4$ and $L=0,1$. Moreover, our mass formula is different from the result of Ref. Brodsky:2016yod due to a correction which takes into account the spontaneous breaking of superconformal symmetry with

[TABLE]

Next we include the finite masses of the constituents which form the tetraquark states. Note that the first inclusion of finite quark mass corrections to the tetraquark spectroscopy has been done in Ref. Brodsky:2016yod . In our case we proceed in analogy with quark-antiquark mesons (see details in Ref. Gutsche:2012ez ), where we derived full wave functions containing transverse wave functions matched from AdS/QCD and longitudinal wave functions encoding the mass effects of tetraquark constituents. In particular, we consider two possibilities:

1. tetraquarks are bound states of two quarks and two antiquarks;
2. tetraquarks are bound states of two mesons (hadronic molecules).

The paper is organized as follows. In Sec. II we present our formalism and at the end derive the master equation of motion for the full wave function of the tetraquark, providing solutions for the mass spectrum including diquark mass effects. In Sec. III we present numerical results and give our final conclusions.

II Approach

The starting point for a discussion of tetraquarks in soft-wall AdS/QCD is the action for spin-$J$ boson fields $\Phi_{J}=\Phi_{M_{1}\cdots M_{J}}(x,z)$, with a negative dilaton in the exponential prefactor, proposed in Refs. Branz:2010ub ; SW6

[TABLE]

This expression is fully equivalent to the action with a positive dilaton Brodsky:2014yha after appropriate dilaton-dependent redefinition of the spin-$J$ boson fields.

The AdS metric is specified as $ds^{2}=e^{2A(z)}(dx_{\mu}dx^{\mu}-dz^{2})$, $g=e^{5A(z)}$, $A(z)=\log(R/z)$ and $R$ is the AdS radius. Here $\Phi_{J}$ is the symmetric, traceless tensor classified by the representation $D(E_{0},J/2,J/2)$ with energy $E_{0}=\Delta=\tau$. $E_{0}$ is related to the bulk mass $\mu_{J}$ as $\mu_{J}^{2}R^{2}=(E_{0}-J)(E_{0}-4+J)$ where $\tau=N+L$ is the twist-dimension and $N$ is number of partons; $\hat{V}_{J}(z)$ is the effective dilaton potential, which has an analytical expression in terms of the field $\varphi(z)$ and the ”metric” field $A(z)$, without referring to any specific form of their $z$ profiles:

[TABLE]

In order to study the bound-state problem it is convenient to make a dilaton-dependent redefinition of the bulk field, $\Phi_{J}\to\Phi_{J}e^{\varphi(z)/2}$. In the case of the action with positive dilaton profile the dilaton-dependent redefinition of the bulk field is $\Phi_{J}\to\Phi_{J}e^{-\varphi(z)/2}$. After the redefinitions $\Phi_{J}\to\Phi_{J}e^{\pm\varphi(z)/2}$ both versions of the soft-wall model reduce to the same no-wall action

[TABLE]

where

[TABLE]

At $d=4$, $A(z)=\log(R/z)$ and $\varphi(z)=\kappa^{2}z^{2}$ one gets

[TABLE]

The potential $U_{J}(z)$ is the confinement potential which breaks both conformal and chiral invariance spontaneously.

Next, using the Kaluza-Klein decomposition

[TABLE]

we derive the Schrödinger-type equation for the bulk profiles $\varphi_{n\tau}(z)$

[TABLE]

where

[TABLE]

is the centrifugal potential. For $d=4$ and in the case of quark-antiquark mesons ($\tau=2+L$), tetraquarks ($\tau=4+L$) and six quarks ($\tau=6+L$), this potential reads

[TABLE]

The hadronic wave functions are identified with the profiles of the AdS modes $\varphi_{n\tau}(z)$ in the $z$ direction:

[TABLE]

They posses the correct behavior in both the ultraviolet and infrared limits, with

[TABLE]

and are normalized according to the condition

[TABLE]

The mass spectrum of multiquark meson states is given by

[TABLE]

For $z\to 0$ the scaling of the bulk profile is identified with the scaling of the corresponding mesonic interpolating operator $\tau$. As we mentioned in the Introduction, $\tau$ depends on $L$ (instead of $J$ as in conformal field theory), because we are modeling QCD and therefore should reproduce the scaling of hadronic form factors. As we stressed before, the dependence on $L$ reflects the a spontaneous breaking of chiral invariance, which is expected, since after the introduction of the dilaton field we break the conformal or gauge invariance acting in AdS space. As we noted before, the chiral group is isomorphic to a subgroup of $SO(4,2)$.

The next step is to include the longitudinal part of the LFWF following the approach used in our paper on mesons Gutsche:2012ez . In the case of mesons the first step in this direction was taken in Refs. Brodsky:2008pg for mesons and in Ref. Brodsky:2016yod for tetraquarks. These authors proposed a factorized form for the mesonic two-parton wave function as a product of transverse $\phi_{nL}(\zeta)$, longitudinal $f(x,m_{1},m_{2})$ and angular $e^{im\phi}$ modes. In Ref. Branz:2010ub we presented a more convenient form, factorizing the additional factor $\sqrt{x(1-x)}$, which is the Jacobian of the coordinate transformation $\zeta\to|{\bf b}_{\perp}|$, with:

[TABLE]

In Ref. Branz:2010ub we used a Gaussian ansatz for the longitudinal part of the LFWF and $m_{1}$ and $m_{2}$ as constituent quark masses. In Ref. Gutsche:2012ez we followed Refs. 'tHooft:1974hx ; Bergknoff:1976xr and considered current quark masses. By an appropriate choice of the longitudinal wave function $f(x,m_{1},m_{2})$, we got consistency with QCD in both sectors of light and heavy quarks. In this way we generate the masses of light pseudoscalar mesons in agreement with the scheme resulting from explicit breaking of chiral symmetry — in the leading order of the chiral expansion the masses of pseudoscalar mesons are linear in the current quark mass Weinberg:1978kz . We also guarantee that the pseudoscalar $\pi$, $K$, and $\eta$ meson masses $M_{\pi}^{2},M_{K}^{2},M_{\eta}^{2}$ satisfy the Gell-Mann-Oakes-Renner

[TABLE]

and the Gell-Mann-Okubo

[TABLE]

relations, where $\hat{m}=(m_{u}+m_{d})/2$ is the average mass of $u$ and $d$ quarks, $B=|\langle 0|\bar{u}u|0\rangle|/F_{\pi}^{2}$ is the quark condensate parameter, and $F_{\pi}$ is the leptonic decay constant. In the sector of heavy quarks we set agreement with both heavy quark effective theory and potential models of heavy quarkonia. In the heavy quark mass limit $m_{Q}\to\infty$ we obtaine the correct scaling of the leptonic decay constants for both heavy-light mesons $f_{Q\bar{q}}\sim 1/\sqrt{m_{Q}}$ and heavy quarkonia $f_{Q\bar{Q}}\sim\sqrt{m_{Q}}$ and $f_{c\bar{b}}\sim m_{c}/\sqrt{m_{b}}$ at $m_{c}\ll m_{b}$. In this limit we also generated the correct expansion of heavy meson masses

[TABLE]

where $\bar{\Lambda}$ is the approximate difference between the masses of the heavy-light meson and the heavy quark, $E$ is the binding energy in heavy quarkonia. The corresponding mass splittings, e.g. between vector and pseudoscalar states of heavy-light mesons, become

[TABLE]

We chose the longitudinal wave function in the form

[TABLE]

where $N$ is the normalization constant fixed from

[TABLE]

and $\alpha_{1},\alpha_{2}$ are parameters that have been fixed in order to get consistency with QCD.

Our master formula for the mass spectrum $M^{2}_{nJ}$ of quark-antiquark mesons in terms of the arbitrary longitudinal wave function $f(x,m_{1},m_{2})$ is given by the expression

[TABLE]

Using our ansatz for the $f(x,m_{1},m_{2})$ (24) we get an analytic expression for the correction to the mass spectrum Branz:2010ub :

[TABLE]

In the case of the tetraquark states we have two possibilities:

1. tetraquarks are compact bound states of two quarks and two antiquarks;
2. tetraquarks are bound states of two mesons (hadronic molecules).

When tetraquarks have the hadronic molecular configuration the mass formula reads

[TABLE]

where $M_{1}$ and $M_{2}$ are the masses of the quark-antiquark clusters forming the hadronic molecule. Note that $M_{1}$ and $M_{2}$ are a bit smaller than the physical masses of the corresponding mesons because of the separation of the contribution of the longitudinal and transverse wave function of tetraquarks. Somehow, one can call $M_{1}$ and $M_{2}$ bare meson masses, because the transverse part of the wave function gives an additional contribution due to the confinement potential.

We will now consider the four-quark structure of tetraquark states. Note that in Ref. Brodsky:2016yod the mass formula for the tetraquark states, in terms of quark degrees of freedom, was obtained using superconformal algebra as

[TABLE]

where

[TABLE]

The difference between the two formulas (28) and (29) in the zero mass limit for the constituents is due to the term which spontaneously breaks superconformal symmetry

[TABLE]

which is zero for tetraquark systems with $J=L+S$.

Notice that our formula gives, in the zero mass limit, an natural explanation of the tetraquark state $a(1420)$ discovered by the COMPASS Collaboration Adolph:2015pws . In our approach, for $n=0$, $J=L=1$, we get

[TABLE]

or $M_{a(1420)}=2\kappa\sqrt{2}$ resulting in $M_{a(1420)}=\sqrt{2}$ GeV $\simeq 1.414$ GeV at $\kappa=0.5$ GeV, which agrees perfectly with the experimental result of $M_{a(1420)}=1.414^{+0.015}_{-0.013}$ GeV. The analysis of Ref. Wang:2014bua within QCD sum rules disfavors assigning the $a_{1}(1420)$ to an axial-vector tetraquark state and proposes that it is a mixed state of the $a_{1}(1260)$ meson and the tetraquark state with the configuration $[su]_{S=1}[\bar{s}\bar{d}]_{S=0}+[su]_{S=0}[\bar{s}\bar{d}]_{S=1}$. On the other hand, the QCD sum rules analysis performed in Ref. Chen:2015fwa confirmed the existence of $a_{1}(1420)$ as a tetraquark state. Questions related to the nature of the $a_{1}(1420)$ meson have also been addressed in other papers. In particular, it has been proposed that this state is a consequence of rescattering effects, and in fact in Ref. Ketzer:2015tqa the $a_{1}(1420)$ was interpreted as a dynamical effect due to a singularity (branch point) in the triangle diagram formed by the processes $a_{1}(1260)\to K^{\ast}\bar{K}$, $K^{\ast}\to K\pi$ and $K\bar{K}\to f_{0}(980)$. In Ref. Basdevant:2015wma it was shown that a single $I=1$ spin-parity $J^{PC}=1^{++}$ $a_{1}$ resonance can manifest itself as two separated mass peaks. One decays into an $S$-wave $\rho\pi$ system and the second decays into a $P$-wave $f_{0}(980)\pi$ system, with a rapid increase of the phase difference between their amplitudes, arising mainly from the structure of the diffractive production process. In Ref. Liu:2015taa it was claimed that resonances such as the $a_{1}(1420)$ could be produced due to the so-called “anomalous triangle singularity”, if it is located in a specific kinematical region. In Ref. Aceti:2016yeb $a_{1}(1420)$ was considered as peak in the $a_{1}(1260)\to\pi f_{0}(980)$ decay mode. In Ref. Wang:2015cis it was proposed to test the possible rescattering nature of the $a_{1}(1420)$ in heavy meson decays.

Our approach gives also a lower limit for the masses of the tetraquarks with $n=J=L=S=0$

[TABLE]

for $\kappa=0.5$ GeV. It means that the lightest possible tetraquark is composed of two diquarks forming a state with spin parity $J^{P}=0^{+}$, and then a possible candidate is the $f_{0}(980)$ consistent with the conclusion of Ref. Brodsky:2016yod . Note that in the massless limit we consider the same universal dilaton parameter $\kappa$ for the constituents of all states forming tetraquarks. When the masses of the tetraquark constituents are taken into account, we assume that the numerical value of $\kappa$ for tetraquarks could change from its original value of $0.5$ GeV.

Next we include quark mass effects, following the approach presented in our paper on applications of holographic QCD to mesons Gutsche:2012ez . We proposed the following form for the longitudinal wave function containing quark masses

[TABLE]

where $N$ is a normalization constant fixed from the condition

[TABLE]

The contribution of the longitudinal wave function (34) to the mass spectrum is

[TABLE]

It should be stressed, as in case of mesons, that by an appropriate choice of the $\alpha_{i}$ parameters we can guarantee the correct behavior of the tetraquark spectrum in both the light and heavy quark sectors. In particular, tetraquarks composed of light nonstrange $(u,d)$ quarks receive the following quark mass correction

[TABLE]

where $\hat{m}=m_{u}=m_{d}$ in the isospin limit. The parameter $\alpha_{q}=3\hat{m}/(2B)$ is fixed from the condition

[TABLE]

By analogy, for light tetraquarks containing single, two and three strange quarks/antiquarks using $\alpha_{q}=3\hat{m}/(2B)$ and $\alpha_{s}=3m_{s}/(2B)$ we get

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We will now derive quark mass corrections for tetraquarks containing single, two, tree and four heavy quarks/antiquarks $Q=b,c$ (in the following index $q$ denotes all light quarks $u,d,s$, and for simplicity we consider heavy and light quarks each of the same flavor):

[TABLE]

To get the correct scaling of heavy tetraquarks for $m_{Q}\to\infty$ with

[TABLE]

where $\bar{\Lambda}_{nq}$ is the approximate difference between the mass of the heavy-light tetraquark and the sum of the masses of its $(4-n)$ constituent heavy quarks, we fix the $\alpha_{Q}$ parameters as follows: $\alpha_{Q}=\alpha$ (independent on the heavy quark flavor) in the case of heavy-light tetraquarks and

[TABLE]

in the case of tetraquarks composed only of heavy quarks. The parameter for light quarks $\alpha_{q}$ occuring in the case of heavy-light tetraquarks is fixed as

[TABLE]

where $n$ is the number of light quark/antiquarks in a tetraquark. In particular,

[TABLE]

in the case of $n=3$,

[TABLE]

in the case of $n=2$,

[TABLE]

in the case of $n=1$.

In the case of tetraquarks with a hadronic molecular configuration it is worthwhile to test the prediction of Ref. Bondar:2011ev about the structure of the $Z_{b}(10610)=(B\bar{B}^{\ast}+{\rm h.c.})$ and $Z_{b}(10650)=(B^{\ast}\bar{B}^{\ast})$ resonances, which are supposed to be $J^{P}=1^{+}$ states. In our approach their masses are given by

[TABLE]

Fixing $\alpha_{P_{q\bar{b}}}$ and $\alpha_{V_{q\bar{b}}}$ as

[TABLE]

results in the expressions

[TABLE]

where $\bar{M}_{B}=\frac{M_{V_{q\bar{b}}}+M_{P_{q\bar{b}}}}{2}$. For the mass splitting $M_{Z_{b}(10650)}^{2}-M_{Z_{b}(10610)}^{2}$ we get

[TABLE]

or

[TABLE]

in agreement with Ref. Bondar:2011ev . By analogy we get a similar conclusion for the related charmed tetraquark states $Z_{c}(3900)=(D\bar{D}^{\ast}+{\rm h.c.})$ and $Z_{c}(4020)=(D^{\ast}\bar{D}^{\ast})$.

Another interesting state is the $X(3872)$. In our approach we consider this state in a mixed configuration of the molecular $(D^{0}\bar{D}^{\ast 0}+\bar{D}^{0}D^{\ast 0})/\sqrt{2}$ and the charmonia $c\bar{c}$ states following Refs. Swanson:2003tb ; Dong:2008gb with

[TABLE]

Using Eq. (55) and

[TABLE]

we get the following mass formula for the $X(3872)$ state

[TABLE]

where by analogy with the bottom sector we have

[TABLE]

Note that the hadronic molecular contribution to the $X(3872)$ state is the same expression as for the $Z_{c}(3900)$ and $Z_{c}(4020)$ states. The values $m_{c}=1.275$ GeV and $E=0.795$ GeV are taken from Ref. Gutsche:2012ez . We find that for $|\theta|=11^{0}$ we could reproduce the experimental value of the $X(3872)$ mass of $M_{X(3872)}=3871.69\pm 0.17$ MeV.

Next we look at the possible $J^{P}=0^{+}$ $Y$–states as hadronic molecules

[TABLE]

studied by us before using a phenomenological Lagrangian approach in Ref. Branz:2009yt . Holographic QCD gives the following results for the masses of $Y(3940)$ and $Y(4140)$ states

[TABLE]

where

[TABLE]

and $\Delta_{Y}$ is phenomenological parameter of order of ${\cal O}(M_{i}^{0})$. With $\Delta_{Y}=96$ MeV we can reproduce the mass data for these states.

Finally, we consider the $Z(4430)^{+}$ state in the hadronic molecular picture, which has the structure Branz:2010sh

[TABLE]

Its mass squared neglecting isospin breaking corrections is given by

[TABLE]

Taking

[TABLE]

we get

[TABLE]

Our predictions for the masses for selected light and heavy tetraquarks are shown in Tables 1-4. For some states, suspected as tetraquarks, we include a comparison with data Olive:2016xmw , while for other states we make mass predictions. In future work we plan to make a more detailed analysis.

For completeness we also consider two candidates on tetraquarks with color diquark-color diquark configuration — $X(3755)$ and $X(3915)$ states, with $\bar{\Lambda}_{2q}=1.07$ GeV we reproduce the data for their masses. Also we suppose that $X(3915)$ is the radial excitation of the $X(3755)$ state.

In the numerical analysis we use the same set of quark masses as for the conventional quark-antiquark mesons:

[TABLE]

We use the universal parameters $\kappa=351/435$ MeV for light and $\kappa=500$ MeV for heavy-light tetraquarks. We suppose that the inclusion of finite mass effects of the constituents in the light tetraquarks shifts the parameter $\kappa$ to lower values than 500 MeV. However, in the massless limit for the tetraquark constituents, the parameter $\kappa$ is universal and the same for all tetraquark states independent on the light/heavy quark/diquark/meson content. The masses of the quark-antiquark clusters for pseudoscalar states are taken as

[TABLE]

As we stressed before, the masses of other spin-parity states are calculated according to the mass splittings

[TABLE]

and etc.

Acknowledgements.

The authors thank Stan Brodsky for useful discussions. This work was funded by the German Bundesministerium für Bildung und Forschung (BMBF) under Project 05P2015 - ALICE at High Rate (BMBF-FSP 202): “Jet- and fragmentation processes at ALICE and the parton structure of nuclei and structure of heavy hadrons”; by CONICYT (Chile) PIA/Basal FB0821, by CONICYT (Chile) Research Projects No. 80140097, and under Grants No. 7912010025, 1140390; by Tomsk State University Competitiveness Improvement Program and the Russian Federation program “Nauka” (Contract No. 0.1764.GZB.2017).