The $X_{0}(2900)$ and its heavy quark spin partners in molecular picture

Introduction.   The conventional quark model GellMann:1964nj ; Zweig:1981pd , which inherits part of the properties of Quantum Chromo-Dynamics (QCD), has made a great success to understand hadrons before 2003. Quark model tells us that hadrons can be classified as either mesons made of $q\bar{q}$ or baryons made of three quarks. However, QCD tells us that any color neutral configuration (especaily exotics) could exist upon the two configurations mentioned above. That leaves us two questions: where to find these exotic candidates and how to understand the underlying mechanism. The observation of the first exotic candidate $X(3872)$ Choi:2003ue in 2003 and the succeed tremendous experimental measurements  Olsen:2017bmm ; Yuan:2019zfo partly answer the first question. Among these, the observation of the first pentaquarks Aaij:2015tga ; Aaij:2019vzc , the first fully heavy four quark states  LHCb-X6900 and the first exotic candidates with four different flavors i.e. the $X_{0}(2900)$ Aaij:2020hon ; Aaij:2020ypa reported by the LHCb Collaboration recently, set milestones from experimental side. Different prescriptions from theoretical side are put forward for understanding the nature of these exotic candidates  Guo:2017jvc ; Chen:2016qju ; Esposito:2016noz ; Ali:2017jda ; Liu:2019zoy ; Brambilla:2019esw ; Bondar:2016hva ; Lebed:2016hpi . Among them, hadronic molecule Guo:2017jvc , as an analogy of deuteron formed by a proton and a neutron, is proposed due to the fact that they are with a few MeV below the nearby $S$-wave threshold.

However, one have to confront one problem that different configurations with the same quantum number can mix with each other and cannot be well isolated. For instance, although the $X(3872)$ is proposed as a hadronic molecule at the very beginning Tornqvist:2004qy due to its closeness to the $D\bar{D}^{*}+c.c.$ threshold, it still could be or mix with the normal charmonium $\chi_{c1}(2P)$ Meng:2007cx ; Kalashnikova:2005ui ; Zhang:2009bv ; Danilkin:2010cc ; Li:2009zu ; Li:2009ad ; Coito:2010if ; Coito:2012vf . Another typical example is the $D^{*}_{s0}(2317)$ and the $D_{s1}(2460)$ which are about $160~{}\mathrm{MeV}$ and $70~{}\mathrm{MeV}$ below the $J^{P}=0^{+}$ and $J^{P}=1^{+}$ $c\bar{s}$ charmed-strange mesons of Godfrey-Isgur quark model Godfrey:1985xj . Meanwhile, they are about $45~{}\mathrm{MeV}$ below the $DK$ and $D^{*}K$, thresholds, respectively, which can be explained naturally if the systems are bound states of the $DK$ and $D^{*}K$ meson pairs  Barnes:2003dj ; vanBeveren:2003kd ; vanBeveren:2003jv ; Kolomeitsev:2003ac ; Guo:2006fu ; Zhang:2006ix ; Guo:2006rp ; Guo:2009id . However, because the light quark and anti-quark in the isosinglet $D^{(*)}K$ system are of the same flavor, despite of those comprehensive studies, one still cannot avoid the possibility of the mixture with the normal $c\bar{s}$ configurations  vanBeveren:2003kd ; vanBeveren:2003jv ; Coito:2011qn ; Hwang:2004cd ; Simonov:2004ar ; Lee:2004gt ; Zhou:2011sp . Fortunately, the LHCb Collaboration reported a $J^{P}=0^{+}$ Aaij:2020hon ; Aaij:2020ypa narrow state $X_{0}(2900)$ with mass $2866\pm 7~{}\mathrm{MeV}$ and width $\Gamma_{0}=57\pm 13~{}\mathrm{MeV}$ as well as another broader $J^{P}=1^{-}$ state with mass $2904\pm 7~{}\mathrm{MeV}$ and width $\Gamma_{1}=110\pm 12~{}\mathrm{MeV}$ in the $\bar{D}K$ invariant mass distribution. They are the first exotic states with four different flavors, which brings us a potential ultimate solution for the problem from different aspects.

In this letter, we solve the Lippmann-Schwinger Equation (LSE) with leading order contact potentials of the $\bar{D}^{(*)}K^{(*)}$ system, in the heavy quark limit, to extract the mass position of the spin partners of the $X_{0}(2900)$. That the $X_{0}(2900)$ exists as a $I(J^{P})=0(0^{+})$ $\bar{D}^{*}K^{*}$ hadronic molecule is an input in our framework. With that assumption, we predict the masses of its heavy quark spin partners. Searching for those spin partners could help to understand the nature of the $X_{0}(2900)$.

Framework.   The heavy quark spin structure Bondar:2011ev could reexpress of the hadron basis by the heavy-light basis. One could find an example for the $Z^{(\prime)}_{c}$ and $Z_{b}^{(\prime)}$ case with two heavy quarks in Refs. Hanhart:2015cua ; Guo:2016bjq ; Wang:2018jlv ; Baru:2019xnh ; Voloshin:2011qa ; Baru:2017gwo . Along the same line, the $S$-wave $\bar{D}^{(*)}K^{(*)}$ system with only one heavy quark can be written in terms of the heavy degree of freedom $\frac{1}{2}$ and light degree of freedom $s_{l}$ as the following Yasui:2013vca

with $\hat{j}=\sqrt{2j+1}$. Here, $j_{1}=\frac{1}{2}$, $j_{2}=\frac{1}{2}$ and $j_{12}=0,1$ are spins of anti-charm quark $\bar{c}$, light quark $q$ and the sum of them in the $\bar{D}^{(*)}$ meson. $j_{3}=0,1$ and $J=0,1,2$ are the spins of the $K^{(*)}$ meson and the total spin of the $\bar{D}^{(*)}K^{(*)}$ system. $s_{l}$ on the right hand side of Eq. (1) is the light degree of freedom of the system, which is the only relevant quantity for the dynamics in the heavy quark limit. Following Eq. (1), one can obtain the decompositions of the $\bar{D}^{(*)}K^{(*)}$ system as

Here the heavy degree of freedom is suppressed due to the same value, leaving only the light degrees of freedom $s_{l}$ in $|\dots\rangle$. Although the $K$ and $K^{*}$ have the same quark content, the light degrees of freedoms in the first two equations and those in the last four equations can be distinguishable due to the large scale separation of the $K$ and $K^{*}$ masses. Analogous to those in Refs. Guo:2013sya , by defining the contact potential

the potentials of the $\bar{D}^{(*)}K$ and and $\bar{D}^{(*)}K^{*}$ systems are

and

respectively. $V_{J^{+}}$ and $V^{*}_{J^{+}}$ are for the potentials of the $\bar{D}^{(*)}K$ and $\bar{D}^{(*)}K^{*}$ systems, respectively. The subindex $J^{+}$ presents the total spin and parity of the corresponding system. The transition between $|l\rangle$ and $|l\rangle^{*}$ is the higher order contribution, which is set to zero in this work ¹¹1The observation of the $X_{0}(2900)$ in the $\bar{D}K$ channel is due to this higher order contribution, i.e. the mixing between the $|\frac{1}{2}\rangle$ and the $|\frac{1}{2}\rangle^{*}$ compoments.. The above decomposition and the corresponding potentials also work for the $D^{(*)}K^{(*)}$ systems, but with different values of $C_{2l}^{(*)}$.

With the above potentials, one can solve LSE

with $V$ the potentials for specific channels of a given quantum number. Here two-body non-relativistic propagator is

with power divergence subtraction Kaplan:1998tg to regularize the ultraviolet (UV) divergence. The value of $\Lambda$ should be smaller enough to preserve heavy quark symmetry, leaving the physics insensitive to the details of short-distance dynamics Guo:2013sya . Here $m_{1}$, $m_{2}$ and $\mu$ are the masses of the intermediated two particles and their reduced mass, respectively. $M$ is the total energy of the system. The expression of the second Riemann sheet $G^{\mathrm{II}}_{\Lambda}(M,m_{1},m_{2})$ can be obtained by changing the sign of the second term of $G_{\Lambda}(M,m_{1},m_{2})$.

Results and Discussions.   Before the numerical results, we estimate the values of the contact potential $C_{1}$. The leading contact terms between heavy-light mesons and Goldstone bosons can be obtained by the following Lagrangian  Guo:2008gp ; Du:2017ttu ; Burdman:1992gh ; Wise:1992hn ; Yan:1992gz ; Yao:2015qia

where

with

and chiral connection

Here the chiral building blocks are

Here $U=\exp\left(i\sqrt{2}\phi/f_{0}\right)$ with $\phi$ the Goldstone boson octet. To the leading order, $f_{0}$ is the pion decay constant. The isospin singlet $J^{P}=0^{+}$ $DK$ and $\bar{D}K$ systems are defined as

which is associated with the $D_{s0}^{*+}(2317)$ and

The definitions of the isospin singlet $J^{P}=1^{+}$ $D^{*}K$ and $\bar{D}^{*}K$ systems are analogous. From Eq. (17), we obtain

which agrees with those obtained from the heavy-light decomposition, i.e. Eqs. (9), (10), and

As the result, the value of $C_{1}$ for the $\bar{D}^{(*)}K$ system is half of that for the $D^{(*)}K$ system. We find that any parameter set, i.e. $(\Lambda,C_{1})$, for the existence of the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ as $DK$ and $D^{*}K$ molecular states (both bound states and virtual states) does not indicate the existence of the analogous $\bar{D}K$ and $\bar{D}^{*}K$ molecules. Here and in what follows, we focus on the discussion of the formation of the $\bar{D}^{(*)}K^{(*)}$ molecule instead of their isospin breaking effect. As the result, the isospin average masses

are implemented in this letter.

For the $\bar{D}^{(*)}K^{*}$ interaction, the $X_{0}(2900)$ recently observed by the LHCb collaboration is assumed to be a $I(J^{P})=0(0^{+})$ $\bar{D}^{*}K^{*}$ molecular state Molina:2010tx . We consider the two cases for the $X_{0}(2900)$

• •

  a bound state with the $X_{0}(2900)$ mass $m_{X_{0}(2900)}$ satisfying

  $$1-C_{1}^{*}G_{\Lambda}(m_{X_{0}(2900)},m_{\bar{D}^{*}},m_{K^{*}})=0$$

  (27)

• •

  a virtual state with the $X_{0}(2900)$ mass $m_{X_{0}(2900)}$ satisfying

  $$1-C_{1}^{*}G^{\mathrm{II}}_{\Lambda}(m_{X_{0}(2900)},m_{\bar{D}^{*}},m_{K^{*}})=0$$

  (28)

We take $\Lambda=0.05~{}\mathrm{GeV}$ and $\Lambda=0.03~{}\mathrm{GeV}$ to illustrate the mass positions of its heavy quark spin partners and the corresponding properties.

For the bound state solution of the $X_{0}(2900)$, $C_{1}^{*}=33.56~{}\mathrm{GeV}^{-2}$ and $C_{1}^{*}=102.09~{}\mathrm{GeV}^{-2}$ correspond to $\Lambda=0.05~{}\mathrm{GeV}$ and $\Lambda=0.03~{}\mathrm{GeV}$, respectively. Fig. 1 shows how the poles move with the variation of the two parameter sets. The blue triangle and green square curves show the pole trajectory of the bound state and resonance in the $1^{+}$ channel. One can see that, with $C^{*}_{3}$ variation between $40~{}\mathrm{GeV}^{-2}$ and $170~{}\mathrm{GeV}^{-2}$, one bound state and one resonance emerge with tens of MeV below the $\bar{D}K^{*}$ and $\bar{D}^{*}K^{*}$ thresholds, respectively. The bound state in the $2^{+}$ channel is more sensitive to the parameter $C_{3}^{*}$. If one would expect that light quark spin symmetry also works here as that for the two $Z_{b}$ states Voloshin:2016cgm , i.e. $C^{*}_{3}=C^{*}_{1}$, we find the pole position of the above three states are

The vanishing imaginary part of the higher $1^{+}$ state is because of the degenerance of the two $1^{+}$ states.

For the virtual state solution of the $X_{0}(2900)$, $C_{1}^{*}=14.24~{}\mathrm{GeV}^{-2}$ and $C_{1}^{*}=19.92~{}\mathrm{GeV}^{-2}$ correspond to $\Lambda=0.05~{}\mathrm{GeV}$ and $\Lambda=0.03~{}\mathrm{GeV}$, respectively. Fig. 2 shows how the poles move with $C^{*}_{3}$ variation between $22~{}\mathrm{GeV}^{-2}$ and $36~{}\mathrm{GeV}^{-2}$ for the two $\Lambda$ values. For the former case, the blue triangles and red circles show the pole trajectories of the bound states for the $1^{+}$ and $2^{+}$ channels, respectively. For the later case, both of them become virtual states. As the result, whether the higher $1^{+}$ $\bar{D}^{*}K^{*}$ molecule exists or not depends on the nature of the $X_{0}(2900)$, i.e. either a bound state or a virtual state, which can be studied by the further detailed scan of its line shape. Thus, searching for these heavy quark spin partners would help to reveal the nature of the $X_{0}(2900)$.

Summary and Outlook.   Under the assumption that the LHCb Collaboration recently reported $X_{0}(2900)$ is a $I(J^{P})=0(0^{+})$ $\bar{D}^{*}K^{*}$ hadronic molecule, we extract the pole trajectories of its heavy quark spin partners with the variation of the parameter $C_{3}^{*}$. The parameter $C_{1}^{*}$ is fixed by the mass position of the $X_{0}(2900)$ (either a bound state or a virtual state). For the bound state case, in the light quark spin symmetry, we extract the mass positions of its heavy quark spin partners, i.e. $2.722~{}\mathrm{GeV}$ and $2.866~{}\mathrm{GeV}$ for $1^{+}$ state, and $2.866~{}\mathrm{GeV}$ for $2^{+}$ state. For the virtual state case, the higher $1^{+}$ state is far away from the physical region and will not have large impact on the physical observables. Searching for those states would help to shed light on the nature of the $X_{0}(2900)$.

During the update of this manuscript, several works He:2020jna ; Liu:2020orv ; Zhang:2020oze ; Lu:2020qmp ; Liu:2020nil ; Chen:2020aos ; Wang:2020xyc ; Huang:2020ptc ; Qin:2020zlg appear to discuss the relevant topics.