The magnetic moment of $Z_{c}(3900)$ as an axial-vector molecular state

$Z_{c}(3900)$, as a good candidate of exotic hadrons, was observed by BESIII collaboration in 2013 in the $\pi^{\pm}J/\psi$ invariant mass distribution of the process $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ at a center-of-mass energy of $4.260\mbox{GeV}$ bes1 . Then the Belle and CLEO collaborations confirmed the existence of $Z_{c}(3900)$ belle ; cleo . In 2017, the BESIII collaboration determined the $J^{P}$ quantum number of $Z_{c}(3900)$ to be $J^{P}=1^{+}$ with a statistical significance larger than $7\sigma$ over other quantum numbers in a partial wave analysis of the process $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ bes2 . Inspired by these experimental progress, there have been plentiful theoretical studies on $Z_{c}(3900)$’s properties through different approaches (see review article H.X.Chen and references therein for details). However, the underlying structure of $Z_{c}(3900)$ is not understood completely and more endeavors are necessary in order to arrive at a better understanding for the properties of $Z_{c}(3900)$.

The electromagnetic multipole moments of hadron encode the spatial distributions of charge and magnetization in the hadron and provide important information about the quark configurations of the hadron and the underlying dynamics. So it is interesting to study the electromagnetic multipole moments of hadron.

The studies on the properties of hadrons inevitably involve the nonperturbative effects of quantum chromodynamics (QCD). The QCD sum rule method SVZ is a nonperturbative analytic formalism firmly entrenched in QCD with minimal modeling and has been successfully applied in almost every aspect of strong interaction physics. In Ref.Balitsky ; Ioffe1 ; Ioffe2 , the QCD sum rule method was extended to calculate the magnetic moments of the nucleon and hyperon in the external field method. In this method, a statics electromagnetic field is introduced which couples to the quarks and polarizes the QCD vacuum and magnetic moments can be extracted from the linear response to this field. Later, a more systematic studies was made for the magnetic moments of the octet baryons octet1 ; octet2 ; octet3 ; octet4 , the decuplet baryons decuplet1 ; decuplet2 ; decuplet3 ; decuplet4 and the $\rho$ meson rho . In the case of the exotic X, Y, Z states, only the magnetic moment of $Z_{c}(3900)$ as an axialvector tetraquark state was calculated through this method wangzhigang .

In this article, we study the magnetic moment of $Z_{c}(3900)$ as an axialvector molecular state with quantum number $J^{P}=1^{+}$ by the QCD sum rule method. The mass and pole residue, two of the input parameters needed to determine the magnetic moment, are calculated firstly including contributions of operators up to dimension 10. Then the magnetic moment is extracted from the linear term in $F_{\mu\nu}$ (external electromagnetic filed) of the correlation function.

The rest of the paper is arranged as follows. In Sec.II, we derive the sum rules for the mass, pole residue and magnetic moment of $Z_{c}(3900)$ state. Sec.III is devoted to the numerical analysis and a short summary is given in Sec.IV. In the Appendix B, the spectral densities are shown.

The starting point of our calculation is the time-ordered correlation function in the QCD vacuum in the presence of a constant background electromagnetic field $F_{\mu\nu}$,

where

is the interpolating current of $Z_{c}(3900)$ as a molecular state with $J^{P}=1^{+}$ cuichunyu . The $\Pi^{(0)}_{\mu\nu}(p)$ term is the correlation function without external electromagnetic field, and give rise to the mass and pole residue of $Z_{c}(3900)$. The magnetic moment will be extracted from the linear response term, $\Pi^{(1)}_{\mu\nu\alpha\beta}(p)F^{\alpha\beta}$.

The external electromagnetic field can interact directly with the quarks inside the hadron and also polarize the QCD vacuum. As a consequence, the vacuum condensates involved in the operator product expansion of the correlation function in the external electromagnetic field $F_{\mu\nu}$ are,

• •

  dimension-2 operator,

  $$F_{\mu\nu},$$

  (3)

• •

  dimension-3 operator,

  $$\langle 0|\bar{q}\sigma_{\mu\nu}q|0\rangle_{F},$$

  (4)

• •

  dimension-5 operators,

  $$\langle 0|\bar{q}q|0\rangle F_{\mu\nu},\langle 0|\bar{q}g_{s}G_{\mu\nu}q|0\rangle_{F},\epsilon_{\mu\nu\alpha\beta}\langle 0|\bar{q}g_{s}G^{\alpha\beta}q|0\rangle_{F},$$

  (5)

• •

  dimension-6 operators,

  $$\langle 0|\bar{q}q|0\rangle\langle 0|\bar{q}\sigma_{\mu\nu}q|0\rangle_{F},\langle 0|g^{2}_{s}GG|0\rangle F_{\mu\nu},\cdots,$$

  (6)

• •

  dimension-7 operators,

  $$\langle 0|g^{2}_{s}GG|0\rangle\langle 0|\bar{q}\sigma_{\mu\nu}q|0\rangle_{F},\langle 0|g_{s}\bar{q}\sigma\cdot Gq|0\rangle F_{\mu\nu},\cdots,$$

  (7)

• •

  dimension-8 operators,

  	$\displaystyle\langle 0|\bar{q}q|0\rangle^{2}F_{\mu\nu},\langle 0|g_{s}\bar{q}\sigma\cdot Gq|0\rangle\langle 0|\bar{q}\sigma_{\mu\nu}q|0\rangle_{F},\langle 0|\bar{q}q|0\rangle\langle 0|\bar{q}g_{s}G_{\mu\nu}q|0\rangle_{F},$
  	$\displaystyle\epsilon_{\mu\nu\alpha\beta}\langle 0|\bar{q}q|0\rangle\langle 0|\bar{q}g_{s}G^{\alpha\beta}q|0\rangle_{F},\cdots,$		(8)

The new vacuum condensates induced by the external electromagnetic field $F_{\mu\nu}$ can be described by introducing new parameters, $\chi$, $\kappa$ and $\xi$, called vacuum susceptibilities as follows,

In order to express the two-point correlation function (1) physically, we expand it in powers of the electromagnetic interaction Hamiltonian $H_{int}=-ie\int d^{4}yj^{em}_{\alpha}(y)A^{\alpha}(y)$,

where $j^{em}_{\alpha}(y)$ is the electromagnetic current and $A^{\alpha}(y)$ is the electromagnetic four-vector.

Inserting complete sets of relevant states with the same quantum numbers as the current operator $J_{\mu}(x)$ and carrying out involved integrations, one has

where we make use of the following matrix elements

with $\lambda_{Z_{c}}$ and $\epsilon_{\mu}(p)$ being the pole residue and polarization vector of $Z_{c}(3900)$, respectively,

with $q=p^{\prime}-p$ and $Q^{2}=-q^{2}$. The Lorentz-invariant functions $G_{1}(Q^{2})$, $G_{2}(Q^{2})$ and $G_{3}(Q^{2})$ are related to the charge, magnetic and quadrupole form-factors,

respective, where $\eta=\frac{Q^{2}}{4m^{2}_{Z_{c}}}$. At zero momentum transfer, these form-factors are proportional to the usual static quantities of the charge $e$, magnetic moment $\mu_{Z_{c}}$ and quadrupole moment $Q_{1}$,

The constant $a$ parameterizes the contributions from the pole-continuum transitions.

On the other hand, $\Pi_{\mu\nu}(p)$ can be calculated theoretically via OPE method at the quark-gluon level. To this end, one can insert the interpolating current $J_{\mu}(x)$ (2) into the correlation function (1), contract the relevant quark fields via Wick’s theorem and obtain

where $S^{(c)}(x)=\langle 0|T[c(x)\bar{c}(0)]|0\rangle$ and $S^{(q)}(x)=\langle 0|T[q(x)\bar{q}(0)]|0\rangle,q=u,d$ are the full charm- and up (down)-quark propagators, whose expressions are given in the Appendix A, $Tr$ denotes the trace of the Dirac spinor indices, and $a$, $b$, $c$ and $d$ are color indices. Through dispersion relation, $\Pi^{OPE}_{\mu\nu}(p)$ can be written as

where $\rho^{i}(s)=\frac{1}{\pi}\mbox{Im}\Pi^{OPE}_{i}(s),i=0,1$ are the spectral densities. The spectral densities $\rho^{i}(s)$ are given in the Appendix B.

Finally, matching the phenomenological side (11) and the QCD representation (17), we obtain

for the Lorentz-structure $(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}})$, and

for the Lorentz-structure $iF_{\mu\nu}$.

According to quark-hadron duality, the excited and continuum states’ spectral density can be approximated by the QCD spectral density above some effective threshold $s^{0}_{Z_{c}}$, whose vale will be determined in Sec.III,

Subtracting the contributions of the excited and continuum states, one gets

In order to improve the convergence of the OPE series and suppress the contributions from the excited and continuum states, it is necessary to make a Borel transform. As a result, we have

where $M^{2}_{B}$ is the Borel parameter and $A=\frac{a}{\lambda^{2}_{Z_{c}}}$. Taking derivative of the first equation in (II) with respect to $-\frac{1}{M^{2}_{B}}$ and dividing it by the original expression, one has

In the next section, (II) and (23) will be analysed numerically to obtain the numerical values of the mass, the pole residue and the magnetic moment of the $Z_{c}(3900)$.

The input parameters needed in numerical analysis are presented in Table 1. For the vacuum susceptibilities $\chi$, $\kappa$ and $\xi$, we take the values $\chi=-(3.15\pm 0.30)\mbox{GeV}^{-2}$, $\kappa=-0.2$ and $\xi=0.4$ determined in the detailed QCD sum rules analysis of the photon light-cone distribution amplitudes P.Ball . Besides these parameters, we should determine the working intervals of the threshold parameter $s^{0}_{Z_{c}}$ and the Borel mass $M^{2}_{B}$ in which the mass, the pole residue and the magnetic moment vary weakly. The continuum threshold is related to the square of the first exited states having the same quantum number as the interpolating field, while the Borel parameter is determined by demanding that both the contributions of the higher states and continuum are sufficiently suppressed and the contributions coming from higher dimensional operators are small.

We define two quantities, the ratio of the pole contribution to the total contribution (RP) and the ratio of the highest dimensional term in the OPE series to the total OPE series (RH), as followings,

where $i=0,1$ and $n=10(8)$ as $i=0(1)$, respectively.

In Fig.1(a), we compare the various terms in the OPE series as functions of $M^{2}_{B}$ with $\sqrt{s^{0}_{Z_{c}}}=4.6\mbox{GeV}$. From it one can see that except the quark condensate $\langle\bar{q}q\rangle$, other vacuum condensates are much smaller than the perturbative term. So the OPE series are under control. Fig.1(b) shows $RP_{0}$ and $RH_{0}$ varying with $M^{2}_{B}$ at $\sqrt{s^{0}_{Z_{c}}}=4.6\mbox{GeV}$. The figure shows that the requirement $RP_{0}\geq 50\%$ ($RP_{0}\geq 40\%$) gives $M^{2}_{B}\leq 3.3\mbox{GeV}^{2}$ ($M^{2}_{B}\leq 3.7\mbox{GeV}^{2}$) and $RH_{0}=5\%$ at $M^{2}_{B}=1.25\mbox{GeV}^{2}$.

From Fig.2(a), we know that the sum rule for the mass $m_{Z_{c}}$ depends strongly on the Borel parameter $M^{2}_{B}$ as $M^{2}_{B}\leq 3\mbox{GeV}^{2}$. Along with the criterions of pole dominance, this fact confines $M^{2}_{B}$ from $3\mbox{GeV}^{2}$ to $3.7\mbox{GeV}^{2}$. In the analysis, we take $RP_{0}\geq 40\%$ so that we can obtain a larger interval of the Borel parameter. Within the interval of $M^{2}_{B}$ determined above, the mass varies weakly with $M^{2}_{B}$ as depicted in Fig.2(b). Fig.2(b) also shows the weak dependence of the mass on the threshold parameter $s^{0}_{Z_{c}}$ as $4.5^{2}\mbox{GeV}^{2}\leq s^{0}_{Z_{c}}\leq 4.7^{2}\mbox{GeV}^{2}$. As a result, we can reliably read the value of the mass, $m_{Z_{c}}=3.97\pm 0.12\mbox{GeV}$, in agreement with the experimental value $m^{exp}_{Z_{c}}=3899.0\pm 3.6\pm 4.9\mbox{MeV}$.

In Fig.3, we show the variation of the pole residue with the Borel parameter $M^{2}_{B}$ in the determined interval at three different values of $s^{0}_{Z_{c}}$. It is obvious that the pole residue depends weakly on $M^{2}_{B}$ and $s^{0}_{Z_{c}}$ and $\lambda_{Z_{c}}=2.1\pm 0.4\times 10^{-2}\mbox{GeV}^{5}$.

The same procedure can be done for the sum rule of the magnetic moment. The results are shown in Fig.4, from which the value of $G_{M}(0)$ can be read as $G_{M}(0)=0.82^{+0.17}_{-0.07}$. Finally, we obtain

where $\mu_{N}$ is the nucleon magneton.

In Ref.wangzhigang , the author gave $\mu_{Z_{c}}=0.47^{+0.27}_{-0.22}\mu_{N}$ assuming $Z_{c}(3900)$ as an axialvector tetraquark state by the same method used in this article. In Ref.U.Ozdem , $\mu_{Z_{c}}=0.67\pm 0.32\mu_{N}$ was predicted using light-cone sum rule under the axialvector tetraquark assumption. In Table 2, we summarize the values of the magnetic moment of $Z_{c}(3900)$ under different assumptions about the quark configuration and with different methods. It is obvious that the magnetic moment of $Z_{c}(3900)$ has different values if $Z_{c}(3900)$ has different quark configurations. The theoretical predictions can be confronted to the experimental data in the future and give important information about the inner structure of $Z_{c}(3900)$ state.

In this paper, we tentatively assign $Z_{c}(3900)$ to be an axialvector molecular state, calculate its magnetic moment using the QCD sum rule method in the external weak electromagnetic field. Starting with the two-point correlation function in the external electromagnetic field and expanding it in power of the electromagnetic interaction Hamiltonian, we extract the mass and pole residue of $Z_{c}(3900)$ state from the leading term in the expansion and the magnetic moment from the linear response to the external electromagnetic field. The numerical values are $m_{Z_{c}}=3.97\pm 0.12\mbox{GeV}$ in agreement with the experimental value $m^{exp}_{Z_{c}}=3899.0\pm 3.6\pm 4.9\mbox{MeV}$, $\lambda_{Z_{c}}=2.1\pm 0.4\times 10^{-2}\mbox{GeV}^{5}$ and $\mu_{Z_{c}}=0.19^{+0.04}_{-0.01}\mu_{N}$ with $\mu_{N}$ the nucleon magneton. The prediction can be confronted to the experimental data in the future and give important information about the inner structure of $Z_{c}(3900)$ state.