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

In this article, we tentatively assign the $Z_{c}(3900)$ to be an axialvector molecular state, calculate its magnetic moment using the QCD sum rule method in 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 the $Z_{c}(3900)$ 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 $m_{Z_{c}}=3.97\pm0.12\mbox{GeV}$ in agreement with the experimental value $m^{exp}_{Z_{c}}=3899.0\pm3.6\pm4.9\mbox{MeV}$, $\lambda_{Z_{c}}=2.1\pm0.4\times10^{-2}\mbox{GeV}^{5}$ and $\mu_{Z_{c}}=0.19^{+0.04}_{-0.01}\mu_{N}$.

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 [6] 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. [7][8][9], 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 [10][11][12][13], the decuplet baryons [14][15][16][17] and the ρ meson [18]. In the case of the exotic X, Y, Z states, only the magnetic moment of the Z c (3900) as an axialvector tetraquark state was calculated through this method [19].
In this article, we study the magnetic moment of the 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 µν (external electromagnetic filed) of the correlation function.
The rest of the paper is arranged as follows. In section II, we derive the sum rules for the mass, pole residue and magnetic moment of the Z c (3900) state. Section III is devoted to the numerical analysis and a short summary is given in section IV. In the Appendix B, the spectral densities are shown.

II. THE DERIVATION OF THE SUM RULES
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 µν , where is the interpolating current of Z c (3900) as a molecular state with J P = 1 + [20]. The Π (0) µν (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 extract from the linear response term, Π 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 µν are dimension-2 operator dimension-5 operators dimension-6 operators dimension-7 operators dimension-8 operators The new vacuum condensates induced by the external electromagnetic field F µν can be described by introducing new parameters called vacuum susceptibilities as follows, There are three main steps in the QCD sum rule calculation, • Representing the correlation function in terms of hadronic parameters, • Calculating the correlator via operator product expansion(OPE) at the quark-gluon level, • Matching the two representation with the help of quark-hadron duality and extracting the needed quantities.
In the last step, Borel transform is introduced to suppress the higher and continuum states' contributions and improve the convergence of the OPE series.
In order to express the two-point correlation function (1) physically, we expand it in powers of the electromagnetic interaction Hamiltonian H int = −ie d 4 yj em α (y)A α (y), where j em α (y) is the electromagnetic current and A α (y) is the electromagnetic four-vector. Inserting complete sets of relevant states with the same quantum numbers as the current operator J µ (x) and carrying out involved integrations, one has where we make use of the following matrix elements with λ Zc and ǫ µ (p) being the pole residue and polarization vector of the Z c (3900) respectively, with q = p ′ − 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,

Zc
. At zero momentum transfer, these form-factors are proportional to the usual static quantities of the charge e, magnetic moment µ Zc and quadrupole moment Q 1 , The constant a parameterizes the contributions from the pole-continuum transitions.
On the other hand, Π µν (p) can be calculated theoretically via OPE method at the quark-gluon level. To this end, one can insert the interpolating current J µ (x) (2) into the correlation function (1), contract the relevant quark fields via Wick's theorem and obtain where S (c) (x) and S (q) (x), q = u, d are the full charm-and up (down)-quark propagators, whose expressions are given in the Appendix A. Through dispersion relation, Π OP E µν (p) can be written as where ρ i (s) = 1 π ImΠ OP E i (s), i = 0, 1 are the spectral densities and the · · · represent other Lorentz structures. The spectral densities ρ 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 µν + pµpν p 2 ), and for the Lorentz-structure iF µν . 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 Zc , whose vale will be determined in section 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 = a In the next section, (22) 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).

III. NUMERICAL ANALYSIS
The input parameters needed in numerical analysis are presented in Table I. For the vacuum susceptibilities χ, κ and ξ, we take the values χ = −(3.15 ± 0.30)GeV −2 , κ = −0.2 and ξ = 0.4 determined in the detailed QCD sum rules analysis of the photon light-cone distribution amplitudes [22]. Besides these parameters, we should determine the working intervals of the threshold parameter s 0 Zc 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, In Fig.1(a), we compare the various OPE contributions as functions of M 2 B with s 0 Zc = 4.6GeV. From it one can see that except the quark condensate qq , other vacuum condensates are much smaller than the perturbative term. So the OPE series are under control. Fig.1(b) shows RP and RH varying with M 2 B at s 0 Zc = 4.6GeV. The figure shows that the requirement RP ≥ 50% (RP ≥ 40%) gives M 2 B ≤ 3.3GeV 2 (M 2 B ≤ 3.7GeV 2 ) and RH = 5% at M 2 B = 1.25GeV 2 . From Fig.2(a), we know that the sum rule for the mass m Zc depends strongly on the Borel parameter M 2 B as M 2 B ≤ 3GeV 2 . Along with the criterions of pole dominance, this fact confines M 2 B from 3GeV 2 to 3.7GeV 2 . In the analysis, we take RP ≥ 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 Zc as 4.5 2 GeV 2 ≤ s 0 Zc ≤ 4.7 2 GeV 2 . As a result, we can reliably read the value of the mass, m Zc = 3.97 ± 0.12GeV, in agreement with the experimental value m exp Zc = 3899.0 ± 3.6 ± 4.9MeV. In Fig.3 where µ N is the nucleon magneton. In Ref. [19], the author gave µ Zc = 0.47 +0.27 −0.22 µ N assuming Z c (3900) as an axialvector tetraquark state by the same method used in this article. In Ref. [23], µ Zc = 0.67 ± 0.32µ N was predicted using light-cone sum rule under the axialvector-tetraquark assumption. The theoretical predictions can be confronted to the experimental data in the future and give important information about the inner structure of the Z c (3900) state.

IV. CONCLUSION
In this article, we tentatively assign the 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 the 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 Zc = 3.97 ± 0.12GeV in agreement with the experimental value m exp Zc = 3899.0 ± 3.6 ± 4.9MeV, λ Zc = 2.1 ± 0.4 × 10 −2 GeV 5 and µ Zc = 0.19 +0.04 −0.01 µ N . The prediction can be confronted to the experimental data in the future and give important information about the inner structure of the Z c (3900) state.