The magnetic moment of P c ( 4312 ) as a D̄ Σ c molecular state

In this article, we tentatively assign the Pc(4312) to be a D̄Σc molecular state with quantum number JP = 12 − , and calculate its magnetic moment using the QCD sum rule method in external weak electromagnetic field. Starting with the two-point correlation function in external electromagnetic field and expanding it in power of the electromagnetic interaction Hamiltonian, we extract the magnetic moment from the linear response to the external electromagnetic field. The numerical value is μPc = 0.59 +0.10 −0.20.

In Ref. [29], we assumed the P c (4312) as aDΣ c molecular state with quantum number 1 2 − and studied the decay of P c (4312) to J/ψp and to η c p with the QCD sum rule method.
The QCD sum rule method [48] 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. [49][50][51], 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 [52][53][54][55], the decuplet baryons [56][57][58][59] and the ρ meson [60]. In Ref. [61,62], the authors calculated the magnetic moment of Z c (3900) as an axialvector tetraquark state and an axialvector molecular state, respectively. In this article, we extend this method to the investigation of the magnetic moment of the P c (4312) state viewed as aDΣ c molecular state with quantum number J P = 1 2 − . Electromagnetic and multipole moments are the major and meaningful parameters of hadrons. Analysis of the electromagnetic and multipole moments of the exotic states can help us get valuable knowledge about the electromagnetic properties of these states, the charge distributions inside them, their charge radius and geometric shapes and finally their internal substructures.
The rest of the paper is arranged as follows. In section II, we derive the sum rule for the magnetic moment of the P c (4312) 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 P c (4312) considered as aDΣ c molecular state with J P = 1 2 − with T denoting the matrix transposition on the Dirac spinor indices, C meaning charge conjugation, and a, b, c being color indices. In the present work, we shall consider the linear response term, Π µν (p)F µν , from which the magnetic moment will be extracted. 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 F µν , dimension-3 operator 0|qσ µν q|0 F , dimension-7 operators dimension-8 operators and so on. The new vacuum condensates induced by the external electromagnetic field F µν can be described by introducing new parameters, χ, κ and ξ, 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 d 4 yj em α (y)A α (y), where j em µ (y) is the electromagnetic current and A µ (y) is the electromagnetic four-vector. Inserting complete sets of relevant states with the same quantum numbers as the current operator J Pc (x) into the second term of (10) and carrying out involved integrals, one has where we make use of the following formulas, and with q = k ′ − k and Q 2 = −q 2 . λ Pc and u(k, s) are the pole residue and Dirac spinor of the P c (4312) state, respectively. The Lorentz invariant form factors F 1 (Q 2 ) and F 2 (Q 2 ) are related to the charge and magnetic form factors by The magnetic moment µ Pc is given by G M (0). On the other hand, Π(p) can be calculated theoretically via OPE method at the quarkgluon level. To this end, one can insert the interpolating current J Pc (x) (2) into the correlation function (1), contract the relevant quark fields by Wick's theorem and find 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 (p) can be written as where ρ i (s) = 1 π ImΠ OP E i (s), i = 1, 2, 3 are the spectral densities. We will choose the Lorentz structure i(p µ γ ν − p ν γ µ )F µν to obtain our sum rule for the magnetic moment µ Pc because of its better convergence. The spectral density ρ 3 (s) is given in the Appendix B.
Finally, we match the phenomenological side (11) and the QCD representation (16) for the Lorentz structure i( The higher resonances contain contributions from two parts, the pole-excited states transition and the excited-excited states transition induced by the external electromagnetic field. According to the quark-hadron duality, the later can be approximated by the QCD spectral density above some effective threshold s Pc 0 , whose vale will be determined in section III, where the constant a is introduced to parameterize the contributions of the pole-excited states transition. Subtracting the contributions of the excited-excited states transitions, In order to eliminate the subtractions, it is necessary to make a Borel transform which can also improve the convergence of the OPE series and suppress the contributions from the excited and continuum states. As a result, we have where A = a λ 2 Pc and M 2 B is the Borel parameter.

III. NUMERICAL ANALYSIS AND THE PARTIAL DECAY WIDTHS
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 [63]. Besides these parameters, we should determine the working intervals of the threshold parameter s 0 Pc and the Borel mass M 2 B in which the magnetic moment is stable. The continuum threshold is related to the square of the first exited states having the same quantum number as the interpolating field and we use the value determined in Ref. [29], 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, Firstly, we determine the working region of the M 2 B . In Fig.1(a), we compare the various OPE contributions as functions of M 2 B with s Pc 0 = 4.8GeV. From it one can see that the OPE has good convergence. Fig.1(b) shows RP and RH varying with M 2 B at s Pc 0 = 4.8GeV. The figure shows that the requirement RP ≥ 50% (RP ≥ 40%) gives  . In order to have a larger working interval of the Borel mass M 2 B , we require RP ≥ 40%. As a result, we limit M 2 B from 4GeV 2 to 4.8GeV 2 . The result is shown in Fig.2(b), from which we can read reliably the value of the magnetic moment, µ Pc = 0.59 +0. 10 −0.20 .

IV. CONCLUSION
In this article, we tentatively assign the P c (4312) to be aDΣ c molecular state with quantum number J P = 1 2 − , calculate its magnetic moment using the QCD sum rule method The full quark propagators are given as for light quarks, and for heavy quarks. In these expressions t a = λ a 2 and λ a are the Gell-Mann matrix, g s is the strong interaction coupling constant, and i, j are color indices, e Q(q) is the charge of the heavy (light) quark and F µν is the external electromagnetic field.