The partial decay width of $P_{c}(4312)$ as a $\bar{D}\Sigma_{c}$ molecular state

In the present work, we investigate the partial decay width of $P_{c}(4312)\rightarrow J/\psi p$ under the assumption that $P_{c}(4312)$ is a $\bar{D}\Sigma_{c}$ molecular state via QCD sum rule method.Firstly,we calculate the spectral parameters,mass and the residual of mass,which are two of the input parameters as we investigate the strong decay form factors in the next step.After obtaining the numerical values of the two form factors,we finally give the partial decay width of $P_{c}(4312)\rightarrow J/\psi p$ which are compatible with the total width of $P_{c}(4312)$.Our results suggest that it is reasonable to assign $P_{c}$ to be a $\bar{D}\Sigma_{c}$ molecular state.


I. INTRODUCTION
As early as the birth of the quark model,the concept of the mutliquark states appeared,whose quark substructure may be qqqq,qqqqq and so on.There have been many theoretical and experimental study on the multiquark states since the observation of X(3872) in 2003 by the Bell Collaboration (see review articles [1] for details).
Recently,a new pentaquark state P c (4312) was discovered by the LHCb Collaboration in the J/ψp invariant mass spectrum of the Λ b → J/ψpK decay [21].Triggered by this observation,there are many theoretical investigation on the properties of P c (4312) [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].However, to gain a deep understanding on their nature and substructure,which are still not certain yet,it is necessary to do more experimental and theoretical investigations which may help us to learn more about their properties.Studying their possible decay channels may provide valuable insights in this respect.
In this article,we study the spectral parameters and strong decay property of P c (4312) in the QCD sum rule method promoted by M.A.Shifman,A.I.Vainshtein and V.I.Zakharov in 1979 [40].The basic idea of QCD sum rule is that we can calculate the correlation function of the interpolating currents of hadrons we are interested in from both phenomenological and QCD sides,then match the two representation and extract the physical quantities of the considered hadron.One of the most important steps during the calculation is making a Borel transform which can simultaneously improve the convergence of OPE on the QCD side and suppress the contributions of higher and continuum states.The QCD sum rule method has extensively been used to investigate the X,Y,Z states(see the review article [41] for the detail).For the ground pentaquark states,we also can use this method to study their relevant properties.In fact,there are some related works on the pentaquark states in the QCD sum rule method [22,38,[42][43][44][45] based on the menson-baryon molecular configuration assumption and [13] assuming the diquark-diquark-antiquark substructure.
The rest of the paper is organized as follows.In section II,we give the sum rules for the spectral parameters and the strong decay form factors of P c (4312).Section III is devoted to the numerical analysis,the partial decay width and a short summary. In this section,we first give the spectral parameters of P c (4312),which are the input parameters of the calculation of the the form factors.
In the QCD sum rule method,the starting point of the calculation of the spectral parameters,mass and residual of mass,is the 2-point correlation function: where J(x) is the interpolating current of P c (4312) viewed as (DΣ c ) molecular states in the present work.According to ref. [43],J(x) can take the form where T denotes the matrix transposition,C means charge conjugation,and i,j,k are color indices. By inserting into the correlation function Π(p) a full set of relevant states having the same quantum numbers as J(x),we get the phenomenological representation of Π(p) in terms of the hadronic parameter, where m Pc is the hadronic mass, λ Pc is defined as 0 | J(0) | P c (p, s) = λ Pc u(p, s) and s Pc 0 is the threshold parameter.
On the other hand, Π(p) can be calculated from QCD via OPE method in terms of the quark propagators.To this end,we insert the interpolating current J(x) into the correlation function1,contract the relevant quark fields and find with S c (x) and S u(d) (x) are the charm-and up(down)-quark propagator,which are given by for light quarks, for heavy quarks,where t a = λ a 2 and λ a are the Gell-Mann matrix,g s is the strong coupling constant,and i, j are color indices.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 are the spectral densities.We will give the explicit expression of ρ 1 (s) up to dimension-9 and α s order in the following.
Matching the phenomenological and QCD representations,using the quark-hadron duality and making a Borel transform,we obtain the sum rules, and whereM 2 B is the Borel parameter.To get the sum rules for the mass and the residual of mass,taking derivative of Eqs. (8) and (9) and Substituting the obtained value of the mass into Eqs. (8) or (9),we can give the value of the residual of mass. Now,we give the explicit expression of the spectral density, where a max = With the above results about the spectral parameters of P c (4312),we now turn to the calculation of the form factors of the strong decay P c (4312) → J/ψp.To this end,we begin with the following 3-point correlation function, where are the interpolating currents of proton and J/ψ respectively.The interpolating currents take the following form, In order to get the physical representation of the 3-point correlation function (21),we insert complete sets of states having the same quantum numbers with the interpolating currents and use the following definitions, where f J/ψ and ǫ µ (q) are the decay constant and polarization vector of the J/ψ state,λ N and u N (p ′ ) are the residual and spinor of the proton,and f 1 and f 2 are the strong decay form factors,respectively.Finally,we obtain the phenomenological side of the sum rules, where we only remain the two Lorentz structures we are interested in, p ′ γ 5 q µ and p ′ γ µ γ 5 .
On the theoretical side,inserting the interpolating currents into the 3-point correlation function and contracting the quark fields,we obtain the following representation of the correlation function, where P 2 = −p 2 . Following the same steps as the two-pint case,we can get the sum rules for the form factors f 1 (P 2 ) and f 2 (P 2 ), where M 2 B 1 and M 2 B 2 are the Borel parameters corresponding to q 2 and p ′2 respectively.The coefficients Γ 1 (P 2 ) and Γ 2 (P 2 ) can be written as via double dispersion relation, where ρ i (P 2 , s, u), i = 1, 2 are the spectral densities whose explicit expressions,up to dimension-6 and α s order,are with and where δ(u) is the Dirac δ-function.

III. NUMERICAL ANALYSIS AND THE PARTIAL DECAY WIDTH
The QCD sum rules for the spectral parameters and the strong decay form factors contain some input parameters which are required to obtain the numerical values of these quantities.We present them in TableI.Beside these input parameters,there are a few auxiliary parameters introduced during the calculations:the continuum thresholds and the Borel parameters.These are not physical quantities, hence the physical observable should be approximately insensitive to them. Therefore, we look for working regions of these parameters such that the dependence of the mass on these parameters are weak.The continuum thresholds are related to the square of the first exited states having the same quantum numbers as the interpolating currents, while the Borel parameters are 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.
Firstly,we analyze the spectral parameters:mass and the residual of mass.In Fig.1,we compare the various OPE contributions as functions of M 2 B with s Pc 0 = 4.8GeV and represent the ratio of the pole and continuum contribution,the the ratio of the qq 3 term and total contribution varying with M 2 B with s Pc 0 = 4.8GeV .From the figure,we can see that it is needed to limit M 2 B from 2.3GeV 2 to 2.7GeV 2 in order to simultaneously satisfy the requirements of pole dominance at the phenomenological side(the pole contribution is bigger than the continuum contribution) and convergence of the operator product expan-  sion(the contribution from the highest-dimensional term is about 30 percent of the total OPE series). After determining the interval of M 2 B ,we can turn to the analysis of the mass and the residual of the mass.The results are represented in Fig.2,from which it is obvious that the sum rules for the mass and residual vary weakly with the continuum threshold parameter s Pc 0 Borel parameter M 2 B in the interval determined above.As a result,we can reliably read the values of the mass and residual:m Pc = 4.1 ± 0.1GeV 2 which is agreement with the exprimental value m Pc = 4311.9 ± 0.7 6.8 −0.6 MeV [21] considering the accuracy of QCD sum rule method and λ Pc = (1.4 ± 0.2) × 10 −3 GeV 6 . Now it is time to study the form factors f 1 (P 2 ) and f 2 (P 2 ) of the strong decay P c (4312) → J/ψp.Simillar to the 2-point case,we should first determine the allowed ranges of the Borel parameters M 2 B 1 and M 2 B 2 .To this end,the various OPE contributions of the Lorentz structure p ′ γ 5 q µ as functions of the Borel parameters M 2   Fig.3(c) and (d) represents the ratio of the pole and total contributions and the ratio of the qq 2 -term contribution and the total OPE series for the same Lorentz structure.Visually,when 1GeV 2 ≤ M 2 B 1 ≤ 3.8GeV 2 and 1GeV 2 ≤ M 2 B 2 ≤ 1.7GeV 2 ,both of the criteria:pole dominance at the phenomenological side and convergence of the operator product expansion can be met.Similar analysis can be done for the structure p ′ γ µ γ 5 .Finnaly,we obtain the following intervals of the Borel parameters:1.6GeV 2 ≤ M 2 B 1 ≤ 3.3GeV 2 and 1GeV 2 ≤ M 2 B 2 ≤ 1.7GeV 2 .In Fig.4,the numerical results of the sum rules for the form factors f 1 and f 2 are presented.It is obvious that our sum rules for the form factors depend weakly on the Borel parameters and threshold parameters,indicating that our results are reliable.
With the working intervals of the auxiliary parameters and other input parameters,we can now obtain the dependence of the form factors on the P 2 .Because of the limitation of the value of P 2 on the OPE side,it is necessary to make a fit in order to obtain the physical values of the form factors.In the present work,we apply the following fit function where f 0 ,a and b are the fit parameters having the values presented in TableII.To show the consistency of the fit function with the QCD sum rule results,we give the dependencies of the form factors on P 2 obtained from both sum rules and fit results in Fig.5,which indicates that our fit function represents QCD sum rule results well in the region where the sum rule results are reliable.Substituting P 2 = −m 2 Pc in the fit functions we can obtain the values of the form factors which are presented in TableIII.
Substituting the values of the parameters involved in the above formula,we find Γ(P c (4312) → J/ψp) = 6.5 +3.7 −0.9 (MeV ), (34) which are compatible with the total width of P c (4312) reported in [21] supporting the assignment of P c (4312) as theDΣ c molecular state.
To sum up,we study the partial decay width of P c (4312) → J/ψp under the assumption that P c (4312) is aDΣ c molecular state.Firstly,we calculate the spectral parameters,mass and the residual of mass,which are two of the input parameters as we investigate the strong decay form factors in the next step.After obtaining the numerical values of the two form factors,we finally give the partial decay width of P c (4312) → J/ψp which are compatible with the total width of P c (4312).Our results suggest that it is reasonable to assign P c to be aDΣ c molecular state.