X(3860) production in association with J/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi $$\end{document} via e+e-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{+}e^{-}$$\end{document} annihilation at Belle

In this paper, we study the X(3860) production associated with J/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi $$\end{document} via e+e-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{+}e^{-}$$\end{document} annihilation at the next-to-leading-order (NLO) accuracy in αs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _s$$\end{document}, within the nonrelativistic QCD (NRQCD) framework. With the hypothesis of JX(3860)PC=0++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{PC}_{X(3860)}=0^{++}$$\end{document}, the predictions of σe+e-→J/ψ+X(3860)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{e^{+}e^{-} \rightarrow J/\psi +X(3860)}$$\end{document} agree well with the Belle measurements, whereas the results following the 2++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{++}$$\end{document} assignment significantly undershoot the data. This is consistent with the Belle’s conclusion that the 0++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0^{++}$$\end{document} hypothesis is favored over the 2++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{++}$$\end{document} hypothesis for X(3860). Despite fitting the data of the total cross section, the NRQCD predictions seem to be incompatible with the measured J/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi $$\end{document} angular distributions. We simultaneously calculate the cross section of e+e-→J/ψ+X(3940)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{+}e^{-} \rightarrow J/\psi +X(3940)$$\end{document} under the assumption of JX(3940)PC=0-+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J^{PC}_{X(3940)}=0^{-+}$$\end{document}, discovering the consistency of the NRQCD predictions with the light-cone results as well as the experiment.

However, this identification has some problems.As the candidate for χ c0 (2P ), the X(3915) is expected to dominantly decay into D D (D = D 0 or D + ), which, however, has not been observed.On the contrary, the decay mode X(3915) → J/ψω that should largely be suppressed by Okubo-Zweig-Liuka [6] is experimentally hunted.In addition, the measured X(3915) width, 20 ± 5 Mev, is much smaller than the theoretical expectations [7].
In 2015, by reanalyzing the BABAR data [4], Zhou et al. indicated that the assignment of J P C X(3915) = 2 ++ is also permitted, provided the λ = 2 assumption is abandoned [8].In view of these points, the X(3915) is no longer identified as the χ c0 (2P ) in the 2016 PDG [9].
In 2017, by analyzing the process of e + e − → J/ψD D based on a 980 fb −1 data sample, the Belle collaboration observed a new charmoniumlike state X(3860) with a significance of 6.5σ [10].The Belle collaboration concluded that the J P C = 0 ++ hypothesis is favored over the 2 ++ hypothesis for X(3860).The X(3860) seems to be a better candidate for the χ c0 (2P ) charmonium state than the X(3915) [10][11][12].For instance, the mass of X(3860) is measured to be (3862 +26+40 −32−13 ) MeV/c 2 , close to potential model expectations of m χ c0 (2P ) ; the observed decay mode of X(3860) → D D coincides with the theoretical expectations that the charmonium state above the D D threshold should primarily decay to D D; moreover, the measured X(3860) width is (201 +154+88 −67−82 ) MeV, resembling the theoretical prediction of Γ = (221 ± 19) MeV [7].From these perspectives, the X(3860) has been assigned to be χ c0 (2P ) by the recent PDG table.
Inspired by the success of NRQCD in describing the double-charmonium production through e + e − annihilation, in this paper we will use the NRQCD factorization to study the process of e + e − → J/ψ + X(3860) at the next-to-leading-order (NLO) QCD accuracy.
Meanwhile, we will also provide the NLO NRQCD predictions of σ e + e − →J/ψ+X(3940) , so as to compare with the existing results given by the light-cone formalism.
The rest of the paper is organized as follows: In Sec.II, we give a description on the calculation formalism.In Sec.III, the phenomenological results and discussions are presented.Section IV is reserved as a summary.
can further be expressed as where L µν and H µν are the leptonic and hadronic tensors, respectively, and dΠ 2 is the standard two-body phase space. with where s = (p e + + p e − ) 2 , p J/ψ is the three momenta of J/ψ, and θ is the angle between p J/ψ and the spatial momentum of e − (or e + ) in the e + e − center-of-mass frame.
(i) for J/ψ + η c (3S), (ii) for J/ψ + χ cJ (2P ), In the NLO calculations, there are 20 counter-term and 60 one-loop diagrams, as typically shown in Fig. 1(b) and Figs.1(c)-1(f), respectively.We utilize the dimensional regularization with D = 4 − 2 to isolate the ultraviolet (UV) and infrared (IR) divergences.The on-mass-shell (OS) scheme is employed to set the renormalization constants for the c-quark mass (Z m ) and heavy-quark filed (Z 2 ); the minimal-subtraction (M S) scheme is adopted for the QCD-gauge coupling (Z g ) and the gluon filed Z 3 .The renormalization constants are taken as [16] δZ where is an overall factor, γ E is the Euler's constant, and After including the QCD corrections, we acquire the NLO-level C 1 and C 2 , which can generally be written as The coefficients a 1(2) , b 1(2) , and c 1(2) , which are functions of r and m c , are summarized in the Appendix (A1-A10).With the further implementation of Eq. ( 6), one could straightforward obtain the NLO-level A, B, and σ.
In our calculations, we use FeynArts [27] to generate all the involved Feynman diagrams and the corresponding analytical amplitudes.Then the package FeynCalc [28] is applied to tackle the traces of the γ and color matrices such that the hard scattering amplitudes are transformed into expressions with loop integrals.In the next step, we utilize our self-written Mathematica codes with the implementations of Apart [29] and FIRE [30] to reduce these loop integrals to a set of irreducible Master Integrals, which could be numerically evaluated by using the package LoopTools [31].

III. PHENOMENOLOGICAL RESULTS
In the calculations, M J/ψ M X(3860) (or M J/ψ M X(3940) ) = 2m c is implicitly adopted to ensure the gauge invariance of the hard scattering amplitude.We choose two typical values for charm-quark mass: 1) m c = 1.4 GeV, which corresponds to the one-loop charm quark pole mass [33]; 2) m c is identical to the average of and m X(3860,3940) 2 , i.e., m c 1.75 GeV.
The values of wave functions at the origin are correspondingly taken as: for m c = 1.4 GeV, to the BT potential model [34,35]; as for the large charm quark mass of m c = 1.75 GeV, the wave functions are given by the Cornell Potential [35], i.e., |R 1S (0 |R 3S (0)| 2 = 0.791GeV 3 , and |R 2P (0)| 2 = 0.186GeV 5 .α = 1/130.9[33] and the two-loop α s running coupling constant is employed.We summarize the NRQCD predictions of σ e + e − →J/ψ+X(3860) in Tab.I. 1 Inspecting the data, one can perceive : 1) The QCD corrections provide significant contributions, especially for the 0 ++ hypothesis.With the inclusion of the high-order terms, the NLO results show a more steady dependence on the renormalization scale, which can clearly be seen by the uncertainties caused by the variation of µ r .
2) The NLO NRQCD predictions built on the 0 ++ hypothesis agree well with the Belle's measurements within uncertainties; however, the theoretical results provided by the 2 ++ hypothesis significantly fall short of the data.This is consistent with the Belle's conclusion that the 0 ++ hypothesis is favored over the 2 ++ hypothesis for X(3860).In Figs. 2 and 3, we compare the measurements of the J/ψ angular distributions with the theoretical results of NLO in α s .It appears that the NRQCD predictions could not give reasonable explanations for the experiment.Considering the insensitivity of dσ d cos θ 1 σ on µ r (as exemplified by the lines for different µ r in Figs. 2 and 3) and its independence on the choice of the LDMEs, the discrepancies concerning the J/ψ angular distributions seem to hardly be cured, at least at the NLO accuracy.
At last, by identifying X(3940) as η c (3S) [24], we provide the comparisons of the NRQCD predictions of σ e + e − →J/ψ+X(3940) with the Belle's data in Tab.II.One can observe that the NLO NRQCD results, which exhibit a good consistence with the light-cone results (i.e., 11 ± 3 fb) [24], are in good agreement with the measurements within the respective theory errors.

IV. SUMMARY
In this manuscript, we applied the NRQCD factorization to study the production of X(3860) plus J/ψ via e + e − annihilation at the NLO QCD accuracy.We found, by assuming J P C X(3860) = 0 ++ , that the NRQCD predictions of σ e + e − →J/ψ+X(3860) match the Belle's measurements at √ s = Υ(4S, 5S) adequately; however, the results given by the 2 ++ hypothesis largely undershoot the data, which coincide with the Belle's conclusion that the 0 ++ hypothesis is favored over the 2 ++ hypothesis for X(3860).In spite of the agreement of the total cross sections, the NRQCD predictions seem hardly to describe the Belle-measured represents the number of the active-quark flavors; n L (= 3) and n H (= 1) denote the number of the light-and heavyquark flavors, respectively.In SU(3), the color factors are given by T F = 1 2 , C F = 4 3 , and C A = 3.

TABLE II :
Comparisons of the predicted total cross sections (in unit: fb) with Belle's measurements of σ e + e − →J/ψ+X(3940) .The theoretical uncertainties are caused by the variation of µ r in [2m c ,