Study of the rare decay $J/\psi\rightarrow e^+e^- \phi$

We study the decay process of $J/\psi\rightarrow e^+e^- \phi$ where the relatively clean electromagnetic (EM) transitions appear at leading order at tree level while the hadronic contributions only emerge via hadronic loop transitions. We include the low-lying scalar $f_0(980)$ and pseudoscalar $\eta/\eta'$ as the dominant contributions in the evaluation of the hadronic loop contributions. It is found that the hadronic effects are negligible comparing with the EM contributions. The decay width of $J/\psi\rightarrow e^+e^- \phi$ is determined to be about $2.12\times 10^{-6}$ keV if there is no any other leading mechanism contributing, and this result will be tested by the BESIII experiment with a large data sample of $J/\psi$.


I. INTRODUCTION
One of the advantages of high-intensity experiments is that by precision measurement one can testify the standard theory to high accuracy and then look for traces of new physics beyond the standard model (BSM).In the relatively low energy regime it is believed that the new physics effects may manifest themselves in rare decays because the standard model contributions are highly suppressed in these channels, so that if the data exhibit an anomaly, it would hint a possible contribution coming from the BSM physics.Even though such results may not pin down what kind of new physics plays a role, it may offer valuable information about the BSM and complement the search for BSM phenomena at high energy frontier experiment such as LHC.For such a purpose the BEPCII/BESIII [1] and BELLE-II [2,3] experiments would provide large databases for ψ and Υ families and enable us to investigate the rare processes which may expose traces of new physics BSM.
The process J/ψ → e + e − φ can be regarded as a rare decay process which can be measured by the BESIII Collaboration.The leading order (LO) contribution to this decay mode comes from the EM transition via J/ψ → e + e − γ * → e + e − φ where the conversion of γ * → φ is described by the vector meson dominance (VMD) transition [4].The hadronic contributions become sub-leading in this process via the hadronic meson loops where the c and c annihilation involves gluon exchanges and production of light flavored quarks at the quark-gluon level.This process should be complicated in terms of quark-gluon degrees of freedom.However, by knowing the radiative decays of J/ψ → γP and γS, φ → γP and φ → γS in experiment, where P and S denote the light-flavored pseudoscalar and scalar mesons, we can extract the relevant coupling vertices from experimental data and provide a reliable estimate of the hadronic contributions at hadronic level.We mention in advance that the hadronic contributions turn out to be smaller than the leading EM contribution and the most important intermediate hadronic contributions would come from the pseudoscalar or scalar mesons of which the masses are close to the final state φ meson.Notice that the intermediate states are produced by the hadronization of gluons.The isospin-1 states will be suppressed.Thus, the most possibly important intermediate states that are allowed would be pseudoscalar η and η ′ and scalar σ(500) and f 0 (980).Taking into account the small coupling for J/ψ → γσ(500) and φ → γσ(500) we can safely neglect the contribution from the intermediate σ(500) in this leading order calculation.
The paper is organized as follows: in Sec.II and III, we derive the formulas for the contributions from the LO EM process and hadronic meson loops, respectively.In Sec.IV, numerical results and analysis are presented.A brief summary will also be given there.
II.THE EM CONTRIBUTION TO J/ψ → e + e − φ It is noted that the direct photon radiation of J/ψ → γφ is strictly forbidden by the C-parity conservation unless there is BSM new physics [5] to break the rule.In fact, such a radiative decay has never been experimentally observed so far.But, when the photon is virtual and later converts into an electron-positron pair, there is no restriction in principle and J/ψ → e + e − φ can be measured.
We first compute the leading EM contribution to J/ψ → e + e − φ for which the corresponding Feynman diagrams are shown in Fig. 1.
In this process the couplings of γ − J/ψ and γ − φ are effective ones which we treat them at leading order in the framework of vector meson dominance (VMD).Those effective vector currents can be expressed as the followings for the γ − J/ψ and γ − φ couplings, respectively, where ε ν ψ and ε µ φ are the polarization vectors of J/ψ and φ, respectively, and g ψγ and g φγ are the corresponding couplings for J/ψ and φ to a virtual photon, respectively, and they can be evaluated by data for J/ψ, and φ → e + e − [6].
The Feynman amplitude corresponding to Fig. 1 can then be obtained III. THE HADRONIC CONTRIBUTION TO THE PROCESS J/ψ → e + e − φ VIA

MESON LOOPS
The major hadronic contributions arise from the hadronic meson loops in J/ψ → e + e − φ as shown by Fig. 2. Such non-perturbation effects can hardly be evaluated from the first principle.However, there are experimental data for J/ψ → γf 0 (980), γη and γη ′ as well as for φ → γf 0 (980), γη and γη ′ .We can then extract the vertex couplings from experimental data and estimate the leading hadronic contributions.The effective couplings for vector meson radiative decays into scalar meson or pseudoscalar mesons are well established in the literature [7].The following couplings are adopted for which the Lorentz gauge invariance is automatically fulfilled: where ε γ is the polarization vector of the photon and the couplings can be extracted from either V → γS or V → γP with S and P denote the final state scalar or pseudoscalar meson, respectively.In principle, one should include all the intermediate scalar and pseudoscalar mesons and even higher spin states in the estimate of the hadronic contributions based on the quark-hadron duality argument [8].However, one notices that the largest contributions are from the low-lying intermediate states with relatively large couplings.Thus, as an estimate of the leading hadronic contribution we only consider the low-lying scalar and pseudoscalar states in the meson loops.We also note that contributions from the intermediate η c are also possible which, however, is negligibly small due to the extremely small exclusive decay of η c → γφ.So it is a reasonable approximation that we only consider the scalar f 0 (980) and pseudoscalars η and η ′ as the leading contributions to the hadronic loops.
The Feynman amplitude induced by the loop involving η is where the coupling constants g ψη and g φη are determined by the experimental data for J/ψ → γη and φ → γη [6], respectively.For η ′ the amplitude has the same expression as that for the η loop.

IV. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we present the numerical results along with all the necessary inputs which are adopted from the Particle Data Group [6].The following mass values are adopted in the calculation, i.e. m ψ = 3.097 GeV, m φ = 1.020GeV, m f 0 (980) = 0.980 GeV, m η = 0.548 GeV, and m η ′ = 0.958 GeV.
For the vector meson and photon coupling in the VMD it can be extracted by the vector meson leptonic decay, i.e.
The experimental data for J/ψ → γf 0 (980) are not available.However, we can estimate the effective coupling constant g ψf 0 with the data for J/ψ → ωf 0 (980) and φf 0 (980) in the VMD.Therefore, the coupling can be expressed as where V = ω, φ as the vector meson fields contributing to J/ψ → γf 0 (980).The strong coupling g J/ψf 0 V is estimated by where q is the three-vector momentum of the final state mesons in the c.m. frame of J/ψ.
For the couplings arising from the pseudoscalar meson loops g ψη and g ψη ′ are obtained from the J/ψ radiative decay ψ → γη and γη ′ as  The hadronic loops are calculated by LoopTools [9] where cut-offs for both ultra-violet (UV) and infra-red (IR) divergence are embedded.In Tab.I the calculated partial width for J/ψ → e + e − φ is listed in comparison with the exclusive contributions from the leading EM process and the subleading hadronic loop transitions.Since the hadronic loop contributions are much smaller than the leading EM contribution, the sum of all the amplitude is saturated by the EM process.It is interesting to note that the results show that the decay of J/ψ → e + e − φ is highly suppressed at leading order and dominated by the EM contributions.In contrast, the hadronic contributions are even further suppressed by at least three orders of magnitude.It makes this process an ideal place where the SM background is rather small.Thus, it may serve as a potential process to probe effects due to BSM mechanisms.We point out such an advantage of this process but would not continue in this direction in this work to discuss expectations arising from BSM mechanism.
In summary, we investigate the decay mechanisms for J/ψ → e + e − φ which appears to be a rare decay process in the SM.In particular, the leading contribution is from the EM transition at tree level of which the partial decay width is about 2.12 × 10 −6 keV.The subleading contributions from the hadronic meson loops are found negligibly small and such an observation may make this decay process a useful probe for the search for BSM sources which can contribute to this process sizeably.

TABLE I :
The decay widths and branching ratios of J/ψ → e + e − φ contributed from the EM and hadronic processes.