Study of the ${\eta}^{\prime}$ ${\to}$ $Ve^{+}e^{-}$ decay with hidden local symmetry model

Within the hidden local symmetry framework, the Dalitz decay ${\eta}^{\prime}$ ${\to}$ $Ve^{+}e^{-}$ is studied with the vector meson dominance model. It is found that the partial width ${\Gamma}({\eta}^{\prime}{\to}{\omega}e^{+}e^{-})$ ${\approx}$ $40$ eV and branching ratio ${\cal B}({\eta}^{\prime}{\to}{\omega}e^{+}e^{-})$ ${\approx}$ $2{\times}10^{-4}$, and ${\Gamma}({\eta}^{\prime}{\to}{\rho}e^{+}e^{-})$ ${\approx}$ $10{\Gamma}({\eta}^{\prime}{\to}{\omega}e^{+}e^{-})$ and ${\cal B}({\eta}^{\prime}{\to}{\rho}e^{+}e^{-})$ ${\approx}$ $10{\cal B}({\eta}^{\prime}{\to}{\omega}e^{+}e^{-})$. The maximum position of the dilepton distribution is $m_{e^{+}e^{-}}$ ${\approx}$ $1.33$ MeV. These decays are measurable with the advent of high statistics ${\eta}^{\prime}$ experiments.


I. INTRODUCTION
The η and η ′ mesons play an important role in understanding of the low energy QCD.
They provide a valuable place for studying three distinct symmetry breaking patterns simultaneously (the explicit symmetry breaking due to finite quark mass, dynamical spontaneous symmetry breaking, and the axial U(1) anomaly) [1]. In addition, they are the eigenstates of G, C, P , namely, I G J P C = 0 + 0 −+ [2], so their decays are also suitable to test the conservation or breaking of these discrete symmetries (such as charge conjugation invariance 1 ) in the strong and electromagnetic interactions [3][4][5].
The η ′ → V e + e − decay (where V = ρ 0 , ω) 2 is interesting in several respects. (1) The η ′ is the most esoteric meson of the light pseudoscalar nonet because it is closely related to the axial U(1) anomaly of the strong interactions [7], which is manifested in its heaviest mass and largest width among the pseudoscalar nonet [2]. The effect of QCD anomaly should be manifested in the η ′ decay modes, besides the η and π 0 decay modes 3 . From the η ′ → V e + e − decay, we may get some phenomenological implication of the anomaly at low energy.
(2) Many model-dependent approaches of low energy mesonic interaction, e.g. whether the vector meson dominance (VMD) is valid in nature, especially, the applicability of the chiral perturbation theory, can be tested via η ′ decays. On one hand, the influence of the light vector mesons on the η ′ → V transition form factor and branching ratios of η ′ decay can be investigated; On the other hand, the electron-positron invariant mass distribution will provide us with some information about the intrinsic structure of η ′ meson and momentum dependence of the transition form factor. (3) Experimental information on the η ′ decays are relatively scarce. The measurement about η ′ → V e + e − decay is not given by the Particle Data Group (PDG) [2] for the moment. However, this situation will be ameliorated with the advent of high statistics η ′ experiments, such as WASA at COSY, Crystal Ball at MAMI, BESIII at BEPCII, KLOE-2 at DAΦNE, and so on [8]. There is a necessity to provide a consistent and uniform theoretical description for the decay η ′ → V e + e − . 1 There are 19 tests of C invariance listed in [2], including eight of the η decays and six of the η ′ decays. 2 The η ′ → φ transition is forbidden by the kinematic constrain. The η ′ → K 0 * e + e − decay is a weak process, and the permitted weak decays of η ′ mesons in the standard model are expected to occur at the level of 10 −11 and below [6]. 3 It is well known that all possible strong and electromagnetic decays of η are highly suppressed by various constraints (such as P , C, G parity) [3,4]. This paper is organized as follows. In the next section we present the effective lagrangian and amplitude of the η ′ → V e + e − decay. Our numerical estimation and the dilepton spectra distribution of the η ′ → V e + e − decay are given in Sec.III. The last section is summary.

II. THEORETICAL FRAMEWORK AND DECAY AMPLITUDES
The hidden local symmetry (HLS) model [9] provides a convenient and constraining QCD-inspired framework for studying the phenomenology of light mesons in low energy regime of strong interactions. In this approach, the pseudoscalar mesons are the Nambu-Goldstone bosons, and the vector mesons are gauge bosons of a spontaneously broken hidden local symmetry that generates their Higgs-Kibble masses. The anomalous sector (also called WZW [10] and FKTUY [11] lagrangian) based on HLS allows one to describe the coupling of the form AAP , AV P , V V P , AP P P , and V P P P , where A, V , P denote the electromagnetic field, vector meson, pseudoscalar, respectively. The explicit expression of the corresponding lagrangian can be found in [12,13], e.g. the triangle anomaly lagrangians can be written as where N c = 3 is the number of color; e 2 = 4πα; g is the universal vector meson coupling constant; f π ≈ 92.4 MeV is the pion decay constant [13]; Q = diag(2/3, −1/3, −1/3) is the quark charge matrix; P is the matrix of pseudoscalar meson -the Goldstone bosons associated with the spontaneous breakdown of G global = U(3) L ⊗U(3) R ; and V is the matrix of vector meson -the gauge bosons of the hidden local U(3) V symmetry [14], The triangle anomaly lagrangians Eqs.(1-3) depend on two parameters c 3 and c 4 . In our calculation, c 3 = c 4 = 1 [12,13] 4 , so one can obtain the same predictions as the VMD models in the triangle anomalous sector, i.e. L AAP and L AV P vanish, and photons can only couple to pseudoscalar mesons via the V -γ transitions.
Phenomenologically, the Dalitz decay η ′ → V e + e − is regard as a sequential two-body In the triangle anomaly lagrangians, terms of L AV P and L V V P contribute to decay η ′ → V γ.The decay amplitude can be written as where ǫ γ (ǫ V ) and p γ (p V ) are the polarization vector and four-momentum of on-shell photon (vector meson ρ 0 and ω), respectively; the coefficient C η ′ V γ contains the information of the η ′ → V mesonic transition form factor, with the singlet η 0 and octet η 8 pseudoscalar meson decay constant f 0 ≃ 1.04f π and f 8 ≃ 1.30f π [15]. Using the form factor |f ωη ′ | = (0.431±0.020) GeV −1 for η ′ → ωγ decay [17], we can get F η ′ ωγ ≃ 3/4. And we will take the approximation F η ′ ργ ≈ F η ′ ωγ for the η ′ → ργ ( * ) decay in our calculation. By the same token, diagrams contributing to the η ′ → V γ * → V e + e − decay are shown in Fig.1. The corresponding decay amplitude can be written as Compared to the two-body decay amplitude Eq. (7), the polarization of the off-shell photon turned into the electromagnetic current j ν =ū e − (−ieγ ν )υ e + , and the VMD factor is dependent on the invariant momentum p 2 γ * = m 2 γ * = (p e + + p e − ) 2 = m 2 e + e − . 4 As the statement given in ref. [12], the relation c 3 = c 4 = 1 cannot be considered as firmly established without a fully comprehensive fit of all relevant measurements. For example, various fits with different data sets are presented in ref. [12], where one global fit with relatively large probability using ND and CMD data sample prefers c 3 = 0.927±0.010.

III. DECAY RATE AND DISCUSSION
The partial width of the two-body η ′ → V γ decay is The differential width of the three-body η ′ → V e + e − decay is where (| p * e + |, Ω * e + ) is the momentum of lepton e + in the rest frame of off-shell photon γ * ; and (| p V |, Ω V ) is the momentum of vector meson in the rest frame of the decaying η ′ meson, where λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2bc − 2ac. Finally, the differential width in terms of the electron-positron invariant mass m e + e − = m γ * can be written as With the input parameters collected in TABLE. I (if not specified explicitly, their central values are taken as the default input), we can get the integrated partial width and the corresponding branching ratio of the two-body electromagnetic radiative η ′ decays as follows Γ(η ′ →ωγ) = 5.426±0.005 m η ′ ±0.010 mω ±0.021 g keV, where the uncertainties come from m η ′ , m V , g and Γ η ′ , respectively. It is clear that (1) there are two proportions, Γ(η ′ →ργ) ≈ 10Γ(η ′ →ωγ) and B(η ′ →ργ) ≈ 10B(η ′ →ωγ), due to the Eq. within one standard deviation.
Our results about η ′ → ρe + e − decay agree basically with the prediction of Γ(η ′ →π + π − e + e − ) = 431 +35 −62 eV and B(η ′ →π + π − e + e − ) = (2.13 +0.17 −0.31 )×10 −3 [18] within the uncertainties, and accord with the recent measurement B(η ′ →π + π − e + e − ) = (2.11±0.12±0.14)×10 −3 reported by BESIII [19] within one standard deviation, where almost all of the final states π + π − will probably come from the resonant ρ 0 meson. 5 In ref. [16], the multiplicative representation of the transition form factor is used to fit the data. That is to say, many vector and/or excited vector meson are taken into account. For example, the transition form factor F ωγπ is written as: where the satisfactory fit quality is achieved at the price of introducing much more resonance parameters related to the corresponding vector mesons. 6 In ref. [17], the form factor of process η ′ → ωγ is written as a function of six parameters (see Eq.(44) and Eq.(45) in ref. [17] for more detail), many inputs may cause large uncertainty.
Although no available measurement of the η ′ → V e + e − decays is enumerated by PDG so far [2], there is renewed experimental interest in η ′ decays with the advent of high statistics η ′ experiments. For example, some 10 5 fully reconstructed η ′ events per day can be reached with WASA at COSY [20]; Approximately 15×10 3 η ′ events per hour are expected with Crystal Ball at MAMI [21]; With one year's luminosity at J/ψ peak, some 60 million η ′ events could be collected by BESIII at BEPCII [22]; KLOE-2 at DAΦNE experiment expects to increase this sample up to about a few f b −1 integral luminosity per year within the next running [23]. We take BESIII as an example to estimate the production rate of η ′ → V e + e − decays. It is estimated that there are more than 5×10 6 η ′ sample, corresponding to the radiative decay J/ψ → γη ′ with some 10 9 J/ψ dataset accumulated at BESIII. Given the detection efficiency is about 17% [19], some 2000 η ′ → ρe + e − and some 200 η ′ → ωe + e − event could be observed at BESIII. The corresponding distribution of dilepton spectra are displayed in Fig.2. Our studies also show that (1) the influence of the mass and width of vector mesons on the normalization distribution of dΓ(η ′ →V e + e − )/dm e + e − is small 7 . (2) The maximum position in the distribution is near dilepton threshold 8 , i.e. m e + e − ≈ 1.33 MeV, the corresponding common momentum of vector mesons ω and ρ in the η ′ rest frame are 159.11 MeV and 164.94 MeV, respectively. This distinctive feature will be helpful to distinguish signals from backgrounds.

IV. SUMMARY
Based on the triangle anomaly HLS effective lagrangian, the interesting η ′ → V e + e − decay is studied with the VDM model. Our study show that the partial width Γ(η ′ →ωe + e − ) ≈ 40 7 Here, the maximum invariant mass of electron-positron pair is m e + e − max = (m η ′ − m V ), so In addition, the ratio Γ V /m V ≈ 1% for ω meson and ≈ 20% for ρ meson, so with m γ * max , the factor |1/(1 − for ω meson and ≈ 1.04 for ρ meson, that is to say, the effects of the mass and width of vector meson are less than 7% for ω meson and less than 4% for ρ meson. At the maximum position in the dilepton distribution m e + e − ≈ 1.33 MeV, the for ω meson and ≈ 96.44% for ρ meson, that is to say, the effects of the mass and width of vector meson are ignorable tiny for ω meson and less than 4% for ρ meson. 8 It is shown from Eq.(14) that the differential width is proportional to 1/m γ * , so the spectra lineshape tends to the maximum with m γ * moving to the dilepton threshold. In addition, because m γ * is far away from the mass of the vector resonance, there is no peak near the tail of the dilepton distribution and the shapeline is falling down smoothly. eV and branching ratio B(η ′ →ωe + e − ) ≈ 2×10 −4 , and Γ(η ′ →ρe + e − ) ≈ 10Γ(η ′ →ωe + e − ) and B(η ′ →ρe + e − ) ≈ 10B(η ′ →ωe + e − ), which are basically consistent with previous estimation and measurement within uncertainties. It is well known that the charged electron and positron are easily identified by the detector saturated with magnetic field. In addition, there is distinctive maximum position m e + e − ≈ 1.33 MeV. It can be expected that the era of accurate measurements on the η ′ → V e + e − decay is coming with the advent of high statistics η ′ experiments. mass of ρ meson m ρ = 775.49±0.34 MeV [2] width of η ′ meson Γ η ′ = 199±9 keV [2] width of ω meson Γ ω = 8.49±0.08 MeV [2] width of ρ meson Γ ρ = 149.1±0.8 MeV [2] vector coupling constant g = 5.568±0.011 [12]