A theoretical analysis of the semileptonic decays $\eta^{(\prime)}\to\pi^0l^+l^-$ and $\eta^\prime\to\eta l^+l^-$

A complete theoretical analysis of the $C$-conserving semileptonic decays $\eta^{(\prime)}\to\pi^0l^+l^-$ and $\eta^\prime\to\eta l^+l^-$ ($l=e$ or $\mu$) is carried out within the framework of the Vector Meson Dominance (VMD) model. An existing phenomenological model is used to parametrise the VMD coupling constants and the associated numerical values are obtained from an optimisation fit to $V\to P\gamma$ and $P\to V\gamma$ radiative decays ($V=\rho^0$, $\omega$, $\phi$ and $P=\pi^0$, $\eta$, $\eta^{\prime}$). The decay widths and dilepton energy spectra for the two $\eta\to\pi^0l^+l^-$ processes obtained using this approach are compared and found to be in good agreement with other results available in the published literature. Theoretical predictions for the four $\eta^{\prime}\to\pi^0l^+l^-$ and $\eta^\prime\to\eta l^+l^-$ decay widths and dilepton energy spectra are calculated and presented for the first time in this work.

On the experimental front, new upper limits have recently been established by the WASA-at-COSY collaboration for the η → π 0 e + e − decay width [10]. This is a useful contribution, as the previous available empirical measurements date back to the 1970s which provided an upper limit for the relative branching ratio of the above process that was many orders of magnitude larger than the corresponding theoretical estimations at the time. In particular, Adlarson et al. [10] found from the analysis of a total of 3 × 10 7 events of the reaction pd → 3 He + η, with a recorded excess energy of Q = 59.8 MeV, that the results are consistent with no C-violating single-photon intermediate state event being recorded. Based on their analysis, the new upper limits Γ(η → π 0 e + e − )/Γ(η → π + π − π 0 ) < 3.28 × 10 −5 and Γ(η → π 0 e + e − )/Γ(η → all) < 7.5 × 10 −6 (CL = 90%) have been established for the C-violating η → π 0 γ * → π 0 e + e − decay. In addition, the WASA-at-COSY Collaboration is currently analysing additional data from the pp → ppη reaction collected over three periods in 2008, 2010 and 2012 which should put more stringent upper limits on the η → π 0 e + e − branching ratio. The experimental state of play is expected to be further improved in the near future with the advent of new experiments such as the REDTOP, which will focus on rare decays of the η and η mesons, providing increased sensitivity in the search for violations of SM symmetries by several orders of magnitude beyond the current experimental state of the art [11].
The present work is structured as follows: In section II, we present the detailed calculations for the decay widths associated to the six η ( ) → π 0 l + l − and η → ηl + l − processes. In section III, numerical results from theory for the decay widths and the corresponding dilepton energy spectra are presented and discussed for the six reactions. Some final remarks and conclusions are given in section IV.
In order to perform the calculations, one first needs to select an effective vertex that contains the appropriate interacting terms. The V Pγ interaction amplitude consistent with Lorentz, P, C and electromagnetic gauge invariance can be written as [13] M where g V Pγ is the coupling constant for the V Pγ transition involving on-shell photons, ε µναβ is the totally antisymmetric Levi-Civita tensor, ε (V ) and p V are the polarisation and 4-momentum vectors of the initial V , ε * (γ) and q are the corresponding ones for the final γ, andF V Pγ (q 2 ) ≡ F V Pγ (q 2 )/F V Pγ (0) is a normalised form factor to account for off-shell photons mediating the transition 5 . In addition to this, the usual QED vertex is used to describe the subsequent 2γ * → l + l − transition. Accordingly, there are six diagrams (two per vector meson) contributing to each one of the six semileptonic decay processes and the corresponding Feynman diagrams are shown in Fig. 1.
The invariant decay amplitude in momentum space can, therefore, be written as follows where q = p + + p − is the sum of lepton-antilepton pair 4-momenta, e is the electron charge, and g V η ( ) γ and g V π 0 (η)γ are the corresponding VMD coupling constants in Eq. (1). Noting that the Levi-Civita tensors are antisymmetric under the substitutions µ ↔ α and ρ ↔ δ , whilst the products of loop momenta k µ k α and k ρ k δ are symmetric under these substitutions, one finds that the terms in Eq. (2) containing these combinations vanish and that the superficial degree of divergence for the loop integrals of the two diagrams in Fig. 1 is −1. Accordingly, both diagrams are convergent individually. 3 It is worth highlighting that contributions from the exchange of scalar resonances can be safely discarded as they ought to be negligible for the first four η ( ) → π 0 l + l − decays and relatively small for the last two η → ηl + l − processes. The interested reader is referred to the in-depth analysis carried out in Ref. [12] where scalar exchanges were introduced under the framework of the Linear Sigma Model for the η ( ) → π 0 γγ and η → ηγγ decays. 4 Note that any C-violating contributions to these processes, such as e.g. the single-photon exchange channel, would be associated to BSM physics. In this work, though, the focus is on the SM contribution from the C-conserving two-photon exchange channel. 5 For simplicity of the calculation, we neglect the q 2 dependence of the transition form factor in Eq. (1). This is not fully rigorous but, we understand, it is a tolerable approximation given that these form factors are usually determined from on-shell photon processes.
The numerator of M can be simplified using the usual Dirac algebra manipulations and the equations of motion. For these calculations, the mass of the leptons are not approximated to zero, as we are interested in both the electron and muon modes for the six decay processes; as a result, the task of manipulating and simplifying the algebraic expressions would be daunting should computer algebra packages not be available. In the present work, use of the Mathematica package FeynCalc 9.2.0 [14,15] is made for this purpose.
Let us now proceed to calculate the loop integral. As usual, one first introduces the Feynman parametrisation and completes the square in the new denominators ∆ iV (i = 1, 2 and V = ρ 0 , ω, φ ) by shifting to a new loop momentum variable [16]. Hence, the denominators become Rewriting the numerators of the Feynman diagrams 1 and 2 (i.e. t-channel and u-channel diagrams, respectively, in Fig. 1) in terms of the new momentum variable , one finds where the explicit expressions for the parameters A i , B i , C i and D i (i = 1, 2) are provided in A. Finally, we perform a Wick rotation and change to four-dimensional spherical coordinates [16,17] to carry out the momentum integral. The following expressions for the amplitudes of the Feynman diagrams are found where m l is the corresponding lepton mass, and the parameters α V , β V , σ V and τ V in Eq. (5) are defined as with x, y and z being the Feynman integration parameters. Therefore, the full amplitude can now be expressed as where Ω and Σ are defined as follows and the unpolarised squared amplitude is Finally, the differential decay rate for a three-body decay can be written as [18] where

III. THEORETICAL RESULTS
Making use of the theoretical expressions that have been presented in section II, one can find numerical predictions for the decay widths of the η ( ) → π 0 l + l − and η → ηl + l − decay processes, as well as their associated dilepton energy spectra. Both, the integral over the Feynman parameters as well as the integral over phase space, must be carried out numerically, as algebraic expressions cannot be obtained. In addition, the numerical integrals over the Feynman parameters are to be performed using adaptive Monte Carlo methods [19]; this is driven by the complexity of the expressions to be integrated and their multidimensional nature 6 .
In the conventional VMD model, pseudoscalar mesons do not couple directly to photons but through the exchange of intermediate vectors. Thus, in this framework, a particular V Pγ coupling constant times its normalised form factor, cf. Eq. (1), is given by 7 where g VV P are the vector-vector-pseudoscalar couplings, g V γ the vector-photon conversion couplings, and M V the intermediate vector masses. In the SU(3)-flavour symmetry and OZI-rule respecting limits, one could express all the g V Pγ in terms of a single coupling constant and SU(3)-group factors [24]. However, to account for the unavoidable SU(3)-flavour symmetry-breaking and OZI-rule violating effects, we make use of the simple, yet powerful, phenomenological quark-based model first presented in Ref. [25], which was developed to describe V → Pγ and P → V γ radiative decays. According to this model, the decay couplings can be expressed as where g is a generic electromagnetic constant, φ P is the pseudoscalar η-η mixing angle in the quark-flavour basis, φ V is the vector ω-φ mixing angle in the same basis, m/m s is the quotient of constituent quark masses, and z NS and z S are the non-strange and strange multiplicative factors accounting for the relative meson wavefunction overlaps [25,26]. By performing an optimisation fit to the most up-to-date V Pγ experimental data [18], one can find values for the above parameters Given the very wide decay width of the ρ 0 resonance, which, in turn, is associated to its very short lifetime, the use of the usual Breit-Wigner approximation for the ρ 0 propagator is not justified. Instead, an energydependent width for the vector propagator ought to be considered, which may be written for a genericq 2 as follows where θ (x) is the Heaviside step function. Strictly, one would now need to plug Eq. (14) into Eq. (2) and perform the loop integral, which represents a computation challenge in its own right and is outside of the scope of the present work. 8 With this in mind, and for the sake of simplicity, we resolve to stick with the Breit-Wigner approximation for the ρ 0 propagator despite being a potential source of error. The energydependent propagator is not needed, though, for the ω and φ resonances, as their associated decay widths are narrow and, therefore, use of the usual Breit-Wigner approximation suffices. Using the most recent empirical data for the meson masses and total decay widths from Ref. [18], together with all the above considerations, one arrives at the decay width results shown in Table I for the six η ( ) → π 0 l + l − and η → ηl + l − processes. The total decay widths associated to the electron modes turn out to be larger than the ones corresponding to the muon modes despite the second and third terms in the unpolarised squared amplitude (cf. Eq. (9)) being helicity suppressed for the electron modes. This suppression, 8 One could write, for example, the ρ 0 energy-dependent propagator f (s) = , where s th is the particle production threshold, in the case at hand s th = 4m 2 π , and s 0 is the subtraction point such that s 0 < s th , e.g. s 0 = 0. One would then perform the loop integral in the usual way, leaving the dispersion integral to the end of the computation. though, does not overcome the phase space suppression for the muon modes, yielding Γ(η ( ) → π 0 e + e − ) > Γ(η ( ) → π 0 µ + µ − ) and Γ(η → ηe + e − ) > Γ(η → η µ + µ − ).
Let us now look at the contributions from the different vector meson exchanges to the total decay widths. For the first decay, i.e. η → π 0 e + e − , we find that the contribution from the ρ 0 exchange is ∼ 26%, the contribution from the ω is ∼ 22%, whilst the one from the φ is negligible, i.e. ∼ 0%. The interference between the ρ 0 and the ω is constructive, accounting for the ∼ 49%; similarly, the interference between the ρ 0 and the ω with the φ is constructive and about ∼ 3%. The contributions to the second decay, i.e. η → π 0 µ + µ − , are ∼ 25%, ∼ 23% and ∼ 0% from the ρ 0 , ω, and φ exchanges, respectively. As before, the interference between the ρ 0 and the ω is constructive, weighing ∼ 48%, and the interference between the ρ 0 and the ω with the φ is constructive and accounts for approximately the ∼ 4%. For the third decay, i.e. η → π 0 e + e − , the contributions from the ρ 0 , ω and φ turn out to be ∼ 16%, ∼ 39% and ∼ 0%, respectively; the interference between the ρ 0 and the ω exchanges is constructive and accounts for the ∼ 47%, whilst the interference between the ρ 0 and ω with the φ is destructive and weighs approximately ∼ 2%. The contributions to the fourth decay, i.e. η → π 0 µ + µ − , from the ρ 0 , ω and φ exchanges are ∼ 20%, ∼ 35% and ∼ 0%, respectively. The interference between the ρ 0 and the ω is constructive, representing a ∼ 53% contribution, whilst the interference between the ρ 0 and the ω with the φ is destructive and accounts for the ∼ 8%. The fifth decay, i.e. η → ηe + e − , gets contributions from the exchange of ρ 0 , ω and φ resonances of approximately ∼ 76%, ∼ 1% and ∼ 2%, respectively; the interference between the ρ 0 and the ω is constructive weighing ∼ 29%, and the interference between the ρ 0 and the ω with the φ is destructive and contributes with roughly the ∼ 8%. Finally, for the sixth decay, i.e. η → η µ + µ − , we find that the contribution from the ρ 0 exchange is ∼ 94%, the contribution from the ω is ∼ 2% and the one from the φ is ∼ 3%; the interference between the ρ 0 and the ω is constructive and accounts for the ∼ 26%, whilst the interference between the ρ 0 and the ω with the φ is destructive weighing close to ∼ 25%. The tiny contribution from the φ exchange to the decay widths of the six processes is explained by the relatively small product of VMD V Pγ coupling constants. Likewise, the comparatively minute contribution from the ω exchange to the decay widths of the last two reactions is down to the significantly smaller product of coupling constants, if compared to that of the ρ 0 exchange.
be h V = 0.035 [13]. The VV P coupling σ V obtained using the ENJL model turns out to be σ V = 0.28. However, σ V can also be obtained from the analysis of the dilepton mass spectrum in ω → π 0 µ + µ − decays, where one finds σ V ≈ 0.58 [29]. Due to the fact that σ V is poorly known and the dispersion of the above estimations is large, we do not consider the q 2 dependence of the form factors in the subsequent calculations. An alternative model to fix g V Pγ , the normalisation of the form factors, is the Hidden Gauge Symmetry (HGS) model [30], where the vector mesons are considered as gauge bosons of a hidden symmetry. Within this model, a V Pγ transition proceeds uniquely through the exchange of intermediate vector mesons. In this sense, it is equivalent to the conventional VMD model with the relevant exception of including direct γP 3 terms (P being a pseudoscalar meson), which are forbidden in VMD [31]. Due to this similarity, we will not make use of the HGS model to assess the systematic model error and refer the interested reader to Ref. [24] for a detailed calculation of the g V Pγ couplings in this model. Next, our results for the semileptonic decays η ( ) → π 0 l + l − and η → ηl + l − in the conventional VMD framework using the V Pγ couplings from the phenomenological quark-based model in Eq. (12) are discussed and, if available, compared with previous literature. These predictions include a first experimental error ascribed to the propagation of the parametric errors in Eq. (13), a second error down to the numerical integration, and a third systematic error associated to the model dependence of our approach. The latter is calculated as the absolute difference between the predicted central values obtained from the VMD and RChT frameworks (cf. Table I).

IV. CONCLUSIONS
In this work, the C-conserving decay modes η ( ) → π 0 l + l − and η → ηl + l − (l = e or µ) have been analysed within the theoretical framework of the VMD model. The associated decay widths and dilepton energy spectra have been calculated and presented for the six decay processes. To the best of our knowledge, the theoretical predictions for the four η → π 0 l + l − and η → ηl + l − reactions that we have provided in this work are the first predictions from theory that have been published.
The decay width results that we have obtained from our calculations, which are summarised in Table I, have been compared with those available in the published literature. In general, the agreement is reasonably good considering that the previous analysis either contain important approximations or consist of unitary lower bounds. Predictions for the dilepton energy spectra have also been presented for all the above processes, cf. Fig. 2.
Experimental measurements to date have provided upper limits to the decay processes studied in this work. These upper limits, though, are still many orders of magnitude larger than the theoretical results that we have presented. For this reason, we would like to encourage experimental groups, such as the WASA-at-COSY and REDTOP Collaborations, to study these semileptonic decays processes, as we believe that they can represent a fruitful arena in the search for new physics beyond the Standard Model.