Study of the electromagnetic Dalitz decays $\psi(\Upsilon) \to \eta_{c}(\eta_{b}) l^{+} l^{-}$

We study the electromagnetic Dalitz (EM) decays, $\psi \to \eta_{c} l^{+} l^{-}$ and $\Upsilon \to \eta_{b} l^{+} l^{-}$ ($l = e$ or $\mu$), in which the lepton pair comes from the virtual photon emitted by the M1 transition from $c\bar{c}$ ($b\bar{b}$) spin triplet state to the spin singlet state. We estimate the partial width of $\psi(\Upsilon) \to \eta_{c}(\eta_{b}) l^{+} l^{-}$, based on the simple pole approximation. Besides, based on different QCD models, the partial width of $\psi(\Upsilon) \to \eta_{c}(\eta_{b}) \gamma$ is determined.


I. INTRODUCTION
The electromagnetic (EM) Dalitz decays, V → P l + l − , where V and P are vectors and pseudo-scalar mesons, and l denotes lepton (e, µ), provide an ideal opportunity to probe the structure of hadronic states and to investigate the fundamental mechanisms of the interactions between photons and hadrons [1,2].The lepton pair l + l − comes from an off-shell photon, radiated from the transition between V and P .Assuming point-like particles, the process can be exactly described by QED [3].Otherwise, the structure-dependent partial width can be modified by transition form factor f VP (q 2 ), which can be estimated based on QCD models [4][5][6][7][8] and provides information of the EM structure arising from the V -P transition.Experimentally, the M1 transition between ψ and η c has been observed with the average branching fraction, B(J/ψ → γη c (1S)) = (1.7 ± 0.4)% [9].In the following ratio of branching fractions many theoretical uncertainties can cancel, therefore it can be used to test theoretical models.Experimentally, the EM Dalitz decays of light unflavored vector mesons (ρ 0 , ω, φ) have been widely observed [9] and several decays of charmonium vector mesons (J/ψ, ψ ′ ) to light pseudo-scalar mesons, which are studied in Ref. [10], have been observed recently by BESIII experiment [9,11,12].
In Table I, we summarize the experimental results of the EM decays for the light unflavored vector mesons (ρ 0 , ω, φ) and charmonium vector mesons (J/ψ, ψ ′ ).The results indicate that the ratios of the EM Dalitz decay to the corresponding radiative decays are suppressed by two orders of magnitude.However, in previous paper [10], which focus on the EM Dalitz decays of J/ψ, it's assumed that the ψ(Υ) is totally unpolarized.In this paper, we investigate the polarization of ψ(Υ) produced in e + e − collisions, then deduce the general form of the decay width, as a func-tion of polarization vector of ψ(Υ), at last apply it to the ψ(Υ) → l + l − η c (η b ) decay to predict its branching fraction.

TABLE I. The branching fractions of EM Dalitz decays
V → P l + l − , and ratios of the EM Dalitz decays to the corresponding radiative decays of the vector mesons.These data are from PDG2018 [9].

Decay mode
Branching fraction The amplitude of the EM Dalitz decay, V → P l + l − , has the Lorentz-invariant form [10] where α is the fine-structure constant, f VP the transition form factor, ǫ µνρσ the Levi-Civita tensor, p µ the momentum of the pseudoscalar meson, and q ν = k 1 + k 2 with k 1 and k 2 the momenta of the l + and l − .After averaging over the spin of leptons, the amplitude squared is where where m V and m l are masses of vector and pseudo-scalar mesons.The differential decay width of V → P l + l − is obtained as where |k * | is the momentum of l + or l − in the rest frame of l + l − system, |p 3 | momentum of the pseudoscalar meson P in the rest frame of V , dΩ 3 = dφ 3 d(cos θ 3 ) is the solid angle of P in the rest frame of V , and ) the solid angle of l + or l − in the rest frame of l + l − system (the z direction is defined as the momentum direction of l + l − system in the rest frame of V ).

A. Differential partial widths for polarized ψ/Υ EM Dalitz decays
There are totally three polarization states for the massive vector mesons, which are defined as ǫ µ LP = (0, 0, 0, 1) where ǫ µ LP , ǫ µ TL and ǫ µ TR are the longitudinal polarization, the left-hand transverse polarization and the righthand transverse polarization component, respectively.Correspondingly, we obtain the angular distribution for each component that has the following form where θ = θ 3 , is the polar angle of the pseudo-scalar P in the rest frame of ψ(Υ).Therefore, one can determine the polarization of ψ(Υ) by a likelihood fit to the cos θ distribution where non-zero α polar means partial polarization of ψ(Υ), α polar = +1 means only transverse polarization of ψ(Υ), and α polar = −1 means only longitudinal polarization of ψ(Υ), respectively.We show the angular distributions of ψ(Υ) → P l + l − with different α polar values in Fig. 1.In ψ(Υ) → P l + l − , an explicit value of α polar from experiment indicates the polarization information of ψ(Υ).In this section we study the production rate for each polarization state of the vector mesons ψ or Υ in electronpositron colliders.In electron-positron collisions, such as BESIII and Belle II experiments, the amplitude squared of e + e − → ψ or Υ can be written as where e c is the electrical charge of the charm quark, f ψ the form factor of cc → ψ, ǫ µ the polarization four-vector of ψ, m ψ and m e the masses of ψ and electron, q = k 1 +k 2 with k 1 and k 2 the momenta of e + and e − , respectively.Then, we obtain the relative probabilities for three polarization states Consequently, we can conclude that the longitudinal polarization can be neglected and ψ is totally polarized in transverse state in unpolarized e + e − colliders.The amplitude squared of ψ → P l + l − in this situation has the following form where And the angular distribution in this situation is It's easy and intuitive to extend the conclusion to Υ case.
Once averaging the polarization of ψ(Υ), the q 2dependent differential decay width of ψ(Υ) → P l + l − can be obtained as [10] dΓ(ψ(Υ) → P l + l − ) dq 2 = 1 3 For the corresponding radiative decay ψ(Υ) → P γ, the partial decay width is Thus the q 2 -dependent differential decay width of ψ(Υ) → P l + l − can be normalized to the width of the corresponding radiative decay ψ(Υ) → P γ, and the ratio can be obtained as where the normalized transition form factor for the ψ(Υ) → P transition is defined as F VP (q 2 ) ≡ f VP (q 2 )/f VP (0), and QED(q 2 ) represents the QED calculations for point-like particles In experiment, by comparing the measured spectrum of the lepton pair in the EM Dalitz decay with the the QED calculation for point-like particle, one can determine the transition form factor in the time-like region of the momentum transfer [2].Namely, the transition factor can modify the lepton spectrum as compared with that obtained for point-like particles.

Decay mode Γ
To study the dependence of the decay rates on the value of the pole mass, we varied the pole mass.The results for ψ ′ → η c µ + µ − are shown in Fig. 2 as an example.The cases for the other decay modes considered in this work are similar.Both the differential and total decay rates are not sensitive to the value of the pole mass, if the pole mass is in the range Λ 2 ≫ q 2 max .The reason can be well understood.The dominant contribution to the decay rate comes from the region of small value of q 2 .For the pole mass with large value, q 2 /Λ 2 is small, thus this term cannot give large effect.Since the decay rates are not sensitive to the pole mass in the transition form factor, the estimated partial decay widths in Table II based on the VDM are reliable.The EM Dalitz decays, ψ(Υ) → η c (η b )l + l − , are related to the radiative decays, ψ(Υ) → η c (η b )γ, by the transition form factor f VP (q 2 ) and VP (0).Models describing ψ(Υ) → η c (η b )γ can provide information for f VP (q 2 ).In Ref. [15], using the theories of NRQCD and pNRQCD, one studied the M1 transitions between two heavy quarkonia and obtained w(α s ), ( 19) where e Q is the electrical charge of the heavy quark (e c = 2/3, e b = −1/3), w(α s ) the function of the strong coupling constant α s .The predicted result for J/ψ → η c γ is consistent with data [15].However, the predicted results for ψ ′ → η c γ and Υ(2S) → η b γ are larger than data by two and one order of magnitude, respectively.And there is no prediction for Υ(3S) → η b γ.For J/ψ → η c γ, the w(α s ) has the following form [15] where C F = 4/3 is the color coefficient, α s (m J/ψ /2) and α s (p J/ψ ) represent the values of α s in corresponding energy scale, respectively.Typically, α s (p J/ψ ) satisfies p J/ψ ≈ m c C F α s (p J/ψ )/2 ≈ 0.8 GeV [15].To study the dependence of Γ(J/ψ → η c γ) on the value of α s (m J/ψ /2) and α s (p J/ψ ) , we varied the α s (m J/ψ /2) and α s (p J/ψ ).
The results are shown in Fig. 3.In this way, a precision measurement of the partial width of J/ψ → η c γ can be used to test QCD model and provide stringent restriction on α s in charm energy scale.With the explicit form factor f VP (q 2 ), one can obtain the partial decay widths of ψ(Υ) → η c (η b )γ from Eq. ( 15).Hence, It is important to measure the q 2dependent form factor f VP (q 2 ) , since it can be used to determine the the partial decay widths of ψ(Υ) → η c (η b )γ at q 2 = 0.
Actually, for the ψ(Υ) → η c (η b )l + l − decay in e + e − collisions at BESIII and Belle experiments, one can just reconstruct the lepton pair and then look at the recoiling mass of the lepton pair to obtain the signal.In this way, the results are irrelevant to the decay modes of η c (η b ).

IV. SUMMARY
In summary, the EM Dalitz decays, ψ(Υ) → η c (η b )l + l − , are studied in this work.We investigate the effect of polarization of ψ(Υ) and estimate the partial decay widths of ψ(Υ) → η c (η b )l + l − (l = e, µ) by assuming the simple pole approximation.We obtain the polarization components for the vector mesons ψ and/or Υ produced in e + e − colliders.We find the transverse polarization states for ψ and/or Υ dominate in unpolarized e + e − conlliders.We obtain the angular distribution of ψ(Υ) → η c (η b )l + l − decays for each polarization state of ψ and/or Υ, which is helpful to determine the polarization of ψ(Υ) in the EM Dalitz decays ψ(Υ) → η c (η b )l + l − in experiment.The decay widths of ψ(Υ) → η c (η b )l + l − are also obtained.Besides, we discuss a QCD model where Γ(J/ψ → η c γ) is related to α s and suggest that absolute partial decay widths of EM Dalitz decays should be measured with reconstruction of only lepton pairs by taking advantage of the e + e − collision with known initial four momentum of the electron and positron beams.