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

TL;DR

This paper investigates electromagnetic Dalitz decays of charmonium and bottomonium states, estimating their partial widths and comparing them with radiative decay widths using QCD models.

Contribution

It provides the first estimates of partial widths for these Dalitz decays and compares them with radiative decays within a theoretical framework.

Findings

Estimated partial widths of $ o o o$ decays using pole approximation.

Calculated radiative decay widths based on QCD models.

Provided theoretical predictions for rare decay processes.

Abstract

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.

Tables

Table 1. Table 1: The branching fractions of EM Dalitz decays V → P ​ l + ​ l − → 𝑉 𝑃 superscript 𝑙 superscript 𝑙 V\to Pl^{+}l^{-} and ratios of the EM Dalitz decays to the corresponding radiative decays of the vector mesons. These data are from PDG2018  Tanabashi:2018oca .

Decay mode	Branching fraction	$\frac{\Gamma \left(V\to P{l}^{+}{l}^{-}\right)}{\Gamma \left(V\to P\gamma \right)}$
${\rho }^0\to {\pi }^0{e}^{+}{e}^{-}$
$\omega \to {\pi }^0{e}^{+}{e}^{-}$	$\left(7.7\pm 0.6\right)\times {10}^{-4}$	$\left(0.91\pm 0.08\right)\times {10}^{-2}$
$\omega \to {\pi }^0{\mu }^{+}{\mu }^{-}$	$\left(1.34\pm 0.18\right)\times {10}^{-4}$	$\left(0.16\pm 0.02\right)\times {10}^{-2}$
$\varphi \to {\pi }^0{e}^{+}{e}^{-}$	$\left({1.33}_{-0.10}^{+0.07}\right)\times {10}^{-5}$	$\left({1.02}_{-0.09}^{+0.07}\right)\times {10}^{-2}$
$\varphi \to \eta e^{+}{e}^{-}$	$\left(1.08\pm 0.04\right)\times {10}^{-4}$	$\left(0.83\pm 0.03\right)\times {10}^{-2}$
$\varphi \to \eta {\mu }^{+}{\mu }^{-}$
$J/\psi \to {\pi }^0{e}^{+}{e}^{-}$	$\left(7.6\pm 1.4\right)\times {10}^{-7}$	$\left({2.18}_{-0.44}^{+0.45}\right)\times {10}^{-2}$
$J/\psi \to \eta e^{+}{e}^{-}$	$\left(1.16\pm 0.09\right)\times {10}^{-5}$	$\left(1.05\pm 0.09\right)\times {10}^{-2}$
$J/\psi \to {\eta }^{\prime }{e}^{+}{e}^{-}$	$\left(5.81\pm 0.35\right)\times {10}^{-5}$	$\left(1.13\pm 0.08\right)\times {10}^{-2}$
${\psi }^{\prime }\to {\eta }^{\prime }{e}^{+}{e}^{-}$	$\left(1.90\pm 0.27\right)\times {10}^{-6}$	$\left(1.53\pm 0.22\right)\times {10}^{-2}$

Table 2. Table 2: The estimated partial decay widths of ψ → η c ​ l + ​ l − → 𝜓 subscript 𝜂 𝑐 superscript 𝑙 superscript 𝑙 \psi\to\eta_{c}l^{+}l^{-} and Υ → η b ​ l + ​ l − → Υ subscript 𝜂 𝑏 superscript 𝑙 superscript 𝑙 \Upsilon\to\eta_{b}l^{+}l^{-} based on Eq. ( 11 ) by assuming the simple pole approximation according to VDM model. Here we take Λ = m ψ ′ , m ψ ​ ( 3770 ) , m Υ ​ ( 3 ​ S ) Λ subscript 𝑚 superscript 𝜓 ′ subscript 𝑚 𝜓 3770 subscript 𝑚 Υ 3 S \Lambda=m_{\psi^{\prime}},m_{\psi(3770)},m_{\Upsilon(\rm{3S})} and m Υ ​ ( 4 ​ S ) subscript 𝑚 Υ 4 S m_{\Upsilon(\rm{4S})} for J / ψ , ψ ′ , Υ ​ ( 2 ​ S ) 𝐽 𝜓 superscript 𝜓 ′ Υ 2 S J/\psi,\psi^{\prime},\Upsilon(\rm{2S}) and Υ ​ ( 3 ​ S ) Υ 3 S \Upsilon(\rm{3S}) EM Dalitz decays, respectively. The uncertainties are from the errors on measured Γ ​ ( ψ → η c ​ γ ) Γ → 𝜓 subscript 𝜂 𝑐 𝛾 \Gamma(\psi\to\eta_{c}\gamma) and Γ ​ ( Υ → η b ​ γ ) Γ → Υ subscript 𝜂 𝑏 𝛾 \Gamma(\Upsilon\to\eta_{b}\gamma) .

Decay mode	${\Gamma }_{e^{+}{e}^{-}}^{VDM}$ (keV)	${\Gamma }_{{\mu }^{+}{\mu }^{-}}^{VDM}$ (keV)
$J/\psi \to {\eta }_c{l}^{+}{l}^{-}$	$\left(9.6\pm 2.3\right)\times {10}^{-3}$	-
${\psi }^{\prime }\to {\eta }_c{l}^{+}{l}^{-}$	$\left(8.9\pm 1.3\right)\times {10}^{-3}$	$\left(8.2\pm 1.2\right)\times {10}^{-4}$
${\rm Y}\left(2S\right)\to {\eta }_b{l}^{+}{l}^{-}$	$\left(10.8\pm 4.2\right)\times {10}^{-5}$	$\left(8.2\pm 3.2\right)\times {10}^{-6}$
${\rm Y}\left(3S\right)\to {\eta }_b{l}^{+}{l}^{-}$	$\left(9.7\pm 1.6\right)\times {10}^{-5}$	$\left(12.6\pm 2.1\right)\times {10}^{-6}$

Equations

$$R\equiv \frac{B(\psi \to e^{+}{e}^{-}{\eta }_c)}{B(\psi \to \gamma {\eta }_c)},$$

$$T(V\to P{l}^{+}{l}^{-})=4\pi \alpha f_{VP}{ϵ}^{\mu \nu \rho \sigma }{p}_{\mu }{q}_{\nu }{ϵ}_{\rho }\frac{1}{q^2}{\bar{u}}_1{\gamma }_{\sigma }{\nu }_2,$$

$$|T(V\to P{l}^{+}{l}^{-}){|}^2=16{\pi }^2{\alpha }^2\frac{|f_{VP}(q^2){|}^2}{q^4}\cdot h,$$

$$\begin{array}{rl}h=& 8m_V^2{m}_l^2(q^2ϵ\cdot {ϵ}^{\ast }-q\cdot ϵq\cdot {ϵ}^{\ast })\\ & -2m_V^2{q}^4(k_1-k_2)\cdot ϵ(k_1-k_2)\cdot {ϵ}^{\ast }\\ & +8m_l^{2}q\cdot p[q\cdot ϵp.{ϵ}^{\ast }-ϵ\cdot {ϵ}^{\ast }q\cdot p]\\ & +2m_l^2(k_1-k_2)\cdot p\\ & \times [ϵ\cdot {ϵ}^{\ast }(k_1-k_2)\cdot p+(k_1-k_2)\cdot ϵ(p\cdot {ϵ}^{\ast })]\\ & +8[(k_2\cdot p)(k_1\cdot ϵ)-(k_1\cdot p)(k_2\cdot ϵ)]\\ & \times [(k_2\cdot p)(k_1\cdot {ϵ}^{\ast })-(k_1\cdot p)(k_2\cdot {ϵ}^{\ast })],\end{array}$$

$$\begin{array}{rl}d\Gamma (V\to P{l}^{+}{l}^{-})=& \frac{1}{(2\pi {)}^5}\frac{1}{16m_V^2}|T(\psi ({\rm Y})\to P{l}^{+}{l}^{-}){|}^2\\ & \times |{\mathbf{k}}^{\ast }||{\mathbf{p}}_3|d{m}_{l^{+}{l}^{-}}d{\Omega }_{3}d{\Omega }_1^{\ast },\end{array}$$

$$\begin{array}{rl}& {ϵ}_{\mathrm{LP}}^{\mu }=(0,0,0,1)\\ & {ϵ}_{\mathrm{TL}}^{\mu }=\frac{1}{\sqrt{2}}(0,1,-i,0)\\ & {ϵ}_{\mathrm{TR}}^{\mu }=\frac{1}{\sqrt{2}}(0,1,i,0)\end{array}$$

$$\begin{array}{rl}& \frac{d\Gamma (\psi ({\rm Y})\to P{l}^{+}{l}^{-}{)}_{\mathrm{LP}}}{d\cos \theta }\sim 1-{\cos }^2\theta \\ & \frac{d\Gamma (\psi ({\rm Y})\to P{l}^{+}{l}^{-}{)}_{\mathrm{TL}}}{d\cos \theta }\sim \frac{1+{\cos }^2\theta }{2}\\ & \frac{d\Gamma (\psi ({\rm Y})\to P{l}^{+}{l}^{-}{)}_{\mathrm{TR}}}{d\cos \theta }\sim \frac{1+{\cos }^2\theta }{2},\end{array}$$

$$\mathcal{L}(\cos \theta )=1+{\alpha }_{\mathrm{polar}}{\cos }^2\theta ,$$

$$\begin{array}{rl}|T(\psi ){|}^2=& \frac{16{\pi }^2{\alpha }^2{e}_c^2}{q^4}|f_{\psi }{|}^2{m}_{\psi }^2{ϵ}_{\mu }^{\ast }{ϵ}_{\nu }\\ & \times (k_1^{\mu }{k}_2^{\nu }+k_2^{\mu }{k}_1^{\nu }-g^{\mu \nu }{k}_1\cdot k_2+g^{\mu \nu }{m}_e^2),\end{array}$$

$$\begin{array}{rl}|T(\psi ){|}_{\mathrm{LP}}^2:|T(\psi ){|}_{\mathrm{TL}}^2:|T(\psi ){|}_{\mathrm{TR}}^2& =m_e^2:m_{\psi }^2:m_{\psi }^2\\ & \approx 2.7\times 10^{-8}:1:1.\end{array}$$

$$\frac{d\Gamma (\psi ({\rm Y})\to P{l}^{+}{l}^{-})}{d{q}^2\Gamma (\psi ({\rm Y})\to P\gamma )}=|F_{VP}(q^2){|}^2\times [\mathrm{QED}({\mathrm{q}}^2)],$$

$$\begin{array}{rl}\mathrm{QED}({\mathrm{q}}^2)=& \frac{\alpha }{3\pi }\frac{1}{q^2}{\left(1-\frac{4m_l^2}{q^2}\right)}^{\frac{1}{2}}\left(1+\frac{2m_l^2}{q^2}\right)\\ & \times {\left[{\left(1+\frac{q^2}{m_V^2-m_P^2}\right)}^2-\frac{4m_V^2{q}^2}{(m_V^2-m_P^2{)}^2}\right]}^{\frac{3}{2}}.\end{array}$$

$$F_{VP}(q^2)=\frac{1}{1-\frac{q^2}{{\Lambda }^2}},$$

$$\Gamma (\psi ({\rm Y})\to {\eta }_c({\eta }_b)\gamma )=\frac{16e_Q^2}{3}\frac{\alpha (m_{\psi ({\rm Y})}^2-m_{{\eta }_c({\eta }_b)}^2{)}^3}{8m_{\psi ({\rm Y})}^3}w({\alpha }_s),$$

$$w({\alpha }_s)=1+C_F\frac{{\alpha }_s(m_{J/\psi }/2)}{\pi }-\frac{2}{3}{\left(C_F{\alpha }_s(p_{J/\psi })\right)}^2,$$

Full text

I Introduction

The electromagnetic (EM) Dalitz decays, $V\to Pl^{+}l^{-}$, where $V$ and $P$ are vectors and pseudoscalar mesons, and $l$ denotes lepton ($e$, $\mu$), provide an ideal opportunity to probe the structure of hadronic states and to investigate the fundamental mechanisms of the interactions between photons and hadrons Landsberg:1986sk ; Landsberg:1986fd . The lepton pair $l^{+}l^{-}$ comes from an off-shell photon, radiated from the transition between $V$ and $P$. Assuming pointlike particles, the process can be exactly described by QED Kroll:1955zu . Otherwise, the structure-dependent partial width can be modified by transition form factor $f_{\rm VP}(q^{2})$, which can be estimated based on QCD models Achasov:1992ku ; Klingl:1996by ; Faessler:1999de ; Terschluesen:2010ik ; Ivashyn:2011hb and provides information of the EM structure arising from the $V$-$P$ transition. The M1 transition between $\psi(\Upsilon)$ and $\eta_{c}(\eta_{b})$ is widely studied in theory Shifman:1979nx ; Khodjamirian:1983gd ; Beilin:1985da ; Zhang:1991et ; Ebert:2002pp ; Lahde:2002wj ; Hwang:2006cua ; Dudek:2006ej ; Ke:2010pp ; Donald:2012ga ; Becirevic:2012dc ; Pineda:2013lta , and the M1 transition between $\psi$ and $\eta_{\rm c}$ has been observed with the average branching fraction, ${\cal B(}J/\psi\to\gamma\eta_{\rm c}(1S))=(1.7\pm 0.4)\%$ Tanabashi:2018oca . In the following ratio of branching fractions:

[TABLE]

many theoretical uncertainties can cancel; therefore, it can be used to test theoretical models. Experimentally, the EM Dalitz decays of light unflavored vector mesons ($\rho^{0},\omega,\phi$) have been widely observed Tanabashi:2018oca and several decays of charmonium vector mesons ($J/\psi,\psi^{\prime}$) to light pseudoscalar mesons, which are studied in Ref. Fu:2011yy , have been observed recently by BESIII experiment Tanabashi:2018oca ; Ablikim:2018xxs ; Ablikim:2014nro . In Table 1, we summarize the experimental results of the EM decays for the light unflavored vector mesons ($\rho^{0},\omega,\phi$) and charmonium vector mesons ($J/\psi,\psi^{\prime}$). The results indicate that the ratios of the EM Dalitz decay to the corresponding radiative decays are suppressed by 2 orders of magnitude.

However, in previous paper Fu:2011yy , which focus on the EM Dalitz decays of $J/\psi$, it is assumed that the $\psi(\Upsilon)$ is totally unpolarized. In this paper, we investigate the polarization of $\psi(\Upsilon)$ produced in $e^{+}e^{-}$ collisions, then deduce the general form of the decay width, as a function of polarization vector of $\psi(\Upsilon)$, at last apply it to the $\psi(\Upsilon)\to l^{+}l^{-}\eta_{c}(\eta_{b})$ decay to predict its branching fraction.

II General formula for the EM Dalitz decay $V\to Pl^{+}l^{-}$

The amplitude of the EM Dalitz decay, $V\to Pl^{+}l^{-}$, has been obtained in the literature previously, which can be written in a Lorentz-invariant form Landsberg:1986sk ; Landsberg:1986fd ; Fu:2011yy

[TABLE]

where $\alpha$ is the fine-structure constant, $f_{V\!P}$ the transition form factor, $\epsilon^{\mu\nu\rho\sigma}$ the Levi-Civita tensor, $p_{\mu}$ the momentum of the pseudoscalar meson, and $q_{\nu}=k_{1}+k_{2}$ with $k_{1}$ and $k_{2}$ the momenta of the $l^{+}$ and $l^{-}$. After summing over the spin of leptons, the amplitude squared is

[TABLE]

where

[TABLE]

where $m_{V}$ and $m_{l}$ are masses of vector and pseudoscalar mesons. The differential decay width of $V\to Pl^{+}l^{-}$ is obtained as

[TABLE]

where $\bf|k^{*}|$ is the momentum of $l^{+}$ or $l^{-}$ in the rest frame of $l^{+}l^{-}$ system, $\bf|p_{3}|$ momentum of the pseudoscalar meson $P$ in the rest frame of $V$, $d\Omega_{3}=d\phi_{3}d(\cos\theta_{3})$ is the solid angle of $P$ in the rest frame of $V$, and $d\Omega^{*}_{1}=d\phi^{*}_{1}d(\cos\theta^{*}_{1})$ 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 $V$).

II.1 Differential partial widths for polarized $\psi$/$\Upsilon$ EM Dalitz decays

There are totally three polarization states for the massive vector mesons, which are defined as

[TABLE]

where $\epsilon^{\mu}_{\rm{LP}}$, $\epsilon^{\mu}_{\rm{TL}}$ and $\epsilon^{\mu}_{\rm{TR}}$ are the longitudinal polarization, the left-hand transverse polarization and the right-hand transverse polarization component, respectively. Correspondingly, it is generally known that the angular distribution for each component that has the following form

[TABLE]

where $\theta=\theta_{3}$ is the polar angle of the pseudoscalar $P$ in the rest frame of $\psi(\Upsilon)$. Therefore, one can determine the polarization of $\psi(\Upsilon)$ by a likelihood fit to the $\cos\theta$ distribution

[TABLE]

where nonzero $\alpha_{\rm{polar}}$ means partial polarization of $\psi(\Upsilon)$, $\alpha_{\rm{polar}}=+1$ means only transverse polarization of $\psi(\Upsilon)$, and $\alpha_{\rm{polar}}=-1$ means only longitudinal polarization of $\psi(\Upsilon)$, respectively. We show the angular distributions of $\psi(\Upsilon)\to Pl^{+}l^{-}$ with different $\alpha_{\rm{polar}}$ values in Fig. 1. In $\psi(\Upsilon)\to Pl^{+}l^{-}$, an explicit value of $\alpha_{\rm{polar}}$ from experiment indicates the polarization information of $\psi(\Upsilon)$.

II.2 Polarization of $\psi(\Upsilon)$ in $e^{+}e^{-}\to\psi(\Upsilon)$

In this section we study the production rate for each polarization state of the vector mesons $\psi$ or $\Upsilon$ in electron-positron colliders. In electron-positron collisions, such as BESIII and Belle II experiments, the amplitude squared of $e^{+}e^{-}\to\psi$ or $\Upsilon$ can be written as

[TABLE]

where $e_{c}$ is the electrical charge of the charm quark, $f_{\psi}$ the form factor of $c\bar{c}\to\psi$, $\epsilon_{\mu}$ the polarization four vector of $\psi$, $m_{\psi}$ and $m_{e}$ the masses of $\psi$ and electron, $q=k_{1}+k_{2}$ with $k_{1}$ and $k_{2}$ the momenta of $e^{+}$ and $e^{-}$, respectively. Then, the relative probabilities for three polarization states can be obtained

[TABLE]

The above result shows that the longitudinal polarization can be neglected and $\psi$ is totally polarized in transverse state in unpolarized $e^{+}e^{-}$ collider, which is consistent with the conclusion in Ref. Richman:1984gh . It is easy and intuitive to extend the conclusion to $\Upsilon$ case.

II.3 Decay rate of $\psi$($\Upsilon$) in $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$

To remove most part of the uncertainty caused by the form factor $f_{V\!P}(q^{2})$, one can consider the ratio of the decay widths of $\psi(\Upsilon)\to Pl^{+}l^{-}$ and $\psi(\Upsilon)\to P\gamma$. The $q^{2}$-dependent differential decay width of $\psi(\Upsilon)\to Pl^{+}l^{-}$ normalized to the width of the corresponding radiative decay $\psi(\Upsilon)\to P\gamma$ has been given in Ref. Fu:2011yy ; here for clearness we quote it in the following:

[TABLE]

where the normalized transition form factor for the $\psi(\Upsilon)\to P$ transition is defined as $F_{V\!P}(q^{2})\equiv f_{V\!P}(q^{2})/f_{V\!P}(0)$, and $\rm{QED}(q^{2})$ represents the QED calculations for point like particles,

[TABLE]

In experiment, by comparing the measured spectrum of the lepton pair in the EM Dalitz decay with the the QED calculation for pointlike particle, one can determine the transition form factor in the timelike region of the momentum transfer Landsberg:1986fd . Namely, the transition form factor can modify the lepton spectrum as compared with that obtained for pointlike particles.

To estimate the partial width of the $\psi(\Upsilon)$ EM Dalitz decay, the vector dominance model (VDM) is adopted, in which the hadronic EM current is proportional to vector meson fields GellMann:1961tg ; Bauer:1977iq . Hence, the transition form factor can be parametrized in the simple pole approximation

[TABLE]

where the pole mass $\Lambda$ should be the mass of the vector resonance near the energy scale of the decaying particle according to the VDM model. In $\psi\to\eta_{c}l^{+}l^{-}$ decays, the pole mass for $J/\psi$ and $\psi^{\prime}$ could be the mass of $\psi^{\prime}$ and $\psi(3770)$, respectively. Similarly, in $\Upsilon\to\eta_{b}l^{+}l^{-}$ decays, the pole mass for $\Upsilon(\rm{2S})$ ($\Upsilon(\rm{3S})$) decay could be the mass of $\Upsilon(\rm{3S})$ ($\Upsilon(\rm{4S})$).

By assuming the simple pole approximation, in which we take $\Lambda=m_{\psi^{\prime}},m_{\psi(3770)},m_{\Upsilon(\rm{3S})},m_{\Upsilon(\rm{4S})}$ for $J/\psi$, $\psi^{\prime}$, $\Upsilon(\rm{2S})$ and $\Upsilon(\rm{3S})$ EM Dalitz decays, respectively; the partial decay widths of $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$ are estimated and presented in Table 2. It is shown that the decay rates for $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$ when $l^{+}l^{-}$ being $\mu^{+}\mu^{-}$ are approximately one-order smaller then the relevant case when the lepton pair is $e^{+}e^{-}$, which is due to the suppression of the phase space and the fast decrease of partial decay rate as the $q^{2}$ raises shown in Fig. 2.

To study the dependence of the decay rates on the value of the pole mass, we varied the pole mass. The numerical result shows both the differential and the total decay rates are not sensitive to the value of the pole mass. The reason can be well understood. The dominant contribution to the decay rate comes from the region of the small value of $q^{2}$ for the sake of phase space suppression in the region of large $q^{2}$ . For the precesses considered in this work, the pole masses are larger then the maximum value of $q^{2}$; therefore, $q^{2}/\Lambda^{2}$ 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 2 based on the VDM are reliable.

III Discussion

III.1 Model predictions of $\psi(\Upsilon)\to\eta_{c}(\eta_{b})\gamma$

The EM Dalitz decays, $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$, are related to the radiative decays, $\psi(\Upsilon)\to\eta_{c}(\eta_{b})\gamma$, by the transition form factor $f_{V\!P}(q^{2})$ and $f_{V\!P}(0)$. Models describing $\psi(\Upsilon)\to\eta_{c}(\eta_{b})\gamma$ can provide information for $f_{V\!P}(q^{2})$. In Ref. Brambilla:2005zw , using the theories of NRQCD and pNRQCD, one studied the M1 transitions between two heavy quarkonia and obtained

[TABLE]

where $e_{Q}$ is the electrical charge of the heavy quark ($e_{c}=2/3$, $e_{b}=-1/3$), and $w(\alpha_{s})$ the function of the strong coupling constant $\alpha_{s}$. The predicted result for $J/\psi\to\eta_{c}\gamma$ is consistent with data Brambilla:2005zw . However, the predicted results for $\psi^{\prime}\to\eta_{c}\gamma$ and $\Upsilon(2S)\to\eta_{b}\gamma$ are larger than data by 2 and 1 order of magnitude, respectively. For $\Upsilon(3S)\to\eta_{b}\gamma$, the prediction of the branching ratio in the relativistic quark model is $Br(\Upsilon(3S)\to\eta_{b}\gamma)=4.0\times 10^{-4}$ in Ref. Ebert:2002pp , which is approximately consistent with experimental measurement $Br(\Upsilon(3S)\to\eta_{b}\gamma)_{\rm EXP}=(5.1\pm 0.7)\times 10^{-4}$ Tanabashi:2018oca . The prediction from the light-front quark model depends on the model-parameters which can be $(1.0\sim 2.5)\times 10^{-4}$ Ke:2010pp . This can also be consistent with experimental data after taking into account both experimental and theoretical errors. For $J/\psi\to\eta_{c}\gamma$, the $w(\alpha_{s})$ has the following form Brambilla:2005zw

[TABLE]

where $C_{F}=4/3$ is the color coefficient, $\alpha_{s}(m_{J/\psi}/2)$ and $\alpha_{s}(p_{J/\psi})$ represent the values of $\alpha_{s}$ in corresponding energy scale, respectively. Typically, $\alpha_{s}(p_{J/\psi})$ satisfies $p_{J/\psi}\approx m_{c}C_{F}\alpha_{s}(p_{J/\psi})/2\approx 0.8~{}\rm{GeV}$ Brambilla:2005zw . To study the dependence of $\Gamma(J/\psi\to\eta_{c}\gamma)$ on the value of $\alpha_{s}(m_{J/\psi}/2)$ and $\alpha_{s}(p_{J/\psi})$ , we varied the $\alpha_{s}(m_{J/\psi}/2)$ and $\alpha_{s}(p_{J/\psi})$. The results are shown in Fig. 3. In this way, a precision measurement of the partial width of $J/\psi\to\eta_{c}\gamma$ can be used to test QCD model and provide stringent restriction on $\alpha_{s}$ in charm energy scale.

III.2 Discussion on the absolute measurement of the partial widths of $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$

With the explicit form factor $f_{V\!P}(q^{2})$, one can obtain the partial decay widths of $\psi(\Upsilon)\to\eta_{c}(\eta_{b})\gamma$ where $q^{2}=0$. Hence, it is important to measure the $q^{2}$-dependent form factor $f_{V\!P}(q^{2})$ , since it can be used to determine the the partial decay widths of $\psi(\Upsilon)\to\eta_{c}(\eta_{b})\gamma$ at $q^{2}=0$.

Actually, for the $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$ decay in $e^{+}e^{-}$ collisions at BESIII Asner:2008nq , Belle  Kou:2018nap , and BaBarLees:2011mx experiments, several hundred million $\psi$ ($\Upsilon$) are collected. The lepton usually could be clearly distinguished from pion, kaon, and proton, so one need only to 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 $\eta_{c}(\eta_{b})$. Hundreds of million of $\Upsilon(2S)$ and $\Upsilon(3S)$ events are collected by Belle and BABAR detector; we estimate that about 30 signal events could be observed for $\eta_{b}e^{+}e^{-}$ mode with typical signal efficiency 4%. Since there are more than 10 billion and 800 million $J/\psi$ and $\psi\prime$ events collected by BESIII detector up to date, we expect $10^{5}$ and $10^{3}$ signal events from $J/\psi$ and $\psi\prime$, respectively, could be observed by BESIII with typical signal efficiency 10%. So it is possible to measure the branching fraction precisely and probe the transition form factor.

IV Summary

In summary, the EM Dalitz decays, $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$, are studied in this work. We investigate the effect of polarization of $\psi(\Upsilon)$ and estimate the partial decay widths of $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}(l=e,\mu)$ by assuming the simple pole approximation. We obtain the polarization components for the vector mesons $\psi$ and/or $\Upsilon$ produced in $e^{+}e^{-}$ colliders. We find the transverse polarization states for $\psi$ and/or $\Upsilon$ dominate in unpolarized $e^{+}e^{-}$ colliders. We obtain the angular distribution of $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$ decays for each polarization state of $\psi$ and/or $\Upsilon$, which is helpful to determine the polarization of $\psi(\Upsilon)$ in the EM Dalitz decays $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$ in experiment. The decay widths of $\psi(\Upsilon)\to\eta_{c}(\eta_{b})l^{+}l^{-}$ are also obtained. Besides, we discuss a QCD model where $\Gamma(J/\psi\to\eta_{c}\gamma)$ is related to $\alpha_{s}$ 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.

V Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under Contracts No. 11335009, No. 11875054, and No. 11875168; the Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contract No. U1532257; CAS under Contract No.QYZDJ-SSW- SLH003; and the National Key Basic Research Program of China under Contract No. 2015CB856700.