Radiative decays of $h_{c}$ to the light mesons $\eta^{(\prime)}$: A perturbative QCD calculation

TL;DR

This paper calculates the radiative decays of $h_c$ to $ ext{eta}^{( extprime)}$ mesons using perturbative QCD, finding insensitivity to light quark masses and a gluonic contribution comparable to quark-antiquark content, and extracting a mixing angle that differs from some previous results.

Contribution

It provides a perturbative QCD calculation of $h_c$ radiative decays including light quark masses and gluonic content, and determines the $ ext{eta}^{( extprime)}$ mixing angle from these decays.

Findings

Branching ratios are insensitive to light quark masses and distribution amplitude shapes.

Gluonic content contribution is comparable to quark-antiquark content in decays.

Extracted mixing angle $ heta=33.8^ ext{o}\pm2.5^ ext{o}$ differs from Feldmann-Kroll-Stech result.

Abstract

We study the radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$ in the framework of perturbative QCD and evaluate analytically the one-loop integrals with the light quark masses kept. Interestingly, the branching ratios $\mathcal{B}(h_{c}\rightarrow\gamma\eta^{(\prime)})$ are insensitive to both the light quark masses and the shapes of $\eta^{(\prime)}$ distribution amplitudes. And it is noticed that the contribution of the gluonic content of $\eta^{(\prime)}$ is almost equal to that of the quark-antiquark content of $\eta^{(\prime)}$ in the radiative decays $h_{c} \rightarrow \gamma\eta^{(\prime)}$. By employing the ratio $R_{h_{c}}=\mathcal{B}(h_{c}\rightarrow\gamma\eta)/\mathcal{B}(h_{c}\rightarrow\gamma\eta^{\prime})$, we extract the mixing angle $\phi=33.8^{\circ}\pm2.5^{\circ}$, which is in clear disagreement with the Feldmann-Kroll-Stech result $\phi=39.0^{\circ}\pm1.6^{\circ}$ extracted from the ratio $R_{J/\psi}$ with nonperturbative matrix elements $\langle 0\mid G^{a}_{\mu\nu}\tilde{G}^{a,\mu\nu}\mid\eta^{(\prime)}\rangle$, but in consistent with $\phi=33.5^{\circ}\pm0.9^{\circ}$ extracted from the asymptotic limit of the $\gamma^{\ast}\gamma-\eta^{\prime}$ transition form factor and $\phi=33.9^{\circ}\pm0.6^{\circ}$ extracted from $R_{J/\psi}$ in perturbative QCD. We also briefly discuss possible reasons for the difference in the determinations of the mixing angle.

Tables

Table 1. Table 1: Gegenbauer coefficients of three sample models with the scale μ 0 = 1 ​ GeV subscript 𝜇 0 1 GeV \mu_{0}=1~{}\mathrm{GeV}

Model	$c_2^q\left({\mu }_0\right)$	$c_4^q\left({\mu }_0\right)$		$c_2^g\left({\mu }_0\right)$
I	$0.10$	$0$.	$10$	$-0.26$
II	$0.20$	$0$.	$00$	$-0.31$
III	$0.25$	$-0$.	$10$	$-0.25$

Table 2. Table 2: The values of ϕ italic-ϕ \phi , f q subscript 𝑓 𝑞 f_{q} and f s subscript 𝑓 𝑠 f_{s} obtained with three phenomenological models  Escribano and Fr e ` ` e \grave{\textrm{e}} re ( 2005 ); Escribano et al. ( 2014 )

	${\varphi }^{\circ }$	$f_q/f_{\pi }$	$f_s/f_{\pi }$
LEPs Escribano and Fr$\stackrel{`}{\text{e}}$re (2005)	$40.6\pm 0.9$	$1.10\pm 0.03$	$1.66\pm 0.06$
$\eta$TFF Escribano et al. (2014)	$40.3\pm 1.8$	$1.06\pm 0.01$	$1.56\pm 0.24$
${\eta }^{\prime }$TFF Escribano et al. (2014)	$33.5\pm 0.9$	$1.09\pm 0.02$	$0.96\pm 0.04$

Table 3. Table 3: The branching ratios ℬ ​ ( h c → γ ​ η ( ′ ) ) ℬ → subscript ℎ 𝑐 𝛾 superscript 𝜂 ′ \mathcal{B}(h_{c}\rightarrow\gamma\eta^{(\prime)}) with only the contributions of the quark-antiquark content

	LEPs	$\eta$TFF	${\eta }^{\prime }$TFF	Exp. Ablikim et al. (2016)
$ℬ\left(h_c\to \gamma {\eta }^{\prime }\right)$	$2.50\times {10}^{-4}$	$2.26\times {10}^{-4}$	$1.32\times {10}^{-4}$	$\left(1.52\pm 0.27\pm 0.29\right)\times {10}^{-3}$
$ℬ\left(h_c\to \gamma \eta \right)$	$6.5\times {10}^{-7}$	$1.1\times {10}^{-6}$	$3.6\times {10}^{-5}$	$\left(4.7\pm 1.5\pm 1.4\right)\times {10}^{-4}$
$R_{h_c}$	$0.3\%$	$0.5\%$	$27.5\%$	$\left(30.7\pm 11.3\pm 8.7\right)\%$

Table 4. Table 4: The branching ratios ℬ ​ ( h c → γ ​ η ( ′ ) ) ℬ → subscript ℎ 𝑐 𝛾 superscript 𝜂 ′ \mathcal{B}(h_{c}\rightarrow\gamma\eta^{(\prime)}) with only the contributions of the gluonic content

	LEPs	$\eta$TFF	${\eta }^{\prime }$TFF	Exp. Ablikim et al. (2016)
$ℬ\left(h_c\to \gamma {\eta }^{\prime }\right)$	$2.50\times {10}^{-4}$	$2.26\times {10}^{-4}$	$1.32\times {10}^{-4}$	$\left(1.52\pm 0.27\pm 0.29\right)\times {10}^{-3}$
$ℬ\left(h_c\to \gamma \eta \right)$	$5.6\times {10}^{-7}$	$1.0\times {10}^{-6}$	$3.1\times {10}^{-5}$	$\left(4.7\pm 1.5\pm 1.4\right)\times {10}^{-4}$
$R_{h_c}$	$0.2\%$	$0.4\%$	$23.8\%$	$\left(30.7\pm 11.3\pm 8.7\right)\%$

Table 5. Table 5: The branching ratios ℬ ​ ( h c → γ ​ η ( ′ ) ) ℬ → subscript ℎ 𝑐 𝛾 superscript 𝜂 ′ \mathcal{B}(h_{c}\rightarrow\gamma\eta^{(\prime)}) with the contributions of both the quark-antiquark content and the gluonic content

	LEPs	$\eta$TFF	${\eta }^{\prime }$TFF	Exp. Ablikim et al. (2016)
$ℬ\left(h_c\to \gamma {\eta }^{\prime }\right)$	$7.06\times {10}^{-4}$	$6.37\times {10}^{-4}$	$3.73\times {10}^{-4}$	$\left(1.52\pm 0.27\pm 0.29\right)\times {10}^{-3}$
$ℬ\left(h_c\to \gamma \eta \right)$	$2.0\times {10}^{-6}$	$3.5\times {10}^{-6}$	$1.1\times {10}^{-4}$	$\left(4.7\pm 1.5\pm 1.4\right)\times {10}^{-4}$
$R_{h_c}$	$0.3\%$	$0.6\%$	$30.1\%$	$\left(30.7\pm 11.3\pm 8.7\right)\%$

Equations

$$A$$

$$f=\frac{K}{2}+q_c,\bar{f}=\frac{K}{2}-q_c$$

$$\chi (K,q_c)=2\pi \delta (q_c^0){\psi }_{1m}({\mathbf{q}}_{\mathbf{c}})\mathcal{P}(K,q_c),$$

$$\mathcal{P}(K,q_c)$$

$$A$$

$${\mathcal{P}}_{\alpha }(K,0)={\frac{\partial }{\partial q_c^{\alpha }}\mathcal{P}(K,q_c)|}_{q_c=0},{\hat{\mathcal{O}}}_{\alpha }(0)={\frac{\partial }{\partial q_c^{\alpha }}\hat{\mathcal{O}}(q_c)|}_{q_c=0}.$$

$$A$$

$$\int \frac{{\mathrm{d}}^3{q}_c}{(2\pi {)}^3}{\psi }_{1m}({\mathbf{q}}_{\mathbf{c}})q_c^{\alpha }=-i\sqrt{\frac{3}{4\pi }}{R}_{h_c}^{\prime }(0){\epsilon }^{\alpha }(K).$$

$$\mathcal{P}(K,0)$$

$${\mathcal{P}}^{\alpha }(K,0)$$

$$\hat{\mathcal{O}}(0)$$

$${\hat{\mathcal{O}}}^{\alpha }(0)$$

$$⟨{\eta }^{(\prime )}(p)|{\bar{q}}_{\alpha }(x)q_{\beta }(y)|0⟩$$

$$⟨0|\bar{q}(0){\gamma }_{\mu }{\gamma }_{5}q(0)|{\eta }^{(\prime )}(p)⟩$$

$${\mathcal{M}}_{\mu \nu }=-i(4\pi {\alpha }_s){\delta }_{ab}{ϵ}_{\mu \nu \rho \sigma }{k}_1^{\rho }{k}_2^{\sigma }{F}_{g^{\ast }{g}^{\ast }-{\eta }^{(\prime )}}(k_1^2,k_2^2),$$

$$F_{g^{\ast }{g}^{\ast }-{\eta }^{(\prime )}}(k_1^2,k_2^2)=\frac{1}{6}\sum _{q=u,d,s}{f}_{{\eta }^{(\prime )}}^q{\int }_0^1\mathrm{d}u{\varphi }^q(u,\mu )\left(\frac{1}{\bar{u}{k}_1^2+u{k}_2^2-u\bar{u}{m}_{{\eta }^{(\prime )}}^2-m_q^2+iϵ}+(u\leftrightarrow \bar{u})\right),$$

$${\varphi }^q(u)$$

$$M_T=T^{\alpha \beta }{\epsilon }_{\alpha }(K){ϵ}_{\beta }^{\ast }(k)=\frac{1}{2}\int \frac{{\mathrm{d}}^4{k}_1}{(2\pi {)}^4}{A}^{\alpha \beta \mu \nu }{M}_{\mu \nu }{\epsilon }_{\alpha }(K){ϵ}_{\beta }^{\ast }(k)\frac{i}{k_1^2+iϵ}\frac{i}{k_2^2+iϵ},$$

$$T^{\alpha \beta }\sim \left(-g^{\alpha \beta }+\frac{k^{\alpha }{K}^{\beta }}{k\cdot K}\right),$$

$$T^{\alpha \beta }$$

$$h^{\alpha \beta }=\left(-g^{\alpha \beta }+\frac{k^{\alpha }{K}^{\beta }}{k\cdot K}\right).$$

$${\mathbb{P}}^{\alpha \beta }=\frac{1}{2}{h}_{{\alpha }^{\prime }{\beta }^{\prime }}\left(-g^{\alpha {\alpha }^{\prime }}+\frac{K^{\alpha }{K}^{{\alpha }^{\prime }}}{M^2}\right)\left(-g^{\beta {\beta }^{\prime }}\right)=\frac{1}{2}\left(-g^{\alpha \beta }+\frac{k^{\alpha }{K}^{\beta }}{k\cdot K}\right),$$

$$H_{QCD}^q$$

$$H_q$$

$$I_q(u)$$

$$D_1$$

$$D_2$$

$$D_3$$

$$D_4$$

$$D_5$$

$$N_1$$

$$N_2$$

$$N_3$$

$$N_4$$

$$N_5$$

Full text

I Introduction

In recent years, the radiative decays of charmonia to the light mesons $\eta^{(\prime)}$ have been revisited in various approaches, such as perturbative quantum chromodynamics (pQCD) Ma (2002); Yang (2004); Li et al. (2005); Li (2008); He and Yang (2019) and phenomenological models G$\acute{\textrm{e}}$rard and Kou (2005); Zhao (2011); G$\acute{\textrm{e}}$rard and Martini (2014) (see He and Yang (2019) and references therein for more details), since they are closely related to the issue of $\eta-\eta^{\prime}$ mixing, which is important ingredient for understanding many interesting phenomena related to $\eta^{(\prime)}$. In Ref. He and Yang (2019), the processes $J/\psi\rightarrow\gamma\eta^{(\prime)}$ were investigated and the $R_{J/\psi}=\mathcal{B}(J/\psi\rightarrow\gamma\eta^{\prime})/\mathcal{B}(J/\psi\rightarrow\gamma\eta)$ was used as the main input to extract the mixing angle $\phi=33.9^{\circ}\pm 0.6^{\circ}$, which is in excellent agreement with the value $\phi=33.5^{\circ}\pm 0.9^{\circ}$ Escribano et al. (2014) extracted from the asymptotic limit of the $\gamma^{\ast}\gamma-\eta^{\prime}$ transition form factor (TFF) at $Q^{2}\rightarrow+\infty$, but in clear disagreement with the Feldmann-Kroll-Stech (FKS) result $\phi=39.0^{\circ}\pm 1.6^{\circ}$ Feldmann et al. (1998) extracted from the ratio $R_{J/\psi}$ with nonperturbative matrix elements $\langle 0\mid G^{a}_{\mu\nu}\tilde{G}^{a,\mu\nu}\mid\eta^{(\prime)}\rangle$. The difference may arise from the $g^{\ast}g^{\ast}-\eta^{(\prime)}$ TFF used in the Ref. He and Yang (2019), in like manner, the $\gamma^{\ast}\gamma-\eta^{\prime}$ TFF used in Ref. Escribano et al. (2014). Anyhow, more investigations are needed to provide a better understanding of the $\eta-\eta^{\prime}$ mixing.

Recently, the branching ratios of $h_{c}\rightarrow\gamma\eta^{\prime}$ and $h_{c}\rightarrow\gamma\eta$ are first measured to be, respectively, $(1.52\pm 0.27\pm 0.29)\times 10^{-3}$ and $(4.7\pm 1.5\pm 1.4)\times 10^{-4}$ with a statistical significance of $8.4\sigma$ and $4.0\sigma$ by the BESIII Collaboration Ablikim et al. (2016), where $h_{c}$ is assigned as $P$-wave charmonium state with the quantum numbers $J^{PC}=1^{+-}$ Tanabashi et al. (2018). In the literature, there are fewer studies on the exclusive radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$ Zhu and Dai (2016); Wu et al. (2017). In Ref. Zhu and Dai (2016), Zhu and Dai investigated these channels in the nonrelativistic QCD (NRQCD), where the contributions from the color-octet state of $h_{c}$ are suppressed by a relative factor $v_{c\bar{c}}^{2}\alpha_{s}$. And they adopted the result of the Buchmuller-Tye potential model Buchmuller and Tye (1981); Eichten and Quigg (1995) for the value of the nonperturbative matrix elements, which can be directly related to the derivative of the nonrelativistic wave function at the origin Bodwin et al. (1995). While, for $\eta^{\prime}$ production, they evaluated the contributions from the gluonic content of $\eta^{\prime}$ by the TFF $F^{(g)}_{\eta^{\prime}g^{\ast}g^{\ast}}$ Ali and Parkhomenko (2002). However, in fact, the TFF $F^{(g)}_{\eta^{\prime}g^{\ast}g^{\ast}}$ enters the decay amplitude of $h_{c}\rightarrow\gamma\eta^{\prime}$ from one-loop processes. It means that Zhu and Dai Zhu and Dai (2016) missed the leading order contributions of the gluonic content of $\eta^{\prime}$, which come from the tree level processes. Besides the QCD approach, Wu et al. studied the two decay processes with an intermediate meson loops model Wu et al. (2017), and their predictions were compatible with the experimental measurements Ablikim et al. (2016).

From a general point of view Appelquist and Politzer (1975); De R$\acute{\textrm{u}}$jula and Glashow (1975); Barbieri et al. (1976a); Novikov et al. (1978); Barbieri et al. (1979); Mackenzie and Lepage (1981); K$\ddot{\textrm{o}}$rner et al. (1983); K$\ddot{\textrm{u}}$hn (1983), the decays of heavy quarkonium can be assumed that the annihilation of the heavy quark and antiquark is a short-distance process which can be described by perturbation theory, while the nonperturbative effect of the bound state could be factorized into its Bethe-Salpeter (B-S) wave function. In this work, we investigate the radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$ in the framework of pQCD and take a nonrelativistic quark model with the zero-binding approximation for the initial bound state $h_{c}$. For the final light mesons $\eta^{(\prime)}$, because of the large momentum transfer, light-cone distribution amplitudes (DAs) are adopted. As mentioned above, the issue of $\eta-\eta^{\prime}$ mixing is also involved in the decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$, and we address this issue along the same line as Ref. He and Yang (2019).

In this paper, we present a detailed calculation of the radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$. The involved five-point and four-point one-loop integrals are decomposed into a sum of three-point integrals and then evaluated analytically with the light quark masses kept. The branching ratios $\mathcal{B}(h_{c}\rightarrow\gamma\eta^{(\prime)})$ are found to be insensitive to the light quark masses and the shapes of the light meson DAs, which is accord with the situation in the decay processes $J/\psi\rightarrow\gamma\eta^{(\prime)}$ He and Yang (2019). And our numerical results show that the contributions from the gluonic content of $\eta^{(\prime)}$ and those from quark-antiquark content of $\eta^{(\prime)}$ are comparable with each other. The possible reasons are that: Firstly, the quark-antiquark contributions are Okubo-Zweig-Iizuka (OZI) suppressed. Specifically, the quark-antiquark contributions, which come from the one-loop OZI-forbidden processes, are suppressed by the factor $\alpha_{s}$ as compared with the gluonic contributions in the decay amplitudes. Secondly, the gluonic DA of $\eta^{(\prime)}$ can mix with their quark-antiquark DA under the QCD evolution due to the $U_{A}(1)$ anomaly Kroll and Passek-Kumeri$\check{\textrm{c}}$ki (2003), which makes the gluonic part of the $\eta^{(\prime)}$ wave functions be not negligible at the charmonium scale. In addition, we should note that, in the radiative decays $J/\psi\rightarrow\gamma\eta^{(\prime)}$ He and Yang (2019); Baier and Grozin (1981); Ma (2002), the gluonic contributions are strongly suppressed by the factor $m^{2}_{\eta^{(\prime)}}/M_{J/\psi}^{2}$ because of the spin structure of their amplitudes. While similar suppressions do not exist in the decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$.

The paper is organized as follows. The formalism for the decay processes $h_{c}\rightarrow\gamma\eta^{(\prime)}$ is presented in section II. In section III we present our numerical results while section IV is our summary. The expressions of the eleven numerators introduced in section II are given in the Appendix.

II The radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$ in pQCD

II.1 The contributions of the quark-antiquark content of $\eta^{(\prime)}$

For the quark-antiquark content of $\eta^{(\prime)}$, the leading order contributions to the radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$ come from one-loop processes, and one of the corresponding Feynman diagrams is depicted in Fig. 1. There are other five diagrams from permutations of the photon and gluon legs. Following the procedure given in Ref. K$\ddot{\textrm{o}}$rner et al. (1983), it is convenient to divide the covariant amplitude of $h_{c}\rightarrow\gamma\eta^{(\prime)}$ into two independent amplitudes. One amplitude describes the effective coupling between $h_{c}$, a real photon and two virtual gluons, and the other describes the effective coupling between $\eta^{(\prime)}$ and two virtual gluons. Then the amplitude of $h_{c}\rightarrow\gamma\eta^{(\prime)}$ can be obtained by multiplying the two amplitudes, inserting the gluon propagators and performing the loop integration.

In the rest frame of $h_{c}$, the amplitude of $h_{c}\rightarrow\gamma g^{\ast}g^{\ast}$ can be given by Guberina and K$\ddot{\textrm{u}}$hn (1981)

[TABLE]

where $\chi(K,q_{c})$ is the B-S wave function of $h_{c}$ and $\hat{\mathcal{O}}(q_{c})$ is the hard-scattering amplitude. Here the factor $\sqrt{3}$ is included to account for the color properties of the quark and antiquark. $K$, $k$, $k_{1}$, $k_{2}$ and $\varepsilon(K)$, $\epsilon(k)$, $\epsilon(k_{1})$, $\epsilon(k_{2})$ are the momenta and polarization vectors of the $h_{c}$, the photon and the two gluons, respectively. The momenta of the quark $c$ and antiquark $\bar{c}$ have the form

[TABLE]

with $q_{c}$ the relative momentum between the quark $c$ and antiquark $\bar{c}$. In a nonrelativistic bound state picture, the B-S wave function $\chi(K,q_{c})$ can be reduced to its nonrelativistic form  K$\ddot{\textrm{u}}$hn et al. (1979); Guberina et al. (1980)

[TABLE]

where $\psi_{1m}(\bm{q_{c}})$ is the bound state wave function of $P$-wave charmonium $h_{c}$. The spin projection operator $\mathcal{P}(K,q_{c})$ can be written in a covariant form Guberina et al. (1980)

[TABLE]

with $m_{c}$ the $c$ quark mass. Since the wave function $\psi_{1m}(\bm{q_{c}})$ is sharply damped for relative momenta which become large at the scale of the $h_{c}$ mass, one can expand the $\mathrm{Tr}[\mathcal{P}(K,q_{c})\hat{\mathcal{O}}(q_{c})]$ to the first order in $q_{c}$ for the $P$-wave charmonium $h_{c}$. Then the amplitude can be rewritten as

[TABLE]

where

[TABLE]

With the zero-binding approximation K$\ddot{\textrm{u}}$hn et al. (1979); Guberina et al. (1980), one can obtain

[TABLE]

where $R^{\prime}_{h_{c}}(0)$ denotes the derivative of the radial wave function of $h_{c}$ evaluated at the origin

[TABLE]

Taking the nonrelativistic approximation $m_{c}\approx M/2$, one can obtain

[TABLE]

and

[TABLE]

At the leading twist level, the light-cone matrix elements of the quark-antiquark content of $\eta^{(\prime)}$ are given by Ball and Jones (2007)

[TABLE]

where the decay constants $f_{\eta^{(\prime)}}^{q}$ are defined as

[TABLE]

Following the conventions in Refs. Chernyak and Zhitnitsky (1984); Muta and Yang (2000); Yang and Yang (2001); Ali and Parkhomenko (2002); Yang (2004), the $g^{*}g^{*}-\eta^{(\prime)}$ TFFs can be parameterized as

[TABLE]

and the TFFs $F_{g^{\ast}g^{\ast}-\eta^{(\prime)}}(k_{1}^{2},k_{2}^{2})$ read

[TABLE]

where $\bar{u}=1-u$, $u$ is the momentum fraction carried by the quark, $m_{q}$ is the mass of the quark ($q=u,d,s$). And the light-cone DA has the form Agaev et al. (2014)

[TABLE]

with $c^{q}_{n}(\mu)$ the Gegenbauer moments. In Table 1, we list the three models of the DAs discussed in Ref. Agaev et al. (2014). The shapes of the corresponding DAs are shown in Fig. 2, where the $c^{q}_{n}(\mu)$ are evaluated at the scale $\mu=M/2$.

As mentioned above, one can obtain the amplitude of $h_{c}\rightarrow\gamma\eta^{(\prime)}$ by multiplying the two part amplitudes, inserting the gluon propagators and integrating over the loop momentum

[TABLE]

where the factor $\frac{1}{2}$ takes into account that the two gluons have already been interchanged both in $A^{\alpha\lambda\mu\nu}$ and $M_{\mu\nu}$. Using parity conservation, Lorentz invariance and gauge invariance, one can prove that

[TABLE]

i.e., there is only one independent helicity amplitude $H_{QCD}^{q}$ K$\ddot{\textrm{o}}$rner et al. (1983)

[TABLE]

where

[TABLE]

With the help of the helicity projector K$\ddot{\textrm{o}}$rner et al. (1983)

[TABLE]

we obtain the helicity amplitude

[TABLE]

where the dimensionless function $H_{q}$ reads

[TABLE]

$I_{q}(u)$ is the summation of the loop integrals of the six Feynman diagrams

[TABLE]

with $l=k_{1}-k_{2}$. Here the expressions of the denominators are given by

[TABLE]

and the seven numerators $N_{i}$ ($i=1\thicksim 7$) read

[TABLE]

with $x=m^{2}/M^{2}$ and $\xi=1-2u$. Before going to calculate the integrals, it is useful and essential to present a short analysis of its infra-red (IR) properties. Obviously, when one of the gluons becomes soft, e.g. $l\rightarrow p$, the propagator denominators $D_{1}$ (with $u=1-m_{q}/m$), $D_{3}$ and $D_{5}$ tend to zero, namely the individual integral may encounter soft singularities according to the conclusion in the Ref. Dittmaier (2003). However, for a given decay process, one need to go a step further and make an analysis of the numerator and denominator in the integrand simultaneously. When $l=p+\lambda$ (the $\lambda$ is a small quantity), one can obtain the numerators

[TABLE]

and the on-shell propagator denominators

[TABLE]

For the ultrasoft gluon ($\lambda\rightarrow 0$), the contributions to the loop function $I_{q}(u)$ have the form

[TABLE]

It means the loop function is IR safe in the potentially dangerous region.

By using the algebraic identities

[TABLE]

and

[TABLE]

with $y=4m_{q}^{2}/M^{2}$, the loop function $I_{q}(u)$ is decomposed into the sum of three-point one-loop integrals, which can be analytically calculated with the technique proposed in Ref. ’t Hooft and Veltman (1979) or the computer program $Package-\mathrm{X}$ Patel (2015, 2017). Performing the convolution integral between the loop function $I_{q}(u)$ and the DA $\phi^{q}(u)$, our results show that the dimensionless function $H_{q}$ is insensitive to the light quark mass $m_{q}$. Specifically, the change of the absolute value of the dimensionless function $H_{q}$ does not exceed $2\%$ when the value of $m_{q}$ varies in the range $0\sim 100\,\mathrm{MeV}$ for all the three kinds of $\eta^{(\prime)}$ DAs in Fig. 2. As a consequence, the light quark mass can be neglected safely and reasonably. As shown schematically in Fig. 3, we present the light quark mass $m_{q}$-dependence of the dimensionless functions $H_{q}^{\eta}=H_{q}{\mid}_{m=m_{\eta}}$ and $H_{q}^{\eta^{\prime}}=H_{q}{\mid}_{m=m_{\eta^{\prime}}}$ with the “narrow” DA (Model I in Fig. 2). The property of the dimensionless function $H_{q}$ is similar with that in the radiative decays $J/\psi\rightarrow\gamma\eta^{(\prime)}$ He and Yang (2019).

For showing the analytical expression of the loop function more clearly, we define

[TABLE]

and then the helicity amplitude $H_{QCD}^{q}$ in Eq. (21) can be simplified to

[TABLE]

with the effective decay constants

[TABLE]

Finally, we present a brief analysis about the loop function $I_{0}(u)$. Taking the substitution $l\rightarrow-l$, one can obtain

[TABLE]

then the loop function $I_{0}(u)$ can be reduced to eleven one-loop integrals

[TABLE]

and the expressions of the eleven numerators $n_{i}$ ($i=1\sim 11$) are given in the Appendix. It is noticed that the loop functions $I^{\eta}_{0}(u)=I_{0}(u){\mid}_{m=m_{\eta}}$ and $I^{\eta^{\prime}}_{0}(u)=I_{0}(u){\mid}_{m=m_{\eta^{\prime}}}$ are quite steady over the most region of $u$ as shown in Fig. 4. Consequently, the convolution integral between the loop function $I_{0}(u)$ and the DA becomes insensitive to the shape of the final meson DA—in other words, it almost becomes the normalization of the DA. Specifically, the change among the dimensionless function $H_{0}$ with the different models of the DAs in Fig. 2 is less than $2\%$.

II.2 The contributions of the gluonic content of $\eta^{(\prime)}$

For the gluonic content of $\eta^{(\prime)}$, the leading order contributions to the radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$ come from the tree level processes. One of the Feynman diagrams is shown in Fig. 5, and the other two diagrams arise from permutations of the photon and gluon legs.

The leading twist in the light-cone expansion of the matrix elements of the meson $\eta^{(\prime)}$ over two-gluon fields is Kroll and Passek-Kumeri$\check{\textrm{c}}$ki (2003); Ball and Jones (2007); Agaev et al. (2014):

[TABLE]

with the effective decay constant $f_{\eta^{(\prime)}}^{1}=\frac{1}{\sqrt{3}}(f_{\eta^{(\prime)}}^{u}+f_{\eta^{(\prime)}}^{d}+f_{\eta^{(\prime)}}^{s})$ and the gluonic twist-$2$ DA Agaev et al. (2014); Ball and Jones (2007); Alte et al. (2016)

[TABLE]

Then we can obtain the corresponding helicity amplitude

[TABLE]

with

[TABLE]

Generally, the contributions of the gluonic content are expected to be small, since the gluonic content can be seen as the higher-order effects from the point of view of the QCD evolution of the gluon DA, which vanishes in the asymptotic limit. However, due to the $U_{A}(1)$ anomaly, the gluonic DA of $\eta^{(\prime)}$ can mix with their quark-antiquark DA, which makes the contributions of the gluonic content become important in the $\eta^{(\prime)}$ production. Moreover, unlike the situation in the radiative decays $J/\psi\rightarrow\gamma\eta^{(\prime)}$ Baier and Grozin (1981); Ma (2002); He and Yang (2019), in which the contributions of the gluonic content are strongly suppressed by a factor $x=m^{2}/M^{2}$, there is no additional suppression factor in the decay processes $h_{c}\rightarrow\gamma\eta^{(\prime)}$, i.e., the gluonic content of $\eta^{(\prime)}$ may play an important role in $h_{c}\rightarrow\gamma\eta^{(\prime)}$.

III Numerical results

In the rest frame of $h_{c}$, the decay width of $h_{c}\rightarrow\gamma\eta^{(\prime)}$ can be given by:

[TABLE]

In the numerical calculations, we employ the data given by the Particle Data Group Tanabashi et al. (2018): $M=3525\,\mathrm{MeV}$, $m_{\eta}=548\,\mathrm{MeV}$, $m_{\eta^{\prime}}=958\,\mathrm{MeV}$, $\Gamma_{h_{c}}=(0.70\pm 0.28\pm 0.22)\,\mathrm{MeV}$ and the decay constant $f_{\pi}=130.2\,\mathrm{MeV}$. The strong coupling constant $\alpha_{s}(M/2)=0.32$, which is calculated through the two-loop renormalization group equation. For the derivative of the radial wave function of $h_{c}$ evaluated at the origin $R_{h_{c}}^{\prime}(0)$, we adopt the result of the Cornell potential model Eichten et al. (1978, 1980); Eichten and Quigg (1995)

[TABLE]

For the Gegenbauer moments $c^{q}_{2}(\mu)$, $c^{q}_{4}(\mu)$, there are still large uncertainties as depicted in Table 1. Fortunately, the dimensionless function $H_{0}$ in Eq. (II.1) is insensitive to the shapes of the $\eta^{(\prime)}$ DAs as we have shown in section II. So in the following numerical calculations, we choose the Model I in Table 1 for the DA with the scale $\mu=M/2$.

For $\eta-\eta^{\prime}$ system, the physical states $|\eta^{(\prime)}\rangle$ are usually treated as the mixing of the flavor states $|\eta_{q}\rangle=1/\sqrt{2}|u\bar{u}+d\bar{d}\rangle$ and $|\eta_{s}\rangle=|s\bar{s}\rangle$ because of the $U_{A}(1)$ anomaly. As a manifestation of the celebrated OZI-rule, one mixing angle is included in the flavor basis, and more details could be found in Refs. Feldmann et al. (1998); Agaev et al. (2014). This is the known FKS scheme Feldmann et al. (1998, 1999); Feldmann (2000), in which the decay constants are parameterized as

[TABLE]

Here the following definitions have been used Feldmann et al. (1998, 1999); Feldmann (2000)

[TABLE]

with the currents $J_{\mu 5}^{q}=1/\sqrt{2}(\bar{u}\gamma_{\mu}\gamma_{5}u+\bar{d}\gamma_{\mu}\gamma_{5}d)$ and $J_{\mu 5}^{s}=\bar{s}\gamma_{\mu}\gamma_{5}s$. The mixing angle $\phi$ and the decay constants $f_{q}$, $f_{s}$ are three phenomenological parameters, which have been determined in different methods Feldmann et al. (1998); Escribano and Fr$\grave{\textrm{e}}$re (2005); Escribano and Nadal (2007); Cao (2012); Escribano et al. (2014). In Table 2, we take the up-to-date values from Refs. Escribano and Fr$\grave{\textrm{e}}$re (2005); Escribano et al. (2014).

The parameters extracted from the low energy processes (LEPs) $V\rightarrow\eta^{(\prime)}\gamma$, $\eta^{(\prime)}\rightarrow V\gamma$ ($V=\rho,\omega,\phi$) are listed in the first line. In the second line, the parameters are extracted with rational approximations for the $\eta$ TFF $F_{\gamma^{\ast}\gamma\eta}(Q^{2}\rightarrow+\infty)$. It is noteworthy that both the parameters in the first line and those in the second line are consistent with the FKS results Feldmann et al. (1998). While in the third line, the parameters are extracted with rational approximations for the $\eta^{\prime}$ TFF $F_{\gamma^{\ast}\gamma\eta^{\prime}}(Q^{2}\rightarrow+\infty)$, which is in accord with the BaBar measurements in the timelike region at $q^{2}=112$ GeV2 Aubert et al. (2006).

For comparison, in Tables 3 and 4, we present the results with only the quark-antiquark contributions and those with only the gluonic contributions, respectively. Here $\Gamma_{h_{c}}=0.70~{}\mathrm{MeV}$ is adopted. The branching ratios $\mathcal{B}(h_{c}\rightarrow\gamma\eta^{\prime})$, $\mathcal{B}(h_{c}\rightarrow\gamma\eta)$ and their ratio $R_{h_{c}}=\mathcal{B}(h_{c}\rightarrow\gamma\eta)/\mathcal{B}(h_{c}\rightarrow\gamma\eta^{\prime})$ are presented in the first, second and third lines of the Tables 3 and 4, respectively. At first glance (Fig. 1 and Fig. 5), it seems that the branching ratios may be dominated by the “unsuppressed” tree level contributions from the gluonic content of $\eta^{(\prime)}$ (at order $\alpha\alpha_{s}^{2}$), rather than the one-loop contributions from the quark-antiquark content of $\eta^{(\prime)}$ (at order $\alpha\alpha_{s}^{4}$). However, from the two tables, one can find that the quark-antiquark content and the gluonic content of $\eta^{(\prime)}$ are of almost equal importance in the radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$.

In Table 5, the results with the contributions from both the quark-antiquark content and the gluonic content are presented. Here we also adopt $\Gamma_{h_{c}}=0.70~{}\mathrm{MeV}$. From Table 5, the branching ratios $\mathcal{B}(h_{c}\rightarrow\gamma\eta^{\prime})$ and $\mathcal{B}(h_{c}\rightarrow\gamma\eta)$ are found to be greatly enhanced, because of the constructive interference of the quark-antiquark contributions and the gluonic contributions. However, both $\mathcal{B}(h_{c}\rightarrow\gamma\eta^{\prime})$ and $\mathcal{B}(h_{c}\rightarrow\gamma\eta)$ are still smaller than their experimental values. In addition, the ratio $R_{h_{c}}$ can be comparable with its experimental value only with the $\eta^{\prime}$TFF set of parameter values.

Considering the large uncertainties from the experimental measurement of the decay width $\Gamma_{h_{c}}$ Ablikim et al. (2012), the derivative of the radial wave function at the origin $R_{h_{c}}^{\prime}(0)$ Eichten and Quigg (1995) and the factor $\alpha_{s}^{4}(\mu)$ involved in the branching ratios, it is very hard to give precise prediction for the individual branching ratio in practice. Even so, their ratio $R_{h_{c}}$ could be predicted much more reliable, since the ratio $R_{h_{c}}$ is independent of the decay width $\Gamma_{h_{c}}$ and the derivative of the radial wave function at the origin $R_{h_{c}}^{\prime}(0)$. Furthermore, the dependence of the ratio $R_{h_{c}}$ on $\alpha_{s}(\mu)$ is also cut down to a large extent. So we are more interested in the ratio $R_{h_{c}}$ rather than the individual branching ratio.

Without inputting any phenomenological parameters, we present a determination of the mixing angle $\phi$ by the ratio $R_{h_{c}}$

[TABLE]

and the ratio

[TABLE]

Employing the experimental data Ablikim et al. (2016); Babusci et al. (2013); Tanabashi et al. (2018)

[TABLE]

one can obtain the mixing angle and the ratio $f_{s}/f_{q}$

[TABLE]

where the big uncertainty comes mainly from $R^{exp}_{h_{c}}$ Ablikim et al. (2016). Schematically, we show the dependence of the ratio $R_{h_{c}}$ on the mixing angle $\phi$ in Fig. 6.

Obviously, the mixing angle $\phi$ determined by the ratio $R_{h_{c}}$ is consistent with the $\eta^{\prime}$TFF result $\phi=33.5^{\circ}\pm 0.9^{\circ}$ Escribano et al. (2014), but in clear disagreement the FKS result $\phi=39.0^{\circ}\pm 1.6^{\circ}$ extracted from the ratio $R_{J/\psi}$ with nonperturbative matrix elements $\langle 0\mid G^{a}_{\mu\nu}\tilde{G}^{a,\mu\nu}\mid\eta^{(\prime)}\rangle$ due to $U_{A}(1)$ anomaly dominance argument Feldmann et al. (1998) and the LEPs result $\phi=40.6^{\circ}\pm 0.9^{\circ}$ extracted from the low energy processes Escribano and Fr$\grave{\textrm{e}}$re (2005). In lattice QCD, the UKQCD collaboration Gregory et al. (2012) presented a value $\phi^{fit}\sim 34^{\circ}$, while the ETM collaboration Ottnad and Urbach (2018) gave $\phi=38.8^{\circ}\pm 3.3^{\circ}$. The difference in the determinations of the mixing angle $\phi$ may arise from the $g^{\ast}g^{\ast}-\eta^{(\prime)}$ TFF used in our calculation, in like manner, the $\gamma^{\ast}\gamma-\eta^{\prime}$ TFF used in Ref. Escribano et al. (2014). What is more interesting is that, in the same framework of pQCD, the mixing angles extracted from the ratio $R_{h_{c}}$ ($\phi=33.8^{\circ}\pm 2.5^{\circ}$) and the ratio $R_{J/\psi}$ ($\phi=33.9^{\circ}\pm 0.6^{\circ}$) He and Yang (2019) are in excellent agreement with each other. In addition, the central value of the ratio $f_{s}/f_{q}$ obtained in both this work and Refs. Escribano et al. (2014); He and Yang (2019) is smaller than unity. While in the LEPs $V\rightarrow\eta^{(\prime)}\gamma$, $\eta^{(\prime)}\rightarrow V\gamma$ ($V=\rho,\omega,\phi$) Feldmann et al. (1998); Escribano and Fr$\grave{\textrm{e}}$re (2005), one usually predicted $f_{s}>f_{q}$, which is different from the result obtained in this work. However, from another point of view, the discrepancies in these determinations may indicate that our understanding of $\eta-\eta^{\prime}$ mixing scheme is incomplete, and the physical picture of the $\eta-\eta^{\prime}$ mixing at the high energy scale may differ from that at the low energy scale. Anyhow, the physics associated with the $\eta-\eta^{\prime}$ mixing is interesting, and it is certainly worthy of further investigations for a better understanding of many phenomena in $\eta^{(\prime)}$ production processes.

IV Summary

In this work, we have calculated the branching ratios of the radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$ in the framework of pQCD. For the initial heavy quarkonium $h_{c}$, we neglect its internal momentum in a nonrelativistic picture. While the final light mesons $\eta^{(\prime)}$ are treated as a light-cone object, and we employ a set of DAs of the quark-antiquark content and those of the gluonic content as nonperturbative inputs. Using some algebraic identities, we decompose the five-point and four-point one-loop integrals into the three-point one-loop integrals and calculate these three-point one-loop integrals analytically. Our results of the branching ratios $\mathcal{B}(h_{c}\rightarrow\gamma\eta^{(\prime)})$ are insensitive to the light quark masses and the shapes of the $\eta^{(\prime)}$ DAs, which is similar to the situation in the decay processes $J/\psi\rightarrow\gamma\eta^{(\prime)}$ He and Yang (2019). Furthermore, we find that the contributions from the quark-antiquark content are almost equal to those from the gluonic content, which are strongly suppressed in the heavy quarkonium decays $V\rightarrow\gamma\eta^{(\prime)}$ Baier and Grozin (1981); Ma (2002); He and Yang (2019). Interestingly enough, only with the set of $\phi$, $f_{q}$ and $f_{s}$ extracted from the asymptotic limit of the $\eta^{\prime}$ TFF, which is in accord with the BaBar measurement at $q^{2}=112$ GeV2 Aubert et al. (2006), our result of the ratio $R_{h_{c}}$ is comparable with its experimental data. As a crossing check, by using the $R_{h_{c}}$, $\Gamma(\eta^{(\prime)}\rightarrow\gamma\gamma)$ and their experimental values, we obtain the mixing angle $\phi=33.8^{\circ}\pm 2.5^{\circ}$, which is in excellent consistent with the $\eta^{\prime}$TFF result $\phi=33.5^{\circ}\pm 0.9^{\circ}$ Escribano et al. (2014) and the result $\phi=33.9^{\circ}\pm 0.6^{\circ}$ extracted by the ratio $R_{J/\psi}$ He and Yang (2019).

For the individual branching ratios $\mathcal{B}(h_{c}\rightarrow\gamma\eta^{\prime})$ and $\mathcal{B}(h_{c}\rightarrow\gamma\eta)$, there are still considerable discrepancies between our results and the experimental measurements. The reason may be due to the sensitivity of the hard-scattering amplitude $\hat{\mathcal{O}}(q_{c})$ to the internal momentum $q_{c}$ in the $P$-wave decays, which can make the convergence of the $q_{c}$-expansion of $\mathrm{Tr}[\mathcal{P}(K,q_{c})\hat{\mathcal{O}}(q_{c})]$ become poor. It is well known that there are IR divergences in the color-singlet state contributions for the inclusive $P$-wave charmonia decays Barbieri et al. (1976b); Bodwin et al. (1992), and these IR divergences can be removed by considering higher-order contributions. While for the exclusive $P$-wave charmonia decays, the same IR divergences always do not appear. However, as pointed out in Refs. Kroll (1998); Wong (1999), the higher-order contributions, such as the higher Fock-state contributions and the relativistic corrections, are still important to the exclusive $P$-wave charmonia decays. It indicates that the nonperturbative effects beyond those contained in $R_{h_{c}}^{\prime}(0)$ may also play an important role in the radiative decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$, even though the results with the zero-binding approximation for these decays are IR safe. Within a B-S equation approach, we would revisit the decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$ by retaining the relative momentum $q_{c}$ in the hard-scattering amplitude (i.e., the relativistic corrections), and this work is under way.

Acknowledgements.

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11675061, 11775092 and 11435003.

Appendix: The expressions of the eleven numerators $n_{i}$ ($i=1\sim 11$)

The expressions of the numerators $n_{i}$ read

[TABLE]