Power corrections to the pion transition form factor from higher-twist distribution amplitudes of photon

TL;DR

This paper calculates power-suppressed contributions from higher-twist photon distribution amplitudes to the pion transition form factor, highlighting their significance at low momentum transfer and implications for pion LCDA parameters.

Contribution

It introduces a detailed analysis of twist-4 photon LCDAs' impact on the pion transition form factor using light-cone sum rules, emphasizing the importance of these corrections.

Findings

Three-particle twist-4 LCDAs contribute significantly at low Q^2.

Power corrections affect the extraction of pion Gegenbauer moments.

Inclusion of these corrections improves theoretical predictions.

Abstract

In this paper we investigate the power suppressed contributions from two-particle and three-particle twist-4 light-cone distribution amplitudes (LCDAs) of photon within the framework of light-cone sum rules. Compared with leading twist LCDA result, the contribution from three-particle twist-4 LCDAs is not suppressed in the expansion by $1/Q^2$, so that the power corrections considered in this work can give rise to a sizable contribution, especially at low $Q^2$ region. According to our result, the power suppressed contributions should be included in the determination of the Gegenbauer moments of pion LCDAs with the pion transition form factor.

Tables

Table 1. Table 1: Numerical values of the nonperturbative parameters entering the photon LCDAs at the scale μ 0 = 1.0 ​ GeV subscript 𝜇 0 1.0 GeV \mu_{0}=1.0\,{\rm GeV} [ 41 , 46 ] .

$\chi \left({\mu }_0\right)$	$⟨\overline{q}q⟩\left({\mu }_0\right)$	$b_2\left({\mu }_0\right)$	$\kappa \left({\mu }_0\right)$	${\kappa }^{+}\left({\mu }_0\right)$	${\zeta }_1\left({\mu }_0\right)$	${\zeta }_1^{+}\left({\mu }_0\right)$	${\zeta }_2^{+}\left({\mu }_0\right)$
$\left(3.15\pm 0.3\right){\mathrm{GeV}}^{-2}$	$-{\left({246}_{-19}^{+28}\mathrm{MeV}\right)}^3$	$0.07\pm 0.07$	$0.2\pm 0.2$	$0$	$0.4\pm 0.4$	$0$	$0$

Table 2. Table 2: The numerical values of Gegenbauer momemts a 2 subscript 𝑎 2 a_{2} and a 4 subscript 𝑎 4 a_{4} in leading twist pion LCDA.

Models	CZ	BMS	KMOW	Holographic	Platykurtic
$a_2$(1GeV)	0.5	${0.20}_{-0.08}^{+0.07}$	$0.17\pm 0.08$	0.15	0.08
$a_4$(1GeV)	0	$-{0.15}_{-0.09}^{+0.10}$	$0.06\pm 0.10$	0.06	-0.02

Equations

$$⟨\pi (p)|j_{\mu }^{\mathrm{em}}|\gamma (p^{\prime })⟩=g_{\mathrm{em}}^2{ϵ}_{\mu \nu \alpha \beta }{q}^{\alpha }{p}^{\beta }{ϵ}^{\nu }(p^{\prime })F_{{\gamma }^{\ast }\gamma \to {\pi }^0}(Q^2),{ϵ}_{0123}=-1$$

$$j_{\mu }^{\mathrm{em}}=\sum _q{g}_{\mathrm{em}}{Q}_q\bar{q}{\gamma }_{\mu }q.$$

$$F_{{\gamma }^{\ast }\gamma \to {\pi }^0}^{\mathrm{LP}}(Q^2)=\frac{\sqrt{2}(Q_u^2-Q_d^2)f_{\pi }}{Q^2}{\int }_0^{1}dx\left[T_2^{(0)}(x)+T_2^{(1),\Delta }(x,\mu )\right]{\varphi }_{\pi }^{\Delta }(x,\mu )+\mathcal{O}({\alpha }_s^2),$$

$$⟨\pi (p)|\bar{\xi }(y)[y,0]{\gamma }_{\mu }{\gamma }_5\xi (0)|0⟩=-i{f}_{\pi }{p}_{\mu }{\int }_0^{1}du{e}^{iup\cdot y}{\varphi }_{\pi }(u,\mu )+\mathcal{O}(y^2),$$

$$G_{\mu }(p^{\prime },q)$$

$$j_{\pi }=\frac{1}{\sqrt{2}}\left(\bar{u}{\gamma }_{5}u-\bar{d}{\gamma }_{5}d\right).$$

$$|n\cdot p|\sim \bar{n}\cdot p\sim n\cdot p^{\prime }\sim \mathcal{O}(\sqrt{Q^2}),$$

$$⟨0|j_{\pi }|\pi (p)⟩=-i{f}_{\pi }{\mu }_{\pi }(\mu ),{\mu }_{\pi }(\mu )\equiv \frac{m_{\pi }^2}{m_u(\mu )+m_d(\mu )},$$

$$G(p^2,Q^2)=\frac{f_{\pi }{\mu }_{\pi }(\mu )}{m_{\pi }^2-p^2-i0}{F}_{{\gamma }^{\ast }\gamma \to {\pi }^0}^{\mathrm{NLP}}(Q^2)+{\int }_{s_0}^{\infty }ds\frac{{\rho }^h(s,Q^2)}{s-p^2-i0}.$$

$$F_{{\gamma }^{\ast }\gamma \to {\pi }^0}^{2PLT}(Q^2)$$

$$\displaystyle\langle 0|\bar{q}(x)[x,0]\sigma_{\alpha\beta}\,\,q(0)|\gamma(p)\rangle=i\,g_{\rm em}\,Q_{q}\,\langle\bar{q}q\rangle(\mu)\,(p_{\beta}\,\epsilon_{\alpha}-p_{\alpha}\,\epsilon_{\beta})\,\int_{0}^{1}\,dz\,e^{i\,z\,p\cdot x}\,\bigg{[}\chi(\mu)\,\phi_{\gamma}(z,\mu)$$

$$\displaystyle\,\,\,\,\,\,\,\,\,\,+{x^{2}\over 16}\,\mathbb{A}(z,\mu)\bigg{]}+{i\over 2}\,g_{\rm em}\,Q_{q}\,{\langle\bar{q}q\rangle(\mu)\over p\cdot x}\,(x_{\beta}\,\epsilon_{\alpha}-x_{\alpha}\,\epsilon_{\beta})\,\int_{0}^{1}\,dz\,e^{i\,z\,p\cdot x}\,h_{\gamma}(z,\mu)\,.$$

$$⟨0|\bar{q}(x)[x,0]{\gamma }_{\alpha }q(0)|\gamma (p)⟩=g_{\mathrm{em}}{Q}_q{f}_{3\gamma }(\mu ){ϵ}_{\alpha }{\int }_0^{1}dz{e}^{izp\cdot x}{\psi }_{\gamma }^{(v)}(z,\mu )$$

$$⟨0|\bar{q}(x)[x,0]{\gamma }_{\alpha }{\gamma }_{5}q(0)|\gamma (p)⟩=\frac{g_{\mathrm{em}}{Q}_q{f}_{3\gamma }(\mu )}{4}{\epsilon }_{\alpha \beta \rho \tau }{p}^{\rho }{x}^{\tau }{ϵ}^{\beta }{\int }_0^{1}dz{e}^{izp\cdot x}{\psi }_{\gamma }^{(a)}(z,\mu ),$$

$$G_{\mu }(p^{\prime },q)$$

$$G_{\mu }^{2PHT}(p,q)$$

$$F_{{\gamma }^{\ast }\gamma \to {\pi }^0}^{2PHT}(Q^2)$$

$$⟨0|T\{q(x),\bar{q}(0)\}|0{⟩}_G$$

$${\Pi }_{\mu }(p,q)$$

$${\rho }^{3PHT}({\alpha }_i,u,\mu )$$

$$\int [\mathcal{D}{\alpha }_i]\equiv {\int }_0^{1}d{\alpha }_q{\int }_0^{1}d{\alpha }_{\bar{q}}{\int }_0^{1}d{\alpha }_g\delta \left(1-{\alpha }_q-{\alpha }_{\bar{q}}-{\alpha }_g\right).$$

$$F_{{\gamma }^{\ast }\gamma \to {\pi }^0}^{3PHT}(Q^2)$$

$$F_{{\gamma }^{\ast }\gamma \to {\pi }^0}^{\mathrm{HT}}(Q^2)=F_{{\gamma }^{\ast }\gamma \to {\pi }^0}^{2PHT}(Q^2)+F_{{\gamma }^{\ast }\gamma \to {\pi }^0}^{3PHT}(Q^2).$$

$$s_0\sim M^2\sim \mathcal{O}({\Lambda }^2),{\bar{u}}_0\sim \mathcal{O}({\Lambda }^2/Q^2).$$

$$\mathbb{A}(z,\mu )$$

$$h_{\gamma }(z,\mu )$$

$$S({\alpha }_i,\mu )$$

$$\tilde{S}({\alpha }_i,\mu )$$

$$S_{\gamma }({\alpha }_i,\mu )$$

$$T_1({\alpha }_i,\mu )$$

$$T_2({\alpha }_i,\mu )$$

$$T_3({\alpha }_i,\mu )$$

$$T_4({\alpha }_i,\mu )$$

$$T_4^{\gamma }({\alpha }_i,\mu )$$

$${\zeta }_1(\mu )+11{\zeta }_2(\mu )-2{\zeta }_2^{+}(\mu )=\frac{7}{2}.$$

$${\kappa }^{+}(\mu )={\left(\frac{{\alpha }_s(\mu )}{{\alpha }_s({\mu }_0)}\right)}^{\left({\gamma }^{+}-{\gamma }_{q\bar{q}}\right)/{\beta }_0}{\kappa }^{+}({\mu }_0),$$

Full text

Power corrections to the pion transition form factor

from higher-twist distribution amplitudes of photon

Yue-Long Shen111shenylmeteor@ouc.edu.cn

College of Information Science and Engineering, Ocean University of China, Qingdao, Shandong 266100, P.R. China

Jing Gao222gaojing@ihep.ac.cn

Institute of High Energy Physics, CAS, P.O. Box 918(4), Beijing 100049, P.R. China

School of Physics, University of China Academy of Sciences, Beijing 100049, P.R. China

Cai-Dian Lü333lucd@ihep.ac.cn

Institute of High Energy Physics, CAS, P.O. Box 918(4), Beijing 100049, P.R. China

School of Physics, University of China Academy of Sciences, Beijing 100049, P.R. China

Yan Miao444782771338@qq.com

College of Information Science and Engineering, Ocean University of China, Qingdao, Shandong 266100, P.R. China

In this paper we investigate the power suppressed contributions from two-particle and three-particle twist-4 light-cone distribution amplitudes (LCDAs) of photon within the framework of light-cone sum rules. Compared with leading twist LCDA result, the contribution from three-particle twist-4 LCDAs is not suppressed in the expansion by $1/Q^{2}$, so that the power corrections considered in this work can give rise to a sizable contribution, especially at low $Q^{2}$ region. According to our result, the power suppressed contributions should be included in the determination of the Gegenbauer moments of pion LCDAs with the pion transition form factor.

1 Introduction

As one of the simplest hard exclusive processes, the pion transition form factor $F_{\gamma^{\ast}\gamma\to\pi^{0}}(Q^{2})$ at large momentum transfer is of great importance in exploring the strong interaction dynamics of hadronic reactions in the framework of QCD, and to determine the parameters in the LCDAs of pion. It is defined via the matrix element

[TABLE]

where $q=p-p^{\prime}$, $p$ and $p^{\prime}$ refer to the four-momentum of the pion and the on-shell photon respectively, the electro-magnetic current

[TABLE]

In collinear factorization theorem, pion transition form factor can be factorized into the convolution of the hard kernel and the leading twist pion LCDA at leading power of $1/Q^{2}$ [1, 2, 3, 4], and the hard kernel has been calculated up to two-loop level [5, 6, 7, 8]. At one-loop level, the factorization formula is written by

[TABLE]

where the leading twist pion LCDA is defined as

[TABLE]

and the superscript “$\Delta$” indicates the scheme to deal with $\gamma_{5}$ in dimensional regularization which is a subtle problem in QCD loop diagrams [9, 10, 11, 12, 13, 14, 15]. Employing the trace technique, the $\gamma_{5}$ ambiguity of dimensional regularization was resolved by adjusting the way of manipulating $\gamma_{5}$ in each diagram to preserve the axial-vector Ward identity [6]. In a recent paper [16], the one loop calculation is revisited by applying the standard OPE technique [17, 18, 19] with the evanescent operator(s) [20, 21], in both the NDR and HV schemes for $\gamma_{5}$ in the $D$-dimensional space. At one-loop level it has been shown explicitly that the scheme dependence of the hard kernel and the twist-two pion LCDA is cancelled out precisely, which guarantees the form factor $F_{\gamma^{\ast}\gamma\to\pi^{0}}(Q^{2})$ to be free from $\gamma_{5}$ ambiguity.

At leading power the pion transition form factor has also been studied with transverse momentum dependent (TMD) factorization approach at one-loop level [22, 23, 24], where the joint resummation of the large logarithms $\ln^{2}{{k_{\perp}^{2}}/Q^{2}}$ and $\ln^{2}x$ was performed in moment and impact-parameter space [25]. The prediction of joint resummation improved TMD factorization approach can accommodate the anomalous BaBar measurements [26] of $F_{\gamma^{\ast}\gamma\to\pi^{0}}(Q^{2})$, which have stimulated intensive theoretical investigations with various phenomenological approaches as well as lattice QCD simulations (see for instance [27, 28, 29]). In Ref. [30, 31], a leading twist pion LCDA with the non-vanishing end-point behavior was proposed to explain the anomalous BaBar data at high $Q^{2}$. Later it was found that this method is able to be achieved by introducing a sizable nonperturbative soft correction from the TMD pion wave function[32].

To achieve more precise theoretical predictions, power corrections need to be taken into account especially at low $Q^{2}$. In [32, 33], the soft correction to the leading twist contribution is evaluated with the dispersion approach and found to be crucial to suppress the contributions from higher Gegenbauer moments of the twist-2 pion LCDAs [34, 25]. Furthermore, the subleading power “hadronic” photon correction can also be taken into account effectively with dispersion approach. Within this method the theoretical accuracy for predicting the pion-photon form factor is improved by including the next-to-next-to-leading order (NNLO) QCD correction to the twist-2 contribution and the finite-width effect of the unstable vector mesons in the hadronic dispersion relation [35, 36, 37, 38, 39]. Another approach to accommodate the contribution from the “hadronic photon” is to introduce the LCDAs of photon. In [16], the QCD factorization of the correlation function for the construction of the LCSRs for the hadronic photon contribution to the pion-photon form factor is established. Both the hard matching coefficient and the leading twist photon LCDAs are independent of the $\gamma_{5}$ prescription in dimensional regularization, and the next-to-leading logarithmic(NLL) resummation of the large logarithms was also perform by solving the renormalization group equations(RGE) in momentum space. The contribution from the twist-4 pion LCDA is also calculated at tree level in [40, 16]. There is strong cancellation between this contribution and the contribution from hadronic structure of photon, which makes the overall power correction not significant. The LCDAs of photon, including both two-particle and three-particle Fock state, have been studied up the twist-4 level [41]. The higher-twist LCDAs are not suppressed in many processes such as radiative leptonic $B$ meson decay $B\to\gamma\ell\nu$ [42, 43]. In this paper we will investigate the contribution from the full set the LCDAs of photon up to twist-4 to the pion transition form factor using LCSRs approach.

The outline of this paper is as follow: in Section 2 we present the analytic calculation of the pion transition form factor from the higher twist photon LCDAs within LCSRs framework. The numerical results and discussions are given in section 3. The last section is closing remark.

2 Power corrections from the hadronic structure of photon

All two-particle and three-particle LCDAs of photon have been defined and classified up to twist-4, and the expressions of the LCDAs have also been obtained through the conformal expansion in the presence of the background field [41]. To evaluate the power suppressed contribution to the pion-photon form factor due to the hadronic photon effect, the following correlation function is employed

[TABLE]

where pion interpolating current $j_{\pi}$ is defined by

[TABLE]

The power counting rule for the external momenta

[TABLE]

will be adopted to determine the perturbative matching coefficient entering the factorization formula of $G_{\mu}(p^{\prime},q)$. Applying the standard definition for the pion decay constant

[TABLE]

we can write down the hadronic dispersion relation of $G(p^{2},Q^{2})$

[TABLE]

The form factor $F^{\rm NLP}_{\gamma^{\ast}\gamma\to\pi^{0}}(Q^{2})$ will be extracted after the correlation function being calculated by OPE in deep Euclidean region. Employing dispersion relation, subtracting the continuum state contribution with the help of quark hadron duality assumption, and performing Borel transformation, the LCSRs for the subleading power contribution to the $\pi^{0}\gamma^{\ast}\gamma$ form factor are derived as

[TABLE]

where the magnetic susceptibility of the quark condensate $\chi(\mu)$ contains the dynamical information of the QCD vacuum, and the spectral functions $\rho^{(0,1)}(s,Q^{2})$ can be found in [16].

Now we will proceed to investigate the contribution from higher twist LCDAs of photon. Up to twist-4, the two-particle LCDAs of photon are defined as

[TABLE]

where $\psi_{\gamma}^{(v)}(z,\mu),\psi_{\gamma}^{(a)}(z,\mu)$ are twist-3 and $\mathbb{A}(z,\mu),h_{\gamma}(z,\mu)$ are twist-4. Employing the light-cone expansion of the $u,d$-quark propagator and keeping the subleading-power contributions to the correlation function (5) leads to

[TABLE]

where $\delta_{u}=1,\delta_{d}=-1$. The above equation indicates that only twist-2 and twist-4 two-particle LCDAs can contribute to pion transition form factor in the LCSRs approach, which is different from the method based on TMD factorization[44]. Making use of the definitions in Eq.(11), it is straightforward to write down

[TABLE]

where the contribution from $h_{\gamma}(z,\mu)$ vanishes due to the anti-symmetric structure. The resulting LCSRs for the two-particle higher-twist hadronic photon corrections to the pion transition form factors can be further derived as follows

[TABLE]

where $u_{0}=Q^{2}/(s_{0}+Q^{2})$.

To compute higher-twist three-particle hadronic photon corrections to the pion transition form factors, the definition of three-particle photon LCDA is required. In the appendix we collect the definition of three-particle twist-4 photon LCDAs for an incoming photon state. Keeping the one-gluon/photon part for the light-cone expansion of the quark propagator in the background gluon/photon field

[TABLE]

where $G^{\mu\nu}=i[D_{\mu},D_{\nu}]$. By evaluating Fig. 2, we obtain

[TABLE]

where

[TABLE]

and the integration measure is defined as

[TABLE]

Taking advantage of quark-hadron duality, we arrive at the LCSRs of the contribution from three-particle photon LCDAs

[TABLE]

where $u_{s}=[s/(s+Q^{2})-\alpha_{q}]/\alpha_{g}$. The overall higher-twist photon LCDAs contribution is written by

[TABLE]

Now we discuss the power behavior of our results. The power counting scheme for the sum rule parameters are given below:

[TABLE]

Employing Eq.(21), one can obtain that the contribution from leading twist LCDA of photon is suppressed by a factor $\Lambda^{2}/Q^{2}$ [16] compared with LP contribution. The higher twist contributions are conjectured to be also suppressed by only one power of $\Lambda^{2}/Q^{2}$ due to the absent correspondence between the twist counting and the large-momentum expansion [32]. For the contribution from two-particle twist-4 LCDAs of photon, the result in Eq.(14) is suppressed by $\Lambda^{4}/Q^{4}$ compared with LP contribution as the power of twist-4 photon LCDAs is suppressed with respect to leading twist one. While for the contribution from three-particle twist-4 LCDAs in Eq.(19), the scaling of $\alpha_{q}$ is ${\cal O}(\Lambda^{2}/Q^{2})$, and $\alpha_{g}$ is ${\cal O}(1)$. Although there is an overall factor $1/Q^{2}$, the result is only suppressed by $\Lambda^{2}/Q^{2}$ for the spectral function $\rho^{\rm 3PHT}$ is not suppressed at endpoint region. This result confirms the conjecture in [32].

3 Numerical analysis

In the following we explore the phenomenological consequences of the hadronic photon correction to the pion-photon form factor, and the most important input is the LCDAs of photon. The models of twist-4 LCDAs of photon used in this paper are written by

[TABLE]

In the above equations, the conformal expansion of the photon LCDAs have been truncated up to the next-to-leading conformal spin. Due to the Ferrara-Grillo-Parisi-Gatto theorem [45], these parameters satisfy the following relations

[TABLE]

The scale evolution of the nonperturbative parameters is given by

[TABLE]

where the anomalous dimensions at one loop read [41]

[TABLE]

Numerical values of the input parameters entering the photon LCDAs up to twist-4 are collected in Table 1, where for the estimates of the twist-4 parameters from QCD sum rules [47] 100 % uncertainties are assigned.

Now we are in the position to investigate the phenomenological significance of the contribution from higher twist photon LCDAs. For the factorization scale in the evaluation of the contribution of higher-twist photon LCDAs, we will take the value $\mu^{2}=\langle x\rangle\,M^{2}+\langle\bar{x}\rangle\,Q^{2}$ as widely employed in the sum rule calculations [32]. The Borel mass $M^{2}$ and the threshold parameter $s_{0}$ can be determined by applying the standard strategies described in [48, 49],

[TABLE]

where the variation ranges of these parameters are set to be large to allow sufficient theoretical uncertainty. It has been checked that the Borel mass and threshold parameter dependence of the contribution of higher-twist photon LCDAs is mild in the intervals in Eq.(26). In Fig. 3 the $Q^{2}$ dependence of the relevant power suppressed contributions is presented. Compared with the contribution from leading-twist photon LCDA, the two-particle twist-4 contribution is obviously suppressed as the curve declines more quickly and approaches zero at large $Q^{2}$. While for the contribution from three-particle twist-4 LCDAs of photon, the result is comparable with that from leading twist photon LCDA, as they are at the same power. As mentioned in [16], there exists strong cancellation effect between the contribution from leading twist photon LCDA and the twist-4 pion LCDA , thus the overall power correction is mainly from the contribution from twist-4 LCDAs of photon.

To obtain the total result of the photon-pion form factor, we will need to specify the non-perturbative models for the twist-2 pion LCDA. In general it is expanded in terms of Gegenbauer polynomials

[TABLE]

where the Gegenbauer moments $a_{n}$ can be determined by the calculation with QCD sum rules or lattice simulation, or by fitting the experimental data. Following [16], we take advantage of the the Chernyak- Zhitnitsky (CZ) model[50], the Bakulev-Mikhailov-Stefanis (BMS) model[51], the platykurtic model (PK)[52], the KMOW model[53], and the holographic model[54] for comparison. The Gegenbauer coefficients in the BMS model and the PK model are computed from the QCD sum rules with non-local condensates, the first and second nontrivial Gegenbauer moments of the KMOW model are determined by comparing the LCSR predictions for the pion electromagnetic form factor with the experimental data at intermediate-$Q^{2}$, and the holographic model of the twist-2 pion LCDA is motivated by the AdS/QCD correspondence. We collect the values of the Gegenbauer moments in different models in Table. 2. The total results including power suppressed contributions are shown in Fig. 4, where the BMS model is employed. It can be seen that the higher power photon LCDAs manifestly modify the LP result especially at “small” $Q^{2}$ region. We note that the photon-LCSRs employed in this paper is valid when $Q^{2}\gg 2GeV^{2}$, thus the prediction of $F_{\gamma^{\ast}\gamma\to\pi^{0}}(Q^{2})$ should not be taken serious below 2 GeV2.

The model dependence of pion-photon form factor on the leading twist pion LCDA is displayed in Fig. 5. As the contribution from higher twist photon LCDA enhances the form factors significantly, the prediction from every models cannot match the experimental data at $Q^{2}<10GeV^{2}$. This result is inconsistent with the predictions from dispersion approach[35, 36, 37, 38, 39], where the BMS and PK models of pion LCDA work well. This discrepancy is not a surprise because the power suppressed contributions considered in both approaches are not from a systematic study based on the effective theory, and what is omitted is not clear. Our result indicates that there exist significant power suppressed contributions, and they should not be neglected in the phenomenological studies. Meanwhile, we cannot draw the conclusion that the models mentioned in this paper should be ruled out, because in our study the QCD corrections are not included, and contributions from the pion and photon LCDA with twist higher than 4 are not considered, let alone the unknown power suppressed contributions. Thus in the present paper we aim at sheding light on the importance of the power corrections, and more efforts must be devoted to the study on power suppressed contributions to obtain more accurate prediction.

We present our final predictions for $Q^{2}F_{\gamma^{\ast}\gamma\to\pi^{0}}(Q^{2})$ with both LP contribution and power corrections included in Fig. 6, where the combined theory uncertainties are due to the variations of the input parameters $a_{2},a_{4}$ of pion LCDA, $\xi,\langle\bar{q}q\rangle,b_{2}$ in twist-2 photon LCDAs, $\kappa,\zeta_{1},\zeta_{2}$ in twist-4 photon LCDAs, quark mass, and factorization scale, etc. Diagram (a), (b) and (c) in Fig. 6 are corresponding to the BMS model, holographic model and KMOW model of pion LCDA respectively. Among all the parameters, the most important uncertainty comes from the shape parameters $a_{2},a_{4}$ of leading twist pion LCDA, which means the pion transition form factor is still sensitive to the Gegenbauer moments of leading twist pion LCDA after the power suppressed contributions considered. Thus the photon-pion transition process provides a good platform to determine the parameters in the LCDAs of pion, which can also be compared with the future lattice simulation with the help of quasi parton distribution amplitude [58, 59].

4 Closing remark

In this paper we performed a study on the power suppressed contributions from higher-twist LCDAs of photon within the LCSRs. The twist-3 LCDAs cannot contribute for their Lorentz structures, thus the contributions from two-particle and three-particle twist-4 LCDAs of photon are considered in this work. According to the power analysis, the three-particle twist-4 contribution is not suppressed compared with the leading twist photon LCDA result, so that the power corrections considered in this work can give rise to sizable contribution, especially at “low” $Q^{2}$ region. In addition, there exists strong cancellation between the contribution from leading twist photon LCDA and the twist-4 pion LCDA, and the importance of the twist-4 photon LCDAs is further highlighted. The numerical result also confirms that after including power corrections, the predicted $Q^{2}F_{\gamma^{\ast}\gamma\to\pi^{0}}(Q^{2})$ is significantly enhanced especially at at “low” $Q^{2}$ region, thus the power suppressed contributions should be included in the determination of the Gegenbauer moments of pion LCDAs. Note that for the higher-twist photon LCDAs contribution, we only presented a tree level calculation, the NLO QCD corrections which might modify the current result to some extent and stablize the factorization scale dependence are not considered. Furthermore, the other power suppressed contributions are also absent in the present study, a more systematic study based on effective theory is necessary for a thorough understanding of the NLP corrections to the pion transition form factor, which can be checked by the (potentially) more accurate experimental measurements at the BEPCII collider and the SuperKEKB accelerator.

Acknowledgements

We thank S. V. Mikhailov and N. G. Stefanis for valuable comments. This work is supported in part by the National Natural Science Foundation of China (NSFC) with Grant No. 11521505 and 11621131001. CDL would like to express a special thanks to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and Support.

Appendix A Definition of three-particle twist-4 LCDAs of photon

In the following, we present the definition of the three-particle photon LCDAs up to twist-4.

[TABLE]

Note that we have employed the following notations for the dual field strength tensor and the integration measure

[TABLE]