Relativistic corrections to light-cone distribution amplitudes of S-wave $B_c$ mesons and heavy quarkonia

TL;DR

This paper calculates relativistic corrections to the light-cone distribution amplitudes of S-wave $B_c$ mesons and heavy quarkonia, which are significant for high-energy exclusive process predictions.

Contribution

It provides the first calculation of ${ m O}(v^2)$ relativistic corrections to twist-2 LCDAs of S-wave $B_c$ mesons, extending previous NLO $\alpha_s$ results.

Findings

Relativistic corrections are sizable and comparable to NLO radiative corrections.

The corrected LCDAs impact the phenomenological analysis of heavy quarkonium and $B_c$ meson production.

Results are applicable to various heavy quarkonia by setting $m_b=m_c$.

Abstract

In collinear factorization, light-cone distribution amplitudes (LCDAs) are key ingredients to calculate the production rate of a hadron in high energy exclusive processes. For a doubly-heavy meson system (such as $B_c, J/\psi, \Upsilon$ etc), the LCDAs contain perturbative scales that can be integrated out and then are re-factorized into products of perturbatively calculable distribution parts and non-relativistic QCD matrix elements. In this re-factorization scheme, the LCDAs are known at next-to-leading order in the strong coupling constant $\alpha_s$ and at leading order in the velocity expansion. In this work, we calculate the ${\cal O}( { v}^2)$ corrections to twist-2 LCDAs of S-wave $B_c$ mesons. These results are applicable to heavy quarkonia like $\eta_{c,b}$, $J/\psi$ and $\Upsilon$ by setting $m_b=m_c$. We apply these relativistically corrected LCDAs to study their inverse moments and a few Gegenbauer moments which are important for phenomenological study. We point out that the relativistic corrections are sizable, and comparable with the next-to-leading order radiative corrections. These results for LCDAs are useful in future theoretical analyses of the productions of heavy quarkonia and $B_c$ mesons.

Tables

Table 1. Table 1: Input parameters

	Meson mass	Quark mass	${\alpha }_s$	$\mu$	$⟨{\text{q}}^2⟩$
$J/\psi$	3.10 GeV	$m_c=1.28$ GeV	${\alpha }_s\left(2m_c\right)=0.26$	$2.56\mathrm{GeV}$	${⟨{\text{q}}^2⟩}_V=0.69{\mathrm{GeV}}^2$
$B_c$	6.27 GeV	$m_c=1.28$ GeV; $m_b=4.18$ GeV	${\alpha }_s\left(m_b+m_c\right)=0.21$	$5.46\mathrm{GeV}$	${⟨{\text{q}}^2⟩}_P=1.59{\mathrm{GeV}}^2$
${\rm Y}$	9.46 GeV	$m_b=4.18$ GeV	${\alpha }_s\left(2m_b\right)=0.19$	$8.36\mathrm{GeV}$	${⟨{\text{q}}^2⟩}_V=4.6{\mathrm{GeV}}^2$

Table 2. Table 2: Inverse moment and Gegenbauer moments of LCDAs for the B c subscript 𝐵 𝑐 B_{c} , J / ψ 𝐽 𝜓 J/\psi and Υ Υ \Upsilon states. Leading-order results, one-loop QCD radiative corrections, relativistic corrections and the total results are shown, respectively.

$⟨x^{-1}⟩$	LO	LO + ${\alpha }_s$	LO+ $v^2$	LO + ${\alpha }_s$+ $v^2$
$B_c$	1.31	1.51	1.53	1.73
$J/\psi$	2.00	2.31	2.28	2.59
${\rm Y}$	2.00	2.22	2.18	2.40
$a_1$	LO	LO + ${\alpha }_s$	LO+ $v^2$	LO + ${\alpha }_s$+ $v^2$
$B_c$	0.89	0.67	0.53	0.32
$a_2$	LO	LO + ${\alpha }_s$	LO+ $v^2$	LO + ${\alpha }_s$+ $v^2$
$B_c$	0.24	0.11	-0.21	-0.34
$J/\psi$	-0.58	-0.26	-0.17	0.14
${\rm Y}$	-0.58	-0.35	-0.33	-0.09

Equations

$$a^{\mu }=n_{+}\cdot a\frac{n_{-}^{\mu }}{2}+n_{-}\cdot a\frac{n_{+}^{\mu }}{2}+a_{\perp }^{\mu }.$$

$$J[\Gamma ](\omega )\equiv (\bar{b}{W}_c)(\omega n_{+})n{/}_{+}\Gamma (W_c^{†}c)(0),$$

$$\displaystyle W_{c}(x)=\textrm{P exp}\Big{(}ig_{s}\int_{-\infty}^{0}ds~{}n_{+}A(x+sn_{+})\Big{)}$$

$$⟨B_c{(}^1{S}_0,P)|J[{\gamma }_5](\omega )|0⟩$$

$$⟨B_c{(}^3{S}_1,P,{ϵ}^{\ast })|J[1](\omega )|0⟩$$

$$⟨B_c{(}^3{S}_1,P,{ϵ}^{\ast })|J[{\gamma }_{\perp }^{\alpha }](\omega )|0⟩$$

$${\int }_0^{1}dx{\hat{\varphi }}_P(x)={\int }_0^{1}dx{\hat{\varphi }}_V^{\parallel }(x)={\int }_0^{1}dx{\hat{\varphi }}_V^{\perp }(x)=1.$$

$$⟨B_c{(}^1{S}_0,P)|Q[{\gamma }_5](x)|0⟩$$

$$⟨B_c{(}^3{S}_1,P,{ϵ}^{\ast })|Q[1](x)|0⟩$$

$$⟨B_c{(}^3{S}_1,P,{ϵ}^{\ast })|Q[{\gamma }_{\perp }^{\alpha }](x)|0⟩$$

$$Q[\Gamma ](x)$$

$$P^{\mu }=(m_{B_c},0,0,0),n_{\pm }^{\mu }=(1,0,0,\pm 1),$$

$$Q[\Gamma ](x,\mu )=\sum _{n=0}^{\infty }\frac{d_n^{\Gamma }(x,\mu )}{M^{n+1}}{O}_{\Gamma ,n}^{\mathrm{NRQCD}},$$

$$⟨B_c|Q[\Gamma ](x,\mu )|0⟩\simeq \sum _{n=0}^{\infty }\frac{d_n^{\Gamma }(x,\mu )}{M^{n+1}}⟨B_c|O_{\Gamma ,n}^{\mathrm{NRQCD}}|0⟩.$$

$$O_0{(}^1{S}_0)$$

$$O_0{(}^3{S}_1)$$

$$O_2{(}^1{S}_0)$$

$$O_2{(}^3{S}_1)$$

$$-i{f}_P{\hat{\varphi }}_P(x)$$

$$f_V{\hat{\varphi }}_V^{\parallel }(x)$$

$$f_V^{\perp }{\hat{\varphi }}_V^{\perp }(x)$$

$$-i{f}_P$$

$$f_V$$

$$f_V^{\perp }$$

$$C_i^{\Gamma }$$

$${\hat{\varphi }}_P(x)$$

$${\hat{\varphi }}_V^{||}(x)$$

$${\hat{\varphi }}_V^{\perp }(x)$$

$$\displaystyle\left\{\begin{array}[]{l}p_{b}^{\mu}=(E_{b},{\bf q})\,,~{}~{}p_{c}^{\mu}=(E_{c},-{\bf q})\,,~{}~{}P^{\mu}=p_{b}^{\mu}+p_{c}^{\mu}=(E,{\bf 0})\\ E_{b}=m_{b}+\frac{{\bf q}^{2}}{2m_{b}}+{\cal O}(v^{4})\,,~{}~{}~{}E_{c}=m_{c}+\frac{{\bf q}^{2}}{2m_{c}}+{\cal O}(v^{4})\,,\\ E=E_{b}+E_{c}=M+\frac{{\bf q}^{2}}{2m_{b}}+\frac{{\bf q}^{2}}{2m_{c}}+{\cal O}(v^{4})\,.\end{array}\right.$$

$$⟨b^a(p_b){\bar{c}}^b(p_c)|Q[\Gamma ](x)|0⟩$$

$$u_b(p_b)$$

$$v_c(p_c)$$

$$⟨0|{\psi }_b^a(0)|b^b(p_b)⟩$$

$$⟨{\bar{c}}^b(p_c)|{\chi }_c^a(0)|0⟩$$

$$⟨b^a(p_b){\bar{c}}^b(p_c)|Q[{\gamma }_5](x)|0⟩=\frac{{\delta }^{ab}}{n_{+}P}\delta \left(x-\frac{n_{+}{p}_b}{n_{+}P}\right){\bar{u}}_b(p_b)n{/}_{+}{\gamma }_5{v}_c(p_c)$$

$$⟨b^a(p_b){\bar{c}}^b(p_c)|Q[{\gamma }_5](x)|0⟩$$

$$⟨b^a(p_b){\bar{c}}^b(p_c)|O_0{(}^1{S}_0)|0⟩$$

$$⟨b^a(p_b){\bar{c}}^b(p_c)|O_2{(}^1{S}_0)|0⟩$$

Full text

I Introduction

A typical high energy process with hadrons involves physics from high scales, such as the center-of-mass collision energy, down to very low energy scales such as the mass of a proton. Without disentangling physics associated with these scales, it is nearly hopeless to obtain reliable theoretical predictions for any such process. The disentanglement is often referred to as factorization.

There exist two sorts of factorization schemes to deal with hard exclusive processes involving heavy quarkonia and $B_{c}$ mesons, if we regard heavy quarkonia and $B_{c}$ mesons as the non-relativistic bound-states of heavy quark and anti-quark. One is the non-relativistic QCD (NRQCD) factorization Bodwin:1994jh , in which the amplitude can be factorized into a product of the perturbatively calculable short-distance coefficient and the non-perturbative NRQCD matrix-element. For recent reviews of this approach, one can refer to Refs. Brambilla:2004jw ; Brambilla:2004wf ; Brambilla:2010cs . The other one is called the collinear factorization, which has already been established for a few decades Lepage:1980fj ; Chernyak:1983ej . In this framework, the amplitudes of hard exclusive processes can be expressed as convolutions of the perturbatively calculable hard-kernels and the universal light-cone distribution amplitudes (LCDAs). It is necessary to mention at this moment that the factorization approaches do not always hold, but instead should be proved in a channel-by-channel manner. Once proved, these approaches can both clearly disentangle the short-distance and long-distance contributions.

In literatures, the collinear factorization and its extension $k_{T}$ factorization have been adopted in Refs. Liu:2009qa ; Liu:2010nz ; Liu:2010da ; Liu:2010kq ; Xiao:2011zz ; Rui:2015iia ; Xiao:2013lia ; Wang:2014yia ; Liu:2017cwl ; Wang:2016qli ; Yang:2010ba to study various $B_{c}$ decays, where the LCDA of $B_{c}$ has been used to parametrize the non-perturbative physics in the system. Meanwhile the NRQCD factorization has been adopted in Refs. Qiao:2012vt ; Qiao:2012hp ; Shen:2014msa ; Wang:2015bka ; Zhu:2017lqu ; Wu:2002ig ; Chang:2005bf ; Chang:2007si ; Zheng:2017xgj , in which the non-perturbative physics reside in a few NRQCD matrix elements. The LHCb collaboration has started comprehensive experimental studies of the decays Aaij:2014jxa , and some phenomenological studies can be found in Refs. Wang:2008xt ; Wang:2009mi ; Sun:2008ew ; Hwang:2008qi ; Jia:2015pxx .

Though the two factorization approaches are different in nature, it has been argued that since the LCDAs of quarkonium and $B_{c}$ meson contain some perturbatively calculable scales like $\mathcal{O}(m^{2})$, it is reasonable to consider the LCDAs can be further factorized into a product of a perturbatively calculable distribution part and a NRQCD matrix-element for the vacuum to quarkonium or $B_{c}$ state transition Ma:2006hc . This idea can be called as “re-factorization”. Some other approaches to bridge NRQCD and collinear factorization are proposed in Refs. Jia:2008ep ; Jia:2010fw .

The re-factorization scheme has been extended to the next-to-leading order ${\cal O}(\alpha_{s})$ accuracy in Refs. Ma:2006hc ; Bell:2008er ; Wang:2013ywc ; Xu:2016dgp , and the relation between the moments of LCDAs for heavy quarkonia and NRQCD matrix elements are known Braguta:2006wr ; Braguta:2007fh ; Braguta:2007tq ; Braguta:2008qe . However, the factorization is only discussed at leading order of the velocity expansion $v$. In fact, the relativistic corrections are also important for the calculation of LCDAs of quarkonium and $B_{c}$ meson. Relativistic corrections are often characterized by the relative velocity of heavy quarks within the bound states. It is estimated as $v^{2}\sim 0.3$ for $\bar{c}c$ system which is sizable. Thus, in order to test the applications of LCDAs in quarkonium and $B_{c}$ meson productions precisely, one should consider the relativistic corrections, which are the main focus of this paper. To do so, we will calculate the three relativistic corrected LCDAs for the S-wave $B_{c}$ mesons, and the corresponding results for quarkonia like $J/\psi$ can be easily deduced by setting $m_{b}=m_{c}$.

The rest of this paper is organized as follows: in section II, we give the notations used in this paper, and present the definitions of the leading-twist LCDAs of the S-wave $B_{c}$ mesons in terms of the matrix-elements of a certain class of non-local QCD operators. In section III, we present our results of the LCDAs at $\mathcal{O}(\alpha_{s}^{0}v^{0})$, $\mathcal{O}(\alpha_{s}^{0}v^{2})$ and $\mathcal{O}(\alpha_{s}^{1}v^{0})$; In section IV, we will apply the results for the LCDAs and study the inverse moment and Gegenbauer moments of LCDAs. A comparison of relativistic corrections and radiative corrections is made. Finally, we summarize our work in section V.

II LCDAs for S-wave $B_{c}$ mesons

II.1 Light-cone coordinates

As we will see below, LCDAs are defined in the light-cone coordinate frame, in which a four-vector $a$ can be expressed as

[TABLE]

Here $n_{+}$ and $n_{-}$ are two light-like vectors, with $n_{\pm}^{2}=0$, $n_{+}\cdot n_{-}=2$ and $n_{\pm}\cdot a_{\perp}=0$.

II.2 Definition of LCDAs

To define the LCDAs, one needs to introduce the gauge invariant non-local quark bilinear operators:

[TABLE]

where $b$ and $c$ are the quark fields. The Wilson-line

[TABLE]

is a path-ordered exponential with the path along the $n_{+}$ direction and $g_{s}$ is QCD coupling constant and $A_{\mu}\equiv A_{\mu}^{a}(x)T^{a}$, $T^{a}$ are the generators of SU(3) group in the fundamental representation.

The leading-twist, i.e. twist-2, LCDAs can be defined from matrix elements of operator $J[\Gamma](\omega)$:

[TABLE]

Here $f_{P}$, $f_{V}$ and $f^{\perp}_{V}$ are decay constants, $P$ is the momentum of the $B_{c}$ meson, $\epsilon^{*}$ is the polarization vector for the vector $B_{c}$ meson. $\hat{\phi}_{P}(x)$, $\hat{\phi}_{V}(x)$ and $\hat{\phi}_{V}^{\perp}(x)$ are LCDAs for pseudo-scalar, longitudinally polarized and transversely polarized vector $B_{c}$ mesons, respectively. $x$ denotes the light-cone momentum fraction and $\bar{x}\equiv 1-x$. $\mu$ is the renormalization scale. Note that the state $|B_{c}\rangle$ in Eq. (3) is relativistically normalized.

The LCDAs are normalized as

[TABLE]

In the following analysis, it is convenient to employ the Fourier transformed form of Eq. (3)

[TABLE]

where the Fourier-transformed operator

[TABLE]

It is worth noting that the operator $Q[\Gamma](x)$ is invariant under the longitudinal boost, i.e. $n_{+}\to\alpha n_{+}$ and $n_{-}\to\alpha^{-1}n_{-}$ where $\alpha$ is an arbitrary positive real number. This leads to the conclusion that both the decay constants and the LCDAs of $B_{c}$ mesons defined in Eq. (5) are also boost-invariant. This allows us to calculate these decay constants and LCDAs in any reference frame. In this paper, we will choose the rest frame of $B_{c}$ mesons, i.e

[TABLE]

which is particularly convenient for matching the decay constants and LCDAs to NRQCD matrix-elements.

II.3 Re-factorization of the LCDAs

The re-factorization idea has been proposed in Refs. Ma:2006hc ; Wang:2013ywc ; Bell:2008er ; Xu:2016dgp , where all of the LCDAs of quarkonia or $B_{c}$ mesons can be factorized into products of perturbatively calculable distribution parts and non-perturbative NRQCD matrix elements. It means that, at the operator level, we have the generic matching equation

[TABLE]

where $n$ denotes the order of $v$-expansion, $d^{\Gamma}_{n}(x,\mu)$ is the short-distance coefficient as a distribution over the momentum fraction $x$, $O_{\Gamma,n}^{\rm NRQCD}$ is the NRQCD operator which scales $\mathcal{O}(v^{n})$ in the power-counting, and scale $M\equiv m_{b}+m_{c}$ is introduced to balance the mass dimensions of the NRQCD operators so that the short-distance coefficients $d^{\Gamma}_{n}(x,\mu)$ are set to be dimensionless. Therefore, the LCDAs of $B_{c}$ meson can be expressed as

[TABLE]

In the present work, since we focus on the LCDAs of the S-wave $B_{c}$ mesons, up to ${\cal O}(v^{2})$, the NRQCD operators that we will consider are

[TABLE]

Here $\psi_{b}$ and $\chi_{c}$ are the two-component effective fields for the $b$ quark and $\bar{c}$ quark in the NRQCD, respectively, and $\psi_{b}^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{\mathbf{D}}}\chi_{c}\equiv\psi_{b}^{\dagger}(\mathbf{D}\chi_{c})-(\mathbf{D}\psi_{b})^{\dagger}\chi_{c}$ with $\mathbf{D}={\boldsymbol{\nabla}}-ig_{s}\mathbf{A}$.

Thus, up to $\mathcal{O}(v^{2})$, we have the matching equations

[TABLE]

Here the $\boldsymbol{\epsilon}^{*}$ is the polarization vector of ${}^{3}S_{1}$ state, and $\boldsymbol{\epsilon}\cdot\boldsymbol{\epsilon}^{*}=1$. $d^{P,V,V_{\perp}}_{i}(x)~{}(i=0,2)$ are perturbatively calculable Wilson coefficients. We should note that the matrix-elements of the NRQCD operators in Eq. (11) are relativistically normalized.

By integrating over $x$ in Eq. (11), and by imposing the normalization conditions given in Eq. (4), we can get the matching equations for the decay constants

[TABLE]

with the short-distance coefficients

[TABLE]

Then, the LCDAs $\hat{\phi}(x)$ can be derived straight-forwardly. In the following, it is more convenient to express the LCDAs in the following expansions

[TABLE]

where the superscript $(i,j)$ denotes the order of $\alpha_{s}$ and $v^{2}$-expansion. $\hat{\phi}^{(0,0)}(x)$ and $\hat{\phi}^{(1,0)}(x)$ have been given in Ref. Xu:2016dgp . The explicit expressions of $\hat{\phi}^{(0,1)}(x)$ which are the main results of this work will be presented in the next section.

III Calculations of LCDAs for the S-wave $B_{c}$ mesons

III.1 Perturbative Matching

As we describe in the previous section, to get the decay constants and LCDAs of S-wave $B_{c}$ mesons up to ${\cal O}(v^{2})$, we need to obtain the short-distance coefficients $d_{i}^{\Gamma}(x)$ in Eq. (11) and $C_{i}^{\Gamma}$ in Eq. (12) from the perturbative matching.

For the perturbative matching, one is allowed to choose any convenient process. After calculating the matrix elements in both full theory and effective field theory, one can derive the short-distance coefficients by solving the matching equations. To do so, in the standard NRQCD matching procedure, we usually use the free $b$-quark and $\bar{c}$-quark pair to replace the corresponding $B_{c}$ meson state. Firstly, we calculate the corresponding on-shell amplitude in full QCD, then we expand the amplitude in terms of the relative momentum $q^{i}$ so that each expanded term has a definite scaling in $v$-expansion. Finally, we can extract the short-distance coefficients by identifying the corresponding NRQCD matrix-elements in the expanded amplitude.

We set the momenta for on-shell $b$-quark and $\bar{c}$-quark as

[TABLE]

where $M\equiv m_{b}+m_{c}$, and we count $|{\bf q}|\sim{\cal O}(v)$.

At tree level, we have the matrix element

[TABLE]

where the supscripts $a$ and $b$ are color indices for the $b$ quark and $\bar{c}$ quark, respectively, and $n_{\pm}^{\mu}$ is set to have the explicit form in Eq. (7).

Here it is worth addressing that the quark-states in Eq. (19) are non-relativistically normalized. Thus, in the Dirac representation for Gamma matrices, the non-relativistically normalized 4-component on-shell spinors for $b$ quark and $\bar{c}$ quark can be written explicitly as

[TABLE]

Here we suppress the helicity indices for spinors. $\xi({\bf q})$ and $\eta({\bf q})$ are the 2-component Pauli spinors for the $b$ quark and $\bar{c}$ quark, respectively, and they are related to the following NRQCD matrix-elements:

[TABLE]

where the supscripts $a,b$ are color indices.

For illustration, we take $\Gamma=\gamma_{5}$ in Eq. (19) as an example to show how to match $Q[\gamma_{5}](x)$ to the NRQCD operators up to ${\cal O}(v^{2})$. Explicitly, we have

[TABLE]

Then we expand the above matrix-element up to the second power of $q^{i}$,

[TABLE]

where $x_{0}\equiv m_{b}/(m_{b}+m_{c})$ and $\bar{x}_{0}\equiv 1-x_{0}=m_{c}/(m_{b}+m_{c})$.

In this work, we are only interested in the S-wave part of the above matrix-element. Due to the standard extraction procedure of the S-wave contribution in the literature, at first we neglect the first order expansion in $q^{i}$ (which is P-wave part that does not contribute to the vacuum to the S-wave state transition when only leading NRQCD interactions are considered), then replace $q^{i}q^{j}$ with ${\bf q}^{2}\delta^{ij}/3$ in the second order expansion.

Thus, by identifying

[TABLE]

we get

[TABLE]

Therefore, we can easily extract the short-distance coefficients for S-wave operators defined in Eq.(11),

[TABLE]

and in turn,

[TABLE]

It is worth noting that the above matching procedure by directly decomposing the 4-component Dirac spinors to Pauli spinors is not very convenient and efficient in the cases that the Dirac structures become complicated. In the literatures, the covariant projection approach are commonly used Kuhn:1979bb ; Guberina:1980dc ; Bodwin:2002hg ; Wang:2015bka . Generally, one applies the replacements

[TABLE]

to project out the spin-singlet and spin-triplet parts, respectively. After expanding the resulting amplitudes in the relative momentum ${\bf q}$, one can extract the S-wave contributions by neglecting the first order terms in ${\bf q}$ and making the replacement $q^{i}q^{j}\to{\bf q}^{2}\delta^{ij}/3$. Finally one can obtain the S-wave short-distance coefficients by identifying the NRQCD matrix elements at tree level

[TABLE]

where $|b\bar{c}(^{2S+1}S_{J})\rangle$ is formally a non-relativistically normalized color-singlet quark-anti-quark pair state. The factor $\sqrt{2N_{c}}$ is due to the spin and color factors of the normalized $b\bar{c}({}^{2S+1}S_{J})$ state.

Since the Dirac structures involved in this work is not very complicated, we use both methods of the direct spinor decomposition and covariant projectors to do the matching for cross-checks. We get the exactly same results as it should be.

III.2 Results of the S-wave LCDAs up to $\mathcal{O}(\alpha_{s}^{0}v^{2})$

Following the matching procedure described in the previous subsection, we get the rest of the necessary short-distance coefficients up to ${\cal O}(v^{2})$ at tree level,

[TABLE]

and consequently,

[TABLE]

In turn, with Eqs. (11) and (12), at tree-level and up to ${\cal O}(v^{2})$, we have the decay constants in terms of NRQCD matrix elements as

[TABLE]

and the LCDAs as

[TABLE]

where $\langle\textbf{q}^{2}\rangle_{P,V}$ are the mean values of $\textbf{q}^{2}$ in scalar and vector $B_{c}$ mesons respectively:

[TABLE]

which characterize ${\cal O}(v^{2})$ relativistic corrections to the LCDAs.

By setting $m_{b}=m_{c}=m$, i.e. $x_{0}=1/2$, we recover the $f,~{}\hat{\phi}(x)$ for heavy quarkonia at tree-level as given in Ref. Wang:2013ywc ; and also Eq.(32c) returns to the result for $f_{J/\psi}^{T}$ in Ref. Bodwin:2014bpa . Note that the matrix elements in (32) are relativistically normalized which are different from the widely-used non-relativistic normalization. To get the results in the latter form, one can multiply the factor $\sqrt{2m_{B_{c}}}$ in the right hand side of Eq. (32) since it is assumed that the states are non-relativistically normalized, as shown by, e.g., Eq. (3) in Ref. Bodwin:2008vp .

III.3 Results of the S-wave LCDAs at $\mathcal{O}(\alpha_{s}^{1}v^{0})$

${\cal O}(\alpha_{s})$ radiative corrections are usually counted as important as ${\cal O}(v^{2})$ relativistic corrections. For completeness, we quote the results of ${\cal O}(\alpha_{s})$ radiative corrections from Ref. Xu:2016dgp .

We have

[TABLE]

for the decay constants, and

[TABLE]

with

[TABLE]

for the LCDAs. Here $\Delta=0$ is for NDR scheme Chanowitz:1979zu , and $\Delta=1$ for the HV scheme tHooft:1972tcz . The $++$- and $+$-distributions are defined as

[TABLE]

where $g(x)$ is a smooth test function.

IV Phenomenological Results

IV.1 Inverse Moment

In the production or decay processes of mesons where collinear factorization can be applied, the inverse moment of the LCDA is crucial because the hard kernel are often functions of $1/x$ and $1/\bar{x}$. Thus it is of phenomenological importance to apply the results in last section to study the inverse moment. The inverse moment of the LCDAs is defined by

[TABLE]

The inverse moment can be calculated from the LCDA as given in the above section. The results are listed below,

[TABLE]

Here $\Delta=0$ is for the NDR scheme, and $\Delta=1$ is for the HV scheme.

IV.2 Gegenbauer moments

The Gegenbauer moments are also commonly used, which are defined by

[TABLE]

With the results of LCDA derived above, one can work out the first two Gegenbauer moments, and the results are given below. For the $a_{1}$, we have

[TABLE]

Notice that for heavy quarkonia, the first Gegenbauer moment $a_{1}$ vanishes, since the $a_{1}$ reflects the asymmetry in the momentum distribution. For the $a_{2}$, we have

[TABLE]

IV.3 Numerical Results

In this subsection, we would like to show that the relativistic corrections to the LCDAs are just as important as the radiative corrections by showing the numerical results of the inverse moments and first two Gegenbauer moments.

In order to get the numerical results, we estimate $\langle{\bf q}^{2}\rangle_{P,V}$ by using the Gremm-Kapuskin (G-K) relation Gremm:1997dq

[TABLE]

which is deduced by implementing the equations of motion from the leading order NRQCD Lagrangian.

The inverse moments and first two Gegenbauer moments of $B_{c},J/\psi,\Upsilon$ are considered, the parameters are collected in Tab. 1. The values of $\langle\textbf{q}^{2}\rangle_{P}$ for $B_{c}$, $\langle\textbf{q}^{2}\rangle_{V}$ for $J/\psi$ and $\Upsilon$ deduced from G-K relation are also listed. Numerical results for the inverse moment and Gegenbauer moments can be calculated with these input parameters, we have used the NDR scheme with $\Delta=0$. They are shown in Tab. 2.

Tab. 2 shows that the relativistic corrections to the inverse moments and first two Gegenbauer moments are comparable in magnitude with the radiative corrections, if not more important. Note that the value of $\langle{\bf q}^{2}\rangle_{P,V}$ deduced from the G-K relation depend on the masses of mesons and quarks. The heavy quark masses adopted here are the running masses $m_{Q}~{}(\mu=m_{Q})$ $(Q=c,b)$ in $\overline{\textrm{MS}}$ scheme, which can be converted to the “pole masses” $m^{\textrm{pole}}_{c}=1.67$ GeV and $m^{\textrm{pole}}_{b}=4.78$ GeV, different choices of quark masses can lead to different values of $\langle{\bf q}^{2}\rangle_{P,V}$. Some researches on values of $\langle\textbf{q}^{2}\rangle_{V}$ for $J/\psi$ and $\Upsilon$ based on potential model can be found in Ref. Bodwin:2007fz and Ref. Chung:2010vz . In Fig. 1, the LCDA $\hat{\phi}_{P}$ for the $B_{c}$ meson is plotted. In this figure, the dashed line denotes the asymptotic form $\phi(x)=6x(1-x)$, while the dot-dashed line denotes the position $x_{0}=m_{b}/(m_{b}+m_{c})=0.77$. The solid line is obtained by including two Gegenbauer moments derived in Eqs. (42a) and (43a).

V Conclusion

A high energy process may involve several perturbative scales. It is very important to handle these different scales, since QCD radiative corrections will induce large logarithms $\alpha_{s}^{n}\ln^{n}m_{Q}^{2}/s$. In collinear factorization, one can relegate the nonperturbative degrees of freedom into LCDAs, while the logarithms $\alpha_{s}^{n}\ln^{n}m_{Q}^{2}/s$ can be handled using the renormalization group equation, running from the scale $\sqrt{s}$ down to $m_{Q}$. For the heavy quarkonium and $B_{c}$ system, the refactorization scheme allows one to further reduce the nonperturbative inputs into only a few NRQCD matrix elements. LCDAs of heavy quarkonia and $B_{c}$ mesons are known at next-to-leading order (NLO) in the strong coupling constant $\alpha_{s}$ and at leading order in the velocity expansion.

In this paper, we have calculated the relativistic corrections of three twist-2 LCDAs for the S-wave $B_{c}$ mesons. The corresponding results for heavy quarkonia such as $J/\psi$ and $\Upsilon$ can be easily deduced by setting $m_{b}=m_{c}$. With the results for these relativistically corrected LCDAs, we have studied a few inverse/Gegenbauer moments. We find that the relativistic corrections are comparable with the next-to-leading order radiative corrections.

Acknowledgement

The authors are grateful to Prof. Hsiang-nan Li and Prof. Xin Liu for useful discussions. This work is supported in part by National Natural Science Foundation of China under Grant No.11275263, 11575110, 11635009, 11655002, Natural Science Foundation of Shanghai under Grant No. 15DZ2272100 and No. 15ZR1423100, and by Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education.