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# Inverse moment of $B$-meson quasi distribution amplitude

Ji Xu  Corresponding author<EMAIL_ADDRESS>School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China  Xi-Ruo
Zhang  Corresponding author<EMAIL_ADDRESS>School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China  Shuai
Zhao  Corresponding author<EMAIL_ADDRESS>Department of Physics, Old
Dominion University, Norfolk, Virginia 23529, USA  Theory Center, Thomas
Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA

###### Abstract

We perform a study on the structure of inverse moment (IM) of quasi
distributions, by taking $B$-meson quasi distribution amplitude (quasi-DA) as
an example. Based on a one-loop calculation, we derive the renormalization
group equation and velocity evolution equation for the first IM of quasi-DA.
We find that, in the large velocity limit, the first IM of $B$-meson quasi-DA
can be factorized into IM as well as logarithmic moments of light-cone
distribution amplitude (LCDA), accompanied with short distance coefficients.
Our results can be useful either in understanding the patterns of perturbative
matching in Large Momentum Effective Theory or evaluating inverse moment of
$B$-meson LCDA on the lattice.

††preprint: JLAB-THY-22-3569

##  I Introduction

The structure of a high energy hadron can be depicted by nonperturbative
functions like parton distribution functions (PDFs), light-cone distribution
amplitudes (LCDA), etc. Because of their nonperturbative nature, parton
distributions cannot be calculated with perturbation theory but should be
either extracted from experimental data or evaluated with nonperturbative
methods like lattice QCD. However, parton distributions are defined with
matrix elements of nonlocal operators located on the light-cone, which cannot
be simulated on the lattice.

In recent years, it was pointed out that the difficulties of simulating parton
physics on a Euclidean lattice can be overcome by employing the Large Momentum
Effective Theory (LaMET) proposed by X. Ji Ji:2013dva . The idea underlying
LaMET is to introduce quasi distributions, which are defined with matrix
elements of equal-time nonlocal operators. The quasi distribution, when
boosting to infinite momentum frame, i.e., $P_{3}\to\infty$ with $P_{3}$ being
the hadron momentum on the third direction (moving direction), can be reduced
to a light-cone distribution. The quasi distribution and its light-cone
counterpart are related by a matching relation and the matching coefficient
can be calculated with perturbative QCD since the difference between quasi and
light-cone distributions is accompanied with a hard momentum scale
$P_{3}\gg\Lambda_{\mathrm{QCD}}$. In recent years, LaMET has been applied on
the lattice calculation of PDFs, LCDAs, etc., for various hadrons. For recent
reviews of LaMET, see e.g. Cichy:2018mum ; Ji:2020ect . Other related
approaches including pseudo distributions Radyushkin:2017cyf ;
Radyushkin:2019mye , lattice cross sections Ma:2017pxb , etc.

It is interesting to study the moments of a quasi distribution. The positive
moments, which are related to local operators, have been discussed in Refs.
Rossi:2017muf ; Ji:2017rah ; Radyushkin:2018nbf . For a quasi-PDF
$\widetilde{f}(x,P_{3})$, the asymptotic behavior at $|x|\to\infty$ is
$~{}1/|x|$, thus the integral $\int dxx^{n}\widetilde{f}(x,P_{3})$ leads to
power divergence if $n$ is a non-negative integer; on the other hand, if $n$
is a negative number, the inverse moment (IM) defined by such integral has no
power divergence. The IM is still an important nonperturbative parameter to
describe a quasi distribution. Till now, there are few studies on IM of quasi
distributions, except the quasi distribution amplitude (quasi-DA) of heavy
quarkonia Jia:2015pxx .

In this work, we will study the IM of quasi distributions. As an example, we
will study the IM of $B$-meson quasi-DA. The $B$-meson LCDA in LaMET is of
particular interest. It is an inherent part of hard-collinear factorization
theorems for many exclusive $B$ decay reactions Beneke:2000ry ; Beneke:2001ev
; Beneke:2001at ; DescotesGenon:2002mw ; Bauer:2002aj ; Beneke:2003zv ;
Becher:2005fg ; moreover, it is also an essential element in the light-cone
sum-rule studies of the $B$-meson decays Khodjamirian:2006st ; Faller:2008tr ;
Gubernari:2018wyi ; Wang:2015vgv ; Wang:2017jow ; Lu:2018cfc ; Gao:2019lta .
The $B$-meson LCDA has been studied in the framework of quasi distribution
amplitude Kawamura:2018gqz ; Wang:2019msf  and the reduced Ioffe-time
distribution Zhao:2020bsx  approaches. It has been proved that the $B$-meson
DA in coordinate space is multiplicatively renormalizable, and the one-loop
matching which links quasi-DA to LCDA is also derived. An ultraviolet (UV)
finite pseudo distribution for $B$-meson is proposed. Lattice simulations on
quasi-DA or pseudo-DA will be useful to extract the $B$-meson LCDA, which is
important in phenomenology.

It has been pointed out in Braun:2003wx  that there are no positive moments
for $B$-meson LCDA, but the IM exists. Moreover, the first IM is an
indispensable part of many factorization theorems in $B$ physics, e.g., the
decay fraction of $B$-meson radiative decay Beneke:2011nf  and $B\to P,V$ form
factors (see Shen:2021yhe  for a recent review). Furthermore, the first IM is
an essential parameter to build models for LCDA. The value of the first IM of
$B$-meson LCDA has been estimated with various approaches, see, e.g., Refs.
Grozin:1996pq ; Korchemsky:1999qb ; Beneke:1999br ; Ball:2003fq ; Braun:2003wx
. However, there are few lattice-based calculations, and the precision has a
large space to be improved.

It will be of phenomenological significance to study the IM of quasi-DA, which
may shed light on the evaluation of LCDA on the lattice. The natural questions
are how the IM of quasi-DA is related to its light-cone counterpart, and how
does IM of quasi-DA evolves with the renormalization scale and velocity of
$B$-meson.

In this paper, we will introduce the IM of $B$-meson quasi DA and perform a
theoretical study on the first IM of $B$-meson quasi-DA in LaMET. We will
investigate the structure of IM of $B$-meson quasi-DA up to one-loop level.
The renormalization, as well as one-loop matching between IMs of LCDA and
quasi-DA, will be investigated. Since IM is a number instead of a
distribution, a naive expectation of matching relation is a multiplication
form; however, we will show that the naive expectation fails. The correct form
of matching will be figured out in this work.

The paper is organized as follows: in Sec. II, we will introduce the concept
of IM and logarithmic moments of $B$-meson quasi-DA; in Sec. III, we will
present the one-loop results, then discuss the renormalization and derive the
renormalization group equation (RGE) in Sec. IV; the one-loop matching will be
discussed in Sec. V; at last, we summarize our results and give an outlook for
future researches in Sec. VI.

##  II quasi distribution amplitude and inverse moment

We follow the notations in Zhao:2020bsx . To start with, let us consider a
nonlocal heavy-light operator in heavy quark effective theory (HQET),

$\displaystyle
O_{\mu}(z,0;v)\equiv\bar{q}(z)\gamma_{\mu}\gamma_{5}h_{v}(0)\,,$ (1)

where $h_{v}$ is a heavy quark field in HQET, with $v$ denoting the velocity
of $B$-meson. $v$ satisfies $v^{2}=1$ and $\not{v}h_{v}=h_{v}$; $\bar{q}(z)$
is a light quark field locating at $z$;
$S(z,0)\equiv\operatorname{P}\exp[-ig\int_{0}^{1}dtz_{\nu}A^{\nu}(tz)]$ is a
Wilson line where $\operatorname{P}$ denotes the path ordering of operators.
By analyzing the Lorentz structure of its meson-to-vacuum matrix element, we
have

$\displaystyle\left\langle
0\left|\bar{q}(z)S(z,0)\gamma_{\mu}\gamma_{5}h_{v}(0)\right|\overline{B}(v)\right\rangle$
$\displaystyle=$ $\displaystyle
iF(\mu)\left[v_{\mu}M_{B,v}(\nu,-z^{2},\mu)+z_{\mu}M_{B,z}(\nu,-z^{2},\mu)\right],$
(2)

where $M_{B,v}(\nu,\mu)$ and $M_{B,z}(\nu,\mu)$ are two scalar functions and
$\nu\equiv v\cdot z$ will be referred to as the “Ioffe-time” of the $B$-meson
111In QCD case, Ioffe-time is the inner product of momentum $p$ and $z$
Ioffe:1969kf ; Braun:1994jq .. $F(\mu)$ is the decay constant of $B$-meson
defined by the matrix element of the local current

$\displaystyle\left\langle
0\left|\bar{q}(0)\gamma_{\mu}\gamma_{5}h_{v}(0)\right|\overline{B}(v)\right\rangle=iv_{\mu}F(\mu)\,.$
(3)

Unlike the QCD case, the decay constant in HQET is scale-dependent.

$M_{B,v}$ term gives the twist-2 distribution when $z^{2}\to 0$ while
$M_{B,z}$ is a higher-twist contribution. We are only interested in the
leading-twist distribution in this work so we rename $M_{B,v}$ as $M_{B}$ for
convenience, and $M_{B}(\nu,-z^{2},\mu)$ is defined as the Ioffe-time
distribution amplitude (ITDA) of the $B$-meson. If $z$ is a light-like vector
with minus component of $z$ being the only nonzero component, then ITDA will
reduce to the light-cone ITDA $\mathcal{I}_{B}^{+}(\nu,\mu)$, i.e.,
$M_{B}(\nu,0,\mu)=\mathcal{I}_{B}^{+}(\nu,\mu)$, which is in fact the LCDA in
coordinate space. The $B$-meson LCDA is defined by the Fourier transform of
$\mathcal{I}_{B}^{+}(\nu,\mu)$ Grozin:1996pq

$\displaystyle\phi^{+}_{B}(\omega,\mu)=\frac{v^{+}}{2\pi}\int_{-\infty}^{\infty}dz^{-}e^{-i\omega
v^{+}z^{-}}{\mathcal{I}_{B}^{+}}(v^{+}z^{-},\mu)\,.$ (4)

It was proposed in Refs. Braun:2007wv ; Ji:2013dva  that one can study equal-
time separations $z=(0,0,0,z_{3})$ on the lattice. The same idea has also been
applied for the $B$-meson LCDA Kawamura:2018gqz ; Wang:2019msf ; Zhao:2020bsx
. In this case, $\nu=-v_{3}z_{3}$ and $z^{2}=-z_{3}^{2}$. The $B$-meson quasi-
DA $\tilde{\phi}_{B}^{+}(\omega,v_{3},\mu)$ can be expressed in terms of ITDA
as Kawamura:2018gqz ; Wang:2019msf

$\displaystyle\widetilde{\phi}_{B}^{+}(\omega,v_{3},\mu)=\frac{|v_{3}|}{2\pi}\int_{-\infty}^{\infty}dz_{3}e^{i\omega
v_{3}z_{3}}M_{B}(-v_{3}z_{3},z_{3}^{2},\mu)\,.$ (5)

The matching relations linking the LCDA and quasi-DA or ITDA were derived in
Refs. Wang:2019msf ; Zhao:2020bsx ; in the pseudo distribution-based approach,
a matching relation that mapping light-cone ITDA to a UV finite reduced ITDA
was also proposed Zhao:2020bsx .

The first IM is defined as

$\displaystyle\lambda_{B}^{-1}(\mu)\equiv\int_{-\infty}^{\infty}d\omega\frac{\phi_{B}^{+}(\omega,\mu)}{\omega}\,.$
(6)

Note that $\phi_{B}^{+}(\omega,\mu)$ only has nonzero support in $[0,\infty)$,
hence the lower limit in Eq. (6) is effectively $0$. Other important
quantities in phenomenology are the logarithmic moments Braun:2003wx

$\displaystyle\sigma_{n}(\mu)=\lambda_{B}(\mu)\int_{-\infty}^{\infty}\frac{d\omega}{\omega}\ln^{n}\frac{\mu}{\omega}\phi_{B}^{+}(\omega,\mu)\,.$
(7)

Similarly, we introduce the IM and logarithmic moments for quasi-DA:

$\displaystyle\widetilde{\lambda}_{B}^{-1}(v_{3},\mu)\equiv$
$\displaystyle\int_{-\infty}^{\infty}d\omega\frac{\widetilde{\phi}_{B}^{+}(\omega,v_{3},\mu)}{\omega}\,,$
(8) $\displaystyle\widetilde{\sigma}_{n}(v_{3},\mu)\equiv$
$\displaystyle\widetilde{\lambda}_{B}(\mu)\int_{-\infty}^{\infty}\frac{d\omega}{\omega}\ln^{n}\frac{\mu}{\omega}\widetilde{\phi}_{B}^{+}(\omega,v_{3},\mu)\,.$
(9)

In this case, however, the region of integration $(-\infty,\infty)$ is
necessary because the support of quasi-DA extends to the whole axis.

In the following, we will explore the structure of the first IM of quasi-DA.
In particular, we will investigate how $\widetilde{\lambda}_{B}(\mu)$ evolve
with energy scale $\mu$, and how $\widetilde{\lambda}_{B}(\mu)$ is related to
its light-cone counterpart $\lambda_{B}(\mu)$. One needs to calculate the
quantum corrections to answer both of these two questions. We will perform an
analysis on the one-loop structure of $\widetilde{\lambda}_{B}$, derive the
renormalization group equation, and give the matching relation between
$\widetilde{\lambda}_{B}$ and the light-cone quantities.

If one boosts the $B$-meson to the infinite momentum frame, i.e.,
$v_{3}\to\infty$, then by definition,
$\widetilde{\lambda}_{B}\sim\lambda_{B}+\mathcal{O}(1/v_{3})$. Within the
spirit of LaMET, a matching relation can be expected between
$\widetilde{\lambda}_{B}$ and $\lambda_{B}$. In principle, this matching
relation should be a direct inference of the matching relation for DAs in
Wang:2019msf ; however, we find that it is not straightforward. The reason is
that one needs further regularization to do the $\omega$-integral; moreover,
the linear divergence of quasi-DA is hidden in dimensional regularization, and
this linear divergence will be weakened to a logarithmic divergence for IM. If
such divergence was neglected, one cannot renormalize IM correctly. For this
reason, we prefer a new calculation from the beginning, instead of using the
matching coefficient reported in Wang:2019msf  or Zhao:2020bsx .

##  III one-loop calculation

To study the renormalization and matching in perturbation theory, one can
replace the hadron state with Fock state $|b\bar{q}\rangle$, as what we have
done for DAs. Let $p$ being the momentum of the light quark,
$p=\omega_{0}v+p_{\perp}$ with $\omega_{0}>0$, where $v$ is the velocity of
$B$-meson. We set $v=(v^{0},0,0,v^{3})$, or in light-cone coordinates,
$v=(v^{+},v^{-},\mathbf{0}_{\perp})$. Assuming that the light quark is
slightly offshell, i.e., $p^{2}<0$, then $-p^{2}$ will work as an infrared
(IR) regulator. At tree level, we have
$\phi_{B}^{+}(\omega,\omega_{0})=\widetilde{\phi}_{B}^{+}(\omega,\omega_{0})=\delta(\omega-\omega_{0})$.
This leads to the tree level result for IM:
$1/\widetilde{\lambda}_{B}=1/\lambda_{B}=1/\omega_{0}$.

To evaluate the one-loop corrections, we work in the Feynman gauge. The
Feynman diagrams are shown in Fig. 1. Dimensional regularization (DR) is
applied to regularize the UV singularities, where the space-time dimension is
set to $d=4-2\epsilon$.

Figure 1: The Feynman diagrams for quasi-DA and LCDA. The horizontal double
line represents the gauge link, while the vertical double line denotes the
heavy-quark in HQET. The single line represents the light quark.

Before starting one-loop calculation, we note that the singularities at
$\omega=0$ in the integrals Eqs. (6)(8) need prescription. For future
convenience, we add a small imaginary part $i\epsilon$ to the denominator,
i.e., $\omega\to\omega+i\epsilon$, to ensure that the IMs are well defined. IM
of LCDA is independent of prescription because
$\phi_{B}^{+}(\omega)\sim\omega$ when $\omega\to 0$ Lange:2003ff .

In fig. 1(a), there is a hard gluon exchange between the heavy quark and
Wilson line. For DAs, one has

$\displaystyle\phi_{B}^{+}(\omega,\omega_{0},\mu)|_{(a)}=-ig^{2}\widetilde{\mu}^{2\epsilon}C_{F}n\cdot
v\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+i\epsilon)(-v\cdot
k+i\epsilon)(n\cdot k+i\epsilon)}\left[\delta\left(\omega-\frac{n\cdot
k}{n\cdot v}-\frac{n\cdot p}{n\cdot v}\right)-\delta\left(\omega-\frac{n\cdot
p}{n\cdot v}\right)\right]\,,$ (10)

where $\widetilde{\mu}=\mu\sqrt{\frac{e^{\gamma_{E}}}{4\pi}}$, $\mu$ is an
energy scale in DR and $\gamma_{E}$ is the Euler-Mascheroni constant; $C_{F}$
is a color factor with the value $4/3$. Since we have written the above
expression in a Lorentz covariant form, it is applicable for both LCDA and
quasi-DA. Then for IMs, after integrating out $\omega$, we have

$\displaystyle\lambda_{B}^{-1}(\omega_{0},\mu)=$ $\displaystyle-
ig^{2}\widetilde{\mu}^{2\epsilon}C_{F}n\cdot
v\int_{-\infty}^{\infty}d\omega\frac{1}{\omega+i\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+i\epsilon)(-v\cdot
k+i\epsilon)(n\cdot k+i\epsilon)}$
$\displaystyle\times\left[\delta\left(\omega-\frac{n\cdot k}{n\cdot
v}-\frac{n\cdot p}{n\cdot v}\right)-\delta\left(\omega-\frac{n\cdot p}{n\cdot
v}\right)\right]$ $\displaystyle=$ $\displaystyle
ig^{2}\widetilde{\mu}^{2\epsilon}C_{F}\frac{n\cdot v^{2}}{n\cdot
p}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+i\epsilon)(-v\cdot
k+i\epsilon)(n\cdot k+n\cdot p+i\epsilon)}\,.$ (11)

This integral can be easily calculated by using Feynman or Schwinger
pamameterizations. For IM of LCDA, we let $n$ being a light-cone vector and
$n\cdot a=a^{+}$ for arbitrary vector $a$, $n^{2}=0$. We have

$\displaystyle\lambda^{-1}_{B}(\omega_{0},\mu)|_{(a)}=$
$\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\frac{1}{\omega_{0}}\bigg{(}\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}\ln\frac{\mu^{2}}{\omega_{0}^{2}}+\frac{1}{2}\ln^{2}\frac{\mu^{2}}{\omega_{0}^{2}}+\frac{3\pi^{2}}{4}\bigg{)}\,.$
(12)

On the other hand, for IM of quasi-DA, we choose $n$ as a unit vector in the
3rd direction of space-time, $n\cdot a=a_{3}$ for an arbitrary vector $a$, and
$n^{2}=-1$. After integration, one has

$\displaystyle\widetilde{\lambda}^{-1}_{B}(\omega_{0},v_{3},\mu)|_{(a)}=$
$\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\frac{1}{\omega_{0}}\bigg{[}2\frac{\ln
2iv_{3}}{\epsilon}+2\ln 2iv_{3}\bigg{(}\ln\frac{\mu^{2}}{\omega_{0}^{2}}-\ln
2iv_{3}\bigg{)}+\frac{\pi^{2}}{3}\bigg{]}\,.$ (13)

In the light-cone case one can observe a double pole $1/\epsilon^{2}$, which
is the indication of cusp singularity; in the IM of quasi-DA, there is only
single pole but accompanied with a meson-velocity-dependent coefficient. There
are single and double logarithmic dependence on $v_{3}$ in the quasi case.

In fig. 1(b), hard gluon exchanges between the light quark and Wilson line.
With the same method described for (a), one has, for IMs,

$\displaystyle iF(\omega_{0},\mu)\lambda^{-1}_{B}(\omega_{0},\mu)|_{(b)}=$
$\displaystyle
ig^{2}\widetilde{\mu}^{2\epsilon}C_{F}\int\frac{d\omega}{\omega+i\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{\bar{v}(p)\not{n}(\not{p}+\not{k})\not{n}\gamma_{5}u_{v}}{(k^{2}+i\epsilon)[(p+k)^{2}+i\epsilon](-n\cdot
k+i\epsilon)}$ $\displaystyle\times\left[\delta\left(\omega-\frac{n\cdot
p}{n\cdot v}\right)-\delta\left(\omega-\frac{n\cdot p}{n\cdot v}-\frac{n\cdot
k}{n\cdot v}\right)\right]\,.$ (14)

On the light-cone side,

$\displaystyle\lambda_{B}^{-1}(\omega_{0},\mu)=$
$\displaystyle\frac{\alpha_{s}}{4\pi}C_{F}\frac{1}{\omega_{0}}\left(\frac{2}{\epsilon}+2\ln\frac{\mu^{2}}{-p^{2}}+4\right)\,,$
(15)

and for the quasi side, one has

$\displaystyle\widetilde{\lambda}_{B}^{-1}(\omega_{0},\mu)=\frac{\alpha_{s}}{4\pi}C_{F}\frac{1}{\omega_{0}}\left(\frac{1}{\epsilon}+2\ln\frac{\mu^{2}}{-p^{2}}+2\ln
2iv_{3}-\ln\frac{\mu^{2}}{\omega_{0}^{2}}+2\right)\,.$ (16)

Both the light-cone and quasi cases have IR divergences regularized by
$\ln(-p^{2})$, with the same coefficients. It indicates that $\lambda_{B}$ and
$\widetilde{\lambda}_{B}$ have the same IR behavior.

For Fig. 1(c), we have

$\displaystyle-
ig^{2}\widetilde{\mu}^{2\epsilon}C_{F}n^{2}\int\frac{d\omega}{\omega+i\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+i\epsilon)(n\cdot
k+i\epsilon)(-n\cdot k+i\epsilon)}\left[\delta\left(\omega-\frac{n\cdot
p}{n\cdot v}-\frac{n\cdot k}{n\cdot v}\right)-\delta\left(\omega-\frac{n\cdot
p}{n\cdot v}\right)\right]$ $\displaystyle=$ $\displaystyle-
ig^{2}\widetilde{\mu}^{2\epsilon}C_{F}n^{2}\frac{1}{\omega_{0}}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+i\epsilon)(n\cdot
k+i\epsilon)(n\cdot k+n\cdot p+i\epsilon)}\,.$ (17)

The IM of LCDA has no such contribution because $n^{2}=0$. For IM of quasi-DA,
the result reads

$\displaystyle\widetilde{\lambda}_{B}^{-1}(\omega_{0},v_{3},\mu)|_{(c)}=$
$\displaystyle\frac{\alpha_{s}}{2\pi}C_{F}\frac{1}{\omega_{0}}\left(-\frac{1}{\epsilon}+2\ln
2iv_{3}-\ln\frac{\mu^{2}}{\omega_{0}^{2}}-2\right)\,.$ (18)

In the calculation of quasi-DA, Wilson line self-energy diagram Fig. 1(c) has
a linear divergence. How to deal with the linear divergence is one of the
major missions in the development of quasi and pseudo distribution approaches
in the last few years. Many schemes, e.g., RI/MOM Constantinou:2017sej ;
Stewart:2017tvs , reduced Ioffe-time distribution Orginos:2017kos ;
Radyushkin:2017lvu , hybrid scheme Ji:2020brr , etc., have been proposed to
renormalize the linear divergence in quasi distributions. However, the IM
involves only logarithmic UV divergence and has no linear divergence problem.
It allows us to renormalize IM of quasi-DA with $\overline{\mathrm{MS}}$
scheme.

For the box diagram fig. 1(d), we have

$\displaystyle\frac{-ig^{2}}{2n\cdot
v}\widetilde{\mu}^{2\epsilon}C_{F}\int\frac{d\omega}{\omega+i\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}}\bar{v}(p)\not{v}(\not{p}+\not{k})\not{n}\gamma_{5}u_{v}\frac{1}{(k^{2}+i\epsilon)[(p+k)^{2}+i\epsilon](-v\cdot
k+i\epsilon)}\delta\left(\omega-\frac{n\cdot p}{n\cdot v}-\frac{n\cdot
k}{n\cdot v}\right)$ $\displaystyle\sim$ $\displaystyle\frac{-ig^{2}}{2n\cdot
v}\widetilde{\mu}^{2\epsilon}C_{F}\int\frac{d\omega}{\omega+i\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}}\operatorname{Tr}[\not{v}(\not{p}+\not{k})\not{n}\gamma_{5}\frac{1+\not{v}}{2}\gamma_{5}]\frac{1}{(k^{2}+i\epsilon)[(p+k)^{2}+i\epsilon](-v\cdot
k+i\epsilon)}\delta\left(\omega-\frac{n\cdot p}{n\cdot v}-\frac{n\cdot
k}{n\cdot v}\right)$ $\displaystyle=$ $\displaystyle
ig^{2}\widetilde{\mu}^{2\epsilon}C_{F}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+i\epsilon)[(p+k)^{2}+i\epsilon](-v\cdot
k+i\epsilon)}\,,$ (19)

where in the second step we have performed the trace with projector
$\frac{1+\not{v}}{2}\gamma_{5}$ Lange:2003ff . This integral does not depend
on $n$, indicating that it contributes the same results to IMs of LCDA and
quasi-DA. Furthermore, the integral is UV finite. Therefore, the box diagram
can be ignored both in renormalization and perturbative matching. In fact, it
has already been shown in Wang:2019msf ; Zhao:2020bsx  that the box diagram
does not contribute, both in the quasi and pseudo distribution approaches.
Thus, it will not contribute to the renormalization and matching of IM as
well. It can also be clarified through a $v_{3}$ power-counting argument.

##  IV Renormalization group equation

We renormalize the IMs of LCDA and quasi-DA in $\overline{\mathrm{MS}}$
scheme:

$\displaystyle\lambda_{B,\mathrm{bare}}^{-1}(\omega_{0})=\bigg{[}1-\frac{\alpha_{s}(\mu)}{4\pi}C_{F}\bigg{(}\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}\ln\frac{\mu^{2}}{\omega_{0}^{2}}-\frac{2}{\epsilon}\bigg{)}\bigg{]}\lambda_{B}^{-1}(\omega_{0};\mu)\,,$
(20)
$\displaystyle\widetilde{\lambda}_{B,\mathrm{bare}}^{-1}(\omega_{0},v_{3})=\bigg{[}1-\frac{\alpha_{s}(\mu)}{4\pi}C_{F}\bigg{(}\frac{1}{\epsilon}+\frac{2}{\epsilon}\ln
2iv_{3}\bigg{)}\bigg{]}\widetilde{\lambda}_{B}^{-1}(\omega_{0};v_{3},\mu)\,.$
(21)

With the the above renormalization equation, we obtain RGE for IM of LCDA,

$\displaystyle\mu\frac{d}{d\mu}\lambda_{B}^{-1}(\mu)=$
$\displaystyle-\frac{\alpha_{s}(\mu)}{2\pi}C_{F}\left[2\lambda_{B}^{-1}(\mu)\sigma_{1}(\mu)-\lambda_{B}^{-1}(\mu)\right]\,.$
(22)

This result repeats the RGE derived in Ball:2003bf ; Lange:2003ff ; it can
also be derived from the Lange-Neubert equation for LCDA Lange:2003ff . On the
other hand, for IM of quasi-DA, one has

$\displaystyle\mu\frac{d}{d\mu}\widetilde{\lambda}_{B}^{-1}(v_{3},\mu)=$
$\displaystyle-\frac{\alpha_{s}(\mu)}{2\pi}C_{F}(2\ln
2iv_{3}+1)\widetilde{\lambda}_{B}^{-1}(v_{3},\mu)\,.$ (23)

One can observe that the IM of LCDA is not multiplicative renormalized, it
will get mixed with logarithmic moment $\sigma_{1}$ at one-loop. However, for
IM of quasi-DA, there is no mixing between 1st IM and logarithmic moments at
one-loop level, which is different from IM of LCDA. If one works at next-to-
leading order accuracy, one can evolve $\widetilde{\lambda}_{B}$ to other
scales without the input of other parameters. However, we are not clear
whether or not the IM of quasi-DA is multiplicatively renormalizable to all
orders of perturbation theory.

##  V Matching relation in LaMET and velocity RGE

In LaMET, light-cone and equal-time (“quasi”) quantities can be linked by a
matching relation, with a perturbatively calculable hard function.

When $v_{3}\to\infty$, the equal-time matrix element that defines the IM of
quasi-DA will become a light-cone matrix element, it indicates that under
large Lorentz boost, $\widetilde{\lambda}_{B}(v_{3},\mu)\to\lambda_{B}(\mu)$.
With the spirit of LaMET, one can expect the IR physics of
$\widetilde{\lambda}_{B}(v_{3},\mu)$ and $\lambda_{B}(\mu)$ are the same and
there is a matching formula between IMs. Since the IMs have no dependence on
$\omega$, a naive expectation for the matching relation is a multiplication
instead of a convolution, i.e.,

$\displaystyle\widetilde{\lambda}_{B}(v_{3},\mu_{Q})=C\left(v_{3},\mu_{Q},\mu_{L}\right)\lambda_{B}(\mu_{L})+\mathcal{O}(1/v_{3})\,,$
(24)

where $C(v_{3},\mu_{Q},\mu_{L})$ is the hard coefficient, which can be
expanded in series of $\alpha_{s}$ as

$C(v_{3},\mu_{Q},\mu_{L})=\sum_{n=0}\left(\frac{\alpha_{s}(\mu)}{2\pi}C_{F}\right)^{n}C^{(n)}(v_{3},\mu_{Q},\mu_{L})\,;$

$\mu_{Q}$ and $\mu_{L}$ are the scales that define the IMs of quasi and light-
cone DAs, respectively.

We note that, a very similar multiplication-type matching relation was
introduced in Pilipp:2007sb , which connects the IMs of LCDAs defined in HQET
and full QCD; it can also be reproduced from the convolution-type matching
between the LCDAs defined in QCD and HQET Ishaq:2019dst ; Zhao:2019elu .
Another example is the matching of IMs of quasi-PDF and the normal PDF. One
can start with the convolution-type matching relation for quasi and normal
PDFs and derive the matching relation for their IMs. In the large $P_{3}$
limit, for quasi-PDF $\widetilde{f}$ and normal PDF $f$, we have Ji:2013dva

$\displaystyle\widetilde{f}(x,P_{3},\mu_{Q})=\int_{-1}^{1}\frac{dy}{|y|}Z\left(\frac{x}{y},P_{3},\mu_{Q},\mu_{L}\right)f(y,\mu_{L})\,,$
(25)

with $Z$ being the hard function. With variable substitution $x=ty$, we get

$\displaystyle\int_{-\infty}^{\infty}\frac{dx}{x}\widetilde{f}(x,P_{3},\mu_{Q})=$
$\displaystyle\int_{-\infty}^{\infty}\frac{dt}{t}Z\left(t,P_{3},\mu_{Q},\mu_{L}\right)\int_{-1}^{1}\frac{dy}{y}f(y,\mu_{L})\,,$
(26)

which indicates that IM of quasi-PDF can be factorized as IM of normal PDF
multiplied by a hard coefficient.

Expanding $\widetilde{\lambda}_{B}(\mu_{Q})$, $\lambda_{B}(\mu_{L})$ and
$C(v_{3},\mu_{Q},\mu_{L})$ in series of $\mathcal{O}(\alpha_{s})$ in Eq. (24),
up to one-loop level, we have $C^{(0)}=1$, and

$\displaystyle C^{(1)}\left(v_{3},\mu_{Q},\mu_{L}\right)=$
$\displaystyle-\ln^{2}\frac{\mu_{L}}{\omega_{0}}+(2\ln
2iv_{3}+3)\ln\frac{\mu_{L}}{\omega_{0}}+(2\ln
2iv_{3}+1)\ln\frac{\mu_{Q}}{\mu_{L}}$ $\displaystyle-\ln 2iv_{3}(\ln
2iv_{3}+3)-\frac{5\pi^{2}}{24}+3\,.$ (27)

On the one hand, the IR singularities in IMs of LCDA and quasi-DA, which are
regularized by $\ln(-p^{2})$, are canceled, so the matching coefficient is IR
free; on the other hand, $C^{(1)}(v_{3},\mu_{Q},\mu_{L})$ depends on
$\omega_{0}$. It means the matching coefficient depends on $\omega_{0}v$—the
momentum of external light quark, which indicates the failure of naive
factorization relation Eq. (24); moreover, the single and double logarithmic
dependence on $\omega_{0}$ should be absorbed into other nonperturbative
quantities when $\omega_{0}$ is small. Thus the matching relation should not
be a multiplication equation, which is different from the PDF case, and also
the case of matching IM in QCD to HQET Pilipp:2007sb .

Noticing that the $\ln^{n}\mu_{L}/\omega_{0}$ terms are related to the
logarithmic moments defined in Eq. (7), we propose a modified factorization
formula as

$\displaystyle\widetilde{\lambda}_{B}(v_{3},\mu_{Q})=C_{0}\left(v_{3},\frac{\mu_{Q}}{\mu_{L}}\right)\lambda_{B}(\mu_{L})+\sum_{n=1}C_{n}\left(v_{3},\frac{\mu_{Q}}{\mu_{L}}\right)\sigma_{n}(\mu_{L})+\mathcal{O}(1/v_{3})\,.$
(28)

Again, the matching coefficients $C_{n}(v_{3},\mu_{Q}/\mu_{L})$ can be
expanded in series of $\alpha_{s}$ as

$\displaystyle
C_{n}\left(v_{3},\frac{\mu_{Q}}{\mu_{L}}\right)=\sum_{m=0}\left(\frac{\alpha_{s}(\mu)}{2\pi}C_{F}\right)^{m}C_{n}^{(m)}\left(v_{3},\frac{\mu_{Q}}{\mu_{L}}\right)\,.$
(29)

Performing the perturbative expansion, at the leading order, it gives
$C_{0}^{(0)}=1$ and $C_{n}^{(0)}=0$ for $n\geq 1$. With one-loop results, we
get the next-to-leading order corrections of the hard coefficients, which read

$\displaystyle C_{0}^{(1)}\left(v_{3},\frac{\mu_{Q}}{\mu_{L}}\right)$
$\displaystyle=(2\ln 2iv_{3}+1)\ln\frac{\mu_{Q}}{\mu_{L}}-\ln 2iv_{3}(\ln
2iv_{3}+3)-\frac{5\pi^{2}}{24}+3\,,$ (30) $\displaystyle
C_{1}^{(1)}\left(v_{3},\frac{\mu_{Q}}{\mu_{L}}\right)$ $\displaystyle=2\ln
2iv_{3}+3\,,$ (31) $\displaystyle
C_{2}^{(1)}\left(v_{3},\frac{\mu_{Q}}{\mu_{L}}\right)$ $\displaystyle=-1\,,$
(32) $\displaystyle C_{n}^{(1)}\left(v_{3},\frac{\mu_{Q}}{\mu_{L}}\right)$
$\displaystyle=0\,~{}~{}(n\geq 3)\ .$ (33)

Coefficient $C_{0}$ has double and single logarithmic dependence on $B$-meson
velocity $v_{3}$, while $C_{1}$ has only single logarithmic dependence on
$v_{3}$, and there is no $v_{3}$ dependence in $C_{2}$ at one-loop level. This
indicates that the momentum evolution of $\widetilde{\lambda}_{B}(v_{3},\mu)$
is only related to itself and the first logarithmic moment. In fact, one can
write down the velocity evolution equation for
$\widetilde{\lambda}_{B}(v_{3},\mu)$ as

$\displaystyle
v_{3}\frac{d}{dv_{3}}\widetilde{\lambda}_{B}(v_{3},\mu)=-\frac{\alpha_{s}}{2\pi}C_{F}\left[(2\ln
2iv_{3}+3)\widetilde{\lambda}_{B}(v_{3},\mu)-2\widetilde{\sigma}_{1}(v_{3},\mu)\right]\,.$
(34)

To evolve $\widetilde{\lambda}_{B}(v_{3},\mu)$ from one velocity to another,
one needs the input of the first IM and logarithmic moment.

At last, one may notice that the matching coefficients $C_{n}$ are complex
numbers, therefore, the IM of quasi-DA is also complex. This is due to the
$+i\epsilon$ prescription we employed at $\omega=0$. One can also get real
values for IM and matching coefficients with other prescriptions, e.g, Cauchy
principal value. Since
$\operatorname{P.V.}(1/\omega)=\frac{1}{2}(\frac{1}{\omega+i\epsilon}+\frac{1}{\omega-i\epsilon})$,
one can get $\widetilde{\lambda}_{B}$ under principal value prescription by
averaging the results under $+i\epsilon$ prescription and their complex
conjugates, and this is equivalent to taking the real parts in our results. It
is verified by a direct calculation under principal value prescription.

##  VI summary and outlook

We have performed a detailed study on the inverse moment of a kind of quasi
distributions: the $B$-meson quasi distribution amplitude. The non-negative
moments of quasi distributions have power divergences; however, the inverse
and logarithmic moments of quasi distribution are free of power divergences.
The inverse moment could be an important parameter that reflects some
interesting features of a quasi distribution.

In this paper, we introduce the inverse moment of $B$-meson quasi-DA and
explore its properties. Its light-cone partner is of great significance in the
phenomenology of $B$ decays. In the factorization of many $B$-meson exclusive
decay processes, the amplitudes at leading order are always in the form of
$1/\omega$. So the IM of LCDA is an important nonperturbative parameter that
gives the leading contribution in the factorization formula. The studies on IM
of quasi-DA can shed light on the properties of IM of LCDA, and also on the
lattice simulation of $B$-meson LCDA.

With a one-loop calculation of IMs, we derive the renormalization and RGE for
the first IM of quasi-DA and figure out the correct form of matching relation
in LaMET. The first IM of quasi-DA is not only factorized into the first IM
but also the logarithmic moments of LCDA, accompanied with hard coefficients
which are free of IR singularities and independent of external states.

The IM of quasi-DA can be simulated on a Euclidean lattice since it is defined
with equal-time matrix elements. Distinguished from non-negative moments, IM
is not defined by a local operator, so the determination of IM needs the
information of the whole quasi-DA. One then may be worried about the
significance of this work to the lattice QCD community at first glance,
because if one calculated IM on the lattice, one should calculate quasi-DA
first, but quasi-DA can be matched onto LCDA directly without calculating IM.
In fact, the lattice simulation of nonlocal HQET matrix element is very
challenging, and evaluating the continuous $\omega$ distribution will be a
very difficult task. A realistic idea will be building proper models for
quasi-DA and fitting the parameters of models with a few lattice data. The IM
of quasi-DA will be an essential parameter for quasi-DA models, just like the
IM for the models of LCDA Grozin:1996pq ; Braun:2003wx . Then, with the fitted
IM of quasi-DA from lattice data, one can map it onto the IM and logarithmic
moments of LCDA. The mixing of IM and logarithmic moments of LCDA in the
matching formula may cause other difficulties. Such difficulties caused by
mixing may be overcome by evaluating $\widetilde{\lambda}_{B}$ with several
$v_{3}$ and extracting the IM and logarithmic moments by fitting the $v_{3}$
dependence. Further lattice simulations on IM of quasi-DA will be crucial to
improve the determination of IM and other parameters of $B$-meson LCDA.

## Acknowledgments

We thank Wei Wang for careful reading of the manuscript and valuable
suggestions. The work of J. X. and X. R. Z. is supported in part by the
National Natural Science Foundation of China under Grant No. 12105247 and No.
12047545, the China Postdoctoral Science Foundation under Grant No.
2021M702957. The work of S. Z. is supported by Jefferson Science Associates,
LLC under U.S. DOE Contract # DE-AC05-06OR23177 and by U.S. DOE Grant # DE-
FG02-97ER41028.

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