# Single Transverse-Spin Asymmetry and Sivers Function
in Large Momentum Effective Theory

Xiangdong Ji  Center for Nuclear Femtography, SURA, 1201 New York Ave. NW,
Washington, DC 20005, USA  Maryland Center for Fundamental Physics, Department
of Physics, University of Maryland, College Park, Maryland 20742, USA
Yizhuang Liu  Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai
200240, China  Institut für Theoretische Physik, Universität Regensburg,
D-93040 Regensburg, Germany  Andreas Schäfer  Institut für Theoretische
Physik, Universität Regensburg, D-93040 Regensburg, Germany  Feng Yuan
Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA

###### Abstract

We apply recent developments in large momentum effective theory (LaMET) to
formulate a non-perturbative calculation of the single-transverse spin
asymmetry in terms of the quasi transverse-momentum-dependent quark
distribution functions from the so-called Sivers mechanism. When the spin
asymmetry is defined as the ratio of the quark Sivers function over the spin
averaged distribution, it can be directly calculated in terms of the relevant
quasi distributions with the soft functions and perturbative matching kernels
cancelling out. Apart from the general formula presented, we have verified the
result in the small transverse distance limit at one-loop order, which reduces
to a collinear expansion at twist-three level.

##  I Introduction

Transverse momentum dependent (TMD) parton distribution functions (PDFs,
TMDPDFs) are one of the important ingredients in nucleon tomography and a
central focus of hadron physics research in recent years and especially at the
future electron-ion collider Accardi  _et al._ (2016); Boer  _et al._ (2011).
TMDPDFs can be experimentally extracted from hard processes in deep inelastic
scattering (DIS) and lepton pair production in hadronic collisions Collins
(2013); Collins and Soper (1981); Ji  _et al._ (2005); Bacchetta  _et al._
(2008). The available experimental data and global analysis have generated
strong interest in the hadron physics community, see, e.g., recent efforts in
Refs. Cammarota  _et al._ (2020); Bacchetta  _et al._ (2020).

The first attempt to compute the moments of TMDPDFs from lattice QCD has been
made in Refs. Musch  _et al._ (2011, 2012); Yoon  _et al._ (2017). Meanwhile,
great progress has been made to compute $x$-dependent parton physics on the
lattice using large momentum effective theory (LaMET) Ji (2013, 2014), see,
some recent reviews on this topic Cichy and Constantinou (2019); Ji  _et al._
(2020a). LaMET is based on the observation that parton physics defined in
terms of lightcone correlations can be obtained from time-independent
Euclidean correlations (called quasi-distributions) through well-defined
effective field theory (EFT) expansion as well as matching and running. LaMET
has been applied to compute various collinear PDFs and distribution amplitudes
Cichy and Constantinou (2019); Ji  _et al._ (2020a). In the last few years, an
important new development has been to apply LaMET to describe TMDPDFs and
associated soft functions Ji  _et al._ (2015, 2019a); Ebert  _et al._ (2019a,
b, 2020a); Shanahan  _et al._ (2020a, b); Ebert  _et al._ (2020b); Ji  _et
al._ (2020b, 2019b); Vladimirov and Schäfer (2020); Zhang  _et al._ (2020). In
this paper, we study single transverse-spin asymmetries in the region where
the transverse momentum is on the order of $\Lambda_{\rm QCD}$, focusing on
the non-perturbative calculation of the relevant TMDPDF—the quark Sivers
function Sivers (1990)— in terms of a Euclidean-space quasi distribution.

The spin-dependent, $k_{\perp}$-even TMDPDFs have been studied in Ref. Ebert
_et al._ (2020b), where similar factorization and matching were found as for
the unpolarized case. Because the quark Sivers function is a $k_{\perp}$-odd
distribution, it has special features different from those of the
$k_{\perp}$-even ones. In particular, in the large $k_{\perp}$ or small
transverse distance limit, the quark Sivers function can be expressed in terms
of the collinear twist-three quark-gluon-quark correlation functions in the
nucleon, whereas the $k_{\perp}$-even TMDPDFs depend on the leading twist
collinear quark distribution functions. Therefore, the EFT matching
calculation in the present case is more involved compared with that in Ref.
Ebert  _et al._ (2020b).

In this paper, we will focus on computing the quark-Sivers function in the
leading-order expansion from large-momentum effective theory. An extension to
the gluon-Sivers function should be possible in a similar manner. The quark-
Sivers function describes a nontrivial correlation between the quark’s
transverse momentum and the nucleon’s transverse polarization vector.
Therefore, it represents a spin asymmetry in the TMDPDF. The quark Sivers
function is non-zero because the gauge link associated with the quark
distribution contributes the phase needed to obtain a single spin asymmetry
Brodsky  _et al._ (2002); Collins (2002); Ji and Yuan (2002); Belitsky  _et
al._ (2003).

The paper is organized as follows. In Sec. II, we present the evolution
equation of the quasi-Sivers function and its matching to the physical Sivers
function. Resummation formulas for the quasi-Sivers function and the matching
kernel are also given. A generic argument to demonstrate the matching between
the quasi-Sivers function and the light-cone Sivers function will be presented
based on the factorization of the hard, collinear and soft gluon radiation
contributions for the TMDPDFs. Because the matching coefficient only concerns
hard gluon radiation, it does not depend on the spin structure of the nucleon.
This is consistent with the observation in Ref. Ebert  _et al._ (2020b). In
Sec. III, we provide detailed derivations of the quasi-Sivers function in
LaMET up to one-loop order at large transverse momentum. Our calculations is
based on the collinear twist-three quark-gluon-quark correlation functions,
and we compute the quasi-Sivers function in terms of the Qiu-Sterman matrix
element Efremov and Teryaev (1982, 1985); Qiu and Sterman (1991, 1992, 1999)
(defined below). This can be compared to the light-cone quark-Sivers function
calculated in the same framework Ji  _et al._ (2006a, b, c); Koike  _et al._
(2008); Kang  _et al._ (2011); Sun and Yuan (2013); Scimemi  _et al._ (2019)
and the associated matching coefficient can be obtained. In Sec. IV, we show
the application of the formalism to experimental and theoretical single spin
asymmetries. Finally, We summarize our paper in Sec. V.

##  II LaMET Expansion of Sivers Function in Euclidean Quasi TMDPDF

Let us start with the transverse-spin dependent quasi-TMDPDF for quarks in a
proton moving along the $+\hat{z}$ direction Ji  _et al._ (2019a, b)

$\displaystyle\tilde{q}(x,k_{\perp},S_{\perp},\mu,\zeta_{z})=\int
d^{2}b_{\perp}~{}\int\frac{d\lambda}{2(2\pi)^{3}}e^{i\lambda
x+i\vec{k}_{\perp}\cdot\vec{b}_{\perp}}$ (1)
$\displaystyle\\!\lim_{L\rightarrow\infty}\frac{\langle
PS|\bar{\psi}\big{(}\frac{\lambda
n_{z}}{2}\\!+\\!\vec{b}_{\perp}\big{)}\Gamma{\cal W}_{z}(\frac{\lambda
n_{z}}{2}\\!+\\!\vec{b}_{\perp};-L)\psi\big{(}\\!-\\!\frac{\lambda
n_{z}}{2}\big{)}|PS\rangle}{\sqrt{Z_{E}(2L,b_{\perp},\mu)}}\ ,$

where $\overline{\rm MS}$ renormalization is implied,
$b_{\perp}=|\vec{b}_{\perp}|$, and the staple-shaped gauge-link ${\cal W}_{z}$
is

$\displaystyle{\cal
W}_{z}(\xi;-L)=W^{\dagger}_{z}(\xi;-L)W_{\perp}W_{z}(-\xi^{z}n_{z};-L)\ ,$ (2)
$\displaystyle W_{z}(\xi;-L)={\cal P}{\rm
exp}\Big{[}-ig\int_{\xi^{z}}^{-L}ds\,n_{z}\cdot
A(\vec{\xi}_{\perp}\\!+\\!n_{z}s)\Big{]}\ .$ (3)

The spin dependence is introduced by the hadron state $|PS\rangle$. $x$ and
$k_{\perp}$ are the longitudinal momentum fraction and the transverse momentum
carried by the quark, and $\zeta_{z}=4x^{2}P_{z}^{2}$ is the rapidity or
Collins-Soper scale. The direction vector of the gauge-link $n_{z}$ is defined
as $n_{z}=(0,0,0,1)$ and all coordinates are 4-vectors, e.g.
$\vec{b}_{\perp}=(0,b_{1},b_{2},0)$. In contrast, $L$ is just a number. $\mu$
is the ultra-violet (UV) renormalization scale. A transverse gauge link was
included to make the gauge links connected. The spin-1/2 proton has momentum
$P^{z}$ and is polarized transversely, with the polarization vector
$\vec{S}_{\perp}$ being perpendicular to its momentum direction. The Dirac
matrix $\Gamma$ can be chosen as $\Gamma=\gamma^{t}$ or $\Gamma=\gamma^{z}$.
As we will show, to leading order in $1/P^{z}$ the two choices are equivalent.
The subtraction factor $Z_{E}(2L,b_{\perp},\mu)$ is the vacuum expectation
value of a rectangular Wilson-loop that removes the pinch-pole singularity at
large $L$ Ji  _et al._ (2019a, b)

$\displaystyle Z_{E}(2L,b_{\perp},\mu)=\frac{1}{N_{c}}{\rm Tr}\langle
0|W_{\perp}{\cal W}_{z}(\vec{b}_{\perp};2L)|0\rangle\ .$ (4)

As emphasized in Ji  _et al._ (2019a, b), The self-interactions of gauge links
are subtracted using $\sqrt{Z_{E}}$ in order to remove the pinch-pole
singularities Ji  _et al._ (2020b) and to guarantee the existence of the large
$L$ limit.

With the above definition, we can express the transverse-spin dependent quasi-
TMDPDF in terms of appropriate Lorentz structures,

$\displaystyle\tilde{q}(x,k_{\perp},S_{\perp},\mu,\zeta_{z})$
$\displaystyle=\tilde{q}(x,k_{\perp},\mu,\zeta_{z})+\frac{\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})\epsilon^{\beta\alpha}S_{\perp\beta}k_{\perp\alpha}}{M_{P}}\
,$ (5)

where $M_{P}$ is the proton mass and $\epsilon^{12}=1$ in our convention. In
the above equation, the first term represents the spin-averaged, unpolarized
quark distribution and the second term is the quark Sivers function in LaMET.
It is also convenient to Fourier transform the $k_{\perp}$ distribution to get
the $b_{\perp}$-space expression,

$\displaystyle\tilde{q}(x,b_{\perp},S_{\perp},\mu,\zeta_{z})=\int
d^{2}k_{\perp}e^{-i\vec{k}_{\perp}\cdot\vec{b}_{\perp}}\tilde{q}(x,k_{\perp},S_{\perp},\mu,\zeta_{z})\
,$ (6)

which is convenient for factorization calculations. We can similarly express
the quasi-TMDPDF in $b_{\perp}$-space as,

$\displaystyle\tilde{q}(x,b_{\perp},S_{\perp},\mu,\zeta_{z})$
$\displaystyle=\tilde{q}(x,b_{\perp},\mu,\zeta_{z})+\epsilon^{\alpha\beta}S_{\perp}^{\beta}\tilde{f}_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta_{z})\
.$ (7)

We would like to point out that $\tilde{q}$ is the Fourier transform of the
spin-average quark TMDPDF in momentum space, but
$\tilde{f}^{\alpha}_{1T}(x,b_{\perp},\mu,\zeta_{z})$ is not a direct Fourier
transform of $\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})$ due to the
presence of $k_{\perp\alpha}$ in Eq. (II). Our focus in this paper is the
large-momentum factorization of
$\tilde{f}_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta_{z})$.

###  II.1 Evolution Equation

We start with the renormalization property of quasi-TMDPDFs. Similar as for
the unpolarized quasi-TMDPDFs, in the numerator of Eq. (1), there are linear
divergences associated to the self-energy of the staple-shaped gauge-link
${\cal W}_{z}$ and logarithmic divergences associated to the quark-link
vertices. In addition, there are cusp-UV divergence associated to the
junctions between longitudinal and transverse gauge-links at $z=-L$. After
subtraction using $\sqrt{Z_{E}}$ in the denominator, the linear divergences
and the cusp-UV divergences all cancel, and one is left with only the
logarithmic divergences for quark-link vertices. The associated anomalous
dimensions are all equal and are known to be equivalent to the anomalous
dimension $\gamma_{F}$ for the heavy-light quark current Falk  _et al._
(1990); Ji and Musolf (1991). It can also be derived as the anomalous
dimension of the quark field in $A^{z}=0$ gauge. Thus, the spin-dependent
quasi-TMDPDF, in particular the quasi-Sivers function satisfies the following
renormalization group equation

$\displaystyle\mu^{2}\frac{d}{d\mu^{2}}\ln\tilde{f}_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta_{z})=\gamma_{F}\
.$ (8)

At one-loop level one has $\gamma_{F}=\frac{3\alpha_{s}C_{F}}{4\pi}$ and high-
order results can be found in references Falk  _et al._ (1990); Ji and Musolf
(1991).

We then come to the evolution equation of quasi-TMDPDFs with respect to
$\zeta_{z}$, i.e. the momentum evolution equation Ji  _et al._ (2020a).
Similar to the case of quasi-PDFs or unpolarized quasi-TMDPDFs, at large
$P^{z}$ there are large logarithms of $P^{z}$ that can be resummed by the
corresponding momentum evolution equation. Using diagramatic methods developed
in Collins and Soper (1981); Collins (2013), it can be shown Collins and Soper
(1981); Ji  _et al._ (2015) that $\tilde{f}_{1T}^{\perp\alpha}$ satisfies the
evolution equation

$\displaystyle
2\zeta_{z}\frac{d}{d\zeta_{z}}\ln\tilde{f}_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta_{z})=K(b_{\perp},\mu)+G(\zeta_{z},\mu)\
,$ (9)

where $K$ is the non-perturbative Collins-Soper kernel Collins and Soper
(1981) and the $G$ is a perturbative part of the evolution kernel. At one-loop
level, one has Collins and Soper (1981); Ji  _et al._ (2015),

$\displaystyle K^{(1)}(b_{\perp},\mu)$
$\displaystyle=-\frac{\alpha_{s}C_{F}}{\pi}L_{b}\,,$ (10) $\displaystyle
G^{(1)}(\zeta_{z},\mu)$
$\displaystyle=\frac{\alpha_{s}C_{F}}{\pi}\left(1-L_{z}\right)\,.$ (11)

Here $L_{b}=\ln\frac{\mu^{2}b_{\perp}^{2}}{c_{0}^{2}}$ with
$c_{0}=2e^{-\gamma_{E}}$ and $L_{z}=\ln\frac{\zeta_{z}}{\mu^{2}}$. These
equations allows the quasi-Sivers function to be re-summed in the form in
which $\mu_{b}=\frac{c_{0}}{b_{\perp}}$

$\displaystyle\tilde{f}_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta_{z})=\tilde{f}_{1T}^{\perp\alpha}(x,$
$\displaystyle b_{\perp},\mu=\sqrt{\zeta_{z}}=\mu_{b})$
$\displaystyle\times\exp\bigg{(}\frac{1}{2}\ln\frac{\zeta_{z}}{\mu_{b}^{2}}K(b_{\perp},\mu)$
$\displaystyle+\int_{\mu_{b}^{2}}^{\zeta_{z}}\frac{d\zeta^{\prime}}{2\zeta^{\prime}}G(\zeta^{\prime},\mu)$
$\displaystyle+\int^{\mu}_{\mu_{b}}\frac{d\mu^{\prime 2}}{\mu^{\prime
2}}\gamma_{F}(\alpha_{s}(\mu^{\prime}))\bigg{)}$ (12)

with $\mu_{b}=\frac{c_{0}}{b_{\perp}}$. Using the renormalization group
equation Collins and Soper (1981); Ji  _et al._ (2019b) for $G$ and $K$,
$\frac{d\ln G}{d\ln\mu}=-\frac{d\ln K}{d\ln\mu}=2\Gamma_{\rm cusp}$ where
$\Gamma_{\rm cusp}$ is the light-like cusp-anomalous dimension, allows a more
refined treatment of resummation for $K$ and $G$.

###  II.2 LaMET Expansion for Sivers Function

Similar to the unpolarized case Ji  _et al._ (2019b), the quasi-TMDPDFs can be
used to calculate the physical TMDPDFs appearing in the factorizations of
experimental cross sections. The LaMET expansion formula requires the off-
light-cone reduced soft function $S_{r}(b_{\perp},\mu)$, the definition and
more properties of which can be found in Ji  _et al._ (2019b, 2020b, 2020a).
In terms of the non-perturbative reduced soft function, the EFT expansion
formula for spin-dependent quasi-TMDPDF reads:

$\displaystyle f_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta)$
$\displaystyle=\frac{e^{-K(b_{\perp},\mu)\ln(\frac{\zeta_{z}}{\zeta})}}{H\left(\frac{\zeta_{z}}{\mu^{2}}\right)}\sqrt{S_{r}(b_{\perp},\mu)}\tilde{f}_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta_{z})+...\
.$ (13)

where $H$ is the perturbative kernel and the higher-order terms in $1/P^{z}$
expansoin have been omitted.

Similar to the unpolarized case Ji  _et al._ (2019b), here we provide the
sketch of a proof for the matching formula (II.2). We also argue that $H$ is
independent of the spin structure, as was recently argued in Ref. Ebert  _et
al._ (2020b).

First of all, one can perform a standard leading region analysis Collins
(2013) for all spin-structures with minor modifications to include the staple-
shaped gauge-links of the quasi-TMDPDF as in Ji  _et al._ (2019b). The leading
region or the reduced diagram for quasi-TMDPDFs is shown in Figure. 1. There
are collinear and soft subdiagrams responsible for collinear and soft
contributions. The collinear contributions are exactly the same as those for
the light-cone TMDPDF defined with light-like gauge-links. The soft radiations
between the fast moving color charges and the staple shaped gauge-links can be
factorized by the off-light-cone soft function.

In addition to the collinear and soft subdiagrams, there are two hard
subdiagrams around the vertices at $0$ and $\vec{b}_{\perp}$. The natural hard
scale $\zeta_{z}$ for the hard diagram is formed by a Lorentz invariant
combination of the parton momenta entering the hard subdiagram and the
direction vector $n_{z}$ for the staple-shaped gauge links. At large $P^{z}$,
small $k_{\perp}$ or large $b_{\perp}$, the hard contributions are confined
within the vicinities of the quark-link vertices around $0$ and $b_{\perp}$,
since any hard momenta flowing between $0$ and $b_{\perp}$ will cause
additional power suppressions in $\frac{1}{P^{z}}$. In another words, there
are two disconnected hard subdiagrams, one containing $0$ and another one
containing $\vec{b}_{\perp}$. Therefore, the momentum fractions carried by the
quasi-TMDPDF and the physical TMDPDF are the same and the matching formula
contains no convolution.

Given the leading region of the quasi-TMDPDF, one can apply the standard Ward-
identity argument of Ref. Collins (2013) to factorize the quasi-TMDPDF and
obtain Eq. (II.2). The reduced soft function, which is actually the inverse of
the rapidity independent part of the off-light-cone soft function Ji  _et al._
(2020b, 2019b), appears to compensate the differences of soft contributions
for quasi-TMDPDFs and physical TMDPDFs. The exponential of the Collins-Soper
kernel can be explained by the emergence of large logarithms for quasi-TMDPDFs
in the form $K(b_{\perp},\mu)\ln\frac{\zeta_{z}}{\mu^{2}}$ generated by
momentum evolution. To match to the physical TMDPDF at rapidity scale $\zeta$,
a factor $e^{-K(b_{\perp},\mu)\ln(\frac{\zeta_{z}}{\zeta})}$ is therefore
needed. Finally, the mismatch between $\tilde{f}$ and $f$ due to the hard
contributions is captured by the hard kernel $H$ that depends on the hard
scale $\zeta_{z}$ and the renormalization scale $\mu$. The above arguments are
similar for the unpolarized case Ji  _et al._ (2019b).

Here we argue that the hard kernel $H$ is independent of the spin structure.
As we already emphasized, the hard-cores around $0$ and $\vec{b}_{\perp}$ are
disconnected. Any momentum that is allowed to flow between the vertices and
sees the transverse separation is either soft or collinear. The hard momenta
have essentially no effects on the other vertex. Therefore, in order to obtain
the matching kernel, it is sufficient to consider only “half” of the quark
quasi-TMDPDF, which one might want to call an “amputated” form factor
containing only an incoming light-quark with momentum $p=xP$ and an “outing
going” gauge-link along $n_{z}$ direction. This form factor is shown in Fig.
2. For this form factor, the generic Lorentz structure can always be written
as

$\displaystyle\Gamma\left(A+B\gamma\cdot n_{z}\gamma\cdot p+C\gamma\cdot
p\gamma\cdot n_{z}\right)u(p,S)\ ,$ (14)

where $\Gamma$ is a generic Dirac matrix at the quark-link vertex, unrelated
to that in Eq.(1), $u(p,S)$ is the Dirac spinor for the incoming quark and
$A$,$B$,$C$ are scalar functions of $p^{2}$, $n_{z}^{2}$ and $n_{z}\cdot p$ .
Using the anti-commutation relation of Dirac matrices and the equation of
motion $\gamma\cdot pu(p,S)=0$, the above equation can be rewritten as

$\displaystyle\Gamma\left(A+2Cn_{z}\cdot p\right)u(p,S)\ ,$ (15)

which depends only on a universal scalar function $A+2Cn_{z}\cdot p$,
independent of the spin $S$ and the Dirac matrix $\Gamma$. As a result, the
matching kernel only depends on these scalar functions but not the spin $S$
and the Dirac matrix $\Gamma$.

Figure 1: The leading regions of the quasi-TMDPDF where $C$ is the collinear
subdiagram, $S$ is the soft subdiagram and $H$’s are hard subdiagrams. The two
hard cores are not connected with each other (but their open Dirac indices are
ontracted), and as a result, the momentum fraction of the quasi-TMDPDF
receives only contributions from collinear modes and there is no convolution
in the matching formula.  Figure 2: The form factor shown here is sufficient
for calculating the matching kernel, which contains an incoming quark with
momentum $p=xP$, spin $S$ and an outgoing gauge-link in $n_{z}$ direction.

The above general results can be verified at one-loop order when $b_{\perp}$
is small and a perturbative QCD calculation is valid. The one-loop calculation
is more complicated compared with that in Ebert  _et al._ (2020b), because it
involves twist-three collinear factorization.

First of all, let us recall that the standard TMDPDF factorization at small
$b_{\perp}$ follows the procedure in Refs. Collins (2013); Catani  _et al._
(2001, 2014); Prokudin  _et al._ (2015). For the quark Sivers function, we
have Sun and Yuan (2013); Scimemi  _et al._ (2019),

$\displaystyle{f}_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta)=\frac{ib_{\perp}^{\alpha}}{2}T_{F}(x,x)$
$\displaystyle+\frac{ib_{\perp}^{\alpha}}{2}\frac{\alpha_{s}}{2\pi}\left\\{\left(-\frac{1}{\epsilon}-L_{b}\right){\cal
P}_{qg/qg}^{T}\otimes T_{F}(x,x)\right.$
$\displaystyle\left.+\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})\left[-\frac{1}{2N_{c}}(1-\xi_{x})+\delta(1-\xi_{x})C_{F}s^{(1)}\right]\right\\}\
,$ (16)

where $\xi_{x}=\frac{x}{x_{q}}$, $T_{F}(x,x)$ is the twist-3 quark-gluon-quark
correlation function (the Qiu-Sterman matrix element) defined below and ${\cal
P}_{qg/qg}^{T}$ is the associated splitting kernel. For the part involved in
the calculations of Sec. III, we have Braun  _et al._ (2009); Schafer and Zhou
(2012); Kang and Qiu (2012); Sun and Yuan (2013); Scimemi  _et al._ (2019)
111Here and in the following calculations, we only keep the so-called soft
gluon pole and hard gluon pole contributions in the twist-three formalism Ji
_et al._ (2006a, b, c). A complete kernel including soft-fermion pole
contributions and other twist-3 matrix elements can be found in Refs. Braun
_et al._ (2009); Scimemi  _et al._ (2019).,

$\displaystyle{\cal P}_{qg/qg}^{T}\otimes T_{F}(x,x)$
$\displaystyle~{}=\int\frac{dx_{q}}{x_{q}}\left\\{T_{F}(x_{q},x_{q})\left[C_{F}\left(\frac{1+\xi_{x}^{2}}{1-\xi_{x}}\right)_{+}-C_{A}\delta(1-\xi_{x})\right]\right.$
$\displaystyle~{}~{}\left.+\frac{C_{A}}{2}\left(T_{F}(x_{q},x)\frac{1+\xi_{x}}{1-\xi_{x}}-T_{F}(x_{q},x_{q})\frac{1+\xi_{x}^{2}}{1-\xi_{x}}\right)\right\\}\
.$ (17)

The contribution $s^{(1)}$ reads

$\displaystyle
s^{(1)}=-\frac{\pi^{2}}{12}+\frac{3}{2}L_{b}-\frac{1}{2}L_{b}^{2}-L_{\zeta}L_{b}\
,$ (18)

where $L_{\zeta}=\ln\frac{\zeta}{\mu^{2}}$. Our definition of the physical
TMD-PDF follows the standard one in Ref. Echevarría  _et al._ (2013); Collins
and Rogers (2013), although the numerical factors $\frac{\pi^{2}}{12}$ depends
on the renormalization schemes due the presence of double
$\frac{1}{\epsilon^{2}}$ poles, see section VI of Ref. Collins and Rogers
(2013) for a discussion. Our results are in the standard $\overline{MS}$
scheme.

In the next section, we show that the quasi-Sivers function has a similar
factorizaiton at small $b_{\perp}$,

$\displaystyle{\tilde{f}}_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta_{z})=\frac{ib_{\perp}^{\alpha}}{2}T_{F}(x,x)$
$\displaystyle+\frac{ib_{\perp}^{\alpha}}{2}\frac{\alpha_{s}}{2\pi}\left\\{\left(-\frac{1}{\epsilon}-L_{b}\right){\cal
P}_{qg/qg}^{T}\otimes T_{F}(x,x)\right.$
$\displaystyle\left.+\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})\left[-\frac{1}{2N_{c}}(1-\xi_{x})+\delta(1-\xi_{x})C_{F}\tilde{s}^{(1)}\right]\right\\}\
.$ (19)

where $\tilde{s}^{(1)}$ reads

$\displaystyle\tilde{s}^{(1)}=$
$\displaystyle-2+\frac{5}{2}L_{b}-\frac{1}{2}L_{b}^{2}-L_{z}L_{b}-\frac{1}{2}L_{z}^{2}+L_{z}\
.$ (20)

Furthermore, the one-loop reduced soft function reads Ji  _et al._ (2020a):

$\displaystyle S_{r}(b_{\perp},\mu)=1-\frac{\alpha_{s}C_{F}}{\pi}L_{b}\ .$
(21)

Combining all of the above and comparing it to the LaMET expansion in Eq.
(II.2), one obtains the one-loop matching kernel

$\displaystyle
H\left(\frac{\zeta_{z}}{\mu^{2}}\right)=1+\frac{\alpha_{s}C_{F}}{2\pi}\bigg{(}-2+\frac{\pi^{2}}{12}-\frac{1}{2}L_{z}^{2}+L_{z}\bigg{)}\
,$ (22)

which is exactly the answer we expected: The matching kernel is independent of
spin-structure and is equal to that of the unpolarized case for
$\Gamma=\gamma^{z}$ and $\Gamma=\gamma^{t}$.

It can be shown Ji  _et al._ (2019b) that the matching kernel satisfies the
renormalization group equation

$\displaystyle\mu\frac{d}{d\mu}\ln
H\left(\frac{\zeta_{z}}{\mu^{2}}\right)=\Gamma_{\rm
cusp}\ln\frac{\zeta_{z}}{\mu^{2}}+\gamma_{C}$ (23)

where $\gamma_{C}$ can be found in Ref. Ji  _et al._ (2019b). The general
solution to the above equation reads

$\displaystyle
H\left(\alpha_{s}(\mu),\frac{\zeta_{z}}{\mu^{2}}\right)=H\left(\alpha_{s}(\sqrt{\zeta_{z}}),1\right)$
(24)
$\displaystyle\times\exp\left\\{\int_{\sqrt{\zeta_{z}}}^{\mu}\frac{d\mu^{\prime}}{\mu^{\prime}}\left[\Gamma_{\rm
cusp}(\alpha_{s}(\mu^{\prime}))\ln\frac{\zeta_{z}}{\mu^{{}^{\prime
2}}}+\gamma_{C}\big{(}\alpha_{s}(\mu^{\prime})\big{)}\right]\right\\}\,.$

This equation allows the determination of the large logarithms for $H$ to all
orders in perturbation theory, up to unknown constants related to the initial
condition $H(\alpha_{s},1)$.

##  III One-loop Calculation for Spin-Dependent quasi-TMPPDFs

In this section we calculate the quasi-Sivers function at one-loop level. The
idea and procedure is the same as for previous examples in the LaMET formalism
Ji  _et al._ (2015, 2019a); Ebert  _et al._ (2019a, b, 2020a, 2020b); Ji  _et
al._ (2020b, 2019b); Vladimirov and Schäfer (2020). An important difference is
that we will not be able to formulate it in terms of a single quark target.
Instead, we need to use the collinear twist-three quark-gluon-quark
correlation description and compute the quark quasi-TMDPDF and Sivers
asymmetry in these collinear quark distributions at small $b_{\perp}\ll
1/\Lambda_{\rm QCD}$.

For the quasi-TMDPDFs, we follow the definition of Eq. (1) Ji  _et al._
(2019b), where a rectangular Wilson-loop was adopted to remove the pinch-pole
singularities. To match to the physical TMDPDFs at leading order in $1/P^{z}$,
one needs the reduced soft function at small $b_{\perp}$, which can also be
extracted from lattice simulations at any $b_{\perp}$ Zhang  _et al._ (2020).

The perturbative quasi-TMDPDFs at small $b_{\perp}\ll 1/\Lambda_{\rm QCD}$ can
be expressed in terms of the collinear parton distribution and/or the twist-
three quark-gluon-quark correlation functions. For the unpolarized quark
distribution, the previous results of Ji  _et al._ (2019a) can be expressed
as,

$\displaystyle\tilde{q}(x,b_{\perp},\mu,\zeta_{z})$
$\displaystyle=f_{q}(x_{,}\mu)+\frac{\alpha_{s}}{2\pi}\left\\{\left(-\frac{1}{\epsilon}-L_{b}\right){\cal
P}_{q/q}\otimes f_{q}(x)\right.$
$\displaystyle\left.+C_{F}\int\frac{dx_{q}}{x_{q}}f_{q}(x_{q})\left[(1-\xi_{x})+\delta(1-\xi_{x})\tilde{s}^{(1)}\right]\right\\}\
,$ (25)

for the leading order plus next-to-leading order result in
${\vec{b}}_{\perp}$-space, where $\xi_{x}=x/x_{q}$, $\mu$ is the
renormalization scale in the $\overline{\rm MS}$ scheme, ${\cal
P}_{q/q}(\xi_{x})=C_{F}\left(\frac{1+\xi_{x}^{2}}{1-\xi_{x}}\right)_{+}$ is
the usual splitting kernel for the quark, and $f_{q}(x)$ represents the light-
cone integrated quark distribution function. The one-loop coefficient in the
subtraction scheme of Ref. Ji  _et al._ (2019a) reads

$\displaystyle\tilde{s}^{(1)}=$
$\displaystyle\frac{3}{2}\ln\frac{b_{\perp}^{2}\mu^{2}}{c_{0}^{2}}+\ln\frac{\zeta_{z}L^{2}}{4c_{0}^{2}}-\frac{1}{2}\left(\ln\frac{\zeta_{z}b_{\perp}^{2}}{c_{0}^{2}}\right)^{2}$
$\displaystyle+2{\cal K}(\xi_{b})-{\cal K}(2\xi_{b})\ ,$ (26)

where the Collins-Soper scale $\zeta_{z}=4x_{q}^{2}P_{z}^{2}$ and the function
${\cal K}$ will be defined later on. At large $L$, all the $L$ dependencies
cancel and we have:

$\displaystyle\tilde{s}^{(1)}=-2+\frac{5}{2}L_{b}-\frac{1}{2}L_{b}^{2}-L_{z}L_{b}-\frac{1}{2}L_{z}^{2}+L_{z}\
,$ (27)

where $L_{b}$ and $L_{z}$ are defined after Eq. (10).

After the renormalization of the integrated quark distribution $f_{q}(x)$ at
$\mu=\mu_{b}$, we can write the quasi-TMD unpolarized quark distribution as

$\displaystyle\tilde{q}(x,b_{\perp},\mu_{b},\zeta_{z})$
$\displaystyle=\int\frac{dx_{q}}{x_{q}}f_{q}(x_{q},\mu_{b})$
$\displaystyle\bigg{(}\delta(1-\xi_{x})+\frac{\alpha_{s}C_{F}}{2\pi}\bigg{[}(1-\xi_{x})$
$\displaystyle+\delta(1-\xi_{x})\tilde{s}^{(1)}\bigg{]}\bigg{)}\ .$ (28)

The goal of the following derivations in this section is to apply the
collinear twist-three formalism and calculate the quasi-Sivers function in
LaMET at leading order and next-to-leading order.

###  III.1 Phase Contribution from the Gauge Link

In Eq. (1), the quasi-TMDPDF contains a gauge link with a finite length,
implying that the eikonal gauge link propagators will be modified. From
previous works Brodsky  _et al._ (2002); Collins (2002); Ji and Yuan (2002);
Belitsky  _et al._ (2003), we know that the gauge link propagators contribute
to the crucial phase which is necessary to generate a non-zero Sivers
function. Therefore, we need to check that the finite length gauge link can
still do so.

Because of the finite length of the gauge links, the eikonal propagator in
these diagrams will be modified according to

$(-ig)\frac{in^{\mu}}{n\cdot k\pm
i\epsilon}\Longrightarrow(-ig)\frac{in^{\mu}}{n\cdot k}\left(1-e^{\pm in\cdot
kL}\right)\ ,$ (29)

where $n^{\mu}$ represents the gauge link direction. In the current case
$n^{\mu}=n_{z}^{\mu}$. In perturbation calculations, we will make use of the
large length limit $|LP_{z}|\gg 1$. By doing so, many previous results can be
applied to our calculations. For example, in the large $L$ limit, we have the
following identity:

$\lim_{L\to\infty}\frac{1}{n\cdot k}e^{\pm iLn\cdot k}=\pm i\pi\delta(n\cdot
k)\ ,$ (30)

which will contribute to the phases needed for a non-zero quark Sivers
function.

In the following calculations, we will take two limits whenever this is
possible: The large $L$ limit and the large $P_{z}$ limit. In certain
diagrams, we have to use finite $L$ and $P_{z}$ to regulate, for example, the
pinch-pole singularity and/or the end-point singularity Ji  _et al._ (2019a).
We will emphasize these important points when we carry out the detailed
calculations.

###  III.2 Leading Order

We carry out the derivations in the twist-three collinear framework, where the
quark Sivers function depends on the so-called twist-three quark-gluon-quark
correlation function, aka, the Qiu-Sterman matrix elements Efremov and Teryaev
(1982, 1985); Qiu and Sterman (1991, 1992, 1999). It is defined as follows,

$\displaystyle
T_{F}(x_{2},x_{2}^{\prime})\equiv\int\frac{d\zeta^{-}d\eta^{-}}{4\pi}e^{i(x_{2}P^{+}\eta^{-}+(x_{2}^{\prime}-x_{2})P_{B}^{+}\zeta^{-})}\epsilon_{\perp}^{\beta\alpha}S_{\perp\beta}$
$\displaystyle\times\,\left\langle PS|\overline{\psi}(0){\cal
L}(0,\zeta^{-})\gamma^{+}g{F_{\alpha}}^{+}(\zeta^{-}){\cal
L}(\zeta^{-},\eta^{-})\psi(\eta^{-})|PS\right\rangle\ ,$ (31)

where $F^{\mu\nu}$ represent the gluon field strength tensor. From the leading
order derivation Boer  _et al._ (2003), we have,

$\frac{1}{M_{P}}\int d^{2}k_{\perp}\,k^{2}_{\perp}\,f_{1T}^{\perp(\rm
SIDIS)}(x,k_{\perp})=-T_{F}(x,x)\ ,$ (32)

where $f_{1T}^{\perp({\rm SIDIS})}$ represents the quark Sivers function for a
SIDIS process with gauge link going to $+\infty$, corresponding to our choice
of $-L$ in Eq. (1).

Figure 3: Leading order diagrams for quasi-Sivers function.

The method for calculating the single transverse-spin asymmetry in the twist-
three formalism has been well developed Qiu and Sterman (1991, 1992, 1999); Ji
_et al._ (2006a, b, c); Kouvaris  _et al._ (2006); Eguchi  _et al._ (2006,
2007); Koike and Tanaka (2007a, b); Koike  _et al._ (2008); Braun  _et al._
(2009); Kang and Qiu (2009); Vogelsang and Yuan (2009); Zhou  _et al._ (2009);
Schafer and Zhou (2012); Kang  _et al._ (2011); Sun and Yuan (2013); Scimemi
_et al._ (2019). There are different approaches to derive the final result, in
the following, we follow the collinear $k_{g\perp}$-expansion method Qiu and
Sterman (1991, 1992, 1999); Ji  _et al._ (2006a, b, c); Kouvaris  _et al._
(2006). In this approach, the additional gluon from the polarized hadron is
associated with a gauge potential $A^{+}$, assuming that the polarized nucleon
is moving along the $+\hat{z}$ direction. Thus, the gluon will carry
longitudinal polarization and its momentum is parameterized as
$x_{g}P+k_{g\perp}$, where $x_{g}$ is the momentum fraction with respect to
the polarized proton and $k_{g\perp}$ is the transverse momentum. The
contribution to the single-transverse-spin asymmetry arises from terms linear
in $k_{g\perp}$ in the expansion of the partonic amplitudes. When combined
with $A^{+}$, these linear terms will yield $\partial^{\perp}A^{+}$, a part of
the gauge field strength tensor $F^{\perp+}$ in Eq. (31). As shown in Fig. 3,
we have $\vec{k}_{g\perp}=\vec{k}_{q2\perp}-\vec{k}_{q1\perp}$. Therefore, the
$k_{g\perp}$ expansion of the scattering amplitudes can be expresssed in terms
of the transverse momenta $k_{q1\perp}$ and $k_{q2\perp}$. The associated
quark momenta are parameterized as,

$k_{q1}=x_{q1}P+k_{q1\perp},~{}~{}~{}k_{q2}=x_{q2}P+k_{q2\perp}\ .$ (33)

We compute the quasi-Sivers function defined in Eq. (1) with the Gamma matrix
$\Gamma=\gamma^{t}$ or $\gamma^{z}$. The results are the same in the leading
power of $1/P_{z}$. The leading order diagrams of Fig. 3 can be calculated
following the above general procedure. The method is similar to that for the
standard quark Sivers function calculation in Ref. Kang  _et al._ (2011); Sun
and Yuan (2013). In particular, the phase comes from the gauge link
propagator,

$\displaystyle\lim_{L\to\infty}\frac{1}{n_{z}\cdot k_{g}}e^{\pm iLn_{z}\cdot
k_{g}}=\pm i\pi\frac{1}{n_{z}\cdot P}\delta(x_{g})\ ,$ (34)

which determines the kinematics for the twist-three Qiu-Sterman matrix element
at $T_{F}(x,x)$. The plus/minus signs correspond to the left and right
diagrams where the gluon attaches to the left and right sides of the cut-line,
respectively. To calculate the Sivers function in $b_{\perp}$-space, we need
to perform a Fourier transformation with respect to the probing quark
transverse momentum $k_{\perp}$ in Fig. 3. Because of momentum conservation,
at leading order, $k_{\perp}=k_{q2\perp}$ for the left diagram and
$k_{\perp}=k_{q1\perp}$ for the right diagram. As shown above, these two
diagrams contribute with opposite sign to the Sivers function. Therefore, the
total contribution is proportional to:

$\left(e^{i\vec{k}_{q2\perp}\cdot\vec{b}_{\perp}}-e^{i\vec{k}_{q1\perp}\cdot\vec{b}_{\perp}}\right)\to
ib_{\perp}^{\alpha}(k_{q2\perp}^{\alpha}-k_{q1\perp}^{\alpha})=ib_{\perp}^{\alpha}k_{g\perp}^{\alpha}\
,$ (35)

in the collinear expansion. As a result, the leading order result for the
quark Sivers function in LaMET reads

$\displaystyle\tilde{f}_{1T}^{\perp\alpha(0)}(x,b_{\perp},\mu,\zeta_{z})=\frac{ib_{\perp}^{\alpha}}{2}T_{F}(x,x)\
.$ (36)

Here, the normalization is consistent with Eq. (32).

###  III.3 One-loop Order from Cut Diagrams

It has been shown that the quark TMDPDFs in LaMET can be evaluated by the cut
diagram approximation Ji  _et al._ (2015, 2019a). In particular, if we focus
on the kinematic region $0<x<1$, the cut diagram approximation leads to the
same results as the complete calculation. In the following, we will apply this
approximation to simplify the derivation of the quark Sivers function in
LaMET.

Figure 4: Cut diagram approximation to calculate the Siver function in LaMET.
A mirror diagram similar to that in Fig. 3 should be included as well. The
longitudinal gluon from the polarized nucleon can attach to any lines
associated with the blob.

In Fig. 4, we show the generic diagrams to calculate the quark Sivers function
in LaMET. The lower part represents the quark-gluon-quark correlation from the
polarized nucleon. We follow the strategy of Ref. Ji  _et al._ (2006b) to
evaluate these diagrams. The radiated gluon carries transverse momentum
$k_{1\perp}$ equal in size but opposite to $k_{\perp}$. Similar as for the
leading diagrams, we need to generate a phase from the gauge link propagators
in these diagrams. This corresponds to the pole contributions to the single
spin asymmetries in the twist-three formalism Ji  _et al._ (2006a, b, c). In
the following calculations, we focus on the so-called soft-gluon pole and
hard-gluon pole contributions. They are characterized by the longitudinal
momentum fraction carried by the gluon attached to the hard partonic part from
the polarized nucleon: $x_{g}=0$ corresponds to the soft-gluon-pole
contribution, while $x_{g}\neq 0$ corresponds to the hard-gluon-pole
contribution. It is straightforward to extend this treatment to other
contributions such as the soft-fermion pole contribution, and those associated
with the twist-three function $\tilde{G}_{F}$ Koike  _et al._ (2008).

Figure 5: Soft-gluonic pole contribution at one-loop order for the real gluon
radiation.

We emphasize again that the quasi-Sivers function defined in Eq. (1) can be
computed with $\Gamma=\gamma^{t}$ or $\gamma^{z}$ and the results are the same
in the TMD limit. The soft gluon pole diagrams are shown in Fig. 5. The pole
contribution is the same as that for the leading order diagrams, i.e.,
$\delta(n_{z}\cdot k_{g})=\frac{1}{n_{z}\cdot P}\delta(x_{g})$. An important
step to obtain the final result is to perform the collinear expansion for the
incoming parton momenta. Therefore, we will keep the transverse momenta for
$k_{q1}$, $k_{q2}$, and $k_{g}$. Because of momentum conservation, we have
$k_{g\perp}=k_{q2\perp}-k_{q1\perp}$. Therefore, there will be two independent
transverse momenta in the expansion. One of the collinear expansion
contributions comes from the on-shell condition of the radiated gluon in the
cut-diagram approximation. This leads to the so-called derivative terms, which
can be easily evaluated Ji  _et al._ (2006b). The final result can be written
as

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\partial}=-\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{1}{2N_{c}}$
$\displaystyle\times\int\frac{dx_{q}}{x_{q}}\left(x_{q}\frac{\partial}{\partial
x_{q}}T_{F}(x_{q},x_{q})\right)\left(1+\xi_{x}^{2}+(1-\xi_{x})^{2}\frac{D-2}{2}\right)\
,$ (37)

where $\xi_{x}=x/x_{q}$ and $D$ represents the dimension for the transverse
plane. In the following we will also use $\epsilon=(2-D)/2$. We have also
applied the following relation between the momentum fractions along the
$\hat{z}$ direction and those along the light-cone plus direction,

$\displaystyle(1-\xi)=(1-\xi_{x})\frac{1+\sqrt{1+r_{\perp}^{2}}}{2}\ ,$ (38)

where $r_{\perp}=|k_{\perp}|/(x_{q}(1-\xi_{x})P_{z})$. In the TMD limit away
from the end-point of $\xi_{x}=1$, we will have $(1-\xi)\to(1-\xi_{x})$. At
the end-point, we will have to keep the full expression in order to derive the
complete result. However, for the above derivative terms, we can simply
substitute $(1-\xi)\to(1-\xi_{x})$. We further notice that the derivative
terms can be transformed into non-derivative terms by performing an integral
by part,

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\partial}=-\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{1}{2N_{c}}\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})$
$\displaystyle\times\left[2\xi_{x}^{2}+2\epsilon\xi_{x}(1-\xi_{x})+2\delta(1-\xi_{x})\right]\
.$ (39)

The last term in the square brackets comes from the boundary.

Now, we turn to the non-derivative terms. Fig. 5(a) is easy to derive because
it does not have end-point singularity, and we find that

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(a)}^{\rm ND}$
$\displaystyle=\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{1}{2N_{c}}\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})$
$\displaystyle\times\frac{(1-\xi)(1-\epsilon)}{(1-\xi_{x})\sqrt{1+r_{\perp}^{2}}}\left[(1-2\xi)(1-\xi)+\frac{k_{\perp}^{2}}{P_{z}^{2}}\right]\
.$ (40)

Taking the TMD limit, and adding the corresponding term from the non-
derivative contribution, we obtain the final result for Fig. 5(a)

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(a)}$
$\displaystyle=\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{1}{2N_{c}}\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})(1-\xi_{x})(1-\epsilon)\
.$ (41)

On the other hand, the diagrams $(b,c)$ of Fig. 5 contribute to the end-point
singularities. The result can be written as

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(b,c)}^{\rm ND}$
$\displaystyle=\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{1}{2N_{c}}\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})$
$\displaystyle\times\frac{2\xi(1-\xi)^{2}}{(1-\xi_{x})^{3}\sqrt{1+r_{\perp}^{2}}}\left[2-\xi+(1-\xi)r_{\perp}^{2}\right]\
.$ (42)

Clearly, the last term in the bracket is power suppressed in the TMD limit.
Furthermore, we can rewrite $[2-\xi]$ as two terms as $1+(1-\xi)$. The first
term will have an end-point singularity, whereas the second term is regular.
It is interesting to find that this regular term cancels the corresponding
term from the derivative contribution derived above. Therefore, there are only
end-point contributions from Fig. 5(b,c),

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(b,c)}$
$\displaystyle=\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{1}{2N_{c}}\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})\frac{2\xi(1-\xi)^{2}}{(1-\xi_{x})^{3}\sqrt{1+r_{\perp}^{2}}}\
.$ (43)

We further notice that $(1-\xi)^{2}$ can be simplified as
$(1-\xi)^{2}=(1-\xi_{x})^{2}(1+\sqrt{1+r_{\perp}^{2}})^{2}/4\approx(1-\xi_{x})^{2}(1+\sqrt{1+r_{\perp}^{2}})/2$
in the TMD limit. With that, we obtain

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(b,c)}$
$\displaystyle=\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{1}{2N_{c}}\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})\frac{2\xi_{x}}{1-\xi_{x}}\frac{1+\sqrt{1+r_{\perp}^{2}}}{2\sqrt{1+r_{\perp}^{2}}}$
$\displaystyle=\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{1}{2N_{c}}\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})$
$\displaystyle\times\left[\frac{2\xi_{x}}{(1-\xi_{x})_{+}}+\delta(1-\xi_{x})\ln\frac{\zeta_{z}}{k_{\perp}^{2}}\right]\
,$ (44)

where $\zeta_{z}=4x^{2}P_{z}^{2}$. The last equation follows from a similar
derivation for the unpolarized TMD quark calculation in Refs. Ji  _et al._
(2019a) in the TMD limit.

Fig. 5(d) is a little more involved, because it has the so-called pinch pole
singularity if we take the limit $L\to\infty$ first,

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(d)}^{L\to\infty}=$
$\displaystyle\frac{M_{p}}{k_{\perp}^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{1}{2N_{c}}\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})\frac{1}{\sqrt{1+r_{\perp}^{2}}}\frac{1}{k_{1z}+i\epsilon}\frac{1}{k_{1z}-i\epsilon}\
,$ (45)

where $k_{1z}=x_{q}(1-\xi_{x})P_{z}$ represents the longitudinal momentum
carried by the radiated gluon crossing the cut line. Because of the pinch-pole
singularity, the above contribution is not well defined around $x=x_{q}$
($\xi_{x}=1$). A finite length of the gauge link will help to regulate the
pinch pole singularity as shown for the unpolarized case. In addition, similar
to the unpolarized case, the contribution is power suppressed when $x\neq
x_{q}$. Therefore, it will only contribute to a Delta function at $\xi_{x}=1$.
Following the same strategy as in Ref. Ji  _et al._ (2019a), we perform the
Fourier transformation with respect to $k_{\perp}$ and carry out the $k_{1z}$
integral to derive the $b_{\perp}$-space expression. Schematically, the quark
Sivers function in $b_{\perp}$-space can be derived as follows,

$\displaystyle\tilde{f}_{1T}^{\perp}(x,b_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(d)}$
$\displaystyle=\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}e^{i\vec{k}_{\perp}\cdot\vec{b}_{\perp}}\left[{\cal
M}_{\rm fig.\ref{softpole}(d)}-{\cal M}_{\rm fig.\ref{softpole}(d)}^{\rm
mirror}\right]\ .$ (46)

Here, the mirror diagram represents the amplitude with gluon attachment to the
right side of the cut-line in Fig. 5(d). We can further apply
$\vec{k}_{\perp}=\vec{k}_{q2\perp}-\vec{k}_{1\perp}^{L}$ for Fig. 5(d) and
$\vec{k}_{\perp}=\vec{k}_{q1\perp}-\vec{k}_{1\perp}^{R}$ for its mirror graph.
Note that $k_{1}^{L}$ and $k_{1}^{R}$ are different because they have
different $k_{q1\perp}$ and $k_{q2\perp}$ dependences. We find that the Sivers
function in $b_{\perp}$-space is proportional to

$\displaystyle\tilde{f}_{1T}^{\perp}(x,b_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(d)}\propto$ $\displaystyle
e^{i\vec{k}_{q2\perp}\cdot\vec{b}_{\perp}}\int\frac{d^{4}k_{1}^{L}}{(2\pi)^{2}}e^{i\vec{k}_{1\perp}^{L}\cdot\vec{b}_{\perp}}\frac{1}{(k_{z}^{L})^{2}}{\cal
R}(k_{1z}^{L})\delta((k_{1}^{L})^{2})$
$\displaystyle-e^{i\vec{k}_{q1\perp}\cdot\vec{b}_{\perp}}\int\frac{d^{4}k_{1}^{R}}{(2\pi)^{2}}e^{i\vec{k}_{1\perp}^{R}\cdot\vec{b}_{\perp}}\frac{1}{(k_{z}^{R})^{2}}{\cal
R}(k_{1z}^{R})\delta((k_{1}^{R})^{2})\ ,$ (47)

where ${\cal R}(k_{z})=\left(1-e^{ik_{z}L}\right)\left(1-e^{-ik_{z}L}\right)$.
Notice that although $k_{1}^{L}$ and $k_{1}^{R}$ are not identical, their
contribution to the above equation is the same. So, we can combine the above
two terms and obtain,

$\displaystyle\tilde{f}_{1T}^{\perp}(x,b_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(d)}\propto\left[e^{i\vec{k}_{q2\perp}\cdot\vec{b}_{\perp}}-e^{i\vec{k}_{q1\perp}\cdot\vec{b}_{\perp}}\right]$
$\displaystyle\times\int\frac{dk_{1z}}{k_{1z}^{2}}\frac{d^{2}k_{1\perp}}{(2\pi)^{2}}e^{i\vec{k}_{1\perp}\cdot\vec{b}_{\perp}}\frac{1}{\sqrt{k_{1z}^{2}+k_{1\perp}^{2}}}{\cal
R}(k_{1z})\ .$ (48)

It is interesting to observe that the first factor is just the leading order
expression and the second factor represents the amplitude without gluon
attachments and is a diagram similar to that for the unpolarized quark
distribution at one-loop order. Applying the result from Ref. Ji  _et al._
(2019a), we have

$\displaystyle\tilde{f}_{1T}^{\perp}(x,b_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{softpole}(d)}$
$\displaystyle=-\frac{ib_{\perp}^{\alpha}}{2}\frac{\alpha_{s}}{2\pi}\frac{1}{2N_{c}}T_{F}(x,x)\int\frac{dk_{z}}{k_{z}^{2}}\frac{d^{2}k_{\perp}}{(2\pi)^{2}}e^{i\vec{k}_{\perp}\cdot\vec{b}_{\perp}}\frac{{\cal
R}(k_{z})}{\sqrt{k_{z}^{2}+k_{\perp}^{2}}}$
$\displaystyle=-\frac{ib_{\perp}^{\alpha}}{2}\frac{\alpha_{s}}{2\pi}\frac{1}{2N_{c}}T_{F}(x,x)2{\cal
K}(\xi_{b})\ ,$ (49)

where $\xi_{b}=L/|{\vec{b}}_{\perp}|$ and the function ${\cal K}$ is defined
as Ji  _et al._ (2019a),

${\cal K}(\xi_{b})=2\xi_{b}\tan^{-1}\xi_{b}-\ln(1+\xi_{b}^{2})\ .$ (50)

At large $\xi_{b}$ the above ${\cal K}(\xi_{b})$ becomes
$\pi\xi_{b}-2\ln\xi_{b}$, while at small $\xi_{b}$ it behaves as
$\xi_{b}^{2}$.

Figure 6: Hard pole contributions at one-loop order for quasi-Sivers function.

Now, let us move to hard-gluon-pole contributions, for which the diagrams are
shown in Fig. 6. First, we notice that the phase contribution comes from the
gauge link propagator,

$\frac{1}{n_{z}\cdot(k_{q1}-k)}=\frac{1}{n_{z}\cdot P}\frac{1}{x_{q1}-x}\ .$
(51)

The pole contribution leads to $\delta(x_{q1}-x)$ which means that
$x_{g}=(1-\xi_{x})x_{q}$. Because the pole is situated at $x_{g}\neq 0$ this
is a hard-gluon pole contribution. Similar as for the standard Sivers
function, these diagrams do not produce derivative terms. Again, we apply the
collinear expansion of the incoming parton transverse momenta and for
convenience we have chosen the physical polarization for the radiated gluon in
these diagrams. The total contribution from Fig. 6 can be separated into two
terms: one contains the pinch-pole singularity and one is free of pinch-pole
singularity.

We can derive the pinch-pole term following the above procedure, giving the
contribution from the soft gluon pole diagrams. Again, we have to use a finite
length to regulate the divergence and the result in $b_{\perp}$-space is the
same as above but with a different color factor,

$\displaystyle\tilde{f}_{1T}^{\perp}(x,b_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{hardpole}}^{\rm pinch~{}p.}$
$\displaystyle=\frac{ib_{\perp}^{\alpha}}{2}\frac{\alpha_{s}}{2\pi}\frac{C_{A}}{2}T_{F}(x,x)\int\frac{dk_{z}}{k_{z}^{2}}\frac{d^{2}k_{\perp}}{(2\pi)^{2}}e^{i\vec{k}_{\perp}\cdot\vec{b}_{\perp}}\frac{{\cal
R}(k_{z})}{\sqrt{k_{z}^{2}+k_{\perp}^{2}}}$
$\displaystyle=\frac{ib_{\perp}^{\alpha}}{2}\frac{\alpha_{s}}{2\pi}\frac{C_{A}}{2}T_{F}(x,x)2{\cal
K}(\xi_{b})\ .$ (52)

After subtracting the pinch-pole contribution, we derive the rest of the hard
gluon pole contribution from Fig. 6,

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{hardpole}}^{\rm no-pinch~{}p.}$
$\displaystyle=-\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{C_{A}}{2}\int\frac{dx_{q}}{x_{q}}T_{F}(x,x_{q})\frac{(1-\xi)^{2}(1+\xi_{x})}{(1-\xi_{x})^{2}\sqrt{1+r_{\perp}^{2}}}\
,$ (53)

where we have neglected power corrections in the TMD limit, and we have
applied the symmetry property of the Qiu-Sterman matrix element
$T_{F}(x,x_{q})=T_{F}(x_{q},x)$ to simplify the final result. In the TMD
limit, the expression above can be further simplified to

$\displaystyle\tilde{f}_{1T}^{\perp}(x,k_{\perp},\mu,\zeta_{z})|_{\rm
fig.\ref{hardpole}}^{\rm no-pinch~{}p.}$
$\displaystyle=-\frac{M_{p}}{(k_{\perp}^{2})^{2}}\frac{\alpha_{s}}{2\pi^{2}}\frac{C_{A}}{2}\int\frac{dx_{q}}{x_{q}}T_{F}(x,x_{q})$
$\displaystyle\times\left[\frac{1+\xi_{x}}{(1-\xi_{x})_{+}}+\delta(1-\xi_{x})\ln\frac{\zeta_{z}}{k_{\perp}^{2}}\right]\
.$ (54)

Similar to the case for the standard quark Sivers function calculated in Ref.
Ji  _et al._ (2006b). We observe the following: There are cancellations
between the hard gluon pole contributions and the soft gluon pole
contributions. In particular, the hard-gluon-pole is proportional to the color
factor $C_{A}/2$ while the soft-gluon-pole to $1/2N_{c}$. Their cancellation
leads to the final result proportional to $C_{F}$ for the end-point
contribution and the pinch-pole contributions. This is consistent with the
soft gluon radiation contribution and the soft factor subtraction.

Since the soft factor and the subtraction is spin-independent, their
contributions will be same as those calculated in Ref. Ji  _et al._ (2019a).
Combining all these terms, we obtain the final result for the quark Sivers
function at one-loop order in LaMET,

$\displaystyle\tilde{f}_{1T}^{\perp\alpha(1)}(x,{\vec{b}}_{\perp},\mu,\zeta_{z})$
$\displaystyle=\frac{ib_{\perp}^{\alpha}}{2}\frac{\alpha_{s}}{2\pi}\left\\{\left(-\frac{1}{\epsilon}-L_{b}\right){\cal
P}_{qg/qg}^{T}\otimes T_{F}(x,x)\right.$
$\displaystyle\left.+\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q})\left[-\frac{1}{2N_{c}}(1-\xi_{x})+\delta(1-\xi_{x})C_{F}\tilde{s}^{(1)}\right]\right\\}\
,$ (55)

where ${\cal P}_{qg/qg}^{T}\otimes T_{F}(x,x)$ has been defined in Eq. (17)
and $\tilde{s}^{(1)}$ is the same as for the unpolarized case of Eq. (27).
Similar to the unpolarized case, after renormalization, we can write the quasi
Sivers function in terms of the collinear twist-three Qiu-Sterman matrix
element,

$\displaystyle\tilde{f}_{1T}^{\perp\alpha}(x,{\vec{b}}_{\perp},\mu_{b},\zeta_{z})$
$\displaystyle=\frac{ib_{\perp}^{\alpha}}{2}\int\frac{dx_{q}}{x_{q}}T_{F}(x_{q},x_{q},\mu_{b})\left\\{\delta(1-\xi_{x})\right.$
$\displaystyle\left.+\frac{\alpha_{s}}{2\pi}\left[-\frac{1}{2N_{c}}(1-\xi_{x})+\delta(1-\xi_{x})C_{F}\tilde{s}^{(1)}\right]\right\\}\
,$ (56)

at one-loop order.

A couple of comments are in order before we close this section. First, the
above result is obtained in the scheme of Ref. Ji  _et al._ (2019b) in which
$\sqrt{Z_{E}}$ was adopted to subtract out the pinch-pole singularity. The
same strategy has already been adopted in Ref. Ji  _et al._ (2019a) in which
it is regarded as a soft factor subtraction.

Second, for the Sivers contribution, we focused our calculations for the soft-
gluon and hard-gluon pole contributions from the Qiu-Sterman matrix element
$T_{F}(x_{1},x_{2})$. Other contributions from the soft-fermion pole and those
from $\tilde{G}_{F}(x_{1},x_{2})$ can be included as well. They contribute to
both the evolution kernel ${\cal P}_{qg/qg}^{T}$ and the finite term of Eq.
(56), see, for example, the recent study for the light-cone Sivers function in
Ref. Scimemi  _et al._ (2019).

##  IV Single Transverse-Spin Asymmetry

In this section, we discuss applications of the results obtained in previous
sections. In particular, we consider the single spin asymmetry at large and
small $b_{\perp}$. We also comment on previous lattice calculations for
moments of the relevant TMDPDFs.

###  IV.1 Single Spin Asymmetry

One of the most important physical application of spin-dependent TMDPDFs are
single-spin asymmetries defined by the ratio of physical cross sections. For
example, in the Drell-Yan lepton pair production, we define,

$\displaystyle
A_{DY}=\frac{\frac{d^{4}\sigma_{+S_{\perp}}}{d^{2}Q_{\perp}dx_{A}dx_{B}}-\frac{d^{4}\sigma_{-S_{\perp}}}{d^{2}Q_{\perp}dx_{A}dx_{B}}}{\frac{d^{4}\sigma_{+S_{\perp}}}{d^{2}Q_{\perp}dx_{A}dx_{B}}+\frac{d^{4}\sigma_{-S_{\perp}}}{d^{2}Q_{\perp}dx_{A}dx_{B}}}\
,$ (57)

where $\frac{d^{4}\sigma_{\pm S_{\perp}}}{d^{2}Q_{\perp}dx_{A}dx_{B}}$ are the
differential cross sections with the transverse spin for the polarized target
being $\pm S_{\perp}$, $x_{A,B}$ denote the momentum fractions of the incoming
hadrons carried by the quark and antiquark, $Q^{2}$ and $Q_{\perp}$ are the
invariant mass and transverse momentum for the lepton pair. The factorization
formula Collins (2013) for the Drell-Yan or SIDIS process in terms of the
physical TMDPDFs reads

$\displaystyle\frac{d^{4}\sigma}{dQ_{\perp}^{2}dx_{A}dx_{B}}=\hat{\sigma}(\frac{Q^{2}}{\mu^{2}})\times$
$\displaystyle\int
d^{2}b_{\perp}e^{i\vec{b}_{\perp}\cdot\vec{Q}_{\perp}}q(x_{A},b_{\perp},S_{\perp},\mu,\zeta_{A})q(x_{B},b_{\perp},\mu,\zeta_{B})$
(58)

where $x_{A}$ is the momentum fraction for the quark parton coming out of the
polarized target and $x_{B}$ is for the unpolarized one.
$\hat{\sigma}(\frac{Q^{2}}{\mu^{2}})$ is the hard-cross section with
$Q^{2}=2x_{A}x_{B}P^{+}P^{-}$ where $P^{\pm}$ are the largest light-front
components of the hadron momenta. Given the factorization formula and
utilizing a decomposition for the TMDPDF similar to Eq. (II), one found that
the single-spin asymmetry for the Drell-Yan process can be written in terms of
an unpolarized TMDPDF and Sivers function:

$\displaystyle A_{DY}=$ $\displaystyle\frac{\int
d^{2}b_{\perp}e^{i\vec{Q}_{\perp}\cdot\vec{b}_{\perp}}\epsilon^{\beta\alpha}S_{\perp}^{\beta}f_{1T}^{\perp\alpha}(x_{A},b_{\perp},\mu,\zeta_{A})q(x_{B},b_{\perp},\mu,\zeta_{B})}{\int
d^{2}b_{\perp}e^{i\vec{Q}_{\perp}\cdot\vec{b}_{\perp}}q(x_{A},b_{\perp},\mu,\zeta_{A})q(x_{B},b_{\perp},\mu,\zeta_{B})}$
(59)

where the hard cross-section induced by unpolarized quark partons cancels
between numerator and denominator. By using the matching relation Eq. (II.2)
and choosing $\zeta_{zA}=\zeta_{A}=2x_{A}^{2}(P^{+})^{2}$,
$\zeta_{zB}=\zeta_{B}=4x_{B}^{2}(P^{-})^{2}$, one has

$\displaystyle A_{DY}=\frac{D}{S}\ ,$ (60) $\displaystyle D=\int
d^{2}b_{\perp}e^{i\vec{Q}_{\perp}\cdot\vec{b}_{\perp}}\epsilon^{\beta\alpha}S_{\perp}^{\beta}\tilde{f}_{1T}^{\perp\alpha}(x_{A},b_{\perp},\mu,\zeta_{zA})$
$\displaystyle\times\tilde{q}(x_{B},b_{\perp},\mu,\zeta_{zB})S_{r}(b_{\perp},\mu)\
,$ (61) $\displaystyle S=\int
d^{2}b_{\perp}e^{i\vec{Q}_{\perp}\cdot\vec{b}_{\perp}}\tilde{q}(x_{A},b_{\perp},\mu,\zeta_{zA})\tilde{q}(x_{B},b_{\perp},\mu,\zeta_{zB})$
$\displaystyle\times S_{r}(b_{\perp},\mu)\ .$ (62)

Notice that the matching kernel all cancel, but the reduced soft function
$S_{r}$ does not cancel between $D$ and $S$. Since $\tilde{q}$,
$\tilde{f}_{1T}^{\perp\alpha}$ and $S_{r}$ can all be extracted from lattice
calculations, Eq. (60) allows to predict the physical observable single-spin
asymmetry from lattice data.

Because $A_{DY}$ is complicated due to the fact that the soft contribution
fails to cancel, it is attractive to mimic $A_{DY}$ by using the following
simplified version for the single-spin asymmetry ratio in Fourier transform
$b_{\perp}$-space:

$\displaystyle{\cal R}_{S_{\perp}}(x,b_{\perp})$
$\displaystyle=\frac{q(x,b_{\perp},S_{\perp},\mu,\zeta)-q(x,b_{\perp},-S_{\perp},\mu,\zeta)}{q(x,b_{\perp},S_{\perp},\mu,\zeta)+q(x,b_{\perp},-S_{\perp},\mu,\zeta)}$
$\displaystyle=\epsilon^{\beta\alpha}S_{\perp}^{\beta}\frac{f_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta)}{q(x,b_{\perp},\mu,\zeta)}\
.$ (63)

Instead of transforming to momentum space, in ${\cal R}_{S_{\perp}}$ we
directly compare the asymmetry point by point in $b_{\perp}$ space. We
emphasize that this asymmetry is not a physical observable, but a ratio
between the quark-Sivers function and the unpolarized quark distribution. An
important feature is that the $\mu$, $\zeta$ dependencies cancel between
numerator and denominator, thus the ${\cal R}_{S_{\perp}}$ is independent of
the renormalization scale $\mu$ and the rapidity scale $\zeta$. This
contribution is proportional to $\vec{S}_{\perp}\times\vec{b}_{\perp}$ and the
coefficient defines the size of the single spin asymmetry in the quark
distribution. The individual TMDPDFs will depend on the rapidity
renormalization scheme. However, the ratio between the quark Sivers function
and the unpolarized quark distribution does not depend on the scheme. In
particular, the scheme dependent soft functions cancel in the ratio of Eq.
(IV.1).

With the relation between the quasi-TMDPDF and physical TMDPDF in Eq. (II.2),
we will be able to study the single spin asymmetry in the quark distribution.
Using the fact that the matching kernel $H$ is independent of the spin-
structure and is the same for both the Sivers function and the un-polarized
quark TMDPDF, we found that by taking the ratio, the soft factor and matching
kernel dependencies cancel:

$\displaystyle\widetilde{{\cal R}}_{S_{\perp}}$
$\displaystyle=\epsilon^{\beta\alpha}S_{\perp}^{\beta}\frac{\tilde{f}_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta_{z})}{\tilde{q}(x,b_{\perp},\mu,\zeta_{z})}$
$\displaystyle=\epsilon^{\beta\alpha}S_{\perp}^{\beta}\frac{f_{1T}^{\perp\alpha}(x,b_{\perp},\mu,\zeta)}{q(x,b_{\perp},\mu,\zeta)}\equiv{\cal
R}_{S_{\perp}}(x,b_{\perp})\ .$ (64)

Therefore, the single-spin asymmetry ratio extracted from the quasi-Sivers
function and the Sivers function are the same. This is one of the major
results of this paper.

At small $b_{\perp}$, the single spin asymmetry ratio in the quark
distribution at one-loop order can be extracted from the perturbative results
of the Sivers function provided in the previous sections

$\displaystyle{\cal
R}_{S_{\perp}}(x,b_{\perp})|_{b_{\perp}\ll\frac{1}{\Lambda_{QCD}}}=\frac{i|b_{\perp}|\sin(\phi_{b})}{2}$
$\displaystyle\times\frac{T_{F}(x,x,\mu_{b})-\frac{\alpha_{s}}{4\pi
N_{c}}(1-x/x_{q})\otimes
T_{F}(x_{q},x_{q},\mu_{b})}{f_{q}(x,\mu_{b})+\frac{\alpha_{s}C_{F}}{2\pi}(1-x/x_{q})\otimes
f_{q}(x_{q},\mu_{b})}\ ,$ (65)

where $\phi_{b}$ is the azimuthal angle between $\vec{b}_{\perp}$ and
$\vec{S}_{\perp}$ and we have performed the renormalization of the quark
distribution at $\mu=\mu_{b}$. As one can see, the soft contribution in
$s^{(1)}$ cancels. The explicit derivations in the previous sections have
confirmed that the single spin asymmetry extracted from quasi-Sivers function
are the same for small $b_{\perp}$.

For the quark Sivers asymmetry, the $\alpha_{s}$ correction is very small. For
example, numerically, the $\alpha_{s}$ corrections for the numerator and
denominator are less than $1\%$ in most of kinematics of the valence quark
distributions at $x\sim 0.2$. Therefore, we can safely neglect these
corrections and interpret the asymmetry as the ratio between the Qiu-Sterman
matrix element and the unpolarized quark distribution at the scale $\mu_{b}$.

###  IV.2 Asymmetry at Large $b_{\perp}$

On the other hand, the spin asymmetry ratio at large-$b_{\perp}$ is determined
by non-perturbative TMDPDFs, for which lattice calculations in terms of the
formalism presented in Sec. II will be very important.

In previous phenomenology studies, a Gaussian distribution in the transverse
momentum space has been assumed for both the unpolarized and Sivers quark
distributions, e.g.,

$\displaystyle q(x,k_{\perp},\mu=\zeta=\mu_{0})$ $\displaystyle\propto$
$\displaystyle e^{-\frac{k_{\perp}^{2}}{Q_{0}^{2}}}\ ,$ (66) $\displaystyle
f_{1T}^{\perp}(x,k_{\perp},\mu=\zeta=\mu_{0})$ $\displaystyle\propto$
$\displaystyle e^{-\frac{k_{\perp}^{2}}{Q_{s}^{2}}}\ ,$ (67)

where $Q_{0}$ and $Q_{s}$ are parameters for the Gaussian distributions.
Because the asymmetry ratio has to decrease at large transverse momentum, the
Gaussian width for the quark Sivers function is smaller than that for the
unpolarized quark distribution, i.e., $Q_{s}<Q_{0}$. If we translate this into
$b_{\perp}$-space, it will generate a significantly increasing function for
${\cal R}_{S_{\perp}}(x,b_{\perp})$ at large $b_{\perp}$,

${\cal R}_{S_{\perp}}(b_{\perp})|_{\rm model}\propto
b_{\perp}e^{\frac{(Q_{0}^{2}-Q_{s}^{2})b_{\perp}^{2}}{4}}\ .$ (68)

For example, the parameterization in Ref. Sun and Yuan (2013) predicts a
factor of $25$ increase from $b_{\perp}=0.2~{}\rm fm$ to $b_{\perp}=1~{}\rm
fm$. Similar predictions exist for other parameterizations, see, some recent
global analyses Cammarota  _et al._ (2020); Bacchetta  _et al._ (2020). It
will be crucially important to check this in lattice simulations.

###  IV.3 Relation to Previous Lattice Simulations

In Refs. Musch  _et al._ (2011, 2012); Yoon  _et al._ (2017), lattice
computations for certain matrix elements in a hadron state have been carried
out. These matrix elements are defined through TMDPDF-like bi-local operators,
which are separated by transverse distance $b_{\perp}$ perpendicular to the
hadron’s momentum direction. It is easy to see that the matrix elements
calculated there do not correspond exactly to the moments of the quasi-TMDPDF
distribution. Therefore, it is hard to interpret them although interesting
results were obtained.

Our results, obtained by using LaMET can help to improve these earlier
results. For example, we can add the explicite $x$ dependence to the matrix
elements calculated in Refs. Musch  _et al._ (2011, 2012); Yoon  _et al._
(2017). This can help to resolve the difference between, e.g.,

$\displaystyle\frac{\int
dxx^{n}\tilde{f}^{\perp\alpha}_{1T}(x,b_{\perp},\mu,\zeta_{z}=4x^{2}(P^{z})^{2})}{\int
dxx^{n}\tilde{q}(x,b_{\perp},\mu,\zeta_{z}=4x^{2}(P^{z})^{2})}$
$\displaystyle\neq$ $\displaystyle\frac{\int
dxx^{n}f^{\perp\alpha}_{1T}(x,b_{\perp},\mu,\zeta)}{\int
dxx^{n}q(x,b_{\perp},\mu,\zeta)}\ .$ (69)

The above point has also been observed in Ref. Ebert  _et al._ (2020b). It
will be useful to have lattice simulations in the LaMET framework to constrain
the quark Sivers functions and compare to phenomenological studies Sun and
Yuan (2013); Cammarota  _et al._ (2020); Bacchetta  _et al._ (2020).

##  V Conclusion

In summary, we have investigated the quark Sivers function in LaMET. A number
of important features have been found for these distribution functions. In our
derivation, we adopted the definition of quasi-TMDPDF in Refs. Ji  _et al._
(2019a, b). We have shown that the quasi-Sivers function can be matched to the
physical Sivers function using Eq. (II.2). The matching kernel and the reduced
soft function are the same as that of the unpolarized case. As a result, in
the single spin asymmetry, the soft function and hard kernels cancel between
the quark Sivers function and the unpolarized quark distribution and we can
extract the physical single spin asymmetry knowing the lattice calculable
quasi-TMDPDFs.

As a byproduct, Eq. (65) provides another useful method to compute twist-three
quark-gluon-quark correlation function, in particular, for those directly
connected to the leading order TMDPDFs. We notice a recent study of twist-
three parton distribution $g_{T}(x)$ in LaMET framework Bhattacharya  _et al._
(2020). These studies demonstrate the powerful reach of the LaMET formalism
and we hope that more studies of this type will become available in the
future.

The methods discussed in this paper can be extended in various directions. An
immediate extension is the analysis of all other $k_{\perp}$-odd quark
TMDPDFs. The large transverse momentum dependence for these distributions has
been derived in Ref. Zhou  _et al._ (2010). In order to study them in LaMET,
we need to translate these results into the LaMET formalism following the
procedure of the current paper for the quark Sivers function. We plan to study
this in a future publication. Together with a recent paper on $k_{\perp}$-even
spin dependent quark TMDPDFs Ebert  _et al._ (2020b), this will complete all
leading quark TMDPDFs in LaMET.

In addition, the method developed in this paper shpold be applied to other
related parton distribution functions, especially for those relevant for the
quantum phase space Wigner distributions. These distribution functions contain
in principle the complete information needed for nucleon tomography and allow
to unveil the origin of the parton orbital angular momentum in nucleons. We
expect more developments on this subject soon.

We thank Jianhui Zhang and Yong Zhao for discussion and suggestions. This
material is based upon work partially supported by the U.S. Department of
Energy, Office of Science, Office of Nuclear Physics, under contract number
DE-AC02-05CH11231 and DE-SC0020682, and Center for Nuclear Femtography,
Southeastern Universities Research Associations in Washington DC, and within
the framework of the TMD Topical Collaboration. AS acknowledges support by
SFB/TRR-55.

## References

  * Accardi  _et al._ (2016) A. Accardi _et al._ , Eur. Phys. J. A 52, 268 (2016), arXiv:1212.1701 [nucl-ex] .
  * Boer  _et al._ (2011) D. Boer _et al._ , (2011), arXiv:1108.1713 [nucl-th] .
  * Collins (2013) J. Collins,  _Foundations of perturbative QCD_ , Vol. 32 (Cambridge University Press, 2013).
  * Collins and Soper (1981) J. C. Collins and D. E. Soper, Nucl. Phys. B 193, 381 (1981), [Erratum: Nucl.Phys.B 213, 545 (1983)].
  * Ji  _et al._ (2005) X.-d. Ji, J.-p. Ma, and F. Yuan, Phys. Rev. D 71, 034005 (2005), arXiv:hep-ph/0404183 .
  * Bacchetta  _et al._ (2008) A. Bacchetta, D. Boer, M. Diehl, and P. J. Mulders, JHEP 08, 023 (2008), arXiv:0803.0227 [hep-ph] .
  * Cammarota  _et al._ (2020) J. Cammarota, L. Gamberg, Z.-B. Kang, J. A. Miller, D. Pitonyak, A. Prokudin, T. C. Rogers, and N. Sato (Jefferson Lab Angular Momentum), Phys. Rev. D 102, 054002 (2020), arXiv:2002.08384 [hep-ph] .
  * Bacchetta  _et al._ (2020) A. Bacchetta, F. Delcarro, C. Pisano, and M. Radici, (2020), arXiv:2004.14278 [hep-ph] .
  * Musch  _et al._ (2011) B. U. Musch, P. Hagler, J. W. Negele, and A. Schafer, Phys. Rev. D 83, 094507 (2011), arXiv:1011.1213 [hep-lat] .
  * Musch  _et al._ (2012) B. Musch, P. Hagler, M. Engelhardt, J. Negele, and A. Schafer, Phys. Rev. D 85, 094510 (2012), arXiv:1111.4249 [hep-lat] .
  * Yoon  _et al._ (2017) B. Yoon, M. Engelhardt, R. Gupta, T. Bhattacharya, J. R. Green, B. U. Musch, J. W. Negele, A. V. Pochinsky, A. Schäfer, and S. N. Syritsyn, Phys. Rev. D 96, 094508 (2017), arXiv:1706.03406 [hep-lat] .
  * Ji (2013) X. Ji, Phys. Rev. Lett. 110, 262002 (2013), arXiv:1305.1539 [hep-ph] .
  * Ji (2014) X. Ji, Sci. China Phys. Mech. Astron. 57, 1407 (2014), arXiv:1404.6680 [hep-ph] .
  * Cichy and Constantinou (2019) K. Cichy and M. Constantinou, Adv. High Energy Phys. 2019, 3036904 (2019), arXiv:1811.07248 [hep-lat] .
  * Ji  _et al._ (2020a) X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang, and Y. Zhao, (2020a), arXiv:2004.03543 [hep-ph] .
  * Ji  _et al._ (2015) X. Ji, P. Sun, X. Xiong, and F. Yuan, Phys. Rev. D 91, 074009 (2015), arXiv:1405.7640 [hep-ph] .
  * Ji  _et al._ (2019a) X. Ji, L.-C. Jin, F. Yuan, J.-H. Zhang, and Y. Zhao, Phys. Rev. D 99, 114006 (2019a), arXiv:1801.05930 [hep-ph] .
  * Ebert  _et al._ (2019a) M. A. Ebert, I. W. Stewart, and Y. Zhao, Phys. Rev. D 99, 034505 (2019a), arXiv:1811.00026 [hep-ph] .
  * Ebert  _et al._ (2019b) M. A. Ebert, I. W. Stewart, and Y. Zhao, JHEP 09, 037 (2019b), arXiv:1901.03685 [hep-ph] .
  * Ebert  _et al._ (2020a) M. A. Ebert, I. W. Stewart, and Y. Zhao, JHEP 03, 099 (2020a), arXiv:1910.08569 [hep-ph] .
  * Shanahan  _et al._ (2020a) P. Shanahan, M. L. Wagman, and Y. Zhao, Phys. Rev. D 101, 074505 (2020a), arXiv:1911.00800 [hep-lat] .
  * Shanahan  _et al._ (2020b) P. Shanahan, M. Wagman, and Y. Zhao, Phys. Rev. D 102, 014511 (2020b), arXiv:2003.06063 [hep-lat] .
  * Ebert  _et al._ (2020b) M. A. Ebert, S. T. Schindler, I. W. Stewart, and Y. Zhao, JHEP 09, 099 (2020b), arXiv:2004.14831 [hep-ph] .
  * Ji  _et al._ (2020b) X. Ji, Y. Liu, and Y.-S. Liu, Nucl. Phys. B 955, 115054 (2020b), arXiv:1910.11415 [hep-ph] .
  * Ji  _et al._ (2019b) X. Ji, Y. Liu, and Y.-S. Liu, (2019b), arXiv:1911.03840 [hep-ph] .
  * Vladimirov and Schäfer (2020) A. A. Vladimirov and A. Schäfer, Phys. Rev. D 101, 074517 (2020), arXiv:2002.07527 [hep-ph] .
  * Zhang  _et al._ (2020) Q.-A. Zhang _et al._ (Lattice Parton), Phys. Rev. Lett. 125, 192001 (2020), arXiv:2005.14572 [hep-lat] .
  * Sivers (1990) D. W. Sivers, Phys. Rev. D 41, 83 (1990).
  * Brodsky  _et al._ (2002) S. J. Brodsky, D. S. Hwang, and I. Schmidt, Phys. Lett. B 530, 99 (2002), arXiv:hep-ph/0201296 .
  * Collins (2002) J. C. Collins, Phys. Lett. B 536, 43 (2002), arXiv:hep-ph/0204004 .
  * Ji and Yuan (2002) X.-d. Ji and F. Yuan, Phys. Lett. B 543, 66 (2002), arXiv:hep-ph/0206057 .
  * Belitsky  _et al._ (2003) A. V. Belitsky, X. Ji, and F. Yuan, Nucl. Phys. B 656, 165 (2003), arXiv:hep-ph/0208038 .
  * Efremov and Teryaev (1982) A. Efremov and O. Teryaev, Sov. J. Nucl. Phys. 36, 140 (1982).
  * Efremov and Teryaev (1985) A. Efremov and O. Teryaev, Phys. Lett. B 150, 383 (1985).
  * Qiu and Sterman (1991) J.-w. Qiu and G. F. Sterman, Phys. Rev. Lett. 67, 2264 (1991).
  * Qiu and Sterman (1992) J.-w. Qiu and G. F. Sterman, Nucl. Phys. B 378, 52 (1992).
  * Qiu and Sterman (1999) J.-w. Qiu and G. F. Sterman, Phys. Rev. D 59, 014004 (1999), arXiv:hep-ph/9806356 .
  * Ji  _et al._ (2006a) X. Ji, J.-W. Qiu, W. Vogelsang, and F. Yuan, Phys. Rev. Lett. 97, 082002 (2006a), arXiv:hep-ph/0602239 .
  * Ji  _et al._ (2006b) X. Ji, J.-w. Qiu, W. Vogelsang, and F. Yuan, Phys. Rev. D 73, 094017 (2006b), arXiv:hep-ph/0604023 .
  * Ji  _et al._ (2006c) X. Ji, J.-W. Qiu, W. Vogelsang, and F. Yuan, Phys. Lett. B 638, 178 (2006c), arXiv:hep-ph/0604128 .
  * Koike  _et al._ (2008) Y. Koike, W. Vogelsang, and F. Yuan, Phys. Lett. B 659, 878 (2008), arXiv:0711.0636 [hep-ph] .
  * Kang  _et al._ (2011) Z.-B. Kang, B.-W. Xiao, and F. Yuan, Phys. Rev. Lett. 107, 152002 (2011), arXiv:1106.0266 [hep-ph] .
  * Sun and Yuan (2013) P. Sun and F. Yuan, Phys. Rev. D 88, 114012 (2013), arXiv:1308.5003 [hep-ph] .
  * Scimemi  _et al._ (2019) I. Scimemi, A. Tarasov, and A. Vladimirov, JHEP 05, 125 (2019), arXiv:1901.04519 [hep-ph] .
  * Falk  _et al._ (1990) A. F. Falk, H. Georgi, B. Grinstein, and M. B. Wise, Nucl. Phys. B 343, 1 (1990).
  * Ji and Musolf (1991) X.-D. Ji and M. Musolf, Phys. Lett. B 257, 409 (1991).
  * Catani  _et al._ (2001) S. Catani, D. de Florian, and M. Grazzini, Nucl. Phys. B 596, 299 (2001), arXiv:hep-ph/0008184 .
  * Catani  _et al._ (2014) S. Catani, L. Cieri, D. de Florian, G. Ferrera, and M. Grazzini, Nucl. Phys. B 881, 414 (2014), arXiv:1311.1654 [hep-ph] .
  * Prokudin  _et al._ (2015) A. Prokudin, P. Sun, and F. Yuan, Phys. Lett. B 750, 533 (2015), arXiv:1505.05588 [hep-ph] .
  * Braun  _et al._ (2009) V. Braun, A. Manashov, and B. Pirnay, Phys. Rev. D 80, 114002 (2009), [Erratum: Phys.Rev.D 86, 119902 (2012)], arXiv:0909.3410 [hep-ph] .
  * Schafer and Zhou (2012) A. Schafer and J. Zhou, Phys. Rev. D 85, 117501 (2012), arXiv:1203.5293 [hep-ph] .
  * Kang and Qiu (2012) Z.-B. Kang and J.-W. Qiu, Phys. Lett. B 713, 273 (2012), arXiv:1205.1019 [hep-ph] .
  * Echevarría  _et al._ (2013) M. G. Echevarría, A. Idilbi, and I. Scimemi, Phys. Lett. B 726, 795 (2013), arXiv:1211.1947 [hep-ph] .
  * Collins and Rogers (2013) J. C. Collins and T. C. Rogers, Phys. Rev. D 87, 034018 (2013), arXiv:1210.2100 [hep-ph] .
  * Boer  _et al._ (2003) D. Boer, P. Mulders, and F. Pijlman, Nucl. Phys. B 667, 201 (2003), arXiv:hep-ph/0303034 .
  * Kouvaris  _et al._ (2006) C. Kouvaris, J.-W. Qiu, W. Vogelsang, and F. Yuan, Phys. Rev. D 74, 114013 (2006), arXiv:hep-ph/0609238 .
  * Eguchi  _et al._ (2006) H. Eguchi, Y. Koike, and K. Tanaka, Nucl. Phys. B 752, 1 (2006), arXiv:hep-ph/0604003 .
  * Eguchi  _et al._ (2007) H. Eguchi, Y. Koike, and K. Tanaka, Nucl. Phys. B 763, 198 (2007), arXiv:hep-ph/0610314 .
  * Koike and Tanaka (2007a) Y. Koike and K. Tanaka, Phys. Lett. B 646, 232 (2007a), [Erratum: Phys.Lett.B 668, 458–459 (2008)], arXiv:hep-ph/0612117 .
  * Koike and Tanaka (2007b) Y. Koike and K. Tanaka, Phys. Rev. D 76, 011502 (2007b), arXiv:hep-ph/0703169 .
  * Kang and Qiu (2009) Z.-B. Kang and J.-W. Qiu, Phys. Rev. D 79, 016003 (2009), arXiv:0811.3101 [hep-ph] .
  * Vogelsang and Yuan (2009) W. Vogelsang and F. Yuan, Phys. Rev. D 79, 094010 (2009), arXiv:0904.0410 [hep-ph] .
  * Zhou  _et al._ (2009) J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D 79, 114022 (2009), arXiv:0812.4484 [hep-ph] .
  * Bhattacharya  _et al._ (2020) S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, A. Scapellato, and F. Steffens, Phys. Rev. D 102, 034005 (2020), arXiv:2005.10939 [hep-ph] .
  * Zhou  _et al._ (2010) J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D 81, 054008 (2010), arXiv:0909.2238 [hep-ph] .