aainstitutetext: Center for Theoretical Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USAbbinstitutetext: Physics Department,
Brookhaven National Laboratory, Bldg. 510A, Upton, NY 11973, USA

# One-loop Matching for Spin-Dependent Quasi-TMDs

Markus A. Ebert a Stella T. Schindler a Iain W. Stewart a,b and Yong Zhao
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Transverse momentum dependent parton distribution functions (TMDPDFs) provide
a unique probe of the three-dimensional spin structure of hadrons. We
construct spin-dependent quasi-TMDPDFs that are amenable to lattice QCD
calculations and that can be used to determine spin-dependent TMDPDFs. We
calculate the short-distance coefficients connecting spin-dependent TMDPDFs
and quasi-TMDPDFs at one-loop order. We find that the helicity and
transversity distributions have the same coefficient as the unpolarized
TMDPDF. We also argue that the same is true for pretzelosity and that this
spin universality of the matching will hold to all orders in $\alpha_{s}$.
Thus, it is possible to calculate ratios of these distributions as a function
of longitudinal momentum and transverse position utilizing simpler Wilson line
paths than have previously been considered.

††preprint:  MIT–CTP 5200

##  1 Introduction

Understanding the internal structure of hadrons has been a decades-long quest
in nuclear and particle physics. Tremendous progress has been made towards the
measurement of momentum distributions of quarks and gluons, or parton
distribution functions (PDFs), in the longitudinal direction. Multiple
experiments that have been carried out recently or that are coming online soon
promise to open up a new window into the full three-dimensional internal
dynamics of nucleons Gautheron:2010wva ; Dudek:2012vr ; Aschenauer:2015eha ;
Accardi:2012qut . One key target of these experiments are transverse momentum
dependent PDFs (TMDPDFs), which provide information about partons in
transverse-momentum space and make accessible new correlations between the
partonic and hadronic spin. Although TMDPDFs are harder to measure
experimentally than their longitudinal counterparts, a significant effort has
built up in recent years to fit these quantities with global data from Drell-
Yan and semi-inclusive deep inelastic scattering processes Bacchetta:2017gcc ;
Scimemi:2017etj ; Bertone:2019nxa ; Scimemi:2019cmh ; Bacchetta:2019sam ;
1793441 . It is crucial to develop a deeper theoretical knowledge of TMDPDFs
to complement ongoing experimental and global fit analyses and to enhance our
understanding of these fundamental hadron properties.

A challenging regime for theoretical calculation of TMDPDFs is at small
transverse momentum $q_{T}\sim\Lambda_{\rm QCD}$, where the TMDPDF is
intrinsically nonperturbative and can only be determined from first-principles
calculations. Lattice gauge theory is the only known systematic approach to
calculate nonperturbative QCD matrix elements, motivating the formulation of
TMDPDFs in a manner that is tractable for a Euclidean lattice. The time-
dependence of the nonlocal Wilson-line operators that define TMDPDFs makes
this task tricky. The large-momentum effective theory (LaMET) has been
proposed as a method to circumvent this problem by calculating general
partonic structures from boosted hadron matrix elements in lattice QCD
Ji:2013dva ; Ji:2014gla ; Ji:2020ect .

This idea motivated investigations into the application of LaMET for TMDPDFs
Ji:2014hxa ; Ji:2018hvs ; Ebert:2018gzl ; Ebert:2019okf ; Ebert:2019tvc ;
Ji:2019sxk ; Ji:2019ewn ; Vladimirov:2020ofp ; Ji:2020ect . Within the
formulation of LaMET, one constructs so-called quasi-TMDPDFs that are
computable on the lattice and that are related to the desired TMDPDFs through
a factorization formula. Currently this factorization formula requires the
lattice calculation of an additional nonperturbative factor $g_{S}$
Ebert:2019okf  or “reduced soft function” Ji:2019sxk , both of which pose
significant challenges and are still not available in the literature. This
complexity can be avoided by forming ratios of quasi-TMDPDFs in different
hadron states or for different spin-dependent structures Ebert:2019okf , an
idea that has already been utilized for a number of years to access ratios of
the $x$-moments of TMDPDFs from similar matrix elements Musch:2010ka ;
Musch:2011er ; Engelhardt:2015xja ; Yoon:2016dyh ; Yoon:2017qzo . In
particular, a method was recently constructed to determine the nonperturbative
Collins-Soper kernel that governs the energy evolution of TMDPDFs from the
ratio of quasi-TMDPDFs at two different hadron momenta Ebert:2018gzl ;
Ebert:2019tvc . A full practical implementation of this proposal has been
realized in exploratory lattice calculations Shanahan:2019zcq ;
Shanahan:2020zxr . A key ingredient in the factorization formula that relates
quasi-TMDPDFs and TMDPDFs is the perturbative short-distance matching kernel,
which Refs. Ji:2018hvs ; Ebert:2019tvc  calculate to one-loop order for the
unpolarized non-singlet case.

This paper generalizes the quasi-TMDPDF analysis to the full set of leading-
power spin structures and determines the corresponding matching kernels at
one-loop order. Our results enable TMDPDF lattice calculations for the ratios
of spin-dependent and unpolarized TMDPDFs. Importantly, we find that the
matching kernel for TMDPDFs at leading power is the same for all spin
structures up to ${\cal O}(\alpha_{s})$, and we provide additional arguments
that this will remain true to all orders in perturbation theory. We also
compare two definitions of the quasi-TMDPDF, namely those in Refs. Ji:2018hvs
; Ebert:2019okf  and in Refs. Ji:2019ewn ; Ji:2020ect , which agree in the
infinite staple length limit, but differ by where the quark fields are
located.

This paper opens with an introduction to spin-dependent quasi-TMDPDFs in Sec.
2. In Sec. 3 we calculate the spin-dependent quasi-TMDPDFs at one-loop,
comparing them with the spin-dependent TMDPDFs to obtain the matching
coefficients. Next, we discuss potential applications of these calculations in
Sec. 4. We conclude in Sec. 5. Appendix A and Sec. 2 discuss the advantages of
the alternative definition of the quasi-TMDPDF. In appendix B we further
detail the one-loop calculation of the spin-dependent quasi-TMDPDFs.

##  2 Definition of spin-dependent TMDPDFs and quasi-TMDPDFs

We begin by reviewing the definitions of the spin-dependent TMDPDFs and their
corresponding spin-dependent quasi-TMDPDFs, following the notation of Ref.
Ebert:2019okf . We focus on the TMDPDF for an energetic hadron $h$ moving
along the $n$-direction, choosing lightlike reference vectors

$\displaystyle n^{\mu}=(1,0,0,1)\,,\qquad{\bar{n}}^{\mu}=(1,0,0,-1)\,,\qquad
n\cdot{\bar{n}}=2\,,$ (1)

such that the hadron momentum is
$P^{\mu}=\frac{1}{2}(P^{-}n^{\mu}+P^{+}{\bar{n}}^{\mu})$ with $P^{-}\gg
P^{+}=M^{2}/P^{-}$, where $M$ is the hadron mass.

Our convention for the lightcone decomposition of momenta is111Another popular
convention in the literature is to choose $n_{+}^{\mu}=n^{\mu}/\sqrt{2}$ and
$n_{-}^{\mu}={\bar{n}}^{\mu}/\sqrt{2}$, such that the lightcone decomposition
reads $k^{\mu}=k^{+}n_{+}^{\mu}+k^{-}n_{-}^{\mu}+k_{\perp}^{\mu}$, where now
$k^{\pm}=(k^{0}\pm k^{z})/\sqrt{2}$.

$\displaystyle
k^{\mu}=k^{-}\frac{n^{\mu}}{2}+k^{+}\frac{{\bar{n}}^{\mu}}{2}+k_{\perp}^{\mu}\,,$
(2)

where $k^{+}=n\cdot k=k^{0}-k^{z}$ and $k^{-}={\bar{n}}\cdot k=k^{0}+k^{z}$.
The transverse component of the four-vector $k^{\mu}$ is
$k_{\perp}^{\mu}=(0,\vec{k}_{T},0)$ with
$k_{\perp}^{2}=-\vec{k}_{T}^{2}\equiv-k_{T}^{2}$. It is also useful to define
the transverse metric $g_{\perp}^{\mu\nu}$ and the transverse antisymmetric
tensor $\epsilon^{\mu\nu}\equiv\epsilon_{\perp}^{\mu\nu}$ as

$\displaystyle
g_{\perp}^{\mu\nu}=g^{\mu\nu}-\frac{1}{2}\bigl{(}n^{\mu}{\bar{n}}^{\nu}+{\bar{n}}^{\mu}n^{\nu}\bigr{)}\,,\qquad\epsilon^{\mu\nu}=\frac{1}{2}n_{\alpha}{\bar{n}}_{\beta}\epsilon^{\alpha\beta\mu\nu}\,,$
(3)

such that $g_{\perp}^{11}=g_{\perp}^{22}=-1$ and
$\epsilon^{12}=-\epsilon^{21}=1$.

###  2.1 Definition of spin-dependent TMDPDFs

The TMDPDF for finding a quark of flavor $q$ inside a hadron $h$ with momentum
$P$ and polarization $S$ is defined as

$\displaystyle f_{q/h_{S}}^{[\Gamma]}(x,\vec{b}_{T},\mu,\zeta)$
$\displaystyle=\lim_{\begin{subarray}{c}\epsilon\to 0\\\ \tau\to
0\end{subarray}}Z^{q}_{\rm
uv}(\mu,\zeta,\epsilon)\,B_{q/h_{S}}^{[\Gamma]}\bigl{(}x,\vec{b}_{T},\epsilon,\tau,xP^{-}\bigr{)}\frac{\sqrt{S^{q}(b_{T},\epsilon,\tau)}}{S_{q}^{0}(b_{T},\epsilon,\tau)}\,,$
$\displaystyle\equiv\lim_{\begin{subarray}{c}\epsilon\to 0\\\ \tau\to
0\end{subarray}}Z^{q}_{\rm
uv}(\mu,\zeta,\epsilon)\,B_{q/h_{S}}^{[\Gamma]}\bigl{(}x,\vec{b}_{T},\epsilon,\tau,xP^{-}\bigr{)}\Delta_{S}^{q}(b_{T},\epsilon,\tau)\,.$
(4)

Here, $x$ is the fraction of the proton momentum $P^{-}$ carried by the struck
quark, $\vec{b}_{T}$ is Fourier-conjugate to the quark’s transverse momentum,
and $\mu$ and $\zeta$ are the renormalization and Collins-Soper scale $\zeta$
Collins:1981va ; Collins:1981uk , respectively. The Dirac structure $\Gamma$
projects out the desired spin-dependent TMD, as discussed below. In eq. (2.1),
$Z^{q}_{\rm uv}$ is the UV counterterm in the ${\overline{\mathrm{MS}}}$
scheme, where UV divergences are regulated by working in $d=4-2\epsilon$
dimensions. The beam function or unsubtracted TMDPDF $B_{q/h_{S}}^{[\Gamma]}$
is a hadronic matrix element. The soft function $S^{q}$ is a vacuum matrix
element and thus is independent of the hadron state, hadron momentum and spin,
and quark flavor $q$ and spin being probed. $S_{q}^{0}$ subtracts the overlap
between the soft and beam functions (and is sometimes referred to as a zero-
bin subtraction Manohar:2006nz ). Its precise definition depends on the choice
of a rapidity regulator $\tau$. The second line of eq. (2.1) incorporates the
soft function and the zero-bin into a combined soft factor $\Delta_{S}^{q}$.

A characteristic feature of TMDPDFs is the appearance of so-called rapidity
divergences in the beam and soft functions, such that both need an additional
rapidity regulator generically denoted as $\tau$ in eq. (2.1). The divergences
cancel in the combination in eq. (2.1), such that the regulator can be removed
as $\tau\to 0$, yielding a regulator-independent definition of the TMDPDF.
Analogous to the appearance of the renormalization scale $\mu$, the rapidity
divergences induce the Collins-Soper scale $\zeta$, which is related to the
momentum of the struck quark by $\zeta\propto(xP^{-})^{2}$. The
proportionality constants depends on the chosen rapidity regulator $\tau$, and
many different schemes are known in the literature, such as taking Wilson
lines off the light cone Collins:1350496 , analytic regulators Beneke:2003pa ;
Chiu:2007yn ; Becher:2011dz , the $\eta$-regulator Chiu:2011qc ; Chiu:2012ir ,
the $\delta$ regulator Chiu:2009yx ; GarciaEchevarria:2011rb , and the
exponential regulator Li:2016axz . (We do not consider the regulators of Refs.
Collins:1981uk ; Ji:2004wu , which induce an additional regularization
parameter $\rho$; using these to calculate cross-sections gives rise to the
product of a TMDPDF and a hard coefficient that are not in the
${\overline{\mathrm{MS}}}$ scheme.)

We define the bare beam and soft functions as

$\displaystyle B_{q/h_{S}}^{[\Gamma]}(x,\vec{b}_{T},\epsilon,\tau,xP^{-})$
$\displaystyle=\int\frac{\mathrm{d}b^{+}}{4\pi}e^{-\mathrm{i}\frac{1}{2}b^{+}(xP^{-})}\Bigl{<}h_{S}(P)\Bigr{|}\Bigl{[}\bar{q}(b^{\mu})W_{\sqsubset}(b^{\mu},0)\frac{\Gamma}{2}q(0)\Bigr{]}_{\tau,\epsilon}\Bigl{|}h_{S}(P)\Bigr{>}\,,$
(5) $\displaystyle S^{q}(b_{T},\epsilon,\tau)$
$\displaystyle=\frac{1}{N_{c}}\bigl{<}0\bigr{|}\mathrm{Tr}\bigl{[}S^{\dagger}_{n}(\vec{b}_{T})S_{\bar{n}}(\vec{b}_{T})S_{T}(-\infty{\bar{n}};\vec{b}_{T},\vec{0}_{T})$
$\displaystyle\hskip 56.9055pt\times
S^{\dagger}_{\bar{n}}(\vec{0}_{T})S_{n}(\vec{0}_{T})S_{T}^{\dagger}\bigl{(}-\infty
n;\vec{b}_{T},\vec{0}_{T}\bigr{)}\bigr{]}_{\tau,\epsilon}\bigl{|}0\bigr{>}\,.$
(6)

Here, we use $[\cdots]_{\tau,\epsilon}$ to denote that the operator in
brackets is rapidity-regulated by $\tau$ and that the invariant mass is
regulated by dimensional regularization. In eqs. (5) and (2.1), we define the
Wilson lines as

$\displaystyle W_{\sqsubset}(b^{\mu},0)$
$\displaystyle=W(b^{\mu})W_{T}\bigl{(}-\infty{\bar{n}};\vec{b}_{T},\vec{0}_{T}\bigr{)}W^{\dagger}(0)\,,$
$\displaystyle W(x^{\mu})$
$\displaystyle=P\exp\biggl{[}-\mathrm{i}g\int_{-\infty}^{0}\mathrm{d}s\,{\bar{n}}\cdot\mathcal{A}(x^{\mu}+s{\bar{n}}^{\mu})\biggr{]}\,,$
$\displaystyle S_{n}(x^{\mu})$
$\displaystyle=P\exp\biggl{[}-\mathrm{i}g\int_{-\infty}^{0}\mathrm{d}s\,n\cdot\mathcal{A}(x^{\mu}+sn^{\mu})\biggr{]}\,,$
$\displaystyle W_{T}(x^{\mu};\vec{b}_{T},\vec{0}_{T})$
$\displaystyle=P\exp\left[\mathrm{i}g\int_{\vec{0}_{T}}^{\vec{b}_{T}}\mathrm{d}\vec{s}_{T}\cdot\vec{\mathcal{A}}_{T}(x^{\mu}+s_{T}^{\mu})\right]=S_{T}(x^{\mu};\vec{b}_{T},\vec{0}_{T})\,.$
(7)

Note that the transverse gauge links at light-cone infinity are required to
obtain the connected Wilson line paths necessary for gauge-invariant matrix
elements. We can often neglect the transverse gauge links in nonsingular
gauges such as Feynman gauge, where the gluon field strength vanishes at
infinity. However, they are important in certain singular gauges, see e.g.
Refs. Ji:2002aa ; Belitsky:2002sm ; Idilbi:2010im ; GarciaEchevarria:2011md ,
and for operators that are formulated for a finite-size lattice. We illustrate
the Wilson paths of the matrix elements in eqs. (5) and (2.1) in Fig. 1.

Figure 1: Graphs of the Wilson line structure of the $n$-collinear beam
function $B_{q}$ (left) and the soft function $S^{q}$ (right), defined in eqs.
(5) and (2.1). The Wilson lines (solid) extend to infinity in the directions
indicated. Adapted from Ref. Li:2016axz .

At leading power (often referred to as twist-2), only three Dirac structures
contribute to the TMDPDF:

$\displaystyle\Gamma\in\\{\,\not{{\bar{n}}}\,,\,\not{{\bar{n}}}\gamma_{5}\,,\,\mathrm{i}\sigma^{\alpha-}\gamma_{5}\,\\}\,,$
(8)

where $\sigma^{\mu\nu}=\frac{\mathrm{i}}{2}[\gamma^{\mu},\gamma^{\nu}]$. The
resulting spin-dependent TMDs for a spin-$\frac{1}{2}$ hadron comprise eight
independent Lorentz structures Ralston:1979ys ; Tangerman:1994eh ; Boer:1997nt
; Mulders:1995dh  (see also Refs. Bacchetta:2004zf ; Goeke:2005hb ;
Bacchetta:2006tn  for a complete classification at higher orders in the power
expansion, twist-3 and twist-4). Following the notation of Ref.
Bacchetta:2006tn , the leading power decomposition in momentum space is222Note
that these expressions are identical to
$\Phi^{[\gamma^{+}]},\Phi^{[\gamma^{+}\gamma_{5}]},\Phi^{[\mathrm{i}\sigma^{\alpha+}\gamma_{5}]}$
in Ref. Bacchetta:2006tn , since the factor $1/\sqrt{2}$ for converting the
different lightcone conventions is already accounted for in eq. (5).

$\displaystyle f_{q/h_{S}}^{[\not{{\bar{n}}}]}(x,\vec{q}_{T})$
$\displaystyle=f_{1}(x,q_{T})-\frac{\epsilon_{\rho\sigma}q_{\perp}^{\rho}S_{\perp}^{\sigma}}{M}f_{1T}^{\perp}(x,q_{T})\,,$
$\displaystyle f_{q/h_{S}}^{[\not{{\bar{n}}}\gamma_{5}]}(x,\vec{q}_{T})$
$\displaystyle=S_{L}\,g_{1L}(x,q_{T})-\frac{q_{\perp}\cdot
S_{\perp}}{M}g_{1T}(x,q_{T})\,,$ $\displaystyle
f_{q/h_{S}}^{[\mathrm{i}\sigma^{\alpha-}\gamma_{5}]}(x,\vec{q}_{T})$
$\displaystyle=S_{\perp}^{\alpha}h_{1}(x,q_{T})+\frac{S_{L}q_{\perp}^{\alpha}}{M}h_{1L}^{\perp}(x,q_{T})$
$\displaystyle\quad+\frac{q_{\perp}^{2}}{M^{2}}\biggl{(}\frac{1}{2}g_{\perp}^{\alpha\rho}-\frac{q_{\perp}^{\alpha}q_{\perp}^{\rho}}{q_{\perp}^{2}}\biggr{)}S_{\perp\,\rho}h_{1T}^{\perp}(x,q_{T})-\frac{\epsilon^{\alpha\rho}q_{\perp\rho}}{M}h_{1}^{\perp}(x,q_{T})\,.$
(9)

Here, we drop the arguments $\mu$ and $\zeta$ for brevity. We also distinguish
the position-space TMD $f_{q/h_{S}}^{[\Gamma]}(x,b_{T})$ and momentum-space
TMD $f_{q/h_{S}}^{[\Gamma]}(x,\vec{q}_{T})$ only by their arguments. Note that
all functions on the right-hand side of eq. (2.1) only depend on the magnitude
$q_{T}=|\vec{q}_{T}|$ of the transverse momentum. In eq. (2.1), $M$ denotes
the nucleon mass, $\alpha$ is a tranverse index, and the spin vector of the
hadron $h$ is decomposed as

$\displaystyle
S^{\mu}=S_{L}\frac{P^{-}n^{\mu}-P^{+}{\bar{n}}^{\mu}}{2M}+S_{\perp}^{\mu}\,,$
(10)

where $-S_{L}^{2}+S_{\perp}^{2}=-1$ such that $S^{2}=-1$. Eq. (2.1) contains
eight distributions, which each correspond to a specific choice of
polarizations for a quark and its parent hadron, as summarized in Table 1. Six
of these distributions are time-reversal even (T-even); namely, the
unpolarized ($f_{1})$, helicity ($g_{1L}$), transversity ($h_{1}$),
pretzelosity ($h_{1T}^{\perp}$) and worm-gear ($g_{1T}$ and $h_{1L}^{\perp}$)
distributions. The other two are T-odd; specifically, the Sivers
($f_{1T}^{\perp}$) Sivers:1989cc  and Boer-Mulders ($h_{1}^{\perp}$)
Boer:1997nt  functions.

| Quark polarization
---|---
|  U |  L |  T
Hadron polarization |  U | $f_{1}$ |  | $h_{1}^{\perp}$
| unpolarized |  | Boer-Mulders
L |  | $g_{1L}$ | $h_{1L}^{\perp}$
|  |  | helicity | worm-gear
|  T | $f_{1T}^{\perp}$ | $g_{1T}$ |  $h_{1}$, $h_{1T}^{\perp}$
|  | Sivers | worm-gear | transversity, pretzelosity
Table 1: Overview of the quark TMDs of a spin-$\frac{1}{2}$ hadron at leading
power in eq. (2.1), organized by the polarization of the struck quark and the
parent hadron. Here U, L, T stand for unpolarized, longitudinally polarized
and transversely polarized, respectively. The Boer-Mulders and Sivers
functions are time-reversal odd, while all other functions are time-reversal
even.

One can obtain the position-space version of eq. (2.1) by a Fourier transform
with respect to $\vec{q}_{T}$. As in Ref. Gutierrez-Reyes:2017glx , we
normalize the position-space functions so that explicit factors of $M$ are
absent. With this choice, we have

$\displaystyle f_{q/h_{S}}^{[\not{{\bar{n}}}]}(x,\vec{b}_{T})$
$\displaystyle=f_{1}(x,b_{T})-\frac{\epsilon_{\rho\sigma}b_{\perp}^{\rho}S_{\perp}^{\rho}}{b_{T}}f_{1T}^{\perp}(x,b_{T})\,,$
$\displaystyle f_{q/h_{S}}^{[\not{{\bar{n}}}\gamma_{5}]}(x,\vec{b}_{T})$
$\displaystyle=S_{L}g_{1L}(x,b_{T})-\frac{b_{\perp}\cdot
S_{\perp}}{b_{T}}g_{1T}(x,b_{T})\,,$ $\displaystyle
f_{q/h_{S}}^{[\mathrm{i}{\bar{n}}_{\beta}\sigma^{\alpha\beta}\gamma_{5}]}(x,\vec{b}_{T})$
$\displaystyle=S_{\perp}^{\alpha}h_{1}(x,b_{T})+\frac{S_{L}b_{\perp}^{\alpha}}{b_{T}}h_{1L}^{\perp}(x,b_{T})$
$\displaystyle\quad+\biggl{(}\frac{1}{2}g_{\perp}^{\alpha\rho}-\frac{b_{\perp}^{\alpha}b_{\perp}^{\rho}}{b_{\perp}^{2}}\biggr{)}S_{\perp\,\rho}h_{1T}^{\perp}(x,b_{T})-\frac{\epsilon^{\alpha\rho}b_{\perp\rho}}{b_{T}}h_{1}^{\perp}(x,b_{T})\,,$
(11)

where $b_{\perp}^{\mu}=(0,\vec{b}_{T},0)$ and $b_{T}=|\vec{b}_{T}|$. The
explicit relations between position-space and momentum-space distributions are
given by (see also Ref. Echevarria:2015uaa )

$\displaystyle F(x,b_{T})$
$\displaystyle=2\pi\int_{0}^{\infty}\mathrm{d}k_{T}\,k_{T}J_{0}(b_{T}k_{T})F(x,k_{T})\,,\hskip
56.9055ptF\in\\{f_{1},g_{1L},h_{1}\\}$ $\displaystyle F(x,b_{T})$
$\displaystyle=2\pi\mathrm{i}\int_{0}^{\infty}\mathrm{d}k_{T}\,k_{T}\frac{k_{T}}{M}J_{1}(b_{T}k_{T})F(x,k_{T})\,,\hskip
38.69576ptF\in\\{f_{1T}^{\perp},g_{1T},h_{1L}^{\perp},h_{1}^{\perp}\\}$
$\displaystyle h_{1T}^{\perp}(x,b_{T})$
$\displaystyle=-2\pi\int_{0}^{\infty}\mathrm{d}k_{T}\,k_{T}\frac{k_{T}^{2}}{M^{2}}J_{2}(b_{T}k_{T})h_{1T}^{\perp}(x,k_{T})\,,$
(12)

where $J_{n}(x)$ is a Bessel function of the first kind. Note that only
$f_{1}$, $g_{1L}$ and $h_{1}$ are directly related by a Fourier transform to
their momentum-space counterparts. The remaining functions require higher-
order Bessel transforms reweighted with an appropriate factor of $(k_{T}/M)$
or $(k_{T}/M)^{2}$, due to the normalization conventions in eq. (2.1).

###  2.2 Definition of spin-dependent quasi-TMDPDFs

Within the framework of LaMET Ji:2013dva ; Ji:2014gla , quasi-TMDPDFs are
defined as equal-time analogs of TMDPDFs, and are thus amenable to lattice
calculations. To define the spin-dependent quasi-TMDPDFs, we extend the setup
in Ref. Ebert:2019okf :

$\displaystyle\tilde{f}_{q/h_{S}}^{[\tilde{\Gamma}]}(x,\vec{b}_{T},\mu,P^{z})=\int\frac{\mathrm{d}b^{z}}{2\pi}\,e^{\mathrm{i}b^{z}(xP^{z})}\,$
$\displaystyle\tilde{Z}^{\prime}_{q}(b^{z},\mu,\tilde{\mu})\tilde{Z}_{\rm
uv}^{q}(b^{z},\tilde{\mu},a)$
$\displaystyle\times\tilde{B}_{q/h_{S}}^{[\tilde{\Gamma}]}(b^{z},\vec{b}_{T},a,L,P^{z})\tilde{\Delta}_{S}^{q}(b_{T},a,L)\,.$
(13)

Here, $h_{S}$ denotes a hadron with spin $S$ and momentum $P$, $q$ is the
flavor of the struck quark, $x$ is the fraction of the $z$-momentum carried by
this quark, and $\vec{b}_{T}$ is Fourier-conjugate to the quark’s transverse
momentum. The scale $\mu$ is the ${\overline{\mathrm{MS}}}$-renormalization
scale, and the $P^{z}$ dependence here plays the role of the Collins-Soper
scale $\zeta$. Moreover, $\tilde{B}_{q/h_{S}}^{[\tilde{\Gamma}]}$ and
$\tilde{\Delta}_{S}^{q}$ denote the quasi beam function and quasi soft factor.

For the purposes of this paper, we need not go into details about the precise
definition of $\tilde{\Delta}_{S}^{q}$; while we must include it in the quasi
beam function to cancel $L/b_{T}$ divergences, it cancels out when taking the
ratio of two quasi-TMDPDFs. In eq. (2.2) the letter $a$ denotes the UV
regulator, resembling the notation for the lattice spacing that acts as a UV
regulator in practical calculations. Here the rapidity regulator is replaced
by the dependence on the length $L$ of the staple-shaped Wilson lines defined
below Ji:2018hvs ; Ebert:2019okf . In contrast to eq. (2.1), we formulate the
UV renormalization in eq. (2.2) in $b^{z}$ space, where the lattice
renormalization factor $\tilde{Z}_{\rm uv}^{q}$ absorbs Wilson-line self
energies proportional to $b^{z}$, with $\tilde{\mu}$ being the corresponding
renormalization scale. 333More precisely, $\tilde{\mu}$ stands for all
parameters induced by the lattice renormalization; see e.g. Ref. Ebert:2019tvc
for details in the RI′/MOM scheme. Finally, $\tilde{Z}_{q}^{\prime}$ converts
this lattice-renormalization scheme into the ${\overline{\mathrm{MS}}}$
scheme. For more details, we refer to Refs. Ebert:2019okf ; Ebert:2019tvc .

We define the quasi beam function in position space as

$\displaystyle\tilde{B}_{q/h_{S}}^{[\tilde{\Gamma}]}(b^{z},\vec{b}_{T},a,L,P^{z})$
$\displaystyle=N_{\tilde{\Gamma}}\,\Bigl{<}h_{S}(P)\Big{|}\bar{q}(b^{\mu})\widetilde{W}_{\sqsubset}(b^{\mu},0;L)\frac{\tilde{\Gamma}}{2}q(0)\Big{|}h_{S}(P)\Bigr{>}\,,$
(14)

where $h_{S}(P)$ denotes a hadron $h$ with polarization vector $S$ and
momentum $P$, and $b^{\mu}=(0,\vec{b}_{T},b^{z})$. Here we follow Ref.
Ebert:2019okf . $\widetilde{W}_{\sqsubset}(b^{\mu},0;L)$ is a staple-shaped
Wilson line path of length $L$ connecting the quark fields, as illustrated in
the left panel of Fig. 2; it is given by

$\displaystyle\widetilde{W}_{\sqsubset}(b^{\mu},0;L)$
$\displaystyle=W_{\hat{z}}(b^{\mu};L-b^{z})W_{T}(L\hat{z};\vec{b}_{T},\vec{0}_{T})W^{\dagger}_{\hat{z}}(0;L)\,,$
(15)

where $W_{T}$ is given in eq. (2.1) and $W_{\hat{z}}$ denotes a Wilson line
oriented along the $z$-direction,

$\displaystyle W_{\hat{z}}(x^{\mu};L)$
$\displaystyle=P\exp\left[\mathrm{i}g\int_{L}^{0}\mathrm{d}s\,\mathcal{A}^{z}(x^{\mu}+s\hat{z})\right]\,.$
(16)

The Dirac structures $\tilde{\Gamma}$ come from the set

$\displaystyle\tilde{\Gamma}\in\\{\,\gamma^{\lambda}\,,\,\gamma^{\lambda}\gamma_{5}\,,\,\mathrm{i}\sigma^{\alpha\lambda}\gamma_{5}\,\\}\,,\qquad\lambda=0,3\,,$
(17)

because they can be boosted to the three corresponding Dirac structures in eq.
(8). The normalization factor $N_{\tilde{\Gamma}}$ in eq. (14) is defined as

$\displaystyle
N_{\gamma^{3}}=N_{\gamma^{0}\gamma_{5}}=N_{i\sigma^{i3}\gamma_{5}}=1\,,\qquad
N_{\gamma^{0}}=N_{\gamma^{3}\gamma_{5}}=N_{i\sigma^{i0}\gamma_{5}}=\frac{P^{z}}{P^{0}}\,.$
(18)

While the choices $\lambda=0$ and $\lambda=3$ are formally equivalent in a
continuum analysis, they induce different operator mixings on a discretized
lattice with broken chiral symmetry Constantinou:2019vyb ; Shanahan:2019zcq ;
Green:2020xco . Hence, we consider both options.

Since all choices in eq. (17) boost onto the corresponding Dirac structures in
eq. (8), we can decompose the quasi-TMDs in the same fashion as the TMDs, up
to power corrections suppressed by large $P^{z}$. Using the conventions of eq.
(2.1), at leading power we find

$\displaystyle\tilde{f}_{q/h_{S}}^{[\gamma^{\lambda}]}(x,\vec{b}_{T})$
$\displaystyle=\tilde{f}_{1}^{\lambda}(x,b_{T})-\frac{\epsilon_{\rho\sigma}b_{\perp}^{\rho}S_{\perp}^{\sigma}}{b_{T}}\tilde{f}_{1T}^{\lambda\perp}(x,b_{T})\,,$
$\displaystyle\tilde{f}_{q/h_{S}}^{[\gamma^{\lambda}\gamma_{5}]}(x,\vec{b}_{T})$
$\displaystyle=S_{L}\tilde{g}_{1L}^{\lambda}(x,b_{T})-\frac{b_{\perp}\cdot
S_{\perp}}{b_{T}}\tilde{g}_{1T}^{\lambda}(x,b_{T})\,,$
$\displaystyle\tilde{f}_{q/h_{S}}^{[\mathrm{i}\sigma^{\alpha\lambda}\gamma_{5}]}(x,\vec{b}_{T})$
$\displaystyle=S_{\perp}^{\alpha}\tilde{h}_{1}^{\lambda}(x,b_{T})+\frac{S_{L}b_{\perp}^{\alpha}}{b_{T}}\tilde{h}_{1L}^{\lambda\perp}(x,b_{T})$
$\displaystyle\quad+\biggl{(}\frac{1}{2}g_{\perp}^{\alpha\rho}-\frac{b_{\perp}^{\alpha}b_{\perp}^{\rho}}{b_{\perp}^{2}}\biggr{)}S_{\perp\,\rho}\tilde{h}_{1T}^{\lambda\perp}(x,b_{T})-\frac{\epsilon^{\alpha\rho}b_{\perp\rho}}{b_{T}}\tilde{h}_{1}^{\lambda\perp}(x,b_{T})\,.$
(19)

By choosing the appropriate projector $\tilde{\Gamma}$ and hadron polarization
$S$, it is straightforward to obtain all individual distributions.

Each Wilson line segment gives rise to a power-law divergence proportional to
its length due to self-energy contributions Dotsenko:1979wb ; Craigie:1980qs ;
Dorn:1986dt . As argued in Ref. Ebert:2019tvc ; Green:2020xco , the nonlocal
Wilson line operator that defines the quasi beam function can be
multiplicatively renormalized by the factor

$\displaystyle Z(b^{z},\tilde{\mu},a,b_{T},L)=Z_{q,\rm
wf}(\tilde{\mu},a)\,e^{\delta m(2L-b^{z}+b_{T})}\,,$ (20)

where $\delta m=\delta m(a)\sim 1/a$ absorbs the power-law dviergence
originating from each Wilson line segment, and $Z_{q,\rm wf}(\tilde{\mu},a)$
renormalizes all logarithmic UV divergences from the wave functions and quark-
Wilson-line vertices. Thus the $Z$ factor is independent of $\tilde{\Gamma}$.
The $b^{z}$ dependence in eq. (20) induces the presence of $b^{z}$ in
$\tilde{Z}_{\rm uv}^{q}(b^{z},\tilde{\mu},a)$ in eq. (2.2), since subtracting
the self energies on the lattice necessitates the use of a $b^{z}$-dependent
counterterm. This also leads to a $b^{z}$-dependent factor
$\tilde{Z}_{q}^{\prime}(b^{z},\mu,\tilde{\mu})$ to convert to the
${\overline{\mathrm{MS}}}$ scheme.

To avoid this complication, we can slightly modify the definition in eq. (15)
so that the length of the Wilson line path is independent of $b^{z}$. We can
achieve this either by shifting $L\to L+b^{z}/2$ in eq. (15), or equivalently
by using the definition of Refs. Ji:2019ewn ; Ji:2020ect ,

$\displaystyle\tilde{B}_{q}^{[\tilde{\Gamma}]}(b^{z},\vec{b}_{T},a,L,P^{z})$
$\displaystyle=N_{\tilde{\Gamma}}\,\Bigl{<}h_{S}(P)\Big{|}\bar{q}\Bigl{(}b_{\perp}+\frac{b^{z}}{2}\hat{z}\Bigr{)}\widetilde{W}_{\sqsubset}\Bigl{(}b_{\perp}+\frac{b^{z}}{2}\hat{z},-\frac{b^{z}}{2}\hat{z};L+\frac{b^{z}}{2}\Bigr{)}\,,$
$\displaystyle\qquad\qquad\qquad\
\times\frac{\tilde{\Gamma}}{2}q\Bigl{(}-\frac{b^{z}}{2}\hat{z}\Bigr{)}\Big{|}h_{S}(P)\Bigr{>}\,.$
(21)

We illustrate this modified path in the right panel of Fig. 2. The operator in
eq. (14) keeps the position of one quark field fixed so that only one Wilson
line is varied for each choice of $b^{z}$ on the lattice, whereas all three of
the Wilson lines are varied for the operator in eq. (2.2).

Importantly, eqs. (15) and (2.2) are equivalent in the limit $L\gg b^{z}$;
thus, all results expanded in this limit apply to both definitions. The
RI/MOM′ renormalization factor calculated at one loop in Ref. Ebert:2019tvc
applies directly to eq. (15), and can be applied to eq. (2.2) after a shift
$L\to L+b^{z}/2$.

A key benefit of using the definition in eq. (2.2) is that the renormalization
factors in eq. (2.2) become independent of $b^{z}$ and can be extracted from
the Fourier transform, giving us

$\displaystyle\tilde{f}_{q/h_{S}}^{[\tilde{\Gamma}]}(x,\vec{b}_{T},\mu,P^{z})$
$\displaystyle=\lim_{\begin{subarray}{c}a\to 0\\\
L\to\infty\end{subarray}}\tilde{Z}_{q}(\mu,a)\tilde{B}_{q/h_{S}}^{[\tilde{\Gamma}]}(x,\vec{b}_{T},a,L,P^{z})\tilde{\Delta}_{S}^{q}(b_{T},a,L)\,,$
$\displaystyle\tilde{B}_{q/h_{S}}^{[\tilde{\Gamma}]}(x,\vec{b}_{T},a,L,P^{z})$
$\displaystyle=\int\frac{\mathrm{d}b^{z}}{2\pi}e^{\mathrm{i}b^{z}(xP^{z})}\tilde{B}_{q/h_{S}}^{[\tilde{\Gamma}]}(b^{z},\vec{b}_{T},a,L,P^{z})\,.$
(22)

Here
$\tilde{Z}_{q}(\mu,a)=\tilde{Z}^{\prime}_{q}(\mu,\tilde{\mu})\tilde{Z}_{\rm
uv}^{q}(\tilde{\mu},a)$ is the $b^{z}$-independent combination of lattice
renormalization and ${\overline{\mathrm{MS}}}$ conversion factors.
Alternatively, one could of course also apply the renormalization and soft
subtraction prior to the Fourier transform. Moreover, $\tilde{Z}_{q}(\mu,a)$
will drop out in the ratios of quasi-TMDPDFs, further simplifying the
calculation. Appendix A discusses using eq. (2.2) for calculating the Collins-
Soper kernel.

Figure 2: Illustration of the Wilson line structure of the quasi beam function
$\tilde{B}_{q}$ as defined in eq. (15) (left) and eq. (2.2) (right).

###  2.3 Relating quasi-TMDPDFs and TMDPDFs

Following the notation of Ref. Ebert:2019okf , we write the relation between
TMDs and quasi-TMDs as

$\displaystyle\tilde{F}_{{\rm ns}/h_{S}}(x,b_{T},\mu,P^{z})$
$\displaystyle=C^{\tilde{F}}_{{\rm
ns}}\bigl{(}\mu,xP^{z}\bigr{)}g^{S}_{q}(b_{T},\mu)\exp\biggl{[}\frac{1}{2}\gamma_{\zeta}^{q}(\mu,b_{T})\ln\frac{(2xP^{z})^{2}}{\zeta}\biggr{]}F_{{\rm
ns}/h_{S}}(x,b_{T},\mu,\zeta)$
$\displaystyle\quad+\mathcal{O}\biggl{(}\frac{b_{T}}{L},\frac{1}{b_{T}P^{z}},\frac{1}{P^{z}L}\biggr{)}\,.$
(23)

Here, $F$ can be any of the spin-dependent TMDs in eq. (2.1), $\tilde{F}$ is
the corresponding quasi-TMD, and the nonsinglet combination $ns=u{-}d$ is
chosen to avoid mixing with gluons. Notably, the relation between TMDs and
quasi-TMDs is multiplicative in $x$ space, with $C_{{\rm ns}}^{\tilde{F}}$
being a perturbative kernel. It also involves the nonperturbative Collins-
Soper evolution to render the right-hand side independent of $\zeta$ for
situations where $\zeta\neq(2xP^{z})^{2}$, such as when the quasi-TMD and TMD
have different values for $P^{z}$. In addition, following Ref. Ebert:2019okf
we have included in eq. (2.3) a nonperturbative factor $g_{q}^{S}(b_{T},\mu)$
which ensures that this relation is independent of the precise choice for the
quasi soft function $\tilde{\Delta}_{S}^{q}(b_{T},a,L)$ in eq. (2.2). (See
Refs. Ji:2019ewn ; Ji:2020ect ; Vladimirov:2020ofp  for discussions of this
structure.) The precise choice for $\tilde{\Delta}_{S}^{q}$, and thus
$g_{q}^{S}$, is irrelevant for ratios of TMDs and quasi-TMDs, and thus does
not impact our results. Eq. (2.3) is correct up to corrections suppressed by
large momenta $P^{z}$ and staple lengths $L$, as indicated.

We consider ratios of eq. (2.3) with different TMDs, such that the spin-
independent Collins-Soper evolution and $g_{q}^{S}$ cancel. For example,
division by the unpolarized TMD gives

$\displaystyle\frac{\tilde{F}_{{\rm
ns}/h_{S}}(x,b_{T},\mu,P^{z})}{\tilde{f}_{{\rm ns}}(x,b_{T},\mu,P^{z})}$
$\displaystyle=\frac{C^{\tilde{F}}_{{\rm
ns}}\bigl{(}\mu,xP^{z}\bigr{)}}{C_{{\rm
ns}}\bigl{(}\mu,xP^{z}\bigr{)}}\frac{F_{{\rm
ns}/h_{S}}(x,b_{T},\mu,\zeta)}{f(x,b_{T},\mu,\zeta)}\,,$ (24)

up to power-suppressed terms. Ref. Ebert:2019okf  gives the kernel appearing
in the denominator:

$\displaystyle C_{{\rm ns}}\bigl{(}\mu,xP^{z}\bigr{)}$
$\displaystyle=1+\frac{\alpha_{s}C_{F}}{4\pi}\biggl{[}-\ln^{2}\frac{(2xP^{z})^{2}}{\mu^{2}}+2\ln\frac{(2xP^{z})^{2}}{\mu^{2}}-4+\frac{\pi^{2}}{6}\biggr{]}+\mathcal{O}(\alpha_{s}^{2})\,.$
(25)

A main goal of our work here will be to determine results for the kernels
$C^{\tilde{F}}_{{\rm ns}}\bigl{(}\mu,xP^{z}\bigr{)}$ appearing in the
numerator.

Another advantage of eq. (24) is that the soft factor $\tilde{\Delta}_{S}^{q}$
contained in the quasi-TMDs, see eq. (2.2), also cancels in the ratio. We may
thus equally well write

$\displaystyle\frac{\tilde{B}_{{\rm
ns}/h_{S}}(x,b_{T},\mu,P^{z})}{\tilde{B}_{{\rm ns}}(x,b_{T},\mu,P^{z})}$
$\displaystyle=\frac{C^{\tilde{F}}_{{\rm
ns}}\bigl{(}\mu,xP^{z}\bigr{)}}{C_{{\rm
ns}}\bigl{(}\mu,xP^{z}\bigr{)}}\frac{F_{{\rm
ns}/h_{S}}(x,b_{T},\mu,\zeta)}{f(x,b_{T},\mu,\zeta)}\,,$ (26)

where $\tilde{B}_{{\rm ns}/h_{S}}$ and $\tilde{B}_{\rm ns}$ are the beam
functions that give rise to the spin-dependent and unpolarized quasi-TMDs
$\tilde{F}_{\rm ns}$ and $\tilde{f}_{\rm ns}$, respectively.

##  3 One-loop calculation

With the required formalism in place, we now turn to the calculation of the
matching kernel $C_{\rm ns}^{\tilde{F}}$ appearing in eq. (2.3). Since the
matching results are independent of the precise choice of states, as long as
they have overlap with the operators, we can carry out the calculation of the
matrix elements in eqs. (5) and (14) using an on-shell quark state $u_{s}(p)$
with spin vector $s$ and momentum $p^{\mu}=p^{-}n^{\mu}/2$, where
$p^{-}=2p^{0}=2p^{z}$. The following on-shell relations greatly simplify the
calculation:

$\displaystyle\not{p}\,u_{s}(p)=\bar{u}_{s}(p)\,\not{p}=0\,,\qquad
P_{n}u_{s}(p)=u_{s}(p)\,,\qquad\bar{u}_{s}(p)P_{\bar{n}}=\bar{u}_{s}(p)\,,$
(27)

where $P_{n}=\not{n}\not{{\bar{n}}}/4$ and
$P_{\bar{n}}=\not{{\bar{n}}}\not{n}/4$ are projectors. We employ Feynman gauge
and regulate both infrared (IR) and UV divergences by working in
$d=4-2\epsilon$ dimensions. We note that in this setup, both the unpolarized
TMDPDF and unpolarized quasi-TMDPDF already exist in the literature, which
provides a useful reference for our analysis.

##### Notation.

We work exclusively in Fourier space, with $b_{\perp}^{\mu}=(0,\vec{b}_{T},0)$
Fourier conjugate to $q_{T}^{\mu}=(0,\vec{q}_{T},0)$, and
$b_{\perp}^{2}=-\vec{b}_{T}^{2}$. We define a shorthand for the canonical
logarithm that appears as

$\displaystyle L_{b}=\ln\frac{\vec{b}_{T}^{2}\mu^{2}}{b_{0}^{2}}\,,\qquad
b_{0}=2e^{-\gamma_{E}}\,,$ (28)

where $\gamma_{E}$ is the Euler-Mascheroni constant. We express all our
results in the ${\overline{\mathrm{MS}}}$ scheme. The associated
renormalization scale $\mu$ is related to the MS scale $\mu_{0}$ by
$\mu^{2}=4\pi e^{-\gamma_{E}}\mu_{0}^{2}$.

###  3.1 Tree-level results

The tree-level results for the TMDPDF and quasi-TMDPDF are

$\displaystyle f_{q/h_{S}}^{[\Gamma]\,(0)}(x,\vec{b}_{T},\mu,\zeta)$
$\displaystyle=\frac{1}{p^{-}}\mathrm{Tr}\biggl{[}u_{s}(p)\bar{u}_{s}(p)\frac{\Gamma}{2}\biggr{]}\delta(1-x)\,,$
$\displaystyle\tilde{f}_{q/h_{S}}^{[\tilde{\Gamma}]\,(0)}(x,\vec{b}_{T},\mu,p^{z})$
$\displaystyle=\frac{N_{\Gamma}}{p^{z}}\mathrm{Tr}\biggl{[}u_{s}(p)\bar{u}_{s}(p)\frac{\tilde{\Gamma}}{2}\biggr{]}\delta(1-x)\,.$
(29)

For the case of a massless spinor $N_{\tilde{\Gamma}}=1$ (i.e. independent of
$\tilde{\Gamma}$), the completeness relation reads:

$\displaystyle
u_{s}(p)\bar{u}_{s}(p)=\frac{1}{2}\,\not{p}\,(1-\Lambda\gamma_{5}+\gamma_{5}\not{s}_{\perp})\,,$
(30)

where $\Lambda$ is the helicity and $s_{\perp}$ is the transverse polarization
vector. (See Ref. Barone:2001sp  for a review of polarization with spinors.)
Thus, the traces in eq. (3.1) give

$\displaystyle\Gamma=\gamma^{\lambda}:\qquad$
$\displaystyle\mathrm{Tr}\Bigl{[}u_{s}(p)\bar{u}_{s}(p)\frac{\gamma^{\lambda}}{2}\Bigr{]}$
$\displaystyle=p^{\lambda}\,,$
$\displaystyle\Gamma=\gamma^{\lambda}\gamma_{5}:\qquad$
$\displaystyle\mathrm{Tr}\Bigl{[}u_{s}(p)\bar{u}_{s}(p)\frac{\gamma^{\lambda}\gamma_{5}}{2}\Bigr{]}$
$\displaystyle=\Lambda p^{\lambda}\,,$
$\displaystyle\Gamma=\mathrm{i}\sigma^{\alpha\lambda}\gamma_{5}:\qquad$
$\displaystyle\mathrm{Tr}\Bigl{[}u_{s}(p)\bar{u}_{s}(p)\frac{\mathrm{i}\sigma^{\alpha\lambda}\gamma_{5}}{2}\Bigr{]}$
$\displaystyle=p^{\lambda}s_{\perp}^{\alpha}\,.$ (31)

Comparing to the decompositions in eqs. (2.1) and (2.2), we see that only the
unpolarized ($f_{1}$ and $\tilde{f}_{1}$), helicity ($g_{1L}$ and
$\tilde{g}_{1L}$) and transversity ($h_{1}$ and $\tilde{h}_{1}$) TMD and
quasi-TMDs are nonzero at tree level. Moreover, by definition these functions
are normalized to $\delta(1-x)$ at tree level. This implies that

$\displaystyle C_{\rm ns}^{\tilde{f}_{1}}=C_{\rm ns}^{\tilde{g}_{1L}}=C_{\rm
ns}^{\tilde{h}_{1}}=1+{\cal O}(\alpha_{s})\,.$ (32)

It is instructive to carry out the same analysis using massive quarks. In this
case, the completeness relation is

$\displaystyle
u_{s}(p)\bar{u}_{s}(p)=(\not{p}+m)\frac{1}{2}(1+\gamma_{5}\not{s})\,,$ (33)

and the Dirac traces in eq. (3.1) evaluate to (in $d=4$ dimensions)

$\displaystyle\Gamma=\gamma^{\lambda}:\qquad$
$\displaystyle\mathrm{Tr}\Bigl{[}u_{s}(p)\bar{u}_{s}(p)\frac{\gamma^{\lambda}}{2}\Bigr{]}$
$\displaystyle=p^{\lambda}\,,$
$\displaystyle\Gamma=\gamma^{\lambda}\gamma_{5}:\qquad$
$\displaystyle\mathrm{Tr}\Bigl{[}u_{s}(p)\bar{u}_{s}(p)\frac{\gamma^{\lambda}\gamma_{5}}{2}\Bigr{]}$
$\displaystyle=ms^{\lambda}\,,$
$\displaystyle\Gamma=\mathrm{i}\sigma^{\alpha\lambda}\gamma_{5}:\qquad$
$\displaystyle\mathrm{Tr}\Bigl{[}u_{s}(p)\bar{u}_{s}(p)\frac{\mathrm{i}\sigma^{\alpha\lambda}\gamma_{5}}{2}\Bigr{]}$
$\displaystyle=p^{\lambda}s^{\alpha}-p^{\alpha}s^{\lambda}=p^{\lambda}s_{\perp}^{\alpha}\,.$
(34)

In the last equation, we used that $p^{\alpha}=0$ since $\alpha$ is a
transverse index. From eq. (10) $ms^{0}=p^{z}$ and $ms^{3}=p^{0}$, and using
our definition of $N_{\tilde{\Gamma}}$ in eq. (18), we find that both the TMD
and the quasi-TMD are again normalized to $\delta(1-x)$ at tree level. This
result confirms the choices made in eq. (18).

###  3.2 TMDPDFs

To carry out the matching calculation to ${\cal O}(\alpha_{s})$ requires
comparing one-loop results for the spin-dependent TMDPDF and quasi-TMDPDF.
Here we present the one-loop results for the spin-dependent TMDPDF, which
requires us to combine the spin-dependent beam function at one-loop with the
standard TMD soft function at the same order. Both the beam and the soft
function are individually rapidity divergent, and we choose to employ the
$\eta$ regulator of Refs. Chiu:2011qc ; Chiu:2012ir  to regulate these
divergences.

(a)

(b)

Figure 3: One-loop diagrams contributing to the spin-dependent beam function
in eq. (5). Mirror diagrams and the scaleless wave function and tadpole
diagrams are not shown. The dashed line indicates the on-shell constraint on
the emitted gluon. The $\otimes$ denotes the two Wilson line operators.

##### Spin-dependent beam function.

There are only two diagrams that do not vanish in dimensional regularization,
as shown in Fig. 3. Note that in the lightlike case, the transverse Wilson
line does not contribute in nonsingular gauges, such as the Feynman gauge we
employ here. Rapidity divergences for the phase space integral over $k$ are
regulated by the factors of $|k^{-}/\nu|^{-\eta}$ appearing from the regulated
Wilson lines, after which the calculation is straightforward. We obtain

$\displaystyle q_{A}^{\Gamma\,(1)}(x,\vec{b}_{T})$
$\displaystyle=\frac{1}{p^{-}}\bigl{[}\bar{u}_{s}^{a}(p)u_{s}^{b}(p)\bigr{]}\,\frac{\alpha_{s}C_{F}}{4\pi}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}e^{\epsilon
L_{b}}\Bigl{(}\gamma^{\mu}\gamma_{\alpha}\frac{\Gamma}{2}\gamma_{\beta}\gamma_{\mu}\Bigr{)}^{ab}$
$\displaystyle~{}\quad\times\left[-\mathrm{i}\epsilon\frac{{\bar{n}}^{\alpha}b_{\perp}^{\beta}+{\bar{n}}^{\beta}b_{\perp}^{\alpha}}{p^{-}b_{\perp}^{2}}+(1-x)\left(\frac{g_{\perp}^{\alpha\beta}}{2}+\epsilon\frac{b_{\perp}^{\alpha}b_{\perp}^{\beta}}{b_{\perp}^{2}}\right)\right]\,,$
(35) $\displaystyle q_{B}^{\Gamma\,(1)}(x,\vec{b}_{T})$
$\displaystyle=\frac{1}{p^{-}}\bigl{[}\bar{u}_{s}^{a}(p)u_{s}^{b}(p)\bigr{]}\frac{\alpha_{s}C_{F}}{4\pi}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}e^{\epsilon
L_{b}}\biggl{\\{}-\bigl{(}P_{\bar{n}}\Gamma+\Gamma
P_{n}\bigr{)}^{ab}+\Bigl{[}\mathcal{L}_{0}(1-x)$
$\displaystyle~{}\quad+\delta(1-x)\Bigl{(}-\frac{1}{\eta}+\ln\frac{p^{-}}{\nu}\Bigr{)}\Bigr{]}\biggl{[}\bigl{(}P_{\bar{n}}\Gamma+\Gamma
P_{n}\bigr{)}-\mathrm{i}\epsilon\frac{\not{{\bar{n}}}\not{b}_{\perp}\Gamma}{p^{-}b_{\perp}^{2}}-\mathrm{i}\epsilon\frac{\Gamma\not{b}_{\perp}\not{{\bar{n}}}}{p^{-}b_{\perp}^{2}}\biggr{]}^{ab}\biggr{\\}}\,,$
(36)

where $\mathcal{L}_{0}(1-x)=[1/(1-x)]_{+}$ is the standard plus
distribution.444Foreshadowing, note that we reproduce eq. (3.2) from the quasi
beam function vertex diagram calculation in Sec. 3.3. Since the graph leading
to eq. (3.2) is rapidity finite it does not depend on the rapidity regulator.
In eq. (3.2) the rapidity divergence is explicit through the $1/\eta$ pole in
the rapidity regulator $\eta$, with $\ln(p^{-}/\nu)$ the associated rapidity
logarithm. Here, we have already expanded in $\eta\to 0$, while keeping the
exact $\epsilon$ dependence.

Both eqs. (3.2) and (3.2) contain Dirac structures that scale as
$1/(p^{-}b_{\perp}^{2})$. We can neglect these terms in the limit of large
$p^{-}\to\infty$, which is also why they do not appear in the decomposition in
eq. (2.1). Using eq. (27), we obtain

$\displaystyle B_{q/h_{S}}^{[\Gamma]}(x,\vec{b}_{T},\epsilon,p^{-}/\nu)$
$\displaystyle=\frac{1}{p^{-}}\bigl{[}\bar{u}_{s}^{a}(p)u_{s}^{b}(p)\bigr{]}\frac{\Gamma^{ab}}{2}\delta(1-x)+\frac{1}{p^{-}}\bigl{[}\bar{u}_{s}^{a}(p)u_{s}^{b}(p)\bigr{]}\,\frac{\alpha_{s}C_{F}}{4\pi}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}e^{\epsilon
L_{b}}$
$\displaystyle\quad\times\biggl{\\{}\frac{\Gamma^{ab}}{2}\Bigl{[}4\mathcal{L}_{0}(1-x)+4\delta(1-x)\Bigl{(}-\frac{1}{\eta}+\ln\frac{p^{-}}{\nu}\Bigr{)}-4\Bigr{]}$
$\displaystyle\qquad+\Bigl{(}\gamma^{\mu}\gamma_{\alpha}\frac{\Gamma}{2}\gamma_{\beta}\gamma_{\mu}\Bigr{)}^{ab}(1-x)\biggl{(}\frac{g_{\perp}^{\alpha\beta}}{2}+\epsilon\frac{b_{\perp}^{\alpha}b_{\perp}^{\beta}}{b_{\perp}^{2}}\biggr{)}\biggr{\\}}+\mathcal{O}(\alpha_{s}^{2})\,.$
(37)

##### Soft function.

From Refs. Chiu:2012ir ; Luebbert:2016itl  the bare soft function with the
$\eta$ regulator is

$\displaystyle
S_{q}(b_{T},\epsilon,\nu)=1+\frac{\alpha_{s}C_{F}}{4\pi}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}e^{\epsilon
L_{b}}\biggl{[}\frac{8}{\eta}+4L_{b}+8\ln\frac{\nu}{\mu}-4\gamma_{E}-4\psi(-\epsilon)\biggr{]}+\mathcal{O}(\alpha_{s}^{2})\,,$
(38)

where $\psi(x)=\Gamma^{\prime}(x)/\Gamma(x)$ is the digamma function.

##### Spin-dependent TMDPDF.

When using the $\eta$ regulator, the zero bin is scaleless and vanishes. Thus,
the bare spin-dependent TMDPDF is

$\displaystyle f_{q/h_{S}}^{[\Gamma]}(x,\vec{b}_{T},\epsilon,\zeta)$
$\displaystyle=B_{q/h_{S}}^{[\Gamma]}\bigl{(}x,\vec{b}_{T},\epsilon,\sqrt{\zeta}/\nu\bigr{)}\sqrt{S_{q}(b_{T},\epsilon,\nu)}$
$\displaystyle=\frac{1}{p^{-}}\bigl{[}\bar{u}^{a}_{s}(p)u_{s}^{b}(p)\bigr{]}\frac{\Gamma^{ab}}{2}\delta(1-x)+\frac{1}{p^{-}}\bigl{[}\bar{u}_{s}^{a}(p)u_{s}^{b}(p)\bigr{]}\,\frac{\alpha_{s}C_{F}}{4\pi}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}e^{\epsilon
L_{b}}$
$\displaystyle\quad\times\biggl{\\{}\frac{\Gamma^{ab}}{2}\Bigl{[}4\mathcal{L}_{0}(1-x)+2\delta(1-x)\Bigl{(}L_{b}+\ln\frac{\zeta}{\mu^{2}}-\gamma_{E}-\psi_{-\epsilon}\Bigr{)}-4\Bigr{]}$
$\displaystyle\qquad+\Bigl{(}\gamma^{\mu}\gamma_{\alpha}\frac{\Gamma}{2}\gamma_{\beta}\gamma_{\mu}\Bigr{)}^{ab}(1-x)\biggl{(}\frac{g_{\perp}^{\alpha\beta}}{2}+\epsilon\frac{b_{\perp}^{\alpha}b_{\perp}^{\beta}}{b_{\perp}^{2}}\biggr{)}\biggr{\\}}+\mathcal{O}(\alpha_{s}^{2})\,.$
(39)

This result agrees with Ref. Gutierrez-Reyes:2017glx  after accounting for
different conventions.

It is convenient to note that eq. (3.2) has a universal structure because
large parts of the NLO correction have the same Dirac structure as the tree
level result. To make this explicit, we write eq. (3.2) as

$\displaystyle f_{q/h_{S}}^{[\Gamma]}(x,\vec{b}_{T},\epsilon,\zeta)$
$\displaystyle=\frac{1}{p^{-}}\Bigl{[}\bar{u}_{s}(p)\frac{\Gamma}{2}u_{s}(p)\Bigr{]}\Bigl{[}\delta(1-x)+\frac{\alpha_{s}C_{F}}{4\pi}f_{b}^{(1)}\Bigr{]}$
$\displaystyle\quad+\frac{\alpha_{s}C_{F}}{4\pi}\frac{1}{p^{-}}\Bigl{[}\bar{u}_{s}(p)\gamma^{\mu}\gamma_{\alpha}\frac{\Gamma}{2}\gamma_{\beta}\gamma_{\mu}u_{s}(p)\Bigr{]}\biggl{[}f^{(1)}_{a1}\frac{g_{\perp}^{\alpha\beta}}{2}+f^{(1)}_{a2}\Bigl{(}\frac{g_{\perp}^{\alpha\beta}}{2}-\frac{b_{\perp}^{\alpha}b_{\perp}^{\beta}}{b_{\perp}^{2}}\Bigr{)}\biggr{]}$
$\displaystyle\quad+\mathcal{O}(\alpha_{s}^{2})\,,$ (40)

where the coefficient functions $f_{i}^{(1)}$, whose arguments we keep
implicit, are given by

$\displaystyle f^{(1)}_{b}$ $\displaystyle=\Bigl{(}\frac{1}{\epsilon_{\rm
IR}}+L_{b}\Bigr{)}\bigl{[}-2P_{qq}(x)+2(1-x)\bigr{]}$
$\displaystyle\quad+\delta(1-x)\biggl{[}\frac{2}{\epsilon^{2}}+\Bigl{(}\frac{1}{\epsilon}+L_{b}\Bigr{)}\Bigl{(}-2\ln\frac{\zeta}{\mu^{2}}+3\Bigr{)}-L_{b}^{2}-\frac{\pi^{2}}{6}\biggr{]}+\mathcal{O}(\epsilon)\,,$
$\displaystyle f^{(1)}_{a1}$
$\displaystyle=-(1-x)\Bigl{[}\frac{1}{\epsilon_{\rm
IR}}+L_{b}+1+\mathcal{O}(\epsilon)\Bigr{]}\,,$ $\displaystyle f^{(1)}_{a2}$
$\displaystyle=(1-x)+\mathcal{O}(\epsilon)\,.$ (41)

Here, $\epsilon_{\rm IR}$ denotes poles of IR origin, while $\epsilon$ denotes
poles of UV origin. The unpolarized splitting function is given by

$\displaystyle
P_{qq}(x)=\biggl{(}\frac{1+x^{2}}{1-x}\biggr{)}_{+}=2\mathcal{L}_{0}(1-x)+\frac{3}{2}\delta(1-x)-(1+x)\,.$
(42)

In eq. (3.2), $f_{b}^{(1)}$ arises entirely from diagram (b) and the soft
subtraction, whereas the two distinct Lorentz structures of $f_{a1}^{(1)}$ and
$f_{a2}^{(1)}$ result from diagram (a). Eqs. (3.2) and (3.2) reveal that at
NLO, the full distributional structure in $x$ of the spin-dependent TMDPDF is
proportional to the tree-level normalization and is thus universal. The true
dependence on $\Gamma$ is always proportional to $(1-x)$. This also explains
the similar kernels that appear when perturbatively matching the spin-
dependent TMD onto spin-dependent PDFs, see Refs. Bacchetta:2013pqa ;
Echevarria:2015uaa ; Gutierrez-Reyes:2017glx .

For any desired Dirac structure $\Gamma$, it is straightforward to insert this
$\Gamma$ into eq. (3.2) and expand in $\epsilon$ to obtain the various desired
spin-dependent TMDPDFs. For the three Dirac structures we consider, we obtain
the following renormalized results:

$\displaystyle f_{q/h_{S}}^{[\gamma^{\lambda}]}(x,\vec{b}_{T},\mu,\zeta)$
$\displaystyle=\delta(1-x)\biggl{\\{}1+\frac{\alpha_{s}C_{F}}{4\pi}\biggl{[}-L_{b}^{2}+L_{b}\Bigl{(}2\ln\frac{\mu^{2}}{\zeta}+3\Bigr{)}-\frac{\pi^{2}}{6}\biggr{]}\biggr{\\}}$
$\displaystyle\quad+\frac{\alpha_{s}C_{F}}{4\pi}\biggl{[}-2\Bigl{(}\frac{1}{\epsilon_{\rm
IR}}+L_{b}\Bigr{)}P_{qq}(x)+2(1-x)\biggr{]}+\mathcal{O}(\alpha_{s}^{2})\,,$
$\displaystyle
f_{q/h_{S}}^{[\gamma^{\lambda}\gamma_{5}]}(x,\vec{b}_{T},\mu,\zeta)$
$\displaystyle=\Lambda
f_{q/h_{S}}^{[\gamma^{\lambda}]}(x,\vec{b}_{T},\mu,\zeta)\,,$ $\displaystyle
f_{q/h_{S}}^{[\mathrm{i}\sigma^{\alpha-}\gamma_{5}]}(x,\vec{b}_{T},\mu,\zeta)$
$\displaystyle=s_{\perp}^{\alpha}\delta(1-x)\biggl{\\{}1+\frac{\alpha_{s}C_{F}}{4\pi}\biggl{[}-L_{b}^{2}+L_{b}\Bigl{(}2\ln\frac{\mu^{2}}{\zeta}+3\Bigr{)}-\frac{\pi^{2}}{6}\biggr{]}\biggr{\\}}$
$\displaystyle\quad+s_{\perp}^{\alpha}\frac{\alpha_{s}C_{F}}{4\pi}\biggl{[}-2\Bigl{(}\frac{1}{\epsilon_{\rm
IR}}+L_{b}\Bigr{)}P^{T}_{qq}(x)\biggr{]}+\mathcal{O}(\alpha_{s}^{2})\,,$ (43)

where in the second equation we use that at one loop the unpolarized and
helicity splitting functions agree, whereas the transverse splitting function
is given by Artru:1989zv

$\displaystyle P^{T}_{qq}(x)$
$\displaystyle=P_{qq}(x)-(1-x)=2\mathcal{L}_{0}(1-x)-2+\frac{3}{2}\delta(1-x)\,.$
(44)

In eq. (3.2), the $\delta(1-x)$ terms agree between all three results because
they arise solely from the sail diagram, which is proportional to the tree-
level Dirac structure and has the same value for different choices of
$\Gamma$. The nontrivial $x$ dependence differs, as it obtains contributions
from the vertex diagram, which agrees between the unpolarized and helicity
structures, but vanishes for the transverse structure.

Finally, we briefly comment in more detail on the calculation of the leading
spin-dependent TMDs in Ref. Gutierrez-Reyes:2017glx , where a similar result
of the combination of eqs. (3.2) and (3.2) was presented, but with bare
results employing the $\delta$-rapidity regulator. See Eq. (14) of Ref.
Gutierrez-Reyes:2017glx  for the unsubtracted bare beam function. (We remark
that their result contains an additional imaginary term proportional to
$\mathrm{i}\pi/2$, which does not contribute to their final result.) We also
note that a different argument to eliminate the $1/(p^{-}b_{T}^{2})$
suppressed terms in eq. (3.2) is given in Ref. Gutierrez-Reyes:2017glx  (the
corresponding term in eq. (3.2) is not present in their result): by choosing a
scheme that imposes the condition
$\not{{\bar{n}}}\Gamma=\Gamma\not{{\bar{n}}}$, these terms immediately vanish,
as required to cancel the rapidity divergences. It is then observed that the
Dirac structures $\Gamma$ allowed by this criterion are precisely those of
leading power. This is consistent with our observation that due to the
$1/p^{-}$, these terms only contribute at higher orders in the power
expansion.555There appears to be a typo in Ref. Gutierrez-Reyes:2017glx  where
the $1/p^{-}$ factor is missing in their result, such that the designation of
this term as power suppressed is not obvious. It is easy to see that this
factor is required to obtain the correction mass dimension of their
$\not{b}_{\perp}/b_{\perp}^{2}$ terms. A key advantage of our observation
compared to Ref. Gutierrez-Reyes:2017glx  is that by immediately discarding
power suppressed terms, we do not need to restrict the allowed schemes for
treating $\gamma^{5}$ in $d$ dimensions (as was done in Gutierrez-
Reyes:2017glx  with the Larin+ scheme). This provides evidence for a more
intricate structure of rapidity divergences at higher order in the power
expansion that are not canceled by the leading power soft function alone.

###  3.3 Quasi-TMDPDFs

In this section, we calculate the quasi-TMDPDFs for various choices of spin
polarizations. The calculation for the unpolarized quasi beam function can be
found in Ref. Ebert:2019okf , which provides a baseline for our analysis. Ref.
Ebert:2019okf  keeps intermediate results exact as long as possible, with
terms suppressed by $1/L$ or $1/(b_{T}P^{z})$ only dropped at the end to
extract the leading-power matching. Here, we instead carry out these
expansions earlier on. As we see below, this allows us to relate the spin-
dependent beam function to the unpolarized case, thus rendering many explicit
calculations unnecessary.

Three types of diagrams contribute in Feynman gauge: the vertex, tadpole, and
sail topologies shown in Fig. 4. We start our analysis of each diagram in
coordinate space. We discuss each of these cases separately.

(a) Vertex diagram

(b) Sail topology

(c) Tadpole

Figure 4: One-loop diagrams contributing to the quasi-TMD beam function of eq.
(15) in Feynman gauge. Mirror diagrams are not shown. The double line
represents the Wilson line $W_{\sqsubset}$.

#### Vertex diagram

The vertex diagram in Fig. 4(a) gives Ebert:2019okf

$\displaystyle\tilde{q}_{A}^{\tilde{\Gamma}\,(1)}(b,p)$
$\displaystyle=\bigl{[}\bar{u}^{a}_{s}(p)u_{s}^{b}(p)\bigr{]}\frac{\alpha_{s}C_{F}}{4\pi}\Bigl{(}\gamma^{\mu}\gamma_{\alpha}\frac{\tilde{\Gamma}}{2}\gamma_{\beta}\gamma_{\mu}\Bigr{)}^{ab}\times\mathcal{I}_{v}^{\alpha\beta}(b,p)\,,$
(45)

where the integral $\mathcal{I}_{v}^{\alpha\beta}$ is given by

$\displaystyle\mathcal{I}_{v}^{\alpha\beta}(b,p)$
$\displaystyle=-4\mathrm{i}\mu_{0}^{2\epsilon}\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d-2}}\frac{k^{\alpha}k^{\beta}}{k^{4}(p-k)^{2}}\,e^{\mathrm{i}k\cdot
b}\,.$ (46)

To evaluate eq. (45), we first evaluate

$\displaystyle\mathcal{I}_{v}(b^{2},p\cdot b)$
$\displaystyle=-4\mathrm{i}\mu_{0}^{2\epsilon}\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d-2}}\frac{e^{\mathrm{i}k\cdot
b}}{k^{4}(p-k)^{2}}$
$\displaystyle=-\frac{1}{4}(\pi\mu_{0}^{2})^{\epsilon}\Gamma(-1-\epsilon)\frac{1+\mathrm{i}{p{\cdot}b}-e^{\mathrm{i}{p{\cdot}b}}}{({p{\cdot}b})^{2}}(-b^{2})^{1+\epsilon}\,,$
(47)

which can be derived using standard techniques. Note that here we have made
explicit that the integral can only depend on the two Lorentz invariants
$b^{2}$ and ${p{\cdot}b}$, as $p^{2}=0$. From here, we note that

$\displaystyle\mathcal{I}_{v}^{\alpha\beta}(b,p)$
$\displaystyle=\left(\mathrm{i}\frac{\partial}{\partial
b_{\alpha}}\right)\left(\mathrm{i}\frac{\partial}{\partial
b_{\beta}}\right)\mathcal{I}_{v}(b^{2},{p{\cdot}b})$
$\displaystyle=-2g^{\alpha\beta}\frac{\partial\mathcal{I}_{v}}{\partial(b^{2})}-2(p^{\alpha}b^{\beta}+b^{\alpha}p^{\beta})\frac{\partial^{2}\mathcal{I}_{v}}{\partial(b^{2})\partial({p{\cdot}b})}-4b^{\alpha}b^{\beta}\frac{\partial^{2}\mathcal{I}_{v}}{\partial^{2}(b^{2})}-p^{\alpha}p^{\beta}\frac{\partial^{2}\mathcal{I}_{v}}{\partial^{2}({p{\cdot}b})}$
$\displaystyle=-2g_{\perp}^{\alpha\beta}\frac{\partial\mathcal{I}_{v}}{\partial(b^{2})}-4b^{\alpha}b^{\beta}\frac{\partial^{2}\mathcal{I}_{v}}{\partial^{2}(b^{2})}+\mathcal{O}(n^{\alpha},n^{\beta})\,.$
(48)

When plugging eq. (3.3) into eq. (45), all terms with $n^{\alpha}$ or
$n^{\beta}$ (or equivalently $p^{\alpha}$ or $p^{\beta}$) vanish due to
$\not{n}u_{s}(p)=0$. This allows us to replace the full metric $g$ with the
transverse metric $g_{\perp}$. It is now straightforward to evaluate eq. (45),

$\displaystyle\tilde{q}_{A}^{\tilde{\Gamma}\,(1)}(b,p)$
$\displaystyle=\bigl{[}\bar{u}^{a}_{s}(p)u_{s}^{b}(p)\bigr{]}\frac{\alpha_{s}C_{F}}{4\pi}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}e^{\epsilon
L_{b}}\Bigl{(}\gamma^{\mu}\gamma_{\alpha}\frac{\tilde{\Gamma}}{2}\gamma_{\beta}\gamma_{\mu}\Bigr{)}^{ab}$
$\displaystyle\quad\times\biggl{(}\frac{g_{\perp}^{\alpha\beta}}{2}+\epsilon\frac{b^{\alpha}b^{\beta}}{b^{2}}\biggr{)}\biggl{(}\frac{-b^{2}}{b_{T}^{2}}\biggr{)}^{\epsilon}\frac{1+\mathrm{i}{p{\cdot}b}-e^{\mathrm{i}{p{\cdot}b}}}{({p{\cdot}b})^{2}}\,.$
(49)

We write Eq. (3.3) in terms of the Lorentz invariants $b^{2}$ and $p\cdot b$
so that it can be used to obtain both the TMDPDF and quasi-TMDPDF vertex
diagrams. For the TMD, ${p{\cdot}b}=\frac{1}{2}p^{-}b^{+}$ and
$b^{2}=b_{\perp}^{2}=-b_{T}^{2}$, whereas for the quasi-TMD we have
${p{\cdot}b}=-p^{z}b^{z}$ and $b^{2}=-b_{T}^{2}-(b^{z})^{2}$. In both cases,
the required Fourier transform can be obtained from

$\displaystyle\int\frac{\mathrm{d}({p{\cdot}b})}{2\pi}e^{-\mathrm{i}x{p{\cdot}b}}\,({p{\cdot}b})^{n}\,\frac{1+\mathrm{i}{p{\cdot}b}-e^{\mathrm{i}{p{\cdot}b}}}{({p{\cdot}b})^{2}}=\mathrm{i}^{n}\frac{\mathrm{d}^{n}}{\mathrm{d}^{n}x}(1-x)\theta(x)\theta(1-x)\,.$
(50)

Note that the physical support $0\leq x\leq 1$ naturally arises in the
integral in eq. (50). In the following we explicitly suppress the $\theta$
functions. When expanding eq. (3.3) in small $(b^{z}/b_{T})^{2}$, only the
$n=0$ term contributes, yielding

$\displaystyle\tilde{q}_{A}^{\tilde{\Gamma}\,(1)}(b,p)$
$\displaystyle=\frac{1}{p^{z}}\bigl{[}\bar{u}^{a}_{s}(p)u_{s}^{b}(p)\bigr{]}\frac{\alpha_{s}C_{F}}{4\pi}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}e^{\epsilon
L_{b}}\Bigl{(}\gamma^{\mu}\gamma_{\alpha}\frac{\tilde{\Gamma}}{2}\gamma_{\beta}\gamma_{\mu}\Bigr{)}^{ab}(1-x)\left(\frac{g_{\perp}^{\alpha\beta}}{2}+\epsilon\frac{b_{\perp}^{\alpha}b_{\perp}^{\beta}}{b_{\perp}^{2}}\right)$
$\displaystyle=\frac{1}{p^{z}}\bigl{[}\bar{u}^{a}_{s}(p)u_{s}^{b}(p)\bigr{]}\frac{\alpha_{s}C_{F}}{4\pi}\Bigl{(}\gamma^{\mu}\gamma_{\alpha}\frac{\tilde{\Gamma}}{2}\gamma_{\beta}\gamma_{\mu}\Bigr{)}^{ab}$
$\displaystyle\quad\times(x-1)\left[\frac{g_{\perp}^{\alpha\beta}}{2}\left(\frac{1}{\epsilon}+L_{b}+1\right)-\left(\frac{g_{\perp}^{\alpha\beta}}{2}-\frac{b_{\perp}^{\alpha}b_{\perp}^{\beta}}{b_{\perp}^{2}}\right)\right]\,.$
(51)

In the same fashion, we can obtain the vertex diagram of the TMDPDF, given
above in eq. (3.2), where the $b_{T}/p^{-}$-suppressed term has $n=1$ and can
be neglected at leading power. The close relationship between the two results
is not surprising, as the two calculations only differ by the definition of
$\tilde{\Gamma}$ and the choice of $b^{\mu}$.

#### Sail diagram

Next we consider the sail diagram in Fig. 4(b) and its mirror diagram.
Extracting the overall Dirac structure, we write the sail diagram as

$\displaystyle\tilde{q}_{B}^{\tilde{\Gamma}\,(1)}(b)$
$\displaystyle=\frac{\alpha_{s}C_{F}}{2\pi}\Bigl{\\{}\bigl{[}\bar{u}_{s}(p)\tilde{\Gamma}\gamma^{\rho}\gamma^{\mu}u_{s}(p)\bigr{]}\tilde{q}^{\tilde{\Gamma}\,(B1)}_{\mu\rho}(b^{\mu})+\bigl{[}\bar{u}_{s}(p)\gamma^{\mu}\gamma^{\rho}\tilde{\Gamma}u_{s}(p)\bigr{]}\tilde{q}^{\tilde{\Gamma}\,(B2)}_{\mu\rho}(b^{\mu})\Bigr{\\}}\,,$
$\displaystyle\tilde{q}^{\tilde{\Gamma}\,(B1)}_{\mu\rho}(b)$
$\displaystyle=-\frac{\mu_{0}^{2\epsilon}}{(2\pi)^{d-2}}\int\\!\mathrm{d}^{d}k\int_{0}^{1}\mathrm{d}s\,\frac{\gamma^{\prime}(s)_{\mu}k_{\rho}}{k^{2}(p-k)^{2}}\,e^{\mathrm{i}p\cdot
b-\mathrm{i}(p-k)\cdot\gamma(s)}\,,$
$\displaystyle\tilde{q}^{\tilde{\Gamma}\,(B2)}_{\mu\rho}(b)$
$\displaystyle=-\frac{\mu_{0}^{2\epsilon}}{(2\pi)^{d-2}}\int\\!\mathrm{d}^{d}k\int_{0}^{1}\mathrm{d}s\,\frac{\gamma^{\prime}(s)_{\mu}k_{\rho}}{k^{2}(p-k)^{2}}\,e^{\mathrm{i}k\cdot
b+\mathrm{i}(p-k)\cdot\gamma(s)}\,,$ (52)

where $\gamma(s)$ is the path of the Wilson lines. To make a connection to the
calculation presented in Ref. Ebert:2019okf , we work with the path given in
eq. (15) and illustrated in the left panel of Fig. 2, but we stress that in
the limit $L\gg b_{T}$ this path is equivalent to the one in eq. (2.2).

We insert $\tilde{q}^{\tilde{\Gamma}\,(B1)}_{\mu\rho}$ into
$\tilde{q}_{B}^{\tilde{\Gamma}\,(1)}$ and simplify the result using the on-
shell condition $\not{n}u_{s}(p)=0$, giving

$\displaystyle\not{k}\not{\gamma}^{\prime}u_{s}(p)$
$\displaystyle=\Bigl{[}k^{-}\frac{\not{n}}{2}+k^{+}\frac{\not{{\bar{n}}}}{2}+\not{k}_{\perp}\Bigr{]}\Bigl{[}\gamma^{\prime-}\frac{\not{n}}{2}+\gamma^{\prime+}\frac{\not{{\bar{n}}}}{2}+\not{\gamma}^{\prime}_{\perp}\Bigr{]}u_{s}(p)$
$\displaystyle=\Bigl{[}k^{-}\gamma^{\prime+}+\frac{\not{{\bar{n}}}}{2}\bigl{(}k^{+}\not{\gamma}^{\prime}_{\perp}-\not{k}_{\perp}\gamma^{\prime+}\bigr{)}+\not{k}_{\perp}\not{\gamma}^{\prime}_{\perp}\Bigr{]}u_{s}(p)\,.$
(53)

Here, we suppress the argument of $\gamma^{\prime}\equiv\gamma^{\prime}(s)$
and employ lightcone coordinates to make the on-shell condition manifest,
noting that in practice $\gamma^{\prime+}=-\gamma^{\prime z}$. The calculation
for $\tilde{q}^{\tilde{\Gamma}\,(B2)}_{\mu\rho}(b^{\mu})$ works similarly.

In eq. (3.3), the contributions with $\not{\gamma}_{\perp}^{\prime}$ arise
from the transverse gauge links. In particular, evaluation of the line
integrals in eq. (3.3) gives

$\displaystyle\int_{0}^{1}\\!\\!\mathrm{d}s\,\gamma^{\prime}(s)_{\mu}e^{\mathrm{i}q\cdot\gamma(s)}$
$\displaystyle=\mathrm{i}\hat{z}_{\mu}\frac{e^{-\mathrm{i}q^{z}L}-1}{q^{z}}+\mathrm{i}b_{\perp\mu}e^{-\mathrm{i}q^{z}L}\frac{e^{-\mathrm{i}\vec{b}_{T}\cdot\vec{q}_{T}}-1}{\vec{b}_{T}\cdot\vec{q}_{T}}+\mathrm{i}\hat{z}_{\mu}\frac{e^{-\mathrm{i}b^{z}q^{z}}-e^{-\mathrm{i}q^{z}L}}{q^{z}}e^{-\mathrm{i}\vec{b}_{T}\cdot\vec{q}_{T}}\,.$
(54)

The perpendicular contribution contains a pure phase $e^{-\mathrm{i}q^{z}L}$,
which oscillates quickly in the limit $L,p^{z}\to\infty$ and makes this term
neglectable. This is analogous to how transverse gauge links do not contribute
in nonsingular gauges. Thus, we can approximate eq. (3.3):

$\displaystyle\not{k}\not{\gamma}^{\prime}u_{s}(p)$
$\displaystyle=\bigl{(}k^{-}-\not{k}_{\perp}\bigr{)}\gamma^{\prime+}u_{s}(p)+\mathcal{O}\Bigl{(}\frac{1}{p^{z}L}\Bigr{)}\,.$
(55)

After Fourier-transforming eq. (3.3) with respect to $b^{z}$,
$\tilde{q}_{B}^{\tilde{\Gamma}\,(1)}$ only depends on $p^{-}$ and $b_{\perp}$.
Thus, the contributions from the Dirac structures $k^{-}$ and
$\not{k}_{\perp}$ in eq. (55) parametrically behave as $p^{-}$ and
$\not{b}_{\perp}/b_{T}^{2}$, respectively. The latter is suppressed in the
limit $b_{T}p^{z}\gg 1$, similar to the subleading-twist terms in the physical
TMDPDF, cf. eq. (3.2). We can thus further expand

$\displaystyle\not{k}\not{\gamma}^{\prime}u_{s}(p)$
$\displaystyle=k^{-}\gamma^{\prime+}u_{s}(p)+\mathcal{O}\Bigl{(}\frac{1}{p^{z}L},\frac{1}{b_{T}p^{z}}\Bigr{)}\,.$
(56)

Overall, in the physical limit eq. (3.3) reduces to the tree-level Dirac
structure, leaving us free to write

$\displaystyle\tilde{q}_{B}^{\tilde{\Gamma}\,(1)}(x,\vec{b}_{T})$
$\displaystyle=\frac{1}{p^{z}}\Bigl{[}\bar{u}_{s}(p)\frac{\tilde{\Gamma}}{2}u_{s}(p)\Bigr{]}\frac{\alpha_{s}C_{F}}{4\pi}\,\tilde{q}_{B}(x,\vec{b}_{T})\,,$
(57)

where $\tilde{q}_{B}(x,\vec{b}_{T})$ is the one-loop coefficient of the
unpolarized quasi beam function, i.e.,

$\displaystyle\frac{1}{2}\sum_{s}\tilde{q}_{B}^{\gamma^{z}\,(1)}(x,\vec{b}_{T})=\frac{\alpha_{s}C_{F}}{4\pi}\,\tilde{q}_{B}(x,\vec{b}_{T})\,.$
(58)

We can also read off from the result in Ref. Ebert:2019okf  that

$\displaystyle\tilde{q}_{B}(x,\vec{b}_{T})$
$\displaystyle=\Bigl{(}\frac{1}{\epsilon}+L_{b}\Bigr{)}\Bigl{[}-4\mathcal{L}_{0}(1-x)-2\delta(1-x)L_{p^{z}}+4\Bigr{]}$
$\displaystyle\quad+\delta(1-x)\Bigl{(}-\frac{2}{\epsilon}+\frac{2}{\epsilon}L_{p^{z}}-L_{b}^{2}-L_{p^{z}}^{2}+2L_{p^{z}}-4\Bigr{)}\,,$
(59)

where $L_{p^{z}}=\ln[(2p^{z}/\mu)^{2}]$. The plus distribution ${\cal
L}_{0}(1-x)$ implicitly only has support for $x\in[0,1]$. Note that the pole
in the first line in eq. (3.3) has an IR origin, and thus must agree with the
corresponding IR pole in the TMDPDF so that the IR divergences cancel each
other out in the matching. By comparing to eq. (3.2), it is easy to see that
this holds upon identifying $\zeta=(2p^{z})^{2}$.

#### Tadpole diagram

The Wilson line self-energy in Fig. 4(c) is given by

$\displaystyle\tilde{q}_{C}^{\tilde{\Gamma}\,(1)}(x,\vec{b}_{T})$
$\displaystyle=\frac{\mathrm{i}}{p^{z}}\Bigl{[}\bar{u}_{s}(p)\frac{\tilde{\Gamma}}{2}u_{s}(p)\Bigr{]}\frac{\alpha_{s}C_{F}}{2\pi}\frac{\mu_{0}^{2\epsilon}}{(2\pi)^{d-2}}\int_{0}^{1}\mathrm{d}s\,\mathrm{d}t\,[\gamma^{\prime}(s)\cdot\gamma^{\prime}(t)]\int\\!\mathrm{d}^{d}k\frac{e^{\mathrm{i}k\cdot[\gamma(s)-\gamma(t)]}}{k^{2}+\mathrm{i}0}$
$\displaystyle\equiv\frac{1}{p^{z}}\Bigl{[}\bar{u}_{s}(p)\frac{\tilde{\Gamma}}{2}u_{s}(p)\Bigr{]}\frac{\alpha_{s}C_{F}}{4\pi}\,\tilde{q}_{C}(x,\vec{b}_{T})\,.$
(60)

Since the Wilson line vertices do not have a Dirac structure, the tadpole
contribution is trivially proportional to the tree level Dirac structure. In
eq. (3.3), $\tilde{q}_{C}(x,\vec{b}_{T})$ is the coefficient of the tadpole
diagram of the unpolarized quasi beam function because

$\displaystyle\frac{1}{2}\sum_{s}\tilde{q}_{C}^{\gamma^{z}\,(1)}(x,\vec{b}_{T})=\frac{\alpha_{s}C_{F}}{4\pi}\,\tilde{q}_{C}(x,\vec{b}_{T})\,.$
(61)

From Ref. Ebert:2019okf  we can read off

$\displaystyle\tilde{q}_{C}(x,\vec{b}_{T})$
$\displaystyle=\delta(1-x)\biggl{[}\frac{6}{\epsilon}+6L_{b}+4+\frac{4\pi
L}{b_{T}}+\mathcal{O}(\epsilon)\biggr{]}\,,$ (62)

up to corrections suppressed in the $P^{z}b_{T}\gg 1$ and $L/b_{T}\gg 1$
limits. We remark upon the explicit $L/b_{T}$ divergence in eq. (62), which
must be canceled by the soft factor in a manner analogous to the cancellation
of rapidity divergences that appears in the TMD case (see Ref. Ebert:2019okf
).

#### Combined result

The sum of the results for the vertex, sail and tadpole diagrams gives the
bare quasi beam function at one loop,

$\displaystyle\tilde{B}_{q/h_{S}}^{[\tilde{\Gamma}]}(x,\vec{b}_{T},\epsilon,p^{z})$
$\displaystyle=\frac{1}{p^{z}}\biggl{[}\bar{u}_{s}(p)\frac{\tilde{\Gamma}}{2}u_{s}(p)\biggr{]}\Bigl{\\{}\delta(1-x)+\frac{\alpha_{s}C_{F}}{4\pi}\bigl{[}\tilde{B}_{b}^{(1)}+\tilde{R}_{B}^{(1)}\delta(1-x)\bigr{]}\Bigr{\\}}$
$\displaystyle\quad+\frac{\alpha_{s}C_{F}}{4\pi}\frac{1}{p^{z}}\Bigl{[}\bar{u}_{s}(p)\gamma^{\mu}\gamma_{\alpha}\frac{\tilde{\Gamma}}{2}\gamma_{\beta}\gamma_{\mu}u_{s}(p)\Bigr{]}\biggl{[}\tilde{B}_{a1}^{(1)}\frac{g_{\perp}^{\alpha\beta}}{2}+\tilde{B}_{a2}^{(1)}\Bigl{(}\frac{g_{\perp}^{\alpha\beta}}{2}-\frac{b_{\perp}^{\alpha}b_{\perp}^{\beta}}{b_{\perp}^{2}}\Bigr{)}\biggr{]}$
$\displaystyle\quad+\mathcal{O}(\alpha_{s}^{2})\,.$ (63)

The coefficient functions, whose arguments we keep implicit, are given by

$\displaystyle\tilde{B}_{b}^{(1)}$
$\displaystyle=\Bigl{(}\frac{1}{\epsilon_{\rm
IR}}+L_{b}\Bigr{)}[-2P_{qq}(x)+2(1-x)]$
$\displaystyle\quad+\delta(1-x)\Bigl{[}\frac{7}{\epsilon}-(L_{b}+L_{P^{z}})^{2}+2L_{P^{z}}+9L_{b}\Bigr{]}+\mathcal{O}(\epsilon)\,,$
$\displaystyle\tilde{B}_{a1}^{(1)}$
$\displaystyle=-(1-x)\Bigl{(}\frac{1}{\epsilon_{\rm
IR}}+L_{b}+1\Bigr{)}+\mathcal{O}(\epsilon)\,,$
$\displaystyle\tilde{B}_{a2}^{(1)}$
$\displaystyle=(1-x)+\mathcal{O}(\epsilon)\,.$ (64)

Note that we separated out the divergent term into $\tilde{R}_{B}^{(1)}=4\pi
L/b_{T}$ and made the origin of all poles as either IR or UV explicit.
Evaluating eq. (3.3) for the three Dirac structures in eq. (17), we obtain the
UV-renormalized quasi beam functions:

$\displaystyle\tilde{B}_{q/h_{S}}^{[\gamma^{\lambda}]}(x,\vec{b}_{T},\mu,p^{z})$
$\displaystyle=\delta(1-x)\Bigl{\\{}1+\frac{\alpha_{s}C_{F}}{4\pi}\Bigl{[}-(L_{b}+L_{P^{z}})^{2}+2L_{P^{z}}+9L_{b}+\tilde{R}_{B}^{(1)}\Bigr{]}\Bigr{\\}}$
$\displaystyle\quad+\frac{\alpha_{s}C_{F}}{4\pi}\Bigl{[}-2\Bigl{(}\frac{1}{\epsilon_{\rm
IR}}+L_{b}\Bigr{)}P_{qq}(x)+2(1-x)\Bigr{]}+\mathcal{O}(\alpha_{s}^{2})\,,$
$\displaystyle\tilde{B}_{q/h_{S}}^{[\gamma^{\lambda}\gamma]}(x,\vec{b}_{T},\mu,p^{z})$
$\displaystyle=\Lambda\tilde{B}_{q/h_{S}}^{[\gamma^{\lambda}]}(x,\vec{b}_{T},\mu,p^{z})+\mathcal{O}(\alpha_{s}^{2})\,,$
$\displaystyle\tilde{B}_{q/h_{S}}^{[\mathrm{i}\sigma^{\alpha\lambda}\gamma_{5}]}(x,\vec{b}_{T},\mu,p^{z})$
$\displaystyle=s_{\perp}^{\alpha}\delta(1-x)\Bigl{\\{}1+\frac{\alpha_{s}C_{F}}{4\pi}\Bigl{[}-(L_{b}+L_{P^{z}})^{2}+2L_{P^{z}}+9L_{b}+\tilde{R}_{B}^{(1)}\Bigr{]}\Bigr{\\}}$
$\displaystyle\quad+s_{\perp}^{\alpha}\frac{\alpha_{s}C_{F}}{4\pi}\Bigl{[}-2\Bigl{(}\frac{1}{\epsilon_{\rm
IR}}+L_{b}\Bigr{)}P^{T}_{qq}(x)\Bigr{]}+\mathcal{O}(\alpha_{s}^{2})\,.$ (65)

Similar to eq. (3.2), the $\delta(1-x)$ terms in eq. (3.3) agree for all
choices of $\tilde{\Gamma}$ as they arise entirely from the sail diagrams,
which are proportional to the tree-level Dirac structure. We have explicitly
validated this in appendix B.2. The differences result entirely from the
vertex diagram, which is absent for the transverse structure, see also
appendix B.1. We also note that all results are independent of the choice
$\lambda=t$ or $\lambda=z$.

###  3.4 One-loop matching

By comparing the results in eqs. (3.2) and (3.3) and noting that the
unspecified quasi soft factor is spin independent, we see that the ratios of
quasi-TMDPDFs and TMDPDFs agree for the unpolarized, helicity and transversity
structures,

$\displaystyle\frac{\tilde{f}_{1}(x,b_{T},\mu,P^{z})}{f_{1}(x,b_{T},\mu,\zeta)}$
$\displaystyle=\frac{\tilde{g}_{1L}(x,b_{T},\mu,P^{z})}{g_{1L}(x,b_{T},\mu,\zeta)}=\frac{\tilde{h}_{1}(x,b_{T},\mu,P^{z})}{h_{1}(x,b_{T},\mu,\zeta)}\,.$
(66)

From eq. (24), it follows that the short-distance matching kernel is identical
for these functions. Using the result from Ref. Ebert:2019okf , we thus obtain

$\displaystyle C_{\rm ns}^{f_{1}}(\mu,xP^{z})$ $\displaystyle=C_{\rm
ns}^{g_{1L}}(\mu,xP^{z})=C_{\rm ns}^{h_{1}}(\mu,xP^{z})$
$\displaystyle=1+\frac{\alpha_{s}C_{F}}{4\pi}\biggl{[}-\ln^{2}\frac{(2xP^{z})^{2}}{\mu^{2}}+2\ln\frac{(2xP^{z})^{2}}{\mu^{2}}-4+\frac{\pi^{2}}{6}\biggr{]}+\mathcal{O}(\alpha_{s}^{2})\,.$
(67)

This is the main result of our analysis. We discuss a reason for this observed
universality in Sec. 3.5, where we also discuss the extension of this
observation to higher orders.

It is interesting to note that the one-loop results of the both the TMDPDF and
the quasi beam functions, see eqs. (3.2) and (3.3), consist of universal
kinematic structures, whereas the dependence on $\Gamma$ and $\tilde{\Gamma}$
only arises through the Dirac traces. The Dirac traces are identical for each
choice for $\Gamma$/$\tilde{\Gamma}$, and thus the spin independence of the
ratios of quasi-TMDs to TMDs can be traced back to the universal kinematic
structure of the one-loop diagrams. Based on this observation, we present more
general arguments for the spin independence of the matching kernel in Sec.
3.5. In addition, since the tree-level Dirac structures are all normalized,
only the Dirac trace arising from the vertex diagram can lead to differences
between different spin-dependent quasi-TMDs, or between different spin-
dependent TMDs, respectively. Since the vertex diagram only encodes collinear
interactions between the external quarks, which are not affected by the
different Wilson line geometries, it obeys the simple boost picture underlying
LaMET and does not affect the matching. In contrast, the sail and tadpole
diagrams, as well as the soft subtraction, resolve the Wilson line structure;
thus, they induce a nontrivial matching.

The $f_{1T}^{\perp}$, $g_{1T}$, $h_{1L}^{\perp}$, and $h_{1}^{\perp}$
distributions, which are proportional to $b_{\perp}^{\alpha}/b_{T}$, cannot be
constrained by the setup used for our calculation. The chosen on-shell quark
state for the one-loop calculation has no overlap with the corresponding
(quasi-)TMDPDF operators. The analysis for these functions is beyond the scope
of the present work.

A special case is the pretzelosity distribution $h_{1T}^{\perp}$, which is
proportional to
$g_{\perp}^{\alpha\beta}/2-b_{\perp}^{\alpha}b_{\perp}^{\beta}/b_{\perp}^{2}$,
cf. eq. (2.1). From eqs. (3.2) and (3.3), it is clear that this contribution
arises first at one loop from the vertex diagram for both the quasi and non-
quasi TMDs. It does not contain any divergences or logarithms and in both
cases is simply proportional to

$\displaystyle\Bigl{[}\bar{u}_{s}(p)\gamma^{\mu}\gamma_{\alpha}\frac{\mathrm{i}\sigma^{\rho\lambda}\gamma_{5}}{2}\gamma_{\beta}\gamma_{\mu}u_{s}(p)\Bigr{]}$
$\displaystyle=p^{\lambda}(d-4)\bigl{(}s^{\rho}g_{\alpha\beta}-s_{\beta}g^{\rho}_{\alpha}-s_{\alpha}g^{\rho}_{\beta}\Bigr{)}\,.$
(68)

Here, we use naive dimensional regularization with anticommuting $\gamma^{5}$
and $p\cdot s=0$, and furthermore that $\alpha$, $\beta$ and $\rho$ are
transverse indices, while $\lambda$ is not. Eq. (68) implies that pretzelosity
formally contributes at NLO but vanishes in dimensional regularization for
$d=4$, so that the $\overline{\rm MS}$-renormalized pretzelosity quasi-TMD
(and TMD) is zero at this order. In fact, for matching using massless quark
states, Refs. Gutierrez-Reyes:2017glx ; Gutierrez-Reyes:2018iod  observe that
the pretzelosity matrix elements vanish through two loops, and Ref.
Chai:2018mwx  argues that this holds to all orders in perturbation theory.
Recall that the requirement to carry out a valid matching calculation is that
the chosen states have overlap with the operators being considered, which is
not achieved with the above choices for pretzelosity. Interestingly, we can
consider matching using massless quark states with $d\neq 4$. In this case the
$(d-4)$ prefactor occurs for both the quasi-TMD and TMD and cancels in their
ratio, so we find

$\displaystyle\tilde{h}_{1T}^{\perp}(x,b_{T},\mu,P^{z})=h_{1T}^{\perp}\bigl{(}x,b_{T},\mu,\zeta=(2xP^{z})^{2}\bigr{)}\times\bigl{[}1+\mathcal{O}(\alpha_{s})]\,.$
(69)

In this matching relation the Wilson coefficient is only obtained at tree
level, since the overlap matrix elements themselves start at one-loop. To
obtain the one-loop matching kernel for the quasi-pretzelosity in this fashion
would require a two-loop calculation, which is beyond our goals here. However,
the line of reasoning in Sec. 3.5 suggests that $C_{\rm
ns}^{h_{1T}^{\perp}}(\mu,xP^{z})$ will also be universal with the same value
as in eq. (3.4). A calculation of the ratio
$\tilde{h}_{1T}^{\perp}/\tilde{f}_{1}$ would determine the size of the
pretzelosity distribution relative to the unpolarized TMDPDF, which is an
interesting target for lattice QCD.

###  3.5 Generalization of matching to all orders

It is interesting to ask whether the observed spin independence of the one-
loop matching kernels obtained in eq. (3.4) will continue to higher loop
orders. In this section we outline an argument that this universality will
continue to hold to all orders, without providing sufficient detail to call it
a complete proof.

In the one-loop matching calculation we observed that it was the graphs
associated to attachments to the Wilson lines that led to the mismatch between
quasi-TMD and TMD contributions, and hence to a nonzero result for the
matching coefficient. At leading order in power counting, these Wilson line
graphs could not modify the spin structure of the initial operator. This is
reminiscent of the spin universality of Soft Collinear Effective Theory (SCET)
diagrams for heavy-to-light and back-to-back light-to-light currents
Bauer:2000yr ; Bauer:2002nz ; SCETlecture , which arise from the spin
universality of Wilson line and self-energy diagrams.

Consider the generic beam function and quasi beam function correlators using
quark fields with open Dirac spin indices $i$ and $j$,

$\displaystyle B_{ij,q/h_{S}}(b^{+},\vec{b}_{T},a,\tau,P^{-})$
$\displaystyle=\Bigl{<}h(P)\Bigr{|}\Bigl{[}\bar{q}_{i}(b^{\mu})W_{\sqsubset}(b^{\mu},0)q_{j}(0)\Bigr{]}_{\tau,a}\Bigl{|}h(P)\Bigr{>}\,,$
$\displaystyle\tilde{B}_{ij,q/h_{S}}(b^{z},\vec{b}_{T},a,L,P^{z})$
$\displaystyle=\Bigl{<}h(P)\Bigr{|}\Bigl{[}\bar{q}_{i}(b^{\mu})\widetilde{W}_{\sqsubset}(b^{\mu},0;L)q_{j}(0)\Bigr{]}_{a}\Bigl{|}h(P)\Bigr{>}\,.$
(70)

The paths for the Wilson lines are shown in Figs. 1 and 2, respectively. We
notice that the vertices on the four corners of the Wilson lines are separated
by long distances, either $\sim b_{T}$ or $\sim L$ for
$\tilde{B}_{ij,q/h_{S}}$, or $\sim b_{T}$ or $\infty$ for
$B_{ij,q/h_{S}}$.666Note that the situation for quasi-TMDs differs from that
for matching spin-dependent quasi-PDFs Xiong:2013bka ; Izubuchi:2018srq ;
Alexandrou:2018eet ; Liu:2018uuj ; Liu:2018hxv , since the quark fields in the
quasi-PDF case are not separated by the same type of long distance scales.
These straight Wilson lines can be rewritten in terms of local operators by
the now standard method of introducing scalar auxiliary fields $X_{v}$ for a
line along the four-vector $v^{\mu}$ Korchemsky:1987wg . For simplicity one
can introduce three different auxiliary fields whose direction is along each
straight Wilson line segment in Fig. 2, and another three auxiliary fields for
the segments in Fig. 1. In this case both the quasi beam function and beam
function calculations become matrix elements of a product of four local
current operators whose spacetime arguments situate them at each of the
corresponding four corners. We can refer to them as the current operators,
such as $X_{T}^{\dagger}X_{n}$ and $X_{n}^{\dagger}q_{i}$, and quasi-current
operators, such as $X_{T}^{\dagger}X_{\hat{z}}$ and
$X_{\hat{z}}^{\dagger}q_{i}$. With this setup, two out of the four current
operators (or quasi-current operators) have open spin indices ($i$ or $j$).

When we carry out the matching in this setup we are matching one nonlocal
product of operators onto another nonlocal product of operators; therefore one
expects that the short-distance contribution is isolated to a matching
coefficient $C$ at each of the currents when taking the boosted limit
$X_{\hat{z}}^{\dagger}q_{i}\to CX_{n}^{\dagger}q_{i}$. This type of matching
is like what occurs in SCET when matching a full theory current in boosted
kinematics onto boosted $n$-collinear fields. Due to the spin universality of
the scalar auxiliary fields, no spin dependence will be introduced for the
resulting short-distance matching coefficients $C$ at any order in
$\alpha_{s}$. These arguments clearly must be extended to account for rapidity
divergences that appear in the light-like limit, as well as non-trivial quasi
soft and soft functions, but these are spin-independent and hence will not
modify the spin universality of the matching coefficient. These rapidity
factors and soft functions cancel out in ratios.

In the situation with infinite staple lengths, an analysis of this type that
uses auxiliary fields was carried out recently in Ref. Vladimirov:2020ofp ,
from which we have drawn inspiration. The goal in that work was to derive the
quasi-TMDPDF to TMDPDF matching relation. Ref. Vladimirov:2020ofp  uses SCET
in a somewhat different fashion than what we envisioned above because they
consider the analogy of the quasi-TMDPDF with a TMD hadronic tensor, and
perform a match up which simultaneously yields $n$-collinear,
${\bar{n}}$-collinear, and soft fields. It is known that a quasi soft function
is needed as part of the definition of the quasi-TMDPDF in order to properly
carry out the quasi-TMDPDF to TMDPDF matching Ji:2014hxa ; Ji:2018hvs ;
Ebert:2019okf , and it is so far not clear how the quasi soft function is
treated by the analysis in Ref. Vladimirov:2020ofp , which is also the case
for our outline above.

##  4 Applications

We now discuss applications of our main finding in eq. (3.4). We find that the
ratios of spin-dependent TMDs and the unpolarized TMD can be directly obtained
from those of the quasi-TMDs, i.e.

$\displaystyle\frac{g_{1L}(x,b_{T},\mu,\zeta)}{f_{1}(x,b_{T},\mu,\zeta)}$
$\displaystyle=\frac{\tilde{g}_{1L}(x,b_{T},\mu,P^{z})}{\tilde{f}_{1}(x,b_{T},\mu,P^{z})}\,,$
$\displaystyle\frac{h_{1}(x,b_{T},\mu,\zeta)}{f_{1}(x,b_{T},\mu,\zeta)}$
$\displaystyle=\frac{\tilde{h}_{1}(x,b_{T},\mu,P^{z})}{\tilde{f}_{1}(x,b_{T},\mu,P^{z})}\,,$
$\displaystyle\frac{h_{1T}^{\perp}(x,b_{T},\mu,\zeta)}{f_{1}(x,b_{T},\mu,\zeta)}$
$\displaystyle=\frac{\tilde{h}_{1T}^{\perp}(x,b_{T},\mu,P^{z})}{\tilde{f}_{1}(x,b_{T},\mu,P^{z})}\,.$
(71)

In these ratios the matching coefficients drop out along with the
nonperturbative soft contributions in the function $g_{q}^{S}$ and the
Collins-Soper evolution factor. The anomalous dimensions for the $\mu$\- and
Collins-Soper evolutions are the same for the TMDs here, so the ratios on both
the left- and the right-hand sides are only dependent on $x$ and $b_{T}$.
These relations have power corrections that are suppressed by
$1/(P^{z}b_{T})$, so one can calculate the ratios on the r.h.s. in lattice QCD
with different hadron momenta and interpolate to $P^{z}\to\infty$ to obtain
the final result.

In addition, we can consider the ratios of the $x$-integrated TMDs which were
studied in a different formalism based on exploiting Lorentz invariance in
Refs. Musch:2010ka ; Musch:2011er ; Engelhardt:2015xja ; Yoon:2016dyh ;
Yoon:2017qzo . According to eq. (2.2),

$\displaystyle\int_{-1}^{1}\\!\mathrm{d}x\,\tilde{f}_{q/h_{S}}^{[\tilde{\Gamma}]}(x,\vec{b}_{T},\mu,P^{z})$
$\displaystyle=\tilde{Z}^{\prime}_{q}(0,\mu,\tilde{\mu})\tilde{Z}_{\rm
uv}^{q}(0,\tilde{\mu},a)\tilde{\Delta}_{S}^{q}(b_{T},a,L)\tilde{B}_{q/h_{S}}^{[\tilde{\Gamma}]}(b^{z}=0,\vec{b}_{T},a,L,P^{z})\,,$
(72)

where $\tilde{B}_{q/h_{S}}^{[\tilde{\Gamma}]}(b^{z}=0,\vec{b}_{T},a,L,P^{z})$
is the bare quasi beam function. The r.h.s. of eq. (72) is finite, so the
$x$-integration of the quasi-TMD is convergent. According to eq. (2.3), we
have

$\displaystyle\frac{\int\\!\mathrm{d}x\,\tilde{F}_{{\rm
ns}/h_{S}}(x,b_{T},\mu,P^{z})}{\int\\!\mathrm{d}x\,\tilde{f}_{1}(x,b_{T},\mu,P^{z})}$
$\displaystyle=\frac{\tilde{B}_{{\rm
ns}/h_{S}}(0,b_{T},a,L,P^{z})}{\tilde{B}_{\rm ns}(0,b_{T},a,L,P^{z})}$
$\displaystyle=\frac{\int\\!\mathrm{d}x\,C_{{\rm
ns}}\bigl{(}\mu,xP^{z}\bigr{)}\>F_{{\rm
ns}/h_{S}}\big{(}x,b_{T},\mu,\zeta=(2xP^{z})^{2}\big{)}}{\int\\!\mathrm{d}x\,C_{{\rm
ns}}\bigl{(}\mu,xP^{z}\bigr{)}\>f_{1}\big{(}x,b_{T},\mu,\zeta=(2xP^{z})^{2}\big{)}}\,.$
(73)

Thus we see that this ratio of $x$-integrated quasi-TMDPDF is not directly
related to the ratio of $x$-integrated TMDs $F_{{\rm ns}/h_{S}}$ and $f_{1}$
in the formalism used here.

##  5 Conclusion

This paper constructs spin-dependent quasi-TMDPDFs that have a straightforward
implementation in lattice calculations. We study the relationship between
these spin-dependent quasi-TMDPDFs and their corresponding TMDPDFs at next-to-
leading order. The nonperturbative soft factor and Collins-Soper evolution
cancel in the ratio of polarized and unpolarized quasi-TMDs, and in the ratio
of the corresponding TMDs. This leads to a simple relationship between the
quasi-TMD and TMD ratios involving only a perturbative matching kernel, see
eq. (24). We calculate these kernels explicitly at one-loop in Sec. 3.

At one-loop order in massless perturbation theory, we have access to the
unpolarized ($f_{1}$), helicity ($g_{1L}$), and transversity ($h_{1}$)
structures, which remarkably have the same perturbative kernel as the
unpolarized case $C_{{\rm ns}}^{f_{1}}=C_{\rm ns}^{g_{1L}}=C_{\rm
ns}^{h_{1}}$. A key reason for this simple result is that all diagrams with
nontrivial $x$ dependence must cancel (up to power corrections) between the
perturbative results of quasi-TMDs and TMDs to give rise to an $x$-independent
matching, as required by consistency Ebert:2019okf . The remaining diagrams
only contain the tree-level Dirac structure (which cancels out in suitable
ratios) because the kinematic structures of all diagrams involving Wilson
lines are independent of spin, up to power corrections.

An interesting feature of the pretzelosity is that its matrix element first
appears at one-loop order, but vanishes in dimensional regularization when
$d-4=-2\epsilon=0$. For $\epsilon\neq 0$ we can carry out matching between the
pretzelosity and quasi-pretzelosity, and they again satisfy the same relation
as the aforementioned spin structures, yielding a simple matching relation at
tree-level. The remaining spin structures, namely the Sivers function
($f_{1T}^{\perp}$), Boer-Mulders function ($h_{1}^{\perp}$) and worm-gear
functions ($g_{1T}$, $h_{1L}^{\perp}$), vanish in massless perturbation theory
and are thus beyond the scope of this paper.

We also consider a definition of the quasi beam function with a staple-shaped
Wilson line whose length remains constant under a Fourier transform. This
avoids the necessity of a $b^{z}$-dependent counterterm to cancel Wilson line
self energies, and thus simplifies one aspect of lattice calculations of
quasi-TMDPDF ratios. We discuss implications of this definition for the
determination of the Collins-Soper kernel in appendix A.

Finally, in Sec. 3.5 we outline a procedure for generalizing the universality
of the spin-dependent matching coefficients to all orders in perturbation
theory. We intend to carry out a more detailed analysis of this procedure in
future work, as well as to obtain matching relations for the Sivers, Boer-
Mulders, and worm-gear functions.

###### Acknowledgements.

We thank Phiala Shanahan and Michael Wagman for useful discussions. This work
was supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, from DE-SC0011090, DE-SC0012704 and within the framework of
the TMD Topical Collaboration. I.S. was also supported in part by the Simons
Foundation through the Investigator grant 327942. M.E. was also supported by
the Alexander von Humboldt Foundation through a Feodor Lynen Research
Fellowship. S.T.S. was partially supported by the U.S. National Science
Foundation through a Graduate Research Fellowship.

##  Appendix A Alternative construction of the quasi beam function

In Sec. 2.2, we mentioned that the definition of the quasi beam function in
eq. (2.2) has the nice feature that its renormalization factor is independent
of $b^{z}$, which drops out in the ratios of quasi-TMDPDFs, simplifying the
lattice calculations. In this appendix we contrast the definitions in eq. (15)
and eq. (2.2) at a more technical level.

Take the Collins-Soper evolution kernel as an example. According to Refs.
Ebert:2018gzl ; Ebert:2019tvc ; Shanahan:2020zxr , we can extract the
nonperturbative Collins-Soper evolution kernel for TMDPDFs from the ratio of
quasi-TMDPDFs at different hadron momenta for the definition in eq. (15),

$\displaystyle\gamma^{q}_{\zeta}(\mu,b_{T})=\frac{1}{\ln(P^{z}_{1}/P^{z}_{2})}$
(74) $\displaystyle\times\ln\frac{C_{\rm
ns}(\mu,xP_{2}^{z})\,\int\\!\mathrm{d}b^{z}\,e^{ib^{z}xP_{1}^{z}}\,\tilde{Z}_{q}^{\prime}(b^{z},\mu,\tilde{\mu})\,\tilde{Z}_{\rm
uv}^{q}(b^{z},\tilde{\mu},a)\,{\tilde{R}}_{B}(b_{T},\tilde{\mu},a,L)\,\tilde{B}_{\rm
ns}(b^{z},\vec{b}_{T},a,P_{1}^{z},L)}{C_{\rm
ns}(\mu,xP_{1}^{z})\,\int\\!\mathrm{d}b^{z}\,e^{ib^{z}xP_{2}^{z}}\,\tilde{Z}_{q}^{\prime}(b^{z},\mu,\tilde{\mu})\,\tilde{Z}_{\rm
uv}^{q}(b^{z},\tilde{\mu},a)\,{\tilde{R}}_{B}(b_{T},\tilde{\mu},a,L)\,\tilde{B}_{\rm
ns}(b^{z},\vec{b}_{T},a,P_{2}^{z},L)}\,,$

where the quasi soft factor $\tilde{\Delta}_{S}^{q}$ cancels out, and the
$b^{z}$-independent factor $\tilde{R}_{B}$ is constructed such that it exactly
removes all divergences that would normally be canceled by
$\tilde{\Delta}_{S}^{q}(b_{T},a,L)$ Ebert:2019tvc , _i.e._ all power-law
divergences not yet absorbed by $\tilde{Z}_{\rm uv}^{q}(b^{z},\tilde{\mu},a)$.
In this way, we can take the $L\to\infty$ and $a\to 0$ limits before forming
the ratio.

If we instead use the alternative definition in eq. (2.2), both
$\tilde{Z}_{\rm uv}^{q}(\tilde{\mu},a)$ and
$\tilde{Z}_{q}^{\prime}(\mu,\tilde{\mu})$ also drop out in the ratios of the
quasi-TMDPDFs. Therefore, if one loosens the requirement that the $L\to\infty$
and $a\to 0$ limits be taken first, the Collins-Soper kernel can be obtained
from the ratio of bare quasi beam functions on the lattice,

$\displaystyle\gamma^{q}_{\zeta}(\mu,b_{T})$
$\displaystyle=\frac{1}{\ln(P^{z}_{1}/P^{z}_{2})}\ln\left[\lim_{\begin{subarray}{c}L\to\infty\\\
a\to 0\end{subarray}}\frac{C_{\rm
ns}(\mu,xP_{2}^{z})\,\int\\!\mathrm{d}b^{z}\,e^{ib^{z}xP_{1}^{z}}\,\tilde{B}_{{\rm
ns}}(b^{z},\vec{b}_{T},a,P_{1}^{z},L)}{C_{\rm
ns}(\mu,xP_{1}^{z})\,\int\\!\mathrm{d}b^{z}\,e^{ib^{z}xP_{2}^{z}}\,\tilde{B}_{{\rm
ns}}(b^{z},\vec{b}_{T},a,P_{2}^{z},L)}\right]\,.$ (75)

If one wants to take the $L\to\infty$ and $a\to 0$ limits before forming the
ratio, then $\tilde{Z}_{\rm uv}^{q}(\tilde{\mu},a)$ and
$\tilde{Z}_{q}^{\prime}(\mu,\tilde{\mu})$ can be included and chosen as
$\tilde{Z}_{\rm uv}^{q}(b^{z}=0,\tilde{\mu},a)$ and
$\tilde{Z}_{q}^{\prime}(b^{z}=0,\mu,\tilde{\mu})$ which have already been
studied in the RI/MOM scheme for the quasi beam function in eq. (14)
Constantinou:2019vyb ; Ebert:2019tvc ; Shanahan:2019zcq . Moreover, to exploit
the $b^{z}$-independence, we might as well divide the bare quasi beam
functions by $\tilde{B}_{{\rm ns}}(b^{z}=0,\vec{b}_{T},a,P^{z},L)$, i.e.,

$\displaystyle\gamma^{q}_{\zeta}(\mu,b_{T})=\frac{1}{\ln(P^{z}_{1}/P^{z}_{2})}$
(76) $\displaystyle\qquad\times\ln\frac{C_{\rm
ns}(\mu,xP_{2}^{z})\,\lim_{\begin{subarray}{c}L\to\infty\\\ a\to
0\end{subarray}}\int\\!\mathrm{d}b^{z}\,e^{ib^{z}xP_{1}^{z}}\,\tilde{B}_{{\rm
ns}}(b^{z},\vec{b}_{T},a,P_{1}^{z},L)\big{/}\tilde{B}_{{\rm
ns}}(0,\vec{b}_{T},a,P^{z},L)}{C_{\rm
ns}(\mu,xP_{1}^{z})\,\lim_{\begin{subarray}{c}L\to\infty\\\ a\to
0\end{subarray}}\int\\!\mathrm{d}b^{z}\,e^{ib^{z}xP_{2}^{z}}\,\tilde{B}_{{\rm
ns}}(b^{z},\vec{b}_{T},a,P_{2}^{z},L)\big{/}\tilde{B}_{{\rm
ns}}(0,\vec{b}_{T},a,P^{z},L)}\,.$

This has the advantage that the errors at $b^{z}=0$ and $b^{z}\neq 0$ are
correlated and can be reduced by the division. Here $P^{z}$ can take on any
value, and to reduce the errors one can simply choose $P^{z}=0$.

As a word of caution, we note that the discussion above did not take into
account the mixing of $\tilde{B}_{{\rm ns}/h_{S}}^{\tilde{\Gamma}}$ with other
Dirac structures on the lattice Constantinou:2019vyb ; Shanahan:2019zcq ;
Green:2020xco . According to the RI/MOM analysis in Ref. Shanahan:2019zcq ,
mixing effects can become considerable for certain lattice ensembles which are
$b^{z}$ dependent. Therefore, it may still be inevitable that one must carry
out the procedure of diagonalizing the renormalization matrix and converting
to the ${\overline{\mathrm{MS}}}$ scheme as in Refs. Shanahan:2019zcq ;
Shanahan:2020zxr  to extract the Collins-Soper kernel or other types of
ratios. Nevertheless, if the lattice parameters could be fine tuned to make
the operator mixing effect less important than other systematic errors, then
we can exploit the above simplification.

##  Appendix B Independent calculation of the spin-dependent quasi-TMDPDFs

In this section we perform an independent calculation of the one-loop quasi-
TMDPDFs by postponing the approximations used in eqs. (55) and (56) after the
loop integration. Our strategy is to express the spin-dependent quasi-TMDPDFs
as a linear combination of the unpolarized one and novel structures, the
latter of which turn out to be two terms.

###  B.1 Vertex diagram

In this subsection, we explicitly evaluate the vertex diagram for all relevant
spin structures. Eq. (3.3) gives the exact result for the vertex diagram as

$\displaystyle\tilde{q}_{A}^{\tilde{\Gamma}\,(1)}(b,p)$
$\displaystyle=\mathrm{Tr}\Bigl{[}\bar{u}_{s}(p)\gamma^{\mu}\gamma_{\alpha}\frac{\tilde{\Gamma}}{2}\gamma_{\beta}\gamma_{\mu}u_{s}(p)\Bigr{]}\frac{\alpha_{s}C_{F}}{4\pi}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}e^{\epsilon
L_{b}}$
$\displaystyle\quad\times\biggl{(}\frac{g_{\perp}^{\alpha\beta}}{2}+\epsilon\frac{b^{\alpha}b^{\beta}}{b^{2}}\biggr{)}\biggl{(}\frac{-b^{2}}{b_{T}^{2}}\biggr{)}^{\epsilon}\frac{1+\mathrm{i}{p{\cdot}b}-e^{\mathrm{i}{p{\cdot}b}}}{({p{\cdot}b})^{2}}\,.$
(77)

Using the completeness relation for polarized spinors given in eq. (30), the
Dirac traces for the different choices in eq. (17) yield

$\displaystyle\tilde{\Gamma}=\gamma^{\lambda}:\quad$
$\displaystyle\mathrm{Tr}\biggl{[}\bar{u}_{s}(p)\gamma^{\mu}\gamma_{\alpha}\frac{\tilde{\Gamma}}{2}\gamma_{\beta}\gamma_{\mu}u_{s}(p)\biggr{]}$
$\displaystyle=(d-2)\bigl{(}p^{\lambda}g_{\alpha\beta}-p_{\beta}g_{\alpha}^{\lambda}-p_{\alpha}g_{\beta}^{\lambda}\bigr{)}\,,$
$\displaystyle\tilde{\Gamma}=\gamma^{\lambda}\gamma_{5}:\quad$
$\displaystyle\mathrm{Tr}\biggl{[}\bar{u}_{s}(p)\gamma^{\mu}\gamma_{\alpha}\frac{\tilde{\Gamma}}{2}\gamma_{\beta}\gamma_{\mu}u_{s}(p)\biggr{]}$
$\displaystyle=\Lambda(d-2)\bigl{(}p^{\lambda}g_{\alpha\beta}-p_{\beta}g_{\alpha}^{\lambda}-p_{\alpha}g_{\beta}^{\lambda}\bigr{)}\,,$
$\displaystyle\tilde{\Gamma}=\mathrm{i}\sigma^{\sigma\lambda}\gamma_{5}:\quad$
$\displaystyle\mathrm{Tr}\biggl{[}\bar{u}_{s}(p)\gamma^{\mu}\gamma_{\alpha}\frac{\tilde{\Gamma}}{2}\gamma_{\beta}\gamma_{\mu}u_{s}(p)\biggr{]}$
$\displaystyle=(d-4)\Big{[}s^{\sigma}_{\perp}\bigl{(}p^{\lambda}g_{\alpha\beta}-p_{\beta}g_{\alpha}^{\lambda}-p_{\alpha}g_{\beta}^{\lambda}\bigr{)}$
$\displaystyle\hskip
51.21504pt-p^{\lambda}\bigl{(}s_{\perp\alpha}g^{\sigma}_{\beta}+s_{\perp\beta}g^{\sigma}_{\alpha}\bigr{)}\Bigr{]}\,.$
(78)

The first two traces are equivalent; thus, the unpolarized and helicity quasi-
TMDs are identical and are given by

$\displaystyle\tilde{q}_{A}^{\gamma^{\lambda}\,(1)}(b,p)$
$\displaystyle=\tilde{q}_{A}^{\gamma^{\lambda}\gamma_{5}\,(1)}(b,p)$
$\displaystyle=\frac{\alpha_{s}C_{F}}{4\pi}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}e^{\epsilon
L_{b}}(d-2)\left(p^{\lambda}-2\epsilon\frac{{p{\cdot}b}b^{\lambda}}{b^{2}}\right)\biggl{(}\frac{-b^{2}}{b_{T}^{2}}\biggr{)}^{\epsilon}\frac{1+\mathrm{i}{p{\cdot}b}-e^{\mathrm{i}{p{\cdot}b}}}{({p{\cdot}b})^{2}}\,.$
(79)

For $\lambda=t$, this reduces to the case already calculated in Ref.
Ebert:2019okf . The difference between $\lambda=t$ and $\lambda=z$ is given by
the second term in eq. (B.1), which is finite and can be evaluated after
setting $\epsilon=0$. Applying the Fourier transform, it evaluates to

$\displaystyle\tilde{q}_{A}^{\gamma^{t}\,(1)}(x,\vec{b}_{T})-\tilde{q}_{A}^{\gamma^{z}\,(1)}(x,\vec{b}_{T})$
$\displaystyle=\frac{\alpha_{s}C_{F}}{\pi}\frac{1}{p^{z}}\int\frac{\mathrm{d}b^{z}}{2\pi}e^{\mathrm{i}xb^{z}p^{z}}\frac{1-\mathrm{i}p^{z}b^{z}-e^{-\mathrm{i}p^{z}b^{z}}}{b_{T}^{2}+b_{z}^{2}}$
$\displaystyle=\frac{\alpha_{s}C_{F}}{2\pi}\frac{1}{b_{T}p^{z}}\Bigl{\\{}[1+b_{T}p^{z}{\rm
sgn}(x)]e^{-|x|b_{T}p^{z}}-e^{-|1-x|b_{T}p^{z}}\Bigr{\\}}\,.$ (80)

This term is exponentially suppressed for large $b_{T}p^{z}\gg 1$ and thus can
be neglected.

The last Dirac structure in eq. (B.1) consists of a piece proportional to the
Dirac structure of the unpolarized TMD and a genuinely new structure, which we
can evaluate for $\epsilon\to 0$ using eq. (50),

$\displaystyle\tilde{q}_{A}^{\mathrm{i}\sigma^{\sigma\lambda}\gamma_{5},(1)}(x,\vec{b}_{T})$
$\displaystyle=\frac{d-4}{d-2}s^{\sigma}_{\perp}\tilde{q}_{A}^{\gamma^{\lambda}\,(1)}(x,\vec{b}_{T})-2s_{\perp}^{\sigma}\frac{\alpha_{s}C_{F}}{4\pi}(1-x)\theta(x)\theta(1-x)=0\,.$
(81)

###  B.2 Sail diagram

The sail diagram is given in eq. (3.3) as

$\displaystyle\tilde{q}_{B}^{\tilde{\Gamma}\,(1)}(b)$
$\displaystyle=-\frac{\alpha_{s}C_{F}}{2\pi}\mathrm{Tr}\bigl{[}\bar{u}_{s}(p)\tilde{\Gamma}\gamma^{\rho}\gamma^{\mu}u_{s}(p)\bigr{]}\frac{\mu_{0}^{2\epsilon}}{(2\pi)^{d-2}}\int\\!\mathrm{d}^{d}k\int_{0}^{1}\mathrm{d}s\,\frac{\gamma^{\prime}(s)_{\mu}k_{\rho}}{k^{2}(p-k)^{2}}\,e^{\mathrm{i}p\cdot
b-\mathrm{i}(p-k)\cdot\gamma(s)}$
$\displaystyle\quad-\frac{\alpha_{s}C_{F}}{2\pi}\mathrm{Tr}\bigl{[}\bar{u}_{s}(p)\gamma^{\mu}\gamma^{\rho}\tilde{\Gamma}u_{s}(p)\bigr{]}\frac{\mu_{0}^{2\epsilon}}{(2\pi)^{d-2}}\int\\!\mathrm{d}^{d}k\int_{0}^{1}\mathrm{d}s\,\frac{\gamma^{\prime}(s)_{\mu}k_{\rho}}{k^{2}(p-k)^{2}}\,e^{\mathrm{i}k\cdot
b+\mathrm{i}(p-k)\cdot\gamma(s)}\,.$ (82)

Evaluating the Dirac trace for the different Dirac structures given in eq.
(17), we have

$\displaystyle\tilde{\Gamma}=\gamma^{\lambda}:\quad$
$\displaystyle\mathrm{Tr}\bigl{[}\bar{u}_{s}(p)\tilde{\Gamma}\gamma^{\rho}\gamma^{\mu}u_{s}(p)\bigr{]}$
$\displaystyle=p^{\mu}g^{\rho\lambda}-p^{\rho}g^{\lambda\mu}+p^{\lambda}g^{\rho\mu}\,,$
(83) $\displaystyle\tilde{\Gamma}=\gamma^{\lambda}\gamma_{5}:\quad$
$\displaystyle\mathrm{Tr}\bigl{[}\bar{u}_{s}(p)\tilde{\Gamma}\gamma^{\rho}\gamma^{\mu}u_{s}(p)\bigr{]}$
$\displaystyle=\Lambda\bigl{(}p^{\mu}g^{\rho\lambda}-p^{\rho}g^{\lambda\mu}+p^{\lambda}g^{\rho\mu}\bigr{)}\,,$
$\displaystyle\tilde{\Gamma}=\mathrm{i}\sigma^{\sigma\lambda}\gamma_{5}:\quad$
$\displaystyle\mathrm{Tr}\bigl{[}\bar{u}_{s}(p)\tilde{\Gamma}\gamma^{\rho}\gamma^{\mu}u_{s}(p)\bigr{]}$
$\displaystyle=s_{\perp}^{\sigma}\bigl{(}p^{\mu}g^{\rho\lambda}-p^{\rho}g^{\lambda\mu}+p^{\lambda}g^{\rho\mu}\bigr{)}-p^{\lambda}\bigl{(}s^{\rho}_{\perp}g^{\mu\sigma}+s^{\mu}_{\perp}g^{\rho\sigma}\bigr{)}\,.$

We obtain the same results for the trace in the second line of eq. (B.2). The
first two Dirac structures in eq. (83) yield the same traces, and thus the
vertex diagram results for the unpolarized and helicity structures are
identical. It is also easy to see from the structure of the integrand that
$\lambda=t$ and $\lambda=z$ yield identical results, as also discussed in Ref.
Ebert:2019okf .

Using eq. (83), we can relate the transversity to the unpolarized TMD by

$\displaystyle\left[s_{\perp}^{\sigma}\tilde{h}_{1}^{\lambda(1)}(x,\vec{b}_{T})+\biggl{(}\frac{g_{\perp}^{\sigma\rho}}{2}-\frac{b_{\perp}^{\sigma}b_{\perp}^{\rho}}{b_{\perp}^{2}}\biggr{)}s_{\perp\,\rho}\tilde{h}_{1T}^{\lambda\perp(1)}(x,\vec{b}_{T})\right]_{\rm
s}=s_{\perp}^{\sigma}\tilde{f}_{1,\rm
s}^{\lambda(1)}(x,\vec{b}_{T})+\Delta\tilde{h}_{\rm
s}^{\lambda\sigma}(x,\vec{b}_{T})\,,$ (84)

where prior to the Fourier transform

$\displaystyle\Delta\tilde{h}_{\rm s}^{\lambda\sigma}(p,b)$
$\displaystyle=\frac{\alpha_{s}C_{F}}{2\pi}p^{\lambda}\bigl{(}s^{\rho}_{\perp}g^{\mu\sigma}+s^{\mu}_{\perp}g^{\rho\sigma}\bigr{)}\int_{0}^{1}\mathrm{d}s\,\gamma^{\prime}(s)_{\mu}$
$\displaystyle\quad\times\frac{\mu_{0}^{2\epsilon}}{(2\pi)^{d-2}}\int\\!\mathrm{d}^{d}k\frac{k_{\rho}}{k^{2}(p-k)^{2}}\,\Bigl{[}e^{\mathrm{i}p\cdot[b-\gamma(s)]}e^{\mathrm{i}k\cdot\gamma(s)}+e^{\mathrm{i}k\cdot[b-\gamma(s)]}e^{\mathrm{i}p\cdot\gamma(s)}\Bigr{]}\,.$
(85)

The two integrals over $\mathrm{d}^{d}k$ can be evaluated using

$\displaystyle\frac{\mu_{0}^{2\epsilon}}{(2\pi)^{d-2}}\int\\!\mathrm{d}^{d}k\frac{k_{\rho}e^{\mathrm{i}k\cdot
x}}{k^{2}(p-k)^{2}}$
$\displaystyle=\frac{\mathrm{i}}{4}\frac{\Gamma(-\epsilon)}{e^{\epsilon\gamma_{E}}}\left(\frac{-\mu^{2}x^{2}}{b_{0}^{2}}\right)^{\epsilon}\left[\frac{2\epsilon
x_{\rho}}{x^{2}}\frac{1-e^{\mathrm{i}p\cdot x}}{p\cdot
x}+p_{\rho}\frac{e^{\mathrm{i}p\cdot x}(1-\mathrm{i}p\cdot x)-1}{(p\cdot
x)^{2}}\right]\,.$ (86)

Since the spin structure in eq. (B.2) is purely transverse, we have
$p_{\rho}=0$, and thus only the first structure in eq. (86) contributes. We
then take $\epsilon\to 0$, which yields

$\displaystyle\Delta\tilde{h}_{\rm s}^{\lambda\sigma}(p,b)$
$\displaystyle=-\frac{\mathrm{i}}{2}\frac{\alpha_{s}C_{F}}{2\pi}p^{\lambda}\bigl{(}s^{\rho}_{\perp}g^{\mu\sigma}+s^{\mu}_{\perp}g^{\rho\sigma}\bigr{)}e^{\mathrm{i}p\cdot
b}\int_{0}^{1}\mathrm{d}s\,\gamma^{\prime}(s)_{\mu}$
$\displaystyle\quad\times\biggl{\\{}\frac{\gamma_{\rho}(s)}{\gamma^{2}(s)}\frac{e^{-\mathrm{i}p\cdot\gamma(s)}-1}{p\cdot\gamma(s)}+\frac{[b-\gamma(s)]_{\rho}}{[b-\gamma(s)]^{2}}\frac{e^{\mathrm{i}p\cdot[\gamma(s)-b]}-1}{p\cdot[b-\gamma(s)]}\biggr{\\}}\,.$
(87)

Since $\gamma^{\prime}_{\mu}$ contracts with a transverse vector, only the
perpendicular Wilson line segment contributes; its parametrization takes the
form

$\displaystyle\gamma(s)^{\mu}=(0,s\vec{b}_{T},L)\,.$ (88)

Again using that both $\rho$ and $\sigma$ are transverse indices, we obtain

$\displaystyle\Delta\tilde{h}_{\rm s}^{\lambda\sigma}(p,b)$
$\displaystyle=-\mathrm{i}\frac{\alpha_{s}C_{F}}{2\pi}p^{\lambda}(s_{\perp}{\cdot}b_{\perp})b_{\perp}^{\sigma}e^{\mathrm{i}p\cdot
b}\int_{0}^{1}\mathrm{d}s\,\biggl{[}\frac{s}{s^{2}b_{\perp}^{2}-L^{2}}\frac{1-e^{\mathrm{i}p^{z}L}}{p^{z}L}+(L\to
b^{z}-L)\biggr{]}$
$\displaystyle=-\frac{\mathrm{i}}{2}\frac{\alpha_{s}C_{F}}{2\pi}p^{\lambda}\frac{(s_{\perp}{\cdot}b_{\perp})b_{\perp}^{\sigma}}{b_{\perp}^{2}}e^{\mathrm{i}p\cdot
b}\biggl{[}\ln\Bigl{(}1-\frac{b_{\perp}^{2}}{L^{2}}\Bigr{)}\frac{1-e^{\mathrm{i}p^{z}L}}{p^{z}L}+(L\to
b^{z}-L)\biggr{]}\,.$ (89)

Even prior to evaluating the Fourier transform, it is clear that this
expression vanishes in the limit of large $p^{z}L,L/b_{T}\gg 1$. Thus, in
conclusion, we have that the sail diagram yields

$\displaystyle\tilde{f}_{1,\rm s}^{\lambda(1)}(x,b_{T})=\tilde{g}_{1L,\rm
s}^{\lambda(1)}(x,b_{T})=\tilde{h}_{1,\rm s}^{\lambda(1)}(x,b_{T})\,,$ (90)

while all other structures vanish.

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