B\sqrt{S}$. Here we have a $\sqrt{S}$ as opposed to the $1/\sqrt{S}$ that is
more commonly encountered in the literature. (This is because the precise form
of this relation depends on the nature of the soft zero-bin subtractions,
which in our notation are contained with $B$ itself. If we define $B^{(u)}$ as
the correlator without soft zero-bin subtractions, then one has $f\sim
B^{(u)}\sqrt{S}/S$ in our setup, for many choices of rapidity regulators. In
SCET, this process of absorbing the (physical) soft function and subtracting
the zero-bin region are commonly performed separately.) While it seems that
our result for the absorption of $S$ agrees with ref. [96], it should be noted
that the soft effects under discussion arise from different momentum regions,
and that we have done a much more extensive analysis of all possible NLP soft
effects to arrive at this conclusion. In particular, ref. [96] argued that a
derivative of a soft Wilson line or the addition of an extra soft gluon leads
to an additional power suppression, whereas we find that such terms in fact
already enter at NLP at the operator level, and could only be shown to vanish
when factorized for the SIDIS process at NLP.

Finally we remark that ref. [96] has carried out a complete NLO calculation
for their analog of our Wilson coefficient $C^{(1)}$, which is given by the
formula for $C_{2}(x_{1,2})$ in their Eq. (6.15). This is a very useful result
for applications of the NLP factorization formula for SIDIS beyond LO.
Furthermore, the explicit calculations done in ref. [96] confirm at one-loop
that the same rapidity anomalous dimension appears as at LP, which is
consistent with the appearance of only the LP soft function.

##  6 Conclusion

We have studied the factorization of azimuthal asymmetries in the SIDIS
process that first appear at subleading power, i.e. are suppressed by
$\mathcal{O}(q_{T}/Q)$ relative to the leading processes. Working under the
assumption that leading power Glauber interactions do not spoil factorization
at NLP, we have used the Soft Collinear Effective Theory to derive the form
that an all orders in $\alpha_{s}$ factorization formulae for power suppressed
hard scattering effects in SIDIS must take. In addition to deriving and
constraining all necessary operators needed for our analysis, we present final
explicit factorization formulae for all SIDIS structure functions at NLP in
sections 5.2.4 and 5.2.5, including the full spin dependence of the target
nucleon.

A summary of the main results achieved by our analysis is as follows

  * •

A derivation of a minimal basis of all ${\rm SCET}_{\rm II}$ hard-scattering
operators that enter for observables with two collinear directions at NLP.
This included matching contributions from both the hard and hard-collinear
regions of momentum space.

  * •

A demonstration that subleading power dynamic ${\rm SCET}_{\rm II}$
Lagrangians do not contribute to SIDIS at NLP.

  * •

At NLP there are subleading hard ${\rm SCET}_{\rm II}$ operators involving
soft gluon fields and soft derivatives. In all cases, we find that these soft
contributions vanish for the NLP structure functions under consideration. In
the most difficult case to rule out, involving the
$\mathcal{B}_{s\perp}^{(n_{i})\mu}$ operator, this followed after
factorization of terms into an NLP vacuum soft matrix elements, which could
then be argued to vanish based in part on the charge conjugation and parity
invariance of the QCD vacuum. Our analysis shows that new soft contributions
will appear at NNLP and beyond. We note that such soft contributions have not
been considered in recent independent approaches to subleading power
factorization of TMDs [95, 96].

  * •

At NLP we demonstrated that all soft effects are given by the same leading
power soft function, $S(b_{T})$, and that in all cases it can be absorbed into
the leading and subleading power TMDs.

  * •

There are three non-zero sources of power corrections to SIDIS at NLP, namely
kinematic corrections from subleading terms in the projectors $P_{i}^{\mu\nu}$
which determine structure functions $W_{i}=P_{i}^{\mu\nu}W_{\mu\nu}$, and hard
scattering operators with $\mathcal{P}_{\perp}$ and $\mathcal{B}_{n_{i}\perp}$
insertions. The first two can be fully expressed in terms of the LP TMDs,
while the $\mathcal{B}_{n_{i}\perp}$ operators give rise to quark-gluon-quark
correlators and thus new TMDs that do not appear at LP. These correlators
dependend on variable in the following form:
$\tilde{B}_{\mathcal{B}\,f/N}^{\rho}(x,\xi,\vec{b}_{T},\mu,\zeta_{a})$ for TMD
PDFs and
$\tilde{\mathcal{G}}_{\mathcal{B}\,h/f}^{\rho}(z,\xi,\vec{b}_{T},\mu,\zeta_{b})$
for TMD FFs. In both cases they depend on two longitudinal momentum fractions,
with the appearance of a new variable $\xi$ determining the split of large
momentum between gluon and quark fields that appear in the same collinear
direction at the hard scale.

  * •

The factorization formulae for all spin polarized NLP structure functions is
proven to have only two independent hard functions, the same function
$\mathcal{H}^{(0)}(q^{+}q^{-},\mu)$ that already appears at LP, and a single
new function $\mathcal{H}^{(1)}(q^{+}q^{-},\xi,\mu)$ which appears along with
the quark-gluon-quark correlators that depend on this same variable $\xi$.
These hard functions encode perturbative $\alpha_{s}$ corrections, and also
determine the structure of large logarithms from the invariant mass
renormalization group evolution in $\mu$ for the NLP structure functions.

  * •

The structure of our factorization formula have important implications for the
resummation of large logarithms, $\ln(q_{T}/Q)$ for SIDIS at NLP. In
particular, we have demonstrated that the same rapidity anomalous dimension
(Collins-Soper equation) governs the evolution in rapidity for all NLP terms.
For the evolution in invariant mass governed by $\mu$, some terms inherit the
known resummation from LP since they involve the same $\mathcal{H}^{(0)}$,
while there is a new type of evolution for the terms involving
$\mathcal{H}^{(1)}$. Results from the SCET literature [70, 75] imply that the
evolution equation for $\mathcal{H}^{(1)}$ is known at NLL order.

  * •

At tree level $\mathcal{H}^{(1)}$ is independent of $\xi$, and integrating
over $\xi$ in the quark-gluon-quark correlators
$\tilde{B}_{\mathcal{B}\,f/N}^{\rho}$ and
$\tilde{\mathcal{G}}_{\mathcal{B}\,h/f}^{\rho}$ gives rise to quark-gluon-
quark correlators of this form that were studied in previous literature.
Accounting for the required absorption of the LP soft function, we find that
at tree level our results for the NLP structure functions reduce to exactly
the momentum space results given by the parton based analysis of ref. [21].
The non-trivial integral over $\xi$ first starts to contribute at NLO.

  * •

We have presented complete factorization formula for the structure functions
$W_{1,2,5,6}^{X}$, where $X$ denotes possible spin polarizations. These
results are given in transverse position space in section 5.2.4 and in
transverse momentum space in section 5.2.5.

An important future direction is to fully demonstrate (or not) whether the
leading power Glauber effects from the SCET Lagrangian ${\cal L}_{G}^{(0)}$,
which potentially spoil factorization, can be eliminated for the NLP SIDIS
analysis in the manner we have assumed. At LP proving the absence of
factorization violating Glauber effects in SIDIS is much simpler than in
Drell-Yan [27], giving us some hope that these effects will indeed not spoil
factorization at NLP. Another indication that points in a positive direction
for a complete factorization proof for SIDIS at NLP, is the cancellation of
various NLP soft contributions demonstrated here. In the CSS approach the soft
and Glauber effects are kept combined until a later step in the analysis than
in SCET, which may indicate that the absence of soft effects other than
through the LP soft function, also indicates that the Glauber contributions at
NLP will follow the same pattern for SIDIS as they do at LP. This may imply
that the same type of steps used to demonstrate the absence of factorization
violating Glauber effects in SIDIS at LP, will also work at NLP for the terms
considered here. We leave such investigations to future work.

Another interesting application of our results is to utilize the factorization
formula for the azimuthal asymmetry structure functions, $W_{1,2,5,6}^{X}$, to
carry out phenomenological analyses beyond LO. Combining our results with what
is known in the literature, such analyses are now possible at LL and NLL
orders.

It should also be evident that the same techniques used here can be adapted to
analyze factorization for NLP corrections for spin-polarized TMD observables
in Drell-Yan and in $e^{+}e^{-}$ annihilation with fragmentation to almost
back-to-back hadrons.

## Acknowledgments

We thank Johannes Michel, Gherardo Vita, Leonard Gamberg, Aditya Pathak, and
Alexey Vladimirov for useful discussions. This work was supported by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, from DE-
SC0011090 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.

##  Appendix A Coordinate Systems

In our treatment of SIDIS, wee have considered various reference frames.
Physical observables are defined in the target rest frame following the Trento
conventions [100], while the hadronic tensor decomposition is more naturally
addressed in the hadronic frame, see also figure -1019 for an illustration.
The derivation of the factorization is carried using lightcone coordinates.
Here, we collect explicit expressions of the unit vectors defining these
different frames, as well as explicit parameterizations of all particle
momenta in these frames.

###  A.1 Rest Frame using Trento Conventions

We construct the target rest frame in terms of orthornormal unit vectors
$n_{i}^{\prime\,\mu}$ satisfying

$\displaystyle n_{t}^{\prime\,2}=1\,,\qquad
n_{x}^{\prime\,2}=n_{y}^{\prime\,2}=n_{z}^{\prime\,2}=-1\,,\qquad
n^{\prime}_{i}\cdot n^{\prime}_{j}=0\quad{\rm for}~{}i\neq j\,.$ (A.1)

Following the Trento conventions [100], the lepton plane coincides with the
$x-z$ plane, such that $q_{z}>0$ and $p_{\ell}$ and $p_{\ell^{\prime}}$ have
positive $x$-component. This uniquely fixes

$\displaystyle n_{t}^{\prime\,\mu}$
$\displaystyle=\frac{P_{N}^{\mu}}{M_{N}}\,,\qquad
n_{z}^{\prime\,\mu}=\frac{-2xP_{N}^{\mu}+\gamma^{2}q^{\mu}}{Q\gamma\sqrt{1+\gamma^{2}}}\,,\qquad
n_{y}^{\prime\,\mu}=\epsilon^{\mu\nu\rho\sigma}n_{t\,\nu}n_{x\,\rho}n_{z\,\sigma}\,,$
$\displaystyle n_{x}^{\prime\,\mu}$
$\displaystyle=\sqrt{\frac{2}{\epsilon}}\biggl{[}\sqrt{1-\epsilon}\frac{p_{\ell}^{\mu}}{Q}-x\sqrt{\frac{1+\epsilon}{1+\gamma^{2}}}\frac{P_{N}^{\mu}}{Q}-\frac{1}{2}\biggl{(}\sqrt{1-\epsilon}+\sqrt{\frac{1+\epsilon}{1+\gamma^{2}}}\biggr{)}\frac{q^{\mu}}{Q}\biggr{]}\,,$
(A.2)

which holds for arbitrary hadron masses, with mass corrections encoded in
$\gamma=2xM_{N}/Q$ and $\epsilon$ as defined in eqs. (2.9) and (2.19), as well
as

$\displaystyle\kappa=\frac{\sqrt{1-(m_{Th}\gamma)^{2}/(Qz)^{2}}}{\sqrt{1+\gamma^{2}}}\,,\qquad
m_{Th}^{2}=M_{h}^{2}+\vec{P}_{hT}^{2}\,.$ (A.3)

In this coordinate system, the lepton and hadron momenta are parameterized as

$\displaystyle p_{\ell}^{\mu}$
$\displaystyle=\frac{Q}{2\gamma}\Bigl{(}\frac{2}{y}\,,\,\sqrt{\frac{2\epsilon}{1-\epsilon}}\gamma\,,\,0\,,\,\sqrt{\frac{1+\epsilon}{1-\epsilon}}+\sqrt{1+\gamma^{2}}\Bigr{)}_{T}\,,$
$\displaystyle p_{\ell^{\prime}}^{\mu}$
$\displaystyle=\frac{Q}{2\gamma}\Bigl{(}\frac{2(1-y)}{y}\,,\,\sqrt{\frac{2\epsilon}{1-\epsilon}}\gamma\,,\,0\,,\,\sqrt{\frac{1+\epsilon}{1-\epsilon}}-\sqrt{1+\gamma^{2}}\Bigr{)}_{T}\,,$
$\displaystyle P_{N}^{\mu}$
$\displaystyle=M_{N}\bigl{(}1,0,0,0\bigr{)}_{T}\,,$ $\displaystyle
P_{h}^{\mu}$
$\displaystyle=\Bigl{(}\frac{Qz}{\gamma}\,,\,P_{hT}\cos\phi_{h}\,,\,P_{hT}\sin\phi_{h}\,,\,\frac{Qz}{\gamma}\kappa\sqrt{1+\gamma^{2}}\Bigr{)}_{T}\,,$
$\displaystyle q^{\mu}$
$\displaystyle=\frac{Q}{\gamma}\Bigl{(}1,0,0,\sqrt{1+\gamma^{2}}\Bigr{)}_{T}\,,$
(A.4)

where the subscript $T$ signals using the rest frame in the Trento
conventions.

###  A.2 Hadronic Breit Frame

The hadronic Breit frame is defined in terms of orthornormal unit vectors
$\tilde{n}_{i}^{\mu}$ obeying eq. (A.1). As defined in section 2.2 and
illustrated in figure -1019, the frame is constructed such that
$q^{\mu}=(0,0,0,-Q)$ and $\vec{P}_{hT}$ is aligned along the $x$ axis. This
uniquely fixes

$\displaystyle\tilde{n}_{t}^{\mu}$
$\displaystyle=\frac{2xP_{N}^{\mu}+q^{\mu}}{Q\sqrt{1+\gamma^{2}}}\,,\qquad\tilde{n}_{y}^{\mu}=\epsilon^{\mu\nu\rho\sigma}\tilde{n}_{t\,\nu}\tilde{n}_{x\,\rho}\tilde{n}_{z\,\sigma}\,,\qquad\tilde{n}_{z}^{\mu}=-\frac{q^{\mu}}{Q}\,,$
$\displaystyle\tilde{n}_{x}^{\mu}$
$\displaystyle=\frac{1}{P_{hT}}\Bigl{(}P_{h}^{\mu}-z\kappa
q^{\mu}+2xz\frac{\kappa-1}{\gamma^{2}}P_{N}^{\mu}\Bigr{)}\,,$ (A.5)

which is valid for arbitrary hadron masses, with $\kappa$ given by eq. (A.3).
In this coordinate system, the lepton and hadron momenta are parameterized as

$\displaystyle p_{\ell}^{\mu}$
$\displaystyle=\frac{Q}{2}\Bigl{(}\sqrt{\frac{1+\epsilon}{1-\epsilon}}\,,\,\sqrt{\frac{2\epsilon}{1-\epsilon}}\cos\phi_{h}\,,\,\sqrt{\frac{2\epsilon}{1-\epsilon}}\sin\phi_{h}\,,\,-1\Bigr{)}_{B}\,,$
$\displaystyle p_{\ell^{\prime}}^{\mu}$
$\displaystyle=\frac{Q}{2}\Bigl{(}\sqrt{\frac{1+\epsilon}{1-\epsilon}}\,,\,\sqrt{\frac{2\epsilon}{1-\epsilon}}\cos\phi_{h}\,,\,\sqrt{\frac{2\epsilon}{1-\epsilon}}\sin\phi_{h}\,,\,+1\Bigr{)}_{B}\,,$
$\displaystyle P_{N}^{\mu}$
$\displaystyle=\frac{Q}{2x}\Bigl{(}\sqrt{1+\gamma^{2}}\,,\,0\,,\,0\,,+1\Bigr{)}_{B}\,,$
$\displaystyle P_{h}^{\mu}$ $\displaystyle=\bigl{(}m_{Th}\cosh
Y_{h}\,,P_{hT}\,,0\,,m_{Th}\sinh Y_{h}\bigr{)}_{B}\,,$ $\displaystyle q^{\mu}$
$\displaystyle=(0,0,0,-Q)_{B}\,,$ (A.6)

where the rapidity of the outgoing hadron is defined as

$\displaystyle\sinh
Y_{h}=\frac{Qz}{m_{Th}\gamma^{2}}\bigl{[}1-(1+\gamma^{2})\kappa\bigr{]}\,.$
(A.7)

The rest frame and Breit frame are related through a longitudinal Lorentz
boost, a rotation about the $y$ axis (to revert the direction of the $z$
axis), and a rotation about the $z$ axis, given by

$\displaystyle\begin{pmatrix}n^{\prime}_{t}\\\ n^{\prime}_{z}\end{pmatrix}$
$\displaystyle=\frac{1}{\gamma}\begin{pmatrix}\sqrt{1+\gamma^{2}}&1\\\
-1&-\sqrt{1+\gamma^{2}}\\\ \end{pmatrix}\begin{pmatrix}\tilde{n}_{t}\\\
\tilde{n}_{z}\end{pmatrix}\,,\quad\begin{pmatrix}n^{\prime}_{x}\\\
n^{\prime}_{y}\end{pmatrix}=\begin{pmatrix}\cos\phi_{h}&\phantom{-}\sin\phi_{h}\\\
\sin\phi_{h}&-\cos\phi_{h}\\\ \end{pmatrix}\begin{pmatrix}\tilde{n}_{x}\\\
\tilde{n}_{y}\end{pmatrix}\,.$ (A.8)

In particular, for $\phi_{h}=0$, the $x$ axis of both frames coincide, while
the $y$ axis has opposite signs reflecting the reversion of the $z$ axis.

##  Appendix B Fourier Transformation

Our convention for the Fourier transform and its inverse are

$\displaystyle\tilde{f}({\vec{b}}_{T})$
$\displaystyle=\int\mathrm{d}^{2}{\vec{k}}_{T}e^{-\mathrm{i}{\vec{k}}_{T}\cdot{\vec{b}}_{T}}f({\vec{k}}_{T})\,,\quad~{}f({\vec{k}}_{T})$
$\displaystyle=\int\frac{\mathrm{d}^{2}{\vec{b}}_{T}}{(2\pi)^{2}}e^{+\mathrm{i}{\vec{k}}_{T}\cdot{\vec{b}}_{T}}\tilde{f}({\vec{b}}_{T})\,,$
$\displaystyle\tilde{D}({\vec{b}}_{T})$
$\displaystyle=\int\mathrm{d}^{2}{\vec{p}}_{T}e^{+\mathrm{i}{\vec{p}}_{T}\cdot{\vec{b}}_{T}}D({\vec{k}}_{T})\,,\quad
D({\vec{p}}_{T})$
$\displaystyle=\int\frac{\mathrm{d}^{2}{\vec{b}}_{T}}{(2\pi)^{2}}e^{-\mathrm{i}{\vec{p}}_{T}\cdot{\vec{b}}_{T}}\tilde{D}({\vec{b}}_{T})\,.$
(B.1)

Here, ${\vec{b}}_{T}$, ${\vec{k}}_{T}$ and ${\vec{p}}_{T}$ are transverse
vectors in Euclidean space, i.e.
$b_{\perp}^{2}=-{\vec{b}}_{T}^{2}=-b_{T}^{2}$,
${\vec{b}}_{T}\cdot{\vec{k}}_{T}=-b_{T}\cdot k_{\perp}$, etc. In this
appendix, for clarity we denote the Fourier transform by a tilde, while in the
main body of this paper we will drop the tilde as it is used for the
subleading TMDs, and the distinction is clear by the arguments. In the first
line of eq. (B), $f({\vec{k}}_{T})$ is a TMDPDF, while in the second line
$D({\vec{p}}_{T})$ is a TMDFF. As discussed in section 5.1, they are defined
with opposite conventions for their Fourier phases. In the following, we only
show explicit results for the Fourier transform of the TMPDF, as the
corresponding results for the TMDFF can be easily obtained by letting
$\mathrm{i}\to-\mathrm{i}$ and $M_{N}\to M_{h}$.

Often, we encounter functions that only depend on the magnitude of
${\vec{b}}_{T}$ or ${\vec{k}}_{T}$, i.e. $f({\vec{k}}_{T})=f(k_{T})$ with
$k_{T}=|{\vec{k}}_{T}|$. In this case, eq. (B) simplifies to

$\displaystyle\tilde{f}(b_{T})=2\pi\int_{0}^{\infty}\mathrm{d}k_{T}\,k_{T}J_{0}(b_{T}k_{T})f(k_{T})\,,\quad
f(k_{T})=\frac{1}{2\pi}\int_{0}^{\infty}\mathrm{d}b_{T}\,b_{T}J_{0}(b_{T}k_{T})\tilde{f}(b_{T})\,,$
(B.2)

where $J_{n}(x)$ is the $n$-th order Bessel function of first kind.

In addition, we frequently require Fourier transforms of functions of the form

$\displaystyle\int\mathrm{d}^{2}{\vec{k}}_{T}e^{-\mathrm{i}{\vec{k}}_{T}\cdot{\vec{b}}_{T}}(k_{\perp}^{\mu}\cdots
k_{\perp}^{\nu})f(k_{T})$
$\displaystyle=(-\mathrm{i}\partial^{\mu})\cdots(-\mathrm{i}\partial^{\nu})f(b_{T})$
$\displaystyle=(-\mathrm{i}\partial^{\mu})\cdots(-\mathrm{i}\partial^{\nu})2\pi\int_{0}^{\infty}\mathrm{d}k_{T}\,k_{T}J_{0}(b_{T}k_{T})f(k_{T})\,.$
(B.3)

Here, $\partial^{\mu}=\partial/\partial b_{\perp\mu}$, and it is crucial to
note that $k_{\perp}^{\mu}$ and $b_{\perp}^{\mu}$ are vectors in Minkowski
space, as opposed to the Euclidean vectors ${\vec{k}}_{T}$ and
${\vec{b}}_{T}$. Using eq. (B) together with the Bessel function identity

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}z^{-m}J_{m}(z)=-z^{-m}J_{m+1}(z)\,,$
(B.4)

one can easily obtain explicit results for the required Fourier transforms.
For example, one obtains

$\displaystyle\int\mathrm{d}^{2}{\vec{k}}_{T}\,e^{-\mathrm{i}{\vec{k}}_{T}\cdot{\vec{b}}_{T}}\,\frac{k_{\perp}^{\mu}}{k_{T}}f(k_{T})$
$\displaystyle=(-\mathrm{i})\frac{b_{T}^{\mu}}{b_{T}}\
2\pi\int_{0}^{\infty}\\!\\!\\!\mathrm{d}k_{T}\,k_{T}\,J_{1}(b_{T}k_{T})f(k_{T})\,,$
(B.5)
$\displaystyle\int\mathrm{d}^{2}{\vec{k}}_{T}\,e^{-\mathrm{i}{\vec{k}}_{T}\cdot{\vec{b}}_{T}}\biggl{(}\frac{g_{T}^{\mu\nu}}{2}+\frac{k_{\perp}^{\mu}k_{\perp}^{\nu}}{k_{T}^{2}}\biggr{)}f(k_{T})$
$\displaystyle=(-\mathrm{i})^{2}\biggl{(}\frac{g_{T}^{\mu\nu}}{2}+\frac{b_{T}^{\mu}b_{T}^{\nu}}{b_{T}^{2}}\biggr{)}\,2\pi\\!\int_{0}^{\infty}\\!\\!\\!\mathrm{d}k_{T}\,k_{T}\,J_{2}(b_{T}k_{T})\,f(k_{T}).$

These relations match the dimensionless prefactors, which in the second line
is also traceless, onto corresponding prefactors in Fourier space. The
integrals themselves have a form very similar to eq. (B.2), up to replacing
$J_{0}$ by $J_{1}$ and $J_{2}$, respectively.

In practice, due to the traditional normalization of correlators as
$k_{\perp}^{\mu}/M_{N}$ rather than $k_{\perp}^{\mu}/k_{T}$, we need slightly
modified transforms. To absorb these mass factors, following ref. [127] we
define the derivative of the Fourier transform as

$\displaystyle\tilde{f}^{(n)}(b_{T})$
$\displaystyle=n!\left(\frac{-1}{M_{N}^{2}b_{T}}\frac{\partial}{\partial
b_{T}}\right)^{n}\tilde{f}(b_{T})$
$\displaystyle=\frac{2\pi\,n!}{(b_{T}M_{N})^{n}}\int_{0}^{\infty}\mathrm{d}k_{T}\,k_{T}\left(\frac{k_{T}}{M_{N}}\right)^{n}J_{n}(b_{T}k_{T})\,f(k_{T})\,.$
(B.6)

Note that $n!$ factor is included for historical reasons for consistency with
moments of TMD distributions with respect to $k_{T}$ [127]. The inverse of eq.
(B) is given by

$\displaystyle f(k_{T})$
$\displaystyle=\frac{M_{N}^{2n}}{2\pi\,n!}\int_{0}^{\infty}\mathrm{d}b_{T}\,b_{T}\left(\frac{b_{T}}{k_{T}}\right)^{n}J_{n}(b_{T}k_{T})\,\tilde{f}^{(n)}(b_{T})\,,$
(B.7)

which follows from the orthogonality relation of Bessel functions,

$\displaystyle\int_{0}^{\infty}\mathrm{d}b_{T}\,b_{T}\,J_{n}(k_{T}b_{T})\,J_{n}(p^{\prime}_{T}b_{T})=\frac{1}{k_{T}}\delta(k_{T}-p^{\prime}_{T})\,.$
(B.8)

Using eqs. (B) and (B), the Fourier transforms required in this paper are
obtained as

$\displaystyle\int\mathrm{d}^{2}{\vec{k}}_{T}\,e^{-\mathrm{i}{\vec{b}}_{T}\cdot{\vec{k}}_{T}}\frac{k_{\perp}^{\mu}}{M_{N}}f(k_{T})$
$\displaystyle=-\mathrm{i}M_{N}b_{\perp}^{\mu}\tilde{f}^{(1)}(b_{T})\,,$
$\displaystyle\int\mathrm{d}^{2}{\vec{k}}_{T}\,e^{-\mathrm{i}{\vec{b}}_{T}\cdot{\vec{k}}_{T}}\frac{k_{\perp}^{\mu}k_{\perp}^{\nu}}{M_{N}^{2}}f(k_{T})$
$\displaystyle=\frac{(-\mathrm{i})^{2}}{2}M_{N}^{2}\,b_{\perp}^{\mu}b_{\perp}^{\nu}\,\tilde{f}^{(2)}(b_{T})+(-\mathrm{i})^{2}g_{\perp}^{\mu\nu}\tilde{f}^{(1)}(b_{T})\,,$
$\displaystyle\int\mathrm{d}^{2}{\vec{k}}_{T}\,e^{-\mathrm{i}{\vec{b}}_{T}\cdot{\vec{k}}_{T}}\frac{k_{\perp}^{\mu}k_{\perp}^{\nu}k_{\perp}^{\rho}}{M_{N}^{3}}f(k_{T})$
$\displaystyle=\frac{(-\mathrm{i})^{3}}{6}M_{N}^{3}\,b_{\perp}^{\mu}b_{\perp}^{\nu}b_{\perp}^{\rho}\,\tilde{f}^{(3)}(b_{T})$
$\displaystyle\quad+\frac{(-\mathrm{i})^{3}}{2}M_{N}\left(g_{\perp}^{\mu\nu}b_{\perp}^{\rho}+g_{\perp}^{\nu\rho}b_{\perp}^{\mu}+g_{\perp}^{\rho\mu}b_{\perp}^{\nu}\right)\tilde{f}^{(2)}(b_{T})\,.$
(B.9)

The following traceless relation is also useful,

$\displaystyle\int\mathrm{d}^{2}{\vec{k}}_{T}\,e^{-\mathrm{i}{\vec{b}}_{T}\cdot{\vec{k}}_{T}}\frac{k_{T}^{2}}{M_{N}^{2}}\left(\frac{1}{2}g_{\perp}^{\mu\nu}+\frac{k_{\perp}^{\mu}k_{\perp}^{\nu}}{k_{T}^{2}}\right)f(k_{T})$
$\displaystyle=-\frac{1}{2}M_{N}^{2}b_{T}^{2}\left(\frac{1}{2}g_{\perp}^{\mu\nu}+\frac{b_{\perp}^{\mu}b_{\perp}^{\nu}}{b_{T}^{2}}\right)\tilde{f}^{(2)}(b_{T})\,.$
(B.10)

At subleading power, we also encounter Fourier transformations of the form

$\displaystyle f^{(0^{\prime})}(b_{T})$
$\displaystyle\equiv\int\mathrm{d}^{2}{\vec{k}}_{T}\,e^{-\mathrm{i}{\vec{b}}_{T}\cdot{\vec{k}}_{T}}\frac{k_{T}^{2}}{M_{N}^{2}}f(k_{T})=2\pi\int_{0}^{\infty}\mathrm{d}k_{T}\,k_{T}J_{0}(b_{T}k_{T})\frac{k_{T}^{2}}{M_{N}^{2}}f(k_{T})$
$\displaystyle=\frac{(-\mathrm{i})^{2}}{2}M_{N}^{2}\,b_{T}^{2}\tilde{f}^{(2)}(b_{T})-2(-\mathrm{i})^{2}\tilde{f}^{(1)}(b_{T})\,,$
$\displaystyle-\mathrm{i}M_{N}b_{\perp}^{\mu}f^{(1^{\prime})}(b_{T})$
$\displaystyle\equiv\int\mathrm{d}^{2}{\vec{k}}_{T}\,e^{-\mathrm{i}{\vec{b}}_{T}\cdot{\vec{k}}_{T}}\frac{k_{\perp}^{\mu}k_{T}^{2}}{M_{N}^{3}}f(k_{T})=-\mathrm{i}\frac{b_{\perp}^{\mu}}{b_{T}}2\pi\int_{0}^{\infty}\mathrm{d}k_{T}\,k_{T}J_{1}(b_{T}k_{T})\frac{k_{T}^{3}}{M_{N}^{3}}f(k_{T})$
$\displaystyle=-\mathrm{i}M_{N}b_{\perp}^{\mu}\Bigl{[}\frac{(-\mathrm{i})^{2}}{6}M_{N}^{2}\,b_{T}^{2}\,\tilde{f}^{(3)}(b_{T})-2(-\mathrm{i})^{2}\tilde{f}^{(2)}(b_{T})\Bigr{]}\,.$
(B.11)

The expressions in terms of the $\tilde{f}^{(n)}$ can easily be obtained from
appropriate contractions of eq. (B).

### Fourier Transforms and Convolutions in Momentum Space

The structure functions require evaluations of the Fourier transform in eq.
(5.1),

$\displaystyle\mathcal{F}[\mathcal{H}\,g^{(n)}\,D^{(m)}]$
$\displaystyle=2z\sum_{f}\mathcal{H}_{f}(q^{+}q^{-})\int\frac{\mathrm{d}^{2}{\vec{b}}_{T}}{(2\pi)^{2}}e^{-\mathrm{i}{\vec{q}}_{T}\cdot{\vec{b}}_{T}}(-\mathrm{i}M_{N}b_{T})^{n}(\mathrm{i}M_{h}b_{T})^{m}\cos[(n+m)\varphi)]$
$\displaystyle\hskip
99.58464pt\times\,g_{f}^{(n)}(x,b_{T})\,D_{f}^{(m)}(z,b_{T})+(f\to\bar{f})\,.$
(B.12)

Here, we work in a frame where ${\vec{q}}_{T}=q_{T}(-1,0)$ and
${\vec{b}}_{T}=b_{T}(\cos\varphi,\sin\varphi)$. Using the covariant vector
$\hat{h}_{\perp}=q_{\perp}^{\mu}/q_{T}$, we can rewrite the $\cos\varphi$
dependence as

$\displaystyle\cos(\varphi)$
$\displaystyle=\frac{b_{\perp}^{\mu}\hat{h}_{\perp\,\mu}}{b_{T}}\,,$
$\displaystyle\cos(2\varphi)$
$\displaystyle=\frac{2b_{\perp}^{\mu}b_{\perp}^{\nu}}{b_{T}^{2}}\Bigl{(}\frac{g_{\perp\,\mu\nu}}{2}+\hat{h}_{\perp\,\mu}\hat{h}_{\perp\,\nu}\Bigr{)}\,,$
(B.13)

and likewise for higher terms. Using this to replace the $\cos$ term in eq.
(B), we can reduce the integrand to the particular combinations in eq. (B),
after which one can easily read off the momentum-space result, which we
express using eq. (5),

$\displaystyle\tilde{\mathcal{F}}[\omega\,\mathcal{H}\,g\,D]$
$\displaystyle=2z\sum_{f}\mathcal{H}_{f}(q^{2})\int\\!\mathrm{d}^{2}{\vec{k}}_{T}\,\mathrm{d}^{2}{\vec{p}}_{T}\,\delta^{(2)}\bigl{(}{\vec{q}}_{T}+{\vec{k}}_{T}-{\vec{p}}_{T}\bigr{)}$
$\displaystyle\qquad\quad\times\omega(\vec{k}_{T},\vec{k}_{T})\,g_{f}(x,k_{T})\,D_{f}(z,p_{T})+(f\to\bar{f})\,.$
(B.14)

The expressions required at LP are

$\displaystyle\mathcal{F}[\mathcal{H}\,g\,D]$
$\displaystyle=\tilde{\mathcal{F}}[\mathcal{H}gD]\,,$ (B.15a)
$\displaystyle\mathcal{F}[\mathcal{H}\,g^{(1)}\,D^{(0)}]$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{\hat{h}_{\perp}\cdot
k_{\perp}}{M_{N}}\mathcal{H}gD\biggr{]}$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{k_{Tx}}{M_{N}}\mathcal{H}gD\biggr{]}\,,$
(B.15b) $\displaystyle\mathcal{F}[\mathcal{H}\,g^{(0)}\,D^{(1)}]$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{\hat{h}_{\perp}\cdot
p_{\perp}}{M_{h}}\mathcal{H}gD\biggr{]}$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{p_{Tx}}{M_{h}}\mathcal{H}gD\biggr{]}\,,$
(B.15c) $\displaystyle\mathcal{F}[\mathcal{H}\,g^{(1)}\,D^{(1)}]$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{2(k_{\perp}{\cdot}\hat{h}_{\perp})(p_{\perp}{\cdot}\hat{h}_{\perp})+k_{\perp}\cdot
p_{\perp}}{M_{N}M_{h}}\mathcal{H}gD\biggr{]}$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{2k_{Tx}p_{Tx}-{\vec{k}}_{T}\cdot{\vec{p}}_{T}}{M_{N}M_{h}}\mathcal{H}gD\biggr{]}\,,$
(B.15d) $\displaystyle\mathcal{F}[\mathcal{H}\,g^{(2)}D^{(1)}]$
$\displaystyle=\tilde{\mathcal{F}}\left[\frac{4(\hat{h}_{\perp}\cdot
k_{\perp})(k_{\perp}\cdot p_{\perp})+2k_{\perp}^{2}(\hat{h}_{\perp}\cdot
p_{\perp})-8(\hat{h}_{\perp}\cdot k_{\perp})^{2}(\hat{h}_{\perp}\cdot
p_{\perp})}{M_{N}^{2}M_{h}}\,\mathcal{H}gD\right]$
$\displaystyle=\tilde{\mathcal{F}}\left[\frac{-4k_{Tx}\,(\vec{k}_{T}\cdot\vec{p}_{T})-2k_{T}^{2}\,p_{Tx}+8k_{Tx}^{2}\,p_{Tx}}{M_{N}^{2}M_{h}}\,\mathcal{H}gD\right]\,.$
(B.15e)

Each result is given in two forms. First, in a manifestly covariation form
with the unit vector $\hat{h}_{\perp}=q_{\perp}^{\mu}/q_{T}$, and secondly by
making use that in the factorization frame $\hat{h}_{\perp}=(0,-1,0,0)$ such
that $\hat{h}\cdot k_{\perp}=k_{Tx}$. At NLP, we need the additional
combinations

$\displaystyle\mathcal{F}[\mathcal{H}\,g^{(1)}\,D^{(0^{\prime})}]$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{-(\hat{h}\cdot
k_{\perp})p_{\perp}^{2}}{M_{N}M_{h}^{2}}\mathcal{H}gD\biggr{]}$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{k_{Tx}p_{T}^{2}}{M_{N}M_{h}^{2}}\mathcal{H}gD\biggr{]}\,,$
(B.16a) $\displaystyle\mathcal{F}[\mathcal{H}\,g^{(0^{\prime})}\,D^{(1)}]$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{-k_{\perp}^{2}(\hat{h}\cdot
p_{\perp})}{M_{N}^{2}M_{h}}\mathcal{H}gD\biggr{]}$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{k_{T}^{2}\,p_{Tx}}{M_{N}^{2}M_{h}}\mathcal{H}gD\biggr{]}\,,$
(B.16b) $\displaystyle\mathcal{F}[\mathcal{H}\,g^{(1)}\,D^{(1^{\prime})}]$
$\displaystyle=\mathcal{F}\Biggl{[}-\frac{2(h_{\perp}\cdot
k_{\perp})(\hat{h}_{\perp}\cdot p_{\perp})+(k_{\perp}\cdot
p_{\perp})}{M_{N}M_{h}}\frac{p_{\perp}^{2}}{M_{h}^{2}}\mathcal{H}gD\biggr{]}$
$\displaystyle=\mathcal{F}\Biggl{[}\frac{2k_{Tx}p_{Tx}-{\vec{k}}_{T}\cdot{\vec{p}}_{T}}{M_{N}M_{h}}\frac{p_{T}^{2}}{M_{h}^{2}}\mathcal{H}gD\biggr{]}\,,$
(B.16c) $\displaystyle\mathcal{F}[\mathcal{H}\,g^{(2)}\,D]$
$\displaystyle=\mathcal{F}\biggl{[}\frac{4(\hat{h}_{\perp}\cdot
k_{\perp})^{2}+2k_{\perp}^{2}}{M_{N}^{2}}\mathcal{H}gD\biggr{]}$
$\displaystyle=\mathcal{F}\biggl{[}\frac{4k_{Tx}^{2}-2k_{T}^{2}}{M_{N}^{2}}\mathcal{H}gD\biggr{]}=\mathcal{F}\biggl{[}\frac{2(k_{Tx}^{2}-k_{Ty}^{2})}{M_{N}^{2}}\mathcal{H}gD\biggr{]}\,,$
(B.16d) $\displaystyle\mathcal{F}[\mathcal{H}\,g^{(2)}\,D^{(0^{\prime})}]$
$\displaystyle=\mathcal{F}\biggl{[}\frac{4(\hat{h}_{\perp}\cdot
k_{\perp})^{2}+2k_{\perp}^{2}}{M_{N}^{2}}\frac{-p_{\perp}^{2}}{M_{h}^{2}}\mathcal{H}gD\biggr{]}$
$\displaystyle=\mathcal{F}\biggl{[}\frac{4k_{Tx}^{2}-2k_{T}^{2}}{M_{N}^{2}}\frac{p_{T}^{2}}{M_{h}^{2}}\mathcal{H}gD\biggr{]}=\mathcal{F}\biggl{[}\frac{2(k_{Tx}^{2}-k_{Ty}^{2})}{M_{N}^{2}}\frac{p_{T}^{2}}{M_{h}^{2}}\mathcal{H}gD\biggr{]}\,,$
(B.16e) $\displaystyle\mathcal{F}[\mathcal{H}\,g^{(2)}\,D^{(1^{\prime})}]$
$\displaystyle=\tilde{\mathcal{F}}\left[\frac{4(\hat{h}_{\perp}\cdot
k_{\perp})(k_{\perp}\cdot p_{\perp})+2k_{\perp}^{2}(\hat{h}_{\perp}\cdot
p_{\perp})-8(\hat{h}_{\perp}\cdot k_{\perp})^{2}(\hat{h}_{\perp}\cdot
p_{\perp})}{M_{N}^{2}M_{h}}\frac{-p_{\perp}^{2}}{M_{h}^{2}}\mathcal{H}gD\right]$
$\displaystyle=\tilde{\mathcal{F}}\left[\frac{-4k_{Tx}\,(\vec{k}_{T}\cdot\vec{p}_{T})-2k_{T}^{2}\,p_{Tx}+8k_{Tx}^{2}\,p_{Tx}}{M_{N}^{2}M_{h}}\frac{p_{T}^{2}}{M_{h}^{2}}\,\mathcal{H}gD\right]\,.$
(B.16f)

In addition, we also require expressions for the $\mathcal{F}^{\prime}$
transform defined in eq. (5.2.4) as

$\displaystyle\mathcal{F}^{\prime}[\mathcal{H}\,g^{(n)}\,D^{(m)}]$
$\displaystyle=2z\sum_{f}\mathcal{H}_{f}(q^{+}q^{-})\int\frac{\mathrm{d}^{2}{\vec{b}}_{T}}{(2\pi)^{2}}e^{-\mathrm{i}{\vec{q}}_{T}\cdot{\vec{b}}_{T}}(-\mathrm{i}M_{N}b_{T})^{n}(\mathrm{i}M_{h}b_{T})^{m}$
$\displaystyle\hskip
99.58464pt\times\,g_{f}^{(n)}(x,b_{T})\,D_{f}^{(m)}(z,b_{T})+(f\to\bar{f})\,.$
(B.17)

Specifically, we encounter the transformations

$\displaystyle\mathcal{F}^{\prime}[\mathcal{H}gD]$
$\displaystyle=\mathcal{F}[\mathcal{H}gD]$
$\displaystyle=\tilde{\mathcal{F}}[\mathcal{H}gD]\,,$ (B.18a)
$\displaystyle\mathcal{F}^{\prime}[\mathcal{H}g^{(0^{\prime})}D]$
$\displaystyle=\mathcal{F}[\mathcal{H}g^{(0^{\prime})}D]$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{k_{T}^{2}}{M_{N}^{2}}\mathcal{H}gD\biggr{]}\,,$
(B.18b) $\displaystyle\mathcal{F}^{\prime}[\mathcal{H}gD^{(0^{\prime})}]$
$\displaystyle=\mathcal{F}^{\prime}[\mathcal{H}gD^{(0^{\prime})}]$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{p_{T}^{2}}{M_{N}^{2}}\mathcal{H}gD\biggr{]}\,,$
(B.18c) $\displaystyle\mathcal{F}^{\prime}[\mathcal{H}g^{(1)}D^{(1)}]$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{-k_{\perp}\cdot
p_{\perp}}{M_{N}M_{h}}\mathcal{H}gD\biggr{]}$
$\displaystyle=\tilde{\mathcal{F}}\biggl{[}\frac{{\vec{k}}_{T}\cdot{\vec{p}}_{T}}{M_{N}M_{h}}\mathcal{H}gD\biggr{]}\,.$
(B.18d)

##  Appendix C Transverse Gauge Links at LP and NLP

In the main text, we have derived factorization formula at NLP for SIDIS where
the various TMD PDFs, TMD FFs and soft functions were considered across any
covariant gauge, which are gauges where the gluon fields vanish at infinity.
This avoided the discussion of Wilson lines in the transverse direction at
light-cone infinity that are needed for full gauge invariance of various
definitions, and are of crucial importance in light-cone gauge [108]. In order
to modify the definitions so that they are gauge invariant across all gauges,
it is natural to add gauge links connecting the points at lightcone infinity
separated in the transverse direction. For the leading power results discussed
in section 4.2 this gives a staple shaped Wilson line path $\sqsupset$ with
ends attached to the quark fields for the matrix elements
$\hat{B}_{f/N}^{\beta\beta^{\prime}}$, two links that extend the Wilson lines
to infinity in the transverse direction for the two matrix elements in
$\hat{\mathcal{G}}_{h/f}^{\alpha\alpha^{\prime}}$, in the form  and  (which
would combine to $\sqsupset$ if they were in the same matrix elements), and a
closed six sided Wilson loop  for the soft function $S(b_{T})$. See ref. [27]
for a review.

We can justify the addition of the extra gauge links by starting from the
factorized results in covariant gauge. Since here the gauge fields vanish at
infinity, all the TMD functions are identical to their original versions after
adding the transverse connecting links (and no matter which specific path we
choose). Since they are now manifestly gauge invariant even at light-cone
infinity, they are invariant even for gauges whose gauge field does not vanish
at infinity, as long as we include a suitable regulator to deal with the
issues rapidity/lightcone divergences consistently. Although different choices
for the path of the transverse Wilson lines should be equivalent, the most
convenient one (and the one considered in literature) involves straight
transverse Wilson lines. We can modify the LP definitions in eq. (4.2) with
these transverse Wilson lines inserted to give

$\displaystyle\hat{B}_{f/N}^{\beta\beta^{\prime}}(x,{\vec{b}}_{T})$
$\displaystyle=\theta(\omega_{a})\,\braket{N}{\bar{\chi}_{n}^{\beta^{\prime}}(b_{\perp})\,T_{n}(b_{\perp},0_{\perp})\,\chi_{n,\omega_{a}}^{\beta}(0)}{N}\,,$
$\displaystyle\hat{\mathcal{G}}_{h/f}^{\alpha^{\prime}\alpha}(z,{\vec{b}}_{T})$
$\displaystyle=\frac{\theta(\omega_{b})}{2zN_{c}}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X_{\bar{n}}}\operatorname{tr}\braket{0}{T_{\bar{n}}(\infty,b_{\perp})\chi_{{\bar{n}}}^{\alpha^{\prime}}(b_{\perp})\,\,}{h,X_{\bar{n}}}\braket{h,X_{\bar{n}}}{\bar{\chi}_{{\bar{n}},-\omega_{b}}^{\alpha}(0)T_{\bar{n}}(0,\infty)}{0}\,,$
(C.9) $\displaystyle S(b_{T},b_{s}^{+}b_{s}^{-})$
$\displaystyle=\frac{1}{N_{c}}\operatorname{tr}\braket{0}{T_{n}^{s}(0,b_{\perp})\,\bigl{(}S_{n}^{\dagger}S_{{\bar{n}}}\bigr{)}(b_{s}^{+},b_{s}^{-},b_{\perp})\,T_{\bar{n}}^{s}(b_{\perp},0)\,\bigl{(}S_{{\bar{n}}}^{\dagger}S_{n}\bigr{)}(0)}{0}\,,$
(C.10)

where $T_{n_{i}}(a_{\perp},b_{\perp})$ and
$T_{n_{i}}^{s}(a_{\perp},b_{\perp})$ are collinear and soft transverse gauge
links connecting two points at infinity, $\infty n_{i}^{\mu}+a_{\perp}^{\mu}$
and $\infty n_{i}^{\mu}+b_{\perp}^{\mu}$.

We may also want to consider such modification at the Lagrangian level, since
for example otherwise the hard scattering operators are not gauge invariant
unless we assume that the gauge fields vanish at infinity. This approach has
been taken in Refs. [110, 109, 29]. There should be some gauge link connecting
the $n_{i}$ (for $S_{n_{i}}$) and ${\bar{n}}_{i}$ (for $W_{n_{i}}$) infinities
($i=1,2$), and we should make sure that the modified operator is invariant for
both collinear and soft gauge transformations.

A possible choice for this construction would be that, we assign a reference
point at infinity, and make it invariant for all kinds of gauge
transformations that we are considering ($n_{i}$ collinear and soft gauge
transformations, which in SCET can be done independently). It can be ensured
that there is no transformation at this special point by using a global SU(3)
transformation. We then connect this point to the endpoint of $W_{n_{i}}$ and
$S_{n_{i}}$ with Wilson lines $Z_{n_{i}}$ involving $n_{i}$-collinear gauge
fields, and Wilson lines $U_{n_{i}}$ involving soft gauge fields,

$\displaystyle\bar{\chi}_{n_{2}}[S^{\dagger}_{n_{2}}S_{n_{1}}]\chi_{n_{1}}\to\bar{\chi}_{n_{2}}Z_{n_{2}}[U^{\dagger}_{n_{2}}S^{\dagger}_{n_{2}}S_{n_{1}}U_{n_{1}}]Z_{n_{1}}^{\dagger}\chi_{n_{1}}$
(C.11)

Using the same logic as in the previous paragraph, the choices of path to the
reference point does not matter and they should all be equivalent. We can then
choose a path consistent with the choice of the transverse gauge link in the
previous paragraph, so that for example
$Z_{n}(b_{\perp})=T_{n}(b_{\perp},\infty)$ and
$Z_{n}^{\dagger}(0_{\perp})=T_{n}(\infty,0_{\perp})$, which give the proper
Wilson lines for the fragmentation matrix element and together combine to
$T_{n}(b_{\perp},0_{\perp})$ for the staple needed in the TMD PDF and the
closed Wilson loop needed for the soft function,

$\displaystyle Z_{n}(b_{\perp})Z_{n}^{\dagger}(0_{\perp})$
$\displaystyle=T_{n}(b_{\perp},0_{\perp})\,,$ $\displaystyle
U_{n}(b_{\perp})U_{n}^{\dagger}(0_{\perp})$
$\displaystyle=T_{n}^{s}(b_{\perp},0_{\perp})\,.$ (C.12)

The analysis in ref. [29] corresponds to the choice of $\infty n^{\mu}+\infty
a_{\perp}^{\mu}$ as the reference point for these steps, and it was argued
there that the choice of $a_{\perp}^{\mu}$ did not matter.

We now consider the generalization of this discussion of transverse Wilson
lines to the operators that give non-zero contributions at NLP. Generalizing
the LP analysis of transverse lines to the $\mathcal{B}_{n_{i}\perp}$
operators given in eq. (4.79) is fairly straightforward. Each of the three
$\omega_{i}$ momenta appearing in this operator can be written as Fourier
transforms of fields at light-cone positions $b_{i}^{+}$ in position space.
The quark and gluon fields that are next to each other and in the same $n_{i}$
collinear direction, get connected by a finite length Wilson line from (for
eg.) $b_{1}^{+}$ to $b_{3}^{+}$. Thus the only transverse lines that are
needed, occur in the parts of the operator that are next to the
$S^{\dagger}_{{\bar{n}}_{1}}S_{n_{1}}$, just like in the LP operator in eq.
(C.11). For this reason, the exact same type of modifications end up holding
for the NLP quark-gluon-quark correlators, giving

$\displaystyle\hat{B}_{\mathcal{B}\,f/N}^{\rho\beta\beta^{\prime}}(x,\xi,{\vec{b}}_{T})$
$\displaystyle=\theta(\omega_{a})\Bigl{\langle}N\Bigr{\rvert}\bar{\chi}_{n}^{\beta^{\prime}}(b_{\perp})\,T_{n}(b_{\perp},0_{\perp})\,g{\mathcal{B}}_{n\perp,\,-\xi\omega_{a}}^{\rho}(0)\,\chi_{n,\,(1-\xi)\omega_{a}}^{\beta}(0)\Bigr{\rvert}N\Bigr{\rangle}\,,$
$\displaystyle\hat{\mathcal{G}}_{\mathcal{B}\,h/f}^{\rho\,\alpha^{\prime}\alpha}(z,\xi,{\vec{b}}_{T})$
$\displaystyle=\frac{1}{2zN_{c}}\theta(\omega_{b})\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X_{\bar{n}}}\operatorname{tr}\Bigl{\langle}0\Bigr{\rvert}Z_{\bar{n}}^{\dagger}(b_{\perp})\chi_{\bar{n}}^{\alpha^{\prime}}(b_{\perp})\Bigr{\rvert}h,X_{\bar{n}}\Bigr{\rangle}$
(C.21)
$\displaystyle\qquad\times\Bigl{\langle}h,X_{\bar{n}}\Bigr{\rvert}\bar{\chi}_{\bar{n},\,-(1-\xi)\omega_{b}}^{\alpha}(0)\,g{\mathcal{B}}_{\bar{n}\perp,\,\xi\omega_{b}}^{\rho}(0)Z_{\bar{n}}(0_{\perp})\Bigr{\rvert}0\Bigr{\rangle}\,.$
(C.22)

For the case of $\mathcal{P}_{\perp}$ operator contribution, we note that
$\mathcal{P}_{\perp}$ must act on a gauge invariant object, so (suppressing
Dirac matrices) we can modify the $\mathcal{P}_{\perp}$ operators as

$\bar{\chi}_{n_{2}}[S^{\dagger}_{n_{2}}S_{n_{1}}]\,\mathcal{P}_{n_{1}\perp}\,\chi_{n_{1}}\to\bar{\chi}_{n_{2}}Z_{n_{2}}[U^{\dagger}_{n_{2}}S^{\dagger}_{n_{2}}S_{n_{1}}U_{n_{1}}]\,\mathcal{P}_{n_{1}\perp}\,Z_{n_{1}}^{\dagger}\chi_{n_{1}}\,,$
(C.23)

with an analogous result for the term involving
$\mathcal{P}_{n_{2}\perp}^{\dagger}$. Then, the modified
$\hat{B}_{\mathcal{P}\,f/N}$ becomes

$\displaystyle\hat{B}_{\mathcal{P}f/N}^{\beta\beta^{\prime}}(x,{\vec{b}}_{T})$
$\displaystyle\equiv\,\frac{\theta({\omega_{a}})}{2Q}\biggl{\\{}\Bigl{\langle}N\Bigr{\rvert}\bar{\chi}_{n}^{\beta^{\prime}}(b_{\perp}^{\mu})Z_{n}(b_{\perp})\,\left[\not{\mathcal{P}}_{\perp}\not{{\bar{n}}}\,Z_{n}^{\dagger}(0_{\perp})\chi_{n,\omega_{a}}(0)\right]^{\beta}\Bigr{\rvert}N\Bigr{\rangle}$
$\displaystyle\qquad\qquad+\Bigl{\langle}N\Bigr{\rvert}\left[\bar{\chi}_{n}(b_{\perp}^{\mu})\,Z_{n}(b_{\perp})\not{{\bar{n}}}\,\not{\mathcal{P}}_{\perp}^{\dagger}\right]^{\beta^{\prime}}\,Z_{n}^{\dagger}(0_{\perp})\chi_{n,\omega_{a}}^{\beta}(0)\Bigr{\rvert}N\Bigr{\rangle}\biggr{\\}}$
$\displaystyle=\,\frac{-\mathrm{i}}{2Q}\frac{\partial}{\partial
b_{\perp}^{\rho}}\left[\gamma_{\perp}^{\rho}\,\not{{\bar{n}}}\,,\,\hat{B}_{f/N}(x,{\vec{b}}_{T})\right]^{\beta\beta^{\prime}}\,.$
(C.24)

The second equality holds since $\mathcal{P}_{\perp}$ acts like a derivative
to the 2-dimensional transverse local points at $0_{\perp}$ or $b_{\perp}$.
After rewriting $\mathcal{P}_{\perp}$ as a derivative, it can be pulled out,
and then $Z_{n}(b_{\perp})$ and $Z_{n}(0_{\perp})$ combine to give the
$T_{n}(b_{\perp},0_{\perp})$ that appears in $\hat{B}_{f/N}$, so it is the
$\hat{B}_{f/N}$ with the transverse link that appears on the last line of eq.
(C). Thus we find that relation between NLP and LP TMD PDFs in eq. (4.234b) is
still valid in the presence of transverse Wilson lines. A similar story also
applies to $\hat{\mathcal{G}}_{\mathcal{P}h/f}$ with transverse lines, and its
relation to $\hat{\mathcal{G}}_{h/f}$ with transverse lines,

$\displaystyle\hat{\mathcal{G}}_{\mathcal{P}\,h/f}^{\alpha^{\prime}\alpha}(z,{\vec{b}}_{T})$
$\displaystyle\equiv\frac{\theta(\omega_{b})}{4zN_{c}Q}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X_{\bar{n}}}\biggl{\\{}\Bigl{\langle}0\Bigr{\rvert}\left[\not{\mathcal{P}}_{\perp}\not{n}\,Z_{\bar{n}}^{\dagger}(b_{\perp})\chi_{\bar{n}}(b_{\perp})\right]^{\alpha^{\prime}}\Bigr{\rvert}h,X_{\bar{n}}\Bigr{\rangle}\Bigl{\langle}h,X_{\bar{n}}\Bigr{\rvert}\bar{\chi}_{\bar{n},-\omega_{b}}^{\alpha}(0)Z_{\bar{n}}(0_{\perp})\Bigr{\rvert}0\Bigr{\rangle}$
(C.33) $\displaystyle\hskip
56.9055pt+\Bigl{\langle}0\Bigr{\rvert}Z_{\bar{n}}(b_{\perp})\chi_{\bar{n}}^{\alpha^{\prime}}(b_{\perp})\Bigr{\rvert}h,X_{\bar{n}}\Bigr{\rangle}\Bigl{\langle}h,X_{\bar{n}}\Bigr{\rvert}\left[\bar{\chi}_{\bar{n},-\omega_{b}}(0)\,Z_{\bar{n}}^{\dagger}(0_{\perp})\not{n}\,\not{\mathcal{P}}_{\perp}^{\dagger}\right]^{\alpha}\Bigr{\rvert}0\Bigr{\rangle}\biggr{\\}}$
$\displaystyle=\frac{\mathrm{i}}{2Q}\frac{\partial}{\partial
b_{\perp}^{\rho}}\left[\gamma_{\perp}^{\rho}\,\not{n}\,,\,\hat{\mathcal{G}}_{h/f}(x,{\vec{b}}_{T})\right]^{\alpha^{\prime}\alpha}\,.$
(C.34)

Hence we find that eq. (4.235k) also remains valid including transverse Wilson
lines.

In conclusion, all the results presented in the main text remain valid with
the inclusion of transverse Wilson lines as presented by the explicit formulas
given here.

##  Appendix D Relation to Correlators in the Literature

Our results heavily make use of the decompositions of quark-quark and quark-
gluon-quark correlators in the literature, which is required to make a
connection to our results for the structure functions. In our work, we use
different conventions for the lightcone notation, and different definitions of
our correlators. In this appendix, we show how to relate our correlators to
those in the literature, and use this to obtain decompositions of our
correlators since the general Lorentz structure of these correlators is
identical to the ones that have been considered in the past. All our reference
results for these decompositions are taken from ref. [21], up to using certain
combinations of the functions therein, which we will point out explicitly.

###  D.1 Lightcone Conventions

Our conventions are based on two reference vectors $n^{\mu}$ and
${\bar{n}}^{\mu}$ with $n^{2}={\bar{n}}^{2}=0$ and $n\cdot{\bar{n}}=2$, and we
define the lightcone coordinates as $p^{-}={\bar{n}}\cdot p$ and $p^{+}=n\cdot
p$, see section 2.3. Ref. [21] uses reference vectors $n_{\pm}^{\mu}$ with
$n_{+}^{2}=n_{-}^{2}=0$ and $n_{+}\cdot n_{-}=1$, and defines lightcone
components as $\tilde{p}^{\pm}=n_{\mp}\cdot p$. To distinguish the two
conventions, in this section we will always denote their lightcone coordinates
using a tilde. The two conventions are related by

$\displaystyle n_{+}^{\mu}=\frac{n^{\mu}}{\sqrt{2}}\,,\quad
n_{-}^{\mu}=\frac{{\bar{n}}^{\mu}}{\sqrt{2}}\,,\quad\tilde{p}^{\pm}=\frac{p^{\mp}}{\sqrt{2}}\,.$
(D.1)

The first two relations are fixed by having the incoming momentum $P^{\mu}$ be
aligned along the $n$ and $n_{+}$ direction, respectively.

###  D.2 Quark-Quark Correlator

##### The quark-quark correlator in ref. [21]

is defined in their Eq. (3.10) as

$\displaystyle\Phi_{ij}(x,{\vec{p}}_{T})$
$\displaystyle=\int\frac{\mathrm{d}\tilde{\xi}^{-}\mathrm{d}^{2}\vec{\xi}_{T}}{(2\pi)^{3}}e^{\mathrm{i}x\tilde{P}^{+}\tilde{\xi}^{-}}e^{-\mathrm{i}{\vec{p}}_{T}\cdot\vec{\xi}_{T}}\braket{N}{\bar{\psi}_{j}(0)\,\mathcal{U}^{n_{-}}_{(0,\infty)}\,\mathcal{U}^{n_{-}}_{(\infty,\xi)}\,\psi_{i}(\xi)}{N}$
$\displaystyle=\int\frac{\mathrm{d}\xi^{+}\mathrm{d}^{2}\vec{\xi}_{T}}{\sqrt{2}(2\pi)^{3}}e^{\frac{\mathrm{i}}{2}xP^{-}\xi^{+}}e^{-\mathrm{i}{\vec{p}}_{T}\cdot\vec{\xi}_{T}}\braket{N}{\bar{\psi}_{j}(0)\mathcal{U}^{n_{-}}_{(0,\infty)}\,\mathcal{U}^{n_{-}}_{(\infty,\xi)}\psi_{i}(\xi)}{N}\,,$
(D.2)

where $ij$ are spinor indices, and the quark flavor is kept implicit. The
Wilson lines $\mathcal{U}$ will be converted to our $W_{n}$ Wilson lines
below. In the first line, we use their lightcone convention, while in the
second line we convert to our notation. Next, we use that the correlator only
depends on the relative position $0-\xi=-\xi$ to shift the positions of the
fields. Defining $b\equiv-\xi$, this yields272727In recent work by Collins
[27], the fields are evaluated as $\bar{\psi}(w/2)W(w/2,-w/2)\psi(-w/2)$,
which by the same argument is identical to our convention.

$\displaystyle\Phi_{ij}(x,{\vec{p}}_{T})$
$\displaystyle=\int\frac{\mathrm{d}b^{+}\mathrm{d}^{2}{\vec{b}}_{T}}{\sqrt{2}(2\pi)^{3}}e^{-\frac{\mathrm{i}}{2}xP^{-}b^{+}}e^{\mathrm{i}{\vec{p}}_{T}\cdot{\vec{b}}_{T}}\braket{N}{\bar{\psi}_{j}(b)\mathcal{U}^{n_{-}}_{(b,\infty)}\,\mathcal{U}^{n_{-}}_{(\infty,0)}\psi_{i}(0)}{N}\,.$
(D.3)

This ensures that the Fourier phase for the transverse integration has the
same sign as in our convention. This yields the position-space correlator

$\displaystyle\Phi_{ij}(x,{\vec{b}}_{T})$
$\displaystyle=\int\mathrm{d}^{2}{\vec{p}}_{T}\,e^{-\mathrm{i}{\vec{b}}_{T}\cdot{\vec{p}}_{T}}\Phi_{ij}(x,{\vec{p}}_{T})$
$\displaystyle=\frac{1}{\sqrt{2}}\int\frac{\mathrm{d}b^{+}}{2\pi}e^{-\frac{\mathrm{i}}{2}xP^{-}b^{+}}\braket{N}{\bar{\psi}_{j}(b)\mathcal{U}^{n_{-}}_{(b,\infty)}\,\mathcal{U}^{n_{-}}_{(\infty,0)}\psi_{i}(0)}{N}\,.$
(D.4)

It remains to rewrite their Wilson lines $\mathcal{U}^{n_{-}}$ in terms of our
$W_{n}$. The relation is

$\displaystyle\mathcal{U}^{n_{-}}_{(\infty,0)}$
$\displaystyle=P\exp\left[-\mathrm{i}g\int_{\infty}^{0}\mathrm{d}s\,{\bar{n}}\cdot
A(s{\bar{n}})\right]$ $\displaystyle=W_{n}^{\dagger}(0)\,,$
$\displaystyle\mathcal{U}^{n_{-}}_{(b,\infty)}$
$\displaystyle=P\exp\left[-\mathrm{i}g\int_{0}^{\infty}\mathrm{d}s\,{\bar{n}}\cdot
A(s{\bar{n}}+b)\right]$ $\displaystyle=W_{n}(b)\,,$ (D.5)

where we note that the $P$s written here would be $\bar{P}$s in our notation,
and we neglected transverse Wilson lines for brevity. The first equality is
the definition of ref. [21], converted to our conventions, and in the second
step we used eq. (3.6). Inserting eq. (D.2) into eq. (D.2) yields

$\displaystyle\Phi_{ij}(x,{\vec{b}}_{T})$
$\displaystyle=\frac{1}{\sqrt{2}}\int\frac{\mathrm{d}b^{+}}{2\pi}e^{-\frac{\mathrm{i}}{2}xP^{-}b^{+}}\braket{N}{\bar{\psi}_{j}(b)W_{n}(b)W_{n}^{\dagger}(0)\psi_{i}(0)}{N}\,,$
(D.6)

where we remind the reader that $b^{\mu}=(b^{+},0,b_{\perp})$.

##### Our beam function

is defined in eq. (4.2) as

$\displaystyle\hat{B}^{ij}(x=\omega_{a}/P_{N}^{-},{\vec{b}}_{T})$
$\displaystyle=\braket{N}{\bar{\chi}_{n}^{j}(b_{\perp})\chi_{n,\omega_{a}}^{i}(0)}{N}\,,$
(D.7)

where here we label the spinor indices as $ij$ instead of
$\beta\beta^{\prime}$. For consistency with eq. (D.2), we also and suppress
the flavor and hadron labels, as well as the explicit $\theta(\omega_{a})$
factor present in eq. (4.2). Inserting the definitions of the collinear quark
fields from eq. (3.3),

$\displaystyle\chi_{n,\omega}(x)$
$\displaystyle=\Bigl{[}\delta(\omega-\overline{\mathcal{P}}_{n})\,W_{n}^{\dagger}(x)\,\xi_{n}(x)\Bigr{]}\,,$
$\displaystyle\bar{\chi}_{n,\omega}(x)$
$\displaystyle=(\chi_{n,-\omega})^{\dagger}\gamma^{0}=\Bigl{[}\bar{\xi}_{n}(x)W_{n}(x)\delta(\omega+\overline{\mathcal{P}}_{n}^{\dagger})\Bigr{]}\,,$
(D.8)

the beam correlator becomes

$\displaystyle\hat{B}^{ij}(x,{\vec{b}}_{T})$
$\displaystyle=\Braket{N}{\bar{\xi}_{n}^{j}(b_{\perp})W_{n}(b_{\perp})\bigl{[}\delta(\omega_{a}-\overline{\mathcal{P}}_{n})\,W_{n}^{\dagger}(0)\,\xi_{n}^{i}(0)\bigr{]}}{N}\,.$
(D.9)

The combination in square brackets has a fixed longitudinal momentum
$\omega_{a}$, which can equivalently be expressed through the Fourier-
transformed field

$\displaystyle\bigl{[}\delta(\omega_{a}-\overline{\mathcal{P}}_{n})\,W_{n}^{\dagger}(0)\,\xi_{n}^{i}(0)\bigr{]}$
$\displaystyle=\int\frac{\mathrm{d}b^{+}}{4\pi}e^{\frac{\mathrm{i}}{2}\omega_{a}(-b^{+})}W_{n}^{\dagger}(-b^{+})\,\xi_{n}^{i}(-b^{+})\,.$
(D.10)

This puts the $\xi_{n}$ field in eq. (D.9) to position $-b^{+}$ which we again
shift into the first $\bar{\xi}_{n}$ field,

$\displaystyle\hat{B}^{ij}(x,{\vec{b}}_{T})$
$\displaystyle=\int\frac{\mathrm{d}b^{+}}{4\pi}e^{-\frac{\mathrm{i}}{2}\omega_{a}b^{+}}\Braket{N}{\bar{\xi}_{n}^{j}(b)W_{n}(b)W_{n}^{\dagger}(0)\,\xi_{n}^{i}(0)}{N}\,,$
(D.11)

where now $b^{\mu}=(b^{+},0,b_{\perp})$. The $n$-collinear fields $\xi_{n}$
only consistent of the “good components” of the quark field, which are
projected out as

$\displaystyle\xi_{n}^{i}=\frac{(\not{n}\not{{\bar{n}}})^{ii^{\prime}}}{4}\psi_{i^{\prime}}\,,\qquad\bar{\xi}_{n}^{j}=\bar{\psi}_{j^{\prime}}\frac{(\not{{\bar{n}}}\not{n})^{j^{\prime}j}}{4}\,,$
(D.12)

where we make the spinor indices explicit. Inserting this into eq. (D.11)
yields

$\displaystyle\hat{B}^{ij}(x,{\vec{b}}_{T})$
$\displaystyle=\frac{(\not{n}\not{{\bar{n}}})^{ii^{\prime}}}{4}\frac{(\not{{\bar{n}}}\not{n})^{jj^{\prime}}}{4}\int\frac{\mathrm{d}b^{+}}{4\pi}e^{-\frac{\mathrm{i}}{2}\omega_{a}b^{+}}\Braket{N}{\bar{\psi}_{j^{\prime}}(b)W_{n}(b)W_{n}^{\dagger}(0)\,\psi_{i^{\prime}}(0)}{N}\,.$
(D.13)

Comparing eqs. (D.2) and (D.13), we finally obtain the desired relation

$\displaystyle\hat{B}^{ij}(x,{\vec{b}}_{T})$
$\displaystyle=\frac{(\not{n}\not{{\bar{n}}})^{ii^{\prime}}}{4}\,\frac{\Phi_{i^{\prime}j^{\prime}}(x,{\vec{b}}_{T})}{\sqrt{2}}\,\frac{(\not{{\bar{n}}}\not{n})^{j^{\prime}j}}{4}$
$\displaystyle=\frac{1}{4\sqrt{2}}\bigl{[}\not{n}_{+}\not{n}_{-}\Phi(x,{\vec{b}}_{T})\not{n}_{-}\not{n}_{+}\bigr{]}^{ij}\,.$
(D.14)

In the second line, we have made use of eq. (D.1). Since the projection
operators and constant factors commute with taking the Fourier transform, one
obtains the same result for the Fourier-transformed correlators.

##### Decomposition.

Eq. (D.2) allows us to easily obtain the spin-dependendent decomposition of
$\hat{B}^{ij}$ from that of $\Phi^{ij}$ given in Eq. (3.15) of ref. [21],

$\displaystyle\Phi(x,{\vec{k}}_{T})$
$\displaystyle=\biggl{(}f_{1}-f_{1T}^{\perp}\frac{\epsilon_{\perp}^{\rho\sigma}k_{\perp\rho}S_{\perp\sigma}}{M}+g_{1s}\gamma_{5}+h_{1T}\gamma_{5}\not{S}_{\perp}+h_{1s}^{\perp}\gamma^{5}\frac{\not{k}_{\perp}}{M}+\mathrm{i}h_{1}^{\perp}\frac{\not{k}_{\perp}}{M}\biggr{)}\frac{\not{n}_{+}}{2}$
$\displaystyle\quad+(\mathrm{higher~{}twist})\,,$ (D.15)

where all functions on the right-hand side are evaluated as $f_{i}\equiv
f_{i}(x,k_{T})$. Inserting this into eq. (D.2) only requires to use

$\displaystyle\frac{1}{4\sqrt{2}}(\not{n}_{+}\not{n}_{-})\frac{\not{n}_{+}}{2}(\not{n}_{-}\not{n}_{+})=\frac{\not{n}_{+}}{2\sqrt{2}}=\frac{\not{n}}{4}\,.$
(D.16)

We arrive at the desired result

$\displaystyle\hat{B}(x,{\vec{k}}_{T})$
$\displaystyle=\biggl{(}f_{1}-f_{1T}^{\perp}\frac{\epsilon_{\perp}^{\rho\sigma}k_{\perp\rho}S_{\perp\sigma}}{M}+g_{1s}\gamma_{5}+h_{1T}\gamma_{5}\not{S}_{\perp}+h_{1s}^{\perp}\frac{\gamma^{5}\not{k}_{\perp}}{M}+\mathrm{i}h_{1}^{\perp}\frac{\not{k}_{\perp}}{M}\biggr{)}\frac{\not{n}}{4}\,.$
(D.17)

Note that all higher-twist terms vanish because $\not{n}_{\pm}^{2}=0$, and
only the leading twist contribution has the $\not{n}_{+}$ structure preventing
one from commuting a $\not{n}_{-}$ past $\Phi$ in eq. (D.2). Since
$h_{1s}^{\perp}$ is multiplied by a term proportional to $\not{k}_{\perp}$ in
eq. (D.17), it contributes a linear and quadratic Lorentz structure. It is
useful to split off a traceless piece, as the tracelessness is preservered by
the Fourier transform we ultimately need to take. This leads to

$\displaystyle B_{f/N}(x,{\vec{k}}_{T})$
$\displaystyle=\biggl{[}f_{1}-f_{1T}^{\perp}\frac{\epsilon_{\perp}^{\rho\sigma}k_{\perp\rho}S_{\perp\sigma}}{M_{N}}+g_{1L}\,S_{L}\gamma_{5}-g_{1T}\frac{k_{\perp}\cdot
S_{\perp}}{M_{N}}\gamma_{5}+h_{1}\,\gamma_{5}\not{S}_{\perp}$
$\displaystyle\qquad+h_{1L}^{\perp}\,S_{L}\frac{\gamma_{5}\,\not{k}_{\perp}}{M_{N}}-h_{1T}^{\perp}\,\frac{k_{\perp}^{2}}{M_{N}^{2}}\Bigl{(}\frac{1}{2}g_{\perp}^{\mu\nu}-\frac{k_{\perp}^{\mu}k_{\perp}^{\nu}}{k_{\perp}^{2}}\Bigr{)}S_{\perp\,\mu}\gamma_{\nu}\gamma_{5}+\mathrm{i}h_{1}^{\perp}\frac{\not{k}_{\perp}}{M_{N}}\biggr{]}\frac{\not{n}}{4}\,.$
(D.18)

Here, we defined the correlator [21]

$\displaystyle
h_{1}(x,k_{T})=h_{1T}(x,k_{T})-\frac{k_{\perp}^{2}}{2M_{N}^{2}}h_{1T}^{\perp}(x,k_{T})\,.$
(D.19)

We have also replaced the shorthand notations $g_{1s}$ and $h_{1s}$ by [21]

$\displaystyle g_{1s}(x,{\vec{k}}_{T})$
$\displaystyle=S_{L}\,g_{1L}(x,k_{T})-\frac{k_{\perp}\cdot
S_{\perp}}{M_{N}}\,g_{1T}(x,k_{T})\,,$ $\displaystyle
h_{1s}^{\perp}(x,{\vec{k}}_{T})$
$\displaystyle=S_{L}\,h_{1L}^{\perp}(x,k_{T})-\frac{k_{\perp}\cdot
S_{\perp}}{M_{N}}\,h_{1T}^{\perp}(x,k_{T})\,.$ (D.20)

###  D.3 Quark-Gluon-Quark Correlator

Here, we repeat the steps shown in appendix D.2 for the quark-gluon-quark
correlators. Most intermediate steps have been discussed in full detail in
there, and here we will limit ourselves to the key steps.

####  D.3.1 TMDPDF

We begin with the quark-gluon-quark correlator for the incoming hadron, i.e.
corresponding to the TMDPDF. In ref. [21], it is defined as

$\displaystyle(\tilde{\Phi}_{A}^{\rho})^{\beta\beta^{\prime}}(x,{\vec{b}}_{T})$
$\displaystyle=\int\frac{\mathrm{d}b^{+}}{\sqrt{2}(2\pi)}e^{-\frac{\mathrm{i}}{2}xP_{N}^{-}b^{+}}\braket{h(P)}{\bar{\psi}^{\beta^{\prime}}(b)gA_{T}^{\rho}(0)\psi^{\beta}(0)}{h(P)}\,,$
(D.21)

where as usual $b=(b^{+},0,b_{\perp})$. Here, we have already converted to our
conventions following the same steps as in appendix D.2. Note that
$A_{T}^{\rho}$ is the transverse part of the gluon field, and hence $\rho$ is
a transverse index. For simplicity, we use light-cone gauge with $A^{+}=0$ but
neglect transverse Wilson lines. Note that ref. [21] also defines a correlator
$(\Phi_{A}^{\rho})^{\beta\beta^{\prime}}$ similar to eq. (D.21) but with a
covariant derivative $\mathrm{i}D^{\mu}$ instead of the gluon field. Since its
large component can be reduced to the leading-power correlator, it is
subtracted out to yield the purely transverse correlator in eq. (D.21). In our
approach, this is built in by using a collinear gluon field from the
beginning.

We contrast this with our correlator as defined in eq. (4.7),

$\displaystyle\hat{B}_{\mathcal{B}}^{\rho\beta\beta^{\prime}}(x,\xi,{\vec{b}}_{T})$
$\displaystyle=\Bigl{\langle}N\Bigr{\rvert}\bar{\chi}_{n}^{\beta^{\prime}}(b_{\perp})\,g{\mathcal{B}}_{n\perp,\,-\xi\omega_{a}}^{\rho}(0)\,\chi_{n,\,(1-\xi)\omega_{a}}^{\beta}(0)\Bigr{\rvert}N\Bigr{\rangle}$
(D.22)
$\displaystyle=\int\frac{\mathrm{d}b_{1}^{+}}{4\pi}\int\frac{\mathrm{d}b_{2}^{+}}{4\pi}e^{\frac{\mathrm{i}}{2}[\xi
b_{1}^{+}+(1-\xi)b_{2}^{+}]xP_{N}^{-}}\Bigl{\langle}N\Bigr{\rvert}\bar{\chi}_{n}^{\beta^{\prime}}(b_{\perp})\,g{\mathcal{B}}_{n\perp}^{\rho}(-b_{1}^{+})\,\chi_{n}^{\beta}(-b_{2}^{+})\Bigr{\rvert}N\Bigr{\rangle}\,.$

Here, we suppressed the overall $\theta(\omega_{a})$ for simplicity, and in
the second step we replaced the fields with fixed momentum fraction by their
Fourier transform as in eq. (D.10). Note that only the $\bar{\chi}_{n}$ field
has a transverse position, while the other fields only have a longitudinal
displacement. In particular, the fields have distinct lightcone positions,
reflecting that one is sensitive to their individual momenta.

To relate eq. (D.22) to eq. (D.21), we integrate over $\xi$ which fixes
$b_{1}^{+}=b_{2}^{+}\equiv-b^{+}$. Shifting this position into the antiquark
field, we obtain

$\displaystyle\int\\!\mathrm{d}\xi\,\hat{B}_{\mathcal{B}}^{\rho\beta\beta^{\prime}}(x,\xi,{\vec{b}}_{T})=\frac{1}{xP_{N}^{-}}\int\frac{\mathrm{d}b^{+}}{4\pi}e^{-\frac{\mathrm{i}}{2}b_{2}^{+}xP_{N}^{-}}\Bigl{\langle}N\Bigr{\rvert}\bar{\chi}_{n}^{\beta^{\prime}}(b)\,g{\mathcal{B}}_{n\perp}^{\rho}(0)\chi_{n}^{\beta}(0)\Bigr{\rvert}N\Bigr{\rangle}\,,$
(D.23)

where now $b^{\mu}=(b^{+},0,b_{\perp})$. By replacing the good components
$\chi_{n}$ of the collinear quarks with the projection onto the full collinear
fields as in eq. (D.12), we arrive at

$\displaystyle\int\\!\mathrm{d}\xi\,\hat{B}_{\mathcal{B}}^{\rho\beta\beta^{\prime}}(x,\xi,{\vec{b}}_{T})=\frac{(\not{n}\not{{\bar{n}}})^{\beta\alpha}}{4}\frac{(\tilde{\Phi}_{A}^{\rho})^{\alpha\alpha^{\prime}}(x,{\vec{b}}_{T})}{\sqrt{2}xP_{N}^{-}}\frac{(\not{{\bar{n}}}\not{n})^{\alpha^{\prime}\beta^{\prime}}}{4}\,.$
(D.24)

We remark that here, we have neglected some Wilson lines, which were dropped
in writing eq. (D.21). In eq. (D.23), collinear quark and gluon fields contain
Wilson lines to yield gauge-invariant building blocks. Likewise, for a general
gauge ref. [21] provides a gauge-invariant definition of eq. (D.21) that
dresses the fields in there with Wilson lines. Since anyway our Wilson line
structure will differ, due to the appearance of the extra momentum fraction
$\xi$, we do not bother to demonstrate explicitly that upon integration over
$\xi$ the Wilson line structures match between the approaches. Both procedures
should anyway be unique, up to choosing the Wilson line directions as outgoing
or incoming as appropriate for SIDIS or Drell-Yan kinematics.

Finally, we remark that the projectors in eq. (D.24) have no practical effect,
as the correlators $\tilde{\Phi}_{A}$ correlators in ref. [21] are
parameterized in momentum space without writing out subleading power terms,
with the form

$\displaystyle\tilde{\Phi}_{A}^{\rho}(x,{\vec{k}}_{T})$
$\displaystyle=\frac{xM_{N}}{2}\bigl{\\{}\cdots\bigr{\\}}\frac{\not{n}_{+}}{2}\,,$
(D.25)

where the ellipses contain the individual TMDs and will be presented in eq.
(5.2.3). Inserting this into eq. (D.24) and using that $n_{+}=n/\sqrt{2}$, we
obtain

$\displaystyle\int\\!\mathrm{d}\xi\,\hat{B}_{\mathcal{B}}^{\rho\beta\beta^{\prime}}(x,\xi,{\vec{b}}_{T})$
$\displaystyle=\frac{M_{N}}{4P_{N}^{-}}\bigl{\\{}\cdots\bigr{\\}}\frac{\not{n}}{2}\,.$
(D.26)

####  D.3.2 TMDFF

Next, we consider the quark-gluon-quark correlator for the fragmenting hadron,
i.e. corresponding to the TMDPDF. In ref. [21], it is defined as282828Note
that the color average is not made explicit in ref. [21]. Following e.g. ref.
[27], we have reinstated it here.

$\displaystyle(\tilde{\Delta}_{A}^{\rho})^{\alpha\alpha^{\prime}}(z,{\vec{b}}_{T})$
$\displaystyle=\frac{1}{2zN_{c}}\int\frac{\mathrm{d}b^{-}}{\sqrt{2}(2\pi)}e^{\frac{\mathrm{i}}{2}b^{-}(P_{h}^{+}/z)}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X}\braket{0}{gA_{T}^{\rho}(b)\psi_{\alpha}(b)}{hX}\braket{hX}{\bar{\psi}_{\alpha^{\prime}}(0)}{0}\,,$
(D.35)

where $b=(0,b^{-},b_{\perp})$, and as before we use lightcone gauge for
simplicity. We will also require the following conjugate operator:

$\displaystyle\bigl{[}\gamma_{0}\tilde{\Delta}_{A}^{\dagger\rho}(z,{\vec{b}}_{T})\gamma_{0}\bigr{]}^{\alpha\alpha^{\prime}}$
$\displaystyle=\frac{1}{2zN_{c}}\int\frac{\mathrm{d}b^{-}}{\sqrt{2}(2\pi)^{2}}e^{-\frac{\mathrm{i}}{2}b^{-}(P_{h}^{+}/z)}$
$\displaystyle\quad\times\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X}\braket{0}{\psi_{\alpha^{\prime}}(0)}{hX}\braket{hX}{gA_{T}^{\rho}(b)\bar{\psi}_{\alpha}(b)}{0}\,.$
(D.44)

Our corresponding correlator is defined in eq. (4.7) as

$\displaystyle\hat{\mathcal{G}}_{\mathcal{B}\,h/f}^{\rho\,\alpha\alpha^{\prime}}(z,\xi,{\vec{b}}_{T})$
$\displaystyle=\frac{1}{2zN_{c}}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X_{\bar{n}}}\operatorname{tr}\Bigl{\langle}0\Bigr{\rvert}\chi_{\bar{n}}^{\alpha}(b_{\perp})\Bigr{\rvert}h,X_{\bar{n}}\Bigr{\rangle}\Bigl{\langle}h,X_{\bar{n}}\Bigr{\rvert}\bar{\chi}_{\bar{n},\,-(1-\xi)\omega_{b}}^{\alpha^{\prime}}(0)\,g{\mathcal{B}}_{\bar{n}\perp,\,\xi\omega_{b}}^{\rho}(0)\Bigr{\rvert}0\Bigr{\rangle}$
(D.53)
$\displaystyle=\frac{1}{2zN_{c}}\int\frac{\mathrm{d}b_{1}^{-}}{4\pi}\int\frac{\mathrm{d}b_{2}^{-}}{4\pi}e^{\frac{\mathrm{i}}{2}[(1-\xi)b_{1}^{-}+\xi
b_{2}^{-}](P_{h}^{+}/z)}$
$\displaystyle\qquad\times\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X_{\bar{n}}}\operatorname{tr}\Bigl{\langle}0\Bigr{\rvert}\chi_{\bar{n}}^{\alpha}(b_{\perp})\Bigr{\rvert}h,X_{\bar{n}}\Bigr{\rangle}\Bigl{\langle}h,X_{\bar{n}}\Bigr{\rvert}\bar{\chi}_{\bar{n}}^{\alpha^{\prime}}(b_{1}^{-})\,g{\mathcal{B}}_{\bar{n}\perp}^{\rho}(b_{2}^{-})\Bigr{\rvert}0\Bigr{\rangle}\,.$
(D.62)

Here, we suppress all $\theta$ functions for brevity, and in the second step
replaced the fields of fixed label momentum by their Fourier representation as
in eq. (D.10).

To relate eq. (D.53) to eq. (D.3.2), we integrate over $\xi$ which fixes
$b_{1}^{-}=b_{2}^{-}\equiv-b^{-}$, and we shift the field positions by
$b^{-}$. This yields

$\displaystyle\int\mathrm{d}\xi\,\hat{\mathcal{G}}_{\mathcal{B}\,h/f}^{\rho\,\alpha\alpha^{\prime}}(z,\xi,{\vec{b}}_{T})$
$\displaystyle=\frac{1}{2zN_{c}}\frac{1}{P_{h}^{+}/z}\int\frac{\mathrm{d}b^{-}}{4\pi}e^{-\frac{\mathrm{i}}{2}b^{-}(P_{h}^{+}/z)}$
$\displaystyle\qquad\times\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X_{\bar{n}}}\operatorname{tr}\Bigl{\langle}0\Bigr{\rvert}\chi_{\bar{n}}^{\alpha}(b)\Bigr{\rvert}h,X_{\bar{n}}\Bigr{\rangle}\Bigl{\langle}h,X_{\bar{n}}\Bigr{\rvert}\bar{\chi}_{\bar{n}}^{\alpha^{\prime}}(0)\,g{\mathcal{B}}_{\bar{n}\perp}^{\rho}(0)\Bigr{\rvert}0\Bigr{\rangle}\,,$
(D.71)

where now $b=(0,b^{-},b_{\perp})$. Comparing this to eq. (D.3.2) by using the
quark projection in eq. (D.12), we find that

$\displaystyle\int\mathrm{d}\xi\,\hat{\mathcal{G}}_{\mathcal{B}\,h/f}^{\rho\,\alpha\alpha^{\prime}}(z,\xi,{\vec{b}}_{T})$
$\displaystyle=\frac{(\not{{\bar{n}}}\not{n})^{\alpha\beta}}{4}\frac{\bigl{[}\gamma_{0}\tilde{\Delta}_{A}^{\dagger\rho}(z,{\vec{b}}_{T})\gamma_{0}\bigr{]}^{\beta\beta^{\prime}}}{\sqrt{2}P_{h}^{+}/z}\frac{(\not{n}\not{{\bar{n}}})^{\beta^{\prime}\alpha^{\prime}}}{4}\,.$
(D.72)

This result is consistent with eq. (D.24) for the $B_{\mathcal{B}}$ operator.
As in that case, here we do not explicitly verify the Wilson structures in eq.
(D.3.2) and the correlator in ref. [21] for a generic gauge.

Finally, we remark that the projectors in eq. (D.72) have no practical effect,
as the correlators $\tilde{\Delta}_{A}$ correlators in ref. [21] are
parameterized in momentum space in a form that does not contain power
suppressed terms,

$\displaystyle\tilde{\Delta}_{A}^{\rho}(z,{\vec{k}}_{T})=\frac{M_{h}}{2z}\bigl{\\{}\cdots\bigr{\\}}\frac{\not{n}_{-}}{2}\,,$
(D.73)

where the ellipses contain the individual TMDs and will be presented in eq.
(5.26). Inserting this into eq. (D.72) and using that
$n_{-}={\bar{n}}/\sqrt{2}$, we obtain

$\displaystyle\int\mathrm{d}\xi\,\hat{\mathcal{G}}_{\mathcal{B}\,h/f}^{\rho\,\alpha\alpha^{\prime}}(z,\xi,{\vec{b}}_{T})$
$\displaystyle=\frac{M_{h}}{4P_{h}^{+}}\bigl{\\{}\cdots\bigr{\\}}^{\dagger}\frac{\not{{\bar{n}}}}{2}\,.$
(D.74)

Here, care has to be taken because $\bigl{\\{}\cdots\bigr{\\}}$ contains
imaginary parts and needs to be conjugated, as indicated.

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