aainstitutetext: Bogoliubov Laboratory of Theoretical Physics, JINR,
141980 Dubna, Russiabbinstitutetext: National Centre for Nuclear Research
(NCBJ),
02-093 Warsaw, Poland

# Archetypal Factorization and Gluon Poles in semi-exclusive reactions

I. V. Anikin b L. Szymanowski<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Within the archetypal factorization procedure, we study the gluon pole
contributions manifesting in the nucleon-lepton hard processes of hadron
production. We prove the dominant role of gluon pole contributions in the
processes of such kind. We analyse the different sources of complexity
associated with the gluon pole functions. As a practical application, we
derive a new single spin asymmetry related directly to the gluon poles which
can be studied experimentally.

##  1 Introduction

The hadron production in nucleon-lepton processes attracts still much
attentions not only of the experimental collaborations (for example, at JLab)
which investigate the composite (spin) structure of hadrons. From the
theoretical viewpoint, this kind of processes is interested because it opens
the window for the studies of the transverse momentum dependent functions, in
particular, appearing in the frame of semi-inclusive kinematics, see Fig. 1.
In this kinematics, the hadron tensor involves both the distribution and
fragmentation functions which are linked by the loop integrations over the
transverse components of momenta (this is a general feature of TMD-
factorizations). Based on the classical formulation of the factorization
theorem Efremov:1976ih ; Efremov:1978rn ; Efremov:1978xm ; Efremov:1978cu ,
the different non-perturbative functions (such as distribution functions and
fragmentation functions) bound by the dimensionful transverse momentum
integration cannot be considered as totally independent ones. Nevertheless,
there are different approaches where the mentioned non-factorized effects can
be diminished by the corresponding summation taking into account the
transverse momenta of partons (see, for example, Barone:2001sp  and the
references therein). Moreover, in the framework of semi-inclusive kinematics,
the photon and produced hadron transverse momenta are related each other with
the necessary condition $\vec{\bf q}_{\perp}=\vec{\bf
P}^{h}_{\perp}/z\sim\vec{\bf k}_{\perp}$, where $\vec{\bf k}_{\perp}$ denotes
the primordial transverse momentum of quark.

On the other hand, the attempts to use the approaches inspired by TMD-
factorization for description of the existing experimental data from JLab are
encountered the difficulties at rather a large value of $\vec{\bf q}_{\perp}$
(sometimes, it is named as the “$q_{T}$ crisis”).

However, the hadron production in nucleon-lepton collision can be also
considered using the semi-exclusive kinematics where the hadron tensor
contains two distribution amplitudes instead of one fragmentation function,
see Fig. 1. In the present paper, alternatively to the semi-inclusive
description, we propose an approach based on the well-defined and archetypal
factorization procedure (see for example Anikin:2020ipg ; Anikin:2017azc ;
Anikin:2017pch ; Anikin:2017vvn ; Anikin:2018fom ) applied to the semi-
exclusive mode of the hadron production processes. Within this approach, the
hadron tensor is given by the corresponding convolution of the distribution
function and the distribution amplitudes if a value of intermediate quark
fraction is large, $x\to 1$. Another feature of the used approach is that the
photon and produced hadron transverse momenta are not tied in contrast to the
approaches based on TMD-factorization. In this context, we calculate a new
single spin asymmetry which is associated with the gluon pole contribution for
a wide region of produced hadron transverse momentum. This asymmetry can be
also a object of experimental measurements at a large value of $\vec{\bf
P}^{h}_{\perp}$ using the existing data of JLab.

In the paper, we also give a proof that, in the limits of $x\to 1$ and
$m_{q}\to 0$, the leading order hadron tensor is actually suppressed in
comparison with the gluon pole contribution to the hadron tensor where the
gluon radiation has been taken into account. Based on this our principle
statement, we concentrate on the manifestation of gluon poles in the lepton-
hadron collision of hadron production.

As shown in Anikin:2017azc ; Anikin:2017pch ; Anikin:2017vvn ; Anikin:2018fom
; Anikin:2021osx , the use of contour gauge conception gives the strong
mathematical evidences for the nature of functional complexity in the
treatment of gluon poles. From the viewpoint of practical uses, the complexity
of the corresponding distribution functions related to the gluon poles is very
important because it leads to the different kinds of asymmetries. In addition
to the contour gauge conception, we investigate the different sources of
functional complexity which are associated with the gluon pole effects.

Within the factorization procedure we adhere in our paper, we introduce a new
kind of the transverse momentum dependent distribution functions which leads
to the different new observables. With the help of these new functions and the
gluon pole contributions it becomes to be possible to study the hadron spin
structure in a different light using the experimental data extracted from the
measurements of processes with a large value of transverse momentum of
produced hadron.

The paper is organized as follows. The kinematics is introduced and the
Sudakov decompositions are presented in Sec. 2. In Sec. 3, we obtain the
leading order hadron tensor and the hadron tensor which involves the gluon
pole contribution. Also, we describe the main stages of our factorization
procedure focusing on its important features. In Sec. 4, we formulate the
principal statement on the dominant role of gluon pole contributions to the
corresponding hadron tensor in comparison with the hadron tensor without the
gluon pole contribution. Sec. 5 is devoted to the derivation of the gauge-
invariant hadron tensor with gluon pole contributions and to the introduction
of the new transverse momentum dependent functions. We also analyse the
different sources of complexity for the function involving the gluon pole. The
new single spin asymmetry is presented and discussed in Sec. 6 . Sec. 7 is
reserved for a summary and outlook. Some technical issues are presented in
appendices. Appendix A contains the discussion on the different choices of the
frame systems. The alternative representation of functions with the on-shell
fields in correlators is presented in Appendix B. The off-shell extension of
the fields in the corresponding correlators is discussed in Appendix C.

##  2 Kinematics

To begin with, we specify our kinematics. We study the hadron (pion,
$\rho$-meson) production in the hard process which can be considered, from the
kinematical point of view, as the intermediate one between the semi-inclusive
and semi-exclusive reactions. Namely, we deal with

$\displaystyle\ell(l_{1})+N^{(\uparrow\downarrow)}(P_{2})\to\ell(l_{2})+h(P_{1}^{h})+\bar{q}(K)+X(P_{X})$
(1)

or

$\displaystyle\gamma^{*}(q)+N^{(\uparrow\downarrow)}(P_{2})\to
h(P_{1}^{h})+\bar{q}(K)+X(P_{X}).$ (2)

In Eqns.(1) and (2), $h$ denotes $\pi$ or $\rho$ triplet of mesons (in
general, any hadron) and the virtual photon produced by the lepton
($l_{1}=l_{2}+q$) has a large mass squared ($q^{2}=-Q^{2}$) while the photon
transverse momenta are small.

The Sudakov decompositions take the standard forms (for the sake of shortness,
we omit the four-dimension indices) Anikin:2017azc ; Anikin:2017pch ;
Anikin:2017vvn ; Anikin:2018fom

$\displaystyle
P_{2}\approx\frac{\tilde{Q}}{y_{B}\sqrt{2}}\,n+P_{2}^{\perp},\quad
P_{1}^{h}\approx\frac{\tilde{Q}}{x_{B}\sqrt{2}}\,n^{*}+P_{1}^{h\,\perp}\,,$
(3) $\displaystyle n^{*}=(1/\sqrt{2},\,{\bf 0}_{T},\,1/\sqrt{2}),\quad
n=(1/\sqrt{2},\,{\bf 0}_{T},\,-1/\sqrt{2})$ $\displaystyle
S\approx\frac{\lambda}{M_{2}}\,P_{2}+S_{\perp}\,$ (4)

for the hadron momenta and spin vector;

$\displaystyle
q=\frac{\tilde{Q}}{\sqrt{2}}\,n^{*}+\frac{\tilde{Q}}{\sqrt{2}}\,n+q_{\perp},\quad
q_{\perp}\ll\tilde{Q}=\sqrt{-Q^{2}},$ (5)

for the photon momentum. The hadron momenta $P_{1}^{h}$ and $P_{2}$ have the
plus and minus dominant light-cone components, respectively, see Fig. 1.

The differential cross-section takes therefore the following form

$\displaystyle d\sigma=\frac{d^{3}\vec{{\bf
l}}_{2}}{(2\pi)^{3}2l_{2}^{0}}\frac{d^{3}\vec{{\bf
P}}^{h}_{1}}{(2\pi)^{3}2E^{h}}{\cal L}_{\mu\nu}{\cal W}_{\mu\nu},$ (6)

where ${\cal L}_{\mu\nu}$ and ${\cal W}_{\mu\nu}$ denote the lepton and hadron
tensors, respectively.

Figure 1:  The hadron production in the semi-exclusive process, see the left
panel, and in the semi-inclusive process, see the right panel.

##  3 The hadron production in semi-exclusive processes

The process under our discussion resembles the standard Drell-Yan process, see
Fig. 1, where one of the initial hadron states is being replaced by the final
hadron state. In this context, we mainly follow the stages of our analysis
that have been used in a series of papers Anikin:2017azc ; Anikin:2017pch ;
Anikin:2017vvn ; Anikin:2018fom . So, by definition, the hadron tensor of (1)
is defined to be given by the following expression

$\displaystyle{\cal
W}_{\mu\nu}=\int(d{\mathbb{P}}{\mathbb{S}})\delta^{(4)}(q+P_{2}-P_{1}^{h}-K-P_{X})\times$
$\displaystyle\langle P_{2}|J^{em}_{\nu}(0)|P_{1}^{h},K,P_{X}\rangle\langle
P_{1}^{h},K,P_{X}|J^{em}_{\mu}(0)|P_{2}\rangle,$ (7)

where the matrix elements are written in the Heisenberg representation and the
phase space measure is defined by

$\displaystyle(d{\mathbb{P}}{\mathbb{S}})=\frac{d^{3}\vec{{\bf
K}}}{(2\pi)^{3}2K_{0}}\frac{d^{3}\vec{{\bf P}}_{X}}{(2\pi)^{3}2E_{X}}.$ (8)

Notice that from the viewpoint of factorization theorem the hadron tensor of
(3) is not yet in a factorized form. To apply the factorization procedure, the
hadron momenta $P_{2}$ and $P^{h}_{1}$ should be associated with the light-
cone minus and plus directions, respectively, which are well-separated. In
other words, there are no the strong interferences between the hadron dominant
directions. With these, the Fock states corresponding to the initial and final
hadrons can be permuted resulting in

$\displaystyle{\cal W}_{\mu\nu}\sim$ $\displaystyle\Big{\\{}\langle
0|\bar{\psi}(0)|P_{1}^{h},K\rangle\,\gamma_{\nu}\,\langle
P_{2}|\psi(0)|P_{X}\rangle\Big{\\}}\times$ (9) $\displaystyle\Big{\\{}\langle
P_{X}|\bar{\psi}(0)|P_{2}\rangle\,\gamma_{\mu}\,\langle
P_{1}^{h},K|\psi(0)|0\rangle\Big{\\}}$

for the matrix elements in (3).

###  3.1 The hadron tensor at leading order

In the paper, our main interest is associated with the gluon pole generated by
the radiative corrections. Nevertheless, we begin our consideration with the
detail analysis of leading order hadron tensor.

Making used the Fourier transform for the delta-function and the translation
transforms in Eqn. (3), the leading order hadron tensor ${\cal
W}_{\mu\nu}^{(0)}$ can readily be written down as

$\displaystyle{\cal W}_{\mu\nu}^{(0)}=\int(d^{4}\xi)e^{iq\xi}{\rm
tr}\big{[}\gamma_{\nu}\,\Phi(\xi)\,\gamma_{\mu}\,\bar{\Phi}(\xi)\big{]},$ (10)

where the hidden spinor indices related to the functions $\Phi$ and
$\bar{\Phi}$ are open, and

$\displaystyle\Phi(\xi)=\sum_{X}\int\frac{d^{3}\vec{{\bf
P}}_{X}}{(2\pi)^{3}2E_{X}}\langle P_{2}|\psi(\xi)|P_{X}\rangle\,\langle
P_{X}|\bar{\psi}(0)|P_{2}\rangle,$ (11)
$\displaystyle\bar{\Phi}(\xi)=\int\frac{d^{3}\vec{{\bf
K}}}{(2\pi)^{3}2K_{0}}\langle 0|\bar{\psi}(\xi)|P_{1}^{h},K\rangle\,\langle
P_{1}^{h},K|\psi(0)|0\rangle.$ (12)

In (10), the functions $\Phi(\xi)$ and $\bar{\Phi}(\xi)$ can be moved apart to
different positions in $x$-space by the integration with the corresponding
delta-functions, i.e.

$\displaystyle{\cal
W}_{\mu\nu}^{(0)}=\int(d^{4}\xi)e^{iq\xi}\int(d^{4}\eta_{1})(d^{4}\eta_{2})\times$
$\displaystyle\delta^{(4)}(\eta_{1}-\xi)\delta^{(4)}(\eta_{2}-\xi){\rm
tr}\big{[}\gamma_{\nu}\,\Phi(\eta_{1})\,\gamma_{\mu}\,\bar{\Phi}(\eta_{2})\big{]}.$
(13)

Afterwards, a series of Fourier transforms gives the following representation
for the hadron tensor through the momentum loop integrations

$\displaystyle{\cal W}_{\mu\nu}^{(0)}$ $\displaystyle=$
$\displaystyle\int(d^{4}k_{1})(d^{4}k_{2})\delta^{(4)}(k_{1}+k_{2}-q)\times$
(14) $\displaystyle{\rm
tr}\big{[}\gamma_{\nu}\,\Phi(k_{2})\,\gamma_{\mu}\,\bar{\Phi}(k_{1})\big{]},$

and

$\displaystyle\Phi(k_{2})=\int(d^{4}\eta_{2})e^{ik_{2}\eta_{2}}\times$ (15)
$\displaystyle\sum_{X}\int\frac{d^{3}\vec{{\bf
P}}_{X}}{(2\pi)^{3}2E_{X}}\langle P_{2}|\psi(\eta_{2})|P_{X}\rangle\,\langle
P_{X}|\bar{\psi}(0)|P_{2}\rangle,$
$\displaystyle\bar{\Phi}(k_{1})=\int(d^{4}\eta_{1})e^{ik_{1}\eta_{1}}\times$
(16) $\displaystyle\int\frac{d^{3}\vec{{\bf K}}}{(2\pi)^{3}2K_{0}}\langle
0|\bar{\psi}(\eta_{1})|P_{1}^{h},K\rangle\,\langle
K,P_{1}^{h}|\psi(0)|0\rangle.$

These functions can be expressed in terms of the corresponding distribution
functions after the use of Lorentz decompositions.

###  3.2 The archetypal (original) factorization procedure

The factorization theorem is the main tool of the hadron tensor calculation.
Generally speaking, the QCD factorization theorem, as one of asymptotic
methods, allows only the estimation of the amplitudes or hadron tensors
instead of exact calculations due to the quark confinement problem.

It is worth to remind that the original factorization theorem states the
following: if the kinematics has a large parameter, the short (hard) and long
(soft) distance dynamics can be independently separated out during the
asymptotical estimation procedure applied to given amplitudes (or hadron
tensors). The final result of such estimation or factorization has to be
presented in the form of the mathematical convolution over the dimensionless
parton fractions Efremov:1976ih ; Efremov:1978rn ; Efremov:1978xm ;
Efremov:1978cu .

To illustrate this typical factorization procedure, we consider the arbitrary
DY-like hadron tensor which involves two non-perturbative blobs, it reads
(here, for the sake of brevity, we omit all possible Lorentz indices)

$\displaystyle
W=\int(d^{4}k_{1})(d^{4}k_{2})E(k_{1},k_{2},q)\Phi_{1}(k_{1})\bar{\Phi}_{2}(k_{2}),$
(17)

where

$\displaystyle E(k_{1},k_{2},q)=\delta^{(4)}(k_{1}+k_{2}-q)\,{\cal
E}(k_{1},k_{2},q)$ $\displaystyle\Phi_{1}(k_{1})\stackrel{{\scriptstyle{\cal
F}_{1}}}{{=}}\langle\bar{\psi}(z_{1})\Gamma_{1}\psi(0)\rangle,$
$\displaystyle\bar{\Phi}_{2}(k_{2})\stackrel{{\scriptstyle{\cal
F}_{2}}}{{=}}\langle\bar{\psi}(0)\Gamma_{2}\psi(z_{2})\rangle$ (18)

and $\stackrel{{\scriptstyle{\cal F}_{i}}}{{=}}$ denotes the corresponding
Fourier transforms. In Eqn. (3.2), ${\cal E}(k_{1},k_{2},q)$ indicates the
product of different propagators appearing after the use of Wick’s theorem,
while the $\delta$-function reflects the corresponding momentum conservation
at the subprocess level, see Eqns. (10)-(14).

We emphasize that the representation of hadron tensor given by Eqn. (17) is
exact, before the factorization procedure has been applied. One of the
features pointing out that we deal with the non-factorized representation is
the presence of dimensionful loop integrations binding the product of
propagators with the non-perturbative functions. The dimensionful loop
integrations do not allow to treat the product of propagators, which pretends
to form the hard part, as independent of the non-perturbative (soft)
functions.

We begin with choosing the dominant directions dictated by the given process
kinematics. In the example of Eqn. (17), we have two dominant directions: one
of them is associated with the plus light-cone direction, the other to the
minus light-cone direction. The important condition for factorization is that
the dominant directions must be separated rather well in order to be
independent ones.

In the next step, we have to introduce the definitions of the dimensionless
parton fractions with the help of the following replacement:

$\displaystyle d^{4}k_{i}\Rightarrow
d^{4}k_{i}\int_{-1}^{+1}dx_{i}\delta(x_{i}-k_{i}^{\pm}/P_{i}^{\pm})$ (19)

depending on the chosen dominant direction. The physical spectral properties
of fractions are determined by the $\alpha$-representation where the
estimation procedure should be established.

Further, we expand the product of propagators (together with the
$\delta$-function, see Eqn. (3.2)) around the chosen dominant directions. As a
result, we obtain that

$\displaystyle W^{(0)}=\int(dx_{1})(dx_{2})E(x_{1}P^{+}_{1},x_{2}P^{-}_{2};q)$
$\displaystyle\times\Big{\\{}\int(d^{4}k_{1})\delta(x_{1}-k_{1}^{+}/P_{1}^{+})\Phi_{1}(k_{1})\Big{\\}}$
$\displaystyle\times\Big{\\{}\int(d^{4}k_{2})\delta(x_{2}-k_{2}^{-}/P_{2}^{-})\bar{\Phi}_{2}(k_{2})\Big{\\}}$
(20)

if we neglect the $k^{\perp}_{i}$-terms in the expansion; and

$\displaystyle
W^{(i,j)}=\int(dx_{1})(dx_{2})\sum_{i,j}E^{(i,j)}(x_{1}P^{+}_{1},x_{2}P^{-}_{2};q)$
$\displaystyle\times\Big{\\{}\int(d^{4}k_{1})\delta(x_{1}-k_{1}^{+}/P_{1}^{+})\,\prod_{i^{\prime}=1}^{i}k_{1\,i^{\prime}}^{\perp}\,\Phi_{1}(k_{1})\Big{\\}}$
$\displaystyle\times\Big{\\{}\int(d^{4}k_{2})\delta(x_{2}-k_{2}^{-}/P_{2}^{-})\,\prod_{j^{\prime}=1}^{j}k_{2\,j^{\prime}}^{\perp}\,\bar{\Phi}_{2}(k_{2})\Big{\\}}$
(21)

if the transverse momentum terms are essential in the expansion. Notice that
in Eqn. (3.2) the $k_{i}^{\perp}$-dependence appears in the form of the
corresponding integral moments only. Moreover, the parts of $W^{(0)}$ and
$W^{(i,j)}$ with $E$-function and $\Phi$-functions are not related by the
integration over $k_{i}^{\perp}$.

Last but not least, in order to achieve the results of Eqns. (3.2) and (3.2),
in the DY-like processes the four-dimensional $\delta$-function with the
argument describing the momentum conservation at the parton level should be
treated as the “hard” parts of hadron tensor. This statement has been proven
with the help of the factorization links introduced in Anikin:2006hm . In
particular, the decomposition of delta function around the hadron dominant
directions takes the form of

$\displaystyle\delta^{(4)}(k_{1}+k_{2}-q)=\delta^{(4)}(x_{1}P_{1}+x_{2}P_{2}-q)+$
$\displaystyle\frac{\partial\delta^{(4)}(k_{1}+k_{2}-q)}{\partial
k_{i}^{\rho}}\Big{|}_{k_{i}=x_{i}P_{i}}\biggr{.}\,k_{i\,\perp}^{\rho}+\ldots$
(22)

Notice that for our purposes it is enough to be limited by the first term of
decomposition in the r.h.s. of (3.2). The other important feature of
$\delta^{(4)}(\langle mom.conserv.\rangle)$ is that the momentum conservation
law is valid for any system chosen for the kinematical reason. In this
context, we may consider this delta function as a kind of the Lorentz
invariant despite the covariant form of momentum conservation.

Thus, once the factorization procedure applied, the leading order hadron
tensor takes the factorized form

$\displaystyle{\cal W}_{\mu\nu}^{(0)}=$ $\displaystyle\delta^{(2)}(\vec{\bf
q}_{\perp})\int(dx)(dy)\delta(xP^{h\,+}_{1}-q^{+})\delta(yP^{-}_{2}-q^{-})$
(23) $\displaystyle\times{\rm
tr}\big{[}\gamma_{\nu}\,\Phi(y)\,\gamma_{\mu}\,\bar{\Phi}(x)\big{]}\,,$

where, as stressed above, the mathematical convolution is given by the
integration over dimensionless fractions $x$ and $y$.

In contrast to the genuine-factorized form presented by Eqn. (23), the
approaches based on the formalism with the nonzero photon transverse momentum
and without the $\delta$-function expansion, see Barone:2001sp , result in the
leading order hadron tensor with the $k^{\perp}$-dependent non-perturbative
functions which has the following form

$\displaystyle\tilde{W}^{(0)}=\int(dx_{1})(dx_{2}){\cal
E}(x_{1}P^{+}_{1},x_{2}P^{-}_{2};q)$
$\displaystyle\times\Big{\\{}\int(d^{2}\vec{\bf k}^{\perp}_{1})(d^{2}\vec{\bf
k}^{\perp}_{2})\delta^{(2)}(\vec{\bf k}^{\perp}_{1}+\vec{\bf
k}^{\perp}_{2}-\vec{\bf q}^{\perp})$
$\displaystyle\times\int(dk^{+}_{1}dk_{1}^{-})\delta(x_{1}-k_{1}^{+}/P_{1}^{+})\Phi_{1}(k_{1})$
$\displaystyle\times\int(dk^{-}_{2}dk^{+}_{2})\delta(x_{2}-k_{2}^{-}/P_{2}^{-})\bar{\Phi}_{2}(k_{2})\Big{\\}},$
(24)

where the functions $\Phi(k_{1})$ and $\bar{\Phi}(k_{2})$ cannot be
independent each others because the two-dimensional integrations with the
$\delta$-function relate them. This leads to the factorization breaking
effects which should be compensated by some theoretical conditions and
assumptions. For example, the corresponding additional summation of gluon
radiations is required in the particular case in order to generate the
functional $k_{\perp}$-dependence à la the Sudakov exponentials of
$e^{-S(\vec{\bf k}^{2}_{\perp}/\Lambda^{2})}$-form which should minimize the
non-factorized effects.

###  3.3 The leading order function $\bar{\Phi}(k_{1})$

Let us now dwell on the function $\bar{\Phi}(k_{1})$ of (16), see Fig. 2, the
right panel. This function contains the integration over the intermediate on-
shell (anti)quark Fock state

$\displaystyle{\cal I}_{K}=\int\frac{d^{3}\vec{{\bf
K}}}{(2\pi)^{3}2K_{0}}|K\rangle\,\langle K|=$
$\displaystyle\int\frac{d^{3}\vec{{\bf
K}}}{(2\pi)^{3}2K_{0}}d^{+}_{\lambda}(K)|0\rangle\,\langle 0|d_{\lambda}(K)$
(25)

which can also be presented as

$\displaystyle{\cal
I}_{K}=\frac{E}{m_{q}}\int(d^{4}P)\delta(P^{2})\int(d^{4}K)\delta(K^{2})\delta^{(3)}(\vec{\bf
P}-\vec{\bf K})\times$
$\displaystyle[d^{+}_{\lambda}(K)v_{\lambda}(K)][d_{\lambda^{\prime}}(P)\bar{v}_{\lambda^{\prime}}(P)]\big{|}_{K_{0}=P_{0}=E},$
(26)

where $E$ and $m_{q}$ denote the energy and mass of the intermediate
(anti)quark, respectively.

In (3.3), $m_{q}$ is a result of the normalization condition

$\displaystyle
v_{\lambda}(K)\bar{v}_{\lambda^{\prime}}(K)=2m_{q}\delta_{\lambda\lambda^{\prime}},$
(27)

and the integral unit given by

$\displaystyle{\bf 1}=\int(d^{3}\vec{\bf P})\,\delta^{(3)}(\vec{\bf
P}-\vec{\bf K})$
$\displaystyle=2E\int(d^{4}P)\,\delta(P^{2})\,\delta^{(3)}(\vec{\bf
P}-\vec{\bf K})$ (28)

has been used. Also, the imposed condition $K_{0}=P_{0}=E$, which is shown in
Eqn. (3.3), can be written in the equivalent form as
$E\delta\left(P_{0}-K_{0}\right)$. All these tricks lead to the following
representation

$\displaystyle{\cal
I}_{K}=\frac{E^{2}}{m_{q}}\int(d^{4}P)\int(d^{4}K)\delta^{(4)}(P-K)\times$
(29) $\displaystyle\Big{\\{}\int(d^{4}x)e^{-iKx}\psi_{\text{on-
sh.}}(x)\Big{\\}}\Big{\\{}\int(d^{4}y)e^{iPy}\bar{\psi}_{\text{on-
sh.}}(y)\Big{\\}}.$

Here, $\psi_{\text{on-sh.}}$ denotes the on-shell particle (similary, for
$\bar{\psi}_{\text{on-sh.}}$), i.e.

$\displaystyle\int(d^{4}x)e^{-iKx}\psi_{\text{on-
sh.}}(x)=\delta(K^{2})[d^{+}_{\lambda}(K)v_{\lambda}(K)].$ (30)

Hence, the function $\bar{\Phi}(k_{1})$ of Eqn. (16) takes the form of

$\displaystyle\bar{\Phi}(k_{1})=\frac{E^{2}}{m_{q}}\,\bar{\Phi}^{(2)}_{\text{o}}(P^{h}_{1}-k_{1})\,\bar{\Phi}^{(1)}_{\text{o}}(k_{1})$
(31)

where

$\displaystyle\bar{\Phi}^{(1)}_{\text{o}}(k_{1})=\int(d^{4}\xi_{1})e^{ik_{1}\xi_{1}}\langle
0|\bar{\psi}(\xi_{1})\psi_{\text{on-sh.}}(0)|P_{1}^{h}\rangle$ (32)

and

$\displaystyle\bar{\Phi}^{(2)}_{\text{o}}(P^{h}_{1}-k_{1})=$
$\displaystyle\int(d^{4}\eta_{1})e^{-i(P^{h}_{1}-k_{1})\eta_{1}}\langle
P_{1}^{h}|\bar{\psi}_{\text{on-sh.}}(\eta_{1})\psi(0)|0\rangle.$ (33)

For the sake of completeness, it is instructive to give the representation of
$\bar{\Phi}(k_{1})$ in Eqn. (31) if the factorization procedure has been
applied. It reads (here, $\bar{x}_{1}=1-x_{1}$)

$\displaystyle\bar{\Phi}(x_{1})=\int(d^{4}k_{1})\delta(x_{1}-k_{1}^{+}/P_{1}^{h\,+})\,\bar{\Phi}(k_{1})=$
$\displaystyle P_{1}^{h,+}\int(dk_{1}^{-})(d^{2}\vec{\bf k}_{1}^{\perp})$
$\displaystyle\times\frac{E^{2}}{m_{q}}\bar{\Phi}^{(2)}_{\text{o}}(\bar{x}_{1},k_{1}^{-},\vec{\bf
k}_{1}^{\perp})\,\bar{\Phi}^{(1)}_{\text{o}}(x_{1},k_{1}^{-},\vec{\bf
k}_{1}^{\perp}).$ (34)

It is worth to notice that there is an alternative way to present the
functions like $\bar{\Phi}(k_{1})$ involving the on-shell fields in
correlators, see App. B for details.

As above mentioned, the intermediate quark (see, for example, Eqn. (3.3)) is
an on-shell particle, i.e.

$\displaystyle(k_{1}-P^{h}_{1})^{2}\equiv(k_{1}^{+}-P^{h\,+}_{1})k_{1}^{-}-(\vec{\bf
k}^{\perp}_{1}-\vec{\bf P}^{h\,\perp}_{1})^{2}=m^{2}_{q}$
$\displaystyle\Longrightarrow(k_{1}^{+}-P^{h\,+}_{1})=\frac{m^{2}_{q}}{k^{-}_{1}},$
(35)

where the condition $|\vec{\bf k}^{\perp}_{1}|\sim|\vec{\bf
P}^{h\,\perp}_{1}|\ll P^{h\,+}_{1}$ has been used. Therefore, using
$k_{1}^{+}=xP^{h\,+}_{1}$, Eqn. (3.3) shows that the on-shell quark condition
corresponds to the limits of $x\to 1$ and $m_{q}\to 0$.

Figure 2:  The hadron production in the semi-exclusive process: the next-to-
leading function $\bar{\Phi}^{(A)}(k_{1},\ell)$ (the left panel) and the
leading order function $\bar{\Phi}(k_{1})$ (the right panel).

###  3.4 Hadron tensor with the radiative correction before factorization

A straightforward extension of Anikin:2017azc ; Anikin:2017pch ;
Anikin:2017vvn ; Anikin:2018fom  gives the sum of standard and non-standard
contributions to the hadron tensor with radiative corrections, see Fig. 2, the
left panel, and Fig. 3.

Before the factorization procedure applied, we have

$\displaystyle{\cal
W}_{\mu\nu}^{(\text{stand.})}=\int(d^{4}k_{1})\,(d^{4}k_{2})\,\delta^{(4)}(k_{1}+k_{2}-q)$
$\displaystyle\times\int(d^{4}\ell)\,{\cal
D}_{\alpha\beta}(\ell)\text{tr}\big{[}\gamma_{\nu}\Gamma\gamma_{\alpha}S(k_{2}-\ell)\gamma_{\mu}\Gamma_{1}\gamma_{\beta}\Gamma_{2}\big{]}$
$\displaystyle\times\Phi^{[\Gamma]}(k_{2})\,\bar{\Phi}_{(2)}^{[\Gamma_{1}]}(k_{1};\ell)\,\bar{\Phi}_{(1)}^{[\Gamma_{2}]}(k_{1})\,\tilde{\delta}\big{(}(P_{1}^{h}-k_{1})^{2}\big{)}\,,$
(36)

for the standard diagram contribution; while, the non-standard diagram
contribution is given the following expression

$\displaystyle{\cal
W}_{\mu\nu}^{(\text{nonstand.})}=\int(d^{4}k_{1})\,(d^{4}k_{2})\,\delta^{(4)}(k_{1}+k_{2}-q)$
$\displaystyle\times\int(d^{4}\ell)\,{\cal
D}_{\alpha\beta}(\ell)\text{tr}\big{[}\gamma_{\nu}\Gamma\gamma_{\mu}S(k_{1})\gamma_{\alpha}\Gamma_{1}\gamma_{\beta}\Gamma_{2}\big{]}$
$\displaystyle\times\Phi^{[\Gamma]}(k_{2})\,\bar{\Phi}_{(2)}^{[\Gamma_{1}]}(k_{1};\ell)\,\bar{\Phi}_{(1)}^{[\Gamma_{2}]}(k_{1})\,\tilde{\delta}\big{(}(P_{1}^{h}-k_{1})^{2}\big{)}.$
(37)

The non-perturbative functions of Eqns. (3.4) and (3.4) are defined as

$\displaystyle\Phi^{[\Gamma]}(k_{2})=\int\sum\nolimits_{X}\,\int(d^{4}\eta_{2})\,e^{-ik_{2}\eta_{2}}\,$
(38) $\displaystyle\times\langle
P_{2},S_{\perp}|\text{tr}\big{[}\psi(0)|P_{X}\rangle\,\langle
P_{X}|\bar{\psi}(\eta_{2})\Gamma\big{]}|S_{\perp},P_{2}\rangle\ $
$\displaystyle\equiv-\int(d^{4}\eta_{2})\,e^{-ik_{2}\eta_{2}}\,\langle
P_{2},S_{\perp}|\bar{\psi}(\eta_{2})\Gamma\psi(0)|S_{\perp},P_{2}\rangle\ $

and

$\displaystyle\bar{\Phi}_{(2)}^{[\Gamma_{1}]}(k_{1};\ell)=$ (39)
$\displaystyle\int(d^{4}\eta_{1})\,e^{-i(P_{1}^{h}-\ell-
k_{1})\eta_{1}}\,\langle
P_{1}^{h}|\bar{\psi}(\eta_{1})\,\Gamma_{1}\psi(0)|0\rangle\,,$
$\displaystyle\bar{\Phi}_{(1)}^{[\Gamma_{2}]}(k_{1})=\int(d^{4}\xi)\,e^{ik_{1}\xi}\,\langle
0|\bar{\psi}(\xi)\,\Gamma_{2}\psi(0)|P_{1}^{h}\rangle\,,$ (40)

where the correlators contain the off-shell fermion operators, cf. Eqns. (32)
and (3.3). Moreover, for the sake of practical application, we introduce the
dimensionless analog of delta-function in the form of (see appendix B)

$\displaystyle\tilde{\delta}\big{(}(P_{1}^{h}-k_{1})^{2}\big{)}=$ (41)
$\displaystyle\lim_{\\{m_{q},\,\bar{x}_{1}\\}\to
0}\,\frac{m_{q}^{2}}{2\bar{x}_{1}P_{1}^{h\,+}}\delta\Big{(}k_{1}^{-}-\frac{(\vec{\bf
k}^{\perp}_{1}-\vec{\bf
P}_{1}^{h\,\perp})^{2}+m_{q}^{2}}{2\bar{x}_{1}P_{1}^{h\,+}}\Big{)}.$

which ensures that one of fermions in the function
$\bar{\Phi}_{(1)}^{[\Gamma_{2}]}(k_{1})$ becomes the on-shell one.

###  3.5 The next-to-leading order function $\bar{\Phi}^{(A)}(k_{1},\ell)$
with gluon radiation

In the standard and non-standard hadron tensors, see Eqns. (3.4) and (3.4),
the next-to-leading function $\bar{\Phi}^{(A)}(k_{1},\ell)$ (see Fig. 2, the
left panel) extends the leading order function $\bar{\Phi}(k_{1})$ (see Fig.
2, the right panel) and includes the functions $\bar{\Phi}_{(2)}(k_{1};\ell)$
and $\bar{\Phi}_{(1)}(k_{1})$ of Eqns. (39) and (40) (here, we omit the
$\Gamma$-structure in each functions). The gluon momentum $\ell$ dependence of
$\bar{\Phi}_{(2)}(k_{1};\ell)$ indicates the fact that we deal with the next-
to-leading order due to the gluon radiation.

To demonstrate this, we begin with the representation of the leading order
function $\bar{\Phi}(k_{1})$ given by Eqn. (31). We now take into account the
first order of interaction in the correlator and write the following

$\displaystyle\bar{\Phi}^{(A)}(k_{1})=\frac{E^{2}}{m_{q}}\,\Big{\\{}\int(d^{4}\xi)e^{-i(P^{h}_{1}-k_{1})\xi}\int(d^{4}z)$
$\displaystyle\times\langle
P_{1}^{h}|\psi(0)\big{[}\bar{\psi}(z)\hat{A}(z)\psi(z)\big{]}\bar{\psi}_{\text{on-
sh.}}(\xi)|0\rangle\Big{\\}}$
$\displaystyle\times\Big{\\{}\int(d^{4}\eta)e^{ik_{1}\eta}\langle
0|\psi_{\text{on-sh.}}(0)\bar{\psi}(\eta)|P_{1}^{h}\rangle\Big{\\}}.$ (42)

Here, the first figure-bracketed correlator with interactions has to be
understood as the correlator written in the interaction representation (the
time-ordering symbol has been omitted). Using Wick’s theorem and Fourier
transformations, we derive that

$\displaystyle\bar{\Phi}^{(A)}(k_{1})=\frac{E^{2}}{m_{q}}\,\Big{\\{}\int(d^{4}z)e^{-i(P^{h}_{1}-k_{1})z}\int(d^{4}\ell)$
$\displaystyle\times\langle
P_{1}^{h}|\psi(0)\bar{S}(k_{1}-P^{h}_{1}+\ell)\hat{A}(\ell)\bar{\psi}_{\text{on-
sh.}}(z)|0\rangle\Big{\\}}$
$\displaystyle\times\Big{\\{}\int(d^{4}\eta)e^{ik_{1}\eta}\langle
0|\psi_{\text{on-sh.}}(0)\bar{\psi}(\eta)|P_{1}^{h}\rangle\Big{\\}},$ (43)

or in the equivalent form it reads

$\displaystyle\bar{\Phi}^{(A)}(k_{1})=\frac{E^{2}}{m_{q}}\,\Big{\\{}\int(d^{4}\ell)\,\bar{\Phi}^{(2)}_{\text{o}}(P^{h}_{1}-k_{1})$
$\displaystyle\times\bar{S}(k_{1}-P^{h}_{1}+\ell)\hat{A}(\ell)\Big{\\}}\,\bar{\Phi}^{(1)}_{\text{o}}(k_{1}).$
(44)

##  4 Dominant role of gluon pole contribution

In this section, we give a proof that the next-to-leading order (gluon pole
contribution) function $\bar{\Phi}^{(A)}(k_{1},\ell)$ is dominating over the
leading order function $\bar{\Phi}(k_{1})$ in the kinematical region of
$x_{1}\to 1$ and $m_{q}\to 0$. We remind that the limits of $x_{1}\to 1$ and
$m_{q}\to 0$ correspond to the fact that the intermediate quark is on-shell,
see Eqn. (3.3). The result of this section can be considered as a principle
finding of our study.

The kinematical and model-independent prefactor $E^{2}/m_{q}$ which appears in
both the leading and next-to-leading functions plays the crucial role in our
analysis. Namely, one can see that $E^{2}/m_{q}$ at the leading function
$\bar{\Phi}(k_{1})$ of Eqn. (31) nullify this contribution in the certain
domain. Indeed, using the kinematical constrains inspired by (3), we have

$\displaystyle\bar{\Phi}(k_{1})\Rightarrow\frac{E^{2}}{m_{q}}=\frac{(k_{1}^{+}-P_{1}^{h,+}+k_{1}^{-})^{2}}{2m_{q}}$
(45)
$\displaystyle\Rightarrow\frac{(\bar{x}_{1}P_{1}^{h,+}+k_{1}^{-})^{2}}{2m_{q}}\Big{|}^{x_{1}\to
1}_{m_{q}\to 0}\sim\frac{\lim_{\epsilon\to 0}\epsilon^{2}}{\lim_{\delta\to
0}\delta}\sim\lim_{\epsilon\to 0}\epsilon\equiv[0]^{1},$

where $\sim$ means “behaves as”.

On the other hand, the situation is changed drastically if we consider the the
next-to-leading function related to the radiative correction. Focusing on the
gluon pole contribution to Eqn. (3.5), which exists at $x_{2}=x_{1}$, we
obtain that

$\displaystyle\bar{\Phi}^{(A)}(k_{1})\Big{|}^{x_{1}\to 1}_{m_{q}\to
0}\Rightarrow$
$\displaystyle\frac{(k_{1}^{+}-P_{1}^{h,+}+k_{1}^{-})^{2}}{2m_{q}}\bar{S}(k_{1}-P^{h}_{1}+\ell)\Big{|}^{x_{1}\to
1}_{m_{q}\to
0}=\frac{(k_{1}^{+}-P_{1}^{h,+}+k_{1}^{-})^{2}}{2m_{q}}\frac{\gamma^{+}}{k^{+}_{1}-P^{h\,+}_{1}+\ell^{+}}\Big{|}^{x_{1}\to
1}_{m_{q}\to 0}$ $\displaystyle\sim\frac{\lim_{\epsilon\to
0}\epsilon^{2}}{\lim_{\delta\to 0}\delta\,\lim_{\eta\to
0}\eta}\sim\lim_{\epsilon\to 0}\frac{\epsilon^{2}}{\epsilon^{2}}\equiv[1],$
(46)

where the representation given by

$\displaystyle
k^{+}_{1}-P^{h\,+}_{1}+\ell^{+}=-\bar{x}_{1}P^{h\,+}_{1}+(x_{2}-x_{1})P^{h\,+}_{1}$
(47)

has been applied. For the function $\bar{\Phi}^{(A)}(k_{1})$, we are
interested in the combination of
$\bar{\Phi}^{(2)\,\text{tw-2}}_{\text{o}}\otimes\bar{\Phi}^{(1)\,\text{tw-3}}_{\text{o}}$
which results in the $\gamma^{+}$-term of the propagator in the second line of
Eqn. (4). Thanks for Eqn. (4), one can see that $[0]$ stemming from the
combination $E^{2}/m_{q}$ is compensated by $[0]$ from the (anti)quark
propagator if we deal with the gluon pole contribution.

It is important to stress that an alternative consideration of function
$\bar{\Phi}^{(A)}(k_{1})$ described in Appendix B leads to the same behaviour
of $\bar{\Phi}^{(A)}(k_{1})$ as in Eqn. (4).

Therefore, we can conclude that the leading order hadron is actually
suppressed in comparison with the gluon pole contribution to the hadron tensor
in the limits of $x_{1}\to 1$ and $m_{q}\to 0$. Based on that, we are able to
concentrate entirely on the contributions associated with the gluon poles.

##  5 Gauge-invariant hadron tensor with the radiative correction in the
factorized form

We are in a position to discuss the final expression for the hadron tensor. We
remind that the concrete $\Gamma$ matrix projections (or the Fierz
projections) in the corresponding functions depend on the considered case. For
example, in order to describe the pion or rho meson production, the
corresponding functions receive the $\gamma$-structure as (see (3.4) and (39),
(40))

$\displaystyle\Gamma\otimes\Phi^{[\Gamma]}\Rightarrow\gamma^{+}\otimes\Phi^{[\gamma^{-}]}\oplus\sigma^{+\perp}\otimes\Phi^{[\sigma^{-\perp}]}$
(48)

for the leading twist two nucleon distribution functions;

$\displaystyle\Gamma_{1}\otimes\bar{\Phi}_{(2)}^{[\Gamma_{1}]}\Rightarrow\gamma^{-}(\gamma_{5})\otimes\bar{\Phi}_{(2)}^{[\gamma^{+}(\gamma_{5})]},$
(49)
$\displaystyle\Gamma_{2}\otimes\bar{\Phi}_{(1)}^{[\Gamma_{2}]}\Rightarrow\gamma_{\rho}^{\perp}(\gamma_{5})\otimes\bar{\Phi}_{(1)}^{[\gamma^{\perp}_{\rho}(\gamma_{5})]}\oplus\sigma^{+-}(\gamma_{5})\otimes\bar{\Phi}_{(1)}^{[\sigma^{-+}(\gamma_{5})]}$
(50)

for the pion and rho meson distribution amplitudes which correspond to the
twist two (49) and the twist three (50).

While the factorization procedure applied, we deal with the gauge invariant
hadron tensor that takes the form of

$\displaystyle{\cal W}^{\mu\nu}=\delta^{(2)}(\vec{\bf
q}_{\perp})\int(dx_{1})\,(dy)\,\delta(x_{1}P^{h,+}_{1}-q^{+})\delta(yP^{-}_{2}-q^{-})$
$\displaystyle\times
F(y)\int(dx_{2})\widetilde{B}(x_{1},x_{2})\,\frac{T^{\nu}}{P^{h}_{1}\cdot
P_{2}}\Big{[}\frac{P^{h\,\mu}_{1}}{y}-\frac{P^{\mu}_{2}}{x_{1}}\Big{]}\,,$
(51)

where

$\displaystyle F(y)=\begin{pmatrix}f_{T}(y)\vspace{0.2cm}\\\
h_{1}(y)\end{pmatrix},\quad
T^{\nu}=\begin{pmatrix}\varepsilon^{P^{h}_{1}+S_{\perp}P_{2}^{\perp}}V^{\nu}_{\perp}\vspace{0.2cm}\\\
\varepsilon^{\nu S_{\perp}P_{2}P^{h}_{1}}_{\perp}\end{pmatrix}$ (52)

with $V^{\nu}_{\perp}=\\{P^{h\,\perp}_{1},\,e^{\perp}\\}^{\nu}$. The
distribution function $F(y)$ represents the $k^{\perp}$-integrated function
Barone:2001sp .

The structure function (or distribution function) $\widetilde{B}(x_{1},x_{2})$
related to gluon pole is defined as Anikin:2017azc

$\displaystyle\widetilde{B}(x_{1},x_{2})=\frac{1}{2}\frac{\bar{\Phi}_{(1)}^{\text{tw-3}}(x_{1})\stackrel{{\scriptstyle\hskip
5.69046pt\vec{\bf
k}_{1}^{\,\perp}}}{{\circledast}}\bar{\Phi}_{(2)}^{\text{tw-2}}(x_{2})}{x_{2}-x_{1}-i\epsilon},$
(53)

where the transverse momentum integration is given by

$\displaystyle\bar{\Phi}_{(1)}^{\text{tw-3}}(x_{1})\stackrel{{\scriptstyle\hskip
5.69046pt\vec{\bf
k}_{1}^{\,\perp}}}{{\circledast}}\bar{\Phi}_{(2)}^{\text{tw-2}}(x_{2})=\int(d^{2}\vec{\bf
k}_{1}^{\,\perp})\,\bar{\Phi}_{(1)}^{\text{tw-3}}(x_{1};\vec{\bf
k}_{1}^{\,\perp})\times$
$\displaystyle\int(d\ell^{-}d^{2}\vec{\ell}_{\,\perp})\frac{\bar{\Phi}_{(2)}^{\text{tw-2}}(\bar{x}_{2};\vec{\bf
k}_{1}^{\,\perp};\ell^{-}\vec{\ell}_{\,\perp})}{\ell^{-}-\vec{\ell}^{\,2}_{\perp}/(2x_{21}P_{1}^{h\,+})+i\,{\rm
sign}(x_{21})\,\tilde{\epsilon}}.$ (54)

Figure 3: The hadron production in the semi-exclusive process: the standard
diagram contribution (the left panel) and the non-standard diagram
contribution (the right panel) to the gauge invariant hadron tensor. The
dominant quark and gluon momenta $k_{1}$ and $\ell$ lie along the plus
direction while the dominant antiquark momentum $k_{2}$ – along the minus
direction.

###  5.1 Hadron tensor with the radiative correction: the case of
$k_{\perp}$-dependent nucleon distribution functions

The gauge invariant hadron tensor of (5) can be readily extended to the case
with the essential $k_{\perp}$-dependence of the nucleon distribution
function. As the first stage, we consider the nucleon matrix element of the
quark vector projection given by (see (48))

$\displaystyle\Phi^{[\gamma^{-}]}(k_{2})=\,\varepsilon^{-+k^{\perp}_{2}\alpha}\begin{pmatrix}f_{(1)}(y,\vec{\bf
k}_{2}^{\perp})\,S_{\alpha}^{\perp}\vspace{0.2cm}\\\ f_{(2)}(y,\vec{\bf
k}_{2}^{\perp})\,s_{\alpha}^{\perp}\vspace{0.2cm}\end{pmatrix},$ (55)

where $S_{\alpha}^{\perp}$ and $s_{\alpha}^{\perp}$ stand for the nucleon and
quark spin covariant vectors, respectively. Notice that the function
$f_{(1)}(y,\vec{\bf k}_{2}^{\perp})$ is nothing but the well-known function
$f_{1\,T}^{\perp}(y,\vec{\bf k}_{2}^{\perp})$, while the $k_{\perp}$-dependent
function $f_{(2)}(y,\vec{\bf k}_{2}^{\perp})$ is a new function which has been
first introduced in Anikin:2021zxl .

Hence, for the gauge invariant hadron tensor we have the following expression

$\displaystyle{\cal W}_{(1),(2)}^{\mu\nu}=\delta^{(2)}(\vec{\bf
q}_{\perp})\int(dx_{1})\,(dy)\,\delta(x_{1}P^{h,+}_{1}-q^{+})$
$\displaystyle\times\delta(yP^{-}_{2}-q^{-})\,\Big{\\{}\int(d^{2}\vec{\bf
k}_{2}^{\perp})F^{\alpha}_{(1),(2)}(y,\vec{\bf
k}_{2}^{\perp})\,k_{2\,\alpha}^{\perp}\Big{\\}}$
$\displaystyle\times\int(dx_{2})\widetilde{B}(x_{1},x_{2})\,\frac{V^{\nu}_{\perp}}{P^{h}_{1}\cdot
P_{2}}\Big{[}\frac{P^{h\,\mu}_{1\,\mu}}{y}-\frac{P^{\mu}_{2}}{x_{1}}\Big{]}\,,$
(56)

where

$\displaystyle F^{\alpha}_{(1),(2)}(y,\vec{\bf
k}_{2}^{\perp})=\begin{pmatrix}f_{(1)}(y,\vec{\bf
k}_{2}^{\perp})\,\varepsilon^{P^{h}_{1}+\alpha S^{\perp}}\vspace{0.2cm}\\\
f_{(2)}(y,\vec{\bf k}_{2}^{\perp})\,\varepsilon^{P^{h}_{1}+\alpha
s^{\perp}}\vspace{0.2cm}\end{pmatrix},$ (57)

with the functions $f_{(1)}(y,\vec{\bf k}_{2}^{\perp})$ and
$f_{(2)}(y,\vec{\bf k}_{2}^{\perp})$ corresponding to the nucleon polarized
and nucleon unpolarized cases. Then, we focus on the quark axial-vector
projection of the nucleon correlator which reads

$\displaystyle\Phi^{[\gamma^{-}\gamma_{5}]}(k_{2})=\,\varepsilon^{-+k^{\perp}_{2}P_{2}^{\perp}}\,f_{(3)}(y,\vec{\bf
k}_{2}^{\perp})$ (58)

and defines the other new $k_{\perp}$-dependent function Anikin:2021zxl  .

The covariant (invariant) integration is one of our essential tool (see, for
example, Efimov:1993zg ). It needs to define the set of exterior Lorentz
tensors which are, in our case, $(P_{2}^{\perp},S^{\perp},s^{\perp})$.
Appendix D summarizes the main stages of this technique.

Notice that the covariant integration of $f_{(3)}(y,\vec{\bf
k}_{2}^{\perp})k^{\perp}_{2\,\alpha}$ over $d^{2}\vec{\bf k}_{2}^{\perp}$ does
not contain the vector $P_{2}^{\perp}$. Instead, the transverse
$k_{\perp}$-dependence should be realized by the quark spin covariant vector
only. Indeed, the Lorentz covariant integration gives us the following

$\displaystyle\int(d^{2}\vec{\bf k}_{2}^{\perp})f_{(3)}(y,\vec{\bf
k}_{2}^{\perp})k_{2\,\alpha}^{\perp}=\varepsilon^{\alpha
s^{\perp}+-}{\mathbb{A}}\big{[}f_{(3)}\big{]}$ (59)

with

$\displaystyle{\mathbb{A}}\big{[}f_{(3)}\big{]}=\int(d^{2}\vec{\bf
k}_{2}^{\perp})f_{(3)}(y,\vec{\bf
k}_{2}^{\perp})\frac{\,\varepsilon^{k_{2}^{\perp}s^{\perp}+-}}{(s^{\perp})^{2}}.$
(60)

Therefore, in the case of axial correlator, the gauge invariant hadron tensor
reads

$\displaystyle{\cal W}_{(3)}^{\mu\nu}=\delta^{(2)}(\vec{\bf
q}_{\perp})\int(dx_{1})\,(dy)\,\delta(x_{1}P^{h,+}_{1}-q^{+})\delta(yP^{-}_{2}-q^{-})$
$\displaystyle\times\int(dx_{2})\widetilde{B}(x_{1},x_{2})|P^{\perp}_{2}|\cos\phi_{PS}\,\varepsilon^{\nu+P^{h}_{1}P^{h}_{1\,\perp}}$
$\displaystyle\times\frac{1}{P^{h}_{1}\cdot
P_{2}}\,\Big{[}\frac{P^{h\,\mu}_{1}}{y}-\frac{P^{\mu}_{2}}{x_{1}}\Big{]}{\mathbb{A}}\big{[}f_{(3)}\big{]},$

where $|P^{\perp}_{2}|\cos\phi_{PS}=(s^{\perp}P^{\perp}_{2})$ provided
$|s^{\perp}|=1$.

To conclude this subsection, one remark is in order. Having restored the
corresponding Wilson line in the nucleon correlators, we may conclude that the
$k_{\perp}$-dependent parton functions possess the non-trivial properties
under the time-reversal transforms because these transforms convert the
future-pointed WL to the past-pointed WL Boer:2003cm .

###  5.2 Kinematical constraints for extractions of $k_{\perp}$-dependent
$f_{(i)}$-functions

The covariant (invariant) integrations of (55) and (58) give the constraint
conditions imposed on the corresponding angles in order to single out the
contribution of given $f_{(i)}$-functions, see Appdendix D.

Let us consider the following integration, which appears in the hadron tensor
after the factorization theorem used,

$\displaystyle{\cal I}_{1}=\int(d^{2}\vec{\bf
k}_{2}^{\perp})f_{(1)}(y,\vec{\bf
k}_{2}^{\perp})\,\,\varepsilon^{-+S_{\perp}k^{\perp}_{2}},$ (62)

where $\varepsilon^{-+S_{\perp}k^{\perp}_{2}}=\vec{\bf
S}_{\perp}\wedge\vec{\bf k}_{2}^{\perp}$. Hence, we have (we omit the unit
vector $\vec{\bf n}$ defining the direction of vector product)

$\displaystyle{\cal I}_{1}=|\vec{\bf S}_{\perp}|\int(d^{2}\vec{\bf
k}_{2}^{\perp})f_{(1)}(y,\vec{\bf k}_{2}^{\perp})\,|\vec{\bf
k}_{2}^{\perp}|\sin(\phi_{S}-\phi_{k_{2}}),$ (63)

where, and in what follows, the angle $\phi_{A}$ is defined in
$(\hat{x},\hat{y})$-plane as the angle between the vector $A$ and
$\hat{x}$-axis with $A=(S^{\perp},s^{\perp},P_{2}^{\perp},k_{2}^{\perp})$. On
the other hand, the integration ${\cal I}_{1}$ can be presented with the help
of the covariant integration as (see Appendix D)

$\displaystyle{\cal I}_{1}=\int(d^{2}\vec{\bf
k}_{2}^{\perp})f_{(1)}(y,\vec{\bf k}_{2}^{\perp})$ (64)
$\displaystyle\times\Big{\\{}\,\varepsilon^{-+S_{\perp}P^{\perp}_{2}}\frac{(\vec{\bf
k}^{\perp}_{2}\vec{\bf P}^{\perp}_{2})}{(\vec{\bf
P}^{\perp}_{2})^{2}}+\,\varepsilon^{-+S_{\perp}\alpha}\varepsilon^{\alpha
s_{\perp}-+}\frac{\varepsilon^{k^{\perp}_{2}s_{\perp}-+}}{\vec{\bf
s}_{\perp}^{2}}\Big{\\}}.$

Comparing the representations with and without the covariant integrations, we
obtain that

$\displaystyle\int(d^{2}\vec{\bf k}_{2}^{\perp})f_{(1)}(y,\vec{\bf
k}_{2}^{\perp})\,|\vec{\bf
S}_{\perp}|\Big{\\{}\sin(\phi_{S}-\phi_{k_{2}})-\sin(\phi_{S}-\phi_{P_{2}})$
$\displaystyle\times\cos(\phi_{P_{2}}-\phi_{k_{2}})-\sin(\phi_{s}-\phi_{k_{2}})\cos(\phi_{s}-\phi_{S})\Big{\\}}=0,$
(65)

where all angles are determined in the two-dimensional Euclidian space forming
the perpendicular plane, i.e.  $\forall\phi\in\mathbb{R}^{2}_{\perp}$. The
equation of (5.2) has a solution given by $\phi_{P_{2}}=\phi_{s}-\pi n$. On
the other hand, this condition can be considered as a necessity condition for
the covariant integration.

In the similar manner, we can analyse the other $f_{(i)}$-functions. Finally,
we can fix all the kinematical restrictions on the angles which allow to
extract the corresponding $f_{(i)}$-function contributions to the hadron
tensor. We have

$\displaystyle f_{(1)}(y,\vec{\bf k}_{2}^{\perp})\,\,\text{extracted
at}\,\,\big{\\{}\phi_{P_{2}}=\phi_{s}-\pi n\big{\\}};$ (66) $\displaystyle
f_{(2)}(y,\vec{\bf k}_{2}^{\perp})\,\,\text{extracted
at}\,\,\big{\\{}\phi_{P_{2}}=\phi_{S}+\pi,\phi_{S}=\phi_{s}-\frac{\pi}{2}\big{\\}};$
$\displaystyle f_{(3)}(y,\vec{\bf k}_{2}^{\perp})\,\,\text{extracted
at}\,\,\big{\\{}\phi_{S(\text{or}\,\,s)}=\phi_{k_{2}},\phi_{s(\text{or}\,\,S)}=\phi_{P_{2}}\big{\\}}.$

The remaining angles, which are not shown in (66), should be considered as
free angles.

On the other hand, the kinematical constraints of Eqn. (66) give a possibility
to express the quark spin angle $\phi_{s}$ through the hadron angles
$\phi_{P_{2}}$ and $\phi_{S}$ which are available in experiments.

###  5.3 On the complexity of $\widetilde{B}(x_{1},x_{2})$-function

The complexity of $\widetilde{B}(x_{1},x_{2})$-function given by the
representation of (53) together with (5) is very important property which
influences on the observables. We concentrate on the analysis of complexity
and we determine the possible reasons of its origin.

From Eqns. (53) and (5), one can immediately see the first source of
complexity which is due to the gluon pole at $x_{2}=x_{1}$ regularized by the
complex prescription in the frame of contour gauge conception Anikin:2017azc ;
Anikin:2021osx .

Then, we discuss the momentum dependence of nonperturbative function related
to the correlator. The exact form of momentum dependence is dictated by the
detailization of interactions in the correlator. Indeed, as an illustrative
example, let us consider the simplest function $\Phi^{[\Gamma]}(k)$ given by
(the symbol of time-ordering is not shown)

$\displaystyle\Phi^{[\Gamma]}(k)=\int(d^{4}z)e^{+ikz}\langle\bar{\psi}(0)\Gamma\psi(z)\mathbb{S}\rangle.$
(67)

Here, we use the interaction representation with the $\mathbb{S}$-matrix for
the correlator. If we neglect the interactions in the correlator, i.e.
$\mathbb{S}\to\mathbb{I}$, we can see that all $k$-dependence is accumulated
in the exponential:

$\displaystyle\Phi^{[\Gamma]}(k)\Big{|}_{\mathbb{S}=\mathbb{I}}=\int(d^{4}z)e^{+ikz}\langle\bar{\psi}(0)\Gamma\psi(z)\rangle.$
(68)

Having integrated the function $\Phi^{[\Gamma]}(k)$ over $k^{-}$ and $\vec{\bf
k}_{\perp}$, we obtain that the coordinate $z$ of $\psi$-function lies on the
light-cone minus direction:

$\displaystyle\int(dk^{-})(d^{2}\vec{\bf
k}_{\perp})\Phi^{[\Gamma]}(k^{+},k^{-},\vec{\bf
k}_{\perp})\Big{|}_{\mathbb{S}=\mathbb{I}}=\int(d^{4}z)e^{+ik^{+}z^{-}}\,\delta(z^{+})\,\delta^{(2)}(\vec{\bf
z}_{\perp})\langle\bar{\psi}(0)\Gamma\psi(z)\rangle.$ (69)

However, the momentum dependence becomes more involved if we take into
account, say, the second order of interactions. Namely, replacing $\mathbb{S}$
on $\mathbb{S}^{(2)}$ in Eqn. (67), we have the following expression

$\displaystyle\Phi^{[\Gamma]}(k)\Big{|}_{\mathbb{S}=\mathbb{S}^{(2)}}=\int(d^{4}z)e^{+ikz}\langle\bar{\psi}(0)\Gamma\psi(z)\Big{[}\int(d^{4}\xi)\bar{\psi}(\xi)\hat{A}(\xi)\psi(\xi)\Big{]}^{2}\rangle=$
$\displaystyle\int(d^{4}\xi)e^{+ik\xi}\langle\bar{\psi}(0)\Gamma\Big{[}S(k)\int(d^{4}\ell)\gamma_{\alpha}S(k-\ell)\gamma_{\beta}{\cal
D}_{\alpha\beta}(\ell)\Big{]}\psi(\xi)\rangle$ $\displaystyle+(\text{the other
contributions from Wick's theorem}).$ (70)

One can see that in Eqn. (5.3) the $k$-dependence of $\Phi^{[\Gamma]}(k)$ is
determined by the exponential and propagators, cf. Eqn. (68). Thus, neglecting
the corresponding interaction, we simplify the momentum dependence of
nonperturbative functions.

In this connection, if we neglect the interaction in the channel defined by
the light-cone minus direction, using Eqns. (39) and (5) we can get that

$\displaystyle\int(d\ell^{-})\frac{e^{+i\ell^{-}\eta_{1}^{+}}}{\ell^{-}-\vec{\ell}^{\,2}_{\perp}/(2x_{21}P_{1}^{h\,+})+i\,\tilde{\epsilon}}=$
$\displaystyle-2\pi
i\,\theta(-\eta_{1}^{+})\,\theta(x_{21})e^{+i\eta_{1}^{+}\big{(}\vec{\ell}^{\,2}_{\perp}/(2x_{21}P_{1}^{h\,+})-i\,\tilde{\epsilon}\big{)}},$
(71)

where $|\vec{\ell}_{\perp}|\ll P_{1}^{h\,+}$ has been always assumed. That is,
in this case the Cauchy theorem use provides the complex $i$ accompanying the
corresponding residue. Hence, the $\widetilde{B}$-function takes the form of

$\displaystyle\widetilde{B}_{(\text{\tiny
appr.})}(x_{1},x_{2})\equiv\widetilde{\bf
B}(x_{1},x_{2})=\frac{i}{x_{2}-x_{1}-i\epsilon}\int(d^{2}\vec{\bf
k}_{1}^{\,\perp})\,\bar{\Phi}_{(1)}^{\text{tw-3}}(x_{1};\vec{\bf
k}_{1}^{\,\perp})$
$\displaystyle\times\int(d^{2}\vec{\ell}_{\,\perp})\bar{\Phi}_{(2)}^{\text{tw-2}}(\bar{x}_{2};\vec{\bf
P}_{1}^{h\,\perp}-\vec{\bf k}_{1}^{\,\perp},\vec{\ell}_{\,\perp}).$ (72)

So, we have demonstrated that the origin of complexity of $\widetilde{B}$ is
being entirely determined by the interaction order. Within our approximation,
the function $\bar{\Phi}_{(2)}^{\text{tw-2}}$ contributing to $\widetilde{B}$
has to be treated as a function with the correlator involving the non-
interacting quarks.

##  6 Spin Asymmetry Parameters

In this section, we concentrate on the differential cross sections and the
several spin asymmetry parameters which are available for the experimental
studies.

First, we want to mention that, in the most general case, the cross section
takes the following standard form (for the sake of shortness the normalization
factors have been absorbed by the definition of integration measure)

$\displaystyle\int
d\sigma=\int(d{\mathbb{P}}{\mathbb{S}})\delta^{(4)}\left(\langle
mom.conserv.\rangle\right)\big{|}{\cal M}\big{|}^{2},$ (73)

where the integrated phase space given by $(d{\mathbb{P}}{\mathbb{S}})$
expressed through the function $\lambda(s,t,u)$ is the Lorentz invariant
object as well as the process amplitude ${\cal M}$. Therefore, the integrated
cross section is a frame-independent function of Lorentz scalars. Moreover,
the contributions of vector components to the given scalars which appear in
the integrated phase space and in the amplitude can be different ones. Namely,
let $s$ be a scalar that parametrizes the phase space and is equal to $(a\cdot
b)_{\text{frame-}1}$ defined by components in the given frame. The same value
of $s$ which is now an argument of the amplitude can be presented by different
product $(a\cdot b)_{\text{frame-}2}$ defined in another frame. It means that
the integrated phase space and the given amplitude as the functions of scalars
can be calculated in the different frames depending on our practical choice.
However, the angular dependence of differential cross-sections under the
experimental studies is a frame-dependent subject.

Further, every of spin asymmetries is given by the hadron tensor that
contracted with the lepton tensor. In our case, we define the lepton tensor as

$\displaystyle{\cal
L}^{\mu\nu}=\big{[}\bar{u}(l_{2})\gamma^{\nu}u(l_{1})\big{]}^{*}\big{[}\bar{u}(l_{2})\gamma^{\mu}u(l_{1})\big{]}=$
$\displaystyle
2\left[l_{1}^{\mu}l_{2}^{\nu}+l_{1}^{\nu}l_{2}^{\mu}-g^{\mu\nu}(l_{1}l_{2})\right]+2i\lambda_{l}\epsilon^{\mu\nu
lq}$ $\displaystyle\equiv{\cal L}^{\mu\nu}_{U}+i{\cal L}^{\mu\nu}_{L},$ (74)

where the first term corresponds to the unpolarized leptons while the second
term appears if the initial lepton is longitudinally polarized.

For the present study, the lepton center-of-mass system as known as the CS-
frame is the most convenient frame for both the considered factorization
procedure and the analysis of spin asymmetries related to the gluon pole
contributions. Indeed, within the CS-frame, (a) the initial and final hadron
dominant momenta are separated well ensuring the needed conditions for
factorization; (b) the hadrons possess the nonzero transverse momentum
components which are necessary to construct the observables associated with
the gluon poles. However, from the experimental point of view, the best choice
for the investigation of several observables with the essential angular
dependence is provided by the the initial hadron rest frame. For example, for
the JLab experiments the hadron target at the rest has being used.

In this connection, we can first calculate the contraction of lepton and
hadron tensors in terms of Lorentz invariant scalar products. Afterwards, all
scalar products which are appeared in according to the considered case are
expressed through the frame-dependent components which match the preferable
phase space system. We refer the reader to Appendix A for all details of the
frame choice and of the transitions between the frames.

###  6.1 Differential cross sections

Let us now go over to the discussion on the differential cross section. At the
begining, we consider the phase space, we have the following

$\displaystyle d{\mathbb{P}}{\mathbb{S}}=\frac{d^{3}\vec{{\bf
l}}_{2}}{(2\pi)^{3}2l_{2}^{0}}\frac{d^{3}\vec{{\bf
P}}^{h}_{1}}{(2\pi)^{3}2E^{h}}=$ $\displaystyle\frac{d^{3}\vec{{\bf
P}}^{h}_{1}}{(2\pi)^{3}2E^{h}}\int(d^{4}q)\delta\left(q^{2}-2(l_{1}q)\right).$
(75)

Hence, the differential cross-section reads

$\displaystyle d\sigma=\frac{d^{3}\vec{{\bf
P}}^{h}_{1}}{(2\pi)^{3}2E^{h}}\int(d^{4}q)\delta\left(q^{2}-2(l_{1}q)\right)\times$
$\displaystyle{\cal
L}_{\mu\nu}T_{\mu\nu}^{\text{GI}}(P^{h}_{1},P_{2}){\mathbb{F}}(x_{B},y_{B})\delta^{(2)}\left(\vec{\bf
q}_{\perp}\right),$ (76)

where the gauge invariant hadron tensor, see (5), has been presented in the
form of

$\displaystyle{\cal
W}_{\mu\nu}=T_{\mu\nu}^{\text{GI}}(P^{h}_{1},P_{2}){\mathbb{F}}(x_{B},y_{B})\delta^{(2)}\left(\vec{\bf
q}_{\perp}\right)$ (77)

with the gauge invariant algebraic tensor given by

$\displaystyle
T_{\mu\nu}^{\text{GI}}(P^{h}_{1},P_{2})=\frac{V^{\perp}_{\nu}}{P^{h}_{1}\cdot
P_{2}}\Big{[}\frac{P^{h}_{1\,\mu}}{y}-\frac{P_{2\,\mu}}{x_{1}}\Big{]}$ (78)

and

$\displaystyle{\mathbb{F}}(x_{B},y_{B})=\int(dx_{1})\,(dy)\,\delta(x_{1}P^{h,+}_{1}-q^{+})\delta(yP^{-}_{2}-q^{-})$
$\displaystyle\times\Big{\\{}\int(d^{2}\vec{\bf k}_{2}^{\perp})F(y;\vec{\bf
k}_{2}^{\perp})w\big{(}\vec{\bf
k}_{2}^{\perp\,2}\big{)}\Big{\\}}\int(dx_{2})\widetilde{B}(x_{1},x_{2}).$ (79)

The functions $F(y;\vec{\bf k}_{2}^{\perp})$ and $w\big{(}\vec{\bf
k}_{2}^{\perp\,2})$ denote the corresponding distribution function and the
weight function

We now assume that the contraction of (6) and (77) is being expressed in terms
of Lorentz scalars. The two-dimension delta function
$\delta^{(2)}\left(\vec{\bf q}_{\perp}\right)$ of Eqns. (5.1) and/or (77) is
appeared thanks for the decomposition of (3.2).

Introducing the photon rapidity as

$\displaystyle 2\,y_{\gamma}=\ln\frac{q^{+}}{q^{-}},$ (80)

the differential cross section takes the form of

$\displaystyle d\sigma=\frac{d^{3}\vec{{\bf P}}^{h}_{1}}{(2\pi)^{3}2E^{h}}\int
dy_{\gamma}{\cal
L}_{\mu\nu}T_{\mu\nu}^{\text{GI}}(P^{h}_{1},P_{2}){\mathbb{F}}(x_{B},y_{B})\Big{|}_{\vec{\bf
q}_{\perp}=0}$ (81)

or

$\displaystyle(2\pi)^{3}d\sigma=\frac{d\tilde{z}}{\tilde{z}}d^{2}\vec{{\bf
P}}^{h\,\perp}_{1}\int dy_{\gamma}{\cal
L}_{\mu\nu}T_{\mu\nu}^{\text{GI}}(P^{h}_{1},P_{2}){\mathbb{F}}(x_{B},y_{B})\Big{|}_{\vec{\bf
q}_{\perp}=0}$ (82)

where

$\displaystyle\frac{d(P^{h}_{1})_{3}}{E^{h}_{1}}\approx\frac{d(P^{h}_{1})_{3}}{(P^{h}_{1})_{3}}\approx\frac{dP^{h\,+}_{1}}{P^{h\,+}_{1}}=\frac{d\tilde{z}}{\tilde{z}},$
$\displaystyle\tilde{z}=\frac{P_{2}\cdot P^{h}_{1}}{P_{2}\cdot
q}\Big{|}_{\text{CS}}=\frac{P^{h\,+}_{1}}{q^{+}}.$ (83)

Following Barone:2001sp , we can also introduce the other fracture parameter
as

$\displaystyle\tilde{y}=\frac{P_{2}\cdot q}{P_{2}\cdot
l_{1}}\Big{|}_{\text{CS}}=\frac{q^{+}}{l^{+}_{1}},$ (84)

and we can readily derive that

$\displaystyle 1-\tilde{y}=e^{2(y_{\gamma}-y_{\ell})}$ (85)

with the lepton rapidity $y_{\ell}$. Hence, if $y_{\gamma}\to 0$ (see Eqn.
(80)) then

$\displaystyle 1-\tilde{y}=e^{-2y_{\ell}}\Rightarrow
d\tilde{y}=2e^{-2y_{\ell}}dy_{\ell}.$ (86)

For the region of small lepton rapidity, we get that $d\tilde{y}\approx
2dy_{\ell}$. The photon momentum integration in the phase space is very
crucial to complete the factorization procedure for the process under our
consideration.

###  6.2 The case of unpolarized leptons and hadrons

Since every of single spin asymmetries has to be normalized by the unpolarized
cross section, we begin with a consideration of the case where both the lepton
and hadron tensors correspond to the unpolarized particles. In Section 4, it
has been proven that if we focus on the particular kinematics where $x\to 1$
and the intermediate (anti)quark is massless one, the leading order hadron
tensor is suppressed in comparison with the next-to-leading hadron tensor
related to the gluon pole contributions.

Hence, the differential cross section is given by the following contractions
of lepton and hadron tensors:

$\displaystyle d\sigma^{unpol.}\sim{\cal L}^{U}_{\mu\nu}\Big{\\{}{\cal
W}^{(2)}_{\mu\nu}\big{(}\Re\text{e}\widetilde{\bf B}\big{)}+{\cal
W}^{(3)}_{\mu\nu}\big{(}\Im\text{m}\widetilde{\bf B}\big{)}\Big{\\}},$ (87)

where the hadron tensor is presented by (5.1) and (5.1), while the lepton
tensor is taken from (6).

In Eqn. (87), the tensor contractions give the following expressions

$\displaystyle{\cal L}^{U}_{\mu\nu}{\cal
W}^{(2)}_{\mu\nu}\big{(}\Re\text{e}\widetilde{\bf B}\big{)}=$
$\displaystyle\int(dx_{1})\,(dy)\,\delta(x_{1}P^{h,+}_{1}-q^{+})\delta(yP^{-}_{2}-q^{-})$
$\displaystyle\times\Big{\\{}\int(d^{2}\vec{\bf
k}_{2}^{\perp})f_{(2)}(y,\vec{\bf
k}_{2}^{\perp})\varepsilon^{P^{h}_{1}+k_{2}^{\perp}s^{\perp}}\Big{\\}}$
$\displaystyle\times\int(dx_{2})\Re\text{e}\widetilde{\bf
B}(x_{1},x_{2})\,\frac{(\hat{x}\cdot P^{h\,\perp}_{1})}{P^{h}_{1}\cdot
P_{2}}\sin 2\theta\,\cos\varphi,$ (88)

and

$\displaystyle{\cal L}^{U}_{\mu\nu}{\cal
W}^{(3)}_{\mu\nu}\big{(}\Im\text{m}\widetilde{\bf B}\big{)}=$
$\displaystyle\int(dx_{1})\,(dy)\,\delta(x_{1}P^{h,+}_{1}-q^{+})\delta(yP^{-}_{2}-q^{-})$
$\displaystyle\times\Big{\\{}\int(d^{2}\vec{\bf
k}_{2}^{\perp})f_{(3)}(y,\vec{\bf
k}_{2}^{\perp})\varepsilon^{P^{h}_{1}+k_{2}^{\perp}P_{2}^{\perp}}\Big{\\}}$
$\displaystyle\times\int(dx_{2})\Im\text{m}\widetilde{\bf
B}(x_{1},x_{2})\,\frac{\hat{y}\wedge P^{h}_{1\,\perp}}{m_{N}\,P^{h}_{1}\cdot
P_{2}}\sin 2\theta\,\sin\varphi.$ (89)

We remind once more that the angular dependences have been fixed in the CS-
frame, see Appendix A.

###  6.3 Single spin asymmetry

The single spin asymmetry can be determined for our process with unpolarized
leptons provided the (target) initial hadron has a transverse polarization.
The practical reason for the experimental study of this asymmetry can be
formulated as the following: the given asymmetry directly probes the gluon
pole, i.e. if the asymmetry is nonzero, the gluon pole exists. Indeed, the
single spin asymmetry is usually related to the imaginary part of the gluon
pole which is encoded in the distribution function
$\widetilde{B}(x_{1},x_{2})$ (see, the prefactor $1/(x_{2}-x_{1}-i\epsilon)$
of Eqn. (53)). Notice that, within our approximation described in subsection
5.3, the single spin asymmetry is given by $\Re\text{e}\widetilde{\bf
B}\sim\delta(x_{2}-x_{1})$.

Thus, we have

$\displaystyle{\cal A}_{UT}\sim{\cal L}^{U}_{\mu\nu}\Big{\\{}{\cal
W}_{\mu\nu}\big{(}\Re\text{e}\widetilde{\bf B}\big{)}+{\cal
W}^{(1)}_{\mu\nu}\big{(}\Re\text{e}\widetilde{\bf B}\big{)},\Big{\\}}$ (90)

where the hadron tensor is determined by Eqns. (5) and (5.1). The expression
for this contraction reads

$\displaystyle{\cal L}^{U}_{\mu\nu}\Big{\\{}{\cal
W}_{\mu\nu}\big{(}\Re\text{e}\widetilde{\bf B}\big{)}+{\cal
W}^{(1)}_{\mu\nu}\big{(}\Re\text{e}\widetilde{\bf B}\big{)},\Big{\\}}=$
$\displaystyle\int(dx_{1})\,(dy)\,\delta(x_{1}P^{h,+}_{1}-q^{+})\delta(yP^{-}_{2}-q^{-})\int(dx_{2})$
$\displaystyle\times\Re\text{e}\widetilde{\bf
B}(x_{1},x_{2})\Big{[}\frac{(\hat{x}\cdot P^{h\,\perp}_{1})}{P^{h}_{1}\cdot
P_{2}}\sin 2\theta\,\cos\varphi$
$\displaystyle\times\Big{\\{}f_{T}(y)\varepsilon^{P^{h}_{1}+S^{\perp}P_{2}^{\perp}}+\int(d^{2}\vec{\bf
k}_{2}^{\perp})f_{(1)}(y,\vec{\bf
k}_{2}^{\perp})\varepsilon^{P^{h}_{1}+k_{2}^{\perp}S^{\perp}}\Big{\\}}$
$\displaystyle+m_{N}\,h_{1}(y)\hat{y}\wedge S_{\perp}\sin
2\theta\,\sin\varphi\Big{]}.$ (91)

###  6.4 Double spin asymmetry

It is important to show that the double spin asymmetry, associated with the
longitudinally polarized initial lepton and the transverse polarized initial
hadron, does not exist within our approach. Indeed, the double spin asymmetry
is related to the following contraction between the lepton and hadron tensors

$\displaystyle{\cal A}_{LT}\sim{\cal L}^{L}_{\mu\nu}\Big{\\{}{\cal
W}_{\mu\nu}\big{(}\Im\text{m}\widetilde{\bf B}\big{)}+{\cal
W}^{(1)}_{\mu\nu}\big{(}\Im\text{m}\widetilde{\bf B}\big{)},\Big{\\}}$
$\displaystyle\sim 2\lambda_{l}\epsilon^{\mu\nu
lq}\Big{[}\frac{P^{h}_{1\,\mu}}{y}-\frac{P_{2\,\mu}}{x_{1}}\Big{]}\Big{\\{}P^{h\,\perp}_{1\,\nu}\varepsilon^{P^{h}_{1}+S_{\perp}P_{2}^{\perp}}\oplus\,m_{N}\varepsilon^{\nu
S_{\perp}P_{2}P^{h}_{1}}_{\perp}\Big{\\}}$
$\displaystyle\sim\epsilon^{\hat{z}\nu
lq}\big{\\{}P^{h\,\nu}_{1\,\perp}\oplus\varepsilon^{\nu
S_{\perp}P_{2}P^{h}_{1}}_{\perp}\big{\\}}=(\hat{z}\cdot q)\vec{\bf
l}_{\perp}\wedge\vec{\bf T}_{\perp}=0$ (92)

due to the gauge invariant combination $(\hat{z}\cdot q)=0$.

##  7 Conclusions

In the paper, we have used the approach which is ground on the archetypal
factorization procedure established in a series of seminal papers
Efremov:1976ih ; Efremov:1978rn ; Efremov:1978xm ; Efremov:1978cu . This
approach has been applied to the semi-exclusive mode of the hadron production
processes where the hadron tensor is given by the corresponding convolution of
the distribution function and the distribution amplitudes provided a large
value of intermediate quark fraction, $x\to 1$. We have singled out the fact
that, in our approach, the transverse momenta of photon and produced hadron
are not tied by any conditions in contrast to the TMD-factorization
approaches. This is definitely a preponderance of our approach which gives a
possibility to obtain a new single spin asymmetry associated with the gluon
pole contribution in a wide region of the transverse momentum of produced
hadron. We have suggested to consider this kind of asymmetry as a object of
experimental measurements at a large value of $\vec{\bf P}^{h}_{\perp}$ using
the existing data of JLab.

In the paper, we have demonstrated that the leading order hadron tensor is
suppressed by the model-independent kinematical factor
$(\bar{x}_{1}P_{1}^{h,+})^{2}/m_{q}$, see Eqn. (45), for $x_{1}\to 1$ and
$m_{q}\to 0$ in comparison with the gluon pole contribution to the hadron
tensor associated with the gluon radiations. Based on this finding, we have
investigated the manifestation of gluon poles in the lepton-hadron collision
of hadron production at large $x$, where the produced hadron can be described
by the corresponding distribution amplitude.

In the frame of the contour gauge conception, it is known that the twist three
distribution function Anikin:2021osx  which arises from the quark-gluon
correlators have to be treated as a complex functions due to the gluon poles.
In this context, we have studied the different sources of functional
complexity which are associated with the gluon pole effects.

We have also introduced the new transverse momentum dependent distribution
functions, see Eqns. (55) and (58), which should extend our knowledge on the
hadron spin structure Anikin:2021zxl . We have demonstrated that these new
functions together with the gluon pole contributions have formed the new
observables which should be accessible in the experimental studies based on
the recent data of JLab.

###### Acknowledgements.

We thank colleagues from the Theoretical Physics Division of NCBJ (Warsaw) for
useful and stimulating discussions. Our special thanks go to Prof. Harut
Avakian of JLab. This work is supported by the Ulam Program of NAWA No.
PPN/ULM/2020/1/00019. The work of L.Sz. is supported by the grant
2019/33/B/ST2/02588 of the National Science Centre in Poland.

##  Appendix A On the frame choice

In this Appendix, we present the Lorentz transforms which are tying the CS-
frame and the initial hadron rest frame. The boost for the initial hadron can
be performed by two steps: (a) the Lorentz rotation in
$\left((P_{2})_{0},(P_{2})_{1}\right)$-plane and, then, (b) the Lorentz
rotation in $\left((P^{\prime}_{2})_{0},(P_{2})_{3}\right)$-plane. As a
result, we have

$\displaystyle
P_{2\,\mu}\Big{|}_{\text{RS}}={\mathbb{B}}_{\mu\nu}\,P_{2\,\nu}\Big{|}_{\text{CS}}$
(A.93)

where

$\displaystyle{\mathbb{B}}_{00}=\cosh\theta_{2}\cosh\theta_{1},{\mathbb{B}}_{01}=\cosh\theta_{2}\sinh\theta_{1},$
$\displaystyle{\mathbb{B}}_{02}=0,{\mathbb{B}}_{03}=\sinh\theta_{2};$
$\displaystyle{\mathbb{B}}_{10}=\sinh\theta_{1},{\mathbb{B}}_{11}=\cosh\theta_{1},{\mathbb{B}}_{12}=0,{\mathbb{B}}_{13}=0;$
$\displaystyle{\mathbb{B}}_{20}=0,{\mathbb{B}}_{21}=0,{\mathbb{B}}_{22}=1,{\mathbb{B}}_{23}=0;$
$\displaystyle{\mathbb{B}}_{30}=\sinh\theta_{2}\cosh\theta_{1},{\mathbb{B}}_{31}=\sinh\theta_{2}\sinh\theta_{1},$
$\displaystyle{\mathbb{B}}_{32}=0,{\mathbb{B}}_{33}=\cosh\theta_{2};$ (A.94)

with

$\displaystyle
2\theta_{1}=\ln\frac{(P_{2})_{0}-(P_{2})_{1}}{(P_{2})_{0}+(P_{2})_{1}},\quad
2\theta_{2}=\ln\frac{(P^{\prime}_{2})_{0}-(P_{2})_{3}}{(P^{\prime}_{2})_{0}+(P_{2})_{3}}$
(A.95)

and

$\displaystyle(P^{\prime}_{2})_{0}=(P_{2})_{0}\cosh\theta_{1}+(P_{2})_{1}\sinh\theta_{1}.$
(A.96)

From Eqn. (A.95), we can readily obtain that

$\displaystyle(P_{2})_{0}\not=\pm(P_{2})_{1},\quad(P^{\prime}_{2})_{0}\not=(P_{2})_{3},\quad
P_{2}^{2}=M_{2}^{2}\not=0.$ (A.97)

Having used the hadron momentum representations in CS-frame given by (cf. (3))

$\displaystyle P_{1}^{h}=\left(E_{1}^{h},|\vec{\bf
P}^{h}_{1}|\sin\alpha,0,|\vec{\bf P}^{h}_{1}|\cos\alpha\right)$ $\displaystyle
P_{2}=\left(E_{2},|\vec{\bf P}_{2}|\sin\alpha,0,-|\vec{\bf
P}_{2}|\cos\alpha\right),$ (A.98)

we derive that

$\displaystyle
2\theta_{1}=\ln\frac{1-\beta_{2}\sin\alpha}{1+\beta_{2}\sin\alpha},\,$
$\displaystyle
2\theta_{2}=\ln\frac{\sqrt{1-\beta_{2}^{2}\sin^{2}\alpha}+\beta_{2}\cos\alpha}{\sqrt{1-\beta_{2}^{2}\sin^{2}\alpha}-\beta_{2}\cos\alpha},$
(A.99)

where

$\displaystyle\beta_{2}=|\vec{\bf P}_{2}|/E_{2},\quad P_{2}^{\perp\,2}\approx
M^{2}_{2}\ll\tilde{Q}^{2}.$ (A.100)

Hence, we have

$\displaystyle\cosh\theta_{1}=\frac{1}{\sqrt{1-\beta_{2}^{2}\sin^{2}\alpha}},\sinh\theta_{1}=-\frac{\beta_{2}\sin\alpha}{\sqrt{1-\beta_{2}^{2}\sin^{2}\alpha}},$
$\displaystyle\cosh\theta_{2}=\sqrt{\frac{1-\beta_{2}^{2}\sin^{2}\alpha}{1-\beta_{2}^{2}}},\sinh\theta_{2}=\frac{\beta_{2}\cos\alpha}{\sqrt{1-\beta_{2}^{2}}}.$
(A.101)

So, we adhere the system where the factorization of hadron tensor is more
simple from the theoretical point of view. As mentioned, this system can be
traced from the CS-frame (the lepton center-of-mass system). Then, we make a
transformation to the initial hadron rest system which is more suitable for
the considered experiment.

Within the CS-frame, the lepton sector is determined as

$\displaystyle
2l_{1\,\mu}=Q\,L_{\mu}(\theta,\varphi;\hat{x},\hat{y},\hat{z})+q_{\mu},$
(A.102) $\displaystyle
2l_{2\,\mu}=Q\,L_{\mu}(\theta,\varphi;\hat{x},\hat{y},\hat{z})-q_{\mu}$

with $-q^{2}=2(l_{1}l_{2})=Q^{2}$ and

$\displaystyle L_{\mu}(\theta,\varphi;\hat{x},\hat{y},\hat{z})=$
$\displaystyle\hat{x}_{\mu}\,\cos\varphi\,\sin\theta+\hat{y}_{\mu}\,\sin\varphi\,\sin\theta+\hat{z}_{\mu}\,\cos\theta,$
(A.103)

where the normalized axes are determined as
$\hat{x}_{\mu}=x_{\mu}/\sqrt{-x^{2}}$ and so on. In this system, we have the
basis formed by (here, symbol $\sim$ means “behaves as" and the hat symbol
means the corresponding normalization)

$\displaystyle\hat{x}_{\mu}\sim\hat{q}^{\perp}_{\mu}\sim\hat{e}^{x}_{\mu}=(0,1,0,0),$
$\displaystyle\hat{y}_{\mu}\sim\epsilon^{\perp}_{\mu\hat{q}}\sim\hat{e}^{y}_{\mu}=(0,0,1,0),$
$\displaystyle\hat{z}_{\mu}\sim\hat{P}^{h}_{1\,\mu}-\hat{P}_{2\,\mu}.$ (A.104)

After the boost transforms from CS-frame to the RS-frame defined by (A), the
scalar product $l_{1}\cdot\hat{z}$ takes the following form

$\displaystyle
l_{1}\cdot\hat{z}\Big{|}_{RS}={\mathbb{A}}(\alpha)+{\mathbb{B}}(\alpha)\cos\varphi\sin\theta+{\mathbb{C}}(\alpha)\cos\theta,$
(A.105)

where

$\displaystyle{\mathbb{A}}(\alpha)=\frac{Q\beta_{2}\cos\alpha}{1-\beta_{2}^{2}},\quad{\mathbb{B}}(\alpha)=-\frac{Q\beta_{2}^{2}\sin
2\alpha}{2(1-\beta_{2}^{2})},$
$\displaystyle{\mathbb{C}}(\alpha)=\frac{Q(1+\beta_{2}^{2}\cos
2\alpha)}{2(1-\beta_{2}^{2})}.$ (A.106)

##  Appendix B The alternative representation of functions with the on-shell
fields in correlators

Now, we give an alternative representations for the functions which contain
the correlators with the on-shell fields. As shown in section 3.3, the leading
order function $\bar{\Phi}(k_{1})$, which stems from the squared amplitude, is
presented in the product form of functions
$\bar{\Phi}^{(1)}_{\text{o}}(k_{1})$ and $\bar{\Phi}^{(2)}_{\text{o}}(k_{1})$.
Besides this representation, it is convenient to present the function
$\bar{\Phi}(k_{1})$ in the equivalent form as

$\displaystyle\bar{\Phi}(k_{1})=\frac{E^{2}}{m_{q}}\,\bar{\Phi}^{(2)}(P^{h}_{1}-k_{1})\,\bar{\Phi}^{(1)}(k_{1})\delta\Big{(}\frac{(k_{1}-P_{1}^{h})^{2}}{m_{q}^{2}}-1\Big{)}$
(B.107)

where

$\displaystyle\bar{\Phi}^{(1)}(k_{1})=\int(d^{4}\xi_{1})e^{ik_{1}\xi_{1}}\langle
0|\bar{\psi}(\xi_{1})\psi(0)|P_{1}^{h}\rangle$ (B.108)

and

$\displaystyle\bar{\Phi}^{(2)}(P^{h}_{1}-k_{1})=$
$\displaystyle\int(d^{4}\eta_{1})e^{-i(P^{h}_{1}-k_{1})\eta_{1}}\langle
P_{1}^{h}|\bar{\psi}(\eta_{1})\psi(0)|0\rangle.$ (B.109)

That is, the functions $\bar{\Phi}^{(1)}(k_{1})$ and
$\bar{\Phi}^{(2)}(P^{h}_{1}-k_{1})$ involve now the correlators with only the
off-shell fields and, at the same time, the on-shellness of the intermediate
(anti)quark with the momentum $k_{1}-P_{1}^{h}$ is ensured by the delta
function.

For the use of the limits given by $m_{q}\to 0$ and $x_{1}\to 1$, the function
$\bar{\Phi}(k_{1})$ can be written as

$\displaystyle\bar{\Phi}(k_{1})=\frac{m_{q}}{4\bar{x}_{1}P_{1}^{h\,+}}\,\big{[}\bar{x}_{1}P_{1}^{h\,+}+k_{1}^{-}\big{]}^{2}$
(B.110) $\displaystyle\times\delta\Big{(}k_{1}^{-}-\frac{(\vec{\bf
k}^{\perp}_{1}-\vec{\bf
P}_{1}^{h\,\perp})^{2}+m_{q}^{2}}{2\bar{x}_{1}P_{1}^{h\,+}}\Big{)}\bar{\Phi}^{(2)}(P^{h}_{1}-k_{1})\,\bar{\Phi}^{(1)}(k_{1}).$

It is important to emphasize that the leading order function
$\bar{\Phi}(k_{1})$ exists only if the intermediate (anti)quark is on-shell.
In other words, the function $\bar{\Phi}(k_{1})$ can be extended beyond the
on-shell condition and it is uniquely related to the square or interference of
two amplitudes which are given by function
$\bar{\Phi}^{(1)}_{\text{o}}(k_{1})$ and
$\bar{\Phi}^{(2)}_{\text{o}}(P^{h}_{1}-k_{1})$. In this connection, the on-
shell delta function in Eqns. (B.107) and (B.110) is mandatory for the
representations.

Let us now go over to the alternative representations for the next-to-leading
function $\bar{\Phi}^{(A)}(k_{1},\ell)$. We remind that in section 3.5 the
representation for this function is obtained by the direct extension of the
leading order function $\bar{\Phi}(k_{1})$. Analogously to the above-mentioned
way, there is also the equivalent and alternative representation of
$\bar{\Phi}^{(A)}(k_{1},\ell)$ which is more convenient to use in the most
cases. In order to demonstrate it, we begin with (the spin $\lambda$-indices
are omitted)

$\displaystyle\bar{\Phi}(k_{1})=\int(d^{4}\xi)e^{-ik_{1}\xi}\,\int\frac{d^{3}\vec{{\bf
K}}}{(2\pi)^{3}2K_{0}}$ $\displaystyle\times\langle
0|\bar{\psi}(0)d^{+}(K)|P_{1}^{h}\rangle\,\langle
P_{1}^{h}|d^{-}(K)\mathbb{S}\psi(\xi)|0\rangle^{I}$ (B.111)

written in the interaction representation. The next stage is a commutation of
$\mathbb{S}$-matrix with the operator $d^{-}(K)$ given the following
expression in the first order of interaction

$\displaystyle\bar{\Phi}^{(A)}(k_{1})=\int(d^{4}\xi)e^{-ik_{1}\xi}\,\int\frac{d^{3}\vec{{\bf
K}}}{(2\pi)^{3}2K_{0}}\int(d^{4}\eta)e^{+iK\eta}$ $\displaystyle\times\langle
0|\bar{\psi}(0)d^{+}(K)|P_{1}^{h}\rangle\,\langle
P_{1}^{h}|\big{[}\bar{\psi}(\eta)\hat{A}(\eta)v(K)\big{]}\psi(\xi)|0\rangle^{I}.$
(B.112)

Having used that

$\displaystyle\int\frac{d^{3}\vec{{\bf
K}}}{(2\pi)^{3}2K_{0}}=\int(d^{4}K)\delta\big{(}K^{2}-m_{q}^{2}\big{)}$
(B.113)

together with Eqn. (30), after some algebra we derive

$\displaystyle\bar{\Phi}^{(A)}(k_{1})=\bar{\Phi}^{(2)}(k_{1})\hat{A}(0)\bar{\Phi}^{(1)}_{\text{o}}(k_{1}-P^{h}_{1})$
(B.114)

where

$\displaystyle\bar{\Phi}^{(2)}(k_{1};\ell)=\int(d^{4}z)\,e^{-ik_{1}z}\,\langle
P_{1}^{h}|\bar{\psi}(0)\,\psi(z)|0\rangle\,,$ (B.115)
$\displaystyle\bar{\Phi}^{(1)}_{\text{o}}(k_{1})=\int(d^{4}\xi)e^{-i(k_{1}-P_{1}^{h})\xi}\langle
0|\bar{\psi}(0)\psi_{\text{on-sh.}}(\xi)|P_{1}^{h}\rangle.$ (B.116)

In Eqn. (B.114) $A_{\alpha}(0)$ denotes the gluon field operator in the
$x$-space (the position space). This gluon operator gives the gluon propagator
in the hadron tensor after paring with the other gluon field operator
appearing in the radiation correction to the corresponding quark field, see
Eqns. (3.4) and (3.4).

In analogy with the leading order function, see Eqn. (B.107), the function
$\bar{\Phi}^{(A)}(k_{1})$ can be presented in the form as

$\displaystyle\bar{\Phi}^{(A)}(k_{1})=\bar{\Phi}^{(2)}(k_{1})\hat{A}(0)\bar{\Phi}^{(1)}(k_{1}-P^{h}_{1})$
$\displaystyle\times\delta\Big{(}\frac{(k_{1}-P_{1}^{h})^{2}}{m_{q}^{2}}-1\Big{)}$
(B.117)

or

$\displaystyle\bar{\Phi}^{(A)}(k_{1})=\bar{\Phi}^{(2)}(k_{1})\hat{A}(0)\bar{\Phi}^{(1)}(k_{1}-P^{h}_{1})$
$\displaystyle\times\frac{m_{q}^{2}}{2\bar{x}_{1}P_{1}^{h\,+}}\delta\Big{(}k_{1}^{-}-\frac{(\vec{\bf
k}^{\perp}_{1}-\vec{\bf
P}_{1}^{h\,\perp})^{2}+m_{q}^{2}}{2\bar{x}_{1}P_{1}^{h\,+}}\Big{)}.$ (B.118)

Notice that in contrast to the leading order function, the next-to-leading
function $\bar{\Phi}^{(A)}(k_{1})$ allow the off-shell extension due to the
presence the interaction vertex between $\bar{\Phi}^{(2)}(k_{1})$ and
$\bar{\Phi}^{(1)}(k_{1}-P^{h}_{1})$.

##  Appendix C On the off-shell extension

In the Appendix, we give a short description of the off-shell extension
method. The necessity of the off-shell extension is dictated by the fact that,
as a rule, the corresponding parton distributions stem from the factorization
procedure applied to the amplitudes involving the non-perturbative correlators
with the off-shell fields.

The principal basis of our method is a statement that the permutation
relations for fields are the translation invariant subjects for both the on-
shell and off-shell particles. We remind that this general statement is
forming one of the fundament axioms.

For a pedagogical reason, let us start from the permutation relations written
for the on-shell scalar particles. We have

$\displaystyle[\phi^{+}_{\text{on-sh.}}(x),\,\phi^{-}_{\text{on-
sh.}}(y)]=D^{+}(x,y).$ (C.119)

In QFT, it is imposed that the function $D^{+}$ must depend on the difference
$x-y$ only, i.e.

$\displaystyle D^{+}(x,y)=D^{+}(x-y).$ (C.120)

Since the Fourier transforms are the linear operations, the translation
invariance property of the permutation relations, see Eqn. (C.120), is
determining the commutation relations for the Fourier images of $\phi^{\pm}$.
Indeed, having used the Fourier transforms, we have the following

$\displaystyle[\phi^{+}_{\text{on-sh.}}(x),\,\phi^{-}_{\text{on-sh.}}(y)]=$
$\displaystyle\int(d^{3}\vec{\bf k})(d^{3}\vec{\bf p})e^{ikx-
ipy}[\tilde{\phi}^{+}_{\text{on-sh.}}(\vec{\bf
k}),\,\tilde{\phi}^{-}_{\text{on-sh.}}(\vec{\bf p})],$ (C.121)

where

$\displaystyle\tilde{\phi}^{\pm}_{\text{on-sh.}}(\vec{\bf
k})\equiv\frac{a^{\pm}_{\text{on-sh.}}(k)}{2E(\vec{\bf
k})}\equiv\frac{a^{\pm}_{\text{on-sh.}}(\vec{\bf k})}{\sqrt{2E(\vec{\bf
k})}},$ $\displaystyle E(\vec{\bf k})=\sqrt{\vec{\bf k}^{2}+m^{2}}.$ (C.122)

From Eqn. (C), one can see that in order to ensure the translation invariance
of the permutation relation, we have to demand the commutation relation
$[\tilde{\phi}^{+}_{\text{on-sh.}}(\vec{\bf k}),\,\tilde{\phi}^{-}_{\text{on-
sh.}}(\vec{\bf p})]$ to be proportional to the three-dimensional delta
function, i.e.

$\displaystyle[\tilde{\phi}^{+}_{\text{on-sh.}}(\vec{\bf
k}),\,\tilde{\phi}^{-}_{\text{on-sh.}}(\vec{\bf p})]=\frac{1}{2E(\vec{\bf
k})}\delta^{(3)}(\vec{\bf k}-\vec{\bf p}).$ (C.123)

Hence, we obtain the standard Paul-Jordan function given by

$\displaystyle[\phi^{+}_{\text{on-sh.}}(x),\,\phi^{-}_{\text{on-
sh.}}(y)]=\int\frac{(d^{3}\vec{\bf k})}{2E(\vec{\bf k})}e^{ik(x-y)}\equiv
D^{+}(x-y).$ (C.124)

Notice that the Fourier transforms of on-shell fields, see Eqn. (C), can be
presented in the four-dimensional forms as

$\displaystyle\phi^{\pm}_{\text{on-sh.}}(x)=\int(d^{4}\vec{\bf k})\,e^{\pm
ikx}\,\delta(k^{2}-m^{2})a^{\pm}_{\text{on-sh.}}(k).$ (C.125)

With the help of this representation we can readily make an extension of the
considered fields beyond the on-shell surface. To this goal, we introduce the
following replacement

$\displaystyle\delta(k^{2}-m^{2})a^{\pm}_{\text{on-sh.}}(k)\Rightarrow
a^{\pm}_{\text{off-sh.}}(k),$ (C.126)

where $a^{\pm}_{\text{off-sh.}}(k)$ denotes the off-shell field operator. As a
consequence, the off-shell commutation relation takes the form of

$\displaystyle[a^{+}_{\text{off-sh.}}(k),\,a^{-}_{\text{off-
sh.}}(p)]=\frac{1}{4E^{2}(\vec{\bf k})}\delta^{(4)}(k-p)$ (C.127)

which results in

$\displaystyle[\phi^{+}_{\text{off-sh.}}(x),\,\phi^{-}_{\text{off-sh.}}(y)]=$
$\displaystyle\int\frac{(d^{4}k)}{4E^{2}(\vec{\bf k})}e^{ik(x-y)}\equiv
D^{+}_{\text{off-sh.}}(x-y),$ (C.128)

where $k_{0}\not=E(\vec{\bf k})$ anymore.

##  Appendix D The covariant (invariant) integrations

In this appendix, we describe the method of the covariant (invariant)
integration which is based on the Lorentz decomposition. We begin with the
simplest example which clearly illustrates the method. We consider an
arbitrary correlator with one Lorentz vector index which corresponds to the
forward hadron matrix element of the quark non-local operator. It reads

$\displaystyle\langle
P|\mathbb{O}_{\alpha}\big{(}\psi(0),\bar{\psi}(z)\big{)}|P\rangle\stackrel{{\scriptstyle{\cal
F}}}{{=}}\int(d^{2}\vec{\bf
k}_{\perp})\,F(x,k_{\perp};P)\,k^{\perp}_{\alpha}\equiv J^{\perp}_{\alpha},$
(D.129)

where $\stackrel{{\scriptstyle{\cal F}}}{{=}}$ denotes the Fourier transform
and defined by the integration measure as

$\displaystyle d\mu^{-}(k)=(dz^{-})e^{-ik^{+}\,z^{-}}\Big{|}_{k^{+}=xP^{+}}$
(D.130)

and $a^{\perp}=(0,\vec{\bf a}_{\perp},0)$ for any four-vector $a$. Here and in
what follows, we omit the covariant and contravariant indices unless it leads
to misunderstandings. In Eqn. (D.129), the hadron momentum $P$ represents the
exterior Lorentz vector characterizing the correlator $\langle
P|\mathbb{O}_{\alpha}|P\rangle$ and we assume that $P=(P^{+},0^{-},\vec{\bf
P}_{\perp})$.

We now perform the Lorentz decomposition of $J^{\perp}_{\alpha}$ over the
exterior Lorentz vector, we obtain that

$\displaystyle J^{\perp}_{\alpha}=P^{\perp}_{\alpha}\,{\cal A}\equiv
P^{\perp}_{\alpha}\,\int(d^{2}\vec{\bf
k}_{\perp})\,F(x,k_{\perp};P)\,\frac{(k^{\perp}\cdot
P^{\perp})}{P_{\perp}^{2}}.$ (D.131)

The covariant spin axial-vector $S_{\alpha}$, which can form the Lorentz
vector $i\varepsilon^{\alpha S_{\perp}+-}$, is not participating in the
decomposition of Eqn. (D.131) even if the spin axial-vector characterizes the
given correlator. Indeed, if $S=(\lambda P^{+}/M,0^{-},\vec{\bf S}_{\perp})$
then $(P\cdot S)=(P_{\perp}\cdot S_{\perp})=0$ and, hence,
$i\varepsilon^{P_{\perp}S_{\perp}+-}\not=0$. So, the Lorentz vector
$i\varepsilon^{\alpha S_{\perp}+-}$ is not an orthogonal basis vector for the
decomposition.

The Eqns. (D.129) and (D.131) correspond, roughly speaking, to two different
vector representations of the single Lorentz vector $J^{\perp}_{\alpha}$. Let
us focus on the scalar product $(J^{\perp}\cdot B^{\perp})$ where an arbitrary
exterior vector $B$ which is not associated with the correlator of Eqn.
(D.129). We have the following expression

$\displaystyle(J^{\perp}\cdot B^{\perp})=\int(d^{2}\vec{\bf
k}_{\perp})\,F(x,k_{\perp};P)\,(k^{\perp}\cdot B^{\perp})=$
$\displaystyle(P^{\perp}\cdot B^{\perp})\,\int(d^{2}\vec{\bf
k}_{\perp})\,F(x,k_{\perp};P)\,\frac{(k^{\perp}\cdot
P^{\perp})}{P_{\perp}^{2}}$ (D.132)

which involves $J^{\perp}_{\alpha}$ written before (see the first line of Eqn.
(D)) and after (see the second line of Eqn. (D)) implementation of the
covariant (invariant) integration.

Figure 4:  The definitions of angles $\varphi_{B}$, $\varphi_{P}$ and
$\varphi_{k}$.

From Eqn. (D), we can immediately derive the necessary condition for the self-
consistency of the Lorentz decomposition of Eqn. (D.131). Indeed, Eqn. (D) can
be equivalently rewritten as

$\displaystyle\int(d^{2}\vec{\bf
k}_{\perp})\,F(x,k_{\perp};P)\,\Big{\\{}(k^{\perp}\cdot
B^{\perp})-(P^{\perp}\cdot B^{\perp})\,\frac{(k^{\perp}\cdot
P^{\perp})}{P_{\perp}^{2}}\Big{\\}}=$ $\displaystyle\int(d^{2}\vec{\bf
k}_{\perp})\,F(x,k_{\perp};P)|k_{\perp}|\,|B^{\perp}|\,\Big{\\{}\cos\varphi_{Bk}-\cos\varphi_{BP}\cos\varphi_{Pk}\Big{\\}}=0,$
(D.133)

where the notation
$\varphi_{\mathbb{A}\mathbb{B}}=\varphi_{\mathbb{A}}-\varphi_{\mathbb{B}}$
with $(\mathbb{A},\mathbb{B})\equiv(B_{\perp},P_{\perp},k_{\perp})$ has been
used for the corresponding angles, see Fig. 4.

One can readily solve Eqn. (D), we have the solution given by
($\varphi_{Bk}=\varphi_{BP}+\varphi_{Bk}$)

$\displaystyle\varphi_{B}=\varphi_{P}\pm n\pi\,,$ (D.134)
$\displaystyle\varphi_{P}=\varphi_{k}\pm n\pi.$ (D.135)

These conditions ensure the self-consistency of the covariant integration (see
Eqn. (D.129)). Moreover, the (anti)collinearity of $P_{\perp}$ and
$B_{\perp}$, see Eqn. (D.134), is a necessary and enough condition for the
mentioned self-consistency, while the condition given by Eqn. (D.135) is not
crucial and can be omitted.

We now dwell on the practical example of the covariant (invariant) integration
applied to Eqn. (55). In this case, we have the following decomposition

$\displaystyle K^{\perp}_{\alpha}\equiv\int(d^{2}\vec{\bf
k}_{\perp})\,f_{(2)}(x,k_{\perp},s_{\perp};P)\,k^{\perp}_{\alpha}=P^{\perp}_{\alpha}\,{\cal
B}+\varepsilon^{\alpha s_{\perp}+-}{\cal C}$ (D.136)

where $s_{\perp}$ implies the covariant quark spin vector and

$\displaystyle{\cal B}=\int(d^{2}\vec{\bf
k}_{\perp})\,f_{(2)}(x,k_{\perp},s_{\perp};P)\,\frac{(k^{\perp}\cdot
P^{\perp})}{P^{2}_{\perp}},$ (D.137) $\displaystyle{\cal C}=\int(d^{2}\vec{\bf
k}_{\perp})\,f_{(2)}(x,k_{\perp},s_{\perp};P)\,\frac{\varepsilon^{k_{\perp}s_{\perp}+-}}{s^{2}_{\perp}}.$
(D.138)

Notice that $\varepsilon^{k_{\perp}s_{\perp}+-}\not=0$ otherwise we deal with
a trivial case of ${\cal C}=0$ provided $s^{2}_{\perp}\not=0$. The important
requirement for the Lorentz decomposition of Eqn. (D.136) is that the vectors
$P^{\perp}$ and $\varepsilon^{\alpha s_{\perp}+-}$ have to be orthogonal ones,
i.e. they are forming the orthogonal system. In this context, we have to
derive the necessary conditions for the orthogonality.

Consider the scalar product $(P_{\perp}\cdot K^{\perp})$ giving

$\displaystyle(P_{\perp}\cdot K^{\perp})=P^{2}_{\perp}\,{\cal
B}+\varepsilon^{P_{\perp}s_{\perp}+-}{\cal C}$ (D.139)

which takes place if and only if

$\displaystyle\varepsilon^{P_{\perp}s_{\perp}+-}\sim\sin\varphi_{Ps}=0\Rightarrow\varphi_{P}=\varphi_{s}\pm
n\pi.$ (D.140)

That is, the vectors $P^{\perp}$ and $s_{\perp}$ have to be collinear (or
anti-collinear) ones. On the other hand, one may contract $K^{\perp}_{\alpha}$
in (D.136), with the vector $\varepsilon^{\alpha s_{\perp}+-}$. This case also
leads to the condition of Eqn. (D.140) .

To conclude, the condition of Eqn. (D.140) is the necessary condition for the
Lorentz decomposition given by Eqn. (D.136).

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