machine learning or integer linear programming. For small imbalance
parameters, generalizing negative cycle detection techniques previously used
for graph partitioning (Sanders and Schulz, 2013) could be interesting.

###### Acknowledgements.

The authors would like to thank the additional co-authors of previous
conference publications, namely Robin Andre, Michael Hamann, Vitali Henne,
Henning Meyerhenke, Daniel Seemaier, Darren Strash, and Dorothea Wagner. This
work was partially supported by DFG grants WA654/19- 2, SA933/10-2,
SA933/11-1, SCHU 2567/1-2, and the Leibniz prize for Peter Sanders. The
authors acknowledge further support by the state of Baden-Württemberg through
bwHPC.

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##  Appendix A Benchmark Statistics

Figure 16. Summary of different properties for benchmark set A, B and C and
the different sources. It shows for each hypergraph (points), the number of
vertices $|V|$, nets $|E|$ and pins $|P|$, as well as the median and maximum
net size ($|\tilde{e}|$ and $\Delta_{e}$ and vertex degree ($\tilde{d}(v)$ and
$\Delta_{v}$).


# Orbital Angular Momentum Distribution of Gluons at Small-$x$
Analytic interpolation between Ji and Jaffe-Manohar

Khatiza Banu  Nahid Vasim  Raktim Abir<EMAIL_ADDRESS>Department of
Physics, Aligarh Muslim University, Aligarh - $202001$, India.

###### Abstract

In this work we first introduced a generalized Wilson line gauge link that
reproduces both staple and near straight links along light cone in different
parameter limits. We then studied the gauge invariant bi-local orbital angular
momentum operator with such a general gauge link, in the framework of Chen et.
al. decomposition of gauge fields. At the appropriate combination of limits,
the operator reproduces both Jaffe-Manohar and Ji’s operator structure and
offers a continuous analytical interpolation between the two.

###### pacs:

12.38.-t

##  I Introduction

It has been more than three decades since the discovery of proton spin crisis
by the European Muon Collaboration (EMC) Ashman:1987hv . By now, a lot of
progress has been achieved both on the theoretical and experimental fronts to
understand and validate the proton spin sum rule. Moreover, a resurgence of
interests have been triggered in the wake of the planned Electron-Ion Collider
(EIC) at Brookhaven National Laboratory (BNL) Aschenauer:2017jsk . The sum
rule for proton spin contains both helicity contribution and orbital angular
momentum contribution each separately for quarks and gluons, as,

$\displaystyle S_{q+\bar{q}}+L_{q+\bar{q}}+S_{G}+L_{G}=\frac{1}{2}~{},$ (1)

This can often be presented in terms of the corresponding distributions as,

$\displaystyle\int_{0}^{1}dx\left[\Delta\Sigma\left(x,Q^{2}\right)+\Delta
G\left(x,Q^{2}\right)+L_{q+\bar{q}}\left(x,Q^{2}\right)+L_{G}(x,Q^{2})\right]=\frac{1}{2}~{},$
(2)

where $\Delta\Sigma\left(x,Q^{2}\right)$, $\Delta G\left(x,Q^{2}\right)$ are
helicity distributions for quarks and gluons whereas
$L_{q+\bar{q}}\left(x,Q^{2}\right)$ and $L_{G}(x,Q^{2})$ are that for angular
momentum respectively.

Gluon orbital angular momentum (OAM) is essentially a two-point correlation
function of the field strength tensors at two space-time points. To make such
bi-local operators gauge invariant, one needs to introduce appropriate gauge
links between the field strength tensors. Two important choices in this regard
are, (a) the staple gauge links and (b) the straight gauge links. The staple
gauge links may show up either as the past pointing or as the future pointing
links. This leads to two main types of OAMs namely the dipole type and the
Weizsäcker-Williams type. The dipole OAM contains both the past and future
links (Fig[1]) while the Weizsäcker-Williams type OAM contains either the past
pointing or the future pointing links along light cone (Fig[2]) - nonetheless
both of them lead to Jaffe-Manohar Jaffe:1989jz  definition of OAM operator.
The gauge links connecting the two space-time points could also be the
straight one as shown in Fig[3], leading to Ji’s Ji:2012sj  definition of OAM.

In this study, we have derived a generalized form of the gauge-invariant OAM
operator, presented in Eq.(42), which accommodates all possible geometrics of
the gauge links. In the appropriate limit the operator reproduces both Jaffe-
Manohar and Ji’s operator structure and offers a continuous analytical
interpolation between the two. Next we jotted down some of the key points that
motivate to have a generalised operator and the continuous interpolation like
the one we studied in this work.

Figure 1: The gauge link structure $[\eta_{1},\eta_{2}]$ equals
$[-\infty,+\infty^{\dagger}]$ and $[+\infty,-\infty^{\dagger}]$. Both
correspond to the dipole type JM orbital angular momentum. Figure 2: The gauge
link structure $[\eta_{1},\eta_{2}]$ equals $[-\infty,-\infty^{\dagger}]$ and
$[+\infty,+\infty^{\dagger}]$. Both correspond to the Weizsäcker-Williams type
JM orbital angular momentum. Figure 3: The straight gauge link structure
$[\eta_{1},\eta_{2}]$ equals $[0,0^{\dagger}]$. This appears in Ji’s orbital
angular momentum.

Potential OAM - the integrated torque accumulated by a gluon :

A straight Wilson line gauge link yields Ji orbital angular momentum while an
infinite staple shaped gauge link yields Jaffe-Manohar orbital angular
momentum. Unlike the Ji OAM, the Jaffe-Manohar decomposition of OAM affected
by the final or/and initial state interactions. Hence the difference between
the two is often interpreted as the integrated torque accumulated by a gluon
struck in a deep inelastic scattering process as it enters or exits the
proton, through initial or final state interactions - exerted on the outgoing
quark or gluon by the chromo-magnetic field produced by the spectators. The
difference between the two OAMs are directly connected to the off-forward
extension of a Qiu-Sterman term and often cited as potential OAM
Burkardt:2012sd .

Quasi PDF in lattice - connection to TMDs in DIS and Drell-Yan processes :

The operator defining the quasi PDF involves a straight line link from $0$ to
$z$ rather than a staple link often appears in the definitions of TMDs to
describe the Drell-Yan and semi-inclusive DIS processes. While the staple link
reflects initial and final state interactions inherent in these processes, the
straight link describes the internal structure of the hadron when it is in the
non-disturbed or primordial state. It is unlike that such a TMD can be
measured in a scattering experiment, however being a well-defined object they
can be measured in the lattice. As the quasi PDF is defined with space
separation connected by straight gauge link a proper synergy of 2d Fourier
transform and interpolation like the one addressed in this work can connect it
to the TMDs that show up in the scattering processes Radyushkin:2017cyf .

TMD PDFs - Weizsäcker-Williams to dipole :

Unintegrated gluon distribution or transverse momentum-dependent gluon
distribution functions are one of the key topics to be fully investigated in
the upcoming EIC. The TMD PDFs can either be probed in quark-antiquark jet
correlation in DIS (the Weizsäcker-Williams distribution) or the direct
photon-jet correlation in $pA$ collision. The various TMD PDFs involved in
different processes specifically in different di jet channels in $pA$ or $eA$
collisions are related by the two universal ones: the Weizsäcker-Williams TMD
and the dipole TMD specifically in small-$x$ at large $N_{c}$ limit. Both the
WW distribution or dipole distribution are essentially dimension four two-
point correlation functions of classical gluon fields. The operator
definitions are different only by the way the gauge links are oriented. An
interpolation of the link, the kind we studied here, lead to a universal
structure that can reproduce WW and Dipole TMD at the appropriate limit.

In Section II, we briefly review the essentials of canonical gauge invariant
decomposition, of parton angular momentum by Chen et. al., where one has to
separate the gauge potential into so called pure part and physical or
dynamical part. In Section III, we studied the transverse derivatives of the
staple gauge links along the light cone. Here we introduced two parameters,
$\eta_{1}$ and $\eta_{2}$ to make the extent of the gauge link arbitrary with
the anticipation that at different combinations of the limit of the
parameters, it would produce the past pointing, the future pointing, or near
straight gauge links. We finally express the transverse derivative of such
general gauge link in terms of the pure and physical gauge components. This
also explicitly contains the gauge link extent parameters. In Section IV, we
recalled the connection between the orbital angular momentum of gluons and the
gluon Wigner distribution and then derive the OAM operators in terms of pure
gauge and the extent parameters. In Section V, we moved to small-$x$ eikonal
limit and went on to integrate both $z^{-}$ and $z_{\perp}$, the position
components, which lead to the general gauge invariant OAM operator structure,
valid for arbitrary geometry of the gauge link at small-$x$. In Section VI, we
briefly noted the results for sub-eikonal cases. We conclude and give our
outlook at the end.

##  II Gauge invariant decomposition of angular momentum

Decomposition of angular momentum in as gauge invariant way can be achieved by
separating the gauge field into the ${\it pure}$ and ${\it physical}$ parts of
the field, first proposed by Chen et. al. Chen:2008ag ; Chen:2009mr ,

$\displaystyle A_{\mu}=A_{\mu}^{\rm pure}+A_{\mu}^{\rm phys}~{}.$ (3)

Pure gauge field as differential $1$-form defined by identically vanishing the
corresponding $2$-form $i.e.$ field strength tensor,

$\displaystyle F_{\mu\nu}^{~{}\rm pure}~{}=~{}\partial_{\mu}~{}A_{\nu}^{~{}\rm
pure}-\partial_{\nu}~{}A_{\mu}^{~{}\rm pure}-ig\left[A_{\mu}^{\rm
pure},~{}A_{\nu}^{\rm pure}\right]~{}=~{}0~{}.$ (4)

One can define covariant derivative as well,

$\displaystyle D_{\mu}^{\rm~{}pure}=\partial_{\mu}-igA_{\mu}^{\rm pure}~{}.$
(5)

The covariant derivative satisfy the commutation relation that vanishes
because field tensor vanishes identically,

$\displaystyle\left[D_{\mu}^{\rm pure},~{}D_{\nu}^{\rm pure}\right]$
$\displaystyle=$ $\displaystyle-igF_{\mu\nu}^{\rm pure}~{}=~{}0~{}.$ (6)

Under gauge transformation the pure part of the gauge field transform as,

$\displaystyle A_{\mu}^{~{}\rm pure}\longmapsto A_{\mu}^{~{}\rm
pure}=U(x)~{}A_{\mu}^{~{}\rm
pure}~{}U^{-1}(x)+\frac{i}{g}~{}U(x)~{}\partial_{\mu}~{}U^{-1}(x)~{},$ (7)

On the other hand physical part of the gauge field transform as,

$\displaystyle A_{\mu}^{\rm phys}~{}\mapsto~{}A_{\mu}^{\rm
phys}=U(x)~{}A_{\mu}^{\rm phys}~{}U^{-1}(x)~{},$ (8)

and makes the field strength tensor non-zero,

$\displaystyle F_{\mu\nu}\equiv{\cal D}_{\mu}^{\rm pure}~{}A_{\nu}^{\rm
phys}-{\cal D}_{\nu}^{\rm pure}A_{\mu}^{\rm phys}-ig~{}[A_{\mu}^{\rm
phys},~{}A_{\nu}^{\rm phys}]~{}.$ (9)

This makes physical part of the gauge field to be dynamical degrees of freedom
of the theory. The covariant derivative in the adjoint representation is
defined as,

$\displaystyle{\cal D}_{\mu}^{\rm~{}pure}~{}A_{\nu}^{\rm~{}phys}$
$\displaystyle=$
$\displaystyle\partial_{\mu}A_{\nu}^{\rm~{}phys}~{}-~{}ig\left[A_{\mu}^{\rm~{}pure},~{}A_{\nu}^{\rm~{}phys}\right]~{}.$
(10)

One often define the natural gauge as the gauge where $A_{\mu}^{~{}\rm
pure}=0$ and $A_{\mu}=A_{\mu}^{~{}\rm phys}$. Now this separation of gauge
field in terms of pure and physical part is not unique as still some gauge
freedom remain. This requires further constrain on $A_{\mu}^{\rm~{}phys}$
which essentially makes it, instead of local, a non-local functional of full
gauge field, $e.g.$ in the $A^{+}_{\rm phys}=0$ gauge, as,

$\displaystyle A^{\mu}_{\rm
phys,\pm}\left(x^{-},~{}x_{\perp}\right)~{}=~{}\int_{\pm\infty^{-}}^{x^{-}}d\omega^{-}~{}U\left(x^{-},~{}\omega^{-};~{}x_{\perp}\right)~{}F^{+\mu}\left(\omega^{-},~{}x_{\perp}\right)U\left(\omega^{-},~{}x^{-};~{}x_{\perp}\right),$
(11)

where,

$\displaystyle U(a^{-},~{}b^{-};~{}x_{\perp})~{}\equiv~{}{\cal
P}\left(ig\int^{a^{-}}_{b^{-}}~{}dx^{-}~{}A^{+}\left(x^{-},~{}x_{\perp}\right)\right).$
(12)

For recent review and a comprehensive study within scalar di-quark model, see
Amor-Quiroz:2020qmw .

##  III Transverse derivative of staple gauge links with finite extent

In this section we review the derivative of a staple gauge link with respect
to transverse coordinate, along light cone,

$\displaystyle{\cal
U}_{[\eta]}\left(0,~{}z\right)=U\left(0,~{}\eta^{-};~{}0_{\perp}\right)U\left(\eta^{-};~{}0_{\perp},~{}z_{\perp}\right)U\left(\eta^{-},~{}z^{-};~{}z_{\perp}\right)$
(13)

In the limit $\eta=+\infty^{-}$ and $\eta=-\infty^{-}$ the link above becomes
past pointing and future pointing staple infinite gauge link along light cone,

$\displaystyle{\cal U}_{[\eta=-\infty]}$ $\displaystyle=$ $\displaystyle
U\left(0,-\infty^{-};~{}0_{\perp}\right)U\left(-\infty^{-};~{}0_{\perp},~{}z_{\perp}\right)U\left(-\infty^{-},~{}z^{-};~{}z_{\perp}\right)~{},$
(14) $\displaystyle{\cal U}_{[\eta=+\infty]}$ $\displaystyle=$ $\displaystyle
U\left(0,\infty^{-};~{}0_{\perp}\right)U\left(\infty^{-};~{}0_{\perp},~{}z_{\perp}\right)U\left(\infty^{-},~{}z^{-};~{}z_{\perp}\right)~{}.$
(15)

Another interesting limit is when $\eta\rightarrow 0$, which is not exactly
the straight gauge link but the staple link with shortest extent. This is
partly motivated by the recent work by M. Engelhardt et. al. where the authors
considered a general form of staple shaped path and makes an interpolation
between both Ji and Jaffe-Manohar orbital angular momentum when calculating
the OAM of quarks on the lattice using direct derivative method
Engelhardt:2020qtg .

We now recall the Wilson line from $y$ to $z$ along some arbitrary path ${\cal
C}$ that can be parametrize via $s^{\mu}$,

$\displaystyle U(z,y)$ $\displaystyle=$
$\displaystyle~{}P\exp\left(ig\int_{y,~{}{\cal
C}}^{z}~{}A_{\mu}\left(s\right)ds^{\mu}\right)~{}.$ (16)

The position derivative of the Wilson line at any point on the path $\cal C$
is given by,

$\displaystyle\left.\frac{\partial}{\partial
x^{\lambda}}U(z,y)~{}=~{}ig~{}U(z,s)A_{\mu}(s)~{}\frac{\partial
s^{\mu}}{\partial
x^{\lambda}}~{}U(s,y)\right|_{s=y}^{s=z}~{}+~{}ig\int_{y}^{z}ds^{\nu}~{}U(z,s)~{}F_{\mu\nu}(s)~{}\frac{\partial
s^{\mu}}{\partial x^{\lambda}}~{}U(s,y)~{}.$ (17)

Eq.(17) can be used to derive the derivative of ${\cal
U}_{[\eta]}\left(0,~{}z\right)$ as given in Eq.(13) with respect to
$z_{\perp}$ as,

$\displaystyle\frac{\partial}{\partial z_{\perp}^{i}}~{}{\cal
U}_{~{}[\eta]}\left(0,~{}z\right)$ $\displaystyle=$
$\displaystyle+~{}ig~{}U\left(0^{-},~{}\eta^{-};~{}0_{\perp}\right)A^{i}\left(\eta^{-},~{}0_{\perp}\right)U\left(\eta^{-};~{}0_{\perp},~{}z_{\perp}\right)U\left(\eta^{-},~{}z^{-};~{}z_{\perp}\right)$
$\displaystyle-~{}ig~{}U\left(0^{-},~{}\eta^{-};~{}0_{\perp}\right)U\left(\eta^{-};~{}0_{\perp},~{}z_{\perp}\right)U\left(\eta^{-},~{}z^{-};~{}z_{\perp}\right)A^{i}\left(z^{-},~{}z_{\perp}\right)$
$\displaystyle+~{}ig~{}U\left(0^{-},~{}\eta^{-};~{}0_{\perp}\right)~{}\int^{0_{\perp}}_{z_{\perp}}d\omega_{\perp}~{}U\left(\eta^{-};~{}0_{\perp},~{}\omega_{\perp}\right)~{}F^{\perp
i}\left(\eta^{-},~{}\omega_{\perp}\right)~{}U\left(\eta^{-};~{}\omega_{\perp},~{}z_{\perp}\right)U\left(\eta^{-},~{}z^{-};~{}z_{\perp}\right)$
$\displaystyle+~{}ig~{}U\left(0^{-},~{}\eta^{-};~{}0_{\perp}\right)U\left(\eta^{-};~{}0_{\perp},~{}z_{\perp}\right)\int^{\eta^{-}}_{z^{-}}d\omega^{-}~{}U\left(\eta^{-},~{}\omega^{-};~{}z_{\perp}\right)~{}F^{+i}\left(\omega^{-},~{}z_{\perp}\right)~{}U\left(\omega^{-},~{}z^{-};~{}z_{\perp}\right)~{}.$

In the limit $z_{\perp}^{i}\rightarrow 0$ the expression simplifies further
as,

$\displaystyle\lim_{z_{\perp}^{i}\rightarrow 0}~{}\frac{\partial}{\partial
z_{\perp}^{i}}~{}{\cal U}_{~{}[\eta]}\left(0,~{}z\right)$ $\displaystyle=$
$\displaystyle
ig~{}U\left(0^{-},~{}\eta^{-};~{}0_{\perp}\right)A^{i}\left(\eta^{-},~{}0_{\perp}\right)U\left(\eta^{-},z^{-};~{}0_{\perp}\right)-~{}ig~{}U\left(0^{-},~{}z^{-};~{}0_{\perp}\right)A^{i}\left(z^{-},~{}0_{\perp}\right)$
$\displaystyle+~{}ig~{}\int^{\eta^{-}}_{z^{-}}d\omega^{-}~{}U\left(0^{-},~{}\omega^{-};~{}0_{\perp}\right)~{}F^{+i}\left(\omega^{-},~{}0_{\perp}\right)~{}U\left(\omega^{-},~{}z^{-};~{}0_{\perp}\right)~{}.$

Special Case: ($\eta\rightarrow\pm\infty$)

$\displaystyle\left.\lim_{z_{\perp}^{i}\rightarrow
0}~{}\frac{\partial}{\partial z_{\perp}^{i}}~{}{\cal
U}_{~{}[\eta=\pm\infty]}\left(0,~{}z\right)\right|_{\eta^{-}\rightarrow\pm\infty}$
$\displaystyle=ig~{}U\left(0^{-},~{}\pm\infty^{-};~{}0_{\perp}\right)A^{i}\left(\pm\infty^{-},~{}0_{\perp}\right)U\left(\pm\infty^{-},~{}z^{-};~{}0_{\perp}\right)-~{}ig~{}U\left(0^{-},~{}z^{-};~{}0_{\perp}\right)A^{i}\left(z^{-},~{}0_{\perp}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+~{}ig~{}\int^{\pm\infty^{-}}_{z^{-}}d\omega^{-}~{}U\left(0^{-},~{}\omega^{-};~{}0_{\perp}\right)~{}F^{+i}\left(\omega^{-},~{}0_{\perp}\right)~{}U\left(\omega^{-},~{}z^{-};~{}0_{\perp}\right)~{}.$
(20)

We can now express Eq.$\eqref{LaLaLand3}$ in terms of the physical gauge field
$A^{i}_{\rm phys,\pm}$ and the pure gauge field $A^{i}_{\rm pure}$,

$\displaystyle\lim_{z_{\perp}^{i}\rightarrow 0}~{}\frac{\partial}{\partial
z_{\perp}^{i}}~{}{\cal U}_{[\eta]}\left(0,~{}z\right)$ (21) $\displaystyle=$
$\displaystyle
ig~{}U\left(0^{-},~{}\eta^{-};~{}0_{\perp}\right)A^{i}\left(\eta^{-},~{}0_{\perp}\right)U\left(\eta^{-},~{}z^{-};~{}0_{\perp}\right)-~{}ig~{}U\left(0^{-},~{}z^{-};~{}0_{\perp}\right)A^{i}\left(z^{-},~{}0_{\perp}\right)$
$\displaystyle-~{}U\left(0^{-},~{}\eta^{-};~{}0_{\perp}\right)A^{i}_{\rm
phys,\pm}\left(\eta^{-},~{}0_{\perp}\right)U\left(\eta^{-},~{}z^{-};~{}0_{\perp}\right)+ig~{}U\left(0^{-},~{}z^{-};~{}0_{\perp}\right)~{}A^{i}_{\rm
phys,\pm}\left(z^{-};~{}0_{\perp}\right)$ $\displaystyle=$ $\displaystyle
ig~{}U\left(0^{-},~{}\eta^{-};~{}0_{\perp}\right)A^{i}_{\rm
pure}\left(\eta^{-},~{}0_{\perp}\right)U\left(\eta^{-},~{}z^{-};~{}0_{\perp}\right)-~{}ig~{}U\left(0^{-},~{}z^{-};~{}0_{\perp}\right)A^{i}_{\rm
pure}\left(z^{-},~{}0_{\perp}\right),$

where we have used the definition of $A^{\mu}_{\rm phys}$ as in Eq.(11) and
also Eq.(3), which are based on the Chen et. al. decomposition Chen:2008ag ;
Chen:2009mr  of gauge fields as reviewed in the previous section.

##  IV Gluon Orbital Angular Momentum

Orbital angular momentum of gluon can be expressed as the phase space average
of the classical orbital angular momentum weighted with the Wigner
distribution of polarized gluons in a longitudinally polarized nucleon. The
Wigner distribution is essentially the phase space distribution of gluons in
transverse momentum ($k_{\perp}$) - impact parameter ($b_{\perp}$) space,

$\displaystyle{\cal W}_{g}\left(x,k_{\perp},b_{\perp}\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\Delta_{\perp}}{(2\pi)^{2}}~{}e^{-i\Delta_{\perp}b_{\perp}}\int\frac{d^{2}z_{\perp}}{(2\pi)^{2}}~{}e^{iz_{\perp}k_{\perp}}\int\frac{dz^{-}}{2(2\pi)~{}xP^{+}}~{}e^{-ixP^{+}z^{-}}$
(22) $\displaystyle~{}~{}~{}~{}~{}\times~{}\left\langle
P^{+},~{}-\frac{\Delta_{\perp}}{2},~{}S\left|~{}{\rm
Tr}~{}F^{+i}\left(0\right)~{}{\cal
U}_{[\eta_{1}]}\left(0~{};z\right)~{}F^{+i}\left(z\right)~{}{\cal
U}_{[\eta_{2}]}\left(z~{};0\right)\right|P^{+},~{}\frac{\Delta_{\perp}}{2},~{}S\right\rangle~{},$

where $S$ is the spin of the target states, and the trace is in fundamental
representation. The cross product of transverse momentum and impact parameter
returns orbital angular momentum of gluons, integrating over them gives
orbital angular momentum distribution,

$\displaystyle L_{g}(x)$ $\displaystyle=$ $\displaystyle\int
d^{2}b_{\perp}~{}d^{2}k_{\perp}~{}\left(b_{\perp}\times
k_{\perp}\right)~{}{\cal W}_{g}\left(x,k_{\perp},b_{\perp}\right)~{},$ (23)
$\displaystyle=$ $\displaystyle\int
d^{2}b_{\perp}~{}d^{2}k_{\perp}~{}\epsilon^{kj}~{}b_{\perp}^{k}k_{\perp}^{j}~{}{\cal
W}_{g}\left(x,k_{\perp},b_{\perp}\right)~{}.$ (24)

This can further be written as,

$\displaystyle L_{g}(x)$ $\displaystyle=$ $\displaystyle\int
d^{2}b_{\perp}~{}\epsilon^{kj}~{}b_{\perp}^{k}~{}\int\frac{d^{2}\Delta_{\perp}}{(2\pi)^{2}}~{}e^{-i\Delta_{\perp}b_{\perp}}\int\frac{dz^{-}}{2(2\pi)~{}xP^{+}}~{}e^{-ixP^{+}z^{-}}~{}$
(25) $\displaystyle~{}~{}~{}~{}~{}\times\lim_{z_{\perp}^{j}\rightarrow
0}\frac{1}{i}~{}\frac{\partial}{\partial z_{\perp}^{j}}~{}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S\left|~{}{\rm
Tr}~{}~{}F^{+i}\left(0\right)~{}{\cal
U}_{[\eta_{1}]}\left(0~{};z\right)~{}F^{+i}\left(z\right)~{}{\cal
U}_{[\eta_{2}]}\left(z~{};0\right)\right|P^{+},+\frac{\Delta}{2},~{}S\right\rangle~{}.$

Now the transverse derivative in Eq.(25) can be performed using Eq.(21)
leading to,

$\displaystyle L_{g}(x)$ $\displaystyle=$ $\displaystyle\int
d^{2}b_{\perp}~{}\epsilon^{kj}~{}b_{\perp}^{k}~{}\int\frac{d^{2}\Delta_{\perp}}{(2\pi)^{2}}~{}e^{-i\Delta_{i}b_{\perp}}\int\frac{dz^{-}}{2(2\pi)~{}xP^{+}}~{}e^{-ixP^{+}z^{-}}\frac{1}{i}~{}\left\langle
P^{+},-\frac{\Delta}{2},S\left|~{}{\rm Tr}~{}\sum_{m=1}^{5}~{}{\cal
O}_{m}^{j}~{}\right|P^{+},+\frac{\Delta}{2},S\right\rangle~{}.$

Here the object ${\cal O}_{m}$ are defined for convenience and are as follows,

$\displaystyle{\cal O}_{1}^{j}$ $\displaystyle=$ $\displaystyle
F^{+i}\left(0\right)~{}ig~{}U\left(0^{-},~{}\eta_{1}^{-};~{}0_{\perp}\right)A^{j}_{\rm
pure}\left(\eta_{1}^{-},~{}0_{\perp}\right)U\left(\eta_{1}^{-},~{}z^{-};~{}0_{\perp}\right)F^{+i}\left(z^{-},~{}0_{\perp}\right){\cal
U}_{[\eta_{2}]}\left(z^{-},~{}0^{-};~{}0_{\perp}\right),$ $\displaystyle{\cal
O}_{2}^{j}$ $\displaystyle=$
$\displaystyle-~{}F^{+i}\left(0\right)~{}ig~{}U\left(0^{-},~{}z^{-};~{}0_{\perp}\right)A^{j}_{\rm
pure}\left(z^{-},~{}0_{\perp}\right)F^{+i}\left(z^{-},~{}0_{\perp}\right){\cal
U}_{[\eta_{2}]}\left(z^{-},~{}0^{-};~{}0_{\perp}\right),$ $\displaystyle{\cal
O}_{3}^{j}$ $\displaystyle=$ $\displaystyle F^{+i}\left(0\right)~{}{\cal
U}_{[\eta_{1}]}\left(0^{-},~{}z^{-};~{}0_{\perp}\right)~{}\left[\partial^{j}F^{+i}\left(z^{-},~{}z_{\perp}\right)\right]_{z_{\perp}=~{}0}~{}{\cal
U}_{[\eta_{2}]}\left(z^{-},~{}0^{-};~{}0_{\perp}\right),$ $\displaystyle{\cal
O}_{4}^{j}$ $\displaystyle=$ $\displaystyle F^{+i}\left(0\right)~{}{\cal
U}_{[\eta_{1}]}\left(0~{},z^{-};~{}0_{\perp}\right)~{}F^{+i}\left(z^{-};~{}0_{\perp}\right)~{}ig~{}A^{j}_{\rm
pure}\left(z^{-},~{}0_{\perp}\right)~{}U\left(z^{-},~{}0^{-};~{}0_{\perp}\right),$
$\displaystyle{\cal O}_{5}^{j}$ $\displaystyle=$
$\displaystyle-F^{+i}\left(0\right)~{}{\cal
U}_{[\eta_{1}]}\left(0~{},z^{-};~{}0_{\perp}\right)~{}F^{+i}\left(z^{-};~{}0_{\perp}\right)~{}ig~{}U\left(z^{-},~{}{\eta}^{-}_{2};~{}0_{\perp}\right)~{}A^{j}_{\rm
pure}\left({\eta_{2}}^{-},~{}0_{\perp}\right)~{}U\left({\eta}_{2}^{-},~{}0^{-};~{}0_{\perp}\right).$

Eq. (LABEL:Inception12) can further be simplified as,

$\displaystyle L_{g}(x)$ $\displaystyle=$
$\displaystyle\epsilon^{kj}~{}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\int\frac{dz^{-}}{2(2\pi)xP^{+}}~{}e^{-ixP^{+}z^{-}}~{}\left\langle
P^{+},-\frac{\Delta}{2},S\left|~{}{\rm Tr}~{}\sum_{m=1}^{5}~{}{\cal
O}_{m}^{j}~{}\right|P^{+},+\frac{\Delta}{2},S\right\rangle$

Until now we have not taken any assumptions specifically for small-$x$. The
problems now reduced to performing the $z^{-}$ integration. Now we expand the
exponential in Eq.(LABEL:Inception120),

$\displaystyle L_{g}(x)$ $\displaystyle=$
$\displaystyle\epsilon^{kj}~{}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\int\frac{dz^{-}}{2(2\pi)xP^{+}}~{}e^{-ixP^{+}z^{-}}~{}\left\langle
P^{+},-\frac{\Delta}{2},S\left|~{}{\rm Tr}~{}\sum_{m=1}^{5}~{}{\cal
O}_{m}^{j}~{}\right|P^{+},+\frac{\Delta}{2},S\right\rangle$ (28)
$\displaystyle=$ $\displaystyle\epsilon^{kj}~{}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\int\frac{dz^{-}}{2(2\pi)}~{}\sum_{n=0}^{\infty}(-i)^{n}(xP^{+})^{n-1}{z^{-}}^{n}~{}\left\langle
P^{+},-\frac{\Delta}{2},S\left|~{}{\rm Tr}~{}\sum_{m=1}^{5}~{}{\cal
O}_{m}^{j}~{}\right|P^{+},+\frac{\Delta}{2},S\right\rangle$
$\displaystyle\equiv$ $\displaystyle\epsilon^{kj}~{}\left[\frac{1}{x}~{}{\cal
L}^{jk}_{g,\rm{0}}\left(x\right)+{\cal
L}^{jk}_{g,\rm{1}}\left(x\right)+x~{}{\cal
L}^{jk}_{g,\rm{2}}\left(x\right)+x^{2}{\cal
L}^{jk}_{g,\rm{3}}\left(x\right)+...\right]$

Here ${\cal L}^{jk}_{g,\rm{0}}\left(x\right)$, ${\cal
L}^{jk}_{g,\rm{1}}\left(x\right)$, ${\cal L}^{jk}_{g,\rm{2}}\left(x\right)$ …
can be interpreted as eikonal, sub-eikonal, sub-sub-eikonal contributions to
the gluon OAM. In this convention, sub-eikonal refers to the term suppressed
by one power of $x$ compared to the eikonal or leading scattering, sub-sub-
eikonal refers to suppression by two power of $x$, and so on. Any dependence
on $x$, for the terms, ${\cal L}^{jk}_{g,\rm{0}}\left(x\right)$, ${\cal
L}^{jk}_{g,\rm{1}}\left(x\right)$, ${\cal L}^{jk}_{g,\rm{2}}\left(x\right)$ …,
can be estimated only by constructing and solving small-$x$ evolution of the
object inside the angle bracket $<...>$. In the following we will study the
operator structure of eikonal and sub-eikonal terms in Eq.(28).

##  V Gluons OAM at eikonal limit

In this leading order we can approximate $\exp\left(-ixP^{+}z^{-}\right)\simeq
1$, and go ahead to perform the $z^{-}$ integration in
Eq.(LABEL:Inception120). Now its also important to isolate only the polarised
contributions, from the object inside the angle bracket $<...>$, that would
give one more $\epsilon^{ij}$, in addition to the one that is already there
originating from the definition of OAM, leading to surviving OAM for gluons.
Now before we proceed, it will be convenient to define the following PT even
and PT odd parts of $A_{\rm phys}$,

$\displaystyle A^{\mu}_{\rm phys,~{}e}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[A^{\mu}_{\rm phys,+}(x)+A^{\mu}_{\rm
phys,-}(x)\right]$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{-\infty^{-}}^{+\infty^{-}}dy^{-}\epsilon(y^{-})~{}U(x^{-},y^{-};~{}x_{\perp})~{}F^{+\mu}(y^{-},~{}x_{\perp})~{}U(y^{-},x^{-};~{}x_{\perp})$
$\displaystyle A^{\mu}_{\rm phys,~{}o}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[A^{\mu}_{\rm phys,+}(x)-A^{\mu}_{\rm
phys,-}(x)\right]$ (30) $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{-\infty^{-}}^{+\infty^{-}}dy^{-}~{}U(x^{-},y^{-};~{}x_{\perp})~{}F^{+\mu}(y^{-},~{}x_{\perp})~{}U(y^{-},x^{-};~{}x_{\perp})$

with $\epsilon(y^{-})$ being the sign function. Now here we list the integral
over $z^{-}$ of the respective ${\cal O}_{m}$, details of $z^{-}$ integrations
are in the appendix,

$\displaystyle\int dz^{-}~{}{\cal O}_{1}^{j}$ $\displaystyle=$
$\displaystyle-2ig~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}U\left(0^{-},~{}\eta_{1}^{-};~{}0_{\perp}\right)~{}A^{j}_{\rm
pure}\left(\eta_{1}^{-},~{}0_{\perp}\right)~{}U\left(\eta_{1}^{-},~{}\eta_{2}^{-};~{}0_{\perp}\right)~{}A_{\rm
phys,~{}o}^{i}\left(\eta^{-}_{2},~{}0_{\perp}\right)~{}U\left(\eta_{2}^{-},~{}0^{-};~{}0_{\perp}\right)$
$\displaystyle\int dz^{-}~{}{\cal O}_{2}^{j}$ $\displaystyle=$ $\displaystyle
2ig~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)A^{j}_{\rm
res}\left(0^{-},~{}0_{\perp}\right)~{}A^{i}_{\rm
phys,~{}o}\left(0^{-},~{}0_{\perp}\right)$ $\displaystyle\int dz^{-}~{}{\cal
O}_{3}^{j}$ $\displaystyle=$
$\displaystyle-2F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}\partial^{j}A^{i}_{\rm
phys,~{}o}\left(0^{-},~{}0_{\perp}\right)$ $\displaystyle\int dz^{-}~{}{\cal
O}_{4}^{j}$ $\displaystyle=$
$\displaystyle-2ig~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}A_{\rm
phys,~{}o}^{i}\left(0^{-},~{}0_{\perp}\right)~{}A_{\rm
res}^{j}\left(0^{-},~{}0_{\perp}\right)$ $\displaystyle\int dz^{-}~{}{\cal
O}_{5}^{j}$ $\displaystyle=$ $\displaystyle
2ig~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)U\left(0^{-},~{}\eta_{1}^{-};~{}0_{\perp}\right)A^{i}_{\rm
phys,~{}o}\left(\eta_{1}^{-},~{}0_{\perp}\right)U\left(\eta_{1}^{-},~{}{\eta_{2}^{-}};~{}0_{\perp}\right)A_{\rm
pure}^{j}~{}\left(\eta_{2}^{-},~{}0_{\perp}\right)~{}U\left(\eta_{2}^{-},~{}0^{-};~{}0_{\perp}\right)$

This results can be supplemented back to Eq.(LABEL:Inception120) to arrive at
the final expression of the Gluon OAM at small-$x$,

$\displaystyle{\cal L}^{jk}_{g,\rm{0}}\left(x\right)$ $\displaystyle=$
$\displaystyle~{}-\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S\left|~{}{\rm
Tr}~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}\partial^{j}A^{i}_{\rm
phys,~{}o}\left(0^{-},~{}0_{\perp}\right)~{}\right|P^{+},+\frac{\Delta}{2},~{}S\right\rangle$
(31) $\displaystyle~{}-\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S\left|~{}{\rm
Tr}~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}ig~{}\left[A^{i}_{\rm
phys,~{}o}~{},~{}A^{j}_{\rm
res}\right]\left(0^{-},~{}0_{\perp}\right)~{}\right|P^{+},+\frac{\Delta}{2},~{}S\right\rangle$
$\displaystyle~{}-\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S\left|~{}igF^{+i}(0^{-},0_{\perp})~{}U\left(0^{-},\eta^{-}_{1};0_{\perp}\right)\left[A^{j}_{\rm
pure}\left(\eta_{1}^{-},0_{\perp}\right)U(\eta_{1}^{-},\eta_{2}^{-};0_{\perp})A^{i}_{\rm
phys}\left(\eta_{2}^{-},0_{\perp}\right)\right.\right.\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.-A^{i}_{\rm
phys}\left(\eta_{1}^{-},0_{\perp}\right)U(\eta_{1}^{-},\eta_{2}^{-};0_{\perp})A^{j}_{\rm
pure}\left(\eta_{2}^{-},0_{\perp}\right)\right]U(\eta_{2}^{-},0^{-};0_{\perp})\right|P^{+},+\frac{\Delta}{2},~{}S\right\rangle$

In the contour gauge e.g. in $A^{+}=0$ gauge, $A^{\rm res}=A^{\rm pure}$,
which leads to,

$\displaystyle\left[A^{i}_{\rm phys,~{}o},~{}A^{j}_{\rm
res}\right]\mapsto\left[A^{i}_{\rm phys,~{}o},~{}A^{j}_{\rm pure}\right]$ (32)

The eikonal contribution to gluon OAM operator, ${\cal L}^{g}_{\rm{0}}(x)$,
using Eq.(30), can then be written as,

$\displaystyle{\cal L}^{jk}_{g,\rm{0}}(x,k_{\perp},\Delta_{\perp},S)$ (33)
$\displaystyle=$ $\displaystyle-~{}\frac{1}{2}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S\left|~{}{\rm
Tr}~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}{\cal D}^{j}_{\rm
pure}\left(A^{i}_{\rm phys,+}-A^{i}_{\rm
phys,-}\right)\left(0^{-},~{}0_{\perp}\right)~{}\right|P^{+},+\frac{\Delta}{2},~{}S\right\rangle$
$\displaystyle-~{}\frac{1}{2}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S\left|~{}igF^{+i}(0^{-},0_{\perp})~{}U\left(0^{-},\eta^{-}_{1};0_{\perp}\right)\left[A^{j}_{\rm
pure}\left(\eta_{1}^{-},0_{\perp}\right)U(\eta_{1}^{-},\eta_{2}^{-};0_{\perp})\right.\right.\right.$
$\displaystyle\left.~{}~{}~{}\left(A^{i}_{\rm phys,+}-A^{i}_{\rm
phys,-}\right)\left(\eta_{2}^{-},0_{\perp}\right)\left.\left.-\left(A^{i}_{\rm
phys,+}-A^{j}_{\rm
phys,-}\right)\left(\eta_{1}^{-},0_{\perp}\right)U(\eta_{1}^{-},\eta_{2}^{-};0_{\perp})A^{j}_{\rm
pure}\left(\eta_{2}^{-},0_{\perp}\right)\right]\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}U(\eta_{2}^{-},0^{-};0_{\perp})\right|P^{+},+\frac{\Delta}{2},~{}S\right\rangle,$

where we have explicitly shown the presence of $A^{i}_{\rm phys,\pm}$ in the
expression. Now under PT transformation,

$\displaystyle\Delta_{\perp}$ $\displaystyle\rightarrow$
$\displaystyle-\Delta_{\perp},$ $\displaystyle S$ $\displaystyle\rightarrow$
$\displaystyle-S,$ $\displaystyle F^{\mu\nu}$ $\displaystyle\rightarrow$
$\displaystyle-F^{\mu\nu},$ $\displaystyle A^{\mu}_{\rm phys,\pm}$
$\displaystyle\rightarrow$ $\displaystyle A^{\mu}_{\rm phys,\mp}~{}.$ (34)

This implies that ${\cal L}^{jk}_{g,\rm{0}}(x,k_{\perp},\Delta_{\perp},S)$ and
its PT transformed pair are identical $i.e.$,

$\displaystyle{\cal L}^{jk}_{g,\rm{0}}(x,k_{\perp},\Delta_{\perp},S)={\cal
L}^{jk}_{g,\rm{0}}(x,-k_{\perp},-\Delta_{\perp},-S),$ (35)

which ensure that the off-forward matrix elements, in ${\cal
L}^{jk}_{g,\rm{0}}(x,k_{\perp},\Delta_{\perp},S)$, do not have any spin
dependent structure of the following form,

$\displaystyle
i\frac{S^{+}}{P^{+}}\epsilon^{lm}\Delta^{l}_{\perp}k^{m}_{\perp}...$ (36)

In absence of another totally antisymmetric tensor $\epsilon$ inherent to
${\cal L}^{jk}_{g,\rm{0}}$, the eikonal contribution in Eq.(28) vanishes as,

$\displaystyle\epsilon^{kj}{\cal L}^{jk}_{g,\rm{0}}=0.$ (37)

Non-vanishing spin effect starts showing up only at sub eikonal level which we
will discuss next.

##  VI Gluon OAM at first sub-eikonal order

When one moves to next non-trivial order in the expansion of the exponential,
as $\exp\left(-ixP^{+}z^{-}\right)\simeq 1-ixP^{+}z^{-}$, the additional $i$
makes all leading T-even terms to be T-odd, leading to non-zero contribution
to gluon OAM for longitudinally polarized targets,

$\displaystyle{\cal L}^{jk}_{g,\rm{1}}\left(x\right)$ $\displaystyle=$
$\displaystyle~{}iP^{+}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S\left|~{}{\rm
Tr}~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}\partial^{j}{\bar{O}}^{i}_{\rm
phys,*}\left(0^{-},~{}0_{\perp}\right)~{}\right|P^{+},+\frac{\Delta}{2},~{}S\right\rangle$
(38) $\displaystyle+iP^{+}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S\left|~{}{\rm
Tr}~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}ig~{}\left[{\bar{O}}^{i}_{\rm
phys,*},~{}A^{j}_{\rm
res}\right]\left(0^{-},~{}0_{\perp}\right)~{}\right|P^{+},+\frac{\Delta}{2},~{}S\right\rangle$
$\displaystyle+iP^{+}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S\left|~{}igF^{+i}(0^{-},0_{\perp})~{}U\left(0^{-},\eta^{-}_{1};0_{\perp}\right)\left[A^{j}_{\rm
pure}\left(\eta_{1}^{-},0_{\perp}\right)U(\eta_{1}^{-},\eta_{2}^{-};0_{\perp})~{}{\bar{O}}^{i}_{\rm
phys,*}\left(\eta_{2}^{-},0_{\perp}\right)\right.\right.\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.-{\bar{O}}^{i}_{\rm
phys,*}\left(\eta_{1}^{-},0_{\perp}\right)U(\eta_{1}^{-},\eta_{2}^{-};0_{\perp})A^{j}_{\rm
pure}\left(\eta_{2}^{-},0_{\perp}\right)\right]U(\eta_{2}^{-},0^{-};0_{\perp})\right|P^{+},+\frac{\Delta}{2},~{}S\right\rangle$

where,

$\displaystyle{\bar{O}}^{i}_{\rm phys,*}(x)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{-\infty^{-}}^{+\infty^{-}}dz^{-}~{}z^{-}~{}U(x^{-},z^{-};~{}x_{\perp})~{}F^{+i}(z^{-},~{}x_{\perp})~{}U(z^{-},x^{-};~{}x_{\perp}).$
(39)

This is essentially the $z^{-}$ moment of the physical gauge $A^{i}_{\rm
phys,o}$ in Eq.(42). However, unlike $A^{i}_{\rm phys,o}$, the object
${\bar{O}}^{i}_{\rm phys,*}$ in the above expression is PT even, similar to
Eq.(V), because of the presence of an extra $z^{-}$, in the integrand, which
can be written as $z^{-}=|z^{-}|~{}{\rm sign}(z^{-})$. This implies that
unlike ${\cal L}^{jk}_{g,\rm{0}}$, the term ${\cal
L}^{jk}_{g,\rm{1}}(x,k_{\perp},\Delta_{\perp},S)$ is odd under PT
transformation as,

$\displaystyle{\cal L}^{jk}_{g,\rm{1}}(x,k_{\perp},\Delta_{\perp},S)=-{\cal
L}^{jk}_{g,\rm{1}}(x,-k_{\perp},-\Delta_{\perp},-S).$ (40)

One would expect to get non-trivial contribution to gluon OAM, in
longitudinally polarized proton, right from this order. As the expression is
originally stems from a Fourier transform, one can get the intermediate time
$z^{-}$ out of the integral to have an overall energy derivative,

$\displaystyle{\bar{O}}^{i}_{\rm phys,*}(x)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}~{}i\partial_{k}^{+}\int_{-\infty^{-}}^{+\infty^{-}}dz^{-}~{}U(x^{-},z^{-};~{}x_{\perp})~{}F^{+i}(z^{-},~{}x_{\perp})~{}U(z^{-},x^{-};~{}x_{\perp}),$
(41) $\displaystyle=$
$\displaystyle-\frac{1}{2}~{}i\partial_{k}^{+}\left(A^{i}_{\rm
phys,+}-A^{i}_{\rm phys,-}\right),$

which leads to,

$\displaystyle{\cal L}^{jk}_{g,\rm{1}}(x,k_{\perp},\Delta_{\perp},S)$ (42)
$\displaystyle=$ $\displaystyle\frac{1}{2}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}\frac{\partial}{\partial
x}~{}\left\langle P^{+},-\frac{\Delta}{2},~{}S_{\perp}\left|~{}{\rm
Tr}~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}{\cal D}^{j}_{\rm
pure}\left(A^{i}_{\rm phys,+}-A^{i}_{\rm
phys,-}\right)\left(0^{-},~{}0_{\perp}\right)~{}\right|P^{+},+\frac{\Delta}{2},~{}S_{\perp}\right\rangle$
$\displaystyle+\frac{1}{2}\lim_{\Delta^{k}\rightarrow
0}\frac{\partial}{\partial\Delta^{k}}\frac{\partial}{\partial
x}~{}\left\langle
P^{+},-\frac{\Delta}{2},~{}S_{\perp}\left|~{}igF^{+i}(0^{-},0_{\perp})~{}U\left(0^{-},\eta^{-}_{1};0_{\perp}\right)\left[A^{j}_{\rm
pure}\left(\eta_{1}^{-},0_{\perp}\right)U(\eta_{1}^{-},\eta_{2}^{-};0_{\perp})\right.\right.\right.$
$\displaystyle\left.~{}~{}~{}\left(A^{i}_{\rm phys,+}-A^{i}_{\rm
phys,-}\right)\left(\eta_{2}^{-},0_{\perp}\right)\left.\left.-\left(A^{i}_{\rm
phys,+}-A^{j}_{\rm
phys,-}\right)\left(\eta_{1}^{-},0_{\perp}\right)U(\eta_{1}^{-},\eta_{2}^{-};0_{\perp})A^{j}_{\rm
pure}\left(\eta_{2}^{-},0_{\perp}\right)\right]\right.\right.$
$\displaystyle\left.\left.~{}~{}~{}U(\eta_{2}^{-},0^{-};0_{\perp})\right|P^{+},+\frac{\Delta}{2},~{}S_{\perp}\right\rangle.$

This also corroborate the fact that in the Taylor expansion of the phase
factor $\exp(ixP^{+}z^{-})$, only the odd terms in $x$ can contribute to the
gluon OAM for longitudinally polarized proton Hatta:2016aoc .

##  VII conclusion

Recently the connection between gluon Wigner distribution and gluon orbital
angular momentum (OAM) have been used to probe the gluon OAM in the hard
scattering process at the Electron - Ion Collider. The single longitudinal
spin asymmetry in hard diffractive dijet production found to be sensitive to
the gluon orbital angular momentum distribution, at least in the moderate
$x$-range Ji:2016jgn . In the theoretical front, specifically at small-$x$,
Hatta presented a general analysis of the orbital angular momentum of
distribution of gluons Hatta:2011ku ; Hatta:2016aoc . Novel operator
representation of $L_{g}(x)$ has been derived at small-$x$ limit and
interestingly found to contain covariant derivatives inserted at some
intermediate time between far past and far future. Moreover, the orbital
angular momentum distribution was found to be proportional to the gluon
helicity distributions $L_{g}(x)\approx-\Delta G(x)$ at small-$x$.
Subsequently, Kovchegov derived the small-$x$ asymptotics of the gluon orbital
angular momentum distribution of proton in the double logarithmic
approximation Kovchegov:2019rrz . The procedure adopted was to start with the
operator definition of gluon OAM, simplifying it at small-$x$ and relating it
to the polarized dipole amplitudes for the gluon helicities. The small-$x$
asymptotic of the later then utilized to derive the small-$x$ asymptotic of
the OAM at large $N_{c}$ limit.

In this work we studied the transverse derivatives of the staple gauge links,
with varying extent along the light cone. This is partly motivated by the
recent work by M. Engelhardt Engelhardt:2020qtg  where the authors considered
a general form of staple shaped path and computed both Ji and Jaffe-Manohar
orbital angular momentum in lattice using the direct derivative method. The
derivative of such gauge link with respect to transverse position at zero
transverse position limit has been calculated and expressed in terms of the
pure gauge components of the gauge fields within the framework of Chen et. al.
decomposition of gauge fields.

Wigner distribution of the gluon is essentially the distribution of gluons in
impact parameter and transverse momentum space. The connection between the two
has been recalled to derive the OAM distribution in terms of the pure or
physical gauge components and the gauge link parameters. After performing the
integration over both the $z^{-}$ and $z_{\perp}$ we derive the general
operator form of the OAM, of gluons in a longitudinally polarized proton, that
is valid for all possible geometrics of the gauge links. At an appropriate
combination of the extent parameters this correctly reproduces both Jaffe-
Manohar and Ji’s OAM, and offers a continuous analytical interpolation between
the two, for gluon OAM distribution in a longitudinally polarised proton.

###### Acknowledgements.

We sincerely acknowledge and thank the anonymous referee for the critical
reading and all the suggestions that substantially improve the article. The
work is supported by the DST - Govt. of India through SERB-MATRICS Project
Grant No. MTR/2019/001551.

##  VIII appendix

###  VIII.1 Integration over $z^{-}$

####  VIII.1.1 For ${\cal O}_{1}^{j}$ and ${\cal O}_{5}^{j}$

$\displaystyle\int dz^{-}~{}{\cal O}_{1}^{j}$ $\displaystyle=$
$\displaystyle\int
dz^{-}F^{+i}\left(0\right)~{}ig~{}U\left(0^{-},~{}\eta_{1}^{-};~{}0_{\perp}\right)A^{j}_{\rm
pure}\left(\eta_{1}^{-},~{}0_{\perp}\right)U\left(\eta_{1}^{-},~{}z^{-};~{}0_{\perp}\right)F^{+i}\left(z^{-},~{}0_{\perp}\right){\cal
U}_{[\eta_{2}]}\left(z^{-},~{}0^{-};~{}0_{\perp}\right)$

Now,

$\displaystyle{\cal
U}_{[\eta_{2}]}\left(z^{-},~{}0^{-};~{}0_{\perp}\right)={U}\left(z^{-},~{}0^{-};~{}0_{\perp}\right)$
(43)

and,

$\displaystyle~{}~{}~{}~{}~{}\int_{-\infty}^{+\infty}dz^{-}~{}U\left(\eta_{2}^{-},~{}z^{-};~{}0_{\perp}\right)F^{+i}\left(z^{-},~{}0_{\perp}\right){U}\left(z^{-},~{}\eta_{2}^{-};~{}0_{\perp}\right)$
$\displaystyle=-\int_{+\infty}^{\eta_{2}^{-}}dz^{-}~{}U\left(\eta_{2}^{-},~{}z^{-};~{}0_{\perp}\right)F^{+i}\left(z^{-},~{}0_{\perp}\right){U}\left(z^{-},~{}\eta_{2}^{-};~{}0_{\perp}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int_{-\infty}^{\eta_{2}^{-}}dz^{-}~{}U\left(\eta_{2}^{-},~{}z^{-};~{}0_{\perp}\right)F^{+i}\left(z^{-},~{}0_{\perp}\right){U}\left(z^{-},~{}\eta_{2}^{-};~{}0_{\perp}\right)$
$\displaystyle=-A_{\rm phys,+}\left(\eta_{2}^{-},~{}0_{\perp}\right)+A_{\rm
phys,-}\left(\eta_{2}^{-},~{}0_{\perp}\right)$ $\displaystyle=-2A_{\rm
phys,~{}o}\left(\eta_{2}^{-},~{}0_{\perp}\right)$

Therefore,

$\displaystyle\int dz^{-}~{}{\cal O}_{1}^{j}$ $\displaystyle=$
$\displaystyle-2ig~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}U\left(0^{-},~{}\eta_{1}^{-};~{}0_{\perp}\right)~{}A^{j}_{\rm
pure}\left(\eta_{1}^{-},~{}0_{\perp}\right)~{}U\left(\eta_{1}^{-},~{}\eta_{2}^{-};~{}0_{\perp}\right)~{}A_{\rm
phys,~{}o}^{i}\left(\eta^{-}_{2},~{}0_{\perp}\right)~{}U\left(\eta_{2}^{-},~{}0^{-};~{}0_{\perp}\right)$

Similarly one can get,

$\displaystyle\int dz^{-}~{}{\cal O}_{5}^{j}$ $\displaystyle=$
$\displaystyle-\int dz^{-}F^{+i}\left(0\right)~{}{\cal
U}_{[\eta_{1}]}\left(0~{},z^{-};~{}0_{\perp}\right)~{}F^{+i}\left(z^{-};~{}0_{\perp}\right)~{}ig~{}U\left(z^{-},~{}{\eta}^{-}_{2};~{}0_{\perp}\right)~{}A^{j}_{\rm
pure}\left({\eta_{2}}^{-},~{}0_{\perp}\right)~{}U\left({\eta}_{2}^{-},~{}0^{-};~{}0_{\perp}\right)$
$\displaystyle=$ $\displaystyle
2ig~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)U\left(0^{-},~{}\eta_{1}^{-};~{}0_{\perp}\right)A^{i}_{\rm
phys,~{}o}\left(\eta_{1}^{-},~{}0_{\perp}\right)U\left(\eta_{1}^{-},~{}{\eta_{2}^{-}};~{}0_{\perp}\right)A_{\rm
pure}^{j}~{}\left(\eta_{2}^{-},~{}0_{\perp}\right)~{}U\left(\eta_{2}^{-},~{}0^{-};~{}0_{\perp}\right)$

####  VIII.1.2 For ${\cal O}_{2}^{j}$ and ${\cal O}_{4}^{j}$

$\displaystyle\int dz^{-}~{}{\cal O}_{2}^{j}$ $\displaystyle=$
$\displaystyle-\int
dz^{-}~{}F^{+i}\left(0^{-},0_{\perp}\right)~{}ig~{}U\left(0^{-},~{}z^{-};~{}0_{\perp}\right)A^{j}_{\rm
pure}\left(z^{-},~{}0_{\perp}\right)F^{+i}\left(z^{-},~{}0_{\perp}\right){\cal
U}_{[\eta_{2}]}\left(z^{-},~{}0^{-};~{}0_{\perp}\right)$

In the first step we perform the following transformation of $A_{\rm pure}$ at
$(z^{-},~{}0_{\perp})$ to the reference point $(0^{-},~{}0_{\perp})$.

$\displaystyle U(0^{-},z^{-};0_{\perp})~{}A^{j}_{\rm
pure}\left(z^{-},0_{\perp}\right)~{}U\left(z^{-},0^{-};~{}0_{\perp}\right)\mapsto
A^{j}_{\rm res}\left(0^{-},0_{\perp}\right)$

and then perform the $z^{-}$ integration,

$\displaystyle\int_{-\infty}^{+\infty}dz^{-}~{}U\left(0^{-},~{}z^{-};~{}0_{\perp}\right)F^{+i}\left(z^{-},~{}0_{\perp}\right)U\left(z^{-},~{}0^{-};~{}0_{\perp}\right)$
$\displaystyle=-2A_{\rm phys,~{}o}\left(0^{-},~{}0_{\perp}\right)$

which leads to,

$\displaystyle\int dz^{-}~{}{\cal
O}_{2}^{j}=2ig~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)A^{j}_{\rm
res}\left(0^{-},0_{\perp}\right)A_{\rm
phys,~{}o}\left(0^{-},~{}0_{\perp}\right),$

Similarly,

$\displaystyle\int dz^{-}~{}{\cal
O}_{4}^{j}=-2ig~{}F^{+i}\left(0^{-},~{}0_{\perp}\right)A_{\rm
phys,~{}o}\left(0^{-},~{}0_{\perp}\right)A^{j}_{\rm
res}\left(0^{-},0_{\perp}\right).$

####  VIII.1.3 For ${\cal O}_{3}^{j}$

$\displaystyle\int dz^{-}~{}{\cal O}_{3}^{j}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dz^{-}~{}F^{+i}\left(0\right)~{}{\cal
U}_{[\eta_{1}]}\left(0^{-},~{}z^{-};~{}0_{\perp}\right)~{}\left[\partial^{j}F^{+i}\left(z^{-},~{}z_{\perp}\right)\right]_{z_{\perp}=~{}0}~{}{\cal
U}_{[\eta_{2}]}\left(z^{-},~{}0^{-};~{}0_{\perp}\right)$

In order to perform integration we shall first change the order of the
integration and differentiation. This is legit as both are linear operators.
This will lead to,

$\displaystyle\int dz^{-}~{}{\cal O}_{3}^{j}$ $\displaystyle=$
$\displaystyle-2F^{+i}\left(0^{-},~{}0_{\perp}\right)~{}\partial^{j}A^{i}_{\rm
phys,~{}o}\left(0^{-},~{}0_{\perp}\right).$

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