would only set $\sum_{i}x_{i}=x_{F}$. , and unless one is considering an
exclusive process with non-zero skewedness444In an exclusive process, $x_{F}$
is related to the skewedness parameter $\xi$, and the relation between $x$ and
$\xi$ obtained from computing the hard part is not always as simple as
$x=\xi=x_{F}$., the hard part imposes that $x=x_{F}$, which is why these two
variables tend to be used interchangeably. For example in the inclusive DIS
cross section considered in this article, $x_{F}$ is given by $x_{F}=x_{{\rm
Bj}}(1+\frac{\boldsymbol{\ell}^{2}}{z\bar{z}Q^{2}})$ and the hard part
provides a $\delta(x-x_{F})$ function. This could allow to mimick an $x$
dependence in the parton distribution by introducing instead an $x_{F}$
dependence through the evolution equation. We will now show that the shock
wave description of the DIS cross section is also incompatible with any
dependence on this $x_{F}$ variable in the distribution, be it from BK
evolution or not.

Let us consider the $\gamma^{\ast}P\rightarrow q\bar{q}X$ amplitude in the
shock wave approximation. It reads Boussarie:2021lkb :

$\displaystyle{\cal A}_{L/T}$
$\displaystyle=z\bar{z}e_{q}Q\delta(p_{q}^{+}+p_{\bar{q}}^{+}-q^{+})\int{\rm
d}^{2}\boldsymbol{x}_{1}{\rm d}^{2}\boldsymbol{x}_{2}{\rm
e}^{-i\left(\boldsymbol{p}_{q}\cdot\boldsymbol{x}_{1}\right)-i\left(\boldsymbol{p}_{q}\cdot\boldsymbol{x}_{2}\right)}$
$\displaystyle\times(1-U_{\boldsymbol{x}_{1}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{2}}^{\dagger})\varphi_{L/T}(z,\boldsymbol{x}_{1}-\boldsymbol{x}_{2}),$
(167)

with

$\varphi_{L}(z,\boldsymbol{x}_{1}-\boldsymbol{x}_{2})=2K_{0}(\sqrt{z\bar{z}Q^{2}\boldsymbol{r}^{2}})(\bar{u}_{p_{q}}\gamma^{+}v_{p_{\bar{q}}})$
(168)

and

$\varphi_{T}(z,\boldsymbol{x}_{1}-\boldsymbol{x}_{2})=-i\frac{\varepsilon_{q\perp\mu}r_{\perp\nu}}{\left|\boldsymbol{r}\right|\sqrt{z\bar{z}}}K_{1}(\sqrt{z\bar{z}Q^{2}\boldsymbol{r}^{2}})\bar{u}_{p_{q}}\left[z(\gamma_{\perp}^{\mu}\gamma_{\perp}^{\nu})-\bar{z}(\gamma_{\perp}^{\nu}\gamma_{\perp}^{\mu})\right]\gamma^{+}v_{p_{\bar{q}}}.$
(169)

The $\gamma^{\ast}P\rightarrow q\bar{q}X$ cross section is thus given by:

$\displaystyle{\rm d}\sigma_{L/T;L/T}^{\gamma^{\ast}P\rightarrow q\bar{q}X}$
$\displaystyle=\frac{{\rm d}p_{q}^{+}{\rm d}^{2}\boldsymbol{p}_{q}{\rm
d}p_{\bar{q}}^{+}{\rm
d}^{2}\boldsymbol{p}_{\bar{q}}}{4(q^{+})^{3}(2\pi)^{6}}\delta(p_{q}^{+}+p_{\bar{q}}^{+}-q^{+})$
(170) $\displaystyle\times z\bar{z}q_{f}^{2}\alpha_{{\rm em}}Q^{2}\int{\rm
d}^{2}\boldsymbol{x}_{1}{\rm d}^{2}\boldsymbol{x}_{2}{\rm
d}^{2}\boldsymbol{x}_{1}^{\prime}{\rm d}^{2}\boldsymbol{x}_{2}^{\prime}{\rm
e}^{i\boldsymbol{p}_{q}\cdot\left(\boldsymbol{x}_{1}^{\prime}-\boldsymbol{x}_{1}\right)+i\boldsymbol{p}_{q}\cdot\left(\boldsymbol{x}_{2}^{\prime}-\boldsymbol{x}_{2}\right)}$
$\displaystyle\times\langle{\rm
tr}_{c}(1-U_{\boldsymbol{x}_{1}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{2}}^{\dagger})(1-U_{\boldsymbol{x}_{2}^{\prime}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{1}^{\prime}}^{\dagger})\rangle_{P}\varphi_{L/T}(z,\boldsymbol{x}_{1}-\boldsymbol{x}_{2})\varphi_{L/T}^{\ast}(z,\boldsymbol{x}_{1}^{\prime}-\boldsymbol{x}_{2}^{\prime}).$

We can now recover the leading perturbative power of the DIS cross section
without using the optical theorem, simply by integrating w.r.t. the quark and
antiquark momenta and summing over the flavors in the loop:

$\displaystyle{\rm d}\sigma_{L/T;L/T}^{\gamma^{\ast}P\rightarrow X}$
$\displaystyle=\sum_{f}\frac{q_{f}^{2}\alpha_{{\rm
em}}Q^{2}}{4(q^{+})^{2}(2\pi)^{2}}\int{\rm d}^{2}\boldsymbol{x}_{1}{\rm
d}^{2}\boldsymbol{x}_{2}\langle{\rm
tr}_{c}(1-U_{\boldsymbol{x}_{1}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{2}}^{\dagger})(1-U_{\boldsymbol{x}_{2}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{1}}^{\dagger})\rangle_{P}$
$\displaystyle\times\int{\rm
d}z(z\bar{z})\varphi_{L/T}(z,\boldsymbol{x}_{1}-\boldsymbol{x}_{2})\varphi_{L/T}^{\ast}(z,\boldsymbol{x}_{1}-\boldsymbol{x}_{2}).$
(171)

The most crucial step here is that the integral w.r.t. $\boldsymbol{p}_{q}$
and $\boldsymbol{p}_{\bar{q}}$ have set
$\boldsymbol{x}_{1}=\boldsymbol{x}_{1}^{\prime}$ and
$\boldsymbol{x}_{2}=\boldsymbol{x}_{2}^{\prime}$. Then the target matrix
element becomes

$\langle{\rm
tr}_{c}(1-U_{\boldsymbol{x}_{1}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{2}}^{\dagger})(1-U_{\boldsymbol{x}_{2}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{1}}^{\dagger})\rangle_{P}=2{\rm
Re}\langle N_{c}-{\rm
tr}_{c}(U_{\boldsymbol{x}_{1}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{2}}^{\dagger})\rangle_{P}.$
(172)

Suppose we had introduced a dependence on $x_{F}$ in the distribution. Given
that
$x_{F}=p_{q}^{-}+p_{\bar{q}}^{-}-q^{-}=\frac{\bar{z}\boldsymbol{p}_{q}^{2}+z\boldsymbol{p}_{\bar{q}}^{2}+z\bar{z}Q^{2}}{2z\bar{z}q^{+}}$
depends on the transverse momenta, we would not have been able to set
$\boldsymbol{x}_{1}=\boldsymbol{x}_{1}^{\prime}$ and
$\boldsymbol{x}_{2}=\boldsymbol{x}_{2}^{\prime}$ using the momentum
integrations. Instead of having a dipole matrix element for the target, we
would have found:

$\langle{\rm
tr}_{c}(1-U_{\boldsymbol{x}_{2}^{\prime}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{1}^{\prime}}^{\dagger}-U_{\boldsymbol{x}_{1}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{2}}^{\dagger}+U_{\boldsymbol{x}_{1}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{2}}^{\dagger}U_{\boldsymbol{x}_{2}^{\prime}}^{\phantom{\dagger}}U_{\boldsymbol{x}_{1}^{\prime}}^{\dagger})\rangle_{P}.$
(173)

In other words, using the dipole operator to describe the inclusive DIS cross
section is inherently incompatible with having any $x_{F}$ dependence in the
non-perturbative matrix element. This was previously observed in a similar
context in Bialas:2000xs , in which the authors found an incompatibility
between the dipole model (the large $N_{c}$ limit of the CGC) and a so-called
exact kinematics small-$x$ factorization scheme based on a perturbative ansatz
for the target where a dependence on $x$ is added by hand. This observation
was then confirmed with a numerical anaysis of this scheme updf . To
summarize, for inclusive DIS gluon distributions are evaluated strictly at
$x=x_{F}=0$ in the Regge limit. Indeed, $x$ is $0$ because we have infinite
Wilson line operators, and $x_{F}$ is $0$ because we have a dipole operator in
particular.

It is thus impossible to correct for Eq. (165) with an $x$ dependence in the
distribution: the DGLAP convolution is irremediably undone by the shock wave
approximation.

Finally, the last “$x$” variable is the Bjorken $x_{{\rm Bj}}$ variable. This
is the variable whose logarithms the small-$x_{{\rm Bj}}$ evolution equations
were initially designed to resum. In all known non-exclusive cases $x_{F}$ and
$x_{{\rm Bj}}$ are proportional to each other, but the proportionality factor
is not suppressed in either the Bjorken regime or the Regge limit: in our
inclusive DIS example, $x_{F}=x_{{\rm
Bj}}(1+\frac{\boldsymbol{\ell}^{2}}{z\bar{z}Q^{2}})$ and
$\boldsymbol{\ell}^{2}/Q^{2}$ is leading power in twist counting555Indeed
$\ell$ is a hard loop momentum so it can be of order $Q$. and in eikonal
counting. However, the logarithms of $x_{F}$ and $x_{{\rm Bj}}$ are the same
at leading logarithm of $s$ accuracy:

$\ln(x_{F})=-\ln(s)+\ln(Q^{2}+\frac{\boldsymbol{\ell}^{2}}{z\bar{z}})\sim-\ln(s)\sim\ln(x_{{\rm
Bj}}).$ (174)

This is finally why if we know the $x$ dependence in the PDF, we know the
$x_{{\rm Bj}}$ dependence of the inclusive DIS cross section: first the hard
part sets $x=x_{F}$, then the leading logarithm of $x_{F}$ is noted to be
equivalent to that of $x_{{\rm Bj}}$ at leading logarithmic accuracy, which
means any power dependence $x^{-\gamma}$ in the distribution will lead in
practice to an $x_{{\rm Bj}}^{-\gamma}$ dependence in the observable, up to
higher logarithmic corrections. The converse is not necessarily true: knowing
the $x_{\rm Bj}$ dependence of an observable does not mean we know the $x$
dependence in the involved distributions without further assumptions. We can,
maybe incidentally, justify the disentanglement of the DGLAP convolution using
an $x_{{\rm Bj}}$ dependence: if $xg(x)\sim_{x=0}x^{-\gamma}$, we have

$\lim_{x_{{\rm Bj}}\rightarrow 0}\int_{x_{{\rm Bj}}}^{1}{\rm
d}y\left[xg(x)\right]_{x=x_{{\rm Bj}}/y}{\cal P}^{qg}(y)=x_{{\rm
Bj}}^{-\gamma}\int_{0}^{1}{\rm d}y[y^{\gamma}{\cal P}^{qg}(y)].$ (175)

The power of $x_{{\rm Bj}}$ obtained via BK evolution yields the $x_{{\rm
Bj}}^{-\gamma}$ factor one would have obtained using the more rigorous
distribution $\left[xg(x)\right]_{x=x_{{\rm Bj}}}$ which, let us insist again,
cannot be computed using infinite Wilson line operators. For a fully
consistent scheme, is it then absolutely necessary to check order by order
that the $\gamma$-th moments of the splitting functions appear in each
observable. It is also worth noting that the ${\cal P}^{gg}$ splitting
function whose integral is divergent will yield serious issues with
disentangling the DGLAP convolution that this consideration cannot fix.

The negative cross section problem for small-$x$ physics was initially noticed
by studies of observables at NLL accuracy. In practice, the BK resummation
schemes mimic a dependence on $x$ in order to catch the leading collinear
divergences by imposing by hand the entanglement of longitudinal and
transverse variables: one can add additional ad hoc kinematic constraints
collinear-logs-Beuf  in the evolution equation or directly include an
additional resummation of logarithms collinear-logs-Edmond-1 ; collinear-logs-
Edmond-2 ; collinear-logs-Edmond-3 ; collinear-logs-Edmond-4 ; collinear-logs-
Edmond-5 ; collinear-logs-zhongbo . Such ad hoc bottom-up modifications of the
resummation scheme, albeit not providing a fully consistent picture of the
collinear phase space at small $x$, have proven to be successful at postponing
the negativity issue of small $x_{{\rm Bj}}$ cross sections to larger values
of the hard scales at a given perturbative order.

Most of such bottom-up approaches to collinear logarithm resummation at small
$x$ rely on imposing either one of two equivalent orderings: $k^{-}$ ordering
in order to reconstruct the DGLAP ladder structure, or light cone time
ordering along said ladder. In effect, such corrections allow to correct how
one treats the kinematical phase space for the emission of gluons whose
transverse momenta are large enough to break longitudinal momentum ordering
along the gluon ladder: if $\boldsymbol{k}^{2}$ is large enough, even with
$k^{+}\gg\ell^{+}$, one may have
$k^{-}\sim\boldsymbol{k}^{2}/(2k^{+})\gg\ell^{-}\sim\boldsymbol{\ell}^{2}/(2\ell^{+})$
which violates the $-$ ordering between $k$ and $\ell$. In fact, it is
possible to argue that the problem with this part of the phase space could
already have been diagnosed at LL accuracy with only quarks. Indeed at leading
twist where $|\boldsymbol{\ell}|\gg|\boldsymbol{k}|$, we can study the exact
same corner of phase space for transverse momenta in the $z\rightarrow 0$
limit. This kinematics correspond precisely to the collinear (to the target)
limit where one would derive the DGLAP kernel for quarks.

As already discussed earlier, there are two potential problems to the r.h.s.
of Eq. (165) that corresponds to the semi-classical small-$x$ scheme. First,
it is correct only if the distribution is a constant at $x=0$. Second, it
diverges if the splitting function is not integrable on $[0,1]$. Given the
fact that this corner of phase space is the one that causes problems for
gluons, and that as we proved earlier the actual $x$ variable appearing in
DGLAP is automatically set to be strictly null by CGC schemes, we can finally
see where the issue originates from. Indeed, the two problematic conditions
are met with gluons: the gluon distribution potentially has a non-trivial
dependence on $x$ around $x=0$ if saturation effects do not completely cancel
out the intercept, and in a more obvious fashion the ${\cal P}^{gg}$ splitting
function has poles at the end points $y=0,1$. This is the reason why
difficulties were encountered for NLL small-$x$ studies.

Our approach has two advantages. First and foremost, the fact that our
distribution is not evaluated in the strict $x=0$ approximation means that the
DGLAP kernel is fully reproduced at least within LL accuracy, as we proved in
Section 5. Second, the ordering in light cone time is built into the framework
because we did not allow the light cone time decoupling which is the starting
point of CGC computations. See the discussions in Section 2.3 and around Eqs.
(89) and (90). Applying our scheme to the evolution equation of our
distribution will naturally incorporate the desired ordering. We traded the
dipole operator which is defined only in the strict $x=0$ limit for the
$x$-dependent gluon distribution Eq. (125) and we thus have one more variable
when compared to standard semi-classical small-$x$ schemes. The distribution
we found depends on a transverse momentum and on the rapidity factorization
scale which allows to separate out classical target fields in $+$ momentum
space using the BK equation, as is the case in these schemes, but also on the
physical DGLAP $-$ momentum variable $x$ that is missing from them:

$\displaystyle\varphi_{{}_{G}}({\boldsymbol{k}};\Lambda^{+})\equiv\frac{1}{N_{c}}\int_{\boldsymbol{r}}\,{\rm
e}^{-i{\boldsymbol{r}}\cdot{\boldsymbol{k}}}\,\langle{\rm
tr}\,U_{\boldsymbol{0}}U^{\dagger}_{\boldsymbol{r}}\rangle_{\Lambda^{+}}\quad\to\quad
G_{\Lambda^{+}}^{ij}(x,{\boldsymbol{k}})$ (176)

One of the main strategies adopted in previous studies to solve the problem of
small-$x$ evolution in the collinear corner of phase-space has been to perform
ad hoc modifications to the BK evolution equation by forcing a relation
between the cutoff variable $\Lambda^{+}>k^{+}$ and momenta along the other
light cone direction $k^{-}=P^{-}$ roughly as follows
$2\Lambda^{+}k^{-}\sim{\boldsymbol{k}}^{2}$, throughout the evolution. In our
case, the $-$ light cone direction is fully accounted for by the presence of
$x$.

What we conclude from the present discussion is that although ad hoc
considerations allow us to extend the validity of the description of
observables in the shock wave approximation, we argued that we should actually
fix what is being evolved instead of fixing how it is evolved. As long as we
have infinite Wilson lines, we have $x=0$. In semi-inclusive cases that
involve jets $x_{F}$ is not integrated out at tree level so one can set a
relation between $\Lambda^{+}$ and $x_{F}$, but it does not fix the collinear
corner of phase space since DGLAP involves $x$ rather than $x_{F}$. Beyond NLL
accuracy, or in any other semi-inclusive case where $x_{F}$ is a convolution
variable because of the presence of fragmentation functions, the relation
between $\Lambda^{+}$ and $x_{F}$ cannot be imposed by hand anymore. Changing
how we evolve the dipole operator or the scale at which we evaluated it does
not solve the $x=x_{F}=0$ problem which is due to the very fact that we are
studying the dipole operator itself. Of course to complete the picture we need
to investigate quantum evolution of the $x$-dependent unintegrated gluon
distribution and compare to the aforementioned schemes, but the distribution
found in this article does not present the fundamental flaw of being set to
$x=0$ from the get-go.

##  8 Conclusion and outlook

In this article, we provided a semi-classical scheme inspired by small-$x_{\rm
Bj}$ frameworks, in view of a first principle interpolation between both major
limits of perturbative QCD. We applied this scheme to the inclusive DIS cross
section and obtained the leading power of $x_{\rm Bj}/Q^{2}$ for this
observable. Our scheme is based on a partial twist expansion (PTE) that resums
subclasses of powers of $s$ and $Q^{2}$ to all orders. We have derived the
leading contribution in the PTE and showed that it encompasses both Bjorken
and Regge regimes, and thus provides the desired interpolation. Our
computation yields a new, unexpected, form for an unintegrated gluon
distribution with correct dependences on both transverse and longitudinal
gluon momentum components. From our interpolating formula, we perform a top-
down analysis of the collinear limit of Regge descriptions of our observable.
We find that the shock wave approximation on which semi-classical
small-$x_{\rm Bj}$ framework rely is inherently incompatible with a proper
account of the DGLAP equation because of its treatment of the $x$ variable.
Much progress has been made towards including all subleading power corrections
to the shock wave approximation, so-called subeikonal corrections subeikonal-
tolga-1 ; subeikonal-tolga-2 ; subeikonal-tolga-3 ; subeikonal-tolga-4 ;
subeikonal-tolga-5 ; subeikonal-tolga-6 ; subeikonal-yuri-1 ; subeikonal-
yuri-2 ; subeikonal-yuri-3 ; subeikonal-yuri-4 ; subeikonal-yuri-5 ;
subeikonal-yuri-6 ; subeikonal-yuri-7 ; subeikonal-yoshitaka ; subeikonal-
giovanni-1 ; subeikonal-giovanni-2 ; subeikonal-jamal-1 ; subeikonal-jamal-2 ;
subeikonal-jamal-3 . In such approaches, some of the subleading terms
correspond to the expansion in the Feynman $x$ variables of the leading twist
distribution. This would yield a correction to the assumption which is made in
Eq. (165): along with $[xg(x)]_{x=0}$, the second term in the Taylor expansion
$[\frac{\rm d}{{\rm d}x}xg(x)]_{x=0}$ hides in the expressions one would find
in the collinear limit of subeikonal results. Although this does not make Eq.
(165) valid in full generality, it would be insightful to study how such
corrections would compare to an expansion of DGLAP from our top-down approach
and whether or not they could allow to reconstruct the full equation and
correct for the dangerous hypothesis in a bottom-up approach.

In this article, we have focused on the leading order diagram in the small $x$
limit. However, since we encountered a quark loop at this order, we have
partially tackled the question of quantum evolution by providing a
prescription for dealing with the longitudinal structure of the target.
Nevertheless, a complete picture would require the derivation of the analog of
the BK equation for the unintegrated gluon distribution which we leave for a
following work.

## Acknowledgements

The work of Y. M.-T. is supported by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics, under contract No. DE- SC0012704. Y. M.-T.
acknowledges support from the RHIC Physics Fellow Program of the RIKEN BNL
Research Center.

##  Appendix A On the inclusion of non-pure gauge transverse gluon fields

This appendix aims at showing that although they contribute as gauge
invariance fixing counterterms to $\partial^{i}A^{-}$ in $F^{i-}$, transverse
gluon fields are not necessary to compute the cross section beyond a
consistency check. In the Regge regime, one always assumes that one can gauge
away transverse gluon fields. In the Bjorken regime, it is not a valid
assumption. However, for inclusive DIS, the leading (collinear) distributions
do not involve a transverse separation. As a result, the only contribution
from $\boldsymbol{A}^{i}$ fields at our perturbative order is contained in the
$F^{i-}$ tensors that define the PDF. Let us write the DIS cross section in
the factorized form

${\rm d}\sigma=\int{\rm d}x{\cal H}(x){\rm e}^{ixP^{-}z^{+}}\langle P|{\rm
tr}F^{i-}(z^{+})[z^{+},0^{+}]F^{i-}(0^{+})[0^{+},z^{+}]|P\rangle,$ (177)

where ${\cal H}(x)$ is the collinear hard part. Let us note that for any
$y^{+},u^{+}$:

$\displaystyle{\rm tr}[y^{+},u^{+}]F^{i-}(u^{+})[u^{+},y^{+}]F^{i-}(y^{+})$
$\displaystyle={\rm
tr}(\partial^{i}A^{-})[u^{+},y^{+}](\partial^{i}A^{-})(y^{+})[y^{+},u^{+}]$
$\displaystyle+\frac{\partial}{\partial u^{+}}\frac{\partial}{\partial
y^{+}}{\rm tr}[y^{+},u^{+}]A^{i}(u^{+})[u^{+},y^{+}]A^{i}(y^{+})$
$\displaystyle-\frac{\partial}{\partial y^{+}}{\rm
tr}(\partial^{i}A^{-})(u^{+})[u^{+},y^{+}]A^{i}(y^{+})[y^{+},u^{+}]$
$\displaystyle-\frac{\partial}{\partial u^{+}}{\rm
tr}[u^{+},y^{+}](\partial^{i}A^{-})(y^{+})[y^{+},u^{+}]A^{i}(u^{+}).$

Once plugged into the convolution, integrations by parts w.r.t. $y^{+}$ in the
penultimate term and w.r.t. $x^{+}$ in the last term allow to cancel the last
two lines. One is left with

$\displaystyle{\rm d}\sigma$ $\displaystyle=\int{\rm d}x{\cal H}(x){\rm
e}^{ixP^{-}z^{+}}\langle P|{\rm
tr}(\partial^{i}A^{-})(z^{+})[z^{+},0^{+}](\partial^{i}A^{-})(0^{+})[0^{+},z^{+}]|P\rangle$
$\displaystyle+\int{\rm d}x{\cal H}(x){\rm
e}^{ixP^{-}z^{+}}x^{2}(P^{-})^{2}\langle P|{\rm
tr}A^{i}(z^{+})[z^{+},0^{+}]A^{i}(0^{+})[0^{+},z^{+}]|P\rangle.$ (178)

The first line of this equation can be computed without any scattering with
transverse gluon fields. We thus have an explicit expression to reconstruct
the full gauge invariant cross section while only computing the hard part with
pure gauge transverse gluons: the term with non-pure gauge transverse gluons
is obtained from the Ward-Takahashi identities and reads
$x^{2}(P^{-})^{2}{\cal H}(x)$. In a nutshell, it is possible to compute the
full cross section while only including pure gauge transverse gluons in the
Bjorken limit as well.

Our purpose is to have exact results in both limits, but we are not so much
interested in the interpolating region. We can therefore conclude that unless
one is trying to get an explicit proof of QCD gauge invariance for the
considered observable, it is not necessary at the considered precision to
explicitly include transverse gluons which are not pure gauge in the
computation.

##  Appendix B Proof of Eq. (73)

The convolution property for scalar propagators Eq. (73) follows from the
Klein-Gordon equations. Let us start by multiplying the two equations

$\displaystyle\left(-\Box_{z}+2igA^{-}(z)\partial_{z}^{+}\right)G_{\rm
scal}(z,x_{0})=\delta^{D}(z-x_{0})$ (179)

and

$\displaystyle G_{\rm
scal}(x,z)\left(\overleftarrow{\Box}_{z}+2ig\overleftarrow{\partial^{+}_{z}}A^{-}(z)\right)=-\delta^{D}(z-x)\,,$
(180)

respectively on the left by $G_{\rm scal}(x,z)$ and on the right by $G_{\rm
scal}(z,x_{0})$. For a given $u^{+}$, let us add them up and integrate over
$z$ while imposing $z^{+}>u^{+}$. This yields

$\displaystyle\int\limits_{z^{+}>u^{+}}\\!\\!\\!\\!{\rm d}^{D}z\,G_{{\rm
scal}}(x,z)\left(\overleftarrow{\Box}_{z}-\Box_{z}+2ig\overleftarrow{\partial_{z}^{+}}A^{-}(z)+2igA^{-}(z)\partial_{z}^{+}\right)G_{{\rm
scal}}(z,x_{0})$ $\displaystyle=\int\limits_{z^{+}>u^{+}}\\!\\!\\!\\!{\rm
d}^{D}z\left[\delta^{D}(z-x_{0})-\delta^{D}(x-z)\right]G_{{\rm
scal}}(x,x_{0}).$ (181)

The third and last terms in the brackets cancel each other after integration
by part over $z^{-}$, neglecting boundary terms at infinity. The same holds
for the transverse part of the d’Alembertian $\Box\equiv
2\partial^{+}\partial^{-}-\boldsymbol{\partial}^{2}$, which cancel out between
the first and second terms. We are thus left with the following:

$\displaystyle\left[\theta(y^{+}-u^{+})-\theta(x^{+}-u^{+})\right]\,G_{\rm
scal}(x,y)$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\int\limits_{z^{+}>u^{+}}\\!\\!\\!\\!{\rm
d}^{D}z\,G_{\rm
scal}(x,z)\left(-2\overset{\rightarrow}{\partial^{+}}\overset{\rightarrow}{\partial^{-}}+2\overset{\leftarrow}{\partial^{+}}\overset{\leftarrow}{\partial^{-}}\right)G_{\rm
scal}(z,y)\,$ $\displaystyle=$
$\displaystyle-\\!\\!\\!\\!\int\limits_{z^{+}>u^{+}}\\!\\!\\!\\!{\rm
d}^{D}z\,G_{\rm
scal}(x,z)\left(2\overset{\rightarrow}{\partial^{+}}\overset{\rightarrow}{\partial^{-}}+2\overset{\rightarrow}{\partial^{+}}\overset{\leftarrow}{\partial^{-}}\right)G_{\rm
scal}(z,y)\,$ $\displaystyle=$
$\displaystyle-2\\!\\!\\!\\!\int\limits_{z^{+}>u^{+}}\\!\\!\\!\\!{\rm
d}^{D}z\,\frac{\partial}{\partial z^{+}}\left(G_{\rm
scal}(x,z)\partial_{z}^{+}G_{\rm scal}(z,y)\right)\,$ $\displaystyle=$
$\displaystyle-2\int_{z^{+}=u^{+}}\\!\\!\\!\\!{\rm
d}^{d}{\boldsymbol{z}}\int{\rm d}z^{-}\,G_{\rm
scal}(x,z)\partial_{z}^{+}G_{\rm scal}(z,y)\,,$

where we have repeatedly neglected boundary terms at $z^{-}=\pm\infty$,
$z^{+}=+\infty$. Finally, for $x^{+}>y^{+}$ only the $x^{+}>u^{+}>y^{+}$
ordering yields a non-vanishing l.h.s:

$\displaystyle G_{\rm scal}(x,y)=2\int_{z^{+}}{\rm d}{\boldsymbol{z}}\int{\rm
d}z^{-}\,G_{\rm scal}(x,z)\partial_{z}^{+}G_{\rm scal}(z,y)\,\,.$ (183)

Similarly, for $y^{+}>u^{+}>x^{+}$ we have

$\displaystyle G_{\rm scal}(x,y)=-2\int_{z^{+}}{\rm d}{\boldsymbol{z}}\int{\rm
d}z^{-}\,G_{\rm scal}(x,z)\partial_{z}^{+}G_{\rm scal}(z,y)\,\,.$ (184)

##  Appendix C Notes on QED gauge invariance

Let us establish a few useful relations as a corollary for the Ward-Takahashi
identities. First of all, the Klein-Gordon equations in momentum space read:

$p^{2}G_{\mathrm{scal}}(p,p_{0})=(2\pi)^{D}\delta^{D}(p-p_{0})-2gp^{+}\\!\int\\!\frac{{\rm
d}^{D}k}{(2\pi)^{D}}A^{-}(k)G_{\mathrm{scal}}(p-k,p_{0})$ (185)

and

$G_{\mathrm{scal}}(p,p_{0})p_{0}^{2}=(2\pi)^{4}\delta^{4}(p-p_{0})-2gp^{+}\\!\int\\!\frac{{\rm
d}^{4}k}{(2\pi)^{4}}G_{\mathrm{scal}}(p,p_{0}+k)A^{-}(k).$ (186)

As a direct consequence of these, one has:

$\not{p}D_{F}(p,p_{0})=i(2\pi)^{D}\delta^{D}(p-p_{0})-ig\gamma^{+}\not{p}_{0}\\!\int\\!\frac{{\rm
d}^{D}k}{(2\pi)^{D}}A^{-}(k)G_{\mathrm{scal}}(p-k,p_{0}),$ (187)

and

$D_{F}(p,p_{0})\not{p}_{0}=i(2\pi)^{D}\delta^{D}(p-p_{0})-ig\not{p}\gamma^{+}\\!\int\\!\frac{{\rm
d}^{D}k}{(2\pi)^{D}}G_{\mathrm{scal}}(p,p_{0}+k)A^{-}(k).$ (188)

Let us check the Ward-Takahashi (WT) identity for the initial photon by
computing $q_{\mu}W^{\mu\nu}$. By writing
$\not{q}=\not{\ell}-(\not{\ell}-\not{q})$, then with the help of Eq. (188) for
the first term and Eq. (187) for the second term, one gets:

$\displaystyle q_{\mu}W^{\mu\nu}=-ie^{2}\sum_{f}q_{f}^{2}\int\\!\frac{{\rm
d}^{D}\ell}{(2\pi)^{D}}\int\\!\frac{{\rm d}^{D}k}{(2\pi)^{D}}\,$ (189)
$\displaystyle\>\times\Bigl{\\{}(2\pi)^{D}\delta^{D}(k)\langle P|{\rm
tr}[\gamma^{\nu}D_{F}(-q+\ell,-q+\ell+k)-\gamma^{\nu}D_{F}(\ell+k,\ell)]|P\rangle$
$\displaystyle-g\\!\int\\!\frac{{\rm d}^{D}k_{0}}{(2\pi)^{D}}\langle P|{\rm
tr}[G_{{\rm
scal}}(\ell+k,\ell+k_{0})A^{-}(k_{0})\gamma^{\nu}(\not{\ell}+\not{k})\gamma^{+}D_{F}(-q+\ell,-q+\ell+k)]|P\rangle$
$\displaystyle+g\\!\int\\!\frac{{\rm d}^{D}k_{0}}{(2\pi)^{D}}\langle P|{\rm
tr}[\gamma^{\nu}D_{F}(\ell+k,\ell)\gamma^{+}(-\not{q}+\not{\ell}+\not{k})A^{-}(k_{0})G_{{\rm
scal}}(-q+\ell-k_{0},-q+\ell+k)]|P\rangle\Bigr{\\}}$

A simple change of variables allows to compensate the first term in the
brackets with the second term. We will now use the explicit expression for the
Dirac propagators in terms of scalar propagators in the remaining terms. Note
that it is greatly simplified by the presence of $\gamma^{+}$ matrices in the
trace:

$\displaystyle q_{\mu}W^{\mu\nu}$
$\displaystyle=ige^{2}\sum_{f}q_{f}^{2}\int\\!\frac{{\rm
d}^{D}\ell}{(2\pi)^{D}}\int\\!\frac{{\rm
d}^{D}k}{(2\pi)^{D}}\,\\!\int\\!\frac{{\rm d}^{D}k_{0}}{(2\pi)^{D}}$
$\displaystyle\times{\rm
tr}_{s}[\gamma^{\nu}(\not{\ell}+\hat{k})\gamma^{+}(-\not{q}+\not{\ell}+\not{k})]$
(190) $\displaystyle\times\langle P|\Bigl{\\{}{\rm tr}_{c}[G_{{\rm
scal}}(\ell+k,\ell+k_{0})A^{-}(k_{0})G_{{\rm scal}}(-q+\ell,-q+\ell+k)]$
$\displaystyle-{\rm tr}_{c}[G_{{\rm scal}}(\ell+k,\ell)A^{-}(k_{0})G_{{\rm
scal}}(-q+\ell-k_{0},-q+\ell+k)]\Bigr{\\}}|P\rangle.$

Finally, we can see that taking the changes of variables
$\ell\rightarrow\ell+k_{0}$ and $k\rightarrow k-k_{0}$ in the second term
casts it in a form where it explicitely cancels the first term. One is left
with $q_{\mu}W^{\mu\nu}$, and the Ward-Takahashi identity thus holds. The
proof for the outgoing photon is identical, and QED gauge invariance is
finally confirmed.

##  Appendix D Cancellation of the monopoles

Let us now prove that the instantaneous terms from the quark propagators do
not contribute to the amplitude. Let us start from the definition of the
hadronic tensor

$\displaystyle W^{\mu\nu}$
$\displaystyle=e^{2}\sum_{f}q_{f}^{2}\int\\!\frac{{\rm
d}^{D}\ell}{(2\pi)^{D}}\int\\!\frac{{\rm d}^{D}k}{(2\pi)^{D}}$ (191)
$\displaystyle\times\langle P|{\rm
tr}\bigl{[}\gamma^{\nu}D_{F}(\ell+k,\ell)\gamma^{\mu}D_{F}(-q+\ell,-q+\ell+k)\bigr{]}|P\rangle.$

With the help of the Ward-Takahashi identity we established in Appendix C, we
can subsitute
$\gamma^{\mu,\nu}\rightarrow\gamma^{\mu,\nu}-n_{2}^{\mu,\nu}\not{q}/q^{+}$ and
restrict ourselves to computing

$\displaystyle W^{\mu\nu}$
$\displaystyle=e^{2}\sum_{f}q_{f}^{2}\int\\!\frac{{\rm
d}^{D}\ell}{(2\pi)^{D}}\int\\!\frac{{\rm d}^{D}k}{(2\pi)^{D}}$ (192)
$\displaystyle\times\langle P|{\rm
tr}\bigl{[}\bigl{(}\gamma^{\nu}-n_{2}^{\nu}\frac{\not{q}}{q^{+}}\bigr{)}D_{F}(\ell+k,\ell)\bigl{(}\gamma^{\mu}-n_{2}^{\mu}\frac{\not{q}}{q^{+}}\bigr{)}D_{F}(-q+\ell,-q+\ell+k)\bigr{]}|P\rangle$

instead. The instaneaneous contribution from the first propagator
$D_{F}(\ell+k,\ell)$ reads:

$\displaystyle W_{{\rm inst}(1)}^{\mu\nu}$
$\displaystyle=e^{2}\sum_{f}q_{f}^{2}\,{\rm Re}\int\\!\frac{{\rm
d}^{D}\ell}{(2\pi)^{D}}\frac{i}{2\ell^{+}}$ (193) $\displaystyle\times\langle
P|{\rm
tr}\bigl{[}\bigl{(}\gamma^{\nu}-n_{2}^{\nu}\frac{\not{q}}{q^{+}}\bigr{)}\gamma^{+}\bigl{(}\gamma^{\mu}-n_{2}^{\mu}\frac{\not{q}}{q^{+}}\bigr{)}D_{F}(-q+\ell,-q+\ell)\bigr{]}|P\rangle.$

When computing the spinor trace, it is worth noticing that the first and third
brackets both have null components along the $+$ direction. This means that in
the final result, the open $+$ index has to be contracted with the $D_{F}$
contribution. In other words, only the $\gamma^{-}$ component of the second
$D_{F}$ propagator contributes. Since there is no momentum transfer in that
propagator, it is easy to identify it:

$\displaystyle W_{{\rm inst}(1)}^{\mu\nu}$
$\displaystyle=e^{2}\sum_{f}q_{f}^{2}\int\\!\frac{{\rm
d}^{D}\ell}{(2\pi)^{D}}\frac{i}{2\ell^{+}}\langle P|{\rm tr}_{c}G_{{\rm
scal}}(-q+\ell,-q+\ell)|P\rangle$ (194) $\displaystyle\times{\rm
tr}_{s}\bigl{[}\bigl{(}\gamma^{\nu}-n_{2}^{\nu}\frac{\not{q}}{q^{+}}\bigr{)}\gamma^{+}\bigl{(}\gamma^{\mu}-n_{2}^{\mu}\frac{\not{q}}{q^{+}}\bigr{)}(-q^{+}+\ell^{+})\gamma^{-}\bigr{]}$

With the spinor trace

$\displaystyle{\rm
tr}_{s}\bigl{[}\bigl{(}\gamma^{\nu}-n_{2}^{\nu}\frac{\not{q}}{q^{+}}\bigr{)}\gamma^{+}\bigl{(}\gamma^{\mu}-n_{2}^{\mu}\frac{\not{q}}{q^{+}}\bigr{)}(-q^{+}+\ell^{+})\gamma^{-}\bigr{]}$
$\displaystyle=4(q^{+}-\ell^{+})g_{\perp}^{\mu\nu},$ (195)

we finally get:

$\displaystyle W_{{\rm inst}(1)}^{\mu\nu}$
$\displaystyle=e^{2}g_{\perp}^{\mu\nu}\sum_{f}q_{f}^{2}\int\\!\frac{{\rm
d}^{D}\ell}{(2\pi)^{D}}\frac{2i(q^{+}-\ell^{+})}{\ell^{+}}\langle P|{\rm
tr}_{c}G_{{\rm scal}}(-q+\ell,-q+\ell)|P\rangle$ (196)

Similar steps lead to the following result for the contribution of the
instantaneous term in the second propagator $D_{F}(-q+\ell,-q+\ell+k)$:

$\displaystyle W_{{\rm inst}(2)}^{\mu\nu}$
$\displaystyle=e^{2}g_{\perp}^{\mu\nu}\sum_{f}q_{f}^{2}\int\\!\frac{{\rm
d}^{D}\ell}{(2\pi)^{D}}\frac{2i\ell^{+}}{(q^{+}-\ell^{+})}\langle P|{\rm
tr}_{c}G_{{\rm scal}}(\ell,\ell)|P\rangle.$ (197)

We finally need to cancel quantities of the form

$\int\frac{\rm d\ell^{-}}{2\pi}\langle P|{\rm tr}_{c}G_{{\rm
scal}}(\ell,\ell)|P\rangle.$ (198)

In coordinate space, the integral sets the propagator to be between two
identical light cone times. The physical argument for the cancellation of the
monopoles thus relies on the fact that there is not a big enough light cone
time window for scatterings with the target to occur, which means monopoles
only contribute to disconnected diagrams. This can be seen mathematically from
the integrated version of the Schrödinger equation:

$\displaystyle G_{{\rm scal}}(x^{\prime},x)-G_{0}(x^{\prime}-x)$ (199)
$\displaystyle=2g\int\\!{\rm d}^{D}y\int\\!\frac{{\rm d}^{D}k}{(2\pi)^{D}}{\rm
e}^{-i(k\cdot y)}\int\frac{{\rm d}^{D}\ell}{(2\pi)^{D}}{\rm
e}^{-i\ell\cdot(x^{\prime}-y)}G_{0}(\ell)\,(k\cdot A)(y)\,G_{{\rm
scal}}(k,x).$

Using the standard Cauchy integral from Eq. (19), the integral over $\ell^{-}$
sets a strict ordering between $x^{\prime}$ and $y$. Similar considerations
show a strict ordering for $x$ as well, and one can easily see that light cone
times for the scatterings are strictly ordered. If the initial and final times
are equal, the r.h.s. cancels due to the absence of such an ordering and thus
$G=G_{0}$. This concludes our proof: in the case of monopoles, the integral
over $\ell^{-}$ sets equal light cone times for the unique scalar propagator,
which sets this propagator to the free propagator. Disconnected contributions
without scatterings with the target being subtracted, monopoles do not
contribute.

##  Appendix E Derivation of the leading $1/Q$ power of the Regge limit

The twist expansion of the dipole scattering matrix element in Eq. (160)
allows us to get the leading $1/Q$ power of Eq. (156) and Eq. (157):

$\displaystyle\lim_{Q^{2}\rightarrow\infty}F_{L}(x_{{\rm Bj}}\rightarrow
0,Q^{2})$ $\displaystyle=8\sum_{f}q_{f}^{2}\int_{0}^{1}{\rm d}z\int{\rm
d}^{d}\boldsymbol{r}\frac{\alpha_{s}\boldsymbol{r}^{2}}{d(2\pi)^{d}}\left[xg(x)\right]_{x=0}$
(200)
$\displaystyle\times\left(\frac{z\bar{z}Q^{2}}{\boldsymbol{r}^{2}}\right)^{\frac{d}{2}-1}z^{2}\bar{z}^{2}Q^{4}\bigl{[}K_{\frac{d}{2}-1}(Q\sqrt{z\bar{z}}|\boldsymbol{r}|)\bigr{]}^{2},$

and

$\displaystyle\lim_{Q^{2}\rightarrow\infty}F_{T}(x_{{\rm Bj}}\rightarrow
0,Q^{2})$ $\displaystyle=2Q^{2}\sum_{f}q_{f}^{2}\int_{0}^{1}{\rm d}z\int{\rm
d}^{d}\boldsymbol{r}\frac{\alpha_{s}\boldsymbol{r}^{2}}{d(2\pi)^{d}}\left[xg(x)\right]_{x=0}$
(201)
$\displaystyle\times\left(1-\frac{4}{d}z\bar{z}\right)\boldsymbol{r}^{2}\left(\frac{z\bar{z}Q^{2}}{\boldsymbol{r}^{2}}\right)^{\frac{d}{2}}\bigl{[}K_{\frac{d}{2}}(Q\sqrt{z\bar{z}}|\boldsymbol{r}|)\bigr{]}^{2}.$

It is not too difficult to take the $\boldsymbol{r}$ and $z$ integrals at this
point. However, it is more instructive to work in momentum space with the
appropriate variables. Let us use Eq. (99) and Eq. (100), then, after an
integration by parts and a few trivial integrals we finally find a form and
Eq. (135):

$\displaystyle\lim_{Q^{2}\rightarrow\infty}F_{L}(x_{{\rm Bj}}\rightarrow
0,Q^{2})$ $\displaystyle=g^{2}Q^{2}\sum_{f}q_{f}^{2}\int{\rm
d}x\left[xg(x)\right]_{x=0}$ (202) $\displaystyle\times\int_{0}^{1}\frac{{\rm
d}z}{2\pi}\int\frac{{\rm
d}^{d}\boldsymbol{\ell}}{(2\pi)^{d}}\frac{1}{d}\frac{16z^{2}\bar{z}^{2}Q^{2}\boldsymbol{\ell}^{2}}{\bigl{(}\boldsymbol{\ell}^{2}+z\bar{z}Q^{2}\bigr{)}^{4}}\delta(x),$

and

$\displaystyle\lim_{Q^{2}\rightarrow\infty}F_{T}(x_{{\rm Bj}}\rightarrow
0,Q^{2})$ $\displaystyle=g^{2}Q^{2}\sum_{f}q_{f}^{2}\int{\rm
d}x\left[xg(x)\right]_{x=0}$ (203) $\displaystyle\times\int_{0}^{1}\frac{{\rm
d}z}{2\pi}\int\frac{d^{d}\boldsymbol{\ell}}{(2\pi)^{d}}\delta(x)\left(1-\frac{4}{d}z\bar{z}\right)$
$\displaystyle\times\frac{1}{d}\left(\frac{d}{\bigl{(}\boldsymbol{\ell}^{2}+z\bar{z}Q^{2}\bigr{)}^{2}}-4\frac{\boldsymbol{\ell}^{2}}{\bigl{(}\boldsymbol{\ell}^{2}+z\bar{z}Q^{2}\bigr{)}^{3}}+4\frac{\boldsymbol{\ell}^{4}}{\bigl{(}\boldsymbol{\ell}^{2}+z\bar{z}Q^{2}\bigr{)}^{4}}\right).$

In the Bjorken regime, we defined the $y\equiv x_{\rm Bj}/x$ variable. Because
of the additional condition that was relating the Feynman $x$ variable to
$x_{\rm Bj}$ and the loop variables, we could have equivalently defined it via

$\boldsymbol{\ell}^{2}=z\bar{z}\left(\frac{1-y}{y}\right)Q^{2}.$ (204)

Let us use the definition above in Eq. (202) and Eq. (203). Given the absence
of angular dependence, we can perform the angular integral for
$\boldsymbol{\ell}$ in a trivial way, by rewriting the integration measure as
follows:

$\displaystyle\int_{0}^{1}{\rm d}z\int{\rm d}^{d}\boldsymbol{\ell}$
$\displaystyle\rightarrow\frac{\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}\int_{0}^{1}{\rm
d}z\int_{0}^{\infty}{\rm
d}\boldsymbol{\ell}^{2}\left(\boldsymbol{\ell}^{2}\right)^{\frac{d}{2}-1}$
(205)
$\displaystyle\rightarrow\frac{\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}\int_{0}^{1}{\rm
d}z\int_{0}^{1}\frac{{\rm
d}y}{y^{2}}\left(z\bar{z}Q^{2}\right)^{\frac{d}{2}}\left(\frac{1-y}{y}\right)^{\frac{d}{2}-1}.$

We find:

$\displaystyle\lim_{Q^{2}\rightarrow\infty}F_{L}(x_{{\rm Bj}}\rightarrow
0,Q^{2})$ $\displaystyle=\frac{\alpha_{s}}{2\pi}\sum_{f}q_{f}^{2}\int{\rm
d}x\int_{0}^{1}{\rm d}y\int_{0}^{1}{\rm d}z\left[xg(x)\right]_{x=0}$
$\displaystyle\times\frac{1}{\Gamma\left(\frac{d}{2}\right)}\left[\frac{Q^{2}(1-y)}{4\pi\mu^{2}y}\right]^{\frac{d}{2}-1}(z\bar{z})^{\frac{d}{2}-1}\delta(x)$
$\displaystyle\times\frac{16}{d}y(1-y),$ (206)

and

$\displaystyle\lim_{Q^{2}\rightarrow\infty}F_{T}(x_{{\rm Bj}}\rightarrow
0,Q^{2})$ $\displaystyle=\frac{\alpha_{s}}{2\pi}\sum_{f}q_{f}^{2}\int{\rm
d}x\int_{0}^{1}{\rm d}y\int_{0}^{1}{\rm d}z\left[xg(x)\right]_{x=0}$ (207)
$\displaystyle\times\frac{1}{\Gamma\left(\frac{d}{2}\right)}\left[\frac{Q^{2}(1-y)}{4\pi\mu^{2}y}\right]^{\frac{d}{2}-1}\left[(z\bar{z})^{\frac{d}{2}-2}-\frac{4}{d}(z\bar{z})^{\frac{d}{2}-1}\right]\delta(x)$
$\displaystyle\times\left[1-\frac{4}{d}y(1-y)\right].$

Note the dimensional regularization parameter $\mu$ whose presence was
implicit until now. Finally taking the $z$ integral and expanding around
$d=2+2\epsilon\simeq 2$, we obtain:

$\displaystyle\lim_{Q^{2}\rightarrow\infty}F_{L}(x_{{\rm Bj}}\rightarrow
0,Q^{2})$
$\displaystyle=4\frac{\alpha_{s}}{\pi}\sum_{f}q_{f}^{2}\left[xg(x)\right]_{x=0}\int_{0}^{1}{\rm
d}yy(1-y)$
$\displaystyle=\frac{2}{3}\frac{\alpha_{s}}{\pi}\sum_{f}q_{f}^{2}\left[xg(x)\right]_{x=0},$
(208)

and

$\displaystyle\lim_{Q^{2}\rightarrow\infty}F_{T}(x_{{\rm Bj}}\rightarrow
0,Q^{2})$ (209)
$\displaystyle=\frac{\alpha_{s}}{\pi}\sum_{f}q_{f}^{2}\left[xg(x)\right]_{x=0}\int_{0}^{1}{\rm
d}y$ $\displaystyle\times\left\\{\frac{1}{\epsilon}\left(\frac{{\rm
e}^{\gamma_{E}}}{4\pi}\right)^{\epsilon}\left[(1-y)^{2}+y^{2}\right]+\left[(1-y)^{2}+y^{2}\right]\ln\left[\frac{Q^{2}(1-y)}{\mu^{2}y}\right]-1+4y(1-y)\right\\}$
$\displaystyle=\frac{\alpha_{s}}{3\pi}\sum_{f}q_{f}^{2}\left[xg(x)\right]_{x=0}\left[\frac{2}{\epsilon}\left(\frac{{\rm
e}^{\gamma_{E}}}{4\pi}\right)^{\epsilon}+2\ln\left(\frac{Q^{2}}{\mu^{2}}\right)-1\right].$

## References

  * (1) L. Gribov, E. Levin and M. Ryskin, Phys. Rept. 100, 1-150 (1983)
  * (2) L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233-2241 (1994) [arXiv:hep-ph/9309289 [hep-ph]] ; L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 3352-3355 (1994) [arXiv:hep-ph/9311205 [hep-ph]] ; L. D. McLerran and R. Venugopalan, Phys. Rev. D 50, 2225-2233 (1994) [arXiv:hep-ph/9402335 [hep-ph]].
  * (3) F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, Ann. Rev. Nucl. Part. Sci. 60 (2010), 463-489 [arXiv:1002.0333 [hep-ph]].
  * (4) I. Balitsky, Nucl. Phys. B 463 (1996), 99-160 [arXiv:hep-ph/9509348 [hep-ph]].
  * (5) J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, Phys. Rev. D 59 (1998), 014014, [arXiv:hep-ph/9706377 [hep-ph]].
  * (6) E. Iancu, A. Leonidov and L. D. McLerran, Nucl. Phys. A 692 (2001), 583-645, [arXiv:hep-ph/0011241 [hep-ph]].
  * (7) Y. V. Kovchegov, Phys. Rev. D 60 (1999), 034008, [arXiv:hep-ph/9901281 [hep-ph]].
  * (8) Y. V. Kovchegov, Phys. Rev. D 61 (2000), 074018, [arXiv:hep-ph/9905214 [hep-ph]].
  * (9) T. Lappi and H. Mäntysaari, Phys. Rev. D 91, no.7, 074016 (2015), [arXiv:1502.02400 [hep-ph]].
  * (10) E. Avsar, A. Stasto, D. Triantafyllopoulos and D. Zaslavsky, JHEP 10, 138 (2011), [arXiv:1107.1252 [hep-ph]].
  * (11) V. S. Fadin and L. Lipatov, Phys. Lett. B 429, 127-134 (1998), [arXiv:hep-ph/9802290 [hep-ph]].
  * (12) M. Ciafaloni and G. Camici, Phys. Lett. B 430, 349-354 (1998) [arXiv:hep-ph/9803389 [hep-ph]].
  * (13) G. Beuf, Phys. Rev. D 89, no.7, 074039 (2014), [arXiv:1401.0313 [hep-ph]].
  * (14) E. Iancu, J. Madrigal, A. Mueller, G. Soyez and D. Triantafyllopoulos, Phys. Lett. B 744, 293-302 (2015), [arXiv:1502.05642 [hep-ph]].
  * (15) E. Iancu, J. Madrigal, A. Mueller, G. Soyez and D. Triantafyllopoulos, Phys. Lett. B 750, 643-652 (2015), [arXiv:1507.03651 [hep-ph]].
  * (16) Y. Hatta and E. Iancu, JHEP 08 (2016), 083, [arXiv:1606.03269 [hep-ph]].
  * (17) B. Ducloué, E. Iancu, A. Mueller, G. Soyez and D. Triantafyllopoulos, JHEP 04, 081 (2019), [arXiv:1902.06637 [hep-ph]].
  * (18) B. Ducloué, E. Iancu, G. Soyez and D. Triantafyllopoulos, Phys. Lett. B 803, 135305 (2020), [arXiv:1912.09196 [hep-ph]].
  * (19) H. Y. Liu, Z. B. Kang and X. Liu, Phys. Rev. D 102 (2020) no.5, 051502, [arXiv:2004.11990 [hep-ph]].
  * (20) R. Boussarie and Y. Mehtar-Tani, [arXiv:2006.14569 [hep-ph]].
  * (21) R. Boussarie and Y. Mehtar-Tani, Phys. Rev. D 103 (2021) no.9, 094012, [arXiv:2001.06449 [hep-ph]].
  * (22) I. Balitsky and A. Tarasov, JHEP 10 (2015), 017, [arXiv:1505.02151 [hep-ph]].
  * (23) T. Altinoluk, N. Armesto, G. Beuf, M. Martínez and C. A. Salgado, JHEP 07 (2014), 068, [arXiv:1404.2219 [hep-ph]].
  * (24) T. Altinoluk, N. Armesto, G. Beuf and A. Moscoso, JHEP 01 (2016), 114, [arXiv:1505.01400 [hep-ph]].
  * (25) T. Altinoluk and A. Dumitru, Phys. Rev. D 94 (2016) no.7, 074032, [arXiv:1512.00279 [hep-ph]].
  * (26) P. Agostini, T. Altinoluk and N. Armesto, Eur. Phys. J. C 79 (2019) no.7, 600, [arXiv:1902.04483 [hep-ph]].
  * (27) P. Agostini, T. Altinoluk and N. Armesto, Eur. Phys. J. C 79 (2019) no.9, 790, [arXiv:1907.03668 [hep-ph]].
  * (28) T. Altinoluk, G. Beuf, A. Czajka and A. Tymowska, Phys. Rev. D 104 (2021) no.1, 014019, [arXiv:2012.03886 [hep-ph]].
  * (29) G. A. Chirilli, JHEP 01 (2019), 118, [arXiv:1807.11435 [hep-ph]].
  * (30) G. A. Chirilli, JHEP 06 (2021), 096, [arXiv:2101.12744 [hep-ph]].
  * (31) Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, JHEP 01 (2016), 072 [erratum: JHEP 10 (2016), 148], [arXiv:1511.06737 [hep-ph]].
  * (32) Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, Phys. Rev. D 95 (2017) no.1, 014033, [arXiv:1610.06197 [hep-ph]].
  * (33) Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, Phys. Rev. Lett. 118 (2017) no.5, 052001, [arXiv:1610.06188 [hep-ph]].
  * (34) Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, JHEP 10 (2017), 198, [arXiv:1706.04236 [nucl-th]]
  * (35) Y. V. Kovchegov and M. D. Sievert, Phys. Rev. D 99 (2019) no.5, 054032, [arXiv:1808.09010 [hep-ph]].
  * (36) Y. V. Kovchegov, JHEP 03 (2019), 174, [arXiv:1901.07453 [hep-ph]].
  * (37) F. Cougoulic and Y. V. Kovchegov, Phys. Rev. D 100 (2019) no.11, 114020, [arXiv:1910.04268 [hep-ph]]
  * (38) Y. Hatta, Y. Nakagawa, F. Yuan, Y. Zhao and B. Xiao, Phys. Rev. D 95 (2017) no.11, 114032, [arXiv:1612.02445 [hep-ph]].
  * (39) J. P. Blaizot and Y. Mehtar-Tani, Int. J. Mod. Phys. E 24 (2015) no.11, 1530012, [arXiv:1503.05958 [hep-ph]].
  * (40) G. Beuf, Phys. Rev. D 85, 034039 (2012), [arXiv:1112.4501 [hep-ph]].
  * (41) S. Caron-Huot, JHEP 05, 093 (2015), [arXiv:1309.6521 [hep-th]]
  * (42) H. Kowalski, L. Motyka and G. Watt, Phys. Rev. D 74 (2006), 074016, [arXiv:hep-ph/0606272 [hep-ph]].
  * (43) D. Kharzeev, Y. V. Kovchegov and K. Tuchin, Phys. Rev. D 68 (2003), 094013, [arXiv:hep-ph/0307037 [hep-ph]].
  * (44) F. Dominguez, C. Marquet, B. W. Xiao and F. Yuan, Phys. Rev. D 83 (2011), 105005, [arXiv:1101.0715 [hep-ph]].
  * (45) A. V. Belitsky, X. Ji and F. Yuan, Nucl. Phys. B 656 (2003), 165-198, [arXiv:hep-ph/0208038 [hep-ph]].
  * (46) V. Gribov and L. Lipatov, Sov. J. Nucl. Phys. 15, 438-450 (1972) IPTI-381-71 ; Y. L. Dokshitzer, Sov. Phys. JETP 46, 641-653 (1977) ; G. Altarelli and G. Parisi, Nucl. Phys. B 126, 298-318 (1977)
  * (47) W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Phys. Rev. D 18 (1978), 3998
  * (48) Y. Hatta, B. W. Xiao and F. Yuan, Phys. Rev. D 95 (2017) no.11, 114026, [arXiv:1703.02085 [hep-ph]].
  * (49) T. Altinoluk and R. Boussarie, JHEP 10 (2019), 208 doi:10.1007/JHEP10(2019)208 [arXiv:1902.07930 [hep-ph]].
  * (50) R. Boussarie, H. Mäntysaari, F. Salazar and B. Schenke, JHEP 09 (2021), 178, [arXiv:2106.11301 [hep-ph]].
  * (51) A. Bialas, H. Navelet and R. B. Peschanski, Nucl. Phys. B 593 (2001), 438-450, [arXiv:hep-ph/0009248 [hep-ph]].
  * (52) K. Kutak and A. Stasto, Eur. Phys. J. C 41, 343-351 (2005), [arXiv:hep-ph/0408117 [hep-ph]].
  * (53) J. Jalilian-Marian, Phys. Rev. D 96 (2017) no.7, 074020, [arXiv:1708.07533 [hep-ph]].
  * (54) J. Jalilian-Marian, Phys. Rev. D 99 (2019) no.1, 014043, [arXiv:1809.04625 [hep-ph]].
  * (55) J. Jalilian-Marian, Phys. Rev. D 102 (2020) no.1, 014008, [arXiv:1912.08878 [hep-ph]].