# Renormalization group improved photon impact factors and the high energy
virtual photon scattering

Dimitri Colferai  Dipartimento di Fisica, Università di Firenze and INFN
Firenze Via Sansone 1, 50019 Sesto Fiorentino, Italy Wanchen Li  Department of
Physics, Penn State University, University Park, PA 16802, USA  Anna M. Staśto
Department of Physics, Penn State University, University Park, PA 16802, USA

###### Abstract

We perform the renormalization group improved collinear resummation of the
photon-gluon impact factors. We construct the resummed cross section for
virtual photon-photon ($\gamma^{*}\gamma^{*}$) scattering which incorporates
the impact factors and BFKL gluon Green’s function up to the next-to-leading
logarithmic accuracy in energy. The impact factors include important
kinematical effects which are responsible for the most singular poles in
Mellin space at next-to-leading order. Further conditions on the resummed
cross section are obtained by requiring the consistency with the collinear
limits. Our analysis is consistent with previous impact factor calculations at
NLO, apart from a new term proportional to $C_{F}$ that we find for the
longitudinal polarization. Finally, we use the resummed cross section to
compare with the LEP data on the $\gamma^{*}\gamma^{*}$ cross section and with
previous calculations. The resummed result is lower than the leading
logarithmic approximation but higher than the pure next-to-leading one, and is
consistent with the experimental data.

##  1 Introduction

High energy particle accelerators, like the Large Hadron Collider (LHC),
opened up a new kinematic regime for particle interactions. Exploration of
this regime is not only important for the phenomenological description of the
scattering processes which occur at these colliders, but also for advancing
our understanding of the theory of strong interactions: Quantum Chromodynamics
(QCD). The high energy limit — also called Regge limit — in QCD is defined
when the center-of-mass energy squared $s$ of the collision is much larger
than other scales in the process, $s\gg-t>\Lambda_{QCD}^{2}$, where $t$ is the
momentum transfer. In the perturbative regime of small coupling $\alpha_{s}\ll
1$, the description of high energy processes has been developed over the
decades, which is based on the high energy factorization (or $k_{T}$
factorization) [1, 2, 3] framework. The cross sections in this limit can be
written in a factorized form with process-dependent impact factors and the
universal gluon Green’s function (GGF) responsible for the exchanges in the
$t$-channel. It is the energy dependence of the latter that controls the high-
energy behavior of the resulting cross section.

The GGF is given by the solution to the Balitsky-Fadin-Kuraev-Lipatov (BFKL)
[4, 5, 6] evolution equation, which resums the powers of $(\alpha_{s}\ln
s/s_{0})$, where $s_{0}$ is some reference energy scale. In the high energy
limit, the logarithms of energy can be very large, and the terms
$(\alpha_{s}\ln s/s_{0})^{n}\sim 1$ even in the perturbative regime of small
coupling. Therefore such terms need to be resummed, and this leads to the
power growth of the gluon Green’s function with the energy, that in turn
translates to the power-like rise of the cross sections with energy. Due to
the latter feature, this solution is traditionally referred to as the BFKL
Pomeron. The BFKL evolution equation is known at the leading logarithmic (LL)
[4, 5] and next-to-leading logarithmic (NLL) accuracy in QCD [7, 8].

As it turned out, the NLL corrections to the BFKL equation are large and
negative, and may lead to instabilities, like oscillating cross section. Thus,
in order to stabilize the BFKL expansion, resummation methods were developed
some time ago [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], and
more recently applied to phenomenology [24, 25, 26]. In the approach developed
in [18, 20, 19, 21], a renormalization group improved (RGI) small-$x$
evolution equation was constructed, which takes into account LL and NLL BFKL
as well as the DGLAP splitting function at lowest order. The consistency of
this formalism is based on the fact that the kernel of the evolution equation
has the correct collinear limits, i.e., the limits of the strong ordering of
the transverse momenta along the ladder of gluon emissions in the $t$-channel.
To be precise, the requirement is that the collinear singularities are single
logarithmic in transverse momenta, which in the Mellin space of the variable
$\gamma$, the variable conjugated to the gluon transverse momentum, manifests
itself as the occurrence of single poles for each power of
$\alpha_{\mathrm{s}}$. The BFKL kernel in Mellin space has only single poles
of type $\sim 1/\gamma,1/(1-\gamma)$ at leading logarithmic order, but at NLL
order quadratic and cubic poles appear. The quadratic poles have been
recognized as originating from the non-singular parts of the LO DGLAP
splitting function which appears in the NLL BFKL kernel, as well as due to the
running coupling. The cubic poles originate from the energy scales which
become relevant at this order. It was demonstrated that the terms with the
quadratic and cubic poles are the ones that are responsible for the major part
of the NLL correction.

In the following, by “resummation” we mean the “collinear resummation” of
DGLAP terms, which coincides with the “renormalization group improvement”
mentioned before. In the Ciafaloni-Colferai-Salam-Stasto (CCSS) resummation
scheme, the cubic poles at the NLL level (and the poles of order $2n+1$ at
NnLL level) are resummed — hence eliminated — by shifting the single poles in
$\gamma$ in the LL kernel eigenvalue. The shift is proportional to the Mellin
variable $\omega$ conjugated to the energy $s$. This shift originates from the
kinematical constraint [27] imposed onto the integrals over the transverse
momenta. The quadratic poles originate from the non-singular part of the DGLAP
splitting function, and again, can be resummed — hence eliminated — by taking
them into account in the leading order kernel with an $\omega$-dependent
redefinition of the coefficients of the single poles.

For a physical process that occurs at high energy, the gluon Green’s function
needs to be supplemented by the process-dependent impact factors, which also
need to be evaluated at the appropriate order of perturbation theory. The NLO
corrections have been calculated for the photon-gluon impact factor [28, 29],
Mueller-Navelet jet vertices [30], Mueller-Tang jets [31, 32, 33], and light
vector mesons [34]. Numerous NLO calculations of impact factors have also been
performed in the context of effective theory for high energy and density, the
Color Glass Condensate, which includes parton saturation. Examples of the
next-to-leading calculations in this framework include inclusive structure
functions [35, 36], also for massive quarks [37], contributions to inclusive
diffraction [38], exclusive vector meson production [39, 40, 41, 42] and for
inclusive dijet [43] and photon+jet [44]. The collinear resummation in the
context of small $x$ evolution with saturation has been also explored, e.g.
[45].

An excellent process for studying the BFKL Pomeron is $\gamma^{*}\gamma^{*}$
scattering, see e.g. [46, 47, 48, 49, 50, 51, 52, 53]. The idea is to select
the events in which the virtualities $Q_{i}^{2}$ of the two photons are
comparable and large and the “rapidity interval” $Y\equiv\log(s/Q_{1}Q_{2})$
between them is very large: $s\gg Q_{1}^{2}\sim
Q_{2}^{2}\gg\Lambda_{QCD}^{2}$. In such kinematics, the DGLAP logarithms are
suppressed and the process should be dominated by the BFKL evolution. This
measurement was performed at LEP $e^{+}e^{-}$ collider [54, 55], through
measurements of events with double tagged leptons. Calculations were performed
to describe this process within the BFKL formalism. In particular, it was
shown that including the partial resummation in the gluon Green’s function
leads to an excellent description of the LEP data [53]. Later on, once the NLO
photon-gluon impact factor became available, the full NLL calculation of the
photon-photon scattering was performed [56, 57]. The numerical difference
between the LL and NLL case was found very large. In fact, the NLL calculation
is not able to describe the LEP data, particularly at the highest rapidity
[57].

Given that the gluon Green’s function required the collinear resummation, it
is important to perform consistently the resummation of the impact factors. As
in the case of the gluon Green’s function it is important to take into account
exact kinematics in the impact factors and analyze the structure of the poles
due to these effects. The exact kinematics was included in the $k_{T}$
factorization formula for the DIS structure functions [58, 59, 60]. Based on
this result, the photon-gluon impact factor with exact kinematics was computed
in Mellin space [61]. Interestingly, these improved impact factors contained a
shift of the poles in the $\gamma$ variable, analogously to what was observed
in the Green’s function with the kinematical constraint. This shift is
proportional to $\omega$, the Mellin variable conjugated to energy.

In this paper, we analyze the photon-photon scattering process at high energy
and perform the resummation of the impact factors in addition to the
resummation of the gluon Green’s function. We construct the renormalization
group improved high-energy factorization formula, where both impact factors
and gluon Green’s function are resummed, hence $\omega$-dependent. A first
consistency condition is imposed to ensure the equivalence of the RGI impact
factors and gluon Green’s function upon expansion in $\omega$ with fixed order
BFKL results up to NLL. The second consistency condition is imposed by
analyzing the cross section in the collinear limit, i.e., assuming strong
ordering of the virtualities of the photons, and consequently in the ladder of
exchanged partons. In that way the coefficients of the highest and next-to-
highest $\gamma$-poles can be fixed both in the gluon Green’s function and in
the impact factors. Since the two consistency conditions do not uniquely
specify all the subleading poles when $\omega\neq 0$, we consider several
resummation schemes, which parametrize the ambiguity due to the unknown lower
order poles.

As previously observed (see eg. [53, 57]), we need to add to the BFKL cross
section other contributions, in particular the one stemming from the quark box
diagram — both photons coupled to the same quark line —, which is dominant for
the lowest rapidities. The results of our calculations are compared with the
experimental data from LEP [54, 55] and an overall agreement is obtained
within the theoretical and experimental uncertainties, with the resummed BFKL
contribution representing the bulk of the cross section at high rapidities.

The paper is organized as follows. In sec. 2 we recall the renormalization
group improved method for the gluon Green’s function, and in particular we
discuss the $\omega$ shifts. In sec. 3 the RGI factorization formula is
introduced and consistency with high-energy factorization is discussed. We
perform the collinear analysis of the transverse-transverse photon cross
section in sec. 4 and construct the resummed impact factor for transversely
polarized photons. In sec. 5 the analogous construction is carried out for the
RGI impact factor in case of longitudinally polarized photons. Numerical
analysis is performed in sec. 6, where we apply the resummed impact factors to
cross sections and compare the results with the experimental data from LEP and
with other theoretical descriptions. Finally, in sec. 7 we state our
conclusions. An appendix contains some formulae on the lowest order cross
sections and structure functions.

##  2 Renormalization group improved gluon Green’s function

The collinear resummation for the evolution of gluon density at small-$x$ was
developed in series of works using slightly different approaches [11, 12, 13,
14, 15], [16, 17, 18, 20, 19, 21], [22], [23]. In this section we shall recap
the essential ingredients of the resummation developed in [16, 17, 18, 20, 19,
21] for the gluon Green’s function. As a result, the renormalization group
improved (RGI) small-$x$ evolution was constructed which contains both DGLAP
and BFKL kernels and satisfies momentum sum rule.

We start from the cross section $\sigma^{(jk)}$ for virtual photon scattering
at high-energy which can be written in a factorized form as the product of
process-dependent impact factors $\phi^{(j)}$ and the universal (energy-
dependent) gluon Green’s function $\mathrm{G}$, as depicted in fig. 1. In
momentum space, the BFKL factorization formula reads

$\sigma^{(jk)}(s,Q_{1},Q_{2})=\int{\rm d}^{2}{\boldsymbol{k}}\;{\rm
d}^{2}{\boldsymbol{k}}^{\prime}\;\phi^{(j)}(Q_{1},{\boldsymbol{k}})\,\mathrm{G}(s,{\boldsymbol{k}},{\boldsymbol{k}}^{\prime})\,\phi^{(k)}(Q_{2},{\boldsymbol{k}}^{\prime})\;,$
(1)

where $j,k\in\\{L,T\\}$ denote the polarizations of the two photons,
$q_{1},q_{2}$ their momenta and $Q_{i}^{2}\equiv-q_{i}^{2}>0:i=1,2$ their
virtualities.

Figure 1: Diagramatic representation of the BFKL factorization formula for the
process of high-energy scattering of two virtual photons.

The gluon Green’s function
$\mathrm{G}(s,{\boldsymbol{k}},{\boldsymbol{k}}_{0})$, which depends on the
transverse gluon momenta ${\boldsymbol{k}}$ and ${\boldsymbol{k}}_{0}$ and
energy squared $s\equiv(q_{1}+q_{2})^{2}$, satisfies the evolution equation
that can be written in the following form

$\frac{\partial}{\partial\log
s}\mathrm{G}(s,{\boldsymbol{k}},{\boldsymbol{k}}_{0})=\int{\rm
d}^{2}{\boldsymbol{k}}^{\prime}\;\mathcal{K}({\boldsymbol{k}},{\boldsymbol{k}}^{\prime})\,\mathrm{G}(s,{\boldsymbol{k}}^{\prime},{\boldsymbol{k}}_{0})\;,$
(2)

where the function $\mathcal{K}$ is the BFKL kernel which has the following
perturbative expansion

$\mathcal{K}=\bar{\alpha}_{s}\mathcal{K}_{0}+\bar{\alpha}_{s}^{2}\mathcal{K}_{1}+\dots\;.$
(3)

In the above equation, we introduced the rescaled strong coupling
$\bar{\alpha}_{s}=\frac{\alpha_{s}N_{c}}{\pi}$ where $N_{c}$ is the number of
colors. In QCD the kernel is known at leading [4, 5] and next-to-leading order
[7, 8], in $\mathcal{N}=4$ super Yang-Mills theory up to next-to-next-to
leading accuracy [62, 63, 64]. It is customary to use the Mellin transform to
obtain the kernel eigenvalue

$\bar{\alpha}_{s}\chi(\gamma)=\int{\rm d}{\boldsymbol{k}}^{\prime
2}\;\bigg{(}\frac{{\boldsymbol{k}}^{\prime
2}}{{\boldsymbol{k}}^{2}}\bigg{)}^{\gamma}\,\mathcal{K}({\boldsymbol{k}},{\boldsymbol{k}}^{\prime})\,,$
(4)

with the corresponding perturbative expansion corresponding to eq. 3

$\chi(\gamma)=\chi_{0}(\gamma)+\bar{\alpha}_{s}\chi_{1}(\gamma)+\dots\;.$ (5)

The leading order kernel’s eigenvalue reads

$\chi_{0}(\gamma)=2\psi(1)-\psi(\gamma)-\psi(1-\gamma)\;,$ (6)

where $\psi(z)=\Gamma^{\prime}(z)/\Gamma(z)$ is the polygamma function, and
$\psi(1)=-\gamma_{E}$. The next-to-leading order eigenvalue is111This
expression for $\chi_{1}$ holds for the scale-invariant part of the NLO kernel
with symmetric energy scale (see eq. 25) and symmetric running coupling
$\alpha_{\mathrm{s}}(|{\boldsymbol{k}}|\,|{\boldsymbol{k}}^{\prime}|)$ in eq.
3.

$\displaystyle\chi_{1}(\gamma)$
$\displaystyle=-\frac{b}{2}\chi_{0}^{2}(\gamma)-\frac{1}{4}\chi_{0}^{\prime\prime}(\gamma)-\frac{1}{4}\left(\frac{\pi}{\sin\pi\gamma}\right)^{2}\frac{\cos\pi\gamma}{3(1-2\gamma)}\left[11+4\frac{T_{R}N_{f}}{N_{c}^{3}}+\frac{(1+2T_{R}N_{f}/N_{c}^{3})\gamma(1-\gamma)}{(1+2\gamma)(3-2\gamma)}\right]$
$\displaystyle\quad+\left(\frac{67}{36}-\frac{\pi^{2}}{12}-\frac{5T_{R}N_{f}}{9N_{c}}\right)\chi_{0}(\gamma)+\frac{3}{2}\xi(3)+\frac{\pi^{2}}{4\sin\pi\gamma}-\Phi(\gamma)\,,$
(7)

where

$b=\frac{11N_{c}-4T_{R}N_{f}}{12\pi}\equiv\frac{C_{A}}{\pi}\bar{b}\,,$ (8)

is the first beta-function coefficient, $N_{f}$ the number of active quark
flavours, $T_{R}=1/2$ and

$\Phi(\gamma)=\sum_{n=0}^{\infty}(-1)^{n}\left[\frac{\psi(n+1+\gamma)-\psi(1)}{(n+\gamma)^{2}}+\frac{\psi(n+2-\gamma)-\psi(1)}{(n+1+\gamma)^{2}}\right]\,.$
(9)

The LO and NLO BFKL eigenvalues contain collinear and anticollinear poles,
i.e. poles when $\gamma\sim 0$ and $\gamma\sim 1$. These correspond to the
strong ordering of the transverse momenta in the $t$ channel, either
${\boldsymbol{k}}^{2}\gg{\boldsymbol{k}}^{\prime 2}$ or
${\boldsymbol{k}}^{2}\ll{\boldsymbol{k}}^{\prime 2}$ respectively.

The LO and NLO eigenvalues have specific pole structures in $\gamma$ variable.
The LO eigenvalue has only single poles, i.e.

$\chi_{0}(\gamma)\,\sim\,\frac{1}{\gamma},\;\;\frac{1}{1-\gamma}\;.$ (10)

The NLO eigenvalue has a more complicated structure, since in addition to the
single poles it also has double and triple collinear poles. It is these higher
order poles that are responsible for the fact that the NLO term is numerically
large, and these terms need to be properly resummed. The double poles stem
from two sources. The first one is the running coupling term

$-\frac{b}{2}\left[\chi_{0}^{2}(\gamma)\right]\,\sim\,-\frac{b}{2\,\gamma^{2}}+(\gamma\to
1-\gamma)\;,$ (11)

which contributes to the poles at $\gamma=0$ and 1. The second is the term

$-\frac{1}{4}\left(\frac{\pi}{\sin\pi\gamma}\right)^{2}\frac{\cos\pi\gamma}{3(1-2\gamma)}\left[11+4\frac{T_{R}N_{f}}{N_{c}^{3}}+\frac{(1+2T_{R}N_{f}/N_{c}^{3})\gamma(1-\gamma)}{(1+2\gamma)(3-2\gamma)}\right]\sim-\frac{11+4T_{R}N_{f}/N_{c}^{3}}{12\;\gamma^{2}}+(\gamma\to
1-\gamma)\;,$ (12)

which originates from the DGLAP anomalous dimension. To be precise, the
coefficient of the double poles is just the non-singular part
$A_{1}(0)=-\frac{11+4T_{R}N_{f}/N_{c}^{3}}{12}$ of the eigenvalue
$\gamma_{+}(\omega)$ of the LO DGLAP anomalous dimension matrix:

$\gamma_{+}(\omega)=P_{gg}(\omega)+\frac{C_{F}}{C_{A}}P_{qg}(\omega)+\mathcal{O}(\omega)=\frac{2C_{A}}{\omega}\left[1+\omega
A_{1}(\omega)\right]\;,$ (13)

where $\omega$ is the Mellin variable conjugated to the energy,
$C_{A}=N_{c}=3$ and $C_{F}=(N_{c}^{2}-1)/(2N_{c})$ and

$A_{1}(\omega)=-\frac{1}{\omega+1}+\frac{1}{\omega+2}-\frac{1}{\omega+3}-\left[\psi(2+\omega)-\psi(1)\right]+\frac{11}{12}-\frac{T_{R}N_{f}}{3N_{c}^{3}}\,,$
(14)

represents the non-singular (for $\omega\to 0$) part of $\gamma_{+}$.

Finally, the triple collinear poles stem from the term

$-\frac{1}{4}\chi_{0}^{\prime\prime}(\gamma)\sim-\frac{1}{2}\frac{1}{\gamma^{3}},\;\;-\frac{1}{2}\frac{1}{(1-\gamma)^{3}}\;.$
(15)

The form of the term above depends on the scale choice for the kernel. Let us
briefly recap the problem of energy scales [9] in the BFKL equation.

Going back to the momentum representation of the BFKL equation (2), we can use
the double Mellin transform to write the azimuthally averaged gluon Green’s
function as

$\mathrm{G}(s,{\boldsymbol{k}},{\boldsymbol{k}}_{0})=\frac{1}{2\pi{\boldsymbol{k}}^{2}}\int\frac{{\rm
d}\omega}{2\pi{\rm
i}}\bigg{(}\frac{s}{{\boldsymbol{k}}{\boldsymbol{k}}_{0}}\bigg{)}^{\omega}\int\frac{{\rm
d}\gamma}{2\pi{\rm
i}}\bigg{(}\frac{{\boldsymbol{k}}^{2}}{{\boldsymbol{k}}_{0}^{2}}\bigg{)}^{\gamma}\mathrm{G}(\omega,\gamma)\,,$
(16)

and the BFKL equation becomes

$\omega\,\mathrm{G}(\omega,\gamma)=1+\bar{\alpha}_{s}\,\chi(\gamma)\,\mathrm{G}(\omega,\gamma)\;.$
(17)

In eq. 16 we are adopting the symmetric energy scale
$s_{0}={\boldsymbol{k}}{\boldsymbol{k}}_{0}$. However, the scale choice can
also be asymmetric, like in the case of the Deep Inelastic Scattering, where
the scales on the virtual photon and the proton side are in principle very
different. In this case, the cross section is dominated by configurations with
${\boldsymbol{k}}\gg{\boldsymbol{k}}_{0}$ so that the proper evolution
variable is ${\boldsymbol{k}}^{2}/s$, corresponding to the asymmetric energy
scale $s_{0}={\boldsymbol{k}}^{2}$. The crucial observation is that such a
change of energy scale in eq. 16 is equivalent to a shift of $\gamma$ by
$\omega/2$:

$\bigg{(}\frac{s}{{\boldsymbol{k}}{\boldsymbol{k}}_{0}}\bigg{)}^{\omega}\bigg{(}\frac{{\boldsymbol{k}}^{2}}{{\boldsymbol{k}}_{0}^{2}}\bigg{)}^{\gamma}=\bigg{(}\frac{s}{{\boldsymbol{k}}^{2}}\bigg{)}^{\omega}\bigg{(}\frac{{\boldsymbol{k}}}{{\boldsymbol{k}}_{0}}\bigg{)}^{\omega}\bigg{(}\frac{{\boldsymbol{k}}^{2}}{{\boldsymbol{k}}_{0}^{2}}\bigg{)}^{\gamma}=\bigg{(}\frac{s}{{\boldsymbol{k}}^{2}}\bigg{)}^{\omega}\bigg{(}\frac{{\boldsymbol{k}}^{2}}{{\boldsymbol{k}}_{0}^{2}}\bigg{)}^{\gamma+\omega/2}\;.$
(18)

The opposite shift is obtained if $s_{0}={\boldsymbol{k}}_{0}^{2}$. Due to
that fact, the gluon Green’s function and thus the kernel in Mellin space gets
$\omega$ dependence; the latter can be written in the form [65]

$\chi_{0}(\omega,\gamma)=2\psi(1)-\psi(\gamma+\frac{\omega}{2})-\psi(1-\gamma+\frac{\omega}{2})\;,$
(19)

for the symmetric scale choice.

Expanding this kernel in $\omega$ and using the solution at lowest order
$\omega=\bar{\alpha}_{s}\chi_{0}(\gamma)$, one obtains for the NLO
contribution

$-\frac{1}{2}\omega\,\psi^{\prime}(\gamma)-\frac{1}{2}\omega\,\psi^{\prime}(1-\gamma)\;\simeq\;-\frac{1}{2}\frac{\bar{\alpha}_{s}}{\gamma^{3}}-\frac{1}{2}\frac{\bar{\alpha}_{s}}{(1-\gamma)^{3}}\;.$
(20)

These terms exactly correspond to the triple collinear poles present in the
NLO kernel, see eq. 15. In other words, the cubic poles in the NLO kernel can
be discarded, since their contribution is taken into account by the
$\omega$-shift in the LO kernel, as in eq. 19.

Next, the collinear term with the non-singular DGLAP splitting function was
included in the form [19]

$\chi_{c}^{\omega}(\gamma)=\omega
A_{1}(\omega)\left(\frac{1}{\gamma+\frac{\omega}{2}}+\frac{1}{1-\gamma+\frac{\omega}{2}}\right)\;,$
(21)

which, when expanded in $\omega$ and retaining the first power in
$\bar{\alpha}_{s}$, gives

$\omega
A_{1}(\omega)\left(\frac{1}{\gamma}+\frac{1}{1-\gamma}\right)\simeq\bar{\alpha}_{s}A_{1}(0)\left(\frac{1}{\gamma^{2}}+\frac{1}{(1-\gamma)^{2}}\right)\;,$
(22)

thus reproducing eq. 12. These terms are then subtracted from the NLO kernel
again.

Actually, the $\omega$-shift predicted by the collinear analysis with upper
and lower energy-scale, leading to eq. 19, allows us to predict the spurious
poles 222It has been verified [66] that the $\omega$-shift correctly
reproduces the highest order poles $\sim 1/\gamma^{5}$ at the NNLO BFKL in the
supersymmetric theory [63, 62, 64]. of the higher order BFKL kernels:

$\chi_{n}(\gamma)\sim\frac{1}{\gamma^{1+2n}}\;,$ (23)

which are more and more singular as the order increases, while only poles of
order $1+n$ are expected from the collinear QCD dynamics. This can be roughly
understood because the $\omega$-shift transforms a LO pole into a series of
spurious poles to all orders, e.g.,

$\frac{1}{\gamma+\omega/2}\sim\frac{1}{\gamma+\bar{\alpha}_{s}\chi_{0}/2}\sim\frac{1}{\gamma(1+\bar{\alpha}_{s}/2\gamma^{2})}\sim\sum_{n=0}^{\infty}\frac{(-\bar{\alpha}_{s})^{n}}{2^{n}\,\gamma^{1+2n}}\;.$
(24)

The occurrence of “spurious” high-order poles in the BFKL approach is
responsible for the bad convergence of the BFKL expansion and the
instabilities of its phenomenological predictions. Therefore, it is compelling
to resum such spurious poles by means of the RGI formulation.

In conclusion, the resummed kernel in the CCSS formalism was constructed by
taking at LO the sum of the $\omega$-shifted kernels in eqs. 19 and 21 and
subtracting the triple and double $\gamma$-poles (eqs. 20 and 22) from the NLO
kernel. In this way the NLO resummed kernel has only simple poles, hence it is
much less singular in the collinear limits $\gamma\to 0,1$ and provides more
stable and reliable phenomenological results.

##  3 High energy factorisation

In this section we recall and compare the factorization formulae in the pure
BFKL formalism and in the RGI approach, with the aim of deriving the
compatibility conditions among the respective impact factors and Green
functions, thus setting the stage for the computation of the RGI impact
factors.

###  3.1 BFKL vs RGI factorization formula

The high-energy factorization formula (1) for $\gamma^{*}\gamma^{*}$
scattering can be rewritten in a more convenient form as a double Mellin
representation with respect to transverse momenta (or virtualities) [cfr. eq.
4] and to energy [cfr. eqs. 16 and 17]:

$\displaystyle\sigma^{(jk)}(s,Q_{1},Q_{2})$ $\displaystyle=\frac{1}{2\pi
Q_{1}Q_{2}}\int\frac{{\rm d}\omega}{2\pi{\rm
i}}\left(\frac{s}{s_{0}(p)}\right)^{\omega}\int\frac{{\rm d}\gamma}{2\pi{\rm
i}}\left(\frac{Q_{1}^{2}}{Q_{2}^{2}}\right)^{\gamma-\mbox{\small$\frac{1}{2}$}}\phi^{(j)}(\gamma;p)\,\mathrm{G}(\omega,\gamma;p)\,\phi^{(k)}(1-\gamma;-p)\,,$
(25a) $\displaystyle\mathrm{G}(\omega,\gamma;p)$
$\displaystyle=\frac{1}{\omega-\bar{\alpha}_{s}\chi(\gamma;p)}\;.$ (25b)

Here we introduced the notation $s_{0}(p)=Q_{1}^{1+p}Q_{2}^{1-p}$ for the
energy-scale. By varying the parameter $p$ we can switch from symmetric scale
$s_{0}=Q_{1}Q_{2}$ ($p=0$), to “upper” scale $s_{0}=Q_{1}^{2}$ ($p=1$) or to
“lower” scale $s_{0}=Q_{2}^{2}$ ($p=-1$).

In eq. 25 both impact factors $\phi$ and eigenvalue function $\chi$ are
perturbative objects that admit a series expansion in $\alpha_{\mathrm{s}}$,
as in eq. 5; from next-to-leading order on, they depend on the choice of the
energy scale:

$\displaystyle\phi^{(j)}(\gamma;p)$
$\displaystyle=\phi^{(j)}_{0}(\gamma)+\bar{\alpha}_{s}\phi^{(j)}_{1}(\gamma;p)+\mathcal{O}(\bar{\alpha}_{s}^{2})\,,$
(26) $\displaystyle\chi(\gamma;p)$
$\displaystyle=\chi_{0}(\gamma)+\bar{\alpha}_{s}\chi_{1}(\gamma;p)+\mathcal{O}(\bar{\alpha}_{s}^{2})\;.$
(27)

On the other hand, the renormalization-group improved (RGI) high-energy
factorization for scattering reads

$\displaystyle\sigma^{(jk)}(s,Q_{1},Q_{2})$ $\displaystyle=\frac{1}{2\pi
Q_{1}Q_{2}}\int\frac{{\rm d}\omega}{2\pi{\rm
i}}\left(\frac{s}{s_{0}(p)}\right)^{\omega}\int\frac{{\rm d}\gamma}{2\pi{\rm
i}}\left(\frac{Q_{1}^{2}}{Q_{2}^{2}}\right)^{\gamma-\mbox{\small$\frac{1}{2}$}}\Phi^{(j)}(\omega,\gamma;p)\,\mathcal{G}(\omega,\gamma;p)\,\Phi^{(k)}(\omega,1-\gamma;-p)$
(28a) $\displaystyle\mathcal{G}(\omega,\gamma;p)$
$\displaystyle=\frac{1}{\omega-\bar{\alpha}_{s}X(\omega,\gamma;p)}\;.$ (28b)

Here, we introduce the new notation $X(\omega,\gamma)$ for the kernel in
Mellin space appearing in RGI factorization (28b) to clearly distinguish it
from the BFKL kernel $\chi(\gamma)$ present in the standard high-energy
factorization, eq. 25b. At variance with the usual BFKL expansion, both impact
factors $\Phi$ and eigenvalue function $X$ are $\omega$-dependent, for a
twofold purpose: (i) to fully agree with the known collinear behaviour at
least in the leading-logarithmic $\log(Q_{1}/Q_{2})$ approximation; (ii) to
resum into a smoother behaviour subleading contributions which are singular in
some region of the complex $\gamma$-plane. Actually, the two issues are
strictly related, as explained in [19] and in sec. 2.

Following the argument leading to eq. 18, a change of the energy scale
$s_{0}$, i.e. a change in $p$, leaves the cross section (28a) invariant
provided

$\displaystyle\Phi^{(j)}(\omega,\gamma;p)=\Phi^{(j)}\Big{(}\omega,\gamma-\frac{\omega}{2}p;0\Big{)}\;,\qquad
X(\omega,\gamma;p)=X\Big{(}\omega,\gamma-\frac{\omega}{2}p;0\Big{)}\;.$ (29)

The corresponding scale change entails more complicated changes in the BFKL
impact factors $\phi$ and eigenvalue function $\chi$ of eq. 25.

Given some choice for the energy scale, the equivalence between the two
factorization formulas (25a) and (28a) is obtained by evaluating the
$\omega$-integrals and requiring the remaining $\gamma$-integrand to be the
same function (up to terms yielding contributions suppressed by powers of
$s$). In eq. 25a the $\omega$ integration is trivial: being $s_{0}<s$ one can
close the $\omega$-integration path to the left and pick up the simple pole at
$\omega=\bar{\alpha}_{s}\chi(\gamma)$, obtaining

$\displaystyle\sigma^{(jk)}(s,Q_{1},Q_{2})$ $\displaystyle=\frac{1}{2\pi
Q_{1}Q_{2}}\int\frac{{\rm d}\gamma}{2\pi{\rm
i}}\left(\frac{s}{s_{0}}\right)^{\bar{\alpha}_{s}\chi(\gamma)}\left(\frac{Q_{1}^{2}}{Q_{2}^{2}}\right)^{\gamma-\mbox{\small$\frac{1}{2}$}}\phi^{(j)}(\gamma)\,\phi^{(k)}(1-\gamma)\;.$
(30)

In eq. 28a there can be many $\omega$-poles. The position of the rightmost
pole — which provides the leading high-energy behaviour of the cross section —
is determined by the implicit equation

$\omega=\bar{\alpha}_{s}X(\omega,\gamma)\equiv\omega^{\mathrm{eff}}(\gamma,\bar{\alpha}_{s})\equiv\bar{\alpha}_{s}\chi^{\mathrm{eff}}(\gamma,\bar{\alpha}_{s})\;,$
(31)

where the last expressions
$\omega^{\mathrm{eff}}=\bar{\alpha}_{s}\chi^{\mathrm{eff}}$ represent such
solution as function of $\gamma$ and $\bar{\alpha}_{s}$. Then the
$\omega$-integral singles out the residue at such pole

$\mathrm{Res}_{\omega=\omega^{\mathrm{eff}}}[\omega-\bar{\alpha}_{s}X(\omega,\gamma)]^{-1}=[1-\bar{\alpha}_{s}\partial_{\omega}X(\omega^{\mathrm{eff}},\gamma)]^{-1}\;,$
(32)

yielding

$\displaystyle\sigma^{jk}(s,Q_{1},Q_{2})$ $\displaystyle=\frac{1}{2\pi
Q_{1}Q_{2}}\int\frac{{\rm d}\gamma}{2\pi{\rm
i}}\left(\frac{s}{s_{0}}\right)^{\bar{\alpha}_{s}X(\omega^{\mathrm{eff}},\gamma)}\left(\frac{Q_{1}^{2}}{Q_{2}^{2}}\right)^{\gamma-\mbox{\small$\frac{1}{2}$}}\frac{\Phi^{(j)}(\omega^{\mathrm{eff}},\gamma)\,\Phi^{(k)}(\omega^{\mathrm{eff}},1-\gamma)}{1-\bar{\alpha}_{s}\partial_{\omega}X(\omega^{\mathrm{eff}},\gamma)}+\cdots\;,$
(33)

where the dots indicate terms suppressed by powers of $s$. Therefore, for any
choice of energy scale,

$\displaystyle\chi(\gamma)$ $\displaystyle=X(\omega^{\mathrm{eff}},\gamma)$
(34a) $\displaystyle\phi^{(j)}(\gamma)\phi^{(k)}(1-\gamma)$
$\displaystyle=\frac{\Phi^{(j)}(\omega^{\mathrm{eff}},\gamma)\Phi^{(k)}(\omega^{\mathrm{eff}},1-\gamma)}{1-\bar{\alpha}_{s}\partial_{\omega}X(\omega^{\mathrm{eff}},\gamma)}\;.$
(34b)

By expanding eq. 34 in $\bar{\alpha}_{s}$ as in eqs. 26 and 27, we obtain the
following equations relating the RGI eigenvalue and impact factors (and their
derivatives) at $\omega=0$ with the BFKL ones:

$\displaystyle\omega^{\mathrm{eff}}$
$\displaystyle=\bar{\alpha}_{s}\chi_{0}(\gamma)+\mathcal{O}(\bar{\alpha}_{s}^{2})$
(35) $\displaystyle\chi_{0}(\gamma)$ $\displaystyle=X_{0}(0,\gamma)$ (36)
$\displaystyle\chi_{1}(\gamma)$
$\displaystyle=X_{1}(0,\gamma)+\chi_{0}(\gamma)\partial_{\omega}X_{0}(0,\gamma)$
(37) $\displaystyle\phi_{0}^{(j)}(\gamma)\phi_{0}^{(k)}(1-\gamma)$
$\displaystyle=\Phi_{0}^{(j)}(0,\gamma)\Phi_{0}^{(k)}(0,1-\gamma)$ (38)
$\displaystyle\phi_{0}^{(j)}(\gamma)\phi_{1}^{(k)}(1-\gamma)+\phi_{1}^{(j)}(\gamma)\phi_{0}^{(k)}(1-\gamma)$
$\displaystyle=\Phi_{0}^{(j)}(0,\gamma)\left[\Phi_{1}^{(k)}(0,1-\gamma)+\chi_{0}(1-\gamma)\partial_{\omega}\Phi_{0}^{(k)}(0,1-\gamma)\right]$
$\displaystyle\quad+\left[\Phi_{1}^{(j)}(0,\gamma)+\chi_{0}(\gamma)\partial_{\omega}\Phi_{0}^{(j)}(0,\gamma)\right]\Phi_{0}^{(k)}(0,1-\gamma)$
$\displaystyle\quad+\Phi_{0}^{(j)}(0,\gamma)\Phi_{0}^{(k)}(0,1-\gamma)\partial_{\omega}X_{0}(0,\gamma)\;.$
(39)

These equations form the first consistency condition of the RGI framework with
the BFKL framework. Equations 35, 36 and 37 are well known from the first
studies on RGI BFKL [19]. Equation 38 implies that
$\phi_{0}^{(j)}(\gamma)=\Phi_{0}^{(j)}(0,\gamma)$ for any polarization $j$. In
particular, eqs. 25a and 28a imply the following normalization for the LO
impact factors, compared to those of refs. [3, 57, 61]:

$\displaystyle\phi_{0}^{(j)}(\gamma)$
$\displaystyle=\frac{2\pi\sqrt{2(N_{c}^{2}-1)}\alpha}{N_{f}}\left(\sum_{q}e_{q}^{2}\right)\;\gamma\,h_{j}(\gamma)$
$\displaystyle(h_{T}=\frac{h_{2}}{\gamma}-h_{L})$ ref [3] Catani et al. (40)
$\displaystyle=\frac{T_{R}\sqrt{2(N_{c}^{2}-1)}}{2}F_{j}(\nu)$
$\displaystyle(\gamma=\mbox{\small$\frac{1}{2}$}+{\rm i}\nu)$ ref [57] Ivanov
et al. (41)
$\displaystyle=\frac{T_{R}\sqrt{2(N_{c}^{2}-1)}}{\pi}\left(\sum_{q}e_{q}^{2}\right)\;S_{j}(N=0,\gamma)$
$\displaystyle(N=\omega)$
$\displaystyle\text{ref~{}\cite[cite]{[\@@bibref{}{Bialas:2001ks}{}{}]} Bia\l
as et al.}\;,$ (42)

where $\sum_{q}$ denotes the sum over quark flavours and $e_{q}$ is the
electric charge of quark $q$ in units of the positron charge. In those papers,
the expressions are often given for $N_{c}=3$ and $T_{R}=1/2$, but it is
better to keep track of such colour structure for the comparison with the
subsequent collinear analysis. Explicitly, the LO impact factors read

$\displaystyle\phi_{0}^{(T)}(\gamma)$
$\displaystyle=\alpha\alpha_{\mathrm{s}}\Big{(}\sum_{q}e_{q}^{2}\Big{)}T_{R}\sqrt{2(N_{c}^{2}-1)}\,\frac{\pi}{2}\;\frac{(1+\gamma)(2-\gamma)\Gamma^{2}(\gamma)\Gamma^{2}(1-\gamma)}{(3-2\gamma)\Gamma(3/2+\gamma)\Gamma(3/2-\gamma)}\,,$
(43a) $\displaystyle\phi_{0}^{(L)}(\gamma)$
$\displaystyle=\alpha\alpha_{\mathrm{s}}\Big{(}\sum_{q}e_{q}^{2}\Big{)}T_{R}\sqrt{2(N_{c}^{2}-1)}\,\pi\;\frac{\Gamma(1+\gamma)\Gamma(2-\gamma)\Gamma(\gamma)\Gamma(1-\gamma)}{(3-2\gamma)\Gamma(3/2+\gamma)\Gamma(3/2-\gamma)}\;,$
(43b)

where $\alpha$ is the electromagnetic coupling. It is apparent from the
$\Gamma$ functions in the numerators that both LO impact factors have poles at
$\gamma=0$ and $\gamma=1$, similarly to the eigenvalue functions $\chi_{0}$
and $\chi_{1}$ in eqs. 10 and 15. This is due to QCD dynamics which, in the
collinear limit $Q_{1}\gg Q_{2}$ ($Q_{1}\ll Q_{2}$), generates logarithmic
term $\sim\log^{n}(Q_{1}^{2}/Q_{2}^{2}):n\geq 0$, corresponding to poles of
order $n+1$ at $\gamma=0$ ($\gamma=1$) in Mellin space. More precisely, the
RGI impact factors and eigenvalue function have poles whose order increases as
the perturbative order:

$\displaystyle\Phi^{(T)}_{n}(\omega,\gamma;1)$
$\displaystyle\sim\frac{1}{\gamma^{2+n}}\;,$
$\displaystyle\Phi^{(L)}_{n}(\omega,\gamma;1)$
$\displaystyle\sim\frac{1}{\gamma^{1+n}}\;,$ $\displaystyle
X_{n}(\omega,\gamma;1)$ $\displaystyle\sim\frac{1}{\gamma^{1+n}}\;,$ (44)

as will be evident from the collinear analysis in the next section. On the
other hand, the corresponding BFKL quantities at symmetric scale
$s_{0}=Q_{1}Q_{2}$ have poles that increase twice as much:

$\displaystyle\phi^{(T)}_{n}(\gamma;0)$
$\displaystyle\sim\frac{1}{\gamma^{2+2n}}\;,$
$\displaystyle\phi^{(L)}_{n}(\gamma;0)$
$\displaystyle\sim\frac{1}{\gamma^{1+2n}}\;,$
$\displaystyle\chi_{n}(\gamma;0)$
$\displaystyle\sim\frac{1}{\gamma^{1+2n}}\;.$ (45)

This has been already observed at leading and next-to-leading order for the
eigenvalue functions $\chi_{0}$ and $\chi_{1}$ in sec. 2 and in particular in
eq. 15. The collinear poles of the NLO impact factors can be derived from the
expressions computed in [57]:

$\displaystyle\frac{\phi^{(T)}_{1}(\gamma;0)}{\phi^{(T)}_{0}(\gamma)}$
$\displaystyle=\frac{\chi_{0}(\gamma)}{2}\ln{\frac{s_{0}}{Q^{2}}}+\bar{b}\ln{\frac{\mu_{R}^{2}}{Q^{2}}}$
$\displaystyle+\frac{3C_{F}}{4N_{C}}-\frac{5}{9}\frac{T_{R}N_{f}}{N_{C}}+\frac{\pi^{2}}{4}+\frac{85}{36}-\frac{\pi^{2}}{\sin^{2}(\pi\gamma)}-\frac{1}{\gamma\left(\gamma-1\right)}+\frac{3\chi_{0}(\gamma)}{2\left(\gamma+1\right)\left(2-\gamma\right)}$
$\displaystyle+\frac{1}{4(1-\gamma)}-\frac{1}{4\gamma}-\frac{7}{36(1+\gamma)}+\frac{5}{3(1+\gamma)^{2}}-\frac{25}{36(\gamma-2)}$
$\displaystyle+\frac{1}{2}\chi_{0}(\gamma)\left[\psi\left(1-\gamma\right)+2\psi\left(2-\gamma\right)-2\psi\left(4-2\gamma\right)-\psi\left(2+\gamma\right)\right],$
(46) $\displaystyle\frac{\phi^{(L)}_{1}(\gamma;0)}{\phi^{(L)}_{0}(\gamma)}$
$\displaystyle=\frac{\chi_{0}(\gamma)}{2}\ln{\frac{s_{0}}{Q^{2}}}+\bar{b}\ln{\frac{\mu_{R}^{2}}{Q^{2}}}$
$\displaystyle+\frac{3C_{F}}{4N_{C}}-\frac{5}{9}\frac{T_{R}N_{f}}{N_{C}}+\frac{\pi^{2}}{4}+\frac{85}{36}-\frac{\pi^{2}}{\sin^{2}(\pi\gamma)}-\frac{1-4\gamma}{2\gamma^{2}(\gamma^{2}-1)}+\frac{1}{1-\gamma^{2}}\chi_{0}(\gamma)$
$\displaystyle+\frac{1}{2}\chi_{0}(\gamma)\left[\psi\left(1-\gamma\right)+2\psi\left(2-\gamma\right)-2\psi\left(4-2\gamma\right)-\psi\left(2+\gamma\right)\right],$
(47)

where $\bar{b}$ is defined in eq. 8 and $\mu_{R}$ is the renormalization
scale. The origin of the higher order poles in impact factors is the same as
that of poles in the BFKL kernel, as explained in sec. 2. Such spurious poles
can be resummed using the $\omega$-shift of poles suggested by the RGI
procedure.

Equation 39 will be used to determine the RGI impact factors at NLO. For this
purpose, we need to know the $\omega$-dependence of the LO eigenvalue and
impact factors. All that will be the subject of the next section.

##  4 RGI impact factor for transverse photons

###  4.1 Lowest order $\boldsymbol{TT}$ cross section in the collinear limits

Further information for the $\gamma^{*}\gamma^{*}$ cross section, somehow
complementary to the multi-Regge kinematics, can be inferred by analyzing the
collinear limit, i.e., by considering two photons with very different
virtualities, say $Q_{1}\gg Q_{2}$. This situation is well described by
effective ladder diagrams, like the one depicted in fig. 2, where the
intermediate propagators are strongly ordered in virtuality (decreasing from
left to right). At each QCD vertex, the strong coupling is evaluated at a
scale given by the largest virtuality of the connected propagators, while a
splitting function $P_{ba}(z_{b}/z_{a})$ describes the fragmentation of the
parent parton $a$ (to the right) into a child parton $b$ (to the left) and an
emitted on-shell parton (vertical line). The integrals over the ordered
longitudinal momentum fractions are convolutions, which can be diagonalized by
a Mellin transform in the Bjorken variable
$1/x_{Bj}=s/Q_{1}^{2}=s/s_{0}(p=1)$:

$\displaystyle\sigma^{(jk)}(s,Q_{1},Q_{2})$ $\displaystyle=\frac{1}{2\pi
Q_{1}Q_{2}}\int\frac{{\rm d}\omega}{2\pi{\rm
i}}\left(\frac{s}{Q_{1}^{1+p}Q_{2}^{1-p}}\right)^{\omega}\;\tilde{\sigma}^{(jk)}(\omega,Q_{1},Q_{2};p)\;.$
(48)

which is exactly the structure of eqs. 25a and 28a.

Figure 2: Diagramatics of collinear limit at lowest order in the BFKL
factorization formula.

The collinear integrand $\tilde{\sigma}^{(TT)}$ for two transverse photons at
$\mathcal{O}(\alpha^{2}\alpha_{\mathrm{s}}^{2})$ — corresponding to the four-
rungs LO BFKL diagram — is given by (cfr. appendix A)

$\displaystyle\tilde{\sigma}^{(TT)}(\omega,Q_{1},Q_{2};1)$
$\displaystyle=(2\pi)^{3}\alpha\Big{(}2\sum_{q\in A}e_{q}^{2}\Big{)}\times$
$\displaystyle\int_{Q_{2}^{2}}^{Q_{1}^{2}}\frac{{\rm
d}l_{1}^{2}}{l_{1}^{2}}\frac{\alpha_{\mathrm{s}}(l_{1}^{2})}{2\pi}P_{qg}(\omega)\int_{Q_{2}^{2}}^{l_{1}^{2}}\frac{{\rm
d}k^{2}}{k^{2}}\frac{\alpha_{\mathrm{s}}(k^{2})}{2\pi}P_{gq}(\omega)\int_{Q_{2}^{2}}^{k^{2}}\frac{{\rm
d}l_{2}^{2}}{l_{2}^{2}}\frac{\alpha}{2\pi}\Big{(}2\sum_{q\in
B}e_{q}^{2}\Big{)}P_{q\gamma}(\omega)\;.$ (49)

where $l_{1}$, $k$ and $l_{2}$ are the momenta of the $t$-channel quark, gluon
and quark respectively, as depicted from left to right in fig. 2, $A$ and $B$
denote the sets of active quarks of momenta $l_{1}$ and $l_{2}$ respectively,
while $P_{ab}(\omega)$ denote Mellin moments of the one-loop splitting
functions. The running coupling at scale $|k|$ is defined in terms of the
renormalized coupling $\alpha_{\mathrm{s}}$ at the renormalization scale
$\mu_{R}$:

$\displaystyle\alpha_{\mathrm{s}}(k^{2})$
$\displaystyle\;\raisebox{0.27127pt}{$:$}=\frac{\alpha_{\mathrm{s}}(\mu_{R}^{2})}{1+\alpha_{\mathrm{s}}(\mu_{R}^{2})b\ln\frac{k^{2}}{\mu_{R}^{2}}}\simeq\alpha_{\mathrm{s}}(\mu_{R}^{2})\left(1-\alpha_{\mathrm{s}}(\mu_{R}^{2})b\ln\frac{k^{2}}{\mu_{R}^{2}}+\cdots\right)\,,$
(50)

where $b$ is defined in eq. 8. Substituting the above expansion for
$\alpha_{\mathrm{s}}(l_{1}^{2})$ and $\alpha_{\mathrm{s}}(k^{2})$ in eq. 49
and switching to logarithmic variables
$L_{i}\,\raisebox{0.37pt}{:}=\ln\frac{Q_{i}^{2}}{\mu_{R}^{2}}$,
$\lambda_{i}\,\raisebox{0.37pt}{:}=\ln\frac{l_{i}^{2}}{\mu_{R}^{2}}$,
$\lambda_{k}\,\raisebox{0.37pt}{:}=\ln\frac{k^{2}}{\mu_{R}^{2}}$, we obtain

$\displaystyle\tilde{\sigma}^{(TT)}$
$\displaystyle=(2\pi)^{3}\frac{\alpha^{2}}{2\pi}\left(\frac{\alpha_{\mathrm{s}}(\mu_{R}^{2})}{2\pi}\right)^{2}\Big{(}2\sum_{q\in
A}e_{q}^{2}\Big{)}\Big{(}2\sum_{q\in
B}e_{q}^{2}\Big{)}P_{qg}(\omega)P_{gq}(\omega)P_{q\gamma}(\omega)$
$\displaystyle\quad\times\int_{L_{2}}^{L_{1}}{\rm
d}\lambda_{1}\int_{L_{2}}^{\lambda_{1}}{\rm
d}\lambda_{k}\int_{L_{2}}^{\lambda_{k}}{\rm
d}\lambda_{2}\;\left[1-\alpha_{\mathrm{s}}(\mu_{R}^{2})b(\lambda_{1}+\lambda_{k})+\mathcal{O}(\alpha_{\mathrm{s}}^{2})\right]\;.$
(51)

The nested integral in the second line of eq. 51 yields

$\displaystyle\iiint$
$\displaystyle=\frac{(L_{1}-L_{2})^{3}}{3!}\left[1-\alpha_{\mathrm{s}}(\mu_{R}^{2})b(L_{1}+L_{2})\right]-\alpha_{\mathrm{s}}(\mu_{R}^{2})b\frac{(L_{1}-L_{2})^{4}}{4!}+\mathcal{O}(\alpha_{\mathrm{s}}^{2})\;.$

By including the overall factor $\alpha_{\mathrm{s}}^{2}(\mu_{R}^{2})$ written
in the first line of eq. 51 and noting that

$\displaystyle\alpha_{\mathrm{s}}^{2}(\mu_{R}^{2})\left[1-\alpha_{\mathrm{s}}(\mu_{R}^{2})b(L_{1}+L_{2})\right]=\alpha_{\mathrm{s}}^{2}(Q_{1}Q_{2})+\mathcal{O}(\alpha_{\mathrm{s}}^{4})\;,$

we get

$\displaystyle\alpha_{\mathrm{s}}^{2}(\mu_{R}^{2})\iiint$
$\displaystyle\simeq\alpha_{\mathrm{s}}^{2}(Q_{1}Q_{2})\left[\frac{(L_{1}-L_{2})^{3}}{3!}-\alpha_{\mathrm{s}}b\frac{(L_{1}-L_{2})^{4}}{4!}+\mathcal{O}(\alpha_{\mathrm{s}}^{4})\right]\;,$
(52)
$\displaystyle\simeq\alpha_{\mathrm{s}}^{2}(Q_{1}Q_{2})\left[\frac{1}{3!}\log^{3}\frac{Q_{1}^{2}}{Q_{2}^{2}}-\alpha_{\mathrm{s}}b\frac{1}{4!}\log^{4}\frac{Q_{1}^{2}}{Q_{2}^{2}}+\mathcal{O}(\alpha_{\mathrm{s}}^{4})\right]\;.$
(53)

By Mellin transforming in $Q_{1}^{2}/Q_{2}^{2}$ the terms in square
brackets333Recall that $(L_{1}-L_{2})^{n}=\ln^{n}\frac{Q_{1}^{2}}{Q_{2}^{2}}$
becomes $\frac{n!}{\gamma^{n+1}}$ under Mellin transform. (while keeping the
strong couplings as factors outside the Mellin transform), we obtain the
corresponding expression in $\gamma$-space:

$\displaystyle\alpha_{\mathrm{s}}^{2}(Q_{1}Q_{2})\,\frac{1}{\gamma^{4}}\left[1-\alpha_{\mathrm{s}}b\frac{1}{\gamma}\right]\;.$
(54)

The first term $\mathcal{O}(\alpha_{\mathrm{s}}^{2}/\gamma^{4})$ could have
been obtained by using a fixed coupling constant in eq. 49. The introduction
of the running coupling is responsible for the second ($b$-dependent) term
$\mathcal{O}(\alpha_{\mathrm{s}}^{3}/\gamma^{5})$, which will be important in
the analysis of the NLO impact factors.444If one chooses a different scale for
the running coupling, the coefficient of the $b$-dependent term in eq. 54
would change accordingly.

Finally, by restoring all the factors of eq. 51, we obtain the Mellin
transform of $\tilde{\sigma}^{(TT)}$ of eq. 49 (with respect to the variable
$Q_{1}^{2}/Q_{2}^{2}$), expanded at order $\alpha_{\mathrm{s}}^{2}$, which is
nothing but the integrand of the RGI factorization formula (28a) in the
collinear limit $\gamma\to 0$:

$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(TT)}(\omega,\gamma;1)\big{|}^{\mathrm{coll}}$
$\displaystyle=\Phi^{(T)}_{0}\,\mathcal{G}_{0}\,\Phi^{(T)}_{0}\big{|}^{\mathrm{coll}}_{p=1}$
$\displaystyle=(2\pi)^{3}\alpha\Big{(}2\sum_{q\in
A}e_{q}^{2}\Big{)}\frac{1}{\gamma}\;\cdot\;\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{qg}(\omega)}{\gamma}\;\cdot\;\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{gq}(\omega)}{\gamma}\;\cdot\;\frac{\alpha}{2\pi}\Big{(}2\sum_{q\in
B}e_{q}^{2}\Big{)}\frac{P_{q\gamma}(\omega)}{\gamma}\;.$ (55)

Some remarks are in order:

  * (i)

Since the collinear analysis of the cross section based on the DGLAP chain
singles out the leading logarithmic behaviour in the ratio $Q_{1}/Q_{2}$, eq.
55 provides just the leading $\gamma$-pole structure of the RGI integrand in
the neighborhood of $\gamma=0$.

  * (ii)

Such pole structure correspond to $p=1$, i.e., energy scale $s_{0}=Q_{1}^{2}$.
Adopting the symmetric energy scale $s_{0}=Q_{1}Q_{2}$ ($p=0$), according to
eq. 29 the pole at $\gamma=0$ is shifted at $\gamma=-\omega/2$, while keeping
the same coefficient.

  * (iii)

In the opposite (anti)collinear limit $Q_{1}\ll Q_{2}$, one obtains the same
result of eq. 55, provided one replaces $\gamma\to 1-\gamma$ and $p\to-1$,
i.e., $s_{0}=Q_{2}^{2}$. At symmetric energy scale $s_{0}=Q_{1}Q_{2}$, the
pole at $\gamma=1$ is shifted at $\gamma=1+\omega/2$. If $p=1$, i.e.,
$s_{0}=Q_{1}^{2}$, the pole at $\gamma=1$ is shifted at $\gamma=1+\omega$.

  * (iv)

The two sums with electric charges are over quark flavours
($q\in\\{u,d,\dots\\}$) and a factor of 2 in front of each sum takes into
account quark+antiquark contributions.

Therefore, with energy scale $s_{0}=Q_{1}^{2}$ ($p=1$) and including both
collinear and anticollinear contributions, the pole structure at LO of the RGI
improved cross section reads

$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(TT)}(\omega,\gamma;1)\big{|}^{2\times\mathrm{coll}}$
$\displaystyle=(2\pi)^{3}\alpha\Big{(}2\sum_{q}e_{q}^{2}\Big{)}\frac{1}{\gamma}\cdot\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{qg}(\omega)}{\gamma}\cdot\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{gq}(\omega)}{\gamma}\cdot\frac{\alpha}{2\pi}\Big{(}2\sum_{q}e_{q}^{2}\Big{)}\frac{P_{q\gamma}(\omega)}{\gamma}$
$\displaystyle\qquad+\Big{(}\gamma\to 1+\omega-\gamma\Big{)}\;.$ (56)

In order to classify terms according to the small-$x$ expansion, and also to
compare with the known expressions (eqs. 40, 41 and 42) of impact factors at
leading order, it is convenient to expand the Mellin transforms of the
splitting functions as powers series in $\omega$:

$\displaystyle P_{qq}(\omega)$
$\displaystyle=C_{F}\left(\frac{5}{4}-\frac{\pi^{2}}{3}\right)\omega+\mathcal{O}(\omega^{2})$
(57) $\displaystyle P_{gq}(\omega)$
$\displaystyle=\frac{2C_{F}}{\omega}\left[1+\omega A_{gq}(\omega)\right]$
$\displaystyle A_{gq}(0)$ $\displaystyle=-\frac{3}{4}$ (58) $\displaystyle
P_{qg}(\omega)$ $\displaystyle=\frac{2}{3}T_{R}\left[1+\omega
A_{qg}(\omega)\right]$ $\displaystyle A_{qg}(0)$
$\displaystyle=-\frac{13}{12}$ (59) $\displaystyle P_{gg}(\omega)$
$\displaystyle=\frac{2C_{A}}{\omega}\left[1+\omega A_{gg}(\omega)\right]$
$\displaystyle A_{gg}(0)$
$\displaystyle=-\frac{11}{6}+\bar{b}\;,\quad\bar{b}=\frac{11}{12}-\frac{T_{R}N_{f}}{3N_{c}}$
(60) $\displaystyle P_{q\gamma}(\omega)$
$\displaystyle=\frac{N_{c}}{T_{R}}P_{qg}(\omega)\;.$ (61)

Note that $P_{qg}$ refers to the process where a gluon produces a single quark
emitting an antiquark, or viceversa. Therefore, a gluon splitting into a quark
or antiquark of a given flavour requires a factor of two. If the (anti-) quark
at some point splits into a gluon, the sum over flavours yields an additional
factor $N_{f}$. On the contrary, if the (anti-)quark couples to a photon, the
sum over flavours yields a factor $\sum_{q}e_{q}^{2}$. Equation 61 stems from
the fact that, if a gluon of colour $c$ splits into a quark-antiquark pair
with colours $a,b$, then the squared matrix element contains
$\sum_{ab}t^{c}_{ab}t^{d*}_{ab}=\mathrm{tr}(t^{c}t^{d})=T_{R}\delta_{cd}$,
while if a photon splits into a quark-antiquark pair, the sum over colours is
$\sum_{ab}\delta_{ab}\delta_{ab}=\sum_{a}\delta_{aa}=N_{c}$.

By taking into account eqs. 58, 59, 60 and 61 and noting that
$C_{F}N_{c}=(N_{c}^{2}-1)T_{R}$, we can rewrite eq. 56 as

$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(TT)}(\omega,\gamma;1)$
$\displaystyle=\left[\alpha\alpha_{\mathrm{s}}\Big{(}\sum_{q}e_{q}^{2}\Big{)}2P_{qg}(\omega)\sqrt{2(N_{c}^{2}-1)}\left(\frac{1}{\gamma^{2}}+\frac{1}{(1+\omega-\gamma)^{2}}\right)\right]^{2}$
$\displaystyle\qquad\times\frac{1}{\omega}\,\Big{(}1+\omega
A_{gq}(\omega)\Big{)}\;+\mathcal{O}(\gamma^{-3})+\mathcal{O}((1+\omega-\gamma)^{-3})\;,$
(62)

where only the quartic poles in $\gamma$ are of our concern.

The term in square brackets is exactly the collinear limit of the
$\omega$-dependent LO impact factor derived from eq. 42; in other words, it
represents the double poles of Białas, Navelet and Peschanski (BNP) impact
factor [61] for a transverse photon with their full $\omega$-dependent
coefficient:

$\displaystyle\Phi_{\mathrm{BNP}}^{(T)}(\omega,\gamma)$
$\displaystyle=\alpha\alpha_{\mathrm{s}}\Big{(}\sum_{q}e_{q}^{2}\Big{)}T_{R}\sqrt{2(N_{c}^{2}-1)}\;\frac{\pi\Gamma(\gamma+\delta)\Gamma(\gamma)}{\Gamma(\omega)}\frac{1}{\left(\delta^{2}-1\right)\left(\delta^{2}-4\right)}\bigg{\\{}\frac{\psi(\gamma+\delta)-\psi(\gamma)}{\delta}\times$
$\displaystyle\qquad\frac{\omega^{2}\left[3(\omega+1)^{2}+9\right]-2\omega\left(\delta^{2}-1\right)+\left(\delta^{2}-1\right)\left(\delta^{2}-9\right)}{4\omega}-\frac{3(\omega+1)^{2}+3+\left(\delta^{2}-1\right)}{2}\bigg{\\}}$
(63) $\displaystyle=C_{0}\left[\frac{1+\omega
A_{qg}}{\gamma^{2}}+\frac{D(\omega)}{\gamma}+\mathcal{O}(\gamma^{0})\right]+(\gamma\to
1+\omega-\gamma)$ (64) $\displaystyle C_{0}$
$\displaystyle=\alpha\alpha_{\mathrm{s}}\Big{(}\sum_{q}e_{q}^{2}\Big{)}\frac{4}{3}T_{R}\sqrt{2(N_{c}^{2}-1)}\;,\qquad
D(\omega)=\frac{7}{6}+\mathcal{O}(\omega)\;,\quad\delta\equiv\omega+1-2\gamma\;.$
(65)

The factor $1/\omega$ — stemming from $P_{gq}(\omega)$ — in the second line of
eq. 62 yields the GGF (28b) at lowest order ($\alpha_{\mathrm{s}}\to 0$),
while the finite part $\propto A_{gq}$ provides a NLL correction, to be
reconsidered later.555In the $\omega\to 0$ limit, eq. 62 reduces to the
product of the LL GGF $1/\omega$ with the LO impact factors
$\phi_{0}^{(T)}(\gamma)\phi_{0}^{(T)}(1-\gamma)$ of Catani et al. as in eq. 40
— restricting $\phi_{0}^{(T)}$ to the double poles in $\gamma$. In conclusion,

$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(TT)}(\omega,\gamma;1)=C_{0}^{2}\frac{1}{\omega}\left[\frac{(1+\omega
A_{qg})^{2}\,(1+\omega
A_{gq})}{\gamma^{4}}+\mathcal{O}(\gamma^{-3})\right]+(\gamma\to
1+\omega-\gamma)\;.$ (66)

###  4.2 LO RGI transverse impact factor

Our first task now is to determine (a possible form of) the LO RGI transverse
impact factor. If we ignore for a moment the factor
$P_{gq}(\omega)/P_{gq}(0)=(1+\omega A_{gq})$, eq. 62 tells us that the LL $TT$
cross section in the collinear limit is given by the collinear limit of the LO
transverse impact factors of BNP [61], times the LL GGF. One could then claim
that the full LL RGI $TT$ cross section is given by the product of the
complete BNP transverse impact factors with the LL GGF, and conclude that the
LO RGI transverse impact factor is just the one provided in ref. [61], i.e.,
section 4.1. More properly, such an impact factor is a perfect candidate,
since it reproduces the LO BFKL cross section in the high-energy limit
($\omega\to 0$) and also the LO DGLAP cross section in the collinear limits
$\gamma\to 0$ and $\gamma\to 1+\omega$.

However, the collinear limit (62) of the cross section has the additional
$\omega$-dependent factor $P_{gq}(\omega)/P_{gq}(0)$. In order to take it into
account, we must modify either the BNP impact factors or the GGF. Since this
factor stems from the quark-gluon interaction, while the LL GGF is determined
by pure gluon dynamics, it is natural to associate such a factor to the impact
factors. The modification of impact factors is ambiguous, since the collinear
analysis just provides constraints for the leading twist poles, i.e., for
$\gamma\simeq 0$ and $\gamma\simeq 1+\omega$, of their products. Let’s
parametrize the leading-twist poles of $\Phi_{0}^{(T)}$ as follows:666We use
the convention of parametrizing coefficients of the collinear and anti-
collinear poles with the same letter, but with a bar over the coefficients of
the anti-collinear poles.

$\Phi_{0}^{(T)}(\omega,\gamma;1)=C_{0}\left[\frac{1+\omega
B(\omega)}{\gamma^{2}}+\frac{D(\omega)}{\gamma}+\frac{1+\omega\bar{B}(\omega)}{(1+\omega-\gamma)^{2}}+\frac{\bar{D}(\omega)}{1+\omega-\gamma}\right]+r(\omega,\gamma)\;,$
(67)

where $r(\omega,\gamma)$ has no leading-twist poles. We then have, for
$\gamma\simeq 0$,

$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(TT)}(\omega,\gamma;1)$
$\displaystyle=\Phi_{0}^{(T)}(\omega,\gamma;1)\;\frac{1}{\omega}\;\Phi_{0}^{(T)}(\omega,1+\omega-\gamma;1)$
$\displaystyle=C_{0}^{2}\frac{1}{\omega}\left[\frac{(1+\omega
B)(1+\omega\bar{B})}{\gamma^{4}}+\mathcal{O}(\gamma^{-3})\right]+(\gamma\to
1+\omega-\gamma)\;.$ (68)

Comparing the above expression with eq. 66 we get

$\displaystyle(1+\omega B)(1+\omega\bar{B})$ $\displaystyle=(1+\omega
A_{qg})^{2}(1+\omega A_{gq})$ (69a) $\displaystyle\Longrightarrow\quad
B+\bar{B}$ $\displaystyle=2A_{qg}+A_{gq}+\mathcal{O}(\omega)\;.$ (69b)

In the following, we often neglect the subleading terms $\mathcal{O}(\omega)$
in eq. 69b.

On the contrary, the coefficients $D(\omega)$ and $\bar{D}(\omega)$ of the
simple poles are out of reach of the present LO collinear analysis, but their
value at $\omega=0$ can be determined from the explicit expression of eq. 43a
in ref. [3]: $D(0)=\bar{D}(0)=7/6$ [cfr. eq. 65]. The simplest and more
natural choice for us is to adopt $D(\omega)=\bar{D}(\omega)$ as in the impact
factor $\Phi_{\mathrm{BNP}}^{(T)}$ of section 4.1.

According to the constraints previously derived, we present some possible
choices of the transverse LO RGI impact factor, whose differences have to be
considered a resummation-scheme ambiguity: 777Other schemes can be considered,
see sec. 4.3.

$\displaystyle\Phi_{0}^{(T)}(\omega,\gamma;1)$
$\displaystyle=\Phi_{\mathrm{BNP}}^{(T)}(\omega,\gamma)\left[1+\frac{\omega}{2}A_{gq}(\omega)\right]$
$\displaystyle(\text{scheme I})$ (70a)
$\displaystyle\Phi_{0}^{(T)}(\omega,\gamma;1)$
$\displaystyle=\Phi_{\mathrm{BNP}}^{(T)}(\omega,\gamma)+C_{0}\frac{\omega}{2}A_{gq}(\omega)\left[\frac{1}{\gamma^{2}}+\frac{1}{(1+\omega-\gamma)^{2}}\right]$
$\displaystyle(\text{scheme II})$ (70b)
$\displaystyle\Phi_{0}^{(T)}(\omega,\gamma;1)$
$\displaystyle=\Phi_{\mathrm{BNP}}^{(T)}(\omega,\gamma)+C_{0}\,\omega
A_{gq}(\omega)\frac{1+\omega A_{qg}}{(1+\omega-\gamma)^{2}}$
$\displaystyle(\text{scheme III})\;.$ (70c)

Scheme I is an overall renormalization of the impact factor. Scheme II just
modifies the coefficient of the (leading-twist) double poles. Scheme III is
motivated by the fact that the $P_{gq}$ vertex is attached to the impact
factor to the right, thus providing a $1/\gamma$ pole only to
$\Phi_{0}(\omega,1-\gamma)$. Note that schemes I and II preserve the
$\gamma\leftrightarrow 1-\gamma$ symmetry of the impact factor, while scheme
III does not. In particular $B=\bar{B}=A_{qg}+A_{gq}/2$ in schemes I and II,
while $B=A_{qg}$, $\bar{B}=A_{qg}+A_{gq}+\omega A_{qg}A_{gq}$ in scheme III
(which fulfills exactly eq. 69a).

###  4.3 NLO $\boldsymbol{TT}$ cross section in the collinear limit

Our next task is to determine the transverse impact factors at NLO.
Specifically, we want to determine a function $\Phi^{(T)}_{1}(\omega,\gamma)$
such that

  * •

the RGI cross section (28a) agrees with the NLL BFKL one (25a);

  * •

the same RGI cross section agrees with the DGLAP cross section in the
collinear limits $Q_{1}\gg Q_{2}$ and $Q_{1}\ll Q_{2}$ — in this case at order
$\alpha_{\mathrm{s}}^{3}$.

The first condition has already been considered, and leads to the constraint
provided by eq. 39 at $\omega=0$.

Figure 3: Ladder diagrams relevant in the collinear limit at next-to-leading
order in the BFKL factorization formula. A third diagram, the left-right
symmetric of (b) with the gluon emitted from the quark line on the right, is
understood.

The second condition determines the structure of the collinear poles
($\gamma\simeq 0$ and $\gamma\simeq 1+\omega$) of the impact factors. We begin
by generalizing eq. 56 at $\mathcal{O}(\alpha_{\mathrm{s}}^{3})$. This amounts
to consider ladder diagrams with five splittings between the photons. The
vertices at the photon legs are necessarily QED couplings as before. The other
three vertices are of QCD type, and lead to the diagrams of fig. 3 (a third
diagram is understood, left-right symmetric with respect to (b) with the
vertical gluon emitted from the quark on the right). Such diagrams, together
with the running-coupling term of eq. 54, provide the integrand of the RGI
factorization formula at $\mathcal{O}(\alpha_{\mathrm{s}}^{3})$ in the
collinear limit

$\alpha_{\mathrm{s}}\,\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{1}^{(TT)}(\omega,\gamma;1)=\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(TT)}(\omega,\gamma;1)\left[\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{gg}}{\gamma}+2\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{qq}}{\gamma}-\frac{\alpha_{\mathrm{s}}b}{\gamma}+\mathcal{O}(\gamma^{0})\right]\;,$
(71)

where $\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(TT)}$ is the
LO (collinear) integrand defined in eq. 55.

The first term of eq. 71 stems from the diagram of fig. 3a which involves a
$P_{gg}$ splitting function. According to the analysis of CCSS [19], this
contribution can be entirely associated to the GGF. Actually, at fixed
$\alpha_{\mathrm{s}}$, the iteration of the $P_{gg}$ splitting function along
the gluon ladder provides a geometric series that is easily summed, yielding

$\displaystyle\omega\,\mathcal{G}(\omega,\gamma;1)^{\mathrm{coll}}$
$\displaystyle=\sum_{n=0}^{\infty}\left(\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{gg}(\omega)}{\gamma}\right)^{n}=\left[1-\frac{\bar{\alpha}_{s}}{\omega}\frac{1+\omega
A_{gg}}{\gamma}\right]^{-1}\;.$ (72)

Since $\omega\mathcal{G}=[1-\frac{\bar{\alpha}_{s}}{\omega}X]^{-1}$ [cfr. eq.
28b], we find $X_{0}^{\mathrm{coll}}(\omega,\gamma;1)=[1+\omega
A_{gg}]/\gamma$.

The second term in eq. 71 stems from the diagram of fig. 3b and its symmetric
counterpart — with the gluon emitted from the quark line on the right — which
involve $P_{qq}$ splitting functions. It is naturally associated to the impact
factors. Actually, since $P_{qq}$ vanishes at $\omega=0$ (cfr. eq. 57), these
contributions are suppressed by two powers of $\omega$ w.r.t. the diagram with
$P_{gg}$, and thus are next-to-next-to-leading in the BFKL hierarchy. However,
we keep them, in the spirit of being accurate in the leading DGLAP evolution.

The third term in eq. 71 is the running coupling ($b$-dependent) contribution
derived in eq. 54, and can be incorporated into either the impact factors or
the GGF, or both. In the following section, we face this situation more
systematically, and propose some possible choices of transverse NLO RGI impact
factor.

###  4.4 NLO RGI transverse impact factor

We now determine the NLO RGI impact factor from the NLO cross section derived
in the previous section. Let’s parametrize the collinear structure of RGI
impact factors and kernel as follows (from now on in this section we suppress
the superscript $(T)$ on the impact factors):

$\displaystyle\Phi(\omega,\gamma;1)$
$\displaystyle=\Phi_{0}(\omega,\gamma;1)\left[1+\bar{\alpha}_{s}\left(\frac{M(\omega)}{\gamma}+\frac{\bar{M}(\omega)}{1+\omega-\gamma}+r_{1}(\omega,\gamma)\right)+\mathcal{O}(\bar{\alpha}_{s}^{2})\right]$
(73) $\displaystyle X(\omega,\gamma;1)$ $\displaystyle=\frac{1+\omega
U(\omega)}{\gamma}+\mathcal{O}(\gamma^{0})+\bar{\alpha}_{s}\left(\frac{V(\omega)}{\gamma^{2}}+\mathcal{O}(\gamma^{-1})\right)+\mathcal{O}(\bar{\alpha}_{s}^{2})\;,$
(74)

where $r_{1}$ is regular at $\gamma=0,1+\omega$ and we have taken into account
that additional powers of $\bar{\alpha}_{s}$ involve additional powers of
$1/\gamma$ and $1/(1+\omega-\gamma)$.888We recall that $\Phi_{0}^{(T)}$ has
collinear poles of second order. Therefore, we expect an improved NL impact
factor with cubic poles. This has to be contrasted with the collinear
behaviour of the BFKL impact factor $\phi_{1}^{(T)}$ featuring cubic and even
quartic poles at $\gamma=0,1$, as it is apparent from eq. 78.  We find

$\displaystyle\alpha_{\mathrm{s}}\,\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{1}(\omega,\gamma;1)$
$\displaystyle=(\Phi_{0}\,\mathcal{G}_{1}\,\Phi_{0}+\Phi_{1}\,\mathcal{G}_{0}\,\Phi_{0}+\Phi_{0}\,\mathcal{G}_{0}\,\Phi_{1})-\Phi_{0}\,\mathcal{G}_{0}\,\Phi_{0}$
$\displaystyle=\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}(\omega,\gamma;1)\;\bar{\alpha}_{s}\left[\frac{\frac{1}{\omega}+M+\bar{M}+U}{\gamma}+\mathcal{O}(\gamma^{0})\right]\;.$
(75)

By comparing eq. 75 with eq. 71 yields

$M+\bar{M}+U=A_{gg}+2\bar{P}_{qq}-\bar{b}\;,$ (76)

having defined $\bar{b}$ in eq. 8, $\bar{P}_{ab}\equiv P_{ab}/(2C_{A})$ and
used $\bar{P}_{gg}=1/\omega+A_{gg}$ [cfr. eq. 60].

In order to check the compatibility of the collinear analysis with the known
BFKL results, and also to further constrain the RGI impact factors, let us
write down the collinear structure of the NLO BFKL transverse impact factor
[eqs. 43a and 46] and kernel [eqs. 6 and 7].999With running-coupling scale
$\mu_{R}^{2}=Q_{1}^{2}$, the double poles of $\chi_{1}(\gamma)$ are
$A_{1}/\gamma^{2}$ and $(A_{1}-\bar{b})/(1-\gamma)^{2}$. With symmetric scale
$\mu_{R}^{2}=Q_{1}Q_{2}$, the coefficients of both poles are equal to
$A_{1}-\bar{b}/2$. At symmetric scales $s_{0}=\mu_{R}^{2}=Q_{1}Q_{2}$, i.e.,
$p=0$:

$\displaystyle\phi_{0}(\gamma)$
$\displaystyle=\phi_{0}(1-\gamma)=C_{0}\left(\frac{1}{\gamma^{2}}+\frac{D(0)}{\gamma}+\frac{1}{(1-\gamma)^{2}}+\frac{\bar{D}(0)}{1-\gamma}+\cdots\right)$
(77) $\displaystyle\phi_{1}(\gamma)$
$\displaystyle=\phi_{0}(\gamma)\left(\frac{-1}{\gamma^{2}}+\frac{\eta}{\gamma}+\frac{-3/2}{(1-\gamma)^{2}}+\frac{\bar{\eta}}{1-\gamma}+\cdots\right)$
(78) $\displaystyle\chi(\gamma)$
$\displaystyle=\frac{1}{\gamma}+\bar{\alpha}_{s}\left(\frac{-1/2}{\gamma^{3}}+\frac{A_{1}(0)-\bar{b}/2}{\gamma^{2}}+\frac{H_{1}}{\gamma}+\cdots\right)$
(79) $\displaystyle\eta=-\frac{11}{6}\;,\quad\bar{\eta}=-\frac{7}{4}\;,\quad
H_{1}=-\frac{T_{R}N_{f}}{N_{c}}\left(\frac{5}{9}+\frac{13}{18N_{c}^{2}}\right)\;,$
(80)

where $C_{0}$, $D(0)$ and $\bar{D}(0)$ were already determined in eq. 65 and
$A_{1}$ was defined in eq. 14. Then, from eq. 37 we find

$A_{1}-b/2=U+V\;.$ (81)

By noting that $\phi_{0}(\gamma)=\phi_{0}(1-\gamma)=\Phi_{0}(0,\gamma;p)$ for
any $p$, eq. 39 can be simplified into

$\displaystyle\phi_{1}(\gamma)+\phi_{1}(1-\gamma)$
$\displaystyle=\Phi_{1}(0,\gamma)+\Phi_{1}(0,1-\gamma)$
$\displaystyle\quad+\chi_{0}(\gamma)[\partial_{\omega}\Phi_{0}(0,\gamma)+\partial_{\omega}\Phi_{0}(0,1-\gamma)]+\phi_{0}(\gamma)\partial_{\omega}X_{0}(0,\gamma)\;,$
(82)

where $\Phi$’s and $X$ must be considered here at $p=0$, i.e., by replacing
$\gamma\to\gamma+\omega/2$ in eqs. 73 and 74. From eq. 78 we can expand the
l.h.s. of eq. 82 around the collinear pole $\gamma=0$:

$\phi_{1}(\gamma)+\phi_{1}(1-\gamma)=\phi_{0}(\gamma)\;\bar{\alpha}_{s}\left[\frac{-5/2}{\gamma^{2}}+\frac{\eta+\bar{\eta}}{\gamma}+\mathcal{O}(\gamma^{0})\right]\;.$
(83)

By expanding the r.h.s. of eq. 82 using eqs. 73 and 74 — with the replacement
$\gamma\to\gamma+\omega/2$ — , the coefficient $-5/2$ of the quadratic pole
within square brackets in eq. 83 is correctly reproduced, while the
coefficients of the simple poles are equal if

$\eta+\bar{\eta}=B+\bar{B}+\mbox{\small$\frac{1}{2}$}D+\mbox{\small$\frac{1}{2}$}\bar{D}+M+\bar{M}+U\big{|}_{\omega=0}\;.$
(84)

This is indeed the case. In fact, by exploiting eq. 69 and eq. 76, we find

$\displaystyle
B+\bar{B}+\mbox{\small$\frac{1}{2}$}D+\mbox{\small$\frac{1}{2}$}\bar{D}+M+\bar{M}+U\big{|}_{\omega=0}$
$\displaystyle=2\bar{P}_{qq}+2A_{qg}+A_{gq}+A_{gg}+\mbox{\small$\frac{1}{2}$}D+\mbox{\small$\frac{1}{2}$}\bar{D}-\bar{b}\big{|}_{\omega=0}$
(85a) $\displaystyle=-\frac{43}{12}=\eta+\bar{\eta}\;,$ (85b)

thus proving the consistency of next-to-leading BFKL and leading-order DGLAP.

Of course, the constraints (81) and (85a) derived from eqs. 37 and 39
respectively, can be fulfilled in many ways. In table 1 we present some
choices that we prefer on physical grounds.

scheme name | $U$ | $V$ | $B+\bar{B}$ | $M+\bar{M}$
---|---|---|---|---
collA | $A_{gg}-\bar{b}$ | $\Delta A+\bar{b}/2$ | $2A_{qg}+A_{gq}$ | $2\bar{P}_{qq}$
collB | $A_{gg}$ | $\Delta A-\bar{b}/2$ | $2A_{qg}+A_{gq}$ | $2\bar{P}_{qq}-\bar{b}$
zVnB | $A_{1}-\bar{b}/2$ | $0$ | $2A_{qg}+A_{gq}$ | $2\bar{P}_{qq}-\Delta A-\bar{b}/2$
zVnM | $A_{1}-\bar{b}/2$ | $0$ | $2A_{qg}+A_{gq}-\Delta A-\bar{b}/2$ | $2\bar{P}_{qq}$
zVzM | $A_{1}-\bar{b}/2$ | $0$ | $2\bar{P}_{qq}+2A_{qg}+A_{gq}-\Delta A-\bar{b}/2$ | $0$
Table 1: Favourite scheme choices for defining the NLO RGI transverse impact
factor.

Schemes “collA” and “collB” are motivated by the collinear analysis that
suggests the value of $B+\bar{B}$ from eq. 69 and the values of $M+\bar{M}$
and $U$ from eqs. 76 and 72. In the former we assign the running-coupling term
$-\bar{b}$ to the kernel, in the latter to the impact factors. For
convenience, we have introduced

$\Delta
A=A_{1}(\omega)-A_{gg}(\omega)=\frac{C_{F}}{C_{A}}2N_{f}\bar{P}_{qg}(0)=\left(1-\frac{1}{N_{c}^{2}}\right)\frac{T_{R}N_{f}}{3N_{c}}\;.$
(86)

In the other three schemes “zV…” we set to zero the coefficient $V$ of the
double pole of $X_{1}$, following the spirit of the RG improvement to transfer
the most singular $\gamma$-poles of NL objects into regular
$\omega$-corrections of leading-order terms. In this way, we assign all the
dependence of the kernel on the gluon anomalous dimension and running coupling
$A_{1}-\bar{b}/2$ to the $\mathcal{O}(\omega)$-term of the leading eigenvalue
$X_{0}$. Scheme “zVnB” adopts the natural (i.e., collinearly motivated) choice
for the $B$’s coefficients; scheme “zVnM” adopts the natural choice for the
$M$’s coefficients; scheme “zVzM” sets to zero the coefficients $M$ of the
cubic poles of the NLO impact factors, thus assigning all the residual
dependence on the anomalous dimensions to the $\mathcal{O}(\omega)$-term of
the leading impact factor $\Phi_{0}$.

Actually, each of the schemes in table 1 can be implemented in many ways,
depending on how $B$, $\bar{B}$, $M$ and $\bar{M}$ are individually defined,
and also because the regular part of impact factors is fully constrained only
at $\omega=0$. Concerning the leading impact factor $\Phi_{0}^{(T)}$, we
propose the three sub-schemes of eq. 70, where $B=\bar{B}$ in the sub-schemes
I and II, while $\bar{B}=B+A_{gq}$ in sub-scheme III.

As for the leading eigenvalue function, we adopt the recipe proposed in ref.
[19]:

$X_{0}(\omega,\gamma;0)=2\psi(1)-\psi(\gamma+\frac{\omega}{2})-\psi(1-\gamma+\frac{\omega}{2})+\omega
U(\omega)\left(\frac{1}{\gamma+\omega/2}+\frac{1}{1-\gamma+\omega/2}\right)\;,$
(87)

where $U(\omega)$, according to table 1, depends on the scheme choice. Then,
according to eq. 37, the next-to-leading improved eigenvalue at $\omega=0$
reads

$X_{1}(0,\gamma)=\chi_{1}(\gamma)+\frac{1}{2}\chi_{0}(\gamma)\frac{\pi^{2}}{\sin^{2}\pi\gamma}-U(0)\left(\frac{1}{\gamma}+\frac{1}{1-\gamma}\right)\chi_{0}(\gamma)\;.$
(88)

The above expression is free of cubic poles, but still contains simple poles
and possibly double poles, depending on the scheme choice:

$X_{1}(0,\gamma)=\left(A_{1}-\frac{\bar{b}}{2}-U(0)\right)\frac{1}{\gamma^{2}}+\left(H_{1}+\frac{\pi^{2}}{6}-U(0)\right)\frac{1}{\gamma}+\cdots\;.$
(89)

According to the RGI method, we require the RGI eigenvalue function
$X_{1}(\omega,\gamma)$ to have poles at the expected $\omega$-shifted
positions. The final expression that we adopt is

$\displaystyle X_{1}(\omega,\gamma)=X_{1}(0,\gamma)$
$\displaystyle+\frac{A_{1}(\omega)-\frac{\bar{b}}{2}-U(\omega)}{(\gamma+\frac{\omega}{2})^{2}}-\frac{A_{1}(0)-\frac{\bar{b}}{2}-U(0)}{\gamma^{2}}+(\gamma\leftrightarrow
1-\gamma)+$
$\displaystyle+\left(H_{1}+\frac{\pi^{2}}{6}-U(0)\right)\left[X_{0}(\omega,\gamma)-\chi_{0}(\gamma)\right]\;.$
(90)

We can now exploit eq. 82 to constrain the NLO improved transverse impact
factor at $\omega=0$ and arbitrary $\gamma$. If we further require such impact
factor to be symmetric in $\gamma\to 1-\gamma$, we obtain101010We note that,
while $\phi_{0}(\gamma)$ is symmetric in $\gamma\to 1-\gamma$, the NLO impact
factor $\phi_{1}(\gamma)$ is not. Actually, since the latter has been derived
[57] from a cross section [67] which depends on the product
$\phi^{(T)}(\gamma)\phi^{(T)}(1-\gamma)$, it is not clear to us how $\phi_{1}$
has been unambiguously derived, without imposing further requirements.

$\displaystyle\Phi_{1}(0,\gamma)$
$\displaystyle=\frac{1}{2}\left[\Phi_{1}(0,\gamma)+\Phi_{1}(0,1-\gamma)\right]$
$\displaystyle=\frac{1}{2}\left[\phi_{1}(\gamma)+\phi_{1}(1-\gamma)-\phi_{0}(\gamma)\partial_{\omega}X_{0}(0,\gamma)-\chi_{0}(\gamma)\big{(}\partial_{\omega}\Phi_{0}(0,\gamma)+\partial_{\omega}\Phi_{0}(0,1-\gamma)\big{)}\right]\;.$
(91)

Its Laurent expansion around $\gamma=0$ reads

$\Phi_{1}(0,\gamma)=C_{0}\left[\frac{M(0)}{\gamma^{3}}+\frac{M_{2}}{\gamma^{2}}+\frac{M_{1}}{\gamma}+\mathcal{O}(\gamma^{0})\right]\;,$
(92)

where $M_{2}$ and $M_{1}$ depend on the scheme choice that defines the
$\omega$-dependence of $\Phi_{0}(\omega,\gamma)$ and $X_{0}(\omega,\gamma)$ in
eqs. 70 and 87.

We extend $\Phi_{1}$ at $\omega\neq 0$ by requiring the collinear poles to be
located at $\gamma=-\omega/2$ and $\gamma=1+\omega/2$ and with
$\omega$-dependent leading coefficients $M(\omega)$ and $\bar{M}(\omega)$ as
in eq. 73. This can be obtained in various ways, and we adopt the following
choice:

$\displaystyle\Phi_{1}(\omega,\gamma;0)$ $\displaystyle=\Phi_{1}(0,\gamma)$
$\displaystyle+C_{0}\left\\{\left[\frac{M(\omega)}{(\gamma+\frac{\omega}{2})^{3}}+\frac{M_{2}}{(\gamma+\frac{\omega}{2})^{2}}+\frac{M_{1}}{\gamma+\frac{\omega}{2}}\right]-\left[\frac{M(0)}{\gamma^{3}}+\frac{M_{2}}{\gamma^{2}}+\frac{M_{1}}{\gamma}\right]+\begin{pmatrix}\gamma\leftrightarrow
1-\gamma\\\ M\to\bar{M}\end{pmatrix}\right\\}\;.$ (93)

Having required $\Phi_{1}$ to be symmetric causes $M(\omega)=\bar{M}(\omega)$
equal to half the expression in the last column of table 1.

##  5 RGI impact factor for longitudinal photons

###  5.1 Cross section and impact factor at leading order

In order to determine the longitudinal RGI impact factor at leading order, we
first consider the cross section $\sigma^{(LT)}(Q_{1},Q_{2})$ where the photon
$Q_{1}$ (on the left) has longitudinal polarization, while the other one
$Q_{2}$ (on the right) is transverse. We are interested in the collinear limit
$Q_{1}^{2}\gg Q_{2}^{2}$, therefore we need the vertices that describe how the
longitudinal photon $Q_{1}$ couples to quarks and gluons $k$ in the collinear
limit $Q_{1}^{2}\gg k^{2}$. They can be derived from the longitudinal
coefficient functions, as explained in appendix A.

Figure 4: Diagramatics of collinear limit at leading order for the
longitudinal impact factor. The blob represents the gluonic contribution to
the longitudinal coefficient function at lowest order in
$\alpha_{\mathrm{s}}$.

The lowest order ladder diagram of this kind involving a high-energy gluon
exchange is depicted in fig. 4. The shaded circle at the left represents the
gluon contribution to the longitudinal coefficient function $C_{L}^{g}$, while
the two vertices on the right side represent two splitting functions, as in
the $TT$-case.

We can then repeat the collinear analysis of sec. 4.1 by replacing in eq. 55
the “transverse” factor eq. (125) with the “longitudinal” factor eq. (127)
(see Appendix), thus obtaining the leading $\gamma$-pole structure of
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(LT)}$:

$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(LT)}(\omega,\gamma;1)_{\mathrm{coll}}$
$\displaystyle=\frac{\alpha\alpha_{\mathrm{s}}\big{(}\sum_{q}e_{q}^{2}\big{)}8T_{R}\sqrt{2(N_{c}^{2}-1)}}{\gamma\,(2+\omega)(3+\omega)}\;\frac{1+\omega
A_{gq}(\omega)}{\omega}\;\frac{\alpha\alpha_{\mathrm{s}}\big{(}\sum_{q}e_{q}^{2}\big{)}2P_{qg}(\omega)\sqrt{2(N_{c}^{2}-1)}}{\gamma^{2}}+\mathcal{O}\left(\gamma^{-2}\right)$
$\displaystyle=\Phi_{\mathrm{BNP}}^{(L)}(\omega,\gamma)\;\frac{1+\omega
A_{gq}(\omega)}{\omega}\;\Phi_{\mathrm{BNP}}^{(T)}(\omega,1+\omega-\gamma)+\mathcal{O}\left(\gamma^{-2}\right)\;,$
(94)

namely the product of the corresponding BNP impact factors with exact
kinematics [61], the LO GGF $1/\omega$ and the same $\mathcal{O}(\omega)$
correction $\propto A_{gq}(\omega)$. The second line of eq. 94 follows from
the collinear structure of the BNP impact factors, reported in eq. 64 for the
transverse polarization and in the following equation for the longitudinal
polarization:

$\displaystyle\Phi_{\mathrm{BNP}}^{(L)}(\omega,\gamma)$
$\displaystyle=\alpha\alpha_{\mathrm{s}}\Big{(}\sum_{q}e_{q}^{2}\Big{)}T_{R}\sqrt{2(N_{c}^{2}-1)}\;4\frac{\pi\Gamma(\gamma+\delta+1)\Gamma(\gamma+1)}{\Gamma(\omega)}\frac{1}{\left(\delta^{2}-1\right)\left(\delta^{2}-4\right)}\times$
$\displaystyle\qquad\left[\frac{\psi(\gamma+\delta)-\psi(\gamma)}{\delta}\cdot\frac{3\omega^{2}-\left(\delta^{2}-1\right)}{2\omega}-3\right]$
(95)
$\displaystyle=C_{0}\left[\frac{1+\omega\Lambda(\omega)}{\gamma}+D_{L}(\omega)+\mathcal{O}(\gamma)\;+\;(\gamma\leftrightarrow
1+\omega-\gamma)\right]$ (96) $\displaystyle 1+\omega\Lambda(\omega)$
$\displaystyle=\frac{6}{(2+\omega)(3+\omega)}\;,\quad\Lambda(0)=-\frac{5}{6}\;,\quad
D_{L}(0)=-\frac{1}{3}\;,\quad\delta\equiv 1+\omega-2\gamma\;,$ (97)

where $C_{0}$ is the same normalization coefficient of the transverse impact
factor, as given in eq. 65. Therefore, eq. 94 can be rewritten as

$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(LT)}(\omega,\gamma;1)=C_{0}^{2}\frac{1}{\omega}\left[\frac{(1+\omega\Lambda)(1+\omega
A_{gq})(1+\omega A_{qg})}{\gamma^{3}}+\mathcal{O}(\gamma^{-2})\right]\;.$ (98)

Taking inspiration from eqs. 67 and 96, we parametrize the collinear structure
of the longitudinal LO RGI impact factor as

$\Phi_{0}^{(L)}(\omega,\gamma;1)=C_{0}\left[\frac{1+\omega
B_{L}(\omega)}{\gamma}+D_{L}(\omega)+\frac{1+\omega\bar{B}_{L}(\omega)}{1+\omega-\gamma}+\bar{D}_{L}(\omega)\right]+r_{L}(\omega,\gamma)\;,$
(99)

where $r_{L}(\omega,\gamma)$ vanishes at $\gamma=0$ and $\gamma=1+\omega$. By
combining the above expression with the analogue one in eq. 67, we find

$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(LT)}_{0}(\omega,\gamma;1)$
$\displaystyle=\Phi_{0}^{(L)}(\omega,\gamma;1)\;\frac{1}{\omega}\;\Phi_{0}^{(T)}(\omega,1+\omega-\gamma;1)$
$\displaystyle=C_{0}^{2}\frac{1}{\omega}\left[\frac{(1+\omega
B_{L})(1+\omega\bar{B})}{\gamma^{3}}+\mathcal{O}(\gamma^{-2})\right]+(\gamma\to
1+\omega-\gamma)\;.$ (100)

If we compare eq. 100 with eq. 98, we obtain a relation among $B_{L}$,
$\bar{B}$ and the known quantities $\Lambda,A_{qg},A_{gq}$. However,
remembering that $B$ and $\bar{B}$ are constrained by eq. 69, we can actually
relate $B_{L}$ and $B$:

$\displaystyle\frac{1+\omega B_{L}}{1+\omega B}$
$\displaystyle=\frac{1+\omega\Lambda}{1+\omega A_{qg}}$ (101)
$\displaystyle\Longrightarrow\quad B_{L}$
$\displaystyle=\Lambda+B-A_{qg}+\mathcal{O}(\omega)\;.$ (102)

The coefficient $\bar{B}_{L}$ of the simple anti-collinear pole can be
determined in an analogous way by considering the cross section for two
longitudinal photons, i.e., by comparing the two expansions for

$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(LL)}_{0}(\omega,\gamma;1)$
$\displaystyle=\Phi_{0}^{(L)}(\omega,\gamma;1)\;\frac{1}{\omega}\;\Phi_{0}^{(L)}(\omega,1+\omega-\gamma;1)$
$\displaystyle=\Phi_{\mathrm{BNP}}^{(L)}(\omega,\gamma)\;\frac{1+\omega
A_{gq}(\omega)}{\omega}\;\Phi_{\mathrm{BNP}}^{(L)}(\omega,1+\omega-\gamma)+\mathcal{O}\left(\gamma^{-1}\right)\;,$
(103)

yielding

$\displaystyle(1+\omega B_{L})(1+\omega\bar{B}_{L})$
$\displaystyle=(1+\omega\Lambda)^{2}(1+\omega A_{gq})$ (104a)
$\displaystyle\Longrightarrow\quad B_{L}+\bar{B}_{L}$
$\displaystyle=2\Lambda+A_{gq}+\mathcal{O}(\omega)\;.$ (104b)

Note that the role played by $A_{qg}$ for $B$ and $\bar{B}$ in eq. 69 is now
played by $\Lambda$ for $B_{L}$ and $\bar{B}_{L}$ in eq. 104.

Just like in the transverse case, we can define the LO RGI longitudinal impact
factor by sharing the $A_{gq}$ correction term between the leading collinear
and anticollinear poles:

$\displaystyle\Phi_{0}^{(L)}(\omega,\gamma;1)$
$\displaystyle=\Phi_{\mathrm{BNP}}^{(L)}(\omega,\gamma)\left[1+\frac{\omega}{2}A_{gq}(\omega)\right]$
$\displaystyle(\text{scheme I})$ (105a)
$\displaystyle\Phi_{0}^{(L)}(\omega,\gamma;1)$
$\displaystyle=\Phi_{\mathrm{BNP}}^{(L)}(\omega,\gamma)+C_{0}\frac{\omega}{2}A_{gq}(\omega)\left[\frac{1}{\gamma}+\frac{1}{1+\omega-\gamma}\right]$
$\displaystyle(\text{scheme II})$ (105b)
$\displaystyle\Phi_{0}^{(L)}(\omega,\gamma;1)$
$\displaystyle=\Phi_{\mathrm{BNP}}^{(L)}(\omega,\gamma)+C_{0}\,\omega
A_{gq}(\omega)\frac{1+\omega A_{qg}}{1+\omega-\gamma}$
$\displaystyle(\text{scheme III})\;.$ (105c)

Schemes I and II implement the choice $B_{L}=\bar{B}_{L}=\Lambda+A_{gq}/2$,
giving rise to symmetric impact factors, while scheme III has $B_{L}=\Lambda$
and $\bar{B}_{L}=\Lambda+A_{gq}+\omega\Lambda A_{gq}$, giving rise to an
asymmetric impact factor, but fulfilling exactly eq. 104a.

###  5.2 Cross section and impact factor at next-to-leading order

The collinear analysis at NLO for the longitudinal-transverse photon cross
section involves the three diagrams depicted in fig. 5 and can be presented in
the following form:

$\alpha_{\mathrm{s}}\,\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{1}^{(LT)}(\omega,\gamma;1)=\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(LT)}(\omega,\gamma;1)\left[\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{gg}}{\gamma}+\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{qq}}{\gamma}+\frac{C_{F}}{T_{R}}\frac{3+\omega}{2}\cdot\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{qg}}{\gamma}-\frac{\alpha_{\mathrm{s}}b}{\gamma}+\mathcal{O}(\gamma^{0})\right]\;,$
(106)

where $\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(LT)}$ is the
LO integrand defined in eq. 98.

Figure 5: Diagramatics of collinear limit at next-to-leading order for the
longitudinal impact factor. (a) Photon-gluon coefficient function and gluon
emission from gluon line; (b) Photon-gluon coefficient function and gluon
emission from quark line. (c) Photon-quark coefficient function and quark
emission from parent gluon;

The first term in the r.h.s. of eq. 106 stems from the diagram of fig. 5(a)
involving a $P_{gg}$ splitting function, and can be entirely associated to the
GGF.

The second term stems from the diagram of fig. 5(b), with a gluon emitted from
the quark line on the right, and it is naturally associated to the impact
factor of the transverse photon $Q_{2}$.

The third term stems from the diagram of fig. 5(c), which is genuinely
different from other diagrams, because it involves a coefficient function
where the longitudinal photon $Q_{1}$ couples to a quark. As explained in
section A.2, the photon-quark coefficient function differs from the photon-
gluon one by the multiplicative factor $C_{F}(3+\omega)/(2T_{R})$ [cfr. eq.
132]; just to the right of the blob, we find the vertex with the $P_{qg}$
splitting function. This contribution is naturally associated to the impact
factor of the longitudinal photon $Q_{1}$.

The fourth and last term in eq. 106 is the running coupling ($b$-dependent)
contribution derived in eq. 54, and can be incorporated into either the impact
factors or the GGF, or both.

In order to determine the NLO RGI longitudinal impact factor from the NLO
cross section, we parametrize the collinear structure of the longitudinal
impact factor exactly as in eq. 73, by appending the subscript $L$ to the
various (unbarred) coefficients, e.g., $M\to M_{L}$. A straightforward
calculation yields

$\alpha_{\mathrm{s}}\,\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{1}^{(LT)}(\omega,\gamma;1)=\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{0}^{(LT)}(\omega,\gamma;1)\;\bar{\alpha}_{s}\left[\frac{\frac{1}{\omega}+M_{L}+\bar{M}+U}{\gamma}+\mathcal{O}(\gamma^{0})\right]\;.$
(107)

which is nothing but the result of eq. 75 with $T\to L$ in the first impact
factor. We then derive [cfr. eq. 76 and the subsequent definitions]

$\displaystyle M_{L}+\bar{M}+U$
$\displaystyle=\mathcal{P}_{L}+\bar{P}_{qq}+A_{gg}-\bar{b}\;,\qquad\mathcal{P}_{L}(\omega)\equiv\frac{C_{F}}{T_{R}}\bar{P}_{qg}\frac{3+\omega}{2}$
(108) $\displaystyle\Longrightarrow\quad M_{L}-M$
$\displaystyle=\mathcal{P}_{L}-\bar{P}_{qq}=\frac{C_{F}}{2C_{A}}+\mathcal{O}(\omega)\;.$
(109)

In order to check the compatibility of the collinear analysis with the known
BFKL results, let us write down the collinear structure of the NLO BFKL
longitudinal impact factors [eqs. 43b and 47]:

$\displaystyle\phi_{0}^{(L)}(\gamma)$
$\displaystyle=\phi_{0}^{(L)}(1-\gamma)=C_{0}\left(\frac{1}{\gamma}+D_{L}(0)+\mathcal{O}(\gamma)\right)$
(110) $\displaystyle\phi_{1}^{(L)}(\gamma)$
$\displaystyle=\phi_{0}^{(L)}(\gamma)\left(\frac{-1/2}{\gamma^{2}}+\frac{\eta_{L}}{\gamma}+\frac{-1}{(1-\gamma)^{2}}+\frac{\bar{\eta}_{L}}{1-\gamma}+\cdots\right)\;,\quad\eta_{L}=-\frac{7}{3}\;,\quad\bar{\eta}_{L}=-\frac{9}{4}\;.$
(111)

At this point we expand eq. 39 around $\gamma=0$ by using eqs. 77, 78, 79, 80,
110 and 111 for the LHS and eqs. 73 and 74 for the RHS. As a result, the
(spurious) leading $\gamma$ poles match, while the subleading ones are equal
if

$\eta_{L}+\bar{\eta}=B_{L}+\bar{B}+\mbox{\small$\frac{1}{2}$}(D_{L}+\bar{D})+M_{L}+\bar{M}+U\big{|}_{\omega=0}\;.$
(112)

Subtracting 85a from the above equation, we obtain a relation among collinear
coefficients:

$\eta_{L}-\eta=B_{L}-B+\mbox{\small$\frac{1}{2}$}(D_{L}-D)+M_{L}-M\;.$ (113)

While the LHS evaluates to $-1/2$, the RHS is equal to $-1/2+C_{F}/(2C_{A})$.
Therefore, we find agreement with the result of ref. [57], were it not for the
presence of a term proportional to $C_{F}$ Casimir in the collinear pole. It
looks like their impact factor misses the contribution from the diagram of
fig. 5c.

Finally, by considering the cross section for two longitudinally polarized
cross section, we obtain the result of eq. 112 with the barred (transverse)
coefficients replaced by their corresponding longitudinal counterparts:

$\eta_{L}+\bar{\eta}_{L}=B_{L}+\bar{B}_{L}+\mbox{\small$\frac{1}{2}$}(D_{L}+\bar{D}_{L})+M_{L}+\bar{M}_{L}+U\big{|}_{\omega=0}\;,$
(114)

which is satisfied only if $M_{L}+\bar{M}_{L}=\mathcal{O}(\omega)$. If we take
$\bar{M}_{L}=M_{L}$, as it is natural to assume in the transverse case, then
we disagree with the result of ref. [57] by a $C_{F}$ term in the $1/\gamma$
pole of the ratio $\phi_{1}^{(L)}/\phi_{0}^{(L)}$.

We collect our results for the longitudinal RGI impact factor in table 2.

scheme name | $U$ | $V$ | $B_{L}+\bar{B}_{L}$ | $M_{L}+\bar{M}_{L}$
---|---|---|---|---
collA | $A_{gg}-\bar{b}$ | $\Delta A+\bar{b}/2$ | $2\Lambda+A_{gq}$ | $2\mathcal{P}_{L}$
collB | $A_{gg}$ | $\Delta A-\bar{b}/2$ | $2\Lambda+A_{gq}$ | $2\mathcal{P}_{L}-\bar{b}$
zVnB | $A_{1}-\bar{b}/2$ | $0$ | $2\Lambda+A_{gq}$ | $2\mathcal{P}_{L}-\Delta A-\bar{b}/2$
zVnM | $A_{1}-\bar{b}/2$ | $0$ | $2\Lambda+A_{gq}-\Delta A-\bar{b}/2$ | $2\mathcal{P}_{L}$
zVzM | $A_{1}-\bar{b}/2$ | $0$ | $2\mathcal{P}_{L}+2\Lambda+A_{gq}-\Delta A-\bar{b}/2$ | $0$
Table 2: Favourite scheme choices for defining the NLO RGI longitudinal impact
factor. The values of $U$ and $V$ are the same as in table 1.

Once a scheme has been chosen, the LO impact factor $\Phi_{0}^{(L)}$ can be
specified according to one of the sub-schemes in eq. 105, with
$B_{L}=\bar{B}_{L}$ in sub-schemes I and II, while $\bar{B}_{L}=B_{L}+A_{gq}$
in sub-scheme III.

The NLO impact factor $\Phi_{1}^{(L)}$ is constructed to be symmetric, as in
the transverse case: at $\omega=0$ eq. 91 holds unaltered, provided we add to
$\phi_{1}$ the contribution

$\Delta\phi_{1}^{(L)}(\gamma)=\mathcal{P}_{L}(0)\left(\frac{1}{\gamma}+\frac{1}{1-\gamma}\right)\phi_{0}^{(L)}(\gamma)\;.$
(115)

The Laurent expansion around $\gamma=0$ shows a double pole

$\Phi_{1}^{(L)}(0,\gamma)=C_{0}\left[\frac{M_{L}(0)}{\gamma^{2}}+\frac{M_{L,1}}{\gamma}+\mathcal{O}(\gamma^{0})\right]\;,$
(116)

where $M_{L,1}$ depends on the scheme choice. We extend $\Phi_{1}$ at
$\omega\neq 0$ by requiring the collinear poles to be located at
$\gamma=-\omega/2$ and $\gamma=1+\omega/2$ and with $\omega$-dependent leading
coefficients $M_{L}(\omega)$ and $\bar{M}_{L}(\omega)$ as we did for the
transverse impact factor:

$\displaystyle\Phi_{1}^{(L)}(\omega,\gamma;0)$
$\displaystyle=\Phi_{1}^{(L)}(0,\gamma)$
$\displaystyle+C_{0}\left\\{\left[\frac{M_{L}(\omega)}{(\gamma+\frac{\omega}{2})^{2}}+\frac{M_{L,1}}{\gamma+\frac{\omega}{2}}\right]-\left[\frac{M_{L}(0)}{\gamma^{2}}+\frac{M_{L,1}}{\gamma}\right]+\begin{pmatrix}\gamma\leftrightarrow
1-\gamma\\\ M\to\bar{M}\end{pmatrix}\right\\}\;.$ (117)

Having required $\Phi_{1}^{(L)}$ to be symmetric causes
$M_{L}(\omega)=\bar{M}_{L}(\omega)$ equal to half the expression in the last
column of table 2.

##  6 Numerical analysis

In this section, we apply the factorization formula with renormalization-group
improved impact factors and Green’s function to compute the
$\gamma^{*}\gamma^{*}$ cross section in phenomenologically relevant
situations. The presented results contain the sum over all combinations of
photon polarizations:
$\sigma=\sigma^{(TT)}+\sigma^{(LT)}+\sigma^{(TL)}+\sigma^{(LL)}$. For the NLL
RGI calculation, the $\sigma^{(TT)}$ is about 56% of the total cross section
on average at $Q^{2}=17\,\rm GeV^{2}$, while both $\sigma^{(TL)}$ and
$\sigma^{(LT)}$ about 19%, $\sigma^{(LL)}$ about 6%. These percentages vary by
about $2\%$ for $\sigma^{(TT)}$, and about $1\%$ for other polarization
combinations upon changes of the scheme and varying rapidity between 2 and 7
units.

The numerical calculation is based on the following formulae: the cross
section is calculated using eq. 33; the leading eigenvalue $X_{0}$ is given in
eq. 87; the NL one $X_{1}$ in eqs. 88 and 90 with $\chi_{1}$ in eq. 7;
$\omega_{\mathrm{eff}}$ in eq. 31; the leading impact factors in eqs. 70 and
105; the NL ones in eq. 91 at $\omega=0$ and eqs. 93 and 117 at $\omega\neq
0$.

We shall compare our results with the experimental measurements of L3 [54] at
$Q^{2}=16\ \text{GeV}^{2}$ and of OPAL [55] at $Q^{2}=17.9\ \text{GeV}^{2}$,
and also with previous calculations of the same cross section. Since the
values of $Q^{2}$ in L3 and OPAL are very close, it is reasonable to compare
the data from both experiments with theoretical predictions at $Q^{2}=17\
\text{GeV}^{2}$.

We adopt the strong coupling value to be $\alpha_{s}\left(Q^{2}=17\
\text{GeV}^{2}\right)\approx 0.229$ as derived from the Particle Data Group
[68].

In fig. 7 we show the results for the NLL RGI cross sections using scheme I
for the LO impact factors eqs. 70 and 105 and the five different schemes from
tables 1 and 2 at NLO, and compare them with the pure LL and NLL cross
sections.

All five NLL RGI cross sections are significantly reduced with respect to the
LL calculation, however they are also significantly above the pure NLL
calculation. We observe that, the different schemes give very similar results.
In order to present the results more intuitively, we incorporate a band to
represent the scheme ambiguity as in fig. 7. The band size is defined as the
standard deviation calculated from the five schemes at each rapidity $Y$. In
the following, if the improved NLL cross section is presented as a single
curve, then the curve is just the average for the five NLL RGI schemes.
Adopting schemes II and III for the LO impact factors does not change
significantly our estimates.

Figure 6: The value of the $\gamma^{*}\gamma^{*}$ cross section contribution
from the BFKL exchange for $Q^{2}=17\ \text{GeV}^{2}$ as a function of
rapidity $Y$. All five schemes (see tables 1 and 2) for the NLL RGI
calculation are shown together with the pure LL calculation (black solid and
rescaled with a factor 0.5) and pure NLL calculation (green dot-dashed).

Figure 7: The value of the $\gamma^{*}\gamma^{*}$ cross section contribution
from the BFKL exchange for $Q^{2}=17\ \text{GeV}^{2}$ as a function of
rapidity $Y$. The scheme average band (blue-solid) represents the average
value and standard deviation of the five resummed schemes. The $\mu_{R}$ band
(yellow-dashed) is computed from average values of the five resummed schemes
with half or double $\mu_{R}^{2}$ respectively.

In fig. 7, we also test the stability of the improved NLL cross section
calculation with respect to the variation of the $\mu_{R}$ scale. The upper
and lower $\mu_{R}$ band is computed from average values of the five resummed
schemes with half or double $\mu^{2}_{R}$ respectively. It turns out that the
$\mu_{R}$ band size is slightly smaller than the scheme ambiguity band size.
It is worth noting that besides the dependence on $\mu_{R}$ of the NLO impact
factor and the running coupling argument, the NLO BFKL eigenfunction would
also rely on $\mu_{R}$ when $\mu_{R}^{2}\neq Q_{1}Q_{2}$,

$\tilde{X}_{1}(\omega,\gamma)=X_{1}(\omega,\gamma)+\bar{b}\,X_{0}(\omega,\gamma)\ln{\frac{\mu_{R}^{2}}{Q_{1}Q_{2}}}.$
(118)

and the resummed effective $\omega$ after the NLO subtraction with $\mu_{R}$
dependency is then the solution of

$\omega=\bar{\alpha}_{s}(\mu_{R}^{2})X_{0}(\omega,\gamma)+\bar{\alpha}_{s}^{2}(\mu_{R}^{2})\left[X_{1}(\omega,\gamma)+\bar{b}\,X_{0}(\omega,\gamma)\ln{\frac{\mu_{R}^{2}}{Q_{1}Q_{2}}}\right].$
(119)

In fig. 9, we compare the pure LL and NLL results (the latter computed using
expressions from refs. [56, 69]), with the improved LL and NLL cross sections.
Note the logarithmic vertical scale, which makes the characteristic
exponential dependence of the cross section on the rapidity clearly visible.
The NLL improved curve is given as the average of different schemes as
explained above.

The improved LL and NLL calculations both tame the quick growth of the pure LL
cross section with rapidity. It is worth noting that the improvement at LL
alone — consisting in the $\omega$ shifted LO eigenfunction and LO impact
factors — brings the curve down significantly. We also observe that, the
improved NLL is higher than the improved LL calculation, mostly because the
improved NLO corrections bring a positive $O(\alpha_{s}^{2})$ term to the
impact factors. Finally we observe that improved calculations (both at LL and
NLL) are above the pure NLL cross section.

Figure 8: The value of the $\gamma^{*}\gamma^{*}$ cross section contribution
from the BFKL exchange for $Q^{2}=17\ \text{GeV}^{2}$ as a function of
rapidity in the logarithmic vertical scale. Pure LL is shown in black-solid,
NLL in green dashed-dotted, LL improved in red-dotted and NLL improved in
blue-dashed. The NLL improved curve is the average of our five resummed NLL
schemes (see text).

Figure 9: The value of the $\gamma^{*}\gamma^{*}$ cross section contribution
from the BFKL exchange from NLL RGI calculation for $Q^{2}=5$ (blue-dashed),
$17$ (yellow-solid), $100$ $\text{GeV}^{2}$ (green-dotted) as a function of
rapidity $Y$ in logarithmic vertical scale.

In fig. 9, we compare NLL RGI cross sections for $Q^{2}=5,\ 17,\ 100\
\text{GeV}^{2}$. The cross section is strongly dependent on $Q^{2}$. The
growth with rapidity is slowed down with increasing $Q^{2}$ due to the smaller
value of the coupling constant, which affects the value of the leading
exponent in the gluon Greens’s function.

So far we have shown the contribution to the $\gamma^{*}\gamma^{*}$ cross
section stemming only from the gluon exchanges, resummed by the BFKL
evolution, which should be the dominant contribution at high energies.
However, at lower energies, another contribution is important, namely the one
from the ‘quark box’ diagram. This contribution decreases with the rapidity,
however it becomes dominant at low rapidities and is important when comparing
with the experimental data. In the following, we evaluate the quark box in the
lowest order [70, 71]. The total $\gamma^{*}\gamma^{*}$ cross section
presented in the following includes both the quark box and the BFKL
contributions.

Figure 10: Cross sections for $Q^{2}=17\ \text{GeV}^{2}$, compared with L3
($Q^{2}=16\ \text{GeV}^{2}$) [54] and OPAL ($Q^{2}=17.9\ \text{GeV}^{2}$) [55]
data. The NLL improved curve is the sum of our averaged NLL BFKL resummed
scheme and LO quark box contribution. The band size represents a combination
of the scheme uncertainty and the $\mu_{R}$ band, i.e.
$\delta_{\text{total}}=\sqrt{\delta_{\text{scheme}}^{2}+\delta_{\mu_{R}}^{2}}$.
The calculation is done for $N_{f}=4$ massless flavours. The Ivanov-Murdaca-
Papa’s (IMP’s) PMS optimized curve (solid-cyan) is from [57]. Separately shown
is the quark box contribution (dashed red).

In fig. 10 we compare the results from NLL improved calculation with the
experimental measurements of L3 [54] at $Q^{2}=16\ \text{GeV}^{2}$ and of OPAL
[55] at $Q^{2}=17.9\ \text{GeV}^{2}$, and also with previous calculations of
the same cross section from [57]. As mentioned before, since the values of
$Q^{2}$ in L3 and OPAL are very close, and the errors on the data points are
such that $Q^{2}$ dependence is not visible, it is reasonable to compare the
data from both experiments with theoretical predictions at $Q^{2}=17\
\text{GeV}^{2}$. We also show the LO quark box contribution in this figure. We
observe from fig. 10 that the RGI NLL improved calculation has a stronger
increase over rapidities than the pure NLL one. We also see that our result is
significantly higher than the calculation from [57], particularly at high
rapidities. The RGI calculation is consistent with the experimental data from
LEP within the theoretical and experimental uncertainties.

In the calculations we assumed $N_{f}=4$ and treated charm as massless. This
is of course an approximation as the mass of the charm is expected to have
some impact and to reduce the theoretical calculations. This effect was for
example studied in detail in [49] and it was shown to decrease the cross
section somewhat. However, this study was performed for the BFKL exchange at
the leading order only. The full calculation would require small-$x$
resummation including the mass effects. This is an interesting problem in
itself but it is beyond the scope of the current work.

##  7 Conclusions

In this paper, we have applied the collinear resummation, based on the
renormalization group improvement (RGI) for high-energy processes, to the
$\gamma^{*}\gamma^{*}$ cross section. The RGI formulation is based on a
factorization formula whose structure is similar to the one in the BFKL
approach, but whose impact factors explicitly depend on the Mellin variable
$\omega$, conjugated to the center of mass energy squared $s$. Firstly, we
have computed RGI impact factors for transverse photons at LO and NLO, which
are consistent with the BFKL one in the high-energy limit and with LO DGLAP in
the collinear limit. We then extended this procedure for the determination of
the RGI impact factor for longitudinal photons.

At LO, the RGI impact factors are consistent with the impact factors with
exact kinematics computed in [61]. This is a non-trivial check, since we
reproduce the shifted position of the collinear poles in $\gamma$ — the Mellin
variable conjugated to the photon virtualities — and also the coefficients of
the leading $\gamma$-poles (apart from a subleading term which is out of
control in the approximations adopted in [61]).

At NLO we predict, and thus resum, the spurious energy-scale dependent quartic
(cubic) $\gamma$-poles of the transverse (longitudinal) BFKL impact factors.
For the transverse impact factors we can predict the cubic $\gamma$-poles,
which have both physical and spurious components. Having identified the
physical component of such poles, stemming from partonic anomalous dimensions
and running of the coupling, we obtain an impact factor which, in the
collinear limit, is less singular than its BFKL counterpart and contains the
full LO DGLAP information.

In the case of the longitudinal photon impact factor the collinear analysis at
NLO predicts a term proportional to the Casimir $C_{F}$ which is enhanced in
the collinear region but is absent in the BFKL impact factors originally
computed in [29] and presented in [56, 57]. This term stems from the
$\mathcal{O}(\alpha_{\mathrm{s}})$ coefficient function $C_{L,q}$ that couples
the longitudinal photon to quarks at first order in QCD. We find full
compatibility with the other colour structures, which quantitatively make up
the bulk of the longitudinal impact factor.

Finally, we compute the $\gamma^{*}\gamma^{*}$ cross section using the NL RGI
impact factors and Green’s function, and compare it with previous calculations
and also with experimental data. Note that the RGI resummation suffers from
some ambiguities, due to the lack of information in the kinematical region of
low-energies (not controlled by BFKL) and comparable virtualities (not
controlled by DGLAP) of the exchanged partons. Therefore, we propose a handful
of physically motivated resummation schemes, and consider the average of the
ensuing cross sections as our best estimate (with the corresponding standard
deviation as resummation-scheme-uncertainty).

According to the expectations, the resummed cross section increases as a
function of $Y=\log(s/Q_{1}Q_{2})$. It is found between the pure LL BFKL
prediction and the pure NLL BFKL one. Note that switching from LL to NLL
reduces the cross section by more than one order of magnitude. The
resummation-scheme uncertainty is about 20%, slightly larger than the
renormalization-scheme uncertainty.

In order to provide a phenomenologically meaningful observable, the BFKL cross
section, which is expected to dominate at large $Y$, must be supplemented with
the so-called quark-box contribution, which dominates at small $Y$ and rapidly
decreases with increasing $Y$. In this way, we are consistent with the
experimental data of OPAL and L3, without the need of particular choices of
running coupling scale fixing. This is a strong indication that the RGI
procedure is the proper context for a correct description of virtual-photon
scattering at large energies.

In the present calculation, we treat all four flavours, including charm, as
massless. A more detailed analysis would need to include the charm mass in the
full scheme of small $x$ resummation, which would require the knowledge of the
massive impact factors at NLL. The mass effects at NLO in DIS structure
functions were in fact computed in the dipole picture of high energy [72, 73,
37], however this result would require extensive calculation (linearization to
two gluon exchange as well as transforming it to momentum space) in order to
extract the impact factors. A more detailed investigation on this issue is
left for the future.

## Acknowledgments

We thank Ian Balitsky and Stefano Catani for the discussions. This project is
supported by the U.S. Department of Energy Grant DE-SC-0002145 and within the
framework of the Saturated Glue (SURGE) Topical Theory Collaboration, by
Polish NCN Grant No. 2019/33/B/ST2/02588, and by the European Union’s Horizon
2020 research and innovation programme under grant agreement No 824093.

##  Appendix A Lowest-order cross sections and structure functions

In this appendix, we sketch the determination of the photon-parton cross
sections at the lowest order in perturbation theory, which is the basis of the
analysis of the photon-photon cross section in the collinear regime
$Q_{1}^{2}\ll Q_{2}^{2}$. Such cross sections are proportional to the
corresponding partonic structure functions, which in turn can be derived by
the DIS coefficient functions.

###  A.1 Transverse photon

The cross section of a virtual photon with polarization $\lambda$ scattering
on a particle of momentum $P$ (e.g., a hadron) is given by (cfr. [61])

$\displaystyle\sigma^{(\lambda)}(P,q)=\frac{4\pi^{2}\alpha}{Q^{2}}F^{(\lambda)}(x,Q^{2})\;,\qquad
x\;\raisebox{0.27127pt}{$:$}=\frac{Q^{2}}{2P\cdot q}\;.$ (120)

where $F^{(\lambda)}(x,Q^{2}):\lambda=L,T,2$ are the standard structure
functions with $F^{(2)}=F^{(L)}+F^{(T)}$. The integrand
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}(\omega,\gamma;p)$ of the
double Mellin representation (28a) can then be written as

$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}(\omega,\gamma;1)$
$\displaystyle=\int_{Q_{1}^{2}}^{\infty}\frac{{\rm
d}s}{s}\left(\frac{Q_{1}^{2}}{s}\right)^{\omega}\int\frac{{\rm
d}Q_{1}^{2}}{Q_{1}^{2}}\left(\frac{Q_{2}^{2}}{Q_{1}^{2}}\right)^{\gamma-\mbox{\small$\frac{1}{2}$}}2\pi
Q_{1}Q_{2}\,\sigma(s,Q_{1}^{2},Q_{2}^{2})$
$\displaystyle=\int_{0}^{1}\frac{{\rm d}x}{x}x^{\omega}\int\frac{{\rm
d}Q_{1}^{2}}{Q_{1}^{2}}\left(\frac{Q_{2}^{2}}{Q_{1}^{2}}\right)^{\gamma}(2\pi)^{3}\alpha
F(x,Q_{1}^{2})\;.$ (121)

In the case of an incoming quark of flavour $a$ and small offshellness
$Q_{2}^{2}\ll Q_{1}^{2}\equiv Q^{2}$, the partonic structure functions at
lowest order are nothing but the corresponding coefficient functions:

$F^{(T,a)}_{0}(x,Q^{2})=F^{(2,a)}_{0}(x,Q^{2})=x\;e_{a}^{2}\;C^{(2,q)}_{0}(x)=e_{a}^{2}\delta(1-x)\;,\qquad
F^{(L,a)}_{0}(x,Q^{2})=0\;.$ (122)

Therefore, at the lowest order only the transverse polarization is effective
and we have

$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(T,a)}_{0}(\omega,\gamma;1)=\int_{0}^{1}\frac{{\rm
d}x}{x}x^{\omega}\int\frac{{\rm
d}Q_{1}^{2}}{Q_{1}^{2}}\left(\frac{Q_{2}^{2}}{Q_{1}^{2}}\right)^{\gamma}(2\pi)^{3}\alpha
F^{(T,a)}_{0}(x,Q_{1}^{2})=(2\pi)^{3}\alpha\,e_{a}^{2}\frac{1}{\gamma}$ (123)

which is the first factor of the collinear chain (55), before summing over
quark and antiquark flavours. By taking the inverse Mellin transform with
respect to $\gamma$, we have

$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(T,a)}_{0}(\omega,Q_{1}^{2},Q_{2}^{2};1)=(2\pi)^{3}\alpha\,e_{a}^{2}\;,$
(124)

representing the first factor in eq. 49 — again, before summing over quark and
antiquark flavours.

The first non-vanishing contribution of the photon-gluon structure functions
starts at $\mathcal{O}(\alpha_{\mathrm{s}})$. In the collinear limit, i.e.,
considering the strong ordering of partons’ momenta, each rung provides a
factor $\int_{k_{i-1}^{2}}^{k_{i}^{2}}\frac{{\rm
d}k^{2}}{k^{2}}\frac{\alpha_{\mathrm{s}}(k^{2})}{2\pi}P_{ab}(\omega)$. With
fixed running coupling such a factor reduces to
$\frac{\alpha_{\mathrm{s}}}{2\pi}\log\frac{k_{i}^{2}}{k_{i-1}^{2}}P_{ab}(\omega)$,
which becomes $\frac{\alpha_{\mathrm{s}}}{2\pi}P_{ab}(\omega)/\gamma$ in
$\gamma$-space. Therefore, at $\mathcal{O}(\alpha_{\mathrm{s}})$, for a
transverse photon we have

$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}_{1}^{(T,g)}(\omega,\gamma;1)=\sum_{a}(2\pi)^{3}\alpha\,e_{a}^{2}\frac{1}{\gamma}\cdot\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{P_{(q=a)g}(\omega)}{\gamma}\;.$
(125)

in agreement with the first factors of eq. 55 since
$\sum_{a}e_{a}^{2}=2\sum_{q}e_{q}^{2}$. The other factors follow from the
remaining two vertices.

###  A.2 Longitudinal photon

The longitudinal structure function starts at
$\mathcal{O}(\alpha\alpha_{\mathrm{s}})$ in perturbation theory, and receive
contributions from gluons and quarks. We start by considering the gluon-
initiated structure function, which is well known in the literature, and can
be read, e.g., from eq. (B.5) of ref. [74]

$\displaystyle
F_{1}^{(L,g)}(x,Q^{2})=x\;\frac{\sum_{a}e_{a}^{2}}{2N_{f}}\;C_{1}^{(L,g)}(x,Q^{2}/\mu_{F}^{2})=\frac{\alpha_{\mathrm{s}}}{2\pi}\big{(}\sum_{a}e_{a}^{2}\big{)}\,T_{R}\,4\,x^{2}(1-x)\;.$
(126)

The corresponding longitudinal photon-gluon cross section in Mellin space can
then be determined from eq. 121 and reads

$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(L,g)}_{1}(\omega,\gamma;1)=\frac{16\pi^{2}\alpha\alpha_{\mathrm{s}}\,\big{(}\sum_{a}e_{a}^{2}\big{)}\,T_{R}}{\gamma\,(2+\omega)(3+\omega)}\;.$
(127)

It is straightforward to check that the r.h.s. of eq. 127 is proportional to
the simple pole at $\gamma=0$ of the BNP longitudinal impact factor with exact
kinematics with its full $\omega$-dependence, just like the r.h.s. of eq. 125
is proportional to the double pole of the BNP transverse impact factor:

$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(T,g)}_{1}(\omega,\gamma;1)$
$\displaystyle=2\pi\big{(}\sum_{a}e_{a}^{2}\big{)}T_{R}\,S_{T}(\omega,\gamma)+\mathcal{O}(\gamma^{-1})$
(128)
$\displaystyle\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(L,g)}_{1}(\omega,\gamma;1)$
$\displaystyle=2\pi\big{(}\sum_{a}e_{a}^{2}\big{)}T_{R}\,S_{L}(\omega,\gamma)+\mathcal{O}(\gamma^{0})\;.$
(129)

The quark-initiated structure function, is also well known in the literature,
and can be read, e.g., from eq. (B.1) of ref. [74]:

$\displaystyle
F_{1}^{(L,a)}(x,Q^{2})=x\;e_{a}^{2}\;C_{1}^{(L,q)}(x,Q^{2}/\mu_{F}^{2})=\frac{\alpha_{\mathrm{s}}}{2\pi}e_{a}^{2}\,C_{F}\,2x^{2}\qquad\text{($a=$
quark or antiquark)}\;.$ (130)

The corresponding longitudinal photon-quark cross section in Mellin space can
then be determined from eq. 121 and reads

$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(L,a)}_{1}(\omega,\gamma;1)=\frac{8\pi^{2}\alpha\alpha_{\mathrm{s}}\,e_{a}^{2}\,C_{F}}{\gamma\,(2+\omega)}\qquad\text{($a=$
quark or antiquark)}\;.$ (131)

Summing over all quarks and antiquarks we get

$\sum_{a}\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(L,a)}_{1}(\omega,\gamma;1)=\frac{C_{F}}{T_{R}}\,\frac{3+\omega}{2}\,\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\sigma}$}}^{(L,g)}_{1}(\omega,\gamma;1)\;.$
(132)

In practice, the blob connecting a longitudinal photon to all quarks and
antiquarks displayed in fig. 5(c) is equal to the blob connecting the
longitudinal photon to a gluon in fig. 5(a),(b) up to the additional
multiplicative factor $C_{F}(3+\omega)/(2T_{R})$.

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