# QGSJET-III model of high energy hadronic interactions: The formalism

Sergey Ostapchenko
Universität Hamburg, II Institut für Theoretische Physik, 22761 Hamburg,
Germany

Abstract

The physics content of the QGSJET-III Monte Carlo generator of high energy
hadronic collisions is described. In particular, a phenomenological
implementation of higher twist corrections to hard parton scattering processes
is discussed in some detail. Additionally addressed is the treatment of the
so-called “color fluctuation” effects related to a decomposition of hadron
wave functions into a number of Fock states characterized by different spatial
sizes and different parton densities. Selected model results regarding the
energy-dependence of the total, elastic, and diffractive proton-proton cross
sections are presented.

##  1 Introduction

Nowadays high energy experiments both in the collider and cosmic ray (CR)
fields imply an extensive use of Monte Carlo (MC) generators of hadronic
interactions. At colliders, the primary goal of such generators is to describe
Standard Model backgrounds for new physics searches and to confront novel
theoretical ideas to experimental data. On the other hand, in the CR field, MC
models of high energy interactions play an important role, when interpreting
experimental data. This is particularly so for investigations of very high
energy CRs, which are performed by indirect methods: studying various
characteristics of extensive air showers (EAS) – huge nuclear-electromagnetic
cascades induced by interactions of primary CR particles in the atmosphere of
the Earth – and reconstructing the properties of those particles, based on the
measured EAS characteristics. Such applications imply a number of requirements
to CR interaction models. Those should be able to provide a reasonable
description of collisions with nuclei of various hadron species, primarily, of
(anti)nucleons, pions, and kaons, over a wide energy range: from fixed target
energies up to some $10^{12}$ GeV laboratory energy. Additionally, the steeply
falling down primary CR flux and the cascade nature of extensive air showers
enhance the importance of forward secondary particle production. Last but nor
least, given the scarcity of available experimental data regarding such
forward production and the lack of possibility to re-tune MC generators, based
on CR data, a substantial predictive power is required from CR interaction
models.

Over the past three decades, the QGSJET [1, 2, 3] and QGSJET-II [4, 5, 6] MC
generators proved to be very successful regarding the analyses and
interpretations of various CR data, notably, from air shower experiments. In
the current work, we report a further development of the model framework,
related to taking into consideration the so-called dynamical higher twist
corrections to hard parton scattering processes and to the implementation of
“color fluctuation” effects in high energy hadronic collisions. While the
treatment of secondary particle production and the application of the model to
calculations of EAS characteristics will be discussed elsewhere [7], we
concentrate here on the description of the model formalism, providing also
some selected results regarding the energy-dependence of the total, elastic,
and diffractive proton-proton cross sections.

The paper is organized as follows. In Section 2, we discuss the Reggeon Field
Theory (RFT) approach to multiple scattering in hadronic collisions. Section 3
is devoted to the treatment of hard parton scattering within the RFT
framework. In Section 4, we address the implementation of color fluctuation
effects. The treatment of nonlinear interaction effects due to Pomeron-Pomeron
interactions is described in Section 5. Section 6 is devoted to
phenomenological implementation of dynamical power corrections to hard parton
scattering. The model generalization to the case of nuclear collisions and the
MC realization of the formalism are described in Section 7. In Section 8, we
present and discuss selected model results. Finally, we conclude in Section 9.

##  2 Multiple scattering in the Reggeon Field Theory

High energy hadronic collisions are predominantly multiple scattering
processes, being mediated by multiple parton cascades developing between the
interacting projectile and target hadrons (nuclei). While a perturbative
description of multiple scattering (so-called multi-parton interactions) is an
actively developing field (see, e.g., [8] for a review), the corresponding
treatment in MC generators has to rely presently on the old RFT formalism [9].

Since the underlying, so to say “elementary”, parton cascades develop, at
least partly, in the nonperturbative domain of low parton virtualities, where
the notion of partons can be used for a qualitative discussion only, one is
forced to rely on an effective macroscopic description for such cascades –
treating them as Pomeron exchanges. The Pomeron exchange eikonal (the
imaginary part111The real part of the Pomeron amplitude can be neglected in
the high energy limit. of the corresponding amplitude) is usually chosen in
the form

$\displaystyle\chi_{hp}^{\mathbb{P}}(s,b)=\frac{\gamma_{h}\,\gamma_{p}\,(s/s_{0})^{\alpha_{\mathbb{P}}(0)-1}}{R_{h}^{2}+R_{p}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\,\ln(s/s_{0})}\,$
$\displaystyle\times\;\exp\\!\left[-\frac{b^{2}/4}{R_{h}^{2}+R_{p}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\,\ln(s/s_{0})}\right],$
(1)

where $\alpha_{\mathbb{P}}(0)$ and $\alpha_{\mathbb{P}}^{\prime}(0)$ are,
respectively, the intercept and the slope of the Pomeron Regge trajectory,
$\gamma_{h}$ is the residue and $R_{h}^{2}$ the slope for the Pomeron coupling
to hadron $h$, $s$ and $b$ are, correspondingly, the center-of-mass (c.m.)
energy squared and impact parameter for the collision, and $s_{0}\simeq 1$
GeV2 – the hadronic mass scale.

Using eikonal description for multiple Pomeron emission vertices, one obtains
the well-known simple expressions for the total and elastic hadron-proton
cross sections:

$\displaystyle\sigma^{\rm
tot}_{hp}(s)=2\int\\!d^{2}b\left[1-e^{-\chi_{hp}^{\mathbb{P}}(s,b)}\right]$
(2) $\displaystyle\sigma^{\rm
el}_{pp}(s)=\int\\!d^{2}b\left[1-e^{-\chi_{hp}^{\mathbb{P}}(s,b)}\right]^{2}.$
(3)

Moreover, considering unitarity cuts of the corresponding elastic scattering
diagrams shown schematically in Fig. 1

Figure 1: General multi-Pomeron contribution to hadron-hadron scattering
amplitude; elementary scattering processes (vertical thick lines) are
described as Pomeron exchanges.

and applying the so-called Abramovskii-Gribov-Kancheli (AGK) cutting rules
[10], one is able to obtain partial cross sections for various inelastic final
states corresponding to having precisely $n$ “elementary” production processes
($n$ cut Pomerons):

$\sigma^{(n)}_{hp}(s)=\int\\!d^{2}b\;\frac{\left[2\chi_{hp}^{\mathbb{P}}(s,b)\right]^{n}}{n!}\;e^{-2\chi_{hp}^{\mathbb{P}}(s,b)}.$
(4)

To describe secondary particle production, one assumes that each cut Pomeron
corresponds to a creation of a pair of strings of color field, stretched
between constituent partons [(anti)quarks or (anti)diquarks] of the
interacting hadrons and models the breakup of those strings by means of
suitable string fragmentation procedures [11, 12]. Importantly, the
corresponding parameters can be expressed via intercepts of secondary Regge
trajectories [13].

##  3 Treating hard scattering within the RFT framework

While the original Gribov’s formulation of RFT relied on the assumption that
the bulk of hadron production is characterized by small transverse momenta
[9], $p_{t}\lesssim 1$ GeV, the contribution of the so-called semihard
processes corresponding to parton cascades developing, at least partly, in the
high $p_{t}$ domain becomes increasingly important in the very high energy
limit. Indeed, the smallness of the corresponding strong coupling,
$\alpha_{s}(p_{t}^{2})$, in such cascades becomes compensated by large
collinear and infrared logarithms and by a high parton density [14]. To treat
such processes within the RFT framework, the so-called “semihard Pomeron”
approach has been proposed [2, 15, 16, 17] (see also [18] for a recent
discussion). The underlying basic idea was to employ the above-discussed
phenomenological Pomeron description for purely “soft” nonperturbative (parts
of) parton cascades: for parton virtualities $|q^{2}|<Q_{0}^{2}$, while
treating parton evolution in the perturbative $|q^{2}|>Q_{0}^{2}$ domain by
means of the Dokschitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) formalism
[19, 20, 21], with $Q_{0}^{2}$ being some chosen virtuality cutoff for the
perturbative quantum chromodynamics (pQCD) to be applicable. This allowed one
to develop a Pomeron calculus, based on a “general Pomeron”, the latter being
a sum of the soft and semihard ones, as shown symbolically in Fig. 2.

Figure 2: A “general Pomeron” (left-hand side) consists of the soft and
semihard ones – correspondingly the first and the second contributions in the
right-hand side.

In particular, Eqs. (2–4) hold upon defining the eikonal
$\chi^{\mathbb{P}}_{hp}$ as a sum $\chi^{{\mathbb{P}}_{\rm
soft}}_{hp}+\chi^{{\mathbb{P}}_{\rm sh}}_{hp}$, where $\chi^{{\mathbb{P}}_{\rm
soft}}_{hp}$ is given by Eq. (1), while for $\chi^{{\mathbb{P}}_{\rm
sh}}_{hp}$ one obtains [16, 17]:

$\displaystyle\chi_{hp}^{\mathbb{P}_{{\rm
sh}}}(s,b)=\frac{1}{2}\sum_{I,J}\int\\!\\!d^{2}b^{\prime}\\!\\!\int\\!\frac{dx^{+}}{x^{+}}\frac{dx^{-}}{x^{-}}$
$\displaystyle\times\;\chi_{Ih}^{\mathbb{P}_{{\rm
soft}}}(\frac{s_{0}}{x^{+}},b^{\prime})\,\chi_{Jp}^{\mathbb{P}_{{\rm
soft}}}(\frac{s_{0}}{x^{-}},|\vec{b}-\vec{b}^{\prime}|)$
$\displaystyle\times\;\sigma_{IJ}^{{\rm
QCD}}(x^{+}\,x^{-}\,s,Q_{0}^{2},Q_{0}^{2})\,.$ (5)

Here $\sigma_{IJ}^{{\rm QCD}}(\hat{s},q_{1}^{2},q_{2}^{2})$ corresponds to the
contribution of the DGLAP parton “ladder”, with the ladder leg partons $I$ and
$J$ [(anti)quarks or gluons222We shall not discuss explicitly the contribution
of hard interactions of valence quarks; see, e.g., [16] for the corresponding
details.] characterized by the virtualities $q_{1}^{2}$ and $q_{2}^{2}$,
respectively:

$\displaystyle\sigma_{IJ}^{{\rm
QCD}}(\hat{s},q_{1}^{2},q_{2}^{2})=K\sum_{I^{\prime},J^{\prime}}\\!\int\\!dz^{+}dz^{-}\int\\!dp_{t}^{2}$
$\displaystyle\times\;E_{II^{\prime}}^{{\rm QCD}}(z^{+},q_{1}^{2},\mu_{{\rm
F}}^{2})\,E_{JJ^{\prime}}^{{\rm QCD}}(z^{-},q_{2}^{2},\mu_{{\rm F}}^{2})$
$\displaystyle\times\;\frac{d\sigma_{I^{\prime}J^{\prime}}^{2\rightarrow
2}(z^{+}z^{-}\hat{s},p_{t}^{2},\mu_{\rm R})}{dp_{t}^{2}}$
$\displaystyle\times\;\Theta(\mu_{F}^{2}-\max[q_{1}^{2},q_{2}^{2}])\,,$ (6)

where $d\sigma_{IJ}^{2\rightarrow 2}/dp_{t}^{2}$ is the Born parton cross
section, $p_{t}$ being the parton transverse momentum in the hard process,
$\mu_{F}$ and $\mu_{R}$ are the factorization and renormalization scales,
respectively (we use $\mu_{F}=\mu_{R}=p_{t}/2$), and the factor $K=1.5$ takes
effectively into account higher order QCD corrections. $E_{II^{\prime}}^{{\rm
QCD}}(z,q^{2},\mu_{{\rm F}}^{2})$ describes parton density evolution from the
scale $q^{2}$ to $\mu_{{\rm F}}^{2}$, subject to the initial condition
$E_{II^{\prime}}^{{\rm
QCD}}(z,q^{2},q^{2})=\delta_{I}^{I^{\prime}}\,\delta(1-z)$.

In turn, the eikonal $\chi^{{\mathbb{P}}_{\rm soft}}_{Ih}$ corresponding to a
soft Pomeron exchange between hadron $h$ and parton $I$ is obtained from Eq.
(1), neglecting the small slope of the Pomeron-parton coupling $R_{I}^{2}\sim
1/Q_{0}^{2}$ and replacing the vertex $\gamma_{p}$ by a parameterized Pomeron-
parton vertex $V^{\mathbb{P}}_{I/h}$:

$\displaystyle\chi_{Ih}^{\mathbb{P}_{{\rm
soft}}}(\hat{s},b)=\frac{\gamma_{h}\,V^{\mathbb{P}}_{I/h}(s_{0}/\hat{s})\,(\hat{s}/s_{0})^{\alpha_{\mathbb{P}}(0)-1}}{R_{h}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\,\ln(\hat{s}/s_{0})}$
$\displaystyle\times\;\exp\\!\left[-\frac{b^{2}/4}{R_{h}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\,\ln(\hat{s}/s_{0})}\right].$
(7)

We use333The ansatz, Eqs. (8-9), differs from the one used in [16, 17] by the
factor $(1+x)^{b_{h}}$ chosen to improve the large $x$ behavior of gluon PDFs.
More sophisticated parameterizations may generally be used for
$V^{\mathbb{P}}_{I/h}$. [17]

$\displaystyle
V^{\mathbb{P}}_{g/h}(x)=r_{g/\mathbb{P}}\,(1-w_{qg})\,(1-x)^{\beta_{g/h}}$
$\displaystyle\times\;(1+x)^{b_{h}}$ (8) $\displaystyle
V^{\mathbb{P}}_{q/h}(x)=r_{g/\mathbb{P}}\,w_{qg}\int_{x}^{1}\\!dz\,z^{\alpha_{\mathbb{P}}(0)-1}$
$\displaystyle\times\;P_{qg}(z)\,(1-x/z)^{\beta_{g/h}}\,(1+x/z)^{b_{h}},$ (9)

where $P_{qg}$ is the usual Altarelli-Parisi splitting kernel for three active
flavors and the parameter $r_{g/\mathbb{P}}$ characterizes gluon density in
the soft Pomeron in the low $x$ limit, when probed at the virtuality scale
$Q_{0}^{2}$. We use $\beta_{g/p}=4$ and $\beta_{g/\pi}=\beta_{g/K}=2$, while
the constants $b_{h}$ are fixed requiring momentum conservation for parton
distribution functions (PDFs).

By construction, the eikonal $\chi_{Ih}^{\mathbb{P}_{{\rm soft}}}$ is related
to the generalized parton distribution (GPD) $G_{I/h}$ at the virtuality scale
$Q_{0}^{2}$ [4, 22]:

$x\,G_{I/h}(x,b,Q_{0}^{2})=\chi_{Ih}^{\mathbb{P}_{{\rm soft}}}(s_{0}/x,b)\,.$
(10)

It is further noteworthy that the above-discussed “semihard Pomeron” approach
largely resembles the “heterotic Pomeron” concept proposed in [23] (see also
[24]). In particular, the relatively large slope
$\alpha_{\mathbb{P}}^{\prime}$ of the soft Pomeron gives rise to a rather fast
transverse expansion of parton “clouds”. On the other hand, the perturbative
($|q^{2}|>Q_{0}^{2}$) parton evolution, being characterized by small
transverse displacements ($\lesssim 1/Q_{0}$), leads to a quick rise of parton
density.

##  4 Treatment of “color fluctuations”

One of the drawbacks of the scheme discussed so far is that it offers no room
for inelastic diffraction. To overcome that, one has to account for
transitions of interacting hadrons into various excited states, after each
“elementary” rescattering process (Pomeron exchange) [25]. This can be
conveniently done following the Good-Walker (GW) approach [26] (see, e.g.,
[27] for a recent discussion): assuming both the original hadron $h$ and its
excited states $h^{*}$ to be represented by a superposition of eigenstates of
the scattering matrix:

$\displaystyle|h\rangle=\sum_{i}\sqrt{C^{(i)}_{h}}\,|i\rangle$ (11)
$\displaystyle|h^{*}\rangle=\sum_{i}\sqrt{C^{(i)}_{h^{*}}}\,|i\rangle,$ (12)

with $C^{(i)}_{h}$, $C^{(i)}_{h^{*}}$ being the corresponding partial weights
($\sum_{i}C^{(i)}_{h}=\sum_{i}C^{(i)}_{h^{*}}=1$).

In such an approach, one arrives to a trivial generalization of Eqs. (2–4),
averaging over different combinations of GW Fock states:

$\displaystyle\sigma^{\rm
tot}_{hp}(s)=2\int\\!d^{2}b\sum_{i,j}C^{(i)}_{h}C^{(j)}_{p}$
$\displaystyle\times\;\left[1-e^{-\chi_{hp(ij)}^{\mathbb{P}}(s,b)}\right]$
(13) $\displaystyle\sigma^{\rm el}_{hp}(s)=\int\\!d^{2}b$
$\displaystyle\times\left[\sum_{i,j}C^{(i)}_{h}C^{(j)}_{p}\,(1-e^{-\chi_{hp(ij)}^{\mathbb{P}}(s,b)})\right]^{2}$
(14)
$\displaystyle\sigma^{(n)}_{hp}(s)=\int\\!d^{2}b\sum_{i,j}C^{(i)}_{h}C^{(j)}_{p}\,$
$\displaystyle\times\;\frac{\left[2\chi_{hp(ij)}^{\mathbb{P}}(s,b)\right]^{n}}{n!}\,\exp(-2\chi_{hp(ij)}^{\mathbb{P}}(s,b))\,.$
(15)

Here the eikonals $\chi^{\mathbb{P}}_{hp(ij)}=\chi^{{\mathbb{P}}_{\rm
soft}}_{hp(ij)}+\chi^{{\mathbb{P}}_{\rm sh}}_{hp(ij)}$ correspond to exchanges
of both soft and semihard Pomerons between GW states $|i\rangle$ and
$|j\rangle$ of the projectile and the target, respectively.

In turn, the inelastic cross section is now split into the “absorptive” and
diffractive parts:

$\sigma^{\rm inel}_{hp}(s)\equiv\sigma^{\rm tot}_{hp}(s)-\sigma^{\rm
el}_{hp}(s)=\sigma^{\rm abs}_{hp}(s)+\sigma^{\rm diffr}_{hp}(s)\,,$ (16)

with the former corresponding to any number ($n\geq 1$) of cut Pomeron
exchanges,

$\displaystyle\sigma^{\rm
abs}_{hp}(s)=\sum_{n=1}^{\infty}\sigma^{(n)}_{hp}(s)$
$\displaystyle=\,\int\\!d^{2}b\sum_{i,j}C^{(i)}_{h}C^{(j)}_{p}\,[1-e^{-2\chi_{hp(ij)}^{\mathbb{P}}(s,b)}]\,,$
(17)

and the latter containing contributions of the projectile, target, and double
diffraction.444See, e.g., [5] for the corresponding partial cross sections.

The corresponding soft Pomeron exchange eikonal is defined as

$\displaystyle\chi_{hp(ij)}^{{\mathbb{P}}_{\rm
soft}}(s,b)=\frac{\gamma_{h(i)}\gamma_{p(j)}(s/s_{0})^{\alpha_{\mathbb{P}}(0)-1}}{R_{h(i)}^{2}+R_{p(j)}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\ln(s/s_{0})}\,$
$\displaystyle\times\,\exp\\!\left[-\frac{b^{2}/4}{R_{h(i)}^{2}+R_{p(j)}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\ln(s/s_{0})}\right]\\!,$
(18)

taking into consideration that different GW Fock states are generally
characterized by different sizes and different couplings to the Pomeron.
Moreover, it is quite reasonable to assume that the coupling $\gamma_{h(i)}$
is approximately proportional to the transverse area of the state [11]:

$\gamma_{h(i)}=g_{0}\,R_{h(i)}^{2},$ (19)

using thus a universal parameter $g_{0}$.

The “semihard Pomeron” eikonal is generalized similarly:

$\displaystyle\chi_{hp(ij)}^{\mathbb{P}_{{\rm
sh}}}(s,b)=\frac{1}{2}\sum_{I,J}\int\\!d^{2}b^{\prime}\int\\!\frac{dx^{+}}{x^{+}}\frac{dx^{-}}{x^{-}}$
$\displaystyle\times\;\chi_{Ih(i)}^{\mathbb{P}_{{\rm
soft}}}(\frac{s_{0}}{x^{+}},b^{\prime})\;\chi_{Jp(j)}^{\mathbb{P}_{{\rm
soft}}}(\frac{s_{0}}{x^{-}},|\vec{b}-\vec{b}^{\prime}|)$
$\displaystyle\times\;\sigma_{IJ}^{{\rm
QCD}}(x^{+}\,x^{-}\,s,Q_{0}^{2},Q_{0}^{2})\,,$ (20)

where the eikonal $\chi_{Ih(i)}^{\mathbb{P}_{{\rm soft}}}$ corresponds to a
soft Pomeron exchange between parton $I$ and GW state $|i\rangle$ of hadron
$h$:

$\displaystyle\chi_{Ih(i)}^{\mathbb{P}_{{\rm
soft}}}(\hat{s},b)=\frac{\gamma_{h(i)}V^{\mathbb{P}}_{I/h(i)}(s_{0}/\hat{s})(\hat{s}/s_{0})^{\alpha_{\mathbb{P}}(0)-1}}{R_{h(i)}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\,\ln(\hat{s}/s_{0})}$
$\displaystyle\times\;\exp\\!\left[-\frac{b^{2}/4}{R_{h(i)}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\,\ln(\hat{s}/s_{0})}\right]\\!,$
(21)

with $V^{\mathbb{P}}_{I/h(i)}$ being defined by Eqs. (8-9), under the
replacement $b_{h}\rightarrow b_{h(i)}$. Here we have an important feature:
assuming a universal gluon density $r_{g/\mathbb{P}}$ in the soft Pomeron in
the low $x$ limit, at the scale $Q_{0}^{2}$, the momentum conservation for
PDFs for individual Fock states,

$\displaystyle
x\,f_{I/h(i)}(x,Q_{0}^{2})=\int\\!d^{2}b\;x\,G_{I/h(i)}(x,b,Q_{0}^{2})$
$\displaystyle=\int\\!d^{2}b\;\chi_{Ih(i)}^{\mathbb{P}_{{\rm
soft}}}(s_{0}/x,b)\,,$ (22)

gives rise to different $b_{h(i)}$ for different states. Consequently, partial
PDFs $f_{I/h(i)}$ vary considerably from one Fock state to another [28]: with
smaller Fock states being characterized by smaller (integrated) parton
densities in the low $x$ limit but having harder PDF shapes.

It is noteworthy, however, that the above-discussed approach is applicable,
strictly speaking, to the treatment of low mass diffraction only. Indeed,
considering hadron $h$ transitions into multiparticle states $h^{*}$ of
arbitrary mass, one can no longer apply the decomposition, Eqs. (11-12), based
on a finite number of GW states:555In principle, one may consider a continuum
of intermediate multiparticle states, postulating some energy and mass
dependence for the decomposition in Eqs. (11-12). with increasing energy,
larger and larger excited states will play an important role.

In the following, we consider equal probabilities for different GW states,
$C^{(i)}_{h}\equiv 1/N_{\rm GW}$, using $N_{\rm GW}=3$ for any hadron666We
verified explicitly that using a larger number of GW states does not modify
our results significantly. and choosing a loguniform distribution for
$R^{2}_{h(i)}$:

$\displaystyle R^{2}_{h(i)}=R^{2}_{h(1)}\,d_{h}^{\frac{i-1}{N_{\rm GW}-1}}.$
(23)

##  5 Enhanced Pomeron diagrams

The major feature inherited from the previous model version, QGSJET-II [4, 5],
is a treatment of nonlinear interaction effects, based on all-order
resummation of the underlying Pomeron-Pomeron interaction diagrams [29, 30,
31]. The simplest examples of such, so-called enhanced, graphs are shown in
Fig. 3,

Figure 3: Simplest examples of enhanced Pomeron diagrams.

which correspond to rescattering of intermediate partons in the “elementary”
parton cascades off the projectile (a, b, f, g) and the target (c, d, f, g)
hadrons or off each other (e). Taking such diagrams into consideration amounts
to replace the eikonal $\chi^{\mathbb{P}}_{hp(ij)}$ in Eqs. (13–15) by
$\chi^{\rm scr}_{hp(ij)}=\chi^{\mathbb{P}}_{hp(ij)}+\chi^{\rm enh}_{hp(ij)}$,
where $\chi^{\rm enh}_{hp(ij)}$ corresponds to the summary contribution of all
significant irreducible enhanced Pomeron graphs exchanged between GW Fock
states $|i\rangle$ and $|j\rangle$ of the projectile and the target,
respectively.

Using eikonal vertices for the transition of $m$ into $n$ Pomerons ($m+n\geq
3$) [32],

$G^{(m,n)}=G\,\gamma_{\mathbb{P}}^{m+n},$ (24)

with the constant $G$ being related to the triple-Pomeron coupling
$r_{3\mathbb{P}}$ as $G=r_{3\mathbb{P}}/(4\pi\gamma_{\mathbb{P}}^{3})$, and
neglecting the small slope of the triple-Pomeron vertex $R^{2}_{3\mathbb{P}}$,
one was able to obtain $\chi^{\rm enh}_{hp(ij)}$ in a relatively compact form
[30, 31, 5]:

$\displaystyle\chi_{hp(ij)}^{{\rm
enh}}(s,b)=G\\!\int_{\xi}^{Y-\xi}\\!dy^{\prime}\\!\int\\!\\!d^{2}b^{\prime}$
$\displaystyle\times\,\\{[(1-e^{-\chi_{h(i)|p(j)}^{{\rm
net}}})\,(1-e^{-\chi_{p(j)|h(i)}^{{\rm net}}})$
$\displaystyle-\,\chi_{h(i)|p(j)}^{{\rm net}}\,\chi_{p(j)|h(i)}^{{\rm
net}}]-[\chi_{h(i)|p(j)}^{{\rm net}}-\chi_{h(i)}^{{\rm loop}}]$
$\displaystyle\times\,[(1-e^{-\chi_{p(j)|h(i)}^{{\rm
net}}})\,e^{-\chi_{h(i)|p(j)}^{{\rm net}}}-\chi_{p(j)|h(i)}^{{\rm net}}]$
$\displaystyle+\,\chi_{p(j)}^{\mathbb{P}}(y^{\prime},b^{\prime})\,[\chi_{h(i)}^{{\rm
loop}}-\chi_{h(i)}^{{\rm loop(1)}}]\\}.$ (25)

Here $Y=\ln(s/s_{0})$, $\xi$ is the minimal rapidity “size” of the Pomeron,
and the omitted arguments of the eikonals read $\chi_{h(i)|p(j)}^{{\rm
net}}=\chi_{h(i)|p(j)}^{{\rm
net}}(Y-y^{\prime},\vec{b}-\vec{b}^{\prime}|Y,\vec{b})$,
$\chi_{p(j)|h(i)}^{{\rm net}}=\chi_{p(j)|h(i)}^{{\rm
net}}(y^{\prime},\vec{b}^{\prime}|Y,\vec{b})$, $\chi_{h(i)}^{{\rm
loop}}=\chi_{h(i)}^{{\rm loop}}(Y-y^{\prime},|\vec{b}-\vec{b}^{\prime}|)$,
$\chi_{h(i)}^{{\rm loop(1)}}=\chi_{h(i)}^{{\rm
loop(1)}}(Y-y^{\prime},|\vec{b}-\vec{b}^{\prime}|)$.

The “net-fan” contributions $\chi_{h(i)|p(j)}^{{\rm net}}$ are defined by a
recursive equation:

$\displaystyle\chi_{h(i)|p(j)}^{{\rm
net}}(y^{\prime},\vec{b}^{\prime}|Y,\vec{b})=\chi_{h(i)}^{{\rm
loop}}(y^{\prime},b^{\prime})+G\\!\int\\!\\!d^{2}b^{\prime\prime}$
$\displaystyle\times\int_{\xi}^{y^{\prime}-\xi}\\!dy^{\prime\prime}\,(1-e^{-\chi^{{\rm
loop}}(y^{\prime}-y^{\prime\prime},|\vec{b}^{\prime}-\vec{b}^{\prime\prime}|)})$
$\displaystyle\times\left[(1-e^{-\chi_{h(i)|p(j)}^{{\rm
net}}(y^{\prime\prime},\vec{b}^{\prime\prime}|Y,\vec{b})})\right.$
$\displaystyle\times\,\exp(-\chi_{p(j)|h(i)}^{{\rm
net}}(Y-y^{\prime\prime},\vec{b}-\vec{b}^{\prime\prime}|Y,\vec{b}))$
$\displaystyle-\left.\chi_{h(i)|p(j)}^{{\rm
net}}(y^{\prime\prime},\vec{b}^{\prime\prime}|Y,\vec{b})\right]\\!,$ (26)

where $\chi_{h(i)}^{{\rm loop}}$ and $\chi_{h(i)}^{{\rm loop(1)}}$ correspond
to general irreducible two-point sequences of Pomerons and Pomeron loops,
exchanged between GW state $|i\rangle$ of hadron $h$ and a multi-Pomeron
vertex, while $\chi^{{\rm loop}}$ and $\chi^{{\rm loop(1)}}$ are contributions
of such sequences exchanged between two multi-Pomeron vertices (see [5] for
more details). Neglecting Pomeron loop insertions, $\chi_{h(i)}^{{\rm loop}}$
and $\chi_{h(i)}^{{\rm loop(1)}}$ reduce to the eikonal
$\chi_{h(i)}^{\mathbb{P}}=\chi^{{\mathbb{P}}_{\rm
soft}}_{h(i)}+\chi^{{\mathbb{P}}_{\rm sh}}_{h(i)}$ describing an exchange of a
soft or semihard Pomeron between the hadron $h$ represented by its GW state
$|i\rangle$ and a multi-Pomeron vertex. Here $\chi^{{\mathbb{P}}_{\rm
soft}}_{h(i)}(y,b)$ is obtained from Eq. (18), for $s=s_{0}\,e^{y}$, replacing
the vertex factor $\gamma_{p(j)}$ by $\gamma_{\mathbb{P}}$ and $R_{p(j)}^{2}$
by $R_{3\mathbb{P}}^{2}\simeq 0$:

$\displaystyle\chi^{{\mathbb{P}}_{\rm
soft}}_{h(i)}(y,b)=\frac{\gamma_{h(i)}\gamma_{\mathbb{P}}\,e^{(\alpha_{\mathbb{P}}(0)-1)y}}{R_{h(i)}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\,y}e^{\frac{-b^{2}/4}{R_{h(i)}^{2}+\alpha_{\mathbb{P}}^{\prime}(0)\,y}}.$
(27)

In turn, $\chi^{{\mathbb{P}}_{\rm sh}}_{h(i)}(y,b)$ is obtained from Eq. (20),
for $s=s_{0}\,e^{y}$, replacing $\chi_{Jp(j)}^{\mathbb{P}_{{\rm soft}}}$ by
$\chi_{J\mathbb{P}}^{\mathbb{P}_{{\rm soft}}}$, with

$\displaystyle\chi_{J\mathbb{P}}^{\mathbb{P}_{{\rm
soft}}}(\hat{s},b)=\frac{\gamma_{\mathbb{P}}\,V^{\mathbb{P}}_{J}(s_{0}/\hat{s})\,(\hat{s}/s_{0})^{\alpha_{\mathbb{P}}(0)-1}}{\alpha_{\mathbb{P}}^{\prime}(0)\ln(\hat{s}/s_{0})}$
$\displaystyle\times\;\exp\\!\left[-\frac{b^{2}}{4\alpha_{\mathbb{P}}^{\prime}(0)\ln(\hat{s}/s_{0})}\right],$
(28)

where $V^{\mathbb{P}}_{J}$ is defined by Eqs. (8-9), for
$\beta_{g/h}=\beta_{g/\pi}$.

In contrast to [4, 5], where $\gamma_{\mathbb{P}}$ was treated as an
adjustable parameter, here we make a specific choice:

$\gamma_{\mathbb{P}}=r_{3\mathbb{P}}/(\alpha_{\mathbb{P}}(0)-1).$ (29)

Taking into account the “renormalization” of the soft Pomeron in the “dense”
(high $s$, small $b$) limit [32], this pushes it into the “critical” regime in
such a limit (see [33] for a recent discussion): with the renormalized Pomeron
intercept

$\alpha_{\mathbb{P}}^{\rm(ren)}(0)=\alpha_{\mathbb{P}}(0)-r_{3\mathbb{P}}/\gamma_{\mathbb{P}}=1\,.$
(30)

In turn, this leads to a “saturation” of both partial gluon and sea
(anti)quark GPDs $G_{I/h(i)}^{\rm scr}$ and of the corresponding total GPDs,

$G_{I/h}^{\rm scr}(x,b,Q_{0}^{2})=\sum_{i}C^{(i)}_{h}\,G_{I/h(i)}^{\rm
scr}(x,b,Q_{0}^{2})\,,$ (31)

at the virtuality scale $Q_{0}^{2}$, in the low $x$ and small $b$ limit, for
any hadron $h$ [34].

Here, taking into account absorptive corrections due to enhanced Pomeron
diagrams, $G_{I/h(i)}^{\rm scr}$ are defined as777In [4, 22], a simplified
expression for $G_{I/h(i)}^{\rm scr}$ had been provided, neglecting Pomeron
loop contributions. [4, 22]

$\displaystyle x\,G_{I/h(i)}^{\rm scr}(x,b,Q_{0}^{2})$
$\displaystyle=\;\chi_{Ih(i)}^{\mathbb{P}_{{\rm
soft}}}(s_{0}/x,b)+G\\!\int\\!\\!d^{2}b^{\prime}$
$\displaystyle\times\;\int_{\xi}^{-\ln
x}\\!dy^{\prime}\left\\{\chi_{I\mathbb{P}}^{\mathbb{P}_{{\rm
soft}}}(s_{0}e^{-y^{\prime}}/x,|\vec{b}-\vec{b}^{\prime}|)\right.$
$\displaystyle\times\left[\chi_{h(i)}^{{\rm
loop}}(y^{\prime},b^{\prime})-\chi_{h(i)}^{{\rm
loop(1)}}(y^{\prime},b^{\prime})\right]$ $\displaystyle+\,\chi_{I}^{\rm
loop}(-\ln x-y^{\prime},|\vec{b}-\vec{b}^{\prime}|)$
$\displaystyle\times\left.\left[1-e^{-\chi_{h(i)}^{{\rm
fan}}(y^{\prime},b^{\prime})}-\chi_{h(i)}^{{\rm
fan}}(y^{\prime},b^{\prime})\right]\right\\}\\!.$ (32)

Here $\chi_{h(i)}^{{\rm fan}}$ is a solution of the “fan” diagram equation
[c.f. Eq. (26)]:

$\displaystyle\chi_{h(i)}^{{\rm fan}}(y^{\prime},b^{\prime})=\chi_{h(i)}^{{\rm
loop}}(y^{\prime},b^{\prime})+G\int\\!d^{2}b^{\prime\prime}$
$\displaystyle\times\;\int_{\xi}^{y^{\prime}-\xi}\\!dy^{\prime\prime}\left[1-e^{-\chi^{\rm
loop}(y^{\prime}-y^{\prime\prime},|\vec{b}^{\prime}-\vec{b}^{\prime\prime}|)}\right]$
$\displaystyle\times\left[1-e^{-\chi_{h(i)}^{{\rm
fan}}(y^{\prime\prime},b^{\prime\prime})}-\chi_{h(i)}^{{\rm
fan}}(y^{\prime\prime},b^{\prime\prime})\right],$ (33)

while $\chi_{I}^{\rm loop}$ is defined as

$\displaystyle\chi_{I}^{\rm
loop}(y^{\prime},b^{\prime})=\chi_{I\mathbb{P}}^{\mathbb{P}_{{\rm
soft}}}(s_{0}e^{y^{\prime}},b^{\prime})+G\int\\!d^{2}b^{\prime\prime}$
$\displaystyle\times\;\int_{\xi}^{y^{\prime}-\xi}\\!dy^{\prime\prime}\;\chi_{I\mathbb{P}}^{\mathbb{P}_{{\rm
soft}}}(s_{0}e^{y^{\prime}-y^{\prime\prime}},|\vec{b}^{\prime}-\vec{b}^{\prime\prime}|)$
$\displaystyle\times\left[1-e^{-\chi^{\rm
loop}(y^{\prime\prime},b^{\prime\prime})}-\chi^{\rm
loop(1)}(y^{\prime\prime},b^{\prime\prime})\right].$ (34)

Applying the AGK cutting rules [10], one was able to obtain the complete set
of unitarity cuts of elastic scattering diagrams for hadron-hadron collisions,
corresponding to the above-discussed resummation scheme, explicitly verifying
the $s$-channel unitarity of the approach, and to derive positive-definite
partial cross sections for all the various configurations of final states,
including diffractive ones [30, 31]. In turn, those allowed one to develop a
MC procedure for generating such configurations both for hadron-proton and for
hadron-nucleus (nucleus-nucleus) scattering events [5].

As discussed in [4, 22], an important feature of the described approach is a
consistency with the collinear factorization of pQCD: by virtue of the AGK
cancellations [10], the inclusive parton jet production cross section is
defined by the usual factorization ansatz:

$\displaystyle\frac{d\sigma_{hp}^{\rm jet}(s,p_{{\rm t}})}{dp_{{\rm
t}}^{2}}=K\\!\int\\!dx^{+}dx^{-}\sum_{I,J}f_{I/h}^{\rm scr}(x^{+},\mu_{{\rm
F}}^{2})$ $\displaystyle\times\,f_{J/p}^{\rm scr}(x^{-},\mu_{{\rm
F}}^{2})\,\frac{2\,d\sigma_{IJ}^{2\rightarrow
2}(x^{+}x^{-}s,p_{t}^{2},\mu_{\rm R})}{dp_{{\rm t}}^{2}}\,.$ (35)

Here the PDFs $f_{I/h}^{\rm scr}$, with the absorptive corrections taken into
account, are expressed via the partial GPDs $G_{I/h(i)}^{\rm scr}$, Eq. (32),
evolving the latter from $Q_{0}^{2}$ to $\mu_{{\rm F}}^{2}$:

$\displaystyle f_{I/h}^{\rm scr}(x,\mu_{{\rm
F}}^{2})=\sum_{i}C^{(i)}_{h}\int\\!d^{2}b\sum_{I^{\prime}}\int_{x}^{1}\\!\frac{dz}{z}$
$\displaystyle\times\,E_{I^{\prime}I}^{{\rm QCD}}(z,Q_{0}^{2},\mu_{{\rm
F}}^{2})\,G^{\rm scr}_{I^{\prime}/h(i)}(x/z,b,Q_{0}^{2})\,.$ (36)

##  6 Dynamical power corrections to hard scattering

As mentioned in Section 5, an important feature of the above-discussed
approach is the consistency with the collinear factorization of pQCD: the
inclusive jet cross section is defined by Eq. (35). However, this creates an
unpleasant sensitivity of the model predictions to the choice of the
“infrared” cutoff $Q_{0}^{2}$: since $d\sigma_{hp}^{\rm jet}(s,p_{{\rm
t}})/dp_{{\rm t}}^{2}$ explodes in the small $p_{t}$ limit. For example, in
case of the QGSJET-II model, a reasonable consistency with collider
measurements is reached for a rather high value of that cutoff, $Q_{0}^{2}=3$
GeV2 [5]. On the other hand, one may expect the pQCD approach to remain
applicable for parton virtualities as small as $\sim 1$ GeV2. It is thus
natural to ask ourselves whether there exists a perturbative mechanism capable
of damping the jet production at small $p_{t}$.

To address this question, it is useful to remind oneself that the collinear
factorization of pQCD is established at the leading twist level [35, 36],
i.e., neglecting the so-called higher twist (HT) corrections suppressed by
powers of the relevant hard scale. One may thus expect that those are such
power corrections which should provide the desirable suppression of low
$p_{t}$ jet production.

Unfortunately it is hardly possible at the present stage to treat HT effects
in hadronic collisions in a systematic way, especially, regarding their
potential implementation in MC event generators: in particular, since this
involves a significant number of unknown multiparton correlators and the
corresponding HT contributions are not generally positive-definite, excluding
thereby a probabilistic interpretation [37, 38, 39]. Therefore, we adopt here
a phenomenological approach [34, 40], concentrating on a particular class of
dynamical power corrections to parton scattering processes, corresponding to
coherent multiple rescattering of $s$-channel partons on virtual “soft”
[characterized by small light cone (LC) momentum fractions, $x_{g}\sim 0$]
gluon pairs [41, 42, 43]. Such contributions have been shown to provide
dominant nuclear size-enhanced power corrections to the low $x$ and low
$Q^{2}$ behavior of structure functions (SFs) in deep inelastic scattering
(DIS) on nuclear targets [41, 42] and to the suppression of jet
$p_{t}$-spectra in high energy proton scattering on heavy nuclei, for
moderately small $p_{t}$ [43].

###  6.1 Resummed $A$-enhanced power corrections [41, 42, 43]

Regarding HT corrections to nuclear SFs, the dominant $A$-enhanced
contributions, in the small Bjorken $x$ limit, were shown to come from
diagrams of the kind depicted in Fig. 4 (left),

Figure 4: Example diagrams for dominant $A$-enhanced power corrections to
nuclear DIS (left) and to jet production in proton-nucleus scattering (right).
Propagators marked by crosses and vertical dashes correspond to pole and
contact terms, respectively.

corresponding to a rescattering of the struck quark on soft, pairwise singlet,
virtual gluon pairs [41, 42]. The perturbative hard part of the graph is
characterized by a specific structure of the $s$-channel quark propagators:
with the gluon fields in a pair being separated by the so-called “contact”
term, implying no propagation along the LC coordinate [44]; propagators which
separate such gluon pairs from the quark field and from each other are
represented, on the other hand, by pole terms corresponding to a propagation
over considerable LC distances. It is such a prescription which gives rise to
the nuclear enhancement: in the low $x$ limit, the struck quark from one
nucleon in the nucleus propagates over large distances $\propto 1/(x\,p^{+})$,
$p^{+}$ being the LC-plus (LC+) momentum of the nucleon, scattering coherently
on many correlated soft gluon pairs from other nucleons [41, 42]. Upon the
full all-order resummation of such contributions (in the LC $A^{+}=0$ gauge),
one obtained the HT correction to the nuclear transverse structure function
$F_{\rm T}^{(A)}$ (c.f. Eq. (8) in [41]):

$\displaystyle\delta_{HT}F_{\rm T}^{(A)}(x,Q^{2})$
$\displaystyle=\;\sum_{q}\frac{e_{q}^{2}}{2}\sum_{n\geq
1}\frac{1}{n!}\left[\frac{4\pi^{2}\alpha_{s}x}{3Q^{2}}\right]^{n}\frac{d^{n}}{d^{n}\\!x}T^{(n)}_{q\underset{n}{\underbrace{gg...}}}(x)\,,$
(37)

where $e_{q}$ is the fractional charge of (anti)quark $q$ and the multiparton
correlators $T^{(n)}_{qgg...}$ are defined as

$\displaystyle
T^{(n)}_{q\underset{n}{\underbrace{gg...}}}(x)=\int\\!\frac{dy^{-}}{4\pi}\;e^{ip^{+}x\,y^{-}}\langle
A|\bar{\psi}(0)\,\gamma^{+}\,$
$\displaystyle\times\,\prod_{i=1}^{n}\left[\int\\!p^{+}dy^{-}_{g_{i}}\,\Theta(y^{-}_{g_{i}})\,\hat{F}^{2}(y^{-}_{g_{i}})\right]\psi(y^{-})|A\rangle,$
(38)

with

$\displaystyle\hat{F}^{2}(y^{-}_{g})=\int\\!\frac{d\tilde{y}^{-}_{g}}{2\pi\,p^{+}}\;F^{+\alpha}(y^{-}_{g})\,F^{+}_{\alpha}(\tilde{y}^{-}_{g})$
$\displaystyle\times\;\Theta(y^{-}_{g}-\tilde{y}^{-}_{g})\,.$ (39)

Here $y^{-}$, $y^{-}_{g_{i}}$ are LC-minus (LC-) coordinates of the fields and
$F^{+}_{\alpha}$ is the projection of the gluon field tensor on the LC+
direction.

Likewise, in case of nucleus-proton scattering, the dominant $A$-enhanced HT
corrections to jet production cross section arise from the diagrams of Fig. 4
(right), corresponding to a rescattering of the nuclear parton (quark or
gluon), participating in the hard scattering, on such soft gluon pairs, with
the same structure of $s$-channel propagators along the struck parton
line888Alternative configurations of the hard scattering part of the graph
contain contact terms leading to a loss of the $A$-enhancement [43]. [43].
Resumming such contributions to all orders, one obtained [43]:

$\displaystyle\delta_{HT}\frac{d\sigma_{pA}^{\rm jet}(s,p_{{\rm t}})}{dp_{{\rm
t}}^{2}}=\int\\!dx^{+}dx^{-}$
$\displaystyle\times\,\sum_{I,J}f_{J/p}(x^{-},\mu_{{\rm F}}^{2})\sum_{n\geq
1}\frac{1}{n!}\left[\frac{-C_{I}\pi^{2}\alpha_{s}x^{+}}{\hat{t}}\right]^{n}$
$\displaystyle\times\,\frac{d^{n}}{d^{n}\\!x}\left[T^{(n)}_{I\underset{n}{\underbrace{gg...}}}(x^{+})\,\frac{2\,d\sigma_{IJ}^{2\rightarrow
2}}{dp_{{\rm t}}^{2}}\right],$ (40)

with $C_{q(\bar{q})}=C_{F}=4/3$, $C_{g}=C_{A}=3$, and $\hat{t}=q^{2}$ being
the momentum transfer squared for parton-parton scattering. In addition to
$T^{(n)}_{qgg...}$ defined by Eqs. (38-39), Eq. (40) involves multigluon
correlators $T^{(n)}_{ggg...}$:

$\displaystyle
T^{(n)}_{g\underset{n}{\underbrace{gg...}}}(x)=\int\\!\frac{dy^{-}}{2\pi
xp^{+}}\;e^{ip^{+}x\,y^{-}}\langle A|F^{+\beta}(0)$
$\displaystyle\times\,\prod_{i=1}^{n}\left[\int\\!p^{+}dy^{-}_{g_{i}}\Theta(y^{-}_{g_{i}})\hat{F}^{2}(y^{-}_{g_{i}})\right]F^{+}_{\beta}(y^{-})|A\rangle.$
(41)

Further, one proposed a model for the multiparton correlators
$T^{(n)}_{Igg...}$, $I=q,g$: assuming that the dominant contributions to Eqs.
(37) and (40) arise from a rescattering of the struck nuclear parton on gluon
pairs belonging to different nucleons and performing the corresponding
factorization [41]:

$\langle A|\hat{O}_{0}\prod_{i=1}^{n}\hat{O}_{i}|A\rangle\propto\langle
p|\hat{O}_{0}|p\rangle\prod_{i=1}^{n}\langle p|\hat{O}_{i}|p\rangle.$ (42)

This allowed one to obtain closed compact results for $F_{\rm T}^{(A)}$ and
$d\sigma^{\rm jet}_{pA}/dp_{t}^{2}$, the power corrections being accounted
for:

$\displaystyle F_{\rm T}^{(A)}(x,Q^{2})\simeq A\sum_{q}\frac{e_{q}^{2}}{2}$
$\displaystyle\times\;f_{q/p}(x+xC_{q}\xi^{2}(A^{1/3}-1)/Q^{2},Q^{2})$ (43)
$\displaystyle\frac{d\sigma_{pA}^{\rm jet}(s,p_{{\rm t}})}{dp_{{\rm
t}}^{2}}\simeq A\int\\!dx^{+}dx^{-}$
$\displaystyle\times\,\sum_{I,J}f_{I/p}(x^{+}-\frac{x^{+}C_{I}\xi^{2}(A^{1/3}-1)}{\hat{t}},\mu_{{\rm
F}}^{2})$ $\displaystyle\times\;f_{J/p}(x^{-},\mu_{{\rm
F}}^{2})\,\frac{2\,d\sigma_{IJ}^{2\rightarrow 2}}{dp_{{\rm t}}^{2}}\,,$ (44)

where $\xi^{2}\propto\langle p|\hat{F}^{2}|p\rangle\propto\lim_{x\rightarrow
0}x\,f_{g}/p$ defines the characteristic scale of the HT corrections.

###  6.2 Phenomenological implementation in QGSJET-III

Since we are going to extrapolate the treatment of [43] to the case of hadron-
proton scattering, we need a different approach for modeling the corresponding
multiparton correlators. Starting with the quark-gluon correlator
$T^{(1)}_{qg}$, a closer look at Eq. (38) reveals that it formally coincides,
up to a factor, with the quark-gluon 2GPD $F^{(2)}_{qg}$ multiplied by the
gluon LC momentum fraction $x_{g}$, in the limit $x_{g}\rightarrow 0$ and for
zero transverse separation between the two partons, $\vec{\Delta}=0$ (see,
e.g. [8] for the corresponding definitions), and similarly for $T^{(1)}_{gg}$
in Eq. (41). This motivated us to employ a probabilistic treatment for
$T^{(1)}_{qg}$ and $T^{(1)}_{gg}$, interpreting them as
$\propto\left.x_{g}\,F^{(2)}_{qg}\right|_{\vec{\Delta}=0}$ and
$\propto\left.x_{g}\,F^{(2)}_{gg}\right|_{\vec{\Delta}=0}$, respectively, and
to proceed in a similar way with all the other correlators $T^{(n)}_{Igg...}$
involving larger numbers of soft gluons.

Here we have to make additional assumptions concerning the relevant virtuality
scales and gluon momentum fractions in the corresponding multiparton GPDs,
e.g., for $Q_{q}^{2},Q_{g}^{2}$, and $x_{g}$ in
$F^{(2)}_{qg}(x,x_{g},Q_{q}^{2},Q_{g}^{2},\vec{\Delta})$. While the natural
choice for $Q_{q}^{2}$ is the factorization scale $\mu_{\rm F}^{2}$ for the
hard process, one usually considers soft gluons to be purely nonperturbative
ones, with $Q_{g}^{2}\sim\Lambda_{\rm QCD}^{2}$ (e.g. [45]). Instead, we set
$Q_{g}^{2}$ equal to our separation scale $Q_{0}^{2}$, in order to describe
the GPDs by soft Pomeron asymptotics, plus absorptive corrections, Eq. (32).

Finally, assuming a finite virtuality for the soft gluons implies that they
have nonzero LC± momentum fractions $x_{g}^{\pm}$:

$|q_{g}^{2}|\sim x_{g}^{+}\,x^{-}_{g}\,s\,.$ (45)

In the factorization procedure which led to Eqs. (37) and (40), one neglected
LC- momentum components for projectile partons (similarly neglecting LC+
momenta of target partons) and considered the limit $x_{g}\rightarrow 0$ for
the soft gluons involved in the process. Here, taking into account the finite
virtuality of such gluons, Eq. (45), and the fact that these gluons belong to
the projectile proton (for the case of rescattering on the projectile soft
gluons), their LC- momentum fractions should be much smaller than the LC-
fraction of the target parton participating in the hard process:

$x^{-}_{g}\sim\frac{|q_{g}^{2}|}{x_{g}^{+}\,s}\ll x^{-}\,.$ (46)

Since we expect a rather weak $x_{g}$-dependence for $x_{g}\,F^{(2)}_{qg}$ and
$x_{g}\,F^{(2)}_{gg}$ in the small $x_{g}$ limit at the low virtuality scale
$Q_{g}^{2}=Q_{0}^{2}$, we set

$x_{g}=\frac{Q_{0}^{2}}{x^{-}\,s}\,.$ (47)

Using these additional assumptions, we obtain the leading power correction to
the inclusive parton jet production cross section as [c.f. Eq. (40)]

$\displaystyle\delta^{(1)}_{HT}\frac{d\sigma_{hp}^{\rm jet}(s,p_{{\rm
t}})}{dp_{{\rm t}}^{2}}$
$\displaystyle=K\int\\!dx^{+}dx^{-}\sum_{I,J}f_{J/p}^{\rm scr}(x^{-},\mu_{{\rm
F}}^{2})$ $\displaystyle\times\;\frac{K_{\rm HT}\,C_{I}\,\pi^{2}\,\alpha_{\rm
s}(\mu_{\rm R}^{2})\,x^{+}}{|\hat{t}|}\,\frac{2\,d\sigma_{IJ}^{2\rightarrow
2}(\hat{s},p_{t}^{2},\mu_{\rm R})}{dp_{{\rm t}}^{2}}$
$\displaystyle\times\left.x_{g}^{+}F^{(2)}_{Ig/h}(x^{+},x_{g}^{+},\mu_{\rm
F}^{2},Q_{0}^{2},\vec{\Delta}=\vec{0})\right|_{x_{g}^{+}=\frac{Q_{0}^{2}}{x^{-}\,s}},$
(48)

where $\hat{s}=x^{+}x^{-}s$ and, in view of the numerous brute force
assumptions made, we introduced an adjustable parameter $K_{\rm HT}$ which
controls the magnitude of the HT corrections in our approach.

Including also higher power corrections, accounting for the GW decomposition
of hadron wave functions, and expressing multiparton GPDs via single parton
ones (thereby neglecting parton-parton correlations at the $Q_{0}^{2}$ scale),

$\displaystyle
F^{(n)}_{I\underset{n}{\underbrace{gg...}}/h(i)}(x,x_{g_{1}},...,Q^{2},Q^{2}_{g_{1}},...,\vec{\Delta}_{g_{1}},...)$
$\displaystyle\simeq\int\\!d^{2}b_{I}\;G^{\rm scr}_{I/h(i)}(x,b_{I},Q^{2})$
$\displaystyle\times\prod_{i=1}^{n}G^{\rm
scr}_{g/h(i)}(x_{g_{i}},|\vec{b}_{I}+\vec{\Delta}_{g_{i}}|,Q^{2}_{g_{i}})\,,$
(49)

considering rescatterings on soft gluon pairs both from the projectile and
from the target (taking into account that those become significant in
different parts of the kinematic space), we finally get

$\displaystyle\frac{d\sigma_{hp}^{\rm jet}(s,p_{{\rm t}})}{dp_{{\rm
t}}^{2}}\simeq\int\\!d^{2}b\left\\{K\\!\int\\!\\!d^{2}b^{\prime}\sum_{i,j}C^{(i)}_{h}C^{(j)}_{p}\right.$
$\displaystyle\times\int\\!dx^{+}dx^{-}\sum_{I,J}G_{I/h(i)}^{\rm
scr}(x^{+}+\tilde{x}^{+},b^{\prime},\mu_{{\rm F}}^{2})$
$\displaystyle\times\left.G_{J/p(j)}^{\rm
scr}(x^{-}+\tilde{x}^{-},|\vec{b}-\vec{b}^{\prime}|,\mu_{{\rm
F}}^{2})\,\frac{2\,d\sigma_{IJ}^{2\rightarrow 2}}{dp_{{\rm t}}^{2}}\right\\},$
(50)

where

$\displaystyle\tilde{x}^{+}=x^{+}\,\frac{K_{\rm
HT}\,C_{I}\,\pi^{2}\,\alpha_{\rm s}(\mu_{\rm R}^{2})}{|\hat{t}|}$
$\displaystyle\times\left.x_{g}^{+}G_{g/h(i)}^{\rm
scr}(x_{g}^{+},b^{\prime},Q_{0}^{2})\right|_{x_{g}^{+}=\frac{Q_{0}^{2}}{x^{-}\,s}}$
(51) $\displaystyle\tilde{x}^{-}=x^{-}\,\frac{K_{\rm
HT}\,C_{J}\,\pi^{2}\,\alpha_{\rm s}(\mu_{\rm R}^{2})}{|\hat{t}|}$
$\displaystyle\times\left.x_{g}^{-}G_{g/p(j)}^{\rm
scr}(x_{g}^{-},|\vec{b}-\vec{b}^{\prime}|,Q_{0}^{2})\right|_{x_{g}^{-}=\frac{Q_{0}^{2}}{x^{+}\,s}}.$
(52)

Like in the original approach of Refs. [41, 42, 43], the effect of the
considered HT corrections amounts to a shift of the LC± momentum fractions of
the active partons $I$ and $J$, participating in the hard scattering. However,
in our case, due to the assumed model for multiparton correlators, the
magnitude of these shifts $\tilde{x}^{\pm}$, Eqs. (51-52), depends on the
collision kinematics. Consequently, the overall strength of the HT corrections
varies with the energy and impact parameter of the collision, increasing in
the “dense” (high $s$, small $b$) limit.

Modifications similar to Eq. (50) apply to all the eikonals describing
“semihard Pomeron” exchanges. For example, $\chi_{hp(ij)}^{\mathbb{P}_{{\rm
sh}}}$, Eq. (20), now takes the form

$\displaystyle\chi_{hp(ij)}^{\mathbb{P}_{{\rm
sh}}}(s,b)=\frac{K}{2}\sum_{I,J}\int\\!d^{2}b^{\prime}\int\\!dx^{+}dx^{-}$
$\displaystyle\times\int\\!dp_{t}^{2}\;G_{I/h(i)}(x^{+}+\tilde{x}^{+},b^{\prime},\mu_{{\rm
F}}^{2})$
$\displaystyle\times\;G_{J/p(j)}(x^{-}+\tilde{x}^{-},|\vec{b}-\vec{b}^{\prime}|,\mu_{{\rm
F}}^{2})$ $\displaystyle\times\;\frac{d\sigma_{IJ}^{2\rightarrow
2}(\hat{s},p_{t}^{2},\mu_{\rm R})}{dp_{{\rm
t}}^{2}}\,\Theta(\mu_{F}-Q_{0})\,,$ (53)

with $\tilde{x}^{\pm}$ as in Eqs. (51-52), and with $G_{I/h(i)}$ being defined
neglecting absorptive corrections [c.f. Eq. (22)]:

$\displaystyle G_{I/h(i)}(x,b,\mu_{{\rm
F}}^{2})=\sum_{I^{\prime}}\int_{x}^{1}\\!dz$
$\displaystyle\times\,E_{I^{\prime}I}^{{\rm QCD}}(z,Q_{0}^{2},\mu_{{\rm
F}}^{2})\,\chi_{Ih(i)}^{\mathbb{P}_{{\rm soft}}}(s_{0}z/x,b)/x\,.$ (54)

Neglecting the HT corrections, i.e., setting $K_{\rm HT}=0$, Eqs. (35) and
(20) are recovered from Eqs. (50) and (53), respectively.

In turn, for the proton structure function $F^{(p)}_{2}$ we obtain, using the
same assumptions:

$\displaystyle F^{(p)}_{2}(x,Q^{2})\simeq
x\sum_{q}e_{q}^{2}\int\\!d^{2}b\,\sum_{i}C^{(i)}_{p}$
$\displaystyle\times\;G^{\rm scr}_{q/p(i)}(x+\tilde{x},b,Q^{2})\,,$ (55)

with

$\displaystyle\tilde{x}=\frac{4K_{\rm HT}\,\pi^{2}\,\alpha_{\rm
s}(Q^{2})\,x}{3Q^{2}}$ $\displaystyle\times\left.x_{g}G_{g/p(i)}^{\rm
scr}(x_{g},b,Q_{0}^{2})\right|_{x_{g}=\frac{x\,Q_{0}^{2}}{Q^{2}}}.$ (56)

At this point, we have to provide some arguments to support our extrapolation
of the treatment of [43] to the case of hadron-proton scattering. In
particular, here we can not use the $A$-enhancement argument to neglect other
potential HT contributions characterized by a different structure of the hard
scattering part, compared to the one in Fig. 5 (left), considered so far.

Figure 5: Left: the structure of the hard “blob” in Fig. 4 (right), for
leading power corrections discussed in the text, for the case of hard
scattering of quarks of different flavors. Right: an alternative leading power
correction to the $qq^{\prime}$ hard scattering, which provides a subleading
contribution in the high energy limit.

To this end, let us remind ourselves that the Lorentz-contraction acts
differently on partons of different momenta, in a fast-moving parton cloud of
a hadron. While fast (large $x$) partons are confined to a narrow “pancake” in
the longitudinal direction, the abundant small $x$ gluons are spread over
longitudinal distances $\propto 1/(xp^{+})$. For the diagram in Fig. 5 (left),
corresponding to the approach of [43], the struck low-$x$ quark propagates
over large distances $\propto 1/(x^{+}p^{+})$ comparable to the longitudinal
size of the gluon cloud and, thus, may scatter coherently on many correlated
soft gluon pairs. In contrast, considering, for example, an alternative
configuration depicted in Fig. 5 (right), the first gluon is separated from
the quark by the contact propagator, which implies there is a very small
distance between the quark and the gluon in the LC- direction. Hence, only a
very small portion of the gluon content of the proton can be involved in that
type of interaction, with the corresponding contribution being a subdominant
one.

##  7 Nuclear collisions and Monte Carlo implementation

The generalization of the model for the treatment of hadron-nucleus and
nucleus-nucleus collisions is performed without introducing any additional
adjustable parameters, similarly to the case of the QGSJET-II model [5]. The
only additional input are nucleon densities for nuclear ground states, chosen
according to the corresponding experimental measurements999For light nuclei,
with mass number $A\leq 10$, Gaussian distributions are used, while densities
of heavier nuclei are described by three parameter Wood-Saxon distributions.
[46].

As discussed in [5], in case of $hA$ and $AA$ collisions, enhanced Pomeron
diagrams account for rescattering of intermediate partons off different
nucleons from the projectile or/and from the target, i.e., multi-Pomeron
vertices generally couple together sequences of Pomerons and Pomeron loops
connected to different nucleons. This gives rise to a dynamical treatment of
the corresponding absorptive corrections: the strength of such nonlinear
effects increases with the collision energy, with the size of the colliding
nuclei, and with the “centrality” of the collision.

The very same tendencies hold also for the HT effects discussed in Section 6,
for $hA$ and $AA$ collisions. Indeed, the multiparton correlators, Eqs. (38)
and (41), involve in those cases soft gluons emitted by different nucleons.
Describing such correlators by multiparton GPDs, the HT corrections rise in
the low $x$ and small $b$ limits, increasing also with the size of the nuclei.

The MC procedure for generating individual inelastic scattering events is
almost identical to the one of the QGSJET-II model [5]: starting from sampling
the impact parameter for a collision, according to the respective interaction
profile, and proceeding to specifying a “macro-configuration” of the event,
i.e., defining the structure of the corresponding cut Pomeron “net”, based on
partial cross sections for such macro-configurations. This is followed by
sharing the energy-momentum between all “elementary” parton cascades (cut
Pomerons) and choosing, with the appropriate weight, whether a particular cut
Pomeron involves a purely “soft” ($|q^{2}|<Q_{0}^{2}$) parton evolution or
corresponds to a semihard parton cascade. In the latter case, one generates
explicitly the initial and final state parton emission, treating the
respective $t$\- and $s$-channel parton cascades, using the DGLAP formalism.

More specifically, the initial ($t$-channel) parton emission is modeled using
a forward evolution algorithm, based on an integral representation of the
DGLAP equations. For any parton (sub)ladder of mass squared $\hat{s}$, with
the ladder “leg” partons $I$ and $J$, characterized by virtualities
$q_{1}^{2}$ and $q_{2}^{2}$, respectively ($q_{1}^{2}=q_{2}^{2}=Q_{0}^{2}$ for
first partons in a particular perturbative cascade), one considers a
successive emission of a parton from any of the ladder “ends”, according to
the probability (see, e.g. [16] for more details):

$\displaystyle
f_{I^{\prime}}(z,q^{2})\propto\frac{\alpha_{s}(q^{2})}{2\pi\,q^{2}}\,P_{I^{\prime}I}(z)\,\Delta^{\rm
S}_{I}(q_{1}^{2},q^{2})$ $\displaystyle\times\;\sigma_{I^{\prime}J}^{{\rm
QCD}}(z\hat{s},q^{2},q_{2}^{2})\,\Theta(1-\varepsilon-z)\,\Theta(q^{2}-q_{1}^{2})$
$\displaystyle\times\;\Theta(z\hat{s}/16-\max[q^{2},q_{2}^{2}]),$ (57)

with $I^{\prime}$, $z$, and $q^{2}$ being, respectively, the type, LC (plus or
minus) momentum fraction, and virtuality of the new $t$-channel parton.
$P_{I^{\prime}I}$ are the usual (unregularized) Altarelli-Parisi splitting
kernels, $\sigma_{I^{\prime}J}^{{\rm QCD}}$ is the contribution of the
remaining subladder, defined by Eq. (6), and $\Delta^{\rm
S}_{I}(q_{1}^{2},q^{2})$ is the so-called Sudakov form factor defining the
probability for no parton emission in the virtuality range
$[q_{1}^{2},q^{2}]$:

$\displaystyle\Delta^{\rm
S}_{q}(q_{1}^{2},q^{2})=\exp\\!\left[-\int_{q_{1}^{2}}^{q^{2}}\frac{d\tilde{q}^{2}}{\tilde{q}^{2}}\,\int_{0}^{1-\varepsilon}\\!dz\right.$
$\displaystyle\times\left.\frac{\alpha_{s}(\tilde{q}^{2})}{2\pi}\,P_{qq}(z)\right]$
(58) $\displaystyle\Delta^{\rm
S}_{g}(q_{1}^{2},q^{2})=\exp\\!\left[-\int_{q_{1}^{2}}^{q^{2}}\frac{d\tilde{q}^{2}}{\tilde{q}^{2}}\,\int_{\varepsilon}^{1-\varepsilon}\\!dz\right.$
$\displaystyle\times\left.\frac{\alpha_{s}(\tilde{q}^{2})}{2\pi}\,\left[\frac{1}{2}P_{gg}(z)+N_{f}\,P_{qg}(z)\right]\right].$
(59)

Here $\varepsilon$ is a small enough technical “resolution” parameter (we use
$\varepsilon=10^{-2}$), $N_{f}=3$ is the number of active quark flavors, and
the last $\Theta$-function in Eq. (57) is to assure that the remaining ladder
is massive enough to allow for a parton-parton scattering with
$p_{t}^{2}/4>\max[q^{2},q_{2}^{2}]$ (for our choice of the factorization
scale, $\mu_{F}=p_{t}/2$).

Here comes an important difference, compared to the corresponding treatment of
[5]. The generation of macro-configurations of collisions and the energy-
momentum sharing procedure are quite similar to the ones of [5], with the only
difference that the respective “general Pomeron” eikonals now contain HT
corrections to hard scattering processes, as discussed in Section 6. On the
other hand, when treating the $t$–channel parton evolution, a two step
procedure is adopted. First, the corresponding parton emission pattern is
generated based on the standard DGLAP evolution, as described above, the HT
corrections being neglected. Next, the obtained parton configuration is
accepted with the probability defined by the ratio of the corresponding
partial cross sections taking the HT corrections into account (i.e.,
accounting for the kinematics-dependent shift of LC± momentum fractions of
active partons) or neglecting them. Otherwise, the configuration is rejected
and the procedure is repeated. In other words, the HT corrections to the hard
scattering pattern are accounted for via a rejection procedure. For brevity,
we omit here the corresponding technical details.

In turn, the modeling of the final ($s$-channel) parton emission follows
closely the procedure described in [47], imposing the final transverse
momentum cutoff $p^{2}_{{t,\rm cut(f)}}=0.15$ GeV2. In particular, one assures
angular ordering of sequential $s$-channel subcascades, resulting from color
coherence effects [48, 49].

As the final step, one considers a formation of strings of color field,
stretched between constituent partons of the interacting hadrons (nuclei)
or/and between the final $s$-channel partons resulting from the above-
discussed treatment of perturbative parton cascades, following the directions
of the corresponding color flows. The breakup and hadronization of such
strings is modeled by means of a string fragmentation procedure, to be
discussed elsewhere [7]. Here two important comments are in order. First, the
color connections between final partons are defined based on the $1/N_{c}$
approximation ($N_{c}$ being the number of colors): following the directions
of the color and “anticolor” flows [50]. For each $s$-channel gluon emission,
there are two alternative ways (taken with equal probabilities) to continue
such flows, such that a diagram with $n$ $s$-channel gluons can produce up to
$2^{n}$ possible patterns for the string configuration (see [18] for a recent
detailed discussion). Secondly, in the hadronization procedure of the QGSJET-
III model, inherited from the original QGSJET [3], one considers a
nonperturbative splitting of final gluons into quark-antiquark pairs, with the
strings having such (anti)quarks at the ends, without gluon “kink”
perturbations. Such an approach is generally inferior in quality, compared to
more advanced “kinky string” hadronization procedures [50, 51] employed in the
PYTHIA [52] and EPOS [53] MC generators, notably, regarding high $p_{t}$
production of relatively heavy hadrons.

##  8 Selected results and discussion

The parameters of the model have been fixed using experimental data on total,
elastic, and diffractive hadron-proton cross sections, on the proton SF
$F_{2}$, and on secondary hadron production in $hp$ and $hA$ collisions101010A
comparison with experimental data on secondary hadron production and a
discussion of model parameters related to the hadronization procedure will be
presented elsewhere [7]., the corresponding values being compiled in Table 1.
As one can see in the Table, replacing the projectile proton by pion or kaon,
subject to a change are only the parameters describing the transverse sizes of
GW Fock states of the hadrons.

$\alpha_{\mathbb{P}}(0)$ | $\alpha^{\prime}_{\mathbb{P}}(0)$ |  $Q_{0}^{2}$ | $r_{3\mathbb{P}}$ |  $K_{\rm HT}$ | $r_{g/\mathbb{P}}$ | $w_{qg}$ |  $\xi$ | $\beta_{p}$ |  $\beta_{\pi}$ | $g_{0}$ |  $d_{p}$ | $d_{\pi}$ |  $d_{K}$ | $R^{2}_{p(1)}$ | $R^{2}_{\pi(1)}$ | $R^{2}_{K(1)}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| GeV-2 |  GeV2 | GeV-1 |  |  |  |  |  |  |  |  |  |  | GeV-2 | GeV-2 | GeV-2
1.21 | 0.21 |  2 | 0.3 |  2.5 | 0.11 | 0.28 |  1.5 | 4 |  2 | 2 |  0.11 | 0.2 |  0.28 | 5.58 | 1.9 | 1.12
Table 1: Model parameters.

In Fig. 6,

Figure 6: Calculated energy-dependence of the total and elastic $pp$ cross
sections – solid lines. The results obtained omitting the HT corrections or
neglecting all nonlinear effects are shown by dashed and dotted lines,
respectively. The experimental data (points) are from Refs. [54, 55].

we compare the calculated energy-dependence of the total $\sigma_{pp}^{\rm
tot}$ and elastic $\sigma_{pp}^{\rm el}$ proton-proton cross sections to
accelerator data. Additionally, we show by dashed lines the corresponding
results obtained neglecting the HT corrections, i.e., setting $K_{\rm HT}=0$,
while keeping all the other parameters unchanged. In turn, dotted lines
correspond to the case of all nonlinear effects being neglected, i.e., setting
also the triple-Pomeron coupling $r_{3\mathbb{P}}=0$. It is easy to see that
the highest impact on the $\sqrt{s}$-dependence of $\sigma_{pp}^{\rm tot/el}$
is due to absorptive corrections generated by enhanced Pomeron diagrams. On
the other hand, the considered HT corrections also affect somewhat the
calculated $\sigma_{pp}^{\rm tot/el}$ at sufficiently high energies.

Even such a moderate effect on $\sigma_{pp}^{\rm tot/el}$ may seem somewhat
surprising since the HT corrections apply to hard scattering processes only.
Such processes can be expected to dominate relatively central (small $b$)
collisions of hadrons, while having a weak impact on the large $b$ behavior of
the scattering amplitude. In that regard, it is useful to remind oneself that
in the “semihard Pomeron” scheme employed here, hard parton scattering is
preceded by a long enough “soft preevolution” [16, 17], as discussed in
Section 3. With increasing energy, such soft parton evolution covers a larger
rapidity interval and the corresponding transverse diffusion gives rise to a
substantial widening of the transverse profile for semihard scattering
processes. This is illustrated in Fig. 7,

Figure 7: Calculated transverse profile for the absorptive part of
$\sigma^{\rm inel}_{pp}$ for $\sqrt{s}=10^{2}$ (left), $10^{3}$ (middle), and
$10^{4}$ GeV (right) – solid lines. Partial contributions of soft and semihard
production processes are shown by dotted and dashed lines, respectively.
Dashed-dotted lines correspond to the latter contribution calculated
neglecting the HT corrections.

where the transverse profile for the absorptive part of the inelastic cross
section, $d^{2}\sigma^{\rm abs}_{pp}/d^{2}b$, for $pp$ collisions at
$\sqrt{s}=10^{2}$, $10^{3}$, and $10^{4}$ GeV is plotted, together with
partial contributions of purely soft (i.e., with only soft Pomerons being cut)
or semihard (with some “real” parton cascades entering the perturbative,
$|q^{2}|>Q_{0}^{2}$, domain) particle production. Additionally shown by
dashed-dotted lines is the semihard contribution calculated without the HT
corrections, i.e., setting $K_{\rm HT}=0$. As is easy to see in the Figure,
not only the normalization of the profile for semihard interactions rises with
energy but also its width increases, corresponding to such interactions
happening at larger and larger impact parameters. Further noteworthy is the
strong damping of purely soft production processes at small $b$ and large
$\sqrt{s}$, caused by absorptive corrections due to virtual semihard
scattering processes111111Roughly speaking, the corresponding damping factor
can be interpreted as the probability to have no semihard production at the
respective $b$ values. – see the dotted lines in Fig. 7.

What may also seem surprising are the relatively similar shapes of the
semihard scattering profiles calculated with and without the HT corrections:
dashed and dashed-dotted lines in Fig. 7. Here we remind ourselves that those
corrections involve soft gluon GPDs $G_{g/p}(x,b,Q^{2}_{0})$ at small $x$.
Since those GPDs are described by soft Pomeron asymptotics (plus the relevant
absorptive corrections), c.f. Eqs. (31-32), the corresponding significant
transverse diffusion gives rise to a rather large slope for these GPDs, in the
low $x$ limit, which exceeds the one for the semihard scattering itself. As a
consequence, these HT corrections mainly reduce the normalization of the
semihard interaction profile, without modifying significantly its shape.

In relation to the model treatment of color fluctuations, it is worth
comparing the same profiles as in Fig. 7, for partial contributions of
different combinations of GW Fock states $|i\rangle$ and $|j\rangle$ of the
projectile and target protons, respectively, as plotted in Fig. 8.

Figure 8: Same as in Fig. 7, for particular combinations of GW states
$|i\rangle$ and $|j\rangle$ of the projectile and target protons, from top to
bottom: $(i,j)=(1,1)$, $(1,2)$, $(1,3)$, $(2,2)$, $(2,3)$, $(3,3)$.

The first thing to notice is the quick energy-rise of the relative importance
of smaller size GW states: both due to increasing opaqueness of the
corresponding profiles at small $b$ and due to a fast transverse broadening of
these profiles. The two effects are due to, respectively, the low $x$ rise of
the (initially small at large $x$) parton densities and due to the quick
transverse expansion of the (initially compact) parton “clouds” of small size
GW states. Secondly, the damping of purely soft hadron production is
substantially stronger for larger size GW states, owing to their larger
(integrated) parton densities. Likewise, the HT corrections to the calculated
transverse profiles are more significant for larger GW Fock states: because of
larger soft gluon densities of those states.

In Fig. 9, we compare the calculated proton SF $F_{2}^{(p)}$,

Figure 9: Calculated $x$-dependence of the proton structure function
$F_{2}^{(p)}(x,Q^{2})$, for different $Q^{2}$, as indicated in the plots,
compared to HERA data [56].

Eq. (55), to the corresponding HERA data, plotting also the results obtained
either by suppressing the corresponding HT corrections (setting $K_{\rm
HT}=0$) or by neglecting all nonlinear effects (setting also
$r_{3\mathbb{P}}=0$). As one can see in the Figure, the considered power
corrections have a significant impact on the calculated proton SF $F_{2}$ at
relatively small values of $Q^{2}$, with the effect vanishing for sufficiently
high $Q^{2}$. Clearly, these data constrain considerably the magnitude of the
HT corrections in the model, i.e., the value of the parameter $K_{\rm HT}$. On
the other hand, much more serious effect is due to the taming of the low $x$
rise of proton PDFs, caused by absorptive corrections due to enhanced Pomeron
graphs. To discuss the latter in more detail, it is worth considering the
$x$-dependence of the gluon GPD $G_{g/p}(x,b,Q^{2}_{0})$, calculated with and
without such corrections, for different $b$ values [57, 34], as plotted in
Fig. 10.

Figure 10: Gluon GPD $x\,G_{g/p}(x,b,Q_{0}^{2})$ for different values of $b$
(as indicated in the plots), calculated with (solid lines) and without (dotted
lines) absorptive corrections due to Pomeron-Pomeron interactions.

Here one clearly sees the strong $b$-dependence of nonlinear effects due to
Pomeron-Pomeron interactions: while being moderate at large $b$, such
corrections damp strongly the low $x$ rise of the gluon GPD in the
$b\rightarrow 0$ limit, causing a saturation of the gluon density.

It is worth considering also the same dependence for partial gluon GPDs of
different GW Fock states, $G_{g/p(i)}(x,b,Q^{2}_{0})$, as plotted in Fig. 11.

Figure 11: Same as in Fig. 10, for partial gluon GPDs,
$x\,G_{g/p(i)}(x,b,Q_{0}^{2})$, of different GW Fock states; from top to
bottom: $i=1,2,3$.

Here we observe the strongest impact of nonlinear effects on the low $x$
behavior of the partial GPDs of the largest Fock states: as a consequence of
their largest soft parton densities.

In Fig. 12,

Figure 12: Calculated differential elastic proton-proton cross section for
different $\sqrt{s}$ in GeV (as indicated in the plot), compared to
experimental data (points) from Refs. [55, 58, 59, 60, 61, 62].

we compare the calculated differential elastic cross section $d\sigma^{\rm
el}_{pp}/dt$ to experimental measurements. While the agreement with the data
is satisfactory for small values of $|t|$, where the bulk of the contribution
to $\sigma^{\rm el}_{pp}$ comes from, this is not the case for larger
$|t|\gtrsim 0.2$. Generally, a better description of the observed
$t$-dependence of $d\sigma^{\rm el}_{pp}/dt$ requires a more sophisticated
choice for the proton form factor (see, e.g. [63]), compared to the simple
Gaussian ansatz used in the current work. Moreover, for relatively large
$|t|$, the simple GW decomposition of the proton wave function, with
$t$-independent partial weights and profiles of GW Fock states, becomes
invalid [64]. Since the presented model aims at describing the bulk of general
hadronic and nuclear scattering processes, such potential developments are
beyond the scope of the current study.

Additionally, in Fig. 13,

Figure 13: Energy-dependence of the calculated forward elastic scattering
slope for $pp$ collisions, compared to experimental data (points) from Refs.
[54, 55, 61, 62, 65, 66, 67].

we compare to experimental data the calculated $\sqrt{s}$-dependence of the
elastic scattering slope $B^{\rm el}_{pp}=d\ln\left.d\sigma^{\rm
el}_{pp}/dt\right|_{t=0}$, which quantifies the energy-dependence of the
average squared impact parameter $\langle b^{2}\rangle$ for $pp$ collisions.
Clearly, the model fails to reproduce the rather large values of $B^{\rm
el}_{pp}$, reported by the TOTEM and ATLAS collaborations at LHC energies. On
the one side, this may be related to additional physics mechanisms missing in
the model, like the pion loop contributions to the Pomeron Regge trajectory
[68] (see [69] for a recent discussion). On the other hand, it may indicate a
certain deficiency of the treatment of color fluctuations in the model: as one
can see in Fig. 13, the increasing relative importance of smaller size GW
states slows down the energy rise of $B^{\rm el}_{pp}$ at $\sqrt{s}\gtrsim 1$
TeV.

Let us now come to the model predictions for the inelastic diffraction. In
view of a certain tension between different LHC results on the cross section
for high mass diffraction, discussed previously in [70, 71], of significant
importance is the recent measurement by the ATLAS experiment of the
differential single diffractive (SD) cross section, $d\sigma^{\rm
SD}_{pp}/dt$, at $\sqrt{s}=8$ TeV, using the Roman Pot technique [72].
Comparing in Fig. 14 the calculated121212The calculation involves a Fourier
transform to impact parameter space, in order to account for all the relevant
absorptive corrections (see, e.g. [73]). $d\sigma^{\rm SD}_{pp}/dt$,

Figure 14: Calculated $d\sigma^{\rm SD}_{pp}/dt$ at $\sqrt{s}=8$ TeV, for
$10^{-4}<\xi<10^{-1.6}$, compared to ATLAS data (points) [72].

for the experimental event selection $10^{-4}<\xi=M_{X}^{2}/s<10^{-1.6}$,
$M_{X}$ being the diffractive state mass, with the ATLAS data, we observe a
rather good agreement for the obtained $t$-dependence, while the magnitude of
the calculated cross section (1.4 mb) is $\sim 25$% below the measured value,
$1.88\pm 0.15$ mb.131313Taking into account that an additional $\sim 40$%
contribution to the measured cross section, for the selected $\xi$-range,
comes from nondiffractive production [7] (so-called random rapidity gaps
[74]), the total SD-like event rate predicted actually exceeds the measurement
by $\sim 20$%.

Finally, to illustrate the impact of the considered HT effects on secondary
hadron production, we compare in Fig. 15

Figure 15: Transverse momentum spectrum for (mini)jet production in $pp$
collisions at $\sqrt{s}=10^{2}$, $10^{3}$, and $10^{4}$ GeV, as indicated in
the plot, calculated with and without HT corrections – solid and dashed lines,
respectively.

the $p_{t}$-dependence of the (mini-)jet production cross section,
$d\sigma^{\rm jet}_{pp}(s,p_{t})/dp_{t}^{2}$, at $\sqrt{s}=10^{2}$, $10^{3}$,
and $10^{4}$ GeV, as calculated with and without the HT corrections, i.e.,
using Eqs. (50) and (35), respectively. It is easy to see that such
corrections reduce $d\sigma^{\rm jet}_{pp}(s,p_{t})/dp_{t}^{2}$ considerably
in the small $p_{t}$ limit, with the suppression becoming more and more
significant at higher energies. On the other hand, as expected, the effect
vanishes for sufficiently high $p_{t}$. Additionally, in Fig. 16,

Figure 16: Transverse momentum spectrum for (mini)jet production in $pp$
collisions at $\sqrt{s}=10^{4}$ GeV, for different impact parameters, as
indicated in the plot, calculated with and without HT corrections – solid and
dashed lines, respectively.

we consider contributions of different impact parameters to these jet spectra,
i.e., we plot $d^{3}\sigma^{\rm jet}_{pp}(s,p_{t})/dp_{t}^{2}/d^{2}b$ defined
by the expression in the curly brackets in Eq. (50). The impact of the
considered HT corrections on low $p_{t}$ (mini-)jet production becomes maximal
for $b\simeq 0$, while decreasing slowly with the increase of $b$.

##  9 Conclusions

We presented here a new model for high energy hadronic scattering, QGSJET-III,
discussing in some detail its underlying theoretical mechanisms. In
particular, a considerable attention has been devoted to a phenomenological
treatment of dynamical power corrections to hard parton scattering processes,
based on the approach of Refs. [41, 43]: with the respective contributions
being related to coherent rescattering of $s$-channel partons on soft gluon
pairs emitted by the colliding hadrons (nuclei). Modeling the corresponding
multiparton correlators as multiparton GPDs, we developed a dynamical scheme:
with the strength of the HT effects increasing both in the very high energy
and small impact parameter limits.

Additionally, we discussed in some detail the model implementation of color
fluctuation effects: based on a decomposition of hadron wave functions in a
number of GW Fock states characterized by different transverse sizes and
different parton densities.

Selected model results regarding the energy dependence of the total and
elastic proton-proton cross sections, the $x$-dependence of the proton SF
$F_{2}$, the $x$\- and $b$-dependence of the gluon GPD
$G_{g/p}(x,b,Q^{2}_{0})$, and the $t$-dependence of single-diffractive $pp$
cross section have been presented and the impact of various nonlinear
corrections to the interaction dynamics has been investigated. Overall, the
most important feature of the model is the microscopic treatment of nonlinear
effects due to Pomeron-Pomeron interactions, which is inherited from the
previous model version, QGSJET-II. On the other hand, the developed
phenomenological treatment of HT effects serves the principal goal: taming the
steep rise of the (mini-)jet production in the small $p_{t}$ limit, thereby
reducing considerably the sensitivity of the model results to the choice of
the “infrared” cutoff $Q^{2}_{0}$, as discussed previously in [34].

The description of secondary hadron production and applications of the model
to modeling the development of CR-induced extensive air showers will be
discussed elsewhere [7].

### Acknowledgments

This work was supported by Deutsche Forschungsgemeinschaft (project number
465275045).

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