# Soft processes at high energy without soft Pomeron:
a QCD motivated model.

A. Kormilitzin and E. Levin
Department of Particle Physics, School of Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Science
Tel Aviv University, Tel Aviv, 69978, Israel Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

In this paper we develop a QCD motivated model for both hard and soft
interactions at high energies. In this model the long distance behavior of the
scattering amplitude is determined by the approximate solution to the non-
linear evolution equation for parton system in the saturation domain. All
phenomenological parameters for dipole-proton interaction were fitted from the
deep inelastic scattering data and the soft processes are described with only
one new parameter, related to the wave function of hadron. It turns out that
we do not need to introduce the so called soft Pomeron that has been used in
high energy phenomenology for four decades. The model described all data on
soft interactions: the values of total, elastic and diffractive cross sections
as well as their $s$ and $t$ behavior. The value for the survival probability
of the diffractive Higgs production is calculated being less 1% for the LHC
energy range.

High density QCD , saturation, single diffraction, Pomeron structure

††preprint:  TAUP -2884-08

##  1 Introduction

For three decades our microscopic theory-QCD, has been unable to describe soft
(long distance) interactions. It happens, partly, because of the embryonic
stage of our understanding of the most challenging problem in QCD: confinement
of quarks and gluons. We believe, however, that the main features of high
energy scattering at long distances can be described in the framework of QCD,
and such a description is based on the phenomenon of saturation of the parton
densities in QCD [1, 2, 3]. Indeed, at short distances the parton densities
increase with the growth of energy due to gluon emission, which is described
by the QCD linear evolution equations [4, 5]. Such an emission is proportional
to the density of emitters (partons). The process of annihilation is
suppressed since it’s strength is proportional to the parton density squared,
reflecting the fact that two partons have to meet each other in the small
volume of interaction. However, at high energy the density of partons becomes
so large that the annihilation term tends to be of the same order as the
emission term. It leads to a slow down in the increase of the parton density,
and the system of partons lives in the dynamical equilibrium with the critical
value of density (it is saturated). We have several theoretical approaches for
a QCD description of the parton system in the saturation domain (see Ref. [1,
2, 3, 6, 7, 8, 9, 10, 11, 12], but the equations that have been proposed turn
out to be so complicated, that our knowledge about this domain mostly stems
from numerical solutions of these equations [13].

The alternative approach that has been developed during the past decade is to
build models [14, 15] that incorporate the main qualitative properties of
these solutions, such as the geometrical scaling behavior of the scattering
amplitude [19, 20, 21] and the existence of the new dimensional scale (the so
called saturation scale) [1, 2, 3], which increases in the region of low $x$
(high energy). Such models successfully described all the data on deep
inelastic scattering, including the behavior of inclusive and diffractive
cross sections, at a low value of photon virtuality [14].

The main goal of this paper is to describe the processes that occur at long
distances (soft processes), where we cannot apply perturbative QCD, using the
saturation model. In this model all the needed phenomenological parameters, in
particular the energy behavior and the value of the saturation scale, were
fitted using the DIS data. The main ingredient of the high energy
phenomenology during the past four decades, namely the soft Pomeron, is not
introduced in this model. In simple words, we replace the soft Pomeron by the
scattering amplitude that describes the behavior of the saturated parton
system. The idea, that the matching between soft and hard interaction occurs
in the saturation region, and can be described by the amplitude that
originates from the high density QCD approach, is not new. This idea has been
in the air for a number of years and have been advocated in several lectures
on the subject [22]. The first attempt of a practical application of this idea
was done in Ref. [24] (see also Ref. [25]). The result of this paper was
encouraging since the estimates gave a reasonably good agreement with the
experimental data on the total cross section of the pion, kaon, and proton
interaction with the proton target. In this paper we continue to develop these
ideas and we give the comparison with the experiment data for the full set of
the experimental data of soft interactions, that include the energy behavior
of the total, elastic and diffraction cross sections as well as a $t$\-
behavior of the elastic cross sections, and the mass behavior for the cross
sections of diffraction production.

##  2 The main idea

As has been mentioned, our main idea is to describe the so called ‘soft’
interaction in the high parton density QCD (hdQCD). In hdQCD we are dealing
with the dense system of partons which cannot be treated in the perturbative
QCD approach. Being non-perturbative in its nature, hdQCD leads to typical
distances $r=1/Q_{s}(x)\ll R\sim 1/\Lambda_{QCD}$ where $R$ is the size of a
hadron. The saturation momentum $Q_{s}(x)$ is a new dimensionful scale which
increases with the growth of the energy ( or at $x\to 0$)[1, 2, 3]. Therefore,
the QCD coupling $\alpha_{S}\left(Q_{s}(x)\right)$ is small and, because of
this, we are able to develop a theoretical approach for such a dense system
having only a limited input from the unknown confinement region. To illustrate
the limitation of our approach, it is enough to look at the expression for the
saturation scale in hdQCD[1, 26]

$\displaystyle Q^{2}_{s}(Y)\,\,$ $\displaystyle=$
$\displaystyle\,\,Q^{2}_{s}(Y_{0})\,\exp\left(\bar{\alpha}_{S}\,\frac{\chi(\gamma_{cr})}{1-\gamma_{cr}}\,\,(Y-Y_{0})\,\,-\,\,\frac{3}{2(1-\gamma_{cr})}\,\ln(Y/Y_{0})-\right.$
$\displaystyle-$
$\displaystyle\,\left.\frac{3}{(1-\gamma_{cr})^{2}}\,\sqrt{\frac{2\,\pi}{\bar{\alpha}_{S}\,\chi^{\prime\prime}(\gamma_{cr})}}\,(\frac{1}{\sqrt{Y}}\,-\,\frac{1}{\sqrt{Y_{0}}}\,)\,,+\,O(\frac{1}{Y})\right)$
(2.1)

In Eq. (2) the energy dependence of $Q_{s}(Y=\ln(1/x))$ is predicted from
hdQCD, ($\chi(\gamma)$ is the Mellin transform of the BFKL kernel) but the
scale $,Q^{2}_{s}(Y_{0})$ stems from the confinement region, demonstrating a
need for some input from this theoretically unknown region. Our hope that such
input will be a limited number of constants.

Let us consider the deep inelastic electron-proton scattering (DIS), to
illustrate our point. If the virtuality of the photon $Q^{2}$ is very large,
$Q^{2}\gg Q^{2}_{s}(x)$, the typical distances $r\approx 1/Q$, and we can
safely apply the perturbative QCD approach based on the operator product
expansion, and the DGLAP evolution equation. However, if $\Lambda^{2}_{QCD}\ll
Q^{2}\ll Q^{2}_{s}(x)$, the situation changes crucially. All terms in the
operator product expansion become of the same order, and we have to use the
hdQCD approach. In this approach, the cross section $\sigma^{\gamma^{*}p}$
shows a geometrical scaling behaviour[27], namely,
$\sigma^{\gamma^{*}p}=\left(1/Q^{2}_{s}(x)\right)\,F\left(Q^{2}/Q^{2}_{s}(x)\right)$
which has been confirmed experimentally. This means that the typical distances
$r\approx 1/Q_{s}(x)\ll 1/\Lambda_{QCD}$, which are short.

Our idea is that we can describe $Q^{2}\approx\Lambda^{2}_{QCD}$, in the
framework of the same hdQCD approach. Having the geometrical scaling behavior
in mind, such an idea does not look crazy. However, in DIS we are dealing with
the total cross sections which are related to the amplitude integrated over
the impact parameters ($b$). For a treatment of the soft interaction
observables, we need to know the $b$ dependence. As was shown in Ref. [28],
hdQCD predicts the power-like decrease at large value of $b$, which
contradicts both the theoretical estimates and experimental observations.
Therefore, we need an input from the confinement region to specify the $b$
dependence of the scattering amplitude. Let us first discuss large $b\geq
1/\Lambda_{QCD}\gg 1/Q_{s}(x)$. The scattering amplitude for DIS
$A\left(Q^{2},W;t\right)$ depends on the photon virtuality $Q^{2}$, energy
$W=\sqrt{s}=Q/\sqrt{x}$ and on the momentum transferred $t=-q^{2}<0$. Using
the dispersion relation in the $t$-channel we have

$A\left(Q^{2},W;t\right)\,\,=\,\,\frac{1}{\pi}\,\int^{\infty}_{4m^{2}_{\pi}}\,\frac{dt^{\prime}\,Im_{t}A\left(Q^{2},W;t^{\prime}\right)}{t^{\prime}+q^{2}}$
(2.2)

Calculating the amplitude in the impact parameter representation we can reduce
Eq. (2.2) to the form

$\displaystyle A\left(Q^{2},W;b\right)\,\,$ $\displaystyle=$
$\displaystyle\,\,\frac{1}{(2\pi)^{2}}\,\int\,d^{2}\vec{q}\,A\left(Q^{2},W;t\right)\,e^{i\vec{q}\cdot\vec{b}}\,\,=\,\,\frac{1}{2\pi^{2}}\,\int^{\infty}_{4m^{2}_{\pi}}\,dt^{\prime}\,Im_{t}A\left(Q^{2},W;t^{\prime}\right)\,\int\,\frac{q\,dq\,\,J_{0}\left(qb\right)}{t^{\prime}+q^{2}}\,$
$\displaystyle=$
$\displaystyle\,\,\frac{1}{2\pi^{2}}\,\int^{\infty}_{4m^{2}_{\pi}}\,dt^{\prime}\,K_{0}\left(\sqrt{t^{\prime}}\,b\right)\,\,Im_{t}A\left(Q^{2},W;t^{\prime}\right)\,\,\
\stackrel{{\scriptstyle b\geq
1/(2m_{\pi})}}{{\longrightarrow}}\,C\,Im_{t}A\left(Q^{2},W;t^{\prime}=4m^{2}_{\pi}\right)\,K_{0}\left(2m_{\pi}b\right)$

where in the constant $C$, all the numerical factors have been absorbed.

In Eq. (2), the $b$-dependence is determined by the mass of the lightest
hadron (pion), and cannot not be reproduced in perturbative or/and high
density QCD. However, one can see from Eq. (2) that we have a factorization of
the $b$ dependence, and the dependence on the photon virtuality and energy.

As it is well known (see Ref. [12] and references therein), in the kinematic
region $\alpha_{S}\,\leq\alpha_{S}\,\ln(1/x)\leq 1/\alpha_{S}$ in the
saturation region the hdQCD approach reduces to the BFKL Pomeron calculus. We
generalize Eq. (2) for the BFKL Pomeron in the following way

$A_{BFKL}\left(Q^{2},x;b\right)\,\,\,=\,\,\,\stackrel{{\scriptstyle\mbox{
\small short
distances}}}{{A_{BFKL}\left(Q^{2},x;t=0\right)}}\,\times\,\stackrel{{\scriptstyle\mbox{\small
long distances}}}{{S(b)}}$ (2.4)

One can see that Eq. (2.4) claims a kind of factorization between short and
long distances. The short distance part we can describe in hdQCD, while the
long distance part has to be taken from the confinement domain. In a more
general way, we can re-write Eq. (2.4) in the form

$A_{BFKL}\left(Q^{2},x;\vec{b}\right)\,\,\,=\,\,\,\int
d^{2}\vec{b^{\prime}}\stackrel{{\scriptstyle\mbox{ \small short
distances}}}{{A_{BFKL}\left(Q^{2},x;\vec{b^{\prime}}\right)}}\,\times\,\stackrel{{\scriptstyle\mbox{\small
long distances}}}{{S(\vec{b}-\vec{b}^{\prime})}}$ (2.5)

where $b^{\prime}$ in the short distance part is of the order of
$b^{\prime}\approx 1/Q_{s}(x)$.

The approach of Eq. (2.4) and Eq. (2.5), at first sight contradicts the high
energy phenomenology based on the soft Pomeron and Reggeons, as well as well
as the lattice calculations [30]. In both approaches, the Pomeron trajectory
has $\alpha^{\prime}_{{I\\!\\!P}}>0$, and at positive $t$ the glueballs lie on
this trajectory. In Eq. (2.4) and Eq. (2.5), the Pomeron slope
$\alpha^{\prime}_{{I\\!\\!P}}=0$. We will show below that we will be able
reproduce the experimental data on energy dependence of the elastic slope
which used to consider as the argument for
$\alpha^{\prime}_{{I\\!\\!P}}=0.25\,GeV^{-2}$. On the theoretical side, the
only theory which can treat at the moment on the same footing both short and
long distances: N=4 SYM with AdS/CFT correspondence, leads to a picture which
is in striking agreement with our approach[29]. Namely, in this approach at
$t>0$ ( resonance region) we have a normal soft Pomeron with
$\alpha^{\prime}_{{I\\!\\!P}}>0$ while at $t<0$ (scattering region)
$\alpha^{\prime}_{{I\\!\\!P}}=0$.

##  3 The model

###  3.1 Motivation

Our building of the model is based on the dipole approach to high energy QCD
[31]. In this approach, the evolution of the parton (colorless dipole) system
can be written in the form of the equation for the generating functional, with
a transparent probabilistic interpretation for them. The generating functional
is defined as [31]

$Z\left(Y\,-\,Y_{0};\,[u]\right)\,\,\equiv\,\,\sum_{n=1}\,\int\,\,P_{n}\left(Y\,-\,Y_{0};\,x_{1},y_{1};\dots;x_{i},y_{i};\dots;x_{n},y_{n}\right)\,\,\prod^{n}_{i=1}\,u(x_{i},y_{i})\,d^{2}\,x_{i}\,d^{2}\,y_{i}$
(3.6)

where $u(x_{i},y_{i})\equiv u_{i}$ is an arbitrary function of $x_{i}$ and
$y_{i}$. The coordinates $(x_{i},y_{i})$ describe the colorless pair of gluons
or a dipole. $P_{n}$ is a probability density to find $n$ dipoles with the
size $x_{i}-y_{i}$, and with impact parameter $(x_{i}+y_{i})/2$. Assuming that
we have only a decay of one dipole to two dipoles, directly from the physical
meaning of $P_{n}$ it follows [31, 32, 33]

$\frac{\partial\,P_{n}(Y;\dots;x_{i},y_{i};\dots;x_{n},y_{n})}{\partial
Y}\,\,\,=$ (3.7) $=\,\,\sum_{i}\ V_{1\to
2}\bigotimes\left(P_{n-1}(Y;\dots;x_{i},y_{i};\dots;x_{n},y_{n})\,-\,P_{n}(Y;\dots;x_{i},y_{i};\dots;x_{n},y_{n})\right)$

where $\bigotimes$ denotes all necessary integrations and $V_{1\to 2}$ is the
vertex for the decay of one dipole to two dipoles. This vertex is equal to

$V_{1\to
2}\left((x,y)\to(x,z)+(z,y)\right)\,\,=\,\,\frac{\bar{\alpha}_{S}}{2\pi}\,\frac{(\vec{x}-\vec{y})^{2}}{(\vec{x}-\vec{z})^{2}\,(\vec{z}-\vec{y})^{2}}$
(3.8)

Eq. (3.7) is a typical Markov’s chain which takes into account the $s$ \-
channel unitarity on each step of evolution since it has two terms: the birth
of the new dipole due to the decay of the parent dipole (positive term in Eq.
(3.7)) and the death of the dipole due to the same process.

Eq. (3.7) can be rewritten as the equation for the generating functional [32,
33], namely,

$\frac{\partial\,Z\,\left(Y-Y_{0};[\,u\,]\right)}{\partial\,Y}\,\,=\,\,\int\,d^{2}x\,d^{2}y\,d^{2}z\,\,V_{1\to
2}\left((x,y)\to(x,z)+(z,y)\right)\,\frac{\partial}{\partial\,u(x,y)}\,\,Z\,\left(Y-Y_{0};[\,u\,]\right)$
(3.9)

$Z\,\left(Y-Y_{0};[\,u\,]\right)$ satisfies the initial and boundary
conditions:

Initial conditions: $\displaystyle Z\,\left(Y-Y_{0}=0;[\,u\,]\right)$
$\displaystyle=\,\,\,u(x,y)\,;$ (3.10) Boundary conditions: $\displaystyle
Z\,\left(Y-Y_{0};[\,u=1\,]\right)$ $\displaystyle=\,\,\,1\,;$ (3.11)

The advantage of the generating functional is that we can write it in terms of
this simple functional formulae for the scattering amplitude, which has an
obvious partonic interpretation. Indeed, the scattering amplitude in the lab.
frame can be written in the form [6, 32, 33]

$\displaystyle N\left(Y-Y_{0};r,b\right)\,\,=$ (3.12)
$\displaystyle\,\,\sum^{\infty}_{n=1}\,\frac{(-1)^{n}}{n!}\,\,\int\,\prod^{n}_{i=1}\,\,d^{2}\,x_{i}\,d^{2}\,y_{i}\,\,\rho_{n}\left(r,b;\\{x_{i},y_{i}\\},Y-Y_{0}\right)\gamma_{n}\left(r,b;\\{x_{i},y_{i}\\},Y_{0}\right)$

where $Y-Y_{0}\,\,\gg\,\,1$ but $Y_{0}\,\,\approx\,\,1$ and

$\rho_{n}\left(Y-Y_{0},\\{x_{i},y_{i}\\}\right)\,\,\,=\,\,\,\prod^{n}_{i=1}\,\frac{\delta}{\delta
u(x_{i},y_{i})}Z\,\left(Y-Y_{0};\\{\,u(x_{i},y_{i})\,\\}\right)|_{u(x_{i},y_{i})=1}$
(3.13)

and $\gamma_{n}$ is the amplitude of interaction of $n$ dipoles at low energy
( small values of rapidity $Y_{0}$) with the coordinates $x_{i},y_{i}$ with
the target with size $r$ and impact parameter $b$. In the parton language the
generating functional determines the parton wave function for which we have
the evolution equation Eq. (3.9) in QCD. $\gamma_{n}$ can have a non-
perturbative origin. We assume that
$\gamma_{n}\left(r,b;\\{x_{i},y_{i}\\},Y_{0}\right)\,\,=\,\,\prod^{n}_{i=1}\gamma(x_{i},y_{i};Y_{0})$.
This assumption means that the dipoles inside the target have no correlation,
which is correct, as far as we know, only for a nucleus target.

In spite of the transparent physics behind our equations, the equation for the
generating functional is difficult to solve analytically, especially if we
generalize Eq. (3.9) to the case of the so called Pomeron loops [35, 36, 37].
As has been mentioned, we wish to find a model solution to Eq. (3.9). We will
do this using two key observations of Ref.[12]. The first one is the fact that
in the huge kinematic region of $\bar{\alpha}_{S}Y\,\leq 1/\bar{\alpha}_{S}$,
the system of partons that we are dealing with turns out to be a system of
non-interactive partons. In other words, it means that the generating
functional of Eq. (3.6) can be rewritten in a simpler form, namely.

$Z\,\left(Y-Y_{0};\\{\,u(x_{i},y_{i})\,\\}\right)\,\,\,=\,\,\,\sum^{\infty}_{n=1}\,\frac{C_{n}}{n!}\,\,\left(\int\,d^{2}x^{\prime}\,d^{2}y^{\prime}\,P(Y-Y_{0};x,y;x^{\prime},y^{\prime})\,\,\left(u(x^{\prime},y^{\prime})\,-\,1\right)\right)^{n}$
(3.14)

where $P(Y-Y_{0};x,y;x^{\prime},y^{\prime})$ is the amplitude for one BFKL
Pomeron exchange, between dipoles $(x,y)$ and $(x^{\prime},y^{\prime})$.

In Ref.[12] it was shown , using the analytical solution of Ref. [34] , that
actually $C_{n}\,=\,1$ for the dipoles with the size $r\sim 1/Q^{2}_{s}$ where
$Q_{s}$ is the saturation momentum.

Therefore, Eq. (3.14) can be simplified and rewritten in the form

$Z\,\left(Y-Y_{0};\\{\,u(x_{i},y_{i})\,\\}\right)\,\,\,=\,\,\,\exp\left(\int\,d^{2}x^{\prime}\,d^{2}y^{\prime}\,P(Y-Y_{0};x,y;x^{\prime},y^{\prime})\,\left(u(x^{\prime},y^{\prime})\,-\,1\right)\right)$
(3.15)

and the amplitude has the form

$N\left(Y-Y_{0};x,y\right)\,\,=\,\,1-\exp\left(-\int\,d^{2}x^{\prime}\,d^{2}y^{\prime}\,\,P\left(Y-Y_{0};x,y;x^{\prime},y^{\prime}\right)\gamma\left(Y_{0};x^{\prime},y^{\prime}\right)\right)$
(3.16)

where $\gamma\left(Y_{0};x^{\prime},y^{\prime}\right)$ is the scattering
amplitude of the dipole $(x^{\prime},y^{\prime})$, with the target at low
energy.

###  3.2 Main formulae and assumptions

Eq. (3.16) is the main formula for constructing our model. The first
ingredient of the model is the choice of the amplitude $\gamma$ and the value
of $Y_{0}$. We choose this amplitude in the form [38]

$\gamma\left(Y_{0};x^{\prime},y^{\prime}\right)\,\,=\,\,\gamma_{mod}\left(Y_{0};x^{\prime},y^{\prime}\right)\,\,=\,\,\,\,\frac{\pi^{2}\,\alpha_{S}(\mu^{2})}{3}\,r^{2}\,x_{0}G^{DGLAP}(x_{0},\mu^{2})\,S(b)$
(3.17)

where $\vec{r}\,=\,\vec{x}-\vec{y}$ and
$\vec{b}=\frac{1}{2}\,(\vec{x}+\vec{y})$ with $x_{0}=10^{-2}$. $xG$ in Eq.
(3.17) is the solution of the DGLAP evolution equation which describes the DIS
experimental data. $\mu$ is equal to

$\mu^{2}\,\,=\,\,\mu^{2}_{0}\,\,+\,\,\frac{C}{r^{2}}$ (3.18)

where $\mu_{0}$ and $C$ are phenomenological parameters which has to be found
fitting the experimental data on DIS (see Ref.[15] for details). Eq. (3.18)
means that $\gamma_{mod}\left(Y_{0};x^{\prime},y^{\prime}\right)$ at large
dipole size $r\mu_{0}\gg 1$ is determined by the following expression

$\gamma_{mod}\left(Y_{0};x^{\prime},y^{\prime}\right)\,\,=\,\,\,\,\frac{\pi^{2}\,\alpha_{S}(\mu^{2}_{0})}{3}\,r^{2}\,x_{0}G^{DGLAP}(x_{0},\mu^{2}_{0})\,S(b)$
(3.19)

Of course Eq. (3.19) has no theoretical basis, and our hope is that the
amplitude at low $x$ will be not very sensitive to this kinematic region, in
the initial condition.

$S(b)$ is the impact parameter profile function, which is chosen as the
Fourier image of the proton form factor, namely,

$S(b)\,\,=\,\,\frac{2}{\pi\,R^{2}}\left(\frac{\sqrt{8}\,b}{R}\right)\,K_{1}\left(\frac{\sqrt{8}\,b}{R}\right)$
(3.20)

where $R$ is the proton radius ($R=0.89\,fm$).

In Eq. (3.16) $P\left(Y-Y_{0};x,y;x^{\prime},y^{\prime}\right)$ stands for the
BFKL Pomeron Green’s function. This Green’s function describes the energy
($x$) evolution of the gluon system in the region of high energy (low $x$).
This evolution allows us to find the dipole scattering amplitude or parton
density at low $x$ from the initial condition: the amplitude at lower
$x=x_{0}$, but for any value of the dipole size ($r$).

However, we know that the BFKL Pomeron alone cannot describe the experimental
data, since it determines the anomalous dimension only at low $x$. The
experimental data shows a good agreement with the DGLAP evolution equation,
which is written as the evolution in $\ln(\mu^{2})$ (see Eq. (3.17)), which
gives us the parton density from the boundary condition: the parton density at
$\mu^{2}=\mu^{2}_{0}$ but at any value of $x$ including the region of low $x$.
Strictly speaking, it cannot be used in Eq. (3.16), but assuming Eq. (3.18)
for the scale $\mu^{2}$, we impose the condition that at long distances, the
typical momentum scale in the linear evolution equation is equal to
$\mu^{2}_{0}$. In the framework of this assumption, it looks reasonable to
assume that

$\displaystyle\Omega(Y,r,b)\,\,\,$ $\displaystyle=$
$\displaystyle\,\,2\,\int\,d^{2}x^{\prime}\,d^{2}y^{\prime}\,\,P\left(Y-Y_{0};x,y;x^{\prime},y^{\prime}\right)\gamma\left(Y_{0};x^{\prime},y^{\prime}\right)\,\,$
(3.21) $\displaystyle=$
$\displaystyle\frac{\pi^{2}\,\alpha_{S}(\mu^{2})}{3}\,r^{2}\,xG^{DGLAP}(x,\mu)\,S(b)$

Finally, we can write the following model expression for the scattering
amplitude;

$N\left(x;r,b\right)\,\,\,=\,\,1\,\,\,-\,\,\exp\left(-\frac{1}{2}\Omega\left(x,r,b\right)\right)$
(3.22)

where $\Omega\left(x,r,b\right)$ is given by Eq. (3.21).

###  3.3 Fixing the values of the phenomenological parameters

In Eq. (3.18) we have introduced two phenomenological parameters:
$\mu^{2}_{0}$ and $C$. The first determines the value of the virtuality of the
probe in DIS processes from which we can start using the perturbative QCD
evolution equations. The second ($C$) gives the relation between scales in the
momentum representation and in the coordinate representation. In the case of
the DGLAP evolution, $C=4$ ( see Ref. [38]), but since the amplitude has a
more complicated form than Eq. (3.22), the value of $C$ can be different.

Two more parameters stem from the initial condition for the gluon density in
Eq. (3.21), namely,

$xG^{DGLAP}\left(x,Q_{0}^{2}\right)\,\,\,=\,\,\frac{A}{x^{\omega_{0}}}(1-x)^{6}\,$
(3.23)

which corresponds to the exchange of the soft Pomeron with the intercept
$\omega_{0}$ at the initial hard scale $\mu^{2}=Q^{2}_{0}$ in traditional high
energy phenomenology, and the factor $(1-x)^{6}$ reflects the valence quark
dependance. In our approach we do not assume that the soft Pomeron exists, and
we view Eq. (3.23) just as the simplest function that reflects the behavior of
the experimental data in DIS.

We use the observation of Ref. [39], that the anomalous dimension in the
leading order can be written in a very simple form, namely,

$\gamma(\omega)\,\,=\,\,\bar{\alpha}_{S}\left(\frac{1}{\omega}\,\,-\,\,1\right)$
(3.24)

The explicit solution to the DGLAP evolution equation with $\gamma(\omega)$ of
Eq. (3.24), and with the initial condition of Eq. (3.23), has been found in
Ref. [15] and looks as follows

$xG^{DGLAP}(x,t)\,=\,\sum_{k=0}^{6}\binom{6}{k}\,(-1)^{k}\,xG^{(k)}(x,t,\omega_{k})$
(3.25)

where

$xG^{(k)}\left(y\equiv\ln(1/x),t,\omega_{k}\right)\,\,=\,\,A\,e^{-t\,+\,\omega_{k}\,y}\,\,\left\\{\int^{y}_{0}\,\,dy^{\prime}\,e^{-\omega_{k}\,y^{\prime}}\,\sqrt{\frac{t}{y^{\prime}}}\,\,I_{1}\left(2\sqrt{t\,y^{\prime}}\right)\,\,\,+\,\,\,1\right\\}$
(3.26)

with

$\omega_{k}=\omega_{0}-k\,;\;\;\;\;t\,\,\,\equiv\,\,\,\frac{4N_{c}}{b_{0}}\,\ln\frac{\ln\left(\mu^{2}/\Lambda^{2}\right)}{\ln\left(\mu^{2}_{0}/\Lambda^{2}\right)}\,;\,\,\,\,\,\,\,\,b_{0}\,\,=\,\,11-\frac{2n_{f}}{3}\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,y\,=\,\ln(1/x)$
(3.27)

In Eq. (3.21) the typical scale of hardness in the gluon structure function is
determined by the process of gluon emission while the factor
$\alpha_{S}\,r^{2}$ takes into account the integration over the wave function
of the dipole with the size $r$. Having this in mind, we introduce a different
scale for $\alpha_{S}$ in Eq. (3.21). Finally, we use

$\Omega(x,r,b)\,\,\,=\,\,\,\frac{\pi^{2}\,\alpha_{S}(\tilde{\mu}^{2})}{3}\,r^{2}\,xG^{DGLAP}(x,\mu)\,S(b)\,\,\,\mbox{with}\,\,\,\tilde{\mu}^{2}\,\,=\,\,\tilde{\mu}^{2}_{0}\,\,+\,\,\frac{C}{r^{2}}$
(3.28)

where $xG^{DGLAP}(x,\mu)$ is given by Eq. (3.26) and the initial hard scale
$\widetilde{\mu}^{2}_{0}$ is different from that of $\mu_{0}^{2}$. We perform
a fit of the saturation model, to the DIS experimental data and fixing the
parameters, we wish to describe the soft data without using the concept of the
”soft” pomeron. The fit procedure was performed by using the minuit routine,
the statistical and systematical errors have been added in quadrature. We have
used all of the recent data on deep inelastic scattering processes, from
different collaborations. The most suitable were H1 [16] and ZEUS [17, 18]. It
was observed, that H1 data for medium and large values of $Q^{2}$ coincides
with that from the ZEUS col., but up to a scaling factor of 1.05. Hence, for
the fitting procedure, we took only the ZEUS data for vary small, medium and
large values of $Q^{2}$. Since we are interested in the high energy
description, we took data below $x<0.01$ and $0.045<Q^{2}<150GeV^{2}$. The
upper limit on virtuality is originated from the upper limit on the $x$
variable. Fitting with H1 scaled data, yields almost the same result. The
total number of experimental points was 170. During the fitting procedure, we
observed a strong correlation between different parameters, so we decided to
fix a number of them. We choose to fix a parameter $C$ and the initial hard
scale $Q^{2}_{0}$. The masses of the quarks were taken as follows:
$m_{light}\;=\;0.25\;GeV$ and $m_{charm}\;=\;1.3\;GeV$. $\chi^{2}$ for this
parametrization is $\thicksim 1.05$. The summary of the fitting procedure is
given in the table 1

Model | $\widetilde{\mu}_{0}^{2}$ | $\mu_{0}^{2}$ | $Q_{0}^{2}$ | $C$ | $\omega_{0}$ | $A$ | $\chi^{2}/d.o.f.$
---|---|---|---|---|---|---|---
A | 0.23 | 1.23 | 1.0 | 1.0 | 0.028 | 1.81 | 1.04
B | 0.069 | 1.23 | 0.58 | 1.0 | 0.087 | 0.6 | 3.08
Table 1: _Resulting parameters from the fit to DIS data Model A gives a very
good description of the DIS data, while model B reproduces quite well the data
on soft processes_.

The bolded font corresponds to the fixed parameters $C$ and $Q^{2}_{0}$, which
were chooses to be equal to $1.0$. The value of the hard scale
$\mu^{2}_{0},\;\widetilde{\mu}_{0}^{2}$. is in the units of energy $GeV$.

As one can see, $\chi^{2}/d.o.f.$ is close to 1 and fit gives very good
description of all experimental data on DIS (see Fig. 1). However, we present
in Table 1 a second fit which leads to worse $\chi^{2}/d.o.f.$ but reproduces
the data on soft interaction quite well as we will see below. It should be
stressed that both fits describe the DIS data at large values of photon
virtualities with small $\chi^{2}/d.o.f.$ . Therefore, we can consider model
the A as an attempt to describe the experimental data on DIS and soft
processes, assuming that Eq. (3.22) is correct in the saturation region. In
model B we explore a different idea: Eq. (3.22) is only approximate formula
that incorporates the main qualitative properties of scattering amplitude for
both DIS and soft scattering in the saturation region. The question which we
try to answer using model B is the following: is it possible to give the
unique description of long distance physics both in DIS and soft interaction
based on the QCD amplitudes at short distances.

|  |
---|---|---
|  |
|  |
Figure 1: Description of the DIS experimental data in our models. Dotted line
describes the model B while solid line corresponds to Model A.

##  4 Cross sections of hadron-hadron interaction at high energy

###  4.1 General approach

Eq. (3.22) gives a smooth continuation to the long distance physics describing
the DIS data at very low values of the photon virtualities. We wish to extend
this description to the hadron-hadron interaction without assuming something
in addition, for example, the existence of the soft Pomeron. The simplest
formula we can write for the total cross section, is a straightforward
generalization of the formula for the DIS cross section with the replacement
of $\Psi_{\gamma^{*}}(r)\to\Psi_{hadron}(\\{r_{i}\\})$ , namely

$\sigma_{tot}\,\,=\,\,2\,\int\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)$
(4.29)

where $n$ is the number of dipoles that we need to introduce to describe a
hadron. For example, for a meson we need only one dipole, while for the proton
we have to introduce at least two colorless dipoles.

The total elastic cross section can be written in the form:

$\sigma_{el}\,\,=\,\,\int\,d^{2}b\left(\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)\right)^{2}$
(4.30)

For the differential elastic cross section we have the following expression

$\frac{d\sigma_{el}}{dt}\,\,=\,\,\left(\frac{1}{(2\pi)^{2}}\,\int
J_{0}(qb)d^{2}b\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i},b)\right)^{2}$
(4.31)

where $J_{0}$ is the Bessel function. It is even easier to calculate the slope
in $t$ at $t=0$. It is equal to

$B_{el}\,\,=\,\,\frac{d\ln
d\sigma_{el}/dt}{dt}|_{t=0}\,\,=\,\,\frac{1}{2}\,\frac{\int\,b^{2}\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)}{\int\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)}$
(4.32)

The process of diffractive dissociation is a more complicated phenomenon.
Indeed, the eikonal type formula of Eq. (3.22) leads to diffractive
dissociation in the state of $n$ free dipoles which, being a system with a
limited number of dipoles, cannot create a system of hadrons with large mass.
In other words, this diffractive production falls down as a function of
produced mass. Therefore, we can calculate diffraction in the region of the
small mass using the Good-Walker formula [41] , namely

$\displaystyle\sigma^{low\;M}_{diff}\,\,\,=\,\,\,$ (4.33)
$\displaystyle\int\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N^{2}(x,r_{i};b)\,\,-\,\,\int\,d^{2}b\left(\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)\right)^{2}$

the first term is the cross section for the production of the system of $n$
free dipoles, while the second is the elastic cross section. We subtracted
this term to find the cross section of the hadron state, which is different
from the initial one.

Figure 2:  Simple Pomeron diagrams for the total cross section (Fig. 2-a) and
for the diffractive production in the region of large mass
($Y-Y_{1}=\ln(M^{2}/s_{0})$, Fig. 2-b). The dashed line shows the cut Pomeron.

. For large mass diffraction, we have to develop a new approach. The simple
form of the generating functional of Eq. (3.15) stems from the fact that the
system of interacting Pomerons can be reduced to the exchange of non-
interacting Pomerons after integration over rapidity $Y_{1}$, as it is shown
in Fig. 2-a. In the case of diffractive production we can also calculate the
Pomeron diagrams with one cut Pomeron (shown by dashed line in Fig. 2-b) but
the value of $Y-Y_{1}$ is fixed, namely, $Y-Y_{1}=\ln(M^{2}/s_{0})$ where $M$
is the mass of the diffractively produced system and $s_{0}$ is the energy
scale ($s_{0}\approx 1GeV^{2}$). In more complicated diagrams, we have to
integrate over rapidities (for example, over $Y^{\prime}_{1}$ and
$Y^{\prime}_{2}$ in Fig. 2-b). For the Pomeron with the intercept larger than
1, as in our case, these integrations result in three diagrams in Fig. 2-b. It
should be stressed that the integral over the produced mass gives the dominant
contribution at $Y-Y_{1}=\ln(M^{2}/s_{0})\,\approx 1/\bar{\alpha}_{S}$.
However, it has not been taken into account in our eikonal - type model, which
only describes the G-W contribution to the region of small mass.

Therefore, for $d\sigma/dM^{2}$ at large $M^{2}\gg s_{0}$, for one dipole
$(x,y)$ we can write a generic formula (see Refs. [44])

$\displaystyle
M^{2}\,\frac{d\sigma^{high\;M}_{diff}}{dM^{2}}\,\,\,=\,\,\,V_{1\to
2}\left((x^{\prime},y^{\prime})\to(x^{\prime},z)+(z,y^{\prime})\right)\,\,\bigotimes\,\,\,\,\exp\left(-\Omega(s;x,y)\right)$
(4.34)
$\displaystyle\left(1-\exp\left(-\tilde{\Omega}(M^{2};x,y;x^{\prime},y^{\prime})\right)\right)\,\left[\exp\left(-\frac{1}{2}\left\\{\Omega(s/M^{2};x^{\prime},z)+\Omega(s/M^{2};z,y^{\prime})\right\\}\right)\,-\,\exp\left(-\frac{1}{2}\Omega(s;x^{\prime},y^{\prime})\right)\right]^{2}$

where $\bigotimes$ denotes all needed integrations, and $\tilde{\Omega}$ is a
new opacity for the interaction of two dipoles: $(x,y)$ and
$(x^{\prime},y^{\prime})$, which we can build using the same approach as in
Eq. (3.21). We start to analyze Eq. (4.34) considering the emission of an
extra gluon (see Fig. 4).

Figure 3: The process of diffractive production in the region of large mass in
perturbative QCD.

Figure 4: Feedback to the process of elastic scattering of dipole$(x,y)$ due
to the emission of an extra gluon.

One can see that our process has three well separated stages. The first one is
a penetration of the dipole $(x,y)$ through the target without inelastic
interaction. We introduce the factor ”$\exp\left(-\Omega\right)$” to describe
this stage. This factor sums all Pomeron diagrams with Pomerons that carry the
total energy of the process( rapidity $Y$). See the second diagram in Fig.
2-b. The second stage is the emission of one extra gluon. The dipole decay is
responsible for this stage with the probability given by Eq. (3.8). The last,
third stage is the interaction of two produced dipoles $(x^{\prime},z)$ and
$z,y^{\prime})$ with the target. This stage is taken into account in Eq.
(4.34) by the factor in the brackets. This factor has been discussed in Ref.
[43].

In the first diagrams (see Fig. 4) the second stage is very simple: it is just
the perturbative emission of one extra gluon. The simplified formula for this
diagram has the same structure as Eq. (4.34), but with a simple expression for
$\exp\left(-\Omega(s;x,y)\right)\left(1-\exp\left(-\tilde{\Omega}(M^{2};x,y;x^{\prime},y^{\prime})\right)\right)$,
namely

$\displaystyle\sigma^{high\;M}_{diff}({Fig.~{}\ref{gem1}})\,\,=\,\,\frac{\bar{\alpha}_{S}\,n}{2}\,\,\int^{\infty}_{M^{2}_{0}}\,\frac{dM^{2}}{M^{2}}\,\,\,\int
d^{2}b\int\prod^{n}_{i=1}\,dr^{2}_{i}\,\,\exp\left(-\sum^{n}_{i=2}\Omega(s/s_{0};r_{i},b)\right)\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,$
$\displaystyle\left\\{\exp\left(-\Omega(s/s_{0};r_{1},b)\right)\,\,r^{2}_{1}\int^{\infty}_{r^{2}_{1}}\frac{dr^{2}}{r^{4}}\,\left(1-\exp\left(-\left\\{\Omega(s/M^{2};r;b)\,-\,\frac{1}{2}\Omega(s/M^{2},r_{1};b)\right\\}\right)\right)^{2}\,\,\right.$
$\displaystyle\left.-\,\,r^{2}_{1}\,\int\frac{d^{2}r}{2\pi\,r^{2}\,(\vec{r}_{1}-\vec{r})^{2}}\,\left(1\,\,-\,\,\exp\left(-\Omega(s/s_{0};r_{1},b)\right)\right)^{2}\right\\}$
(4.35)

where $n$ is the number of dipoles in a proton and $\vec{r}=\vec{x}-\vec{z}$ .
In Eq. (4.1) we consider the region of integration where $r_{i}\ll r$. The
second term in curly brackets takes into account a change for elastic
scattering of the dipole $r_{1}$ due to the emission of one extra gluon (see
Fig. 4.) Eq. (4.1) is written in the leading log approximation of perturbative
QCD in which we consider $\bar{\alpha}_{S}\ln(M^{2}/s_{0})\,\approx\,1$ while
$\alpha_{S}\ll 1$. Factor
$\left(1-\exp\left(-\Omega(s/M^{2};r)+\frac{1}{2}\Omega(s/M^{2},r_{l};b)\right)\right)$
is the amplitude of gluon-dipole scattering that has been discussed in Ref.
[43]. The formulae for diffraction production is well known (see Refs. [44]
for details). In the same approximation Eq. (4.34) has the form

$\displaystyle
M^{2}\,\frac{d\sigma^{high\;M}_{diff}}{dM^{2}}\,\,\,=\,\,\,\frac{\bar{\alpha}_{S}\,n}{2}\,\,\,\int\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\left(\exp\left(-\Omega(s/s_{0};r_{i},b)\right)\right)\,$
(4.36) $\displaystyle\int\,d^{2}r\,\,r^{2}\,\,\int
d^{2}b^{\prime}\,\,\,\left(1-\exp\left(-\Omega^{BFKL}\left(M^{2}/s_{0};r_{i},r\right)\right)\right)\,\,\int^{\infty}_{r^{2}_{i}}\frac{dr^{\prime
2}}{r^{\prime
4}}\,\,\,\left(1-\exp\left(-\Omega(s/M^{2};r^{\prime};b)\right)\right)^{2},$

where $\Omega^{BFKL}\left(M^{2}/s_{0};r_{i},r\right)$ is the solution of the
BFKL equation with the initial condition:

$\frac{d\,\,\Omega^{BFKL}\left(M^{2}/s_{0}=1;r_{i},r\right)}{d\ln(M^{2}/s_{0})}\,\,\,=\,\,\,\frac{\bar{\alpha}_{S}}{2}\,\delta(\vec{r}-\vec{r}^{\prime})$
(4.37)

This $\Omega$ is equal to [5]

$\displaystyle\Omega^{BFKL}\left(M^{2}/s_{0}=1;r_{i},r\right)\,\,$
$\displaystyle=$
$\displaystyle\,\,\int\frac{d\gamma}{2\pi\,i\,\omega(\gamma)}\,e^{\omega(\gamma)\ln(M^{2}/s_{0})\,\,+\gamma\ln(r^{2}_{i}/r^{2})}\,\,$
$\displaystyle=$
$\displaystyle\,\,\sqrt{\frac{r_{i}^{2}}{r^{2}}}\,\frac{1}{\omega(0)}\,\sqrt{\frac{2\pi}{\omega"(0)\ln(M^{2}/s_{0})}}\,\,e^{\omega(0)\ln(M^{2}/s_{0})\,\,-\,\,\frac{\ln^{2}(r^{2}/r^{2}_{i})}{2\omega"(0)\,\ln(M^{2}/s_{0})}}$

where the eigenvalues of the BFKL equation $\omega(\nu)$ can be found in Ref.
[5].

The factor $1-\exp\left(-\Omega^{BFKL}\right)$ describes the inelastic cross
section which in terms of the Pomeron diagrams of Fig. 2, reflects the
possibility that in the fourth of Fig. 2-b, we can have two cut upper
Pomerons.

We would like to stress again that Eq. (4.1) describes the dependence of the
diffractive production cross section, in the region of large mass, while Eq.
(4.33) is written for the low mass diffractive cross section. Therefore, to
calculate the cross section for diffractive production, we need to calculate

$\sigma_{diff}\,\,\,=\,\,\sigma^{low\;M}_{diff}\left({Eq.~{}(\ref{SDXSSM})}\right)\,\,+\,\,\sigma^{high\;M}_{diff}({Eq.~{}(\ref{SDLM})})$
(4.39)

###  4.2 Hadronic wave functions

As seen from the formulae in the previous subsection, we need to know the
hadronic wave functions. We have made an assumption in writing these formulae,
that the correct degrees of freedom at long distances are the colorless
dipoles at least at high energy[23, 24]. This is one of the strongest
assumptions in our approach. It is enough to recall that for a long time the
constituent quarks have been considered as a good candidate for the correct
degrees of freedom in the entire range of energy. The only support for such an
approach, can be seeen in the success of the Heildelberg group (see Refs. [23]
and references therein), in the description of soft interactions using this
ansatz.

In the case of mesons, we have only one colorless dipole and we take the
transverse wave function in the form of a simple Gaussian, namely

$|\Psi_{meson}\left(r\right)|^{2}\,\,\,=\,\,\frac{1}{\pi
S^{2}_{M}}\,e^{-\frac{r^{2}}{S^{2}_{M}}}$ (4.40)

where $S_{M}$ is a parameter that can be found from the electromagnetic radii,
namely, $R_{\pi}=0.66\pm 0.01fm$ and $R_{K}=0.58\pm 0.04fm$ [40]. Using
$S_{M}=\sqrt{\frac{8}{3}}R_{M}$ we obtain $S_{\pi}=1.08fm$ and $S_{K}=0.95fm$.

Figure 5: Electromagnetic (Fig. 5-a) and two gluon (Fig. 5-b) form factors.
for a meson. Fig. 5-c shows the interaction of a virtual photon with a parton.

However, electromagnetic form factors gives us the space distribution of the
electric charge inside of the hadron, while in our case the distribution of
the density of partons (dipoles) is probed by the two gluon interaction (see
Fig. 5-b). This distribution could be different from the charged one. We
prefer to find the value of $S_{M}$ from the experimental data and, therefore,
$S_{M}$ as well as $S_{p}$ (see below), will be the only fitting parameters in
our description of the soft experimental data.

For the baryon, the situation is more complicated: we can have two or even
three dipoles. Follow Ref. [23], we choose the proton wave function in the
simple form for the two dipole model

$|\Psi_{proton}(r_{1},r_{2})|^{2}\,\,=\,\,\frac{1}{(\pi\,S_{p})^{2}}\,\,e^{-\frac{r^{2}_{1}+r^{2}_{2}}{S^{2}_{p}}}$
(4.41)

where two dipoles are defined as $\vec{r}_{1}=\vec{R}_{1}-\vec{R}_{2}$ and
$\vec{r}_{2}=\vec{R}_{3}-\frac{1}{2}\left(\vec{R}_{1}+\vec{R}_{2}\right)$
where $\vec{R}_{i}$ is the position of the constituent quark $i$. In Eq.
(4.41) $S_{P}=\sqrt{\frac{3}{2}}\,R_{p}\,=\,1.05fm$ for $R_{p}=0.862\pm
0.012fm$[40].

However, for large $N_{c}$ it is proven that the baryon consists of $N_{c}$
dipoles [42]. Therefore, we consider the alternative assumption for
$\Psi_{proton}(r_{1},r_{2},r_{3})$, namely

$|\Psi_{proton}(r_{1},r_{2},r_{3})|^{2}\,\,=\,\,\frac{1}{(\pi\,S_{p})^{3}}\,\,e^{-\frac{r^{2}_{1}+r^{2}_{2}\,+\,r^{2}_{3}}{S^{2}_{p}}}$
(4.42)

where dipoles have the size
$\vec{R}_{i}-\frac{1}{3}(\vec{R}_{1}+\vec{R}_{2}+\vec{R}_{3})$. In this case,
from the electromagnetic radius of the proton, follows the value of
$S_{p}=R_{p}=0.862\pm 0.012fm$.

In Eq. (4.41), we take the size of two dipoles to be equal. We did this for
simplicity, since even in the constituent quark model (CQM),
$<|r^{2}_{2}|>=4/3<|r^{2}_{1}|>$. Our hope is that the hadron interaction will
be determined by the behavior of the dipole amplitude in the saturation
domain, where the sensitivity to the size of dipoles is expected to be weak.
On the other hand, the experimental ratio
$\sigma_{tot}(\pi-p)/\sigma_{tot}(p-p)\,\approx\,2/3$ in the CQM stems from
quark counting. In our approach, the dipole counting leads to the value of
this ration $1/2$ (for the two dipole model for a proton) or even 1/3 (in the
three dipole model) which contradicts the experimental data. The only way to
obtain a reasonable description, is to hope that the perturbative QCD
dependence of the dipole cross section on the size of the dipoles
$\sigma\propto r^{2}$ will remain in the entire accessible range of energies.
Fortunately, Ref. [24] demonstrates that this is the case, and this result
encourages us to search for the description of the soft processes using the
dipole hypothesis for the hadronic wave functions.

###  4.3 Energy variable in the model

In section 2.2 and 2.3 we have discussed our approach introducing the typical
energy variable for deep inelastic scattering: $x_{Bjorken}\equiv x=Q^{2}/s$
where $Q^{2}$ is the photon virtuality and $s$ is the energy. However, with
this variable we cannot discuss the soft processes which have $Q^{2}=0$. We
reconsider the derivation of the Bjorken variable to introduce a new energy
variable which will have a limit $x_{soft}\to x$ at $Q^{2}\gg\mu^{2}$ where
$\mu$ is the scale for the soft processes. From Fig. 5-c we have a relation

$\displaystyle(q+k)^{2}$
$\displaystyle=-Q^{2}+x_{soft}s\,\,+\,\,k^{2}\,\,=\,\,0;$ $\displaystyle
x_{soft}$ $\displaystyle=\left(Q^{2}+k^{2}_{\perp}\right)/s$ (4.43)

where $q$ is the momentum of virtual photon ($q^{2}-Q^{2}$).

In Eq. (4.43) we used the fact that at high energy, $k^{2}=-k^{2}_{\perp}$.
Since in the saturation domain $k^{2}_{\perp}\,\,=\,\,Q^{2}_{s}(x)$, we obtain
the final expression for our energy variable

$\displaystyle x_{soft}$
$\displaystyle\,\,=\,\,\frac{Q^{2}+Q^{2}_{s}\left(x_{soft}\right)}{s}\,\,\xrightarrow{Q^{2}\rightarrow
0}\,\,\frac{Q^{2}_{s}\left(x_{soft}\right)}{s};$ $\displaystyle x_{soft}$
$\displaystyle\,\,=\,\,\frac{Q^{2}+Q^{2}_{s}\left(x_{soft}\right)}{s}\,\,\xrightarrow{Q^{2}\gg
Q^{2}_{s}}\,\,x_{Bjorken};$ (4.44)

We believe that one of the main advantages of the saturation approach is the
natural choice of the energy variable given by Eq. (4.44), which depends on
the new scale: the saturation momentum.

The value of the saturation momentum in our model, we find by resolving the
following equation:

$\Omega\left(x_{soft},r^{2}_{sat}\,b=0\right)\,\,=\,\,1\,\,\,\mbox{with}\,\,r^{2}_{sat}=4/Q^{2}_{S}\left(x_{soft}\right)$
(4.45)

Partons (dipoles) with size $r_{sat}$ are populated densely in the hadron
disc, and $Q_{s}$ gives the new scale which shows that at $Q^{2}<Q^{2}_{s}$
the hard process reaches the saturation domain.

###  4.4 Final formulae for proton (antiproton) - proton scattering

Using the form of the proton wave function given by Eq. (4.41), we can rewrite
Eq. (4.29) - Eq. (4.32) in a more accurate form, taking into account the
simultaneous interaction of two dipoles (see Eq. (6)).

Figure 6:  Total cross section for proton( antiproton) -proton collision
taking into account the interaction of two dipoles.

For the total cross section, the general formula has the form which follows
directly from Fig. 6, namely

$\displaystyle\sigma_{tot}\,\,=\,\,2\,\int\,d^{2}b\,d^{2}r_{1}\,d^{2}r_{2}\,\,|\Psi_{proton}\left(r_{1},r_{2}\right)|^{2}$
(4.46)
$\displaystyle\,\,\left(N(x,r_{1};\vec{b}-\frac{1}{3}\vec{r}_{2})\,\,+\,\,N(x,r_{2};\vec{b}-\frac{1}{6}\vec{r}_{2})\,\,-\,\,N(x,r_{1};\vec{b}+\frac{1}{3}\vec{r}_{2})\,N(x,r_{2};\vec{b}+\frac{1}{6}\vec{r}_{2})\right)$

where $\vec{b}$ is the impact parameter or the distance between
$\frac{1}{3}(\vec{R}_{1}+\vec{R}_{2}+\vec{R}_{3})$ and the position of the
target nucleon. One can see that we need to replace

$\displaystyle\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)\,\,\longrightarrow\,\,\,|\Psi_{hadron}(r_{1},r_{2})|^{2}$
(4.47)
$\displaystyle\left(N(x,r_{1};\vec{b}-\frac{1}{3}\vec{r}_{2})\,\,+\,\,N(x,r_{2};\vec{b}-\frac{1}{6}\vec{r}_{2})\,\,-\,\,N(x,r_{1};\vec{b}+\frac{1}{3}\vec{r}_{2})\,N(x,r_{2};\vec{b}+\frac{1}{6}\vec{r}_{2})\right)$

in all of the equations of section 3.1.

Using the explicit form of the proton wave function of Eq. (4.41), we can
easily rewrite the integrals in Eq. (4.29) - Eq. (4.32), in the form

$\displaystyle\int\,d^{2}b\,d^{2}r_{1}\,d^{2}r_{2}\,\,|\Psi_{proton}\left(r_{1},r_{2}\right)|^{2}$
$\displaystyle\left(N(x,r_{1};\vec{b}-\frac{1}{3}\vec{r}_{2})\,\,+\,\,N(x,r_{2};\vec{b}-\frac{1}{6}\vec{r}_{2})\,\,-\,\,N(x,r_{1};\vec{b}+\frac{1}{3}\vec{r}_{2})\,N(x,r_{2};\vec{b}+\frac{1}{6}\vec{r}_{2})\right)$
$\displaystyle=$ $\displaystyle\int
dr^{2}_{1}\,db^{2}_{1}\,db^{2}_{2}\,\frac{1}{S_{p}^{2}}\,\exp\left(-\frac{r^{2}_{1}+4\,b^{2}_{1}+4\,b^{2}_{2}}{S^{2}_{p}}\right)\,I_{0}\left(\frac{4}{S^{2}_{p}}b_{1}\,b_{2}\right)\,\,$
$\displaystyle\times$
$\displaystyle\left(N(x,r_{1};b_{1})\,\,\,+\,\,\,N(x,r_{1};b_{2})\,\,\,-\,\,N(x,r_{1};b_{1})\,\,N(x,r_{1};b_{2})\right)$

In Eq. (4.4), we used the formula 8.431(3) of Ref. [61].

Since the typical impact parameter increases with energy, (see the next
section) and the $r^{2}/3$ is small (about $1/3\,fm$ ), we can safely neglect
the shift in the definition of the impact parameters in different amplitudes,
replacing

$\displaystyle\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)\,\,\longrightarrow\,\,\,$
(4.49)
$\displaystyle|\Psi_{hadron}(r_{1},r_{2})|^{2}\,\,\left(N(x,r_{1};\vec{b})\,\,+\,\,N(x,r_{2};\vec{b})\,\,-\,\,N(x,r_{1};\vec{b})\,N(x,r_{2};\vec{b})\right)$
$\displaystyle|\Psi_{hadron}(r_{1},r_{2})|^{2}\,\,\left(1\,\,\,-\,\,\,\exp\left[-\frac{1}{2}\left(\Omega\left(x,r_{1};\vec{b}\right)+\Omega\left(x,r_{2};\vec{b}\right)\right)\right]\right)$
(4.50)

This simplified formula works quite well in all the observables of section
3.1, except for $d\sigma_{el}/dt$ at large values of $t$. The formula of Eq.
(4.50) has a simple generalization to the case of the three dipole proton
model . In the three dipole model, for a proton the cross section has the
form:

$\displaystyle\sigma_{tot}\,\,=$ (4.51) $\displaystyle
2\,\int\,d^{2}b\,d^{2}r_{1}\,d^{2}r_{2}\,\,|\Psi_{proton}\left(r_{1},r_{2},r_{3}\right)|^{2}\,\,\left(1\,\,\,-\,\,\,\exp\left[-\frac{1}{2}\left(\Omega\left(x,r_{1};\vec{b}\right)+\Omega\left(x,r_{2};\vec{b}\right)+\Omega\left(x,r_{3};\vec{b}\right)\right)\right]\right)$

It should be stressed that the second term in Eq. (4.46) plays a very
important role at high energies, since it provides the observation that the
cross section approaches a black disc limit. Indeed, only with this non-linear
term in the region where the dipole amplitude tends to unity, ($N\to 1$) the
factor in parenthesis ($2N-N^{2}$) also approaches 1 leading in the black disc
regime.

###  4.5 Description of the soft cross sections

In the previous sections we have built our model and fitted all needed
parameters using the DIS experimental data. In this section, we compare the
model with the experimental data without any additional fitting parameters.

####  4.5.1 Total cross sections

Using Eq. (4.29) , Eq. (4.40) and Eq. (4.41) we calculate the cross sections
of pion-proton, kaon -proton, proton-proton and antiproton - proton
scattering. The results are presented in Fig. 7 and Fig. 8. In addition to Eq.
(4.29), we add a contribution of the secondary Regge trajectories using the
Donnachie and Landshoff parametrization [45]. The agreement with the data is
good, and shows that the saturation can replace the soft Pomeron, which has
been used for fitting the soft experimental data. However, as has been
mentioned, we still have one fitting parameter, namely, $S_{M}$ and $S_{p}$ in
the hadron wave function. The values of $S_{\pi}$,$S_{K}$ and $S_{p}$ is given
in Table 2. It is interesting to notice that the relation between the fitted
values of $S$, is the same since it stems from the electromagnetic radii.

Model |  $S_{\pi}$ (fm) |  $S_{K}$ (fm) |  $S_{p}$(fm)
---|---|---|---
A(2D) | 0.515 | 0.552 | 0.561
A(3D) | 0.515 | 0.552 | 0.458
B(2D) | 0.556 | 0.597 | 0.615
B(3D) | 0.556 | 0.597 | 0.519
Table 2: _Values of parameters in hadronic wave function that give description
of the soft data_.

One can see that we are able to describe both the value and the energy
behavior of the total cross section for meson-proton and baryon - proton
interaction in our two models. However,the model B leads to a better
description demonstrating, in our opinion, that the main idea to replace the
phenomenological soft Pomeron by the behavior of the ’hard’ amplitude in the
saturation region, is fruitful . As has been discussed in the previous
subsection, the most surprising fact is that the model describes both the
proton-proton and meson-proton cross sections.

|
---|---
Fig. 7 \- a | Fig. 7 -b
Figure 7:  Total cross section for $\pi-p$ (Fig. 7-a) and $K-p$ (Fig. 7-b) as
a function of energy. The solid line (DIS) corresponds to the model A which
describes the DIS data with $\chi^{2}/d.o.f.\,<\,1$, while the dotted line
presents the model B (soft) with a worse $\chi^{2}/d.o.f.\,\approx 3$, but
which leads to a better description for DIS, and soft processes data together.
The number of dipoles in the proton wave function is shown in the legend.

Figure 8:  Total cross section for $p\bar{p}$ scattering versus energy in our
approach. Solid lines correspond to the model A, which describes the DIS data
with $\chi^{2}/d.o.f.\,<\,1$, while the dotted lines present the model B
(soft) with worse $\chi^{2}/d.o.f.\,\approx 3$ but which leads to a better
description for DIS, and soft processes data together.

This fact can be explained only by the dependence of the dipole cross section,
on the size of the dipole which the model reproduces. Since such a sensitivity
can be only outside the saturation region, or close to the boundary between
the saturation domain, and the perturbative QCD region. Therefore, we conclude
that the soft data allow us to obtain new information on this very important
transition region between the saturation and perturbative QCD region. It
should be noticed that we also reproduce the difference between the
interaction of pions and kaons. Since our QCD dipole cross section does not
depend on the mass of the quarks, this difference also reflects the difference
in the size of the dipoles, inside pion and kaon.

Figure 9:  Elastic cross section for proton-proton scattering versus energy in
our approach. Solid line shows the comparison with the model A with 3 dipoles
in the proton wave function which described DIS with $\chi^{2}/d.o.f.\,<\,1$
while in the dotted line we plot the comparison with the experimental data the
model B with $\chi^{2}/d.o.f.\,\approx\,3$ for DIS data for the two dipole
model of proton.

Figure 10:  Slope for the cross section for proton-proton scattering versus
energy in our approach. Notations are the same as in Fig. 9.

####  4.5.2 Elastic cross section: energy dependence

We concentrate our efforts on the proton(antiproton) interactions since meson-
proton collisions have been studied experimentally only in the limited energy
range (see Fig. 8). The energy dependence of the elastic cross section is
presented in Fig. 9. We use Eq. (4.30) for performing our calculations. In
Fig. 9 \- a we compare with the experimental data our model B, while in Fig. 9
\- b, is plotted our prediction for model A (three quarks). One can see that
we obtain a good agreement with the experimental data, without any fitting
parameter.

####  4.5.3 Elastic cross section: t-dependence

Using Eq. (4.32) we calculate the slope for the elastic cross section (see
Fig. 10). One can see that in the model B, we reproduce the value and energy
behavior of the slope at high energy, but overshoot the experimental value of
this slope at sufficiently low energy. We did not include the secondary
Reggeons, since we need a piece of information on the slope of the
contribution of the secondary Reggeons. We also need to take into account the
fact that our profile function $S(b)$ corresponds to a power-like form factor
in the $t$ representation, and the value of the slope for such a function is
sensitive on the range of $t$, where the slope was found experimentally. We
calculate the slope at $t=0$, while the slope at an average value of
$|t|=|t_{0}|>0$ is less than at $t=0$. Therefore, it is better to compare with
the experimental data the $t$ dependence of the differential elastic cross
section, given by Eq. (4.31). Such a comparison, one can see in Fig. 11, where
the data at small $t$ is described by our model even at low energies
($\sqrt{s}=23.5GeV\,\,\mbox{and}\,\,62.5GeV$). Notice that at such low
energies, the slope, that we predict, exceeds the experimental one (see Fig.
10). However, in model A, the slope is too small and, actually, this is the
key difference between the two approaches. The origin for such a difference in
the framework of our approach is clear: a different energy dependence of the
dipole-proton amplitude in the transition region between the saturation
domain, and the perturbative QCD region. Indeed, Fig. 1 shows that model B has
a steeper behavior in this region in comparison with model A.

In Fig. 10 we compare our prediction with the standard phenomenological
parametrization for the slope in the soft Pomeron model:
$B_{el}=B_{0}+2\alpha^{\prime}\ln(s/s_{0})$ with
$\alpha^{\prime}=0.25\,GeV^{-2}$ [45] and $B_{0}=9\,GeV^{-2}$. One can see
that our prediction in model B is in perfect agreement with this model at high
energies, while we predict a large slope at low energies. However, model A
leads to a smaller slope than is seen in experiment. It should be noticed, the
comparison with the experimental data (see Fig. 11) shows that model A fits
the experimental data.

Figure 11:  $t$-dependence of the elastic cross section for different energies
at $t\,\leq\,0.6GeV^{2}$. The solid line corresponds to Model A predictions,
while the dotted line describes Model B behavior of the differential cross
section. Both models reproduce small $t$ behavior of the experimental data in
spite of the poor description of the elastic slope in model A.

In Fig. 11 we plot the $t$-dependence for the different energy. We would like
to draw your attention to two interesting features of our model B: it
describes the experimental data quite well at small values of ${t}\leq
0.6\,GeV^{2}$; and it predicts the minimum at the LHC energy at rather small
values of $t$ ($|t|\approx 0.45\,GeV^{2}$). The appearance of this minimum
shows that our model reproduces the effective shrinkage of the diffractive
peak which results in moving the typical diffractive minima in the region of
smaller $t$, in comparison with lower energies. Since we did not include the
real part of our scattering amplitude, we see a deep minimum. However, the
real part of the amplitude will make our minima more shallow. Model _A_ leads
to a smaller value of $B_{el}$, which actually leads to a dip at large values
of $t$, as compared to model _B_. For our rather crude models, such a
description looks successful and , we hope, will encourage others to look
seriously into attempts involving the description of the experimental, data
based on the saturation regime of QCD.

####  4.5.4 Diffraction production: energy behavior of the cross section

Figure 12: Total cross section for meson-proton and proton(antiproton)-proton
scattering scattering at high energies (extrapolation). Notations are the same
as in Fig. 9.

In Fig. 13 the total cross section (integrated over the masses of the produced
particles) of diffractive production is plotted. We use Eq. (4.34) to
calculate this cross section. However, we multiply this formula by 2 since the
two protons: projectile and target, can dissociate diffractively. One can see
from this picture that we reproduce values and energy dependence in agreement
with the experimental data. The qualitative behavior of the experimental data
on $\sigma_{diff}$ versus energy is characterized by the slow dependence on
energy in the region of high energies. This fact is perfectly reproduced by
our model.

Figure 13:  Cross section for single diffraction production for
proton(antiproton)-proton scattering versus energy. The upper curves are the
total cross section for diffractive production while the low curves show the
energy behavior of the low mass contribution to the diffractive cross section
from Eq. (4.33). Notations are the same as in Fig. 9.

However, one can see from Fig. 13 that our approach cannot reproduce
$\sigma_{SD}$ in the region of low energies (small values of produced mass).
There are at least two reasons for such a failure: first, we do not take into
account the exchange of the secondary Reggeons whose contribution is
essential, as we see on the example of the energy behavior of the total cross
section, and, second, the contribution of the one extra gluon emission has
been calculated in leading log(1/x) order, which cannot describe the low mass
diffraction. The experimental data on single diffraction (see Refs. [53, 52])
can be described assuming, in addition to the triple Pomeron vertex, that we
modeled by extra gluon emission, a significant contribution from the Reggeon-
Pomeron-Reggeon vertex in Regge phenomenology, which we cannot include in our
approach.

###  4.6 Predictions for the LHC range of energies

As we have seen, the model is able to describe the available experimental data
quite satisfactorily (and Model B even quite well) both for soft and hard
interactions. Therefore, we can rely on the model B for the predictions in the
LHC energy range. From Fig. 8-Fig. 13 we see that at the LHC energy we have
$\sigma_{tot}=101.3\,mb$, $\sigma_{el}=28.84\,mb$ , $B_{el}=18.4\,GeV^{-2}$
and $\sigma_{diff}=10.5\,mb$ .

Model | $\sigma_{tot}$ | $\sigma_{el}$ | $\sigma_{diff}$ | $B_{el}$ | $A_{el}(b=0)$ | $A_{el}(b=0)$
---|---|---|---|---|---|---
| mb | mb | mb | $GeV^{-2}$ | (Tevatron) | (LHC)
Our A(3D) | 83.0 | 23.54 | 10 | 16.67 | 0.94 | 0.98
Our B(2D) | 101.3 | 28.84 | 10.5 | 18.4 | 0.94 | 0.98
GLM1[25] | 110.5 | 25.3 | 11.6 | 20.5 | 0.6 | 0.7
GLM2[56] | 91.7 | 20.9 | 11.8 | 17.3 | 0.94 | 0.95
RMK[55] | 88.0 (86.3) | 20.1 (18.1) | 13.3 (16.1) | 19 | 0.89 | 0.92
Table 3: Cross sections and elastic slope at the LHC energy in different
models. For the RMK model, we put two parameterizations and the value for
$B_{el}$ directly from the curve in Fig.18 of Ref.[55]

Table 3 shows that our prediction for LHC energy is close to ones that are
given by the models that fitted the experimental data. The RMK and GLM2 models
are based on soft Pomeron phenomenology while the GLM1 model is close to our
approach: in this model the existence of soft Pomeron is not assumed and the
parametrization was chosen for from the same ideas as our approach here.
However, in the GLM1 model all parameters are found from fitting soft data. In
spite the fact that the values of cross section are close the different models
are different in more detailed characteristics. For example, one can see from
the Table 1 they predict different values for $A_{el}(b=0)$ . From unitarity
$A_{el}(b=0)\,\leq\,1$. Our model predicts that at the LHC energy our proton-
proton collision is close to the black disc regime. The same we can see in
GLM2 model, but in the RMK model we are not so close to this regime at the
LHC, while in the GLM1 models the proton-proton interaction is far away from
the black disc limit.

As you can see from Eq. (8)-c we have some information on the total cross
section from the cosmic ray experiment. However, to extract the value of the
total cross section from such an experiment, we need to know the meson-proton
cross section as well. In Fig. 12, we plotted our predictions which we hope
will be useful for discussing the cosmic ray experiment*** We thank S.
Nussinov for drawing our attention to the necessary knowledge of the meson-
proton total cross sections at high energy..

##  5 Survival probability for diffractive Higgs production

Figure 14:  Diffractive Higgs production.

The diffractive Higgs production, is the reaction which has the best
experimental signature for the discovery of the Higgs boson at the LHC. At
fist sight this process occurs at short distances of the order of $1/M_{H}$,
where $M_{H}$ is the mass of Higgs boson ($M_{h}\geq 100GeV$) and can be
calculated in perturbative QCD (see Fig. 14-a and Ref.[46] for calculations).
However, as was noticed a long ago [47, 48] that it is not enough to calculate
the diagram of Fig. 14-a which describes the Higgs boson production from one
parton shower. We have to multiply this cross section by a probability that
two parton showers (or more) will not interact with the target since they
could produced hadrons that will fill up the large rapidity gaps between Higgs
and protons in the final state (see Fig. 14-b). This probability we call the
survival probability. In the case of our model with the eikonal type formula
for the scattering amplitude, the expression for the survival probability is
very simple (see Fig. 14-c), namely,[47, 48]

$\langle|S^{2}|\rangle\,\,\,=\,\,\frac{\int d^{2}b\,\,\left(\int
d^{2}r_{1}d^{2}r_{2}|\Psi_{proton}(r_{1},r_{2})|^{2}\,e^{-\frac{1}{2}\Omega(s,r_{1},b)}\right)^{4}\,\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)}{\int
d^{2}b\,\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)}$ (5.52)

The appearance of the factor $\exp\left(-\frac{1}{2}\Omega(s,r_{1},b)\right)$
in Eq. (5.52) is clear from Fig. 14-c) but the power of 4 requires discussion.
Actually this power reflects the fact that we have to find the probability
that neither dipole 1 nor dipole 2 could scatter inelastically. The
probability that one dipole does not scatter inelastically is equal to

$\left(\int
d^{2}r_{1}d^{2}r_{2}|\Psi_{proton}(r_{1},r_{2})|^{2}\,e^{-\frac{1}{2}\Omega(s,r_{1},b)}\right)^{2}\,\,\equiv\,\,\left(\langle|e^{-\frac{1}{2}\Omega(s,r_{1},b)}|\rangle\right)^{2}$

Therefore for the probability that two dipoles cannot scatter inelastically we
obtain $\left(\langle|e^{-\frac{1}{2}\Omega(s,r_{1},b)}|\rangle\right)^{4}$.

Figure 15: The value of the survival probability for diffractive Higgs
production versus energy. Solid lines show the survival probability in Model A
that describes the DIS data with a very good $\chi^{2}/d.o.f<1$, while dashed
lines correspond to $<S^{2}>$ in Model B which gives a good description of all
soft data.

We can see from Eq. (5.52) that for the calculation of the survival
probability we need to know the $b$ \- dependence of the hard cross section.
This dependence has been discussed in detail in Ref.[49], where it was
extracted from the process of diffraction production of J$/\Psi$ with the
proton (elastic) and the state with mass larger than the proton (inelastic).
These two processes have two different slopes in the $t$ behavior:
$B_{el}=4\,GeV^{-2}$ and $B_{in}=1.86\,GeV^{-2}$[50] . The dependence of the
vertex for photon to J/$\Psi$ transition was extracted from these reactions in
Ref. [51] leading to the values of slopes for the vertices of the transition
of proton to proton (elastic), and proton to non-proton final state
(inelastic): $B_{el}=3.6\,GeV^{-2}$ and $B_{in}=1.46\,GeV^{-2}$. In our model,
the projectile proton first goes to the state of two free dipoles (see Fig.
14-c) , these two dipoles can produce two and more parton showers; and finally
two free dipoles in hard processes create a proton. Therefore, at least the
upper vertex is the same as in the inelastic diffractive production of
J$/\Psi$ in DIS. Considering the lower vertex being elastic, we can write
$\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)$ in the form

$\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)\,\,\,=\,\,\,\frac{\sigma_{0}}{\pi
R_{H}}\,e^{-\frac{b^{2}}{R^{2}_{H}}}$ (5.53)

with $R^{2}_{H}=B_{in}+B_{el}=5.06\,GeV^{-2}$. However, it is our model
disadvantage that we describe differently projectile and target protons. In
any case the eikonal type model that we use, does not mean that only the
elastic rescattering contributes to the shadowing corrections. Treating both
protons on the same ground we take $R^{2}_{H}=2B_{in}=2.92\,GeV^{-2}$.

Since our model naturally includes both soft and hard interactions we can hope
that the hard process of diffractive Higgs production can be described in the
same way as we did in the model using the same amplitude $\Omega$. However, we
need to take into account that in this case

$\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)\,\,\,=\,\,\sigma_{0}\,S^{2}(b)$ (5.54)

where $S(b)$ is given by Eq. (3.20).

Our results on the survival probability is shown in Fig. 15, and in Table 2.
Table 2 demonstrates that our value for the survival probability turns out to
be much smaller than the values in the phenomenological models, that fitted
the experimental data using the soft Pomeron approach (see Table 2, the review
of [54] and Ref. [55]). We think that the difference stems not from the
details of the models, but from the key ingredient of our model: the fact that
we took into account the interaction at short distances. Therefore, we confirm
that the short distances give a substantial contribution to the value of
survival probability, as has been noticed in Refs. [57, 58]. The Table 2 also
shows that the value of the survival probability crucially depends on the
model for the $b$-dependence of the hard cross section. Our model with Eq.
(5.54) for the hard cross section is similar to the one that was used in the
GLM model [25]. It is interesting that our model confirms the general tendency
to obtain a smaller value for the survival probability advocated in Ref. [25].

Model | $\sigma_{H}(b)$ | $\langle|S^{2}|\rangle$
---|---|---
Our Model A(3D) | Eq. (5.53) $R^{2}_{H}=5.06\,GeV^{-2}$ | 0.24%
| Eq. (5.53) $R^{2}_{H}=2.92\,GeV^{-2}$ | 0.02%
| Eq. (5.54) | 0.89%
Our Model B(2D) | Eq. (5.53) $R^{2}_{H}=5.06\,GeV^{-2}$ | 0.24%
| Eq. (5.53) $R^{2}_{H}=2.92\,GeV^{-2}$ | 0.096%
| Eq. (5.54) | 0.57%
GLM1[25] | Two channel model for $\sigma_{H}$ | 2% (0.7%)
GLM2[56] | Two channel model for $\sigma_{H}$ | 0.21%
RMK[55] | Eq. (5.53) $R^{2}_{H}=11\,GeV^{-2}$ | 3.2% (2.3%)
| Eq. (5.53) $R^{2}_{H}\,=\,8\,GeV^{-2}$ | 1.7% (1.2%)
Table 4:  The survival probability for diffractive Higgs production at the LHC
energy, in different models. In the GLM model, for $\sigma_{H}(b)$ is used in
the two channel model, with the same $B_{in}$ and $B_{el}$ for the diffractive
production of J/$\Psi$ in DIS.

##  6 Lessons from the model

Long distance physics is very complicated, non-perturbative phenomenon, and we
certainly do not pretend that we are able to describe it in its full richness.
However, we demonstrate in this paper that the gap between this physics and
the short distance physics which is under full control of perturbative QCD, is
not so huge that it would be a hopeless task to build a bridge. Comparing this
with the experimental data, we showed that the soft data depends on the
transition region between saturation domain and perturbative QCD region and,
therefore, can give valuable information on this transition, checking our
theoretical approaches to it. The widely used, phenomenological soft Pomeron
do not appear in our approach, and we hope that the reader will ask the
question: do we need a soft Pomeron, having in mind our negative answer.

Our description of the experimental data, is not worse (in the case of Model
B) than the one in the models which fitted the data on the basis of the soft
Pomeron phenomenology (see Refs. [60, 59, 55, 25]). Recalling that we fitted
all parameters of our model from DIS processes, we interpret this success in
the way that the high energy soft scattering processes are determined by QCD,
at short distances of the order of $1/Q_{s}$, where $Q_{s}$ is the saturation
momentum. Model A describes the data worse than Model B, but we consider this
description quite satisfactory remembering the crude character of this
approach. It should be stressed, once more that the difference between our
model A and model B, is in our attitude to the simple formula of Eq. (3.22):
in model A we trust Eq. (3.22) in the entire kinematic region of accessible
distances, while in model B we view this formula as a kind of qualitative
description, that includes the main features of the saturation regime.
Therefore, we were searching for the parameters of model B, in the way that
describes all the data both on DIS, and on soft interactions, in the best
possible way.

In our model, we used several assumptions which have a different theoretical
status. The first assumption is the exponential form of the dipole scattering
amplitude (see Eq. (3.16) and Eq. (3.22). We have discussed the theoretical
arguments for such an assumption, namely, this form has been proven for the
transition region between perturbative QCD, and the saturation domain (see
also Ref. [12]). It should be stressed that the soft data, are sensitive to
the transition region as we have discussed.

The second assumption, is the expression of Eq. (3.21) for
$\Omega\left(x,r,b\right)$. This assumption is a compromise between what we
should do, and what we can do. We can check it by describing the DIS
experimental data, at large values of the photon virtualities, where the
difference between our formula for $\Omega$, and the DGLAP expression for it
should be large.

The third assumption is the impact parameter dependence of
$\Omega\left(x,r,b\right)$ (see Eq. (3.19) and Eq. (3.20)). This is a pure
phenomenological ansatz, since the current stage of our theory does not allow
us to find $b$ dependence [28]. Actually, only soft data allows us to check
this dependence. Indeed, our $\Omega$ describes the $t$-dependence of the
elastic cross section while , for example, the Gaussian $b$ dependence results
in the appearance of the structure of maxima and minima, at small values of
$t$, which contradicts the experimental behavior of $d\sigma_{el}/dt$.

In Fig. 16-a we plot the dipole amplitude averaged with the parton wave
function

$\langle|N|\rangle|\,\,\,\equiv\,\,\,\int
d^{2}r_{1}d^{2}r_{2}|\Psi_{proton}(r_{1},r_{2})|^{2}N(x,r_{1},\vec{b}=0)$
(6.55)

One can see, that this amplitude increases and it approaches 1. However, it
happens at ultra high energies and at the Tevatron energy, for example, this
amplitude is only $0.8$ at $b=0$. Such an average scattering amplitude, leads
to the elastic amplitude of proton-proton scattering $2N-N^{2}=0.96$. The
averaged slowly increases with energy, and reaches 1 at energy. At the LHC we
expect $\langle|N|\rangle=0.915$ and the average elastic amplitude is close to
unity $\langle|2N-N^{2}|\rangle=0.99$. Such a behavior shows, that the values
as well as the dependence on energy crucially depends on the behavior of our
dipole amplitude, in the vicinity of the saturation scale. To illustrate the
strength of the saturation effect, we plot the average $\Omega$ at $b=$ as a
function of energy (see Fig. 16-b), which is a considerably overshoots the
average scattering amplitude. It should be stressed, that in spite of the fact
that the elastic amplitude is very close to unity at the LHC energy, one can
see from Fig. 16-a that $\langle|N|\rangle$ is only 0.9. and the asymptotic
behavior with $\langle|N|\rangle$ close to 1, starts from
$s=10^{12}\,GeV^{2}$.

|
---|---
Fig. 16-a | Fig. 16-b
Figure 16:  Average dipole amplitude ($\langle{N}\rangle$ see Eq. (6.55)),
average proton- proton amplitude (both in Fig. 16-a) and average opacity
$\Omega$(see Fig. 16-b) at $b=0$ as a function of energy in our model.

The principle difference between our models and the models of soft
interactions, is the fact that we predict all observables fitting all
parameters from the DIS data. The GLM1 model[25] is close to our approach
ideologically, because it does not assume the existence of the soft Pomeron.
However, at first sight this model has two major shortcomings: minima at small
$t$ which have not been seen experimentally; and the slow fall down of the
cross section of diffractive production, as a function of the mass of produced
system of hadrons. Our model shows, that the $t$ dependence can be easily
heeled by assuming the exponential form for the profile function, instead of
the Gaussian one that has been used in the GLM1 model. As far as large mass
diffraction that has been neglected in the GLM1 model[25], we show that this
diffraction is essential. In this respect, we are close to the RMK model[55]
and to the GLM2 model [56] which are based on the soft Pomeron exchange. Our
model can be considered as an argument that the multi Pomeron exchanges, and
the Pomeron interactions could be essential for high mass diffraction.

In general we demonstrated in this paper that the distances, essential in so
called soft interactions, is not so long, but rather about $1/Q_{s}\ll R_{h}$,
where $R_{h}$ is the hadron radius. This conclusion stems both from the
success of our model in the description of the data, using the parameters
fitted in DIS, and from the use of the energy variable $x_{soft}$, which is
determined by the saturation scale.

We hope that our model will generate deeper theoretical ideas on the matching
between soft and hard interactions, based on high parton density QCD.

## Acknowledgements

We are grateful to Jochen Bartels, Errol Gotsman, Lev Lipatov, Uri Maor and
Misha Ryskin for fruitful discussions on the subject. This research was
supported in part by the Israel Science Foundation, founded by the Israeli
Academy of Science and Humanities, by BSF grant $\\#$ 20004019 and by a grant
from Israel Ministry of Science, Culture and Sport and the Foundation for
Basic Research of the Russian Federation.

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