# A QCD motivated model for soft interactions at high energies

E. Gotsman , E. Levin , U. Maor and J.S. Miller
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>levin@mail.desy.de.Email:maor<EMAIL_ADDRESS>
###### Abstract:

In this paper we develop an approach to soft scattering processes at high
energies, which is based on two mechanisms: Good-Walker mechanism for low mass
diffraction and multi-Pomeron interactions for high mass diffraction. The
pricipal idea, that allows us to specify the theory for Pomeron interactions,
is that the so called soft processes occur at rather short distances
($r^{2}\propto 1/<p_{t}>^{2}\propto\alpha^{\prime}_{I\\!\\!P}\approx
0.01\,GeV^{-2}$), where perturbative QCD is valid. The value of the Pomeron
slope $\alpha^{\prime}_{I\\!\\!P}$ was obtained from the fit to experimental
data. Using this theoretical approach we suggest a model that fits all soft
data in the ISR-Tevatron energy range, the total, elastic, single and double
diffractive cross sections, including $t$ dependence of the differential
elastic cross section, and the mass dependence of single diffraction. In this
model we calculate the survival probability of diffractive Higgs production,
and obtained a value for this observable, which is smaller than 1% at the LHC
energy range.

Soft Pomeron, BFKL Pomeron, Diffractive Cross Sections, Survival Probability

††dedicated: PACS: 13.85.-t, 13.85.Hd, 11.55.-m, 11.55.Bq††preprint: TAUP
-2878-08
0805.2799 [hep-ph]

##  1 Introduction

The goal of this paper is to construct a QCD motivated model for the strong
interactions at high energy, and to elucidate the model’s predictions and
implications at LHC and Cosmic Rays energies.

The difficulties and challenges of a theoretical approach to the strong
interaction are well known: a qualitative understanding of the confinement of
quarks and gluons in QCD. For high energy scattering the situation is even
more difficult than for the hadron structure, since approximate methods such
as QCD sum rules and/or effective theories, as well as the lattice QCD
approach, cannot be used to calculate the high energy amplitudes. Today, and
for the past four decades, the accepted method of describing soft interactions
at high energy, is the phenomenology based on the soft Pomeron and secondary
Reggeons (see Refs. [1, 2, 3] for details). All parameters related to the
Pomeron and Reggeons, which are assumed to be simple poles in the J-plane,
such as the intercepts and slopes of the trajectories, the vertices and their
dependence on the impact parameter, have to be deduced from the experimental
data. However, the key problem of all approaches based on the soft Pomeron
hypothesis, is that there is no theory which can specify what kind of
interactions between Pomerons have to be taken into account, as well as the
values of the multi-Pomeron vertices. Due to this problem we are doomed to
build basically unreliable models, since the only criteria is that they
describe the experimental data (see Refs. [4, 5, 6, 7]).

In this paper we attempt to overcome this difficulty by using high energy
perturbative QCD approach. At first sight, this appears unrealistic, since the
pQCD approach is based on the smallness of the running QCD coupling at short
distances, while the high energy interaction is a typical example of long
distance non-perturbative QCD. We wish to question this widely held prejudice.
Indeed, the only microscopic explanation for the Pomeron structure is given in
the partonic approach [8], in which the slope of the Pomeron trajectory is
related to the mean transverse momentum of the exchanged partons
($\alpha^{\prime}_{I\\!\\!P}\,\propto 1/<p_{t}>$, where $<p_{t}>$ is the mean
parton momentum). The commonly held view in high energy phenomenology, is that
$\alpha^{\prime}_{I\\!\\!P}\,=\,0.25\,GeV^{-2}$[9], which implies that in the
Pomeron the typical momenta of partons are high (especially if you compare
this value with the value of $\alpha^{\prime}_{{I\\!\\!R}}\,=\,1\,GeV^{-2}$
for the secondary Reggeons). In the next section, we extend our output up to
the Planck mass, and will show that the fit to the experimental data based on
our model with power-like dependence of the Pomeron-proton vertices on energy,
validates the consistency of our calculations with unitarity. The momentum
transferred $t$ behaviour leads to $\alpha^{\prime}_{I\\!\\!P}<0.02$
$GeV^{-2}$ . This result, together with the fit of Ref. [7], where the data
were fitted with $\alpha^{\prime}_{I\\!\\!P}=0$, lends support to our
assumption that, the typical parton momentum is large (approximately
$<p_{t}>=1/\sqrt{\alpha^{\prime}_{I\\!\\!P}}\geq 7\,GeV$). Therefore, the
running QCD coupling
$\alpha_{S}=\pi/b\ln\left(<p^{2}_{t}>/\Lambda^{2}_{QCD}\right)\,\ll\,1$
(approximately 0.18), and we can consider it as our small parameter, when
applying perturbative QCD estimates to the Pomeron-Pomeron interaction
vertices.

The theoretical scheme that emerges from such an approach will be discussed in
section 3. Using perturbative QCD we select the essential Pomeron-Pomeron
vertices, and develop the method for summation of all these diagrams. We also
give the partonic interpretation of our approach, and develop an approximate
method for the solution of the long standing problem of the high energy
asymptotic behaviour of the scattering amplitude. Based on this analysis we
derive formulae for all cross sections taking into account the Pomeron
interactions.

In section 4 we fit the available experimental data using our formalism. This
fit is based on an updated data base which includes the published $p$-$p$ and
$\bar{p}$-$p$ data points of $\sigma_{tot}$, the integrated values of
$\sigma_{el}$, $\sigma_{sd}$, $\sigma_{dd}$ and the forward elastic slope
$B_{el}$, in the ISR-Tevatron energy range.
$\rho=\frac{Re\,a_{el}(t=0,s)}{Im\,a_{el}(t=0,s)}$ and the forward slope of
the SD and DD final states are predictions of the model. The additional data
that we have used to validate our present model, are the $t$-dependence of the
differential elastic cross section, and the mass dependence of the single
diffractive production. We successfully describe this data. Based on this fit
we give reliable predictions of the quantities to be measured at the LHC c.m.
energy of $14\,TeV$. Our output also covers the broad Cosmic Rays energy range
up to the GZK limit.

The fifth section is devoted entirely to a calculation of the survival
probability ($\langle\mid S^{2}\mid\rangle$) for exclusive central diffractive
Higgs production at the LHC, and a discussion on the reliability of these
calculations. We confirm the tendency in which $\langle\mid S^{2}\mid\rangle$
becomes small (less that 1%), noticed in our previous paper [6]. From a
practical point of view, this is the most salient feature of this paper.

In the conclusions, we compare critically our results with other approaches,
and summarize our findings. We list a few experimental signatures, which
should enable us to differentiate between alternative theoretical options and
phenomenological models.

##  2 The two channel model and the value of $\alpha^{\prime}_{I\\!\\!P}$

###  2.1 GLM two channel model

The GLM two channel model has been described in our previous publications (see
Refs.[4, 10, 11, 6, 12], and references therein). In this formalism,
diffractively produced hadrons at a given vertex are considered as a single
hadronic state described by the wave function $\Psi_{D}$, which is orthonormal
to the wave function $\Psi_{h}$ of the incoming hadron (proton in the case of
interest), $<\Psi_{h}|\Psi_{D}>=0$. We introduce two wave functions $\psi_{1}$
and $\psi_{2}$ that diagonalize the 2x2 interaction matrix ${\bf T}$

$A_{i,k}=<\psi_{i}\,\psi_{k}|\mathbf{T}|\psi_{i^{\prime}}\,\psi_{k^{\prime}}>=A_{i,k}\,\delta_{i,i^{\prime}}\,\delta_{k,k^{\prime}}.$
(2.1)

In this representation the observed states are written in the form

$\psi_{h}=\alpha\,\psi_{1}+\beta\,\psi_{2}\,,$ (2.2)
$\psi_{D}=-\beta\,\psi_{1}+\alpha\,\psi_{2}\,,$ (2.3)

where, $\alpha^{2}+\beta^{2}=1$. Using Eq. (2.1), we can rewrite the unitarity
constraints in the form

$Im\,A_{i,k}\left(s,b\right)=|A_{i,k}\left(s,b\right)|^{2}+G^{in}_{i,k}(s,b),$
(2.4)

where $G^{in}_{i,k}$ is the contribution of all non diffractive inelastic
processes, i.e. it is the summed probability for these final states to be
produced in the scattering of particle $i$ off particle $k$.

A simple solution to Eq. (2.4) has the same structure as in a single channel
formalism,

$A_{i,k}(s,b)=i\left(1-\exp\left(-\frac{\Omega_{i,k}(s,b)}{2}\right)\right),$
(2.5) $G^{in}_{i,k}(s,b)=1-\exp\left(-\Omega_{i,k}(s,b)\right).$ (2.6)

From Eq. (2.6) we deduce, that the probability that the initial projectiles
$(i,k)$ reach the final state interaction unchanged, regardless of the initial
state rescatterings, is $P^{S}_{i,k}=\exp\left(-\Omega_{i,k}(s,b)\right)$.

For the opacities $\Omega_{i,k}$ we use the expression

$\Omega_{i,k}\left(s,b\right)\,=\,g_{i}\,g_{k}\left(\frac{s}{s_{0}}\right)^{\Delta_{I\\!\\!P}}\,S\left(b;m_{i},m_{k};\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})\right)$
(2.7)

which differs from the expression that we used in our previous models. The
profile function
$S\left(b,\alpha^{\prime}_{I\\!\\!P};m_{i},m_{k};\ln(s/s_{0})\right)$ at
$s=s_{0}$, corresponds to the power-like behaviour of the Pomeron-hadron
vertices,

$\displaystyle
S\left(b;m_{i},m_{k};,\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})=0\right)\,\,=\,\,\int\,\frac{d^{2}q}{(2\pi)^{2}}\,g_{i}(q)\,g_{k}(q)\,e^{i\vec{q}_{\perp}\cdot\vec{b}},$
(2.8) $\displaystyle\mbox{with a
normalization}\,\int\,d^{2}b\,S\left(b;m_{i},m_{k};,\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})=0\right)\,\,=1\,.$

In this paper we choose

$g_{i}(q)\,\,=\,\,\frac{1}{(1\,\,+\,\,q^{2}/m^{2}_{i})^{2}}$ (2.9)

The arguments we list in favour of this choice are: (i) pQCD leads to
$g_{i}(q)\,\,\rightarrow\,\,\alpha_{S}^{2}(q)/q^{4}$ [13]; (ii) in some models
(for example in the constituent quark model) $g_{i}(q)$ is equal to the
electro-magnetic form factor of the proton, which has the form of Eq. (2.9);
and (iii) Eq. (2.9) reproduces the experimental elastic differential cross
section for p-p (p-${\bar{p}})$ data in the range of $t$, up to 1 to 1.5
$\,GeV^{2}$ [9], while a Gaussian parametrization fits the data only for very
small $t\leq 0.1\,GeV^{2}$.

Using Eq. (2.8) we obtain for
$S\left(b;m_{i},m_{k};,\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})=0\right)$,

$\displaystyle\frac{1}{(1\,\,+\,\,q^{2}/m^{2}_{i})^{2}}\times\frac{1}{(1\,\,+\,\,q^{2}/m^{2}_{k})^{2}}\,\,\,\Longrightarrow\,\,\,S\left(b;m_{i},m_{k};,\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})=0\right)\,\,=\,\,\,$
(2.10)
$\displaystyle=\,\,\frac{m^{3}_{i}\,m^{3}_{k}}{4\pi\,(m^{2}_{i}-m^{2}_{k})^{3}}\,\,\left\\{4m_{i}\,m_{k}\left(K_{0}\left(m_{i}b\right)\,-\,K_{0}\left(m_{k}b\right)\right)\,\,+\,\,(m^{2}_{i}-m^{2}_{k})b\,\left(m_{k}K_{1}\left(m_{i}b\right)\,+\,m_{i}K_{1}\left(m_{k}b\right)\right)\right\\}.$

For energies $s>s_{0}$, we need to take into account the observed shrinkage of
the diffraction peak. To this end we replace $g_{i}(q)\,g_{k}(q)$ by
$g_{i}(q)\,g_{k}(q)\,\exp\left(-\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})q^{2}\right)$
in Eq. (2.8). To simplify our calculations we replace

$m^{2}_{i}\,\,\,\Longrightarrow\,\,\,m^{2}_{i}(s)\,\,\equiv\,\,\,\frac{m^{2}_{i}}{1\,\,+\,\,\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})/4m^{2}_{i}}.$
(2.11)

It is easy to check that
$g_{i}(q;m_{i})\,g_{k}(q;m_{k})\,\exp\left(-\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})q^{2}\right)$
and $g_{i}(q;m_{i}(s))\,g_{k}(q;m_{k}(s))$ have the same behaviour for
$\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})q^{2}\,\ll\,1$. When
$\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})q^{2}\,\gg\,1$ these two expressions
are different. Note that the Regge factor
$\exp\left(-\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})q^{2}\right)$ cannot be
justified in this kinematic region. In the region of
$\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})q^{2}\,\sim\,1$ it is preferable to use
the Regge factor. We assume that $\alpha^{\prime}_{I\\!\\!P}$ is small enough,
so that this region gives a negligible contribution to all experimental
observables.

In general, we have to consider four possible re-scattering processes in Eq.
(2.1). However, in the case of $p$-$p$ ( $\bar{p}$-$p$) the two non-diagonal
amplitudes are equal $A_{1,2}=A_{2,1}$, and we end up with three rescattering
amplitudes. These amplitudes are presented in our two channel formalism in the
following form[4, 12, 11]

$a_{el}(s,b)=i\\{\alpha^{4}A_{1,1}+2\alpha^{2}\beta^{2}A_{1,2}+\beta^{4}{\cal
A}_{2,2}\\},$ (2.12)
$a_{sd}(s,b)=i\alpha\beta\\{-\alpha^{2}A_{1,1}+(\alpha^{2}-\beta^{2})A_{1,2}+\beta^{2}A_{2,2}\\},$
(2.13) $a_{dd}=i\alpha^{2}\beta^{2}\\{A_{1,1}-2A_{1,2}+A_{2,2}\\}.$ (2.14)

It should be stressed that in this approach diffraction dissociation, appears
as an outcome of the Good and Walker mechanism[14] (G-W) or, in other words,
the elastic, single diffraction and double diffraction processes occur due to
elastic scatterings of $\psi_{1}$ and $\psi_{2}$, the correct degrees of
freedom.

The corresponding cross sections are given by

$\sigma_{tot}(s)=2\int d^{2}b\,a_{el}\left(s,b\right),$ (2.15)
$\sigma_{el}(s)=\int d^{2}b\,|a_{el}\left(s,b\right)|^{2},$ (2.16)
$\sigma_{sd}(s)=\int d^{2}b\,|a_{sd}\left(s,b\right)|^{2},$ (2.17)
$\sigma_{dd}(s)=\int d^{2}b\,|a_{dd}\left(s,b\right)|^{2}.$ (2.18)

###  2.2 $\alpha^{\prime}_{I\\!\\!P}\,\,\longrightarrow\,\,0$

Using Eq. (2.1) - Eq. (2.7) we fit the experimental data, so as to find the
value of $\alpha^{\prime}_{I\\!\\!P}$. However, most of the data are available
at rather low energies $W=\sqrt{s}=20$ to $600\,GeV$, where the contributions
of the secondary Reggeon have to be included. For this we replace
$\Omega_{i,k}\left(s,b\right)$ given by Eq. (2.7) by the sum of Pomeron and
Reggeon contributions,

$\Omega_{i,k}\left(s,b\right)\,\,\,\,=\,\,\,\Omega^{I\\!\\!P}_{i,k}\left(s,b\right)\,\mbox{(
see
{Eq.~{}(\ref{OMEGA})})}\,\,\,+\,\,\,\Omega^{{I\\!\\!R}}_{i,k}\left(s,b\right),$
(2.19)

with

$\Omega^{{I\\!\\!R}}_{i,k}\left(s,b\right)\,\,=\,\,\frac{g^{{I\\!\\!R}}_{i}\,g^{{I\\!\\!R}}_{k}\,\eta\,\left(\frac{s}{s_{0}}\,\right)^{\Delta_{{I\\!\\!R}}}}{\pi
R^{2}_{i,k}\left(s\right)}\exp\left(-\frac{b^{2}}{R^{2}_{i,k}\left(s\right)}\right).$
(2.20)

The signature factor is

$\eta\,\,\,=\,\,\,\frac{1\pm
e^{i\pi\alpha_{{I\\!\\!R}}(q)}}{\sin\left(\pi\alpha_{{I\\!\\!R}}(q)\right)}$
(2.21)

corresponding to a Reggeon trajectory
$\alpha_{{I\\!\\!R}}(q)\,=\,1+\,\Delta_{{I\\!\\!R}}\,+\,\alpha^{\prime}_{{I\\!\\!R}}\,q^{2}$
, and

$R^{2}_{i,k}\left(s\right)=R^{2}_{0,i}+R^{2}_{0,k}+4\alpha^{\prime}_{{I\\!\\!R}}\ln(s/s_{0}),=R^{2}_{0;i,k}+4\alpha^{{}^{\prime}}_{{I\\!\\!R}}\ln(\frac{s}{s_{0}}).$
(2.22)

$g^{R}_{i}$,$R^{2}_{0,i}$ , $\Delta_{{I\\!\\!R}}$ and
$\alpha^{{}^{\prime}}_{{I\\!\\!R}}$ are fitted parameters. For the Regge
sector we know the natural values for $\Delta_{{I\\!\\!R}}$ and
$\alpha^{{}^{\prime}}_{{I\\!\\!R}}$ from the behaviour of the Regge trajectory
at $q^{2}<0$ ($t>0$) , since all hadrons lie on these trajectories,
$\Delta_{{I\\!\\!R}}\approx-0.5$ and $\alpha^{\prime}_{{I\\!\\!R}}\approx
1\,GeV^{-2}$.

Eq. (2.19) - Eq. (2.22) specify our model. In this model we neglect the
interactions between Pomerons, but include eikonal type rescatterings. This
means that we assume only a G-W origin for all quasi-elastic processes. The
fitted parameters are shown in Table 1. The quality of the fit is illustrated
in Fig. 2 and Fig. 2, where our fit is shown by dashed lines. The fit is very
good with $\chi^{2}/d.o.f.=0.87$. However, this parametrization fails to fully
describe the cross sections for single and double diffractive production,
yielding values for these cross sections which are approximately two times
smaller than the experimental values.

From this fit we have two conclusions: i) we cannot describe the diffractive
production in the framework of the G-W mechanism if we assume an exponential
parameterization for the proton profile function; and (ii) the value of
$\alpha^{\prime}_{{I\\!\\!P}}$ turns out to be very small
$\alpha^{\prime}_{{I\\!\\!P}}=0.012\,GeV^{-2}$. As we have discussed in the
introduction, we conclude from the smallness of
$\alpha^{\prime}_{{I\\!\\!P}}$, that the hard processes which occur at short
distances are responsible for the Pomeron structure.

$\Delta_{I\\!\\!P}$ | $\beta$ | $\alpha^{\prime}_{{I\\!\\!P}}$ | $g_{1}$ | $g_{2}$ | $m_{1}$ | $m_{2}$
---|---|---|---|---|---|---
0.120 | 0.46 | 0.012 $GeV^{-2}$ | 1.27 $GeV^{-1}$ | 3.33 $GeV^{-1}$ | 0.913 $GeV$ | 0.98 $GeV$
$\Delta_{I\\!\\!R}$ | $\beta$ | $\alpha^{\prime}_{{I\\!\\!R}}$ | $g^{{I\\!\\!R}}_{1}$ | $g^{{I\\!\\!R}}_{2}$ | $R^{2}_{0,1}$ | $\chi^{2}/d.o.f.$
-0.438 | 0.46 | 0.60 $GeV^{-2}$ | 4.0 $GeV^{-1}$ | 118.4 $GeV^{-1}$ | 4.0 $GeV^{-2}$ | 0.87
Table 1: Fitted parameters for two channel (eikonal) model

Figure 1: Energy dependence of $\sigma_{tot}$. The solid line shows the fit
with taking into account all Pomeron interactions while the dashed line
corresponds to two channel (eikonal) model.

Figure 2: Energy dependence of the slope for the differential elastic cross
section. All notation are the same as in Fig. 2

##  3  Pomerons Interactions

###  3.1 QCD input

Our main hypothesis is that the typical distances for a Pomeron exchange are
short, and we can use pQCD as a guide for building a theory for Pomeron
interactions, based on the small value of $\alpha^{\prime}_{{I\\!\\!P}}$, the
Pomeron slope that we obtained in the previous section. We recall that the
value of $\alpha^{\prime}_{{I\\!\\!P}}$ in the parton model [8] can be written
as

$R^{2}(s)\,\,=\,\,\langle b^{2}\rangle_{n}\,\,\,=\,\,\,\langle\Delta
b^{2}\rangle\,n\,\,\,=\,\,\frac{4}{<p^{2}_{t}>}\,\rho\,\ln(s/s_{0})\,\,=\,\,\alpha^{\prime}_{{I\\!\\!P}}\ln(s/s_{0})$
(3.23)

where $n$ is the number of partons at rapidity $Y=\ln(s/s_{0})$ and $\rho$ is
theb density of partons in a unit of rapidity. In Eq. (3.23) the radius of
interaction, is determined as the average distance in impact parameter space
that a ‘wee’ parton can reach at given enegy.

In Eq. (3.23) $R^{2}(s)$ is the radius of interaction at energy $W=\sqrt{s}$,
which is the distance in impact parameter $b$ space, that is reached by
partons in a two dimensional diffusion, after $n$ steps. $\langle\Delta
b^{2}\rangle$ is the mean displacement during one diffusion step, which can be
estimated using the uncertainty principle as $\langle\Delta
b^{2}\rangle\,=\,4/<p^{2}_{t}>$. $<p_{t}>$ is the average parton transverse
momentum which we wish to evaluate. The number of diffusion steps is equal to
the number of partons, since at each step one parton is emitted. At a given
energy the average number of emitted partons is equal to $\rho\ln(s/s_{0})$,
where $\rho$ is the parton density in units of rapidity. Comparing $n$ with
the hadron multiplicities, we conclude that $\rho\approx 1$ or larger.
Therefore, Eq. (3.23) leads to $<p^{2}_{t}\,\geq\,20\,GeV$ for
$\alpha^{\prime}_{{I\\!\\!P}}=0.01\,GeV^{-2}$ . The typical value of the
running QCD coupling, which corresponds to this value of the parton momentum,
is $\alpha_{S}\approx 0.15$, which is small enough to be used in pQCD
estimates.

In our procedure we only use general results of the perturbative QCD approach
to high energy scattering [15]. Consequently:

1.) In the leading order approximation of pQCD, only a Pomeron splitting into
two Pomerons and two Pomeron merging into one Pomeron, should be taken into
account [16, 17], while all other vertices are small. Therefore, using this
input from pQCD, we restrict ourselves by summing Pomeron diagrams with triple
Pomeron vertices only.

2) Since $4\alpha^{\prime}_{{I\\!\\!P}}ln(s)\,\,\ll\;1$ over the entire
kninematical range, we have investigated, we can neglect the value of
$\alpha^{\prime}_{{I\\!\\!P}}$, and consider the theory with
$\alpha^{\prime}_{{I\\!\\!P}}\,\,=\,\,0$.

3) We obtain the anticipated values for all ingredients of the Pomeron
interaction approach: the intercept of the Pomeron $\Delta_{I\\!\\!P}$ above
unity $\,\propto\,\alpha_{S}$, and the triple Pomeron vertex coupling
$g_{3{I\\!\\!P}}\,\propto\,\alpha_{S}^{2}$.

The theory that includes all the above ingredients can be formulated in
functional integral form[17],

$Z[\Phi,\Phi^{+}]\,\,=\,\,\int\,\,D\Phi\,D\Phi^{+}\,e^{S}\,\,\,\mbox{with}\,S\,=\,S_{0}\,+\,S_{I}\,+\,S_{E}\,,$
(3.24)

where $S_{0}$ describes free Pomerons, $S_{I}$ corresponds to their mutual
interaction and $S_{E}$ relates to the interaction with the external sources
(target and projectile). Since $\alpha^{\prime}_{{I\\!\\!P}}\,=\,0$, $S_{0}$
has the form

$S_{0}\,=\,\int
dY\Phi^{+}(Y)\,\left\\{-\,\frac{d}{dY}\,\,+\,\,\Delta\,\right\\}\Phi(Y).$
(3.25)

$S_{I}$ includes only triple Pomeron interactions and has the form

$S_{I}\,=\,g_{3{I\\!\\!P}}\int
dY\,\left\\{\Phi(Y)\,\Phi^{+}(Y)\,\Phi^{+}(Y)\,\,+\,\,h.c.\right\\}.$ (3.26)

For $S_{E}$ we have local interactions both in rapidity and in impact
parameter space,

$S_{E}\,=\,-\,\int
dY\sum_{i=1}^{2}\,\left\\{\Phi(Y)\,g_{i}(b)\,\,+\,\,\Phi^{+}(Y)\,g_{i}(b)\right\\},$
(3.27)

where $g_{i}(b)$ stands for the interaction vertex with the hadrons at fixed
$b$.

In the next sections, after specifying this theory, we solve it. Indeed, this
theory as any theory of Pomeron interactions, is written in such a way that an
high energy amplitude satisfies $t$-channel unitarity. However, $s$-channel
unitarity remains a problem. In the next subsection we will change the
interaction term ($S_{I}$), in such a way, that the theory will have a clear
partonic interpretation, and will, also, satisfy $s$-channel unitarity.

###  3.2 Generating function and a partonic interpretation

To find a reformulation given by the functional of Eq. (3.24), we consider a
system of partons***The partons for high energy QCD are the colourless
dipoles, as was shown in Ref. [18]. that can decay and merge: one parton to
two partons, and two partons into one parton, with probabilities $\Gamma(1\to
2)$ and $\Gamma(2\to 1)$, respectively. For such a system of partons, we can
write a simple equation. Indeed, let $P_{n}(y)$ be the probability to find
$n$-parton (dipoles) with rapidity $y$ in the wave function of the fastest
(parent) parton (dipole), moving with rapidity $Y\,>\,y$. For $P_{n}(y)$, we
can easily write down a recurrence equation (see Refs. [20, 21])

$-\,\frac{\partial\,P_{n}(y)}{\partial\,y}\,\,=\,\,\Gamma(1\to
2)\,\left\\{-\,n\,P_{n}\,+\,\,(n-1)\,P_{n-1}\right\\}\,\,\,+\,\,\,\Gamma(2\to
1)\,\left\\{-\,n\,(n-1)\,P_{n}\,+\,\,(n+1)\,n\,P_{n+1}\right\\}.$ (3.28)

In each bracket the first term on the r.h.s., can be viewed as a probability
of a dipole annihilation in the rapidity range $(y$ to $y-dy)$ (death term).
The second is a probability to create one extra dipole (birth term). Note the
negative sign in front of $\partial P_{n}(y)/\partial y$. It appears due to
our choice of the rapidity evolution which starts at the largest rapidity
$y=Y$, of the fastest dipole and then decreases. The first two terms are
responsible for the process of parton decay, while the last two terms describe
the contribution of partons merging.

It is useful to introduce the generating function [18, 19, 21]

$Z(y,\,u)\,\,=\,\,\sum_{n}\,\,P_{n}(y)\,\,u^{n}\,$ (3.29)

At rapidity $y\,=\,Y$ there is only one fastest parton (dipole), which is
$P_{1}(y\,=\,Y)\,=\,1$, while $P_{n>1}(y\,=\,Y)\,=\,0$. This is the initial
condition for the generating function

$Z(y\,=\,Y)\,=\,u\,.$ (3.30)

At $u=1$

$Z(y,\,u\,=\,1)\,\,=\,\,1,$ (3.31)

which follows from the physical meaning of $P_{n}$ as a probability.

Eq. (3.28) can be rewritten as an equation in partial derivatives for the
generating function $Z(y,u)$,

$\,\,-\frac{\partial\,Z(y,\,u)}{\partial\,y}\,\,=\,\,-\,\Gamma(1\to
2)\,u\,(1\,-\,u)\,\,\frac{\partial\,Z(y,\,u)}{\partial\,u}\,\,\,+\,\,\,\Gamma(2\to
1)\,u\,(1\,-\,u)\,\,\frac{\partial^{2}\,Z(y,\,u)}{\partial^{2}\,u}.$ (3.32)

The description of the parton system given by Eq. (3.32), is equivalent to the
path integral of Eq. (3.24). Indeed, the general solution of Eq. (3.32) has a
form

$Z(Y;u)\,\,=\,\,e^{H(u)\,(Y-Y_{0})}Z(Y_{0};u),$ (3.33)

with the operator $H$ defined as

$H(u)\,\,=\,\,-\,\Gamma(1\to 2)\,u(1-u)\,\frac{\partial}{\partial
u}+\,\Gamma(2\to 1)\,u(1-u)\,\frac{\partial^{2}}{(\partial u)^{2}},$ (3.34)

and

$Z(Y_{0};u)\,\,=\,\,e^{g(b)(u-1)}.$ (3.35)

We introduce operators of creation ($a^{+}$) and annihilation ($a$) that
satisfy $[\hat{a},\hat{a}^{+}]=1$, at fixed Y.

$\hat{a}\,=\,\frac{\partial}{\partial u}\,,\,\hat{a}^{+}\,=\,u\,.$ (3.36)

In this formalism

$H\,\,=\,\,-\,\Gamma(1\to
2)\,\hat{a}^{+}\,(1-\hat{a}^{+})\,\hat{a}\,\,+\,\,\Gamma(2\to
1)\,\hat{a}^{+}\,(1-\hat{a}^{+})\,\hat{a}^{2},$ (3.37)

and the initial state at $Y=Y_{0}$ is defined as

$|Y_{0}>\,\,=\,\,e^{g(b)(\hat{a}^{+}\,-\,1)}|0>,$ (3.38)

with the vacuum defined as $\hat{a}|0>=0$.

The theory with the Hamiltonian of Eq. (3.37), has an equivalent description
using the path integral of Eq. (3.24) (see the detailed derivation in Ref.
[24]) with

$\displaystyle S\,$ $\displaystyle=$
$\displaystyle\,\int\left(\Phi^{+}\frac{d}{dY}\Phi+H(\Phi^{+}+1,-\Phi)\,\right)dY\,\,=\,\,\int
dY\,\,\left(\Phi^{+}\frac{d}{dY}\Phi\,\,\right.$ $\displaystyle-$
$\displaystyle\left.\Gamma(1\rightarrow 2)\Phi^{+}\Phi+\Gamma(1\rightarrow
2)\Phi^{+}(\Phi)^{2}+\Gamma(2\rightarrow
1)(\Phi^{+})^{2}\Phi-\Gamma(2\rightarrow
1)(\Phi^{+})^{2}\,(\Phi)^{2}\,\right).$

Comparing Eq. (3.2) with Eq. (3.25) - Eq. (3.27), one can see that they are
similar if we put $\Gamma(1\to 2)=\Delta$ and $\Gamma(2\to
1)\,=\,g^{2}_{3{I\\!\\!P}}/\Delta$. However, Eq. (3.2) has an additional term
($-\Gamma(2\rightarrow 1)(\Phi^{+})^{2}\,(\Phi)^{2}$ ) which describes the two
Pomeron to two Pomeron transition (four Pomeron interaction). This term
ensures that our approach satisfies $s$-channel unitarity (see Ref.[25]),
where it is shown that without this term there is no probabilistic
interpretation of the Pomeron interaction theory, and $s$-channel unitarity is
violated. We also need to renormalize
$\Phi^{+}\,\to\,(g_{3{I\\!\\!P}}/\Delta)\,\Phi^{+}$ and
$\Phi\,\to\,(\Delta/g_{3{I\\!\\!P}})\,\Phi$.

Using the functional $Z$, we find the scattering amplitude [22], using the
following formula:

$N\left(Y\right)\,\,\,\equiv\,\,\,\mbox{Im}A_{el}\left(Y\right)\,\,\,=\,\,\,\sum^{\infty}_{n=1}\frac{(-1)^{n}}{n!}\,\,\,\frac{\partial^{n}\,Z(y,\,u)}{\partial^{n}\,u}|_{u=1}\,\gamma_{n}(Y=Y_{0},b),$
(3.40)

where $\gamma_{n}(Y=Y_{0},b)$ is the scattering amplitude of $n$-partons
(dipoles) at low energy. These amplitudes depend on the impact parameters
which are the same for all $n$ partons, since
$\alpha^{\prime}_{I\\!\\!P}\,\,=\,\,0$, and we neglect the diffusion of
partons in impact parameter space. Eq. (3.40) corresponds to the partonic
approach [8, 23], in which a high energy scattering can be viewed as a two
stage process. The first stage is a development of the partonic wave function,
which we consider by introducing the generating function $Z$. In Eq. (3.40),
the second stage is the interaction of the lowest energy partons (‘wee’
partons) with the target, which is described by the amplitudes
$\gamma_{n}(Y=Y_{0},b)$. Assuming that there are no correlations between the
interacting partons (dipoles) at low energy, we can consider
$\gamma_{n}(Y=Y_{0},b)\,\,=\,\,\gamma^{n}_{1}(Y=Y_{0},b)$ [22], which we
include, choosing $S_{E}$ in the form of Eq. (3.27) and of Eq. (3.38).

The generating function approach given by Eq. (3.29), Eq. (3.32) and Eq.
(3.40), has the advantage that it can be solved analytically (see Ref. [26]).
This solution leads to a constant cross section at high energy, while the
interaction without the four Pomeron term, decreases at high energy [27]. This
fact emphasizes the importance of $s$-channel unitarity, in finding the
asymptotic behaviour of the scattering amplitude at high energy. Having an
exact solution we are able to develop approximate methods, which we can check
against the exact solution.

###  3.3 Improved Mueller-Patel-Salam-Iancu approximation

Calculating the high energy amplitude in the generating function approach, we
use the approximation which allows us to write a simple analytical formulae
for the physical observable. The main idea of this approximation, which we
call the improved Mueller-Patel-Salam-Iancu approximation, is the following:
we claim that at high energy, in the kinematical region

$Y\,\,\leq\,\,\frac{\Delta^{2}_{I\\!\\!P}}{g^{2}_{3{I\\!\\!P}}}\,\,\equiv\,\,\,\,\frac{1}{\gamma},$
(3.41)

only large Pomeron loops, with a rapidity size of the order of $Y$, contribute
to the high energy asymptotic behaviour of the scattering amplitudes. This
approximation has been discussed in detail in Refs. [28, 29, 30], and here we
illustrate this idea using the example of the first Pomeron loop diagram given
in Fig. 3. Using Eq. (3.24) or the generating function approach, we obtain the
contribution of this diagram in the form†††In Eq. (3.42) we redefine the
hadron-Pomeron vertices denoting them by $g(b)\,\sqrt{\gamma}$. We will
clarify our reason for doing so below. For simplicity we use $g(b)$ for this
vertex, neglecting the different hadronic states in our two channel model.

Figure 3:  The different contribution to the first enhanced diagram and its
renormalization procedure .

$\displaystyle A\left({Fig.~{}\ref{denre}}\right)\,=$ (3.42)
$\displaystyle=\,g^{2}(b)\,\gamma\,G(Y-0)\,\,-\,g^{2}(b)\,\Gamma(1\to
2)\,\Gamma(2\to
1)\,\gamma\,\int^{Y}_{0}\,d\,y_{1}\,\int^{y_{1}}_{0}\,d\,y_{2}\,G(Y-y_{1})\,G^{2}(y_{1}-y_{2})\,G(y_{2}-0)$
(3.43) $\displaystyle=\,\,\,g^{2}(b)\,\gamma\,G(Y-0)\,-\,g^{2}(b)\,\Gamma(1\to
2)\,\Gamma(2\to
1)\,\gamma\,\int^{Y}_{0}\,d\,y_{1}\,\int^{y_{1}}_{0}\,d\,y_{2}\,\,e^{\Delta_{I\\!\\!P}\,(Y+y_{1}-y_{2})}$
$\displaystyle=\,\,g^{2}(b)\,\gamma\,e^{\,\Delta\\_{I\\!\\!P}\,Y}\,-\,g^{2}(b)\,\Gamma\left(\,1\to\,2\right)\,\Gamma(2\to
1)\,\gamma\left(\frac{1}{\Delta^{2}_{I\\!\\!P}}\,e^{2\,\Delta_{I\\!\\!P}\,Y}\,\,-\,\,\frac{1}{\Delta^{2}_{I\\!\\!P}}\,e^{\,\Delta_{I\\!\\!P}\,Y}\,\,-\,\,\frac{Y}{\Delta_{p}om}\,e^{\,\Delta_{I\\!\\!P}\,Y}\right)$
$\displaystyle=\,\,g^{2}(b)\,\left\\{\gamma\,e^{\,\Delta_{I\\!\\!P}\,Y}\,\,-\,\,\gamma^{2}\,e^{2\,\Delta_{I\\!\\!P}\,Y}\,\,+\,\,\gamma^{2}\,e^{\Delta_{I\\!\\!P}\,Y}\,\,+\,\,\gamma\left(\Gamma(2\to
1)\,Y\right)\,e^{\Delta_{p}om\,Y}\right\\}$
$\displaystyle\longrightarrow\,\,\,\,g^{2}(b)\,\left\\{\gamma_{R}\,\,e^{\Delta_{R}\,Y}\,\,-\,\,\gamma^{2}_{R}\,\,e^{2\,\,\Delta_{I\\!\\!P}\,Y}\right\\},$

with $\gamma_{R}\,=\,\gamma+\gamma^{2}$, and the renormalized intercept of the
Pomeron $\Delta_{RE}\,=\,\Delta_{I\\!\\!P}+\,\Gamma(2\to
1)\,=\,\Delta_{I\\!\\!P}\,+\,\,\Delta_{I\\!\\!P}\,\gamma$. In the kinematic
region of Eq. (3.41) we can neglect the renormalization of the intercept,
since in this region $\Delta_{I\\!\\!P}\,\gamma\,Y\,\ll\,1$.

The general procedure for summing large Pomeron loops was suggested by
Mueller, Patel, Salam and Iancu (MPSI) in Ref. [28]. In this approach, the
scattering amplitude is calculated using the unitarity constraints in the
$t$-channel (assuming that the amplitudes at high energy are purely imaginary,
namely $N\,=\,Im\,A$),

$N([\dots]|Y)\,\,=\,\,N([\dots]|Y-Y^{\prime};P\to
nP)\,\bigotimes\,N([\dots]|Y^{\prime};P\to nP).$ (3.44)

$\bigotimes$ stands for all necessary integrations, while $[\dots]$ describes
all quantum numbers. The amplitude on the LHS of Eq. (3.44), describes all the
enhanced diagrams, while the amplitude on the RHS of this equation,
corresponds to the splitting of one Pomeron to $n$ Pomerons. The precise
meaning of this equation, will become clear in the next equation. The
convenient form of Eq. (3.44), has been written [20, 28, 26] in terms of the
generating functional of Eq. (3.29), and it takes the form

$N\left(Y\right)\,\,\,=\,\,\sum^{\infty}_{n=1}\,\,\frac{(-1)^{n}}{n!}\,\,\,\gamma^{n}\,\,\frac{\partial^{n}\,Z^{p}(Y-Y^{\prime},\,u_{p})}{\partial^{n}\,u_{p}}|_{u_{p}=1}\,\,\frac{\partial^{n}Z^{t}(Y^{\prime},\,u_{t})}{\partial^{n}\,u_{t}}|_{u_{t}=1}.$
(3.45)

In Eq. (3.45) we denote by $Z^{p}$ and $Z^{t}$ the generating functions that
describe projectile and target respectively.

Eq. (3.45) shows that each dipole with rapidity $Y^{\prime}$ from the target,
can interact with any dipole from the projectile, (see Fig. 4) with the
scattering amplitude $\gamma$. The factor $1/n!$ in Eq. (3.45) appears due to
the identity of Pomerons. Eq. (3.45) is defined in the kinematic region of Eq.
(3.41), and has a clear physical meaning, being the scattering amplitude of
two partons (two dipoles) at low energy $Y_{0}\,\leq\,1/\Delta_{I\\!\\!P}$.
Eq. (3.45) gives a natural generalization of Eq. (3.40), with an obvious
physical interpretation, that it is the sum of terms, and each of these terms
is the product of probabilities to find $n$-dipoles in the projectile and
target, multiplied by the scattering amplitude. Since we are discussing the
generating functional that satisfies Eq. (3.30) as the initial condition, Eq.
(3.45) gives the sum of enhanced diagrams or, in other words, at high energy
it leads to a new resulting Green’s function of the Pomeron.

Figure 4: An example of enhanced diagrams, that contribute to the unitarity
constraint in the $t$-channel. Wave lines denote the Pomerons. $\gamma$ is the
amplitude of the dipole-dipole interaction at low energies (at rapidity
$Y_{0}\approx 1/\bar{\alpha}_{S}$ ). The particular set of diagrams shown in
this figure, corresponds to the MPSI approach [28, 26].

The generating functions for the projectile $Z^{p}\left(Y-Y^{\prime}\right)$
and for the target $Z^{t}\left(Y^{\prime}\right)$ in Eq. (3.45), satisfy a
very simple equation that describes the parton cascades, in which a parton can
only decay into two partons. This equation has the form

$\,\,-\frac{\partial\,Z(y,\,u)}{\partial\,y}\,\,=\,\,-\,\Delta_{I\\!\\!P}\,u\,(1\,-\,u)\,\,\frac{\partial\,Z(y,\,u)}{\partial\,u}.$
(3.46)

The above equation has the solution

$Z\left(y,u\right)\,\,=\,\,\frac{u}{u\,\,+\,\,(1-u)\,e^{\Delta_{I\\!\\!P}y}}\,\,=\,\,\frac{1}{1\,\,+\,\,\gamma_{R}\,e^{\Delta_{I\\!\\!P}y}},$
(3.47)

where $u=1/(1+\gamma_{R})$. Eq. (3.47) satisfies the initial and boundary
conditions of Eq. (3.30) and Eq. (3.31). Using Eq. (3.47) and Eq. (3.40) it is
easy to show that the amplitude is equal to

$N(Y)\,\,=\,\,1-Z\left(u=1/\gamma_{R}\right)\,\,=\,\,\sum_{n=1}\,(-1)^{n}\,\,\gamma^{n}_{R}\,e^{n\,\Delta_{I\\!\\!P}Y}.$
(3.48)

Eq. (3.48) sums the ‘fan’ Pomeron diagrams and corresponds to the mean field
approximation (MFA) of our problem.

###  3.4  High energy amplitude

Using MFA and Eq. (3.48), we can rewrite Eq. (3.45) as

$\displaystyle N^{MPSI}_{el}\left(Y\right)\,\,\,$ $\displaystyle=$
$\displaystyle\,\,\sum^{\infty}_{n=1}\,\,(-1)^{n+1}\frac{1}{n!}\,\,\,\gamma^{n}\,\,\frac{\partial^{n}\,N^{MFA}(Y-Y^{\prime},\,\gamma^{p}_{R})}{\partial^{n}\,\gamma^{p}_{R}}|_{\gamma^{p}_{R}=0}\,\,\frac{\partial^{n}\,N^{MFA}(Y^{\prime},\,\gamma^{t}_{R})}{\partial^{n}\,\gamma^{t}_{R}}|_{\gamma^{t}_{R}=0}$
$\displaystyle=$
$\displaystyle\,\,1\,\,\,-\,\,\,\left\\{\exp\left(-\,\gamma\,\frac{\partial}{\partial\,\gamma^{p}_{R}}\,\frac{\partial}{\partial\,\gamma^{t}_{R}}\right)\,\,\,N^{MFA}\left(Y-Y^{\prime},\,\gamma^{p}_{R}\right)\,\,N^{MFA}_{0}\left(Y^{\prime},\gamma^{t}_{R}\right)\,\right\\}\mid_{\gamma^{p}_{R}=0;\,\gamma^{t}_{R}=0}.$

Substituting Eq. (3.47) into Eq. (3.4), we obtain that [33, 34]

$N^{MPSI}_{el}\left(Y\right)\,\,\,=\,\,\,1\,\,-\,\,\exp\left(\frac{1}{T(Y)}\right)\,\frac{1}{T(Y)}\,\,\Gamma\left(0,\frac{1}{T(Y)}\right),$
(3.50)

where $\Gamma\left(0,x\right)$ is the incomplete gamma function (see
formsph1.epsulae 8.350 - 8.359 in Ref. [36]) and

$T\left(Y\right)\,\,\,=\,\,\gamma\,e^{\Delta_{I\\!\\!P}\,Y}.$ (3.51)

In Ref. [34], the solution given by Eq. (3.50), is compared with the exact
solution (see Fig.13 of Ref. [34]). It turns out to within a 5% accuracy, that
the MPSI approximation describes the high energy behaviour of the amplitude.
At high energy Eq. (3.50) gives $N^{MPSI}_{el}\left(Y\right)\,\,\to\,\,1$.

As has been mentioned, since Eq. (3.45) satisfies the initial condition of Eq.
(3.30), it describes, as well, Eq. (3.50), the set of enhanced diagrams, and
gives the resulting Pomeron Green’s function

$G^{MPSI}_{I\\!\\!P}\left(Y-Y^{\prime}\right)\,\,\,=\,\,\,N^{MPSI}_{el}\left(Y-Y^{\prime}\right).$
(3.52)

Replacing the bare Pomeron Green’s function by Eq. (3.52), we can use Eq.
(2.12), Eq. (2.15) and Eq. (2.16), to calculate elastic and total cross
sections.

###  3.5 Diffractive production processes

The conclusion derived from the previous discussion, is that we cannot use the
formulae of Eq. (2.13) and Eq. (2.14), for calculating the single and double
diffraction cross sections. Indeed, these two equations describe the
diffraction production due to the G-W mechanism, while the sum of all enhanced
diagrams leads to a new source of diffractive production (see Fig. 5 for
examples of such processes).

Figure 5: Several examples of the Pomeron diagrams that lead to a different
source of the diffractive dissociation that cannot be described in the
framework of the G-W mechanism. Fig. 5-a is the simplest diagram that
describes the process of diffraction in the region of large mass
$Y-Y_{1}=\ln(M^{2}/s_{0})$. Fig. 5-b and Fig. 5-c give examples of more
complicated diagrams in the region ofsph1.eps large mass. The dashed line
shows the cut Pomeron, which describes the production of hadrons (see Fig. 5
-a which illustrates this point).

We use the MPSI approximation to obtain the expression for the additional
contribution to G-W mechanism. The main idea is shown in Fig. 6. As was
discussed in Ref. [29] , in order to apply the MPSI approach to diffractive
production, we need to consider the generating function of three variables:
$w$,$\bar{w}$ and $v_{in}$,

$Z\left(w,\bar{w};v_{in};Y\right)\\\
,\,\,=\,\,\sum^{\infty}_{n=0;m=0;k=0}\,P\left(n,n,k|Y\right)\,\,w^{n}\,\bar{w}^{m}\,v^{k}_{in}.$
(3.53)

$P\left(n,n,k|Y\right)$ is the probability to find $n$ and $m$ Pomerons in the
amplitude and conjugated amplitude respectively, while $k$ is the number of
cut Pomerons (see Ref. [37]). $Z\left(w,\bar{w};v_{in};Y\right)$ has been
found in Ref. [29] for the parton cascade, with a decay of one parton to two
partons,

$\displaystyle sph1.eps$ $\displaystyle
Z\left(w,\bar{w},v_{in}|Y-Y_{M}=\ln(M^{2}/s_{0})\equiv Y_{m}\right)\,\,=\,\,$
$\displaystyle\frac{w\,e^{-\Delta_{I\\!\\!P}Y}}{1+w(e^{-\Delta_{I\\!\\!P}Y_{m}}-1)}\,\,+\,\,\frac{\bar{w}\,e^{-\Delta_{I\\!\\!P}Y}_{m}}{1+\bar{w}(e^{-\Delta_{I\\!\\!P}Y_{m}}-1)}\,\,-\,\,\frac{(w+\bar{w}-v_{in})e^{-\Delta_{I\\!\\!P}Y_{m}}}{1+(w+\bar{w}\,-v_{in})(e^{-\Delta_{I\\!\\!P}\,\,Y_{m}}-1)}.$

To find the cross section for single diffractive production, we need to
calculate the term which is proportional to $v_{in}$, and to replace $v_{in}$
by $2\Delta_{I\\!\\!P}w\bar{w}$. Indeed, this term means that at $Y=Y_{m}$ we
have only one cut Pomeron, while all other cut Pomerons at this rapidity have
decayed to Pomerons without cuts. For simplicity we choose
$Y^{\prime}=Y-Y_{M}=Y_{m}$ (see Fig. 6). Replacing
$v_{in}\,\to\,2\Delta_{I\\!\\!P}w\bar{w}$, means that at $Y=Y_{M}$, the last
cut Pomeron splits into two Pomerons (see Fig. 6). Therefore,

$N^{MFA}_{sd}\left(w,\bar{w};Y-Y_{M}=\ln(M^{2}/s_{0})\equiv
Y_{m}\right)\,\,=\,\,2\,\Delta_{I\\!\\!P}w\,\bar{w}\,\frac{e^{\Delta_{I\\!\\!P}Y_{m}}}{\left(1\,\,+\,\,(w\,+\,\bar{w})\,(e^{\Delta_{I\\!\\!P}Y_{m}}\,-\,1)^{2}\right)^{2}}.$
(3.55)

Figure 6: The MPSI approximation for the cross section of single diffractive
production of mass ($M^{2}$,$Y-Y_{M}=\ln(M^{2}/s_{0})$). The dashed lines
shows the cut Pomerons. All other notations, are as in Fig. 4.

The general Eq. (3.45) can be rewritten in this case in the form

$\displaystyle N^{MPSI}_{sd}(Y,Y_{m}=\ln(M^{2}/s_{0}))\,\,=$ (3.56)
$\displaystyle\,\,\sum^{\infty}_{n=1;m=1}\,\frac{(-1)^{n+m}}{\,n!\,m!}\,\gamma^{n+m}\frac{\partial^{n}\,}{\partial^{n}\,w^{p}}\frac{\partial^{m}\,}{\partial^{n}\,\bar{w}^{p}}\,\,N^{MFA}_{sd}\left(w^{p},\bar{w}^{p};Y-Y_{M}=\ln(M^{2}/s_{0})\equiv
Y_{m}\right)|_{w=1;\bar{w}=1}\,$
$\displaystyle\times\,\,\frac{\partial^{n}\,N^{MFA}\left(w^{t},Y-Y_{m}\right)}{\partial^{n}\,w^{t}}|_{w^{t}=1}\frac{\partial^{m}\,N^{MFA}\left(\bar{w}^{t},Y-Y_{m}\right)}{\partial^{n}\,\bar{w}^{t}}|_{\bar{w}=1}$
$\displaystyle=\,1\,-\,\left\\{\exp\left(-\gamma\left(\frac{\partial\,}{\partial\,w^{p}}\,\frac{\partial\,}{\partial\,w^{t}}\,\,+\,\,\frac{\partial\,}{\partial\,\bar{w}^{p}}\,\frac{\partial\,}{\partial\,\bar{w}^{t}}\right)\right)\,N^{MFA}_{sd}\left(w,\bar{w};Y_{m}\right)\,\right.$
$\displaystyle\left.\times\,\,N^{MFA}\left(\bar{w}^{t},Y-Y_{m}=Y_{M}\right)\,N^{MFA}\left(\bar{w}^{t},Y-Y_{m}=Y_{M}\right)\right\\}|_{w^{p}=w^{t}=\bar{w}^{p}=\bar{w}^{t}=1}$
$\displaystyle=\frac{\Delta_{I\\!\\!P}\gamma^{2}}{6}\,\frac{e^{\Delta_{I\\!\\!P}(2\,Y-Y_{m})}}{L^{2}\left(Y,Y_{m}\right)}\,G\left(L\left(Y,Y_{m}\right)\right),$
(3.57)

where

$G\left(L\right)\,\,=\,\,L\,((L-1)^{2}-2)+e^{1/L}(1+3L)\,\Gamma_{0}(1/L)$
(3.58)

and

$L\left(Y,Y_{m}\right)\,\,=\,\,\gamma\,\exp\left(\Delta_{I\\!\\!P}(Y-Y_{m})\right)\,\left\\{\exp\left(\Delta_{I\\!\\!P}Y_{m}\right)-1\right\\}.$
(3.59)

$N^{MPSI}_{sd}(Y,Y_{m}=\ln(M^{2}/s_{0}))$ describes the differential single
diffraction cross section for the production of mass $M$. We can calculate the
integrated diffraction cross section

$\displaystyle N^{MPSI}_{diff}\left(Y;M_{max},M_{min}\right)\,\,$
$\displaystyle\equiv$
$\displaystyle\,\,\int^{y_{m}(max)}_{y_{m}(min)}\\!\\!\\!\\!\\!\\!\\!\\!d\,y_{m}\,N^{MPSI}_{sd}(Y,Y_{m}=\ln(M^{2}/s_{0}))\,\,$
(3.60) $\displaystyle=$
$\displaystyle\,\,\frac{\gamma}{6}\,e^{\Delta_{I\\!\\!P}Y}\,\left(B\left(L\left(Y,Y_{m}(max)\right)\right)\,\,-\,\,B\left(L\left(Y,Y_{m}(min)\right)\right)\right),$

where $Y_{m}(max)=\ln(M^{2}_{max}/s_{0})$ ,
$Y_{m}(min)=\ln(M^{2}_{min}/s_{0})$ and

$B\left(L\right)\,\,=\,\,2\,+\,\frac{1}{L^{2}}\,-\,\frac{1}{L}\,\,-\,\,\frac{e^{\frac{1}{L}}\Gamma_{0}\left(1/L\right)}{L^{3}}.$
(3.61)

$M_{max}$ and $M_{min}$ denote the largest and the smallest masses which are
produced in the diffractive process.

The expression for the integrated cross section for double diffraction can be
obtained directly from the unitarity constraint of Eq. (2.4), as the diagrams
that describe the elastic and single diffraction cross sections, do not
contribute to the set of Pomeron diagrams, that describe the exact Pomeron
Green’s function (see Fig. 7 which contains examples of the diagrams that
contribute to double diffractive production).

Figure 7: Several examples of the Pomeron diagrams that lead to the double
diffractive production. Fig. 7-a is the simplest diagram that describes the
process of double diffraction in the regions of large mass
$Y-Y_{1}=\ln(M^{2}_{1}/s_{0})$ and $Y_{2}=\ln(M^{2}_{2}/s_{0})$. Fig. 5-b
contains examples of more complicated diagrams in the region of large masses.
The dashed line indicates the cut Pomeron which describes the production of
hadrons (see Fig. 7-a).

The unitarity constraint is given by

$2\,N^{MPSI}\,\,=\,\,N^{MPSI}_{dd}\,\,+\,\,N^{MPSI}_{in},$ (3.62)

where $N^{MPSI}_{in}$ stands for the inelastic cross section. It was shown
that $N^{MPSI}_{in}$ is equal to $N^{MPSI}\left(2T(y)\right)$ (see Refs. [38,
39, 29]). Therefore, the integrated double diffraction cross section can be
written in the form

$N^{MPSI}_{dd}\left(Y\right)\,\,\,=\,\,\,2\,N^{MPSI}\left(T(Y)\right)\,\,\,-\,\,\,N^{MPSI}\left(2\,T(Y)\right).$
(3.63)

###  3.6 Our approach

In our approach we combine the G-W mechanism with the exact Pomeron Green’s
function of Eq. (3.52). First, we replace the bare Pomeron Green’s function
$G(Y-Y^{\prime})\,=\,\,\exp\left(-\Delta_{I\\!\\!P}\,(Y-Y^{\prime})\right)$ in
Eq. (2.7) by $G^{MPSI}_{I\\!\\!P}\left(Y-Y^{\prime}\right)$ of Eq. (3.52),
and, obtain

$\Omega_{i,k}\left(s,b\right)\,=\,g_{i}\,g_{k}\,\,G^{MPSI}_{I\\!\\!P}\left(Y-Y^{\prime}\right)\,\,\,S\left(b;m_{i},m_{k};\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})=0\right),$
(3.64)

with a profile function
$\,S\left(b;m_{i},m_{k};\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})\right)$
determined by Eq. (2.10).

Using Eq. (2.12), Eq. (2.15) and Eq. (2.16) with
$\Omega_{i,k}\left(s,b\right)$ from Eq. (3.64), we can calculate elastic and
total cross sections. These formulae provide a correct description of low mass
diffractive production due to the G-W mechanism. However, to include the
Pomeron interactions in processes of diffractive production, we need to change
Eq. (2.13) and Eq. (2.14). We first introduce (see Fig. 5-a)

$\Omega^{sd}_{i,k}\left(s,M;b\right)\,\,\,=\,\,g_{i}\,g^{2}_{k}\,N^{MPS}_{sd}\left(Y\,=\,\ln(s/s_{0}),Y_{m}=\ln(M^{2}/s_{0})\right)\,S^{sd}\left(b;m_{i},m_{k}\right),$
(3.65)

where the new profile function $S^{sd}\left(b;m_{i},m_{k}\right)$ is the
Fourier transform of

$\frac{1}{1+q^{2}/m^{2}_{i})^{2}}\,\int\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{1+(\vec{q}+\vec{k})^{2}/m^{2}_{i})^{2}}\,\frac{1}{1+(\vec{q}-\vec{k})^{2}/m^{2}_{i})^{2}}\,\,\Longrightarrow\,\,\,S^{sd}\left(b;m_{i},m_{k}\right).$
(3.66)

We found to within an accuracy of around 5%,that the profile function
$S^{sd}\left(b;m_{i},m_{k}\right)$ can be approximated by

$S^{sd}\left(b;m_{i},m_{k}\right)\,\,\,=\,\,\,\frac{m^{2}_{k}}{12\,\pi}\,S\left(b;m_{i},\bar{m}_{k};\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})=0\right)\,\,\,\,\,,$
(3.67)

where $\bar{m}^{2}_{k}\,\,=\,\,2\,\sqrt{3}\,m^{2}_{k}$ .

For the calculation of the integrated cross section of single diffractive
production we define

$\displaystyle\Omega^{sd}_{i,k}\left(s,M_{max},M_{min};b\right)\,\,\,=$ (3.68)
$\displaystyle\,\,g_{i}\,g^{2}_{k}\,N^{MPS}_{diff}\left(Y\,=\,\ln(s/s_{0}),Y_{m}(max)=\ln(M^{2}_{max}/s_{0}),Y_{m}(min)=\ln(M^{2}_{min}/s_{0})\right)\,S^{sd}\left(b;m_{i},m_{k}\right).$

For the integrated cross section of the single diffraction channel, we obtain
the following expression which takes into account both the G-W mechanism, and
the enhanced Pomeron diagrams:

$\displaystyle\sigma_{diff}=\int^{M^{2}_{max}}_{M^{2}_{min}}\\!\\!\\!\frac{dM^{2}}{M^{2}}\,\frac{d\sigma^{sd}(s,M)}{dM^{2}}$
$\displaystyle=$ $\displaystyle\int
d^{2}b\left\\{\,\,\alpha^{2}\,\beta^{2}\left(\alpha^{2}\,e^{-\frac{\Omega_{1,1}}{2}}\,-\,(\alpha^{2}-\beta^{2})\,e^{-\frac{\Omega_{1,2}}{2}}\,-\,\beta^{2}\,e^{-\frac{\Omega_{2,2}}{2}}\right)^{2}\right.$
(3.69) $\displaystyle+$
$\displaystyle\left.\left\\{\alpha^{2}\,\Omega^{diff}_{1,1}e^{-\Omega_{1,1}}\,+\,\beta^{2}\,\Omega^{diff}_{2,2}\,e^{-\Omega_{2,2}}\,\,+\,2\,\alpha^{2}\,\beta^{2}\,\Omega^{diff}_{1,2}e^{-\Omega_{1,2}}\right\\}\right\\},$
(3.70)

where $\Omega^{diff}_{i,k}$ are given by Eq. (3.68). The differential cross
section for single diffraction in the region $M\;\;>\;\;M_{min}$ is

$M^{2}\,\frac{d\sigma^{sd}(s,M)}{dM^{2}}\,=\,\int
d^{2}b\left\\{\alpha^{2}\,\Omega^{sd}_{1,1}e^{-\Omega_{1,1}}\,+\,\beta^{2}\,\Omega^{sd}_{2,2}\,e^{-\Omega_{2,2}}\,\,+\,2\,\alpha^{2}\,\beta^{2}\,\Omega^{sd}_{1,2}e^{-\Omega_{1,2}}\right\\},$
(3.71)

where $\Omega^{sd}_{i,k}$ are given by Eq. (3.65). In Eq. (3.69) and Eq.
(3.71) we use $\Omega_{i,k}$ of Eq. (3.64).

For the double diffractive cross section we introduce

$\Omega^{dd}_{i,k}\left(s,b\right)\,=\,g_{i}\,g_{k}\,\,N^{MPSI}_{dd}\left(Y\right)\,\,\,S\left(b;m_{i},m_{k};\alpha^{\prime}_{I\\!\\!P}\ln(s/s_{0})=0\right).$
(3.72)

Using this $\Omega^{dd}_{i,k}$ we obtain for the integrated DD cross section

$\displaystyle\sigma_{dd}$ $\displaystyle=$
$\displaystyle\int^{\infty}_{0}\\!\\!\\!\frac{dM^{2}_{1}}{M^{2}_{1}}\,\int^{\infty}_{0}\\!\\!\\!\frac{dM^{2}_{2}}{M^{2}_{2}}\frac{d\sigma^{sd}(s,M_{1},M_{2})}{dM^{2}_{1}\,dM^{2}_{2}}=\int
d^{2}b\left\\{\,\,\alpha^{4}\,\beta^{4}\left(e^{-\frac{\Omega_{1,1}}{2}}\,-\,2\,\,e^{-\frac{\Omega_{1,2}}{2}}\,+\,\,e^{-\frac{\Omega_{2,2}}{2}}\right)^{2}\right.$
(3.73) $\displaystyle+$
$\displaystyle\left.\,\left\\{\alpha^{4}\,\Omega^{dd}_{1,1}e^{-\Omega_{1,1}}\,+\,\beta^{4}\,\Omega^{dd}_{2,2}\,e^{-\Omega_{2,2}}\,\,+\,2\,\alpha^{2}\,\beta^{2}\,\Omega^{dd}_{1,2}e^{-\Omega_{1,2}}\right\\}\right\\}.$
(3.74)

For the integrated single and double diffractive production, the expressions
each contains two terms: the first is responsible for G-W mechanism for these
processes, while the second originates from the large mass diffraction of the
enhanced Pomeron diagrams.

Eq. (2.12), Eq. (2.15) and Eq. (2.16) with $\Omega_{i,k}\left(s,b\right)$ from
Eq. (3.64) and Eq. (3.65) - Eq. (3.74) give the full list of formulae in our
approach. We wish to emphasize that our approach is based on the pQCD input
for calculating the enhanced Pomeron diagrams, and on the G-W mechanism for
low mass diffractive production.

##  4 Results of the fit

###  4.1 Cross sections and elastic slope

We have adjusted the parameters of our model which are listed in Table 2,
using the formulae of section 3.6. The fit is based on 55 experimental data
points, which includes the $p$-$p$ and $\bar{p}$-$p$ total cross sections,
integrated elastic cross sections, integrated single and double diffraction
cross sections, and the forward slope of the elastic cross section in the ISR-
Tevatron energy range. The model gives a good reproduction of the data, with a
$\chi^{2}/d.o.f.\,\approx 1.25$. The quality of description of the
experimental data is shown in Fig. 2, Fig. 2, Fig. 9, Fig. 9 and Fig. 11. A
significant contribution to $\chi^{2}/d.o.f.$ stems from the uncertainty for
the value of two single diffraction cross sections, and of the total cross
section at the Tevatron. The $\chi^{2}/d.o.f.$ in Table 2 is calculated
neglecting the contribution of the CDF measurement [41] ($\sigma_{tot}$ = 80
mb. at the Tevatron energy). The important advantage of our approach, is that
the model provides a very good reproduction of the DD data points. In our
previous attempt to describe the DD data [6] within a G-W approach, it was
necessary to assume a non-factorizable contribution for the Pomeron exchange,
resulting in marginally acceptable results.

Figure 8: Energy dependence of $\sigma_{el}$.

Figure 9: Energy dependence of the cross section $\sigma_{sd}$.

Figure 10: Energy dependence of $\sigma_{dd}$.

Figure 11: $t$ dependence of $d\sigma_{el}/dt$ at the Tevatron
($W=1800\,GeV$).

$\Delta_{I\\!\\!P}$ | $\beta$ | $\alpha^{\prime}_{{I\\!\\!P}}$ | $g_{1}$ | $g_{2}$ | $m_{1}$ | $m_{2}$
---|---|---|---|---|---|---
0.335 | 0.339 | 0.012 $GeV^{-2}$ | 5.82(0.90) $GeV^{-1}$ | 239.6(37.27) $GeV^{-1}$ | 1.54 $GeV$ | 3.06 $GeV$
$\Delta_{{I\\!\\!R}}$ | $\gamma$ | $\alpha^{\prime}_{{I\\!\\!R}}$ | $g^{{I\\!\\!R}}_{1}$ | $g^{{I\\!\\!R}}_{2}$ | $R^{2}_{0,1}$ | $\chi^{2}/d.o.f.$
\- 0.60 | 0.0242 | 0.6 $GeV^{-2}$ | 13.22 $GeV^{-1}$ | 367.8 $GeV^{-1}$ | 4.0 $GeV^{-2}$ | 1.0
Table 2: Fitted parameters for our model which includes G-W mechanism for
diffractive production, as well as the diffractive processes that stem from
enhanced Pomeron diagrams (Pomeron loops). In our approach we have extracted a
factor of $\gamma$ from the product $g_{i}g_{k}$. Therefore, the value of the
Pomeron-hadron vertex is equal to $g_{i}=g_{i}(\mbox{from the
table)}\times\sqrt{\gamma}$, and this value is shown in parentheses.

###  4.2 $t$ \- dependence of the differential elastic cross section

The behaviour of the elastic cross section in the region of small $t$, is
characterized by $B_{el}$ and $\sigma_{tot}$, which are included in the set of
experimental data that we use for our fit. However, we wish to know the
scattering amplitude at a relatively high value of $t\geq 0.1\,GeV^{2}$, where
the simple exponential $t$ behaviour of the input elastic amplitude does not
describe the data [6]. In Fig. 11 we plot our prediction with the parameters
of Table 2 for the $t$-behaviour of the elastic cross section at the Tevatron
energy $W=1800\,GeV$. We reproduce the data quite well, and it is a
considerable improvement over the results obtained in [6]. ‡‡‡We thank M.
Ryskin who pointed out that the model of Ref. [6], gives a minimum at
$|t|\approx 0.1\,GeV^{2}$ which contradicts the experimental data.

###  4.3 Mass dependence of the diffractive cross section

In Fig. 12, we plot the cross section of single diffraction as a function of
$1-x_{l}=M^{2}/s$ at the Tevatron energy. For this cross section, in the
region of high mass, we use Eq. (3.71), while for diffraction in the region of
low mass, we need to make some assumptions regarding the dependence of this
cross section on $M$. Following Refs. [42, 7] we assume that the main
contribution for the G-W part of single diffractive production stems, from the
${I\\!\\!R}{I\\!\\!P}{I\\!\\!R}$ term, which does not depend on $x_{L}$.
Therefore, the resulting contribution has the following form:

$\displaystyle M^{2}\frac{d\sigma_{sd}\left(s,M\right)}{dM^{2}\,dt}=$ (4.75)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,M^{2}\frac{d\sigma^{\mbox{Low
M}}_{sd}\left(s,M\right)}{dM^{2}\,dt}\left({I\\!\\!R}{I\\!\\!P}{I\\!\\!R}\,\mbox{term
; G-W contribution}\right)+M^{2}\frac{d\sigma^{\mbox{High
M}}_{sd}\left(s,M\right)}{dM^{2}\,dt}\left({Eq.~{}(\ref{DXSSD})}\right)\,$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,B_{sd}e^{-B_{sd}\,|t|}\left\\{\sigma^{\mbox{High
M}}_{sd}\,\frac{N^{MPSI}_{sd}\left(Y,Y_{m};{Eq.~{}(\ref{SD1})}\right)}{N^{MPSI}_{diff}\left({Eq.~{}(\ref{TSD})}\right)}\,+\,\sigma^{\mbox{Low
M}}_{sd}\right\\}.$

In our parametrization, the scale of the second term is determined by the G-W
mechanism. The fact that the ${I\\!\\!R}{I\\!\\!P}{I\\!\\!R}$ term is
responsible for $x_{L}$ behaviour, is an additional independent input.
However, an argument for the ${I\\!\\!R}{I\\!\\!P}{I\\!\\!R}$ term , is that
if $\Delta_{I\\!\\!P}=0$, then both ${I\\!\\!P}{I\\!\\!P}{I\\!\\!P}$ and
${I\\!\\!R}{I\\!\\!P}{I\\!\\!R}$ lead to a diffractive cross section, which is
constant as a function of energy.

Figure 12: Dependence of single inclusive cross section on $1-x_{L}=M^{2}/s$,
where $M$ is the mass of the diffractively produced system. Data are taken
from Refs. [41, 42].

##  5 Predictions for LHC and Cosmic Rays Energies: $b$-dependence of the
amplitudes

Fig. 13 shows our prediction for high energy behaviour of the total, elastic
and diffractive cross sections as well as of the elastic slope.

|
---|---
Fig. 13-a | Fig. 13-b
Figure 13: The energy dependence of cross sections (Fig. 13 -a) and elastic
slope $B_{el}$(Fig. 13-b) for elastic scattering and diffractive production.
We have plotted $0.25\,\sigma_{tot}$, so as to show all the predictions on the
same figure.

At W=14 $TeV$ (LHC energy): $\sigma_{tot}=92.1\,mb$, $\sigma_{el}=20.9\,mb$,
$\sigma_{sd}=11.8\,mb$, $\sigma_{dd}=6.08\,mb$ and $B_{el}=20.6\,GeV^{-2}$.
Comparing these results with the prediction of our previous model
($\sigma_{tot}=110.5\,mb$, $\sigma_{el}=25.3\,mb$, $\sigma_{sd}=11.6\,mb$,
$\sigma_{dd}=4.9\,mb$ and $B_{el}=20.5\,GeV^{-2}$ at the LHC energy) we see
that the above comparison

|
---|---
Fig. 14-a | Fig. 14-b
Figure 14: Impact parameter dependence of $A_{i,k}$ and $a_{el}$ at different
energies.

leads to a few general observations, on which we will elaborate in the
Discussion section.
1) The predictions of our Ref. [6] for $\sigma_{tot}$ and $\sigma_{el}$ at the
LHC are considerably higher than the corresponding predictions obtained in the
present analysis. The difference between the two sets of predictions grow
monotonically with energy. Note, that the moderate growth of $\sigma_{tot}$
and $\sigma_{el}$ obtained in the present study continues up to energies as
high as $10^{5}$ GeV, and probably higher. See Fig.13 and Table 3. We
attribute this behaviour to the fact, that in our fit,
$\alpha^{{}^{\prime}}_{{I\\!\\!P}}$ is small but not zero.
2) These features reflect the fact that whereas our previous model has an
output compatible with $\Delta_{eff}\approx 0.08$ up to exceedingly high
energies, the present model (in which Pomeron enhanced diagrams play a
significant role), has an output $\Delta_{eff}$ which decreases monotonically
with energy.
3) The qualitative features of $\sigma_{tot}$ and $\sigma_{el}$ in our two
models are also seen in $\sigma_{sd}$. However, $\sigma_{dd}$ behaves
differently, slowly approaching a constant value above LHC energies.
4) Our $B_{sd}$ slope is approximately two times less $B_{el}$. At high energy
we have only the results of fit to the diffraction experimental data given in
Ref. [41]. The values of $B_{sd}$ from this fit are $B_{sd}=7.7\pm
0.6\,(4.2\pm 0.5)\,GeV^{-2}$ for W = 546 (1800) GeV. Our values (see Fig.
13-b) are $B_{sd}=8.5(9.9)\,GeV^{-2}$ at these energies. The large difference
in the experimental slopes have been discussed in Ref. [41].

To investigate the quantitative features of our present model further, we plot
in Fig.14 the $b$ dependence of $A_{i,k}(s,b)$ and $a_{el}(s,b)$ output
amplitudes, at Tevatron and LHC energies. As seen, $A_{2,2}$ and $A_{1,2}$
reach the black bound at relatively low energies, whereas $A_{1,1}$ is below
the black bound, approaching it very slowly.

As noted in Ref.[6], the output cross sections reach the unitarity bound when,
and only when, $A_{1,1}\;=\;A_{1,2}\;=\;A_{2,2}\;=\;1$. Since, $A_{1,1}$ grows
very slowly with energy, we conclude that the very slow approach to the
unitarity bound observed in Ref.[6], also occurs in the present model. Despite
the fact that the two models have very different $t$ dependences, we hope that
such structure of the amplitude does not depend on the model assumptions, but
reflects the principle features of the hadron scattering at high energy. We
shall expand on this issue in the Discussion section.

##  6 Survival probability of diffractive Higgs production

In the following we limit our discussion to the survival probability of Higgs
production, in an exclusive central diffractive process. Most estimates of the
values of survival probability have been made in the framework of G-W
mechanism, in two channel eikonal models. A general review of such survival
probability calculations can be found in Ref.[11]. The general formulae for
the calculation of the survival probability for diffractive Higgs boson
production, have been discussed in Refs.[10, 11, 12]. The structure of the
survival probability expression is shown in Fig. 15.-a. Accordingly,

$\langle\mid S^{2}_{2ch}\mid\rangle=\frac{N(s)}{D(s)},$ (6.76)

where,

$\displaystyle N(s)=\int
d^{2}\,b_{1}\,d^{2}\,b_{2}\left[\sum_{i,k}\,<p|i>^{2}<p|k>^{2}\,A^{i}_{H}(s,b_{1})\,A^{k}_{H}(s,b_{2})(1-A^{i,k}_{S}\left((s,(\mathbf{b}_{1}+\mathbf{b}_{2})\right))\right]^{2},$
(6.77) $\displaystyle
D(s)=\int\,d^{2}\,b_{1}\,d^{2}\,b_{2}\left[\sum_{i,k}<p|i>^{2}<p|k>^{2}\,A^{i}_{H}(s,b_{1})\,A^{k}_{H}(s,b_{2})\right]^{2}.$
(6.78)

$<p|i>$ is equal to $\langle\Psi_{proton}\mid\Psi_{i}\rangle$ and , therefore,
$<p|1>=\alpha$ and $<p|2>=\beta$.

$A_{s}$ denotes the soft strong interaction amplitude given by Eq. (2.7),Eq.
(2.12),Eq. (2.13) and Eq. (2.14). The form of $A_{H}(s,b)$ has been discussed
in Refs.[10, 11]. In our model we assume an input Gaussian $b$-dependence for
the hard amplitudes.

${A_{i,k}^{H}}=A_{H}(s)\,\Gamma_{i,k}^{H}(b),$ (6.79)

where $A_{H}(s)$ is an $s$\- dependent arbitrary function which does not
depend on $i,k$, and
$\Gamma_{i,k}^{H}(b)=\frac{1}{\pi(R^{H}_{i,k})^{2}}\,e^{-\frac{2\,b^{2}}{(R^{H}_{i,k})^{2}}}$.
The hard vertices and radii ${R_{i,k}^{H}}^{2}$, are constants derived from
HERA $J/\Psi$ elastic and inelastic photo and DIS production[40].

Following Refs.[12, 11] we have introduced in the above, two hard $b$-profiles

$\displaystyle A^{pp}_{H}(b)$ $\displaystyle=$ $\displaystyle\frac{V_{p\to
p}}{2\pi B_{el}^{H}}\exp\left(-\frac{b^{2}}{2\,B_{el}^{H}}\right),$ (6.80)
$\displaystyle A^{pd}_{H}(b)$ $\displaystyle=$ $\displaystyle\frac{V_{p\to
d}}{2\pi B_{in}^{H}}\exp\left(-\frac{b^{2}}{2B_{in}^{H}}\right).$ (6.81)

The values $B_{el}^{H}$=3.6 $GeV^{-2}$ and $B_{in}^{H}$=1 $GeV^{-2}$, have
been taken from the experimental ZEUS data on $J/\Psi$ production at HERA (see
Refs.[11, 43]).

Using Eq. (2.12)-Eq. (2.14), the integrands of Eq. (6.77) and Eq. (6.78) are
reduced by eliminating common $s$-dependent expressions,

$\displaystyle N(s)$ $\displaystyle=$
$\displaystyle\int\,d^{2}b_{1}d^{2}b_{2}[A_{H}(s,b_{1})\,A_{H}(s,b_{2})(1-A_{S}\left(\mathbf{b}=\mathbf{b}_{1}+\mathbf{b}_{2}\right))]^{2}$
(6.82) $\displaystyle=$ $\displaystyle\int
d^{2}b_{1}d^{2}b_{2}\,[(1-a_{el}(s,b))A^{pp}_{H}(b_{1})A^{pp}_{H}(b_{2})-a_{sd}(s,b)\left(A^{pd}_{H}(b_{1})A^{pp}_{H}(b_{2})\right.$
$\displaystyle+$
$\displaystyle\left.A^{pp}_{H}(b_{1})A^{pd}_{H}(b_{2})\right)-a_{dd}(s,b)A^{pd}_{H}(b_{1})A^{pd}_{H}(b_{2})]^{2},$
$D=\int
d^{2}b_{1}d^{2}b_{2}\left[A^{pp}_{H}(b_{1})A^{pp}_{H}(b_{2})\right]^{2}.$
(6.83)

Figure 15: Survival probability for exclusive central diffractive production
of the Higgs boson. Fig. 15-a shows the contribution to the survival
probability in the G-W mechanism, while Fig. 15-b illustrates the origin to
the additional factor $\langle\mid S^{2}_{enh}\mid\rangle$.

Eq. (6.76) does not give a correct estimate for the survival probability, and
should be multiplied by a factor ($\langle\mid S^{2}_{enh}\mid\rangle$) that,
incorporates the possibility for the Higgs boson to be emitted from the
enhanced diagrams (see Fig. 15-b). Therefore, the resulting survival
probability can be written as

$\langle\mid S^{2}\mid\rangle\,\,\,=\,\,\langle\mid
S^{2}_{enh}\mid\rangle\left(\mbox{{Eq.~{}(\ref{SPE})}}\right)\,\times\,\langle\mid
S^{2}_{2ch}\mid\rangle\left(\mbox{{Eq.~{}(\ref{SP})}}\right).$ (6.84)

The first attempt to find $\langle\mid S^{2}_{enh}\mid\rangle$ was made in
Refs.[44, 34], where this factor was calculated neglecting the fact that the
Higgs boson could be produced from the two gluon scattering with a difference
in rapidity $\delta Y_{H}=\ln(M^{2}_{H}/s_{0})$ (see for example Refs. [45,
44]). The MPSI approach for this case is shown in Fig. 17, and it leads to the
result:

$\langle\mid
S^{2}_{enh}\mid\rangle\left(Y\right)\,\,=\,\,\frac{\partial}{\partial\,T}N^{MPSI}_{el}\left(T\right)\,=\,\frac{1}{T^{3}(Y)}\left\\{-T(Y)\,+\,e^{\frac{1}{T(Y)}}\,\left(1+T(Y)\right)\,\Gamma_{0}\left(\frac{1}{T(Y)}\right)\right\\},$
(6.85)

where $N^{MPSI}_{el}\left(T\right)$ is given Eq. (3.50). It was originally
suggested to divide this factor by $\langle\mid
S^{2}_{enh}\mid\rangle\left(Y=\delta Y_{H}\right)$,

$\langle\mid S^{2}_{enh}\mid\rangle\left(Y\right)\,\,=\,\,\frac{\langle\mid
S^{2}_{enh}\mid\rangle\left(Y;{Eq.~{}(\ref{SPE1})}\right)}{\langle\mid
S^{2}_{enh}\mid\rangle\left(Y=\delta Y_{H};{Eq.~{}(\ref{SPE1})}\right)}.$
(6.86)

In this paper we take into account $\delta Y_{H}$ in a consistent way, which
was outlined in Ref. [34], and presented in Fig. 17 [16]. At rapidity
$Y-Y^{\prime}-\frac{1}{2}\delta Y_{H}$, one of the partons (Pomerons) will
produce a Higgs boson, and it should be removed from the cascade evolution.
Therefore, those partons which will participate in the evolution will be
characterized by a new generating function,

$\displaystyle\tilde{Z}\left(Y-Y^{\prime}-\frac{1}{2}\delta
Y_{H},u\right)\,\,\,=$ (6.87) $\displaystyle
e^{-\Delta_{I\\!\\!P}(Y-y^{\prime}-\frac{1}{2}\delta Y_{H})}\,\frac{\partial
Z^{MFA}\left(Y-Y^{\prime}-\frac{1}{2}\delta
Y_{H},u\right)}{\partial\,u}\,\,=\,\,\frac{1}{\left(u\,+\,(1-u)\,e^{-\Delta_{I\\!\\!P}(Y-Y^{\prime}-\frac{1}{2}\delta
Y_{H})}\right)^{2}}.$

$\tilde{Z}\left(Y-Y^{\prime}-\frac{1}{2}\delta Y_{H};u\right)$ should be
evolved to rapidity $Y-Y^{\prime}$ using Eq. (3.46). This evolution results in
an improved generating function

$\widetilde{\widetilde{Z}}\left(Y-Y^{\prime},u\right)\,\,=\,\,\frac{\left(u+(1-u)\,e^{\frac{1}{2}\Delta_{I\\!\\!P}\delta
Y_{H}}\right)^{2}}{\left(u+(1-u)\,e^{\Delta_{I\\!\\!P}(Y-Y)}\right)^{2}}.$
(6.88)

Note that for central Higgs boson production $\frac{1}{2}Y=Y^{\prime}$.

Symmetrically, using Eq. (6.87) we need to find
$\tilde{Z}\left(Y^{\prime}-\frac{1}{2}\delta Y_{H},u\right)$, and from Eq.
(6.88), $\tilde{Z}\left(Y^{\prime},u\right)$. The result for this function is

$\widetilde{\widetilde{Z}}\left(Y^{\prime},u\right)\,\,\,=\,\,\,\frac{\left(u+(1-u)\,e^{\frac{1}{2}\Delta_{I\\!\\!P}\delta
Y_{H}}\right)^{2}}{\left(u+(1-u)\,e^{\Delta_{I\\!\\!P}Y^{\prime}}\right)^{2}}.$
(6.89)

Using these generating functions we obtain

$\displaystyle\langle\mid
S^{2}_{enh}\left(MPSI\right)\mid\rangle\left(Y\right)\,\,\,=$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\sum^{\infty}_{n=1}\,\,\frac{(-1)^{n+1}}{n!}\,\,\,\gamma^{n}\,\,\frac{\partial^{n}\,\widetilde{\widetilde{Z}}(Y-Y^{\prime},u^{p})}{\partial^{n}\,u^{p}}|_{u^{p}=1}\,\,\frac{\partial^{n}\,\widetilde{\widetilde{Z}}(Y^{\prime},u)\,u^{t})}{\partial
u_{t}}|_{u^{t}=1}$ (6.90)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,S\left({\cal
T}(Y)\right)\,\,-\,\,2\,e^{-\Delta_{I\\!\\!P}(Y-\delta
Y_{H})/2}\,S1\left({\cal T}(Y)\right)\,\,+\,\,e^{-2\Delta_{I\\!\\!P}(Y-\delta
Y_{H})/2}\,S2\left({\cal T}(Y\right);$
$\displaystyle\,\,\,\,\,\,\,\,\,\,S(T)=\frac{1}{T^{3}}\left\\{-T\,+\,e^{\frac{1}{T}}\,\left(1+T\right)\,e^{\frac{1}{T}}\,\Gamma_{0}\left(\frac{1}{T}\right)\right\\};$
(6.91)
$\displaystyle\,\,\,\,\,\,\,\,\,S1(T)=\frac{1}{T^{3}}\left\\{-T(1+T)\,+\,(1+2T)\,e^{\frac{1}{T}}\,\Gamma_{0}\left(\frac{1}{T}\right)\right\\};$
(6.92)
$\displaystyle\,\,\,\,\,\,\,\,\,S2(T)=\frac{1}{T^{3}}\left\\{T\left[\left(T-1\right)^{2}-2\right]\,+\,(1+3T)\,e^{\frac{1}{T}}\,\Gamma_{0}\left(\frac{1}{T}\right)\right\\}.$
(6.93)

where

${\cal
T}\left(Y\right)\,\,\,=\,\,\gamma\left(e^{\Delta_{I\\!\\!P}(Y-Y^{\prime})}\,-\,1\right)\,\left(e^{\Delta_{I\\!\\!P}Y^{\prime}}\,-\,1\right).$
(6.94)

Figure 16: MPSI approach for $\langle\mid S^{2}_{enh}\mid\rangle$ in the case
when we neglect $\delta Y_{H}=\ln(M^{2}_{H}/s_{0})$.

Figure 17: MPSI approach for $\langle\mid S^{2}_{enh}\mid\rangle$ in the case
when we take into account $\delta Y_{H}=\ln(M^{2}_{H}/s_{0})$.

Figure 18: Energy dependence of centrally produced Higgs survival probability.

Using Eq. (6.84)-Eq. (6.93), we calculate the survival probability
$\langle\mid S^{2}\mid\rangle$ for exclusive Higgs production in central
diffraction. Our results are plotted in Fig. 18. In the following we focus on
a detailed discussion of our LHC predictions based on the above. Calculating
in the framework of a two amplitude eikonal model, $\langle\mid
S^{2}_{2ch}\mid\rangle$ for exclusive Higgs production in central diffraction
is equal to 2.35%, which is close to the value of this part of the survival
probability estimated by the Durham group [5, 7], and by our group [11].
However, the additional factor $\langle\mid S^{2}_{enh}\mid\rangle$ = 0.063 as
derived from Eq. (6.90), leads to a final $\langle\mid
S^{2}\mid\rangle=0.15\%$. This result reflects a tendency for the value of the
survival probability to be much smaller, than when evaluated in the two
channel models [44, 6]. Note, that the approximate expression for $\langle\mid
S^{2}_{enh}\mid\rangle$ derived from Eq. (6.85) with our parameters, turns out
to be three times smaller. In Ref. [7] the Pomeron diagrams were taken into
account, but nevertheless, their contribution in the calculation of
$\langle\mid S^{2}\mid\rangle$ was omitted, without any explanation.

|  Tevatron |  LHC |  $W=10^{5}GeV$
---|---|---|---
| GLM KMR | GLM KMR | GLM KMR
$\sigma_{tot}$( mb ) | 73.29 74.0 | 92.1 88.0 | 108.0 98.0
$\sigma_{el}$(mb) | 16.3 16.3 | 20.9 20.1 | 24\. 22.9
$\sigma_{sd}$(mb) | 9.76 10.9 | 11.8 13.3 | 14.4 15.7
$\sigma^{\mbox{low M}}_{sd}$ | 8.56 4.4 | 10.52 5.1 | 12.2 5.7
$\sigma^{\mbox{high M}}_{sd}$ | 1.2 6.5 | 1.28 8.2 | 2.2 10.0
$\sigma_{dd}$(mb) | 5.36 7.2 | 6.08 13.4 | 6.29 17.3
$\left(\sigma_{el}+\sigma_{sd}+\sigma_{dd}\right)/\sigma_{tot}$ | 0.428 0.464 | 0.421 0.531 | 0.412 0.57
$S^{2}_{2ch}(\%)$ | 3.2 2.7 - 4.8 | 2.35 1.2-3.2 | 2.0 0.9 - 2.5
$S^{2}_{enh}(\%)$ | 28.5 100 | 6.3 100 | 3.3 100
$S^{2}(\%)$ | 1.2 2.7 - 4.8 | 0.21 1.2-3.2 | 0.05 0.9 - 2.5
Table 3: Comparison of the GLM ( this paper) and KMR[7] models.

##  7 Discussion

In this paper we have presented an approach for soft interactions at high
energies based on two ingredients:
1) The Good-Walker mechanism for elastic and low mass diffraction.
2) Pomeron enchanced contributions which leads to the exact Pomeron Green’s
function, which is significantly different from one Pomeron exchange. This
component provides the main contribution to high mass diffraction.

Our enhanced Pomeron formalism, is based on the observation that the soft
scattering cross sections and slopes, can be reproduced with
$\alpha^{{}^{\prime}}_{{I\\!\\!P}}\approx\;0.01\;GeV^{-2}$, rather than
$\alpha^{{}^{\prime}}_{P}=\;0.25\;GeV^{-2}$ typical of conventional Regge
phenomenology. Our result of a very small $\alpha^{{}^{\prime}}_{{I\\!\\!P}}$,
implies that the observed shrinkage of the forward differential cross
sections, traditionally associated with
$\alpha^{{}^{\prime}}_{{I\\!\\!P}}\;>\;0$, can be also reproduced with
$\alpha^{{}^{\prime}}_{{I\\!\\!P}}\;\approx\;0$ coupled to strong screening,
which produces the desired shrinkage.

In the following we discuss further the properties of our model, by comparing
it with the KMR model [7], which is conceptually similar to ours, having the
same two mechanisms: G-W and multi-Pomeron interactions. In the KMR model
$\alpha^{\prime}_{{I\\!\\!P}}\;\equiv\;0$ is an input assumption. In the
present GLM model $\alpha^{\prime}_{{I\\!\\!P}}$ is a fitted parameter, and
its value $\alpha^{\prime}_{{I\\!\\!P}}\;=\;0.012$ is an output. As we have
discussed the small value of $\alpha^{\prime}_{{I\\!\\!P}}$, led us to our key
hypothesis that we advocate in this paper: the soft processes are not so soft,
but stem from short distances, where the QCD coupling is small
($\alpha_{S}\approx 0.12$ to $0.16$). Using this hypothesis we built our
theoretical approach. This approach is self consistent based on pQCD. This
enables us to restrict our summation only to triple Pomeron vertices. KMR
summation is based on an ad hoc assumption

$\Gamma\left(n\,\,\,Pomerons\to
m\,\,\,Pomerons\right)\,\,\propto\,\,n\,m\,\lambda^{m+n-2}.$ (7.95)

Further, KMR claim that Eq. (7.95) leads to their main equation (Eq.(26) of
Ref.[7]), which corresponds to the parton model. Both claims maybe correct,
but they have not been proven in the KMR paper. A formal difficulty also noted
by the KMR authors (at the beginning of their section 4.3) is that the
formulae for diffractive production are ad hoc, and are not actually
compatible with our Eq. (7.95) and Eq.(26) of Ref.[7].

In spite of the fact, that we do not think that KMR model is able to provide
reliable estimates, it is interesting to compare our results, since both are
based on the same physics.
1) The introduction of Pomeron induced interactions to the calculation,
results in the accumulation of Pomeron loops along the initial Pomeron
propogator, which lead to a monotonic reduction of the output $\Delta_{eff}$
with energy. In the KMR model, where additional diagrams, and not only
diagrams with triple Pomeron interactions have been included, this process
occurs more rapidly than in our model. Accordingly, they choose a very high
input value, $\Delta_{in}\;=\;0.55$. In our fit we have
$\Delta_{in}\;=\;0.335$. Both in the KMR model and in our model the effective
shrinkage of the diffraction peak, stems from the Pomeron interactions. The
difference is that $R^{2}(s)$ in the GLM model grows as $ln^{2}s$ with a
coefficient proportional to $\alpha^{\prime}_{P}$. Since in evaluating our
summations, we have made an approximation in which
$\alpha^{\prime}_{{I\\!\\!P}}$ = 0, our calculations are only trustworthy up
to $W\approx 10^{5}\,GeV$. The high energy output of GLM and KMR models are
presented in Table 3. For $\sigma_{dd}$ the contribution to the diffractive
channels coming from the Pomeron enhanced diagrams, are larger in KMR than in
GLM. Note that $\sigma_{dd}$, as calculated by GLM, saturates just above LHC
energies, while in KMR it continues growing even at energies of W = $10^{5}$
GeV, where it is predicted to be much larger than the $\sigma_{sd}$.
2) In Table 3 we define $\sigma^{\mbox{Low M}}_{sd}$ as the contribution of
G-W mechanism, while in the KMR model low and high mass diffraction is
allocated to mass values $M<M_{0}=2.5\,GeV$ and $M>M_{0}=2.5\,GeV$.
3) The behaviour of the ratio
$R_{D}\;=\;\frac{(\sigma_{el}+\sigma_{sd}+\sigma_{dd})}{\sigma_{tot}}$ conveys
information regarding the onset of unitarity constraints at high enough
energy. In the G-W model the Pumplin bound $R_{D}\;\leq\;0.5$ is relevant. The
multi-Pomeron induced contributions are not included in this bound. In the GLM
model, $R_{D}\;\ <\;0.5$ and decreases very slowly with energy. It’s
corresponding $R^{G-W}_{D}\approx\;0.35$, is a constant similar to the output
of Ref.[6]. In KMR, $R_{D}\;\ >\;0.5$ and grows with energy. The difference
between GLM and KMR appears to be due to the different summations of the
Pomeron induced diagrams, incorporated in the models.
4) The difference in the values of the survival probabilities calculated in
the two channel models are significant. The KMR estimate should be reduced by
$S^{2}_{enh}$ which is ignored in their calculation. Including this factor
brings the GLM and KMR numbers closer.
5) The data analysis aimed at determining the three opacities $\Omega_{i,k}$,
is conceptually different in the GLM and KMR models. In principle this
information should be obtained utilizing Eq.(2.12)-Eq.(2.18). However, the
experimental information we have at the UA(4)-Tevatron energies is not
sufficient to constrain the Pomeron parameters. In the GLM model we, thus,
included both the Pomeron and Reggeon trajectories, covering also the
extensive ISR data. This enabled us to extract the Pomeron (fitted) parmeters.
KMR have adopted a different strategy, obtaining the Pomeron opacities from a
good fit to $\frac{d\sigma_{el}}{dt}$ in UA(4)-Tevatron energy range. This
implies a good reproduction of $\sigma_{tot}$, $\sigma_{el}$ and $B_{el}$.
From their paper [7], it is not clear whether KMR also fit the diffractive
channels. This is not the only ambiguity in the KMR presentation of their
results. Some of their parameters are explicitly presented and fitted, some
are presented and ”tuned”, some are assumed and some are implied, but not
explicitly presented.
6) As noted, we attribute our dynamical result to the output of our fit, in
which $\alpha^{\prime}_{{I\\!\\!P}}$ is small but not zero! Both our model and
KMR, predict total and elastic cross sections which are significantly smaller
than in two channel eikonal estimates, which do not include the Pomeron
enhanced contributions (see Refs. [4, 5, 6]). The differences are sufficiently
large, so that measurements at LHC and Auger should be able to discrimenate
between the various approaches. See details below.
7) The most practical, and perhaps the most interesting result we have
obtained, is the small value (about 0.15%) for the survival probability of
central diffractive Higgs production at the LHC. The very small value of the
final $S^{2}_{H}$, is due to the smallnes of $S^{2}_{2ch}$, multiplied by the
small $S^{2}_{enh}$. Our calculation does not include further reductions of
$S^{2}_{H}$, due to additional short distance processes (see Refs.[46, 44]).
These may further reduce the result we obtained for the survival probability.

The region of applicability for our formulae is given by Eq. (3.41), in which,
$Y\ll 1/\gamma=41$, and by the fact that in our procedure for summing Pomeron
diagrams we considered $\alpha^{\prime}_{{I\\!\\!P}}=0$. The first restriction
leads to a very large kinematic range for rapidity. The seco

that $\alpha^{\prime}_{I\\!\\!P}\,\ln(s/s_{0})\,<\,0.25/m^{2}_{1}$ in our
parameterization, leads to $Y<25$ . This region, is marked as questionable in
Fig. 13-a. The set of formulae in section 3.6 do not include the contribution
of the ’fan’ diagram to the elastic amplitude. In our context, it is important
that the contribution of such diagrams which determines the high mass
diffraction, turns out to be small. In the following we list a few
experimental signatures, which should be measured at the LHC in the near
future. These will give us a clue regarding the veracity of the models
discussed in the paper:
i) Measurements of $\sigma_{tot}$ and $\sigma_{el}$ should serve as a critical
test for the relevance of Pomeron enhanced diagrams. The difference in
predictions for models including (excluding) Pomeron enhanced diagrams,
becomes even more significant at Cosmic Ray energies. The Auger experiment,
where we expect results for cross sections in the near future, at energies in
the $10^{5}$ GeV range, should allow us to discriminate between the
alternative approaches.
ii) The Pomeron enhanced contribution to the diffractive channels, as
calculated by KMR, is considerably larger than our predictions. This is
significant for $\sigma_{dd}$, which acquires a large value in the KMR
approach.
iii) An early estimate of the value of $S^{2}_{H}$ should be obtained by an
LHC measuremnt of the role of hard central LRG dijet production in a GJJG
configuration, when compared to the pQCD prediction.

To summarize, we developed an approach which is self consistent, and is based
on a perturbative QCD input. We hope that a more microscopic approach with
roots in QCD saturation, can be built and we believe that this paper will
contribute to such an effort.

## Acknowledgements

We are grateful to Omry Netzer and Andrey Kormilitzin for useful discussions
on the subject. We thank Michail Ryskin for an interesting correspondance and
criticism. 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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