# Mechanisms of proton-proton inelastic cross-section growth in multi-
peripheral model within the framework of perturbation theory. Part 3

I.V. Sharf  Odessa National Polytechnic University, Shevchenko av. 1, Odessa,
65044, Ukraine.  G.O. Sokhrannyi  Odessa National Polytechnic University,
Shevchenko av. 1, Odessa, 65044, Ukraine.  A.V. Tykhonov  Odessa National
Polytechnic University, Shevchenko av. 1, Odessa, 65044, Ukraine.  Department
of Experimental Particle Physics, Jozef Stefan Institute,Jamova 39, SI-1000
Ljubljana, Slovenia.  K.V. Yatkin  Odessa National Polytechnic University,
Shevchenko av. 1, Odessa, 65044, Ukraine.  N.A. Podolyan  Odessa National
Polytechnic University, Shevchenko av. 1, Odessa, 65044, Ukraine.  M.A.
Deliyergiyev  Odessa National Polytechnic University, Shevchenko av. 1,
Odessa, 65044, Ukraine.  Department of Experimental Particle Physics, Jozef
Stefan Institute,Jamova 39, SI-1000 Ljubljana, Slovenia.  V.D. Rusov
<EMAIL_ADDRESS>Odessa National Polytechnic University, Shevchenko av. 1,
Odessa, 65044, Ukraine.  Department of Mathematics, Bielefeld University,
Universitatsstrasse 25, 33615 Bielefeld, Germany.

###### Abstract

We develop a new method for taking into account the interference contributions
to proton-proton inelastic cross-section within the framework of the simplest
multi-peripheral model based on the self-interacting scalar ${\phi^{3}}$ field
theory, using Laplace‘s method for calculation of each interference
contribution.

We do not know any works that adopted the interference contributions for
inelastic processes. This is due to the generally adopted assumption that the
main contribution to the integrals expressing the cross section makes multi-
Regge domains with its characteristic strong ordering of secondary particles
by rapidity. However, in this work, w e find what kind of space domains makes
a major contribution to the integral and these space domains are not multi-
Regge. We demonstrated that because these interference contributions are
significant, so they cannot be limited by a small part of them. With the help
of the approximate replacement the sum of a huge number of these contributions
by the integral were calculated partial cross sections for such numbers of
secondary particles for which direct calculation would be impossible.

The offered model qualitative agrees with experimental dependence of total
scattering cross-section on energy $\sqrt{s}$ with a characteristic minimum in
the range $\sqrt{s}\approx 10$ GeV. However, quantitative agreement was not
achieved; we assume that due to the fact that we have examined the simplest
diagrams of $\phi^{3}$ theory.

inelastic scattering cross-section, total scattering cross-section, Laplace
method, virtuality, multi-peripheral model, Regge theory

††preprint: AIP/123-QED

##  I Introduction

This paper is the sequel to [Sharf and Rusov, 2006; Sharf, Rusov  _et al._ ,
2007], where to calculate proton-proton scattering partial cross-sections
within the framework of multi-peripheral model the Laplace method was applied.

The inelastic scattering amplitude with production of a specified multiplicity
of secondary particles, in framework of the multi-peripheral model can be
represented as a sum of diagrams demonstrated on Fig.1.

Figure 1:  Diagram representation of an inelastic scattering amplitude when
the $n$ secondary particles are formed. Here $P_{1}$ and $P_{2}$ are the four-
momenta of primary particles before scattering; $P_{3}$ and $P_{4}$ are the
four-momenta of primary particles after scattering;
${p_{{i_{1}}}},{p_{{i_{2}}}},\cdots,{p_{{i_{n}}}}$ are the four-momenta of
secondary particles. Symbol
$\sum\limits_{\hat{P}({i_{1}},\;{i_{2}},...,\,{i_{n}})}{}$ denote a sum over
all permutations of indices ${i_{1}}=1,{i_{2}}=2,...,{i_{n}}=n$.

To calculate the partial cross-section ${\sigma_{n}}$ is necessary to evaluate
an integral of the squared modulus of a sum of contributions shown in Fig.1.
After simple transformations [Sharf, Rusov  _et al._ , 2007], the expression
for the partial cross-section can be represented as a sum of “cut” diagrams in
Fig.2. We call summands entering into the sum Fig.2 the interference
contributions. Approximate calculation of their sum is the purpose of this
paper.

At present time the inelastic scattering processes are considered without the
interference contributions [Kuraev, Lipatov, and Fadin, 1976; Bartels,
Lipatov, and Vera, 2009]. This due to the generally adopted assumption that
the main contribution to the integrals expressing an inelastic processes makes
multi-Regge domains [Kuraev, Lipatov, and Fadin, 1976; Bartels, Lipatov, and
Vera, 2009; Kozlov, Reznichenko, and Fadin, 2007; Danilov and Lipatov, 2006]
with its characteristic strong ordering of secondary particles by rapidity.
This means that the rapidity of neighboring particles on the “comb” should be
different from each other by a large value. Thus the amplitude of the right-
hand and left-hand parts of the diagram on Fig.2 for different orders of
connecting lines would be significantly different from zero to almost non-
overlapping regions of phase space and integral of their product would be a
small quantity.

However, as it was shown in [Sharf and Rusov, 2006] near the threshold of the
$n$ particles production at the maximum point of the scattering amplitude
Fig.1 difference between neighboring particle‘s of rapidities is close to zero
and at higher energies increases logarithmically with energy $\sqrt{s}$
growth. This difference has factor $1/(n+1)$, so for high numbers of secondary
particles it increases slowly with energy. Moreover, even if each of
interference terms is insignificant, all of them are positive and a huge
amount $n!$ of them not only makes it impossible to discard them, but also
leads to the conclusion that the contribution of a “ladder” diagram Fig.2,
which is usually only taken into account, is negligibly small compared with
the sum of the remaining interference terms. This was shown in [Sharf, Rusov
_et al._ , 2007]. For the relatively small number of secondary particles
($n\leq 8$) we are able to calculate all the interference contributions in the
direct way without any approximations.

Further in this paper we will demonstrate method for approximate calculation
of the sum of the interference contributions for large numbers of secondary
particles, when direct numerical calculation is not feasible.

##  II Method description

Figure 2: Representation of the partial cross-section as a sum of “cut”
diagrams. The order of joining of lines with four-momenta $p_{k}$ from the
left-hand side of the cut is as following: the line with $p_{1}$ is joined to
the first vertex, the lines with $p_{2}$ is joined to the second vertex, etc.
The order of joining of lines from the right side of cut corresponds to one of
the $n!$ possible permutations of the set of numbers $1,2,\ldots,n$. Where
$\hat{P}_{j}(k),k=1,2,\ldots,n$ denote the number into which a number $k$ goes
due to permutation $\hat{P}_{j}$. An integration is performed over the four-
momenta $p_{k}$ for all “cut lines” taking into account the energy-momentum
conservation law and mass shell condition for each of $p_{k}$.

Using the Laplace‘s method we have found [Sharf and Rusov, 2006; Sharf, Rusov
_et al._ , 2007] the mechanism of partial cross-section growth, which was not
taken into account in the previously known variants of multi-peripheral model.
This mechanism may be responsible for the experimentally observed increase of
hadron-hadron total cross-section. However, in this approach based on the
Laplace‘s method, it was found out that the calculation of partial cross-
sections in the multi-peripheral model can be limited just to contributions
from the “cut ladder diagram”. Because for any number of the secondary
particles $n$ there is the wide range of energies $\sqrt{s}$, where such
contribution is negligibly small compared to the sum of $n!$ positive
interference contributions. At the same time, as we will demonstrated further,
the allowance for the interference contributions results in the appearance of
multipliers in expression for the partial cross-section, which are decrease
with the energy $\sqrt{s}$ rise (see below Eq.7). Thereupon the question
arises: “Will the sum of partial cross-sections increase with energy rise if
we take interference summands into account?”

As shown in [Sharf and Rusov, 2006], each term in sum shown in Fig.1 with
accuracy up to the fixed factor is a function with real and positive values,
which has a constrained maximum if its arguments satisfy the mass-shell
conditions and energy-momentum conservation law. Therefore, in the c.m.s. of
initial particles function corresponding to the left-hand part of cut diagram
in Fig.2 can be rewritten in the neighborhood of maximum point in the form
[Sharf and Rusov, 2006; Sharf, Rusov  _et al._ , 2007]

$\displaystyle
A\left({\hat{X}}\right)=A\left({{{\hat{X}}^{\left(0\right)}}}\right)$
$\times\exp\left({-\frac{1}{2}{{\left({\hat{X}-{{\hat{X}}^{\left(0\right)}}}\right)}^{T}}\hat{D}\left({\hat{X}-{{\hat{X}}^{\left(0\right)}}}\right)}\right)$
(1)

where $\hat{X}$ is the column composed of $3n+2$ independent variables, on
which the scattering amplitude depends after consideration of mass-shell
conditions and energy-momentum conservation law; the first $n$ components of
column are the rapidities of secondary particles; the next $n$ components are
the $x$ components of transversal momenta of secondary particles (it is
supposed that the reference system is chosen so that $Z$-axis is directed in
the line of the three-dimensional momentum $P_{1}$ of initial particle in
Fig.1), the $y$ \- components of secondary particle transversal momenta and
the two last variables are the antisymmetric combinations of particle
transversal momenta $P_{3}$ and $P_{4}$, i.e.,
${X_{3n+1}}=\frac{1}{2}\left({{P_{3\bot x}}-{P_{4\bot x}}}\right)$
${X_{3n+2}}=\frac{1}{2}\left({{P_{3\bot y}}-{P_{4\bot y}}}\right)$ We denote
the column of the values of variables in a maximum point through
${\hat{X}^{\left(0\right)}}$ and a matrix with the elements

${D_{ab}}=-{\left.{\frac{{{\partial^{2}}}}{{\partial{X_{a}}\partial{X_{b}}}}\left({\ln\left({A\left({\hat{X}}\right)}\right)}\right)}\right|_{\hat{X}={{\hat{X}}^{\left(0\right)}}}}$
(2)

where

$a=1,2,\cdots,3n+2,b=1,2,\ldots,3n+2$ (3)

are the coefficients of the Taylor series expansion of amplitude logarithm in
the neighborhood of maximum point. As it was shown in [Sharf and Rusov, 2006],
if we do our computations in the c.m.s.of initial particles, the maximum is
reached when transversal momenta is zero and secondary particle rapidities are
close to numbers that formed an arithmetic progression.

If we denote the difference of this progression through $\Delta
y\left({n,\sqrt{s}}\right)$ and the value of particle‘s rapidity to which the
line attached to the $k$-th vertex of diagram in Fig.1 corresponds, through

$\Delta
y\left({n,\sqrt{s}}\right)=y_{k}^{\left(0\right)}-y_{k+1}^{\left(0\right)},\quad
k=1,2,\cdots,n-1$ (4)

we get [Sharf and Rusov, 2006]:

$y_{k}^{\left(0\right)}=\left({\frac{{n+1}}{2}-k}\right)\Delta
y\left({n,\sqrt{s}}\right),\quad k=1,2,\cdots,n$ (5)

The form of the function $\Delta y\left({n,\sqrt{s}}\right)$ has been
discussed in [Sharf and Rusov, 2006]. For further consideration, it is
important that it is a slowly increasing function on $s$ and decreasing
function on the number $n$ of the secondary particles and vanishes when $s$ is
equal to the threshold of $n$ particle production. Thus, the column
${\hat{X}^{\left(0\right)}}$ contains only the first nonzero $n$ rapidity
components, which are defined by Eq.5.

The following expression corresponds to the right-hand part of cut diagram in
Fig.2:

${{\hat{P}}_{j}}\left({A\left({\hat{X}}\right)}\right)=A\left({{{\hat{X}}^{\left(0\right)}}}\right)$
$\displaystyle\times\mbox{$\exp\left({-\frac{1}{2}{{\left({{{\hat{P}}_{j}}\hat{X}-{{\hat{X}}^{\left(0\right)}}}\right)}^{T}}\hat{D}\left({{{\hat{P}}_{j}}\hat{X}-{{\hat{X}}^{\left(0\right)}}}\right)}\right)$
}$ (6)

The interference contribution corresponding to whole “cut” diagram, which
correlates with the $j$-th summand in Fig.2, is proportional to an integral of
the product of functions Eq.1 and Eq.6 over all variables. Denoting an
interference summand corresponding to the permutation ${\hat{P}_{j}}$ through
${\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)$ and calculating its
Gaussian integral (at the same time, other multipliers besides the squared
modulus of scattering amplitude in an integrand are approximately replaced by
their values at the maximum point [Sharf, Rusov  _et al._ , 2007]), we get

$\displaystyle{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)=\frac{{{{\left({A\left({{{\hat{X}}^{\left(0\right)}}}\right)}\right)}^{2}}v\left({\sqrt{s}}\right)}}{{\sqrt{\det\left({\frac{1}{2}\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}}$
$\displaystyle\times\exp\left({-\frac{1}{2}\left({{{\left({\Delta\hat{X}_{j}^{\left(0\right)}}\right)}^{T}}{{\hat{D}}^{\left(j\right)}}\Delta\hat{X}_{j}^{\left(0\right)}}\right)}\right)$
(7)

where we use the following notations:

$\displaystyle\Delta\hat{X}_{j}^{\left(0\right)}={\hat{X}^{0}}-{\hat{P}_{j}}^{-1}\left({{{\hat{X}}^{\left(0\right)}}}\right)$
(8a)
$\displaystyle{\hat{D}^{\left(j\right)}}={\left({{{\hat{D}}^{-1}}+{{\hat{P}}_{j}}^{T}{{\hat{D}}^{-1}}{{\hat{P}}_{j}}}\right)^{-1}}$
(8b)
$v\left({\sqrt{s}}\right)=\frac{1}{2}\frac{1}{{\sqrt{s}\sqrt{s/4-{M^{2}}}\left({E_{P}/2}\right)\sqrt{{{\left({E_{P}/2}\right)}^{2}}-{M^{2}}}}}$
(8c) $\displaystyle{E_{P}}=\sqrt{s}-\sum\limits_{k=1}^{n}{{\mathop{\rm
ch}\nolimits}\left({y_{k}^{\left(0\right)}}\right)}$ (8d)

$M$ is the mass of initial particle, which is made dimensionless by the mass
of secondary particle (it is supposed that the energy $\sqrt{s}$ is also made
dimensionless by the mass of the secondary particle).

Note, that here and in the following sections we will use the “prime” sign in
ours notation to indicate that we use a dimensionless quantity that
characterized the dependence of the cross-sections on energy, but not their
absolute values.

The value of amplitude at the maximum point
$A\left({{{\hat{X}}^{\left(0\right)}}}\right)$ increases with the $\sqrt{s}$
growth due to mechanism of virtuality reduction [Sharf and Rusov, 2006].
However, the distance $\Delta\hat{X}_{j}^{\left(0\right)}$ between maximum
points of “cut” diagram also increases with the $\sqrt{s}$ growth. Therefore,
the exponential factor entering in Eq.7 can decrease with energy growth. This
makes considered above question. How competition of these two multipliers will
result on the dependence of the sum of partial cross-sections on $\sqrt{s}$?

Thus, each interference contribution can be computed numerically. However due
to the huge number of contributions and large number of secondary particles
$n$ the direct numerical calculation of the sum of interference terms in Fig.2
is impossible.

Figure 3:  The interference contributions dependence on ${z_{l}}$ at
$\sqrt{s}=1000$ GeV: (a) $n=8$, (b) $n=9$. Here and in subsequent figures the
interference contributions divided by the common multiplier
$\exp\left({-\sum\limits_{a=1}^{3n+2}{\sum\limits_{b=1}^{3n+2}{X_{a}^{\left(0\right)}{D_{ab}}X_{b}^{\left(0\right)}}}}\right)$
are indicated on the $Y$-axis. Obviously, that to the one value of $z_{l}$
correspond a lot of different contributions, as well as that the average
values of the logarithms of these contributions are placed approximately on
a straight line (see below Eq.26 and Fig.4).

We can avoid this difficulty in the following way. The maximum in the right
part of cut diagram in Fig.2 is attained at
$\hat{X}=\hat{P}_{j}^{-1}\left(\hat{X}^{(0)}\right)$. In other words, a
maximum of function, which is associated with the right-hand part of cut
diagram, can be obtained from a maximum of function, which maps with the left-
hand part of cut diagram, by the rearrangement of arguments. Then the value of
each interference contribution is determined by the distance between points of
maximum in the right-hand and left-hand part of cut diagram as well as by the
relative position of these maximum points, since in different directions
contributions to scattering amplitude fall off with distance from point of
maximum, in general, with different rate, and also by the relative position of
proper directions of the matrices $\hat{D}$ and
$\hat{P}_{j}^{T}\hat{D}\hat{P}_{j}$. In other words, multiplying Gaussian
functions corresponding to the right-hand and to the left-hand part of
interference diagrams in Fig.2 each time we will obtain as a result Gaussian
function, which has the proper value at the maximum point (which we call the
“height” of the maximum) and the proper multidimensional volume cutout by
resulting Gaussian function from an integration domain (which we call the
“width” of the maximum).

Figure 4: Comparing the values of
$\ln\left({\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle}\right)$
obtained by a direct numerical calculation with consideration of all
interference contributions (circles) and by approximation Eq.25 (straight
line) at $n=8$, $\sqrt{s}=10$ GeV (a); $n=9$, $\sqrt{s}=10$ GeV (b); $n=8$,
$\sqrt{s}=100$ GeV (c); $n=9$, $\sqrt{s}=100$ GeV (d).

Figure 5:  The values of $\left\langle{w\left({{z_{l}}}\right)}\right\rangle$
obtained by direct calculation values Eq.25 for all interference contributions
for $n=8$ and $n=9$ at $\sqrt{s}=10$ GeV 5,5 accordingly; for the same number
$n$, but at $\sqrt{s}=100$ GeV 5,5 and the ratio
$\left\langle{w({z_{l}}}\right\rangle/\left\langle{w({z_{l}}_{0}}\right\rangle$
for $n=8$ and $n=9$ at $\sqrt{s}=10$ GeV 5,5; $\sqrt{s}=100$ GeV 5,5.

We assume that summands in Fig.2 are arranged in ascending order of the
distance between the maximum points in the right-hand part and left-hand part
of cut diagram (we denote this distance through $r$) so that ”cut” diagram
with the initial attachment of lines to the right-hand part of diagram
corresponds to $j=1$. In other words, the line of secondary particle with the
four-momentum $p_{i}$ is attached to the $i$-th top in the right-hand part of
cut diagram in Fig.2. As follows from Eq.7, the interference contributions
exponentially decrease with the $r^{2}$ growth. However, in spite of this the
interference contributions do not become negligible due to their huge number,
which, as discussed below, are increases very rapidly with $r^{2}$ growth. The
value of $r^{2}$ is proportional to the square of magnitude $\Delta
y(n,\sqrt{s})$, which, as was noted above, is zero on the threshold of $n$
particle production and slowly increases with distance from this threshold.
Therefore, for each number $n$ there is the fairly wide range of energies
close to the threshold, in which the sharpness of decrease of the interference
contributions with the $r^{2}$ increase is small in the sense that it is less
important factor than the increase in their number. At such energies, which we
call “low”, the partial cross-section $\sigma^{\prime}_{n}$ is determined by
the sum of huge number of small interference contributions. When the magnitude
$\Delta y(n,\sqrt{s})$ is increased with the further growth of energy
$\sqrt{s}$, the decrease rate of interference contributions increases, while
the growth rate of their number with the $r^{2}$ increase does not change with
energy. At such energies, which we call “high”, the main contribution to the
partial cross-section is made by the relatively small number of interference
terms corresponding to the small $r^{2}$, which can be calculated by Eq.7.

If we compose the $n$-dimensional vector (we denote it through
$\vec{y}^{(0)}$) from the particle rapidities Eq.5, which constrainedly
maximizes the function associated with the diagram with the initial
arrangement of momenta in Fig.2, vectors maximizing the functions with another
momentum arrangement will differ from the initial vector only by the
permutation of components, i.e., these vectors have the same length. Consider
two such $n$-dimensional vectors, one of which corresponds to the initial
arrangement, and another - to some permutation, then in the $n$-dimensional
space it is possible to “pull on” a two-dimensional plane on them (as a set of
their various linear combinations), where two-dimensional geometry takes
place. Therefore, the distance $r$ will be determined by cosine of an angle
between the considered equal on length $n$-dimensional rapidity vectors in the
two-dimensional plane, “pulled” on them. An angle corresponding to the
$\hat{P}_{j}$ permutation we designate through $\theta_{j}$,
$0\leq\theta_{j}\leq\pi$.

Thus, each of the terms in the sum Fig.2 can be uniquely matched to its angle
$\theta_{j}$. At the same time the variable $z=cos(\theta)$ is more handy for
consideration than an angle $\theta_{j}$. Using Eq.5, can be shown that the
variable $z$ can take discrete set of values:

$\displaystyle{z_{l}}=1-\frac{{12}}{{\left({n-1}\right)n\left({n+1}\right)}}l\quad$
(9) $\displaystyle
l=0,1,\cdots,\frac{{\left({n-1}\right)n\left({n+1}\right)}}{6}$ (10)

Note that although the relation Eq.5 for the rapidities of secondary particles
is satisfied with high accuracy at the maximum point, it is still approximate.
This means that those contributions, to which matched one and the same value
of variable $z$ in Eq.5, in fact, matched a slightly different from each other
values of $z$.

Consequently, to such contributions correspond a similar but unequal to each
other distances between maximum points in a “cut” diagram. In addition, this
distance, as was discussed above, is not a unique factor affecting to the
value of interference contribution. Therefore, if each interference
contribution is associated with the value of variable $z$ by the approximation
Eq.5, it appears, that the different values of interference contributions
correspond to the one and the same value of $z_{l}$ (see Fig.3).

Thus, while each contribution is associated to some value of variable $z$ in
the approximation Eq.5, the value of contribution is not the unique function
of $z$. However, the sum expressing the partial cross-section
$\sigma^{\prime}_{n}$ can be written in the following way

$\displaystyle{\sigma^{\prime}_{n}}=\sum\limits_{l=0}^{\frac{{\left({n-1}\right)n\left({n+1}\right)}}{6}}{\Delta{N_{l}}\left({\frac{{\sum\limits_{{z_{j}}={z_{l}}}{{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)}}}{{\Delta{N_{l}}}}}\right)}$
(11)

where $\Delta{N_{l}}$ the number of summands to which the value
${z_{j}}={z_{l}}$ is corresponds in the approximation Eq.5. The average value
of all interference contributions in Eq.11 is already the unique function of
$z_{l}$. Therefore, we introduce notation

$\displaystyle\frac{{\sum\limits_{{z_{j}}={z_{l}}}{{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)}}}{{\Delta{N_{l}}}}=\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle$
(12)

where $\left\langle\sigma^{\prime}_{n}(z_{l})\right\rangle$ is some function,
whose form at “low” energies can be determined from the following
considerations.

For any multiplicity $n$ when the values of parameter $l$ in Eq.10 are small
and when number of corresponding interference contributions is relatively
small, we can directly calculate these elements and their sum. Denote the
maximum value $l$, for which all interference contributions are calculated
through $l_{0}$. In particular, in this paper we managed to calculate the
interference contributions up to $l_{0}=6$. Partial cross-section can be
written as

$\displaystyle{\sigma^{\prime}_{n}}=\sigma_{n}^{\prime(h)}+\sigma_{n}^{\prime(l)}=$
$=\sum\limits_{\scriptstyle{z_{j}}={z_{l}},\hfill\atop\scriptstyle
l=0,1,\cdots{l_{0}}\hfill}{{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)}+\sum\limits_{l={l_{0}}+1}^{\frac{{\left({n-1}\right)n\left({n+1}\right)}}{6}}{\Delta{N_{l}}\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle}$
(13)

where $\sigma_{n}^{\prime(h)}$ is the sum of contributions sufficient at
“high” energies, and $\sigma_{n}^{\prime(l)}$ is the sum of contributions
sufficient at “low” energies. Thus, the difficulties in the calculations of
the huge number of interference contributions mainly relates to the range of
“low” energies and can be reduced to the approximate calculation of
$\left\langle\sigma^{\prime}_{n}(z_{l})\right\rangle$ and $\Delta N_{l}$.

##  III The approximate calculation of
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle$.

As follows from Eq.7, the exponential factor exerts the most significant
effect on the dependence of
$\left\langle\sigma^{\prime}_{n}(z_{l})\right\rangle$ on $z_{l}$. Note that
the expression
$\left(\Delta\hat{X}_{j}^{(0)}\right)^{T}\hat{D}^{(j)}\Delta\hat{X}_{j}^{(0)}$
entering into the exponent in Eq.7 depends only on those matrix
$\hat{D}^{(j)}$ components, which are at the intersection of the first $n$
rows and first $n$ columns, since all column $\Delta\hat{X}_{j}^{(0)}$
components starting with $n+1$ are zero, because they are the particle
momentum transverse components at the maximum point. If we denote the matrix
composed of elements located at the intersection of the first $n$ rows and
first $n$ columns of the matrix $\hat{D}^{(j)}$ through $\hat{D}_{y}^{(j)}$
and a matrix, which is obtained from the matrix $\hat{D}$ in analogy, through
$\hat{D}_{y}$, we have

$\displaystyle{\hat{D}^{\left(j\right)}}:\hat{D}_{y}^{\left(j\right)}={\left({\hat{D}_{y}^{-1}+{{\hat{P}}_{j}}^{T}\hat{D}_{y}^{-1}{{\hat{P}}_{j}}}\right)^{-1}}$
(14)

The matrices $\hat{D}_{y}^{-1}$ and
${\hat{P}_{j}}^{T}\hat{D}_{y}^{-1}{\hat{P}_{j}}$ have one and the same
eigenvalues, but they correspond to different eigenvectors. We denote the
normalized to unit eigenvector corresponding to the minimal eigenvalue of
matrix $\hat{D}_{y}^{-1}+{\hat{P}_{j}}^{T}\hat{D}_{y}^{-1}{\hat{P}_{j}}$
through ${\hat{u}_{\min}}$ and the eigenvalue itself - through
${\lambda_{\min}}$. This implies

${\lambda_{\min}}=\hat{u}_{\min}^{T}\hat{D}_{y}^{-1}{{\hat{u}}_{\min}}+\hat{u}_{\min}^{T}{{\hat{P}}_{j}}^{T}\hat{D}_{y}^{-1}{{\hat{P}}_{j}}{{\hat{u}}_{\min}}$
(15)

Since the minimum eigenvalue of matrix $\hat{D}_{y}^{-1}$ is equal to the
minimum values of quadratic form ${\hat{u}^{T}}\hat{D}_{y}^{-1}\hat{u}$ for
the unit vectors $\hat{u}$, the magnitude
$\hat{u}_{\min}^{T}\hat{D}_{y}^{-1}{\hat{u}_{\min}}$ is not less than the
minimum eigenvalue of matrix $\hat{D}_{y}^{-1}$. By analogy the magnitude
$\hat{u}_{\min}^{T}{\hat{P}_{j}}^{T}\hat{D}_{y}^{-1}{\hat{P}_{j}}{\hat{u}_{\min}}$
is not less than the minimum eigenvalue of matrix
${\hat{P}_{j}}^{T}\hat{D}_{y}^{-1}{\hat{P}_{j}}$, which coincides with the
minimal eigenvalue of matrix $\hat{D}_{y}^{-1}$ and is reciprocal of the
maximum eigenvalue of matrix ${\hat{D}_{y}}$ denoted through $d_{y}^{\max}$.
Thus, ${\lambda_{\min}}\geq\frac{2}{{d_{y}^{\max}}}$. From this it follows
that, the maximum eigenvalue of matrix
${\left({\hat{D}_{y}^{-1}+{{\hat{P}}_{j}}^{T}\hat{D}_{y}^{-1}{{\hat{P}}_{j}}}\right)^{-1}}$
does not exceed $d_{y}^{\max}/2$. By analogy we obtain that the minimum
eigenvalue of matrix
${\left({\hat{D}_{y}^{-1}+{{\hat{P}}_{j}}^{T}\hat{D}_{y}^{-1}{{\hat{P}}_{j}}}\right)^{-1}}$
is no smaller than $d_{y}^{\min}/2$, where $d_{y}^{\min}$ is the minimum
eigenvalue of matrix ${\hat{D}_{y}}$. Thus, an interval enclosing the
eigenvalues of matrix $\hat{D}_{y}^{\left(j\right)}$ is, at least, twice
smaller than an interval enclosing the eigenvalues of matrix ${\hat{D}_{y}}$.
We can demonstrate that at approximation of an equal denominators [Sharf and
Rusov, 2006] the value of $d_{y}^{\max}$ can be estimated in the following way

$\displaystyle d_{y}^{\max}\approx\frac{2}{{4{{{\mathop{\rm
sh}\nolimits}}^{2}}\left({\frac{{\Delta y\left({n,s}\right)}}{2}}\right)+1}}$
(16)

i.e., an interval enclosing the eigenvalues of matrix
$\hat{D}_{y}^{\left(j\right)}$ at any energies and number of particles is less
than unity, whereas at the considerable values of $\Delta
y\left({n,s}\right)$, i.e. at a distance from the threshold, this interval is
much less than unity.

Therefore, if we reduce matrix $\hat{D}_{y}^{\left(j\right)}$ to diagonal
form, it will be close to a matrix multiple of unit matrix. If we represent
this matrix in the form

$\displaystyle\hat{D}_{y}^{\left(j\right)}=\frac{1}{n}Sp\left({\hat{D}_{y}^{\left(j\right)}}\right)\hat{E}+\Delta\hat{D}_{y}^{\left(j\right)}$
(17)

where $\hat{E}$ is unit matrix, the eigenvalues of the traceless matrix
$\Delta\hat{D}_{y}^{\left(j\right)}$ will be small. Then

$\frac{1}{2}{\left({\Delta\hat{X}_{j}^{\left(0\right)}}\right)^{T}}{{\hat{D}}^{\left(j\right)}}\Delta\hat{X}_{j}^{\left(0\right)}=\frac{1}{n}Sp\left({D_{y}^{\left(j\right)}}\right){\left|{{{\vec{y}}^{\left(0\right)}}}\right|^{2}}\left({1-\cos\left({{\theta_{j}}}\right)}\right)+\frac{1}{2}\sum\limits_{k=1}^{n}{\Delta
d_{y,k}^{\left(j\right)}}{\left({{V_{kn}}\left({y_{n}^{\left(0\right)}-{{\hat{P}}_{j}}^{-1}\left({y_{n}^{\left(0\right)}}\right)}\right)}\right)^{2}}$
(18)

where $\Delta d_{y,k}^{\left(j\right)}$ are the eigenvalues of matrix
$\Delta\hat{D}_{y}^{\left(j\right)}$, ${V_{kn}}$ is the transformation matrix
to the basis composed from the eigenvectors of matrix
$\Delta\hat{D}_{y}^{\left(j\right)}$ (the summation over reheated indices is
supposed). The second term in this sum is small in comparison with the first
one due to the smallness of eigenvalues $\Delta d_{y,k}^{\left(j\right)}$ as
well as due to their different signs (since the trace of matrix
$\Delta\hat{D}_{y}^{\left(j\right)}$ is zero, the different terms over $k$
partially compensate each other). Therefore, we can adopt the following
approximation:

$\frac{1}{2}{\left({\Delta\hat{X}_{j}^{\left(0\right)}}\right)^{T}}{\hat{D}^{\left(j\right)}}\Delta\hat{X}_{j}^{\left(0\right)}\approx$
$\approx\frac{1}{n}Sp\left({\hat{D}_{y}^{\left(j\right)}}\right)\times{\left|{{{\vec{y}}^{\left(0\right)}}}\right|^{2}}\left({1-\cos\left({{\theta_{j}}}\right)}\right)$
(19)

To approximately calculate the trace of matrix $\hat{D}_{y}^{\left(j\right)}$
we select the spherically symmetric part of matrix ${\hat{D}_{y}}$
representing it in the form

$\displaystyle{\hat{D}_{y}}=\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)\hat{E}+\Delta{\hat{D}_{y}}$
(20)

The results of numeral calculation of the eigenvalues of matrix
${\hat{D}_{y}}$ (which are denoted through
$d_{k}^{\left(y\right)},k=1,2,\cdots,n$) are shown in Table.1. It is obvious
that most eigenvalues are close between themselves with the exception of a few
eigenvalues, which are substantially smaller. Therefore, these smallest
eigenvalues have the highest absolute value of deviations from mean eigenvalue
$\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)$. Since all the eigenvalues of
matrix ${\hat{D}_{y}}$ are positive, the deviation of eigenvalues from average
value is less than this average in absolute value (see Table.1). Note that the
matrix $\hat{D}_{y}^{\left(j\right)}$ can be represented in the following
form:

$\hat{D}_{y}^{\left(j\right)}=\frac{1}{{2n}}Sp\left({{{\hat{D}}_{y}}}\right)\left({\hat{E}+\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}}\right){\left({\hat{E}+\frac{{\Delta{{\hat{D}}_{y}}+{{\hat{P}}_{j}}^{T}\Delta{{\hat{D}}_{y}}{{\hat{P}}_{j}}}}{{\frac{2}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}}\right)^{-1}}\left({\hat{E}+\frac{{{{\hat{P}}_{j}}^{T}\Delta{{\hat{D}}_{y}}{{\hat{P}}_{j}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}}\right)$
(21)

By analogy we can conclude that the minimum eigenvalue of matrix

$\displaystyle\Delta{\hat{D}_{y}}+{\hat{P}_{j}}^{T}\Delta{\hat{D}_{y}}{\hat{P}_{j}}$
(22)

(which is maximum in absolute value, see Table.1) is greater than the doubled
minimum eigenvalue of matrix $\Delta{\hat{D}_{y}}$. This means that all the
eigenvalues of matrix
$\frac{{\Delta{{\hat{D}}_{y}}+{{\hat{P}}_{j}}^{T}\Delta{{\hat{D}}_{y}}{{\hat{P}}_{j}}}}{{\frac{2}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$
are less than unity in absolute value. It applies equally to the eigenvalues
of matrices
$\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$
and
${\hat{P}_{j}}^{T}\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}{\hat{P}_{j}}$.
Therefore, we can represent the matrix $\hat{D}_{y}^{\left(j\right)}$ as the
expansion in powers of
$\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$.
Since matrix
$\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$
is traceless by definition, then a nonzero contribution to
$Sp\left({\hat{D}_{y}^{\left(j\right)}}\right)$ in addition to the term of
“zero” order $\frac{1}{{2n}}Sp\left({{{\hat{D}}_{y}}}\right)\hat{E}$ can give
terms starting with the second-order. As it follows from Table.1, the maximum
in absolute value eigenvalue of matrix
$\frac{{\Delta{{\hat{D}}_{y}}}}{{\frac{1}{n}Sp\left({{{\hat{D}}_{y}}}\right)}}$
increases with the energy growth. Therefore, we can expect that at “low”
energies higher-order terms will make negligibly small contributions. In such
an approximation we have:

$\displaystyle
Sp\left({\hat{D}_{y}^{\left(j\right)}}\right)\approx\frac{1}{2}Sp\left({{{\hat{D}}_{y}}}\right)$
(23)

Let Eq.7 is taken in place of Eq.12 in approximation Eq.23, then we have

$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle={\left({A\left({{{\hat{X}}^{\left(0\right)}}}\right)}\right)^{2}}v\left({\sqrt{s}}\right)$
$\times\exp\left({-\frac{{{{\left|{{{\vec{y}}^{\left(0\right)}}}\right|}^{2}}Sp\left({{{\hat{D}}_{y}}}\right)}}{{2n}}\left({1-{z_{l}}}\right)}\right)$
$\times\frac{1}{{\Delta{N_{l}}}}\sum\limits_{{z_{j}}={z_{l}}}{\frac{1}{{\sqrt{\det\left({\frac{1}{2}\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}}}$
(24)

Let us introduce the following notation

$\left\langle{w\left({{z_{l}}}\right)}\right\rangle=\frac{1}{{\Delta{N_{l}}}}\sum\limits_{{z_{j}}={z_{l}}}{\frac{1}{{\sqrt{\det\left({\frac{1}{2}\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}}}$
(25)

If we assume that multiplier
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ is weakly dependent on
${z_{l}}$, we obtain

$\displaystyle\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle=\left\langle{{\sigma^{\prime}_{n}}\left({{z_{{l_{0}}}}}\right)}\right\rangle\exp\left({\frac{{{{\left|{{{\vec{y}}^{\left(0\right)}}}\right|}^{2}}Sp\left({{{\hat{D}}_{y}}}\right)}}{{2n}}\left({{z_{l}}-{z_{{l_{0}}}}}\right)}\right)$
(26)

where ${z_{l}}$ is the minimum value of ${z_{l}}$ for which can be numerically
calculated all interference contributions. Therefore, the magnitude
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{{l_{0}}}}}\right)}\right\rangle$
can be directly calculated numerically. The results of numerical calculation
of
$\ln\left({\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle}\right)$
over all interference contributions in comparison with the results obtained by
Eq.25 are demonstrated on Fig.4, it follows that such an approximation is
acceptable at “low” energies.

Results shown in Fig.4 confirm also our assumption that
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ weakly depends on
${z_{l}}$. To analyze this dependence we turn to Fig.5. It is obvious, that
the magnitude $\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ takes small
values at “low” energies. This means that

$\displaystyle\det\left({\frac{1}{2}\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)$
(27)

takes large values at the same energies. Indeed, as it follows from the
expression for the matrix $\hat{D}$, Eq.27 tends to infinity on the threshold
of $n$ particle production, and this means that at threshold the phase space
of physical area of the inelastic process with $n$ particles production takes
place is equal to zero.

Because of symmetry with respect to direction inversion in a plane of
transversal momenta the mixed second derivatives with respect to rapidities
and transversal momentum components are zeros. As a consequence, the
determinant Eq.27 is equal to the product of the three determinants, first of
which is composed from second derivatives with respect to rapidities, the
second is composed from the second derivatives with respect to the transversal
momentum $x$-components and the third one is composed from derivatives with
respect to the transversal momentum $y$-components. All the three factors tend
to infinity at the threshold energy. As it follows from a numerical
calculation, a matrix determinant composed from the second derivatives with
respect to rapidities reduced quite rapidly with energy growth. Matrix
determinants composed from the second derivatives with respect to transversal
momentum components also reduced, but in a wide energy range, they remain
quite large. Therefore, the value of Eq.27 is great at all $j$. Since the
function
${1\mathord{\left/{\vphantom{1{\sqrt{x}}}}\right.\kern-1.2pt}{\sqrt{x}}}$
varies slightly at the great values of argument, the function
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ weakly depends on
${z_{l}}$.

To estimate roughly the function
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ we can replace it by the
Taylor expansion taking into account just linear contributions. The expansion
coefficients are found by the calculating of
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ for ${z_{l}}$ close to
$1$ and $(-1)$. In these cases the values of

$\displaystyle{1/\sqrt{\det\left({1/2\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}$
(28)

were obtained directly for all proper interference contributions, and after
that we obtain the values of
$\left\langle{w\left({{z_{l}}}\right)}\right\rangle$ by averaging using Eq.25.

Figure 6: A sphere ${S_{2}}$ and figure ${F_{4!}}$ (is shown by points). Basis
in the four-dimensional space is chosen so that the one of vectors coincides
with the vector
${\vec{e}_{4}}=\left({\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}}\right)$,
and the three basis vectors of three-dimensional subspace, into which depicted
sphere is embedded, are perpendicular to ${\vec{e}_{4}}$.

Figure 7: The partition of sphere ${S_{2}}$ by shortest arcs joining the
points of figure ${F_{4!}}$ into the two “hexagonal” and one “tetragonal”
regions 7; (b) areas, which is located on the borders of 4 or 6 points
belonging to figure ${F_{4!}}$ can be divided between those points into
figures of equal area; (c) whole sphere $S_{2}$ is divided into figures of
equal area, each of which contains the one point of figure ${F_{4!}}$ one of
these shapes are painted in white.

The values in Fig.5 have been obtained by the direct calculation of

$\frac{1}{{\Delta{N_{l}}}}\sum\limits_{{z_{j}}={z_{l}}}{1/\sqrt{\det\left({1/2\left({\hat{D}+\hat{P}_{j}^{T}\hat{D}{{\hat{P}}_{j}}}\right)}\right)}}$
(29)

with consideration of all interference contributions at different $\sqrt{s}$.

So, we have the following expression instead of Eq.26

$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle=\left\langle{{\sigma^{\prime}_{n}}\left({{z_{{l_{0}}}}}\right)}\right\rangle\left({{w_{0}}+{w_{1}}\left({1-{z_{l}}}\right)}\right)\times\exp\left({\frac{{{{\left|{{{\vec{y}}^{\left(0\right)}}}\right|}^{2}}Sp\left({{{\hat{D}}_{y}}}\right)}}{{2n}}\left({{z_{l}}-{z_{{l_{0}}}}}\right)}\right)$
(30)

where the coefficients ${w_{0}}$ and ${w_{1}}$ are found by above mentioned
method.

##  IV Approximate calculation of the $\Delta{N_{l}}$ values

Let us turn to the new variables

$\displaystyle Y_{k}^{\left(0\right)}=\frac{{y_{k}^{\left(0\right)}}}{{\Delta
y\left({n,\sqrt{s}}\right)\sqrt{\frac{{\left({n+1}\right)n\left({n-1}\right)}}{{12}}}}}$
(31)

where $y_{k}^{\left(0\right)}$ are determined by Eq.5,
$Y_{k}^{\left(0\right)},k=1,2,\cdots,n$ are considered as the components of
vector ${\vec{Y}^{\left(0\right)}}$, which, as it follows from Eq.31 is of
unit length.

Thus, the angle ${\theta_{j}}$ between the vector
${\vec{y}^{\left(0\right)}}=\left({y_{1}^{\left(0\right)},y_{2}^{\left(0\right)},\cdots,y_{n}^{\left(0\right)}}\right)$
and vector $\hat{P}_{j}^{-1}\left({{{\vec{y}}^{\left(0\right)}}}\right)$
obtained by the permutation of corresponding components is the same as the
angle between the vector
${\vec{Y}^{\left(0\right)}}=\left({Y_{1}^{\left(0\right)},Y_{2}^{\left(0\right)},\cdots,Y_{n}^{\left(0\right)}}\right)$
and vector $\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$.
Moreover, as it follows from Eq.5

$\displaystyle
y_{1}^{\left(0\right)}=-y_{n}^{\left(0\right)},y_{2}^{\left(0\right)}=-y_{n-1}^{\left(0\right)},\cdots,y_{k}^{\left(0\right)}=-y_{n-k+1}^{\left(0\right)};$
$\displaystyle k=1,2,\cdots,n$ (32)

It follows that all vectors $\hat{P}_{j}^{-1}\left(\vec{Y}^{(0)}\right)$ are
orthogonal to vector

$\displaystyle\vec{e}_{n}=\left(\underbrace{1/\sqrt{n},1/\sqrt{n},\ldots,1/\sqrt{n}}_{n\quad\scriptsize{components}}\right)$
(33)

Figure 8: Diagrams, which correspond $(n-1)$ vectors
$\hat{P}_{l}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$ closest to
vector ${\vec{Y}^{\left(0\right)}}$.

Figure 9: Comparison of the interference contribution distribution by the
variable $z=\cos\left(\Theta\right)$ (histogram) and the plot of function
$\rho\left(z\right)=\frac{{dN\left({z,z+dz}\right)}}{{dz}}$ (solid line) at
$n=8$, $\Delta z=0.1$ (a); $n=9$, $\Delta z=0.1$ (b); $n=9$, ${\Delta z}=0.05$
(c). Here $\Delta N$ is the number of interference contributions corresponding
to value of $z$ in the proper interval of $\Delta z$ width.

Therefore, considering vectors $\hat{P}_{j}^{-1}\left(\vec{Y}^{(0)}\right)$ as
the elements of $n$-dimensional euclidean space, which we denote through
$E_{n}$, then the ends of all vectors
$\hat{P}_{j}^{-1}\left(\vec{Y}^{(0)}\right)$ are lie on the unit sphere
embedded into the $(n-1)$-dimensional subspace of $E_{n}$. We denote this
sphere through $S_{n-2}$ and shape formed by the set of points in which the
ends of vectors $\hat{P}_{j}^{-1}\left(\vec{Y}^{(0)}\right)$
($j=1,2,\ldots,n!$) come, denote through $F_{n!}$. In particular, when $n=4$
the sphere $S_{2}$ and figure $F_{4!}$ graphically look like in Fig.6.

Figure 10:  Comparison of the values of right-hand side and left-hand side of
approximate equality Eq.42 at $n=8$ (a, b) and $n=9$ (c, d). Circles are the
values of $\Delta{N_{l}}$ calculated with consideration of for all
interference contributions; crosses are the values of function
$\rho\left({{z_{l-1}}}\right)\Delta z$ from Eq.40.

We examine some geometrical properties of figure ${F_{n!}}$ at arbitrary $n$.
If we apply the permutation transformation component to all vectors in the
$n$-dimensional space, where the vectors ${\vec{Y}^{\left(0\right)}}$ are
primordially defined, the examined $(n-1)$-dimensional subspace as well as a
sphere ${S_{n-2}}$ and figure ${F_{n!}}$ go into themselves. As it follows
from the group properties of permutation group, the each point of figure
${F_{n!}}$ can be obtained from any other point by some transformation
$\hat{P}_{j}^{-1}$. This means that the configuration of the points of figure
${F_{n!}}$ relative to each of these points must be identical, that can be
clearly seen in Fig.7.

As it follows from Eq.32, besides the end of each vector
$\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$ a figure
${F_{n!}}$ contains also the end of vector
$\left({-\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)}\right)$,
i.e., a figure ${F_{n!}}$ has a center of symmetry, which coincides with the
center of sphere ${S_{n-2}}$. In this case, if we using point of ${F_{n!}}$
form path from the point
$\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$ to the point
$\left({-\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)}\right)$,
then it will be simultaneously formed a centro-symmetrical path, that leads
from
$\left({-\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)}\right)$
to $\hat{P}_{j}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)$ of figure
${F_{n!}}$.

Joining these paths we will obtain the closed path, which “girdles” the sphere
${S_{n-2}}$. If we assume that there is such a “girdling” path, inside of
which are concentrated all points of figure ${F_{n!}}$, we would find that the
figure ${F_{n!}}$ has a “boundary” and “internal” points, that would
contradict the fact that spacing of all points relative to each point of the
${F_{n!}}$ should be the same. In other words, the points of figure ${F_{n!}}$
must “crawl away” all over the sphere ${S_{n-2}}$ and can not be concentrated
on some area of the sphere.

If we consider a vector ${\vec{Y}^{\left(0\right)}}$, then closest to it are
the vectors corresponding to permutations $\hat{P}_{l}^{-1},l=1,2,\cdots,n-1$
defined by the following relation

${\left({\hat{P}_{l}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)}\right)_{k}}=\left\\{\begin{array}[]{l}Y_{k}^{\left(0\right)},{\rm{if}}\quad
k<l,\\\ Y_{l+1}^{\left(0\right)},{\rm{if}}\quad k=l,\\\
Y_{l}^{\left(0\right)},{\rm{if}}\quad k=l+1,\\\
Y_{k}^{\left(0\right)},{\rm{if}}\quad k>l+1.\\\ \end{array}\right.$ (38)

The type of “cut” diagrams corresponding to such permutations is shown on
Fig.8.

At the same time, all the components of vector
$\hat{P}_{l}^{-1}\left({{{\vec{Y}}^{\left(0\right)}}}\right)-{\vec{Y}^{\left(0\right)}}$
, except the $l$-th and $l+1$, are zero, whereas these two components take on
the least values in modulus
$\sqrt{\frac{{12}}{{\left({n+1}\right)n\left({n-1}\right)}}}$ and
$\left({-\sqrt{\frac{{12}}{{\left({n+1}\right)n\left({n-1}\right)}}}}\right)$,
respectively.

Thus, we can conclude that the each point of figure ${F_{n!}}$ has $(n-1)$
nearest neighboring points, which lying at distance of from it:

$\displaystyle{r_{n}}=\sqrt{\frac{{24}}{{\left({n+1}\right)n\left({n-1}\right)}}}$
(39)

Connecting the each point of figure ${F_{n!}}$ with its (n-1) nearest
neighbors points by shortest arc thereby we divide the sphere ${S_{n-2}}$ into
closed regions as is shown in Fig.7. Indeed, let us choose the some point
${A_{0}}$ of figure ${F_{n!}}$, and will move from it to the nearest point
$A_{1}$ along a shortest arc, then we move from the point $A_{1}$ to the
nearest point $A_{2}$ etc. At the same time, motion in a backward direction is
prohibited. Thus, there are $(n-1)$ paths going out from each point, and
$(n-2)$ paths are allowed at each step. But since figure ${F_{n!}}$ has the
finite number of points at some step we will surely come back to the point
${A_{0}}$.

Moreover, since shortest arcs joining two nearest points are subtended by
equal chords ${r_{n}}$ in length (see Eq.39), this arcs are of the same
length. Let us consider any two neighboring points ${A_{i}}$ and ${A_{i+1}}$
of figure ${F_{n!}}$. Under any transformation $\hat{P}_{j}^{-1}$ the shortest
arc, which joins the points ${A_{i}}$ and ${A_{i+1}}$, and an arc joining the
points $\hat{P}_{j}^{-1}\left({{A_{i}}}\right)$ and
$\hat{P}_{j}^{-1}\left({{A_{i+1}}}\right)$ are of the same length. This means
that the boundaries of closed regions formed by shortest arcs, which join
neighboring points, replaced into one another under any transformation
$\hat{P}_{j}^{-1}$. It follows that, if we examine closed areas which include
any point of figure ${F_{n!}}$, then the adjacent areas to all points of this
figure will have the same “area”.

Figure 11:  Comparison of the values of
$\sum\limits_{{z_{j}}={z_{l}}}{{\sigma_{n}}\left({{{\hat{P}}_{j}}}\right)}$
obtained with consideration of all interference contributions (circles) and
the approximate values of
$\left\langle{{\sigma_{n}}\left({{z_{l}}}\right)}\right\rangle\rho\left({{z_{l-1}}}\right)\Delta
z$ (crosses) for 11 \- for $n=8$ at $\sqrt{s}=10$ GeV, 11 \- for $n=8$ at
$\sqrt{s}=100$ GeV, 11 \- for $n=9$ at $\sqrt{s}=10$ GeV, 11 \- for $n=9$ at
$\sqrt{s}=100$ GeV.

Figure 12:  The partial cross-section dependence on energy $\sqrt{s}$
calculated over all interference contributions (solid line) and by Eq.13 with
the application of approximations Eqs.30, 40, 42 (dashed line): 12 \-
$\sigma^{\prime}_{8}(\sqrt{s})$; 12 \- $\sigma^{\prime}_{9}(\sqrt{s})$; 12 \-
$\sigma^{\prime}_{10}(\sqrt{s})$; 12 \- $\sigma^{\prime}_{11}(\sqrt{s})$; 12
\- $\sigma^{\prime}_{11}(\sqrt{s})$. This approximation is acceptable at least
in the range of parameters in which they are can be verified.

Figure 13:  Theoretical dependences of the $\sigma^{\prime I}(\sqrt{s})$ 13
and $\sigma^{\prime\Sigma}(\sqrt{s})$ 13 obtained for the energy range
$\sqrt{s}=1\div 100$ Gev at $L=5.51$. First minimum for the total cross-
section can be obtained only when we take into account contributions from the
high multiplicities. Experimental data for the inelastic 13 and for the total
13 pp scattering cross-section [Nakamura and Group, 2010; Aad  _et al._ ,
2011] presented for qualitative comparison with the prediction from our model.
Note: data-points for the inelastic cross-section, obtained from the
definition $\sigma_{inel}=\sigma_{total}-\sigma_{elastic}$.

There is one more requirement, to which the areas obtained by partition of the
sphere ${S_{n-2}}$ must satisfy: they must not overlap, i.e., these regions do
not have common internal points. Indeed, otherwise, at least any two of the
examined arcs would intersect in some internal point of these arcs. As it
follows from Eq.39, when $n$ is large the value of ${r_{n}}$ is small. This
means that when we join the each point of figure ${F_{n!}}$ with its nearest
neighbors by the shortest arcs of sphere ${S_{n-2}}$, these arcs practically
coincide with chords, which tights them.

If we assume, that any two chords ${A_{{i_{1}}}}{A_{{i_{1}}+1}}$ and
${A_{{i_{2}}}}{A_{{i_{2}}+1}}$ intersect in an internal point, then it is
possible “to pull” on them a two-dimensional plane. Then we get a flat
rectangle ${A_{{i_{1}}}}{A_{{i_{2}}}}{A_{{i_{1}}+1}}{A_{{i_{2}}+1}}$, which
has at least one angle no smaller than $90^{\circ}$. This means that square of
diagonal lying opposite it is not less than sum of squares of the parties that
make up the corner. Denoting the lengths of these sides through $a$ and $b$,
we have ${a^{2}}+{b^{2}}\leq r_{n}^{2}$. In this case, either $a$ or $b$ would
not exceed ${r_{n}}/\sqrt{2}$, i.e., the figure ${F_{n!}}$ contains points,
which are at distance less then ${r_{n}}$ but that cannot happen due to
minimality of this distance.

Thus, we can conclude that at an arbitrary $n$ a sphere ${S_{n-2}}$ can be
divided into the parts of equal area, each of which contains only one point of
figure ${F_{n!}}$, as it shown in Fig.7-7.

Let us introduce a multidimensional spherical coordinate system so that the
end of vector ${\vec{Y}^{\left(0\right)}}$ is the “north pole” of sphere
${S_{n-2}}$. Then the number of points of figure ${F_{n!}}$, to which the
values of variable $z=\cos\left(\Theta\right)$ in the interval $[z,z+dz]$
correspond, is equal

$\displaystyle dN\left({z,dz}\right)=\rho\left(z\right)dz$ (40)

where

$\displaystyle\rho\left(z\right)=\frac{{n!}}{{\sqrt{\pi}}}\frac{{\Gamma\left({\frac{{n-1}}{2}}\right)}}{{\Gamma\left({\frac{{n-2}}{2}}\right)}}{\left({1-{z^{2}}}\right)^{\frac{{n-4}}{2}}}$
(41)

$\Gamma$ is the Euler gamma function.

To verify the validity of Eq.40 we can calculate all interference
contributions and corresponding values of $z$ at $n=8$ and $n=9$ (since for
the larger number of particles this can not be realized).

The distributions of interference contribution from the variable
$z=\cos\left(\Theta\right)$ and the graphs of function
$\rho\left(z\right)=\frac{{dN\left({z,z+dz}\right)}}{{dz}}$ from Eq.40 are
shown in Fig.9. Obtained results of numerical calculation of interference
contributions and by Eq.40 are in a good agreement.

Moreover, as it follows from Fig.9 and from Fig.9 this fitness is improved
with increasing number of particles $n$, i.e., Eq.40 is suitable for large
$n$, when the direct numerical calculation of all interference contributions
is impossible.

Taking Eq,40 and Eq.10 into account we obtain the following the approximate
equality

$\displaystyle\Delta{N_{l}}\approx\rho\left({{z_{l-1}}}\right)\Delta z$ (42)

where

$\displaystyle\Delta z=\frac{{12}}{{\left({n-1}\right)n\left({n+1}\right)}}$
(43)

Verification results of Eq.42 at $n=8$ and $n=9$ are presented in Fig.10.

Another verification of considered above equations is presented in Fig.11,
where the values of
$\sum\limits_{{z_{j}}={z_{l}}}{{\sigma^{\prime}_{n}}\left({{{\hat{P}}_{j}}}\right)}$
and approximating magnitudes
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle\rho\left({{z_{l-1}}}\right)\Delta
z$ (here
$\left\langle{{\sigma^{\prime}_{n}}\left({{z_{l}}}\right)}\right\rangle$ is
calculated by Eq.30) are compared.

From results demonstrated on Fig.4 and Fig.10-12, we can conclude that the at
least for those numbers of particles for which it can be directly tested Eq.13
with Eq.30, Eqs.40 \- 42 yields an acceptable approximation. As is obvious
from Fig.4, than closer energy to the threshold of $n$ particle production,
the better approximation Eq.30. Therefore, if we choose the range of low
energies, for example, up to 100 GeV, because in this range total cross-
section growth is observed, it is expected that the considered approximations
will be acceptable for the large numbers of particles than those for which
they were tested. In addition, as it follows from Fig.10-10, the accuracy of
approximation Eq.42, as expected, increases with the growth of $n$.

Thus, within the framework of examined approximations is possible to calculate
the interference contributions at sufficiently large $n$, and we can consider
the dependence of total inelastic cross-section on energy $\sqrt{s}$ in the
simplest case of multi-peripheral model taking into account all significant
interference contributions.

##  V The model of dependence of hadron inelastic scattering total cross-
section on energy $\sqrt{s}$

Let us consider the magnitude

$\displaystyle{\sigma^{\prime\Sigma}}\left({\sqrt{s}}\right)=\sum\limits_{n=1}^{{n_{\max}}}{{L^{n}}{\sigma^{\prime}_{n}}\left({\sqrt{s}}\right)}$
(44)

which within the framework of the discussed above model is an analogue of
total inelastic scattering cross-section. Here ${n_{\max}}$ is the maximum
number of secondary particles allowed by energy-momentum conservation law and
$L$ is the dimensionless coupling constant, which we considered as a fitting
parameter (see Eq.32 [Sharf, Rusov  _et al._ , 2007]). Since the calculation
of ${\sigma^{\prime}_{n}}$ up to $n={n_{\max}}$ takes a long time, so in
practice we restrict the upper bound of summation by those values of $n$,
beyond which the neglected contributions known to be smaller than the
experimental error of cross-section measurements.

The constant $L$ can be fitted so that the dependence
${\sigma^{\prime}_{\Sigma}}\left({\sqrt{s}}\right)$ looks like the behavior of
total hadron-hadron scattering cross-section with a minimum about
$\sqrt{s}=10$ GeV. The result of such a fitting is shown in Fig.13 (in that
calculations we take proton mass as mass of primary particles and pion mass as
mass of secondary particles).

Quantitative comparison with experimental data requires the consideration of
more realistic model than the self-interacting scalar ${\phi^{3}}$ field
model.

##  VI Conclusions

From obtained result, one might conclude that the considered in [Sharf and
Rusov, 2006] mechanism of virtuality reduction at the constrained maximum
point of multi-peripheral scattering amplitude may be responsible for proton-
proton total cross-section growth when all the considerable interference
contributions are taken into account.

Just the revelation of mechanism of cross-section growth we consider as the
main result of earlier papers [Sharf and Rusov, 2006; Sharf, Rusov  _et al._ ,
2007] and present work, since this mechanism is intrinsic not only to the
diagrams of the “comb” type, but also to different modifications of considered
model.

Application the Laplace method allow to calculate another types of diagrams
corresponding to various scenarios of hadron-hadron inelastic scattering and
compare it with experimental data.

REFERENCES

## References

  * Sharf and Rusov (2006) I. Sharf and V. Rusov, “Mechanism of hadron inelastic scattering cross-section growth in the multiperipheral model within the framework of perturbation theory. part 1,” (2006), arXiv:0605110 [hep-ph] .
  * Sharf, Rusov  _et al._ (2007) I. Sharf, V. Rusov, _et al._ , “Mechanism of hadron inelastic scattering cross-section growth in the multiperipheral model within the framework of perturbation theory. part 2,” (2007), arXiv:0711.3690 [hep-ph] .
  * Kuraev, Lipatov, and Fadin (1976) E. Kuraev, L. Lipatov, and V. Fadin, “Multi reggeon processes in the yang-mills theory,” Sov. Phys. JETP. 44, 443–450 (1976).
  * Bartels, Lipatov, and Vera (2009) J. Bartels, L. N. Lipatov, and A. S. Vera, “Bfkl pomeron, reggeized gluons, and bern-dixon-smirnov amplitudes,” Phys. Rev. D 80, 045002 (2009), arXiv:0802.2065 [hep-th] .
  * Kozlov, Reznichenko, and Fadin (2007) M. G. Kozlov, A. V. Reznichenko, and V. S. Fadin, “Quantum chromodynamics at high energies,” Vestnik NSU 2, 3–31 (2007).
  * Danilov and Lipatov (2006) G. S. Danilov and L. N. Lipatov, “BFKL pomeron in string models,” Nucl. Phys. B754, 187–232 (2006), arXiv:hep-ph/0603073 .
  * Nakamura and Group (2010) K. Nakamura and P. D. Group, “Review of particle physics,” Journal of Physics G: Nuclear and Particle Physics 37, 075021 (2010).
  * Aad  _et al._ (2011) G. Aad _et al._ (ATLAS Collaboration), “Measurement of the Inelastic Proton-Proton Cross-Section at $\sqrt{s}=7$ TeV with the ATLAS Detector,” (2011), * Temporary entry *, arXiv:1104.0326 [hep-ex] .

Table 1:  Results of numerical calculations of the eigenvalues of matrix
${\hat{D}_{y}}$. $n=20$
---
$\sqrt{s}=10$ GeV |  $\sqrt{s}=300$ GeV |  $\sqrt{s}=10$ TeV
$d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$ | $d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$ | $d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$
1.317 | -0.417 | 0.181 | -0.864 | 0.064 | -0.928
3.078 | 0.352 | 0.551 | -0.586 | 0.227 | -0.746
3.006 | 0.321 | 0.878 | -0.34 | 0.421 | -0.527
1.883 | -0.173 | 1.099 | -0.174 | 0.604 | -0.321
2.53 | 0.111 | 1.238 | 0.342 | 0.745 | -0.163
2.527 | 0.11 | 1.785 | 0.342 | 0.849 | -0.047
2.401 | 0.055 | 1.785 | -0.07 | 1.26 | 0.415
2.399 | 0.054 | 1.324 | -0.005 | 1.26 | 0.415
2.061 | -0.094 | 1.38 | 0.037 | 0.92 | 0.033
2.312 | 0.016 | 1.416 | 0.064 | 0.967 | 0.087
2.311 | 0.015 | 1.441 | 0.083 | 1.001 | 0.124
2.124 | -0.067 | 1.573 | 0.183 | 1.022 | 0.147
2.248 | -0.012 | 1.573 | 0.183 | 1.037 | 0.164
2.247 | -0.013 | 1.458 | 0.096 | 1.046 | 0.175
2.152 | -0.055 | 1.47 | 0.105 | 1.053 | 0.183
2.161 | -0.05 | 1.478 | 0.111 | 1.06 | 0.19
2.203 | -0.032 | 1.485 | 0.116 | 1.065 | 0.196
2.203 | -0.032 | 1.483 | 0.115 | 1.057 | 0.188
2.174 | -0.045 | 1.504 | 0.131 | 1.075 | 0.207
$n=10$
$\sqrt{s}=10$ GeV |  $\sqrt{s}=300$ GeV |  $\sqrt{s}=10$ TeV
$d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$ | $d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$ | $d_{k}^{\left(y\right)}$ | $\frac{{d_{k}^{\left(y\right)}-\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}{{\frac{1}{n}Sp\left({{{\hat{D}}^{\left(y\right)}}}\right)}}$
0.955 | -0.457 | 0.147 | -0.809 | 0.037 | -0.901
2.124 | 0.207 | 0.435 | -0.433 | 0.13 | -0.65
2.121 | 0.205 | 0.665 | -0.133 | 0.242 | -0.351
1.529 | -0.131 | 0.794 | 0.036 | 0.34 | -0.087
1.707 | -0.03 | 0.855 | 0.115 | 0.412 | 0.107
1.77 | 0.006 | 0.882 | 0.15 | 0.46 | 0.236
1.805 | 0.026 | 0.893 | 0.164 | 0.503 | 0.352
1.891 | 0.075 | 1.052 | 0.372 | 0.489 | 0.313