The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering
We present the new method for the calculation of inelastic scattering cross-section, which doesn't require the use of any additional Regge-like assumptions and accurately accounts for energy-momentum conservation law. This leads to a new mechanism of cross-section growth, which has not been con...
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Odessa National Polytechnic University
2012
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nasplib_isofts_kiev_ua-123456789-1069742025-02-23T18:01:16Z The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering Проблемы учета закона сохранения энергии-импульса и интерференционных эффектов в адрон-адронных взаимодействиях Проблеми врахування закону збереження енергії-імпульсу та інтерференційних ефектів в адрон-адронних взаємодіях Sharf, I.V. Tykhonov, A.V. Deliyergiyev, M.A. Sokhrannyi, G.O. Podolyan, N.A. Rusov, V.D. Section A. Quantum Field Theory We present the new method for the calculation of inelastic scattering cross-section, which doesn't require the use of any additional Regge-like assumptions and accurately accounts for energy-momentum conservation law. This leads to a new mechanism of cross-section growth, which has not been considered before, and is related to the behavior of hadrons longitudinal momenta. Second, it has been shown that interference diagrams, originating from the identity of final-state hadrons, put a significant contribution to the cross-section. The approximate method for taking into account such interference contributions has been developed. Altogether, this results in a fact that the dependence of the total and inelastic scattering cross-section on energy can be qualitatively reproduced by fitting only one single parameter of the model. Предложен новый метод расчета сечения неупругого адрон-адронного рассеяния, не основанный на предположениях Редже-кинематики и точно учитывающий закон сохранения энергии-импульса. В результате расчетов, основанных на методе, был выявлен новый механизм роста сечений, связанный с вкладом продольных компонент импульса в амплитуду неупругого рассеяния. Было показано, что интерференционные слагаемые, возникающие вследствие тождественности адронов в конечном состоянии, вносят значительный вклад в сечение, и могут быть посчитаны в рамках аналитического приближения, представленного в работе. В итоге, путем подгонки только одного свободного параметра модели было получено качественное согласие с экспериментальными данными по неупругому и полному сечениям протон-протонного рассеяния. Запропоновано новий метод розрахунку перерізу непружного адрон-адронного розсіяння, не заснований на припущеннях Редже-кінематики, такий що точно враховує закон збереження енергії-імпульсу. У результаті розрахунків, заснованих на методі, було виявлено новий механізм зростання перерізів, що пов'язаний з внеском поздовжніх компонент імпульсу в амплітуду непружного розсіяння. Було показано, що інтерференційні доданки, що виникають внаслідок тотожності адронів в кінцевому стані, вносять значний внесок в переріз, і можуть бути розраховані в рамках аналітичного наближення, представленого у роботі. У підсумку, шляхом підгонки лише одного вільного параметра моделі було отримано якісне узгодження з експериментальними даними по непружному і повному перерізам протон-протонного розсіяння. 2012 Article The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering / I.V. Sharf, A.V. Tykhonov, M.A. Deliyergiyev, G.O. Sokhrannyi, N.A. Podolyan, V.D. Rusov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 21-26. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS: 11.15.-q, 11.30.-j, 12.40.Nn,13.85.-t https://nasplib.isofts.kiev.ua/handle/123456789/106974 en Вопросы атомной науки и техники application/pdf Odessa National Polytechnic University |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Section A. Quantum Field Theory Section A. Quantum Field Theory |
| spellingShingle |
Section A. Quantum Field Theory Section A. Quantum Field Theory Sharf, I.V. Tykhonov, A.V. Deliyergiyev, M.A. Sokhrannyi, G.O. Podolyan, N.A. Rusov, V.D. The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering Вопросы атомной науки и техники |
| description |
We present the new method for the calculation of inelastic scattering cross-section, which doesn't require the use of any additional Regge-like assumptions and accurately accounts for energy-momentum conservation law. This leads to a new mechanism of cross-section growth, which has not been considered before, and is related to the behavior of hadrons longitudinal momenta. Second, it has been shown that interference diagrams, originating from the identity of final-state hadrons, put a significant contribution to the cross-section. The approximate method for taking into account such interference contributions has been developed. Altogether, this results in a fact that the dependence of the total and inelastic scattering cross-section on energy can be qualitatively reproduced by fitting only one single parameter of the model. |
| format |
Article |
| author |
Sharf, I.V. Tykhonov, A.V. Deliyergiyev, M.A. Sokhrannyi, G.O. Podolyan, N.A. Rusov, V.D. |
| author_facet |
Sharf, I.V. Tykhonov, A.V. Deliyergiyev, M.A. Sokhrannyi, G.O. Podolyan, N.A. Rusov, V.D. |
| author_sort |
Sharf, I.V. |
| title |
The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering |
| title_short |
The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering |
| title_full |
The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering |
| title_fullStr |
The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering |
| title_full_unstemmed |
The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering |
| title_sort |
problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering |
| publisher |
Odessa National Polytechnic University |
| publishDate |
2012 |
| topic_facet |
Section A. Quantum Field Theory |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/106974 |
| citation_txt |
The problems of the account for energy-momentum conservation law and interference effects at hadron-hadron scattering / I.V. Sharf, A.V. Tykhonov, M.A. Deliyergiyev, G.O. Sokhrannyi, N.A. Podolyan, V.D. Rusov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 21-26. — Бібліогр.: 17 назв. — англ. |
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Вопросы атомной науки и техники |
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| fulltext |
THE PROBLEMS OF THE ACCOUNT FOR
ENERGY-MOMENTUM CONSERVATION LAW AND
INTERFERENCE EFFECTS AT HADRON-HADRON
SCATTERING
I.V. Sharf 1, A.V. Tykhonov 1,2,3, M.A. Deliyergiyev 1,2,3, G.O. Sokhrannyi 1,
N.A. Podolyan 1, and V.D. Rusov 1∗
1Odessa National Polytechnic University, 65044, Odessa, Ukraine
2University of Ljubljana, SI-1000 Ljubljana, Slovenia
3Jozef Stefan Institute, SI-1001, Ljubljana, Slovenia
(Received October 25, 2011)
We present the new method for the calculation of inelastic scattering cross-section, which doesn’t require the use of
any additional Regge-like assumptions and accurately accounts for energy-momentum conservation law. This leads
to a new mechanism of cross-section growth, which has not been considered before, and is related to the behavior of
hadrons longitudinal momenta. Second, it has been shown that interference diagrams, originating from the identity
of final-state hadrons, put a significant contribution to the cross-section. The approximate method for taking into
account such interference contributions has been developed. Altogether, this results in a fact that the dependence of
the total and inelastic scattering cross-section on energy can be qualitatively reproduced by fitting only one single
parameter of the model.
PACS: 11.15.-q, 11.30.-j, 12.40.Nn,13.85.-t
1. INTRODUCTION
At the calculation of inelastic scattering cross-section
(Fig. 1) one usually deals with the assumption that,
at high energies, the main contribution to integral
for the cross-section comes from multi-Regge do-
main (see e.g. [1–9]. The approximations applied
to a delta-function which is responsible for energy-
momentum conservation law and the neglect of de-
pendence of integrand on longitudinal momentums is
based on this assumption. As the result, different
points of phase space correspond to different values
of total energy-momentum of final-state particles.
In this paper we present a new method for the cal-
culation of inelastic scattering cross-section, which is
based on the following concerns. It is shown that
the there is a class of Feynman diagrams, for which
the absolute values of corresponding scattering am-
plitudes have conditional maximum upon the condi-
tion of accurate account for the entanglement of scat-
tering amplitude arguments, caused by the energy-
momentum conservation law. This fact enables us
to apply the well-known Laplace [10] method for the
calculation of cross-section.
As the result, the new mechanism of cross-section
growth is discovered, related to the behavior of lon-
gitudinal momentum components of final-state parti-
cles with the energy growth.
At the same time, according to the Wick’s theo-
rem, the scattering amplitude is the sum of diagrams
of all possible orders of external lines attaching to the
diagram in Fig. 1, b (interference terms). In order to
take into account these interference contributions one
needs to modify the aforementioned method in the
way, which will be outlined further in the paper.
Finally, we obtain the qualitative agreement with
experiment for the total and inelastic scattering
cross-section dependence on the energy by fitting only
one free model parameter (coupling constant).
The paper is structured as follows. In Section 2 we
set ourselves the problem of finding the constrained
maximum point of multi-peripheral scattering am-
plitude under the condition of energy-momentum
conservation. Furthermore, the analytical solution
of this problem is outlined. Section 3 represents the
application of Laplace’s method for cross-section cal-
culation, assuming scattering amplitude has a point
of constrained maximum. The method for taking
into account the interference contributions to cross-
section at high final state multiplicity is outlined in
Section 4. The comparison of calculated total scat-
tering cross-section with the experimental results is
presented in Section 5. Finally, summary and con-
clusions are given in Section 6.
∗Corresponding author E-mail address: siiis@te.net.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 21-26.
21
P
1
P
2
P
3
P
4
p
1
p
2
p
n
P
1
P
2
P
3
P
4
1
2
n
p
1
p
2
p
n
a b
Fig. 1. A general view of an inelastic scattering dia-
gram (a); an elementary inelastic scattering diagram
in the multi-peripheral model (“the comb”) (b)
2. THE CONSTRAINED MAXIMUM
PROBLEM
The starting point of the new approach to the cal-
culation of inelastic scattering cross-section is the
fact that scattering amplitude has a point of con-
strained maximum upon the condition of the account
for energy-momentum conservation law. Consider
the scattering amplitude corresponding to a comb di-
agram on Fig. 1b.
T =
(
−ig(2π)4
)2(
−iλ(2π)4
)n
(
−i
(2π)4
)n+1
A (1)
with
A = 1
m2−(P1−P3)2−iε
1
m2−(P1−P3−p1)
2−iε
· · ·
· · · 1
m2−(P1−P3−p1−p2−···−pn−1−pn)2−iε
,
(2)
where g is a coupling constant in the outermost ver-
tices of the diagram; λ is a coupling constant in all
other vertices; m is the mass of virtual particle field
and also secondary particles. As in the original ver-
sion of multi-peripheral model [2], pions are taken
both as virtual and secondary particles.
Energy of each particles in the finite state can be
expressed by their momentum using the mass shell
conditions, having n + 2 particles in finite state, that
give us 3(n+2) momentum components of these parti-
cles. Moreover, taking into account four relations, ex-
pressing the energy-momentum conservation law will
result in the fact that amplitude Eq. (2) can be repre-
sented as a function of 3n + 2 independent variables.
The first 3n variables we choose are longitudinal and
transverse components of momentums �p1, �p2, . . . , �pn
of particles produced along the“comb“. The other
two variables are the transverse components of mo-
mentum �P3⊥.
If z-axis coincides with momentum direction �P1
in c.m.s. and x and y axes are the coordinate axes in
the plane of transverse momentums, the conservation
laws look like
P30 + P40 =
√
s − (p10 + p20 + ... + pn0) ,
P3‖ + P4‖ = − (p1‖ + p2‖ + ... + p‖
)
,
P4⊥x = − (p1⊥x + p2⊥x + ... + pn⊥x + P3⊥x) ,
P4⊥y = − (p1⊥y + p2⊥y + ... + pn⊥y + P4⊥y) .
(3)
As was shown in [13], the scattering amplitude as
a function of aforementioned independent 3(n + 2)
variables has a point of maximum, which is attained
at zero transverse momentums of secondary particles.
Moreover, scattering amplitude in the point of max-
imum can be expressed as follows [13]
A(0),n = (1 + a(
√
s, n))−2 ×
× (1 + b(
√
s, n))−(n−1) exp (c(
√
s, n)) ,
(4)
where
a(
√
s, n) =
(
1
(√s/M)
2
n+1 −1
)2
,
b(
√
s, n) =
(
(√s/M)
1
n+1
(√s/M)
2
n+1 −1
)2
,
c (
√
s, n) = 2
(
a−1 (
√
s, n) + (
√
s/M)
2
n+1
)−1
×
×
(
1 − (n − 1) (
√
s/M)−
n
n+1 a− 1
2 (
√
s, n)
)
.
(5)
The a(
√
s, n) and b(
√
s, n) determine the characteris-
tic value of virtuality at the maximum point of scat-
tering amplitude and c(
√
s, n) determines the varia-
tion of virtuality along the “comb“. In other words,
the following estimate takes place
a(
√
s, n) ≤
∣∣∣∣(q(j)
)2
∣∣∣∣ ≤ b(
√
s, n), (6)
where
∣∣∣(q(j)
)2∣∣∣ is the absolute value of virtuality
corresponding to j-th internal line on the “comb”
(Fig. 1b) in the point of constrained maximum.
It is useful to rewrite the energy term, which en-
ters both sides of Eq. (6) in the form
(√
s/M
) 1
n+1 = exp
(
1
n + 1
ln
(√
s/M
))
. (7)
It is obvious that the growth of exponent with energy√
s is much weaker than the corresponding decrease
with the growth of number of particles n. Thus,
one can see that at not very small n the value of
(
√
s/M)
1
n+1 ∼ 1 even at high energies (
√
s >> M).
As the result, the difference of energy and longitudi-
nal momentum squares is at least not negligible with
respect to transverse momentum for each virtuality
on the “comb”. This result comes in contradiction
with the statement that virtualities can be reduced
to transverse momentum squares, which is usually
claimed in the standard approach [1, 3–9]. Taking
into account the growth of (
√
s/M)
2
n+1 with energy√
s growth, we see that virtuality at the maximum
point really decreases and the maximum value of am-
plitude grows with the growth of energy
√
s. Note
also that at not very small n the (
√
s/M)
1
n+1 is close
to unity at rather wide energy range which results
in the much steeper growth than the one which is at-
tained in Regge-based theories [2,4] and described by
factor of lnn−2 (
√
s/M). Moreover, the higher n, the
wider is the energy range. Thus the asymptotic be-
havior for different n is reached at different
√
s which
enables to doubt the validity of the asymptotic for-
mulas of multi-Regge kinematics.
22
3. ON THE LAPLACE’S METHOD
In this section we will outline the calculation of
inelastic scattering cross-section with the Laplace’s
method. The cross-section is described by the multi-
dimensional integral of scattering amplitude squared
modulus over the phase volume of finite state:
σn = (2π)8g4λ2n
4n!
√
(P1P2)2−(M1M2)2
×
× ∫ d�P3
2P30(2π)3
d�P4
2P40(2π)3
n∏
k=1
d�pk
2p0k(2π)3
×
×|A(n, p1, p2, ..., pn, P1, P2, P3, P4)|2×
×δ(4)
(
P3 + P4 +
n∑
k=1
pk − P1 − P2
)
.
(8)
Since scattering amplitude is not a product of func-
tions of some variables, and also due to the complex-
ity of integration domain, the multidimensional inte-
gral in Eq. (8) is not a product of smaller-dimensional
ones. In considered inelastic process this domain of
phase space of finite state particles is determined by
the energy-momentum conservation law. As a re-
sult, the integration limits for one variable depend
on the values of others. However, the existence of
constrained maximum point of scattering amplitude
enable to overcome these problems.
As was shown in the previous section, an in-
tegrand A(n, P3, P4, p1, p2, . . . , p2, P1, P2) in Eq. (8),
expressed as a function of independent integration
variables, has a maximum point in the domain of inte-
gration. At the neighborhood of this maximum point
it can be represented in the form
A (n, P3, P4, p1, p2, ..., pn, P2, P1) = A(0),n
(√
s
)
(9)
×exp
(
−1
2
3n+2∑
a=1
3n+2∑
b=1
Dab
(
Xa−X(0)
a
)(
Xb−X
(0)
b
))
,
where A(0),n(
√
s) is the value of the function Eq. (2)
at the point of constrained maximum; Dab =
− ∂2
∂Xa∂Xb
ln(A); the derivatives are taken at the con-
strained maximum point of scattering amplitude. In
other words, the real and positive magnitude A deter-
mined by Eq. (2) is represented as A = exp ln(A), and
the power of the exponential function is expanded
into the Taylor series in the neighborhood of the max-
imum point with an accuracy up to the second-order
summands.The applicability of such approximation
has been verified in [14].
Up to this moment we ignored the interference
effects. Nevertheless, according to Wick’s theorem,
the scattering amplitude is the sum of diagrams with
all possible orders of external lines attaching to the
“comb”. Therefore, partial cross-section σn can be
represented [14, 15] as a sum of n! cut diagrams on
Fig. 2, and we can write down instead of Eq. (8)
σn = ((2π)4)2
g4λ2n
4
√
(P1P2)
2−(M1M2)2
×
× ∫ d�P3
2P30(2π)3
d�P4
2P40(2π)3
n∏
k=1
d�pk
2p0k(2π)3
×
×δ(4)
(
P3 + P4 +
n∑
k=1
pk − P1 − P2
)
×
×Φ (n, P3, P4, p1, p2, ..., pn, P2, P1) ,
(10)
where
Φ (n, P3, P4, p1, ..., pn, P2, P1) =
A (n, P3, P4, p1, ..., pn, P2, P1)×
× ∑
P (j1,j2,...,jn)
A (n, P3, P4, pj1 , ..., pjn , P2, P1).
(11)
Here, the summation is assumed over all possible n!
permutations of n indices. Each integral in Eq. (10)
can be analytically evaluated using Laplace’s method
[10]. Finally one gets partial cross-section is ex-
pressed as follows
σ′
n =
(
A
(
X̂(0)
))2
v (
√
s)×
× ∑
P (j1,...,jn)
exp
�
− 1
2
��
ΔX̂
(0)
j
�T
D̂(j)ΔX̂
(0)
j
��
�
det( 1
2 (D̂+P̂ T
j D̂P̂j))
,
(12)
where we use the following designations: ΔX̂
(0)
j =
X̂(0) − P̂−1
j (X̂(0)), D̂(j) =
(
D̂−1 + P̂T
j D̂−1P̂j
)−1
,
v(
√
s) ≡
(
2
√
s
√
s/4 − M2
(
EP
2
)√(
EP
2
)2 − M2
)−1
(here A
(
X̂(0)
)
≡ A(0),n is the value of scattering
amplitude at the constrained maximum point), P̂j is
the permutation matrix.
Note, that here and further we will use the
“prime” sign in our notation to indicate that we use a
dimensionless quantity that characterized the depen-
dence of the cross-sections on energy, but not their
absolute values.
4. ANALYTICAL APPROXIMATION FOR
INTERFENCE CONTRIBUTIONS
Each interference contribution in Eq. (12) can be cal-
culated numerically, for instance by the Lagrange
method. However, the large number of terms in
Eq. (12) represents the severe computational diffi-
culty, which we are able to overcome only for the
number of particles n ≤ 8. Therefore, we introduce
an analytical approximation for calculating the sum
of interference contributions at any multiplicity of
final-state particles.
The essence of our method is as follows. mul-
tidimensional volume cutout by resulting Gaussian
function from an integration domain (which we call
the “width” of the maximum). If we compose the n-
dimensional vector (we denote it through �y(0)) from
the particle rapidities, maximizing the function asso-
ciated with the diagram with the initial arrangement
of momenta (left-hand side part of “cut” diagram on
Fig. 2), then vectors maximizing the functions with
another momentum arrangement will differ from the
initial vector only by the permutation of components
(right-hand side part on Fig. 2), i.e., these vectors
have the same length.
23
P
1
P
2
P
3
P
4
P
1
P
2
P
3
P
4
p
j
1 (n)
p
j
1 (1)
p
j
1 (2)
n
1
2
p
1
p
2
p
n
1
2
n
p
j
(1)
p
2
p
1
p
n
^
^
^
^
p
j
(n)^
p
j
(2)^
j= 1
n!
Fig. 2. Representation of the partial cross-section
as a sum of “cut” diagrams. The order of joining of
lines with four-momenta pk from the left-hand side
of the cut is as following: the line with p1 is joined
to the first vertex, the lines with p2 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! pos-
sible permutations of the set of numbers 1, 2, . . . , n.
Here P̂j(k), k = 1, 2, . . . , n denote the number into
which a number k goes due to permutation P̂j. An
integration is performed over the four-momenta pk
for all “cut lines” taking into account the energy-
momentum conservation law and mass shell condi-
tion for each of pk
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 “stretch” a two-
dimensional plane on them (as a set of their various
linear combinations), where two-dimensional geom-
etry takes place. Therefore, the distance r will be
determined by cosine of an angle between the consid-
ered equal on length n-dimensional rapidity vectors
in the two-dimensional plane, “stretched” on them.
An angle corresponding to the P̂j permutation we
designate through θj , 0 ≤ θj ≤ π.
Thus, each of the terms in the sum Fig. 2 can be
uniquely match to its angle θj . At the same time the
variable z = cos(θ) is more handy for consideration
than an angle θj .
It has been shown [15] that the value of each in-
terference contribution can be approximately repre-
sented as a unique function σ′
n (z) of z in the following
way
σ′
n (z) = σ′
n (1) exp
⎛
⎝
∣∣�y(0)
∣∣2Tr
(
D̂y
)
2n
(z − 1)
⎞
⎠ . (13)
Here σ′
n (1) is the interference contribution, corre-
sponding to a “cut” diagram with initial line re-
arrangement of momenta (“ladder” type diagram).
Thus, now we have the dependence for the value
of interference contribution on z. The only thing left
is to find out, how many contributions correspond to
some given interval [z; z + dz] or, in other words, the
intereference contribution density function.
It turns out [15] that the ends of vectors
P̂−1
j
(
�y(0)
)
are uniformly lying on the sphere in n− 1
dimensional space. The interval [z; z + dz] corre-
sponds to a belt on this sphere, and one gets the
density function [15]
ρ (z) =
n!√
π
Γ
(
n−1
2
)
Γ
(
n−2
2
)(1 − z2
)n−4
2 , (14)
where Γ is the Euler’s gamma function. The num-
ber of interference contributions corresponding to
[z; z + dz] is equal to
dN (z, dz) = ρ (z)dz. (15)
And the partial cross-section can be approximately
represented as
σ′
n =
1∫
−1
σ′
n (z)ρ (z) dz. (16)
Substituting Eqs. (13), (14) into Eq. (16) we get an
analytical approximation for calculating the partial
cross-section as a sun af all interference contributions.
The comparison of this approximation with the “ex-
act” cross-sections, for which all the interference con-
tributions were calculated directly (Eq. (12)) is given
in Fig. 11 of [15] for relatively small n (n = 8, 9).
5. TOTAL AND INELASTIC
SCATTERING CROSS-SECTION
Finally, since we have an expression for calculating
partial cross-sections σ′
n which can be evaluated at
any multiplicity of final-state particles, let’s proceed
to the expression for the total and inelastic scattering
cross-sections:
σ′Σ (√s
)
=
nmax∑
n=0
Lnσ′
n
(√
s
)
, (17)
σ′I (√s
)
=
nmax∑
n=1
Lnσ′
n
(√
s
)
. (18)
Within the framework of the examined φ3 model
σ′Σ (
√
s) is the analogue of total scattering cross-
section. Here nmax is the maximum number of sec-
ondary particles allowed by energy-momentum con-
servation law and L is the dimensionless coupling
constant, which we considered as an adjustable para-
meter. Fitting the constant L we achieve a qualita-
tive agreement σ′I(
√
s) and σ′Σ(
√
s) with observed in
proton-proton collisions [16, 17] dependences on
√
s.
The result of such a fitting presented in Fig. 3 and it
qualitatively agrees with experimental data not only
at the high energies that is usually accepted in the
Regge based theories, but also near the threshold of
two-particle production (the first minimum of the to-
tal cross section Fig. 3, c). This is due to the fact
that the proposed method of calculation does not re-
quire any approximations, based on the asymptoti-
cally large energies. This may indicate that the ex-
perimentally observed behavior of cross sections is de-
termined not by high energy asymptotic of the scat-
tering amplitude as it is assumed in the contemporary
approaches [8, 9, 11, 12].
24
However, the quantitative agreement with the ex-
perimental results requires the application of more
realistic model than the self-acting scalar φ3 field
model.
a
b
c
d
Fig. 3. Theoretical dependences of the σ′I(
√
s) (a)
and σ′Σ(
√
s) (c) obtained for the energy range
√
s =
1 ÷ 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 multiplici-
ties. Experimental data for the inelastic (b) and for
the total (d) pp scattering cross-section Ref. [16,17]
are presented for qualitative comparison with the pre-
diction from our model. Note: data-points for the in-
elastic cross-section are obtained from the definition
σinel = σtotal − σelastic
6. CONCLUSIONS
A new method for the calculation of partial inelastic
scattering cross-section which, contrary to the state-
of-the-art approaches, takes into account the energy-
momentum conservation law is presented.
It has been shown that the main contribution to
integral expressing inelastic scattering cross-section
comes not from multi-Regge domain.
The results for calculated total and inelastic scat-
tering cross-section qualitatively agree with experi-
ment.
References
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3. E. Byckling and K. Kajantie. Particle kinemat-
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4. P.D.B. Collins. An introduction to Regge the-
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25
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