ABSTRACT

We demonstrate a new technique for calculating proton-proton inelastic cross-section, which allows one by application of the Laplace’ method replace the integrand in the integral for the scattering amplitude in the vicinity of the maximum point by expression of Gaussian type. This, in turn, allows us to overcome the computational difficulties for the calculation of the integrals expressing the cross section to sufficiently large numbers of particles. We have managed to overcome these problems in calculating the proton-proton inelastic cross-section for production (500476\8ddf0bce-a6ef-4cca-b1bd-410ac368b0c7.jpg width=46.169998550415 height=22.2299996376038  />) number of secondary particles in within the framework of 500476\21f97ecd-9cba-4ea8-a8b0-dcbd516cec51.jpg width=25.4599992752075 height=31.825  /> model. As the result the obtained dependence of inelastic cross-section and total scattering cross-section on the energy 500476\d04fc221-9e4c-4531-a25e-67dcb5b442e5.jpg width=30.2099992752075 height=30.2099992752075  /> are qualitative agrees with the experimental data. Such description of total cross-section behavior differs considerably from existing now description, where Reggeons exchange with the intercept greater than unity is considered.

1. Introduction

The problems of the inelastic scattering cross-sections calculation have been discussed in details in [1]. As the result of approximations, which are usually made to overcome these difficulties [2-5], are obtained the integral over the areas of phase space, where different points correspond to different values of the energy-momentum, but at the same time they come to the equation with equal weights. Therefore, the energy-momentum conservation law does not consider reasonably.

Besides, the virtualities, in the equation for the amplitude, reduced to values of the square of transversal components particles momentums [6], meanwhile the rest components of virtualities are not insignificant and appear as quite essential [1].

These approximations are based on the assumption that the main contribution to the integral makes the multi-Regge domain [7]. This assumption is crucial for the modern approaches to the description of inelastic scattering processes [8]. However, the obtained results in [1] lead to the conclusion that main contribution in the integral does not make the multi-Regge domain.

The aim of this paper is to propose an alternative method for calculating inelastic scattering cross-sections based on well-known Laplace’ method for the multidimensional integral [9]. In order to apply this method it is required the element of integration has the point of maximum within integration domain. It has been shown [1] that for the diagrams of “comb” type with the accurate energy-momentum conservation law calculation the square of scattering amplitude module is really has that maximum.

Analysis of the properties of this maximum led to the conclusion that there is the mechanism of cross-section growth. This mechanism has not been considered previously, due to the above approximations associated with the multi-Regge kinematics. Now we would like to show that this mechanism can be responsible for the experimentally observed behavior of cross sections dependence with energy500476\45a5c49f-e929-4e69-81ee-7498af2788b0.jpg width=30.2099992752075 height=28.5950003623962  />. However, application of Laplace’s method to the processes of production of large number of secondary particles faces the challenge of accounting the vast amount of interference contributions, which will be discussed in detail in Section 3 of this work. In fact, there are 500476\d50fddc1-6ab5-4055-8b80-c56fd92a9580.jpg width=22.2299996376038 height=22.2299996376038  /> of these contributions for the process with production of n secondary particles. Therefore, in the present paper we were able to calculate all interference contributions to the production of 500476\035e2cc9-4026-449f-89c3-220f3c819432.jpg width=46.169998550415 height=22.2299996376038  /> secondary particles.

Typically, these contributions are underestimate, because according to the considering assumption that particles on the “comb” strongly ordered in rapidity [3] or strongly ordered in Sudakov’s parameters [4], they should be negligible.

However, later in this paper, we show that these contributions are significant and the contribution from the square modulus of just one “comb” type diagram with initial particles arrangement, which are usually limited, is only a small fraction of the sums of all interference contributions. Despite the fact that it was possible to calculate the partial cross sections only for small amounts of secondary particles, we can achieve qualitative agreement with experimental results.

2. On the Need of Consideration of Diagrams with the Different Sequence of the Attaching External Lines to the “Comb”

An inelastic scattering cross-section, which is interesting for us, is described by the following equation:

500476\d1236e4c-8cb4-4513-ab43-0979a965dd02.jpg width=373.825 height=119.319998550415 />(1)

where

500476\7513d5ef-6f0b-45e5-a748-6ec1fee572c4.jpg width=194.08500289917 height=41.325 /> (2a)

500476\eba7a39d-8b8c-4464-9fea-1c8edfd8c5d1.jpg width=291.07999420166 height=39.805001449585 /> (2b)

The scattering amplitude in this equation will be considered within framework of the multi-peripheral model, i.e., for the diagrams of “comb” type. However, here we will make the important remark.

According to the Wick theorem, the scattering amplitude is the sum of diagrams with all possible orders of external lines attaching to the “comb”. In the terms of diagram technique it looks as follows. Plotting the multi-peripheral diagram of the scattering amplitude (as it is shown in Figure 1 of [1]) at first we have adequate number of vertices with three lines going out of it and n lines corresponding to the secondary particles as it is shown in Figure 1(a).

“Pairing” some lines Figure 1(a) in order to obtain the “comb”, we will get a situation shown in Figure 1(b). The weighting coefficient appearing from this procedure is included to the coupling constant. Finally we have to “pair” the appropriate lines of particles in the final state with the remaining unpaired internal lines in the diagram of Figure 1.

If we marked by500476\47636fa0-65ed-454a-8ece-2bd811d8c2a3.jpg width=15.8650007247925 height=28.5950003623962  />—the external line, paired with the first vertex;500476\03ce10b9-7ab3-4322-abbe-e8e12338e051.jpg width=17.4799996376038 height=28.5950003623962  />—the external line, paired with the second vertex and etc.; then 500476\f68f2795-f0b6-4073-aaba-203399889145.jpg width=17.4799996376038 height=28.5950003623962  /> is an external line, which is paired with k-th vertex, so every diagram will be characterized by sequence500476\ad27887c-5c89-4fea-a3a1-1419a0488e32.jpg width=17.4799996376038 height=28.5950003623962  />500476\b127696a-28f4-409c-8bae-4758e33ae9fa.jpg width=52.5349992752075 height=28.5950003623962  />500476\01e2cb1a-754b-4b55-9b3b-6d74e7d1721b.jpg width=22.2299996376038 height=28.5950003623962  />. And in this case the total amplitude is expressed by the sum of 500476\defe8665-797d-436e-a667-3937c752c82a.jpg width=22.2299996376038 height=22.2299996376038  /> terms, each of them corresponds to one of 500476\c8db35cc-88e9-4a46-b4c3-a396a917be1f.jpg width=22.2299996376038 height=22.2299996376038  /> possible index sequences and therefore the inelastic scattering cross-section can be written as

500476\35cf34f3-4d5a-427d-b380-52774e467c19.jpg width=345.13498840332 height=124.07000579834 /> (3)

500476\1450628a-b7c6-460b-8d0b-7d4a129b472b.jpg width=227.42999420166 height=84.2649971008301 /> (4a)

500476\342a6fc8-21ee-4633-9b97-059026f86dab.jpg width=407.26501159668 height=138.41499710083  />(4b)

Here, as well as in [1], 500476\9b630a90-8c6e-4efd-b7ac-cf22716ff9fd.jpg width=28.5950003623962 height=28.5950003623962  />and 500476\eb777281-47b5-41fd-a416-46ee814a818c.jpg width=31.825 height=28.5950003623962  /> are the masses of particles in initial state and we assume that

500476\662728eb-7d3f-4234-be1f-60bce602adc7.jpg width=116.08999710083 height=28.5950003623962  />, where M is proton mass. Moreover, 500476\92b084a3-68cc-444e-b4d8-895d35252969.jpg width=20.7099992752075 height=28.5950003623962  />and 500476\43e9c296-aecb-46d7-9c58-030e1f937c20.jpg width=22.2299996376038 height=28.5950003623962  /> are four-momenta of initial protons; 500476\361fae5b-d0ca-4a56-9eb2-2bb6fb833139.jpg width=22.2299996376038 height=28.5950003623962  />and 500476\4fdc7b16-334b-4ead-b290-56fe4d9bcf7c.jpg width=22.2299996376038 height=28.5950003623962  /> are four-momenta of protons in the finite state;500476\13aacee3-06d5-467d-a4b0-7d69d522b518.jpg width=25.4599992752075 height=28.5950003623962  />,

500476\c4bf59aa-fa49-4235-b1e6-f56bad6481ef.jpg width=100.225 height=27.075  />are four-momenta of secondary particles (pions of mass m). As the virtual particles we understand the quanta of real scalar field with pion mass m. A coupling

constant in vertexes, in which the pion lines join with proton lines, is denote as g and 500476\1b8945f5-1d74-4774-89b4-5eb331d6f497.jpg width=19.0950003623962 height=22.2299996376038  /> is coupling constant in vertexes, where three pion lines meet. The function A is defined by:

500476\8c5de5b9-92c8-4c7d-a24f-3b1fcb9b0c1e.jpg width=372.20998840332 height=300.675 />(5)

Moreover, as it was shown in [1], the function A is real and positive therefore sign of complex conjugation in Equation (3) can be dropped and we can rewrite this expression in the following form:

500476\b972a909-eb39-4546-87fc-cdb735aefada.jpg width=419.9 height=217.92999420166  />(6)

where 500476\726e9e68-deb6-4f03-aa3f-2cb2dc318cdf.jpg width=20.7099992752075 height=22.2299996376038  /> defined by Equation (4a).

Notation 500476\d280e7c9-f92e-45d5-b388-790e5d2e7808.jpg width=70.0149971008301 height=39.805001449585  /> means that we consider sum of terms corresponding to all possible permutations of indices500476\6d7abebb-e91e-4dcb-89af-ba4310687a84.jpg width=79.5149971008301 height=28.5950003623962  />. Let us note that the integration variables in each of term of considered sum can be renaming, so that the indexes 500476\9402a1bb-df9f-4f3b-aded-1d17fbbc01b4.jpg width=79.5149971008301 height=28.5950003623962  /> formed the original placing500476\543de05b-1e93-4e8c-8caa-7af30bf3d459.jpg width=70.0149971008301 height=25.4599992752075  />. At the same time the indexes 500476\658b5943-c9d7-4c81-ab36-d466dfa60641.jpg width=87.494998550415 height=28.5950003623962  /> will run through all possible permutations and summation must be carried over all these permutations. Taking into account this, we get instead of Equation (6):

500476\f50f649f-7642-4511-87eb-70b73e2a8a07.jpg width=349.97999420166 height=163.875 /> (7)

where

500476\7072bc50-2e47-4878-8179-44b77b6409c3.jpg width=221.06499710083 height=87.494998550415 /> (8)

500476\4115289a-e11f-42ee-9c77-d93e11a988d3.jpg width=378.575 height=120.93500289917 /> (9)

Now we can use the fact that the amplitudes A in Equation (9) have the points of constrained maximum [1].

3. Computation of Contributions to Inelastic Scattering Cross-Sections Corresponding to the Multi-Peripheral Diagrams by Laplace Method

For further analysis, we examine expression Equation (6) in c.m.s. separating the longitudinal and transverse to the collision axis components of three-dimensional particle momenta

500476\b2531849-63e2-4889-a11e-86534de50f5b.jpg width=438.99500579834 height=440.60998840332 /> (10)

where 500476\1b0e23b6-bf5e-43ec-b0d3-b32697448ad5.jpg width=22.2299996376038 height=22.2299996376038  /> defined by Equation (8) and

500476\567a1434-3e7e-47b4-8778-a68a9c453887.jpg width=167.01000289917 height=39.805001449585 /> (11a)

500476\57630d5d-0152-4e3a-8a81-f2064596934d.jpg width=168.625 height=39.805001449585 /> (11b)

The last three 500476\4daf845f-d824-4228-933d-3218b98ce50d.jpg width=19.0950003623962 height=22.2299996376038  />-functions in Equation (10), whose arguments are linear with respect to integration variables, we can vanish by the integration over500476\b6e9af11-b830-4a23-b327-ce79cb645bf9.jpg width=27.075 height=30.2099992752075  />, 500476\497549a3-a178-4212-9983-36a441ad2619.jpg width=38.1900007247925 height=28.5950003623962  />,500476\2141ad3b-36d7-4141-a16c-4e0e22ddbf81.jpg width=39.805001449585 height=30.2099992752075  />. In order to take into account the rest δ-function, which expresses the energy conservation law let’s replace 500476\70b5c516-1f19-4598-992f-0a0801dbbbd0.jpg width=27.075 height=30.2099992752075  /> by new integration variable

500476\d3cbd725-b2ac-4e46-a000-0a38d1e1e9fb.jpg width=376.95998840332 height=108.205001449585 />(12)

Those, making the following replacement we must express 500476\39740436-6063-4449-91b5-85239fb758b0.jpg width=27.075 height=30.2099992752075  /> through500476\5af85167-7c32-48cb-ae32-e51f77555a10.jpg width=27.075 height=30.2099992752075  />. The corresponding expression will coincide with Equation (8) in [1] with the positive sign in front of the square root. Moreover, introduce the rapidities instead of longitudinal momenta:

500476\c49af62f-c7fc-4e8b-b1dc-cc6dc86f6e37.jpg width=176.60499420166 height=31.825 /> (13a)

500476\077914f5-342d-4a0e-9fe1-3a89f4065098.jpg width=171.76000289917 height=38.1900007247925 /> (13b)

After these transformations we get:

500476\fd3b0496-492c-4f6b-9544-7b2cfc86fdf8.jpg width=381.80499420166 height=240.16000289917 />(14)

where

500476\653ac67b-b425-4bdd-a769-fc5bdaeda313.jpg width=314.925 height=66.7850028991699 /> (15)

with

500476\e50942ff-e34b-4229-b248-dc1ba4352aac.jpg width=271.98498840332 height=58.9 /> (16a)

500476\9c96aafa-0bb2-4c28-b330-bc6425ed257f.jpg width=182.97000579834 height=58.9 /> (16b)

Note that the500476\7086a109-2cae-41dd-8ba2-c2cd4b4c2a37.jpg width=27.075 height=30.2099992752075  /> is expressed through the other variables of integration by Equation (8) in [1].

Now turn to the dimensionless integration variables and made the following replacement500476\fa635262-cd64-427a-890a-5f7de48d0c59.jpg width=116.08999710083 height=28.5950003623962  />,500476\80d08e75-4081-40d3-bf96-495f7492c6a7.jpg width=111.33999710083 height=28.5950003623962  />. We designate the new dimensionless integration variables just as the old variables, for short. Moreover, replace expression for 500476\d88f3e1a-d103-4b7d-af34-93c13fa94f0a.jpg width=27.075 height=30.2099992752075  /> by the same expression divided by m. The same concerns the constants in expressions for cross-section, i.e., the designations M and 500476\8884b400-f626-45db-8778-c94c4a538674.jpg width=30.2099992752075 height=28.5950003623962  /> are used for proton mass and energy of colliding particles in c.m.s., which is made dimensionless with the pion mass m.

Now introduce the following notations of integration variables in Equation (14) and designate the rapidities 500476\032e5ef6-bbad-4a13-a253-0232b3369365.jpg width=103.36000289917 height=28.5950003623962  /> as500476\d8c60ea1-4b7d-4f40-b96d-7dbe736a900e.jpg width=111.33999710083 height=28.5950003623962  />; x-components of transverse momenta of secondary particles 500476\423b575c-a2ad-4fc4-af6b-a069e6dce28f.jpg width=149.52999420166 height=28.5950003623962  /> as500476\baa3cff9-4d64-4dbf-b6c8-73a6e8598236.jpg width=155.89500579834 height=30.2099992752075  />; y-components of transverse momenta of secondary 500476\af224a9d-cd7c-464f-8b44-ef7c40152330.jpg width=157.51000289917 height=30.2099992752075  /> as 500476\8704890c-7c4d-4c39-92f8-c5eec79bb340.jpg width=168.625 height=30.2099992752075  />. Moreover, designate 500476\34eabbdf-2c8b-4a6c-b483-4906d9ae166c.jpg width=47.6900007247925 height=30.2099992752075  /> as 500476\411ae6df-7f6d-4fa9-92d6-d3d1d59b008f.jpg width=38.1900007247925 height=28.5950003623962  /> and 500476\ae3fc361-fb4d-4f6b-8ff2-39c487ce2b8c.jpg width=49.305001449585 height=30.2099992752075  /> as500476\b6f44753-ebdd-4233-91b4-a428fd941f70.jpg width=38.1900007247925 height=30.2099992752075  />.

As it was shown in the previous sections, that an integrand 500476\e9f27526-deef-454e-bd85-ae88b9bda656.jpg width=243.39001159668 height=31.825  /> in Equation (14), expressed as a function of independent integration variables, has a maximum point in the domain of integration. At the neighborhood of this maximum point it can be represented in the form

500476\6552c380-a08e-4ed7-ac3a-f515bee07cd3.jpg width=373.825 height=130.43500289917 />(17)

where 500476\1b03e903-b2bf-4b2d-b1ef-18fcd5bc7fb4.jpg width=87.494998550415 height=39.805001449585  />is the value of function (see Equation (4) in [1]) at the point of constrained maximum;500476\aaf80bdd-a5a1-4fcd-a795-577e0b80c3b5.jpg width=116.08999710083 height=50.919998550415  />, where derivatives are taken at the constrained maximum point of scattering amplitude; 500476\ac254404-b01b-4e88-a25e-cd8dd187039c.jpg width=38.1900007247925 height=34.9599992752075  />- value of variables, maximize the scattering amplitude. That is, the real and positive value A defined by Equation (4) (see [1]) is represented as500476\9448db16-a7ec-4848-8262-1ab788f7431c.jpg width=128.82000579834 height=36.575  />, and exponential function is expanded into the Taylor series in the neighborhood of the maximum point with an accuracy up to the second-order summands.

An accuracy of approximation Equation (17) can be numerically verified in the following way. Function A, defined by Equation (4) (see [1]) can be written as

500476\dfb6ef01-0ae4-486b-8952-104628e371cb.jpg width=213.17999420166 height=31.825 /> (18)

Let us introduce also the notation:

500476\497d395c-360f-4937-ab3e-6e4c7237fc88.jpg width=438.99500579834 height=95.475  />(19)

Now let us examine functions:

500476\5def2f54-a0ad-460e-80fd-0a966bd3d75c.jpg width=400.9 height=74.7649971008301  /> (20)500476\276ac01a-1ff1-46b7-915d-a795b976b589.jpg width=419.9 height=77.9  />

(21)

Three-dimensional curves of these functions can be easily plotted in the vicinity of the maximum point (i.e., at the neighborhood of zero of variables a and b). The typical examples of such curves are shown in Figure 2 and Figure 3, where it is easy to see that the approximation Equation (19) works well in the wide energy range. The results similar to ones in Figures 2-3 were obtained

at different values of 500476\7899a617-8110-477a-b9aa-f28d74c02ecd.jpg width=62.0350028991699 height=31.825  /> and n. As one can see from Figure 2 and Figure 3, that true amplitude and its Gaussian approximation Equation (19) start differ visibly only in the parameter region, which makes a insignificant contribution to the integral.

Now let us proceed with identification500476\aba3794b-8f08-418f-82f3-eadcad7a0536.jpg width=245.00499420166 height=31.825  />in Equation (9) and define all possible arrangements of indices 500476\c1447b80-149f-45b9-af6b-3a3a68dda590.jpg width=70.0149971008301 height=25.4599992752075  /> by500476\868c17af-b4f4-4a70-bc22-d5c697f8c740.jpg width=138.41499710083 height=34.9599992752075  />. The function of variables500476\2b59eba0-f120-4943-9650-68ab7801e94f.jpg width=30.2099992752075 height=28.5950003623962  />, where500476\c4087d8e-bed2-470a-bc35-7f406b1844ca.jpg width=136.8 height=27.075  />, which corresponds to arrangement500476\0a0d2c04-c30b-4ed6-975e-a345e6699c49.jpg width=31.825 height=28.5950003623962  />, define as500476\c690897e-ca7c-4ff9-b0b0-d6574d36d455.jpg width=202.06499710083 height=36.575  />. It differs from the function Equation (18) just by renaming the arguments, and therefore it also has a point of constrained maximum under condition of the energy-momentum conservation. The value of this function at the constrained maximum point is equal to the value of function Equation (18), i.e.,

and it is equal to 500476\3900385a-8691-4a9d-bc1c-c5a7672007e3.jpg width=87.494998550415 height=39.805001449585  /> according to the replacement made above. Thus, if 500476\5f0ef4e4-746e-4c7b-83f2-cb4dc8be4257.jpg width=144.77999420166 height=34.9599992752075  /> are the values of variables 500476\4a602135-76ff-4f45-9c54-01033f5785b7.jpg width=111.33999710083 height=28.5950003623962  /> at the maximum pointnow same values 500476\93a52941-49e5-4226-a37c-9dd7a598b61a.jpg width=144.77999420166 height=34.9599992752075  /> will be the values of the variables 500476\1343225c-4db8-4f18-abb7-7b30b5a7a5f0.jpg width=128.82000579834 height=31.825  /> at the maximum point.

Analogously 500476\c35d7824-0d09-49be-b03c-7b87c63f4290.jpg width=149.52999420166 height=34.9599992752075  />are the values of variables 500476\68fcd91e-eb34-4ecc-a5e4-264c106e7af8.jpg width=171.76000289917 height=31.825  />at the maximum point, and500476\9d3436bf-3db1-4bfb-abea-ad405031f446.jpg width=163.875 height=34.9599992752075  />for500476\cef0efe0-59eb-4bb7-a440-ef20292a43f5.jpg width=189.33500289917 height=31.825  />. For short we label the index of variable, into which the variable a goes at given rearrangement, as 500476\89d81064-803e-4583-b9fe-a800b50f07bc.jpg width=60.419998550415 height=36.575  /> i.e.the variable 500476\4ae02d47-0c2b-49a4-ae0a-42f571281c8e.jpg width=28.5950003623962 height=28.5950003623962  /> replaced by the variable500476\26232e2d-3f48-4af4-afa7-bcb2687a7b13.jpg width=71.5350028991699 height=36.575  />.

If we denote the matrix of second derivatives of the logarithm of the function 500476\7c81475c-6be6-4a03-8562-ed30eacf0495.jpg width=38.1900007247925 height=34.9599992752075  />at the maximum point by500476\474a5b1d-3614-4de1-b4be-5b77fd7a688c.jpg width=41.325 height=30.2099992752075  />, we will get the following approximation for the function500476\99df7a5b-d8ee-44b1-9ecd-147e18d6e2ac.jpg width=38.1900007247925 height=34.9599992752075  />:

500476\22612f8e-802a-434f-a04a-852776713411.jpg width=392.92000579834 height=54.055001449585 />(22)

where

500476\243860c1-5921-494e-86ed-b1f083bbe518.jpg width=427.87999420166 height=55.669998550415  />(23)

Taking into account that Equation (22) depends on variables 500476\cce99f40-1dcf-44b0-a7a7-28dbf1447234.jpg width=57.2849992752075 height=38.1900007247925  /> and 500476\3a1eee49-9e1b-48e2-88a8-12847724dc2f.jpg width=57.2849992752075 height=38.1900007247925  /> just as a function A depends on variables 500476\b2f241bb-c21f-4cde-85c1-75979160f444.jpg width=28.5950003623962 height=28.5950003623962  /> and 500476\0cf93ae7-00f8-42af-9b96-d79545b5d942.jpg width=28.5950003623962 height=28.5950003623962  /> and the second derivative is taken at the same values of arguments, we have

500476\b5deac8e-95d7-4f9e-8bfe-0e41e8d07a57.jpg width=143.16499710083 height=46.169998550415 /> (24)

Using Equation (24) rewrite Equation (22) in more convenient form. For this purpose introduce the matrices500476\2ead8773-7820-4e7f-8d34-72828d8c9626.jpg width=31.825 height=28.5950003623962  />, 500476\0c0aae31-03a8-4cf0-81df-e54376457d78.jpg width=103.36000289917 height=27.075  />and by multiplying it with the column 500476\59a6e2a5-2352-489b-a33a-04a5fcb683d5.jpg width=22.2299996376038 height=27.075  /> of initial variables in Equation (19), we get a column in which the variables are arranged in that way so that in place of variable 500476\87c8b737-dcd1-465d-bf0b-e18eee5c41b5.jpg width=28.5950003623962 height=28.5950003623962  /> became a variable500476\1338b749-3c54-480b-8553-66ac6feb648a.jpg width=57.2849992752075 height=38.1900007247925  />. At next iteration taking into account Equation (24) one can rewrite Equation (22) in a matrix form in the following way:

500476\403e76de-0e23-4066-af8b-e21b4b3da257.jpg width=426.26501159668 height=184.48999710083  />(25)

where 500476\67d0f016-9dfe-4059-9f11-a55863fc5b68.jpg width=38.1900007247925 height=28.5950003623962  /> is a column whose elements are the numbers500476\3d104c91-9370-4823-b743-8f0ae7b4de8f.jpg width=38.1900007247925 height=34.9599992752075  />, 500476\8e189567-09f5-4a66-91c7-c6c5e13ed41e.jpg width=136.8 height=25.4599992752075  />in the initial arrangement. And now we can rewrite Equation (9) in the form:

500476\aac44c69-6348-498e-8574-f818da8cc0ea.jpg width=324.52000579834 height=162.26000289917  />      (26)

where

500476\5ce1a1d1-23a9-441c-a3d5-e89a7c501ec9.jpg width=186.10499420166 height=34.9599992752075 /> (27a)

500476\b58272ab-954c-4753-9970-c15c04906b14.jpg width=128.82000579834 height=34.9599992752075 /> (27b)

If now we take up the further consideration of Equation (14), we can see that all the other coefficients (except500476\5e49a8d0-2a05-4e31-ba1b-b451c8fcf2a0.jpg width=27.075 height=22.2299996376038  />) under the integral don’t change the values under the permutation of arguments. We replace these expressions by their values at the maximum point and take them out from integral. From this, we introduce the following notation:

500476\05b24726-3a64-4faf-942e-d289f67346c4.jpg width=257.73498840332 height=74.7649971008301 /> (28)

where 500476\a64dde91-2367-4d0a-bdaa-470e70f6472a.jpg width=34.9599992752075 height=36.575  /> is the value of expression (see Equation 8 in [1]), corresponding to particle momenta, for which the scattering amplitude has maximum, i.e., at the 500476\4a5245c5-6a8b-44a3-a5fe-924ec3dadbc5.jpg width=38.1900007247925 height=34.9599992752075  /> and undimensionalized by m.

The expression for cross-section in this case can be written in the form:

500476\004c6335-05ea-4323-8347-c85a1202bcca.jpg width=438.99500579834 height=273.6  />(29)

As the value 500476\ea61162c-6aa5-4129-8782-2362d9da380c.jpg width=138.41499710083 height=55.669998550415  /> in Equation (29) is the negative value of longitudinal component of momentum500476\0f392b86-7be0-4d4d-947f-44deb79db92e.jpg width=34.9599992752075 height=36.575  />taken at the maximum point, it can be replaced by500476\da07af78-cd2c-4781-9cb1-41ca9a9bdc1c.jpg width=34.9599992752075 height=36.575  />due to the symmetry properties that have been discussed above.

Multidimensional integrals under the summation sign can be calculated by diagonalizing of quadratic form in the exponent of each of them. Such diagonalization can be numerical realized, for instance, by the Lagrange method. The large number of terms in Equation (29) is substantial computational difficulty, which we overcame only for the number of particles500476\9e346a3e-5604-453e-ad0c-7918b6fbe8a6.jpg width=42.9400007247925 height=22.2299996376038  />. To represent results of numerical computations, it is useful to decompose Equation (29) in the following way:

500476\a3486546-4b97-4b3e-8dbb-eb20dfbb014e.jpg width=397.67000579834 height=146.3 />(30)

500476\4ce95a26-fc2a-43cd-b863-1bd3bef2e648.jpg width=335.63498840332 height=96.994998550415 /> (31)

500476\deae1044-e689-4035-ae33-9a3979fad97b.jpg width=139.93500289917 height=65.2649971008301 /> (32)

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 Equation (31) differs from the inelastic scattering cross-section 500476\d83c9f59-7d9b-4f19-8822-a7de5b0296a4.jpg width=66.7850028991699 height=39.805001449585  />only by the absence of factor500476\6d4f055c-8758-47c2-8d0a-63391e8ebe2e.jpg width=227.42999420166 height=66.7850028991699  />, which is energy independent and its consideration allow us to trace the dependence of inelastic scattering cross-section on energy 500476\6122a064-ace3-4aff-bc45-556917be6269.jpg width=30.2099992752075 height=28.5950003623962  /> (Figures 4-5).

From Figure 4 it is obvious that derivatives of crosssections with respect to energies along the real axis are equal to zero at points corresponding to the threshold energy of n particle production, i.e., though the threshold values of energy are branch points for the cross-sections, they have continuous first derivative along the real axis at the branch points. This can be illustrated in the following way.

In the examined approximation of equal denominators, for the even number of particles value of square of scattering amplitude at the maximum point can be written like:

500476\df9128af-49cf-44b8-8c9d-3399b5135e56.jpg width=245.00499420166 height=79.5149971008301 />(33)

where 500476\3cfbbaf2-f731-4ada-b72e-2f40cae08903.jpg width=34.9599992752075 height=30.2099992752075  /> defined by (see [1]):

500476\0ae014a9-989d-405f-ac8d-1c1a2d75986b.jpg width=225.91000289917 height=62.0350028991699 /> (34)

Derivative from Equation (34) along the real axis at the threshold branching-point is infinite. However, cause at this point value of 500476\3c59a569-531b-4fef-8bd5-3f366343bf49.jpg width=34.9599992752075 height=30.2099992752075  /> is zero, than from Equation (33) it is obvious that derivative of 500476\2ddbd8bd-335b-44dc-b337-008b42418e55.jpg width=42.9400007247925 height=28.5950003623962  /> will be converge to threshold along the real axis tends to zero.

As it follows from Figures 4-5, 500476\00272e6a-f80f-479f-999b-148a3829a215.jpg width=66.7850028991699 height=39.805001449585  />monotone increases in the all considered energy range. At the same time from Figure 6 one can see that 500476\b52ecb31-711f-4c68-97f3-f4f3ccc8829e.jpg width=76.380001449585 height=39.805001449585  /> has dropdown sections. Moreover, even on those sections, where500476\02c0d425-760f-4bb1-8ac6-bd7db3638ee7.jpg width=77.9 height=39.805001449585  />,500476\1dd209eb-2df5-4e68-9b23-10150b700a3f.jpg width=71.5350028991699 height=22.2299996376038  /> increase, corresponding 500476\9f4053c6-a8db-4407-8da1-684938eba630.jpg width=66.7850028991699 height=39.805001449585  />decrease. It makes possible to conclude, that amplitude growth at maximum point (which is the consequence of virtuality reduction) is generally responsible for the growth of inelastic scattering cross-section.

As it evident from Figures 4-5 for some values of energy Equation (31) has a positive energy derivative and for some values of energy Equation (31) has a negative energy derivative. This makes a question? If we form from them a quantities

500476\6518f55a-ff1b-4521-ae53-99b58381bc71.jpg width=192.47000579834 height=57.2849992752075 /> (35)

the range of threshold energies for 1, 2, ∙∙∙, 8 particle productions. Via 500476\6dd562bd-937f-4c1d-96e1-fa4fc243e708.jpg width=60.419998550415 height=38.1900007247925  />was denoted one of the contributions from the diagram (shown on the right) to inelastic scattering cross-section.

500476\ad265e73-7494-4f3a-a668-e58ff3e59fe8.jpg width=189.33500289917 height=55.669998550415 /> (36)

where L is defined by Equation (32), is it possible to choose the “coupling constant” L so that the value of Equation (35) has a characteristic minimum for the total proton-proton scattering cross-section? Answer for this question is positive (see Figure 7), i.e., the curves agree qualitatively at the close values of L. The energy range shown in Figure 7 takes into account all the inelastic contributions. We find indeed very interesting result that curves presented on Figures 7-8, where calculated values of Equations (35), (36) are given at500476\a5a1a77f-7bc1-4a34-b323-8b0cb0844b81.jpg width=63.65 height=22.2299996376038  />, qualitatively agree with experimental data [10,11].

Let us point to the fact that in Figures 7-8 the minimum at higher energies 500476\4815fe17-9fc9-4329-b457-9f60dc2f6adc.jpg width=30.2099992752075 height=28.5950003623962  /> than in the experiment. We believe that the accounting contributions with higher number of secondary particles n to 500476\c664e77a-18c1-443f-9b65-f77329079455.jpg width=66.7850028991699 height=39.805001449585  /> and the

corresponding change of constant L will “move” a maximum to a required area.

Moreover, in this paper we have examined the simplest diagrams of 500476\166ad288-d904-49eb-a3ef-72547fda889d.jpg width=22.2299996376038 height=31.825  /> theory and we intend to compare the qualitative form of these cross-sections with experimental data, but do not claim quantitative agreement. It is possible to hope that the application of similar computation method to more complicated diagrams in more realistic models will lead to correct outcome.

As known, within the framework of Reggeon theory the drop-down part of total cross-section is described by the Reggeons exchanges with interception less than unity [12,13]. The cuts concerned with multi-Reggeon exchanges with participation of Reggeons with intercept greater than unity are responsible for the cross-section growth after the reaching the minimum [5].

As will be shown further, the accounting of 500476\2e1ab450-5e3b-4c2a-ad02-a045f9bc442e.jpg width=66.7850028991699 height=39.805001449585  /> at 500476\eb84ccb9-bc67-4f4e-9c39-151459e75d70.jpg width=42.9400007247925 height=22.2299996376038  /> will not change the behavior of function 500476\50629d4e-aef5-4770-99d3-8d4796b6f6db.jpg width=74.7649971008301 height=39.805001449585  />Equation (35). This means that within the framework of given model the summation of multi-peripheral diagrams, when we compute the imaginary part of elastic scattering amplitude, will not result in power dependence on energy, since this dependence is monotonic. This, in turn, will mean that the appropriate partial amplitude has no pole singularity! And this obviously differs from the results of standard approach in calculations of multi-peripheral model and from the results of Reggeon theory (see f.ex. [2]).

Another argument in favor of this hypothesis are the results of the “multiplicity distribution” shown in Figure 9, where axis of ordinates designates the number of particles n and abscissa axis designates the value of:

500476\6906622f-07f7-48c8-a2d5-67659fde06c9.jpg width=127.3 height=79.5149971008301 /> (37)

The Poisson distribution for the same average like for distribution Equation (37) is given for comparison. The energy 500476\29d04bab-450a-47d5-9c38-8c765f9bc066.jpg width=30.2099992752075 height=28.5950003623962  /> = 15 GeV is chosen for example, because at higher energies all distributions is no longer fit in the range from 0 to 8 particles. As is obvious from Figure 9, the distribution Equation (37) significantly differs from the Poisson distribution, which, as it is known, lead to powerlaw behavior of the imaginary part of inelastic scattering amplitude and, consequently, to the pole singularity of partial amplitude [5,14].

The described differences from a Regge theory are caused, apparently, by different physical mechanisms determing the inelastic scattering cross-section growth. In our model, a reduction of virtualities at the point of constrained maximum of inelastic scattering amplitude play a role of such mechanism. Consideration of similar diagrams in [2] lead to

500476\3e95f2e5-f3fe-4ff3-a5cc-7698c18fa738.jpg width=117.705001449585 height=50.919998550415 /> (38)

At the same time a similar result is obtained in [15] by the calculating of phase space with “cutting” of transversal momenta, i.e. authors ignore the dependence of inelastic scattering amplitude on rapidity, and its role is reduced only to the cutting of integration over transversal momenta. Similar results are obtained in [5,14], where examined diagrams of same type, but with the exchange of Reggeons instead of virtual scalar particles was considered. In [5,14] as a result of approximation authors totally ignored the dependence of expression under the integral sign for cross-section on particle rapidity in the finite state, thus obtained results include the dependence on energy 500476\f678a385-f81a-4980-8292-5c958b8e011d.jpg width=30.2099992752075 height=28.5950003623962  /> only through the rapidity phase space. At the same time, as it evident from previous argumentations, the dependence of scattering amplitude on longitudinal momenta or rapidity is essential, because it is responsible for the certain mechanism of inelastic cross-sections growth and their sum.

Moreover, Equation (38) has positive derivative with respect to energy at sufficiently great n in sufficiently wide energy range. At the same time, sum of such expressions in [2] results in the cross-section, which decreases monotonically with energy growth. The reason for this may be apparent from Equation (38) factorial suppression of contributions with large n, which provide the positive contributions to derivative with respect to energy.

In the presented model such suppression disappears at transition from Equation (3) to Equation (6) due to taking into account diagrams, with different order of attachment of external lines to the “comb”. The fact that the inclusion of such diagrams is essential as it seen from Figure 10, where the ratio of contribution 500476\76a33eea-bd40-461c-a6b9-82d8e80932aa.jpg width=77.9 height=39.805001449585  /> from a diagram with the initial arrangement of momenta (see Figure 2 in [1]) corresponding to the first summand in a sum Equation (30) to all sum500476\e7f7d259-daca-4276-80a0-cf22d82da3d8.jpg width=77.9 height=39.805001449585  />is given.

As seen from Figure 10 contribution from a diagram with the initial arrangement of external lines in the wide energy range is small fraction of the total sum Equation (30), which was natural to expect since sum Equation (30) has enormous number of positive summands. For the same reason, as was shown on Figure 10, the quota of contribution from a diagram with the initial arrangement of particles decreases sharply with increasing number of particles n in a “comb”.

At the same time, as it follows from Equation (31), the growth of scattering amplitude at the maximum point related with the mechanism of reduction of virtualities can cause the growth of inelastic scattering cross-sections500476\8cdb850b-e445-479f-b429-a3989d9c6dfe.jpg width=66.7850028991699 height=39.805001449585  />and, consequently, the growth of total crosssection. As an argument we can show results of numerical calculation of the function Equation (39), which are listed in Table 1.

500476\0b1ad3ad-a0c6-489a-bec7-def4c217ff21.jpg width=314.925 height=90.630001449585 /> (39)

This function is the ratio of increasing amplitude at the maximum point to the multipliers, which “working” on lowering of the total cross-sections with energy growth.

Submitted data shows that the mechanism of virtuality reduction is “stronger” than multipliers, which “working” on lowering of the total cross-sections with energy growth.

From Equation (33) follows that with increasing of n amplitude at the maximum point will increase sharply with energy growth. Thus, we can expect that factor500476\f718cc96-2845-4072-8131-cd63dcdb373f.jpg width=77.9 height=39.805001449585  />, which besides of 500476\f3c0393a-e11c-4c72-96f6-900ba6b1d67c.jpg width=63.65 height=39.805001449585  /> also inters into the expression of cross-section will decrease, but quite slowly. As is obvious from Equation (9), the possible decrease of 500476\6ba77dda-a7e6-4c30-982f-8c998c038328.jpg width=77.9 height=39.805001449585  /> is caused due to the fact that 500476\e7268c4b-ec15-4c31-80c0-6bd8290f45c6.jpg width=77.9 height=39.805001449585  /> include the product of terms, corresponding to diagrams in which external lines with the same momenta can be attached to the different vertices of the diagram. As result, the momentum of such line cannot have a value that simultaneously set maximum for both vertices. Moreover, with energy growth distance between rapidities corresponding to particles, which providing maximum at the different vertices of the diagram, increase. This can lead to decreasing of value 500476\42d4090a-a603-4a85-a0d1-88cc96afc894.jpg width=77.9 height=39.805001449585  /> with energy. However, as it obvious from relations (see Equation (75) and Equation (81) [1]), we write them here:

500476\be5ec396-9255-4605-ac0c-ef61172239cb.jpg width=216.31499710083 height=62.0350028991699 /> (40)

500476\64168cf7-0695-460a-8d2e-abd43e6ee627.jpg width=227.42999420166 height=62.0350028991699 /> (41)

the difference of these rapidities decreases with increase of particle’s number on the diagram. Therefore, it is hoped that decrease of500476\925f56b6-d98c-4513-b59f-5056e5b469f0.jpg width=77.9 height=39.805001449585  />, even if it will take place, will be not too sharp and cross-sections for high multiplicities of particles will also grow at least in the certain energy range. This will lead to the amplification of contributions with positive derivative with respect to energy into the total scattering cross-section.

As it follows from Equation (33), that at sufficiently high energies the amplitude at the maximum point tends to a constant value and mechanism of the reduction of virtualities become exhausted. This, however, can be avoided if we consider model in which the virtual particles on the diagram of the “comb” type are field quanta with zero mass. Then amplitude at the maximum point will tends to infinity at the infinite increase of energy. All computation in this case can be done similarly to what was described above. In this case, when calculating the first eight inelastic contributions in the wide range of energies does not give us contributions with negative derivative with respect to energy. Therefore we inclined to believe that such model can describe total cross-section growth to arbitrary large energies.

4. Conclusions

From demonstrated results it can be conclude that replacing of the “true” scattering amplitude associated to the multi-peripheral processes within the framework of perturbation theory by its Gaussian approximation is an acceptable approximation. The main conclusion is, that the mechanism of virtuality reduction (considered in [1]) may play a major role in ensuring the experimentally observed increase of the total cross-section [10,11], at least in some range of energies. This growth was obtained with allowance for 500476\28f749c5-6d16-4546-9889-9d522cc8ef64.jpg width=25.4599992752075 height=28.5950003623962  /> at500476\8b739280-dfc4-4ceb-93d2-32450ef0910d.jpg width=42.9400007247925 height=22.2299996376038  />. However, as it follows from500476\78de3376-7263-49ab-9e3a-0f7945b5eab3.jpg width=66.7850028991699 height=39.805001449585  />dependences, the maximum point of cross-section is shifted toward to higher energies with increase of n. We can therefore expect that in the consider energy range accounting of 500476\535b0f74-4895-4b0e-992d-4dd3dcef6bee.jpg width=66.7850028991699 height=39.805001449585  /> will add summands with positive derivative with respect to energy to expression for the total scattering cross-section, which leads to the fact that at least in the considered energy range obtained growth will only intensify.

Discussed above differences from the Reggeon theory suggest that our model is not a model of Reggeon with intercept high than unity and increase of the cross section is occurred in different way. This is also evident from the fact that in the model with a nonzero mass of virtual particles cross-section 500476\39642839-c65d-496a-baf8-2364e8c293ad.jpg width=25.4599992752075 height=28.5950003623962  /> at 500476\92e16d4e-7b7e-40bf-8d4a-f1cc7633e095.jpg width=70.0149971008301 height=28.5950003623962  /> tends to zero. This is a consequence of the fact that the absolute value of virtualities cannot decrease indefinitely, because it is bounded below by zero. Therefore, for sufficiently low coupling constant, when for however-anything high multiplicities do not contribute to the total cross section, at sufficiently high energies the total cross section should begin to decrease.

An additional conclusion is the necessity of accounting the sum of all diagrams with all the permutations of external lines for the scattering amplitude. Although with energy growth the fraction of contribution to the cross section of the diagram with an initial arrangement of the lines of the final particles increases and with 500476\d0aef7eb-eeb9-4d3e-86cc-c2dc64766ffa.jpg width=70.0149971008301 height=28.5950003623962  /> will tends to unity. In a wide range of energies, this fraction is small and decreases with multiplicity increase, which can be easily understood on the basis of the positivity of the amplitudes in the multi-peripheral model.

Note that the application of Laplace method is not limited by simplest diagrams. Therefore, our goal is further consideration of the more realistic models using same method, especially in terms of the law of conservation of electric charge.

REFERENCES

NOTES

^*Corresponding author.