competition of these two multipliers will result on the dependence of the sum of partial cross-sections on?
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 Figure 2 is impossible. We can avoid this difficulty in the following way. The maximum in the right part of cut diagram in Figure 2 is attained at.
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 and. In other words, multiplying Gaussian functions corresponding to the right-hand and to the left-hand part of interference diagrams in Figure 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).
We assume that summands in Figure 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 righthand part of diagram corresponds to j = 1. In other words, the line of secondary particle with the four-momentum pi is attached to the i-th top in the right-hand part of cut diagram in Figure 2. As follows from Equation (7), the interference contributions exponentially decrease with the r2 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 r2 growth. The value of r2 is proportional to the square of magnitude, 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 r2 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 is determined by the sum of huge number of small interfereence contributions. When the magnitudeis increased with the further growth of energy, the decrease rate of interference contributions increases, while the growth rate of their number with the r2 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 r2, which can be calculated by Equation (7). If we compose the n-dimensional vector (we denote it through from the particle rapidities Equation (5), which constrainedly maximizes the function associated with the diagram with the initial arrangement of momenta in Figure 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 permutation we designate through,.
Thus, each of the terms in the sum Figure 2 can be uniquely matched to its angle. At the same time the variable is more handy for consideration than an angle Using Equation (5), can be shown that the variable z can take discrete set of values:
Note that although the relation Equation (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 Equation (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 Equation (5), it appears, that the different values of interference contributions correspond to the one and the same value of (see Figure 3).
Thus, while each contribution is associated to some value of variable z in the approximation Equation (5), the value of contribution is not the unique function of z. However, the sum expressing the partial cross-section can be written in the following way
where the number of summands to which the value is corresponds in the approximation Equation (5). The average value of all interference contributions in Equation (10) is already the unique function of. Therefore, we introduce notation
Figure 3. The interference contributions dependence on at = 100 GeV: (a) n = 8; (b) n = 9. Here and in subsequent figures the interference contributions divided by the common multiplier are indicated on the y-axis. Obviously, that to the one value of 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 Equation (25) and Figure 4).
where 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 Equation (9) 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 l0. In particular, in this paper we managed to calculate the interference contributions up to l0 = 6. Partial cross-section can be written as
where is the sum of contributions sufficient at “high” energies, and 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 and.
3. The Approximate Calculation of
As follows from Equation (7), the exponential factor exerts the most significant effect on the dependence of on. Note that the expression entering into the exponent in Equation (7) depends only on those matrix components, which are at the intersection of the first n rows and first n columns, since all column components starting with are zerobecause 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 through and a matrix, which is obtained from the matrix in analogy, through, we have
The matrices and 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 matrixthrough and the eigenvalue itself —through. This implies
Since the minimum eigenvalue of matrix is equal to the minimum values of quadratic form for the unit vectors, the magnitude is not lessthan the minimum eigenvalue of matrix. By analogy the magnitude is not less than the minimum eigenvalue of matrix, which coincides with the minimal eigenvalue of matrix and is reciprocal of the maximum eigenvalue of matrix denoted through. Thus,. From this it follows that, the maximum eigenvalue of matrixdoes not exceed. By analogy we obtain that the minimum eigenvalue of matrixis no smaller than, where is the minimum eigenvalue of matrix. Thus, an interval enclosing the eigenvalues of matrix is, at least, twice smaller than an interval enclosing the eigenvalues of matrix. We can demonstrate that at approximation of an equal denominators  the value of can be estimated in the following way
i.e., an interval enclosing the eigenvalues of matrix at any energies and number of particles is less than unitywhereas at the considerable values of, i.e. at a distance from the threshold, this interval is much less than unity.
Therefore, if we reduce matrix to diagonal form, it will be close to a matrix multiple of unit matrix. If we represent this matrix in the form
where is unit matrix, the eigenvalues of the traceless matrix will be small. Then
where are the eigenvalues of matrix is the transformation matrix to the basis composed from the eigenvectors of matrix (the summation over repeated indices is supposed). The second term in this sum is small in comparison with the first one due to the smallness of eigenvalues as well as due to their different signs (since the trace of matrix is zero, the different terms over k partially compensate each other). Therefore, we can adopt the following approximation:
To approximately calculate the trace of matrix we select the spherically symmetric part of matrix representing it in the form
The results of numeral calculation of the eigenvalues of matrix (which are denoted through ) 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. Since all the eigenvalues of matrix are positive, which means that their deviation from average value is less than this average in absolute value (see Table 1).
Note that the matrix can be represented in the following form:
by analogy we can conclude that the minimum eigenvalue of matrix
(which is maximum in absolute value, see Table 1) is greater than the doubled minimum eigenvalue of matrix. This means that all the eigenvalues of matrixare less than unity in absolute value. It applies equally to the eigenvalues of matrices and. Therefore, we can represent the matrix as the expansion in powers of.Since matrix is traceless by definition, then a nonzero contribution to in addition to the term term of “zero” order can give terms starting with the second-order. As it follows from Table 1, the maximum in absolute value eigenvalue of matrixincreases with the energy growth. Thereforewe can expect that at “low” energies higher-order terms will make negligibly small contributions. In such an ap-
Table 1. Results of numerical calculations of the eigenvalues of matrix.
proximation we have:
Let Equation (7) is taken in place of Equation (11) in approximation Equation (22), then we have
Let us introduce the following notation
If we assume that multiplier is weakly dependent on, we obtain
where is the minimum value of for which can be numerically calculated all interference contributions. Therefore, the magnitude can be directly calculated numerically. The results of numerical calculation ofover all interference contributions in comparison with the results obtained by Equation (24) are demonstrated on Figure 4, it follows that such an approximation is acceptable at “low” energies.
Results shown in Figure 4 confirm also our assumption that weakly depends on. To analyze this dependence we turn to Figure 5. It is obvious, that the magnitude takes small values at “low” energies.
Figure 4. Two results of evolution by it a direct numerical calculation with consideration of all interference contributions (circles) and by it approximation Equation (24) (straight line) at n = 8, = 10 GeV (a); n = 9, = 10 GeV (b); n = 8, = 100 GeV (c); n = 9, = 100 GeV (d).
Figure 5. The values of obtained by direct calculation values Equation (24) for all interference contributions for n = 8 and n = 9 at= 10 GeV (a), (e) accordingly; for the same number n, but at = 100 GeV (c), (g) and the ratio for n = 8 and n = 9 at = 10 GeV (b), (f); = 100 GeV (d), (h).
This means that
takes large values at the same energies. Indeed, as it follows from the expression for the matrix, Equation (26) tends to infinity on the threshold of n particle production, and this means that at threshold the volume of phase space with n particles production in the inelastic process 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 Equation (26) 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 Equation (26) is great at all. Since the function varies slightly at the great values of argument, the function weakly depends on.
To estimate roughly the function we can replace it by the Taylor expansion taking into account just linear contributions. The expansion coefficients are found by the calculating of for close to 1 and (−1). In these cases the values of
were obtained directly for all proper interference contributions, and after that we obtain the values of by averaging using Equation (24).
The values in Figure 5 have been obtained by the direct calculation of
with consideration of all interference contributions at different.
So, we have the following expression instead of Equation (25)
where the coefficients and are found by above mentioned method.
4. Approximate Calculation of the Values
Let us turn to the new variables
where are determined by Equation (5), are considered as the components of vector, which, as it follows from Equation (30) is of unit length.
Thus, the angle between the vectorand vector obtained by the permutation of corresponding components is the same as the angle between the vectorand vector. Moreover, as it follows from Equation (5)
It follows that all vectors are orthogonal to vector
Therefore, considering vectors as the elements of n-dimensional Euclidean space, which we de note through, then the ends of all vectors are lie on the unit sphere embedded into the -dimensional subspace of. We denote this sphere through and shape formed by the set of points in which the ends of vectors (,) come, denote through. In particular, when the sphere and figure graphically look like in Figure 6.
We examine some geometrical properties of figure at arbitrary n. If we apply the permutation transformation component to all vectors in the n-dimensional space, where the vectors are primordially defined, the examined -dimensional subspace as well as a sphere and figure go into themselves. As it follows from the group properties of permutation group, the each point of figure can be obtained from any other point by some transformation. This means that the configuration of the points of figure relative to each of these points must be identical, that can be clearly seen in Figure 7(a).
Figure 6. A sphere S2 and figure F4!, which is demonstrated by points. Basis in the four-dimensional space is chosen so that the one of vectors coincides with the vector and the three basis vectors of three-dimensional subspace, into which depicted sphere is embedded, are perpendicular to e4.
Figure 7. The partition of sphere S2 by shortest arcs joining the points of figure F4! into the two “hexagonal” and one “tetragonal” regions (a); (b) areas, which is located on the borders of 4 or 6 points belonging to figure F4! can be divided between those points into figures of equal area; (c) whole sphere S2 is divided into figures of equal area, each of which contains the one point of figure F4! one of these shapes are painted in white.
As it follows from Equation (31), besides the end of each vector a figure contains also the end of vector, i.e., a figure has a center of symmetry, which coincides with the center of sphere. In this case, if we using point of form path from the point to the point, then it will be simultaneously formed a centro-symmetrical path, that leads from to of figure.
Joining these paths we will obtain the closed path, which “girdles” the sphere. If we assume that there is such a “girdling” path, inside of which are concentrated all points of figure, we would find that the figure has a “boundary” and “internal” points, that would contradict the fact that spacing of all points relative to each point of the should be the same. In other words, the points of figure must “crawl away” all over the sphere and cannot be concentrated on some area of the sphere.
If we consider a vector, then closest to it are the vectors corresponding to permutations defined by the following relation
The type of “cut” diagrams corresponding to such permutations is shown on Figure 8. At the same time, all the components of vector, except the l-th and l + 1, are zero, whereas these two components take on the least values in modulus and, respectively.
Thus, we can conclude that the each point of figure has nearest neighboring points, which lying at distance of from it:
Connecting the each point of figure with its nearest neighbors’ points by shortest arc thereby we divide the sphere into closed regions as is shown in Figure 7(a). Indeed, let us choose the some point A0 of figure and will move from it to the nearest point A1 along a shortest arc, then we move from the point A1 to the nearest point A2 etc. At the same time, motion in a backward direction is prohibited. Thus, there are paths going out from each point, and paths are allowed at each step. But since figure has the finite number of points at some step we will surely come back to the point A0.
Moreover, since shortest arcs joining two nearest points are subtended by equal chords in length (see Equation (34)), this arcs are of the same length. Let us consider any two neighboring points and of figure. Under any transformation the shortest arc, which joins the points and, and an arc joining the points and 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. It follows that, if we examine closed areas which include any point of figure, then the adjacent areas to all points of this figure will have the same “area”.
There is one more requirement, to which the areas obtained by partition of the sphere 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 Equation (34), when n is large the value of is small. This means that when we join the each point of figure with its nearest neighbors by the shortest arcs of sphere, these arcs practically coincide with chords, which tights them.
If we assume, that any two chords andintersect in an internal point, then it is possible “to pull” on them a two-dimensional plane. Then we get a flat rectangle, which has at least one angle no smaller than 90˚. 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. In
Figure 8. Diagrams, which correspond (n – 1) vectors closes to vector.
this case, either a or b would not exceed, i.e., the figure contains points, which are at distance less then but that cannot happen due to minimality of this distance.
Thus, we can conclude that at an arbitrary n a sphere can be divided into the parts of equal area, each of which contains only one point of figure, as it shown in Figures 7(b), (c).
Let us introduce a multidimensional spherical coordinate system so that the end of vector is the “north pole” of sphere. Then the number of points of figure to which the values of variable in the interval correspond, is equal
is the Euler gamma function.
To verify the validity of Equation (35) 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 cannot be realized). The distributions of interference contribution from the variable and the graphs of function from Equation (35) are shown in Figure 9. Obtained results of numerical calculation of interference contributions and by Equation (35) are in a good agreement.
Moreover, as it follows from Figure 9(b) and from Figure 9(c) this fitness is improved with increasing number of particles n, i.e., Equation (35) is suitable for large n, when the direct numerical calculation of all interference contributions is impossible.
Taking Equation (35) and Equation (9) into account we obtain the following the approximate equality
Figure 9. Comparison distribution of the interference contribution by the variable (histogram) and function (solid line) at (a) n = 8, = 0.1; (b) n = 9, = 0.1; (c) n = 9, = 0.05. Here is the number of interference contributions corresponding to value of in the proper interval of width.
Verification results of Equation (37) at n = 8 and n = 9 are presented in Figure 10.
Another verification of considered above equations is presented in Figure 11, where the values of and approximating magnitudes (here is calculated by Equation (29)) are compared.
From results demonstrated on Figure 4 and Figures 10-12, we can conclude that the at least for those numbers of particles for which it can be directly tested Equation (12) with Equation (29), Equations (35)-(37) yields an acceptable approximation. As is obvious from Figure 4, than closer energy to the threshold of n particle production, the better approximation Equation (29). 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 Figures 10(b-d), the accuracy of approximation Equation (37), 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 in the simplest case of multi-peripheral model taking into account all significant interference contributions.
5. The Model of Dependence of Hadron Inelastic Scattering Total Cross-Section on Energy
Let us consider the magnitude
which within the framework of the discussed above model is an analogue of total inelastic scattering cross-section. Here is the maximum number of secondary particles allowed by energy-momentum conservation law and is the dimensionless coupling constant, which we considered as a fitting parameter (see Equation (32) ). Since the calculation of up to 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 can be fitted so that the dependencelooks like the behavior of total hadron-hadron scattering cross-section with a minimum about =10 GeV. The result of such a fitting is shown in Figure 13 (in that calculations we take proton mass as mass of primary particles and pion mass as mass of secondary particles).
Figure 10. Comparison of the values of right-hand side and left-hand side of approximate equality Equation (37) at n = 8 (a, b) and n = 9 (c, d). Circles are the values of calculated with consideration of for all interference contributions; crosses are the values of function from Equation (35).
Figure 11. Comparison of the values of obtained with consideration of all interference contributions (circles) and the approximate values of (blue crosses) for (a) For n = 8 at = 10 GeV; (b) For n = 8 at= 100 GeV; (c) For n = 9 at = 10 GeV; (d) For n = 9 at = 100 GeV.
Figure 12. The partial cross-section dependence on energy calculated over all interference contributions (solid line) and by Equation (12) with the application of approximations Equations (29), (35), (37) (red circles): (a); (b); (c); (d); (e). This approximation is acceptable at least in the range of parameters in which they are can be verified.
Figure 13. Theoretical dependences of the (a) and (c) obtained for the energy range = 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 multiplicities. Experimental data for the inelastic (b) and for the total (d) pp scattering cross-section Reference [7,8] presented for qualitative comparison with the prediction from our model. Note: data-points for the inelastic cross-section, obtained from the definition.
Quantitative comparison with experimental data requires the consideration of more realistic model than the self-interacting scalar field model.
From obtained result, one might conclude that the considered in  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 [1,2] 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 allows to calculate another types of diagrams corresponding to various scenarios of hadron-hadron inelastic scattering and compare it with experimental data.