Laplace method for the simplest diagrams of elastic scattering of scalar particles: https://arxiv.org/abs/2003.07321

We propose an algorithm for the application of the Laplace method for the calculation of the simplest Feynman diagram with a single loop in the scalar {\phi}^3 theory. The calculation of the contribution of such a diagram to the scattering amplitude requires the calculation of a fourfold integral over the four-momenta components circulating in a loop. The essence of the Laplace method for the calculation of multiple integrals lies in the fact that if the module of an integrand has a point of sufficiently sharp maximum inside the integration domain, then the integral can be replaced by a Gaussian integral by representing the integrand in the form of an exponent from the logarithm and expanding this logarithm into Taylor series in the vicinity of a maximum point up to the second degree terms. We show that there are two-dimensional and non-intersecting surfaces inside the four-dimensional region of integration, on which the maximum of the module of integrand is reached. This leads to a problem that the integrand is non-analytically dependent on the parameters responsible for bypassing the poles. Also the derivatives of logarithm of the scattering amplitude are non-analytically dependent on these parameters. However, in the paper we show that these non-analyticities compensate each other. As a result of such a procedure, three of the four integrations can be done analytically, and the calculation of the contribution of the diagram to the scattering amplitude is reduced to a numerical calculation of the single integral in finite bounds from an expression that does not contain non-analyticities. The described calculation method is used to construct a model dependence of elastic scattering differential cross section d{\sigma}_elastic/dt on the square of the transmitted four-momentum t (Mandelstam variable).

Comments: 31 pages, 10 figures, in Russian