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We propose an algorithm for the application of the Laplace method for thecalculation 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 thescattering amplitude requires the calculation of a fourfold integral over thefour-momenta components circulating in a loop. The essence of the Laplacemethod for the calculation of multiple integrals lies in the fact that if themodule of an integrand has a point of sufficiently sharp maximum inside theintegration domain, then the integral can be replaced by a Gaussian integral byrepresenting the integrand in the form of an exponent from the logarithm andexpanding this logarithm into Taylor series in the vicinity of a maximum pointup to the second degree terms. We show that there are two-dimensional andnon-intersecting surfaces inside the four-dimensional region of integration, onwhich the maximum of the module of integrand is reached. This leads to aproblem that the integrand is non-analytically dependent on the parametersresponsible for bypassing the poles. Also the derivatives of logarithm of thescattering amplitude are non-analytically dependent on these parameters.However, in the paper we show that these non-analyticities compensate eachother. As a result of such a procedure, three of the four integrations can bedone analytically, and the calculation of the contribution of the diagram tothe scattering amplitude is reduced to a numerical calculation of the singleintegral in finite bounds from an expression that does not containnon-analyticities. The described calculation method is used to construct amodel dependence of elastic scattering differential cross sectiond{\sigma}_elastic/dt on the square of the transmitted four-momentum t(Mandelstam variable).

Laplace method for the simplest diagrams of elastic scattering of scalar particles