标量粒子弹性散射最简图解的Laplace方法
Laplace method for the simplest diagrams of elastic scattering of scalar particles
摘要：
我们提出了一种算法，用于拉普拉斯方法在标量理论中计算最简单的单圈费曼图。计算这样一个图对散射振幅的贡献，需要计算在环中循环的四动量分量上的四重积分。计算多重积分的拉普拉斯方法的本质在于，如果被积函数的模在积分域内具有足够尖锐的最大值点，则该积分可以由高斯积分代替，通过将被积函数表示为对数的指数形式，并将该对数在最大值点附近展开为泰勒级数直到第二次届我们证明了在四维积分区域内存在二维不相交的曲面，在这些曲面上被积函数的模达到最大值。这导致了一个问题，被积函数是非解析依赖于负责绕过极点的参数。此外，散射振幅的对数的导数非解析地依赖于这些参数。然而，在本文中，我们表明，这些非解析性相互补偿。作为这样的程序的结果，四个积分中的三个可以解析地完成，并且图对散射振幅的贡献的计算被减少到从不包含非解析性的表达式在有限界中的单个积分的数值计算。利用所述计算方法，建立了弹性散射微分截面dσ_elastic/dt与透射四动量t（Mandelstrom变量）平方的关系模型。
英文摘要：
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 ϕ^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 dσ_elastic/dt on the square of the transmitted four-momentum t (Mandelstam variable).
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