Toward Automatic Regularization for Feynman Loop Integrals in Perturbative Quantum Field Theory

Abstract

Themotivations of high energy physics collider experiments include the precise measurement of parameters in the standard model (Passarino & Veltman, 1979; van Oldenborgh & Vermaseren, 1990) and beyond. In recent years, the precision of high energy experiments at colliders has increased significantly as a result of developments in electronics. The detection of any deviations of the experimental data from the theoretical predictions may lead to the study of new phenomena. In modern physics, everything is composed of elementary particles, and there are four basic interactions acting on particles: gravitation, and the weak, electromagnetic and strong interactions. When we consider a scattering process of elementary particles, the cross section reflects the dynamics which govern the motion of the particles, caused by the interaction. All information pertaining to a particle interaction is contained in the amplitude according to the (Feynman) rules of Quantum Field Theory. Generally, with a given particle interaction, a large number of configurations (represented by Feynman diagrams) are associated. Each diagram represents one of the possible configurations of the virtual processes, and it describes a part of the total amplitude. The square sum of the amplitudes delivers the probability or cross section of the process (by integration). Based on the Feynman rules, it is the goal to obtain the amplitude using the steps listed in Figure 1. Feynman diagrams are constructed in such a way that the initial state particles are connected to the final state particles by propagators and vertices. Particles meet at vertices according to a coupling constant g which indicates the strength of the interaction. The amplitude is expanded as a perturbation series in g, where the leading (lowest) order of approximation corresponds to the tree level of the Feynman diagrams. The evaluation of tree diagrams is well known and analytical formulations exist, which have been developed into automatizations of Figure (1) and are heavily used in high energy physics. For the tree level, packages such as GRACE, COMPHEP, CALCHEP, FEYNARTS/FEYNCALC/FORMCALC, MADGRAPH, FDC, and so on, are available. 14