Gauge invariant decomposition of 1-loop multiparticle scattering amplitudes

A simple algorithm is presented to decompose any 1-loop amplitude for scattering processes of the class 2 fermions→ 4 fermions into a fixed number of gauge-invariant form factors. The structure of the amplitude is simpler than in the conventional approaches and its numerical evaluation is made faster. The algorithm can be efficiently applied also to amplitudes with several thousands of Feynman diagrams. PACS: 12.15.L, 02.70 The complete calculation of the electroweak radiative corrections to the class of processes ee → 4f is still missing, for several reasons of theoretical but also especially technical origin. At 1-loop the probability amplitude is given by the sum of several thousands of Feynman diagrams, that we try to organize in a sensible way. The algebraic programs which should perform the simplifications find severe obstacles, because they have to deal with huge expressions: it is indeed very difficult to look for simplification patterns, taking the amplitude as a whole. We propose the opposite approach: having a physically motivated structure in mind, we can apply it systematically to every single Feynman diagram. The simplification of a small expression is very efficient and the bookkeeping of the various contributions follows from the beginning a precise pattern. The physical picture we are thinking of is the following: the interaction of elementary fermionic neutral and charged currents, which are factorized in the amplitude, is described by a rank-3 Lorentz tensor, which can be evaluated either at treeor at 1-loop level and can be decomposed in a gauge-invariant way. The paper is organized in the following way. In section 1 the present approaches and their efficiency in dealing with the scattering amplitudes are briefly described. In section 2 the decomposition proposed in this paper is formulated, proving the gauge invariance of the coefficients. In section 3 the algorithm to reduce any 1-loop Feynman diagram into the proposed form is described, and in section 4 we make some final remarks. 1 Present approaches The number of Feynman diagrams which contribute to the probability amplitude of a process of the class ee → 4f is very large. In the following we consider only processes with massless external fermions; in table 1 we list the number of diagrams which are part of the virtual corrections to some representative processes, omitting the tadpoles contributions. One first comment is that it does not make sense to consider, pictorially, Feynman diagrams as building blocks of the calculations. As the well known example of on-shell W -pair production shows, individual Feynman diagrams contain unitarity-violating terms, which cancel in the sum at the level of the amplitude; the interference between different diagrams yields most of the physical contributions. process diagrams spinor products form factors form factor + eq. of motion ee → μνμud 1907 2069 64 16 ee → eedd 8522 4056 128 34 ee → eeee 21444 9157 384 104 Table 1: Some representative processes of the class ee → 4f and the number of Feynman diagrams due to their 1-loop virtual corrections, excluding tadpoles contributions, in the limit of massless external particles. [email protected]