Gauge invariance, infrared/collinear singularitiesand tree level matrix element for $e^ + e^- \to \nu_e \bar{\nu}_e \gamma \gamma$

Abstract

One of the necessary steps in constructing a high-precision option of KKMC, a Monte Carlo program for the high-precision simulation of fermion pair production at LEP and Linear Collider energies, was to make a careful study of the appropriate matrix elements calculated from QED and the complete standard model. In particular, the installation of the double bremsstrahlung matrix element for the process $e^ + e^- \to \nu_e \bar{\nu}_e $ into the scheme of coherent exclusive exponentiation (CEEX) was necessary. In the CEEX scheme one has to define an extrapolation and/or reduction procedure to enable the use of the matrix elements for kinematical configurations with a large number of outgoing particles. The process under study is particularly interesting because of the gauge cancellation of contributions for photon emission from incoming fermion lines and t-channel W. The QED U(1) gauge properties require terms of the triple and quartic gauge couplings to be taken into consideration as well. A natural separation of the complete amplitude into gauge invariant parts was found and is among the main results of the paper. Each part has a well defined physical interpretation, which after partial integration over phase space provides infrared singular, leading-log, next-to-leading-log, and other terms. Contributions related to the triple and quartic gauge coupling of W (extracted with the help of an expansion around the contact W-interaction) have been ordered as well. The separation can be of broader interest; it originates from the rigorous calculation of matrix elements; it visualizes, in the language of spin amplitudes, the properties of factorization necessary for the common multi-process picture. For example, the multiple photon algorithm of PHOTOS, based on the parton shower-like approach, profits from similar considerations. These somewhat speculative aspects of the calculation will be mentioned in the paper as well.