Projection scheme for handling large-number cancellation related to gauge invariance.

Abstract

A scheme, the so-called "projection," for handling singularities in processes such as ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}t\overline{b}{e}^{\ensuremath{-}}\overline{\ensuremath{\nu}}$ (or ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}u\overline{d}{e}^{\ensuremath{-}}\overline{\ensuremath{\nu}}$) is proposed. In the scheme, with the help of gauge invariance, the large power quantities ${(\frac{s}{{m}_{e}^{2}})}^{n}$ ($n>~1$; $s\ensuremath{\rightarrow}\ensuremath{\infty}$) are removed from the calculation totally, while in the usual schemes the large quantities appear and only will be canceled at last. The advantages of the scheme in numerical calculations are obvious; thus, we focus our discussions mainly on the advantages of the scheme in the special case where the absorptive part for some propagators relevant to the process could not be ignored, and a not satisfactory but widely adopted approximation is made; i.e., a finite constant "width" is introduced to approximate the absorptive part of the propagators phenomenologically even though QED gauge invariance is violated.