Gauge Invariance in the Process $e^{+}{e}^{-}\to {\bar{\nu }}_e{e}^{-}{W}^{+}\to {\bar{\nu }}_e{e}^{-}u\bar{d}$

Abstract

The process $e^{+}{e}^{-}\to {\bar{\nu }}_e{e}^{-}{W}^{+}\to {\bar{\nu }}_e{e}^{-}u\bar{d}$ is considered as an example of the problems associated with maintaining gauge invariance in matrix elements involving unstable particles. It is shown how to construct a matrix element that correctly treats width effects for the intermediate unstable $W$ boson and that is both $SU(2{)}_L$ and $U(1{)}_{e.m.}$ gauge-invariant. $SU(2{)}_L$ gauge-invariance is maintained by Laurent expansion in kinematic invariants and $U(1{)}_{e.m.}$ gauge-invariance is enforced by means of a projection operator under which the exact matrix element is invariant.

Comments: Case of $SU(2{)}_L$ gauge-invariance addressed. Substantial changes in text. 9 pages LaTeX, also available from via anonymous ftp from ftp://feynman.physics.lsa.umich.edu/pub/preprints/UM-TH-96-02.ps