 
Summary: Nuclear PhysicsB229 (1983) 372380
~3NorthHollandPublishingCompany
THE BACKGROUND FIELD METHOD AND THE SMATRIX
L.F. ABBOTI"1, M.T. GRISARU2 and R.K. SCHAEFER1
Physics Department, Brandeis University, Waltham, MA 02254, USA
Received 26 May 1983
We provethat the Smatrixcan be correctlyobtainedfromthe gaugeinvarianteffectiveaction
in the backgroundfieldapproachto gaugetheories. In addition,we present a computationof the
twoloopfermioniccontributionsto the YangMills/3function.
1. Introduction
It is well established that the background field method [16] greatly simplifies
calculations of renormalization factors in gauge theories [4, 710]. Less appreciated
is the fact that the method can also be useful for calculating Smatrix elements. The
Smatrix is constructed from trees with vertices and propagators given by the 1PI
Green functions. When constructing these trees, it is sometimes convenient to use
a gauge, which is particularly well suited to the physical process being considered.
This is often a Lorentz noncovariant gauge. In the usual formulation of gauge
theories, the 1PI Green functions must also be computed in this noncovariant
gauge making loop calculations extremely difficult. In the background field approach
the gauge used for calculating 1PI graphs and the gauge used for constructing trees
