For some cases such as the Supersymmetric (Nf = 2 + Ns) or pure Yang-Mills
(Nf = N, = 0) the summands in the expressions for the Y's do not change sign. When
Nf < 8, so that these cases are asymptotically free, the summands on the right sides
of (3-64) and (3-66) have a sign which works against asymptotic freedom. Since the
full sum exhibits asymptotic freedom for each polarization, this means that that the
complementary time orderings, (3-63) and (3-67), must contribute more than their share
to asymptotic freedom. This fact may be useful for approximations involving selective
summation.
3.5 Details of the Loop Calculation
We present here some of the calculational details that were omitted from the above
sections for clarity.
3.5.1 Feynman-diagram Calculation: Evaluation of p^"V
In the calculation of F"' we start by integrating by parts. This transfers the
derivative to the factor (e-H e-Ho). For definiteness take the case I > 0. Then we
compute
a H Ho) H H TIIP Ho Ho TIP* HK (M )TIT2
9T2 T T2 M, (MT2 + MITI) 2
(3-70)
The first two terms on the r.h.s. partly cancel after integration over T1, T2. This is because
the integrals are separately finite, so one can change variables T = T2T in each term
separately. For the first term we find
dT dT2(T1, T2) H(TI, T2) TIlI P c-H(T1,T2
dTdT2I(T, 1) [H(T, 1) TIP*] -H(T) dTI(T, 1) ,(3-71)