where
ST 2 Mi(M1 1)2TK2 A' IT(Ho H)(M1 -1 )M1A'
'M /:H(T,1)(M1T + M)3 / -'.H(T + 1. + 1)(M+ MIT)
IT(Ho H)(M1- l)MA (3 37)
3 /-H(M + MIT)2
S1 = S )M[-(, )-- a l- )2- ff () (3-38)
I2 I TP2,P1, S2= S 1-, M1, (3-39)
and where we r
call, for convenience, our definitions (appropriate to the case kl, I > 0)
M2 f2 M2 V2 (IL. + 1)2 (1 1)2
A M+ M + +
12(1 1)2 l 2( + 1)2 (Mi- 1)2 (22 + 1)2
A' M2 22 2 (3[ + 1)2 (M1 13
[M (3[. +1)2 1MiM(MV- -1) MAM{(3[. +1)2
B (3 )3 (M1 1)3 MMI
B ( + +
1/-(Mi-1) /-(.V+1) 1(Mi-1)
(M1 1)T K2
H = H(T, 1) = (. + l)P* + lTP* + (- )T K2
M + MIT M1M
Ho Ho(T, 1) (1[. + )PI* + TP*
a M1 (FMp + 1)
l3 1IM
To complete the continuum limit we assume M, M1, 31 f large and attempt to replace
the sums over 1 by integrals over a continuous variable ( A l/MI, with 0 < ( < 1. This
procedure is obstructed by singular behavior of the integrand for near 0 or 1. When this
occurs, we introduce a cutoff c << 1, and only do the replacement for c < < 1 e,
dealing with the sums directly in the singular regions. The detailed analysis is presented
in the appendices of the paper on which this chapter is mostly based [6]. Referring to
-40)