integrals. We shall denote the contributions from these terms by F A. Then using the
above identities, an integration by parts (for which the surface term vanishes) makes the
integrand similar to that in Fj^" and simplifications can be achieved. For details see Sec
3.5.
pAAV g3 K M J -_.9l3MI-1 0 TK2(MI 1)2A'
AV B1 4x72To MI 1 Jo L-H(M + MIT)3
lT(Ho H)(M1 l)MA' IT(Ho H)(M1 l)MIA]
1/- H(IT + + 1)(M + MIT) / -'H(M + MIT)2
+ (1 2 ) (3-26)
where
m 2M + 1)2 [ (M. 1)3
A' M /2].2 (3-27)
/IMI(3[M. + 1)2 MIM(M1 -1) MIM(. + 1)2 ( 27)
Since the integrand of (3-26) is a rational function of T the last integral can also be done.
The evaluation is sketched in Sec 3.5.
There remains the contribution of the term e-Ho which would give a divergent
integral. However, because the TI, T2 (T1, T3) dependence in the exponential is disentangled
by our choice of Ho, the sums can be directly analyzed in the a -+ 0 limit, giving an
explicit expression for the divergent part in terms of the lattice cutoff. We denote this
contribution, containing the ultraviolet divergence of the triangles, by AAV. Referring to
Sec 3.5 for details we obtain
AAV g3K^A M [M-1 I (N/l N2([. +1) ( 2p+1 a
pV + In + f-
B2 20 M81M.1- 1 MM, M M ap
]My1 1 Nil N2(. +l)- I\
M M/M, M, M ) 3K 3 ( f
g3K A M '-1 B/' 2p(+ a A.V[ (M- 1)2a fa]
472ToMl. l [