encounters no obstacle, and we obtain (displaying licitlyy the contribution for k, > 0)
IgAAV K^ M All (A1 1,1 iT2(1Ti I (/2 + l)T2)K2 A _TT2)
,q,, 9 rl2 11j, 1T1I VT T2) 4
( 1 +-+ 2)
)4 + ( 1. .- 2 ) (3 19)
It will be useful to note that H can be written in the alternative forms
H I= + 1) T22 [ )T, 2 3-20)
A1 (l 1,2 A] 1 1
Kp2 (,T T )'}'
(i + 1')T P* + I'T1 p* + (3 21)
where the first is useful when k ,l > 0 and the second for kA,l < 0.
However the B terms produce logarithmically divergent integral with this procedure,
so 1.. must be handled differently. To deal with these logarithmically divergent terms,
we first note the identities:
11 3 (MI 1) 7
SM ( 13 (3 22)
(Ti7+ T1 + T1)3 12 2M (1 2 +, T)2
S ((3 )
i-l 2M (MTf+ T) f ')
T2 A ( MM -3) Mi, i 2 ,
2 3-24)
( T I T % I I )3 0T ( 2 I M 1)2
(Ti I T I T3s)3 T (AT3 1T1i)2
where the partial derivatives are taken with T7 fixed.
Because of the divergences we can't immediately write the continuum limit of the B
terms as an integral. However we can make the substitution e-11 (e-Hf e-10) +
where H0 is chosen to be an appropriate simplified version of H, which coincides with 7H
at T = 0. For ki, I > 0, it is convenient to choose Ho (IT1 + (' 1) 2) P1*, whereas for
ki, < 0 ( (] + :' + '1i ) T) is more convenient. ii :: the factor (-c e-HII
regulates the integrand at small TI so that the sums then safely be .1 ',