and
M1 M2 2 (1 12
A + + + (3 14)
12 1 ) 2( 4( + )2 1 ,+ 21 )
B, 12( +2 ( -2 (3-15)
M 2 + 1)2f
1)2 ( 1-)2 M( 2 (
,M, 1)2 M:1 1(f 1)2 11"( I )2
Note the constraint IT, I (1" ) )T2 -(Mi-)T = 0, which implies that for fixed 1, o::1 two
of the T's are independent. Also, momentum conservation implies that K,j is cyclically
--mmetric and we therefore use K K112 Ks2 Al31. -' introduce the following
notation that will help streamline some of the formulae: P1* = /' P* /M P*
For example,
K2 = -Mi 'P1 PP + ) = M1' M(P P P*). (3-18)
We are now dealing with potentially ultraviolet divergent diagrams. To reveal the
ultraviolet structure we consider the continuum limit in the order a -+ 0 followed .
m -- 0. Recall that a :/ 0 serves as our ultraviolet cutoff. In the a -- 0 limit we can
attempt to replace the sums over kA, k' (k'A, k2) for kA > 0 (kA < 0) '. integral over T, and
T2 (Ts-). Since we wish to keep 1 fixed in this first step, for the case kA > 0 we express eT
in terms of T\ and T1: T3 (1\ + ( i: + 1)7')/(Mi 1). For the case kf < 0, it is more
convenient to express T2 in terms of T, and 3: T ( (I'j +1'))/( +- 1'). find
ET (MfT2 I '. T) / ( 1: 1) = (MTs I Ir T,)I/( 1. 1'). For the A term, this ccdurc