In the .. '.,egral only the terms "oportional to q2 times the Caussian will exhibit a -# 0
divergences so we retain only those, i ::' general loop integral is given by
f ,q ktI 2 + t2 + t3\
2Jdqx1 3 \2 2" -,/a
( /a)2 (M {MI r Kr i 3 1 2 'k k 3 k
2(1, 1 12 1 13)3 yk r e K "
(3-131)
(T. arrow means that a -- 0 finite terms have been 1 .: ..1 ) Some simplification
can be done right away, for example the term -: :tional to Tr (7"' y" "" 2 -t) after
contracting with the momentum integral is :tional to
Z Tr (' /q' rY r ( K)) =(4- Do) Tr (n /f77
u2(- -K))
(3-132)
where D0 is the .... -..... dimensionality of the loop momentum integral, that is the
reduced dimension DIo 4 so this term vanishes. Further simplifications can be seen when
a particular external polarization is chosen.
S: detailed a -- 0 behavior becomes 'ent when the sum over A and A is done.
With a little work it can be shown that
1 i H Iln(1 /a)
'; 21 3 ) 2
(e11 hn1 1 2 11
,C (tI ( /a)
13 e In)2 ( a) 1
k (11 1 12 1 )2
13 )2
kjk (11 1 1,2 1 s "
Carrying this through for the polarization na =
Nfag K"- ln(I /a) 1f 2
32712m MiM"' r
2A, 1)( 1f 1)
2M, 1
= = n yields
-- )Mj I -2- (
(3 133)
M[ 1 1 ( 1
+ (3 134)
(3 1
-i)]}
(3
(3 1 ..)
In the continuum limit we have Z7 f1(1/M) [ K. "/ dx f(x) for any continuous
function f. Ti. :'clore, after adding the ki < 0 contribution and mun 1 by 2 for the
-_