Putting Eqs.3-118,3-120,3-123 together, some simplification occurs and we obtain
- S(1 /M1 + IdT d IdT + dS(
M1d S !Jo/ Jo dJ(l
Mijl.T2 2._[ 2Mip2 Mlp2 -1-/ i 2
+ M 211 + (in M, + 7) 2ML I + + In
MMp2 Mp Mp2 + __I 1, Mp_
+ M- 2 In3 -
K2 MiP + 3_ in K2 + MI[ 31,2
[ M 21 6Mlp- 3_
n (n M +7)+ In2( ) + C (3-124)
When we add the contribution with 1 +-+ 2, the antisymmetry of some of the coefficients
leads to further simplification as well as a reduction in the degree of divergence of some of
the terms
t1- 1S(I/Mi) + IdT + (1 2 )
j1 do
1-e o 1- 2, pp+2 p+ 2 +
d IdT + d/S() + (1 2 )-- In 2 2PI P12I
(1JO J c P1P2+P2P1 P2
2p+ 2 + 2 +2 p p-2 + 2 2p+2 + 2 p2
+ In In + p I 2 in + In 6+
L+P2 +- 2 pi p pi 2p + p2p pIp2 p+p2 p+p p+p 3]
Sp_ \ P- 1_ p2 1 P2 P2- P22P 12
2+ 2 +2 +2 2 + + + 2 + 2
p+ K2 +2 + p+pp K2 2 31 + +p 2
(inP P2 2 1)3 ,\-12 1 p2 P2P1
27 2M1 2Mlp2 Mp 31 Mnp2 2
3M + (In M1 + ) Mp + 1 In 3 +
3M2 Mip +31 2 Mp 6
( 2 _2 M 2 ? 22 1
2_(I, MIPM in 3 + (3 125)
-+ L1,2 Mp M- p + -2 Mp 6
As c -- 0 the first two lines on the r.h.s. approach a finite c independent answer. The
third line is explicitly finite. All divergences are shown in the last two lines. As .3 oo
there is a leading linear divergence as well as a single logarithmic sub-leading divergence.
3.5.5 Details of SUSY Particle Calculation
We consider the diagram depicted in Fig. 3-2 with fermions on internal lines and
gluons with polarizations nl, n2 and n3 on external lines. In section 2.2 the method of
constructing the world sheet vertices is outlined and referring to the paper itself [7] for the