Eq. B.56 of that paper, we see that we can write
16w2T0 M1 [. 12 2
( np2p+ \_ M M 22 p2p f M 22
+ In--- 4 1n + +in-- -[4in + } 1 +
p+p M, 9 p+p 1. 9
g3A^ 2 2 (M M p2 + Mp Mip p\ Mip2 2
+42T L 3 M M1p2 Mp Mp + MIj Mp 6-
_I 1.' + Mp2 Mip2 1]2 _2_11
+(In *. + 7) 2 M In 3 + (3-41)
M_,2- Mp- Mip2 + M1,2 Mp 6
Comparing the zeroth order vertex, -2gKAM/MIM_.To, to Eq. 3-41, we see that the
ultraviolet divergence of the triangle is contained in the multiplicative factor
1 + 9ln- 4 (lnM + nMi + ln.- + 37) (3-42)
t672 a 3
Note the entanglement of ultraviolet (In(1/a)) and infrared (In .[) divergences, typical of
Lightcone gauge. The In M's multiplying In(1/a) must cancel to give the correct charge
renormalization. To see how this happens, do the I sum in the gluon wave function
renormalization factor from before
Z(Q) 1 2 8(In M + 7)- n2Q 4 (3-43)
t672 3 aQ2 3
Thus the appropriate wave function renormalization factor for the triangle, //Z(pi)Z(p2)Z(p),
contains the ultraviolet divergent factor
g2N 2
1 [4(lnMMi..+37) 11] In -, (3-44)
167w2 a
so the divergence for the renormalized triangle is contained in the multiplicative factor
+ g 2 ln (3-45)
3 167 2 a
implying the correct relation of renormalized to bare charge
9R =g + 1 asNc in (3-46)
2471 a