We write out the results for the renormalized vertex Y where a subscript refers to
the two different time orderings, k1 > 0, (1 < MI) or ki < 0, (1 > Mi) respectively.
y^AA^ Y=VVV (3-61)
yAAV g KA M-1 (1/.) F(/M) (3 62)
YAV 7"2aln(l/a) (3-62)
82 P 1P2 1- (1 MJ I
g3 1p+ ^ MI-1 F(1/M1.) F(/ )63)
YAVA aln(1/a)- (3-63)
8712 P lP 3 M
g3 p7+ K^A MI-1 F(1/1) F(1/.)
YV --a1n(1/a)- Ey (3-64)
872 P2 P3 1- 1 11. ) I
g3 K^ M3-1
V aln(l/a)PK F( /) F( (3-65)
l MI+1
yDAVA h 3 c g aln(t/a)PK M3-' (1 -A-) F (_ __
YAaA I n(1 A 1 F(a[.) /[.) F(3[-. (3-66)
1 P3 Ihl MI+1
P2 P3 / M1+1
Y ^ a In(1/a)Uln3 ) (3-67)
The expressions for the Y's with A 4 V are the same with K^A K. We stress that the
summands in the above expressions for Y are exactly contribution of the three diagrams in
Fig. 3 3 with the loop fixed at 1.
Define the coupling constant renormalization A(Nf, NI) by:
Yn'n2n3 -Y41n2n3 + Y'1n2n3 1Fl2n3A(Nf, NS). (3-68)
We then have in the limit 3[ -+ +oo:
A(Nf, Ns) 8,2 ln(l/a) 3 3 6 (3-69)
which is the well known result. In particular we have asymptotic freedom when A > 0,
and A vanishes for the particle content of A/ 4 Supersymmetric Yang-Mills theory,
Nf 8 and N 6.