102
4 Four-Dimensional Superfield Supersymmetry
d
6 z
1
(16π 2 ) 2
2
(z) log(−
μ 2 ))(z).
(4.200)
Carrying out the calculations for all supergraphs above we conclude that the twoloop low-energy leading chiral correction in the N = 1 SYM theory with chiral
matter is nonlocal. Its complete form, after subtracting of the corresponding counterterms, is
˜
(2)
c =
g 2
(4π) 4 {ζ(3)[6 5 g 2 (λ imn (T K T I T K ) n
k (T I ) m
j + λ lnp (T I T K )
p
k (T I ) l
i (T K ) n
j +
+ λ imn (T I T K ) n
k {T K , T I } m
j ) + 4 × 6 3 (T I T J ) l
i (T J ) m
k (λ lmp (T I )
p
j +
+ λ l j p (T I )
p
m )] + 3 × 6 4 [
λ iml g 2
3!
(
1
64
− 4)(T K ) a
b (T I ) c
d (T K ) m
j (T I ) l
k ×
× [(T L ), (T N )] b
a [(T L ), (T N )] d
c +
λ rnl λ rns λ skm
(3!) 3
(T I ) l
i (T I ) m
j +
+ λ snp g 2 (T K ) s
i (T K )
p
k (T I ) n
m (T I ) m
j +
+ λ ir p g 2 (T I ) m
n (T K ) n
m (T I )
p
k (T K ) r
j ](1 − γ)C 0 +
+ 2 × 6 4 λ iml
2!3!
g 2 (T K ) s
p (T K )
p
k (T I ) l
s (T I ) m
j (1 −
γ
2
C 0 )} ×
×
d 6 z i (z)) j (z)) k (z) −
−
2g 4
(4π) 4 6 4 λ iml
2!3!
(T K ) s
p (T K )
p
k (T I ) l
s (T I ) m
j
d 6 z i (z)) j (z)[ln
−
μ 2
] k (z) −
− 3 × 6 4 {
d 2 θ
d 4 p 1 d 4 p 2
(2π) 8 [
λ iml g 4
3!
(
1
64
− 4)(T K ) a
b (T I ) c
d (T K ) m
j (T I ) l
k ×
× [(T L ), (T N )] b
a [(T L ), (T N )] d
c + λ ir p g 4 (T I ) m
n (T K ) n
m (T I )
p
k (T K ) r
j +
+
λ rnl λ rns λ skm g 2
(3!) 3
(T I ) l
i (T I ) m
j + λ smp g 4 (T I ) m
n (T I ) n
j (T K )
p
k (T K ) s
i ]} ×
×
1
0
dβdα[ln
p 2
1 β(1 − β) + p 2
2 α(1 − α) − 2 p 1 p 2 αβ
μ 2 (1 − α − β)
] ×
(4.201)
×
( p 1 + p 2 ) 2
p 2
1 β(1 − β) + p 2
2 α(1 − α) − 2 p 1 p 2 αβ
i (− p 1 , θ)) j (− p 2 , θ)) k ( p 1 + p 2 , θ).
This result is found for the SYM theory with an arbitrary non-Abelian gauge group.
In the particular case of the SU (N ) group, the result reduces to