4.8 The Wess-Zumino Model and a Problem of the Chiral Effective Potential
103
˜
(2)
c = a 1 λ i jk g
4
d
6 z
i
(z))
j
(z))
k
(z) +
+ a 2 λ i jk g
4
d
6 z
i
(z))
j
(z)[ln(−
μ 2 )]
k
(z) +
+ λ rnj λ rns λ ski g
2
a 3
d
6 z
i
(z))
j
(z))
k
(z) +
+ a 4
d
6 z
i
(z))
j
(z)[ln(−
μ 2 )]
k
(z)
.
(4.202)
Here the constants a 1 , a 2 , a 3 , a 4 are equal to
a 1 =
1
(4π) 4 {ζ(3)[6
5
(−
N − 1
2N 2 +
1
4
(1 −
1
N
)
2
+
1
4
(N + 1 −
2
N
)) +
= 2 × 6
3
(1 −
1
N
)
2
] + 3 × 6
4
[
1
3!
N − 1
2
(
1
256
− 1) +
+
(N
2
− 1)(N − 1)
4N 2
+
N − 1
4N
](1 −
γ
2
)C 0 +
+ 6
3 (N
2
− 1)(N − 1)
4N 2
(1 −
γ
2
)C 0 } −
−
C
(4π) 4 3 × 6
4
{
1
3!
(
1
256
− 1)
N − 1
2
+
(N − 1)(N
2
− 1)
4N 2
+
N − 1
4N
};
a 2 = −6
3 1
(4π) 4
(N − 1)(N
2
− 1)
4N 2
−
(4.203)
−
C 0
(4π) 4 3 × 6
4
{
1
3!
(
1
256
− 1)
N − 1
2
+
(N − 1)(N
2
− 1)
4N 2
+
N − 1
4N
};
a 3 = (1 −
1
N
)
9
(4π) 4 (C + (1 −
γ
2
)C 0 );
a 4 = (1 −
1
N
)
9
(4π) 4 C 0 ,
and
C =
1
0
dβdα[ln
p
2
1 β(1 − β) + p
2
2 α(1 − α) − 2 p 1 p 2 αβ
( p 1 + p 2 ) 2 (1 − α − β)
] ×
×
( p 1 + p 2 )
2
p
2
1 β(1 − β) + p
2
2 α(1 − α) − 2 p 1 p 2 αβ
| p 1 , p 2 →0 .
(4.204)
We note that nonlocal corrections in the SYM theory arise also in the pure gauge
sector [78].
This nonlocality is a natural consequence of the presence of UV divergences in
a theory with massless fields. However, treating the problem of the possible renormalization of the chiral effective potential (to the best of our knowledge, it was not