96
4 Four-Dimensional Superfield Supersymmetry
in terms of dilogarithms of some combinations of spinor supercovariant derivatives
of background superfields.
Now we turn to calculating of the chiral effective potential. It differs from zero
for massless theories. Really, as it was noted by West [67], the mechanism of arising
chiral corrections is the following one. If the theory describes dynamics of chiral and
antichiral superfields, then quantum correction of the form
d
8 z f (()(−
D
2
4
)g(()
(4.184)
can be rewritten as
d
6 z f (()g(().
(4.185)
Here we used properties
d
8 z =
d
6 z(−
¯
D
2
4
) and ¯
D
2 D
2
= 16 (the last identity is valid for any chiral superfield ), and f ((), g(() are arbitrary functions of
the chiral superfield . However, the presence of the factor
−1 is characteristic
for massless theories, whereas in massive theories we have ( − m
2
)
−1 instead of
−1 , and this mechanism of arising contributions to the chiral effective potential
does not work. We note that this situation is rather generic, that is, the perturbative
contributions to the chiral effective potential can arise only if all propagators are
massless (except of very rare situations where the integral over massive propagators
completely factorizes out giving only a constant with no dependence on external
momenta). Actually, namely this effect is crucial to prove the Goldstone theorem
in noncommutative superfield theories [68] which is related with the statement of
absence of
2 corrections in theories with chiral self-couplings.
In the case of the massless theory we can find the matrix Green function (4.157)
exactly: first,
G
ψ
v (z 1 , z 2 ) = ( +
1
4
¯
D
2
)
−1
δ
8
(z 1 − z 2 ) =
1
1
δ
8
(z 1 − z 2 ) −
−
1
1
(z 1 )
¯
D
2
1
4 1
δ
8
(z 1 − z 2 ).
(4.186)
The higher terms in this expansion are equal to zero because they are proportional to ¯
D
2
= 0 or ¯
D
4
= 0. It follows straightforwardly from this expression
that Tr ln G
ψ
v = 0, therefore, the one-loop effective potential in the Wess-Zumino
model identically vanishes. The components of matrix superpropagator (4.157) look
like
G ++ = 0; G +− = G
∗
−+ =
¯
D
2
1 D
2
2
16
δ
8
(z 1 − z 2 );
G −− = −
D
2
1
4 1
[(z 1 )
¯
D
2
1 D
2
2
16
δ
8
(z 1 − z 2 )].
(4.187)
4 Four-Dimensional Superfield Supersymmetry
in terms of dilogarithms of some combinations of spinor supercovariant derivatives
of background superfields.
Now we turn to calculating of the chiral effective potential. It differs from zero
for massless theories. Really, as it was noted by West [67], the mechanism of arising
chiral corrections is the following one. If the theory describes dynamics of chiral and
antichiral superfields, then quantum correction of the form
d
8 z f (()(−
D
2
4
)g(()
(4.184)
can be rewritten as
d
6 z f (()g(().
(4.185)
Here we used properties
d
8 z =
d
6 z(−
¯
D
2
4
) and ¯
D
2 D
2
= 16 (the last identity is valid for any chiral superfield ), and f ((), g(() are arbitrary functions of
the chiral superfield . However, the presence of the factor
−1 is characteristic
for massless theories, whereas in massive theories we have ( − m
2
)
−1 instead of
−1 , and this mechanism of arising contributions to the chiral effective potential
does not work. We note that this situation is rather generic, that is, the perturbative
contributions to the chiral effective potential can arise only if all propagators are
massless (except of very rare situations where the integral over massive propagators
completely factorizes out giving only a constant with no dependence on external
momenta). Actually, namely this effect is crucial to prove the Goldstone theorem
in noncommutative superfield theories [68] which is related with the statement of
absence of
2 corrections in theories with chiral self-couplings.
In the case of the massless theory we can find the matrix Green function (4.157)
exactly: first,
G
ψ
v (z 1 , z 2 ) = ( +
1
4
¯
D
2
)
−1
δ
8
(z 1 − z 2 ) =
1
1
δ
8
(z 1 − z 2 ) −
−
1
1
(z 1 )
¯
D
2
1
4 1
δ
8
(z 1 − z 2 ).
(4.186)
The higher terms in this expansion are equal to zero because they are proportional to ¯
D
2
= 0 or ¯
D
4
= 0. It follows straightforwardly from this expression
that Tr ln G
ψ
v = 0, therefore, the one-loop effective potential in the Wess-Zumino
model identically vanishes. The components of matrix superpropagator (4.157) look
like
G ++ = 0; G +− = G
∗
−+ =
¯
D
2
1 D
2
2
16
δ
8
(z 1 − z 2 );
G −− = −
D
2
1
4 1
[(z 1 )
¯
D
2
1 D
2
2
16
δ
8
(z 1 − z 2 )].
(4.187)