2 Effective Action and Loop Expansion: General Formulation
7
It is clear that the first block,
i
√
(S
[] + J )φ, can lead only to one-particle-reducible
Feynman (super)graphs since its contribution with one quantum field φ can form
only one propagator when the Feynman diagrams are constructed, thus it does not
contribute to the effective action. Hence we can omit this term. Then we can expand
the exponent into power series in :
e
i
¯
[]
=
Dφe
i
2 S
[]φ
2
1 +
i
√
3!
S
(3)
[]φ
3
+
i
4!
S
(4)
[]φ
4
+
+(1/2)
i
√
3!
2
(S
(3)
[]φ
3
)
2
+ . . .
⎞
⎠ .
(2.6)
At the same time, after substituting the expansion of ¯
in the left-hand side of (2.5)
into power series in , we get:
exp
i
¯
[]
= e
i((
(1) []+
(2) []+...)
= e
i
(1) []
(1 + i
(2)
[] + . . .).
Substituting this expansion into (2.6) and comparing equal powers of we see that any
correction
(n) corresponds to a some functional integral. For example, the correction
of the first order in (i.e., as we will show further, the one-loop correction) is defined
from the paradigmatic integral
exp(i
(1)
[]) =
Dφ e
i
2 S
[]φ
2 ,
(2.7)
and the correction of the second order in —from the relation
(2)
[] =
1
i
Dφe
(
i
2 S
[]φ
2 )
i
4!
S
(4)
[]φ
4
−
1
2(3!) 2 (S
(3)
[]φ
3
)
2
Dφ exp(
i
2
S
[]φ 2 )
.
(2.8)
Here, as usual, integration over coordinates in expressions of the form S
(n)
[]φ
n is
assumed. The denominator of this expression serves to eliminate the one-particlereducible contributions.
We can see that:
(i) All fractional degrees of like
n+1/2 , with integer n, vanish since they accompany integrals like
Dφφ
2n+1 exp(
i
2
S
[]φ
2
). By the symmetry reasons such
an expression is equal to zero.
(ii) All terms beyond the first order in are expressed in the form of some functional
integrals.
(iii) The one-loop correction (2.7) can be expressed in the form of a functional determinant since