136
GAUGE FIELDS AND STRINGS
Up to now everything has mimicked precisely the consideration of
Section 8.1. Now we come to the crucial difference. Namely, in spite of
the N"^^^-factor before higher powers in v, the nonlinearity in (8.40) is
absolutely important, because this smallness is compensated by summation over isotopic indices in Feynman diagrams. Let us prove this
and simultaneously separate the relevant set of diagrams. According to
(8.40):
ffablcd(^) ^
d^k
1
' (2ny (k^ + ^^)({k + qf H -
^ li^ac^bd + ^ad^bc)
=
\{^aAd + ^ad^bc)
(This follows from the matrix equation:
Tr Iog((-r^ -f //^)/ + v)
1
(8.41)
= Tr log(
1
-f /r) -f Tr log / -h
2
//' - r '
(8.42)
where Tr is understood both in coordinate and isotopic indices).
The propagator for the r-field is given by:
^^^ab(q)^\d(~ q)y — li^ac^bd ^ad^bc)
1
^(q )
(8.43)
The isotopic structure in (8.43) is conveniently represented by the
picture
h -d b^d
i^'ab^'cd) - ^ ^
(8.44)
where each line corresponds to a ¿-symbol. Let us now consider the
nonlinear correction to this propagator, coming from the cubic term in
(8.40). The structure of T, obtained in the same way as in (8.42) is:
+ permutations')
(8.45)
(here single lines are ordinary propagators
Therefore,
the first correction to
contains a term which has the form:
(8.46)