186
GAUGE FIELDS AND STRINGS
uniquely is gauge invariance of the effective action. For the actual
computation it is convenient to introduce complex notation:
=/ci ±ifc2 = |fc|e^‘“
and to consider first the component:
d^k kl(q^ + k^Ÿ
(9.159)
_ 9 r
Cd^k k^(q^+k^)
2 J ' (271)2
= ^C{q^)q%
(9.160)
with as yet unknown C(q^); we have used the 0(2) invariance of the
integral and the fact that l/k_ transforms as fc + . It is tempting to split
the integral (9.160) into ( + ) and ( —) factors since 27iid^k = dk+ dk_.
This is almost possible but requires some extra care. Oddly, one has to
go to Minkowskian space. The propagators and momenta change as:
1
1
kl
>ko±ki
1
k^ -h ie
k.
(9.161)
1
k^ -h ie /c_ -h ie sign k+
So our integral has the form:
au
dfc + k^(q^ 4- k^) 16in^
T ao
d/c_
(/c_ -f- ie sign k+)(q_ 4- /c_ + k sign( (9.162)
If we take q+ <0 then the singularities in k-space are on opposite sides
of the real axis only if 0 < k+ < —q + . Otherwise the fc_-integral gives
zero. We have:
^ 1 f
fI++,+ + “
8 ^ J
+
+
^ S’ q \
487T q_ ’
which is the desired answer.
(9.163)