Selecting quadratic terms in (p we get:
28
GAUGE FIELDS AND STRINGS
1
{{d,cp^ - Afcp^f + B;bI(c p ^(p^ -
d^x
2elj
(B:)^d^x
(2.47)
The logarithmic contribution to the renormalization of el comes from
the second term in (2.47). Integrating over q> with momenta
A < Ip I < A
gives:
A
< = '5‘''’[ 1 - ( A f - 1)]
°
(2nr
.
el
A
¿‘"’(2 - /V) ^ log ^
1 _ 1 N -2. A
c n C fi
l o g -
^0 *^0
Recalling the arguments of the preceeding section we obtain:
P(e^) = -
N - 2
2n '
e\p) =
1 -h
e\p) \og(p^/n^)
The correlation function
mx -y ) =
(2.49)
(2.50)
and the function y(e^) are also easily computed to leading order. We
have:
^ ^ ^)/io(y)>A(i - <^^>a,a)
/
N - 1 ,
A \
= ( 1 ----- ^ W lo g ^ |
(2.51)
and
2n
y(e^) =
N - 1
2n
1 /
N -2
,
p
®(p) = ^ ( 1 + ^ e (/^) log
p^ \
4n
/I
2 w N - D K N - 2 )
(2.52)
28
GAUGE FIELDS AND STRINGS
1
{{d,cp^ - Afcp^f + B;bI(c p ^(p^ -
d^x
2elj
(B:)^d^x
(2.47)
The logarithmic contribution to the renormalization of el comes from
the second term in (2.47). Integrating over q> with momenta
A < Ip I < A
gives:
A
<
°
(2nr
.
el
A
¿‘"’(2 - /V) ^ log ^
1 _ 1 N -2. A
c n C fi
l o g -
^0 *^0
Recalling the arguments of the preceeding section we obtain:
P(e^) = -
N - 2
2n '
e\p) =
1 -h
e\p) \og(p^/n^)
The correlation function
mx -y ) =
(2.49)
(2.50)
and the function y(e^) are also easily computed to leading order. We
have:
^ ^ ^)/io(y)>A(i - <^^>a,a)
/
N - 1 ,
A \
= ( 1 ----- ^ W lo g ^ |
(2.51)
and
2n
y(e^) =
N - 1
2n
1 /
N -2
,
p
®(p) = ^ ( 1 + ^ e (/^) log
p^ \
4n
/I
2 w N - D K N - 2 )
(2.52)