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tr()'v)'a)'.)'1.)'a)'p)'p)'1.)Iap.,(k, q)
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(8-115)
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dd p pp(p + k)a(P + k + q).(p + q), (2n)d p2(p + kf(P + q + W(P + q)2
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The p integration will be first carried out. We introduce Feynman parameters (Xl,"" (X4 for the lines of momentum p, p + q, p + q + k, p + k respectively. We get
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Jap., =
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ddp (2n)d pp(p + k)a(P + k + q).(P + q),
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(8-116)
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x exp - ((X23q 2 + (X 34 k 2 + 2(X3k q +
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2>2 + 2(X23q P + 2(X34k p)
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where we have used the shorthand notations (X23 = (X2 + (X3, etc., I = (Xl + (X2 + (X3 + (X4' It is convenient to shift the integration variable
p' = P +
(X23
(X34
k == p + Q
and to rewrite the numerator of (8-116) as pp(p + k)a(P + k + q).(p + q), = (p' - Q)p(p' + k - Q)a(P' + k + q - Q).(p' + q - Q), We then have to deal with integrals of the form
(8-103), we find
ddp' (2n)d exp ( - Ip,2)
1 (4n Dd/2
(8-117)
ddp' 1 (2n)d p~p~ exp (- Ip'2) = 2D4nI)d/2 D ., 1
Ip'2)
(2~dP~P~P~P'p exp( -
4I2(4nDd/2 (D.,DaP
+ D.aD,p + D.pD,a)
while the integrals involving odd powers of p' vanish. We also need the trace of )' matrices. From tr ()',)'a)'.)' 1.)'a)' p)'p)' 1.) -4(d - 2) [Dap(D,aD.p - D,.DaP + D,pD a.) + DpAD,aD.a - D,.D aa + D,aDa.) - Dap(D,aD.p - D,.Dpa + D,pD a.)]
-4(6 - d)[D,p(DaaD.p- DpaD.a)
+ D,a(DpaD.p -
DapD. p) + D,p(DapD.a - DaaD.p)]
(8-118)
QUANTUM FIELD THEORY
The first bracket is symmetric under the interchange of IX and {3, whereas the second one is antisymmetric. A tedious but straightforward algebra yields
(4n2)
[1X23IX14q2
+ 1X12 1X 34k2 + 2(1X1 1X3 -
1X21X4)q' kJ
Jap v tr (JivJiaJi.Ji,JiaJipJiPJi,)
4(d - 2){ QpQa(k 2 - 2q2)
+ kpqaQ'(k - 2q) + 4(6 - d){(b pak 2 4(d - 2)
+ QAa[ - k (k + 2q) + 4q. QJ + Qpka[(k + 2q)' q - 2k' QJ kpkaQ'(q - Q) + 2MaQ '(k - Q) - bp.(k + q - Q)'(q - Q)Q'(k -
kpka)Q (Q - q) - (bp,k' Q - Qpk,)[b'ak'(Q - q) - k,(Q - q)aJ}
- - - {(2 - d)(2qpqa - kpk a) + (d - 2)b pa [(k
Q)'(q - Q) - Q'(k - Q)]
+ 2b pa(k 4(6 - d)
2Q)'(k
+ 2q -
2Q)} 4(d - 2)
--2-
+ - - {[k 2bpa ==
-4(d - 2)Al
kpkaJ(d - 2)} -
[bpad(d
+ 2)J
4(d - 2) - - - C1
(8-119)
+ 4(6 -
4(d - 2) d)A2 - - - - B l
+ ---B2 2I
4(6 - d)
"'-. where the origin and the denomination of each term is clear and we have used the symmetry between p and (J. The q integration must now be carried out. We write
(8-120)
As the integrand involves
times powers of q, we perform a new shift of integration variable
q=q'-lXk
(8-121)
where Accordingly,
(8-122)
with Also, we call y the coefficient of q2 in the exponential
y==(x+ 1) - - = = - - -
1X231X14
1 1X23 1X14
(8-123)
Therefore, the desired amplitude r\,bJ(k) reads
RENORMALlZATlON
(8-124)
It is then a pure matter of patience to compute
( 4n
'11/2
dd q'
e- yq"
1X23 1X 14
+ 2(1X1 Z)(IX
1X3)(1X4 -
1X2)
(d -
(2n)d
2yI2 [(k 2f Z(1 { , ,
+ b p
k +2
+ z)(l
+ 2z -
IX -
21X23(1 -
1X23)[1X
2z(1X
+ z)] + 2(1X1 - 1X3)(1X4 - 1X2)} -'--"---'-'-'----=
2 - k d 2y
[(IX
+ z)(l
IX -
Z)&~3 + (1 -
&d 2 z(1
+ d(d + 2) 1X23(1 - - 24y
1X23)
Consequently,
(:n)d 2(6 - d)(d - 2)(k 2 b p
k p k.)(k 2 )d-4q4 - d)
1X3 1X4)
,d/2-2 fl dlXl ... d1X4 b(1 - 1X1 - 1X2 l dx x (1 - x) 3d/2-5 fo 0 (1X141X23)
X [1X121X231X341X41 -
X'(1X11X3 -
1X21X4)2Jd/2-4
After the change of variables,
1X1 =
1X2 = (1 -
f3)v
1X14
1X3 =
(1 - f3)(1 - v)
1X4 =
f3(1 - u)
= f3
1X21X4fJ =
[1X121X23C(341X41 -
X'(1X11X3 -
f3(1 - f3){[f3(.1 - u) + (1 x [f3u
f3)(1 - v)]
+ (1 -
f3)vJ - x'f3(l - f3)(u - vf}
d1X1" 'dIX4 b(1 -
1X1 -
1X2 -
1X3 -
1X4)F(lXj) =
df3 f3(1 - f3)
dv F(lXj)
QUANTUM FIELD THEORY
it is easy to check that this integral is convergent at d = 4. Therefore, neglecting as before constant terms in w(k 2 ): 2 r(A,+B,) :0= ~(~ - In k + ... )(k k - 0 k 2) (8-125) 2 PG d=4-e 4n e}l2 P pa
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