= det At [det QJ- 1 / 2 exp {Hu + ~At-11X in .NET

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= det At [det QJ- 1 / 2 exp {Hu + ~At-11X
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+ (PAt-l~ JQ-l[U + (~At-1IXl + PAt-l~J + ~At-l~}
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The matrices Q and At have commuting elements, while those of the matrices IX and Pbelong to the Grassmann algebra; Qdenotes the symmetric matrix
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Q - PAt- 11X - (PAt- 1IXl
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Equation (12-101) results from the repeated application of the preceding formulas. An alternative way to recover it is to remember that the saddle-point method is exact for gaussian integrals.
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These formulas are applied to the quadratic form (12-99) where
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(QA')Il
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== [Dv, [DV, A'!'J - [D!', A,vJJ - g[PV, A,vJ + A8!'8' A'
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+ g{ 8!'ij, 11/}
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+ g{ 8!'ij', 11}
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(12-102)
NON ABELIAN GAUGE FIELDS
and where r/, and if' are obtained from the equations
AI'/, == (\[DI', 11']
[DI', 0l'if'] If g = 0, A and
g0I'[A'I', 11] -g[ol'if, A~]
(12-103)
Qreduce to
Ao= 0
0 - (1 - Je)o
the inverse of which are the free propagators. The one-loop effective action, normalized to r[11(0) = 0 is therefore
ir[ll(A, if, 11) = Tr [in (Ao 1A) -
i In (QQo 1)]
(12-104)
The trace is performed on internal and Lorentz indices and on space-time variables. It is not possible to obtain a more explicit expression. As the matrices Q and A act in the adjoint representation, it is more convenient to consider AI' and 11 as matrices in that representation [compare with Eq. (12-9c)]. Explicitly,
(Al'hc = (l1)bc
A~Tabc
CbacA~
(12-105)
Cbacl1a
Therefore, instead of the normalization (12-26), we have tr (TaT b) = c,d
L CcdaCcdb =
- C(jab
(12-106)
For SU(N), C = N. With these notations, we write
Ao 1A(1, 2) = (j(1, 2) -
QQo 1(1,2) = (j(1, 2) + g
x {(-ik A
f f(2n~4:~2)2
(2n)4
d4k e- ik . (XI-X2) k2 k A(X2)
+ o A -
gA 2 )[k 2 gI'I1'2 - (1 - Je-l)kl'lkI'2]
(12-107)
+ [k 2 A1'2 2
(1- Je-l)k AkI'2](ikl'l
+ gAl')
- [k 01'2 - (1- Je-l)kI'2k o]AI'I
+ [k 2 F1'11'2 -
(1 - Je-l)FI'IP kPk]} XI e- ik '(XI- X 2) 1'2
In the curly brackets, the argument of A or F is Xl, and the derivatives act on all xl-dependent terms to their right. The kernel Qdiffers from Q by ghost terms, not made explicit here. In the sequel, we use the same notation for a function and its Fourier transform:
A(x) =
(2n)4
d k e- ik . x A(k)
We shall now focus on the superficially divergent functions.
QUANTUM FIELD THEORY
12-3-2 Two-Point Function
From the expansion ofTr In (Ao 1A) to the two-point function (Fig. 12-1):
i Tr In (Qo1Q)we get three contributions
~ f [dk] tr [A(k)r(k)A( -k)]
J + J
g2 2
[dk][dq] q2(q
+ k tr [(q + k)' A(k)q' A( -k)]
[dk][dq] (q!)2 tr {q2(d - 1)A(k)' A( -k)
+ (1- Je-1)[q A(k)q A(-k)- q2A(k) A(-k)]}
g2 -4
[dk] [dq] (q2)2(q'2 tr {(q
+ q')' A[q2 -
(1 - Je -l)q q]
- q [q2 A - (1 - Je - l)q. Aq] - A
[q2 q' - (1 - Je -1)q- q' q]
Je-1)q" Aq']
+ q2(k A - A k) - (1 - Je -1 )(kq' A - q' kA) qh x {(q + q')' A[q,2 - (1 - Je-1)q' q'] - q' [q,2 A - (1 - A
[q'2 q - (1 - Je-1)q.q'q'] - q,2(k A - A k)
- Je-1)[(kq" A - k q'A) q']}-k
(12-108)
+ (1
In the last trace, q' == q + k, and the argument in the first curly bracket is k and in the second one - k. Tensor notations are adopted for the sake of brevity. A Wick rotation has been carried out on the time components of both momenta and vector fields:
We use dimensional regularization to preserve gauge invariance and [dk] stands for the measure ddk/(2rrY Needless to say, the above expression may also be derived directly from the Feynman rules of Eqs. (12-91) and (12-92). As explained in Chap. 8, it is consistent, within the dimensional regularization, to consider that the second integral of the right-hand side of (12-108) vanishes
Figure 12-1 One-loop contributions to the vector self-energy. The broken lines stand for ghost propagators.
NON ABELIAN GAUGE FIELDS
identically. The ghost contribution, i.e., the first term on the right-hand side of (12-108) is easy to compute in terms of Euler's functions:
r gh (k) =
~ B (~ ~)(k2)dI2-2 [lk 2T' (1 -~) - k '><' kT' (2 - ~)J (4n)dI2 2' 2 "2" 2 2
\C;I
(12-109)
(12-110)
In these expressions, T'(k) must be regarded as a matrix in the adjoint representation, proportional to the unit matrix. The ghost contribution (12-109) was crucial to achieve the transversity in k, since separate contributions do not satisfy it. The expression (12-110) is suited for the extraction of the divergent part as d - 4 = - 10 --+ O. Recalling from Chap. 8 that g2 should actually be written g2 Ji,', where J1 is an arbitrary mass scale, we have
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