QUANTUM FIELD THEORY in .NET framework

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QUANTUM FIELD THEORY
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dimensions). The virtue of this method is that it automatically preserves internal symmetries that do not involve the Ys matrix. Technically, all the manipulations that we perform to check Ward identities (see Chap. 7 or Sec. 8-4) such as shifts of integration variables, contractions of Lorentz indices, etc., are consistent with the regularization.
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The analytic continuation to d-dimensional space-time is most easily performed, after a Wick rotation to the euclidean region, on the parametric form derived in Chap. 6. Indeed, we may compute the amplitude of Eq. (6-94) in an arbitrary integer dimension d (we drop henceforth the caret notation for euclidean momenta):
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The method of Sec. 6-2-3 is easily seen to go through, and the result reads
_ fOO
lo(P) 0
l~ 1 dal
exp - [Qo(P, a) + L.almfJ [(4n)L.9'G(a)]d/2
(8-5)
with the same functions QG and .9'G as those defined in Eqs. (6-86) and (6-87). The integration over the homogeneity parameter .Ie of the a is convergent at .Ie = 0 if 1 - dL/2 = L(l - d/2) + V-I> 0:
(8-6) For instance, for p, n ~ 1,
The whole dependence on d may be explicited by performing the a integration. For low enough d, the result is finite. When d ..... 4, ultraviolet divergences may arise from the Euler function in front of (8-6), or from the a integration, or from both. However, for one-loop diagrams such as (8-7), only 1(1 - dL/2) diverges. If 1 - 2L is a nonpositive integer, then
dL) [' ( 1 -
(_1)2L-1 2 d:4 (2L- 1)! (4 - d)L
(8-8)
and IG(P) has a simple pole at d = 4. When internal divergences coming from the a integration are also present, this pole may become a multiple one (compare with the examples below). For completeness, we have to say how to deal with integrals involving Lorentz vectors and/or spinors. The former case does not offer any difficulty. Four-vectors are transformed into d-vectors, the integrations are performed, and the result continued to arbitrary dimensions. For instance, again in euclidean space,
ddk (2n)d [(P - k)2
+ mIJ"(k 2 + m~Y =
p. 1(n + p - d/2) (4n)d/2 1(n)1(p)
da a"(1 - ay-l[a(1 - a)p2
+ amI + (1- a)m~]d/2-"-p
(8-9a)
RENORMALIZATION
ddk (2n)d [(p - k
k"k,
{f f
+ mnn(k2 + mW
(4n)d/2f'(n)f'(p)
da an- 1(1 - a)p-l[a(1- a)p2
+ amI +
(1- a)m~]d/2+1-n-p Or
(8-9b)
daa n+ 1(1- a)P-l[a(1- a)p2
+ amI + (1-
a)mnd,2-n-pp"p,r(n
+ p -~)}
In this expression,
0", is the unit tensor in d-dimensional space, satisfying
(8-10)
This condition is maintained in the continuation to noninteger dimensions. Such a prescrIptIOn ensures the consistency of this continuation with algebraic manipulations such as contractions or shifts of integration variables. For instance, it is easy to check that
ddk k"k,o"' - m~ (2n)d Up - k - mnn(k2 - m~)p
ddk 1 (2n)d [(P - k; - mnn(k2 - m~)p
Spinors require more care. First, in the original four-dimensional Feynman integrand, we distinguish l' matrices of fermion loops from those pertaining to fermion lines connected with external lines. The latter are reduced by means of projection operators in four dimensions. We therefore have to consider for each diagram a collection of form factors involving only l' matrices pertaining to loops. They are supposed to satisfy the rules
(8-11a)
(remember that we have performed a Wick rotation and hence our l' matrices are now antihermitian) tr (odd number of l' matrices) tr J
f(d)
(8-11 b) (8-11 c)
wheref(d) is an arbitrary smooth function satisfyingf(4) = 4, for instance,f(d) = 4 or f(d) = d. The form off is irrelevant for the procedure of regularization. (This would not be irrelevant if we really wanted to define the theory in d dimensions.) From the rules (8-11), we may reconstruct the whole set of identities for contractions and traces of products of l' matrices that we listed in the Appendix for d = 4 in the Minkowskian case. For instance,
tr 1'"1', =
f(d)o",
1'"1'" = -dJ 1'"1',1'" = (d - 2)1', We have not defined the d-dimensional analog of 1'5' This is because the usual definition of 1'5 appeals to the existence of B",pu, the completely antisymmetric tensor which is specific to d = 4. We conclude that we cannot continue fermion loops containing an odd number of 1'5 matrices. This innocent-looking limitation is the manifestation, in the framework of this dimensional regularization, of a serious problem, namely, the possible appearance of chiral anomalies (Chaps. 11 and 12). This set of prescriptions may look like a rule of thumb to a skeptical reader. Its consistency, though quite likely and checked in everyday computations, has to the best of our knowledge never been completely proved. Especially embarrassing is the case of massless theories. For instance, we encounter integrals of the form
that do not depend on any scale. The analytic continuation of such an integral is ill defined since there is no dimension d where it is meaningful. It is either infrared or ultraviolet divergent according
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