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QUANTUM FIELD THEORY
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to whether d + p - 2n ~ 0 or 2: o. We shall disregard these problems and adhere, whenever necessary, to the prescription that such massless tadpole integrals vanish in the dimensional regularization. In computations with this regularization, we must remember that in d dimensions some coupling constants may acquire a dimension. For instance, the electric charge, the coefficient of
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in the dimensionless action (h
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1) has a dimension in a mass scale
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d - (d - 1) - - - = - -
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since [1/1] = (d - 1)/2 and [A] = (d - 2)/2. We therefore replace e by J.l(4-d)12 e" where e' is dimensionless and J.l is either one of the masses of the problem or an arbitrary energy scale if all particles are massless. When expanding the result near d = 4, we will therefore generate logarithms of this scale. Whether in the neighborhood of d = 4 we also expand the factors (4n)-Ld I2 in Eq. (8-6) is a matter of taste, as is the definition of f(d) in (8-11c). The important point is that in computing counterterms in a gauge-invariant way, or in comparing diagrams to check Ward identities, we always consider classes of diagrams with the same total number of loops L, and hence the same power of 4n, and the same number of fermion loops, and hence the same power of f(d). A change of prescription will not affect the validity of Ward identities and will only modify counterterms by a finite amount. As an illustration consider the vacuum polarization in scalar electrodynamics. Feynman rules for this theory have been given in Chap. 6. After a Wick rotation (for an euclidean external momentum p) the two diagrams of Fig. 8-2 give the contributions
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r:(a)
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= -2e 2
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2 ddk ~= _ 2e o", 2 + m2 (2n)d k (4n)d I2
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r(1-~)(m2)dI2-1
ddk (2k + pM2k + p), (2n)d [(p + k)2 + m2] (k2 + m2)
(4:;d I2
{2o",r (1 -~) [ drx [rx(l - rx)p2 + m2]dI2-1
+ p"p,r(2 -
drx (1 - 2rx)2[rx(1 - rx)p2
r(a)
+ m2]dI2-2}
According to Eq. (8-8), let us isolate the pole at d = 4 in the sum
+ r(b):
(8-12)
r", + r "' (a)
(b) _
--2 - - -
e 2 1 2 (p"p, - po",) (4n) 4 - d 3
+ regular terms
We verify that (a) the divergent terms proportional to m2 have cancelled in the sum, and (b) the tensor structure of the divergence, and hence of the counterterm, is transverse to p. Both results are in agreement with what was expected from gauge invariance. Computation of the finite part is left to the reader.
P +k
Figure 8-2 One-loop vacuum polarization in scalar electrodynamics.
RENORMALIZA nON
Other regularization schemes may be invented. The important point is that the final renormalized result does not depend on the choice of regularization. Strictly speaking, finding a regularization and showing that it makes all diagrams finite is not sufficient. We should also prove that the structure of divergences is such that they may be removed by acceptable counterterms, i.e., by local and hermitian polynomials in the fields. Divergences of the form log p2 log A2, for instance, or with a complex coefficient would be catastrophes. At the very end we shall, however, proceed in a different way. We shall prove that sufficiently subtracted Feynman integrands lead to a finite theory. Since these subtractions do correspond to the introduction of acceptable counterterms, the result will follow a posteriori. In other words, no regularization will be involved in the forthcoming proof of finiteness. However, its outcome and meaning would be obscured without recourse to an implicit regularization, which is therefore a convenient and very common tool.
8-1-3 Power Counting
We have already used the concept of superficial degree of divergence whenever we have based arguments about the convergence of Feynman integrals on dimensional considerations. Here we develop this concept in a systematic way. In this section and the following, we shall only face the problem of ultraviolet divergences, and postpone to a further study the possible infrared troubles arising from the masslessness of some particles. To be specific, we assume for the time being that all fields are massive. A naive way to estimate whether a given Feynman diagram G is convergent is to dilate simultaneously its internal momenta by a common factor A, kl --+ Akl and to look for its behavior h '" A'" as A goes to infinity. In the parametric representation, this amounts to investigating the integrand when all the (X tend to zero at the same rate (Xl--+ A-2(X1. We expect, and will soon prove, that if the overall power of A, called the superficial degree of divergence and denoted w, is nonnegative, w ;:::: 0, the integral is generally divergent. If it is negative, some subintegrations may still be divergent, and the integral is said to be superficially convergent. We consider a theory involving spin or 1 boson fields and spin i fermion fields. The fermion propagators behave for large momentum Ak as A-1 and the boson ones as A- 2. Here we assume that massive vector fields have a propagator in the Stueckelberg gauge [Eq. (3-147)]. We shall reexamine in Chap. 12 what happens to massive fields coupled to nonconserved currents. The case of higherspin fields will not be considered here. We use the notations of Chap. 6. A vertex v of the diagram G carries a power Ali, if the corresponding term in the interaction lagrangian involves field derivatives. Each integration d4 q over the loop momenta contributes a power A4. If L denotes the number of independent loops, IE and IF the number of internal boson and fermion lines respectively, and V the total number of vertices, the
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