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THIRTEEN
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Relations between large-momentum or short-distance behavior and renormalizability properties have been discussed in the early 1950s by Stueckelberg and Peterman and by Gell-Mann and Low. In the 1970s, under the lead of Wilson, Symanzik, and Callan, this subject blossomed. We should not underestimate the flow of new ideas which resulted 'in the confluence with the study of critical phenomena, to ..yhich are associated the names of Kadanoff, Fisher, and Wilson. The troublesome short-distance singularities, which forced upon us the great machinery of renormalization, turn out to be the key to scaling properties of operators. A set of differential equations may be written expressing the nontrivial effect of an infinitesimal change in the scale. Upon integration they reveal that the short-distance behavior acquires, in favorable cases, a universal character, with fields and composite operators being assigned anomalous dimensions. These ideas find successful applications to deep inelastic lepton scattering, electron-positron annihilation, and other high-energy processes. The concepts of short-distance expansion, asymptotic freedom, enrich the theorist's arsenal, and motivate hopes of developing a fundamental theory of strong interactions. The parallel development in statistical mechanics has been tremendous, and has achieved notable successes in the comparison with experimental facts. This chapter can only be an introduction to this vast subject. We shall avoid intricate mathematical proofs and rely mostly on heuristic arguments and examples, following the historical development at the price of some repetition.
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13-1 EFFECTIVE CHARGE IN ELECTRODYNAMICS
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To introduce some of the ideas we begin with the case of electrodynamics originally studied by Gell-Mann and Low. The fundamental quantity is the electric charge or rather the fine-structure constant a = 1/137 measured in low-energy experiments, in atomic physics, say, i.e., over distances much larger than the charged fermion Compton wavelength (for short we refer to electrons) which fixes the fundamental scale. In high-energy experiments we are rather interested in local, quasi-instantaneous properties of the interaction. We would expect naively that this regime is dictated by the bare parameters occurring in the hamiltonian. Unfortunately, as a result of renormalization the relation between bare and renormalized charges is plagued by infinities, at least perturbatively. The manner in which these infinities have to compensate imposes constraints which are reflected in the asymptotic behavior. The meaning of the word asymptotic will be clarified as we proceed.
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13-1-1 The Gell-Mann and Low Function
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Consider the photon's propagator and its relation with vacuum polarization
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Gpv(q)
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-i q2[1
!P~(q2)] + G;v(q)
(13-1)
For simplicity no fictitious photon mass has been introduced. The longitudinal part in qpqv plays no role in the following. Normalization of charge is insured by the condition 6)(0) = 0, and is therefore fitted to low-energy interaction or soft photon emission. The vacuum polarization is a function of q2, a, and m the electron mass. We define the effective charge at momentum q, d(a, q2, m2), by considering the combination aG pv associated to photon propagation
a 2 d(a, q2, m ) = -1---(--2-2-)
+wa,q,m
(13-2)
d(a, q, m2 )
Subtractions are usually carried out at q2 = O. Consider the effect of picking a different subtraction point q2 = A < 0 (in the euclidean region away from singularities). The perturbation series will be defined in terms of a parameter a" equal to (13-3) In terms of a" the effective charge will be a function D such that D(a", q2, m 2, ,F) = d(a, q2, m 2) and
(13-4) (13-5)
634 QUANTUM FIELD THEORY
Equation (13-4) expresses the apparently empty fact that the physical content is not modified by a mere change in the conventions. A different subtraction point leads to a modification of a;. preserving the function D. The discussion is simplified here by the Ward identities of electrodynamics Zl = Z2, a = Z3aO, implying a consistent treatment by referring only to the vacuum polarization. Since the D function is dimensionless we may write
q2 m2) ( q2) D ( a;., ),z' - y = d a, - -;:;;z a;.
D(a;., 1, -
::)=
d(a, -
(13-6)
and exploit these relations in the deep euclidean region where - q21m 2 becomes large. We know in principle the perturbative expansion of the vacuum polarization (ad) -1 - 1. Order by order we extract its asymptotic behavior by neglecting terms behaving as (m 2Iq2) [In (_q2Im 2)Y This defines a function das(a, _q2Im 2) which would also result from massless quantum electrodynamics with m2 as a scale parameter. We therefore explore the domain - q21m 2 --+ 00, a In ( - q21m 2) --+ 0, hoping that neglected terms do not sum up to a nonnegligible contribution. Such an hypothesis cannot be seriously analyzed at this stage. The right-hand side of Eq. (13-6) becomes das(a, x) with x = - q21m 2. If we choose the subtraction point such that m 2 - . 1, 2 ;s - q2, we can neglect the dependence on m 2 of the left-hand side, since we know from Chap. 8 that the corresponding massless limit exists. In the limit, D is replaced by Das with ..1,2 the scale parameter. Setting y = - A21m 2 we find (13-7) with (13-8)
These relations imply that d as may be considered as a function of a single variable. To see this we define, following Gell-Mann and Low, the function
!/J(z) =
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