Dimensional regularization and renormalization of QED

We give an -space definition of dimensional regularization suited to the tree expansion method of renormalization. We apply the dimensionally regularized tree expansion to QED, obtaining sharp bounds on the size of a renormalized graph. Subtractions are made with the Lagrangian counterterms of the tree expansion, not by minimal subtraction techniques, and so do not entail a knowledge of the meromorphic structure of a graph as a function of dimension. This renormalization procedure respects the Ward identities, and the counterterms required are gauge invariant.     

Techniques of dimensional regularization and the two-point functions of QCD and QED

The necessary and useful tools of dimensional regularization (and renormalization), the so-called ?-scheme, are reviewed. A survey on a comparison of various renormalization schemes is done. The applications of the ?-scheme to the two-point funtions of quantum chromodynamics (QCD) and quantum electrodynamics (QED) are given. In these applications it is shown explicitly how to compute Feynman diagrams and how to use the renormalization group equation (RGE) for the prediction of some terms induced by higher order diagrams. Some phenomenological uses of the two-point functions are briefly discussed. These include the quark mass, the spectral function sum rules in QCD and the control of the asymptotic SU(n)×SU(n) flavour chiral symmetry, the proton-neutron electromagnetic mass difference in the light of QCD and the running electromagnetic charge of QED. We also confront the operator product expansion (OPE) results of the anomalous dimension of non-singlet operators to the result obtained from the method of factorization of mass singularities.     

Renormalization constants and asymptotic behaviour in quantum electrodynamics

Using dimensional regularization, a field theory contains at least one parameter less than usual with the dimension of mass. The Callan-Symanzik equations for the renormalization constants then become solvable entirely in terms of the coefficient functions. Explicit expressions are obtained for all the renormalization constants in quantum electrodynamics. At non-exceptional momenta the infrared behaviour and the three leading terms in the asymptotic expansion of any Green function are controlled by the Callan-Symanzik equations. For the propagators the three leading terms are computed explicitly. The gauge dependence of the asymptotic electron propagator in momentum space is calculated in all orders of perturbation theory.     

Current correlators to all orders in the quark masses

The contributions to the coefficient functions of the quark and the mixed quark-gluon condensate to mesonic correlators are calculated for the first time to all orders in the quark masses, and to lowest order in the strong coupling constant. Existing results on the coefficient functions of the unit operator and the gluon condensate are reviewed. The proper factorization of short- and long-distance contributions in the operator product expansion is discussed in detail. It is found that to accomplish this task rigorously the operator product expansion has to be performed in terms ofnon-normal-ordered condensates. The resulting coefficient functions are improved with the help of the renormalization group. The scale invariant combination of dimension 5 operators, including mixing with the mass operator, which is needed for the renormalization group improvement, is calculated in the leading order.Supported by the German Bundesministerium für Forschung und Technologie, under the contract 06 TM 761     

Feynman Amplitudes as Tempered Distributions

Feynman amplitudes (FA) are analyzed with the help of the method which employs as much as possible the analogy between ultraviolet (UV) and infrared (IR) divergences and is based on a modified α-representation. It is proved that an analytically and/or dimensionally regularized FA is a tempered distribution of external momenta and a meromorphic function of regularizing parameters with UV and IR poles. An absolutely convergent α-representation of analytically and/or dimensionally regularized FA is derived in various analyticity domains. This representation is obtained from the α-representation in the initial domain of convergence by inserting the operator which has the same structure as the R*-operation that is a generalization of the dimensional renormalization.     

New renormalization group results for scaling of self-avoiding tethered membranes

The scaling properties of self-avoiding polymerized two-dimensional membranes are studied via renormalization group methods based on a multilocal operator product expansion. The renormalization group functions are calculated to second order. This yields the scaling exponent ν to order ε² Our extrapolations for ν agree with the Gaussian variational estimate for large space dimension d and are close to the Flory estimate for d = 3. The interplay between self-avoidance and rigidity at small d is briefly discussed.     

Continuum regularization of quantum field theory

As a complement to our earlier study of renormalization at the Langevin regularized level, we report here on equivalent renormalization programs for regularized Schwinger-Dyson systems. Both one-loop and iterated loop renormalizations of the Green functions of QCD₄ are given, and are shown to be equivalent to the Langevin results. The optional apparent ?-renormalization discussed in IV is shown to apply as well to Schwinger-Dyson systems as to Langevin systems.     

Local operator products in gauge theories,II

We study the operator product expansion of two gauge-invariant currents. We discuss the operators that may appear in the expansion. We study the hadronic matrix elements of gauge-variant operators that may appear in the operator product expansion and show that in the physical applications contemplated so far their hadronic physical matrix elements either vanish or cannot be isolated from strong interactions. We also study the gauge dependence of the Green's functions of the product of two (weak or e.m.) currents and show that is corresponds to the gauge dependence of the additional renormalization counterterms due to weak and electromagnetic interactions.     

Higher-order effects in asymptotically free gauge theories: The anomalous dimensions of Wilson operators

We calculate the anomalous dimensions of the lowest twist, flavour non-singlet operators in the Wilson expansion to two loops. The calculation is performed using dimensional regularization and the minimal subtraction renormalization scheme. The physical relevance of our results in deep inelastic scattering is discussed.     

Supersymmetry and its spontaneous breaking in the random field Ising model

We provide a resolution of one of the long-standing puzzles in the theory of disordered systems. By reformulating the functional renormalization group for the critical behavior of the random field Ising model in a superfield formalism, we are able to follow the associated supersymmetry and its spontaneous breaking along the functional renormalization group flow. Breaking is shown to occur below a critical dimension d(DR) ? 5.1 and leads to a breakdown of the "dimensional reduction" property. We compute the critical exponents as a function of dimension and give evidence that scaling is described by three independent exponents.     

Renormalization and symmetry conditions in supersymmetric QED

For supersymmetric gauge theories a consistent regularization scheme that preserves supersymmetry and gauge invariance is not known. In this article we tackle this problem for supersymmetric QED within the framework of algebraic renormalization. For practical calculations, a non-invariant regularization scheme may be used together with counterterms from all power-counting renormalizable interactions. From the Slavnov–Taylor identity, expressing gauge invariance, supersymmetry and translational invariance, simple symmetry conditions are derived that are important in a twofold respect: they establish exact relations between physical quantities that are valid to all orders, and they provide a powerful tool for the practical determination of the counterterms. We perform concrete one-loop calculations in dimensional regularization, where supersymmetry is spoiled at the regularized level, and show how the counterterms necessary to restore supersymmetry can be read off easily. In addition, a specific example is given how the supersymmetry transformations in one-loop order are modified by non-local terms. Received: 23 July 1999 / Published online: 14 October 1999     

Functional renormalization group for quantized anharmonic oscillator

Functional renormalization group methods formulated in the real-time formalism are applied to the O(N) symmetric quantum anharmonic oscillator, considered as a 0 + 1 dimensional quantum field-theoric model, in the next-to-leading order of the gradient expansion of the one- and two-particle irreducible effective action. The infrared scaling laws and the sensitivity-matrix analysis show the existence of only a single, symmetric phase. The Taylor expansion for the local potential converges fast while it is found not to work for the field-dependent wavefunction renormalization, in particular for the double-well bare potential. Results for the gap energy for the bare anharmonic oscillator potential hint on improving scheme-independence in the next-to-leading order of the gradient expansion, although the truncated perturbation expansion in the bare quartic coupling provides strongly scheme-dependent results for the infrared limits of the running couplings.     

Continuum regularization of quantum field theory

As an extension of our earlier one-loop renormalization studies at the regularized Schwinger-Dyson level, we report here on equivalent renormalization programs for regularized Langevin systems. Proper structure is discussed, and proper one-loop renormalizations of the Green functions of φ ₆ ³ and QCD₄ are given. An optional apparent?-renormalization is discussed as a technical simplificaiton for gauge theories with Zwanziger's gauge-fixing.     

Phase transitions and dimensional reduction

Using field theoretic methods a formalism is presented within which the critical behaviour of a system undergoing a dimensional reduction may be investigated. As a paradigm we study an Ising-like system on S¹ × R^3−ε. If the size of the system is L, and the correlation length ξ, then as L/ξ varies it is possible to get critical behaviour associated with two different fixed points. By exploiting a set of renormalization schemes which lead to manifest dimensional reduction in the loop expansion, and utilizing the renormalization group and an expansion about the fixed point of the finite system, we quantitatively investigate such crossover behaviour in its entirety. In particular, effective susceptibility and correlation length exponents are defined and computed. These exponents interpolate between those associated with a (4 − ε)-dimensional and a (3 − ε)-dimensional Ising model.     

The Operator Product Expansion Converges in Perturbative Field Theory

We show, within the framework of the massive Euclidean j⁴{\varphi^4} -quantum field theory in four dimensions, that the Wilson operator product expansion (OPE) is not only an asymptotic expansion at short distances as previously believed, but even converges at arbitrary finite distances. Our proof rests on a detailed estimation of the remainder term in the OPE, of an arbitrary product of composite fields, inserted as usual into a correlation function with further “spectator fields”. The estimates are obtained using a suitably adapted version of the method of renormalization group flow equations. Convergence follows because the remainder is seen to become arbitrarily small as the OPE is carried out to sufficiently high order, i.e. to operators of sufficiently high dimension. Our results hold for arbitrary, but finite, loop orders. As an interesting side-result of our estimates, we can also prove that the “gradient expansion” of the effective action is convergent.     

Interpolation parameter and expansion for a three-dimensional nontrivial scalar infrared fixed point

We compute a nontrivial infraredϕ ₃ ⁴ -fixed point by means of an interpolation expansion in fixed dimension. The expansion is formulated for an infinitesimal momentum-space renormalization group. We choose a coordinate representation for the fixed-point interaction in derivative expansion, and compute its coordinates to high orders by means of computer algebra. We compute the series for the critical exponentv up to order 25 of interpolation expansion in this representation, and evaluate it using Padé, Borel-Padé, Borel-conformal-Padé, andD log -Padé resummation. The resummation returns 0.6262(13) as the value ofv. Our renormalization group uses canonical resealing, for whichη = 0     

On the equivalence between implicit regularization and constrained differential renormalization

Constrained differential renormalization (CDR) and the constrained version of implicit regularization are two regularization independent techniques that do not rely on dimensional continuation of the space-time. These two methods, which have rather distinct bases, have been successfully applied to several calculations, which show that they can be trusted as practical, symmetry invariant frameworks (gauge and supersymmetry included) in perturbative computations even beyond one-loop order. In this paper, we show the equivalence between these two methods at one-loop order. We show that the configuration space rules of CDR can be mapped into the momentum-space procedures of implicit regularization, the major principle behind this equivalence being the extension of the properties of regular distributions to regularized ones. PACS 11.10.Gh; 11.15.Bt; 11.15.-q     

The renormalization of several composite operator insertions,Wilson expansions of composite operators,and the applications to short-distance behaviour

We examine the renormalization of composite operator insertions within a normal product framework, and prove various identities between composite operator insertions. In particular a proof of the Wilson expansion of composite operators in the presence of other composite operators is given. The analysis necessitates a re-examination of several physical processes, and it is shown in particular why the results for deep inelastic scattering still hold.