A note on the dimensional regularization method in the presence of infrared divergences

We indicate a practical way of applying the dimensional regularization scheme in the presence of infrared divergences, consisting of a finite subtraction procedure.     

Supersymmetric dimensional regularization

Being inconsistent in superfield language the regularization by dimensional reduction is reformulated unambiguously in terms of the component fields. It is found to violate the supersymmetric Ward identities in higher orders. For several models the corresponding domains of invariance are determined and shown to embrace the range of the calculations done so far (for details see Avdeev L. V., Chochia G. A., Vladimirov A. A.: Phys. Letters (in press); preprint JINR Dubna, E2-81-370, 1981).Abstract of the paper presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–21, 1981.     

Equivalence of dimensional reduction and dimensional regularisation

For some years there has been uncertainty over whether regularisation by dimensional reduction (DRED) is viable for non-supersymmetric theories. We resolve this issue by showing that DRED is entirely equivalent to standard dimensional regularisation (DREG), to all orders in perturbation theory and for a general renormalisable theory. The two regularisation schemes are related by an analytic redefinition of the couplings, under which the -functions calculated using DRED transform into those computed in DREG. TheS-matrix calculated using DRED is numerically equal to the DREG version, ensuring that both schemes give the same physics.     

Haag''s theorem and dimensional regularization

It is shown that dimensionally regularized field theories are free from the difficulty in Haag's theorem, which states that there exists no relativistic theory with interaction in which the Fock representation is used.     

Infrared singularities in the dimensional regularization scheme

Infrared singularities arising in some renormalized amplitudes of quantum electrodynamics are analyzed using the dimensional regularization method. We define infrared and ultraviolet convergent regions in the ν complex plane (ν is the number of dimensions of space time). It turns out that these regions do not overlap for quantum electrodynamics. Nevertheless, it is shown that there exists a unique analytic continuation from the infrared convergent region which allows us to interpret the infrared divergence in the renormalized electron self-energy amplitude as an isolated singularity at ν = 4. This statement seems to be true at all orders of perturbation theory. We also prove that the double limit μ → 0, ν → 4 (μ is the auxiliary photon mass) does not exist in quantum electrodynamics and we conjecture that this lack of uniformity provides theoretical support for the ansatz of Marciano and Sirlin.     

Ambiguities in the infrared regularization of QCD

The next-to-leading corrections of leading-log asymptotic freedom are determined in the known infrared (IR) regularization schemes. On- and off-shell calculation leads to different answers; the n-dimensional regularization of IR and mass singularities agrees with the on-shell results. The non-log corrections in Drell-Yan processes are important for Q² ? 10³ GeV².     

Physical model of dimensional regularization

We explicitly construct fractals of dimension \(4{-}\varepsilon \) on which dimensional regularization approximates scalar-field-only quantum-field theory amplitudes. The construction does not require fractals to be Lorentz-invariant in any sense, and we argue that there probably is no Lorentz-invariant fractal of dimension greater than 2. We derive dimensional regularization’s power-law screening first for fractals obtained by removing voids from 3-dimensional Euclidean space. The derivation applies techniques from elementary dielectric theory. Surprisingly, fractal geometry by itself does not guarantee the appropriate power-law behavior; boundary conditions at fractal voids also play an important role. We then extend the derivation to 4-dimensional Minkowski space. We comment on generalization to non-scalar fields, and speculate about implications for quantum gravity.     

The Callan-Symanzik equation and dimensional regularization

A study is made of the Callan-Symanzik equation using the dimensional regularization method. The final equation is far more general than in the usual derivation in that the effect of mass terms is explicitly included; moreover the momenta need not be restricted to the deep-Euclidean region when considering asymptotic limits.     

Cut-off versus dimensional regularization in the light-cone gauge

The conventional cut-off method is applied to massless light-cone gauge Feynman integrals. Despite the presence of non-local terms in the unintegrated expression for the Yang-Mills self-energy, the cut-off procedure yields the same ultra-violet behaviour as the lengthier technique of dimensional regularization.     

Dimensional regularization of infrared divergences

An analysis of the application of dimensional regularization to infrared divergences in lowest order radiative corrections is presented. The main emphasis of the paper is to show explicitly how dimensional regularization can lead in some cases of considerable interest to very simple and elegant evaluations of infrared divergent contributions and their associated finite parts, and to pinpoint the mathematical reason for the equivalence with the traditional method of regularization.     

On the analytic computation of massless propagators in dimensional regularization

We comment on the algorithm to compute periods using hyperlogarithms, applied to massless Feynman integrals in the parametric representation. Explicitly, we give results for all three-loop propagators with arbitrary insertions including order ε⁴${\epsilon }^4$ and show examples at four and more loops.     

Dimensional regularization and the self-energy of a massive quark in a cavity

Self-energies of massive quarks are calculated in the framework of cavity QCD to order _s for the excited modes of a spherical cavity. The calculation, closely related to Schwinger's proper time technique, is based on the ms dimensional regularization scheme in free space.     

A practicableγ 5-scheme in dimensional regularization

We present a new simpleγ ₅ regularization scheme. We discuss its use in the standard radiative correction calculations including the anomaly contributions. The new scheme features an anticommutingγ ₅ which leads to great simplifications in practical calculations. We carefully discuss the underlying mathematics of ourγ ₅-scheme which is formulated in terms of simple projection operations.     

Two-loop parameter relations between dimensional regularization and dimensional reduction applied to SUSY-QCD

The two-loop relations between the running gluino–quark–squark coupling, the gluino and the quark mass defined in dimensional regularization (DREG) and dimensional reduction (DRED) in the framework of SUSY-QCD are presented. Furthermore, we verify with the help of these relations that the three-loop β-functions derived in the minimal subtraction scheme combined with DREG or DRED transform into each other. This result confirms the equivalence of the two schemes at the three-loop order, if applied to SUSY-QCD.     

A local and integrable lattice regularization of the massive Thirring model

The light-cone lattice approach to the massive Thirring model is reformulated using a local and integrable lattice hamiltonian written in terms of discrete Fermi fields. Several subtle points concerning boundary conditions, normal ordering, continuum limit, finite renormalizations and decoupling of fermion doublers are elucidated. The relations connecting the six-vertex anisotropy and the various coupling constants of the continuum are analyzed in detail.     

On mass-independence of the minimal subtraction scheme in dimensional regularization II

A self-contained argument is given for the mass independence of the renormalization constants in the minimal subtraction scheme in dimensional regularization in a two mass theory (Yukawa theory). An extension to a theory containing more mass parameters seems straightforward.     

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.