Dimensional Regularization Technique Research Articles

Large-N expansion, conformal field theory and renormalization-group flows in three dimensions

I study a class of interacting conformal field theories and conformal windows in three dimensions, formulated using the Parisi large-N approach and a modified dimensional-regularization technique. Bosons are associated with composite operators and their propagators are dynamically generated by fermion bubbles. Renormalization-group flows between pairs of interacting fixed points satisfy a set of non-perturbative g <-> 1/g dualities. There is an exact relation between the beta function and the anomalous dimension of the composite boson. Non-Abelian gauge fields have a non-renormalized and quantized gauge coupling, although no Chern-Simons term is present. A problem of the naive dimensional-regularization technique for these theories is uncovered and removed with a non-local, evanescent, non-renormalized kinetic term. The models are expected to be a fruitful arena for the study of odd-dimensional conformal field theory.

Pairing in low-density Fermi gases

We consider pairing in a dilute system of Fermions with a short-range interaction. While the theory is ill-defined for a contact interaction, the BCS equations can be solved in the leading order of low-energy effective field theory. The integrals are evaluated with the dimensional regularization technique, giving analytic formulas relating the pairing gap, the density, and the energy density to the two-particle scattering length.

Sp(2) regularization and genuine anomalies

An Sp(2) formulation of the dimensional regularization technique is proposed, in order to calculate higher loop genuine anomalies and effective action in the extended framework. The extended anomalous master equations and corresponding action are generalized to the BRST–anti-BRST symmetry generating two towers of equations that encode the entire proposed algorithm and give the well defined, non-trivial quantities involved.

Probing negative-dimensional integration: two-loop covariant vertex and one-loop light-cone integrals

The negative-dimensional integration method (NDIM) seems to be a very promising technique for evaluating massless and/or massive Feynman diagrams. It is unique in the sense that the method gives solutions in different regions of external momenta simultaneously. Moreover, it is a technique whereby the difficulties associated with performing parametric integrals in the standard approach are transferred to a simpler solving of a system of linear algebraic equations, thanks to the polynomial character of the relevant integrands. We employ this method to evaluate a scalar integral for a massless two-loop three-point vertex with all the external legs off-shell, and consider several special cases for it, yielding results, even for distinct simpler diagrams. We also consider the possibility of NDIM in non-covariant gauges such as the light-cone gauge and do some illustrative calculations, showing that for one-degree violation of covariance (i.e. one external, gauge-breaking, light-like vector n μ ) the ensuing results are concordant with the ones obtained via either the usual dimensional regularization technique, or the use of the principal value prescription for the gauge-dependent pole, while for two-degree violation of covariance—i.e. two external, light-like vectors n μ , the gauge-breaking one, and (its dual) n μ ∗ —the ensuing results are concordant with the ones obtained via causal constraints or the use of the so-called generalized Mandelstam-Leibbrandt prescription.

The three-point function in split dimensional regularization in the Coulomb gauge

We use a gauge-invariant regularization procedure, called split dimensional regularization, to evaluate the quark self-energy Σ( p) and quark-quark-gluon vertex function Λ μ ( p′, p) in the Coulomb gauge, ∇ · A a = 0 . The technique of split dimensional regularization was designed to regulate Coulomb-gauge Feynman integrals in non-Abelian theories. The technique which is based on two complex regulating parameters, ω and σ, is shown to generate a well-defined set of Coulomb-gauge integrals. A major component of this project deals with the evaluation of four-propagator and five-propagator Coulomb integrals, some of which are non-local. It is further argued that the standard one-loop BRST identity relating Σ and Λ μ , should by rights be replaced by a more general BRST identity which contains two additional contributions from ghost vertex diagrams. Despite the appearance of non-local Coulomb integrals, both Σ and Λ μ are local functions which satisfy the appropriate BRST identity. Application of split dimensional regularization to two-loop energy integrals is briefly discussed.

One-loop renormalization of the QCD Schrödinger functional

In a previous publication we have constructed the Schrödinger functional in Wilson's lattice QCD. It was found that the naive continuum limit leads to a well-defined classical continuum theory. Starting from the latter, a formal continuum definition of the Schrödinger functional is given and its saddle point expansion is carried out to one-loop order of perturbation theory. Dimensional regularization and heat kernel techniques are used to determine the one-loop divergences. These are partly canceled by the usual renormalizations of the quark mass and the coupling constant in QCD. An additional divergence can be absorbed in a multiplicative renormalization of the quark boundary fields. The corresponding boundary counterterm is a local polynomial in the fields, so that we confirm a general expectation of Symanzik.

Axial anomaly and dimensional regularization (A review)

This is a brief review of some theoretical and practical aspects of the technique of dimensional regularization. A detailed general overview of theγ5 problem and its solutions within various schemes of dimensional regularization is given and, as an example of practical calculation, we explicitly compute the VVA anomaly using differentγ5 prescriptions. Apart from this topic, we present an elementary introduction to the subject of ultraviolet divergences in perturbative quantum field theory, regularization procedures, Ward identities and anomalies with a special stress on the axial anomaly in order to cover the background necessary for the rest of the review and to make it readable for non-specialists.

Dimensional versus lattice regularization with Lüscher's Yang mills theory

It is pointed out that the coefficients of Lüscher's effective model space Hamiltonian, which is based upon dimensional regularization techniques, can be reproduced by applying folded diagram perturbation theory to the Kogut Susskind Hamiltonian and by performing a lattice continuum limit (keeping the volume fixed). Alternative cutoff regularizations of the Hamiltonian are in general inconsistent, the critical point beeing the correct prediction for Lüscher's tadpole coefficient which is formally quadratically divergent and which has to become a well defined (negative) number.

Decoherence and back reaction in quantum cosmology: Multidimensional minisuperspace examples.

We study some aspects of the semiclassical limit of quantum cosmology using multidimensional minisuperspace models with scalar perturbations. Regarding the minisuperspace coordinates as the relevant degrees of freedom and the scalar field modes as the irrelevant ones, we study the reduced density matrix to analyze the existence of decoherence and correlations. For a Bianchi type-I model we show that the correlations predicted from the peak of the Wigner function associated with a decohered WKB branch are the same as those determined by the semiclassical Einstein equations. We discuss the renormalization of the divergent quantities that appear in the calculation and show that usual dimensional-regularization techniques only allow us to regulate the phase of the reduced density matrix. This illustrates the fundamental difference that exists between the divergences found in the back-reaction equations and the ones that affect the quantum coherence. We analyze a class of models with adiabatic behavior in which there are no problematic divergences affecting the decoherence factor. Finally, we discuss some aspects of the relation between our approach, based on the use of the reduced density matrix, and that formulated in terms of spacetime histories.

The gluon self-energy in cavity QCD

A cavity regularization method, which is closely related to the dimensional regularization technique, is applied to the gluon self-energy in cavity QCD. The values obtained for the self-energies of the low-lying gluon modes are positive for both the quark- and the gauge-loop diagrams.

Chiral symmetry and nuclear interaction

After summing the bubble diagrams of pion meson, σ meson and ω meson, the nucleon-nucleon interaction of Chiral σ - ω model is investigated. The technique of dimensional regularization and the counterterms for renormalization are introduced. The nuclear force of chiral σ - ω model is given.

The one-loop Green’s functions of dimensionally reduced gauge theories

We apply the dimensional regularization technique as well as that by dimensional reduction to the calculation of the regularized one-loop Green’s functions ind0-dimensional Yang-Mills theory with real massless scalars and spinors in arbitrary (real) representations of a gauge groupG. As a particular example, the super-symmetrically regularized one-loop Green’s functions of theN=4 supersymmetric Yang-Mills model are derived.

Charge renormalization of the Yang-Mills theory up to fourth order using dimensional regularization

We use the technique of dimensional regularization with Wilson loops to do explicitly the calculations of the renormalization of the charge in the Yang-Mills theory up to fourth order. We show that in dimensions different from four the results cannot be interpreted as charge renormalization.

Wave vector dependent susceptibility of a free electron gas in D dimensions and the singularity at 2kF

The static susceptibility of a free electron gas in D dimensions at T=0 is obtained by techniques of dimensional regularization. Our solutions for the susceptibility χ(k,D) are given in terms of the hypergeometric function. For any integer dimensions analytic expressions are possible. The high- and low-k series solutions are shown to be related by an analytic continuation if D is an odd integer, but not related if D is an even integer. The singularity at 2kF is a branch point, whereupon the series solutions are absolutely convergent, yielding χ(k=2kF,D)=(D−1)−1. The relationship of χkD has the appearance of a PVT diagram.

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.

Decay constants and 2γ decay amplitudes of massive pseudoscalar bosons

Expressions for decay constantsfP(q2) and 2γ decay amplitudesFP2γ(q2) of massive pseudoscalar mesons P are given in terms of effective quark massmq assuming that the decay amplitudes can be evaluated from the lowest-order quark-line diagrams with the effective meson-quark vertexGiγ5, and, for logarithmically divergent Feynman amplitudes, only the finite terms which are evaluated by use of the technique of dimensional regularization have physical meaning. A new relationfπmπ2)=\(\left( {\sqrt {6/2\pi } } \right)K^{^{\frac{1}{2}} } m_q \), whereK=ln2π-γ=1.261, is obtained, and it predictsmu=md≅300 MeV for input datafπ(mπ2)=132 MeV. It is shown thatfP(q2)/fP(0) andFP2γ(0) are almost independent ofz2=q2/4mq2 for 0≤z≲0.9. At the same time, it is pointed out that those critically depend on the value ofz2 atz2≅1, so that use of the precise value ofz2 is important for reliable estimates of those for heavy mesons.

Gauge theories in external electromagnetic fields: The standard SU(2) × U(1) model

Spontaneously broken SU(2) × U(1) electroweak theory in a constant background electromagnetic field is shown to admit a phase transition to a restored symmetry phase for large values of the external field. Allowing for two arbitrary parameters, the Higgs mass and the expectation value of the Higgs field, a lower bound is derived on the critical magnetic field required for restoration. The configuration space heat kernel method adapted to dimensional regularization techniques is used allowing calculations to be performed in arbitrary renormalizable gauges.

Exact one-gluon corrections for inclusive weak processes

First-order gluon radiative corrections to the rate of weak decay of a quark into three quarks are calculated taking full account of all the masses involved. Using dimensional regularization techniques, the infrared divergences inherent to the problem cancel out, and the finite parts in convenient analytic forms which tend to the correct zero-mass limits are obtained. The total correction to the decay rate always exceeds its (negative) zero-mass limit and changes sign as the masses increase. Thus, except for small mass values, it is positive and acts in a direction opposite to that expected when the masses vanish.

Evaluation of Effective Potential for Gross-Neveu Model with Tadpole Diagram Method

The effective potential of Gross-Neveu model with corrections is calculated by using tadpole diagrams and dimensional regularization technique. After the high order of the effective potential is taken into account the continuous symmetry will not be violated while the discrete chiral symmetry still remains broken. This provides an interesting example of Coleman's theorem in two-dimensional field theory. The calculation results are in agreement with those of R.G. Root who had summed up a sort of infinite diagrams. Our conclusion is also consistent with that of the variational method based on BCS wave functions.

An electrostatic example to illustrate dimensional regularization and renormalization group technique

An easy and intuitive introduction to the regularization and renormalization techniques in elementary particle physics is given. It is based on the use of a simple electrostatic problem.