Showing papers on "Dimensional regularization published in 1997"

Regularization, Scale-Space, and Edge Detection Filters

Abstract: Computational vision often needs to deal with derivatives of digital images. Such derivatives are not intrinsic properties of digital data; a paradigm is required to make them well-defined. Normally, a linear filtering is applied. This can be formulated in terms of scale-space, functional minimization, or edge detection filters. The main emphasis of this paper is to connect these theories in order to gain insight in their similarities and differences. We do not want, in this paper, to take part in any discussion of how edge detection must be performed, but will only link some of the current theories. We take regularization (or functional minimization) as a starting point, and show that it boils down to Gaussian scale-space if we require scale invariance and a semi-group constraint to be satisfied. This regularization implies the minimization of a functional containing terms up to infinite order of differentiation. If the functional is truncated at second order, the Canny-Deriche filter arises. It is also shown that higher dimensional regularization boils down to a rotated version of the one dimensional case, when Cartesian invariance is imposed and the image is vanishing at the borders. This means that the results from 1D regularization can be easily generalized to higher dimensions. Finally we show how an efficient implementation of regularization of order n can be made by recursive filtering using 2n multiplications and additions per output element without introducing any approximation.

Non-perturbative regularization and renormalization: simple examples from non-relativistic quantum mechanics

Abstract: We examine several zero-range potentials in non-relativistic quantum mechanics. The study of such potentials requires regularization and renormalization. We contrast physical results obtained using dimensional regularization and cutoff schemes and show explicitly that in certain cases dimensional regularization fails to reproduce the results obtained using cutoff regularization. First we consider a delta-function potential in arbitrary space dimensions. Using cutoff regularization we show that for $d \ge 4$ the renormalized scattering amplitude is trivial. In contrast, dimensional regularization can yield a nontrivial scattering amplitude for odd dimensions greater than or equal to five. We also consider a potential consisting of a delta function plus the derivative-squared of a delta function in three dimensions. We show that the renormalized scattering amplitudes obtained using the two regularization schemes are different. Moreover we find that in the cutoff-regulated calculation the effective range is necessarily negative in the limit that the cutoff is taken to infinity. In contrast, in dimensional regularization the effective range is unconstrained. We discuss how these discrepancies arise from the dimensional regularization prescription that all power-law divergences vanish. We argue that these results demonstrate that dimensional regularization can fail in a non-perturbative setting.

Two-loop euler-heisenberg lagrangian in dimensional regularization

Abstract: By calculating the two-loop QED Euler-Heisenberg Lagrangian in dimensional regularization, we clarify a discrepancy between two previous calculations of this quantity performed in proper-time regularization.

The Lamb Shift in Dimensional Regularization

Abstract: We present a simple derivation of the Lamb shift using effective field theory techniques and dimensional regularisation.

Dimensional Regularization in Quarkonium Calculations

Abstract: Dimensional regularization is incompatible with the standard covariant projection methods that are used to calculate the short-distance coefficients in inclusive heavy-quarkonium production and annihilation rates. A new method is developed that allows dimensional regularization to be used consistently to regularize the infrared and ultraviolet divergences that arise in these perturbative calculations. We illustrate the method by calculating the leading color-octet terms and the leading color-singlet terms in the gluon fragmentation functions for arbitrary quarkonium states. We resolve a discrepancy between two previous calculations of the gluon fragmentation functions for the spin-triplet P-wave quarkonium states. {copyright} {ital 1997} {ital The American Physical Society}

Algebraic algorithms for multiloop calculations The first 15 years. What's next?

Abstract: The ideas behind the concept of algebraic (“integration-by-parts”) algorithms for multiloop calculations are reviewed. For any topology and mass pattern, a finite iterative algebraic procedure is proved to exist which transforms the corresponding Feynman-parametrized integrands into a form that is optimal for numerical integration, with all the poles in D — 4 explicitly extracted.

Two-loop QCD corrections to charged-Higgs-mediated $b\to s\gamma$ decay

Abstract: The charged-Higgs-mediated contribution to the Wilson coefficient of the $b\to s\gamma$ magnetic penguin is expected to be one of the more promising candidates for a supersymmetric effect in B physics, probably the only one in gauge-mediated models. We compute the two-loop QCD correction to it. With naive dimensional regularization and MSbar subtraction, for reasonable values of the charged Higgs mass and for mu-bar = m_top, we find a (10--20)% reduction of the corresponding one-loop effect.

Nonequilibrium dynamics: A renormalized computation scheme

Abstract: We present a regularized and renormalized version of the one-loop nonlinear relaxation equations that determine the nonequilibrium time evolution of a classical (constant) field coupled to its quantum fluctuations. We obtain a computational method in which the evaluation of divergent fluctuation integrals and the evaluation of the exact finite parts are cleanly separated so as to allow for a wide freedom in the choice of regularization and renormalization schemes. We use dimensional regularization here. Within the same formalism we analyze also the regularization and renormalization of the energy-momentum tensor. The energy density serves to monitor the reliability of our numerical computation. The method is applied to the simple case of a scalar {phi}{sup 4} theory; the results are similar to the ones found previously by other groups. {copyright} {ital 1997} {ital The American Physical Society}

Reduction of One-loop Tensor Form-Factors to Scalar Integrals: A General Scheme

Abstract: A general method for reducing tensor form factors, that appear in one-loop calculations in dimensional regularization, to scalar integrals is presented. The method is an extension of the reduction scheme introduced by Passarino and Veltman and is applicable in all regions of parameter space including those where kinematic Gram determinant vanishes. New relations between the the form factors that valid for vanishing Gram determinant play a key role in the extended scheme.

The Two-Loop Euler-Heisenberg Lagrangian in Dimensional Renormalization

Abstract: We clarify a discrepancy between two previous calculations of the two-loop QED Euler-Heisenberg Lagrangian, both performed in proper-time regularization, by calculating this quantity in dimensional regularization.

Regularization, renormalization, and range: The nucleon-nucleon interaction from effective field theory

Abstract: Regularization and renormalization is discussed in the context of low energy effective field theory treatments of two or more heavy particles (such as nucleons). It is desirable to regulate the contact interactions from the outset by treating them as having a finite range. The low energy physical observables should be insensitive to this range provided that the range is of a similar or greater scale than that of the interaction. Alternative schemes, such as dimensional regularization, lead to paradoxical conclusions such as the impossibility of repulsive interactions for truly low energy effective theories where all of the exchange particles are integrated out. This difficulty arises because a nonrelativistic field theory with repulsive contact interactions is trivial in the sense that the S matrix is unity and the renormalized coupling constant zero. Possible consequences of low energy attraction are also discussed. It is argued that in the case of large or small scattering lengths, the region of validity of effective field theory expansion is much larger if the contact interactions are given a finite range from the beginning. {copyright} {ital 1997} {ital The American Physical Society}

Renormalization of nonequilibrium dynamics in FRW cosmology

Abstract: We derive the renormalized nonequilibrium equations of motion for a scalar field and its quantum back reaction in a conformally flat Friedmann-Robertson-Walker universe. We use a fully covariant formalism proposed by us recently for handling numerically and analytically nonequilibrium dynamics in the one-loop approximation. The system is assumed to be in a conformal vacuum state initially. We use dimensional regularization; we find that the counterterms can be chosen independent of the initial conditions though the divergent leading order graphs do depend on them. {copyright} {ital 1997} {ital The American Physical Society}

One-loop counterterms for higher derivative regularized Lagrangians

Abstract: We explicitly calculate one-loop divergences for an arbitrary field theory model using the higher derivative regularization and nonsingular gauge condition. They are shown to agree with the results found in the dimensional regularization and do not depend on the form of regularizing term. So, the consistency of the higher derivative regularization is proven at the one-loop level. The result for the Yang-Mills theory is reproduced.

Nonequilibrium dynamics: Preheating in the SU(2) Higgs model

Abstract: The term ``preheating'' has been introduced recently to denote the process in which energy is transferred from a classical inflaton field into fluctuating field (particle) degrees of freedom without generating yet a real thermal ensemble. The models considered up to now include, in addition to the inflaton field, scalar or fermionic fluctuations. On the other hand, the typical ingredient of an inflationary scenario is a non-Abelian spontaneously broken gauge theory. So the formalism should also be developed to include gauge field fluctuations excited by the inflaton or Higgs field. We have chosen here, as the simplest non-Abelian example, the SU(2) Higgs model. We consider the model at temperature zero. From the technical point of view we generalize an analytical and numerical renormalized formalism developed by us recently to coupled channnel systems. We use the 't Hooft--Feynman gauge and dimensional regularization. We present some numerical results but reserve a more exhaustive discussion of solutions within the parameter space of two couplings and the initial value of the Higgs field to a future publication.

Effects of Self-Avoidance on the Tubular Phase of Anisotropic Membranes

Abstract: We study the tubular phase of self-avoiding anisotropic membranes. We discuss the renormalizability of the model Hamiltonian describing this phase and derive from a renormalization group equation some general scaling relations for the exponents of the model. We show how particular choices of renormalization factors reproduce the Gaussian result, the Flory theory and the Gaussian Variational treatment of the problem. We then study the perturbative renormalization to one loop in the self-avoiding parameter using dimensional regularization and an epsilon-expansion about the upper critical dimension, and determine the critical exponents to first order in epsilon.

Differential regularization of topologically massive Yang-Mills theory and Chern-Simons theory

Abstract: We apply a differential renormalization method to the study of three-dimensional topologically massive Yang-Mills and Chern-Simons theories. The method is especially suitable for such theories as it avoids the need for dimensional continuation of a three-dimensional antisymmetric tensor and the Feynman rules for three-dimensional theories in coordinate space are relatively simple. The calculus involved is still lengthy but not as difficult as other existing methods of calculation. We compute one-loop propagators and vertices and derive the one-loop local effective action for topologically massive Yang-Mills theory. We then consider Chern-Simons field theory as the large mass limit of topologically massive Yang-Mills theory and show that this leads to the famous shift in the parameter k. Some useful formulas for the calculus of differential renormalization of three-dimensional field theories are given in an Appendix.

NLO Production and Decay of Quarkonium

Abstract: We present a calculation of next-to-leading-order (NLO) QCD corrections to total hadronic production cross sections and to light-hadron-decay rates of heavy quarkonium states. Both colour-singlet and colour-octet contributions are included. We discuss in detail the use of covariant projectors in dimensional regularization, the structure of soft-gluon emission and the overall finiteness of radiative corrections. We compare our approach with the NLO version of the threshold-expansion technique recently introduced by Braaten and Chen. Most of the results presented here are new. Others represent the first independent reevaluation of calculations already known in the literature. In this case a comparison with previous findings is reported.

Supersymmetric Yang-Mills-Chern-Simons theory

Abstract: We prove that three-dimensional N=1 supersymmetric Yang Mills-Chern-Simons theory is finite to all loops. This leaves open the possibility that different regularization methods give different finite effective actions. We show that for this model dimensional regularization and regularization by dimensional reduction yield the same effective action.

Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion

Abstract: A method of calculating Feynman diagrams from their small momentum expansion [1] is extended to diagrams with zero mass thresholds We start from the asymptotic expansion in large masses [2] (applied to the case when all $M_i^2$ are large compared to all momenta squared) Using dimensional regularization, a finite result is obtained in terms of powers of logarithms (describing the zero-threshold singularity) times power series in the momentum squared Surprisingly, these latter ones represent functions, which not only have the expected physical “second threshold” but have a branchcut singularity as well below threshold at a mirror position These can be understood as pseudothresholds corresponding to solutions of the Landau equations In the spacelike region the imaginary parts from the various contributions cancel For the two-loop examples with one mass M, in the timelike region for q2 ≈ M2 we obtain approximations of high precision This will be of relevance in particular for the calculation of the decay Z → bbin the m b = 0 approximation

The potential of effective field theory in NN scattering

Abstract: We study an effective field theory of interacting nucleons at distances much greater than the pion's Compton wavelength. In this regime the NN potential is conjectured to be the sum of a delta function and its derivatives. The question we address is whether this sum can be consistently truncated at a given order in the derivative expansion, and systematically improved by going to higher orders. Regularizing the Lippmann-Schwinger equation using a cutoff we find that the cutoff can be taken to infinity only if the effective range is negative. A positive effective range---which occurs in nature---requires that the cutoff be kept finite and below the scale of the physics which has been integrated out, i.e. O(m_\pi). Comparison of cutoff schemes and dimensional regularization reveals that the physical scattering amplitude is sensitive to the choice of regulator. Moreover, we show that the presence of some regulator scale, a feature absent in dimensional regularization, is essential if the effective field theory of NN scattering is to be useful. We also show that one can define a procedure where finite cutoff dependence in the scattering amplitude is removed order by order in the effective potential. However, the characteristic momentum in the problem is given by the cutoff, and not by the external momentum. It follows that in the presence of a finite cutoff there is no small parameter in the effective potential, and consequently no systematic truncation of the derivative expansion can be made. We conclude that there is no effective field theory of NN scattering with nucleons alone.

Massive Two-Loop Integrals and Higgs Physics

Abstract: We describe in some detail the present features of an automatic loop calculation program as well as the integration techniques that go into it. The program, called XLOOPS 1.0, allows one to calculate massive one- and two-loop Feynman diagrams in the Standard Model including their tensor structure. UV divergences in UV divergent integrals are explicitly computed in dimensional regularization. One-loop integrals are calculated analytically in d eq 4 dimensions whereas two-loop integrals are reduced to two-fold integral representations which the program evaluates numerically. We discuss Higgs decay at the two-loop level as a first application of the novel integration techniques that are incorporated into XLOOPS.

Asymptotic expansions of two-loop Feynman diagrams in the Sudakov limit

Abstract: Recently presented explicit formulae for asymptotic expansions of Feynman diagrams in the Sudakov limit are applied to typical two-loop diagrams. For a diagram with one non-zero mass these formulae provide an algorithm for analytical calculation of all powers and logarithms, i.e. coefficients in the corresponding expansion $(Q^2)^{-2} \sum_{n,j=0} c_{nj} t^{-n} \ln^j t$, with $t=Q^2/m^2$ and $j \leq 4$. Results for the coefficients at several first powers are presented. For a diagram with two non-zero masses, results for all the logarithms and the leading power, i.e. the coefficients $c_{nj}$ for n=0 and j=4,3,2,1,0 are obtained. A typical feature of these explicit formulae (written through a sum over a specific family of subgraphs of a given graph, similar to asymptotic expansions for off-shell limits of momenta and masses) is an interplay between ultraviolet, collinear and infrared divergences which represent themselves as poles in the parameter $\eps=(4-d)/2$ of dimensional regularization. In particular, in the case of the second diagram, which is free from the divergences, individual terms of the asymptotic expansion involve all the three kinds of divergences resulting in poles, up to $1/\eps^4$, which are successfully canceled in the sum.