Gauge invariance and renormalization schemes in quantum chromodynamics

A general analysis of the Slavnov-Taylor identity connecting the triple gluon and ghost-ghost-gluon vertices and its consequences for two momentum subtraction (symmetric and asymmetric) renormalization schemes are given. It is shown that in the asymmetric scheme proposed in this paper the relation follows directly from the identity for a simple and natural definition of the renormalization constants. Explicit one-loop expressions for the renormalization constants in an arbitrary covariant gauge, including quark masses are given in support of the general analysis.     

The use of dimensional renormalization schemes in unified theories

We present details and clarify some points in a recently proposed modification of the minimal subtraction adapted to unified theories. As an application we give a precise value of the superheavy gauge boson masses in SU(5).     

Improved Monte Carlo renormalization group method

An extensive program to analyze critical systems using an improved Monte Carlo renormalization group method (IMCRG),⁽¹⁾ being undertaken at LANL and Cornell, is described. Here we first briefly rview the method and then list some of the topics being investigated.     

Running coupling at low momenta, renormalization schemes, and renormalons

We propose a method for summing a perturbation-theory asymptotic series that is related to infrared (IR) renormalons in QCD using special renormalization schemes in which the running coupling constant can be integrated over the small momenta. For our method to work, we should consider higher order perturbation-theory corrections to the standard bubble-chain diagrams. High-order corrections allow one to choose a scheme in which the coupling-constant evolution can be smoothly extrapolated to low momenta. In these schemes, the sum of an (extended) IR-renormalon asymptotic series is defined as an integral of the running coupling constant over the IR region. We present explicit examples of renormalization schemes of QCD that can be used to sum IR-renormalon asymptotic series according to our definition.     

Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings

We present a new efficient analytical approximation scheme to two-point boundary value problems of ordinary differential equations (ODEs) adapted to the study of the derivative expansion of the exact renormalization group equations. It is based on a compactification of the complex plane of the independent variable using a mapping of an angular sector onto a unit disc. We explicitly treat, for the scalar field, the local potential approximations of the Wegner–Houghton equation in the dimension d=3$d=3$ and of the Wilson–Polchinski equation for some values of d∈]2,3]$d\in ]2,3]$. We then consider, for d=3$d=3$, the coupled ODEs obtained by Morris at the second order of the derivative expansion. In both cases the fixed points and the eigenvalues attached to them are estimated. Comparisons of the results obtained are made with the shooting method and with the other analytical methods available. The best accuracy is reached with our new method which presents also the advantage of being very fast. Thus, it is well adapted to the study of more complicated systems of equations.     

Axial and vector anomalies in a broad class of invariant regularization and renormalization schemes

Within the framework of the spinor QED we describe a broad class of gauge invariant ultraviolet cut-off schemes, which provide a unifying view of the regularization of a Feynman graph in general and which may also be generalized to other gauge theories. Extending slightly this class of regularizations, we recover the standard anomalies in the axial-vector and vector Ward identities for the VVA triangle diagram. A cancellation of ultraviolet divergences for this graph is demonstrated, using an explicit-parametric representation, for a broad sub-class of our regularizations. Renormalization of the triangle graph in terms of an auxiliary soft mass is also discussed. Some earlier results concerning properties of the VVA triangle graph and perturbative aspects of the corresponding anomalies are thereby generalized considerably.We are indebted to A. Pogrebkov for numerous discussions. One of us (J. H) thanks J. Novotny for an independent check of a part of calculations.     

Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation

The relation between the Wilson–Polchinski and the Litim optimized ERGEs in the local potential approximation is studied with high accuracy using two different analytical approaches based on a field expansion: a recently proposed genuine analytical approximation scheme to two-point boundary value problems of ordinary differential equations, and a new one based on approximating the solution by generalized hypergeometric functions. A comparison with the numerical results obtained with the shooting method is made. A similar accuracy is reached in each case. Both two methods appear to be more efficient than the usual field expansions frequently used in the current studies of ERGEs (in particular for the Wilson–Polchinski case in the study of which they fail).     

Improved renormalization group equations with dimensional regularization and the ε expansions

Due to the absence of dimensional cut-off parameters in the dimensional regularization scheme, vanishing of the renormalized mass of the scalar boson implies vanishing of its renormalized mass; thus the masses of both bosons and fermions in renormalizable field theories can be made finite by multiplicative mass renormalizations. The improved renormalization group equations in D dimensions are derived in such a way that both the large (or the small) momentum limits and the Wilson ? expansions can be uniformly treated for the fermion as well as the boson cases. We discuss the improved equations for φ₆³ theory, φ₄⁴ theory, quantumelectrodynamics, massive vector-gluon model, and non-Abelian guage theories incorporating fermions. For the latter three classes of theories, the gauge dependent problem of the coefficient functions in the improved renormalization group equations is discussed.     

The\overline {MS} and on-shell renormalization schemes in the analysis of neutrino scattering experiments

The relationship between the \(\overline {MS} \) and on-shell renormalization schemes is discussed and the correction, for finite top quark mass, to the formula connecting sin² θ _W =1?M _W ² /M _Z ² and sin² \(\widehat\theta _W (M_W )\) is given. A table is presented to allow easy conversion. The relative sensitivity, to the top quark and Higgs masses, of the two definitions, when extracted from semi-leptonic neutrino scattering experiments is considered.     

Entanglement renormalization

We propose a real-space renormalization group (RG) transformation for quantum systems on a D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining it into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the entanglement of a block.     

The equations of Wilson's renormalization group and analytic renormalization

In the present series of two papers we solve exactly Wilson's equations for a long-range effective hamiltonian. These equations arise when one seeks a fixed point of the Wilson's renormalization group transformations in the formulation of perturbation theory. The first paper has a general character. Wilson's renormalization transformation and its modifications are defined and the group property for them is established. Some topological aspects of the renormalization transformations are discussed. A space of projection hamiltonians is introduced and a theorem on the invariance of this space with respect to the renormalization transformations is proved.     

Rapidity renormalization group

We introduce a systematic approach for the resummation of perturbative series which involves large logarithms not only due to large invariant mass ratios but large rapidities as well. A series of this form can appear in a variety of gauge theory observables. The formalism is utilized to calculate the jet broadening event shape in a systematic fashion to next-to-leading logarithmic order. An operator definition of the factorized cross section as well as a closed form of the next-to-leading-log cross section are presented. The result agrees with the data to within errors.     

Summed analytic renormalization

With a small modification, analytic renormalization is shown to be multiplicative and the renormalization group equation is derived. For quantum electrodynamics the method is compatible with gauge invariance at the one-loop level.     

Mixing-matrix renormalization revisited

We study the renormalization of normal mixing matrices, which include hermitian and unitary matrices as particular cases. We give a minimal, multiplicative parameterization of counterterms, and compute the renormalized Lagrangian to one-loop order in several simple models with N species of fermions, both in on-shell and [`(MS)]\overline{\mathrm{MS}} schemes. In the on-shell scheme the mass-degenerate case is considered separately.     

On several parameter renormalization group equations and renormalization scheme specifications

An example of the explicit realization of the full renormalization group for physical quantities is investigated. It introduces two-index beta functions β_ij(g_j): (i) the first index specifying the renormalization scheme chosen (including the scale μ); (ii) the second index specifying the coupling of which the appropriate beta function is the differential transformation. The treated example is presented in a 4 — ? dimensional O(n) massless (?·?)² theory with emphasis on the calculation of the fixed point.     

Mixing-matrix renormalization revisited

Eur. Phys. J. C 20, 239-252 (2001) - DOI 10.1007/s100520100663 Published online: 11 May 2001 / Erratum published online: 7 March 2003     

Monte Carlo renormalization group

Monte Carlo computer simulations have long been used to obtain information on the behavior of thermodynamic systems. The method has the advantages of being applicable to a very large class of models and of using only systematically improvable approximations (finite size of system, statistical errors, etc.). However, in the critical region, finite-size effects mask the critical singularities, and put severe practical limits onto the accuracy to which the true critical behavior can be determined. By combining Monte Carlo simulations with a real-space renormalization-group analysis, a large increase in efficiency and accuracy can be achieved—without the uncertainties of the usual truncation approximations. The methods are illustrated by explicit calculations on models exhibiting critical and tricritical behavior.