What can we learn from the finite temperature renormalization group?

The renormalization group for finite temperature quantum field theories is studied, in particular for λ?⁴. It is shown that the “high” temperature limit can only be discussed perturbatively ifT dependent renormalization schemes are implemented. Zero temperature renormalization schemes or renormalization at some fixed reference temperatureT _o are both inadequate as they imply perturbative expansions about fixed points of the renormalization group which are associated with a zero temperature system and a system at temperatureT _o respectively.T dependent schemes give rise to an expansion about the true fixed point of the system, the resulting renormalization group allows the entire crossover between high and low temperature behaviour to be investigated.     

Serious inconsistency implied in conventional on-shell renormalization and a new renormalization concept involving bound-state problems in QED

In an earlier Invited Comment, the present author, as all other authors, advocated the generally accepted conception about the applicability of the conventional on-shell renormalization (elimination of the free-electron pole and residue corrections) for the renormalization of the complete electron propagator in QED. In the present Letter, for the first time, it is proved rigorously that this renormalization procedure, applied for more than 50 years in the literature, drastically violates the U-matrix boundary condition (U(t,t)=1) and, therefore, its application for the renormalization of the electron propagator is not permitted. As a consequence, a new consistent renormalization concept is formulated and applied for the derivation of, for the first time, correctly renormalized bound-electron propagator with the corresponding self-energy function and Feynman diagrams.     

Renormalization Group Invariance in Pionless Effective Field Theory for the NN System

We consider the NN interaction in pionless effective field theory (EFT) up to next-to-next-to-leading order (NNLO) and use a recursive subtractive renormalization scheme to describe NN scattering in the ¹ S ₀ channel. We fix the strengths of the contact interactions at a reference scale, chosen to be the one that provides the best fit for the phase-shifts, and then slide the renormalization scale by evolving the driving terms of the subtracted Lippmann?CSchwinger equation through a non-relativistic Callan?CSymanzik equation. The results show that such a systematic renormalization scheme with multiple subtractions is fully renormalization group invariant.     

The running gauge coupling in the exact renormalization group approach

We discuss the perturbative running Yang-Mills coupling constant in the Wilsonian exact renormalization group approach, and compare it to the running coupling in the more conventional MS scheme. The exact renormalization group approach corresponds to a particular renormalization scheme, and we relate explicitly the corresponding Λ parameters. The unambiguous definition of an exact renormalization group scheme requires, however, the use of a one-loop improved high energy effective action.     

A renormalization approach to many-atom radiation processes

We present a second-order many-body perturbation approach to coherent radiation processes. The theory gives a unified view of several many-atom effects as eigenmodes of the interacting atom-field system. Two renormalization procedures are essential to perform our program; (complex) energy renormalization using a noncanonical transformation and light-velocity renormalization using Bogoliubov transformation for field operators.     

The renormalization group and the ϵ expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ? = 4?d is explained [d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.     

Renormalization and renormalization group behaviour of the O(N) non-linear sigma-model in stochastic quantization

The renormalization and renormalization group behaviour of the two-dimensional O(N) non-linear sigma-model has been discussed in the framework of stochastic quantization. For this purpose the path integral representation of the generating functional of the extended stochastic correlation function for constrained systems has been given. The renormalization is formulated by use of the stochastic effective action which can be defined from this generating functional. The renormalization group behavior of the correlation function with different fictitious time arguments has also been investigated. It reduces to the normal one if all external fictitious times are taken to be equal.     

On the limit cycle for the 1/r potential in momentum space

The renormalization of the attractive 1/r² potential has recently been studied using a variety of regulators. In particular, it was shown that renormalization with a square well in position space allows multiple solutions for the depth of the square well, including, but not requiring a renormalization group limit cycle. Here, we consider the renormalization of the 1/r² potential in momentum space. We regulate the problem with a momentum cutoff and absorb the cutoff dependence using a momentum-independent counterterm potential. The strength of this counterterm is uniquely determined and runs on a limit cycle. We also calculate the bound state spectrum and scattering observables, emphasizing the manifestation of the limit cycle in these observables.     

An introduction to the fundamentals of the renormalization group in critical phenomena

The fundamental concepts underlying the application of the renormalization group and related techniques to critical phenomena are reviewed at an elementary level. Topics discussed include: the definition of the renormalization group as a functional integral over high momentum components of the spin field, the behaviour of the renormalization group near the fixed point and the derivation of scaling, Wilson's approximate recursion relation, trivial and non-trivial fixed points of isotropic spin systems near d = 4, Feynman graph expansions for critical exponents, ? = 4 ? d and 1/n-expansions, the derivation of exact recursion relations and co-ordinate space transformations for d = 2 Ising systems     

Perturbative renormalization of the CPN model

Within an unusual but very convenient parametrization of the fields we show the perturbative renormalizability of the CP^N model. This is first worked out with raditional dimensional renormalization techniques: one finds there is a unique divergence related to an equal coupling constant and mass renormalization while the field and its transformation law are not renormalized. In a second part we use BPHZ renormalization, strengthened by BRS consistency constraints, to obtain the relevant Ward identity for the non-linear symmetry of the model.     

Anomalous dimensions and the renormalization group equations

An interpretation is given of scale and anomalous dimensions in the framework of the renormalization group, and the renormalization group equations are derived which are regarded to represent the conservation of these scale dimensions. By the use of continuous dimensional regularization all coefficient functions appearing in these equations and in the Callan-Symanzik equations are explicitly expressed in terms of the residues of the single poles at n = 4 as well as the finite part of renormalization counter-terms.     

Two-parameter expansion in the renormalization-group analysis of turbulence

The renormalization of the solution of the Navier-Stokes equation for randomly stirred fluid with long-range correlations of the driving force is analysed near two dimensions. It is shown that a local term must be added to the correlation function of the random force for the correct renormalization of the model at two dimensions. The interplay of the short-range and long-range terms in the large-scale behaviour of the model is analysed near two dimensions by the field-theoretic renormalization group. A regular expansion in 2ε=d-2 and δ=2-λ is constructed, whered is the space dimension and λ the exponent of the powerlike correlation function of the driving force. It is shown that in spite of the additional divergences, the asymptotic behaviour of the model near two dimensions is the same as in higher dimensions, contrary to recent conjectures based on an incorrect renormalization procedure.     

θ renormalization, electron-electron interactions and super universality in the quantum Hall regime

The renormalization theory of the quantum Hall effect relies primarily on the non-perturbative concept of θ renormalization by instantons. Within the generalized non-linear σ model approach initiated by Finkelstein we obtain the physical observables of the interacting electron gas, formulate the general (topological) principles by which the Hall conductance is robustly quantized and derive—for the first time—explicit expressions for the non-perturbative (instanton) contributions to the renormalization group β and γ functions. Our results are in complete agreement with the recently proposed idea of super universality which says that the fundamental aspects of the quantum Hall effect are all generic features the instanton vacuum concept in asymptotically free field theory.     

Mixing renormalization in Majorana neutrino theories

The renormalization of general theories with inter-family mixing of Dirac and/or Majorana fermions is studied at the one-loop electroweak order. The phenomenological significance of the mixing-matrix renormalization is discussed, within the context of models based on the SU(2)_L⊗U(1)_Y gauge group. The effect of radiative neutrino masses present in these models is naturally taken into account in this formulation. As an example, charged-lepton universality in pion decays is investigated in the heavy-neutrino limit. Non-decoupling heavy-neutrino effects induced by mixing renormalization are found to considerably affect the predictions in these new-physics scenarios.     

Renormalization of Yang-Mills superfields in noncovariant supergauges

As a sequel to previous work of the authors in which general homogeneous non-supersymmetric choices ofN=1 supergauges were considered, we now proceed to the renormalization programme, remaining consistently within the superfield formalism. The possibility of a renormalization of the supersymmetric BRS-transformation is discussed.     

Temperature renormalization of the magnon energies of FeBr2

The temperature renormalization of the magnon energies is investigated in FeBr₂, a S = 1 Heisenberg ferromagnet with rather large single-ion and exchange anisotropies. Two theories are applied: a self-consistently-renormalized (SCR) spin-wave theory (which is not affected by “kinematic” inconsistencies) and a calculation of the self-energy in the T-matrix formalism. At low temperature both theories are found to give a small renormalization of the energy gap in agreement with neutron scattering experiments. On the other hand the two approximations give significantly different renormalization at higher temperature. Measurements in this temperature range should be able to put in evidence non linear effects in the anisotropy.     

Gauge dependence of renormalization group parameters in ghost-free non-abelian gauge theories

We discuss the gauge dependence of the renormalization group parameters in a class of ghost-free non-abelian gauge theories. We show, using the n-dimensional regularization with the “minimal” renormalization procedure, that these parameters are gauge independent.     

Microscopic optical model analysis of elastic scattering cross sections for nucleon-40Ca systems at low energy

Neutron and proton-⁴⁰Ca elastic scattering angular distributions and total reaction cross sections are calculated between 10 and 40 MeV using the microscopic optical potential derived by Bouyssy et al. within the nuclear structure approach. Direct comparison with experiment confirms that our calculation reproduces the imaginary potential at low energy for protons, but it gives insufficient absorption above the deuteron pick-up threshold. A renormalization of both the real and the imaginary parts of the potential leads to good agreement with the data. For both parts of the potential this renormalization is compatible with the renormalization factors obtained by Bouyssy et al. from a comparison of calculated and phenomenological volume integrals.     

The XY model and Kadanoff's variational real space renormalization group

The XY model in d dimensions is studied by means of a variational real space renormalization group transformation. Contrary to an earlier computation in the same framework for the d = 2 case, we find that a low order operator basis truncation is highly unstable. For certain values of the variational parameter p the renormalization group flow can display period doubling sequences towards a chaotic regime. The behavior in the d = 3 case is very similar.