Renormalization group equations for effective field theories

We derive the renormalization group equations for a generic non-renormalizable theory. We show that the equations allow one to derive the structure of the leading divergences at any loop order in terms of one-loop diagrams only. In chiral perturbation theory, e.g., this means that one can obtain the series of leading chiral logs by calculating only one-loop diagrams. We discuss also the renormalization group equations for the subleading divergences, and the crucial role of counterterms that vanish at the equations of motion. Finally, we show that the renormalization group equations obtained here apply equally well also to renormalizable theories.Received: 5 September 2003, Published online: 20 November 2003     

New mathematical structures in renormalizable quantum field theories

Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments.     

Renormalization and continuum limit of composite operators in lattice gauge theories

The gluon condensate of dimension 4 is extracted from different operators in a pureSU(2) lattice gauge theory. Multiplicative finite renormalization effects are observed, which are in qualitative agreement with one loop perturbative calculations. Asymptotic scaling is found in the range 2.45≦β≦2.85.     

Renormalization group equations in lattice gauge theories

New recursion equations for renormalization group transformations of the Migdal-Kadanoff type are obtained for gauge systems including fermion variables on a d-dimensional Euclidean space-time lattice. It is shown that in the weak gauge coupling region these equations have β-functions similar to those of continuum field theories in the case of U(1), SU(2) gauge groups (QED, QCD). On the other hand in the strong-coupling limit there is an infrared attractive fixed point corresponding to a color-confining effective system in both groups. A possible entire trajectory of the non-Abelian system is briefly conjectured.     

Universality of the continuum limit of lattice quantum theories

The lattice approximation of the naïve continuum action in quantum mechanics or in field theory is not uniquely determined. We investigate to what extent corrections to the lattice action, which vanish in the naïve continuum limit, affect the continuum limit when taking quantum fluctuations into account. In the quantum mechanical case, modifications of the lattice action may induce non-trivial corrections to the potential of the system and thereby change the structure of the theory completely. We verify this statement analytically as well as numerically by performing a Monte Carlo simulation. In the field theoretical case we argue that the lattice corrections considered do not affect the physics of the continuum limit, at least not for asymptotically free gauge field theories. In four dimensions, one might encounter finite renormalization of CP violating terms.     

On the predictivity of the non‐renormalizable quantum field theories

Following a Four Dimensional Renormalization approach to ultraviolet divergences (FDR), we extend the concept of predictivity to non‐renormalizable quantum field theories at arbitrarily large perturbative orders. The idea of topological renormalization is introduced, which keeps a finite value for the parameters of the theory by trading the usual order‐by‐order renormalization procedure for an order‐by‐order redefinition of the perturbative vacuum. One additional measurement is then sufficient to systematically compute quantum corrections at any loop order, with no need of absorbing ultraviolet infinities in the Lagrangian.     

Renormalization group approach to lattice gauge field theories

The fluctuation field integral, constructed in Part I, is represented by the exponentiated cluster expansion. It is proved that the terms of the expansion satisfy the inductive assumptions. This completes the construction of the sequence of effective actions in the small field approximation.Work supported in part by the Air Force under Grant AFOSR-86-0229 and by the National Science Foundation under Grant DMS-86-02207     

Renormalization group approach to lattice gauge field theories

We study four-dimensional pure gauge field theories by the renormalization group approach. The analysis is restricted to small field approximation. In this region we construct a sequence of localized effective actions by cluster expansions in one step renormalization transformations. We construct also -functions and we define a coupling constant renormalization by a recursive system of renormalization group equations.     

Renormalization equations for models with renormalizable and nonrenormalizable interactions

We study models including renormalizable and nonrenormalizable polynomial interactions. We derive the partial differential equations, which are relevant for the variation of parameters of the model. A supersymmetric model is considered as example.     

Renarks on the classical limit of quantum field theories

Recently, there has been an increasing interest in computing quantum mechanical corrections to solutions of classical field equations. In this note, we want to proceed in the opposite way and we summarize theorems about the classical limit of relativistic quantum field models. These results are a byproduct of the so called constructive approach to quantum field theory.After a section on generalities, we discuss in Section 2 the situation where no phase transitions occur in the limith0 and in Section 3 we reformulate one result in the case where such a transition occurs (Glimmet al. [7]). We discuss the validity of the loop expansion. It seems however that the tools to show the rigorous validity of soliton calculations are not yet prepared.     

Renormalization group in the large N limit

The normalization group in the large N limit is described and its fixed point and eigenvalues determined for 2 < d < 4. N is the number of components of the order parameter and d is the dimension.     

Renormalization group fixed points in Yukawa theories

The Callan-Symanzik functions of Yukawa theories in which the scalar mesons transform as the regular representation of SU(3), SU(2) or U(1) are calculated in two-loop order. An attractive renormalisation group fixed point away from the origin is found, but at a distance such that perturbation theory can not be considered reliable.     

Renormalization group constraints on unified gauge theories

Renormalization group constraints on the behavior of Yukawa and scalar quartic couplings in unified gauge theories are examined. Yukawa couplings are generally asymptotically free whenever the gauge couplings are, but scalar quartic couplings can be asymptotically free only for simple scalar multiplets in large groups with large fermion content. The infrared behavior of Yukawa and scalar quartic couplings implied by the renormalization group equations has interesting and phenomenologically useful consequences: infrared fixed points (or quasifixed points) lead to bounds on masses of fermions and scalars, while scalar quartic couplings can be driven out of the domain of positivity of the classical potential, with possible implications for patterns of symmetry breaking.     

Renormalization group equations of Briot-Bouquet type

The purpose of this paper is to investigate the renormalization group equations for massless models with two couplings and vanishing lowest order contribution of the β-function. It is shown that in this case the renormalization group equations can be reduced to others of Briot-Bouquet type. This observation permits to study in a simple and obvious manner the stability and asymptotic behaviour of solutions in the weak coupling limit.     

Renormalization group equations for the Kobayashi-Maskawa matrix

The renormalization group equations for the parametrization-convention independent quadratic parameters |V _ij |² of the KM matrix are derived. Numerical analysis of these equations shows that the heavy quark family (t, b) tends to mix with the lighter families (c, s) and (u, d) with increasing energy, although the variation is very much slow. The CP-nonconservation effects are shown to get larger with energy.     

Renormalization group equations and the gaussian spin model

Exact renormalization group equations are constructed for the one and two dimensional gaussian spin model by direct elimination of spins on a sublattice. They lead to the correct values of the critical exponents α, γ, ν and η.