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 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.     

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 of gauge theories

Gauge theories are characterized by the Slavnov identities which express their invariance under a family of transformations of the supergauge type which involve the Faddeev Popov ghosts. These identities are proved to all orders of renormalized perturbation theory, within the BPHZ framework, when the underlying Lie algebra is semisimple and the gauge function is chosen to be linear in the fields in such a way that all fields are massive. An example, the SU2 Higgs Kibble model is analyzed in detail: the asymptotic theory is formulated in the perturbative sense, and shown to be reasonable, namely, the physical S operator is unitary and independent from the parameters which define the gauge function.     

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 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     

Renormalization group and dynamical symmetry breakdown in abelian gauge theories

Generalized renormalization group equations are used to analyze the dynamical mechanism of particle mass generation in terms of the Cornwall-Norton model both with and without cut-off. We look for solutions which contain non-zero physical masses of the two fermions (m₁, m₂) and of one of the vector bosons (μ) when the bare masses m_1Λ, m_2Λ, μ_Λ approach zero. For a theory without cut-off we obtain results which are similar to those of Cornwall. For a theory with cut-off the mass generation mechanism may only occur when a bare coupling constant α_Λ of the A_μ vector boson, which remains massless, exceeds some critical value α_c. In this case the fermion masses turn out to be of the superconductivity type.The model's “memory” of the nature of spontaneous symmetry breaking lim_{m1Λ, m2Λ → 0}m_1Λ/m_2Λ ≠ 1 is an indispensible factor for the vector boson to acquire a mass.     

Renormalization in non-Abelian gauge theories

The renormalization is performed in a manifestly covariant approach. The simplest form of the Ward identities z₁=z₂=…=z_n=… is fulfilled automatically in every gauge. In the Yang-Mills theory the counter-terms are gauge-invariant and depend on the charge renormalization constant only. In pure gravitation the analysis of all divergences is reduced in the present approach to some special classification of the kth order scalar densities in a Riemannian space.     

Propagators for lattice gauge theories in a background field

We prove regularity and decay properties for propagators connected with the renormalization group method in lattice gauge theories. These propagators depend on an external gauge field configuration, called a background field.Research supported in part by the National Science Foundation under Grant PHY-82-0369     

Ultraviolet stability of three-dimensional lattice pure gauge field theories

We prove the ultraviolet stability for three-dimensional lattice gauge field theories. We consider only the Wilson lattice approximation for pure Yang-Mills field theories. The proof is based on results of the previous papers on renormalization group method for lattice gauge theories.Work partially supported by the National Science Foundation under Grant PHY 82-03669 and DMS 84-01989On leave of absence, Postal address: Department of Physics, Harvard University, Cambridge, MA 02138, USA     

Higher order mean field expansions for lattice gauge theories

The leading contribution to the free energy of lattice gauge theories is evaluated in the mean field expansion to the two-loop level. The methods are general but we only deal with theU(1) case in this paper. The corrections improve the agreement with Monte Carlo calculations. We show that in order to obtain a satisfactory formalism it is necessary to include a new redundant parameter, γ, in the mean field expansion. For γ→0 we recover the usual mean field expansion whereas for γ→∞ we obtain the weak coupling expansion. Thus γ measures the amount of resummation that is done by the mean field formalism.     

A well-defined transition from continuum to lattice formulation of gauge field theories

In this paper it is shown that the well-known expressions for the action of lattice gauge theories (abelian and nonabelian) can be derived in a well-defined manner from corresponding nonlocal expressions once we take the local limit. First, a nonlocal formulation of the spinorial part of the QED action, accomplished 30 years ago, is extended to the Maxwell field part. Subsequently, the nonlocal action is discretized in the conventional way yielding, in the local limit, the well-known form of the lattice action. The above procedure is successfully repeated for nonabelian gauge fields.     

Renormalization of composite operators in gauge theories

We give a logically complete treatment of the renormalization of composite operators in gauge theories.     

Renormalization of gauge theories in nonlinear gauges

The usual rules for constructing the Lagrangian of fictitious particles in gauge theories is generalized. The gauge invariant and multiplicative renormalizability of the theory with generalized action in nonlinear gauges is proved.Translated from Izvestiya Vysshikh, Uchebnykh Zavedenii, Fizika, No. 6, pp. 11–15, June, 1981.The author thanks I. A. Batalin and B. L. Voronov for discussing the problems considered here.     

Renormalization of gauge theories: Non-linear gauges

The renormalization action for gauge theories quantized with non-linear gauge conditions is written down in its most general form. A natural interpretation of the well-known quartic ghost term generated by renormalization is given in terms of a generalized Faddeev-Popov determinant.     

Renormalization of Wilson operators in gauge theories

The operators in a Wilson expansion are not in general multiplicatively renormalized in non-Abelian gauge theories. This is because of the renormalization of the gauge transformations themselves. Renormalized fields may be defined, which have the old gauge transformations. Alternatively, a special choice of gauge may be made, in which the gauge transformations are unchanged on renormalization. In any case, one gauge invariant factor appears in the renormalization of the Wilson operators.     

On-shell improved lattice gauge theories

By means of a spectrum conserving transformation, we show that one of the 3 coefficients in Symanzik's improved action can be chosen freely, if only spectral quantities (masses of stable particles, heavy quark potential etc.) are to be improved. In perturbation theory, the other 2 coefficients are however completely determined and their values are obtained to lowest order.Heisenberg foundation fellow