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     

Renormalizable asymptotically free quantum theory of gravity

We prove that higher derivative quantum gravity is asymptotically free in all essential coupling constants by the calculation of one-loop counterterms (correcting the previous result of Julve and Tonin) and the solution of the corresponding renormalization group (RG) equations. Strong arguments are presented in favour of the possibility that renormalizable asymptotically free gravity establishes asymptotic freedom for the effective mass parameters and non-gauge couplings in grand unified gauge theories. We also analyse the RG equations in the Einstein theory with Λ term and in the higher derivative conformal invariant theories. Among other topics discussed are the algorithm for the divergences of the determinant of the fourth-order differential operator, the consistent renormalization of the boundary terms in the action, the one-loop β-function in the fourth derivative vector gauge theory and the RG equations in the $\text{“gφ}{}^4\text{+ ηRφ}{}^2\text{”}$ theory.     

Renormalized Normal Product and equations of motion of Φ4-coupling

The notion of a Renormalized Normal Product (RNP) in Euclidean space of 1 ≤ r ≤ 4 dimensions is introduced for a Φ⁴-model in a nonperturbative approach. The essential ingredients used for this purpose are the composite operators defined in perturbation theory and the renormalized G-convolution product constructed in the axiomatic field theory framework in Euclidean momentum space. Convergent equations of motion for the connected Green's functions are established where the interaction term is represented by the RNP. The corresponding renormalization constants are defined as boundary values of the RNP by imposing “physical” renormalization conditions. In the special case of 2-dimensions it is proved that these equations conserve analyticity and algebraic properties (in complex Minkowski space of 2-momenta) coming from the first principles of general local field theory, together with properties of asymptotic behaviour at infinity (in Euclidean space of 2-momenta).     

Small distance behaviour in field theory and power counting

For infinitesimal changes of vertex functions under infinitesimal variation of all renormalized parameters, linear combinations are found such that the net infinitesimal changes of all vertex functions are negligible relative to those functions themselves at large momenta in all orders of renormalized perturbation theory. The resulting linear first order partial differential equations for the asymptotic forms of the vertex functions are, in quantum electrodynamics, solved in terms of one universal function of one variable and one function of one variable for each vertex function whereby, in contrast to the renormalization group treatment of this problem, the universal function is obtained from nonasymptotic considerations. A relation to the breaking of scale invariance in renormalizable theories is described.     

The asymptotic D-state to S-state ratio of triton

At low energies, an effective field theory (EFT) with only contact interactions as well as three-body forces allow a detailed analysis of renormalization in a non-perturbative context and uncovers novel asymptotic behaviour. Triton as a three-body system, based on the EFT have been previously shown to provide representative binding energies, charge radii, S-wave scattering amplitude and asymptotic normalization constants for the ³H bound state system. Herein, EFT predictions of the asymptotic D-state to S-state ratio of triton are calculated to more fully evaluate the adequacy of the EFT model. Manifestly model-independent calculations can be carried out to high orders, leading to high precision.     

Asymptotic Freedom in the Conformal Quantum Gravity with Matter

The grand unified theory (GUT) based on the O(N) and SU(N)-gauge groups after the conformally invariant gravity being included is investigated. We calculate the one-loop gravity contributions into the renormalization group equations and study their solutions. The analysis performed show that the asymptotic freedom behaviour early established for all GUT's coupling constants is not broken by taking into account this kind of gravity. However all restrictions imposed on the GUT multiplet composition become less firmly and the physical content of the constructed models is more realistic.     

Renormalization of N = 2, 4 Yang-Mills theories with soft breaking of an extended supersymmetry by N = 1 mass term

N = 2, 4 Yang-Mills theories with soft breaking of an extended supersymmetry by mass terms are considered. It is proved that for N = 4 there are no ultraviolet divergences in the mass renormalization constants to all orders of perturbation theory. For N = 2 our two-loop calculations show that the charge and mass renormalization constants contain only one-loop divergences and are the same in this order. It is shown by direct calculation that mass terms can acquire finite quantum corrections starting from the two-loop approximation. The renormalization scheme dependence of N = 4 renormalization group functions is investigated. We have found that unlike renormalization schemes with minimal subtractions of divergences other renormalization schemes give a nonzero β-function. At nonzero masses the β-function in MOM schemes is not zero even at the one-loop level. In the massless case β≠0 beginning from the two-loop approximation.     

Finite quantum field theory and neutrinodynamics

In quantum neutrinodynamics (photon-neutrino weak coupling) all the renormalization constants vanish and therefore the field equations cannot be expressed in terms of unrenormalized field quantities. This helps us to formulate quantum neutrinodynamics as a convergent quantum field theory. It is also pointed out that from the viewpoint of the unified model of weak and electromagnetic interaction as developed on the basis of the photon-neutrino weak coupling by Bandyopadhyay, quantum electrodynamics also manifests itself as a convergent field theory.     

Renormalization-group improvement of effective actions beyond summation of leading logarithms

Invariance of the effective action under changes of the renormalization scale μ leads to relations between those (presumably calculated) terms independent of μ at a given order of perturbation theory and those higher-order terms dependent on logarithms of μ. This relationship leads to differential equations for a sequence of functions, the solutions of which give closed form expressions for the sum of all leading logs, next to leading logs, and subsequent subleading logarithmic contributions to the effective action. The renormalization group is thus shown to provide information about a model beyond the scale dependence of the model's couplings and masses. This procedure is illustrated using the φ₆³ model and Yang–Mills theory. In the latter instance, it is also shown by using a modified summation procedure that the μ dependence of the effective action resides solely in a multiplicative factor of g²(μ) (the running coupling). This approach is also shown to lead to a novel expansion for the running coupling in terms of the one-loop coupling that does not require an order-by-order redefinition of the scale factor ΛQCD. Finally, logarithmic contributions of the instanton size to the effective action of an SU(2) gauge theory are summed, allowing a determination of the asymptotic dependence on the instanton size ρ as ρ goes to infinity to all orders in the SU(2) coupling constant.     

The renormalized σ model

A renormalization procedure of the boson σ model based on the finite field equations of Brandt-Wilson is given. We first show that the current operators appearing in the field equations, which are finite local limit of sums of nonlocal field products and suitable subtraction terms, can be chosen to be the same form as the one given for the symmetric limit except for the symmetry breaking constant source term itself. The set of integral equations derived from the field equations is shown to be equivalent to the usual Bogoliubov-Parasiuk-Hepp renormalization theory, and gives us immediately all the renormalized Green's functions in each order of perturbation theory in clear and straightforward fashion. We then analyze the structures of the model in detail. In particular, Ward identities are shown to be satisfied to all orders of perturbation theory. The Goldstone theorem is a particular consequence of these identities.     

Wilsonian RG Analysis of the P-wave Nucleon–Nucleon Scattering Including Pions

We perform a Wilsonian renormalization group analysis for the nucleon–nucleon scattering in the P waves in the nuclear effective field theory including pions, in a similar way to the one done for the S-waves in our previous paper. We emphasize that the one-pion exchange interaction with large momentum transfer is of the same order as the leading contact interaction, so that there is no mismatch of the power counting. It is explicitly shown by obtaining consistent sets of renormalization group equations, that the cutoff dependence generated by the loop diagrams containing pion exchanges can be compensated by the cutoff dependence of the coupling constants of the contact interactions.     

Renormalized field theory of critical dynamics

We formulate a Gell'Mann-Low-type renormalization group approach to the critical dynamics of stochastic models described by Langevin or Fokker-Planck equations including mode-coupling terms.Dynamical correlation and response functions are expressed in terms of path integrals, which are investigated by well-known methods of renormalized perturbation theory.Dynamical scaling laws and relations between static and dynamic critical exponents are derived. The leading temperature-dependence of correlation and response functions is obtained from the Kadanoff-Wilson short-distance expansion. We also consider corrections to dynamic scaling which are due to a finite lattice constant.     

Scaling and the approach to scaling in Reggeon field theory

We address ourselves to the problem of the approach to asymptotic scaling in Reggeon field theory. We present a method of integration of the renormalization group equations and obtain the asymptotic scaling functions, as well as the corrections which occur at non-asymptotic energies.     

Cookbook asymptotics for spiral and scroll waves in excitable media

Algebraic formulas predicting the frequencies and shapes of waves in a reaction-diffusion model of excitable media are presented in the form of four recipes. The formulas themselves are based on a detailed asymptotic analysis (published elsewhere) of the model equations at leading order and first order in the asymptotic parameter. The importance of the first order contribution is stressed throughout, beginning with a discussion of the Fife limit, Fife scaling, and Fife regime. Recipes are given for spiral waves and detailed comparisons are presented between the asymptotic predictions and the solutions of the full reaction-diffusion equations. Recipes for twisted scroll waves with straight filaments are given and again comparisons are shown. The connection between the asymptotic results and filament dynamics is discussed, and one of the previously unknown coefficients in the theory of filament dynamics is evaluated in terms of its asymptotic expansion. (c) 2002 American Institute of Physics.     

Ultraviolet renormalons in abelian gauge theories

We analyze the large-order behaviour in perturbation theory of classes of diagrams with an arbitrary number of chains (i.e. photon lines, dressed by vacuum polarization insertions). We derive explicit formulae for the leading and subleading divergence as n, the order in perturbation theory, tends to infinity, and a complete result for the vacuum polarization at the next-to-leading order in an expansion in l / N f, where N f is the number of fermion species. In general, diagrams with more chains yield stronger divergence. We define an analogue of the familiar diagrammatic R-operation, which extracts ultraviolet renormalon counterterms as insertions of higher-dimension operators. We then use renormalization group equations to sum the leading (in n/ N f ) k corrections to all orders in l INf and find the asymptotic behaviour in n up to a constant that must be calculated explicitly order by order in 1/N_f.     

Renormalization of Quantum Field Theory in Curved Space-Time and Renormalization Group Equations

The structure of quantum field theory renormalization in curved space-time is investigated. The equations allowing us to investigate the behaviour of vacuum energy and vertex functions in the limit of small distances in the external gravitational field are established. The behaviour of effective charges corresponding to the parameters of nonminimal coupling of the matter with the gravitational field is studied and the conditions under which asymptotically free theories become asymptotically conformally invariant are found. The examples of asymptotically conformally invariant theories are given. On the basis of a direct solution of renormalization group equations the effective potential in the external gravitational field and the effective action in the gravity with the high derivatives are obtained. The expression for the cosmological constant in terms of R²-gravity Lagrangian parameters is given which does not contradict the observable data. Renormalization and renormalization group equations for the theory in curved space-time with torsion are investigated.     

Renormalization in the theory of a quantized scalar field interacting with a robertson-walker spacetime

We demonstrate the possibility of removing the divergences in the energy-momentum tensor by identifying divergent terms with renormalizations of the coupling constants in the gravitational field equation, modified to include a cosmological term and terms quadratic in the curvature. The model studied is that of a classical Robertson-Walker metric and a quantized minimally coupled neutral scalar field. The theory is constructed first with an ultraviolet cutoff as a phenomenological ansatz. The cutoff is then removed in an attempt to obtain a more fundamental theory, whereupon the question arises of the covariance and uniqueness of the resulting renormalized energy-momentum tensor. In the case of a massless field in a spatially flat universe, an apparent infrared divergence is discussed from the point of view of operational determination of the renormalized coupling constants. In the other cases, although the divergences are successfully accounted for by renormalization, we are left with finite leading terms which do not appear to be identifiable with geometrical tensors; the significance of this result is under investigation. If these anomalous terms are dropped, the renormalized energy-momentum tensor agrees with that defined by adiabatic regularization, provided that the limit of slow time variation taken in that method is generalized to a limit of “spacetime flatness.”     

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.