Renormalizable Field Theory Research Articles

Interfacial Roughening in Field Theory

In the rough phase, the width of interfaces separating different phases of statistical systems increases logarithmically with the system size. This phenomenon is commonly described in terms of the capillary wave model, which deals with fluctuating, infinitely thin membranes, requiring ad hoc cut-offs in momentum space. We investigate the interface roughening from first principles in the framework of the Landau-Ginzburg model, that is renormalized field theory, in the one-loop approximation. The interface profile and width are calculated analytically, resulting in finite expressions with definite coefficients. They are valid in the scaling region and depend on the known renormalized coupling constant.

Dimensional Regularization and Renormalization of Non-Commutative Quantum Field Theory

Using the recently introduced parametric representation of non-commutative quantum field theory, we implement here the dimensional regularization and renormalization of the vulcanized $\Phi^{\star 4}_4$ model on the Moyal space.

AN EQUIVALENCE OF TWO MASS GENERATION MECHANISMS FOR GAUGE FIELDS

Two mass generation mechanisms for gauge theories are studied. It is proved that in the Abelian case the topological mass generation mechanism introduced in Refs. 4, 12 and 15 is equivalent to the mass generation mechanism defined in Refs. 5 and 20 with the help of "localization" of a nonlocal gauge invariant action. In the non-Abelian case the former mechanism is known to generate a unitary renormalizable quantum field theory, describing a massive vector field.

Quantum evolution across singularities

Attempts to consider evolution across space-time singularities often lead to quantum systems with time-dependent Hamiltonians developing an isolated singularity as a function of time. Examples include matrix theory in certain singular time-dependent backgounds and free quantum fields on the two-dimensional compactified Milne universe. Due to the presence of the singularities in the time dependence, the conventional quantum-mechanical evolution is not well-defined for such systems. We propose a natural way, mathematically analogous to renormalization in conventional quantum field theory, to construct unitary quantum evolution across the singularity. We carry out this procedure explicitly for free fields on the compactified Milne universe and compare our results with the matching conditions considered in earlier work (which were based on the covering Minkowski space).

FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES

A Rota–Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota–Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota–Baxter algebras have been for commutative algebras. Two constructions of free commutative Rota–Baxter algebras were obtained by Rota and Cartier in the 1970s and a third one by Keigher and one of the authors in the 1990s in terms of mixable shuffles. Recently, noncommutative Rota–Baxter algebras have appeared both in physics in connection with the work of Connes and Kreimer on renormalization in perturbative quantum field theory, and in mathematics related to the work of Loday and Ronco on dendriform dialgebras and trialgebras.This paper uses rooted trees and forests to give explicit constructions of free noncommutative Rota–Baxter algebras on modules and sets. This highlights the combinatorial nature of Rota–Baxter algebras and facilitates their further study. As an application, we obtain the unitarization of Rota–Baxter algebras.

Hopf algebra of non-commutative field theory

We construct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory.

Charge renormalization in quantum field theory by the unitary clothing transformation method

The clothing procedure in several lowest orders in the coupling constant has been implemented using the unitary clothing transformation method. Within a simple field theory model, including interaction of charged spinless nucleons and scalar mesons, an expression for the charge shift is obtained, which is determined by the operators beyond the energy shell.

Recursion and Growth Estimates in Renormalizable Quantum Field Theory

In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. In the nonnegative case the radius of convergence turns out to be min{ρ,1/b 1}, where ρ is the bound on the convergent ones, the instanton radius, and b 1 the first coefficient of the β-function, while in general it is bounded by the above.

Parametric Representation of “Covariant” Noncommutative QFT Models

We extend the parametric representation of renormalizable non commutative quantum field theories to a class of theories which we call critical, because their power counting is definitely more difficult to obtain. This class of theories is important since it includes gauge theories, which should be relevant for the quantum Hall effect.

Noncommutative induced gauge theories on Moyal spaces

Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is then closely examined. The computation in 4-D of the one-loop effective gauge theory generated from the integration over a scalar field appearing in a renormalisable theory minimally coupled to an external gauge potential is presented. The gauge invariant effective action is found to involve, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic term, which for the noncommutative φ4-theory on Moyal space ensures renormalisability. A class of possible candidates for renormalisable gauge theory actions defined on Moyal space is presented and discussed.

Exorcizing the Landau ghost in noncommutative quantum field theory

We show that the simplest noncommutative renormalizable field theory, the ø4 model on four dimensional Moyal space with harmonic potential is asymptotically safe to all orders in perturbation theory

Differential Algebraic Birkhoff Decomposition and the renormalization of multiple zeta values

In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is viewed as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multiple zeta values.

Renormalization: A Number Theoretical Model

We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.

Pion form factor in the Kroll-Lee-Zumino model

The renormalizable Abelian quantum field theory model of Kroll, Lee, and Zumino is used to compute the one-loop vertex corrections to the tree-level, vector meson dominance pion form factor. These corrections, together with the known one-loop vacuum polarization contribution, lead to a substantial improvement over vector meson dominance. The resulting pion form factor in the spacelike region is in excellent agreement with data in the whole range of accessible momentum transfers. The timelike form factor, known to reproduce the Gounaris-Sakurai formula at and near the rho-meson peak, is unaffected by the vertex correction at order O(g{sub {rho}}{sub {pi}}{sub {pi}}{sup 2})

Finite-size scaling of directed percolation in the steady state

Recently, considerable progress has been made in understanding finite-size scaling in equilibrium systems. Here, we study finite-size scaling in nonequilibrium systems at the instance of directed percolation (DP), which has become the paradigm of nonequilibrium phase transitions into absorbing states, above, at, and below the upper critical dimension. We investigate the finite-size scaling behavior of DP analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d-dimensional hypercube of finite edge length L with periodic boundary conditions near the bulk critical point. In particular, we study the order parameter and its higher moments using renormalized field theory. We derive finite-size scaling forms of the moments in a one-loop calculation. Moreover, we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous Binder cumulant in equilibrium systems and that, in particular, provides a signature of the DP universality class. To complement our analytical work, we perform Monte Carlo simulations which confirm our analytical results.

The Interfacial Profile in Two-Loop Order

The profile of interfaces separating different phases of statistical systems is investigated in the framework of renormalized field theory. The profile function is calculated analytically in the local potential approximation, using the effective potential to two loops. It can be interpreted as an intrinsic interfacial profile. The loop corrections to the leading tanh-type term turn out to be small. They yield a broadening of the interface.

Master singular behavior for the Sugden factor of one-component fluids near their gas-liquid critical point

We present the master (i.e., unique) behavior of the squared capillary length-the so-called Sugden factor-as a function of the temperaturelike field along the critical isochore, asymptotically close to the gas-liquid critical point of about twenty (one-component) fluids. This master behavior is obtained using the scale dilatation of the relevant physical fields of the one-component fluids. The scale dilatation method introduces the fluid-dependent scale factors in a manner analog to the linear relations between physical fields and scaling fields needed by the renormalization theory applied to any physical system belonging to the Ising-like universality class. The master behavior for the Sugden factor satisfies hyperscaling. It can be asymptotically fitted by the leading terms of the theoretical crossover functions for the correlation length and the susceptibility in the homogeneous domain, recently obtained from massive renormalization in field theory. In the absence of corresponding estimation of the theoretical crossover functions for the interfacial tension, we define the range of the temperaturelike field where the master leading power law can be practically used to predict the singular behavior of the Sugden factor, in conformity with the theoretical description provided by the massive renormalization scheme within the extended asymptotic domain of the one-component fluid "subclass."

On the stability of some groups of formal diffeomorphisms by the Birkhoff decomposition

Let G ∞ be the group of one parameter identity-tangent diffeomorphisms on the line whose coefficients are formal Laurent series in the parameter ε with a pole of finite order at 0. It is well known that the Birkhoff decomposition can be defined in such a group. We investigate the stability of the Birkhoff decomposition in subgroups of G ∞ and give a formula for this decomposition. These results are strongly related to renormalization in quantum field theory, since it was proved by A. Connes and D. Kreimer that, after dimensional regularization, the unrenormalized effective coupling constants are the image by a formal identity-tangent diffeomorphism of the coupling constants of the theory. In the massless ϕ 6 3 theory, this diffeomorphism is in G ∞ and its Birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants.

A renormalization group analysis of leading logarithms in ChPT

We give strong evidence that the linear sigma model at small external momenta is an effective theory for the leading logarithms of chiral perturbation theory. Based on this evidence an attempt is made to sum the leading logarithms of chiral perturbation theory to all orders. We illustrate why this summation nonetheless fails when one uses standard renormalization group techniques of renormalizable quantum field theories.

Power counting for nuclear effective field theory and Wilsonian renormalization group

We consider an application of Wilsonian Renormalization Group (RG) to the power counting issue in Nuclear Effective Field Theory(NEFT). After reviewing the relevance of scaling dimensions in determining the power counting, we consider the pionless NEFT as the simplest example. The inclusion of the so-called “redundant operators” in a Wilsonian analysis is emphasized.