Nonperturbative Scheme Research Articles

Flow equations for Yang-Mills theories in general axial gauges

We present a formulation of non-Abelian gauge theories in general axial gauges using a Wilsonian (or 'Exact') Renormalisation Group. No 'spurious' propagator divergencies are encountered in contrast to standard perturbation theory. Modified Ward identities, compatible with the flow equation, ensure gauge invariance of physical Green functions. The axial gauge nA=0 is shown to be a fixed point under the flow equation. Possible non-perturbative approximation schemes and further applications are outlined.

Solving a coupled set of truncated QCD Dyson—Schwinger equations

Truncated Dyson—Schwinger equations represent finite subsets of the equations of motion for Green's functions. Solutions to these nonlinear integral equations can account for nonperturbative correlations. A closed set of coupled Dyson—Schwinger equations for the propagators of gluons and ghosts in Landau gauge QCD is obtained by neglecting all contributions from irreducible 4-point correlations and by implementing the Slavnov—Taylor identities for the 3-point vertex functions. We solve this coupled set in an one-dimensional approximation which allows for an analytic infrared expansion necessary to obtain numerically stable results. This technique, which was also used in our previous solution of the gluon Dyson—Schwinger equation in the Mandelstam approximation, is here extended to solve the coupled set of integral equations for the propagators of gluons and ghosts simultaneously. In particular, the gluon propagator is shown to vanish for small spacelike momenta whereas the previously neglected ghost propagator is found to be enhanced in the infrared. The running coupling of the nonperturbative subtraction scheme approaches an infrared stable fixed point at a critical value of the coupling, α c ⋍ 9.5.

A Solution to Coupled Dyson–Schwinger Equations for Gluons and Ghosts in Landau Gauge

A truncation scheme for the Dyson–Schwinger equations of QCD in Landau gauge is presented which implements the Slavnov–Taylor identities for the 3-point vertex functions. Neglecting contributions from 4-point correlations such as the 4-gluon vertex function and irreducible scattering kernels, a closed system of equations for the propagators is obtained. For the pure gauge theory without quarks this system of equations for the propagators of gluons and ghosts is solved in an approximation which allows for an analytic discussion of its solutions in the infrared: The gluon propagator is shown to vanish for small spacelike momenta whereas the ghost propagator is found to be infrared enhanced. The running coupling of the non-perturbative subtraction scheme approaches an infrared stable fixed point at a critical value of the coupling,αc≃9.5. The gluon propagator is shown to have no Lehmann representation. The results for the propagators obtained here compare favorably with recent lattice calculations.

Pauli-Villars regulator as a nonperturbative ultraviolet regularization scheme in discretized light-cone quantization

We propose a solution to the problem of renormalizing light-cone Hamiltonian theories while maintaining Lorentz invariance and other symmetries. The method uses generalized Pauli-Villars regulators to render the theory finite. We discuss the method in the context of Yukawa theory at one loop and for a soluble model in $3+1$ dimensions. The model is studied nonperturbatively. Numerical results obtained with use of discretized light-cone quantization, special integration weighting factors, and the complex symmetric Lanczos diagonalization algorithm compare well with the analytic answers.

Nonlinear operators and their propagators

Mathematical physicists are familiar with a large set of tools designed for dealing with linear operators, which are so common in both the classical and quantum theories; but many of those tools are useless with nonlinear equations of motion. In this work a general algebra and calculus is developed for working with nonlinear operators: The basic new tool being the “slash product,” defined by A(1+εB) =A+εA/B+O(ε2). For a generic time development equation, the propagator is constructed and then there follows the formal version of time dependent perturbation theory, in remarkable similarity to the linear situation. A nonperturbative approximation scheme capable of producing high accuracy computations, previously developed for linear operators, is shown to be applicable as well in the nonlinear domain. A number of auxiliary mathematical properties and examples are given.

Rayleigh-Ritz variational approximation and symmetry nonrestoration

The investigation of symmetry nonrestoration scenarios has led to a controversy, with certain nonperturbative approximation schemes giving indications in sharp disagreement with those found within conventional perturbation theory. A Rayleigh-Ritz variational approach to the problem, which might be useful in bridging the gap between perturbative and nonperturbative viewpoints, is here proposed. As a first application, this approach is used in the investigation of a Z 2 × Z 2-invariant thermal field theory with two scalar fields, placing particular emphasis on the region of parameter space that has been claimed to support symmetry nonrestoration.

ELECTRON-POSITRON PRODUCTION IN STRONG FIELD PULSES

A short critical review is presented concerning the physical interpretation of electron and positron production data at the GSI accelerator in Darmstadt. The only reliable conclusions seem to be that the creation mechanism is a strong field effect and that the phenomenon is largely independent of the characteristic time of the collision. Before looking for new (exotic) physics it seems reasonable to inquire whether non-perturbative QED is fully understood. Following this motivation a rigorous but, at the same time, sufficiently explicit nonperturbative calculation scheme is developed, for QED in strong fields of short duration. The possible occurrence of correlated e+e− narrow structures is discussed in this framework.

THE EXACT RENORMALIZATION GROUP AND APPROXIMATE SOLUTIONS

We investigate the structure of Polchinski’s formulation of the flow equations for the continuum Wilson effective action. Reinterpretations in terms of I.R. cutoff Green’s functions are given. A promising nonperturbative approximation scheme is derived by carefully taking the sharp cutoff limit and expanding in “irrelevancy” of operators. We illustrate with two simple models of four-dimensional λφ4 theory: the cactus approximation, and a model incorporating the first irrelevant correction to the renormalized coupling. The qualitative and quantitative behaviour give confidence in a fuller use of this method for obtaining accurate results.

Crossing-symmetric two-particle reduction of four-point vertex.

We develop a nonperturbative reduction scheme for the four-point vertex [ital T] of a generic field theory with interactions among bosons, fermions, and antifermions. We exhibit integral equations which express, in a manifestly crossing-symmetric way, the full vertex in terms of its two-particle irreducible part, together with the dressed three-point vertices and two-point propagators. This scheme generalizes the usual summation of ladder or bubble diagrams, thus providing for the consistent summation of a larger subset of all diagrams.

A theoretical discussion of strong laser field effects: intense short pulses interacting with atoms and molecules

A non-perturbative scheme for the interaction of strong laser pulses with atoms and molecules is discussed and model calculations are presented. A semiclassical analysis leads to a clear-cut distinction of four categories of reactions. Separatrix crossing mechanisms, the dissolution of initial tori and classical adiabatic motion are seen to play a central role in a non-perturbative picture of interactions with strong laser fields.

RENORMALIZATION GROUP FLOWS IN NONPERTURBATIVE 2D QUANTUM GRAVITY

We study the existence of solutions of the string equations which are pole-free on the positive real axis and exhibit such a solution following the work of Boutroux. We further study, using perturbation theory, the flows among different models and we find that there is no flow between the k=2 and k=3 theories. Moreover, we develop a nonperturbative renormalization-group scheme to extract the scaling behavior of the partition function of the k=2 model under perturbations. The key idea is to reinterpret the recent formulation, as Virasoro constraints, of the string-loop equations as generalized Callan-Symanzik renormalization-group equations.

Axial anomaly on a momentum lattice.

We study the axial anomaly in a noncompact formulation on a momentum lattice. We compute the continuum limit for the one-loop diagram from {ital k}-lattice perturbation theory. Starting from {ital k}-lattice regularization, we obtain the standard result for the axial anomaly, if we renormalize by introducing nonlocal subtractions which correspond to nonlocal Lagrangian counterterms. In the worst case this could imply that perturbative renormalization would require an infinite number of counterterms; i.e., it would be nonrenormalizable perturbatively. We have undertaken this study in order to investigate properties of {ital k}-lattice regularization and in particular to exhibit the nonlocalities. We consider this as a starting point in the search for a nonperturbative renormalization scheme, spurred in particular by the recent ideas of Wilson and co-workers on the renormalization of infrared singularities of light-cone field theories, using nonlocal counterterm functions.

Relevance of operators in modified lattice Hamiltonians.

The modified Hamiltonians with an exact ground state for lattice gauge theory and the nonlinear $\ensuremath{\sigma}$ model are analyzed in the nonperturbative subtraction scheme. By subtracting operators with lower dimensions, we obtain a subtracted operator $\stackrel{-}{\ensuremath{\Delta}H}$ which is divergence-free. For a modified Hamiltonian with an exact ground state, $\stackrel{-}{\ensuremath{\Delta}H}$ is in general relevant. The vacuum functional and low-energy spectrum calculations are discussed.

Electroweak physics

The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation of Wilson’s RG, which allows one to set up nonperturbative approximation schemes that go beyond the standard perturbative RG approaches. The FRG is based on an exact functional flow equation of a coarse-grained effective action (or Gibbs free energy in the language of statistical mechanics). We review the main approximation schemes that are commonly used to solve this flow equation and discuss applications in equilibrium and out-of-equilibrium statistical physics, quantum many-particle systems, high-energy physics and quantum gravity.

A novel method to solve functional differential equations application to quantum field theory

We present a method to solve differential equations containing the variational operator as the derivation operation. We call them variational differential equations (VDE). The solution to a VDE should be a function containing the derivatives, with respect to the base space coordinates, of the fields up to a generic orders: as-th-order function. The variational operator doubles the order of the function on which it acts. Therefore, in order to make compatible the orders of the different terms appearing in a VDE, the solution should be a function containing the derivatives of the fields at all orders. But this takes us again back to functional methods. In order to avoid this, one must restrict the considerations, in the case of second-order VDEs, to the space ofs-th-order functions on which the variational operator acts transitively. We characterize these functions for a one-dimensional base space for the first-and second-order cases. These functions turn out to be polynomial in the highest-order derivatives of the fields with functions of the lower-order derivatives as coefficients. The VDEs reduce to a system of coupled partial differential equations for the coefficients we mentioned above. The importance of the method lies on the fact that the solution to VDEs are in a one-to-one correspondence with the solutions of functional differential equations. The previous method finds direct applications in quantum field theory, where the Schrodinger equation plays a central role. Since the Schrodinger equation is reduced to a system of coupled partial differential equations, this provides a nonperturbative scheme for quantum field theory. As an example we consider the massless scalar field.

Divergences in the iterative and perturbative methods for computing Hannay's angle.

A classical analog is obtained for Berry's nonperturbative scheme of adiabatic iteration which computes corrections to Berry's phase (or Hannay's angle) for finite values of the adiabatic parameter \ensuremath{\epsilon}. The iterative method is compared to the Lie version of adiabatic perturbation theory. Both approaches show a divergence of k!${\mathrm{\ensuremath{\epsilon}}}^{\mathit{k}}$, where k is the order of iteration. It is argued that the divergences are a mathematical artifact of the asymptotic methods used, not related to the physical effect of transitions, nor to the nonconservation of the action variables.

Dirac electron in axial-wiggler fields

Relativistic electron motion in a guiding and wiggler magnetic field configuration is considered in the framework of Dirac theory and the modification of the Landau level distribution due to the guiding and wiggler fields has been worked out in a nonperturbative scheme. The energy surfaces now consist of two sets of hyperbolic sheets in the space of wiggler and axial momentum wave vectors.

Topological expansion, pomeron, and odderon

An inconsistency of applying perturbative QCD to soft physics is pointed out and the need for a unified treatment of both elastic and inelastic production in a nonperturbative QCD scheme with confinement built-in is emphasized.

Families in the nonperturbative unification scheme

Within the nonperturbative unification framework of Maiani, Parisi and Petronzio, we examine the influence of the number of fermion and Higgs families, when they are grouped in representations of horizontal family groups, on the low energy couplings of the standard model. In this way we find a number of new phenomenologically acceptable solutions for the standard model's low energy couplings.

Number of light families and non-perturbative unification of technicolor

Abstract It is shown that the non-perturbative unification scheme of technicolor (TC) recently proposed by the author, which incorporates TC in the scenario of Maiani, Parisi and Petronzio and assumes it becomes strong along with the standard-model forces near a common high-energy cut-off M GUT , gives constraints on the number n of technisinglet generations. Only n ⩾2 is consistent with M GUT M Planck . A theoretical upper bound n ⩽6 is also obtained. Phenomenologically acceptable values of the low-energy couplings are obtained only for n =2, 3, thus disfavoring the existence of a fourth light family.