Effective potential for the massless KPZ equation

In previous work we have developed a general method for casting a classical field theory subject to Gaussian noise (that is, a stochastic partial differential equation (SPDE)) into a functional integral formalism that exhibits many of the properties more commonly associated with quantum field theories (QFTs). In particular, we demonstrated how to derive the one-loop effective potential. In this paper we apply the formalism to a specific field theory of considerable interest, the massless KPZ equation (massless noisy Burgers equation), and analyze its behavior in the ultraviolet (short-distance) regime. When this field theory is subject to white noise we can calculate the one-loop effective potential and show that it is one-loop ultraviolet renormalizable in 1, 2, and 3 space dimensions, and fails to be ultraviolet renormalizable in higher dimensions. We show that the one-loop effective potential for the massless KPZ equation is closely related to that for λφ⁴ QFT. In particular, we prove that the massless KPZ equation exhibits one-loop dynamical symmetry breaking (via an analog of the Coleman–Weinberg mechanism) in 1 and 2 space dimensions, and that this behavior does not persist in 3 space dimensions.     

Small-scale properties of the KPZ equation and dynamical symmetry breaking

A functional integral technique is used to study the ultraviolet or short distance properties of the Kardar–Parisi–Zhang (KPZ) equation with white Gaussian noise. We apply this technique to calculate the one-loop effective potential for the KPZ equation. The effective potential is (at least) one-loop ultraviolet renormalizable in 1, 2, and 3 space dimensions, but non-renormalizable in 4 or higher space dimensions. This potential is intimately related to the probability distribution function (PDF) for the spacetime averaged field. For the restricted class of field configurations considered here, the KPZ equation exhibits dynamical symmetry breaking (DSB) via an analog of the Coleman–Weinberg mechanism in 1 and 2 space dimensions, but not in 3 space dimensions.     

One-loop counterterms for dimensionally reduced quantum gravity

The technique of regularization by dimensional reduction is applied to source-free quantum gravity. The one-loop counterterms for the effective gravity-matter system are calculated in the background field formalism. The ersatz matter fields which arise in this regularization scheme are found to have no effect on the renormalizability of the theory.     

Improved methods for supergraphs

We introduce some new techniques into superfield perturbation theory which allow considerable simplifications in calculations. As a result, we show that all contributions to the effective action can be written as integrals over a single d⁴θ. We also give the background group field formalism for supersymmetric non-abelian gauge theories. To illustrate our methods, we give examples of loop calculations: in particular, we show that in O(4) extended supersymmetric non-abelian gauge theories all one-loop propagator corrections cancel identically (both infinite and finite parts) and that these theories, at one loop, are finite and have no renormalizations (in the Fermi-Feynman gauge).     

Semiclassical calculation of decay rates

Several relevant aspects of quantum-field processes can be well described by semiclassical methods. In particular, the knowledge of non-trivial classical solutions of the field equations, and the thermal and quantum fluctuations around them, provide non-perturbative information about the theory. In this work, we discuss the calculation of the one-loop effective action from the semiclasssical viewpoint. We intend to use this formalism to obtain an accurate expression for the decay rate of non-static metastable states.     

Higgs and Wilson lines in field and string theory

This is a short review on basics of the use of the Wilson line to break gauge symmetry in theories with compact extra dimensions. We show how the computation of the one-loop effective field theory leads to a finite result. We then explain the realization of this breaking and the effective potential computation in an open string theory framework with D-branes. To cite this article: K. Benakli, C. R. Physique 8 (2007).     

Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains

With focus on anharmonic chains, we develop a nonlinear version of fluctuating hydrodynamics, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added. The required model-dependent parameters are written in such a way that they can be computed numerically within seconds, once the interaction potential, pressure, and temperature are given. In principle the theory is applicable to any one-dimensional system with local conservation laws. The resulting nonlinear stochastic field theory is handled in the one-loop approximation. Some of the large scale predictions can still be worked out analytically. For more details one has to rely on numerical simulations of the corresponding mode-coupling equations. In this way we arrive at detailed predictions for the equilibrium time correlations of the locally conserved fields of an anharmonic chain.     

Quantum fluctuations and dynamical chaos: An effective potential approach

We discuss the intimate connection between the chaotic dynamics of a classical field theory and the instability of the one-loop effective action of the associated quantum field theory. Using the example of massless scalar electrodynamics, we show how the radiatively induced spontaneous symmetry breaking stabilizes the vacuum state against chaos, and we speculate that monopole condensation can have the same effect in non-Abelian gauge theories.     

Renormalization group flow,entropy and eigenvalues

The irreversibility of the renormalization group flow is conjectured to be closely related to the concept of entropy. In this paper, the variation of eigenvalues of the Laplacian in the Polyakov action under the renormalization group flow will be studied. Based on the one-loop approximation to the effective field theory, we will use the heat kernel method and zeta function regularization. In even dimensions, the variation of eigenvalues is given by the top heat kernel coefficient, and the conformal anomaly is relevant. In odd dimensions, we will conjecture a formula for the variation of eigenvalues through the holographic renormalization in the setting of geometric AdS/CFT correspondence.     

Beyond Poisson-Boltzmann: Fluctuation effects and correlation functions

We formulate the exact non-linear field theory for a fluctuating counter-ion distribution in the presence of a fixed, arbitrary charge distribution. The Poisson-Boltzmann equation is obtained as the saddle-point of the field-theoretic action, and the effects of counter-ion fluctuations are included by a loop-wise expansion around this saddle point. The Poisson equation is obeyed at each order in this loop expansion. We explicitly give the expansion of the Gibbs potential up to two loops. We then apply our field-theoretic formalism to the case of a single impenetrable wall with counter ions only (in the absence of salt ions). We obtain the fluctuation corrections to the electrostatic potential and the counter-ion density to one-loop order without further approximations. The relative importance of fluctuation corrections is controlled by a single parameter, which is proportional to the cube of the counter-ion valency and to the surface charge density. The effective interactions and correlation functions between charged particles close to the charged wall are obtained on the one-loop level. Received 8 February 1999 and Received in final form 15 May 1999     

Supersymmetry under the influence of a magnetic field and high temperature

We evaluate the one-loop effective potential in the presence of a strong magnetic field and high temperature for a supersymmetric non-abelian gauge theory and study SUSY breaking at the one-loop level.     

Three-loop corrections in a new variational treatment ofλφ 4

We present the three-loop calculation of the effective potential forλφ ⁴ in 3+1 dimensions, based on a new variational approach to quantum field theory. This formulation is based on a 2PPI loop expansion of an effective action for local composite operators and yields non-perturbative results. Renormalisation is carried out for both the trivial and the precarious phase using dimensional regularisation. The three-loop result is seen to be similar to the one-loop result (which coincides with the G.E.P.) and only one numerical constant in the effective potential has to be changed.     

Connecting T-duality invariant theories

We show that the vanishing of the one-loop beta-functional of the doubled formalism (which describes string theory on a torus fibration in which the fibres are doubled) is the same as the equation of motion of the recently proposed generalised metric formulation of double field theory restricted to this background: both are the vanishing of a generalised Ricci tensor. That this tensor arises in both backgrounds indicates the importance of a new doubled differential geometry for understanding both constructions.     

Violation of finite-size scaling in three dimensions

We reexamine the range of validity of finite-size scaling in the lattice model and the field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the theory do not rule out the possibility of a violation of finite-size scaling due to a finite lattice constant and a finite cutoff. For a confined geometry of linear size L with periodic boundary conditions we analyze the approach towards bulk critical behavior as at fixed for where is the bulk correlation length. We show that for this analysis ordinary renormalized perturbation theory is sufficient. On the basis of one-loop results and of exact results in the spherical limit we find that finite-size scaling is violated for both the lattice model and the field theory in the region . The non-scaling effects in the field theory and in the lattice model differ significantly from each other. Received 5 February 1999     

Optimized Rayleigh–Schrödinger expansion of the effective potential

An optimized Rayleigh–Schrödinger expansion scheme of solving the functional Schrödinger equation with an external source is proposed to calculate the effective potential beyond the Gaussian approximation. For a scalar field theory whose potential function has a Fourier representation in a sense of tempered distributions, we obtain the effective potential up to the second order, and show that the first-order result is just the Gaussian effective potential. Its application to the λφ⁴ field theory yields the same post-Gaussian effective potential as obtained in the functional integral formalism.     

Scattering amplitudes in open superstring theory

This review is concerned with scattering amplitudes in open superstring theories. In particular, we introduce two different formalisms to compute tree level amplitudes – the Ramond Neveu Schwarz‐ (RNS‐) and the Pure Spinor (PS‐) formalism. The RNS approach proves to be flexible in describing compactifications from ten to four flat spacetime dimensions. We solve the technical problems due to the underlying interacting conformal field theory on the worldsheet. This is exploited to extract phenomenologically relevant scattering amplitudes of gluons and quarks as well as production‐ and decay rates of massive vibration modes which have already been identified as virtual exchange particles at the massless level. In case of a TeV string scale, string specific signatures in parton collisions might be observed at the LHC experiment in the near future and constitute the first experimental evidence for string theory. These statements apply to a wide class of string vacua and therefore bypass the so‐called landscape problem of string theory. The PS formalism allows for a manifestly supersymmetric treatment of scattering amplitudes in ten spacetime dimensions with sixteen supercharges. We introduce a family of superfields which arises in tree amplitudes of massless open string states and can be naturally identified with diagrams made of cubic vertices. We firstly achieve a compact superspace representation of multiparticle field theory amplitudes and moreover express the complete n point superstring amplitude as a minimal linear combination of partial field theory amplitudes and hypergeometric functions. The latter carry the stringy effects and are analyzed from different perspectives.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Ward Identities in CP1 Nonlinear σ Model with Maxwell – Chern – Simons Term

In (2+1) space-time dimensions, CP¹ nonlinear σ model with Maxwell–Chern–Simons (MCS) term is studied by Ward identities. Firstly we revised, in the Coulomb gauge, the system is quantized in Faddeev–Senjanovic (FS) path integral quantization formalism. The canonical Ward identities are then given. Based on the Ward identities, the relations of the generating functional of proper vertex can be derived, and be expressed in Feynman rules with one-loop graphs.     

Relaxation to Stationary Nonequilibrium States in Stochastic Ginzburg–Landau Models

We propose an approach to investigate properties of the time relaxation to stationary nonequilibrium states of correlation functions of stochastic Ginzburg–Landau models with noise (temperature of the reservoirs in contact with the system) changing in space. The formalism relates the stochastic expectations to correlation functions of an imaginary time field theory, and it allows us to study the nonlinear dynamics in terms of a field theory given by a perturbation of a Gaussian measure related to the (easier) linear dynamical problem. To show the usefulness of the formalism, we argue that a perturbative analysis within the integral representation is enough to give us the time relaxation rates of the correlations in some situations.