Exact Renormalization Group Equation Research Articles

Anomalous dimensions determine the power counting: Wilsonian RG analysis of nuclear EFT

The Legendre flow equation, a version of exact Wilsonian renormalization group (WRG) equation, is employed to consider the power counting issues in nuclear effective field theory. A WRG approach is an ideal framework because it is nonperturbative and does not require any prescribed power counting rule. The power counting is determined systematically from the scaling dimensions of the operators at the nontrivial fixed point. The phase structure is emphasized and the inverse of the scattering length, which is identified as a relevant coupling, is shown to play a role of the order parameter. The relations to the work done by Birse, McGovern, and Richardson and to the Kaplan–Savage–Wise scheme are explained.

Differential renormalization-group approach to the layered sine-Gordon model

A new qualitative view of vortex length–scale dependence has been found in recent electrical transport measurements performed on strongly anisotropic Bi2Sr2CaCu2O8 single crystals in zero magnetic field. This indicates the need for a better description of the 3D/2D crossover in vortex dimensionality. The vortex-dominated properties of high transition temperature superconductors with extremely high anisotropy (layered systems) are reasonably well described in the framework of the layered XY model which can be mapped onto the layered sine-Gordon model. For the latter we derive an exact renormalization group (RG) equation using Wegner's and Houghton's approach in the local potential approximation. The agreement of the UV scaling laws found by us by linearizing the RG equations with those obtained previously in the literature in the dilute gas approximation makes apparent the improvement which can be achieved by solving our RG equations numerically.

Wilson–Polchinski exact renormalization group equation forO (N) systems: leading and next-to-leading orders in the derivative expansion

With a view to study the convergence properties of the derivative expansion of the exactrenormalization group (RG) equation, I explicitly study the leading and next-to-leadingorders of this expansion applied to the Wilson–Polchinski equation in the case of theN-vector model with thesymmetry O (N). As a test,the critical exponents η and ν as well as thesubcritical exponent ω (and higher ones) are estimated in three dimensions for values ofN rangingfrom 1 to 20. Icompare the results with the corresponding estimates obtained in preceding studies or treatments ofother O (N) exact RG equations at second order. The possibility of varyingN allows the derivative expansion method to be better valued. The values obtained from theresummation of high orders of perturbative field theory are used as standards to illustratethe eventual convergence in each case. Particular attention is drawn to the preservation (ornot) of the reparameterization invariance.

Proper Time Flow Equation for Gravity

We analyze a proper time renormalization group equation for Quantum Einstein Gravity in the Einstein-Hilbert truncation and compare its predictions to those of the conceptually different exact renormalization group equation of the effective average action. We employ a smooth infrared regulator of a special type which is known to give rise to extremely precise critical exponents in scalar theories. We find perfect consistency between the proper time and the average action renormalization group equations. In particular the proper time equation, too, predicts the existence of a non-gaussian fixed point as it is necessary for the conjectured nonperturbative renormalizability of Quantum Einstein Gravity.

Power-Counting Theorem for Non-Local Matrix Models and Renormalisation

Solving the exact renormalisation group equation à la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties—typically arising from orthogonal polynomials—which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative ℝ D in matrix formulation.

An algebraic Birkhoff decomposition for the continuous renormalization group

This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing us to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited.

Running Newton constant, improved gravitational actions, and galaxy rotation curves

A renormalization group (RG) improvement of the Einstein-Hilbert action is performed which promotes Newton's constant and the cosmological constant to scalar functions on spacetime. They arise from solutions of an exact RG equation by means of a ``cutoff identification'' which associates RG scales to the points of spacetime. The resulting modified Einstein equations for spherically symmetric, static spacetimes are derived and analyzed in detail. The modifications of the Newtonian limit due to the RG evolution are obtained for the general case. As an application, the viability of a scenario is investigated where strong quantum effects in the infrared cause Newton's constant to grow at large (astrophysical) distances. For two specific RG trajectories exact vacuum spacetimes modifying the Schwarzschild metric are obtained by means of a solution-generating Weyl transformation. Their possible relevance to the problem of the observed approximately flat galaxy rotation curves is discussed. It is found that a power law running of Newton's constant with a small exponent of the order ${10}^{\ensuremath{-}6}$ would account for their non-Keplerian behavior without having to postulate the presence of any dark matter in the galactic halo.

Antiferromagnetic gap in the Hubbard model

We compute the temperature-dependence of the antiferromagnetic order parameter and the gap in the two-dimensional Hubbard model at and close to half filling. Our approach is based on truncations of an exact functional renormalization group equation. The explicit use of composite bosonic degrees of freedom permits a direct investigation of the ordered low temperature phase. We show that the Mermin–Wagner theorem is not practically applicable for the spontaneous breaking of the continuous spin symmetry in the antiferromagnetic state. The critical behavior is dominated by the fluctuations of composite Goldstone bosons.

Renormalization flow of Yang-Mills propagators

We study Landau-gauge Yang-Mills theory by means of a nonperturbative vertex expansion of the quantum effective action. Using an exact renormalization group equation, we compute the fully dressed gluon and ghost propagators to lowest nontrivial order in the vertex expansion. In the mid-momentum regime, $p^2\sim\mathcal{O}(1)\text{GeV}^2$, we probe the propagator flow with various {\em ans\atze} for the three- and four-point correlations. We analyze the potential of these truncation schemes to generate a nonperturbative scale. We find universal infrared behavior of the propagators, if the gluon dressing function has developed a mass-like structure at mid-momentum. The resulting power laws in the infrared support the Kugo-Ojima confinement scenario.

Stochastic formulation of the renormalization group: supersymmetric structure and topology of the space of couplings

The exact or Wilson renormalization group equations can be formulated as a functional Fokker–Planck equation in the infinite-dimensional configuration space of a field theory, suggesting a stochastic process in the space of couplings. Indeed, the ordinary renormalization group differential equations can be supplemented with noise, making them stochastic Langevin equations. Furthermore, if the renormalization group is a gradient flow, the space of couplings can be endowed with a supersymmetric structure a la Parisi–Sourlas. The formulation of the renormalization group as supersymmetric quantum mechanics is useful for analysing the topology of the space of couplings by means of Morse theory. We present simple examples with one or two couplings.

Infrared Behavior and Fixed Points in Landau-Gauge QCD

We investigate the infrared behavior of gluon and ghost propagators in Landau-gauge QCD by means of an exact renormalization group equation. We explain how, in general, the infrared momentum structure of Green functions can be extracted within this approach. An optimization procedure is devised to remove residual regulator dependences. In Landau-gauge QCD this framework is used to determine the infrared leading terms of the propagators. The results support the Kugo-Ojima confinement scenario. Possible extensions are discussed.

The Wilson–Polchinski exact renormalization group equation

The critical exponent η is not well accounted for in the Polchinski exact formulation of the renormalization group (RG). With a particular emphasis laid on the introduction of the critical exponent η, I re-establish (after Golner, hep-th/9801124) the explicit relation between the early Wilson exact RG equation, constructed with the incomplete integration as cutoff procedure, and the formulation with an arbitrary cutoff function proposed later on by Polchinski. I (re)-do the analysis of the Wilson–Polchinski equation expanded up to the next to leading order of the derivative expansion. I finally specify a criterion for choosing the “best” value of η to this order. This Letter will help in using more systematically the exact RG equation in various studies.

Exact renormalization group equation for the Lifshitz critical point

An exact renormalization equation (ERGE) accounting for an anisotropic scaling is derived. The critical and tricritical Lifshitz points are then studied at leading order of the derivative expansion which is shown to involve two differential equations. The resulting estimates of the Lifshitz critical exponents compare well with the O ( ε 2 ) calculations. In the case of the Lifshitz tricritical point, it is shown that a marginally relevant coupling defies the perturbative approach since it actually makes the fixed point referred to in the previous perturbative calculations O ( ε ) finally unstable.

APPLICATION OF EXACT RENORMALIZATION GROUP TECHNIQUES TO THE NON-PERTURBATIVE STUDY OF SUPERSYMMETRIC FIELD THEORY

A simple form of the exact renormalization group method is proposed for the study of supersymmetric gauge field theory. The method relies on the existence of ultraviolet-finite four dimensional gauge theories with extended supersymmetry. The resulting exact renormalization group equation crucially depends on the Konishi anomaly of N=1 super Yang–Mills. We illustrate our method by dealing with the NSVZ exact relation for the beta functions, the N=2 super Yang–Mills effective potential and the N=1 super Yang–Mills gluon superpotential (the so-called Veneziano–Yankielowicz potential).

Gravity and matter with asymptotic safety

Building a consistent Quantum Theory of Gravity is one of the most challenging aspects of modern theoretical physics. In the past couple of years, new attempts have been made along the path of ``asymptotic safety'' through the use of Exact Renormalisation Group Equations, which hinge on the existence of a non-trivial fixed point of the flow equations. We will first summarize the major results that have been obtained along these lines, then we will consider the effect of introducing matter fields into the theory. Our analyses show that in order to preserve the existence of the fixed point one must satisfy some constraints on the matter content of the theory.

Critical phenomena in continuous dimension

We present a calculation of critical phenomena directly in continuous dimension d employing an exact renormalization group equation for the effective average action. For an Ising-type scalar field theory we calculate the critical exponents ν(d) and η(d) both from a lowest-order and a complete first-order derivative expansion of the effective average action. In particular, this can be used to study critical behavior as a function of dimensionality at fixed temperature.

Subleading critical exponents from the renormalisation group

We study exact renormalisation group equations for the 3d Ising universality class. At the Wilson–Fisher fixed point, symmetric and antisymmetric correction-to-scaling exponents are computed with high accuracy for an optimised cutoff to leading order in the derivative expansion. Further results are derived for other cutoffs including smooth, sharp and background field cutoffs. An estimate for higher order corrections is given as well. We establish that the leading antisymmetric corrections to scaling are strongly subleading compared to the leading symmetric ones.

Exact multilocal renormalization of the effective action: application to the random sine Gordon model statics and nonequilibrium dynamics.

We extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the eta exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent z are recovered and several results are obtained, such as the equilibrium two-point scaling functions. The nonequilibrium, finite momentum, two-time t,t' response and correlations are computed. They are shown to exhibit scaling forms, characterized by exponents lambda(R) not equal lambda(C), as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be nontrivial and of the form X[q(z)(t-t'),t/t']. Analogies and differences with pure critical models are discussed.

LOOP VARIABLES IN STRING THEORY

The loop variable approach is a proposal for a gauge-invariant generalization of the sigma-model renormalization group method of obtaining equations of motion in string theory. The basic guiding principle is space–time gauge invariance rather than world sheet properties. In essence it is a version of Wilson's exact renormalization group equation for the world sheet theory. It involves all the massive modes and is defined with a finite world sheet cutoff, which allows one to go off the mass-shell. On shell the tree amplitudes of string theory are reproduced. The equations are gauge-invariant off shell also. This paper is a self-contained discussion of the loop variable approach as well as its connection with the Wilsonian RG.

SOLUTIONS OF THE POLCHINSKI ERG EQUATION IN THE O(N) SCALAR MODEL

Solutions of the Polchinski exact renormalization group equation in the scalar O(N) theory are studied. Families of regular solutions are found and their relation with fixed points of the theory is established. Special attention is devoted to the limit N = ∞, where many properties can be analyzed analytically.