Exact Renormalization Group Equation Research Articles

Exact renormalization group and loop equation

We propose a gauge invariant formulation of the exact renormalization group equation for nonsupersymmetric pure U(N) Yang-Mills theory, based on the construction by Tim Morris. In fact we show that our renormalization group equation amounts to a regularized version of the loop equation, thereby providing a direct relation between the exact renormalization group and the Schwinger-Dyson equations. We also discuss a possible implication of our formulation to the holographic correspondence of the bulk gravity and the boundary gauge theory.

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t0=t is naturally understood where t0 is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.

The chiral phase transition at high baryon density from nonperturbative flow equations

We investigate the QCD chiral phase transition at high baryon number density within the linear quark meson model for two flavors. The method we employ is based on an exact renormalization group equation for the free energy. Truncated nonperturbative flow equations are derived at nonzero chemical potential and temperature. Whereas the renormalization group flow leads to spontaneous chiral symmetry breaking in vacuum, we find a chiral symmetry restoring first order transition at high density. Combined with previous investigations at nonzero temperature, the result implies the presence of a tricritical point with long--range correlations in the phase diagram.

Scaling critical behavior of superconductors at zero magnetic field

We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The first part of the paper is concerned with the Ginzburg-Landau model in the zero-magnetic-field Meissner phase. We discuss the scaling behavior of the superfluid density and we give an alternative proof of Josephson's relation for a charged superfluid. This proof is obtained as a consequence of an exact renormalization group equation for the photon mass. We obtain Josephson's relation directly in the form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{t}^{\ensuremath{\nu}};$ that is, we do not need to assume that the hyperscaling relation holds. Next, we give an interpretation of a recent experiment performed in thin films of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.$ We argue that the measured mean-field-like behavior of the penetration depth exponent ${\ensuremath{\nu}}^{\ensuremath{'}}$ is possibly associated with a nontrivial critical behavior and we predict the exponents $\ensuremath{\nu}=1$ and $\ensuremath{\alpha}=\ensuremath{-}1$ for the correlation length and specific heat, respectively. In the second part of the paper we discuss the scaling behavior in the continuum dual Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau model, we discuss the continuum dual version by considering a family of scalings characterized by a parameter $\ensuremath{\zeta}$ introduced such that ${m}_{h,0}^{2}\ensuremath{\sim}{t}^{\ensuremath{\zeta}},$ where ${m}_{h,0}$ is the bare mass of the magnetic induction field. We discuss the difficulties in identifying the renormalized magnetic induction mass with the photon mass. We show that the only way to have a critical regime with ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3$ is having $\ensuremath{\zeta}\ensuremath{\approx}4/3,$ that is, with ${m}_{h,0}$ having the scaling behavior of the renormalized photon mass.

Non-Trivial Ultraviolet Fixed Point in Quantum Gravity

The non-trivial ultraviolet fixed point in quantum gravity is calculated by means of the exact renormalization group equation in d-dimensions $(2\simeq d\leq4)$. It is shown that the ultraviolet non-Gaussian fixed point which is expected from the perturbatively $\epsilon$-expanded calculations in $2+\epsilon$ gravity theory remains in d=4. Hence it is possible that quantum gravity is an asymptotically safe theory and renormalizable in 2<d.

Renormalization group at finite temperature in quantum mechanics

We establish the exact renormalization group equation for the potential of a one quantum particle system at finite and zero temperature. As an example we use it to compute the ground state energy of the anharmonic oscillator. We comment on an improvement of the Feynman–Kleinert's variational method by the renormalization group.

Wilson-renormalization-group approach of the principal chiral model around two dimensions

We study the Principal Chiral Ginzburg-Landau-Wilson model around two dimensions within the Local Potential Approximation of an Exact Renormalization Group equation. This model, relevant for the long distance physics of classical frustrated spin systems, exhibits a fixed point of the same universality class that the corresponding Non-Linear Sigma model. This allows to shed light on the long-standing discrepancy between the different perturbative approaches of frustrated spin systems.

Exact renormalization group equation in presence of rescaling anomaly

Wilson's approach to renormalization group is reanalyzed for supersymmetric Yang-Mills theory. Usual demonstration of exact renormalization group equation must be modified due to the presence of the so called Konishi anomaly under the rescaling of superfields. We carry out the explicit computation for N=1 SUSY Yang-Mills theory with the simpler, gauge invariant regularization method, recently proposed by Arkani-Hamed and Murayama. The result is that the Wilsonian action S_M consists of two terms, i.e. the non anomalous term, which obeys Polchinski's flow equation and Fujikawa-Konishi determinant contribution. This latter is responsible for Shifman-Vainshtein relation of exact beta-function.

Field strength correlator and an infrared fixed point of the Wilsonian exact renormalization group equations

The correlator of two field strengths is computed from an effective action for Yang Mills theories, which contains both gluons and an auxiliary antisymmetric tensor field for the field strength as local variables. This action allows to relate explicitly many different approaches to confinement, and is computed using Wilsonian renormalization group equations with the bare Yang Mills action as starting point. Due to the inclusion of a higher dimensional operator in the ghost sector the running gauge coupling becomes vanishingly small at a critical scale kc, and the resulting low energy action resembles the action of a confining string theory proposed by Polyakov.

Attractor properties of physical branches of the solution to the renormalization group equation

We investigate the global phase-portrait structure of a local version of the exact renormalization group (RG) equation for a fluctuating scalar field of the order parameter. All the physical branches of the RG equation solution for the fixed points belong to the attractor subspace to which the local density of the Ginzburg-Landau-Wilson functional is attracted for largely arbitrary initial configurations. The solution of the RG equation corresponding to the nontrivial fixed point determining the critical behavior under the second-order phase transition is a fixed saddle point of this attractor subspace separating the attraction domains of two stable solutions corresponding to the high- and low-temperature thermodynamic regimes.

Critical behavior of ϕ4-theory from the thermal renormalization group

We discuss the universal critical behavior of a selfinteracting scalar field theory at finite temperature as obtained from approximate solutions to nonperturbative renormalization group (RG) equations. We employ a formulation of the RG-equations in real-time formalism which is particularly well suited for a discussion of the thermal behavior of theories which are weakly coupled at T=0. We obtain the equation of state and critical exponents of the theory with a few percent accuracy even for a relatively simple approximation to the exact renormalization group equations.

Field strength correlator and an infrared fixed point of the Wilsonian exact renormalization group equations

The correlator of two field strengths is computed from an effective action for Yang Mills theories, which contains both gluons and an auxiliary antisymmetric tensor field for the field strength as local variables. This action allows to relate explicitly many different approaches to confinement, and is computed using Wilsonian renormalization group equations with the bare Yang Mills action as starting point. Due to the inclusion of a higher dimensional operator in the ghost sector the running gauge coupling becomes vanishingly small at a critical scale k_c, and the resulting low energy action resembles the action of a confining string theory proposed by Polyakov.

Exact renormalization group equation for systems of arbitrary symmetry free of redundant operators

A generalized exact renormalization group (RG) equation for the Ginzburg–Landau–Wilson functional of an arbitrary symmetry, containing invariants of all orders, with nonlocal vertices is derived. Unlike other RG equations which contain an infinite number of redundant operators, the equation derived is free of them.

Exact Renormalization Group Approach in Scalar and Fermionic Theories

The Polchinski version of the exact renormalization group equation is discussed and its applications in scalar and fermionic theories are reviewed. Relation between this approach and the standard renormalization group is studied, in particular the relation between the derivative expansion and the perturbation theory expansion is worked out in some detail.

Rapidly Converging Truncation Scheme of the Exact Renormalization Group

The truncation scheme dependence of the exact renormalization group equations is investigated for scalar field theories in three dimensions. The exponents are numerically estimated to the next-to-leading order of the derivative expansion. It is found that the convergence property in various truncations in the number of powers of the fields is remarkably improved if the expansion is made around the minimum of the effective potential. It is also shown that this truncation scheme is suitable for evaluation of infrared effective potentials. The physical interpretation of this improvement is discussed by considering O(N) symmetric scalar theories in the large-N limit.

Effective quark interactions and QCD propagators

We compute the momentum dependence of the effective four quark interaction in QCD after integrating out the gluons. Our method is based on a truncation of exact renormalization group equations which should give reasonable results for momenta above the confinement scale. The difference between the four quark interaction and the heavy quark potential can be minimized for an optimal renormalization scheme in Landau gauge. Within the momentum range relevant for quarkonia our results agree with phenomenological potentials. We also calculate the propagators for gluons and ghosts in Landau-gauge.

Nonperturbative Flow Equations in QCD

We review the formalism of the effective average action in quantum field theory which corresponds to a coarse grained free energy in statistical mechanics. The associated exact renormalization group equation and possible nonperturbative approximations for its solution are discussed. This is applied to QCD where one observes the consecutive emergence of mesonic bound states and spontaneous chiral symmetry breaking as the coarse graining scale is lowered. We finally present a study of the chiral phase transition in two flavor QCD. A precision estimate of the universal critical equation of state for the three-dimensional 0( 4) Heisenberg model is presented. We explicitly connect the 0( 4) universal behavior near the critical temperature and zero quark mass with the physics at zero temperature and a realistic pion mass. For realistic quark masses the pion correlation length near Tc turns out to be smaller than its zero temperature value. Quantum chromodynamics (QCD) describes qualitatively different physics at different length scales. At short distances the relevant degrees of freedom are quarks and gluons which can be treated perturbatively. At long distances we observe hadrons, and an essential part of the dynamics can be encoded in the masses and interactions of mesons. Any attempt to deal with this situation analytically and to predict the meson properties from the short distance physics (as functions of the strong gauge coupling o: 8 and the current quark masses mq) has to bridge the gap between two qualitatively different effective descriptions. Two basic problems have to be mastered for an extrapolation from short distance QCD to mesonic length scales: • The effective couplings change with scale. This does not only concern the running gauge coupling, but also the coefficients of non-renormalizable opera­ tors as, for example, four quark operators. Typically, these non-renormalizable terms become important in the momentum range where 0:8 is strong and deviate substantially from their perturbative values. Consider the four-point function which results after integrating out the gluons. For heavy quarks it contains the information about the shape of the heavy quark potential whereas for light quarks the complicated spectrum of light mesons and chiral symmetry breaking are encoded in it. At distance scales around 1 fm one expects that the effec­ tive action resembles very little the form of the classical QCD action which is relevant at short distances. • Not only the couplings, but even the relevant variables or degrees of freedom are different for long distance and short distance QCD. It seems forbiddingly

Renormalization of the Topological Charge in Yang–Mills Theory

The conditions leading to a nontrivial renormalization of the topological charge in four-dimensional Yang–Mills theory are discussed. It is shown that if the topological term is regarded as the limit of a certain nontopological interaction, quantum effects due to the gauge bosons lead to a finite multiplicative renormalization of the θ-parameter while fermions give rise to an additional shift of θ. A truncated form of an exact renormalization group equation is used to study the scale dependence of the θ-parameter. Possible implications for the strong CP-problem of QCD are discussed.

Quantum Liouville field theory as solution of a flow equation

A general framework for the Weyl-invariant quantization of Liouville field theory by means of an exact renormalization group equation is proposed. This renormalization group, or flow equation, describes the scale dependence of the effective average action which has a built-in infrared cutoff. For c < 1 it is solved approximately by a truncation of the space of action functionals. We derive the Ward identities associated to Weyl transformations in the presence of the infrared cutoff. They are used to select a specific universality class for the renormalization group trajectory which is found to connect two conformal field theories with central charges 25 - c and 26 - c, respectively.

The renormalization group equation in N = 2 supersymmetric gauge theories

We clarify the mass dependence of the effective prepotential in N = 2 supersymmetric SU( N c ) gauge theories with an arbitrary number N f < 2 N c of flavors. The resulting differential equation for the prepotential extends the equations obtained previously for SU(2 and for zero masses. It can be viewed as an exact renormalization group equation for the prepotential, with the beta function given by a modular form. We derive an explicit formula for this modular form when N f = 0, and verify the equation to 2-instanton order in the weak-coupling regime for arbitrary N f and N c We also extend the renormalization group equation to the case of other classical gauge groups.