International Journal of Modern Physics A

Exact Renormalization Group techniques are applied to supersymmetric models in order to get some insights into the low energy effective actions of such theories. Starting from the ultra-violet finite mass deformed N=4 supersymmetric Yang Mills theory, one varies the regularising mass and compensates for it by introducing an effective Wilsonian action. (Polchinski's) renormalization group equation is modified in an essential way by the presence of rescaling (a.k.a. Konishi) anomaly, which is responsible for the beta-function. When supersymmetry is broken up to N=1 the form of effective actions in terms of massless fields is quite reasonable, while in the case of the N=2 model we appear to have problems related to instantons.

After a brief presentation of the exact renormalization group equation, we illustrate how the field theoretical (perturbative) approach to critical phenomena takes place in the more general Wilson (nonperturbative) approach. Notions such as the continuum limit and the renormalizability and the pressure of singularities in the perturbative series are discussed.

The Wilson renormalization group formulation of gauge theories is reviewed. In particular, the fine tuning procedure needed to recover the gauge invariance broken by the infrared cutoff is discussed. When the cutoff is larger than any physical scale, this procedure determines the finite non-invariant couplings of the ultraviolet action. These couplings are used to build up a local field transformation which allows to write a BRS invariant ultraviolet action.

We discuss questions related to renormalization group and to nonperturbative aspects of non-Abelian gauge theories with N=2 and/or N=1 supersymmetry. Results on perturbative and nonperturbative β functions of these theories are reviewed, and new mechanisms of confinement and dynamical symmetry breaking recently found in a class of SU(n_c), US_p(2n_c) and SO(n_c) theories are discussed.

Some correlation functions of critical models, like the anisotropic spin chain with nearest and next-to-nearest neighbor interaction, or the eight vertex model, are computed as a corollary of the study of the XYZ model in [2].

We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the complete flow from the bare action of the microscopic model to the effective low-energy action is obtained from a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signalled by the diverging interactions we compute the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find antiferromagnetic order or d-wave superconductivity as leading symmetry breaking instability.

A gauge invariant Wilsonian effective action is constructed for pure SU(N) Yang-Mills theory by formulating the corresponding flow equation. Manifestly gauge invariant calculations can be performed i.e. without gauge fixing or ghosts. Regularisation is implemented in a novel way which realises a spontaneously broken SU(N|N) supergauge theory. As an example we sketch the computation of the one-loop β function, performed for the first time without any gauge fixing.

The advantageous points of ERG in applications to non-perturbative analyses of quantum field theories are discussed in comparison with the Schwinger-Dyson equations. First we consider the relation between these two formulations specially by examining the large N field theories. In the second part we study the phase structure of dynamical symmetry breaking in three dimensional QED as a typical example of the practical application.

Studies of first-order phase transitions through the use of the exact renormalization group are reviewed. In the first part the emphasis is on universal aspects: We discuss the universal critical behaviour near weakly first-order phase transitions for a three-dimensional model of two coupled scalar fields – the cubic anisotropy model. In the second part we review the application of the exact renormalization group to the calculation of bubble-nucleation rates. More specifically, we concentrate on the pre-exponential factor. We discuss the reliability of homogeneous nucleation theory that employs a saddle-point expansion around the critical bubble for the calculation of the nucleation rate.

A method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous unitary transformation is reviewed. We will outline (i) the procedure for the elimination of the electron-phonon interaction and the construction of the effective attractive electron-electron interaction, and (ii) the application to some systems with electron-electron interaction (n-orbital model and Hubbard model).

An exact renormalization group equation describes the dependence of the free energy on an infrared cutoff for the quantum or thermal fluctuations. It interpolates between the microphysical laws and the complex macroscopic phenomena. We present a simple unified description of critical phenomena for O(N)-symmetric scalar models in two, three or four dimensions, including essential scaling for the Kosterlitz-Thouless transition.

In this talk we describe a novel method for the renormalization of the Hamiltonian operator in Quantum Field Theory in the spirit of the Wilson renormalization group¹. By a series of unitary transformations that successively decouple the high-frequency degrees of freedom and partially diagonalizes the high-energy part, we obtain the effective Hamiltonian for the low energy degrees of freedom. Using this technique we study λϕ⁴ theory at two loops and QED and Yang-Mills theory at one loop.

A gauge invariant regularisation for dealing with pure Yang-Mills theories within the exact renormalization group approach is proposed. It is based on the regularisation via covariant higher derivatives and includes auxiliary Pauli-Villars fields which amounts to a spontaneously broken SU(N|N) super-gauge theory. We demonstrate perturbatively that the extended theory is ultra-violet finite in four dimensions and argue that it has a sensible limit when the regularization cutoff is removed.

By subdividing a 2d square lattice into plaquettes containing 4 lattice sites each, the Hubbard Hamiltonian is reformulated in terms of different "colored" fermion species. This makes it possible to describe e.g. antiferromagnetic order or d-wave superconductivity in terms of expectation values of composite scalar fields. A suitable mean field approximation indeed reproduces qualitatively the measured phase diagram and shows a competition between phases with antiferromagnetic ordering and d_x²-y²-superconductivity at low temperatures.

Three-dimensional spin models of the Ising and XY universality classes are studied by a combination of high-temperature expansions and Monte Carlo simulations applied to improved Hamiltonians. The critical exponents and the critical equation of state are determined to very high precision.

Within the framework of the perturbative renormalization-group approach, we discuss the case when perturbation theory is affected by infrared divergences well inside a stable phase of matter. Contrary to the case of critical phenomena, a cancellation of the singularities, rather than a resummation, has to take place, in order to obtain finite response functions, as implied by the stability conditions. We show how symmetries, and related Ward identities, can be implemented to relate the singular parts of the different running coupling constants, and reduce the number of independent parameters, thus allowing for a closure of the renormalization-group equations, and for the asymptotical solution of the problem. Applications to the cases of Fermi and Bose systems in low spatial dimensions are discussed.

In the AdS/CFT correspondence motion in the radial direction of the AdS space is identified with renormalization group flow in the field theory. For the N=4 Yang-Mills theory this motion is trivial. More interesting examples of renormalization group flow occur when the N=4 theory is deformed. Aspects of the flows are discussed for the N=4 theory on its moduli space, and deformed to N=2 in the infra-red within the context of 5d SUGRA. 10d lifts and brane probing are crucial tools for linking the spacetime backgrounds to the dual field theory.

In this talk the gauge symmetry for Wilsonian flows in pure Yang-Mills theories is discussed. The background field formalism is used for the construction of a gauge invariant effective action. The symmetries of the effective action under gauge transformations for both the gauge field and the auxiliary background field are separately evaluated. Modified Ward-Takahashi and background field identities are used in my study. Finally it is shown how the symmetry properties of the full theory are restored in the limit where the cut-off is removed.

The coarse-graining operation in hydrodynamics is equivalent to a change of scale which can be formalized as a renormalization group transformation. In particular, its application to the probability distribution of a self-gravitating fluid yields an "exact renormalization group equation" of Fokker-Planck type. Since the time evolution of that distribution can also be described by a Fokker-Planck equation, we propose a connection between both equations, that is, a connection between scale and time evolution. We finally remark on the essentially non-perturbative nature of astrophysical problems, which suggests that the exact renormalization group is the adequate tool for them.

We show that the BRS symmetry is present even with the finite IR cutoff in the Wilsonian RG approach. Based on the antifield formalism, we explain how the symmetry is realized on the RG flow.

Finite Unified Theories are N=1 supersymmetric GUT's that can be made finite beyond the unification point, including the softly broken sector. The new characteristic predictions of FUTs are: 1) The lightest Higgs boson mass is predicted to be in the window 120-130 GeV, in case the LSP is neutralino, while in case the LSP is the (which can be consistently accommodated in presence of bilinear R-parity violating terms) it can be as light as 111 GeV. 2) The s-spectrum starts above several hundreds of GeV.

We investigate the Polchinski ERG equation for d-dimensional O(N) scalar field theory. In the context of the non-pertubative derivative expansion we find families of regular solutions and establish their relation with the physical fixed points of the theory. Special emphasis is given to the limit N=∞ for which many properties can be studied analytically.

The path integral formulation of Quantum Field Theory implies an infinite set of local, Schwinger-Dyson-like relation. Exact renormalization group equations can be cast as a particular instance of these relations. Furthermore, exact scheme independence is turned into a vector field transformation of the kernel of the exact renormalization group equation under field redefinitions.

We propose a self-consistent renormalization group prescription for the scalar field theory. The formalism coincides with the local potential approximation in the sharp cutoff limit but differ with a smooth cutoff. The dependence of the critical exponent ν on the smoothness parameter and the field of expansion is explored. We employ an optimization scheme based on the minimum sensitivity principle to ensure the most rapid convergence of ν with the polynomial truncation level.

We discuss an optimisation criterion for the exact renormalisation group based on the inverse effective propagator, which displays a gap. We show that a simple extremisation of the gap stabilises the flow, leading to better convergence of approximate solutions towards the physical theory. This improves the reliability of truncations, most relevant for any high precision computation. These ideas are closely linked to the removal of a spurious scheme dependence and a minimum sensitivity condition. The issue of predictive power and a link to the Polchinski RG are discussed as well. We illustrate our findings by computing critical exponents for the Ising universality class.

The BRST algebraic proofs of the nonrenormalization theorems for the β-functions of N=2 and N=4 Super Yang-Mills theories are reviewed.

The convergence of the derivative expansion of the exact renormalisation group is investigated via the computation of the β function of massless scalar λφ⁴ theory. The derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. Convergence of the derivative expansion of the Legendre flow equation is trivial at one loop, but also can occur at two loops and in particular converges for an exponential cutoff.

The Exact Renormalization Group (ERG) approach to massive gauge theories in the axial gauge is studied and the smoothness of the massless limit is analysed for a formally gauge invariant quantity such as the Euclidean Wilson loop.

An Exact Renormalisation Group (ERG) approach to non-Abelian gauge theories is discussed. We focus on the derivation of universal beta-functions and the choice of the initial effective action, the latter being a key input in the approach. To that end we establish the map between Gell-Mann–Low scaling of the full theory and the scaling in an ERG approach. Then this map is used to derive the 2-loop β-function within a simple straightforward calculation. The implications for the choice of the initial effective action are discussed.

A renormalization group transformation suitable for spin glasses, and more generally for disordered systems, is presented. The approach is non-standard for two reasons. 1. The coarse graining transformation is performed on the distribution of site overlaps, which clearly is not a Boltzmann-Gibbs distribution. 2. The additional interactions are realized introducing a gauge-field disorder distribution, while the Hamiltonian is kept fixed. Universality classes are thus naturally defined on a large set of models, going from ℤ₂ and Gaussian spin glasses to Ising and fully frustrated models, and others. The role of frustration and of the deconfinement transition can also be analysed in terms of fixed points. The proposed analysis is tested numerically on the ℤ₂ Edwards-Anderson model in d=4. Good estimates of the critical index ν and of T_c are obtained, and an RG flow diagram is sketched for the first time.

Self-consistent new renormalization group flow equations for an O(N)-symmetric scalar theory are approximated in next-to-leading order of the derivative expansion. The Wilson-Fisher fixed point in three dimensions is analyzed in detail and various critical exponents are calculated.

I study a class of Wilsonian formulations of non-Abelian gauge theories in algebraic noncovariant gauges where the Wilsonian infrared cutoff Λ is inserted as a mass term for the propagating fields. In this way the Ward-Takahashi identities are preserved to all scales. Nevertheless the BRS-invariance in broken and the theory is gauge-dependent and unphysical at Λ≠ 0. Then I discuss the infrared limit Λ→0. I show that the singularities of the axial gauge choice are avoided in planar gauge and in light-cone gauge. Finally the rectangular Wilson loop of size 2L×2T is evaluated at lowest order in perturbation theory and a noncommutativity between the limits Λ→0 and T→∞ is pointed out.

Frustrated magnets are a notorious example where usual perturbative methods fail. Having recourse to an exact renormalization group approach, one gets a coherent picture of the physics of Heisenberg frustrated magnets everywhere between d=2 and d=4: all known perturbative results are recovered in a single framework, their apparent conflict is explained while the description of the phase transition in d=3 is found to be in good agreement with the experimental context.