Is there a c-theorem in four dimensions?

The difficulties of extending Zamolodchikov's c-theorem to dimensions d ≠ 2 are discussed. It is shown that, for d even, the one-point function of the trace of the stress tensor on the sphere, S^d, when suitably regularized, defines a c-function, which, at least to one loop order, is decreasing along RG trajectories and is stationary at RG fixed points, where it is proportional to the usual conformal anomaly. It is shown that the existence of such a c-function, if it satisfies these properties to all orders, is consistent with the expected behavior of QCD in four dimensions.     

A resummable β-function for massless QED

Within the set of schemes defined by generalized, manifestly gauge invariant exact renormalization groups for QED, it is argued that the β-function in the four-dimensional massless theory cannot possess any nonperturbative power corrections. Consequently, the perturbative expression for the β-function must be resummable. This argument cannot be extended to flows of the other couplings or to the anomalous dimension of the fermions and so perturbation theory does not define a unique trajectory in the critical surface of the Gaussian fixed point. Thus, resummability of the β-function is not inconsistent with the expectation that a non-trivial fixed point does not exist.     

What can we learn from the finite temperature renormalization group?

The renormalization group for finite temperature quantum field theories is studied, in particular for λ?⁴. It is shown that the “high” temperature limit can only be discussed perturbatively ifT dependent renormalization schemes are implemented. Zero temperature renormalization schemes or renormalization at some fixed reference temperatureT _o are both inadequate as they imply perturbative expansions about fixed points of the renormalization group which are associated with a zero temperature system and a system at temperatureT _o respectively.T dependent schemes give rise to an expansion about the true fixed point of the system, the resulting renormalization group allows the entire crossover between high and low temperature behaviour to be investigated.     

Vanishing β-function for Grosse–Wulkenhaar model in a magnetic field

We prove that the β-function of the Grosse–Wulkenhaar model including a magnetic field vanishes at all order of perturbations. We compute the renormalization group flow of the relevant dynamic parameters and find a non-Gaussian infrared fixed point. Some consequences of these results are discussed.     

Three-loop β-functions for the superstring and heterotic string

The β-function for the (1,1) superstring is calculated explicitly up to three loops, using two different choices of the fermion quantum field. In both cases the bosonic contribution is exactly cancelled by the fermionic contribution. The calculation is repeated for the heterotic string and the non-vanishing β-function is shown to be compatible with an O(α′²) effective action in which there are no terms cubic in the Riemann tensor- or gauge-field strength. A renormalization group equation for the gauge field is constructed and its β-function is shown to vanish at the two-loop level.     

Perturbative series and the 1/N expansion for the QED β-function

A comparison of the 5-loop perturbative series and the 1/N expansion for the QED renormalization group β-function in the Minimal Subtraction scheme is performed. The good agreement between two expansions is found which proves that the MS β-function is adequately described by both series.     

On Kadanoff's approximate renormalization group transformation

The fixed point structure resulting from the approximate renormalization group equations obtained by shifting bonds on the square Ising lattice is considered as a function of a free parameter h appearing in the definition of these equations. Next to the fixed point S considered by Kadanoff which is located in a symmetry plane two other “critical” fixed points A and B are found for h0.726. At the value h = 0.741, A crosses the fixed point S and vanishes together with the fixed point B at h = 0.726. Furthermore correction terms to the eigenvalues of the linearized renormalization group equations as obtained by Kadanoff are considered which arise if one chooses h to be optimal at all points of the coupling parameter space.     

C function representation of the Local Potential Approximation

Within the Local Potential Approximation to Wilson's, or Polchinski's, exact renormalization group, and for general spacetime dimension, we construct a function, c, of the coupling constants; it has the property that (for unitary theories) it decreases monotonically along flows, and is stationary only at fixed points — where it ‘counts degrees of freedom’, i.e. is extensive, counting one for each Gaussian scalar. Furthermore, by choosing restrictions to some sub-manifold of coupling constant space, we arrive at a very promising variational approximation method.     

Analytical results for the four-loop RG functions in the 2D non-linear O(n) σ-model on the lattice

We recalculate four-loop renormalization group functions in 2-dimensional nonlinear O(n) σ-model using coordinate-space method. The high accuracy of calculation allow us to find the analytical form of β- and γ-function (anomaluos dimension).     

An introduction to the fundamentals of the renormalization group in critical phenomena

The fundamental concepts underlying the application of the renormalization group and related techniques to critical phenomena are reviewed at an elementary level. Topics discussed include: the definition of the renormalization group as a functional integral over high momentum components of the spin field, the behaviour of the renormalization group near the fixed point and the derivation of scaling, Wilson's approximate recursion relation, trivial and non-trivial fixed points of isotropic spin systems near d = 4, Feynman graph expansions for critical exponents, ? = 4 ? d and 1/n-expansions, the derivation of exact recursion relations and co-ordinate space transformations for d = 2 Ising systems     

The method of uniqueness,a new powerful technique for multiloop calculations

We present a new method for multiloop calculations that provides an “algebraic” procedure to evaluate the renormalization group functions up to five loops. As an example, a final analytical expression for the five-loop β-function in the ?⁴ theory is given.     

A hierarchical model exhibiting the Kosterlitz-Thouless fixed point

We present a new hierarchical model for 2D Coulomb gases and investigate its behavior under renormalization group transformations. Unlike all previous hierarchical models (including the Migdal-Kadanoff version) for this system our model displays a line of stable fixed points describing the Kosterlitz-Thouless phase transition. For Coulomb gases corresponding to Z_N models these fixed points are stable in an intermediate range of temperatures.     

Perturbative relations between e+e− annihilation and τ decay observables including resummation effects

By exploiting the analyticity properties of the two-point current-current correlator we obtain numerical predictions for the e⁺e⁻ moments in terms of the τ decay rate. We perform a partial resummation of the pertinent perturbative series expansion by solving the renormalization group equation for Adler's function. Our predictions are renormalization scheme independent but depend on the order of the perturbative β-function expansion. The analysis involves the unknown five-loop coefficient k₃ for which we give some new estimates.     

Can renormalization group flow end in a Big Mess?

The field theoretical renormalization group equations have many common features with the equations of dynamical systems. In particular, the manner how Callan–Symanzik equation ensures the independence of a theory from its subtraction point is reminiscent of self-similarity in autonomous flows towards attractors. Motivated by such analogies we propose that besides isolated fixed points, the couplings in a renormalizable field theory may also flow towards more general, even fractal attractors. This could lead to Big Mess scenarios in applications to multiphase systems, from spin-glasses and neural networks to fundamental string (M?) theory. We consider various general aspects of such chaotic flows. We argue that they pose no obvious contradictions with the known properties of effective actions, the existence of dissipative Lyapunov functions, and even the strong version of the c-theorem. We also explain the difficulties encountered when constructing effective actions with chaotic renormalization group flows and observe that they have many common virtues with realistic field theory effective actions. We conclude that if chaotic renormalization group flows are to be excluded, conceptually novel no-go theorems must be developed.     

A nonconventional approach to describe classical-quantum crossover phenomena near criticality

A nonconventional renormalization group procedure, which involves fixed points and eigenvectors depending on temperature T, is proposed to describe classical-quantum crossover phenomena for quantum systems near criticality. In this new picture, T-dependent critical exponents occur, where T assumes the role of a natural crossover parameter.     

Block spin renormalization group for dipole gas and (∇φ)4

A rigorous, non-perturbative analysis of the block spin (BS) renormalization group is performed for a large class of massless lattice models such as the dipole gas and (?φ)⁴. Effective Hamiltonians are shown to be driven to the line of Gaussian fixed points by iteration of the BS-transformation. Analyticity of the presure and the dielectric constant in the (weak) coupling is proven. In particular, the perturbation expansion in activity for the dipole gas is convergent.     

The a-theorem

The c-theorem is a profound result applying to statistical mechanical theories or quantum field theories in two dimensions. Such theories may be described by a set of parameters which vary as we increase or decrease the scale on which we observe the system, until we reach a fixed point or critical point where the couplings have fixed values. The c-theorem defines a quantity (the c-function) which always increases (or is constant) with increasing scale and thereby gives a valuable insight into the ‘flows’ of the couplings between fixed points. The a-theorem is a proposed generalisation of the c-theorem to higher dimensions, especially four. In this article, we describe the c-theorem, starting in the simpler statistical mechanical context and then showing how in quantum field theory the theorem is most easily formulated in a curved spacetime. We then sketch how these concepts are applied in the more technically complex scenario of four dimensions.     

Callan-Symanzik and renormalization group equation in a model with spontaneous breaking of the symmetry

In a simple model with spontaneous breaking of the axialU(1)-symmetry via the Higgs mechanism we construct the Callan-Symanzik and renormalization group equation in the Goldstone mode. Aiming at questions of renormalization group improvement and the like we compare two different parametrizations the model can be described with. We show that in the presence of fermions a β-function for a physical mass or some equivalent of it enters unavoidably the Callan-Symanzik equation, which leads to significant differences to the symmetric theory starting with two loops. On the other hand in the asymptotic region the equivalence to the symmetric theory is manifest.     

An investigation of the phase structure of the two-dimensional O(2) and O(4) symmetric nonlinear sigma-models

We investigate the phase structure of the two-dimensional O(2) and O(4) lattice σ-models by means of a Monte Carlo simulation using Binder's phenomenological renormalization group. For the O(2) model the transition temperature β_c⁻¹ is estimated and the expected line of critical points is found. For the O(4) model no signal of a phase transition is found in the range of couplings considered. This is in contradiction to a recent claim that the O(4) model has a critical point at finite β.