Constraints on Beta Functions from Duality

We analyze the way in which duality constrains the exact beta function and correlation length in single-coupling spin systems. A consistency condition we propose shows very concisely the relation between self-dual points and phase transitions, and implies that the correlation length must be duality invariant. These ideas are then tested on the 2-d Ising model, and used towards finding the exact beta function of the $q$-state Potts model. Finally, a generic procedure is given for identifying a duality symmetry in other single-coupling models with a continuous phase transition.Comment: LaTeX, 6 page

Differential Regularization of Topologically Massive Yang-Mills Theory and Chern-Simons Theory

We apply differential renormalization method to the study of three-dimensional topologically massive Yang-Mills and Chern-Simons theories. The method is especially suitable for such theories as it avoids the need for dimensional continuation of three-dimensional antisymmetric tensor and the Feynman rules for three-dimensional theories in coordinate space are relatively simple. The calculus involved is still lengthy but not as difficult as other existing methods of calculation. We compute one-loop propagators and vertices and derive the one-loop local effective action for topologically massive Yang-Mills theory. We then consider Chern-Simons field theory as the large mass limit of topologically massive Yang-Mills theory and show that this leads to the famous shift in the parameter $k$. Some useful formulas for the calculus of differential renormalization of three-dimensional field theories are given in an Appendix.Comment: 25 pages, 4 figures. Several typewritten errors and inappropriate arguments are corrected, especially the correct adresses of authors are give

RG Flow Irreversibility, C-Theorem and Topological Nature of 4D N=2 SYM

We determine the exact beta function and a RG flow Lyapunov function for N=2 SYM with gauge group SU(n). It turns out that the classical discriminants of the Seiberg-Witten curves determine the RG potential. The radial irreversibility of the RG flow in the SU(2) case and the non-perturbative identity relating the $u$-modulus and the superconformal anomaly, indicate the existence of a four dimensional analogue of the c-theorem for N=2 SYM which we formulate for the full SU(n) theory. Our investigation provides further evidence of the essentially topological nature of the theory.Comment: 9 pages, LaTeX file. Discussion on WDVV and integrability. References added. Version published in PR

Relative entropy in 2d Quantum Field Theory, finite-size corrections and irreversibility of the Renormalization Group

The relative entropy in two-dimensional Field Theory is studied for its application as an irreversible quantity under the Renormalization Group, relying on a general monotonicity theorem for that quantity previously established. In the cylinder geometry, interpreted as finite-temperature field theory, one can define from the relative entropy a monotonic quantity similar to Zamolodchikov's c function. On the other hand, the one-dimensional quantum thermodynamic entropy also leads to a monotonic quantity, with different properties. The relation of thermodynamic quantities with the complex components of the stress tensor is also established and hence the entropic c theorems are proposed as analogues of Zamolodchikov's c theorem for the cylinder geometry.Comment: 5 pages, Latex file, revtex, reorganized to best show the generality of the results, version to appear in Phys. Rev. Let

On the equivalence between Implicit Regularization and Constrained Differential Renormalization

Constrained Differential Renormalization (CDR) and the constrained version of Implicit Regularization (IR) are two regularization independent techniques that do not rely on dimensional continuation of the space-time. These two methods which have rather distinct basis have been successfully applied to several calculations which show that they can be trusted as practical, symmetry invariant frameworks (gauge and supersymmetry included) in perturbative computations even beyond one-loop order. In this paper, we show the equivalence between these two methods at one-loop order. We show that the configuration space rules of CDR can be mapped into the momentum space procedures of Implicit Regularization, the major principle behind this equivalence being the extension of the properties of regular distributions to the regularized ones.Comment: 16 page

Implicit Regularization and Renormalization of QCD

We apply the Implicit Regularization Technique (IR) in a non-abelian gauge theory. We show that IR preserves gauge symmetry as encoded in relations between the renormalizations constants required by the Slavnov-Taylor identities at the one loop level of QCD. Moreover, we show that the technique handles divergencies in massive and massless QFT on equal footing.Comment: (11 pages, 2 figures

Quantum equivalence of sigma models related by non Abelian Duality Transformations

Coupling constant renormalization is investigated in 2 dimensional sigma models related by non Abelian duality transformations. In this respect it is shown that in the one loop order of perturbation theory the duals of a one parameter family of models, interpolating between the SU(2) principal model and the O(3) sigma model, exhibit the same behaviour as the original models. For the O(3) model also the two loop equivalence is investigated, and is found to be broken just like in the already known example of the principal model.Comment: As a result of the collaboration of new authors the previously overlooked gauge contribution is inserted into eq.(43) changing not so much the formulae as part of the conclusion: for the models considered non Abelian duality is OK in one loo

O(d,d) invariance at two and three loops

We show that in a two-dimensional sigma-model whose fields only depend on one target space co-ordinate, the O(d,d) invariance of the conformal invariance conditions observed at one loop is preserved at two loops (in the general case with torsion) and at three loops (in the case without torsion).Comment: 21 pages. Plain Tex. Uses Harvmac ("b" option). Revised Version with references added and minor errors correcte

Comparing Implicit, Differential, Dimensional and BPHZ Renormalisation

We compare a momentum space implicit regularisation (IR) framework with other renormalisation methods which may be applied to dimension specific theories, namely Differential Renormalisation (DfR) and the BPHZ formalism. In particular, we define what is meant by minimal subtraction in IR in connection with DfR and dimensional renormalisation (DR) .We illustrate with the calculation of the gluon self energy a procedure by which a constrained version of IR automatically ensures gauge invariance at one loop level and handles infrared divergences in a straightforward fashion. Moreover, using the $\phi^4_4$ theory setting sun diagram as an example and comparing explicitly with the BPHZ framework, we show that IR directly displays the finite part of the amplitudes. We then construct a parametrization for the ambiguity in separating the infinite and finite parts whose parameter serves as renormalisation group scale for the Callan-Symanzik equation. Finally we argue that constrained IR, constrained DfR and dimensional reduction are equivalent within one loop order.Comment: 21 pages, 2 figures, late