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

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

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

Supergravity corrections to (g−2)l(g-2)_l in differential renormalization

The method of differential renormalization is extended to the calculation of the one-loop graviton and gravitino corrections to $(g-2)_l$ in unbroken supergravity. Rewriting the singular contributions of all the diagrams in terms of only one singular function, U(1) gauge invariance and supersymmetry are preserved. We compare this calculation with previous ones which made use of momentum space regularization (renormalization) methods.Comment: 23 pages, LaTeX, 4 PostScript figures embedded with epsf ver. 1.13. Complete ps paper and figures available also at ftp://ftae3.ugr.es/pub/rmt/ugft71.p

The beta function of N=1 SYM in Differential Renormalization

Using differential renormalization, we calculate the complete two-point function of the background gauge superfield in pure N=1 Supersymmetric Yang-Mills theory to two loops. Ultraviolet and (off-shell) infrared divergences are renormalized in position and momentum space respectively. This allows us to reobtain the beta function from the dependence on the ultraviolet renormalization scale in an infrared-safe way. The two-loop coefficient of the beta function is generated by the one-loop ultraviolet renormalization of the quantum gauge field via nonlocal terms which are infrared divergent on shell. We also discuss the connection of the beta function to the flow of the Wilsonian coupling.Comment: 20 pages, 2 figures. Reference added, minor correction

Techniques for one-loop calculations in constrained differential renormalization

We describe in detail the constrained procedure of differential renormalization and develop the techniques required for one-loop calculations. As an illustration we renormalize Scalar QED and show that the two-, three- and four-point Ward identities are automatically satisfied.Comment: LaTex, 29 pages with 4 Postscript figure

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

Differential Equations for Definition and Evaluation of Feynman Integrals

It is shown that every Feynman integral can be interpreted as Green function of some linear differential operator with constant coefficients. This definition is equivalent to usual one but needs no regularization and application of $R$-operation. It is argued that presented formalism is convenient for practical calculations of Feynman integrals.Comment: pages, LaTEX, MSU-PHYS-HEP-Lu2/9