NSVZ Beta Function Research Articles

Study of the renormalization-group evolution ofN=1supersymmetric gauge theories using Padé approximants

We study asymptotically free SU($N_c$) gauge theories with ${\cal N}=1$ supersymmetry, including the purely gluonic theory and theories with $N_f$ copies of a pair of massless chiral superfields in the respective representations $R$ and $\bar R$ of SU($N_c$). The cases in which $R$ is the fundamental representation and the symmetric and antisymmetric rank-2 tensor representation are considered. We calculate Pad\'e approximants to the beta functions for these theories in the $\overline{\rm DR}$ scheme up to four-loop order for the gluonic theory and up to three-loop order for the theories with matter superfields and compare results for IR zeros and poles with results from the NSVZ beta function. Our calculations provide a quantitative measure, for these theories, of how well finite-order perturbative results calculated in one scheme reproduce properties of a known beta function calculated in a different scheme.

N=(0,2)deformation of (2, 2) sigma models: Geometric structure, holomorphic anomaly, and exactβfunctions

We study N=(0,2) deformed (2,2) two-dimensional sigma models. Such heterotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional N=1 theories. We study geometric aspects and holomorphic properties of these models, and derive a number of exact expressions for the beta functions in terms of the anomalous dimensions analogous to the NSVZ beta function in four-dimensional Yang-Mills. Instanton calculus provides a straightforward method for the derivation. The anomalous dimensions are calculated up to two loops implying that one of the beta functions is explicitly known up to three loops. The fixed point in the ratio of the couplings found previously at one loop is not shifted at two loops. We also consider the N=(0,2) supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, as well as the "Konishi anomaly." This gives us another method for finding exact $\beta$ functions. We prove that despite the chiral nature of the models under consideration quantum loops preserve isometries of the target space.

BRANE TILINGS: THE NSVZ BETA FUNCTION

In honor of Misha Shifman's 60th birthday this note will start with one of Misha's many contributions, the famous NSVZ beta function, and study how it affects the construction of Brane Tilings. The main features of Brane Tilings for 3+1 dimensions SCFT's are reviewed and the generalization to 2+1 dimensional SCFT's is discussed.

Space-time dependent couplings in Script N = 1 SUSY gauge theories: anomalies and central functions

We consider N=1 supersymmetric gauge theories in which the couplings are allowed to be space-time dependent functions. Both the gauge and the superpotential couplings become chiral superfields. As has recently been shown, a new topological anomaly appears in models with space-time dependent gauge coupling. Here we show how this anomaly may be used to derive the NSVZ beta function in a particular, well-determined renormalisation scheme, both without and with chiral matter. Moreover we extend the topological anomaly analysis to theories coupled to a classical curved superspace background, and use it to derive an all-order expression for the central charge c, the coefficient of the Weyl tensor squared contribution to the conformal anomaly. We also comment on the implications of our results for the central charge a expected to be of relevance for a four-dimensional C-theorem.

Holomorphy, rescaling anomalies and exact β functions in supersymmetric gauge theories

There have been known "exact" beta functions for the gauge coupling in N=1 supersymmetric gauge theories, the so-called NSVZ beta functions. Shifman and Vainshtein (SV) further related these beta functions to the exact 1-loop running of the "Wilsonian" gauge coupling. All these results, however, remain somewhat mysterious. We attempt to clarify these issues by presenting new perspectives on the NSVZ beta function. Our interpretation of the results is somewhat different than the one given by SV, having nothing to do with the distinction between "Wilsonian" and "1PI" effective actions. Throughout we work in the context of the Wilsonian Renormalization Group; namely, as the cutoff of the theory is changed from M to M', we determine the appropriate changes in the bare couplings needed to keep the low energy physics fixed. The entire analysis is therefore free of infrared subtleties. When the bare Lagrangian given at the cutoff is manifestly holomorphic in the gauge coupling, we show that the required change in the holomorphic gauge coupling is exhausted at 1-loop to all orders of perturbation theory, and even non-perturbatively in some cases. On the other hand, when the bare Lagrangian has canonically normalized kinetic terms, we find that the required change in the gauge coupling is given by the NSVZ beta function. The higher order contributions in the NSVZ beta function are due to anomalous Jacobians under the rescaling of the fields done in passing from holomorphic to canonical normalization. We also give prescriptions for regularizing certain N=1 theories with an ultraviolet cutoff M preserving manifest holomorphy, starting from finite N=4 and N=2 theories. It is then in principle possible to check the validity of the exact beta function by higher order calculations in these theories.

On the beta function in supersymmetric gauge theories

We re-examine perturbative and nonperturbative aspects of the beta function in N=1 and N=2 supersymmetric gauge theories, make comments on the recent literature on the subject and discuss the exactness of several known results such as the NSVZ beta function.