Abstract
Working with a gauge coupling field in a linear superfield, we construct effective Lagrangians for N=1 super-Yang–Mills theory fully compatible with the expected all-order behavior or physical quantities. Using the one-loop dependence on its ultraviolet cutoff and anomaly matching or cancellation of R and dilatation anomalies, we obtain the Wilsonian effective Lagrangian. With similar anomaly matching or cancellation methods, we derive the effective action for gaugino condensates, as a function of the real coupling field. Both effective actions lead to a derivation of the NSVZ β function from algebraic arguments only. The extension of results to N=2 theories or to matter systems is briefly considered. The main tool for the discussion of anomalies is a generic supercurrent structure with 16B+16F operators (the S multiplet), which we derive using superspace identities and field equations for a fully general gauge theory Lagrangian with the linear gauge coupling superfield, and with various U(1)R currents. As a byproduct, we show under which conditions the S multiplet can be improved to contain the Callan–Coleman–Jackiw energy-momentum tensor whose trace measures the breaking of scale invariance.
Highlights
The approach which identifies coupling constants with background values of fields and superfields has proved, following Seiberg [1], a useful and powerful tool in the study of perturbative and nonperturbative properties of supersymmetric gauge theories. It has been successful for N = 2 theories [2,3], using the factorization “theorem” of hypermultiplet and vector multiplet scalars and special Kähler geometry formulated in terms of a holomorphic prepotential
The situation changes if one introduces a field, the gauge coupling field, to describe the gauge coupling constant in an N = 1 supersymmetric gauge theory, in agreement with the fact that, as shown for instance in [5,6,7], the holomorphic dependence on the gauge coupling in N = 1 super Yang–Mills theory is anomalous
We will show that the anomalous dependence on the gauge coupling creates an obstruction to analytically perform the duality transformation, that it provides the adequate information to write all-order effective actions with the linear superfield and how the obstruction disappears with extended N = 2 supersymmetry, where holomorphicity is relevant
Summary
The approach which identifies coupling constants with background values of fields and superfields has proved, following Seiberg [1], a useful and powerful tool in the study of perturbative and nonperturbative properties of supersymmetric gauge theories. We establish two effective Lagrangians with the gauge coupling field in the linear superfield: the all-order perturbative Wilsonian Lagrangian for super-Yang–Mills theory and the effective action determining the gaugino condensate In both cases, anomaly matching or compensation is sufficient to derive the all-order renormalization-group (RG) equation and β function originally found by Novikov, Shifman, Vainshtein and Zakharov (NSVZ) [12]. Similar anomaly matching/cancellation arguments can be used to derive an effective Lagrangian describing gaugino condensates in N = 1 super-Yang–Mills theory, as a function of the real gauge coupling field C.3. In both cases, the all-order NSVZ β function is derived, using anomaly matching/cancellation only. In Appendix E, we collect some useful formulas for the Legendre transformation which appears in linear-chiral duality
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