N=(0,2) Deformation of CP(1) Model: Two-dimensional Analog of N=1 Yang-Mills Theory in Four Dimensions

TL;DR

This paper derives all-order beta functions for two-dimensional (1) models with (0,2) supersymmetry, revealing parallels with four-dimensional super-Yang-Mills theories and predicting a conformal window at large fermion flavors.

Contribution

It provides the first all-orders beta function expressions for (0,2) CP(1) models, establishing their relation to four-dimensional supersymmetric gauge theories and identifying a conformal window.

Findings

Exact  functions similar to NSVZ form.

Existence of a conformal window at large N_f.

Reduction to =(2,2) model at N_f=1.

Abstract

We consider two-dimensional $\mathcal{N}=(0,2)$ sigma models with the CP(1) target space. A minimal model of this type has one left-handed fermion. Nonminimal extensions contain, in addition, $N_f$ right-handed fermions. Our task is to derive expressions for the $\beta$ functions valid to all orders. To this end we use a variety of methods: (i) perturbative analysis; (ii) instanton calculus; (iii) analysis of the supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, and some other arguments. All these arguments, combined, indicate a direct parallel between the heterotic $\mathcal{N}=(0,2)$ CP(1) models and four-dimensional super-Yang-Mills theories. In particular, the minimal $\mathcal{N}=(0,2)$ CP(1) model is similar to ${\mathcal N}=1$ supersymmetric gluodynamics. Its exact $\beta$ function can be found; it has the structure of the Novikov-Shifman-Vainshtein-Zakharov (NSVZ) $\beta$ function of supersymmetric gluodynamics. The passage to nonminimal $\mathcal{N}=(0,2)$ sigma models is equivalent to adding matter. In this case an NSVZ-type exact relation between the $\beta$ function and the anomalous dimensions $\gamma$ of the "matter" fields is established. We derive an analog of the Konishi anomaly. At large $N_f$ our $\beta$ function develops an infrared fixed point at small values of the coupling constant (analogous to the Banks-Zaks fixed point). Thus, we reliably predict the existence of a conformal window. At $N_f=1$ the model under consideration reduces to the well-known $\mathcal{N}=(2,2)$ CP(1) model.