Effective Lagrangians for Orientifold Theories

TL;DR

The authors construct Veneziano–Yankielowicz–type effective Lagrangians for orientifold theories, showing that at $N oty$ the bosonic sector matches the VY description of SYM and yields vanishing vacuum energy with scalar–pseudoscalar degeneracy. They then introduce finite-$N$ corrections via non-holomorphic modifications to saturate scale and chiral anomalies, predicting a negative vacuum energy and a gluon condensate, and they analyze the resulting spectrum, including a mass splitting between $M_\sigma$ and $M_{\eta'}$ and the impact of a nonzero fermion mass and $ heta$ angle. The antisymmetric representation at $N=3$ maps to one-flavor QCD, where the results imply a heavier scalar relative to the pseudoscalar, and a mass-term lifts vacuum degeneracy and introduces a $ heta$-dependent vacuum structure; the symmetric representation follows a parallel program with nuanced $1/N$ corrections. Overall, the work provides a coherent low-energy framework to study vacuum structure, condensates, and hadron spectra in orientifold theories, testing planar equivalence and yielding concrete predictions for QCD-like theories with a single Dirac fermion.

Abstract

We construct effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric theories which are orientifold daughters of supersymmetric gluodynamics (containing one Dirac fermion in the two-index antisymmetric or symmetric representation of the gauge group). Since the parent and daughter theories are planar equivalent, at N\to\infty the effective Lagrangians in the orientifold theories basically coincide with the bosonic part of the VY Lagrangian. We depart from the supersymmetric limit in two ways. First, we consider finite (albeit large) values of N. Then 1/N effects break supersymmetry. We suggest seemingly the simplest modification of the VY Lagrangian which incorporates these 1/N effects, leading to a non-vanishing vacuum energy density. We analyze the spectrum of the finite-N non-supersymmetric daughters. For N=3 the two-index antisymmetric representation (one flavor) is equivalent to one-flavor QCD. We show that in this case the scalar quark-antiquark state is heavier than the corresponding pseudoscalar state, `` eta' ''. Second, we add a small fermion mass term. The fermion mass term breaks supersymmetry explicitly. The vacuum degeneracy is lifted. The parity doublets split. We evaluate the splitting. Finally, we include the theta-angle and study its implications.