Compositeness, Triviality and Bounds on Critical Exponents for Fermions and Magnets

TL;DR

This paper explores universal features and bounds on critical exponents in fermionic theories with chiral symmetry breaking, highlighting differences from scalar theories and implications for lattice simulations and triviality of certain models.

Contribution

It identifies universal features and bounds on critical indices in fermionic theories, contrasting them with scalar theories, and discusses implications for lattice simulations and triviality.

Findings

Universal bounds on critical indices $\delta$ and $\eta$ for fermionic theories.

Logarithmic scaling violations differ between scalar and fermionic models.

$\lambda\phi^4$ theory cannot reliably indicate triviality of spinor QED.

Abstract

We argue that theories with fundamental fermions which undergo chiral symmetry breaking have several universal features which are qualitatively different than those of theories with fundamental scalars. Several bounds on the critical indices $\delta$ and $\eta$ follow. We observe that in four dimensions the logarithmic scaling violations enter into the Equation of State of scalar theories, such as $\lambda\phi^4$, and fermionic models, such as Nambu-Jona-Lasinio, in qualitatively different ways. These observations lead to useful approaches for analyzing lattice simulations of a wide class of model field theories. Our results imply that $\lambda\phi^4$ {\it cannot} be a good guide to understanding the possible triviality of spinor $QED$.