S-matrix Unitarity and Renormalizability in Higher Derivative Theories

TL;DR

This paper explores the connection between $S$-matrix unitarity and renormalizability in higher derivative theories with negative norm states, demonstrating that unitarity can impose meaningful constraints even when the unitarity bound fails.

Contribution

The study introduces scalar field models with higher derivative kinetic terms to analyze the $S$-matrix unitarity and its relation to renormalizability, providing positive results.

Findings

$S$-matrix unitarity can constrain renormalizability in higher derivative theories.

The unitarity relation holds in scalar models with higher derivatives.

Negative norm states do not necessarily invalidate the unitarity-renormalizability connection.

Abstract

We investigate the relation between the $S$-matrix unitarity ($SS^{\dagger}=1$) and the renormalizability, in theories with negative norm states. The relation has been confirmed in many theories, such as gauge theories, Einstein gravity and Lifshitz-type non-relativistic theories by analyzing the unitarity bound, which follows from the $S$-matrix unitarity and the norm positivity. On the other hand, renormalizable theories with a higher derivative kinetic term do not necessarily satisfy the unitarity bound essentially because the unitarity bound does not hold due to the negative norm states. In these theories, it is not clear if the $S$-matrix unitarity provides a nontrivial constraint related to the renormalizability. In this paper we introduce scalar field models with a higher derivative kinetic term and analyze the $S$-matrix unitarity. We have positive results of the relation.