Abstract
Abstract We investigate the relation between $S$-matrix unitarity ($SS^\dagger=1$) and renormalizability in theories with negative-norm states. The relation has been confirmed in many field theories, including gauge theories and Einstein gravity, 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 because of the negative-norm states. In these theories it is not known whether the $S$-matrix unitarity provides a nontrivial constraint related to the renormalizability. In this paper, by relaxing the assumption of norm positivity we derive a bound on scattering amplitudes weaker than the unitarity bound, which may be used as a consistency requirement for $S$-matrix unitarity. We demonstrate in scalar field models with a higher-derivative kinetic term that the weaker bound and the renormalizability imply identical constraints.
Highlights
Unitarity and renormalizability are fundamental principles in quantum field theories (QFTs)
We can see the relation by the four-point vertex in the simple model that we introduce in Sect
We show the quantization for the fields ψ1, ψ2, and σ
Summary
Unitarity and renormalizability are fundamental principles in quantum field theories (QFTs). In the present paper we consider a scalar field model with a modified propagator as a toy model for quadratic gravity and demonstrate that S-matrix unitarity and renormalizability lead to identical constraints on this model. We introduce a scalar field model with a higher-derivative kinetic term as the simplest example of theories with negative-norm states. Due to the fourth-order derivative term in the kinetic term of the scalar field φ, its propagator is modified as ∼p−4 at UV and negative-norm (ghost) states appear. This model can be thought of as a toy model for quadratic gravity. Based on the extended PCR conditions, we introduce two marginal interaction terms
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