Unitarity violation in noninteger dimensional Gross-Neveu-Yukawa model

Yao Ji,Michael Kelly

doi:10.1103/physrevd.97.105004

Yao Ji, Michael Kelly

Open Access

https://doi.org/10.1103/physrevd.97.105004

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Abstract

We construct an explicit example of unitarity violation in fermionic quantum field theories in non-integer dimensions. We study the two-point correlation function of four-fermion operators. We compute the one-loop anomalous dimensions of these operators in the Gross-Neveu-Yukawa Model. We find that at one loop order, the four-fermion operators split into three classes with one class having negative norms. This implies that the theory violates unitarity following the definition in arXiv:1512.00013 [hep-th].

Highlights

Conformal field theories (CFTs) have always been an area of active research due to their rich mathematical structure and physical applications
We have demonstrated the existence of negative norm states in the Gross-Neveu-Yukaw model in d 1⁄4 4 − 2ε dimensions through the study of the two-point correlation functions of four-fermion operators and their one-loop anomalous dimension matrix
The negative norm states we found are unavoidable, as the two-point correlation functions are an integral part of the theory

Summary

INTRODUCTION

Conformal field theories (CFTs) have always been an area of active research due to their rich mathematical structure and physical applications. [1,24], where unitarity violation in φ4 theory was demonstrated by constructing states (operators) with negative norm. The aim of this article is to demonstrate the existence of the negative norm states in the d 1⁄4 4 − 2ε dimensional Gross—Neveu—Yukawa (GNY) model [26] It was argued in [1] that unitarity implies the positiveness of the coefficient C in the correlator hO†ðxÞOð0Þi 1⁄4 C=x2Δ; ð1Þ where O is a conformal operator with scaling dimension Δ. In an integer dimensional CFT, violation of this condition indicates the presence of negative norm states in the theory [1]. Since the canonical dimension of these operators is not large, Δcan 1⁄4 6, one can wonder about the effect of negative norm operators to the conformal bootstrap technique. We show that the negative norm states are generated by one of these two classes, depending on the number of fermion flavors of the theory

FOUR-FERMION CORRELATION FUNCTION IN NONINTEGER DIMENSIONS

ANOMALOUS DIMENSIONS AND UNITARITY IN THE GNY MODEL

CONCLUSION