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Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions

Matthijs Hogervorst, Slava Rychkov, Balt C. van Rees

TL;DR

The paper demonstrates that extending free and interacting scalar theories to non-integer dimensions introduces evanescent operators that generate negative-norm states, thereby violating unitarity. At the Wilson-Fisher fixed point in $d=4-\epsilon$, these evanescent sectors yield complex anomalous dimensions at leading order in $\epsilon$, even though normal operators retain real dimensions at this order. The authors develop a framework based on radial quantization Gram matrices and an OPE-driven one-loop anomalous-dimension analysis to reveal the persistence and impact of these non-unitary features, and they compare their findings with earlier work that neglected evanescent operators. The results imply that unitarity-violating effects propagate into high-dimension sectors and can influence higher-loop corrections and non-perturbative aspects, with implications for spectrum continuity and bootstrap approaches in non-integer dimensions. This work highlights fundamental limitations of analytic continuation in QFT and motivates refined treatments of non-unitary spectra in dimensional regularization contexts.

Abstract

We consider the continuation of free and interacting scalar field theory to non-integer spacetime dimension d. We find that the correlation functions in these theories are necessarily incompatible with unitarity (or with reflection positivity in Euclidean signature). In particular, the theories contain negative norm states unless d is a positive integer. These negative norm states can be obtained via the OPE from simple positive norm operators, and are therefore an integral part of the theory. At the Wilson-Fisher fixed point the non-unitarity leads to the existence of complex anomalous dimensions. We demonstrate that they appear already at leading order in the epsilon expansion.

Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions

TL;DR

The paper demonstrates that extending free and interacting scalar theories to non-integer dimensions introduces evanescent operators that generate negative-norm states, thereby violating unitarity. At the Wilson-Fisher fixed point in , these evanescent sectors yield complex anomalous dimensions at leading order in , even though normal operators retain real dimensions at this order. The authors develop a framework based on radial quantization Gram matrices and an OPE-driven one-loop anomalous-dimension analysis to reveal the persistence and impact of these non-unitary features, and they compare their findings with earlier work that neglected evanescent operators. The results imply that unitarity-violating effects propagate into high-dimension sectors and can influence higher-loop corrections and non-perturbative aspects, with implications for spectrum continuity and bootstrap approaches in non-integer dimensions. This work highlights fundamental limitations of analytic continuation in QFT and motivates refined treatments of non-unitary spectra in dimensional regularization contexts.

Abstract

We consider the continuation of free and interacting scalar field theory to non-integer spacetime dimension d. We find that the correlation functions in these theories are necessarily incompatible with unitarity (or with reflection positivity in Euclidean signature). In particular, the theories contain negative norm states unless d is a positive integer. These negative norm states can be obtained via the OPE from simple positive norm operators, and are therefore an integral part of the theory. At the Wilson-Fisher fixed point the non-unitarity leads to the existence of complex anomalous dimensions. We demonstrate that they appear already at leading order in the epsilon expansion.

Paper Structure

This paper contains 22 sections, 70 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The free boson in non-integer $d$
  3. Evanescent operators
  4. Negative-norm states
  5. Primary decomposition and negative norms
  6. The radial quantization Gram matrix
  7. General tensorial operators
  8. Review of the Wilson-Fisher theory at one-loop
  9. Anomalous dimension generalities
  10. $\Gamma$ at one-loop from the OPE
  11. Examples
  12. Non-unitarity at the Wilson-Fisher fixed point
  13. Robustness of negative-norm states
  14. Complex anomalous dimensions
  15. Comparison to the work of Kehrein et al.
  16. ...and 7 more sections

Figures (2)

  • Figure 1: The schematic behavior of $c(R_n)$, $n\geqslant 4$, as a function of $d$ in the range $d\geqslant2$. The norm vanishes for $d=2,\ldots,n-1$, is negative in the interval $n-2<d<n-1$, and is positive otherwise.
  • Figure 2: Expected interpolating behavior for the low-dimension scalar operators. The exact 2d dimensions and the approximate 3d dimensions are given in parentheses. The operator $\phi^3$ is a primary in 4d but is the descendant $\partial^2\sigma$ below 4d. Other descendant states are not shown.