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Non-Hermitian Structure and Exceptional Points in Yang-Mills Theory from Analytic Continuation of Nc

Qingjun Jin, Ke Ren, Gang Yang, Rui Yu

Abstract

We show that analytic continuation of the number of colors, Nc, naturally endows Yang-Mills theory with a non-Hermitian structure. By examining the spectrum of the dilatation operator as a function of complex Nc, we identify a network of Exceptional Points (EPs) -- non-Hermitian degeneracies where anomalous dimensions degenerate and operator eigenstates coalesce. We demonstrate that these EPs act as topological defects in complex Nc-space, generating non-Abelian geometric phases and enforcing nontrivial monodromies among gauge-invariant operators. Moreover, we establish a correspondence between the spontaneous breaking of an emergent PT symmetry of the dilatation operator and the fundamental spacetime PT symmetry of the underlying gauge theory. In the vicinity of EPs, the resulting non-Hermitian dynamics produces logarithmic scaling behavior in correlation functions, characteristic of logarithmic conformal field theories. Our results place conventional unitary Yang-Mills theory within a broader complexified parameter space possessing rich topological structure, suggesting a new interface between non-Hermitian physics and quantum field theory.

Non-Hermitian Structure and Exceptional Points in Yang-Mills Theory from Analytic Continuation of Nc

Abstract

We show that analytic continuation of the number of colors, Nc, naturally endows Yang-Mills theory with a non-Hermitian structure. By examining the spectrum of the dilatation operator as a function of complex Nc, we identify a network of Exceptional Points (EPs) -- non-Hermitian degeneracies where anomalous dimensions degenerate and operator eigenstates coalesce. We demonstrate that these EPs act as topological defects in complex Nc-space, generating non-Abelian geometric phases and enforcing nontrivial monodromies among gauge-invariant operators. Moreover, we establish a correspondence between the spontaneous breaking of an emergent PT symmetry of the dilatation operator and the fundamental spacetime PT symmetry of the underlying gauge theory. In the vicinity of EPs, the resulting non-Hermitian dynamics produces logarithmic scaling behavior in correlation functions, characteristic of logarithmic conformal field theories. Our results place conventional unitary Yang-Mills theory within a broader complexified parameter space possessing rich topological structure, suggesting a new interface between non-Hermitian physics and quantum field theory.
Paper Structure (27 sections, 110 equations, 10 figures, 2 tables)

This paper contains 27 sections, 110 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Color-evanescent operators
  3. Operator setup
  4. Color-evanescent operators
  5. Computational Framework: Gram and dilatation matrices
  6. Gram matrices
  7. Dilatation matrices
  8. One-loop dilatation matrices
  9. Two-loop dilatation matrix
  10. Non-hermiticity and exceptional points
  11. Negative-norm states
  12. Exceptional points in spectrum
  13. Interplay of color- and dimension-evanescent operators
  14. Physical effects of exceptional points
  15. EP and PT-symmetry
  16. ...and 12 more sections

Figures (10)

  • Figure 1: Operator spectrum and exceptional point for the YM operators of dimension-8 length-4 sector.
  • Figure 2: The one-loop cut.
  • Figure 3: Two-loop master integrals. Thick blue legs represent the operator legs carrying off-shell momentum $q$.
  • Figure 4: Cuts for computing two-loop form factors.
  • Figure 5: The one-loop spectrum of dim-8 length-4 sector. The green line denotes a complex conjugate pair. (a) The $(-)^4$ sector. (b) The $(-)^2(+)^2$ sector.
  • ...and 5 more figures