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On the mass of the world-sheet `axion' in SU(N) gauge theories in 3+1 dimensions

Andreas Athenodorou, Michael Teper

TL;DR

The paper investigates whether the world-sheet axion mass on confining flux tubes in D=3+1 SU(N) gauge theories vanishes in the planar limit, which would allow integrability. It uses lattice gauge theory to compute the axion mass across SU(2)–SU(12) by comparing the lowest 0++ and 0-- flux-tube energies, extracting M_A from their gap. The results show a finite, nonzero M_A at N→∞ with M_A/√σ ≈ 1.713(14) + 2.74(7)/N^2, indicating the axion does not provide the required massless mode for planar integrability. The work also carefully assesses lattice-spacing and finite-volume effects and discusses topological ergodicity, concluding that the findings robustly exclude the axion as the sole route to planar integrability and pointing to the need to explore other world-sheet excitations.

Abstract

There is numerical evidence that the world sheet action of the confining flux tube in D=3+1 SU(N) gauge theories contains a massive excitation with 0- quantum numbers whose mass shows some decrease as one goes from SU(3) to SU(5). It has furthermore been shown that this particle is naturally described as arising from a topological interaction term in the world-sheet action, so that one can describe it as being `axion'-like. Recently it has been pointed out that if the mass of this `axion' vanishes as N -> oo then it becomes possible for the world sheet theory to be integrable in the planar limit. In this paper we perform lattice calculations of this `axion' mass from SU(2) to SU(12), which allows us to make a controlled extrapolation to N=oo and so test this interesting possibility. We find that the `axion' does not in fact become massless as N -> oo. So if the theory is to possess planar integrability then it must be some other world sheet excitation that becomes massless in the planar limit.

On the mass of the world-sheet `axion' in SU(N) gauge theories in 3+1 dimensions

TL;DR

The paper investigates whether the world-sheet axion mass on confining flux tubes in D=3+1 SU(N) gauge theories vanishes in the planar limit, which would allow integrability. It uses lattice gauge theory to compute the axion mass across SU(2)–SU(12) by comparing the lowest 0++ and 0-- flux-tube energies, extracting M_A from their gap. The results show a finite, nonzero M_A at N→∞ with M_A/√σ ≈ 1.713(14) + 2.74(7)/N^2, indicating the axion does not provide the required massless mode for planar integrability. The work also carefully assesses lattice-spacing and finite-volume effects and discusses topological ergodicity, concluding that the findings robustly exclude the axion as the sole route to planar integrability and pointing to the need to explore other world-sheet excitations.

Abstract

There is numerical evidence that the world sheet action of the confining flux tube in D=3+1 SU(N) gauge theories contains a massive excitation with 0- quantum numbers whose mass shows some decrease as one goes from SU(3) to SU(5). It has furthermore been shown that this particle is naturally described as arising from a topological interaction term in the world-sheet action, so that one can describe it as being `axion'-like. Recently it has been pointed out that if the mass of this `axion' vanishes as N -> oo then it becomes possible for the world sheet theory to be integrable in the planar limit. In this paper we perform lattice calculations of this `axion' mass from SU(2) to SU(12), which allows us to make a controlled extrapolation to N=oo and so test this interesting possibility. We find that the `axion' does not in fact become massless as N -> oo. So if the theory is to possess planar integrability then it must be some other world sheet excitation that becomes massless in the planar limit.

Paper Structure

This paper contains 7 sections, 4 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Lattice calculation
  3. lattice setup
  4. calculating flux tube energies
  5. Results
  6. Topology and non-ergodicity
  7. Conclusions

Figures (3)

  • Figure 1: Energies of the low-lying flux tube states with the quantum numbers shown, versus the length of the flux tube. Results for $SU(3)$ at $\beta=6.0635$ taken from AABBMT_d4, compared to our new results for the $J^{P_{\shortparallel}P_{\perp}} = 0^{--}$ flux tube at $\beta=5.825$ ($\circ$).
  • Figure 2: The volume dependence of the energies of the $J^{P_{\shortparallel}P_{\perp}} = 0^{++}$ ($\times$, blue) and $0^{--}$ ($+$, red) flux tube ground states. The bands correspond to the energy levels obtained with the largest volume i.e. $8 \times 24 \times 24 \times 32$.
  • Figure 3: The 'axion' mass, using eqn(\ref{['eqn_MA']}), in units of the string tension, versus $1/N^2$ together with a linear extrapolation in $1/N^2$ to $N=\infty$.