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$η_{\rm w}$-meson from topological properties of the electroweak vacuum

Gia Dvali, Archil Kobakhidze, Otari Sakhelashvili

TL;DR

The paper argues that the electroweak sector naturally contains a pseudo-scalar $η_{\rm w}$ whose mass arises from the topological susceptibility of the EW vacuum, by linking the elimination of $θ_{\rm w}$ through the anomalous $B+L$ symmetry to a condensate of the fermion determinant. It develops a consistent picture where $η_{\rm w}$ emerges as the phase of the $B+L$-violating fermion condensate, demonstrated via explicit instanton calculations and a multi-$W$ process interpretation that preserves EFT control. A key result is the nonzero fermion condensate ⟨$(qqql)^{{N_f}}$⟩ and its connection to the $t$ Hooft vertex, with UV-insensitivity explained through re-summation and microstate bounds; the mass and decay constant of $η_{\rm w}$ can be related to the topological scale via a Witten– Veneziano-like relation $m_{η}^2 f_{η}^2 \sim Λ^4$. These findings suggest $η_{\rm w}$ is a dynamical consequence of SM topology (or a weakly coupled new degree of freedom) with potential cosmological and phenomenological implications, while leaving open questions about the exact EFT validity range and the possible necessity of gravity for a complete description.

Abstract

We further scrutinize the evidence for a recently suggested pseudo-scalar particle, the electroweak $η_{\rm w}$-meson. Its existence is demanded by matching the removal of the weak vacuum angle $θ_{\rm w}$ by the anomalous $B+L$ - symmetry with a massive pole in the topological susceptibility of the vacuum. We specifically focus on the possibility of the emergence of $η_{\rm w}$ as a collective excitation of the phase of the condensate of the 't Hooft fermion determinant, generated by the electroweak instantons, which breaks the $B+L$ - symmetry spontaneously. We argue that the generation of the 't Hooft vertex is in one-to-one correspondence with its non-zero vacuum expectation value which is cutoff insensitive. We outline certain puzzles about the nature of the emergent $η_{\rm w}$ which require further investigations.

$η_{\rm w}$-meson from topological properties of the electroweak vacuum

TL;DR

The paper argues that the electroweak sector naturally contains a pseudo-scalar whose mass arises from the topological susceptibility of the EW vacuum, by linking the elimination of through the anomalous symmetry to a condensate of the fermion determinant. It develops a consistent picture where emerges as the phase of the -violating fermion condensate, demonstrated via explicit instanton calculations and a multi- process interpretation that preserves EFT control. A key result is the nonzero fermion condensate ⟨⟩ and its connection to the Hooft vertex, with UV-insensitivity explained through re-summation and microstate bounds; the mass and decay constant of can be related to the topological scale via a Witten– Veneziano-like relation . These findings suggest is a dynamical consequence of SM topology (or a weakly coupled new degree of freedom) with potential cosmological and phenomenological implications, while leaving open questions about the exact EFT validity range and the possible necessity of gravity for a complete description.

Abstract

We further scrutinize the evidence for a recently suggested pseudo-scalar particle, the electroweak -meson. Its existence is demanded by matching the removal of the weak vacuum angle by the anomalous - symmetry with a massive pole in the topological susceptibility of the vacuum. We specifically focus on the possibility of the emergence of as a collective excitation of the phase of the condensate of the 't Hooft fermion determinant, generated by the electroweak instantons, which breaks the - symmetry spontaneously. We argue that the generation of the 't Hooft vertex is in one-to-one correspondence with its non-zero vacuum expectation value which is cutoff insensitive. We outline certain puzzles about the nature of the emergent which require further investigations.

Paper Structure

This paper contains 11 sections, 37 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Evidence from TSV and anomalous $B+L$-symmetry
  3. TSV in theory without fermions
  4. $B+L$ - symmetry
  5. Emergence of $\eta_{\rm w}$
  6. Fermion Condensate
  7. Fermion condensate from instantons
  8. Resolving instantons as multi-$W$ processes
  9. Where is $\eta_{\rm w}$?
  10. Parameters from first principles
  11. Outlook

Figures (4)

  • Figure 1: Graphical representation of the t'Hooft vertex.
  • Figure 2: Graphical representation of the Condesate.
  • Figure 3: Representation of the instanton part of the condensate via $W$-bosons.
  • Figure 4: An example of a multi-particle process, with many virtual $W$-bosons and intermediate fermion loops. Even for an under-saturated value of the 't Hooft coupling, $\alpha N \ll 1$, the multiplicity of diagrams grows factorially with the number of participating quanta $n$. However, this enhancement signals the breakdown of perturbation theory and cannot be trusted beyond the point of optimal truncation $n \sim 1/\alpha$.