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Gauge Theories on an Interval: Unitarity without a Higgs

C. Csaki, C. Grojean, H. Murayama, L. Pilo, J. Terning

TL;DR

This work analyzes whether unitarity in gauge theories can be maintained on an interval without a Higgs, by examining the consequences of general boundary conditions that break gauge symmetry. It shows that the leading energy-growing pieces in longitudinal gauge-boson scattering cancel for consistent BCs, with cancellations driven by KK exchange rather than scalar exchange, and that mixed BCs generally require boundary scalars (e.g., a Higgs) to restore full unitarity. The authors provide explicit 5D toy models illustrating Higgsless unitarity and discuss a realistic-looking, Higgs-free electroweak-like model, highlighting both promise and challenges such as the EW-precision constraints and fermion masses. The results point toward a viable UV completion in higher dimensions where gauge symmetry breaking is achieved via BCs, though substantial work remains to realize a fully realistic and experimentally consistent theory. Overall, the paper advances the case that Higgsless unitarity can arise from extra-dimensional BCs and KK gauge towers, offering a new paradigm for electroweak symmetry breaking with rich phenomenology and clear UV-structure questions.

Abstract

We consider extra dimensional gauge theories on an interval. We first review the derivation of the consistent boundary conditions (BC's) from the action principle. These BC's include choices that give rise to breaking of the gauge symmetries. The boundary conditions could be chosen to coincide with those commonly applied in orbifold theories, but there are many more possibilities. To investigate the nature of gauge symmetry breaking via BC's we calculate the elastic scattering amplitudes for longitudinal gauge bosons. We find that using a consistent set of BC's the terms in these amplitudes that explicitly grow with energy always cancel without having to introduce any additional scalar degree of freedom, but rather by the exchange of Kaluza--Klein (KK) gauge bosons. This suggests that perhaps the SM Higgs could be completely eliminated in favor of some KK towers of gauge fields. We show that from the low-energy effective theory perspective this seems to be indeed possible. We display an extra dimensional toy model, where BC's introduce a symmetry breaking pattern and mass spectrum that resembles that in the standard model.

Gauge Theories on an Interval: Unitarity without a Higgs

TL;DR

This work analyzes whether unitarity in gauge theories can be maintained on an interval without a Higgs, by examining the consequences of general boundary conditions that break gauge symmetry. It shows that the leading energy-growing pieces in longitudinal gauge-boson scattering cancel for consistent BCs, with cancellations driven by KK exchange rather than scalar exchange, and that mixed BCs generally require boundary scalars (e.g., a Higgs) to restore full unitarity. The authors provide explicit 5D toy models illustrating Higgsless unitarity and discuss a realistic-looking, Higgs-free electroweak-like model, highlighting both promise and challenges such as the EW-precision constraints and fermion masses. The results point toward a viable UV completion in higher dimensions where gauge symmetry breaking is achieved via BCs, though substantial work remains to realize a fully realistic and experimentally consistent theory. Overall, the paper advances the case that Higgsless unitarity can arise from extra-dimensional BCs and KK gauge towers, offering a new paradigm for electroweak symmetry breaking with rich phenomenology and clear UV-structure questions.

Abstract

We consider extra dimensional gauge theories on an interval. We first review the derivation of the consistent boundary conditions (BC's) from the action principle. These BC's include choices that give rise to breaking of the gauge symmetries. The boundary conditions could be chosen to coincide with those commonly applied in orbifold theories, but there are many more possibilities. To investigate the nature of gauge symmetry breaking via BC's we calculate the elastic scattering amplitudes for longitudinal gauge bosons. We find that using a consistent set of BC's the terms in these amplitudes that explicitly grow with energy always cancel without having to introduce any additional scalar degree of freedom, but rather by the exchange of Kaluza--Klein (KK) gauge bosons. This suggests that perhaps the SM Higgs could be completely eliminated in favor of some KK towers of gauge fields. We show that from the low-energy effective theory perspective this seems to be indeed possible. We display an extra dimensional toy model, where BC's introduce a symmetry breaking pattern and mass spectrum that resembles that in the standard model.

Paper Structure

This paper contains 19 sections, 102 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gauge Theories on an Interval: Consistent BC's
  3. Bulk scalar
  4. Pure gauge theory in the bulk
  5. Unitarity of the Elastic Scattering Amplitudes for Longitudinal Gauge Bosons
  6. The elastic scattering amplitude
  7. Cancellation of the terms that grow with energy for the simplest BC's
  8. Sum rule with boundary terms
  9. A 4D Effective Theory Without a Higgs
  10. Examples of Unitarity in Higher Dimensional Theories
  11. SU(2)$\to$ U(1) by BC
  12. SU(2)$\to$ U(1) by BC
  13. Completely broken SU(2) by mixed BC's and the need for a boundary Higgs field
  14. A Toy Model for Electroweak Symmetry Breaking via BC's
  15. Conclusions
  16. ...and 4 more sections

Figures (3)

  • Figure 1: Elastic scattering of longitudinal modes of KK gauge bosons, $n+n\to n+n$, with the gauge index structure $a+b\to c+d$.
  • Figure 2: The four gauge diagrams contributing at tree level to the gauge boson elastic scattering amplitude.
  • Figure 3: Higgs exchange contributing at tree level to the gauge boson elastic scattering amplitude. The Higgs boson is localized at $y=\pi R$ and the vertex Higgs-Gauge-Gauge is proportional to $f_n^2(\pi R)$.