[Paper Review] Lectures on perturbative unitarity and decoupling in Higgs physics

Heather E. Logan|arXiv (Cornell University)|Jul 3, 2022

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper provides a pedagogical introduction to perturbative unitarity in Higgs physics, deriving the Standard Model Higgs boson mass upper bound via longitudinal vector boson scattering and extending the analysis to extended Higgs sectors like the two-Higgs-doublet model, scalar septet, and Georgi-Machacek models. It establishes coupling sum rules and mass bounds, emphasizing the role of unitarity in constraining new physics and linking perturbative unitarity to the decoupling and alignment limits.

ABSTRACT

These lectures are a pedagogical introduction to the application of perturbative unitarity to Higgs physics within and beyond the Standard Model (SM). I begin with a review of how perturbative unitarity arises from quantum mechanical scattering theory and apply it to the classic problem of longitudinal vector boson scattering in the SM to derive the famous upper bound on the Higgs boson mass. I then consider extended Higgs sectors, using the two-Higgs-doublet model, the scalar septet model, and the Georgi-Machacek model as case studies. I discuss the resulting Higgs coupling sum rules as well as the intertwined constraints on masses and couplings of the additional Higgs bosons in these models, highlighting the connection between perturbative unitarity and the decoupling limit. I finish with a direct review of the decoupling limit in the two-Higgs-doublet model and a digression on the alignment limit. These notes are based on lectures delivered over the past few years in a graduate-level course on beyond-the-SM phenomenology and are meant as a companion volume to my TASI 2013 lectures on Higgs physics.

Motivation & Objective

• To provide a pedagogical foundation for understanding perturbative unitarity in the context of Higgs physics within and beyond the Standard Model.
• To derive the famous upper bound on the SM Higgs boson mass (≈712 GeV) using longitudinal vector boson scattering and unitarity constraints.
• To extend the unitarity analysis to extended Higgs sectors, including the two-Higgs-doublet model, scalar septet, and Georgi-Machacek models, deriving coupling sum rules and mass-coupling bounds.
• To clarify the connection between perturbative unitarity, the decoupling limit, and the alignment limit in multi-Higgs models.
• To highlight how precise Higgs coupling measurements can test models that lack a decoupling limit, enabling definitive exclusion or discovery of new physics.

Proposed method

• Applies partial wave unitarity from quantum mechanics to quantum field theory amplitudes, focusing on $W_L^+W_L^- \to W_L^+W_L^-$ scattering.
• Uses the Goldstone boson equivalence theorem to relate gauge boson scattering to scalar (Goldstone) scattering amplitudes in the high-energy limit.
• Performs coupled-channel analysis of $2\to2$ scattering amplitudes involving physical and unphysical scalar states to derive unitarity bounds.
• Derives coupling sum rules for Higgs-Vector-Vector interactions in models like the 2HDM, scalar septet, and Georgi-Machacek model.
• Applies the perturbative unitarity bound to derive upper limits on Higgs boson masses and couplings in extended models.
• Analyzes the decoupling and alignment limits in the two-Higgs-doublet model, showing how unitarity constrains the parameter space.

Experimental results

Research questions

• RQ1 What is the upper bound on the Standard Model Higgs boson mass derived from perturbative unitarity in longitudinal vector boson scattering?
• RQ2 How do coupling sum rules emerge from unitarity constraints in extended Higgs sectors such as the two-Higgs-doublet model and scalar septet model?
• RQ3 What are the implications of perturbative unitarity for the decoupling limit in multi-Higgs models, and how does it constrain the masses and couplings of additional Higgs states?
• RQ4 Can models without a decoupling limit be definitively ruled out by precision Higgs coupling measurements, and how does unitarity inform this?
• RQ5 How does the alignment limit in the two-Higgs-doublet model relate to perturbative unitarity and the decoupling mechanism?

Key findings

• The perturbative unitarity bound on the SM Higgs boson mass is $m_h^2 \leq 8\pi v^2/3 \approx (712~\text{GeV})^2$, derived from $W_L^+W_L^- \to W_L^+W_L^-$ scattering amplitudes.
• In the two-Higgs-doublet model, a coupling sum rule relates the Higgs-Vector-Vector couplings and constrains the masses and couplings of additional Higgs states.
• For the scalar septet model, the paper derives sum rules for couplings and upper bounds on masses and couplings of the exotic Higgs states, including doubly charged scalars.
• In the Georgi-Machacek model, the analysis yields similar sum rules and mass-coupling bounds, with constraints arising from unitarity in the $W_L^+W_L^- \to W_L^+W_L^-$ channel.
• Models without a decoupling limit—where the lightest Higgs cannot approach the SM Higgs in the heavy mass limit—can be excluded by precision Higgs coupling measurements.
• The alignment limit in the 2HDM, where the lightest Higgs couples like the SM Higgs, is shown to be distinct from the decoupling limit, and unitarity provides constraints on its viability.