[Paper Review] Unitarity bound in the most general two Higgs doublet model

Shinya Kanemura, Kei Yagyu|arXiv (Cornell University)|Sep 20, 2015

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper derives analytic unitarity bounds in the most general two Higgs doublet model (2HDM) without $Z_2$ symmetry or CP conservation by computing S-wave scattering amplitudes of Nambu-Goldstone and physical Higgs bosons at high energies. By classifying scattering states via conserved quantum numbers (hypercharge, weak isospin, third isospin component), the scattering matrix is block-diagonalized into at most $4\times4$ submatrices, enabling precise eigenvalue evaluation. The key result is a model-independent constraint on the mass of the second-lightest Higgs boson, $M_{\text{2nd}}$, which can be bounded if deviations in the $hVV$ coupling from the SM are detected in future experiments.

ABSTRACT

We investigate unitarity bounds in the most general two Higgs doublet model without a discrete $Z_2$ symmetry nor CP conservation. S-wave amplitudes for two-body elastic scatterings of Nambu-Goldstone bosons and physical Higgs bosons are calculated at high energies for all possible initial and final states (14 neutral, 8 singly-charged and 3 doubly-charged states). We obtain analytic formulae for the block-diagonalized scattering matrix by the classification of the two body scattering states using the conserved quantum numbers at high energies. Imposing the condition of perturbative unitarity to the eigenvalues of the scattering matrix, constraints on the model parameters can be obtained. We apply our results to constrain the mass range of the next--to--lightest Higgs state in the model.

Motivation & Objective

• To derive model-independent unitarity constraints in the most general two Higgs doublet model (2HDM) without $Z_2$ symmetry or CP conservation.
• To compute high-energy S-wave scattering amplitudes for all 25 possible two-body initial and final states (14 neutral, 8 singly-charged, 3 doubly-charged) involving Nambu-Goldstone and physical Higgs bosons.
• To classify scattering states using conserved quantum numbers (hypercharge, weak isospin, $I_3$) to block-diagonalize the scattering matrix for analytical and numerical evaluation.
• To derive constraints on model parameters—particularly the mass of the second-lightest Higgs boson—by requiring perturbative unitarity of the scattering matrix eigenvalues.
• To demonstrate that a deviation in the $hVV$ coupling from the SM value could constrain $M_{\text{2nd}}$ even without direct discovery of the second Higgs state.

Proposed method

• Calculating S-wave scattering amplitudes for all 25 two-body scattering channels of Nambu-Goldstone and physical Higgs bosons at high energies in the most general 2HDM.
• Using conserved quantum numbers—hypercharge, weak isospin, and third component of weak isospin—to classify scattering states and achieve block-diagonalization of the scattering matrix.
• Expressing the scattering matrix in terms of $4\times4$ submatrices (or smaller) to simplify eigenvalue computation and ensure perturbative unitarity.
• Applying the unitarity condition that eigenvalues of the scattering matrix must not exceed $1/2$ to derive bounds on model parameters such as Higgs masses and mixing angles.
• Using the analytic form of the block-diagonalized scattering matrix to validate consistency with previous results (e.g., Ginzburg, 2013).
• Performing numerical analysis to constrain $M_{\text{2nd}}$ under the assumption of a non-zero deviation in the $hVV$ coupling from the SM value, with $m_h = 125$ GeV.

Experimental results

Research questions

• RQ1 What are the unitarity constraints on the most general 2HDM without $Z_2$ symmetry or CP conservation?
• RQ2 How can the scattering matrix for all 25 two-body Higgs and Nambu-Goldstone states be block-diagonalized using conserved quantum numbers?
• RQ3 What is the analytic form of the S-wave scattering matrix in this general 2HDM framework?
• RQ4 How can deviations in the $hVV$ coupling from the SM value be used to constrain the mass of the second-lightest Higgs boson?
• RQ5 What is the upper bound on the mass $M_{\text{2nd}}$ of the second-lightest Higgs boson if the $hVV$ coupling deviates from the SM prediction?

Key findings

• The scattering matrix is block-diagonalized into at most $4\times4$ submatrices by classifying states using conserved quantum numbers, enabling efficient eigenvalue evaluation.
• The analytic form of the block-diagonalized S-wave scattering matrix is consistent with previous results, such as those by Ginzburg (2013), validating the approach.
• Perturbative unitarity imposes constraints on model parameters, including Higgs masses and mixing angles, by requiring eigenvalues of the scattering matrix to remain below $1/2$.
• If a deviation in the $hVV$ coupling from the SM value is observed in future experiments, an upper bound on $M_{\text{2nd}}$ can be derived even without direct discovery of the second Higgs state.
• The upper bound on $M_{\text{2nd}}$ is sensitive to the magnitude of the $hVV$ coupling deviation, with tighter constraints expected from higher-precision measurements at future colliders.
• The method provides a general framework to constrain parameter spaces in the most general 2HDM, applicable to any analysis involving Higgs couplings or decay rates.