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Next-to-Leading Order Unitarity Fits in the Extended Georgi-Machacek Model

This paper establishes next-to-leading order unitarity bounds and bounded-from-below conditions for the Georgi-Machacek and extended Georgi-Machacek models, using these improved theoretical constraints to perform global fits with LHC Higgs data that disfavor specific coupling regions and set upper limits on scalar quartic couplings and heavy Higgs mass differences.

Original authors: Debtosh Chowdhury, Poulami Mondal, Subrata Samanta

Published 2026-09-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Debtosh Chowdhury, Poulami Mondal, Subrata Samanta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is held together by a set of invisible rules that dictate how particles interact and how forces behave. At the heart of this system is a field that fills all of space, giving mass to the fundamental particles that make up matter. We know this field exists because we found the particle associated with it, the Higgs boson, over a decade ago. This discovery confirmed the standard model of particle physics, our best current map of the subatomic world. However, this map is not complete. Scientists have long suspected that the Higgs field might be more complex than the simple version we have observed so far. It is possible that there are other, heavier particles connected to this field, waiting to be discovered. These hidden particles would change how the known Higgs boson behaves, perhaps making it interact with other particles slightly differently than our current theories predict.

A team of researchers in India has taken a fresh look at two specific ideas about what these hidden particles might be. They focused on models that add extra groups of particles, called triplets, to the existing theory. One model, known as the Georgi-Machacek model, has been studied for decades, while a newer variation, called the extended Georgi-Machacek model, allows for even more flexibility in how these particles interact. The researchers wanted to know if these models could still be true given the latest data from the Large Hadron Collider, the massive machine that smashes protons together to create new particles. To do this, they had to solve a difficult problem: ensuring that the mathematical descriptions of these models do not break down when pushed to their limits. They calculated how these particles would scatter off one another at extremely high energies, a process that acts as a stress test for the theory. If the math predicts impossible outcomes, like probabilities greater than one hundred percent, the model is invalid.

The team performed these calculations with a level of precision that had not been achieved before for these specific models. Instead of just looking at the simplest version of the interactions, they included corrections that account for the complex, temporary fluctuations that happen in the quantum world. This allowed them to set much stricter limits on the strength of the forces between these hypothetical particles. They also checked that the energy of the system would not run away to infinity, a condition required for the universe to remain stable. By combining these rigorous theoretical checks with the latest measurements of how the known Higgs boson behaves, they were able to map out exactly which versions of these models are still possible.

Their findings paint a clear picture of what is allowed and what is not. They discovered that large parts of the parameter space that were previously thought to be safe are actually ruled out by these new, more precise calculations. The models are forced into a much narrower range of possibilities. For instance, the strength of the interactions between these new particles cannot be too strong; if they were, the theory would collapse. The researchers found that the strength of these interactions must stay below a specific threshold, roughly two to three times the strength of the standard interactions, depending on the model. They also determined that the masses of the new, heavy particles cannot be too far apart from one another. In the older model, the difference in mass between these heavy particles must be less than 410 gigaelectronvolts, a unit of energy used to measure particle mass. In the newer, more flexible model, this difference can be slightly larger, up to 520 gigaelectronvolts, but still tightly constrained.

The study also looked at how the known Higgs boson talks to other particles. In these extended models, the Higgs boson interacts with force-carrying particles and matter particles in ways that are slightly different from the standard prediction. The researchers found that the data from the collider strongly disfavors scenarios where these interactions are too different from the standard model. Specifically, the way the Higgs boson couples to force-carrying particles cannot be more than five percent stronger or weaker than expected, and its interaction with matter particles cannot be more than eight percent weaker. This means the new particles, if they exist, must be arranged in a way that keeps the Higgs boson's behavior very close to what we have already observed.

One of the most significant outcomes of this work is the validation of a simpler way to check for stability. Checking the stability of these complex models usually requires testing every possible combination of fields, a task that is computationally overwhelming. The researchers showed that testing just a few specific combinations of three fields is enough to give an accurate picture of the whole system. This shortcut saves a tremendous amount of computing power without sacrificing accuracy, making it easier for other scientists to test these ideas in the future.

The implications of these results are direct for future experiments. If the Large Hadron Collider or a future machine discovers these heavy particles, they will have to fit within the narrow mass and interaction ranges defined by this study. The models do not allow for wild variations; the new particles must be relatively close in mass to each other, and their interactions must be moderate. This gives experimentalists a clearer target. They know that if these particles exist, they are likely to be found within a specific energy window, and their properties will be tightly linked to the stability of the universe itself. The work does not prove that these extra particles exist, but it firmly establishes the boundaries within which they must hide if they are to be real.

Technical Summary: Next-to-Leading Order Unitarity Fits in the Extended Georgi-Machacek Model

Problem Statement
The Standard Model (SM) Higgs mechanism remains the primary explanation for electroweak symmetry breaking (EWSB), yet the origin of this breaking is not fully settled. While current LHC data allows for $\sim 10\%$ deviations in Higgs signal strengths, the existence of additional scalar multiplets, specifically $SU(2)_L$ triplets, remains a viable Beyond Standard Model (BSM) scenario. The Georgi-Machacek (GM) model and its extension, the extended GM (eGM) model, preserve the custodial symmetry required to keep the $\rho$ parameter close to unity at the tree level. However, previous phenomenological studies have largely relied on tree-level perturbative unitarity bounds and bounded-from-below (BFB) conditions. These leading-order (LO) constraints may be insufficient to fully constrain the parameter space, particularly given the potential for large quartic couplings and the need for higher-order precision in light of Run 2 LHC data. Furthermore, the eGM model, which relaxes the global $SU(2)_L \otimes SU(2)_R$ symmetry of the GM model, introduces new mixing patterns and mass splittings that require rigorous theoretical constraints.

Methodology
The authors perform a comprehensive global fit of the GM and eGM models incorporating next-to-leading order (NLO) theoretical constraints and the latest LHC Higgs signal strength data.

Theoretical Framework: The study utilizes the generalized two-triplet scalar extension of the SM, which includes a real triplet ( $\xi$ , $Y=0$ ) and a complex triplet ( $\chi$ , $Y=1$ ) alongside the SM doublet ( $\phi$ ). The eGM model imposes specific conditions to maintain $\rho=1$ at the tree level without requiring the full global $SU(2)_L \otimes SU(2)_R$ symmetry, allowing for mass degeneracy lifting among custodial multiplets.
Bounded-from-Below (BFB) Conditions: The authors derive strict copositivity criteria for the scalar potential. They compute BFB conditions considering all combinations of three simultaneously non-vanishing scalar fields. They demonstrate that these "3-field" conditions provide a computationally efficient and highly accurate approximation (within $\sim 1.4\%$ discrepancy) of the full 13-field BFB conditions, which are analytically complex and computationally expensive.
NLO Unitarity Constraints: The paper computes one-loop corrections to all $2 \to 2$ bosonic scattering amplitudes in the high-energy limit ( $s \gg |\lambda_i|v^2$ ). Using the Goldstone boson equivalence theorem and the $\overline{MS}$ renormalization scheme, they derive the eigenvalues of the S-matrix at NLO. The unitarity bounds are applied to these NLO eigenvalues, demanding that the real part satisfies $|Re(a_\ell)| \leq 1/2$ and that the NLO corrections remain perturbative ( $R'_1 < 1$ ). The analysis employs two-loop Renormalization Group Equations (RGEs) to evolve couplings to the 1 TeV scale.
Global Fit: The authors employ the HEPfit framework with Markov Chain Monte Carlo (MCMC) simulations to perform Bayesian fits. They incorporate the derived NLO unitarity bounds and 3-field BFB conditions as theoretical priors. The experimental inputs consist of the latest Run 2 Higgs signal strength measurements ( $\mu$ ) from ATLAS and CMS at $\sqrt{s} = 13$ TeV, covering various production and decay channels ( $\gamma\gamma, ZZ, WW, \tau\tau, bb, \mu\mu, Z\gamma$ ).

Key Contributions

First NLO Unitarity Calculation: This work presents the first computation of one-loop corrections to all $2 \to 2$ bosonic scattering amplitudes in the generalized two-triplet model, recasting results specifically for the GM and eGM models.
Efficient BFB Approximation: The authors establish that 3-field BFB conditions are a robust and computationally superior approximation for the full 13-field stability conditions in both models.
Renormalization of the $\rho$ Parameter: The study clarifies that in the eGM and GM models, the $\rho$ parameter (or T parameter) cannot be treated as a pure prediction at one-loop due to the need for four input parameters for renormalization (versus three in the SM). Consequently, it is treated as an independent input, and the global electroweak fit value determines the finite part of the custodial symmetry breaking parameter.
Updated Global Fits: The paper provides the first global fits for the eGM model and updated fits for the GM model, combining NLO theoretical constraints with the latest LHC signal strength data.

Results

Quartic Couplings: The NLO unitarity bounds significantly tighten the allowed parameter space compared to LO bounds. The absolute values of the scalar quartic couplings are constrained to be less than 1.91 in the GM model and 3.0 in the eGM model.
Signal Strengths and Couplings: The global fit disfavors regions where the vector boson coupling modifier $\kappa_V > 1.05$ or $\kappa_V < 0.95$ , and the fermion coupling modifier $\kappa_f > 1.05$ or $\kappa_f < 0.92$ at a 95.4% probability for both models.
Triplet VEV and Mixing: The triplet vacuum expectation value (VEV) is constrained to $v_\chi \lesssim 30$ GeV, and the mixing angle $|\alpha|$ is restricted to $< 0.45$ . This represents a significant improvement over previous fits (e.g., Ref. [32] which allowed $v_\Delta \approx 45$ GeV).
Mass Splittings:
- In the GM model, the mass differences between heavy Higgs bosons are restricted to be less than 410 GeV (assuming individual masses $< 1.1$ TeV).
- In the eGM model, where tree-level mass degeneracy is lifted, the absolute mass differences between heavy Higgs bosons are constrained to be less than 520 GeV.
- The inclusion of NLO unitarity and signal strength data reduces the maximal allowed mass splitting by approximately 100 GeV compared to earlier studies.
Loop-Mediated Decays: The maximal deviations of the loop-mediated decay ratios $r_{\gamma\gamma}$ and $r_{Z\gamma}$ from their SM values are found to be approximately 25% and 10%, respectively.

Significance
The paper claims that the inclusion of NLO unitarity bounds and improved BFB conditions is imperative for a reliable perturbative analysis of the GM and eGM models. The results demonstrate that a large portion of the parameter space previously considered viable under LO unitarity is ruled out when NLO constraints are applied. The study provides a refined theoretical framework for interpreting LHC data in the context of triplet extensions, offering tighter constraints on the scalar sector's stability and unitarity. These findings directly impact search strategies for heavy Higgs bosons at the LHC and future colliders by defining the theoretically allowed regions for mass splittings and coupling strengths. The authors note that while their analysis improves theoretical bounds, a comprehensive inclusion of all electroweak precision observables (specifically S and U parameters correlated with T) and direct search limits remains a subject for future work.

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