Double SM-like Higgs Production at future $e^+ e^-$ colliders in the 3-Higgs Doublet Model under the $S_{3}$ symmetry

This paper demonstrates that the $S_3$-symmetric three-Higgs-doublet model can produce measurable deviations in double SM-like Higgs production at future $e^+e^-$ colliders, potentially differing from Standard Model predictions by several orders of magnitude.

The Gist

The Cosmic "Secret Sauce": Explaining the S3-3H Higgs Model

Imagine you are a master chef trying to perfect the ultimate recipe for a cake—this cake represents our entire Universe. For decades, scientists have been studying the "secret ingredient" that makes the cake hold its shape instead of collapsing into a puddle of batter. In physics, this ingredient is called the Higgs Boson.

We discovered this ingredient back in 2012, and it worked! But there’s a catch: the recipe we have (the Standard Model) feels a bit too simple. It explains the basic texture, but it doesn't explain why some ingredients are so heavy, why others are so light, or why the kitchen stays stable.

This paper explores a "Premium Recipe" called the S3-3H Model. Here is the breakdown of what the researchers found.

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1. The Concept: From One Chef to Three

In our current understanding (the Standard Model), there is essentially one "Higgs field" acting like a single chef in the kitchen.

The S3-3H model suggests that instead of one chef, there are actually three chefs (three Higgs doublets) working together. To keep things from getting chaotic, they follow a strict set of rules called S3 Symmetry. Think of this like a choreographed dance troupe: even though there are three dancers, they move in such a synchronized way that the performance looks harmonious rather than messy.

2. The Experiment: The "Double Cake" Test

If you want to know if there are really three chefs in the kitchen, you don't just look at one cake. You look for moments when they produce two cakes at the exact same time (Double Higgs Production).

The paper looks at two ways this could happen at future "super-kitchens" (high-energy particle colliders like the ILC or CLIC):

• The "Side-Car" Method (Higgs-strahlung): A particle bumps into a Higgs, and they pop out together like a sidecar attached to a motorcycle.
• The "Collision" Method (Vector Boson Fusion): Two particles smash into each other so hard that they fuse together to create two Higgs bosons.

3. The Discovery: The "Resonance" Boom

This is the most exciting part. The researchers found that if this "Three-Chef" model is true, we won't just see a tiny bit more cake; we might see a massive explosion of cake.

In certain scenarios, the extra "chefs" (additional heavy Higgs particles) act like a magnifying glass. When the energy hits just the right level, these extra particles "resonate"—imagine pushing a child on a swing at exactly the right moment to make them go higher and higher. This resonance can make the production of double Higgs bosons hundreds or even thousands of times more frequent than what the old, simple recipe predicts.

4. Why Does This Matter?

If we build these future colliders and suddenly see a "double Higgs" signal that is much louder than expected, it’s a "smoking gun." It would prove that:

1. The Standard Model is incomplete.
2. There is a much richer, more complex "scalar sector" (a bigger team of chefs) governing the universe.
3. We are finally uncovering the deeper architecture of reality.

Summary in a Nutshell

The Old Way: One chef makes one cake. It’s predictable and a bit boring.
The S3-3H Way: Three chefs dance in sync. Occasionally, their teamwork causes a massive, spectacular "double-cake" event that is much bigger than anything we've ever seen. This paper provides the mathematical map to help us find that spectacular event in the future.

Technical Summary

Technical Summary: Double SM-like Higgs Production in the $S_3$-3HDM

1. Problem Statement

While the discovery of the 125 GeV Higgs boson confirmed the mechanism of electroweak symmetry breaking (EWSB), the precise structure of the scalar potential remains unknown. Specifically, the trilinear Higgs self-coupling ($\lambda_{hhh}$) has not been measured with sufficient precision to determine if the Higgs sector is minimal (as in the Standard Model) or extended. Furthermore, the hierarchy problem and the nature of the electroweak phase transition suggest the existence of additional scalar states. The paper seeks to determine if an extended Higgs sector, specifically one governed by the $S_3$ discrete symmetry, can produce measurable deviations in double Higgs production at future $e^+e^-$ colliders.

2. Methodology

The study focuses on the $S_3$-symmetric Three-Higgs-Doublet Model ($S_3$-3H) without CP violation.

• Model Framework: The model employs three $SU(2)_L$ Higgs doublets: two transforming as a doublet under the $S_3$ permutation group and one as a singlet. This results in nine physical scalar bosons, including one SM-like Higgs ($h$).
• Constraints: The authors restrict the parameter space by applying:
  • Theoretical Bounds: Perturbative unitarity and vacuum stability.
  • Experimental Bounds: Data from the Tevatron and LHC, specifically using the HiggsBounds and HiggsSignals packages to ensure compatibility with observed SM-like Higgs signal strengths.
• Observables: The study calculates the production cross-sections ($\sigma$) for two primary channels at future $e^+e^-$ colliders (such as ILC and CLIC):
  1. Higgs-strahlung (HS): $e^+e^- \to hhZ$
  2. Vector Boson Fusion (VBF): $e^+e^- \to hh\nu_e\bar{\nu}_e$
• Numerical Analysis: The authors perform scans over heavy scalar masses ($m_H, m_{A2}$) and the trilinear coupling modifier ($\kappa_{hhh}$) to identify resonant enhancements and interference patterns.

3. Key Contributions

• Phenomenological Mapping: The paper provides a detailed mapping of the $S_3$-3H parameter space, identifying "benchmark points" (BP1, BP2) that satisfy all current theoretical and experimental constraints.
• Channel Comparison: It distinguishes the sensitivities of the HS and VBF channels. It demonstrates that while both are useful, the VBF channel is a more sensitive probe for the trilinear Higgs self-coupling.
• Resonant Identification: The work identifies that deviations in double Higgs production are not merely due to modified self-couplings but are significantly driven by the exchange of additional heavy scalars ($H$ and $A_2$).

4. Results

• Large Deviations: The $S_3$-3H model predicts massive deviations from the SM. The ratio $R = \sigma_{S_3\text{-}3H}/\sigma_{SM}$ can exceed the SM prediction by several orders of magnitude ($O(10^2)$ to $O(10^3)$) in specific regions of the parameter space.
• Higgs-strahlung ($hhZ$):
  • The cross-section shows a clear resonant peak when the CP-odd scalar mass $m_{A2}$ is around 300–350 GeV.
  • The sensitivity to the trilinear coupling $\kappa_{hhh}$ is relatively weak in this channel because gauge-mediated diagrams dominate.
• Vector Boson Fusion ($hh\nu\bar{\nu}$):
  • The cross-section increases with the center-of-mass energy ($\sqrt{s}$), making high-energy colliders (like CLIC) essential.
  • This channel exhibits a characteristic non-linear dependence on $\kappa_{hhh}$ due to the interplay (interference) between self-coupling and gauge-mediated diagrams.
• Sensitivity: The VBF channel is identified as the superior tool for testing the Higgs self-interaction compared to the HS channel.

5. Significance

This research underscores the importance of high-precision $e^+e^-$ colliders in the search for New Physics. By demonstrating that the $S_3$-3H model produces "spectacular" signatures (orders-of-magnitude increases in cross-sections), the paper provides a clear roadmap for experimentalists. If future colliders observe such enhancements in double Higgs production, it would serve as a definitive hint of an extended scalar sector and potentially validate the $S_3$ symmetry as a fundamental principle governing the Higgs potential.