Light states in real 3HDMs with spontaneous CP violation and softly broken symmetries

This paper demonstrates that in real 3-Higgs-doublet models with spontaneous CP violation and softly broken discrete symmetries, perturbativity constraints on quartic couplings prevent new scalar masses from significantly exceeding the electroweak scale, even when arbitrary large mass terms are present, a finding supported by both analytic and numerical analyses of the scalar sector's phenomenological consequences.

The Gist

Imagine the universe is built on a set of invisible rules, like the laws of physics that govern how particles interact. The Standard Model is our current "rulebook," but scientists know it's incomplete. To fix it, they often propose adding new "players" to the game: new types of particles called Higgs bosons.

This paper investigates a specific scenario where we add three of these Higgs doublets (think of them as three distinct teams of particles) instead of just the one we've already found. The researchers are asking a very specific question: If we add these new teams, how heavy can they be?

Here is the breakdown of their findings, using simple analogies:

1. The "Heavy" Expectation vs. The "Light" Reality

Usually, when physicists add new particles to a theory, they imagine they can make them as heavy as they want. It's like building a skyscraper; you can keep adding floors as high as you like, provided your foundation (the math) holds up.

In this paper, the researchers found a surprising twist. Even if you try to build a "super-heavy" tower of new particles, nature forces at least some of them to stay light.

• The Analogy: Imagine you are trying to build a tower of blocks. You have a rule that says the blocks can't be too "wobbly" (this is the perturbativity rule, a mathematical safety check to keep the theory stable). You also have a heavy foundation (mass terms) that you can make as heavy as you want.
• The Surprise: No matter how heavy you make the foundation, the rules of the game force at least one charged particle and two neutral particles to remain relatively light (close to the weight of the Higgs boson we already know, about 125 GeV). You cannot hide them in the "heavy" zone.

2. The "Mirror World" Trick

Why does this happen? The paper explains it using a concept called Spontaneous CP Violation.

• The Analogy: Imagine you are standing in a room with a mirror. You (the vacuum of space) have chosen to stand on the left side of the room. However, the mirror shows a version of you standing on the right side.
• In this theory, the "mirror version" is just as valid as the real version.
• If you try to make the new particles extremely heavy, the math gets confused. The "real" you and the "mirror" you become indistinguishable to the heavy parts of the equation. This confusion creates "ghost" particles that must be massless.
• When you turn the "volume" back up on the interaction rules (the quartic couplings), these ghost particles gain a tiny bit of weight, but not enough to become heavy. They are stuck at the "electroweak scale" (the weight of our current known particles).

3. The "A4" Symmetry (The Dance Floor)

To make the math easier to understand, the authors focused on a specific type of symmetry called A4.

• The Analogy: Think of the three new Higgs doublets as three dancers on a floor. The A4 symmetry is like a specific dance routine where the dancers must move in a coordinated, triangular pattern.
• The researchers set up the "dance floor" (the potential energy) so that the dancers follow this routine. They found that even with this strict choreography, the "light particle" rule still holds true.
• They also looked at other dance routines (like $\Delta(27)$), and the result was the same: you can't make all the new dancers heavy. Some must stay light.

4. The Numerical Experiment (The Simulation)

Since the math gets very complicated (like trying to solve a puzzle with 10,000 pieces), the authors ran a computer simulation to see what happens in the real world.

• The Setup: They generated millions of random scenarios, ensuring the math stayed stable and the particles behaved like our known universe (specifically, that the lightest particle looks like our 125 GeV Higgs boson).
• The Results:
  • The Light Ones: They confirmed that there are always new particles (one charged, two neutral) that stay below about 800 GeV. They are "light" enough that our current particle colliders (like the Large Hadron Collider) could potentially find them soon.
  • The Heavy Ones: The other new particles can be very heavy (thousands of GeV), effectively hiding from us.
  • The Connection: The light particles are tightly linked to the known Higgs boson. They interact with it in specific ways that we can measure.

5. Why This Matters

The paper concludes that if the universe follows these specific rules (Real 3HDM with Spontaneous CP Violation), we cannot ignore the possibility of finding new, relatively light particles.

• The Takeaway: You can't just say, "Oh, the new particles are so heavy we'll never see them." In this specific scenario, the laws of physics force at least a few of them to be light enough to be discovered. It's a "guaranteed" signal for future experiments.

Summary

This paper is a mathematical detective story. The detectives (the authors) looked at a theory with three Higgs bosons and asked, "Can we hide all the new particles in the heavy zone?" They proved that no, the rules of the game (specifically the symmetry between a particle and its mirror image) force at least three new particles to stay light. This gives experimentalists a clear target: look for these light particles, because if this theory is right, they are there.

Technical Summary

Technical Summary: Light States in Real 3HDMs with Spontaneous CP Violation and Softly Broken Symmetries

Problem Statement
In extensions of the Standard Model (SM) featuring extended scalar sectors, theoretical constraints such as perturbativity are often invoked to bound the masses of new particles. While it is well-established that in Two-Higgs-Doublet Models (2HDMs) with spontaneous CP violation (SCPV), perturbativity constraints on quartic couplings force the entire scalar spectrum to remain near the electroweak scale, the situation in models with more than two doublets ($n > 2$) was less clear. Specifically, in $n$-Higgs-Doublet Models ($n$HDMs), the number of unconstrained quadratic (mass) parameters increases quadratically with $n$. This suggested the possibility of a "decoupling regime" where new scalars could be arbitrarily heavy, even if quartic couplings remained perturbative.

However, a prior theoretical argument [30] suggested that in real $n$HDMs (CP-invariant potentials with SCPV), a subset of new scalars—one charged and two neutral states—must remain light (masses $\sim$ electroweak scale) regardless of the magnitude of the free quadratic parameters. This paper aims to provide a detailed analytic and numerical verification of this counter-intuitive property in the minimal case of three Higgs doublets (3HDM), where one unconstrained quadratic parameter remains after imposing stationarity conditions.

Methodology
The authors analyze real 3HDMs with a CP-invariant scalar potential and spontaneous CP violation. To maintain analytic control while reducing the parameter space, they impose discrete symmetries on the quartic part of the potential, treating the quadratic part as "softly broken" (i.e., the symmetry does not constrain the mass terms). Two specific symmetry scenarios are examined:

1. $A_4$ Symmetry: The three doublets transform as an irreducible triplet. With the additional requirement of CP invariance, the effective symmetry of the quartic potential is $S_4$.
2. $\Delta(27)$ Symmetry: The three doublets transform as an irreducible triplet under this group.

The analysis proceeds in two stages:

• Analytic Derivation: The authors derive the stationarity conditions to identify the unconstrained quadratic parameter. They then construct the charged and neutral mass matrices ($M^2_\pm$ and $M^2_0$). By analyzing the limit where quartic couplings ($\lambda$) are treated as perturbations to large quadratic parameters ($\mu^2$), they investigate the eigenvalue structure. A key theoretical step involves recognizing that in the limit $\lambda \to 0$, the mass matrices cannot distinguish between the chosen vacuum and its CP-conjugate, leading to a doubling of null eigenvectors (Goldstone modes). Reintroducing $\lambda$ lifts the degeneracy of non-Goldstone states, bounding their masses.
• Numerical Exploration: A comprehensive numerical scan is performed for the $A_4$-shaped scenario. The procedure ensures the potential is bounded from below, satisfies stationarity conditions, and yields a physical minimum. Constraints are applied including:
  • The lightest neutral scalar mass $m_h \approx 125$ GeV.
  • The Higgs coupling to vector bosons $K_{hVV} \geq 0.90$.
  • Electroweak precision observables (S, T, U parameters) within 90% CL.
  • Perturbative unitarity constraints ($2 \to 2$ scattering amplitudes).
  • A kinematic constraint $M_{H^\pm_1} > m_h/2$ to avoid unobserved $h \to H^+ H^-$ decays.

Key Contributions and Results

1. Analytic Confirmation of Light States:

   • Charged Sector: In the $A_4$ scenario, the charged mass matrix eigenvalues are derived in closed form. The spectrum consists of a massless Goldstone boson, a light charged state with squared mass $M^2 \propto (\lambda_4 - \lambda_3)v^2$, and a heavy state $M^2 \propto \mu^2 + \mathcal{O}(v^2)$. Crucially, the light charged mass is independent of the large free quadratic parameter $\mu^2$, confirming it is bounded by the electroweak scale if quartic couplings are perturbative.
   • Neutral Sector: While a full analytic solution for the $6 \times 6$ neutral mass matrix is not feasible, the authors identify an intermediate regime where the commutator of the quadratic and quartic mass matrices vanishes (under specific parameter relations). In this limit, three neutral states are shown to have masses bounded by the electroweak scale, while two can become arbitrarily heavy.
   • $\Delta(27)$ Case: Although less transparent analytically, the charged sector analysis confirms that one charged scalar remains light ($M^2 \sim \lambda v^2$) while the other becomes heavy ($M^2 \sim \mu^2$) in the decoupling limit.

2. Numerical Validation:

   • The numerical scan confirms the analytic predictions: in the $A_4$ scenario, the lightest charged scalar ($H^\pm_1$) and the two lightest neutral scalars ($H^0_1, H^0_2$) are strictly bounded, typically remaining below $\sim 800$ GeV even when the heavy states ($H^\pm_2, H^0_3, H^0_4$) are pushed to multi-TeV scales.
   • In the decoupling regime (heavy states $> 1$ TeV), the heavy scalars become degenerate, and the light states exhibit specific correlations in their couplings to vector bosons.

3. Phenomenological Implications:

   • Couplings: The study maps the allowed ranges for couplings of the light scalars to vector bosons ($H^0_{1,2}VV$) and to the Higgs/vector boson pairs ($H^0_{1,2}Zh$, $H^\pm_1 W^\pm h$).
   • Decoupling Patterns: In the regime where heavy scalars are decoupled, a simple correspondence emerges between the mixing matrices and the mass eigenvalues, leading to specific patterns in the coupling strengths (e.g., $|C_{H^0_2 Zh}|^2 \approx |K_{H^0_1 VV}|^2$).
   • Decay Widths: The authors calculate maximal decay widths for the light scalars into gauge boson pairs and fermions. They find that even with a reduced parameter space dictated by $A_4$ symmetry, the scalar sector alone does not force a specific pattern in the dominant decay channels; the couplings can be significantly suppressed or enhanced within the allowed parameter space.

Significance and Claims
The paper claims to provide a "transparent analytic understanding" and detailed numerical illustration of a generic property in real multi-Higgs models with SCPV: the existence of light new scalars that cannot be decoupled, regardless of the size of the unconstrained quadratic mass terms.

The authors emphasize that this result holds specifically because the CP-conjugate vacuum is also a valid minimum of the potential. In the limit of large quadratic parameters, the mass matrices lose the ability to distinguish between these vacua, resulting in extra massless states that acquire only electroweak-scale masses once quartic interactions are restored.

The work is modest in its scope, focusing strictly on the scalar sector properties and gauge interactions without incorporating specific Yukawa textures or flavor constraints. The authors conclude that while the mass spectrum is constrained (forcing light states), the scalar sector alone is insufficient to dictate a unique phenomenological pattern for the decay channels of these light states. The results serve to refine the theoretical expectations for the mass spectra of 3HDMs and provide a benchmark for future experimental searches for light charged and neutral scalars in extended Higgs sectors.