Scalar contributions from $331RHN$ minimal model to oblique parameters

This paper analyzes scalar-sector contributions to oblique parameters within the minimal 331RHN model, demonstrating that the parameter $T$ imposes a dominant constraint that limits the symmetry-breaking scale to approximately 10 TeV.

The Gist

Imagine the universe as a giant, incredibly complex machine. For decades, physicists have been trying to understand how this machine works using a blueprint called the Standard Model. This blueprint explains how tiny particles like electrons and quarks interact. However, the blueprint has some missing pages and unanswered questions, so scientists are constantly trying to build "extensions" or add-ons to it.

One popular add-on is called the 3-3-1 model. Think of this as a new architectural plan for the universe's machinery. It suggests that there are extra layers of symmetry and new types of particles we haven't seen yet. Specifically, this paper looks at a "minimal" (simplified) version of this plan that includes right-handed neutrinos (ghostly particles that barely interact with anything).

Here is what the paper does, broken down into simple concepts:

1. The Problem: The "Tension" in the Machine

Physicists have very precise measurements of how the machine behaves right now. They call these measurements Oblique Parameters (S, T, and U). You can think of these as the machine's "stress gauges."

• If you add new parts to the machine (new particles), it might change how the machine vibrates or holds together.
• If the new parts are too heavy or too different from the old ones, the stress gauges (S, T, U) will go into the red, telling us the blueprint is wrong.

2. The Investigation: Adding New "Weights"

In this specific 3-3-1 model, the scientists added a new Scalar Sector.

• Analogy: Imagine the Standard Model is a balanced scale. The new 3-3-1 model adds new weights to the scale. These weights are new particles called scalars (specifically, a heavy neutral one and a heavy charged one).
• The paper asks: If we add these specific weights, does the scale tip too far? Does the "stress gauge" (the T parameter) break?

3. The Discovery: The "T" Gauge is the Strictest Boss

The researchers ran a massive computer simulation, testing millions of different combinations of how heavy these new particles could be. They looked at three stress gauges: S, T, and U.

• The Result: The T gauge turned out to be the strictest boss. It is the most sensitive to the new weights.
• The Analogy: Imagine you are trying to sneak a heavy suitcase into a hotel room. The S and U guards are asleep, but the T guard is wide awake and checking the weight limit very carefully. If your suitcase is too heavy, the T guard stops you immediately.

4. The Limit: The "Speed Limit" of the Universe

The paper found that for the model to work without breaking the laws of physics (specifically, without making the T gauge go into the red), there is a strict limit on how heavy the new particles can be.

• The Scale ($\omega$): This represents the "energy level" or the size of the new symmetry breaking. Think of it as the "height" of a new floor being added to the building.
• The Finding: The T guard says, "You can add this new floor, but it cannot be higher than 10 TeV (about 10,000 times the mass of a proton)."
• If the new particles are heavier than this limit, the model breaks the rules of the universe as we currently understand them.

5. The Conclusion

The paper concludes that while the 3-3-1 model is a clever idea, it is very fragile. The "T" parameter acts like a strict gatekeeper.

• It doesn't completely kill the model, but it puts a ceiling on how big the new physics can be.
• The model is still "viable" (it can work), but only if the new particles are light enough to pass the T guard's inspection.

In short: The scientists took a simplified version of a new universe blueprint, added some new heavy particles, and checked if the universe's stress sensors would explode. They found that the sensors would explode if the particles were too heavy, so they set a strict speed limit: the new physics must stay below a certain energy level (10 TeV) to keep the universe stable.

Technical Summary

Technical Summary: Scalar Contributions from the 331RHN Minimal Model to Oblique Parameters

Problem Statement
Electroweak precision observables, specifically the oblique parameters $S$, $T$, and $U$, serve as stringent constraints on extensions of the Standard Model (SM). While the 3-3-1 framework based on the gauge symmetry $SU(3)_C \times SU(3)_L \times U(1)_Y$ offers compelling theoretical features—such as the quantization of electric charge and an explanation for the replication of fermion generations—its phenomenological viability depends on satisfying these precision bounds. Previous work [19] analyzed the scalar-sector contributions to these parameters within the standard 331RHN model (which includes right-handed neutrinos), finding that the parameter $T$ imposes the most restrictive constraints on the scalar mass spectrum and symmetry-breaking scales. However, the specific impact of the minimal 331RHN model, which features a reduced scalar sector consisting of only two triplets, on these parameters had not been fully investigated. The central problem addressed is whether the current experimental bounds on the oblique parameter $T$ can constrain the symmetry-breaking scale $\omega$ in this minimal realization.

Methodology
The authors perform a systematic analysis of the scalar sector of the minimal 331RHN model. The methodology proceeds as follows:

1. Model Definition: The study utilizes the minimal 331RHN framework where spontaneous symmetry breaking occurs in two stages via two scalar triplets, $\chi$ and $\phi$. The vacuum expectation values (VEVs) are hierarchical: $v$ (electroweak scale) from $\phi$, and $u$ (GeV scale) and $\omega$ (TeV scale) from the neutral components of $\chi$.
2. Scalar Spectrum Derivation: The authors derive the mass matrices for the CP-even neutral scalars, CP-odd scalars, and charged scalars. They identify the physical spectrum, which consists of four states: a SM-like Higgs boson ($h$), a heavy CP-even neutral scalar ($H_m$), and heavy charged scalars ($H^\pm_m$). Notably, the model lacks trilinear scalar interactions and axion-like pseudoscalars, simplifying the potential compared to the non-minimal 331RHN model.
3. Formalism for Oblique Parameters: To calculate contributions to $S$, $T$, and $U$, the authors employ the Higgs basis formalism. This involves rotating the scalar doublets such that the electroweak VEV is aligned with a single doublet ($\Phi_2$), while the orthogonal doublet ($\Phi_1$) contains the new physical scalar states. This separation allows for a clear identification of new-physics contributions to gauge boson vacuum polarizations.
4. Calculation of Loop Corrections: The scalar contributions to the vacuum polarization functions are calculated at the one-loop level. Due to the specific particle content of the minimal model, the only non-vanishing contributions relevant to the oblique analysis arise in the charged-current sector via the polarization function $\Pi_{WW}$. The authors derive analytical expressions for $\alpha S$, $\alpha T$, and $\alpha U$ in terms of the masses of the heavy scalars ($m_{H_m}$ and $m_{H^\pm_m}$).
5. Numerical Analysis: A numerical scan is performed over the multidimensional parameter space $(u, \omega, \lambda_1, \lambda_2, \lambda_3, \lambda_4)$. The SM-like Higgs mass is fixed at $125.3$ GeV. The parameter $T$ is constrained using the latest global electroweak fit value ($T = 0.01 \pm 0.12$).

Key Contributions

• Derivation of Scalar Contributions: The paper provides the first detailed derivation of the scalar-sector contributions to the oblique parameters specifically for the minimal 331RHN model.
• Simplification of the Formalism: By utilizing the Higgs basis and exploiting the absence of certain scalar states (like the axion-like pseudoscalar) present in the non-minimal version, the authors demonstrate a significantly simplified structure for the vacuum polarization corrections, isolating the contributions to the $T$ parameter.
• Constraint on the Symmetry-Breaking Scale: The analysis establishes a direct link between the experimental bounds on $T$ and the symmetry-breaking scale $\omega$ in the minimal model.

Results

• Dominance of the $T$ Parameter: Consistent with findings in the non-minimal 331RHN model, the oblique parameter $T$ is identified as the dominant constraint. The contributions to $S$ and $U$ are found to be negligible across the allowed parameter space.
• Mass Degeneracy: The results confirm that in the limit of degenerate scalar masses ($m_{H_m} = m_{H^\pm_m}$), the scalar contributions to the oblique parameters vanish, reflecting the suppression of precision effects in the presence of degenerate multiplets.
• Upper Bound on $\omega$: The numerical scan reveals that current experimental bounds on $T$ impose a nontrivial upper limit on the symmetry-breaking scale. While the scan allows $\omega$ up to 20 TeV, the requirement to satisfy the $T$ constraint effectively restricts the viable region to $\omega \lesssim 10$ TeV.
• Scalar Mass Spectrum: Within the viable domain, the masses of the heavy scalar states ($H_m$ and $H^\pm_m$) remain bounded by the symmetry-breaking scale ($m \lesssim \omega$).

Significance and Claims
The paper claims that its findings highlight the sensitivity of electroweak precision data to the scalar sector of 3-3-1 models. The primary significance lies in demonstrating that even in a "minimal" realization with a reduced scalar sector, the model is not free from stringent electroweak constraints. Specifically, the work establishes that the oblique parameter $T$ serves as a powerful probe, capable of constraining the symmetry-breaking scale $\omega$ to the order of 10 TeV. This reinforces the role of electroweak precision tests as essential tools for assessing the viability of 3-3-1 extensions of the Standard Model. The authors do not claim to resolve the hierarchy problem or propose new experimental signatures beyond these precision bounds, but rather delineate the theoretical limits of the model's parameter space based on existing data.