Scalar contributions to the S, T, U parameters in a 3-3-1 model

This paper systematically investigates the scalar sector contributions to the electroweak oblique parameters $S$, $T$, and $U$ within the 3-3-1 model with right-handed neutrinos, demonstrating that the $T$ parameter imposes stringent constraints on the model's scalar masses and energy scales.

The Gist

Imagine the Standard Model of particle physics as a perfectly tuned orchestra. For decades, scientists have been listening to this orchestra, and every note (every particle interaction) has matched the sheet music perfectly. But physicists suspect there might be a hidden second orchestra playing quietly in the background, adding new instruments we haven't heard yet. This "hidden orchestra" is what we call New Physics.

This paper is like a detective story where the authors are trying to find out if a specific type of hidden orchestra, called the 3-3-1 Model, is actually playing. They are listening for a specific sound: the S, T, and U parameters.

The Three "Ear" Tests (S, T, and U)

Think of the S, T, and U parameters as three different ways to check if the orchestra is in tune:

• S checks if the rhythm is slightly off.
• T checks if the pitch is perfectly balanced between the "left" and "right" sections of the orchestra (this is called custodial symmetry).
• U checks for subtle timing glitches.

If the hidden orchestra is too loud or playing the wrong notes, these tests will scream "Something is wrong!"

The Hidden Orchestra: The 3-3-1 Model

The 3-3-1 model is a specific theory that suggests the universe has a more complex structure than we thought. It predicts:

1. New Heavy Musicians: New, heavy particles (gauge bosons) that we haven't seen yet.
2. New Instruments: A whole new family of scalar particles (Higgs-like particles).

Previous detectives (other scientists) had already checked the "heavy musicians" (the gauge bosons). They found that even if these heavy musicians are playing, they are so quiet (or their notes cancel each other out) that the S, T, and U tests barely notice them.

But here is the twist in this paper: The authors decided to check the new instruments (the scalar particles) that the previous detectives ignored. They asked: "What if the new Higgs-like particles are the ones messing up the tune?"

The Plot Twist: The "T" Parameter is the Strict Conductor

The authors ran a massive simulation, like a sound engineer testing thousands of different volume levels and instrument combinations. They found that the T parameter is the strictest conductor in the room.

Here is the analogy:
Imagine the scalar particles are a group of twins. In a perfect world (the Standard Model), the twins are identical in weight. But in the 3-3-1 model, the "T" parameter gets very angry if the twins have different weights (mass splittings).

The paper discovered that for the 3-3-1 model to sound "in tune" with our current measurements:

1. The Twins Must Be Close in Weight: The new heavy particles cannot be too different in mass from each other. If they are too different, the "T" parameter blows a whistle and says, "This model is impossible!"
2. The Volume Knob (Parameter 'f') Must Be Low: There is a specific control knob in the model called $f$ (a trilinear coupling). The authors found that this knob must be turned down very low (less than 10 GeV). If you turn it up, the twins get too heavy and too different, and the model breaks the rules.
3. The Stage Size (Energy Scale $v_{\chi'}$) is Limited: The "stage" where these new particles live cannot be too huge. It's limited to about 14 TeV (a specific energy scale). If the stage is bigger than that, the new particles become too heavy and too different, and the "T" parameter rejects the model.

The Verdict

The authors concluded that the Scalar Sector (the new Higgs-like particles) is actually the biggest threat to the 3-3-1 model's survival, not the new heavy gauge bosons.

• The Good News: The model isn't dead yet! It just has very strict rules.
• The Catch: The new particles must be relatively light (under 800 GeV for some, under 400 GeV for others) and very similar in mass to each other.
• The Exciting Part: Because these particles are predicted to be "light" (in the grand scheme of particle physics), they might actually be light enough to be found by the Large Hadron Collider (LHC) right now or in the near future.

Summary in One Sentence

This paper is a warning label for a specific theory of new physics, saying: "If you want this theory to work, your new particles must be light, similar in weight, and the 'volume knob' on their interactions must be turned down very low, or the strict 'T' test will disqualify them."

This gives experimentalists a clear target: Look for these specific, relatively light particles at the LHC, because if they aren't there, this version of the 3-3-1 model is likely wrong.

Technical Summary

Here is a detailed technical summary of the paper "Scalar contributions to the S, T, U parameters in a 3-3-1 model" by A. Doff and C. A. de S. Pires.

1. Problem Statement

Electroweak precision tests (EWPT), encapsulated in the oblique parameters $S$, $T$, and $U$, are among the most stringent probes for physics beyond the Standard Model (SM). The 3-3-1 model based on the gauge symmetry $SU(3)_C \times SU(3)_L \times U(1)_N$ (specifically the version with right-handed neutrinos, 331RHN) introduces a rich spectrum of new particles, including new gauge bosons and scalars.

While previous studies (e.g., Ref. [18]) established that the contributions of new gauge bosons to the oblique parameters are negligible at the TeV scale, the impact of the scalar sector in the 331RHN model had not been systematically addressed. The authors identify a gap in the literature regarding how the specific scalar spectrum of the 331RHN model constrains the model's energy scales and coupling parameters through EWPT.

2. Methodology

The authors perform a systematic calculation of the scalar contributions to the oblique parameters within the 331RHN framework.

• Model Setup:

  • The scalar sector consists of three $SU(3)_L$ triplets: $\eta$, $\rho$, and $\chi$.
  • The scalar potential includes trilinear couplings ($f$) and quartic couplings ($\lambda_i$).
  • Symmetry breaking occurs in two stages: $G_{331} \to G_{SM}$ (at scale $v_{\chi'}$) and $G_{SM} \to U(1)_{em}$ (at scale $v \approx 246$ GeV).
  • The authors assume a decoupling limit where the heavy scalar $R_{\chi'}$ decouples from the lighter scalars, reducing the system to a 3-Higgs Doublet Model (3HDM) structure plus inert singlets.

• Basis Transformation:

  • To facilitate the calculation, the authors rotate the scalar fields into the Higgs basis. In this basis, only one linear combination of doublets carries the full electroweak vacuum expectation value (VEV), while the orthogonal combinations have vanishing VEVs.
  • This allows for a clear separation of the SM-like Higgs ($h$) from the new heavy scalars ($H, A, h^\pm_1, H'', h^\pm_2$).

• Calculation of Oblique Parameters:

  • The authors compute the vacuum polarization functions $\Pi_{WW}(q^2)$ and $\Pi_{ZZ}(q^2)$ at the one-loop level.
  • They derive the contributions to the oblique parameters $S$, $T$, and $U$ using the standard definitions involving the transverse parts of the gauge boson self-energies.
  • The scalar contributions are expressed in terms of loop functions $F(m_1^2, m_2^2)$ and $G(m_1^2, m_2^2)$, which depend on the mass splittings between the charged and neutral scalar states.

• Numerical Analysis:

  • A parameter scan is performed over the multidimensional space $(f, v_{\chi'}, \lambda_2, \lambda_6, \lambda_7, \lambda_8, \lambda_9)$.
  • Constraints are applied based on the latest global fits for the $T$ parameter ($T = 0.01 \pm 0.12$) and the $S$ parameter.
  • The analysis compares two scenarios: a "decoupled" scenario where the inert doublet ($\Phi_3$) contributions are negligible, and a "non-decoupled" scenario where all scalar states contribute.

3. Key Contributions

• First Systematic Scalar Analysis: This work provides the first comprehensive calculation of the scalar sector's contribution to $S, T, U$ specifically for the 331RHN model.
• Identification of Dominant Constraints: The authors demonstrate that while gauge boson contributions are negligible, the scalar sector imposes severe constraints, primarily driven by the $T$ parameter.
• Correlation Discovery: The study establishes a direct, stringent correlation between the symmetry-breaking scale ($v_{\chi'}$) and the trilinear coupling parameter ($f$).
• Inert Doublet Impact: The paper quantifies how the inclusion of the inert doublet ($\Phi_3$) significantly tightens the bounds on the model's energy scale compared to scenarios where it is decoupled.

4. Key Results

The numerical analysis yields the following specific constraints:

• Dominance of the $T$ Parameter: The $T$ parameter is the most restrictive observable. It is highly sensitive to custodial symmetry breaking caused by mass splittings within the scalar doublets. The $S$ and $U$ parameters impose much weaker constraints.
• Constraints on Coupling $f$:
  • The trilinear coupling $f$ must be very small to satisfy the $T$ constraint.
  • The analysis finds $f \lesssim 10$ GeV. Larger values of $f$ induce large mass splittings, leading to excessive contributions to $T$.
• Constraints on Energy Scale $v_{\chi'}$:
  • In the non-decoupled scenario (where the inert doublet contributes), the symmetry-breaking scale is bounded by $v_{\chi'} \lesssim 14$ TeV.
  • In the decoupled scenario, the bound on $v_{\chi'}$ is significantly weaker, as the heavy scalars from the inert sector do not contribute to $T$.
• Scalar Mass Bounds:
  • The $T$ constraint forces the new scalar masses to remain relatively close to the electroweak scale.
  • SM-like Higgs ($h$): Fixed at 125.1 GeV.
  • Heavy CP-even ($H$): Bounded by $m_H \lesssim 770$ GeV.
  • Charged Scalars ($h^\pm_1$): Bounded by $m_{h^\pm_1} \lesssim 800$ GeV.
  • CP-odd Pseudoscalar ($A$): Bounded by $m_A \lesssim 400$ GeV.
  • Heavier States ($H'', h^\pm_2$): Their masses are also constrained by the $v_{\chi'}$ limit, generally remaining below $\sim 14$ TeV but effectively bounded by the $T$ parameter to be lighter than the scale allows.

5. Significance

• Testability: The results imply that the scalar sector of the 331RHN model is not arbitrarily heavy. The derived mass bounds ($m \lesssim 800$ GeV) place these new particles within the reach of current and future high-energy colliders, such as the LHC.
• Model Viability: The study highlights that the viability of the 331RHN model is heavily dependent on the trilinear coupling $f$. The requirement of $f \lesssim 10$ GeV suggests a specific "fine-tuning" or specific mechanism is required to keep custodial symmetry breaking low.
• Phenomenological Guidance: By establishing the correlation between $v_{\chi'}$ and $f$, the paper provides a clear roadmap for experimental searches. It suggests that if the 331RHN model is realized in nature, the new scalars should be relatively light, and the symmetry-breaking scale should not exceed $\sim 14$ TeV.
• Comparison with Minimal 3-3-1: The results are consistent with findings in the minimal 3-3-1 model but refine the constraints specifically for the version with right-handed neutrinos, showing that the scalar sector alone can rule out large regions of the parameter space previously thought viable.

In conclusion, the paper shifts the focus of constraints in the 331RHN model from the gauge sector to the scalar sector, demonstrating that electroweak precision data, particularly the $T$ parameter, acts as a powerful filter for the model's scalar spectrum and energy scales.