A theoretical account of tiny multi-Higgs vacuum expectation values from non-invertible symmetry

This paper proposes a novel mechanism utilizing non-invertible symmetries within the minimal Fibonacci fusion rule to naturally generate small, radiative vacuum expectation values for exotic multi-Higgs fields, thereby satisfying experimental constraints and enabling viable neutrino mass models without requiring additional loop-inducing particles.

The Gist

Imagine the universe is built on a set of invisible rules, like a giant, cosmic game of Lego. For decades, physicists have known that the "standard" Lego bricks (the particles we know, like electrons and the Higgs boson) work perfectly. But to explain some mysteries—like why neutrinos have such tiny masses—scientists have wanted to add some "exotic" new bricks.

The problem is, these exotic bricks are supposed to be very heavy and rare. However, if they were to "settle down" and take up space (a process physicists call getting a Vacuum Expectation Value, or VEV), they would mess up the delicate balance of the universe's forces. It's like trying to build a delicate sandcastle; if you drop a heavy bowling ball on it, the whole thing collapses. Experiments tell us these exotic bricks must stay "light" in their influence, with a value roughly 100 to 1,000 times smaller than the standard Higgs brick.

The Problem: How to keep them light?
Usually, to keep these exotic bricks from getting too heavy, physicists have to invent complicated new rules or add even more invisible particles to the game. It's like trying to balance a seesaw by adding a whole new playground structure just to keep one kid from falling off. It works, but it's messy and not very elegant.

The Solution: A "Non-Invertible" Magic Rule
This paper proposes a clever, minimal trick using a concept called Non-Invertible Symmetry, specifically a rule known as the Fibonacci Fusion Rule (FFR).

Think of the universe's rules as a recipe book.

• The Old Way: To stop the exotic bricks from settling, you had to write a new, complex chapter in the recipe book that explicitly banned them.
• The New Way: The authors introduce a "magic rule" (the Fibonacci rule) that acts like a strict bouncer at a club.
  • At the "Tree Level" (The Main Entrance): The bouncer says, "No exotic bricks allowed to sit down here!" Because of this rule, the exotic Higgs fields (the quadruplet and quintet) are strictly forbidden from getting a value at the start. They are kept at zero.
  • At the "Loop Level" (The Back Door): However, the universe is quantum, meaning things wiggle and fluctuate. The paper shows that once the symmetry is slightly "broken" (like the bouncer taking a coffee break), these exotic fields can sneak in through a back door. But here's the catch: they can only enter through a one-loop process.

The "One-Loop" Analogy
Imagine trying to get a heavy box into a room.

• Tree-level: You just walk in and put it down. (This is forbidden).
• One-loop: You have to carry the box, go out the door, walk around the block, and come back in. This extra effort naturally makes the box much lighter when it finally arrives.

In physics terms, this "extra effort" is a quantum loop. Because the exotic fields only get their value through this loop, their final value is naturally tiny—suppressed by a factor of roughly $10^{-3}$ to $10^{-2}$ (0.001 to 0.01). This happens without needing to add any new particles to the universe. It's a self-contained trick using the existing rules.

The Results: Three New Scenarios
The authors tested this "magic bouncer" rule in three different scenarios for how neutrinos get their mass:

1. Type-III Seesaw: They added new heavy fermions (particles like electrons but heavier). The math shows this setup works perfectly up to incredibly high energy scales (even higher than the Planck scale), requiring only reasonable interaction strengths.
2. Dirac Seesaw: They used a different set of particles. Here, the "magic rule" keeps the exotic Higgs values small enough that the difference between how heavy electrons are and how light neutrinos are isn't as extreme as in other theories. It's a more "mild" difference.
3. Inverse Seesaw: This is the most complex setup. The authors found that the "magic rule" works, but the universe runs out of "room" for these rules at a lower energy (around 5 to 10 TeV). To make the numbers work, they had to tweak the parameters slightly, but it remains a viable, testable theory.

Why This Matters
The paper claims this is a highly minimal solution. Instead of cluttering the universe with new particles just to keep the exotic Higgs fields light, they used a fundamental symmetry rule (Fibonacci) to do the job.

• The Outcome: The exotic Higgs fields get values between 0.007 and 0.07 GeV.
• The Check: This is safely below the experimental limit (a few GeV) set by the "rho parameter" (a measure of how well the W and Z bosons balance each other).
• The Future: Because these new particles are predicted to be at the "TeV scale" (the energy range of the Large Hadron Collider and future colliders), this theory is testable. We don't need to wait for a new universe; we might be able to see the effects of these tiny, loop-generated values in upcoming experiments at the LHC, FCC, or CEPC.

In short, the paper says: "We found a way to keep the exotic Higgs fields naturally small using a Fibonacci symmetry rule. It's a clean, minimal trick that explains why these fields are light without needing extra clutter, and it fits perfectly with what we know about neutrinos."

Technical Summary

1. Problem Statement

The Standard Model (SM) successfully describes particle physics, but extensions involving large Higgs multiplets (e.g., quartets and quintets) are often proposed to explain phenomena like neutrino masses. However, a significant theoretical and phenomenological challenge arises:

• The $\rho$ Parameter Constraint: The introduction of large Higgs multiplets with non-zero vacuum expectation values (VEVs) typically causes a deviation in the $\rho$ parameter ($\rho = m_W^2 / (m_Z^2 \cos^2 \theta_W)$). Experimental data constrains $\rho \approx 1$ with deviations $< 2.5 \times 10^{-4}$. This implies that the VEVs of any new exotic multiplets must be suppressed to the order of GeV or lower (roughly two orders of magnitude smaller than the SM Higgs VEV, $v_H \approx 246$ GeV).
• Theoretical Challenge: While small VEVs are experimentally required, explaining their natural smallness without fine-tuning is difficult. Previous models often rely on:
  • Inert bosons.
  • New gauge symmetries (e.g., $U(1)_{B-L}$).
  • Specific loop-induced mechanisms that require additional particles solely to induce the loop.
    These approaches often lack minimality or introduce unnecessary complexity if the sole goal is VEV suppression.

2. Methodology

The authors propose a novel, minimal mechanism utilizing Non-Invertible Symmetries, specifically the Fibonacci Fusion Rule (FFR).

• Symmetry Framework:

  • The model employs the FFR algebra generated by two commutable elements: Identity ($I$) and a non-invertible element ($\tau$).
  • Multiplication rules: $\tau \otimes \tau = I \oplus \tau$, $I \otimes \tau = \tau$, $I \otimes I = I$.
  • Charge Assignment: The SM Higgs doublet ($H_2$) is assigned charge $I$ (neutral), while the exotic scalar fields—an $SU(2)_L$ quartet ($H_4$, $Y=1/2$) and a quintet ($H_5$, $Y=1$)—are assigned charge $\tau$.

• Scalar Potential Construction:

  • The scalar potential $V = V_2 + V_3 + V_4$ is constructed such that the FFR symmetry strictly forbids tree-level VEVs for $H_4$ and $H_5$.
  • Specifically, terms like $[H_2^\dagger H_4^* H_2^T H_2]$ that would generate a VEV for $H_4$ are forbidden at the tree level because the fusion of charges does not yield the identity ($I$).
  • Mass parameters $\mu_4^2$ and $\mu_5^2$ are chosen to be positive, ensuring $\langle H_4 \rangle = \langle H_5 \rangle = 0$ at tree level.

• Radiative Generation Mechanism:

  • The FFR symmetry is dynamically broken at the one-loop level.
  • The forbidden quartic coupling term $[H_2^\dagger H_4^* H_2^T H_2]$ is generated radiatively via interactions involving $\lambda_0$ and $\lambda_{H_2 H_4}$ couplings (see Fig. 1 in the paper).
  • This generates an effective coupling $\delta \lambda_{ij}$, which induces small VEVs $v_4$ and $v_5$ through the stationary conditions of the potential.
  • Key Advantage: This mechanism requires no additional loop-inducing particles; the necessary loops are formed by the existing scalar fields and their interactions.

3. Key Contributions

1. Minimal Mechanism for Tiny VEVs: The paper demonstrates that non-invertible symmetries can naturally suppress VEVs of large multiplets to the $10^{-3} - 10^{-2}$ GeV range without introducing extra matter fields or complex gauge sectors.
2. Renormalization Group Evolution (RGE) Analysis: The authors perform a rigorous RGE analysis of the $SU(2)_L$ gauge coupling ($g_2$) to determine the validity of the models. They calculate the new beta function contributions ($\Delta b$) introduced by the exotic multiplets ($H_4, H_5$) and fermions in specific neutrino models.
3. Application to Neutrino Mass Models: The framework is applied to three distinct neutrino mass generation scenarios, demonstrating its versatility:
   • Type-III Seesaw
   • Dirac Seesaw
   • Inverse Seesaw

4. Results

A. General VEV Suppression

Numerical analysis shows that for benchmark cut-off scales ($\Lambda$) ranging from $10^5$ GeV to the Planck scale ($10^{19}$ GeV), and assuming TeV-scale mass parameters:

• $v_4$ is suppressed to 0.0068 GeV – 0.0676 GeV.
• $v_5$ is further suppressed relative to $v_4$ (by a factor of $v_2/\mu_5$), reaching $\sim 10^{-4}$ GeV.
• These values satisfy the experimental bounds from the $\rho$ parameter (limiting VEVs to $\lesssim$ few GeV) by two to three orders of magnitude.

B. Application to Neutrino Models

Model	Key Features	Perturbativity Limit ($g_2 \le 1$)	Yukawa Coupling ($y_\nu$)	Validity Scale
Type-III Seesaw	Uses $H_4$ and triplet fermions $\Sigma_R$. $H_5$ not needed.	$g_2$ diverges at $\sim 10^{23}$ GeV (well above Planck scale).	$10^{-3} - 10^{-1}$	Up to Planck scale.
Dirac Seesaw	Uses $H_4, H_5$ and vector-like quartet fermions $\Sigma$.	$g_2$ exceeds 1 at $\sim 3 \times 10^5$ GeV.	$|y_\nu| \approx 2.08 \times 10^{-7}$	$\sim 10^5$ GeV.
Inverse Seesaw	Uses $H_4, H_5$, quartet $\psi$, and septet $\Sigma_R$.	$g_2$ exceeds 1 at 5 TeV (3 families) or 10 TeV (2 families).	Requires $|y_\nu| \sim 4\pi$ (marginal) or large $\mu_0$.	$\sim 5-10$ TeV.

• Type-III: Highly viable and perturbative up to very high scales.
• Dirac: Offers a milder hierarchy between charged lepton and neutrino Yukawa couplings compared to standard tree-level Dirac models.
• Inverse: The introduction of large multiplets (septet) accelerates the running of $g_2$, limiting the model's validity to the TeV scale. Achieving correct neutrino masses here requires careful tuning of parameters to remain within perturbative limits.

5. Significance

• Theoretical Novelty: This work is among the first to apply non-invertible symmetries (specifically FFR) to the problem of Higgs VEV suppression in particle phenomenology. It provides a rigorous group-theoretic justification for why exotic VEVs are zero at tree level but non-zero (and small) at one loop.
• Minimality: Unlike previous approaches that require "inert" sectors or new gauge groups to suppress VEVs, this mechanism relies solely on the symmetry properties of the existing scalar potential.
• Testability: The models predict new physics at the TeV scale. The exotic Higgs multiplets ($H_4, H_5$) and associated fermions are within the reach of future colliders such as the High-Luminosity LHC, FCC, CEPC, and Muon Colliders.
• Phenomenological Viability: The framework successfully reconciles the need for small VEVs (to satisfy $\rho$ constraints) with the generation of neutrino masses, offering a unified and testable extension of the Standard Model.

In conclusion, the paper establishes a robust theoretical foundation where non-invertible symmetries naturally explain the hierarchy of Higgs VEVs, opening new avenues for constructing minimal, testable models of neutrino mass and beyond-Standard-Model physics.