Grand Unified Origin of Enhanced Scalar Couplings: Connecting Radiative Electroweak Symmetry Breaking to SO(10) Dynamics

This paper proposes that the enhanced Higgs quartic coupling required for radiative electroweak symmetry breaking naturally emerges from SO(10) grand unification, where portal couplings to heavy GUT scalars generate the necessary enhancement factor to align the Standard Model's UV Landau pole with the GUT scale, thereby providing a dynamical origin for both symmetry breaking and the electroweak-GUT hierarchy.

The Gist

Imagine the universe as a giant, complex machine. For decades, physicists have been trying to understand how the "engine" of this machine works, specifically how particles get their mass. A key part of this engine is a field called the Higgs field, and the particle associated with it is the Higgs boson.

This paper proposes a bold new idea: The Higgs boson isn't just a lonely particle; it's actually a tiny window looking out into a much larger, hidden world called Grand Unification.

Here is the story of the paper, broken down into simple concepts and analogies:

1. The Problem: A Wobbly Table

In our current understanding of physics (the Standard Model), the Higgs field is like a table that is balancing very precariously.

• The Issue: If you look at the math, the table is so unstable that it might tip over at very high energies (like those found just after the Big Bang). This suggests our current theory is incomplete or "metastable" (like a pencil balanced on its tip).
• The Clue: Some scientists noticed that if you make the Higgs field slightly "stiffer" (stronger), the table becomes rock-solid. But this "stiffness" factor was surprisingly high—about 6 to 7 times stronger than what we usually expect. Why would nature make it so stiff?

2. The Coincidence: The "Landmark"

The authors noticed something strange. When they calculated where this "stiff" Higgs field would break down (hit a wall), they found it happens at a specific energy level: $10^{16}$ GeV.

• The Analogy: Imagine you are driving a car and your speedometer hits a red line exactly at the same spot where a famous landmark (a mountain peak) is located. You wouldn't think it's a coincidence; you'd think the car was designed to stop there.
• The Reality: That "red line" (where the math breaks) matches the scale of Grand Unification, a theoretical energy level where all the forces of nature (electromagnetism, weak, and strong forces) merge into one single force. The paper argues this isn't a coincidence; it's a sign that the Higgs field is connected to this Grand Unification.

3. The Solution: The "Hidden Room"

The paper suggests that the Higgs boson we see is just the tip of the iceberg.

• The Metaphor: Imagine the Higgs boson is a small, light doorbell on the front of a massive, ancient castle (the Grand Unified Theory).
• The Mechanism: Inside the castle, there are heavy, giant sculptures (heavy particles from the Grand Unified Theory) that we can't see directly. However, the doorbell is connected to these sculptures by invisible wires (called "portal couplings").
• The Effect: When the heavy sculptures inside the castle wiggle, they pull on the wires, making the doorbell ring much louder and stiffer than it would on its own. This explains why the Higgs field is so "stiff" (enhanced). The "stiffness" isn't a mystery; it's the echo of the heavy physics happening inside the Grand Unified castle.

4. The Origin Story: A Self-Starting Engine

How did the universe get started? Usually, we think things need a push to start moving.

• The Analogy: The authors propose a self-starting engine (called the Coleman-Weinberg mechanism).
• How it works: Imagine a ball sitting perfectly flat on a hill. It shouldn't roll. But, if you add a tiny bit of wind (quantum fluctuations/radiative corrections), the ball suddenly finds a spot to roll down.
• The Result: This mechanism explains two things at once:
  1. Why the Grand Unified forces broke apart to create the forces we see today.
  2. Why the Higgs field got its specific "stiffness" to create the mass of particles.
     It turns the "accidental" stability of the universe into a designed feature.

5. How Do We Know? (The Experiments)

The authors don't just tell a story; they give us a checklist to prove it.

• The "Knock" Test (Higgs Self-Coupling): If you hit two Higgs bosons together, they should bounce off each other in a very specific, strong way. The paper predicts this "bounce" will be about 6 times stronger than the standard prediction. The Large Hadron Collider (LHC) is currently checking this. If the number is around 6, the theory wins.
• The "Ghost" Test (Proton Decay): Grand Unified theories predict that protons (the building blocks of atoms) might eventually decay. The paper suggests this will happen at a rate that future giant detectors (like Hyper-Kamiokande) might catch.
• The "Echo" Test (Gravitational Waves): The breaking of these giant forces in the early universe would have created ripples in space-time (gravitational waves). The paper predicts we might hear these ripples as a "hum" from cosmic strings, detectable by future telescopes.

Summary

This paper argues that the Higgs boson is not an isolated oddity. Its unusual strength is a direct message from a hidden, massive world of Grand Unification.

• The "Stiffness" of the Higgs is caused by heavy, invisible particles pulling on it.
• The "Breakdown" of the math happens exactly where the Grand Unified forces begin.
• The Proof lies in measuring how Higgs bosons interact with each other, listening for proton decay, and detecting ancient gravitational waves.

If the experiments confirm these predictions, we will have taken the first real step toward understanding the "source code" of the universe, connecting the tiny world of particles to the grandest scale of cosmic forces.

Technical Summary

1. Problem Statement

The Standard Model (SM) faces a critical theoretical puzzle regarding vacuum stability. The measured Higgs quartic coupling, $\lambda(M_t) \approx 0.126$, runs negative at energy scales around $\mu \sim 10^{10}$ GeV, rendering the electroweak vacuum metastable.

• The RBEWS Scenario: Previous work by Steele and Wang proposed a Radiatively Broken Electroweak Symmetry (RBEWS) scenario where the Higgs quartic coupling is significantly enhanced ($k \approx 7.2$) to ensure absolute vacuum stability.
• The Coincidence: In this enhanced scenario, the coupling grows rapidly, leading to a Landau pole (UV divergence) at $\Lambda_{UV} \approx 1.5 \times 10^{16}$ GeV. This scale is remarkably close to the Grand Unified Theory (GUT) scale ($M_{GUT} \sim 2 \times 10^{16}$ GeV) where gauge couplings unify.
• The Gap: It was previously unclear whether this proximity was accidental or if the enhanced coupling had a dynamical origin within a GUT framework. Furthermore, the raw RBEWS prediction ($k \approx 7.2$) needed rigorous translation from the Coleman-Weinberg (CW) scheme to the $\overline{\text{MS}}$ scheme used in experimental constraints to determine if it was viable against current LHC data.

2. Methodology

The author employs a multi-step theoretical framework combining Renormalization Group (RG) evolution, effective field theory (EFT) matching, and GUT model building:

1. Scheme Conversion and RG Evolution:

   • The paper re-evaluates the Steele-Wang enhancement factor $e_{125} = 7.2$ (defined in the CW scheme at the electroweak VEV $v$).
   • It applies two corrections to translate this to the $\overline{\text{MS}}$ scheme at the top mass scale ($M_t$):
     • Scheme Conversion: A residual difference between CW and $\overline{\text{MS}}$ couplings ($\sim 3\text{--}5\%$ reduction).
     • Scale Running: The ratio $k(\mu) = \lambda_{\text{enh}}(\mu)/\lambda_{\text{SM}}(\mu)$ evolves due to the non-linear $12\lambda^2$ term in the beta function. Running from $v$ down to $M_t$ reduces the ratio by $\Delta k \approx -0.5 \text{ to } -0.8$.
   • This yields a corrected enhancement factor: $k(M_t) \approx 6.0\text{--}6.4$.

2. SO(10) Threshold Corrections:

   • The paper constructs an SO(10) Grand Unified Theory containing scalar representations $\mathbf{10}_H$, $\mathbf{126}_H$, and $\mathbf{45}_H$.
   • It calculates threshold corrections to the light Higgs quartic coupling generated when heavy GUT scalars are integrated out at the matching scale ($M_{GUT}$).
   • The mechanism relies on portal couplings (e.g., $\lambda_{12}|\mathbf{10}_H|^2|\mathbf{126}_H|^2$) between the light Higgs doublet and heavy GUT scalars.

3. Coleman-Weinberg (CW) Dynamics:

   • The paper advocates for a classically scale-invariant SO(10) scalar potential (Gildener-Weinberg approach).
   • Symmetry breaking is driven entirely by radiative corrections (dimensional transmutation) at both the GUT scale (breaking SO(10)) and the electroweak scale (RBEWS).

3. Key Contributions and Results

A. Resolution of the "Accidental" Coincidence

The paper demonstrates that the UV Landau pole at $\Lambda_{UV} \sim 1.5\text{--}2 \times 10^{16}$ GeV is not a pathology but a physical signal. It marks the scale where the SM effective description breaks down and must be embedded into the full SO(10) structure. The proximity of $\Lambda_{UV}$ to $M_{GUT}$ is interpreted as a direct consequence of the unified dynamics.

B. Quantitative Generation of Enhancement

The author derives that portal couplings between the light Higgs and heavy scalars ($\mathbf{126}_H$, $\mathbf{45}_H$) naturally generate the required enhancement.

• Mechanism: One-loop diagrams involving heavy scalars contribute to the Higgs quartic coupling:
  $$\Delta \lambda \approx \frac{1}{16\pi^2} \sum_{\Phi} n_\Phi \lambda_\Phi^2 \ln\left(\frac{M_{GUT}}{M_\Phi}\right)$$
• Result: For portal couplings $\lambda_\Phi \sim 0.5\text{--}1.0$ and mass hierarchies $M_{GUT}/M_\Phi \sim 10\text{--}100$, the threshold correction $\Delta \lambda \sim 0.15\text{--}0.35$. This is sufficient to boost the coupling from the SM value to the required $\lambda(M_{GUT}) \sim 0.25\text{--}0.40$, corresponding to $k(M_t) \approx 6.0\text{--}6.4$.

C. Experimental Viability (Higgs Self-Coupling)

A critical result is the reconciliation of the RBEWS prediction with current LHC data.

• The corrected enhancement factor $k(M_t) \approx 6.0\text{--}6.4$ predicts a trilinear coupling modifier $\kappa_\lambda \approx 6.0\text{--}6.4$.
• This is consistent with the latest ATLAS constraint ($95\%$ CL) from Run 2+3 data ($b\bar{b}\gamma\gamma$ channel): $-1.7 < \kappa_\lambda < 6.6$.
• The paper clarifies that GUT threshold corrections to the trilinear coupling are power-suppressed ($\sim 10^{-32}$) and negligible, while corrections to the quartic coupling are logarithmically enhanced. Thus, the tree-level relation $\kappa_\lambda \approx k$ remains robust, with only minor $\mathcal{O}(10\%)$ corrections from the CW potential structure.

D. Dynamical Origin of Hierarchies

The framework utilizes the CW mechanism to explain the hierarchy between the electroweak scale ($v$) and the GUT scale ($M_{GUT}$).

• $M_{GUT}$ is generated dynamically via gauge loops.
• $v$ is generated via RBEWS acting on the light Higgs sector, inheriting the enhanced coupling from GUT threshold effects.
• The ratio $v/M_{GUT} \sim 10^{-14}$ is naturally produced through dimensional transmutation.

4. Phenomenological Implications

The paper outlines a multi-pronged experimental test program:

1. High-Luminosity LHC (HL-LHC): The prediction $\kappa_\lambda \approx 6.0\text{--}6.4$ is a definitive target for the HL-LHC (3 ab$^{-1}$), which aims for $\sim 50\%$ precision on $\kappa_\lambda$.
2. Proton Decay: The SO(10) framework with intermediate symmetry breaking (e.g., Pati-Salam) predicts proton decay lifetimes ($\tau_p \sim 10^{35}$ years) within the reach of Hyper-Kamiokande.
3. Gravitational Waves (GW):
   • Cosmic Strings: Breaking of $U(1)_{B-L}$ at an intermediate scale ($M_I \sim 10^{11}\text{--}10^{14}$ GeV) generates cosmic strings. The resulting stochastic GW background ($G\mu \sim 10^{-11}\text{--}10^{-10}$) is detectable by Pulsar Timing Arrays (NANOGrav, EPTA) and LISA.
   • Electroweak Phase Transition: The enhanced quartic coupling strengthens the electroweak phase transition to a strong first-order transition, producing a distinct GW signal at millihertz frequencies (detectable by LISA/Einstein Telescope), distinct from the cosmic string signal.

5. Significance

This work provides a unified theoretical origin for three seemingly disparate phenomena:

1. The 125 GeV Higgs mass and the requirement for vacuum stability.
2. The enhanced Higgs self-coupling predicted by RBEWS.
3. Grand Unification in SO(10).

By interpreting the UV Landau pole as a physical threshold rather than a failure of the theory, the paper bridges the gap between low-energy precision Higgs physics and high-energy GUT dynamics. It transforms the RBEWS scenario from a phenomenological fix into a testable prediction of SO(10) physics, offering a coherent narrative where the enhanced scalar coupling is a direct remnant of GUT-scale portal interactions. The framework is falsifiable through upcoming measurements of Higgs self-coupling, proton decay, and gravitational wave backgrounds.