Vacuum Stability in the Standard Model and Beyond

This paper re-evaluates Standard Model vacuum stability using high-order perturbations and precise parameters to show that improved measurements of the top mass and strong coupling could confirm stability at the $5\sigma$ level, while also exploring viable singlet scalar extensions that remain stable up to the Planck scale and predicting significant deviations in Higgs couplings testable at current and future colliders.

The Gist

Imagine the universe is built on a giant, invisible trampoline. This trampoline represents the vacuum of space—the lowest energy state where everything sits comfortably. In our current understanding of physics (the Standard Model), this trampoline isn't perfectly flat; it has a small dip where we live (the "electroweak vacuum"), but there's a terrifyingly deep, bottomless pit far away in the distance.

The big question this paper asks is: Are we safe? Is our little dip stable, or could the universe suddenly fall into that deep pit, destroying everything?

Here is a breakdown of the paper's findings using everyday analogies:

1. The "Wobbly Trampoline" Problem

For a long time, physicists have suspected that our universe's vacuum is metastable. Think of it like a ball sitting in a small valley on a hillside. It looks safe, but if you push it hard enough (or if the ground shifts), it could roll over the edge and plummet into the deep pit below.

The authors of this paper took a fresh look at this problem using the most precise measurements and advanced math available today. They found that the stability of our universe depends heavily on two specific "knobs" on the physics machine:

• The Top Quark Mass: Think of this as the weight of a heavy boulder sitting on the trampoline.
• The Strong Coupling Constant: Think of this as the "stickiness" or tension of the trampoline fabric itself.

The Finding: Currently, our measurements of these two knobs suggest the ball is very close to the edge of the cliff. If the top quark is just a tiny bit lighter, or the trampoline tension is just a tiny bit different, we might actually be safe. But right now, the data says we are teetering on the edge.

The Solution: The authors say we need to measure these two knobs much more precisely (about 2 to 3 times better than we can now). If we do, we will finally know for sure if the universe is safe or if we are living on borrowed time.

2. The "Safety Net" (Higgs Portals)

Since we can't change the laws of physics to make the trampoline safer, the authors ask: Can we add a safety net?

They explored adding new, invisible particles to the universe (called "Singlet Scalars"). These particles act like a Higgs Portal.

• The Analogy: Imagine the trampoline is sagging. If you attach a new, strong rope (the portal) to a heavy anchor (the new particle) underneath it, that rope pulls the trampoline up, making the valley deeper and the cliff edge further away.
• The Result: They found that if these new particles exist with the right weight and connection strength, they can completely stabilize the universe, making the "deep pit" disappear. The universe becomes "Planck-safe," meaning it's stable all the way up to the highest energy levels imaginable.

They tested many different types of these "safety nets":

• Simple nets: Just one new particle.
• Complex nets: Families of particles that interact with each other.
• The "Bad" Nets: They found that if the connection (the portal) pulls the wrong way (negative), it actually makes the trampoline less stable, causing it to snap. So, the safety net must pull up, not down.

3. Catching the "Safety Net" in Action

If these new particles exist to save the universe, how do we find them? They would be hiding, but they would leave footprints on the Higgs boson (the particle that gives other particles mass).

The authors calculated how these new particles would change the behavior of the Higgs:

• The "Shake": The Higgs boson would interact slightly differently with other particles (like the Z boson).
• The "Bounce": The Higgs would interact with itself in a new way.

Where to look:

• The HL-LHC (High-Luminosity Large Hadron Collider): This is our current super-collider. It might be able to see the "Shake" (changes in how the Higgs talks to other particles).
• The FCC-hh (Future Circular Collider): This is a proposed, massive future collider. It is powerful enough to see the "Bounce" (how the Higgs interacts with itself). The authors say this is crucial because the "Bounce" could be huge—like the Higgs suddenly becoming 10 times more energetic in its self-interactions.

Summary

• The Problem: Our universe might be unstable, like a ball on a cliff edge, depending on two poorly measured numbers (Top Quark Mass and Strong Coupling).
• The Fix: We need better measurements to know if we are safe.
• The Backup Plan: If we are not safe, new invisible particles (Singlet Scalars) could act as a safety net to stabilize the universe.
• The Hunt: We can find these particles by looking for subtle changes in how the Higgs boson behaves. Some changes might be seen soon at the LHC, but the most dramatic changes will require a massive future collider.

In short: We are checking if the universe is about to fall off a cliff. If it is, there might be a hidden safety net holding us up, and we are building bigger telescopes to find it.

Technical Summary

Here is a detailed technical summary of the paper "Vacuum Stability in the Standard Model and Beyond" by Hiller et al.

1. Problem Statement

The Standard Model (SM) of particle physics, while experimentally successful, faces a theoretical crisis regarding the stability of the electroweak vacuum. Current measurements of the Higgs boson mass ($M_h$) and the top quark mass ($m_t$) suggest that the SM Higgs potential may become negative at high energy scales (around $10^{10}$ GeV), implying a metastable vacuum rather than an absolutely stable one. While the tunneling rate to a deeper vacuum is currently estimated to be longer than the age of the universe, the margin of stability is extremely sensitive to input parameters.

The paper addresses three core questions:

1. What is the precise status of SM vacuum stability given the most up-to-date experimental inputs and highest-order theoretical calculations?
2. Can minimal extensions of the SM (specifically via the Higgs portal) render the vacuum stable ("Planck-safe") up to the Planck scale ($M_{Pl} \approx 10^{19}$ GeV)?
3. What are the phenomenological signatures of such stable extensions at current and future colliders?

2. Methodology

A. Re-evaluating SM Stability

• Input Parameters: The authors utilize the 2024 PDG central values and uncertainties for key observables: Higgs mass ($M_h$), top mass ($m_t$, including pole, Monte-Carlo, and $\overline{MS}$ definitions), and the strong coupling constant ($\alpha_s^{(5)}(M_Z)$).
• Theoretical Precision: They employ the highest available orders in perturbation theory:
  • Effective Potential: Calculated using a renormalization group (RG) improved ansatz. The effective quartic coupling $\alpha_{\lambda, \text{eff}}$ is determined by matching fixed-order calculations (up to 4-loop QCD, 3-loop Yukawa, and 2-loop electroweak) to the RG-improved potential.
  • Resummation: Large logarithms ($\ln(h/\mu)$) are resummed using 5-loop QCD beta functions, 4-loop Yukawa/quartic beta functions, and 3-loop anomalous dimensions.
  • Gauge Independence: Calculations are performed in Landau gauge, ensuring the depth of the potential minimum is gauge-independent.

B. BSM Stability Analysis (Higgs Portals)

• Models: The study investigates scalar singlet extensions of the SM, including:
  • Real/Complex singlets with $O(N)$ or $U(N)$ symmetry.
  • Flavorful matrix scalars $S_{ij}$ with $SU(N_F) \times SU(N_F)$ symmetry.
• Mechanism: The "Higgs Portal" interaction $\delta (H^\dagger H)(S^\dagger S)$ is introduced. This coupling modifies the RG evolution of the Higgs quartic coupling $\lambda$.
• RG Evolution: Using the tool ARGES, the authors perform 2-loop RG running from the new physics scale ($\mu_0 \sim M_S \gtrsim 1$ TeV) up to $M_{Pl}$.
• Criteria for "Planck-Safety":
  • Strict: $\alpha_\lambda(\mu) > 0$ for all $\mu \le M_{Pl}$ and no Landau poles.
  • Soft: $\alpha_\lambda(\mu)$ may dip slightly negative (but not more than in the SM) at intermediate scales, provided $\alpha_\lambda(M_{Pl}) > 0$.
• Constraints: The analysis incorporates tree-level perturbative unitarity bounds and experimental limits on Higgs-BSM mixing angles.

C. Phenomenology

• The authors calculate the impact of scalar mixing on Higgs couplings:
  • Trilinear self-coupling ($\kappa_3$).
  • Quartic self-coupling ($\kappa_4$).
  • Coupling to electroweak bosons ($ZZh$).
• They compare these deviations against projected sensitivities of the HL-LHC, FCC-ee, and FCC-hh.

3. Key Contributions and Results

A. Status of SM Vacuum Stability

• Critical Dependence: The stability of the vacuum is centrally determined by the top mass ($m_t$) and the strong coupling ($\alpha_s$). The Higgs mass uncertainty is sub-dominant due to the smallness of the Higgs quartic coupling itself.
• Current Status: Using PDG 2024 central values, the SM vacuum is metastable. The effective quartic coupling turns negative at $\Lambda_{0, \text{eff}} \approx 5.3 \times 10^{11}$ GeV.
• Required Shifts: To achieve absolute stability ($\alpha_{\lambda, \text{eff}} > 0$) at the $5\sigma$ level:
  • $m_t$ would need to decrease by $\sim 1.9\sigma$ (if using pole mass).
  • $\alpha_s$ would need to increase by $\sim 3.7\sigma$.
  • Alternatively, a combination of shifts within $2\sigma$ (uncorrelated) or specific correlated shifts (as suggested by CMS data) could resolve the tension.
• Precision Goal: The authors estimate that reducing uncertainties in $m_t$ and $\alpha_s$ by a factor of 2–3 (e.g., $m_t$ uncertainty to $\sim 200-300$ MeV) would be sufficient to definitively confirm or refute SM stability at the $5\sigma$ level.

B. BSM Stability via Higgs Portals

• Stabilization Mechanism: The portal coupling $\delta$ provides a positive contribution to the beta function of the Higgs quartic ($\beta_\lambda \propto +\delta^2$), counteracting the negative contribution from the top quark.
• Parameter Space:
  • $O(N_S)$ Models: A wide range of parameters allows for Planck safety. For $M_S \sim 1$ TeV, a portal coupling $\alpha_\delta \sim 10^{-3} - 10^{-2}$ is sufficient.
  • Indirect Stabilization: Even if $\delta$ is very small, a large pure BSM quartic coupling ($\alpha_v$) can drive $\delta$ to larger values via RG mixing, indirectly stabilizing the Higgs potential.
  • Walking Regimes: The authors identify "walking" regimes where couplings remain near a pseudo-fixed point for orders of magnitude, significantly expanding the viable parameter space for stability.
• Negative Portals: The study proves that negative portal couplings ($\delta < 0$) are not Planck-safe. They inevitably lead to Landau poles or vacuum instabilities below $M_{Pl}$ due to the technical naturalness of the coupling (it runs towards more negative values).
• Flavorful Models: In $SU(N_F) \times SU(N_F)$ models, negative BSM quartics ($\alpha_u$ or $\alpha_v$) are allowed and can lead to stable vacua, offering explanations for flavor anomalies (e.g., lepton flavor universality violation).

C. Phenomenological Signatures

• Mixing Effects: The mixing between the SM Higgs and the BSM scalar suppresses the Higgs coupling to $ZZ$ by $\cos \beta$ and modifies self-couplings.
• Trilinear Coupling ($\kappa_3$): Deviations are typically within 10–20% for TeV-scale BSM masses. These are within reach of the HL-LHC.
• Quartic Coupling ($\kappa_4$): This is the most sensitive probe. The quartic coupling can be enhanced by factors of 2 to 10 (or more) in stable models. This requires a high-energy, high-precision collider like the FCC-hh to detect.
• Diboson Coupling ($ZZh$): Current limits constrain the mixing angle $|\sin \beta| \le 0.2$. Future colliders (FCC-ee) could improve this sensitivity by an order of magnitude.

4. Significance

• Precision Benchmark: The paper establishes the current theoretical and experimental precision limits on SM vacuum stability, highlighting that the top mass and strong coupling are the bottlenecks. It provides a clear roadmap for future $e^+e^-$ colliders to resolve the stability question.
• Model Building Guide: It demonstrates that "Planck-safe" extensions are not only possible but generic in scalar singlet models. It identifies specific "safe" regions in parameter space that avoid Landau poles and vacuum decay.
• Negative Portals Excluded: A significant theoretical result is the exclusion of negative Higgs portal couplings for achieving absolute stability up to the Planck scale, narrowing the scope of viable BSM models.
• Collider Strategy: The work delineates a clear experimental strategy:
  • Near-term (HL-LHC): Probe Higgs trilinear couplings and $ZZh$ deviations to constrain mixing.
  • Long-term (FCC-hh): Essential for probing the Higgs quartic coupling, which is the smoking gun for the stabilization mechanism in many Planck-safe scenarios.
• Flavor Connection: By including flavorful matrix scalars, the paper bridges the gap between vacuum stability and flavor anomalies, suggesting that the mechanism stabilizing the vacuum could also explain observed deviations in B-physics.

In conclusion, the paper argues that while the SM vacuum is currently metastable, it is on the verge of being confirmed as stable or unstable with modest improvements in precision. Furthermore, minimal scalar extensions via the Higgs portal offer a robust, testable pathway to a fundamentally stable universe up to the Planck scale.