On the renormalization-group analysis of the SM: loops, uncertainties, and vacuum stability

Technical Summary: Renormalization-Group Analysis of the SM: Loops, Uncertainties, and Vacuum Stability

Problem Statement
The discovery of the Higgs boson has renewed interest in the high-energy behavior of the Standard Model (SM), particularly regarding the stability of the electroweak vacuum. This stability depends critically on the running of the Higgs self-coupling constant, $\lambda(\mu)$, which is governed by Renormalization Group Equations (RGEs). While multi-loop RGEs are essential for precise predictions, the treatment of perturbative orders (loop counting) and the matching of experimental observables to $\overline{\text{MS}}$ parameters introduce significant theoretical and parametric uncertainties. Specifically, there is an ongoing debate regarding "diagonal" loop configurations (where all couplings are calculated at the same loop order $L$) versus "non-diagonal" configurations (inspired by Weyl consistency conditions, where gauge, Yukawa, and scalar couplings are calculated at orders $L, L-1, L-2$, respectively). This paper aims to systematically compare these approaches, quantify the resulting uncertainties in running couplings, and assess their impact on the estimated probability of electroweak vacuum decay.

Methodology
The authors employ a comprehensive numerical and analytical framework to study the SM RGEs:

1. RGE Analysis: The study reviews one-loop to four-loop RGEs for dimensionless couplings (gauge $g, g', g_s$, Yukawa $y_t, y_b$, and Higgs self-coupling $\lambda$). The authors analyze the phase space of these couplings, identifying fixed points and separatrices to understand asymptotic behaviors (e.g., asymptotic freedom vs. Landau poles).
2. Loop Configurations: The paper compares "diagonal" configurations (e.g., 333, 444) with "non-diagonal" configurations (e.g., 321, 432) derived from Weyl consistency conditions and the existence of a monotonic $\tilde{A}$-function.
3. Matching Procedure: To determine initial conditions at the electroweak scale ($\mu_0 = 173.22$ GeV), the authors utilize the SMDR (Standard Model Dimensional Regularization) code. They relate physical on-shell observables (masses, Fermi constant, fine-structure constant) to $\overline{\text{MS}}$ parameters via a "matching" procedure.
   • Theoretical Uncertainty: Estimated by varying the matching scale $\bar{\mu}$ within the range $[\mu_0/2, 2\mu_0]$ and observing the shift in the extracted $\overline{\text{MS}}$ parameters.
   • Parametric Uncertainty: Estimated by propagating experimental errors from PDG2024 data through the matching relations. To facilitate this, the authors derived polynomial approximations (fits) for the dependence of $\overline{\text{MS}}$ parameters on experimental inputs, validated against the full SMDR calculation.
4. Vacuum Stability Calculation: The running of $\lambda(\mu)$ is extended to high scales (up to the Planck scale). The authors calculate the scale $\mu^*$ where $\lambda(\mu)$ is minimal and the scale $\mu_r$ where $\lambda(\mu_r)=0$. These are used to compute the vacuum decay probability $P$ via the bounce action formalism, $P \propto \exp(-1/6|\lambda^*|)$.

Key Contributions

• Systematic Uncertainty Quantification: The paper provides a detailed breakdown of theoretical vs. parametric uncertainties for initial conditions across various loop configurations (111, 222, 333, 321, 432, 433).
• Non-Diagonal vs. Diagonal Comparison: A rigorous comparison is made between standard diagonal loop counting and the non-diagonal approach suggested by consistency conditions.
• Polynomial Fits for Matching: The authors provide explicit polynomial coefficients (Tables III–VIII) that approximate the conversion from on-shell to $\overline{\text{MS}}$ parameters, allowing for efficient estimation of parametric uncertainties without repeated heavy numerical integration.
• Vacuum Decay Probability: The study computes $\log_{10} P$ for different loop configurations, explicitly showing how the choice of loop order affects the predicted stability of the vacuum.

Results

• Initial Conditions: The central values of the running couplings at $\mu_0$ show dependence on the loop configuration, but the parametric uncertainties (driven by experimental errors) remain largely independent of the loop order.
• Theoretical Uncertainties:
  • Theoretical uncertainties decrease as the number of loops increases for diagonal configurations.
  • Crucially, the authors find that "non-diagonal" configurations (e.g., 321, 432) yield significantly larger theoretical uncertainties compared to their "diagonal" counterparts (e.g., 333, 433).
  • For diagonal configurations starting at three loops, theoretical uncertainties become smaller than parametric uncertainties.
• Vacuum Stability:
  • The running of $\lambda(\mu)$ is highly sensitive to higher-order corrections in the top Yukawa ($\beta_t$) and strong gauge ($\beta_3$) beta functions.
  • The predicted vacuum decay probability varies drastically with loop order. For instance, the 111 configuration suggests a decay probability near unity (instability), while higher-order diagonal configurations (333, 433) suggest a much more stable vacuum (though still metastable).
  • The non-diagonal 321 configuration yields a decay probability compatible with the 222 diagonal case but with a much larger theoretical error band.

Significance and Claims
The paper argues that while Weyl consistency conditions provide a theoretical motivation for non-diagonal loop counting, their practical application in the context of vacuum stability analysis leads to larger theoretical uncertainties than the traditional diagonal approach. The authors conclude that for a consistent and precise analysis of the SM vacuum stability, one should employ "diagonal" beta functions coupled with "diagonal" matching (i.e., using the same loop order for both the RGEs and the extraction of initial conditions).

The authors emphasize that different observables have varying sensitivities to specific parameters, and the theoretical uncertainty is often dominated by the lowest available loop order for the most sensitive parameter. However, given the technical challenges of computing full higher-order terms, they advocate for the common approach of utilizing all state-of-the-art results available in the literature (as implemented in tools like SMDR and mr) while maintaining consistency in the loop counting between the evolution equations and the boundary conditions. The work serves as a critical reference for understanding the limitations and precision of current SM vacuum stability predictions.