Standard Model Vacuum Stability and Weyl Consistency Conditions

Abstract: At high energy the standard model possesses conformal symmetry at the classical level. This is reflected at the quantum level by relations between the different beta functions of the model. These relations are known as the Weyl consistency conditions. We show that it is possible to satisfy them order by order in perturbation theory, provided that a suitable coupling constant counting scheme is used. As a direct phenomenological application, we study the stability of the standard model vacuum at high energies and compare with previous computations violating the Weyl consistency conditions.

• The paper demonstrates that applying Weyl consistency conditions organizes the renormalization group analysis by linking the β-functions of gauge, Yukawa, and Higgs quartic couplings.
• The paper introduces a 3-2-1 loop-counting scheme that preserves conformal symmetry by treating each SM coupling at appropriate perturbative orders.
• The paper reveals that refined perturbative corrections shift the Higgs quartic coupling negativity to lower scales, indicating a metastable vacuum state.

Standard Model Vacuum Stability and Weyl Consistency Conditions

The paper under discussion presents a detailed theoretical investigation into the stability of the Standard Model (SM) vacuum by harnessing Weyl consistency conditions. This study, authored by Antipin, Gillioz, Krog, Mølgaard, and Sannino, revisits the calculation of the renormalization group (RG) flow of the SM couplings, advocating for a perturbative expansion scheme that respects the conformal symmetry of the model at high energies.

Key Contributions

1. Weyl Consistency Conditions: At the heart of this work is the recognition of the symmetry properties inherent in the SM and the application of Weyl consistency conditions. These conditions, pivotal in ensuring the conformal invariance of the theory at the quantum level, impose strict relations between the β-functions of different couplings. The authors demonstrate that these relations can be properly satisfied through an organized loop counting method, facilitating a more coherent RG trajectory.
2. Perturbative Counting Scheme: The researchers propose a loop-counting methodology where gauge couplings, Yukawa couplings, and the Higgs quartic coupling are treated at distinct levels of loop accuracy. Specifically, they advocate for a 3-2-1 loop counting scheme for gauge, Yukawa, and quartic interactions respectively, aligned with the principle of maintaining the conformal symmetry encoded in the Weyl conditions.
3. Vacuum Stability Analysis: The paper robustly addresses the vacuum stability of the SM, a topic of significant interest given the measured Higgs mass and other parameter constraints. The study identifies a precarious state where the SM lies at the boundary between stability and instability near the Planck scale. Through the revised loop-counting procedures, the authors find that the perturbative corrections significantly impact the stability predictions, indicating the model enters a metastable region.

Numerical Results and Implications

The study leverages recent advances in multi-loop β-function calculations to evaluate the stability domain of the SM. Their precise computations affirm that under the new counting scheme, the scale at which the Higgs quartic coupling becomes negative shifts down relative to prior estimates. This renders the possibility of a metastable vacuum more pronounced unless certain conditions on parameters, such as the top-quark mass, are experimentally or theoretically revised.

Importantly, this work challenges earlier conclusions drawn from computations that might inadvertently break the Weyl consistency conditions by not faithfully adhering to the arranged loop orders. The results emphasize the necessity for high-order perturbative calculations, extending to a 4-3-2 or 5-4-3 loop counting, optimizing theoretical predictions vital for understanding the SM’s high-energy behavior and its potential deviations or extensions.

Future Directions

This paper lays the groundwork for further exploration into high-precision computations required for beyond-the-SM scenarios and offers a blueprint for simulating and analyzing these theories within a stringent theoretical framework. The implications of vacuum metastability, as revealed through this study, signal a compelling avenue for both experimental collaboration and theoretical refinement, with the potential to uncover new physics domains or validate the SM's efficacy up to Planck-scale energies.

In conclusion, by aligning loop orders with conformal invariance requirements, this paper provides a pivotal contribution to theoretical SM physics, carrying profound implications for the quest to delineate the ultimate fate of the vacuum and, consequently, the universe.