Scale Invariant Instantons and the Complete Lifetime of the Standard Model

TL;DR

The paper resolves long-standing issues in calculating vacuum decay rates for classically scale-invariant quantum field theories by regularizing the dilatation zero mode, summing leading higher-loop contributions, and deriving exact, gauge-invariant functional determinants for scalar, vector, and fermion fluctuations around the scale-invariant bounce. These developments yield a finite, RG-consistent lifetime for our universe within the Standard Model, with a predicted lifetime around 10^161 years and quantified uncertainties dominated by the top-quark mass, α_s, and threshold corrections. The work delivers a complete NLO tunneling-rate framework, phase diagrams in key SM parameter planes, and a robust discussion of mass corrections and constrained instantons, clarifying how high-scale physics could affect metastability. Overall, the results provide a precise, gauge-invariant portrait of SM vacuum stability and its profound cosmological implications, while highlighting the sensitivity to fundamental SM inputs and the potential impact of new physics at very high scales.

Abstract

In a classically scale-invariant quantum field theory, tunneling rates are infrared divergent due to the existence of instantons of any size. While one expects such divergences to be resolved by quantum effects, it has been unclear how higher-loop corrections can resolve a problem appearing already at one loop. With a careful power counting, we uncover a series of loop contributions that dominate over the one-loop result and sum all the necessary terms. We also clarify previously incomplete treatments of related issues pertaining to global symmetries, gauge fixing and finite mass effects. In addition, we produce exact closed-form solutions for the functional determinants over scalars, fermions and vector bosons around the scale-invariant bounce, demonstrating manifest gauge invariance in the vector case. With these problems solved, we produce the first complete calculation of the lifetime of our universe: 10^139 years. With 95% confidence, we expect our universe to last more than 10^58 years. The uncertainty is part experimental uncertainty on the top quark mass and on $αs$ and part theory uncertainty from electroweak threshold corrections. Using our complete result, we provide phase diagrams in the $mt/mh$ and the $mt/αs$ planes, with uncertainty bands. To rule out absolute stability to $3σ$ confidence, the uncertainty on the top quark pole mass would have to be pushed below 250 MeV or the uncertainty on $αs(mZ)$ pushed below 0.00025.

Figures (3)

• Figure 1: (Left) The real part of the action along family of field configurations $\phi(z)$ parameterized by a complex parameter $z$. $z$ is chosen so that real $z$ passes through ${\color{darkgreen} \phi_\text{FV}}$ (green dot), ${\color{darkred} \phi_b}$ (red dot), and ${\color{darkblue} \phi_\text{shot}}$ (blue dot). (Right) shows a top-down view. Integrating along real field configurations only (black dashed contour) makes the path integral real. The decay rate must be calculated by integrating along steepest descent contours (red and green contours) which involve complex field configurations.
• Figure 2: (Top) phase diagram for stability in the $m_t^{\text{pole}}/m_h^{\text{pole}}$ plane and closeup of the SM region. Ellipses show the 68%, 95% and 99% contours based on the experimental uncertainties on $m_t^{\text{pole}}$ and $m_h^{\text{pole}}$. The shaded bands on the phase boundaries, framed by the dashed lines and centered on the solid lines, are combinations of the $\alpha_s$ experimental uncertainty and the theory uncertainty. (Bottom) phase diagram in the $m_t^{\text{pole}}/\alpha_s(m_Z)$ plane, with uncertainty on the boundaries given by combinations of uncertainty on $m_h^{\text{pole}}$ and theory. The dotted line on the right plots is the naive absolute stability prediction using Eq. \ref{['lostab']}.
• Figure 3: Phase diagram for stability in the $m_t^{\text{pole}}/m_h^{\text{pole}}$ plane with dotted lines indicating the scale at which the addition of higher-dimension operators could stabilize the SM. Note that the curves accumulate on the stability/metastability boundary. $\Lambda_{\text{NP}}$ curves in the $\alpha_s/ {m_t^{\text{pole}}}$ plane (not shown) are similar.