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

TL;DR

The paper tackles how the SM couplings behave at high energy through renormalization-group equations, contrasting diagonal and non-diagonal loop orderings under Weyl consistency, and examining how matching and running influence vacuum stability. It employs state-of-the-art multi-loop RGEs, SM-to-observable matching at the electroweak scale, and polynomial fits to map on-shell inputs to MS parameters, enabling fast, reliable RG evolution. A key finding is that diagonal loop configurations generally reduce theoretical uncertainties, while non-diagonal configurations—though motivated by Weyl consistency—tend to increase predicted uncertainties in quantities like the vacuum decay probability. The study highlights the importance of a consistent loop-order treatment for both RGEs and matching when assessing SM vacuum stability and high-scale behavior, with implications for precision tests and beyond-Standard-Model explorations.

Abstract

Renormalization-group equations (RGE) is one of the key tools in studying high-energy behavior of the Standard Model (SM). We begin by reviewing one-loop RGE for the dimensionless couplings of the SM and proceed to the state-of-the-art results. Our study focuses on the RGE solutions at different loop orders. We compare not only the standard (``diagonal'') loop counting when one considers gauge, Yukawa, and scalar self-coupling beta functions at the same order but also ``non-diagonal'' ones, inspired by the so-called Weyl consistency conditions. We discuss the initial conditions for RGE (``matching'') for different loop configurations and study the uncertainties of running couplings both related to the limited precision of the experimental input (``parametric'') and the missing high-order corrections (``theoretical''). As an application of our analysis we also estimate the electroweak vacuum decay probability and study how the uncertainties in the running parameters affect the latter. We argue that ``non-diagonal'' beta functions, if coupled with a more consistent ``non-diagonal'' matching, lead to larger theoretical uncertainty than ``diagonal'' ones.

Figures (24)

• Figure 1: Renormalization group flow in variables $x$ (gauge) and $y$ (yukawa) defined by the first two equations \ref{['eq:dot00111']}. The flow is directed towards the UV region. Linear phase trajectories ① $x=0$, ② $y = 2/9 x$, and ③ $y=0$ are marked dividing the phase plane into two regions Ⓐ and Ⓑ.
• Figure 2: The direction field for equations \ref{['eq:RGE_YZ']} for $x\neq 0$ (left) and \ref{['eq:RGE_XZ']} for $y\neq0$ (right). The singular points $P_{i,j}$ corresponding to the rays passing through the origin in the phase space $(x,y,z)$ of the system \ref{['eq:dot00111']} are indicated. The projection of the rays $P_{1,i}$ onto the plane $(x,y)$ is the separatrix ① in Fig. \ref{['fig:00111x3y']}, while $P_{2,i}$ give rise to the separatrix ②, and $P_{3,i}$ to the separatrix ③ . Regions $\text{A}_{1,2}$ correspond to the condition $9 y>2 x$ Ⓐ, and $\text{B}_{1,2}$ to the condition $9 y< 2 x$ Ⓑ, (see Fig. \ref{['fig:00111x3y']}).
• Figure 3: Phase curves in the plane $x=0$ (left), $y=2/9 x$ (center), and $y=0$ (right) (see the corresponding separatrices ①, ②, and ③ in Fig. \ref{['fig:00111x3y']}) for the one-loop equations \ref{['eq:dot00111']}. The rays $P_{i,j}$ corresponding to the singular points in Fig. \ref{['fig:1L_XZ_YZ']} and defined by equations \ref{['eq:XZ_P_1_1']} - \ref{['eq:YZ_P_3_2']} are indicated.
• Figure 4: Direction fields for the cases $a_1=a_2=0$ (left), $a_1=0$, $a_2=42/19 a_3$ (middle) and $a_2=a_3=0$ (right), demonstrating the stability of some rays \ref{['eq:rays_1L_SM']} under varying the initial conditions for the Yukawa coupling constants of the $t$- and $b$-quarks. The pictures are qualitatively similar: $a_t=a_b=0$ is a UV stable node and $a_t=a_b\neq0$ is unstable. The rays on which one of the Yukawa constants vanishes turn out to be saddles. Note the absence of a linear trajectory passing through $a_\lambda = 0$ for the case $a_1=0$, $a_2=42/19 a_3$. This means that only when $a_1 = a_2 = a_3 = 0$ does the negative initial $a_\lambda$ tend from below to zero in the UV. Otherwise, $a_\lambda \to + \infty$.
• Figure 5: Solution of the one-loop RGE \ref{['eq:dot111111']} through \ref{['eq:fixed_initial_condition']}. The line $M_P$ denotes $t$, corresponding to the Planck scale ($1.2\cdot10^{19}~\text{GeV}$).