The effective potential in the presence of several mass scales

TL;DR

The paper tackles computing the effective potential in mass-independent schemes when multiple φ-dependent mass scales are present. It introduces a decoupling-based strategy using field-dependent scales $M_i(\\phi)$ and a threshold-free one-loop form, with the optimal evaluation scale $\\mu^* = \\min_i{M_i(\\phi_c)}$ that minimizes $\\mu$-dependence and aligns $V_{eff}$ with the improved tree-level EFT. The authors illustrate the method in a Higgs–Yukawa model, showing no one-loop thresholds at $\\mu^*$ and that the potential reduces to the tree-level form in the EFT, then extend the analysis to two loops, where decoupling scales $\\mu_B^{(2)}$ and $\\mu_F^{(2)}$ suppress large logarithms and preserve perturbativity. They discuss the relation to EJ/FW multi-scale renormalization, demonstrating an exact correspondence with a FW-like scheme under suitable redefinitions. The work provides a practical, scalable framework for multi-scale effective potentials relevant to the SM and MSSM Higgs sector and stability analyses.

Abstract

We consider the problem of improving the effective potential in mass independent schemes, as e.g. the $\MSbar$ or $\DRbar$ renormalization scheme, in the presence of an arbitrary number of fields with $φ$-dependent masses $M_i(φ_c)$. We use the decoupling theorem at the scales $μ_i=M_i(φ_c)$ such that the matching between the effective (low energy) and complete (high energy) one-loop theories contains no thresholds. We find that for any value of $φ_c$, there is a convenient scale $μ^*\equiv\min_i\{M_i(φ_c)\}$, at which the loop expansion has the best behaviour and the effective potential has the least $μ$-dependence. Furthermore, at this scale the effective potential coincides with the (improved) tree-level one in the effective field theory. The decoupling method is explicitly illustrated with a simple Higgs-Yukawa model, along with its relationship with other decoupling prescriptions and with proposed multi-scale renormalization approaches. The procedure leads to a nice suppression of potentially large logarithms and can be easily adapted to include higher-loop effects, which is explicitly shown at the two-loop level.

Figures (4)

• Figure 1: a) Plot of the $m$ parameter as a function of $\mu$ for several values of $\phi_c$. b) Plot of the $m(\mu^*)$ parameter as a function of $\phi$, where $\mu^* = \min\{M_F,M_B\}$. In both figures, $\lambda=0.1$, $g=1$ and $m=1$ TeV at high scale $\mu_M \simeq 2\times 10^{4}$ GeV.
• Figure 2: Plot of the $\lambda(\mu^*)$ parameter as a function of $\phi$, where again $\mu^* = \min\{M_F,M_B\}$ and the initial condition are the same as in Fig. \ref{['fig:masa']}.
• Figure 3: Plot of the total one-loop potential (solid line) and the tree level potential (dotted line) as a function of the scale $\mu$ for two different values of $\phi_c$. In the left panel $\phi_c = 3\times 10^3$ GeV, so that $M_F>M_B$, while in the right panel $\phi_c = 200$ GeV, so that $M_B>M_F$. The decoupling scales, corresponding to $M_B$ and $M_F$, are marked by arrows. The vertical axis variable, $y$, represents the scalar potential in a convenient choice of units as described in the text.
• Figure 4: Plot of the $\mu^*$ as a function of $\phi$, according to the two choices (a) and (b) explained in the text.