Baryon Washout, Electroweak Phase Transition, and Perturbation Theory

TL;DR

This work identifies a fundamental gauge-dependence in the conventional perturbative treatment of electroweak baryogenesis washout, showing that Tc and the standard BNPC can be contaminated by gauge artifacts. It develops a gauge-invariant approach based on the $ħ$-expansion and a gauge-independent sphaleron scale $ar{v}(T)$, derived via dimensional reduction, to determine Tc and estimate the sphaleron rate without gauge ambiguities. The authors propose a ring-sum prescription and discuss higher-order effects, concluding that two-loop finite-temperature potentials and a full gauge-invariant sphaleron calculation are likely necessary for reliable predictions in the SM and many BSM scenarios. They apply the method to the Standard Model as an illustrative case, finding that perturbative Tc values can be substantially sensitive to higher-order corrections and that non-perturbative lattice results still provide essential benchmarks for baryon washout viability. Overall, the paper provides a principled perturbative framework that yields gauge-independent criteria and underscores the need for higher-order and non-perturbative inputs to robustly assess baryon number preservation during the electroweak phase transition.

Abstract

We analyze the conventional perturbative treatment of sphaleron-induced baryon number washout relevant for electroweak baryogenesis and show that it is not gauge-independent due to the failure of consistently implementing the Nielsen identities order-by-order in perturbation theory. We provide a gauge-independent criterion for baryon number preservation in place of the conventional (gauge-dependent) criterion needed for successful electroweak baryogenesis. We also review the arguments leading to the preservation criterion and analyze several sources of theoretical uncertainties in obtaining a numerical bound. In various beyond the standard model scenarios, a realistic perturbative treatment will likely require knowledge of the complete two-loop finite temperature effective potential and the one-loop sphaleron rate.

Figures (9)

• Figure 1: a. Gauge-dependent scalar-QED type graph describing $\phi\phi$-scattering that contributes to $\Gamma[\phi,A]$. b. External-leg corrections necessary to remove gauge-dependence from $\phi\phi$-scattering amplitude, but absent from $\Gamma[\phi,A]$.
• Figure 2: A schematic illustration of the behavior of the exact effective potential as the gauge parameter $\xi$ is varied according to Nielsen's identity. The values of the potential at its extrema stay unchanged but the fields extremizing the potential are gauge-dependent.
• Figure 3: Value of $V_\text{eff}(\phi_\text{min})$ for three different phases as a function of $T$ as determined by (\ref{['Vcurve']}) in a hypothetical theory. Evolution of the universe proceeds from right to left, following the direction of the arrows. Intersections of the two lowest curves define the critical temperatures (there are two in this example).
• Figure 4: A representative ring diagram contributing to $\Delta V_\text{ring}(\phi,T)$. The Matsubara $n=0$ mode propagates in the lower loop. The smaller loops are hard thermal loops, each of which contributes a factor of $g^2T^2/m^2$.
• Figure 5: A comparison of the critical temperatures as computed using the following methods: the standard method (solid line); the gauge-independent methods described in the text, derived from the full theory (dashed lines); and by performing lattice simulations (arrow). Note that the lattice result is much higher than the perturbative estimations and is displayed on a separate scale.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof