The Sphaleron Rate in the Minimal Standard Model (1404.3565v1)

Abstract: We use large-scale lattice simulations to compute the rate of baryon number violating processes (the sphaleron rate), the Higgs field expectation value, and the critical temperature in the Standard Model across the electroweak phase transition temperature. While there is no true phase transition between the high-temperature symmetric phase and the low-temperature broken phase, the cross-over is sharply defined at $T_c = (159\pm 1)$\,GeV. The sphaleron rate in the symmetric phase ($T> T_c$) is $\Gamma/T^4 = (18\pm 3)\alpha_W^5$, and in the broken phase in the physically interesting temperature range $130\mbox{\,GeV} < T < T_c$ it can be parametrized as $\log(\Gamma/T^4) = (0.83\pm 0.01)T/{\rm GeV} - (147.7\pm 1.9)$. The freeze-out temperature in the early Universe, where the Hubble rate wins over the baryon number violation rate, is $T_* = (131.7\pm 2.3)$\,GeV. These values, beyond being intrinsic properties of the Standard Model, are relevant for e.g. low-scale leptogenesis scenarios.

• The paper presents a quantitative determination of the sphaleron rate using reduced three-dimensional SU(2) lattice simulations to capture essential non-perturbative dynamics.
• It accurately computes key thermal parameters, finding a critical temperature of approximately 159±1 GeV and a freeze-out temperature of 131.7±2.3 GeV for baryon number violation.
• The methodology integrates high-temperature scales perturbatively while focusing on infrared physics, offering actionable insights for refining baryogenesis models and new physics explorations.

Insights into the Sphaleron Rate in the Minimal Standard Model

The paper "The Sphaleron Rate in the Minimal Standard Model" by D'Onofrio, Rummukainen, and Tranberg offers an in-depth quantitative exploration of baryon number violating processes within the framework of the Standard Model. Within this context, the authors utilize large-scale lattice simulations to compute the sphaleron rate, the Higgs field expectation value, and the critical temperature across the electroweak phase transition. These computations are pivotal in understanding the dynamics of baryon number non-conservation, which has implications for early Universe phenomenology, especially concerning scenarios like low-scale leptogenesis.

Methodology and Observations

The methodology leverages a reduced three-dimensional lattice simulation approach to effectively capture the non-perturbative dynamics of the Standard Model at high temperatures. This approach simplifies the full gauge theory while retaining the infrared dynamics crucial to sphaleron rate computation. The effective action employed selectively integrates high-temperature scales perturbatively and accounts for non-perturbative interactions through an infrared-sensitive, three-dimensional SU(2) gauge theory coupled with a scalar Higgs field. This technique bypasses the complexities of incorporating the full Standard Model directly on the lattice, focusing instead on capturing essential non-perturbative physics.

Key results showcased include:

• Critical Temperature (T_c): The cross-over temperature between the symmetric and broken phases was determined to be approximately $159 \pm 1$ GeV.
• Sphaleron Rate (Γ/T^4): In the symmetric phase, the rate is approximately $(18 \pm 3)\alpha_W^5$, a result derived through well-established classical real-time dynamics applied to the effective theories on the lattice.
• Freeze-out Temperature (T_*): The temperature at which the baryon number violation freezes out is identified as $131.7 \pm 2.3$ GeV. This freeze-out temperature is of particular interest as it indicates when baryon number violating processes were significant before being suppressed below detectability by the expansion of the Universe.

Implications and Speculations

The determined sphaleron rate's significance extends beyond mere theoretical interest—these findings provide critical input for baryogenesis calculations, impacting especially models positing new physics that operate just above the electroweak scale. A robust understanding of the rate is necessary for refining our calculations of baryon asymmetry observed in today's Universe.

Furthermore, the exploration of the sphaleron rate in the full Minimal Standard Model context forms a basis for analyzing rate modifications in extended theories, such as those incorporating additional scalar fields or supersymmetry. Such future pursuits will involve leveraging similar non-perturbative methodological frameworks adapted to these complex landscapes, allowing predictions about possible first-order phase transitions.

Concluding Remarks

This research successfully leverages sophisticated lattice simulation techniques to deliver a comprehensive account of baryon number violation rates within the Standard Model framework. The precise determination of the sphaleron rate across different phases underscores the existing theoretical toolkit's efficacy while hinting at necessary adaptations when exploring physics beyond the Standard Model. This paper thereby contributes significantly to the nuanced understanding of electroweak phase transitions and the associated dynamics critical to early Universe cosmology.