Enhancing Phase Transition Calculations with Fitting and Neural Network

Abstract

The computation of bounce action in a phase transition involves solving partial differential equations, inherently introducing non-negligible numerical uncertainty. Deriving characteristic temperatures and properties of this transition necessitates both differentiation and integration of the action, thereby exacerbating the uncertainty. In this work, we fit the action curve as a function of temperature to mitigate the uncertainties inherent in the calculation of the phase transition parameters. We find that, after extracting a factor, the sixth-order polynomial yields an excellent fit for the action in the high temperature approximated potential. In a realistic model, the singlet extension of the Standard Model, this method performs satisfactorily across most of the parameter space after trimming the fitting data. This approach not only enhances the accuracy of phase transition calculations but also systematically reduces computation time and facilitates error estimation, particularly in models involving multiple scalar fields. Furthermore, we discussed the possible of using multiple neural networks to predict the action curve from model parameters.

Figures (9)

• Figure 1: Top: The variation of $S_{E}/T$ and $\beta/H$ with temperature for BP1 in the SSM, where the dotted line represent the raw data calculated using PhaseTracer2 and the $\beta/H$ values calculated from the raw $S_{E}/T$ data using \ref{['eq: derivative']}. Bottom: Same as top figure but for BP2. The gray dashed lines represent the nucleation temperatures for each BP.
• Figure 2: Fitting results for the one-dimensional high temperature approximated potential. Top left: $S/T$ as a functions of $T$ for different benchmark points. The dots represent raw data from the shooting method, while the curves show the fitted results. Top right: the RMSE for different fitting orders. The dotted lines correspond to direct fits of $S_E$ directly, while the solid lines represent fits to $S_E(T-T_C)^2$. Bottom left: the distribution of the RMSE from a random parameter scan. Bottom right: the correlation between the RMSE and the temperature interval $\Delta T$.
• Figure 3: Fitting results for the SSM model. The left panel displays the distribution of the RMSE from a random parameter scan. The blue and red histograms represent the distributions before and after data trimming, respectively. The right panel illustrates the relationship between RMSE and the temperature range $\Delta T$, with color indicating the Transition $T_C$.
• Figure 4: The action versus temperature for benchmark points in the SSM model. The dotted points represent raw data obtained from PhaseTracer, and the curves correspond to the fitted functions. The left panel displays cases with small RMSE, while the right panel shows those with large RMSE.
• Figure 5: Distribution of prediction errors for the critical temperature $T_C$ (blue) and the minimum temperature $T_{\rm min}$ (orange).