Thermal Resummation and Phase Transitions

Abstract

The consequences of phase transitions in the early universe are becoming testable in a variety of manners, from colliders physics to gravitational wave astronomy. In particular one phase transition we know of, the Electroweak Phase Transition (EWPT), could potentially be first order in BSM scenarios and testable in the near future. If confirmed this could provide a mechanism for Baryogenesis, which is one of the most important outstanding questions in physics. To reliably make predictions it is necessary to have full control of the finite temperature scalar potentials. However, as we show the standard methods used in BSM physics to improve phase transition calculations, resumming hard thermal loops, introduces significant errors into the scalar potential. In addition, the standard methods make it impossible to match theories to an EFT description reliably. In this paper we define a thermal resummation procedure based on Partial Dressing (PD) for general BSM calculations of phase transitions beyond the high-temperature approximation. Additionally, we introduce the modified Optimized Partial Dressing (OPD) procedure, which is numerically nearly as efficient as old incorrect methods, while yielding identical results to the full PD calculation. This can be easily applied to future BSM studies of phase transitions in the early universe. As an example, we show that in unmixed singlet scalar extensions of the SM, the (O)PD calculations make new phenomenological predictions compared to previous analyses. An important future application is the study of EFTs at finite temperature.

Figures (6)

• Figure 1: Various scalar mass contributions in $\phi^4$ theory: (a) one-loop mass correction, which is quadratically divergent at zero temperature, (b) higher-loop daisy contributions which are leading order in $T$ and $N$ at high temperature, (c) the two-loop "lollipop" contribution which is subleading in $T$ and $N$ to the two-loop daisy, (d) the three-loop superdaisy contribution, which is subleading in $T$ but of equal order in $N$ to the three-loop daisy.
• Figure 2: Complete set of 1- and 2- loop contributions to the scalar mass, as well as the most important higher loop contributions, in $\phi^4$ theory. The scaling of each diagram in the high-temperature approximation is indicated, omitting symmetry- and loop-factors. Diagrams to the right of the vertical double-lines only contribute away from the origin when $\langle \phi \rangle = \phi_0 > 0$. We do not show contributions which trivially descend from e.g. loop-corrected quartic couplings. Lollipop diagrams (in orange) are not automatically included in the resummed one-loop potential.
• Figure 3: Effective Higgs Potential (left) and mass corrections $\delta m_i^2$ (right) for the physical Higgs ($h$), Goldstones ($G$), and singlets ($S$) at $T = T_c$ as a function of $h$. Evaluated in TFD, PD and OPD resummation schemes for $N_S = 3$ and $(m_S, \lambda_{hSS}^\mathrm{loop}/v, \lambda_{S}^\mathrm{loop}) = (300 \;\mathrm{GeV}, 1.52, 0.5)$. In the right plot, $\delta m_h^2 = \delta m_G^2$ in the TFD scheme. The dots correspond to $\delta m_i^2$ in the PD scheme, with gaps indicating regions of the $h$-axis where no exact solution to the gap equation can be found, and the $\delta m_i^2(h, T)$ functions used to evaluate the potential are obtained by linearly interpolating between the obtained $\delta m_i^2$ solutions as a function of $h$. This gives nearly the same $V(h, T_c)$ as OPD. Note that the approximate equality of the three (O)PD mass corrections at the origin is a numerical coincidence for this parameter point. Furthermore, the differences in $v_c$, $T_c$ between TFD and (O)PD are modest here, but for other choices they can be much more pronounced. This is very important when $T_c \sim T_S$ and the predicted nature of the transition can change from one-step to two-step, as we discuss in Section \ref{['s.physical']}.
• Figure 4: Comparison of one-step phase transition in the new PD (blue) vs the standard TFD (red) calculation, for $N_S = 6$ and $(m_S, \lambda_{S}^\mathrm{loop}) = (150 \;\mathrm{GeV}, 1.0)$. The renormalization scale is set to $\mu_R = m_S$ (solid lines). Dashed (dotted) lines correspond to $mu_R = 2 m_S$ ($m_S/2$) to demonstrate the effect of scale variation. To the left of the curves, the PT is one-step and weakly first order or second order. To the right of the curves, $T_S > T_c$ and the transition is two-step for $\lambda_{hSS}(v)^\mathrm{loop} < \lambda_{hSS}^\mathrm{max}$. This upper bound is is set by the condition that EWSB vacuum is preferred and depends on $N_S, m_S, \lambda_S^\mathrm{loop}, \mu_R$ but not the choice of thermal resummation scheme.
• Figure 5: SM + $N_S \times S$ parameter space with a strong EWPT for $N_S = 3$ and $m_S = 250 \;\mathrm{GeV}$. Region between solid red (blue) lines: regions with strong one-step PT satisfying $v_c/T_c > 0.6$ for the standard TFD (new PD) calculation on the left, and $v_c/T_c > 1.0$ on the right. To the left of these lines, the PT is weakly first order or second order. Between the red (blue) lines and the green line, the PT is two-step in TFD (PD) calculation. To the right of the green line, the EWSB vacuum is not preferred at zero temperature (this does not depend on the thermal mass resummation scheme). Varying $\mu_R$ between 0.5 and 2 $m_S$ gives the variation indicated by the red/blue/green shading. To the right of the red (blue) dashed lines, $\Delta^\mathrm{TFD}$ ($\Delta^\mathrm{PD}$) $> 0.1$ for $\mu_R = m_S$. To the left of the black dotted line, the singlet is stable at the origin at zero temperature for $\mu_R = 1$.