# Symmetry Restoration and Breaking at Finite Temperature: An Introductory Review

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## Abstract

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## 1. Introduction

## 2. Symmetry Restoration and Order of Phase Transitions

## 3. Breakdown of Perturbative Expansion and Thermal Resummation

## 4. Electroweak Phase Transition

#### 4.1. Standard Model

- The above demonstration does not take the thermal resummation into consideration. If one performs it by either the Parwani or AE scheme, E would be diminished by the resummation effect as discussed in Section 3.
- The effective potential is the sum of 1-particle-irreducible diagrams by definition, which is inherently gauge dependent, and so the VEV defined by the minimum of the effective potential depends on a choice of specific gauge. This is natural consequence since the normalization of the scalar fields (wavefunction renormalization) is missing. Notwithstanding, energies at stationary points are gauge independent obeyed by the NFK identity [109,110].
- Since the phase transition is generically non-perturbative phenomenon, lattice calculations are more suitable to obtain robust results. It is shown in Refs. [140,141,142,143] that EWPT turns into smooth crossover for ${m}_{h}\gtrsim 73$ GeV. After the Higgs boson discovery, EWPT is re-analyzed in Ref. [144] in a different context. The “critical temperature" is found to be ${T}_{C}=159.5\pm 1.5$ GeV.

#### 4.2. Standard Model with a Real Scalar

#### 4.3. Perturbative Gauge-Invariant Treatment for the Thermal Phase Transitions

#### 4.4. Comparisons among Various Calculation Schemes

## 5. Summary and Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Thermally Corrected Field-Dependent Masses in the rSM

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**Figure 1.**Two types of phase transitions. (Upper) Case of the first-order phase transition; shapes of the effective potential at $T>{T}_{C}$, $T={T}_{C}$ and $T<{T}_{C}$ [left panel] and the temperature evolution of the VEV of scalar [right panel]. (Lower) Counterparts in the case of the second-order phase transition.

**Figure 2.**2 patterns of the phase transitions in the presence of the SM Higgs and real scalar fields: (i) $O\to \mathrm{EW}$ (1-step transition) and (ii) $O\to \mathrm{I}\to \mathrm{EW}$ (2-step transitions). In the latter case, the ${Z}_{2}$ symmetry is spontaneously broken in the intermediate phase (I phase) while restored in the EW phase, and S can become DM.

**Figure 3.**Comparisons among the four calculation schemes. The OS-like and $\overline{\mathrm{MS}}$ schemes are gauge-dependent while the PRM and HT schemes are gauge independent. For the PRM and $\overline{\mathrm{MS}}$ schemes, we take ${m}_{t}/2\le \overline{\mu}\le 2{m}_{t}$. We take ${m}_{S}={m}_{h}/2$ and ${\lambda}_{S}={\lambda}_{S}^{\mathrm{min}}+0.1$, where ${\lambda}_{S}^{\mathrm{min}}$ is determined by the vacuum condition. The plots are taken from Ref. [56].

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Senaha, E.
Symmetry Restoration and Breaking at Finite Temperature: An Introductory Review. *Symmetry* **2020**, *12*, 733.
https://doi.org/10.3390/sym12050733

**AMA Style**

Senaha E.
Symmetry Restoration and Breaking at Finite Temperature: An Introductory Review. *Symmetry*. 2020; 12(5):733.
https://doi.org/10.3390/sym12050733

**Chicago/Turabian Style**

Senaha, Eibun.
2020. "Symmetry Restoration and Breaking at Finite Temperature: An Introductory Review" *Symmetry* 12, no. 5: 733.
https://doi.org/10.3390/sym12050733