Symmetry Nonrestoration in the Pati-Salam Model

Technical Summary: Symmetry Nonrestoration in the Pati–Salam Model

Problem Statement
In standard cosmological scenarios involving grand unified theories (GUTs), a significant constraint arises from the thermal history of the universe following inflation. If the reheating temperature ($T_R$) after inflation exceeds the temperature at which a specific gauge symmetry is restored ($T_{res}$), that symmetry is thermally restored. Upon subsequent cooling, the symmetry breaks again, potentially producing topological defects such as 't Hooft-Polyakov monopoles.

For the Pati–Salam (PS) model, based on the gauge group $G_{PS} = SU(2)_L \otimes SU(2)_R \otimes SU(4)_c$, the breaking to the Standard Model (SM) group $G_{SM}$ typically implies a symmetry restoration temperature $T_{res}$ comparable to the symmetry breaking scale. Current observational bounds on the tensor-to-scalar ratio ($r < 0.036$) from Planck 2018 and BICEP/Keck 2018 suggest a reheating temperature $T_R \lesssim 5.8 \times 10^{15}$ GeV. Future experiments may probe down to $T_R \simeq 2.4 \times 10^{15}$ GeV. If the PS symmetry breaking scale is intermediate ($\sim 10^{12} - 10^{14}$ GeV) or lower, the condition $T_R > T_{res}$ would likely be met, leading to an overproduction of monopoles and ruling out these lower-scale models. The paper addresses whether this constraint is unavoidable or if the symmetry can remain broken even at temperatures significantly above the nominal breaking scale.

Methodology
The authors analyze the thermal behavior of the PS model using a specific Higgs sector and finite-temperature field theory techniques:

1. Higgs Sector: The model utilizes Higgs fields in the representations $(1, 2, 4)$, $(1, 2, 4)$, and $(2, 2, 1)$ of $G_{PS}$. The first two fields ($\Phi$ and $\Xi$) drive the breaking of $G_{PS}$ to $G_{SM}$ at a high scale $v$, while the third ($H$) breaks $G_{SM}$ to the low-energy group at a much smaller scale $\epsilon$.
2. Potential Construction: The authors construct the most general $G_{PS}$-invariant scalar potential up to quartic order. They focus on the tree-level potential for the fields $\Phi$ and $\Xi$, incorporating leading-order finite-temperature corrections ($O(T^2)$) in the high-temperature expansion.
3. Stability Analysis:
   • Zero Temperature: They derive conditions on the quartic coupling constants ($a_1$ through $a_{10}$) required to ensure the local stability of the vacuum that breaks $G_{PS}$ to $G_{SM}$. This involves analyzing the mass matrix eigenvalues and ensuring the vacuum expectation values (VEVs) are non-zero.
   • Finite Temperature: The leading thermal corrections are calculated by taking the trace of the second derivatives of the potential with respect to the Higgs fields. These corrections effectively shift the quadratic mass terms ($a_1, a_2$) to temperature-dependent coefficients $\tilde{a}_1(T)$ and $\tilde{2}(T)$.
4. Symmetry Nonrestoration Condition: The authors investigate whether the broken-phase vacuum persists at high temperatures. They derive inequalities relating the quartic couplings such that the effective quadratic coefficients remain negative even as $T \to \infty$.

Key Contributions and Results

• Derivation of Stability Inequalities: The paper establishes a set of inequalities for the quartic couplings (specifically involving $a_4, a_5, a_6, a_7, a_8, a_9, a_{10}$) that guarantee the stability of the symmetry-broken vacuum at zero temperature.
• Mechanism for Nonrestoration: The authors demonstrate that for specific choices of these couplings, the thermal corrections to the mass terms can be negative. Specifically, if the combination of couplings satisfies:
  $$a_6 + 4a_7 + a_{10} < -18a_4 - 12a_8$$
  and
  $$a_6 + 4a_7 + a_{10} < -18a_5 - 12a_9$$
  then the effective mass squared terms remain negative for all temperatures $T$.
• Absence of Phase Transition: Under these conditions, the Pati–Salam symmetry does not restore at any temperature, even if $T \gg v$. Consequently, no phase transition occurs after inflation that would regenerate 't Hooft-Polyakov monopoles.
• Robustness: The inclusion of the $(2, 2, 1)$ Higgs field and the subsequent breaking to the SM group introduces only negligible corrections ($O(\epsilon^2/v^2)$) to the primary results derived from the $\Phi$ and $\Xi$ sector.

Significance
The paper claims that the mechanism of symmetry nonrestoration, previously demonstrated for minimal $SU(5)$ GUTs, operates consistently within the Pati–Salam model. The primary significance of this result is the removal of the monopole-based constraints on the Pati–Salam symmetry breaking scale.

By showing that suitable quartic couplings can prevent symmetry restoration at any temperature, the authors argue that:

1. Intermediate-scale ($\sim 10^{12} - 10^{14}$ GeV) and low-scale (TeV-PeV) Pati–Salam models are not ruled out by the requirement that $T_{res} > T_R$.
2. Even if the reheating temperature is very high (approaching the GUT scale), the universe can avoid the thermal production of monopoles.
3. The constraints on the Pati–Salam scale arising from the reheating temperature are effectively lifted, provided the scalar sector parameters satisfy the derived inequalities.

The authors conclude that the large freedom in the Higgs sector of the model allows for this nonrestoration, though they note that further extensions (including full sets of invariants, higher-order corrections, gauge/Yukawa effects, and supersymmetric variants) remain to be explored.