Interplay between Electroweak Symmetry Breaking and Higgs Portal Dark Matter

1. Problem Statement

The standard calculation of Dark Matter (DM) relic density in thermal freeze-out scenarios typically assumes that the universe evolves entirely within the electroweak broken phase. In this approach, particle masses and interaction cross-sections are treated as constant (defined by the vacuum expectation value, $v \approx 246$ GeV) throughout the entire thermal history, from the early universe to the present day.

However, Electroweak Symmetry Breaking (EWSB) is a phase transition occurring at a critical temperature $T_{EWSB} \approx 160$ GeV. Above this temperature, the Higgs field has a zero vacuum expectation value (unbroken phase), and particle masses are determined solely by their bare mass parameters.

• The Conflict: For heavy DM candidates ($m_{DM} \gtrsim 4$ TeV), the freeze-out temperature ($T_f \approx m_{DM}/25$) can be higher than $T_{EWSB}$.
• The Consequence: If DM freezes out before EWSB, the standard approach incorrectly uses the broken-phase masses and couplings for the entire evolution. This ignores the fact that in the unbroken phase:
  1. DM masses are different (often lighter or defined by bare parameters).
  2. The spectrum of available annihilation channels changes (e.g., the Higgs boson is not a physical mass eigenstate, and mixing angles differ).
  3. The thermally averaged cross-section $\langle \sigma v \rangle$ is structurally different.

The authors argue that neglecting this phase transition leads to significant errors in predicting the relic density, potentially including viable models that should be excluded or excluding viable ones.

2. Methodology

The paper proposes and implements an "Improved Approach" to calculate the relic density, contrasting it with the "Standard Approach."

A. Theoretical Framework

The authors utilize the Boltzmann equation to track the DM yield $Y = n/s$ (number density over entropy density).

• Standard Approach: Assumes the universe is always in the broken phase. The integration of the Boltzmann equation uses $\langle \sigma v \rangle_{broken}$ and $m_{broken}$ from the freeze-out instant $y_f$ to the present.
• Improved Approach: Explicitly accounts for the phase transition at $T_{EWSB}$.
  • Before EWSB ($T > T_{EWSB}$): Uses the unbroken phase potential, masses ($\tilde{m}$), and cross-sections ($\langle \sigma v \rangle_{bEWSB}$).
  • After EWSB ($T < T_{EWSB}$): Uses the broken phase potential, masses ($m$), and cross-sections ($\langle \sigma v \rangle_{aEWSB}$).
  • The calculation is split into two integration steps, matching the yield at $T_{EWSB}$.

B. Analytical Quantification

The authors derive a model-independent measure of the deviation between the two approaches, denoted as $\delta \Omega h^2$:
$$\delta \Omega h^2 \equiv \frac{\Omega h^2|_I - \Omega h^2|_S}{\Omega h^2|_I} = Y^\infty_S (\alpha - \beta)$$
Where:

• $\alpha$ represents the difference in inverse yields at the freeze-out instant.
• $\beta$ represents the difference in the integrated annihilation history (dependent on the cross-sections and masses in both phases).
  This allows for a prediction of the sign and magnitude of the error without full numerical simulation.

C. Numerical Implementation

• Model: A specific extension of the Standard Model (SM) with two real scalar singlets ($\phi, \chi$) stabilized by $Z_2 \times Z'_2$ symmetries.
  • $\chi$ is the DM candidate.
  • $\phi$ acts as a mediator that acquires a VEV after EWSB, breaking the $Z'_2$ symmetry and allowing $\phi$ to decay, leaving $\chi$ as the sole stable DM.
• Tools: The authors used micrOMEGAs 6.0.5 to numerically solve the Boltzmann equation.
  • Standard: Single integration in the broken phase.
  • Improved: Two-step integration (Unbroken $\to$ Broken), using the yield at $T_{EWSB}$ as the initial condition for the second step.
• Scan: A random scan of $10^6$ benchmark points over a parameter space including $m_\chi \in [1, 100]$ TeV, mixing angles, and quartic couplings.

3. Key Contributions

1. Identification of a Structural Error: The paper highlights that the discrepancy is not merely a higher-order thermal correction but a structural change in the degrees of freedom and interaction channels available to DM depending on whether freeze-out occurs in the unbroken or broken phase.
2. Threshold Definition: They establish that for scalar DM, the standard approach breaks down for masses $m_{DM} \gtrsim 4$ TeV (where $T_f > T_{EWSB}$).
3. Analytical Metric: The derivation of Eqs. (21)–(23) provides a way to quantify the deviation ($\delta \Omega h^2$) based on the interplay between freeze-out yields and the integrated cross-sections, validated against numerical results.
4. Model-Independent Insight: While tested on a specific two-singlet model, the methodology and the derived metric are applicable to any Higgs-portal model where DM freezes out before EWSB.

4. Results and Observations

• Magnitude of Deviation:
  • For $m_\chi < 4$ TeV, the deviation is negligible.
  • For $m_\chi > 4$ TeV, deviations become significant.
  • In the heavy mass regime ($m_\chi > 10$ TeV), the relative difference $|\delta \Omega h^2|$ can reach O(100%).
• Sign of Deviation: The deviation can be positive or negative.
  • Case A (False Exclusion): The standard approach predicts a relic density too high (overabundant), excluding a parameter point. The improved approach, accounting for different early-universe physics, yields the correct relic density ($\Omega h^2 \approx 0.12$), validating the model.
  • Case B (False Inclusion): The standard approach predicts a correct relic density, but the improved approach shows the model is overabundant, correctly excluding it.
• Direct Detection (DD) Constraints:
  • For $m_\chi \lesssim 10$ TeV, current LUX-ZEPLIN (LZ) limits are strong. The authors found that points satisfying the improved relic density often require small couplings to pass DD limits, resulting in small deviations ($\sim 1\%$).
  • For $m_\chi \gtrsim 10$ TeV, DD sensitivity drops significantly. In this regime, the deviation between approaches is large, and the standard approach fails to accurately map the viable parameter space.
• Cross-Section Evolution: The analysis of $\langle \sigma v \rangle$ shows that the dominant annihilation channels change across the phase transition (e.g., from $\chi\chi \to \phi\phi$ in the unbroken phase to $\chi\chi \to h_1 h_1, h_1 h_2$ in the broken phase).

5. Significance

• Re-evaluation of Heavy DM Models: The paper necessitates a re-evaluation of all Higgs-portal DM models with masses above $\sim 4$ TeV. Previous studies using the standard approach may have incorrectly ruled out viable heavy DM candidates or retained invalid ones.
• Model Building: It suggests that for heavy DM, the "correct" relic density is highly sensitive to the unbroken phase physics. This opens up new regions of parameter space that were previously thought to be excluded.
• Future Colliders: The authors argue that these heavy DM scenarios, while difficult to probe via direct detection, could be accessible at future high-energy colliders (like FCC-hh) through the reconstruction of the extended scalar sector.
• Methodological Shift: The work establishes that accurate relic density calculations for heavy particles must treat the thermal history as a multi-phase evolution, rather than a single-phase approximation.

In conclusion, the paper demonstrates that ignoring the electroweak phase transition in the thermal history of heavy Dark Matter leads to order-of-magnitude errors in relic density predictions, fundamentally altering the viability of specific Higgs-portal models.