Prominent enhancement of axion thermalization rate from axion-kaon interactions

Technical Summary: Prominent Enhancement of Axion Thermalization Rate from Axion-Kaon Interactions

Problem Statement
The thermalization rate of axions in the early Universe is a critical parameter for determining the hot dark matter (HDM) bound on axion properties. This rate dictates the axion decoupling temperature ($T_D$), which subsequently influences the effective number of extra relativistic degrees of freedom ($\Delta N_{\text{eff}}$). Previous literature has predominantly assumed that the $a\pi \leftrightarrow \pi\pi$ scattering channel is the sole dominant thermalization mechanism below the QCD critical temperature ($T_c \approx 160$ MeV). This assumption relies on the premise that heavier hadrons are exponentially suppressed due to lower number densities and that no significant resonance enhancements exist for axion interactions involving heavier mesons compared to the pion channel. However, this assumption has not been rigorously tested against non-perturbative effects, particularly regarding the potential contribution of the axion-kaon ($aK$) interaction.

Methodology
The authors employ a systematic calculation of axion-light flavor meson scattering amplitudes within the framework of the chiral unitarization approach.

• Theoretical Framework: The study utilizes $SU(3)$ Chiral Perturbation Theory ($\chi$PT) to derive the leading-order (LO) axion-meson interaction Lagrangian. The axion is treated as a pseudo-Nambu-Goldstone boson arising from the spontaneous breaking of the global $U(1)_{PQ}$ symmetry.
• Unitarization: To account for non-perturbative hadron resonances that are missed by LO $\chi$PT, the authors apply a unitarization procedure. This method resums the scattering amplitudes using inputs from LO and next-to-leading order (NLO) $\chi$PT, specifically incorporating low-energy constants (LECs) and a two-point one-loop function ($G$).
• Channels Analyzed: The study explicitly calculates amplitudes for $a\pi \leftrightarrow \pi\pi$ and $aK \leftrightarrow \pi K$ processes. It considers various isospin and angular momentum channels, including coupled states (e.g., $\pi\pi$ coupled to $K\bar{K}$, and $\pi K$ coupled to $\eta K$).
• Parameter Determination: The subtraction constants and LECs in the unitarized amplitudes are determined by fitting to experimental meson-meson scattering data (phase shifts and inelasticities) using a bootstrap approach to quantify uncertainties.
• Thermalization Calculation: The thermalization rates ($\Gamma_{ch}$) are computed by integrating the squared scattering amplitudes over the thermal phase space, weighted by Bose-Einstein distributions.

Key Contributions and Results

1. Resonance Enhancement in $aK \leftrightarrow \pi K$: The unitarized amplitudes reveal that the $aK \leftrightarrow \pi K$ scattering cross-sections are significantly enlarged by prominent hadron resonances, specifically the $K^*(892)$ and $K^*_0(700)$ (or $\kappa$). These resonances are absent or less effective in the $a\pi \leftrightarrow \pi\pi$ channel in the relevant energy regimes.
2. Cross-Section Comparison: Contrary to the assumption that pion channels dominate due to pion abundance, the sum of various $aK \leftrightarrow \pi K$ cross-sections is found to be substantially larger than that of $a\pi \leftrightarrow \pi\pi$ for center-of-mass energies $\sqrt{s}$ above approximately 1 GeV.
3. Thermalization Rate Enhancement: The inclusion of the $aK \leftrightarrow \pi K$ channel leads to a substantial enhancement of the total axion thermalization rate for temperatures $T \gtrsim 100$ MeV.
   • At $T \gtrsim 110$ MeV, the $aK \leftrightarrow \pi K$ channel contributes more than 40% to the total rate.
   • At $T \simeq 130$ MeV, the contribution from $aK \leftrightarrow \pi K$ exceeds that of the $a\pi \leftrightarrow \pi\pi$ channel.
4. Revised Cosmological Constraints: The enhanced thermalization rate implies that axions decouple at lower temperatures than previously estimated when only the pion channel is considered. When confronting these results with the Planck'18 measurement of $\Delta N_{\text{eff}}$, the lower bound on the axion decay constant ($f_a$) is tightened.
   • Previous Bound (Pion only): $f_a \geq 2.45^{+0.03}_{-0.02} \times 10^7$ GeV.
   • New Bound (Pion + Kaon): $f_a \geq 3.18^{+0.04}_{-0.03} \times 10^7$ GeV.
   • This represents a tightening of the lower limit by approximately 30%.
   • Consequently, the upper bound on the axion mass ($m_a$) is correspondingly reduced.

Significance
The paper demonstrates that the common approximation of considering only the $a\pi \leftrightarrow \pi\pi$ channel for axion thermalization below $T_c$ is significantly underestimated for temperatures above 100 MeV. By systematically incorporating non-perturbative resonance effects in the $aK \leftrightarrow \pi K$ channel, the authors show that heavier hadrons play a dominant role in axion thermalization under specific conditions. This finding necessitates a revision of the hot dark matter bounds on axion parameters derived from cosmological observations, specifically leading to more stringent constraints on the axion decay constant and mass when compared against Planck'18 data. The work underscores the importance of including sub-dominant hadronic channels and resonance dynamics in early Universe axion phenomenology.