Axion Mixing in the String Axiverse

Technical Summary: Axion Mixing in the String Axiverse

Problem Statement
The "String Axiverse" framework, particularly within Type IIB string compactifications (e.g., the Large Volume Scenario or LVS), predicts a rich spectrum of axion-like particles (ALPs) alongside the QCD axion. While the cosmological evolution of single axions or simple two-axion systems (1+1 mixing) has been studied, the dynamics of multi-axion systems involving one QCD axion and multiple ALPs (1+N mixing) remain less explored. Specifically, there is a need to understand the conditions under which these systems exhibit "maximal mixing"—defined as the scenario where axions experience the largest degree of effective mixing—and how the energy density transfers between these multiple mass eigenstates during the QCD phase transition.

Methodology
The authors adopt a bottom-up perspective to investigate axion mass mixing within the context of the Type IIB string axiverse. The study proceeds through the following steps:

1. Model Construction: The paper reviews specific LVS axiverse models (including anisotropic and isotropic configurations) that yield one QCD axion and varying numbers of ALPs (1, 2, or $N$). These models determine the decay constants ($f_a$) and mass hierarchies.
2. Effective Lagrangian Formulation: The authors construct low-energy effective Lagrangians for three specific mixing scenarios:
   • 1 + 1 Mixing: One QCD axion ($a_0$) and one ALP ($A_1$).
   • 1 + 2 Mixing: One QCD axion and two ALPs ($A_1, A_2$), analyzed in both isotropic (similar decay constants) and anisotropic (divergent decay constants) cases.
   • 1 + N Mixing: One QCD axion and $N$ ALPs ($N > 2$).
3. Dynamical Analysis: The equations of motion (EOMs) for the axion fields are derived, incorporating temperature-dependent QCD axion masses and constant ALP masses. The authors analyze the mass mixing matrices ($M^2$) and diagonalize them to find mass eigenvalues ($m_h, m_l$) and mixing angles.
4. Level Crossing Analysis: The study focuses on the phenomenon of "level crossing," where mass eigenvalues cross as the universe cools through the QCD phase transition temperature ($T_{QCD}$). The authors derive the level crossing temperatures ($T_\times$) and analyze the adiabatic transfer of axion energy density between eigenstates.
5. Quantitative Criteria: The paper establishes quantitative conditions for "maximal mixing" by counting the number of level crossings ($n_\times$) and analyzing the relationship between ALP decay constants and the QCD axion decay constant.

Key Contributions and Results

• Conditions for Maximal Mixing: The paper identifies two critical conditions required for a multi-axion system to exhibit maximal mixing (defined as the maximum number of effective level crossings, $n_\times = N$):

  1. Mass Hierarchy: The masses of all ALPs must be strictly smaller than the zero-temperature mass of the QCD axion ($m_{A_i} < m_{a_0,0}$), and no two ALP masses can be equal. A distinct mass hierarchy among ALPs is necessary to prevent interference between mixing dynamics.
  2. Decay Constant Uniformity: The decay constants of all ALPs must be uniformly either smaller than or larger than the decay constant of the QCD axion ($f_{A_i} < f_{a_0}$ for all $i$, or $f_{A_i} > f_{a_0}$ for all $i$). If the decay constants are mixed (some smaller, some larger), the system splits into independent sub-systems (e.g., a 1+6 and 1+4 mixing scenario), preventing global maximal mixing.

• Energy Transfer Dynamics:

  • In the 1 + 1 scenario, energy transfers once between the QCD axion and the ALP.
  • In the 1 + 2 isotropic scenario, the authors identify a novel sequence where the mass eigenstate initially associated with the QCD axion transitions to the lighter ALP, then back to the QCD axion, and finally to the heavier ALP.
  • In the 1 + N scenario, the energy transfer occurs sequentially through a chain of level crossings. A key finding is that the transfer of axion energy density ultimately takes place only between the two axions with the closest masses at any given level crossing.

• Scenario Classification:

  • Light QCD Axion Scenario: Occurs when ALP decay constants are smaller than the QCD axion's. This typically suppresses the QCD axion relic density.
  • Heavy QCD Axion Scenario: Occurs when ALP decay constants are larger. This can enhance the QCD axion relic density.
  • The paper demonstrates that anisotropic 1+2 mixing effectively reduces to two separate 1+1 scenarios (one light, one heavy) rather than a unified 1+2 maximal mixing scenario.

• Quantitative Analysis: Through numerical examples involving ten ALPs, the authors show that random distributions of decay constants often fail to produce maximal mixing, instead fragmenting the system into smaller mixing groups. Maximal mixing is a specific, non-generic outcome requiring strict uniformity in the decay constant hierarchy.

Significance and Claims
The paper claims to provide critical insights into axion dynamics within Type IIB string axiverse models and broader multi-axion frameworks. By defining and analyzing "maximal mixing," the authors offer a framework to determine whether a given string model can naturally account for the overproduction or underproduction of dark matter abundance through mass mixing mechanisms.

The findings suggest that the cosmological evolution of axion energy density is highly sensitive to the specific hierarchy of decay constants and masses. The identification of the "closest mass" rule for energy transfer refines the understanding of how axion relic densities are established in the early universe. Furthermore, the work sets the stage for understanding the implications of these mixing dynamics on isocurvature fluctuations, dark energy, domain walls, gravitational waves, and primordial black holes, though the paper notes that detailed cosmological implications require further model-specific analysis.