Brownian axionlike particles

We study the nonequilibrium dynamics of a pseudoscalar axionlike particle (ALP) weakly coupled to degrees of freedom in thermal equilibrium by obtaining its reduced density matrix. Its time evolution is determined by the in-in effective action which we obtain to leading order in the ALP coupling but to all orders in the couplings of the bath to other fields within or beyond the standard model. The effective equation of motion for the ALP is a Langevin equation with noise and friction kernels obeying the fluctuation dissipation relation. A “misaligned” initial condition yields damped coherent oscillations, however, the ALP population increases towards thermalization with the bath. As a result, the energy density features a mixture of a cold component from misalignment and a hot component from thermalization with proportions that vary in time $(\text{cold})e^{-\mathrm{\Gamma }t}+(\text{hot})(1-e^{-\mathrm{\Gamma }t})$, providing a scenario wherein the “warmth” of the dark matter evolves in time from colder to hotter. As a specific example we consider the ALP-photon coupling $ga\overrightarrow{E}·\overrightarrow{B}$ to lowest order, valid from recombination onwards. For $T\gg m_a$ the long-wavelength relaxation rate is substantially enhanced ${\mathrm{\Gamma }}_T=\frac{g^2{m}_a^{2}T}{16\pi }$. The ultraviolet divergences of the ALP self-energy require higher-order derivative terms in the effective action. We find that at high temperature, the finite-temperature effective mass of the ALP is $m_a^2(T)=m_a^2(0)[1-(T/T_c{)}^4]$, with $T_c\propto \sqrt{m_a(0)/g}$, suggesting the possibility of an inverted phase transition, which when combined with higher derivatives may possibly indicate exotic new phases. We discuss possible cosmological consequences on structure formation, the effective number of relativistic species and birefringence of the cosmic microwave background.

Figure 1

A graphical depiction of $\mathcal{I}[a^{+},a^{-}]$. The black circle denotes the correlation functions $G^{\lessgtr }$ to all orders in the couplings to degrees of freedom other than the ALP.

Figure 2

$\mathcal{I}[a^{+},a^{-}]$ for the ALP-photon interaction in (2.2). The one-loop diagram features free photon propagators, the two-loop diagram includes the polarization from $e^{+}{e}^{-}$ pairs in the thermal plasma, etc.

Figure 3

One loop ALP self-energy from the coupling ${\mathcal{L}}_I=-ga(x)\overrightarrow{E}(x)·\overrightarrow{B}(x)$.

Figure 4

Ratio ${\mathrm{\Gamma }}_T/\mathrm{\Gamma }$ vs $T/m_a,k/m_a$.

Figure 5

Finite temperature correction to the dispersion relation $({\omega }_T-{\omega }_a)/g^2$ in units of $m_a^3$ vs $T/m_a,k/m_a$.