Axion-pion scattering at finite temperature in chiral perturbation theory and its influence in axion thermalization

Axion-pion scattering amplitudes at finite temperatures are calculated within chiral perturbation theory up to the one loop level. Unitarization procedure is implemented to these amplitudes in order to extend the applicable range of energy and temperature. The influence of the thermal axion-pion scattering amplitudes on the $a\pi \to \pi \pi$ cross sections and the axion thermalization rate is investigated, with the emphasis on the comparison with the zero-temperature-amplitude case. A brief discussion on the cosmological implication of the axion thermalization rate, which is calculated by using the $a\pi \to \pi \pi$ amplitudes at finite temperatures, is also given. The thermal corrections to the axion-pion scattering amplitudes can cause around a 10% shift of the determination of the axion decay constant $f_a$ and its mass $m_a$, comparing with the results by using the $a\pi \to \pi \pi$ amplitudes at zero temperature.

Figure 1

Feynman diagrams of the amputated four-point Green functions in Eqs. (13) and (14). The dashed line stands for the axion field and the solid line corresponds to the pion field. The number $n$ for each vertex denotes its chiral order of $\mathcal{O}(p^n)$. Diagrams 1 denote $A_{a\pi ;\pi \pi }^{(2,4)}$, and diagram 1 corresponds to $A_{\pi \pi ;\pi \pi }^{(2)}$ which will be multiplied by the NLO $a\text{−}{\pi }^0$ mixing to give one type of contributions to the $a\pi \to \pi \pi$ scattering amplitudes at $\mathcal{O}(p^4)$, i.e., the last terms in Eqs. (13) and (14).

Figure 2

Magnitudes of the various $a\pi \to \pi \pi$ partial-wave amplitudes in the isospin bases. The curves labeled as NLO include both the contributions at $\mathcal{O}(p^2)$ and $\mathcal{O}(p^4)$.

Figure 3

Cross sections in the charged bases. For the notaions of ${\sigma }_{\mathrm{LO},\mathrm{NLO},{\mathrm{NLO}}^{\prime },\mathrm{IAM}}$, see the text for details. Notice that the LO contributions to the $a{\pi }^0\to {\pi }^0{\pi }^0$ amplitude and cross section vanish.

Figure 4

The symbols of ZT and FT indicate that the results are calculated by taking the zero-temperature and finite-temperature scattering amplitudes, respectively. (a) The curves for the $h_{\text{lo,nlo,IAM}}(m_{\pi }/T)$ functions are defined in Eqs. (36) and (37). (b) The axion rates in this figure are calculated by taking the perturbative $a\pi \to \pi \pi$ scattering amplitudes upto NLO, with $f_a=1.5\times {10}^7\mathrm{GeV}$ for illustration. ${\mathrm{\Gamma }}_a^{\mathrm{LO}}$ only includes the contribution from $h_{\mathrm{lo}}$, and ${\mathrm{\Gamma }}_a^{\mathrm{NLO}}$ contains the contribution both from $h_{\mathrm{lo}}$ and $h_{\mathrm{nlo}}$. (c) Ratio between the NLO parts and LO parts of the axion thermal averaged rates. ${\mathrm{\Gamma }}_a^{\mathrm{lo}}$ denotes the LO part of the perturbative rate and ${\mathrm{\Gamma }}_a^{\mathrm{nlo}}$ denotes the part of perturbative rate contributed only by $h_{\mathrm{nlo}}$.

Figure 5

The symbols of ZT and FT indicate that the results are calculated by taking the zero-temperature and finite-temperature scattering amplitudes, respectively. (a) Axion thermalization rates calculated with the IAM amplitudes as a function of $T$. The solid lines denote the results by taking the amplitudes at zero temperature as an approximation and the dashed lines correspond to the updated results by including the thermal corrections to the $a\pi \to \pi \pi$ amplitudes. (b) Axion decoupling temperature as a function of the axion decay constant $f_a$. (c) Extra effective thermal relativistic d.o.f. as a function of $f_a$. (d) The bounds on the axion masses $m_a$. The notations of $m_a^{\mathrm{LO}}$ and $m_a^{\mathrm{NLO}}$ refer to the situations by using the LO/NLO expressions for the axion masses.

Figure 6

The ratios of the axion and pion masses at finite and zero temperatures predicted by the NLO $\chi \mathrm{PT}$.

Figure 7

Coordinate system chosen for the evaluation of the phase space integral.