Axion cosmology, lattice QCD and the dilute instanton gas

TL;DR

The study addresses the need for a precise determination of the QCD topological susceptibility $χ(T)$ to predict axion dark matter via $m_A^2(T)=χ(T)/f_A^2$. It computes $χ(T)$ and the anharmonic coefficient $b_2(T)$ in quenched lattice QCD over $0.9T_c$–$4T_c$ with controlled continuum extrapolation and compares the results to the dilute instanton gas approximation (DIGA), including a two-loop renormalization-group improved form. The lattice shows the correct temperature dependence but requires a large normalization factor $K$ (of order ten) to match DIGA in the studied range; this factor depends on $T_c/Λ_{ ext{MS}}$ and tends to unity at very high $T$. When extended to full QCD with light quarks, the fitted $K$ (~9.22) implies an axion mass range of roughly $40–930 μeV$ for a scenario with at least 10% of DM from axions, highlighting the importance of including quark effects for definitive DM predictions and motivating further lattice-DIGA analyses.

Abstract

Axions are one of the most attractive dark matter candidates. The evolution of their number density in the early universe can be determined by calculating the topological susceptibility $χ(T)$ of QCD as a function of the temperature. Lattice QCD provides an ab initio technique to carry out such a calculation. A full result needs two ingredients: physical quark masses and a controlled continuum extrapolation from non-vanishing to zero lattice spacings. We determine $χ(T)$ in the quenched framework (infinitely large quark masses) and extrapolate its values to the continuum limit. The results are compared with the prediction of the dilute instanton gas approximation (DIGA). A nice agreement is found for the temperature dependence, whereas the overall normalization of the DIGA result still differs from the non-perturbative continuum extrapolated lattice results by a factor of order ten. We discuss the consequences of our findings for the prediction of the amount of axion dark matter.

Figures (7)

• Figure 1: Demonstration that volume is sufficiently large to have negligible finite volume corrections on $Q^2$. The data for $T/T_c=2$ is multiplied by 3 for better visibility of the comparison.
• Figure 2: Demonstration that Wilson flow/lattice renormalization are under control.
• Figure 3:
• Figure 4: Lattice data on the anharmonicity coefficient $b_2$ of the axion potential compared to its DIGA prediction. The data points are shifted a bit horizontally for better visibility.
• Figure 5: Prediction of the topological susceptibility in the DIGA: comparison between one-loop and two-loop RGI results. We used the four-loop expression for the running coupling in the modified minimal subtraction scheme as given in the appendix of Ref. Chetyrkin:1997un and the central value of $T_c/\Lambda^{(n_f=0)}_{\overline{\rm MS}}=1.26(7)$ as determined from the lattice in Ref. Borsanyi:2012ve.