The QCD axion, precisely

TL;DR

The paper demonstrates that key QCD axion properties can be computed from first principles with percent-level precision by combining NLO chiral perturbation theory with lattice QCD. It provides a precise determination of the zero-temperature mass m_a, full potential V(a) including the self-coupling λ_a and domain-wall tension σ_a, and the photon coupling g_{aγγ}, along with reliable nucleon couplings via lattice inputs. It extends the analysis to finite temperature, showing that low-temperature behavior is well captured by ChPT while high-temperature predictions from instanton methods are unreliable and require non-perturbative input, with significant implications for the axion relic abundance. The work delivers robust, first-principles benchmarks that can guide experimental searches and cosmological inferences, and highlights where upcoming lattice and quark-mass refinements will further sharpen the predictions.

Abstract

We show how several properties of the QCD axion can be extracted at high precision using only first principle QCD computations. By combining NLO results obtained in chiral perturbation theory with recent Lattice QCD results the full axion potential, its mass and the coupling to photons can be reconstructed with percent precision. Axion couplings to nucleons can also be derived reliably, with uncertainties smaller than ten percent. The approach presented here allows the precision to be further improved as uncertainties on the light quark masses and the effective theory couplings are reduced. We also compute the finite temperature dependence of the axion potential and its mass up to the crossover region. For higher temperature we point out the unreliability of the conventional instanton approach and study its impact on the computation of the axion relic abundance.

Figures (6)

• Figure 1: Comparison between the axion potential predicted by chiral Lagrangians, eq. (\ref{['eq:pota']}) (continuous line) and the single cosine instanton one, $V^{inst}(a)=-m_a^2 f_a^2 \cos(a/f_a)$ (dashed line).
• Figure 2: Result of the fit of the 3-flavor couplings $\tilde{C}^W_{7,8}$ from the decay width of $\pi\to\gamma\gamma$ and $\eta\to\gamma\gamma$, which include the experimental uncertainties and a 30% systematic uncertainty from higher order corrections.
• Figure 3: The relation between the axion mass and its coupling to photons for the three reference models with $E/N=0$, $8/3$ and $2$. Notice the larger relative uncertainty in the latter model due to the cancellation between the UV and IR contributions to the anomaly (the band corresponds to $2\sigma$ errors.). Values below the lower band require a higher degree of cancellation.
• Figure 4: The temperature dependent axion mass normalized to the zero temperature value (corresponding to the light quark mass values in each computation). In blue the prediction from chiral Lagrangians. In different shades of red the lattice data from ref. Buchoff:2013nra for different lattice volumes, and in shades of green the preliminary lattice data from Trunin:2015yda for different lattice spacings. The dotted grey curve shows the interacting instanton liquid model (IILM) result Wantz:2009mi.
• Figure 5: Values of $f_a$ such that the misalignment contribution to the axion abundance matches the observed dark matter one for different choices of the parameters of the axion mass dependence on temperature. For definiteness the plot refers to the case where the PQ phase is restored after the end of inflation (corresponding approximately to the choice $\theta_0=2.15$). The temperatures where the axion starts oscillating, i.e. satisfying the relation $m_a(T)=3H(T)$, are also shown. The two points corresponding to the dilute instanton gas prediction and the recent preliminary lattice data are shown for reference.