Finite T topological Susceptibility with heavy Quarks

TL;DR

The paper investigates how a dynamical bottom quark alters the finite-temperature QCD topological susceptibility, a key input for axion cosmology. It combines a caloron-gas (HS caloron) framework with small-$m$ Taylor expansions and large-$m$ heat-kernel expansions of the fermionic determinant, bridged by a Padé-like interpolation to cover the full mass range. By IR-matching the running coupling and comparing theories with 4 light flavors plus a bottom quark to those with an asymptotically heavy bottom, the authors compute the ratio $\kappa$ of susceptibilities across temperatures relevant for axion dynamics. The main finding is that in the 400 MeV to 1.1 GeV window, the bottom quark’s effect on $\chi_{\text{top}}$ remains below ~5%, supporting the sufficiency of 2+1+1 lattice simulations for axion-related estimates and quantifying the quark-mass dependence with a controlled analytic framework.

Abstract

Axion cosmology needs the QCD topological susceptibility between 400 and 1100 MeV. In this range the bottom quark is inconvenient to include in lattice simulations, but not heavy enough to ignore. We estimate its effect on the susceptibility by computing the ratio of the 4-quark susceptibility and the 4+1-quark susceptibility in the caloron gas approximation. We do so by computing small-mass and large-mass expansions of the finite mass and temperature fluctuation determinant and connecting them with a Padé approximant.

Figures (16)

• Figure 1: The ratio of topological susceptibilities $\kappa(m_b/T, 4, 1, 3)$ (\ref{['eq:top_suscep_ratio']}) comparing a theory with four light and a physical $b$ quark in $SU(3)$ gauge theory to lattice QCD, where the $b$ quark is asymptotically heavy. Due to the modification of the running coupling (\ref{['eq:coupling_modified']}), $\kappa$ depends only on the physical quark mass $m_b$. The interesting mass range $3\lesssim m_b/T \lesssim 12$, marked in purple, is chosen to be slightly wider than the physically relevant range, which is determined by the temperature range $400\,\text{MeV}\lesssim T\lesssim 1.1\,\text{GeV}$ and the bottom mass $m_{b}\approx 4.2\,\text{GeV}$particle_data. (This is the $\overline{\mathrm{MS}}$ mass at the renormalization point $\overline{\mu} = m_b$, which is the quantity relevant in a perturbative calculation. The pole mass is somewhat heavier.)
• Figure 2: The running coupling $g^{-2}(\lambda)$ for a theory with four light and a heavy $b$ quark. At large energy scales $\gg m_b$ one has a 5$\,$-$\,$ flavor running () which switches to 4$\,$-$\,$ flavor running at the energy scale $m_b$ (). For a theory with an asymptotically heavy $b$ quark, the switch occurs at the UV scale $m_{b\text{, asy}}$ () and the two theories disagree in the IR, with the asymptotic $b$$\,$-$\,$ theory failing to describe known 4$\,$-$\,$ flavor QCD/IR theory. In order to compare the $m_b\,$- and $m_{b\text{, asy}}$$\,$-$\,$ theories with matching IR physics, we modify the coupling $g_\text{asy}$ and describe it in terms of $g_\text{phys}$ (\ref{['eq:coupling_modified']}) for scales $>m_b$ (). Overall, $g_\text{asy}$ is thus given by (). While this description disagrees with the physical description in the UV, it agrees in the IR and thus corresponds better to what happens in a $2+1+1$$\,$-$\,$ flavor lattice calculation.
• Figure 3: Some closed loop$\,$-$\,$ propagators in the periodic spacetime $\mathbb{R}^3\times S^1_{\text{rad}{}^{\,}={}^{\,1\!}/{}_{2\pi}}$ as they appear in (\ref{['eq:eff_act_Tay_ansatz']}). The periodicity of the spacetime is made explicit by showing all the time copies of the boson and BPST instanton making up the thermal boson and HS caloron, respectively. The anti$\,$-$\,$ periodic boson copies are located at ($\bullet$) $x+j\hat{e}_{4}$, $j\in\mathbb{Z}$ and are connected by closed loops; the solid lines () show "aperiodically closed loops" which do not encounter the spacetime periodicity, the dash$\,$-$\,$ dotted lines ($\,\boldsymbol{\cdot}\,$$\,\boldsymbol{\cdot}\,$) show loops which encounter the periodicity $j$ times and close (anti$\,$-)periodically for $j$ even (odd). The caloron is made up of periodic instanton copies located at ($\blacksquare$) $0+j\hat{e}_{4}$. All boson copies and all connecting, closed loop$\,$-$\,$ propagators are affected by all periodic instanton copies; this is symbolized by the dashed red, green, and blue lines ( ) connecting the instanton and boson copies.
• Figure 4: Left: $m^2$$\,$-$\,$ coefficient $-2 \gamma_{\text{s,}-}^{\text{small}}$ as a function of the caloron size $\varrho$. Right: percent relative error of our fitting function, (\ref{['eq:m^2_coeff']}). For the physically most important region $0.2 < \varrho < 0.6$, the fit is accurate to better than $1\%$.
• Figure 5: