Quark mass thresholds in QCD thermodynamics

TL;DR

The paper addresses how finite quark masses modify high-temperature QCD thermodynamics and the Standard Model expansion rate. It employs dimensionally reduced effective theories to compute quark-mass effects up to ${\mathcal{O}}(g^2)$ in the MSbar scheme, and provides mass-dependent expressions for the perturbative coefficients $\alpha_{E1},$ $\alpha_{E2},$ $\alpha_{E7}$ along with numerical evaluations of the relevant integrals. It demonstrates that mass corrections to the pressure are sizable at ${\mathcal{O}}(g^2)$ (roughly 20–30%), but the induced changes in the mass-correction factors are modest (about 5% for $N_f=3$ and smaller for $N_f=4$); it also develops a phenomenological interpolation for the QCD pressure between the QCD and electroweak scales by matching to lattice data and hadron-resonance gas models, and extends the analysis to the SM weak sector. The results have implications for cosmological expansion and decoupling calculations, while highlighting the need for ${\mathcal{O}}(g^6)$ determinations and transition-region lattice simulations to reduce uncertainties.

Abstract

We discuss radiative corrections to how quark mass thresholds are crossed, as a function of the temperature, in basic thermodynamic observables such as the pressure, the energy and entropy densities, and the heat capacity of high temperature QCD. The indication from leading order that the charm quark plays a visible role at surprisingly low temperatures, is confirmed. We also sketch a way to obtain phenomenological estimates relevant for generic expansion rate computations at temperatures between the QCD and electroweak scales, pointing out where improvements over the current knowledge are particularly welcome.

Figures (5)

• Figure 1: Left: the pressure for $N_{\rm f} = 0$, 3, 4, at ${\mathcal{O}}(g^0)$ and ${\mathcal{O}}(g^2)$. Right: the "correction factors" accounting for the effects of quarks, at ${\mathcal{O}}(g^0)$ and ${\mathcal{O}}(g^2)$ (cf. Eq. (\ref{['corrfac']})). They grey bands indicate the effect of ${\overline{\hbox{\rm MS}}}$ scheme scale variations by a factor 0.5 ... 2.0 around the "optimal" value. It is observed that while the ${\mathcal{O}}(g^2)$ corrections are of order 20...30% in the pressure, they are of order 5% in the correction factors for $N_{\rm f} = 3$, and even less for the physical case $N_{\rm f} = 4$.
• Figure 2: A phenomenological interpolating curve (solid line) for the QCD pressure at $N_{\rm f} = 0$. In the perturbative curve (grey band) the unknown ${\mathcal{O}}(g^6)$ constant has been adjusted so that lattice data (closed squares bi1) is matched, once $T > 3.6 {\Lambda_{\overline{\hbox{\tiny\rm{MS}}}}}$.
• Figure 3: Left: Phenomenological interpolating curves (solid lines) for the QCD pressure at $N_{\rm f} = 3, 4$. The shaded interval corresponds to the transition region where the results can be reliably determined with lattice simulations only. Middle: $g_{\hbox{\scriptsize eff}}, h_{\hbox{\scriptsize eff}}, i_{\hbox{\scriptsize eff}}$ as defined in Eq. (\ref{['ieff']}), for $N_{\rm f} = 4$. Right: the equation-of-state parameter $w$ and the speed of sound squared $c_s^2$, for the same system.
• Figure 4: Left: The Standard Model pressure for $m_H =$ 150 GeV, 200 GeV. The shaded intervals correspond to the QCD and electroweak transition regions. Middle: $g_{\hbox{\scriptsize eff}}, h_{\hbox{\scriptsize eff}}, i_{\hbox{\scriptsize eff}}$ as defined in Eq. (\ref{['ieff']}), for $m_H =$ 150 GeV. Right: the equation-of-state parameter $w$ and the speed of sound squared $c_s^2$, for the same system. Various sources of uncertainties are discussed in the text.
• Figure 5: The functions defined in Eqs. (\ref{['F1def']})--(\ref{['F4def']}), for $\hat{\mu} = 0.0$ (left) and $\hat{\mu} = 1.0$ (right) (all functions are even in $\hat{\mu}$). Note that the ranges of the vertical axes are different in the two plots.