An effective field theory for thermal QCD with 2+1 flavours

TL;DR

The paper develops a bottom-up, finite-temperature EFT for QCD with $N_f=2+1$ flavors, matched to lattice data to capture long-distance thermodynamics just below the crossover. By extending the two-flavor EFT to include strange quarks and performing a Hartree–Fock treatment, it derives a two-condensate structure and a pseudo-Goldstone-boson sector whose low-energy constants encode the effects of the UV theory; the low-energy theory exhibits UV insensitivity and can be reduced to a pion-only EFT for practical predictions. The framework yields a phase diagram with an approximately elliptical chiral boundary and makes quantitative predictions for the phase transition temperatures ($T_{co}$ and $T_c$), as well as static pion properties and the pressure, with substantial agreement with lattice results for both $N_f=2$ and $N_f=2+1$. It also provides predictions for the real-time kinetic mass $m_\pi^K$ and outlines how NLO corrections and direct lattice tests can refine the approach, establishing a bridge between lattice data and continuum thermal QCD phenomenology.

Abstract

We write a long-distance effective field theory (EFT) for QCD at finite temperature just below the crossover temperature $T_c$. The low energy constants (LECs) of this EFT are obtained from lattice measurements of the screening mass of pions at two temperatures for $N_f=2+1$ using lattice results obtained at physical values of pion and Kaon masses, and $N_f=2$ where the lattice simulations were performed with a heavier pion mass. The EFT gives good predictions for other static pion properties for $N_f=2$, where lattice results are available. We show the corresponding predictions for $N_f=2+1$, where they are not yet measured. We demonstrate that EFT gives excellent predictions for the phase diagram in $N_f=2+1$. The predictions for the pressure are investigated, and predictions are also given for a Wick-rotated real-time quantity called the kinetic mass.

Figures (14)

• Figure 1: The first four terms organized by the number of insertions of $\bf M$ into the one-loop expression for $L_{{pGB}}(3)$. Each of these can be expanded in powers of the $\phi^i$s. If we retain only terms up to the fourth power, then these are the only insertions which need to be considered.
• Figure 2: The pion EFT is obtained by integrating over all hard modes in an energy shell between $\Lambda_{2+1}$ and $\Lambda_2$. The main constraints on the former is that it must lie between the proton and $\eta$ masses. On the other hand, $\Lambda_2$ lies below the Kaon mass and must be larger than $T_{co}$ so that it catches thermal physics in this range. The main corrections to the pion 2-point and 4-point functions are shown. The one loop correction is resummed using a Dyson-Schwinger formulation.
• Figure 3: We compare extractions of the LECs $d_3$ and $d_6$ at $T=0$ and finite $T$. The best fit values of the LECs are indicated by a dot, and the successive contours enclose the 68%, 95% and 99% CLs. For the finite $T$ theory the best fit value of $d_4=1.21^{+0.09}_{-0.07}$.
• Figure 4: Predictions for $T_{co}$ and $T_c$ using the fitted LECs shown as histograms obtained by sampling the 90% CLs of the fits. The median value and the limits of the 68% band for the EFT predictions are shown with broken vertical lines. The continuous vertical line shows the best fit value of the LECs, and the gray band is the lattice extraction of the corresponding quantity Brandt:2014qqa.
• Figure 5: Predictions for static pion properties from the EFT using the LECs determined by the new scheme. The gray bands show the 68% (darker colour) and 95% (lighter colour) CLs on the predictions. The vertical bands are the predicted value of $T_{co}$ in the EFT, and the corresponding lattice determination Brandt:2014qqa. Two values of $m_\pi^D$ which are inputs to the fits are shown as the filled points (the black line shows the predictions from the best fit LECs). Clearly the prediction of the temperature dependence of $m_\pi^D/T$, $f_\pi/T$ and $u_\pi$ are as good as the data can support. The pion self-coupling $c_{41}$ has not yet been measured on the lattice; we show the EFT prediction for it.