The hyperfine splitting in QCD mesonic screening masses at asymptotically large temperatures

TL;DR

This paper derives the leading hyperfine (spin-dependent) correction to flavour non-singlet mesonic screening masses at asymptotically high temperatures using a dimensionally reduced effective theory (EQCD with NRQCD for heavy quarks). The authors compute the spin-dependent potential from ultrasoft gluon exchange and solve a 2+1D Schrödinger problem to obtain the leading result ${\Delta m_{VP}^{\text{lo}}}/(2\pi T) = g^4\left(\frac{N_c}{3}+\frac{N_f}{6}\right) \frac{C_F}{8\pi^4} |\hat{\psi}_0(0)|^2$, numerically $0.002376\, g^4$, highlighting that non-perturbative effects must enter at higher orders (e.g., $O(g^3)$ in the static potential) to explain lattice data up to electroweak scales. The work connects perturbative EFT predictions with precise lattice results, showing that substantial higher-order (and non-perturbative) contributions are needed to describe the temperature dependence of the screening masses. This underscores the importance of non-perturbative 3D theory computations to fully match QCD at very high temperatures and motivates further cross-checks with lattice QCD and complementary approaches at the electroweak scale.

Abstract

We determine the hyperfine splitting in the QCD flavour non-singlet mesonic screening masses at asymptotically large temperatures. The analytic calculation is carried out in the dimensionally-reduced effective theory where the first non-zero contribution is of $O(g^4)$ in the QCD coupling constant $g$. Apart for its own theoretical interest, this result provides instrumental information to interpret and to parameterize non-perturbative data that are being produced at very high temperatures by numerical simulations of lattice QCD. Indeed, the comparison with existing non-perturbative results shows that higher order (non-perturbative) contributions in $g$ are needed to explain the data up to the highest temperatures explored, which is of the order of the electroweak scale.

Figures (3)

• Figure 1: Probability density for the quark-antiquark pair as a function of the distance $\hat{r}$. The probability for the quark and the antiquark to be at distances $\hat{r} \leq \hat{r}_{\rm{np}}$, see eq. (\ref{['eq:probability']}), is indicated by the blue area for different values of the coupling constant (temperature), with the latter increasing from left to right. As the temperature decreases, the blue area becomes larger and larger.
• Figure 2: Left: sum of the pseudoscalar and the vector screening masses, normalized to $4\pi T$, as a function of $\hat{g}^2$. Lattice data are obtained from Ref. DallaBrida:2021ddx, the red band represents the best parameterization in eq. \ref{['eq:parameterization_sum']} with its error, while the black dashed line is the analytically known spin-independent contribution in eq. \ref{['eq:m_nlo']}. Right: same as in the left panel but with the lattice data, subtracted by the analytically known contribution, plotted versus $\hat{g}^4$.
• Figure 3: Difference of the vector and pseudoscalar screening masses normalized to $2\pi T$ versus $\hat{g}^4$. The perturbative result in eq. \ref{['eq:HF-splitting']} is the dashed black line, while the red curve represents the fit in eq. \ref{['eq:DeltaM_fit']}.