Full $\mathcal{O}(a)$ improvement in electrostatic QCD

Electrostatic QCD (EQCD) is a three-dimensional bosonic theory containing SU(3) and an adjoint scalar, which efficiently describes the infrared, nonperturbative sector of hot QCD and which is highly amenable to lattice study. We improve the matching between lattice and continuum EQCD by determining the final unknown coefficient in the $\mathcal{O}(a)$ matching, an additive scalar mass renormalization. We do this numerically by using the symmetry-breaking phase transition line of EQCD as a line of constant physics. This prepares the ground for a precision study of the transverse momentum diffusion coefficient $C(q_{\perp })$ within this theory. As a by-product, we provide an updated version of the EQCD phase diagram.

Figure 1

$\mathrm{Tr}{\mathrm{\Phi }}^3$ and $\mathrm{Tr}{\mathrm{\Phi }}^2$, integrated over transverse directions, as a function of the $z$ direction in a $36^3\times 96$ box at $g_{3\mathrm{d}}^{2}a=\frac{1}{3}$, $x=0.08896$, and $y=0.47232$. There is a region near 0 (periodically identified with 96) which is in the symmetric phase, and a region near 50 which is in the broken phase, as well as two phase boundaries. The phases are visible in either order parameter but the fluctuations in $\mathrm{Tr}{\mathrm{\Phi }}^2$ are smaller.

Figure 2

Fits of $\mathcal{O}(g_{3\mathrm{d}}^{2}a)$ behavior for different $x$.

Figure 3

Grand fit of $\frac{\delta y_{3\mathrm{loop}}}{g_{3\mathrm{d}}^{2}a}(x)$ with error band.

Figure 4

Updated version of the phase diagram of EQCD. The phase below the data points is the ${\mathbb{Z}}_3$-broken phase, and the one above is ${\mathbb{Z}}_3$-symmetric. The dashed line is a fit to the data points, to help guide the eye.

Figure 5

Continuum extrapolation of $\frac{\mathrm{\Delta }\mathrm{Tr}{\mathrm{\Phi }}^2}{g_{3\mathrm{d}}^2}$, the difference of the broken and symmetric phase values of $\frac{\mathrm{Tr}{\mathrm{\Phi }}^2}{g_{3\mathrm{d}}^2}$, at different $x$. The intercept in (f) was determined analytically in [28] and was incorporated into that plot.

Figure 6

Fits of the symmetric phase value of $\frac{\mathrm{Tr}{\mathrm{\Phi }}^2}{g_{3\mathrm{d}}^2}(g_{3\mathrm{d}}^{2}a)$ at different $x$ and grand fit.

Figure 7

Diagrams generating $Z_m$ the multiplicative $\mathrm{Tr}{\mathrm{\Phi }}^2$ operator renormalization. Heavy crosses are the renormalized operator (with $Z_m$ factor), while light crosses are the bare operator.