3d SU(N) + adjoint Higgs theory and finite temperature QCD

TL;DR

This work assesses whether a three-dimensional SU($N$) gauge theory with an adjoint Higgs can serve as an effective description of finite-temperature SU($N$) gauge theory for $N=2,3$. By performing 2-loop matching to relate 4d temperature data to 3d parameters $(g_3^2, y, x)$ and computing the 2-loop effective potential for $N=2$, the authors map out the perturbative regime and identify where nonperturbative effects become dominant. Lattice simulations for SU(2) reveal a first-order phase line terminating near $x_{\rm end}\approx0.3$, with large nonperturbative Debye masses and significant deviations from heavy-quark expectations along the dimensional-reduction curve. The findings indicate that dimensional reduction captures high-temperature static correlations but fails to fully describe the confinement-deconfinement transition, emphasizing the need for a Z($N$)-symmetric extension or more complete 3d dynamics to connect with real QCD near $T_c$.

Abstract

We study to what extent the three-dimensional SU(N)+adjoint Higgs theory can be used as an effective theory for finite temperature SU(N) gauge theory, with N=2,3. The parameters of the 3d theory are computed in 2-loop perturbation theory in terms of T/Lambda_MSbar,N,N_f. The perturbative effective potential of the 3d theory is computed to two loops for N=2. While the Z(N) symmetry probably driving the 4d confinement-deconfinement phase transition (for N_f=0) is not explicit in the effective Lagrangian, it is partly reinstated by radiative effects in the 3d theory. Lattice simulations in the 3d theory are carried out for N=2, and the static screening masses relevant for the high-temperature phase of the 4d theory are measured. In particular, we measure non-perturbatively the O(g^2 T) correction to the Debye screening mass. We find that non-perturbative effects are much larger in the SU(2) + adjoint Higgs theory than in the SU(2) + fundamental Higgs theory.

Figures (9)

• Figure 1: The critical curve $y=y_c(x)$ (multiplied by $x$) computed from 1- and 2-loop potentials in the SU(2)+adjoint Higgs theory. Two of the curves are plotted for the scale choice $\mu=g_3^2$; $\mu$ for the second 2-loop curve is determined by optimization perturbative. The scale dependence should give an estimate of higher order corrections. The thin curve is $y_{4d\to3d}(x)$ from eq. (\ref{['ydrntwo']}). For $x\to0$ all curves approach $2/(9\pi^2)= 0.0225$. The non-perturbative critical curve computed numerically is given in Fig. \ref{['ycdata']}.
• Figure 2: The $y_c$ extrapolation to $V\rightarrow\infty$ for $x=0.20$. Within the statistical errors, there is no lattice spacing $\beta_G$ dependence.
• Figure 3: Data points in the limit $V\to\infty,a\to0$ for the critical curve $y=y_c(x)$ (multiplied by $x$). The thick line is the 4th order fit (\ref{['fit']}) to the $V=\infty$ extrapolated data. The dashed line marks the region where the transition turns into a cross-over. The straight lines are the 4d$\to$3d curves of eq. (\ref{['y_dr2']}) marked by the value of $N_f$. The top scale shows the values of $T/\Lambda_{\overline{\rm MS}}$ corresponding to the values of $x$ for $N=2,N_f=0$, obtained using eq. (\ref{['xT']}). The physical implications of the figure are discussed in Sec. 6.
• Figure 4: The probability distributions of $A_0^a A_0^a/2g_3^2 = {\frac{1}{N_S^3}} \sum_x A_0^a A_0^a/2g_3^2$, measured at various locations on the transition line $y=y_c(x)$.
• Figure 5: The behaviour of $\langle A_0^a A_0^a\rangle/2g_3^2$ in continuum normalization (see eq. (\ref{['condvalue']})) when crossing the phase transition at $x=0.12$ and the cross-over at $x=0.35$.