Effective Polyakov line action from the relative weights method

We apply the relative weights method [J. Greensite, Phys. Rev. D 86, 114507 (2012)] to determine the effective Polyakov line action for SU(2) lattice gauge theory in the confined phase, at lattice coupling $\beta =2.2$ and $N_t=4$ lattice spacings in the time direction. The effective action turns out to be bilinear in the fundamental representation Polyakov line variables, with a rather simple expression for the finite range kernel. The validity of this action is tested by computing Polyakov line correlators, via Monte Carlo simulation, in both the effective action and the underlying lattice theory. It is found that the correlators in each theory are in very close agreement.

Figure 1

Derivatives of the PLA $L^{-3}\partial S_P/\partial a_{\mathit{k}}$ evaluated at $a_{\mathit{k}}=\alpha =0.05$, vs lattice momenta $k_L$. Also shown is a linear best fit to the data at $k_L>0.7$.

Figure 2

Derivatives $L^{-3}(\partial S_P/\partial a_{\mathit{k}}{)}_{\alpha }$ divided by $\alpha$, vs lattice momenta $k_L$, for $\alpha =0.05$, 0.10, 0.15, 0.20. It is clear that the derivatives of $S_P$ depend linearly on $\alpha$.

Figure 3

The derivatives of $S_P$ with respect to the amplitude of the zero mode, evaluated at several values of $\alpha$. The slope of this data is used to determine $r_{\max }$ of the bilinear kernel $Q(\mathit{x}-\mathit{y})$, as explained in the text.

Figure 4

A test of Eq. (21) at $\alpha =0.05$. The derivative data of Fig. 1 is plotted against the conjectured fitting function $\alpha (\frac{1}{2}{c}_1-2c_2\tilde{Q}(k_L))$ with $r_{\max }=3$.

Figure 5

A comparison of the Polyakov line correlation functions $G(|\mathit{x}-\mathit{y}|)=⟨P_{\mathit{x}}{P}_{\mathit{y}}⟩$ as computed via lattice Monte Carlo simulation of the underlying gauge theory on a $L^3\times 4$ lattice at coupling $\beta =2.2$, and via Monte Carlo simulation of the corresponding effective action $S_P$ of Eq. (19). Lattices are of spatial extension $L=12$, 16, 20, 24 lattice spacings. Note that off-axis displacements are included.

Figure 6

A high-statistics comparison of the Polyakov line correlation function $G(|\mathit{x}-\mathit{y}|)=⟨P_{\mathit{x}}{P}_{\mathit{y}}⟩$ computed for the lattice gauge and effective theories, for displacements $\mathit{x}-\mathit{y}$ parallel to the $x$, $y$, or $z$ axes, and spatial volume ${24}^3$.

Figure 7

Plaquette energy vs gauge-Higgs coupling $\kappa$ at fixed $\beta =2.2$, for the SU(2) gauge-Higgs theory with fixed Higgs modulus on a ${16}^4$ lattice volume, showing a sharp crossover at $\kappa \approx 0.84$.

Figure 8

The derivatives of $S_P$ with respect to the amplitude of the zero mode in the gauge-Higgs theory, evaluated at positive and negative values of $a_0=\alpha$. Part (a) shows the full range of the data; (b) is a close-up near $\alpha =0$. The $y$ intercept of this data is nonzero, and determines the coefficient $c_0$ of the linear, $Z_2$-symmetry breaking term in the effective PLA (30).

Figure 9

Same as Fig. 4 for the gauge-Higgs theory. We plot the data for the derivative $L^{-3}\partial S_P/\partial a_{\mathit{k}}$ vs $k_L$ against the conjectured fitting function $\alpha (\frac{1}{2}{c}_1-2c_2\tilde{Q}(k_L))$ with $r_{\max }=3.2$.

Figure 10

A comparison of the Polyakov line correlation functions $G(|\mathit{x}-\mathit{y}|)=⟨P_{\mathit{x}}{P}_{\mathit{y}}⟩$ as computed via lattice Monte Carlo simulation of the underlying gauge-Higgs theory (black diamonds) on a ${24}^3\times 4$ lattice, at couplings $\beta =2.2$, $\kappa =0.75$, and via Monte Carlo simulation of the corresponding effective action $S_P$ of Eq. (30) (blue triangles, $c_0=0.0236$). Also shown is a simulation of the effective action with a slightly different value of $c_0=0.02165$ (red circles).