Modeling pion physics in the $ϵ$-regime of two-flavor QCD using strong coupling lattice QED

In order to model pions of two-flavor QCD we consider a lattice field theory involving two flavors of staggered quarks interacting strongly with $U(1)$ gauge fields. For massless quarks, this theory has an $S{U}_L(2)\times S{U}_R(2)\times U_A(1)$ symmetry. By adding a four-fermion term we can break the $U_A(1)$ symmetry and thus incorporate the physics of the QCD anomaly. We can also tune the pion decay constant $F$, to be small compared to the lattice cutoff by starting with an extra fictitious dimension, thus allowing us to model low energy pion physics in a setting similar to lattice QCD from first principles. However, unlike lattice QCD, a major advantage of our model is that we can easily design efficient algorithms to compute a variety of quantities in the chiral limit. Here we show that the model reproduces the predictions of chiral perturbation theory in the $ϵ$-regime.

Figure 1

An example of a $2-d$ lattice configuration as discussed in the text.

Figure 2

Examples of the active update at a site. The absence of an outgoing arrow implies that the update remains on the site.

Figure 3

Examples of the passive update at a site. The absence of an ingoing or an outgoing arrow implies that the update is on the site.

Figure 4

Vector current susceptibility $Y_c$ as a function of $L$ at $T=1.733$, $c=0.3$, and $m=0$ [$Y_c=Y_v$]. The solid line shows the fit with $F=0.0992$ and $a^{\prime }=2.7$. The dashed line shows the same curve but with $a^{\prime }=0$. The dotted line shows the infinite volume limit $F^2/2$.

Figure 5

Chiral condensate susceptibility ${\chi }_{\sigma }$ as a function of lattice size $L$ at $T=1.733$, $c=0.3$, and $m=0$ [${\chi }_{\sigma }={\chi }_{\pi }$]. The solid line is the plot of Eq. (13) with $\Sigma =0.1866$, $F=0.0992$, and $a=3.0$. The dashed line shows the same curve but with $a=0$. The dotted line shows the infinite volume constant ${\Sigma }^2/4$.

Figure 6

Plot of $Y_c^{(2)}$ and $Y_v^{(2)}$, evaluated in the two monomer sector as a function of $L$ at $T=1.733$, $c=0.3$, and $m=0$. The solid lines are fits to Eq. (21) discussed in the text.

Figure 7

$F$ and $\Sigma$ plotted as a function of $T$. The solid lines are fits to Eq. (22) with $A_F=0.943$, $A_{\Sigma }=1.769$, and $T_c=1.73779$.