Quenched Lattice QCD with Domain Wall Fermions and the Chiral Limit

TL;DR

The paper demonstrates that quenched lattice QCD with domain wall fermions can realize a controlled chiral limit, quantified by a residual mass $m_{ m res}$ that decreases with the fifth-dimensional extent $L_s$ and is small enough to recover near-continuum chiral behavior at moderate lattice spacings. It analyzes topological near-zero modes, Banks-Casher relations, and the chiral condensate in quenched QCD, showing how volume and $L_s$ mitigate associated pathologies; it also validates chiral observables through consistent $f_ ext{π}$ and hadron mass scaling across $a^{-1}$ between 1 and 2 GeV. The study reports a nontrivial interplay of zero modes, quenched chiral logs, and finite-volume effects, yet finds that with sufficiently large volumes and $L_s$, domain wall fermions reproduce expected chiral properties and yield physically reasonable values for $f_ ext{π}$, $m_N/m_ ho$, and $\langle\overline{q}q\rangle$. Overall, the work supports domain wall fermions as a robust framework for exploring chiral dynamics in quenched QCD and provides a baseline for future dynamical-fermion studies.

Abstract

Quenched QCD simulations on three volumes, $8^3 \times$, $12^3 \times$ and $16^3 \times 32$ and three couplings, $β=5.7$, 5.85 and 6.0 using domain wall fermions provide a consistent picture of quenched QCD. We demonstrate that the small induced effects of chiral symmetry breaking inherent in this formulation can be described by a residual mass ($\mres$) whose size decreases as the separation between the domain walls ($L_s$) is increased. However, at stronger couplings much larger values of $L_s$ are required to achieve a given physical value of $\mres$. For $β=6.0$ and $L_s=16$, we find $\mres/m_s=0.033(3)$, while for $β=5.7$, and $L_s=48$, $\mres/m_s=0.074(5)$, where $m_s$ is the strange quark mass. These values are significantly smaller than those obtained from a more naive determination in our earlier studies. Important effects of topological near zero modes which should afflict an accurate quenched calculation are easily visible in both the chiral condensate and the pion propagator. These effects can be controlled by working at an appropriately large volume. A non-linear behavior of $m_π^2$ in the limit of small quark mass suggests the presence of additional infrared subtlety in the quenched approximation. Good scaling is seen both in masses and in $f_π$ over our entire range, with inverse lattice spacing varying between 1 and 2 GeV.

Figures (34)

• Figure 1: $\langle \overline{q} q \rangle$ for quenched simulations done on $8^3 \times 32$ lattices ($\circ$) and $16^3 \times 32$ lattices ($\Box$) at $\beta = 5.7$ with $L_s = 32$. The smaller volume shows a pronounced rise as $m_f \rightarrow 0$ as is expected if unsuppressed zero modes are present. For the larger volume, the effect of topological near zero modes is reduced if not eliminated. This is expected since $\langle | \nu | \rangle/V$ should fall as $1/\sqrt{V}$.
• Figure 2: The pion mass squared versus $m_f$ from $\langle \pi^a(x) \pi^a(0) \rangle$ ($\Box$), $\langle A_0^a(x) A_0^a(0) \rangle$ ($\circ$) and $\langle \pi^a(x) \pi^a(0) \rangle + \langle \sigma(x) \sigma(0) \rangle_{\rm c}$ ($\Diamond$) for quenched simulations done on $8^3 \times 32$ lattices at $\beta = 5.7$ with $L_s = 48$. For $m_f = 0.0$, the correlators all give different masses due to the differing topological near-zero mode contributions for each one. For large enough $x$, all the correlators should give the same mass. However, this limit requires a large volume which is expected to suppress such zero-mode effects. The dotted line is the fit of Eq. \ref{['eq:b5_7_8nt32_0.02_0.1_pp_fit']}, the solid line is from Eq. \ref{['eq:b5_7_8nt32_0.02_0.1_aa_fit']} and the dashed line is from Eq. \ref{['eq:b5_7_8nt32_0.02_0.1_pp+ss_fit']}.
• Figure 3: The pion mass squared versus $m_f$ from $\langle \pi^a(x) \pi^a(0) \rangle$ ($\Box$), $\langle A_0^a(x) A_0^a(0) \rangle$ ($\circ$) and $\langle \pi^a(x) \pi^a(0) \rangle + \langle \sigma(x) \sigma(0) \rangle_{\rm c}$ ($\Diamond$) for quenched simulations done on $16^3 \times 32$ lattices at $\beta = 5.7$ with $L_s = 48$. The star is the value of $m_{\rm res}$ as measured from Eq. \ref{['eq:mres_ratio']} and its error bar in the horizontal axis is too small to show on this scale. The solid line is the fit to the $\langle A_0^a(x) A_0^a(0) \rangle$ correlator for $m_f = 0.02$ to 0.08 given in Eq. \ref{['eq:large_vol_0.02_0.08_fit']}, while the dotted line is for the $\langle \pi^a(x) \pi^a(0) \rangle$ correlator for $m_f = 0.0$ to 0.08 as given in Eq. \ref{['eq:large_vol_pp_fit']}.
• Figure 4: The $L_s$ dependence of the residual mass for $16^3 \times 32$ lattices at $\beta = 6.0$. The long-dashed line is the fit given in Eq. \ref{['eq:mres_exp']}, the short-dashed line is the fit from Eq. \ref{['eq:mres_exp_const']} and the solid line is the fit given in Eq. \ref{['eq:mres_exp_exp']}. Each of the three fits is made to all of the $L_s$ points shown. We have employed an intermediate non-perturbative renormalization to convert the plotted values of $m_{\rm res}$ into the $\overline{\rm MS}$ scheme at $\mu = 2$ GeV.
• Figure 5: Results for $f_\pi$ at $\beta=6.0$ with a $16^3 \times 32$ lattice and $L_s=16$ plotted as a function of $m_f$. The open circles are obtained from the $\langle A_0^a(x) A_0^a(0) \rangle$ correlator, while the open diamonds are obtained from the $\langle \pi^a(x) \pi^a(0) \rangle$ correlator. We also show the linear fits which are used to determine our estimate for $f_\pi$ and $f_K$. The vertical dashed lines identify the values for $m_f$ which locate the chiral limit, $m_f=-m_{\rm res}$ and give the physical ratio for $m_K/m_\rho$. The solid symbols represent the extrapolations to the point $m_f=-m_{\rm res}$ and interpolations to the kaon mass.