On Scale Determination in Lattice QCD with Dynamical Quarks

Dependence of $a/r_c$ (inverse Sommer parameter in units of lattice spacing $a$) on $am_q$ (quark mass in lattice unit) has been observed in all lattice QCD simulations with sea quarks including the ones with improved actions. How much of this dependence is a scaling violation has remained an intriguing question. Our approach has been to investigate the issue with an action with known lattice artifacts, i.e., the standard Wilson quark and gauge action with $\beta=5.6$ and 2 degenerate flavors of sea quarks on $ 16^3 \times 32 $ lattices. In order to study in detail the sea quark mass dependence, measurements are carried out at eight values of the Wilson hopping parameter $\kappa$ in the range 0.156 - 0.158 corresponding to PCAC quark mass values $am_q$ from about 0.07 to below 0.015. We analyze the static potential by fitting to the familiar phenomenological form and extract $a/r_c$. Though scaling violations may indeed be present for relatively large $am_q$, a consistent scenario at sufficiently small $am_q$ seems to emerge in the mass-independent scheme where for a fixed $\beta$, $1/r_0$ and $\sqrt{\sigma}$ have linear dependence on $m_q$ as physical effects similar to the quark mass dependence of the rho mass. We present evidence for this scenario and accordingly extract the lattice scale $a$ by chiral extrapolation to the physical point. Care has been exercised to determine optimal values of all fitting parameters and accuracy of the chiral extrapolation. An independent determination of the scale $a$ by chiral extrapolation of the rho mass is consistent with the scale obtained above ($a$ = 0.08041(12)(77) fm, $a^{-1}$ = 2.454(4)(23) GeV).