Errors in Lattice Extractions of alpha_s Due to Use of Unphysical Pion Masses

Abstract

We investigate the errors due to the use of unphysical values of light quark masses in lattice extractions of $\alpha_s$. A functional form for the pion mass dependence of the quarkonium mass splittings ($\Delta m$) is given as an expansion in $m_\pi/(4\pi f_\pi)$ and $m_\pi r_B$, where $r_B$ is the quarkonium Bohr radius. We find that, to lowest order,$\Delta m\simeq A+B m_\pi^2$, where the scale of $B$ is given by $f_\pi^2 r_B^3$. To order $m_\pi^4$ there are four unknown coefficients, however, utilizing multipole and operator product expansions, symmetry arguments eliminate one of the four unknowns. Using the central values for the lattice spacings which were extracted using two different, unphysical values for the pion mass, we find that the errors introduced by extrapolating to the physical regime are comparable to the errors quoted due to other sources. Extrapolation to physical values of the pion mass {\it increases} the value of $\alpha_s(M_Z)$, bringing its value closer to the high energy extractions.