Light quark masses beyond leading order.

Abstract

We describe the measurement of the light quark mass ratios when one calculates to second order in the quark masses. At this order there is an ambiguity in the meaning of the quark mass, which afflicts the past attempts to provide a model-independent measurement of the ratios. We argue that this is similar to the regularization-scheme dependence of coupling constants. We study the anomalous Ward identities and the effects of strong $\mathrm{CP}$ violation in an attempt to resolve the ambiguity. The ambiguity persists even with singlet fields, such as the ${\ensuremath{\eta}}^{\ensuremath{'}}$, but can be resolved by observing the $\ensuremath{\theta}$ dependence of the theory. Since matrix elements of $F\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}$ are related to $\frac{\ensuremath{\partial}{\mathcal{L}}_{\mathrm{QCD}}}{\ensuremath{\partial}\ensuremath{\theta}}$, they are useful probes of quark masses. We give a procedure by which quark mass ratios can be measured in a model-independent way through the matrix elements $〈0|F\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}|{\ensuremath{\pi}}^{0}〉$, $〈0|F\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}|\ensuremath{\eta}〉$, and $〈0|F\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}|3\ensuremath{\pi}〉$, which in turn are observable in ${V}^{\ensuremath{'}}\ensuremath{\rightarrow}V{\ensuremath{\pi}}^{0}(\ensuremath{\eta},3\ensuremath{\pi})$, with $V$ being $\ensuremath{\psi}$ or $\ensuremath{\Upsilon}$, when analyzed using a heavy-quark multipole expansion. Present data are not sufficient to complete this program, but we use available results to provide the value $[\frac{({m}_{d}\ensuremath{-}{m}_{u})}{({m}_{d}+{m}_{u})}]\frac{({m}_{s}+\stackrel{^}{m})}{({m}_{s}\ensuremath{-}\stackrel{^}{m})}=0.59\ifmmode\pm\else\textpm\fi{}0.07\ifmmode\pm\else\textpm\fi{}0.08 (\mathrm{i}.\mathrm{e}., \frac{{m}_{u}}{{m}_{d}}=0.30\ifmmode\pm\else\textpm\fi{}0.05\ifmmode\pm\else\textpm\fi{}0.05)$, where the first error is experimental and the second is our estimate of the remaining theoretical model dependence.