The ratios of the light quark masses

The paper collects the various pieces of information concerning the relative size of mu, md and ms. A coherent picture results, which constrains the mass ratios to a rather narrow range: mu/md=0.553± 0.043, ms/md=18.9± 0.8. CERN-TH/96-44 February 1996 Work supported in part by Schweizerischer Nationalfonds 1. I wish to show that recent results of chiral perturbation theory allow a rather accurate determination of the relative size of mu, md and ms. The paper amounts to an update of earlier work [1]–[6] based on the same method. Sum rules and numerical simulations of QCD on a lattice represent alternative approaches with a broader scope – they permit a determination not only of the ratios mu : md : ms, but also of the individual quark masses, including the heavy ones. The sum rule results for the ratios are subject to comparatively large errors [7, 8]. Concerning the lattice technique, considerable progress has been made [9, 10]. It is difficult, however, to properly account for the vacuum fluctuations generated by quarks with small masses. Further progress with light dynamical fermions is required before the numbers obtained for mu/md or ms/md can be taken at face value. 2. The quark masses depend on the renormalization scheme. Chiral perturbation theory treats the mass term of the light quarks, muuu +md dd + msss, as a perturbation [11, 12]. It exploits the fact that, for mass independent renormalization schemes, the operators uu, dd and ss transform as members of the representation (3, 3)+(3, 3). Since all other operators with this transformation property are of higher dimension, the normalization conventions for mu, md and ms then differ only by a flavour-independent factor. The factor, in particular, also depends on the renormalization scale, but in the ratios mu/md and ms/md, it drops out – these represent conventionindependent pure numbers. 3. The leading order mass formulae for the Goldstone bosons follow from the relation of Gell-Mann, Oakes and Renner. Disregarding the electromagnetic interaction, they read M π+ = (mu + md)B, M 2 K+ = (mu + ms)B, M K0 = (md +ms)B, where the constant of proportionality is determined by the quark condensate: B= |〈0|uu|0〉|/F 2 π . Solving for the quark masses and forming ratios, this constant drops out, so that mu/md and ms/md may be expressed in terms of ratios of meson masses. Current algebra also shows that the mass difference between the π and the π is almost exclusively due to the electromagnetic interaction – the contribution generated by md 6= mu is of order (md − mu) and therefore tiny. Moreover, the Dashen theorem states that, in the chiral limit, the electromagnetic contributions to M K+ and to M π+ are the same, while the self energies of K 0 and π vanish. Using these relations to correct for the electromagnetic self energies, the above