Up and Down Quark Masses and Corrections to Dashen’s Theorem from Lattice QCD and Quenched QED

In a previous Letter [Borsanyi et al., Phys. Rev. Lett. 111, 252001 (2013)] we determined the isospin mass splittings of the baryon octet from a lattice calculation based on $N_f=2+1$ QCD simulations to which QED effects have been added in a partially quenched setup. Using the same data we determine here the corrections to Dashen’s theorem and the individual up and down quark masses. Our ensembles include 5 lattice spacings down to 0.054 fm, lattice sizes up to 6 fm, and average up-down quark masses all the way down to their physical value. For the parameter which quantifies violations to Dashen’s theorem, we obtain $ϵ=0.73(2)(5)(17)$, where the first error is statistical, the second is systematic, and the third is an estimate of the QED quenching error. For the light quark masses we obtain, $m_u=2.27(6)(5)(4)$ and $m_d=4.67(6)(5)(4)\mathrm{MeV}$ in the modified minimal subtraction scheme at $2\mathrm{G}\mathrm{e}\mathrm{V}$ and the isospin breaking ratios $m_u/m_d=0.485(11)(8)(14)$, $R=38.2(1.1)(0.8)(1.4)$, and $Q=23.4(0.4)(0.3)(0.4)$. Our results exclude the $m_u=0$ solution to the strong $CP$ problem by more than 24 standard deviations.

Figure 1

Example of a fit of the dependence of $\mathrm{\Delta }{M}_K^2$ on $\alpha$ and $\mathrm{\Delta }{M}^2$ to the expression of Eq. (2). Here, $\mathrm{\Delta }{M}_K^2$ is plotted as a function of $\mathrm{\Delta }{M}^2$. The dependence of the lattice results on all other variables has been subtracted using the fit. The fit has a correlated ${\chi }^2/\mathrm{dof}$ equal to $61.57/45=1.37$, corresponding to a $p$ value of 5%. It is plotted as a solid curve, with its $1\sigma$ band.

Figure 2

Same fit as in Fig. 1. Here ${\mathrm{\Delta }}_{\mathrm{QED}}{M}_K^2$ is plotted as a function of $1/L$. The grayed symbols show the full volume dependence of the data. For the plain symbols, the universal $O(1/L)$ and $O(1/L^2)$ finite volume effects from Eq. (3) have been subtracted and the fit to the $O(1/L^3)$ correction is plotted.

Figure 3

Same fit as in Fig. 1. Here ${\mathrm{\Delta }}_{\mathrm{QED}}{M}_K^2$ is plotted as a function of $M_{{\pi }^{+}}^2$.