Lattice determination of Sigma-Lambda mixing

Isospin breaking effects in baryon octet (and decuplet) masses are due to a combination of up and down quark mass differences and electromagnetic effects and lead to small mass splittings. Between the Sigma and Lambda this mass splitting is much larger, this being mostly due to their different wave functions. However when isospin is broken, there is a mixing between these states. We describe the formalism necessary to determine the QCD mixing matrix and hence find the mixing angle and mass splitting between the Sigma and Lambda particles due to QCD effects.

Figure 1

The lowest octet for the spin $\frac{1}{2}$ baryons plotted in the $I_3-Y$ plane.

Figure 2

A sketch of the heavy, $H$, and light, $L$, baryon ${(\text{masses})}^2$ against $m_u+m_d-2m_s$ for fixed $m_u-m_d$. The mass splitting between the Sigma and Lambda masses in the isospin limit ($m_u=m_d$) is given by the difference between the (red) circles; if $m_u\ne m_d$ then there is an additional mass difference due to mixing, the filled (blue) circles. Further explanation of the figure is given in the text.

Figure 3

$S_{\mathrm{\Sigma }\mathrm{\Lambda }}$ versus $\delta {\mu }_a$ [$S_{\mathrm{\Sigma }\mathrm{\Lambda }}$ is defined in Eq. (77)], together with a fit also given in Eq. (77). Points used in the fit are denoted by filled circles (those outside the fit range are given by open circles).

Figure 4

$D_{\mathrm{\Sigma }\mathrm{\Lambda }}^{\text{sym}}$ versus $\delta {\mu }_a+\delta {\mu }_b$ [$D_{\mathrm{\Sigma }\mathrm{\Lambda }}^{\text{sym}}$ is defined in Eq. (78)], together with the fit also given in Eq. (78). The same notation as in Fig. 3 is used.

Figure 5

The baryon fan plot for the $\mathrm{\Sigma }$- and $\mathrm{\Lambda }$-type particles ${\tilde{M}}_O^2$ ($O=N$, $\mathrm{\Lambda }$, $\mathrm{\Sigma }$, $\mathrm{\Xi }$, ${\mathrm{\Lambda }}_{2sl}$, $N_s$) versus $\delta m_l$. Filled up triangles, squares, left triangles, right triangles, diamonds and down triangles are the $N(ll{l}^{\prime \prime })$, $\mathrm{\Lambda }(l{l}^{\prime }s)$, $\mathrm{\Sigma }(lls)$, $\mathrm{\Xi }(ssl)$, ${\mathrm{\Lambda }}_{l2s}(s{s}^{\prime }l)$ and $N_s(ss{s}^{\prime \prime })$ results respectively using $32^3\times 64$ sized lattices. The common symmetric point is the filled circle. The open up triangles, left triangles, right triangles and down triangles are from comparison $24^3\times 48$ sized lattices (and not used in the fits here). The vertical dashed line from Eq. (79) is the $n_f=2+1$ pure QCD physical point, with the open circles being the numerically determined pure QCD hadron mass ratios for $2+1$ quark flavors. For comparison, the stars represent the average of the ${(\text{mass})}^2$ of $M_N^{*2}(ll{l}^{\prime \prime })=(M_n^{\exp 2}(ddu)+M_p^{\exp 2}(uud))/2$, $M_{\mathrm{\Lambda }}^{*2}(lls)=M_{{\mathrm{\Lambda }}^0}^{\exp 2}(uds)$, $M_{\mathrm{\Sigma }}^{*2}(lls)=(M_{{\mathrm{\Sigma }}^{-}}^{\exp 2}(dds)+M_{{\mathrm{\Sigma }}^{+}}^{\exp 2}(uus))/2$ and $M_{\mathrm{\Xi }}^{*2}(ssl)=(M_{{\mathrm{\Xi }}^{-}}^{\exp 2}(ssd)+M_{{\mathrm{\Xi }}^0}^{\exp 2}(ssu))/2$.