Coulomb effects and charge symmetry breaking for theA=4hypernuclei

The effect Δ$B_c$ of the Coulomb interaction on the Λ separation energy $B_{\Lambda }$ of ${}_{\mathrm{\Lambda }}{}^4{}_{}{}^{}\mathrm{He}$ was obtained by variational calculations made for ${}_{\mathrm{\Lambda }}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. These calculations were made for several values of $q^2$ in the range 0≤$q^2$≤9 where qe is the proton charge, i.e., the Coulomb repulsion was artificially boosted. For $q^2$≲3, the dependence on $q^2$ is linear, and interpolation to $q^2$=1 gives the physical values with improved accuracy: -Δ$B_c$=0.05±0.02 and 0.025±0.015 MeV for the ground and excited state, respectively. This procedure also gives more accurate values for the differences between the proton and neutron radii of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The corresponding differences of $B_{\Lambda }$ between ${}_{\mathrm{\Lambda }}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{\mathrm{\Lambda }}{}^4{}_{}{}^{}\mathrm{H}$, to be attributed to charge symmetry breaking effects, are then 0.40±0.06 and 0.27±0.06 MeV. From these values we obtain a phenomenological charge symmetry breaking potential which is effectively spin independent. An examination of meson-exchange charge symmetry breaking models shows that these are consistent with the phenomenological charge symmetry breaking potential for the triplet but not for the singlet case.