Improved calculation of the muonic-helium Lamb shift

Several small, previously uncalculated contributions to the $2s_{\frac{1}{2}}-2\mathrm{}{p}_{\frac{3}{2}}$ splitting $S_1$ in muonic ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ are calculated, and previously calculated contributions are verified. The improved theoretical result is $1817.5\pm 0.6+0.4{〈r^2〉}^{\frac{1}{2}}-106.2\mathrm{}〈r^2〉+1.4\mathrm{}{〈r^2〉}^{\frac{3}{2}}\simeq 1527.3-102.6(〈r^2〉=2.80)$ meV, where $〈r^2〉$ is the mean-square charge radius of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ in ${\mathrm{fm}}^2$. Using the most recent value of the rms charge radius we get $S_{1\mathrm{th}}=1527.1\mathrm{}\pm 4.2$ meV, in excellent agreement with the experimental result $S_{1\exp }=1527.5\mathrm{}\pm 0.3$ meV.