Oblate-prolate shape mixing and E0 transition in 28Si

Background: oblate-prolate shape coexistence in $^{28}$Si has been discussed for decades, but the degree of shape mixing between these configurations remains poorly constrained. Purpose: We constrain the oblate-prolate mixing amplitudes in $^{28}$Si using available experimental information and discuss the inter-band E0 transition strength. Methods: Oblate and prolate $0^+$ and $2^+$ configurations are obtained by antisymmetrized molecular dynamics combined with the generator coordinate method. Using these configurations as the basis states, we constrain the mixing amplitudes by simultaneously reproducing the measured charge radius, the quadrupole moment of the $2_1^+$ state, and the in-band and inter-band $B(\mathrm{E2})$ values. The strength of the density-dependent term in the Gogny interaction is also varied within a reasonable range. Results: In the ground state, the oblate component is dominant, and the prolate component in the ground state is limited to less than about $20\%$. For the $2_1^+$ state, the allowed prolate component is smaller than that in the ground state. The present analysis does not tightly constrain the corresponding E0 transition strength, but an upper limit of $\rho^2(\mathrm{E0};0_3^+\rightarrow0_1^+) \lesssim 0.206$ is obtained. Conclusions: The low-lying $0^+$ states of $^{28}$Si may exhibit substantial oblate-prolate mixing. A measurement of the inter-band E0 transition strength would provide a quantitative determination of the mixing amplitude.