Simple analytical expression for vector hypernuclear asymmetry in nonmesonic decay of $^5_Lambda$He and $^{12}_\Lambda$C

We present general explicit expressions for a shell-model calculationof the vector hypernuclear parameter in nonmesonic weak decay. We use a widely accepted effective coupling Hamiltonian involving theexchange of the complete pseudoscalar and vector meson octets(p, h, K, r, w, K*). In contrast to the approximated formula widely used in the literature,we correctly treat the contribution of transitions originated fromsingle-proton states beyond the s-shell. Exact and simple analytical expressions are obtained for the particularcases of $^5_Lambda$He and $^{12}_\Lambda$C, within theone-pion-exchange model. Numerical computations of the asymmetry parameter, $a_\Lambda$, are presented. Our results show a qualitative agreementwith other theoretical estimates but also acontradiction with recent experimental determinations. Our simple analytical formulas provide a guide in searchingthe origin of such discrepancies, andthey will be useful for helping to solve the hypernuclearweak decay puzzle