Solar models with protosolar accretion and turbulent mixing

(abridged) Recent analyses have reported low lithium but high beryllium abundances on the solar surface; however, standard solar models (SSMs) predict Li abundances that are ~30$\sigma$ away from the observed value. In this study, we aim to develop solar models and compare them with the Li and Be abundance constraints. We examine the effect of protosolar accretion and turbulent mixing below the base of the surface convective zone. We compute ~200 solar evolutionary models for each case to optimize input parameters using target quantities, similar to the SSM framework. We confirm that turbulent mixing helps reproduce the surface Li and Be abundances within ~0.6$\sigma$ by enhancing burning. It suppresses gravitational settling, leading to a better matching of the He surface abundance ($\lesssim$0.3$\sigma$) and a smaller compositional gradient. We derive a new protosolar helium abundance $Y_{\rm proto}=0.2651\pm0.0035$. Turbulent mixing decreases the central metallicity ($Z_{\mathrm{center}}$) by $\approx$4.4%, even though accretion increases $Z_{\rm center}$ by $\approx$4.4%, as suggested by our previous study. Unfortunately, the reduction in $Z_{\rm center}$ implies that our models do not reproduce constraints on observed neutrino fluxes by $6.2\sigma$ for $^8{\rm B}$ and $2.7\sigma$ for CNO. Including turbulent mixing in solar models appears indispensable to reproduce the observed atmospheric abundances of Li and Be. However, the resulting tensions in terms of neutrino fluxes, even in the models with the protosolar accretion, show that the solar modeling problem remains, at least partly. We suggest that improved electron screening, as well as other microscopic properties, may help alleviate this problem. An independent confirmation of the neutrino fluxes measured by the Borexino experiment would also be extremely valuable.