Accurate boundary bootstrap for the three-dimensional O($N$) normal universality class

The three-dimensional classical O($N$) model with a boundary has received renewed interest due to the discovery of the extraordinary-log boundary universality class for $2\leq N< N_c$. The critical value $N_c$ and the exponent of the boundary correlation function are related to certain amplitudes in the normal universality class. To determine their precise values, we revisit the 3d O($N$) boundary conformal field theory for $N=1, 2, 3, 4, 5$. After substantially improving the accuracy of the boundary bootstrap, our determinations are in excellent agreement with the Monte Carlo results, resolving the previous discrepancies due to low truncation orders. We also use the recent bulk bootstrap results to deduce highly accurate Ising data. Many bulk and boundary predictions are obtained for the first time. Our results demonstrate the great potential of the $\eta$ minimization method for many unexplored bootstrap problems in which positivity constraints are absent.