The Standard Model of particle physics has been very successful in describing fundamental interactions up to the highest energies currently probed in particle accelerator experiments. However, the Standard Model is incomplete and currently exhibits tension with experimental data for interactions involving $B$~mesons. Consequently, $B$-meson physics is of great interest to both experimentalists and theorists. Experimentalists worldwide are studying the decay and mixing processes of $B$~mesons in particle accelerators. Theorists are working to understand the data by employing lattice gauge theories within and beyond the Standard Model. This work addresses the theoretical effort and is divided into two main parts.
In the first part, I present a lattice-QCD calculation of form factors for exclusive semileptonic decays of $B$~mesons that are mediated by both charged currents ($B \to \pi \ell \nu$, $B_s \to K \ell \nu$) and neutral currents ($B \to \pi \ell^+ \ell^-$, $B \to K \ell^+ \ell^-$). The results are important for constraining or revealing physics beyond the Standard Model. This work uses MILC's $(2+1+1)$-flavor ensembles, with the HISQ action for the up, down, strange, and charm sea and valence quarks, and the clover action in the Fermilab interpretation for the $b$~quark. Simulations are carried out on five ensembles at three lattice spacings down to $\approx 0.088$~fm, with both physical and unphysical sea-quark masses. I present results for correlation-function fits and describe the chiral-continuum extrapolation and $z$~expansion of the form factors.
In the second part, I present the status of an ongoing lattice calculation of $SU(3)$ gauge theory with eight and $12$ fundamental fermions, in which I search for the endpoint of the line of first-order phase transitions in the mass--coupling plane. The results are important for understanding the phase diagram of these models, with possible applications to physics beyond the Standard Model. This work uses the staggered action and the nHYP-improved staggered action for the fermions, as well as the Wilson action in the fundamental and fundamental--adjoint representations for the gauge bosons. With the unimproved actions, I present results for a small lattice and describe the ongoing effort on larger lattices. I also discuss the shift symmetry on lattices with and without improved actions.