The physics of neutrino-nucleus cross sections is a critical probe of the Standard Model and beyond. A precise understanding is also needed to accurately deduce astrophysical neutrino spectra. At energies above $sim 5$ GeV, the cross section is dominated by deep inelastic scattering, mediated by weak bosons. In addition, there are subdominant processes where the hadronic coupling is through virtual photons, $gamma^ast$: (on-shell) $W$-boson production (e.g., where the underlying interaction is $ u_ell + gamma^ast rightarrow ell^- + W^+$) and trident production (e.g., where it is $ u + gamma^ast rightarrow u + ell_1^- + ell_2^+$). These processes become increasingly relevant at TeV--PeV energies. We undertake the first systematic approach to these processes (and those with hadronic couplings through virtual $W$ and $Z$ bosons), treating them together, avoiding common approximations, considering all neutrino flavors and final states, and covering the energy range $10,$--$10^8$ GeV. In particular, we present the first complete calculation of $W$-boson production and the first calculation of trident production at TeV--PeV energies. When we use the same assumptions as in prior work, we recover all of their major results. In a companion paper, we show that these processes should be taken into account for IceCube-Gen2.
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