This chapter explains how neutrinos with energies of a few to a few tens of MeV interact with nuclear matter. At the simplest level, low‑energy neutrino interactions with single protons or neutrons are well described by Fermi’s effective theory of the weak force. Extending that simple picture to dense nuclear matter is hard. The authors provide a concise introduction to the many‑body theory needed to compute the nuclear response to neutrinos and they discuss how those results affect the neutrino mean free path in dense matter.

The chapter first shows how the neutrino–nucleon cross section is derived in free space. Because the W and Z bosons that carry the weak force are very heavy (about 80 and 91 GeV), low‑energy weak interactions can be treated as a short‑range, four‑particle contact interaction. The derivation uses standard constants such as the Fermi constant GF = 1.166×10−5 GeV−2 and nucleon couplings gV = 1 and gA = 1.27. In the low‑energy, nonrelativistic limit the nucleon current simplifies and the scattering rate acquires a simple angular dependence that separates vector and axial contributions.

The main part of the chapter moves from single nucleons to nuclear matter. The authors introduce a nuclear many‑body formalism that builds the nuclear transition amplitude from sums of one‑body currents acting on all nucleons. They use the independent‑particle or “mean‑field” picture as a baseline, and then add the two main kinds of corrections. Short‑range correlations arise from strong, close encounters between nucleons; these push some nucleons to high momentum and reduce the occupation of low‑momentum states, which in turn quenches weak transition strengths. Long‑range correlations involve many nucleons acting together and can produce collective excitations that dominate the response at low momentum transfer.

Why this matters is explained in concrete terms. Neutrino cross sections in nuclear matter matter for supernova explosions, for how neutron stars cool, and for mergers of neutron star binaries. The neutrino mean free path — how far a neutrino travels between interactions — controls how energy and lepton number are transported in these dense environments. The chapter shows how different nuclear effects change the computed response functions and therefore the mean free path.