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beta decay is a weak charged current process. At quark level $d\to u{W}^{-\ast }\to u{e}^{-}{\bar{\nu }}_e$, with amplitude proportional to $G_F{V}_{ud}$ at low momentum transfer. A neutron in a nucleus can thereby become a proton:The second reaction denotes an energetically allowed nuclear conversion, not a decay of an isolated free proton. In a nuclear beta transition the mass number stays fixed and the atomic number changes by one. baryon number and electric charge are conserved; the neutrino or antineutrino supplies the missing energy, momentum and angular momentum. The three-body final state produces a continuous electron spectrum rather than a single electron energy. Neglecting neutrino mass and recoil, an allowed spectrum in electron total energy isThe Fermi function accounts for the electron's Coulomb interaction with the daughter; $E_0$ is the electron total-energy endpoint. forbidden beta decay adds momentum-dependent nuclear shape factors. The V-A interaction violates parity: it couples left-chiral fermion fields, not an equal mixture of left and right currents.

For allowed beta decay the leading leptons have zero orbital angular momentum. Fermi transitions have $\Delta J=0$; Gamow-Teller transitions have $\Delta J=0,1$ with $0\to 0$ excluded. Both preserve nuclear parity. Higher lepton partial waves permit forbidden transitions with extra suppression. Nuclear matrix elements and phase space, as well as these selection rules, determine lifetimes. Particle-physics examples include free-neutron decay and pion beta decay, ${\pi }^{+}\to {\pi }^0{e}^{+}{\nu }_e$. semileptonic decays test quark mixing and the universality of the weak charged current. beta decay changes quark flavour through virtual W exchange; its continuous spectrum reflects an additional lepton in the final state.

Beta decay converts a neutron to a proton or vice versa through the weak interaction, leaving the mass number fixed:Inside a nucleus the full initial and final nuclear masses determine whether either process is possible. Electron capture, $p+e^{-}\to n+{\nu }_e$, is the related bound-electron process. The emitted charged-lepton spectrum is continuous because energy is shared with the neutrino and nuclear recoil.

For allowed beta decay, the lepton wavefunctions' leading spatial terms are constant over the nucleus. A Fermi transition has $\Delta J=0$ and no parity change. A Gamow-Teller transition has $\Delta J=0,1$ and no parity change, excluding $0\to 0$. Thus a same-parity transition with nonzero initial and final spins and $\Delta J=0$ can contain both contributions. First-forbidden transitions change parity and permit $|\Delta J|=0,1,2$; the $|\Delta J|=2$ case is unique forbidden beta decay. Higher forbiddenness requires higher powers of the small lepton momentum times nuclear size. “Forbidden” denotes suppression of the leading allowed operator, not an impossible decay.^[ref]

Fermi's golden rule expresses a rate as a squared nuclear matrix element times leptonic phase space. Neglecting charged-lepton mass and Coulomb distortion, a constant allowed matrix element givesThis is the approximate Sargent rule, $\lambda \propto Q^5$ for comparable allowed strengths; it is not an exact law across arbitrary transitions. Finite electron mass, the Fermi function, forbidden shape factors and nuclear structure change the comparison. A microscopic first-forbidden unique operator introduces additional momentum factors.

Using bare nuclear masses $M_i,M_f$, the kinetic-energy release isThe daughter is different in the two cases: its charge is $Z+1$ for beta minus and $Z-1$ for beta plus. The relevant mass difference must exceed one electron mass. For neutral atomic masses, to leading order in small electronic binding corrections, the corresponding expressions are $Q_{{\beta }^{-}}=(M_i^{\mathrm{at}}-M_f^{\mathrm{at}})c^2$ and $Q_{{\beta }^{+}}=(M_i^{\mathrm{at}}-M_f^{\mathrm{at}}-2m_e)c^2$. Electron capture is energetically possible approximately when $M_i^{\mathrm{at}}>M_f^{\mathrm{at}}$, with available shell binding energy and daughter excitation included. Nuclear masses and atomic masses must not be mixed.

Assume allowed beta decay, so the leading matrix element has negligible lepton-momentum dependence. Neglect nuclear recoil and neutrino mass. Let $Q$ be the kinetic energy available to the electron and antineutrino, $E_0=m_e+Q$, and $E=\sqrt{p^2+m_e^2}$. The neutrino energy and momentum magnitude are $k=E_0-E$.

Fermi's golden rule weights accessible final states. After angular integration, the momentum-space factors are $p^{2}dp$ and $k^{2}dk$. The energy-conserving Dirac delta function removes the neutrino integral:The constant includes the allowed nuclear matrix element, weak-coupling and angular factors. Including the usual Fermi function for electron-nucleus Coulomb distortion givesFor the kinetic-energy spectrum, $dp/dT=E/p$ instead yieldsThe two distributions differ by a Jacobian; forgetting it changes the shape.

A pure fifth-power law additionally neglects electron mass and Coulomb distortion and keeps the nuclear matrix element fixed. Then $E=p$, $E_0\simeq Q$, andTwo momentum powers come from each final particle and one from the remaining integration. This is the origin of the Sargent rule. Exact $Q^5$ dependence requires the relativistic-electron, constant-matrix-element limit. For $Q\ll m_e$, the same approximation without Coulomb distortion has $pE\simeq m_e\sqrt{2m_{e}T}$ and instead scales as $m_e^{3/2}{Q}^{7/2}$. Forbidden-transition factors, nuclear-structure changes and Coulomb distortion also modify the simple law.

If a beta spectrum has nonconstant shape factor $C(T)$, divide by it as well as by the Coulomb and phase-space factors before taking a square root. A leading unique first-forbidden shape contains $p_e^2+p_{\nu }^2$ in natural units, with Coulomb and nuclear corrections. Dividing only by a Coulomb Fermi function does not remove that shape. A straight uncorrected line must therefore be treated as an approximation, not an exact consequence of forbidden-transition selection rules.