= Solution
<beta decay> is a <weak charged current> process. At <quark> level $d\to uW^{-*}\to ue^-\bar\nu_e$, with amplitude proportional to $G_FV_{ud}$ at low <momentum transfer>. A <neutron> in a <nucleus> can thereby become a <proton>:
$$
n\to pe^-\bar\nu_e,\qquad p\to ne^+\nu_e,\qquad pe^-\to n\nu_e.
$$
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> is
$$
\frac{d\Gamma}{dE_e}\propto F(Z,E_e)\,p_eE_e(E_0-E_e)^2
\left(g_V^2|M_F|^2+g_A^2|M_{GT}|^2\right).
$$
The <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 $0\to0$ 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>, $\pi^+\to\pi^0e^+\nu_e$. <semileptonic decays> test <quark> mixing and the universality of the <weak charged current>. \b[<beta decay> changes <quark flavour> through virtual W exchange; its continuous spectrum reflects an additional <lepton> in the final state.]
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