Particle decay

A particle decay into three daughter particles subject to four-momentum conservation and the relativistic energy-momentum relation. The daughter energies generally vary continuously even in the parent's rest frame. Grouping two daughters into a recoil system of variable invariant mass turns the remaining kinematics into a relativistic two-body decay.

A boundary of phase space approached as a massless outgoing particle's energy tends to zero. A zero four-vector is not a physical positive-energy one-particle state. An energy bound requiring such a particle is a supremum or infimum, even when the closed kinematic interval is conventionally described using minimum and maximum. This distinction can matter when applying three-body formulas at mixed massive-massless recoil thresholds.

Use natural units $c=1$ and assume open phase space, $M>m_1+m_2+m_3$. In the parent's rest frame, define $s_{23}=(p_2+p_3{)}^2$. Four-momentum conservation givesThe upper recoil-mass limit follows from $E_1\ge m_1$. For positive $m_2,m_3$, the lower recoil-mass limit follows in their pair rest frame and is attained when their relative momentum vanishes. If both recoil particles are massless, $s_{23}=2E_2{E}_3(1-\cos {\theta }_{23})$ reaches zero with nonzero parallel momenta. If exactly one is massless, $s_{23}$ approaches the nonzero daughter's squared mass only when the massless particle becomes soft. Thus that upper $E_1$ bound is a supremum for a genuine three-body event with all massless particles of positive energy. If $m_1=0$, the lower endpoint is likewise a soft limit. Multiplying the mass expressions for $E_1$ by $c^2$ restores energy units.

A weak interaction decay of a baryon into final hadrons without final leptons. A weak charged current changes quark flavour, while spectator quarks and the produced quarks undergo hadronization. A quark-flow Feynman diagram specifies a possible weak topology; it does not calculate the full hadronic decay amplitude.

A nonleptonic meson decay contains final hadrons and no final leptons. Weak quark transitions set flavour factors, but strong hadronization and additional hadronic matrix elements are needed to determine exclusive rates.

For Ds meson spectator decays, $c\to s{W}^{+}$ followed by $W^{+}\to u\bar{d}$ carries CKM magnitude $|V_{cs}{V}_{ud}|$ and can form either $\varphi {\pi }^{+}$ or $K^{+}{\bar{K}}^0$. Replacing $c\to s$ by $c\to d$ carries $|V_{cd}{V}_{ud}|$ and can form $K^{+}{\pi }^0$ through the $-d\bar{d}/\sqrt{2}$ pion component. With otherwise equal reduced hadronic strengths, the last channel's rate is suppressed relative to the neutral-antikaon channel by $|V_{cd}/V_{cs}{|}^2/2$. Actual branching fractions also depend on phase space, colour contractions and strong final-state interactions; the CKM factors do not determine exact exclusive rates.

A leptonic decay has only leptons in its non-radiative final state. For example, a charged pion can decay as ${\pi }^{+}\to {\mu }^{+}{\nu }_{\mu }$. A semileptonic decay instead retains at least one hadron alongside the leptons. This classification concerns the final particles, not whether the initial particle is a hadron or a lepton.

A particle decay cascade follows successive unstable daughters to longer-lived or stable particles. Probabilities of specified independent decay paths multiply their branching fractions. Waiting times are governed by each particle's mean lifetime, with time dilation relating proper and laboratory times. A particle which appears stable on one observation timescale can be an intermediate on another.

A semileptonic decay contains both final-state hadrons and leptons, typically through a weak charged current. This distinguishes it from a purely leptonic decay.

Pion beta decay changes a charged pion into a neutral pion and a lepton pair through the weak charged current. The small charged-neutral pion mass gap severely restricts phase space, giving an approximately fifth-power dependence on that gap. It does not have the charged-lepton mass suppression of direct leptonic pion decay.

A Dalitz decay replaces a radiated real photon with a virtual photon converting to a charged-lepton pair. The neutral-pion example is ${\pi }^0\to \gamma e^{+}{e}^{-}$. Double Dalitz decay converts both virtual photons in the two-photon mode.

Double Dalitz decay converts both virtual photons from a two-photon hadronic transition into charged-lepton pairs. The neutral-pion four-electron final state avoids the specific two-lepton helicity suppression of ${\pi }^0\to e^{+}{e}^{-}$, despite its more restricted phase space.