Particle Decays and Two-Body Kinematics

A decay is the simplest relativistic reaction: one particle in, two or more out, with no external forces. The conservation of total four-momentum and the invariant $p \cdot p = - m^{2} c^{2}$ are enough to fix every energy and momentum from the particle masses, without any dynamical detail of the interaction. This lesson works out the two-body case completely — daughter energies, the shared momentum magnitude, the lab-frame opening angle, and the invariant-mass reconstruction that turns raw detector hits back into a parent mass.

The signature is $(-, +, +, +)$ , so for a system of particles the total four-momentum $P^{μ} = \sum_{i} p_{i}^{μ}$ has $P \cdot P = - M^{2} c^{2}$ , where $M$ is the invariant mass of the system.

The invariant mass and the center-of-momentum frame

For any collection of particles the total four-momentum is a four-vector, and its Minkowski length defines a mass:

(M c^{2})^{2} = - (P \cdot P) c^{2} = (i \sum E_{i})^{2} - (i \sum p_{i})^{2} c^{2} .

Unlike the individual energies and momenta, $M$ is the same in every inertial frame. The center-of-momentum (CM) frame is the frame in which the total three-momentum vanishes, $\sum_{i} p_{i} = 0$ ; there the total energy is purely the invariant mass energy,

E_{CM} = i \sum E_{i}^{*} = M c^{2} .

For a single particle $M$ is just its rest mass and the CM frame is its rest frame. For a system, $M$ exceeds the sum of the constituent masses by the kinetic and binding energy carried in the CM frame. A decay conserves $P^{μ}$ , so the invariant mass of the final state equals the parent's rest mass.

Two-body decay in the CM frame

A parent of mass $M$ at rest decays into daughters of masses $m_{1}$ and $m_{2}$ . Because the parent is at rest, this is already the CM frame. Conservation of energy and momentum reads

E_{1}^{*} + E_{2}^{*} = M c^{2}, p_{1}^{*} + p_{2}^{*} = 0 .

The momenta are equal and opposite, so the daughters emerge back to back with a common momentum magnitude $p^{*} = ∣ p_{1}^{*} ∣ = ∣ p_{2}^{*} ∣$ . Each daughter is on shell, $E_{i}^{*} = (p^{*} c)^{2} + (m_{i} c^{2})^{2}$ . Two equations in the two unknowns $E_{1}^{*}, E_{2}^{*}$ (with $p^{*}$ shared) solve cleanly. Subtract the squared on-shell relations to eliminate $p^{*}$ ,

(E_{1}^{*})^{2} - (E_{2}^{*})^{2} = (m_{1}^{2} - m_{2}^{2}) c^{4},

and combine with $E_{1}^{*} + E_{2}^{*} = M c^{2}$ to get the daughter energies

E_{1}^{*} = \frac{M ^{2} + m _{1}^{2} - m _{2}^{2}}{2 M} c^{2}, E_{2}^{*} = \frac{M ^{2} + m _{2}^{2} - m _{1}^{2}}{2 M} c^{2} .

The heavier daughter takes the larger share; the split is fixed entirely by the three masses. The shared momentum follows from either on-shell relation and can be written with the Källén triangle function $λ (a, b, c) = a^{2} + b^{2} + c^{2} - 2 ab - 2 b c - 2 c a$ :

p^{*} = \frac{c}{2 M} λ (M^{2}, m_{1}^{2}, m_{2}^{2}) = \frac{c}{2 M} [M^{2} - (m_{1} + m_{2})^{2}] [M^{2} - (m_{1} - m_{2})^{2}] .

The decay is kinematically allowed only when $p^{*}$ is real, which requires $M \geq m_{1} + m_{2}$ : a particle cannot decay into daughters heavier than itself. When $M = m_{1} + m_{2}$ exactly, $p^{*} = 0$ and the daughters are produced at rest — the threshold of the decay.

In the parent rest frame the two daughters leave back to back with equal momentum magnitude p-star and energies fixed by the masses; the heavier daughter carries the larger energy.

The neutral pion to two photons

The neutral pion decays to two photons, $π^{0} \to γ γ$ , more than $98%$ of the time.¹ Both daughters are massless, $m_{1} = m_{2} = 0$ , so the symmetric formulas give each photon half the parent rest energy in the CM frame:

E_{γ}^{*} = \frac{M c ^{2}}{2} = \frac{m _{π^{0}} c ^{2}}{2} = \frac{134.98 MeV}{2} = 67.49 MeV,

with the two photons back to back. Their invariant mass is

(M c^{2})^{2} = (E_{1} + E_{2})^{2} - (p_{1} + p_{2})^{2} c^{2} = 2 E_{1} E_{2} (1 - cos θ),

using $E = ∣ p ∣ c$ for each photon and $θ$ for the opening angle between them. In the pion rest frame $θ = 18 0^{\circ}$ and $E_{1} = E_{2} = m_{π^{0}} c^{2} /2$ reproduces $M c^{2} = m_{π^{0}} c^{2}$ . In the lab the pion moves with Lorentz factor $γ = E_{π} / (m_{π^{0}} c^{2})$ , and the two photons are beamed forward into a cone. The opening angle has a minimum, reached when the decay is symmetric (each photon perpendicular to the boost in the CM frame):

sin \frac{θ _{min}}{2} = \frac{1}{γ} = \frac{m _{π^{0}} c ^{2}}{E _{π}} .

A fast pion produces two nearly collinear photons; measuring their opening angle and energies recovers the pion energy and confirms the parent mass. This is the basic signature by which the $π^{0}$ is detected in a calorimeter.

The neutral-pion decay in the rest frame (photons back to back, each carrying half the rest energy) and boosted to the lab, where the same two photons are beamed forward into a cone of opening angle at least 2 arcsin of one over gamma.

Invariant-mass reconstruction

A detector measures the daughters, not the parent. The parent mass is recovered by building the invariant mass of the observed daughters, which is frame- and boost-independent. For two daughters of measured energies $E_{1}, E_{2}$ , momenta $p_{1}, p_{2}$ , and opening angle $θ$ ,

(M c^{2})^{2} = m_{1}^{2} c^{4} + m_{2}^{2} c^{4} + 2 (E_{1} E_{2} - ∣ p_{1} ∣∣ p_{2} ∣ c^{2} cos θ) .

Every genuine decay of a given parent lands at the same $M$ regardless of how the parent was moving, so plotting the reconstructed $M$ for many events produces a sharp peak at the parent mass sitting on a smooth combinatorial background from uncorrelated pairs. The peak's width is set by detector resolution and, for short-lived states, by the natural decay width; its position is the mass. This is how resonances are discovered: the $ρ$ , the $J / ψ$ , and the Higgs boson all first appeared as invariant-mass peaks in exactly this construction.

Reconstructed invariant mass of many daughter pairs: genuine decays pile up in a peak at the parent mass M, while random uncorrelated pairs form a smooth background beneath it.

A heavy two-body hadronic decay

The lambda baryon decays to a proton and a negative pion, $Λ^{0} \to p + π^{-}$ , about $64%$ of the time.¹ With $m_{Λ} c^{2} = 1115.68 MeV$ , $m_{p} c^{2} = 938.27 MeV$ , $m_{π} c^{2} = 139.57 MeV$ ,² the shared momentum is

p^{*} c = \frac{1}{2 m _{Λ}} [m_{Λ}^{2} - (m_{p} + m_{π})^{2}] [m_{Λ}^{2} - (m_{p} - m_{π})^{2}] c^{4} = 101 MeV,

and the daughter energies are

E_{p} = \frac{m _{Λ}^{2} + m _{p}^{2} - m _{π}^{2}}{2 m _{Λ}} c^{2} = 943.7 MeV, E_{π} = \frac{m _{Λ}^{2} + m _{π}^{2} - m _{p}^{2}}{2 m _{Λ}} c^{2} = 172.0 MeV .

They sum to $E_{p} + E_{π} = 1115.7 MeV = m_{Λ} c^{2}$ , as required. The proton, being much heavier, carries $84%$ of the total energy but shares the same $101 MeV / c$ momentum as the pion, which is highly relativistic ( $γ_{π} = E_{π} / m_{π} c^{2} = 1.23$ ) while the proton is barely moving ( $γ_{p} = 1.006$ ). The unequal energy split with equal momentum magnitude is the generic signature of a two-body decay, and inverting the relations from measured daughters is the standard test of the parent hypothesis.

Energy sharing in a two-body decay: both daughters carry the same momentum magnitude, but the heavier proton takes far more of the released energy than the light pion, with the split fixed by the masses.

Decay in flight and the lab energy spectrum

A parent moving in the lab with Lorentz factor $γ$ and speed $β c$ decays isotropically in its own rest frame. A daughter emitted at CM angle $θ^{*}$ has rest-frame energy $E^{*}$ and momentum $p^{*}$ ; boosting along the flight direction gives its lab energy

E_{lab} = γ (E^{*} + β p^{*} c cos θ^{*}) .

Isotropy in the rest frame means $cos θ^{*}$ is distributed uniformly on $[- 1, 1]$ , so $E_{lab}$ is uniform between the endpoints

E_{\pm} = γ (E^{*} \pm β p^{*} c) .

The lab energy spectrum of the daughter is a flat box: constant probability $d N / d E_{lab} = 1/ (E_{+} - E_{-}) = 1/ (2 γ β p^{*} c)$ across the allowed band and zero outside it. The box is centered on $γ E^{*}$ and widens with the parent's speed. A massless daughter ( $E^{*} = p^{*} c$ ) has $E_{-} = γ E^{*} (1 - β)$ and $E_{+} = γ E^{*} (1 + β)$ , so the two-photon energies of a fast $π^{0}$ spread over a wide flat band whose edges fix the pion energy without any angular measurement.

The lab energy of a daughter from an isotropic decay in flight is uniformly distributed in a flat box between the minimum and maximum boosted energies, centered on gamma times the rest-frame energy.

The parent's finite lifetime turns into a measurable flight distance. If the rest-frame mean lifetime is $τ_{0}$ , the parent travels a mean decay length

L = γ β c τ_{0}

before decaying, since the lab time is dilated to $γ τ_{0}$ and the parent covers $β c$ per unit lab time. A relativistic beam therefore leaves a decay vertex displaced from the production point, and measuring the displacement distribution recovers $τ_{0}$ ; this is how short-lived hadrons with $c τ_{0}$ of tens of microns are timed in a silicon vertex detector.

The same conservation law, applied to two particles colliding rather than one decaying, gives the threshold energies of particle production and the reason colliders beat fixed targets. That is the subject of the next lesson, where the invariant mass of the initial state becomes the available energy $s$ .

Particle Data Group, Review of Particle Physics, Kinematics review, two-body decay formulas; branching fractions from the particle listings. https://pdg.lbl.gov/2023/reviews/rpp2023-rev-kinematics.pdf ↩ ↩²
Particle masses from the Particle Data Group summary tables. https://pdg.lbl.gov/ ↩