Map a three-body decay onto invariant masses

For an on-shell parent P with $P^2=M^2$, define $s_{ij}=(p_i+p_j)^2$ using the module's natural units and positive-time metric. Expand the three pair invariants and use $P=p_1+p_2+p_3$. Their sum is fixed by the parent and daughter masses, leaving two independent pair coordinates. The energy of daughter 3 in the parent rest frame follows by squaring $P-p_3=p_1+p_2$. Thus one pair mass directly determines the energy carried by the recoiling daughter.

Each pair mass must be at least the sum of that pair's masses, and at most the parent mass minus the third daughter mass. These one-coordinate limits are necessary but do not define a rectangle of physical events. At a fixed pair mass, the other coordinate must also correspond to a real relative angle. For nonzero daughter masses the resulting boundary is generally curved. Checking all three pair thresholds while ignoring the angle condition can still admit impossible configurations.

Choose a value of $s_{12}$ and transform to the rest frame of particles 1 and 2. Particle 2 then has energy $(s_{12}-m_1^2+m_2^2)/(2\sqrt{s_{12}})$, while particle 3 has energy $(M^2-s_{12}-m_3^2)/(2\sqrt{s_{12}})$. Their momenta follow from their mass shells. The second pair invariant depends linearly on the cosine of the angle between those two momenta. Parallel and antiparallel configurations therefore set the allowed extrema.

For three massless distinguishable daughters the geometry simplifies: all pair invariants are nonnegative and their sum is $M^2$. In the $(s_{12},s_{23})$ plane this is a triangle bounded by zero and $s_{12}+s_{23}=M^2$. For a spinless parent and constant squared amplitude, the density is uniform in this triangle because the phase-space Jacobian is constant in the squared-invariant coordinates. A plot in pair masses rather than their squares has additional Jacobians and need not look uniform even for the same dynamics.

A resonance in one pair produces enhanced amplitude when that pair's invariant mass approaches the resonant mass. The corresponding band is tied to an invariant coordinate, while its intensity and interference pattern depend on the full coherent amplitude. Several pathways to the same final state must generally be added before squaring. Adding independent event histograms can miss interference and does not reproduce the same amplitude model. The physical boundary remains fixed by the external masses even when resonant structures change the interior distribution.

Identical daughters require consistent labeling or an explicitly folded region. Symmetrizing the amplitude and avoiding duplicate phase-space counting are separate operations. In data, efficiency, resolution, and mistaken particle assignments can distort both bands and apparent endpoints. A phase-space baseline should be passed through the relevant selection model or compared with appropriately corrected observations. A kinematically allowed point is not necessarily likely, and a sparsely populated part of the allowed region is not necessarily forbidden by a conservation law.