Separate decay directions from laboratory collimation

Use natural units and the module's positive-time metric. In a parent's rest frame, the daughters have equal and opposite three-momenta, but their energies need not be equal when their masses differ. Squaring $P-p_1=p_2$ determines the first energy directly. The Källén momentum follows by imposing its mass shell. These relations are exact for an on-shell parent decaying to two on-shell daughters and depend only on masses. They do not determine how likely a particular direction is.

For a spin-zero parent, no external direction is available in its rest frame, so the spin-summed decay is isotropic if no external field or correlated selection supplies one. A rotationally unpolarized ensemble likewise has no preferred absolute orientation when unobserved daughter spin states are summed. A polarized parent or a distribution measured relative to an additional spin analyzer can be anisotropic. Conservation laws constrain allowed amplitudes, but assigning a cosine-shaped law requires more information than simply naming the parent spin.

Take a parent moving along positive z with speed $\beta$ in the laboratory. To transform a rest-frame daughter into that laboratory frame, the energy and longitudinal momentum receive plus-sign boost terms; the passive transformation into a frame moving along positive z would instead use minus signs. Transverse momentum stays unchanged. Keeping this distinction explicit prevents a common sign error in decay reconstruction. A massless daughter has equal rest-frame energy and momentum magnitude, making its angle transformation especially simple.

The resulting aberration formula maps a rest-frame right angle into a forward laboratory angle. Forward daughters are also more energetic, while backward daughters are redshifted. For a massless isotropic decay, transforming the uniform rest-frame distribution in $\cos\theta^*$ requires a Jacobian, giving a laboratory density proportional to $(1-\beta\cos\theta)^{-2}$. It is therefore incorrect to infer a preferred rest-frame direction from a forward laboratory cone before undoing the boost and acceptance. Massive daughters need their actual energy-to-momentum ratio in the transformation.

An angular distribution is a density in a declared measure. An isotropic distribution is uniform in solid angle, hence uniform in cosine of the polar angle after azimuthal integration. It is not uniform in the polar angle itself, because the solid-angle element contains a sine factor. Normalize the density before integrating a detector region. If the efficiency also depends on energy, boost-induced angle-energy correlations can make a simple angular acceptance insufficient even for a perfectly isotropic rest-frame decay.

The total width is obtained by integrating over all physically distinct directions and spin outcomes. A fitted angular shape can instead use a density normalized to one, which contains no lifetime information. Keeping those quantities separate is essential when an analysis measures polarization from a selected sample and branching fractions from event yields. Detector cuts, a chosen spin axis, and the frame used to define it should accompany every reported angular coefficient. A labeled daughter provides the reference direction; exchanging identical daughters can introduce another counting ambiguity.