Cascade Decay Frames

Decay Description

Kind of Decay

The mother 0 decays into

Momenta

Energy and momentum components of the mother 0 and of the final states 1, 2, … in the laboratory frame, in any common unit. The momentum of the mother is the sum of the final states; the mass of every particle follows from its energy and momentum.

Topology

Name of the chain

The topology is an ordered binary tree over the numbers 1, 2, … of the final states. Every pair stands in brackets, its two parts are separated by a space, and a part is a number or another pair:

(1 2) for two final states,
((1 2) 3): the pair of 1 and 2 first, then 3,
((1 3) (2 4)): two pairs.

The first part of a pair is the reference particle, the second the spectator. Every number occurs exactly once.

Reference Topology (optional)

A second topology is the reference topology and gets the label I; the chain of the topology above then gets the name II, which can be changed in the field next to “Topology”. Without a second topology the topology above is the reference topology, named I (the field is grey).

Resolve undefined axes by the canonical fallback

Loading event

Notation of the frame labels

A quantity annotated by [v|K] is expressed in the frame v|K, for example p₁^[(12)|I], the spatial momentum of particle 1 in the frame [(12)|I].

v: The vertex whose parent is at rest in this frame. The frame is named by that vertex: (12) is the rest frame of the intermediate state (12)_i, the system of the particles 1_p and 2_p. R₀ is the rest frame of the mother 0_p with the common root axes.
K: The topology, labeled by a Roman numeral, whose ordered transports lead to the frame. [R₀|I] and [R₀|II] denote identical axes, whereas [(12)|I] and [(12)|II] may differ.
lab: [lab] is the specified starting frame of the event, not an absolute rest frame.
Axes: The root z axis is the laboratory flight direction of the mother. The root x axis lies in the plane of this direction and the laboratory z axis, with a negative z component (towards larger polar angle), and y completes a right-handed system; these are the unit vectors of the polar coordinates of the flight direction. For a mother that flies along the laboratory z axis the azimuth is not defined and is set to zero. Every topology uses these root axes, they do not depend on the decay. Every later frame inherits its z axis from the momentum direction in its parent frame; the azimuth is not reset.
Signs: Rotations turn vectors and states, with positive angles. A coordinate system is parametrised by the inverse rotation: to express the momenta in axes that are turned by φ about z, their coordinates are transformed with R_z(−φ), and in the same way with R_y(−ϑ); for a frame that moves with the rapidity η they are boosted with B_z(−η). This is why the steps of the dynamic view carry negative angles and rapidities. The rotation R = R_z(α) R_y(β) R_z(γ) of the page “Next step” is a rotation of vectors and therefore has the positive angles. The section “Euler rotations” on the page “Amplitude construction” explains the convention.
Angles: For a decay A → B + C, the first daughter B is the reference and C the spectator (the upper and the lower line of the vertex in the sketch of the topology). A particle is written 1_p, an intermediate state (12)_i, a vertex (12)_v, named after everything that is produced there. The angles φ_B and ϑ_B are the azimuth and polar angle of B, and η denotes a rapidity, all measured in the parent frame. A subscript such as 12 denotes the subsystem sum p₁₂ = p₁ + p₂ in one common frame.