Particle Physics Booklet - Particle Data Group - Lawrence Berkeley ...

40. Kinematics 289
where dΩ =dφ1d(cos θ1) is the solid angle of particle 1. The invariant mass
M can be determined from the energies and momenta using Eq. (40.2)
with M = Ecm.
40.4.3. Three-body decays :
P, M
p 1 , m 1
p 2 , m 2
p 3 , m 3
Figure 40.2: Definitions of variables for three-body decays.
Defining pij = pi + pj and m2 ij = p2 ij , then m212 + m223 + m213 =
M 2 + m2 1 + m2 2 + m23 and m212 =(P − p3) 2 = M 2 + m2 3 − 2ME3, where
E3 is the energy of particle 3 in the rest frame of M. Inthatframe,
the momenta of the three decay particles lie in a plane. The relative
orientation of these three momenta is fixed if their energies are known.
The momenta can therefore be specified in space by giving three Euler
angles (α, β, γ) that specify the orientation of the final system relative to
the initial particle [1]. Then
dΓ = 1
(2π) 5
1
16M |M |2 dE1 dE2 dα d(cos β) dγ .
Alternatively
(40.18)
dΓ = 1
(2π) 5
1
16M 2 |M |2 |p ∗ 1 ||p3| dm12 dΩ ∗ 1 dΩ3 , (40.19)
where (|p∗ 1 |, Ω∗1 ) is the momentum of particle 1 in the rest frame of 1
and 2, and Ω3 is the angle of particle 3 in the rest frame of the decaying
particle. |p∗ 1 | and |p3| are given by
|p ∗ ��
m2 12 − (m1 + m2)
1 | =
2��m2 12 − (m1 − m2) 2�� 1/2
, (40.20a)
2m12
and
��
M 2 − (m12 + m3)
|p3| =
2��M2 − (m12 − m3) 2��1/2 2M
[Compare with Eq. (40.16).]
. (40.20b)
If the decaying particle is a scalar or we average over its spin states,
then integration over the angles in Eq. (40.18) gives
dΓ = 1
(2π) 3
1
8M |M |2 dE1 dE2
= 1
(2π) 3
1
32M 3 |M |2 dm 2 12 dm223 . (40.21)
This is the standard form for the Dalitz plot.