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

40. Kinematics 291
40.4.5. Multibody decays : The above results may be generalized to
final states containing any number of particles by combining some of the
particles into “effective particles” and treating the final states as 2 or 3
“effective particle” states. Thus, if p ijk... = pi + pj + p k + ...,then
mijk... =
�
p2 ijk... , (40.25)
and mijk... maybeusedinplaceofe.g., m12 in the relations in Sec. 40.4.3
or Sec. 40.4.4 above.
40.5. Cross sections
p 1 , m 1
p 2 , m 2
.
p 3, m 3
p n+2 , m n+2
Figure 40.5: Definitions of variables for production of an n-body
final state.
The differential cross section is given by
(2π)
dσ =
4 |M | 2
�
4 (p1 · p2) 2 − m2 1m2 2
× dΦn(p1 + p2; p3, ..., pn+2) .
[See Eq. (40.11).] In the rest frame of m2(lab),
�
(40.26)
(p1 · p2) 2 − m2 1m2 2 = m2p1lab ; (40.27a)
while in the center-of-mass
�
frame
(p1 · p2) 2 − m2 1m2 √
2 = p1cm s. (40.27b)
40.5.1. Two-body reactions :
p 1 , m 1
p 2 , m 2
p 3 , m 3
p 4 , m 4
Figure 40.6: Definitions of variables for a two-body final state.
Two particles of momenta p1 and p2 and masses m1 and m2 scatter
to particles of momenta p3 and p4 and masses m3 and m4; the
Lorentz-invariant Mandelstam variables are defined by
s =(p1 + p2) 2 =(p3 + p4) 2
= m 2 1 +2E1E2 − 2p1 · p2 + m 2 2 , (40.28)
t =(p1− p3) 2 =(p2 − p4) 2
= m 2 1 − 2E1E3 +2p1 · p3 + m 2 3 , (40.29)