Pictures Paths Particles Processes

102 November 7, 2013
in which available energy turns into particle-antiparticle pairs : γ γ → e − e + .
3.3.5 Counting states : the phase-space integration element
The treatment of the previous section is also useful in that it provides a hint on
how to count the wave-vector states. For on-shell particles of mass 23 m we use
the integration element
1
(2π) 3
d 3⃗ k
ω( ⃗ k)
, ω( ⃗ k) =
√
⃗ k2 + m 2 .
This object has dimension L −2 . It is not explicitly Lorentz-covariant, but we
can write it also in the more attractive form
1
(2π) 3
d 3⃗ k
ω( ⃗ k)
=
1
(2π) 3 d4 k δ ( k 2 − m 2) θ(k 0 ) . (3.42)
Note that if k 0 is positive for an on-shell particle in any given inertial frame, it is
positive in all intertial frames that can be reached by Lorentz transformations
from the first one. This ensures that the step function θ(k 0 ) always has the
same value, irrespective of any Lorentz boosts we may care to make. Lorentz
covariance of the phase space integration element is thus guaranteed. We shall
use the density of states (3.42) for all on-shell particles in the calculation of
cross sections and lifetimes.
If, for a given scattering process, the final state contains N particles with
masses m j , j = 1, 2, . . . , N, and wavevectors p µ 1 , pµ 2 , . . . , pµ N
, the combined phasespace
integration element is
dV (P ; p 1 , p 2 , . . . , p N ) ≡
⎛
⎞ ⎛ ⎞
N∏
⎝
1
N∑
(2π) 3 d4 p j δ(p 2 j − m 2 j ) ⎠ (2π) 4 δ 4 ⎝P − p j
⎠ , (3.43)
j=1
where P µ is the total wavevector of the scattering system. The four-dimensional
Dirac delta forces the overall conservation of wavevectors 24 . The condition
θ(p 0 > 0) imposing positive energy for the outgoing particles is, here and in the
following, always understood.
j=1
resulting photon is to be on its mass shell. On the other hand, an single off-shell photon can
be produced, but such a photon must immediately decay again, in for instance a particleantiparticle
pair of some kind.
23 We shall use the term ‘mass’ also for m, although strictly speaking it has the wrong
dimensionality ; the actual mass is, of course, M. Confusion will not readily arise. For the
same reason, we shall occasionally call the wave-vector the momentum.
24 Conservation of total energy and momentum.