Pictures Paths Particles Processes

112 November 7, 2013
The n-particle phase-space integration element dV n has dimensionality L 4−2n
as we have seen. Taking into account that the flux factor Φ Γ = 1/2m must have
the dimensionality of 1/m, that is, L, the dimensionality of the decay width of
a single particle into n particles is given by
[ ] [ ]
]
1
dim Γ(1 → n) = dim
m (M n+1) 2 dV n = dim
[L −1
, (4.20)
as required. Similarly, for the cross section of two particles going into n particles
we have
[ ] [ ( ]
2 ]
1
dim σ(2 → n) = dim (M n+2 )
m) 2 dV n = dim
[L 2 , (4.21)
again as required. Note that the above analysis is kept simple because we have
restricted ourselves to the use of wavevectors rather than mechanical momenta,
which would introduce additional factors of ¯h in the calculation. The other
natural constant, c, need not enter here.
4.3.4 Crossing symmetry
In our treatment of antimatter in the previous chapter we have seen that the
production (absorption) of a particle is, in a sense, æquivalent to the absorption
(production) of its antiparticle. We can make this even more specific as a relation
between various scattering amplitudes : this goes by the name of crossing
symmetry. Consider a generic 2 → 2 scattering process :
a(p 1 ) + b(p2) → c(q1) + d(q2)
where we have indicated the momenta of the particles. Let us write the corresponding
amplitude as M(p 1 , p 2 , q 1 , q 2 ). By moving particles from the initial to
the final state 10 , or vice versa, we can then find the amplitudes for the crossingrelated
processes, for example :
a + b → c + d : M(p 1 , p 2 , q 1 , q 2 ) ,
a + ¯c → ¯b + d : M(p 1 , −p 2 , −q 1 , q 2 ) ,
a + ¯d → ¯b + c : M(p 1 , −p 2 , q 1 , −q 2 ) ,
¯c + ¯d → ā + ¯b : M(−p 1 , −p 2 , −q 1 , −q 2 ) . (4.22)
Since the momenta of all (anti)particles have positive energy, the minus signs
yield momenta with negative energy. Depending on the type of the particle 11 ,
this may involve an analytic continuation of the amplitude function M.
10 You can visualize this by taking an outgoing particle, say, and dragging its external leg
from the final to the initial state.
11 Especially for Dirac particles.