## Cross Sections

In many experimental situations, we are interested in the scattering of two particles with momenta k1 and k2 to a state with n particles with momenta p1, p2, . . . , pn. We denote by

∫ d3 k1˜1(k1)|k1 > and

∫ d3 k2˜2(k2)|k2 > the

wave functionals of the incoming particles. This is to describe the more realistic case of a wave packet. The amplitude for scattering to take place is hence given by

A = −i(2π )4 ∫

d3 k1d3 k2 δ4

( ∑n i=1 pi − k1 − k2

)M2+n({−pi }, {k j })√∏n i=1 2p

(i) 0 ( pi )(2π )3

∏2 j=1 2k

× ˜1(k1)˜2(k2)e−i[E0+ p (i) 0 ]T . (10.1)

If we define the wave function in coordinate space as usual

j (x) = ∫

d3 k√ 2k0(k)(2π )3

e−ikx˜ j (k), (10.2)

we can compute the overlap of the two wave functions

∫ d4x 1(x)2(x)eipx = (2π )4

∫ d3 k1√

2k0(k1)(2π )3 d3 k2√

2k0(k2)(2π )3

× δ4(k1 + k2 − p)˜1(k1)˜2(k2) . (10.3)

We assume that over the range of momenta in the wave packets, the reduced matrix elements are constant (which can be achieved with arbitrary precision for arbitrarily narrow wave packets in momentum space). This allows us to write for the scattering probability of two particles into n particles, with momenta in between pi and pi + dpi ,

dW = |M({−pi }, {k¯ j })|2 f ( p) n∏

d3 pi 2p0( pi )(2π )3 ,

f ( p) ≡ ∫

ip(x−y) , (10.4)

A in

where p = ∑ni=1 pi = k1 + k2. The momenta k¯i in the reduced matrix element are the central values of the wave packet in momentum space for the two incoming particle beams. Under the same assumption that the momentum spread in the beams is very small, the function f ( p) will be highly peaked around p = k¯1 + k¯2, such that

f ( p) ≈ δ4( p − k¯1 − k¯2) ∫

d4 p f ( p) = (2π )4δ4( p − k¯1 − k¯2)

× ∫

d4x|1(x)|2|2(x)|2. (10.5)

The quantities | j (x)|2 are of course related to the probability densities of the two particles in their respective beams,

ρ j (x) ≡ i ( ∗j (x)∂t j (x) − j (x)∂t∗j (x)

) ≈ 2k¯( j)0 | j (x)|2, (10.6) again using the fact that the wave packet is highly peaked in momentum space. Putting these results together we find

dW = (2π )4δ4 (∑

pi − k¯1 − k¯2 )

|M({−pi }, {k¯ j })|2

× n∏

d3 pi 2p0( pi )(2π )3

∫ d4x

ρ1(x)ρ2(x)

. (10.7)

Since ρ j (x) will depend on the experimental situation, we should normalise with respect to the total number of possible interactions in the experimental setup, also called the integrated luminosity L .∫

dt L(t) ≡ ∫

d4x ρ1(x)ρ2(x)|v1 − v2|. (10.8)

Here ∫

d3x ρ1(x)ρ2(x) is the number of possible interactions per unit volume at a given time and |v1 − v2| is the relative velocity of the two beams. We have assumed that one of the velocities is zero (fixed target) or that the two velocities are parallel (colliding beams). Hence, L(t) = ∫ d3x ρ1(x, t)ρ2(x, t)|v1 − v2| is a flux, typically of the order of 1028 − 1033cm−2s−1. To consider the general case we note that we can also write∫

dt L(t) = √

(k¯1 · k¯2)2 − m21m22 ∫

d4x ρ1(x)ρ2(x)

. (10.9)

After all, for a fixed target situation k¯2 = (m2, 0), such that√ (k¯1 · k¯2)2 − m21m22

= √

= |v1|, (10.10)

whereas for colliding beams of particles and antiparticles with mass m, where k¯1 = (E, k) = E(1, v) and k¯2 = (E, −k) = E(1, −v), one finds√

(k¯1 · k¯2)2 − m21m22 k¯(1)0 k¯

= √

(E2 + k 2)2 − m4 E2

= 2|k|/E = 2|v|, (10.11)

such that both expressions reduce to |v1 − v2|. We leave it as an exercise to prove the result for the general case of parallel beams.