## Quantum Field Theory

### Scattering Cross Section and Decay Width

Experiments are essential in checking out theoretical predictions. The measurements are usually taken in terms of scattering cross sections between two particles initially and decay width for an unstable particle. As shown in Figure 01o, the laboratory setup often involves a target with area A and NT particles bombarded by a beam of NB particles. If the effective interacting area is , then :

Chance of hitting the target particles = NT /A.

Since there are NB particles in the incident beam, the number of events N such as scattering would be :

N = NB(NT /A),       or = (N/NBNT)A. Furthermore, if the beam is moving at velocity v toward a stationary target for a duration of t, then NB = nBvtA, where nB is the number density of the beam. Thus, = P/Lint ,

where P = N/NT is the transition probability per target particle, Lint = nBvt is the integrated (over time, in unit of barn b = 10-24 cm-2) luminosity of the incident beam.

#### Figure 01o Scattering Cross Section [view large image]

It is the P and Lint measured in experiments to obtain , which is in turn used to check out the theoretical formulation.

In non-relativistic quantum mechanics, the scattering process is described by the Schrodinger equation. The following is a simple case of elastic scattering with no change in neither energy nor number of particles. The incoming particle is just deflected to various angular directions (Figure 01p). With the target particle represented by a potential V(r) depending only on the radial direction, the Schrodinger equation is in the form :   #### Figure 01q Scattering Zones [view large image]

See Fabulous Formulas for a portrait of the spherical coordinates, and the expression for 2. Also see some forms of the Legendre Polynomial. See "Spherical Bessel Functions" for the Hankel function and Plane Wave Expansion, and "Scattering Theory" for more examples.

In high energy collisions, the process would involve pair creation/annihilation, particle decay, and other weird processes beside scattering. Thus, the total cross section usually is the sum of various cross sections, i.e.,   #### Figure 01r Decay Width [view large image]

Figure 01r shows the decay width of the Z boson in weak interaction decay at about 91 Gev with a lifetime of 1/ ~ 2.5x10-25 sec (see "Physics of W and Z Bosons", and natural unit conversion to cgs).

.