Lifetimes and Cross SectionsSpecifically, for a decay of initial state i (an unstable particle) to final state f containing nf particles, we can write
Here is our set of rules for estimating matrix elements for Feynman diagrams:
Consider a situation where you throw darts at a dart board, while blindfolded! Not knowing the precise location of the dart board, you throw the darts in a uniformly random pattern across the wall the board is mounted on. While chance will cause fluctuations in the outcome, on average the fraction of darts that hit the board is proportional to the size of the board; the bigger the board, the more darts that will strike it. The relevant parameter is the area of the board, or in particle physics lingo, the cross section.
This is roughly the situation that occurs in particle physics. It is impossible to accurately locate a nucleus, let alone a proton or an electron. Therefore experimenters aim a beam or particles (typically of order 109 particles in a few nanosecond burst) at a target. The target could be a collection of atoms (solid, liquid, or gas), or another beam of particles. For simplicity, let's assume the target to be a thin foil of material, for instance, gold.
The cross section for the dart board example is easily visualized. After all, a dart either hits the board or misses it -- the board is a "hard" target. When scattering electrons off protons, say, can we define the cross section in a similar way? The proton is more of a "soft" target. The electron need only get close enough for electromagnetic interaction to occur, causing it to be deflected. We can define an "effective" cross section, σ, for a "soft" target, one that is proportional to the probability for an interaction to occur.
For elementary particles, σ depends on the target and the projectile. The cross section for electrons scattering off protons is orders of magnitude larger than the cross section for neutrinos scattering off protons and orders of magnitude smaller than the cross section for pions scattering off protons. So whenever you quote a cross section, you must specify both target and projectile.
Also, in partile physics, the final state particles can be different from the initial state particles. To help keep track of all the possibilities, several types of cross section are commonly used:
Imagine the situation of scattering from a fixed source (i.e. a heavy charged nucleus). We define the scattering angle θ as the angle between the initial and final momenta. The scattering angle depends on the impact parameter, b, the minimum distance between the projectile and target if no scattering occurred. The relation between impact parameter and scattering angle, θ(b), depends on the type of interaction involved. (We normally deal with potentials that are symmetric in φ. This is the situation that I will assume throughout.)
Consider the elastic collision between two spheres of radius R, one fixed rigidly in place. From the figure we see that b=R sinα, and also that 2α + θ = π. Use the second expression to eliminate sinα from the first. We have sinα = sin(π/2 -θ/2) = cos(θ/2), so that
If the impact parameter lies in the ring between b and b+db, then it will have a scattering angle between θ and θ+dθ. The initial ring the particle passes through has area dσ = 2πb db, and it scatters into a solid angle dΩ = 2πsinθ d&theta = -2πd(cosθ). (θ lies between 0 and π such that dΩ is always positive.) We call the ratio dσ/dΩ the differential cross section, and in this example it is
Differentiating the expression for b(θ) we have
The classic(al) scattering experiment is Rutherford's scattering experiment where α particles (4He nuclei) emitted by radioactive decay are collimated into a narrow beam impinging on a thin gold foil. The α particles scatter from the gold nuclei and are detected. In previous courses you have probably worked out the 1/sin4(θ/2) distribution of scattered particles. Let's review how this is done to make connection with quantities we will use.
The relation between impact parameter and scattering angle for a charge q1 with initial kinetic energy T scattering in the electrostatic potential from a fixed charge q2 is
In scattering experiments we characterize the beam by its luminosity, L, defined as the number of beam particles per unit area per second passing through an imaginary plane. The number of interactions per second is N = Lσ. The number of interactions scattering into solid angle dΩ per second is dN = L(dσ/dΩ)dΩ. We can also write dσ/dΩ = (1/L)dN/dΩ.
In the general case of particles 1 and 2 colliding and producing particles 3, 4, ... , N
Consider a two-body to two-body scattering process
In the CM, p1 = -p2, therefore p1μp2μ = E1E2 + p12. After some massaging,
Consider a two-body to two-body scattering process
Consider the target to be of thickness dx, and to have nB B particles per unit volume. The flux of incoming particles (particles/cm2/sec) is Φ = nAvi where nA is the density of A particles in the beam and vi is their velocity. For a single nucleus interaction cross section, σ, the fraction of the target area covered by the cross section of B particles is σ nB dx. The scattering rate is the rate that A particles will hit this area