The Yukawa Theory of leads to the following form for a boson propagator:

f(q^{2}) = g g_{0} / (m^{2} - q^{2})

Now that we've seen how to draw Feynman diagrams, let's learn how to use them to extract additional information about a process. Recall that the 3 ways of learning about elementary particles are by studying bound states, decays, and scattering events. The following information applies to decays and scattering events.

Most particles studies are unstable and decay in a short time. They don't all decay at the same time, rather the process is random (stochastic). For a large collection of identical particles, the number that will decay in a short time interval is proportional the number that exist at the beginning of the time interval. We write this in terms of the change in the number of initial state particles,

dN = -GN dt

where G is a constant for a particular particle called the decay constant, or decay rate, or transition rate.
This expression is readily integrated to yield the exponential decay law
N(t) = N(0)e^{-Gt}

where N(0) is the number of undecayed particles at time t=0, and N(t) is the number undecayed at a later time t.
We define the lifetime, t, of a particle to be the average lifetime for a large collection of identical particles.
Using the exponential decay law we can show that
t = <t> = 1/G

It is usually the case that a particle can decay to several final states (modes).
Then each final state, f, has it own decay constant, G_{f}, and the total decay constant for the particle is the sum of all the G_{f}'s:

G_{tot} = S_{f} G_{f}

The fraction of decays that yield a specific final state is called the branching ratio for that final state.
For initial state i decaying to final state f, the branching ratio is
Br(i-->f) = G_{f}/G_{tot}

Generally, our task is to calculate G_{f}.
Fermi's Golden Rule gives us a general prescription for calculating decay constants based on two ingredients:

- an amplitude, M
_{if}for the process, and - factors for the phase space available.

Fermi's Golden Rule relates the transition rate to the amplitude and phase space factors as:

transition rate = G_{f} = (2p/hbar) |M_{if}|^{2} × (phase space)

Specifically, for a decay of initial state i (an unstable particle) to final state f containing n
dG_{f} = |M_{if}|^{2} (S/2hbar m_{i}) {P_{j=1}^{nf} [(c d^{3}**p**_{j})/((2p)^{3} 2E_{j})]} (2p)^{4} d^{4}(p_{i} - S_{j} p_{j})

where S is a statistical factor to account for cases with identical particles in the final state, and to account for different spin states.
The delta function to the fourth power enforces conservation of energy and momentum.
The energies EThe quantities M_{if} are called amplitudes or matrix elements.
How to calculate them is fully covered in the course of relativistic quantum mechanics or quantum field theory.
Here, we will learn how to do a partial calculation of matrix elements, determining many of the factors that go into them, and allowing us to compare rates for similar decays (where the detailed part of the matrix element can be factored out of the problem).

Here is our set of rules for estimating matrix elements for Feynman diagrams:

**Momenta**: Label incoming and outgoing 4-momenta p_{1}, p_{2}, ... Label the internal 4-momenta q_{1}, q_{2}, ... Put an arrow on each line (internal and external) to keep track of the "positive" direction (the choice is arbitrary for internal lines).**Vertex factors**: Each vertex gets a factor -ig where g is the total coupling constant at that vertex. For QED vertices, g is the electric charge coupling to the photon (q sqrt(α_{EM}). For QCD vertices, g is the strong charge coupling constant (sqrt(α_{S}). For charged weak vertices, g is the weak charge times the appropriate CKM matrix element. For neutral weak vertices, g is the weak charge.**Propagator**: Each internal line gets a factor i/(q^{2}_{j}- m^{2}_{j}). Recall that these virtual particles can be "off the mass shell" so that q^{2}_{j}is not required to be equal to m^{2}_{j}.**Energy and momentum conservation**: For each vertex, write a delta function of the form(2π)where the k's are the three 4-momenta coming^{4}δ^{4}(k_{1}+k_{2}+k_{3})*into*the vertex (if the arrow on a line points away from the vertex, then k is the*negative*of the 4-momentum assigned to that line. This factor imposes energy momentum conservation at each vertex.**Integration over internal momenta**: For each internal line, write down a factor(2π)and integrate over all internal momenta.^{-4}d^{4}q_{j}**Cancel the delta functions**: The result will include a delta function for overall energy momentum conservation:(2π)Remove this factor and what remains is -iM^{4}δ^{4}(p_{i}- Σ_{j}p_{j})_{if}.

The classic(al) scattering experiment is Rutherford's scattering experiment where α particles (^{4}He) 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/sin^{4}θ distribution of scattered particles.
Let's review how this is done to make connection with quantities we will use.

Consider a two-body to two-body scattering process

A B --> C D

in which a parallel beam of particles A, moving with speed vConsider the target to be of thickness dx, and to have n_{B} B particles per unit volume.
The flux of incoming particles (particles/cm^{2}/sec) is f = n_{A}v_{i} where n_{A} is the density of A particles in the beam and v_{i} is their velocity.
We call the effective area around a B particle which will result in an interaction the cross section, s.
Then the fraction of the target area covered by the cross section of B particles is s n_{B} dx.
The scattering rate is the rate that A particles will hit this area

f s n_{B} dx

The reaction rate per target particle is
W = f s

Copyright © Robert Harr 2003