Recall from last lecture:

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

f(q2) = g g0 / (m2 - q2)

Lifetimes and Cross Sections

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.


What is lifetime?

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, Gf, and the total decay constant for the particle is the sum of all the Gf's:

Gtot = Sf Gf
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) = Gf/Gtot

Fermi's Golden Rule for decays

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

The amplitude is a dynamical quantity and is calculated from Feynman diagrams following a set of rules. The phase space factors are kinematical quantities.

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

transition rate = Gf = (2p/hbar) |Mif|2 × (phase space)
Specifically, for a decay of initial state i (an unstable particle) to final state f containing nf particles, we can write
dGf = |Mif|2 (S/2hbar mi) {Pj=1nf [(c d3pj)/((2p)3 2Ej)]} (2p)4 d4(pi - Sj pj)
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 Ej of the final state particles is shorthand for (pj2 + mj2)½. This is the differential rate to a final state with particular momenta pj. We integrate this expression of all the final state momenta to get the full decay rate, Gf.

Example: π0 --> γγ

Example: spin averaged two-body decay

Calculating Mif

The quantities Mif 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:

  1. Momenta: Label incoming and outgoing 4-momenta p1, p2, ... Label the internal 4-momenta q1, q2, ... Put an arrow on each line (internal and external) to keep track of the "positive" direction (the choice is arbitrary for internal lines).
  2. 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(αEM). For QCD vertices, g is the strong charge coupling constant (sqrt(αS). For charged weak vertices, g is the weak charge times the appropriate CKM matrix element. For neutral weak vertices, g is the weak charge.
  3. Propagator: Each internal line gets a factor i/(q2j - m2j). Recall that these virtual particles can be "off the mass shell" so that q2j is not required to be equal to m2j.
  4. Energy and momentum conservation: For each vertex, write a delta function of the form
    (2π)4 δ4(k1+k2+k3)
    where the k's are the three 4-momenta coming 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.
  5. Integration over internal momenta: For each internal line, write down a factor
    (2π)-4 d4qj
    and integrate over all internal momenta.
  6. Cancel the delta functions: The result will include a delta function for overall energy momentum conservation:
    (2π)4 δ4(pi - Σj pj)
    Remove this factor and what remains is -iMif.

Scattering Cross Sections

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/sin4θ 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 vi, impinges on a target containing particles of type B. For instance, A could be a beam of protons, pions, electrons, or neutrinos, and B is some material containing nuclei with protons and neutrons. (When the typical interaction energies exceed the binding energy of protons and neutrons in nuclei (about 10MeV) then we can consider the nucleus as a collection of quasi-free protons and neutrons.) The outgoing particles are called C and D. If the scattering is elastic then C and D are the same as A and B. If the scattering is inelastic then C and D are different from A and B (different types of particles; particles of the same type are indistinguishable).

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 f = nAvi where nA is the density of A particles in the beam and vi 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 nB dx. The scattering rate is the rate that A particles will hit this area

f s nB dx
The reaction rate per target particle is
W = f s


Copyright © Robert Harr 2003