Fermi's Golden Rule for Scattering

In the general case of particles 1 and 2 colliding and producing particles 3, 4, ... , N

1 2 --> 3 4 ... N

the differential cross section is given the formula
dσ = |Mif|^{2} hbar^{2}S/4Sqrt{(p_{1}·p_{2})^{2} - (m_{1}m_{2}c^{2})^{2}} {Π_{j=1}^{nf} [(c d^{3}**p**_{j})/((2π)^{3} 2E_{j})]} (2π)^{4} δ^{4}(p_{i} - Σ_{j} p_{j})

You have noticed that Feynman diagrams where an electron (or any charged fermion) emits and reabsorbs a photon are common at second order.
Similarly a "loop" can be added to a photon line; just let the photon convert to a virtual e^{+}e^{-} pair which then annihilate back into a photon.
I told you that these diagram contributes a small correction to the first order diagram.
This is both correct and a bold faced lie.

Following the Feynman rules for the electron line diagram, we have a pair of coupling factors yielding -α, and integrals for the virtual photon and electron lines of dq/q for all values of q. These integrals diverge logarithmically! Therefore the contribution of this diagram is infinite.

This was a problem with quantum field theory for some time.
It was found that infinities of this sort can be removed (for all diagrams and all orders) by redefining the mass and charge.
If we call the bare mass m_{0} (mass without virtual particles around, an unmeasurable quantity) and the bare charge e_{0}, then in calculations, these quantities always appear multiplied by the same infinite quantity.
We simply replace the product of m_{0} times infinite factor by m, the physical mass of the particle, or e_{0} times infinite factor by e, its actual charge.
Formally, this procedure is called renormalization, and the cartoon I handed out is suggested by the explanation that renormalization is a process of sweeping the infinities in the theory under the rug.

It's interesting to note that when this process is performed on the photon diagrams, the photon mass and charge are unaffected! Massless particles begin and end as massless particles.

When renormalization is applied to the vertex of a photon and electron,
we find that the coupling constant, α, is replaced by a quantity that varies with energy scale.
The measurements of α at low energies (fine structure, Lamb shift, or Hall effect) give a value of approximately 1/137.
At an energy equivalent to the mass of the Z^{0}, the value of α is measured to be approximately 1/128.

This phenomenon is known as running of the coupling constant, and occurs for the electromagnetic, weak, and strong coupling constants.
The electromagnetic and weak coupling constants become larger with energy scale, while the strong coupling constant becomes smaller.
At an energy scale of approximately 10^{16} GeV, the three are approximately equal.
A recent calculation offered in support of a model called supersymmetry indicates that the equality is improved if modifications dictated by supersymmetry are included in the calculation.

Gauge invariance in particle physics means that the physical result should not depend on a redefinition of the phase of a particle. Imposing gauge invariance on quantum field theory yields conservation of charge. This is similar to conservation of energy resulting from time invariance, conservation of linear momentum resulting from invariance under displacement, and conservation of angular momentum resulting from invariance under rotation.

The ideas of renormalizability, running coupling constants, and gauge invariance are required of all workable field theories. The examples given here apply to QED, but the same applies to QCD and to the electroweak theory.

There are literally hundreds of different particle detectors out there, far too many for me to discuss in this course. Luckily, they are based on a small number of possible interactions of particles with bulk matter, and a small number of quantities that can be measured. I will focus on these interactions and how they are used to measure quantities about particles.

Process | Description |
---|---|

ionization | |

electromagnetic showers | |

hadronic showers | |

Cerenkov radiation | |

transition radiation |

Copyright © Robert Harr 2003