|PHY304||Particle Physics||Dr C N Booth|
Consider the electrostatic potential about a charged point particle. This is given by ∇2φ = 0, which has the solution
For a particle with mass, the relativistic equation E2 = p2c2 + m2c4 can be converted into a wave equation by the substitutions
For a point source with spherical symmetry, the differential operator can be written as
where g is a constant (the coupling strength) and R = ħ/mc is the range of the force. This is known as the Yukawa form of the potential, and was originally introduced to describe the nuclear interaction between protons and neutrons due to pion exchange.
Using this form of potential and the Born approximation leads, after some manipulation (see the homework!), to a matrix element given by
Returning to our normal convention of setting c = 1, the terms in the denominator give and this is called the propagator term. It arises from the exchange of a virtual boson whose rest mass (as a physical particle) is m.
The cross-section is proportional to .
For further, non-technical reading, you might like to consult The Ideas of Particle Physics, chapter 7.