PHY304 Particle Physics Dr C N Booth

Yukawa Potential and the Propagator Term


Consider the electrostatic potential about a charged point particle.  This is given by ∇2φ = 0, which has the solution φ = e/4πε0r. This describes the potential for a force mediated by massless particles, the photons.

For a particle with mass, the relativistic equation  E2  =  p2c2 + m2c4  can be converted into a wave equation by the substitutions

Ei ħ ∂/∂t;   px → −i ħ ∂/∂x etc.
Hence,
ħ22φ/∂t2 = (m2c4ħ2c22)φ.
Or, in the static, time independent case, this leads to
 ,

(which gives ∇2φ = 0 for the massless case, as required.)
 

For a point source with spherical symmetry, the differential operator can be written as

2φ → 1/r2 d/dr (r2 dφ/dr) ≡ 1/r d2/dr2 (rφ),   so   d2/dr2 (rφ) = (m2c2/ħ2) rφ
with solution

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 .


Supplementary Material
For further, non-technical reading, you might like to consult The Ideas of Particle Physics, chapter 7.


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