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 2  = 0, which has the solution
.
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

etc.

Hence,

.

Or, in the static, time
independent case, this leads to
,
(which gives 2  = 0 for the massless case, as required.)

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

so

with solution

where g is a constant (the coupling strength) and  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
.

Setting c = 1, the denominator  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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