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}φ = 0, which has the solution
φ = *e*/4πε_{0}*r*.
This describes the potential for a force mediated by massless particles,
the photons.

For a particle with mass, the relativistic equation *E*^{2}
= *p*^{2}*c*^{2} + *m*^{2}*c*^{4}
can be converted into a wave equation by the substitutions

*E* → *i ħ* ∂/∂*t*; *p*_{x} → −*i ħ* ∂/∂*x* etc.
Hence,
−*ħ*^{2}∂^{2}φ/∂*t*^{2} =
(*m*^{2}*c*^{4} − *ħ*^{2}*c*^{2}∇^{2})φ.
Or, in the static, time independent case, this leads to
,

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

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

∇^{2}φ → 1/*r*^{2} d/d*r* (*r*^{2} dφ/d*r*) ≡
1/*r* d^{2}/d*r*^{2} (*r*φ),
so d^{2}/d*r*^{2} (*r*φ) = (*m*^{2}*c*^{2}/*ħ*^{2}) *r*φ
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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