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176 November 7, 2013
µ ↔ iQ¯h γµ QED vertex
Feynman rules, version 7.1 (7.1)
Here Q is the strength of the fermion-photon coupling : the charge of the
fermion 2 . By dimensional analysis, we see that is has dimension
dim [ Q ] [
= dim ¯h −1/2] . (7.2)
The Dirac delta function imposing momentum conservation is implied. As is
conventional, we shall employ wavy lines to indicate photons. As stressed in the
previous chapter, this choice of vertex can only been argued to be reasonable if
the photon current is conserved ; this we shall show in what follows.
7.2.2 Handlebars : a first look
Let us now start to investigate the requirements of current conservation for our
theory. One of the simplest possible processes is the decay of a photon into a
fermion-antifermion pair, shown below :
p
1
q
p 2
Of course the photon has to be off-shell here, but that is no problem since also
off-shell photons must obey current conservation. The part of the amplitude
depicted is given by
M = −Q u(p 1 )γ µ v(p 2 ) , (7.3)
where the index µ of the photon is coupled to a corresponding index somewhere
else in the larger Feynman diagram. Let us now attach the handlebar, so that
we get
p
1
q
p 2
With the convention, to which we shall try to adhere, that the momentum
assigned in the handlebar must be counted outgoing from the vertex, so in this
case should read −q, the handlebarred M becomes
M⌋ = Q u(p 1 ) /q v(p 2 ) . (7.4)
2 Or, rather, it is related to the charge. The precise form of this relation must, of course,
be established by investigating the coupling in a well-defined physical situation.