Pictures Paths Particles Processes

November 7, 2013 201
If we now iterate the SDe cleverly for the first two of these four diagrams, we
obtain
= +
−
−
+ −
since we do have
= 0 , (7.107)
= , (7.108)
owing to the simple, momentum-independent structure of the seagull vertex.
Comparing the lines of the proof for sQED with that of regular QED, the general
proof strategy becomes clear : if in a diagram a slashed propagator occurs as
one of the indicated lines of a (semi-)connected graph, we must iterate de SDe
for that line, and then we can collect the various canceling contributions.
7.4.3 The Gordon decomposition
Consider a charged Dirac particle that scatters by emitting (or absorbing) a
single photon. The corresponding current reads
J µ = iQ¯h u(q) γµ u(p) , (7.109)
where p is the incoming, and q the outgoing momentum. By the properties of
the Dirac spinors we can write this as
Now,
J µ =
iQ
2m¯h u(q) ( /qγ µ + γ µ /p ) u(p) . (7.110)
/qγ µ = q µ + 1 2 q α[γ α , γ µ ] = q µ − iq α σ αµ ,
γ µ /p = p µ + 1 2 p α[γ µ , γ α ] = p µ + ip α σ αµ , (7.111)
and the current takes the form
J µ =
iQ
2m¯h u(q) ( (p + q) µ + i(p − q) α σ αµ) u(p) . (7.112)
This is called the Gordon decomposition : the vertex is split up into a piece that
we recognize as the sQED vertex, which is called the convection term, and a
tensorial part, called the spin term. Both terms vanish individually under the
handlebar operation.