QED running coupling definition.

[center o bass]

Often when one speaks about the effective QED coupling one defines it as

$$e = \frac{Z_2 Z_3^{1/2}}{Z_1} e_0 \ \ \ \ (*)$$

when $Z_1 = Z_2$ by the Ward identity this turns out to be $Z_3^{1/2}e_0$ and some authors just define the coupling to be this right away.
So why do some make a point that the effective coupling is really defined according to (*). In what sense is this the 'natural definition'?

I have thought about it, and the best answer I have come up with is the following:
For an effective coupling one wants a definition which captures as much information about the interaction as possible so that when one has a large effective coupling, one can also say that the probability for an interaction is large. This thus requires us to take in the effects from the propagator of the electron (Z_2), the propagator of the photon (Z_3) and the vertex function (Z_1). Since there are two electron propagators connected to each vertex one gets two factors of $Z_2^{1/2}$ while one gets just one factor of $Z_3^{1/2}$ from the photon propagator.

Any insights would be appreciated!

 

[DarMM]

Basically, when you replace the electron and photon field with their renormalised versions you get:

$e_0 Z_2 Z_3^{1/2} \bar{\psi}\gamma^{\mu}\psi A_{\mu}$

this should be well-defined as the divergences in the bare charge and the field strength coefficients should cancel the divergences in the operator product.

However the physical charge is not manifest when you write the product this way. So you can write:

$e Z_1 \bar{\psi}\gamma^{\mu}\psi A_{\mu}$

Where $Z_1$ is just defined to be the constant that cancels all divergences in the operator product divided by $e$. It can be computed by calculating the three-point function.

Since these two expressions are just different ways of writing the same object, you have the relation:
$e = \frac{Z_2 Z_3^{1/2}}{Z_1} e_0$

 

[vanhees71]

The important point, howeverl, is local gauge invariance, which means that to any order in perturbation theory all field derivatives must come as gauge-covariant derivatives. The Ward-Takahashi identities reflect this demand of gauge invariance, and restrict the counterterms in such a way that at any loop order $Z_1=Z_2$. This is most easily seen in the background field gauge, which leads to a manifestly gauge invariant effective action. See my notes on QFT about this (Sect. 6.6.3):

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf