Ward identities in Minimal Subtraction Scheme

[gheremond]

In QED, the Ward identities set $$Z_1=Z_2$$ and $$Z_1 - 1 = \left. {\frac{{d\Sigma \left( p \right)}}{{dp}}} \right|_{p = m}$$. This can be shown explicitly for the 1-loop calculations if one uses an on-shell subtraction scheme, where the renormalized mass and charge are identical to the experimentally measured ones. What happens if one uses instead a subtraction scheme like MS or $$\overline {MS}$$? Arguably the Ward identities should hold again, but in calculating the various Z functions, one finds that $$Z_1 - 1 = \left. {\frac{{d\Sigma \left( p \right)}}{{dp}}} \right|_{p = m}$$ only holds at the level of the infinite contributions (poles), while there are some finite differences. Is that supposed to happen? Do the identities only refer to the infinite parts of the renormalization functions?

 

[Avodyne]

This is a very good question. Astoundingly, the complete answer was unclear until quite recently:

http://arxiv.org/abs/hep-th/0512187v2

 

[gheremond]

Whoa! I didn't expect it would get that deep! I would assume that things would be pretty clear for a theory like QED, which has be beaten into a pulp over decades of work. Great reference too! Thanks!