# Mandl & Shaw eqn (15.28)

1. Aug 23, 2013

### Vic Sandler

In the second edition of Mandl & Shaw, QFT, on page 334, eqn (15.28) says:
$$G^{\alpha\beta}_r(k^2) = \frac{-ig^{\alpha\beta}}{k^2 + i\epsilon + e^2_r\Pi_r(k^2)} + O(e_r^4)$$

Note that I consider the parameter $k^2$ on the lhs to be a typo because of eqn (15.22) on page 333, and because of something else which I will get to later. I think it should read $G^{\alpha\beta}_r(k)$.

In eqn (15.29) on the next page, he gives the relation between $G_r$ and $G$ where $G$ is given in eqn (15.27) back on page 334. However, this relation does not seem to be correct as there is a factor of $k^2$ in the last term of the denominator in (15.27) that is missing from the last term of the denominator in (15.28). Since $Z_3$ does not depend on $k$, I don't see how this can be. I would like to put this down as a typo in eqn (15.28). However, there is a problem with doing that.

Eqns (15.87) and (15.88) on page 349 have the same form as eqns (15.29) and (15.28) except for the following differences.
1. The parameter on the lhs is $k$, not $k^2$. However, as I wrote above, I consider this a typo in eqns (15.27-29).
2. The coupling constant is $g_r$ instead of $e_r$.
3. There is a factor of $\delta_{il}$ in the numerator.

By the way, the text above eqn (15.88) says that $G_r$ is given by eqns (15.25) and (15.27) although it is clear that he means eqns (15.27) and (15.29). This is the reason that I am reluctant to put down eqn (15.28) as a typo. Should there be a factor of $k^2$ in the last term of the denominator of eqns (15.28) and (15.88) or are they both correct as written?

Last edited: Aug 23, 2013