Pauli- Villars regularization in the Electron Vertex Function: Evaluation I'm studying one loop contribution for electron vertex function form Peskin and Schroeder's book " An introduction to quantum field theory " Section: 6.3. I have some troubles with Pauli- Villars regularization and getting the final results, so any help will be appreciated .. 

Starting from:
$I = \delta\Gamma^\mu(p',p)= 2 i e^2 \int \frac{d^4l}{(2\pi)^4} \int^1_0 dx dy dz \delta (x+y+z-1) \frac{2}{D^3} ~ \times \bar{u}(p') \Big[\gamma^\mu . \Big(-\frac{1}{2} l^2 +(1-x)(1-y)q^2+(1-4z+z^2)m^2\Big) + i \frac{\sigma^{\mu\nu}q_\nu}{2m} (2 m^2 z (1-z)) \Big]u(p)~~~~~~~~~(6.47)$ 
where $D= l^2-\Delta+i\epsilon, ~~~~~~~~\Delta=-xyq^2+(1-z)^2m^2$, while :
$ \Delta_\Lambda = -xyq^2+(1-z)^2m^2+ z \Lambda^2 $.
After momentum integration and Pauli- Villars regularization, this equals:
$I= \frac{\alpha}{2\pi} \int^1_0 dx dy dz \delta (x+y+z-1) ~ \times \bar{u}(p') \Big(\gamma^\mu . \Big[ \log \frac{z\Lambda^2}{\Delta} + \frac{1}{\Delta}\Big((1-x)(1-y)q^2+(1-4z+z^2)m^2\Big)\Big] + i \frac{\sigma^{\mu\nu}q_\nu}{2m\Delta} \Big[2 m^2 z (1-z)\Big] \Big)u(p)~~~~~~~~~(6.54)$ 


*

*Here it's suppose it substitute by $\log\frac{\Delta_\Lambda}{\Delta} = \log\frac{-xy q^2+(1-z)^2m^2+z\Lambda^2}{\Delta}$ , but why instead it substitute only by  $\log\frac{z\Lambda^2}{\Delta}$ ?

*Then how can we reach for :
$ F_1(q^2) = 1 + \frac{\alpha}{2\pi} \int^1_0 dx dy dz \delta (x+y+z-1) ~ \times
\Big[ \log \Big( \frac{m^2(1-z)^2}{m^2(1-z)^2-q^2xy}\Big) + \frac{m^2(1-4z+z^2)+q^2(1-x)(1-y)}{m^2(1-z)^2-q^2xy+\mu^2z} - \frac{m^2(1-4z+z^2)}{m^2(1-z)^2+\mu^2 z}\Big] ~~~~~~~ (6.56)$
In deed I'm little bit confused, how did we get $1$ term ?, where $\log z\Lambda^2$ had gone ? now $\log m^2(1-z)^2$ in the nominator, the part of $\Delta_\Lambda$ which didn't written in the previous equation , also in this $\log$ part of the nominator, where's the $q^2$ term ?
I tried to read the book explanation, but I can not understand too much so have any one made this exercise before ?
 A: All they've done is taken the limit of large $\Lambda$. Then 
\begin{align}
\Delta_\Lambda=-xyq^2+(1-z)^2m^2+z\Lambda^2\rightarrow z\Lambda^2
\end{align}
and 
\begin{align}
\log\left(\frac{\Delta_\Lambda}{\Delta}\right)\rightarrow \log\left(\frac{z\Lambda^2}{\Delta} \right)+\mathcal{O}(\Lambda^{-2}).
\end{align}
This is what they mean when they say just after equation (6.53) that "the convergent terms are modified by terms of $\mathcal{O}(\Lambda^{-2})$, which we ignore."
A: I only recently stumbled over the same issue, so this answer might come a bit late:
The first term in 6.56 - the 1 - appears due to the fact, that here $F_1(q^2)$  includes all corrections (to all orders), indicated by the last term, which represents terms of second order and higher in the electric coupling constant by the Bachmann-Landau-Symbol O($\alpha^2$). Hence, the 1 represents the contribution of the first order of pertubation theory to the Form Factor, as shown on in section 6.2, page 184.
Now, in order to clear up the confusion around the devious log term, one only has to write out the executed subtraction $\delta F_1(q^2) - \delta F_1(0)$ as this will then give 
$$
\delta F_1(q^2) - \delta F_1(0) = log ( \frac{z \Lambda^2}{\Delta(q^2)}) - log ( \frac{z \Lambda^2}{\Delta(0)}) + rest= log (\frac{z \Lambda^2}{\Delta(q^2)}\cdot ( \frac{z \Lambda^2}{\Delta(0)})^{-1}) + rest 
= log ( \frac{\Delta(0)}{\Delta(q^2)}) + rest = log ( \frac{2m^2(1-z)^2}{m^2(1-z)^2-q^2xy} ) +rest \ \ \ \ .
$$
