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\noindent PHY--396~L.\quad
Problem set \#20.\quad
Due March 24, 2005.
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\pointbegin
First, solve problem {\bf1}(d) of the previous homework set.

\point %2
Second, an easy reading assignment: The 4--page note I distributed in class
(also available on-line at the homework page).
Make sure you understand all the algebra: You may need to perform
a similar calculation in the future.

\point %3
And now, a calculational exercise:
Verify that at the one-loop level $\delta_1=\delta_2$, including
the finite parts of both counterterms.

\spointbegin
Continue the classroom calculation of one-loop vertex correction
for the special case of $q=0$.
Derive an exact formula for the $F_1^{\rm 1\,loop}(q^2=0)$ --- and hence
for the $\delta_1$ counterterm --- in $D$ dimensions.
Do not take the $D\to4$ limit.
\subpar
To regularize the IR divergence, assume the photon to have a tiny but non-zero
mass $m_\gamma\ll M_e$, hence photon propagator
$$
{-ig^{\mu\nu}\over k^2-m_\gamma^2+i0}\,.
\eqno\eq
$$

\spoint %b
And now calculate the  $\Sigma^{\rm 1\,loop}(\D p)$ for the electron
in $D$ dimensions, without taking the $D\to4$ limit.
Then evaluate the derivative $d\Sigma^{\rm 1\,loop}/d\!\D p$ for $\D p=M_e$
and hence the $\delta_2$ counterterm.
\subpar
Note that for $\D p=m$, the derivative develops an infrared divergence.
To regularize this divergence, use a tiny photon mass, exactly as
in part~(a).

\spoint %c
Verify that $\delta_1=\delta_2$ in any dimension $D$.
If you do not achieve this equality, check your calculations for mistakes.

\point %4
And finally, another reading assignment:
\S6.1 of the {\it Peskin \& Schroeder} textbook.
The soft-photon bremsstrahlung discussed there is important for understanding
the infra-red divergences of QED.


\bye
