\input phyzzx
\input pstricks97

\hsize=6.6in \hoffset=-0.1in
\vsize=8.25in \overfullrule=0pt
\noindent PHY--396~L.\quad
Problem set \#19.\quad
Due March 10, 2005.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
This homework is about renormalization of the {\sl scalar QED} ---
the theory of  a {\sl charged} scalar field $\Phi(x)$ coupled to
the EM field $A_\mu(x)$.

\spointbegin
Write down all counterterms needed to cancel all the divergences
of the scalar QED.
Start with the superficial divergences, and then use Ward identities
to narrow down the list of actually divergent amplitudes.

\spoint %b
Show that all the requisite counterterms are contained in a renormalizable
bare Lagrangian.

\spoint %c
Write down renormalization conditions for the finite parts of the counterterms.

\spoint %d
In ordinary QED, gauge invariance of the counterterms requires
$\delta_1=\delta_2$.
In the scalar QED, gauge invariance requires two identities of this kind.
Write them down, and prove them using Ward identities of
the un-amputated amplitudes, \cf~homework set~\#15.

\bigskip
\point %2
And now consider the electric charge renormalization in the scalar QED.
At the one-loop level, there are two Feynman diagrams contributing to
the 1PI two-photon amplitude, namely
$$
-i\Sigma_{\rm 1\,loop}^{\mu\nu}\ =
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=
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    \pscircle*(+10,0){0.5}
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+
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 (11.169881,1.233922)(9.983902,0.437048)(8.706936,-0.284442)%
 (7.961796,0.978870)(7.333145,2.261972)(6.007403,1.729091)%
 (4.608335,1.288863)(4.142136,2.679492)(3.793994,4.065259)%
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$$

\spointbegin
Evaluate the two diagrams using dimensional regularization and verify that
the net amplitude has form
$$
\Sigma_{\rm 1\,loop}^{\mu\nu}(k)\
=\ \bigl(k^2g^{\mu\nu}\,-\,k^\mu k^\nu\bigr)\times \Pi_{\rm 1\,loop}(k^2)
\eqno\eq
$$
\par\goodbreak

\spoint %b
Calculate the $\Pi(k^2)$ due to the above diagrams, determine the $\delta_3$
counterterm (at the one-loop level), and write down the {\sl net} $\Pi(k^2)$
as a function of $k^2$.

\spoint %c
Finally, consider the effective coupling $\alpha_{\rm eff}^{}(k^2)$
of the scalar QED at high momenta.
Show that at the one-loop level,
$$
{1\over \alpha_{\rm eff}(k^2)}\ \approx\ {1\over\alpha(0)}\
-\ {1\over 12\pi}\left(\log{-k^2\over m^2}\,-\,{8\over3}\right).
\eqno\eq
$$


\bye

