\input phyzzx

%\def\dim{\mathop{\rm dim}\nolimits}

\hsize=6.7in	\hoffset=-0.1in
\vsize=8.5in    \voffset=0in
%\overfullrule=0pt
\noindent PHY--396~K/L.\qquad
Problem set \#24, the last for the course!.\quad
Due May 2, 2001.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
First, a reading assignment:
\S16.5 of the {\sl Peskin \& Schroeder} textbook.
Make sure you understand the calculation of the (infinite parts of) the counterterms
$\delta_1$, $\delta_2$ and $\delta_3$ {\sl in detail}.

\medskip
\point %2
Next, another reading assignment:
\S20.1 of the {\sl Peskin \& Schroeder} textbook, which introduces the {\it Higgs mechanism}.
The abelian version of the Higgs mechanism --- the photon becomes massive when a charged
scalar field acquires a non-zero vacuum expectation value --- was discussed in class in the
Fall semester, but please refresh your memory.
I will discuss the non-abelian version during the last week of this semester, but I'ld like
you to read about it first so it wouldn't come as a shock.
Pay attention to the Georgi--Glashow model and make sure you understand why --- and how ---
two of the three gauge bosons become massive while the third remains massless. 

\medskip
\point %3
Finally, an exercise:
Consider the pseudoscalar Yukawa theory.
In a previous homework (set\#18, problem 2),
 you (should) have calculated to one-loop order
the infinite parts of all the counterterms in this theory.
Let us now use the results of all this work to calculate the one-loop $\beta$--functions
$$
\partder{g(M)}{\log M}\ =\ \beta_g(g,\lambda)
\quad{\rm and}\quad
\partder{\lambda(M)}{\log M}\ =\ \beta_\lambda(g,\lambda).
\eqn\BetaFunctions
$$

\spointbegin
Argue that for off-shell renormalization conditions at some renormalization scale
$M\gg m_f,m_s$, each of the logarithmically divergent counterterms $\delta_Z^\phi$,
$\delta_Z^\psi$, $\delta_g$ and $\delta_\lambda$ is
$$
\delta\ =\ C\log{(\Lambda^{\rm UV}_{\rm cutoff})^2\over M^2}\ +\ \rm finite
\eqno\eq
$$
or
$$
\delta\ =\ C\left({1\over \epsilon}\,+\,\log{\mu^2\over M^2}\right)\ +\ \rm finite
\eqno\eq
$$
in dimensional regularization,
where the coefficient $C$ of the UV divergence has the same value for all renormalization
schemes (on-shell or off-shell) and the `finite' terms are not only finite but also
$M$ independent.

\spoint %b
Show that
the bare and the renormalized couplings of the Yukawa theory are related to each other as
$$
\lambda\,+\,\delta_\lambda\ =\ \lambda_0 Z_\phi^2\,,\qquad
g\,+\,\delta_g\ =\ g_0Z_\psi\sqrt{Z_\phi}\,,
\eqn\BR
$$
and use these relations to derive
$$
\eqalign{
\beta_\lambda\ &
=\ 2(\lambda+\delta_\lambda)\,\partder{\log Z_\phi}{\log M}\ -\ \partder{\delta_\lambda}{\log M}\,,\cr
\beta_g\ &
=\ (g+\delta_g)\left( \partder{\log Z_\psi}{\log M}\,+\,{1\over2}\,\partder{\log Z_\phi}{\log M}\right)\
     -\ \partder{\delta_g}{\log M}\,.\cr
}\eqn\RG
$$
At the one-loop level, these formul\ae\ simplify to
$$
\eqalign{
\beta_\lambda\ &
=\ 2\lambda\,\partder{\delta_Z^\phi}{\log M}\ -\ \partder{\delta_\lambda}{\log M}\,,\cr
\beta_g\ &
=\ g\left( \partder{\delta_Z^\psi}{\log M}\,+\,{1\over2}\,\partder{\delta_Z^\phi}{\log M}\right)\
     -\ \partder{\delta_g}{\log M}\,.\cr
}\eqn\RGol
$$

\spoint %c
Use the results of homework set \#18 to evaluate the right hand sides of eqs.~\RGol\
and write down the $\beta$--functions~\BetaFunctions\ as functions of the renormalized
couplings $g$ and $\lambda$.


\bye
