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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 12}\\
%{\Large DRAFT/not a final version}

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%\item In addition to the residual $U(1)_{em}$ gauge symmetry
%associated with the electromagnetic interaction, the standard model
%has some global symmetries (phase transitions related to lepton and
%baryon number) surviving the spontaneous breaking of $SU(2)_L \times
%U(1)_Y$. Find them, and interpret them. Is it possible to break them?

%\item Derive the criterion $\frac 32 N_F+N_B+N_D-4 \le 0$ (9.1.10),
%where $N_B$ is the number of boson lines, $N_F$ is the number of
%fermion lines and $N_D$ is the number of derivatives for a particular
%type of vertex, for renormalizable interactions involving derivatives,
%where bosons have propagators with the high momentum behaviour $(p
%^2)^{-1}$. (From this it should follow directly that if in the bare
%Lagrangian there are only renormalizable interactions then the only
%(ultraviolet) divergences are those associated with the renormalizable
%interaction vertices.)

\item Write down the renormalized and bare Lagrangian of QED, as well
as the counter-term Lagrangian. Calculate the renormalization constant
$Z_2$ for the electron self-energy in QED in mass-shell, $MS$ and
$\overline{MS}$ schemes. (See e.g. Itzykson-Zuber p. 319--335,
Bailin-Love, subsection 7.5 and chapter 11 (note that there might be
some misprints around eq. (11.58))) \\ Hint: 
\begin{center}
   \begin{fmfgraph*}(60,20)
     \fmfleft{i1}
     \fmfright{o1}
     \fmflabel{$e^-$}{i1}
     \fmflabel{$e^-$}{o1}
     \fmf{fermion}{i1,v1}
     \fmf{fermion,label=$e^-$}{v1,v2}
     \fmf{fermion}{v2,o1}
     \fmffreeze
     \fmf{photon,label=$\gamma$,left}{v1,v2}
%     \fmfdot{v1,v2}
   \end{fmfgraph*}
\end{center}
Use dimensional regularization, Feynman gauge and results of problem 5
of exercise 11.

%\item Calculate the anomalous magnetic moment of electron to one-loop
%order in QED. \\ Hint: See Bailin-Love, section 11.3.

%\item a) Consider supersymmetric Wess-Zumino model with (four-component)
%Majorana spinor field $\psi$ and real scalar fields $A$ and $B$ having
%Lagrangian
%\begin{eqnarray}
%{\cal L} = &\frac 12 \partial_\mu A \partial^\mu A+ \frac 12
%\partial_\mu B \partial^\mu B + \frac 12 i \overline \psi \gamma^\mu
%\partial_\mu \psi- \frac 12 m^2 (A^2+B^2) - \frac 12 m \overline \psi
%\psi-g m A (A^2+B^2) \nonumber \\ & - \frac 12 g^2 (A^2+B^2)^2-g
%\overline \psi (A-iB \gamma^5) \psi . \nonumber
%\end{eqnarray}
%Show that at one-loop order the quadratic divergences vanish from the
%self-energy of scalar B. (In non-supersymmetric model, like SM, there
%would be quadratic divergences in corresponding self-energy graphs for
%Higgs boson.)

%b) Assume that you have experimentally measured the scalar mass
%$m_R (\mu_0)$ at some renormalization scale $\mu_0$. Find the scalar
%self-energy counterterms for the above supersymmetric model in
%on-shell sheme, $MS$ scheme and ${\overline{MS}}$ scheme. Scetch a
%figure (you don't have to do the actual calculation, just see the
%proportionalities right) how the mass would change as a function of
%the renormalization scale in the above supersymmetric model and
%non-supersymmetric model (above model without the fermion field) in
%on-shell sheme, $MS$ scheme and ${\overline{MS}}$ scheme. \\ Hint: See
%the enclosed copy of Haber, Kane, Physics Reports 117 (1985) 75.

%\item Consider processes
%\begin{displaymath}
%ud \rightarrow ds, \nu_e b \rightarrow \nu_e b,t \rightarrow b u
%\overline d,b\rightarrow g s, gg\rightarrow gg,ec \rightarrow ecg, ud
%\rightarrow ud.
%\end{displaymath}
%Determine which of the processes are allowed in SM. Draw all Feynman
%graphs in leading order. Write explicitly all colour factors appearing
%in the cross sections or decay rates.

\item Calculate the $\beta$-function of QCD, and discuss its physical
content. The $\beta$-function is defined as
\begin{displaymath}
\beta \equiv \lim_{2 \omega \rightarrow 4^-} \frac{d g_R}{d \ln \mu} ,
\end{displaymath}
where the renormalized coupling constant is
\begin{displaymath}
g_R= g_B \mu^{\omega-2} Z_1^{-1} Z_2 Z_3^{\frac 12} 
\end{displaymath}
when the renormalized fields are defined as
\begin{displaymath}
\psi_R = Z_2^{-\frac 12} \psi_B, A^\mu_R=Z_3^{-\frac 12} A^\mu_B .
\end{displaymath}
The $Z$'s can be shown to be
\begin{eqnarray}
Z_1 = & 1-\frac{\alpha_C}{4 \pi} \frac{16}3 \frac{1}{2-\omega} + C_1 ,
\nonumber \\
Z_2 = & 1-\frac{\alpha_C}{4 \pi} \frac 83 \frac{1}{2-\omega} + C_2 ,
\nonumber \\
Z_3 = & 1+\frac{\alpha_C}{4 \pi} \left(10-\frac 43 N_g \right) \frac{1}{2-\omega} + C_3 ,
\nonumber 
\end{eqnarray}
where $C$ are finite constants that are fixed by the renormalization
scheme. $2 \omega$ denotes the number of space-time dimensions.

\item Calculate the color factors for the cross section of process $s
\overline s \rightarrow \gamma g$. \\ Hint: You can assume the
fermions to be massless. You might want to notice that apart from the
color factors the differential cross section is of the same form as in
the pair annihilation process from exercise 9, problem 2.

\end{list}

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Return the solved problems to the downstairs box in Laurikainen
building on Monday December 7th at 15 o'clock at latest. There are
two problems solving sessions: Tuesdays 16-18 in the Exercise room
(harjoitussali) in the basement of the Laurikainen bldg [English] and
Fridays 10-12 in Hall 5, Tallqvist bldg [Finnish]. \\ The www-page of
this course is: http://www.physics.helsinki.fi/{\%}7Epuolamak/hitu/

This is the last problem sheet. There are two (still...) preliminary
choises for the exam date: December 17th and January 22nd. The first
one has got more votes this far, the final date will be decided on the
Wednesday lecture. I will put a note on the date to the web page.


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