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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 10}\\
%{\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 (a) How is the SM modified if one adds a $SU(2)_L$ singlet
%neutrino?  Derive all additional terms. (b) Show that you can make the
%neutrino massive by not adding any particles, but just by giving a
%Majorana mass term for the left-handed neutrino.

%\item Define
%\begin{displaymath}
%R \equiv \frac{ \sigma_{SM} (e^+e^- \rightarrow \overline f f )
%}{\sigma_{QED} ( e^+ e^- \rightarrow \mu^+ \mu^- ) } .
%\end{displaymath}
%Determine the value of $R$ at $Z$-pole ($\sqrt s =m_Z$). \\ Hint: Use
%$s-m_Z^2+ i \Gamma_Z m_Z$ in the Z-propagator (instead of $s-m_Z^2$). 

\item Derive the differential cross section (8.1.6) for the process
$e^+ + e^- \rightarrow \mu^+ + \mu^-$ (Feynman diagrams fig 8.2 at
page 147. See also Okun p. 202, Halzen-Martin p.305). You can assume
that the leptons are massless. Draw the additional graphs you would
need to calculate if you would take the lepton masses into
account. (Note misprint in page 146, in the formula for a the last
parenthesis should be squared.)

\item
Estimate the forward-backward asymmetry in the scattering
$e^+e^-\rightarrow \mu^+\mu^-$
$$
A_{\rm FB} \equiv 
\frac{\int^{\pi/2}_0{\rm d}\sigma-\int^\pi_{\pi/2}{\rm d}\sigma}
{\int^\pi_0{\rm d}\sigma}\, .
$$
Hint: Solve the previous problem properly.

%\item Show that (9.2.30) is invariant in the Becchi-Rouet-Stora
%transformations (9.2.31).

\item Consider a general Fierz transformation:
\begin{displaymath}
\overline u_3 \Gamma_i u_2 \overline u _1 \Gamma_i u_4 = \sum_{j=1}^5
\lambda_{ij} 
{ \overline u_1 \Gamma_j u_2 \overline u_3 \Gamma_j u_4 } ,
\end{displaymath}
where $\Gamma_i=1,\gamma^\mu,\sigma^{\mu \nu}, \gamma^\mu \gamma^5,
\gamma^5$. Find $\lambda_{ij}$ and show especially that
\begin{displaymath}
\overline u_3 \gamma_\mu (1-\gamma^5) u_2 \overline u _1 \gamma^\mu (1-\gamma^5) u_4 =
\overline u_1 \gamma_\mu (1-\gamma^5) u_2 \overline u_3 \gamma^\mu (1-\gamma^5) u_4  .
\end{displaymath}
See e.g. Itzykson-Zuber p. 162, Bailin appendix, Peskin-Schoroeder
p. 75. As application see the book p. 156 (8.2.15) and beyound:
\begin{displaymath}
(\overline e \gamma_{\mu L} \nu_e)(\overline \nu_e \gamma_L^\mu e) =
(\overline \nu _e \gamma_{\mu L} \nu_e)(\overline e \gamma_L^\mu e) .
\end{displaymath}

%\item
%Show that the standard model has no triangle anomaly, by
%showing that 
%$$
%\sum_{i,j,k=q,\ell} \left(
%\begin{picture}(10200,3800)
%\drawline\photon[\W\REG](4000,2334)[4]
%\global\advance\pmidy by 600
%\global\advance\pmidx by -250
%\put(\pmidx,\pmidy){$W^-_V$}
%\drawline\fermion[\SE\REG](\photonfrontx,\photonfronty)[3000]
%\drawarrow[\SE\ATBASE](\pmidx,\pmidy)
%\global\advance\pmidy by 150
%\global\advance\pmidx by 400
%\put(\pmidx,\pmidy){$j$}
%\drawline\photon[\W\REG](4000,-1900)[4]
%\global\advance\pmidy by 600
%\global\advance\pmidx by -250
%\put(\pmidx,\pmidy){$W^+_V$}
%\drawline\fermion[\N\REG](\photonfrontx,\photonfronty)[4242]
%\drawarrow[\N\ATBASE](\pmidx,\pmidy)
%\global\advance\pmidy by -300
%\global\advance\pmidx by 200
%\put(\pmidx,\pmidy){$i$}
%\drawline\fermion[\NE\REG](\photonfrontx,\photonfronty)[3000]
%\drawarrow[\SW\ATBASE](\pmidx,\pmidy)
%\global\advance\pmidy by -450
%\global\advance\pmidx by 400
%\put(\pmidx,\pmidy){$k$}
%\drawline\photon[\E\REG](\fermionbackx,\fermionbacky)[4]
%\global\advance\pmidy by 700
%\global\advance\pmidx by -250
%\put(\pmidx,\pmidy){$Z_A$}
%\end{picture}
%\right) = 0,
%$$
%where $W^\pm_V$ and $Z_A$ denote the vector and axial parts in the 
%coupling of the respective gauge boson and the fermion, and
%$i,j,k$ are all the quarks and leptons. \\
%Hint: it is enough to show that
%$$
%\sum_{i=q,\ell} g(W_V^+ ii)g(W_V^- ii)g(Z_A ii)=0,
%$$
%where $g$ is the respective coupling constant.
%Note that the cancellation occurs separately for each family.
%
%For further hints, see:  e.g. Cheng and Li or Itzykson and
%Zuber.

%\item Calculate the following divergent integral that appears in
%dimensional regularization method:
%\begin{displaymath}
%\int {\frac{d^{2 \omega}k}{(2 \pi)^{2 \omega}} (k^2+2 p \cdot k-m^2+i
%\epsilon)^{-n}} = i (-1)^n \frac{\Gamma(n-\omega)}{\Gamma(n)}
%\frac{(m^2+p^2)^{\omega-n}}{(4 \pi)^\omega} ,
%\end{displaymath}
%where $2 \omega$ denote space-time  dimensions.
%Calculate the above integral explicitly in the case $n=2$. (You should
%get something proportional to $1/(2-\omega)+\Gamma'(1)-\ln 4 \pi$.)
%Differentiate the integral with respect to $p_\mu$ to get
%\begin{displaymath}
%\int {\frac{d^{2 \omega} k}{(2 \pi)^{2 \omega}} k_\mu (k^2+2 p \cdot
%k-m^2+i \epsilon)^{-n}} .
%\end{displaymath}
%You could similarily get the integrals proportional to $k_\mu k_\nu$
%and $k^2$. Look them out from the book and list them (e.g. Bailin-Love
%(7.21b,23)).  \\ Hint: Bailin-Love, page 81 and onwards.



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Please find enclosed information on a memorial mass for Tomasz
Kaszuba.

Return the solved problems to the downstairs box in Laurikainen
building on Monday November 23rd 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/

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