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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 11}\\
%{\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 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 Consider the decay of $W$ boson. Show that in rest frame the
decay width can be written as
\begin{displaymath}
\frac{d \Gamma}{d\Omega} (W \rightarrow f_1 f_2)= \frac
{|T|^2}{64 \pi^2} \frac{\sqrt{\lambda(m_W^2,m_1^2,m_2^2)}}{m_W^3}  .
\end{displaymath}
Calculate the decay width for the process $W^- \rightarrow e^-
\overline \nu_e$. For unpolarized $W^-$ the decay width can be written
as
\begin{displaymath}
\frac{d \Gamma}{d \Omega} (W^- \rightarrow e^- \overline \nu_e) =
\frac 13 \sum_{\lambda=-1}^1 {\frac{d\Gamma}{d\Omega} (W_\lambda)} ,
\end{displaymath}
where the subscripts label the helicity of $W$. Show that the
differential decay widths are
\begin{eqnarray}
\frac{d\Gamma}{d\Omega} (W_0)= \frac{G_F M_W^3}{16 \pi^2 \sqrt{2}}
\sin^2 \theta , \nonumber \\
\frac{d\Gamma}{d\Omega} (W_{+1})= \frac{G_F M_W^3}{32 \pi^2 \sqrt{2}}
(1-\cos \theta)^2 , \nonumber \\
\frac{d\Gamma}{d\Omega} (W_{-1})= \frac{G_F M_W^3}{32 \pi^2 \sqrt{2}}
(1+\cos \theta)^2 , \nonumber 
\end{eqnarray}
where $\theta$ is the angle between electron momentum and the
polarization of $W$. The respective polarization vectors in the given
frame of reference are
\begin{displaymath}
\varepsilon_0^\mu=(0,0,0,1),\varepsilon_{+1}^\mu=\frac
1{\sqrt{2}}(0,-1,-i,0),\varepsilon_{-1}^\mu=\frac
1{\sqrt{2}}(0,-1,i,0).
\end{displaymath}
Use the above result to calculate the numerical value for the total
decay width for $W$ boson in SM. Use Particle Data Group book to get
the latest numerical values (European Physical Journal C3 (1998)
1--794, can be found from the reference library (RT:Pa), or from the
web http://pdg.lbl.gov/). The PDG book contains all experimental data
on particle physics. Spend a silent moment browsing the book and
contemplating the meaning of thousands of numerical values for all
imaginable and unimaginable measurements and particle
properties. (Historical note: The first version of Particle Data Book
was written as a table by professor emeritus Matts Roos.) \\ Hint:
Remember that there are three colours of each quarks. See section 4.4
of Halzen-Martin. You can ignore the masses of final-state fermions
lighter than $W$.

\item Show that (9.2.30) is invariant in the Becchi-Rouet-Stora
transformations (9.2.31). Write down the corresponding Ward
identities. \\ Hint: See the lectures. Alternative derivation is given
in Bailin-Love, appendix B. 

\item
Show that the standard model has no triangle anomaly, by
showing that 
$$
\sum_{i,j,k=q,\ell}{ \left(
\parbox{50mm}{
  \begin{fmfgraph*}(50,30) 
    \fmfleft{i1,i2}
    \fmfright{o1}
%    \fmflabel{$W^-_V$}{i1}
%    \fmflabel{$W^+_V$}{i2}
%    \fmflabel{$Z_A$}{o1}
    \fmf{boson,label=$W^-_V$}{i1,v1}
    \fmf{boson,label=$W^+_V$}{i2,v2}
    \fmf{boson,label=$Z_A$}{v3,o1}
    \fmf{fermion,label=$i$,tension=0.3}{v2,v1}
    \fmf{fermion,label=$j$,tension=0.3}{v1,v3}
    \fmf{fermion,label=$k$,tension=0.3}{v3,v2}
    \fmfdot{v1,v2,v3}
  \end{fmfgraph*}
}
\right)} = 0,
$$
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, or
section 15.5 of Bailin-Love (1993 rev. ed.). In-depth discussion of
anomalies can be found from Weinberg: tQToF2, chapter 22, and
especially section 22.4.


%\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 $(p
%^2)^{-1}$.

%\item Write down the renormalized and bare Lagrangian of QED, as well
%as the counter-term Lagrangian. Calculate the renormalization constant
%$Z_3$ ($Z_A$ in Bailin-Love) for the photon self-energy in QED in
%mass-shell, $MS$ and $\overline{MS}$ schemes. (See e.g. Itzykson-Zuber
%p. 319--335, Bailin-Love chapter 11) \\ Hint: In QED you can show that
%there is only one contributing graph at leading order, namely the
%electron loop (Bailin-Love eq. (11.36)). Use dimensional
%regularization and Feynman gauge. The loop integral should give
%\begin{displaymath}
%I_{\mu \nu}= \frac{i e^2}{2 \pi^2} (q^2 g_{\mu \nu}-q_\mu q_\nu)
%\left[ \frac 16 \left( \frac 1{2-\omega}+\Gamma'(1)+\ln 4 \pi
%\right)-\int_0^1 { x(1-x) \ln {\frac{m^2-x(1-x)^2}{\mu^2}}} \right] .
%\end{displaymath}

%\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 Consider the mixing of photon and $Z$-boson in SM in one-loop
order, where the fermion loop is summed over all quarks and
leptons:     \fmf{fermion,label=$\overline f$,tension=0.5}{v2,v1}

\begin{center}
   \begin{fmfgraph*}(100,50)
     \fmfleft{i1}
     \fmfright{o1}
     \fmflabel{$\gamma$}{i1}
     \fmflabel{$Z$}{o1}
     \fmf{photon}{i1,v1}
%     \fmf{phantom}{v1,v2}
     \fmf{photon}{v2,o1}
%     \fmf{photon}{v3,o2}
%     \fmffreeze
     \fmf{fermion,label=$f$,left,tension=0.9}{v1,v2,v1}
%     \fmf{fermion,label=$\overline f$,left=90,tension=0.3}{v2,v1}
%     \fmf{fermion,label=$F$,tension=0.5}{v3,v1}
     \fmfdot{v1,v2}
   \end{fmfgraph*}\\[3\baselineskip]
\end{center}


Show that the mixing is zero when $\sin^2 \theta_W=3/8$. This number
can also be obtained from grand unified theories a bit differently. \\
Hint: You do not have to calculate the integral, unless you
desperately want to do it. The result is independent of number of
families (why?), so it is enough to just show this for one family of
quarks and leptons. Make up a convincing argument why the fermion
masses can be ignored in the integral.

\item Derive the following integral that appears in the 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
$2 \omega$ denotes the number of space-time dimensions. Calculate the
above integral explicitly in the limit $\omega \rightarrow 2^-$
keeping all divergent and finite terms. Write down the result
explicitly in 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}} A (k^2+2 p \cdot k-m^2+i
\epsilon)^{-n}} \end{displaymath} where $A$ denotes $k_\mu$, $k_\mu
k_\nu$ or $k^2$. Convince yourself that you can make all momentum
integrals to be of this form by using Feynman trick (eq. (9.2.11) of
Chaichian-Nelipa). \\ Hint: Bailin-Love, page 81 and around. You need
only take into account cases when $n \ge 1$ is integer.

\end{list}

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Return the solved problems to the downstairs box in Laurikainen
building on Monday November 30th 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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