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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 8}\\
%{\Large DRAFT/not a final version}

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%\item
%
%Beginning from eq. (4.2.17) of Chaichian and Nelipa derive
%eq. (4.2.19).


\item Adding to the Lagrangian the term
\begin{displaymath}
-\frac 1{2 \alpha} \left( n_\mu A^\mu \right) ^2
\end{displaymath}
which fixes the axial gauge, derive formula (5.3.16) for the
propagator.  \\ Hint: In addition to the lectures the literature might
be also helpful (e.g. Bailin and Love, around eq. (10.57)).

\item By choosing appropriately the vector $n_\mu$ derive from the
propagator in the axial gauge the propagator in the
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\item Hamilton gauge
\item Coulomb gauge
\item Landau gauge
\item Feynman gauge.
\end{list}

\item Derive, using the path integral formalism, the Feynman rules for
a scalar theory with a Lagrangian
\begin{displaymath}
{\cal{L}} = \frac 12 \partial_\mu \phi(x) \partial^\mu \phi(x)-\frac
12 m^2 \phi (x) ^2-\frac {\lambda}{3!} \phi(x) ^3- \frac g{4 !} \phi(x)
^4 .
\end{displaymath}
Draw the connected diagrams up to one loop level for the reaction $a+b
\rightarrow c+d$. \\ Hint: See Chaichian and Nelipa section 5.2.

%\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 Show that
%\begin{displaymath}
%(2 \pi) ^4 \delta ^4 (p_3+p_4-p) \frac{d^3 p_3}{(2 \pi)^3 2 E_3}
%\frac{d^3 p_4 }{(2 \pi)^3 2 E_4 }
%\end{displaymath}
%is Lorentz invariant. \\ Hint: Show $d^3p/(2E)=\delta(p^2-m^2)
%\Theta(p_0) d^4 p$ (Renton p. 152).

\item Show
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\item $\overline u (p,s) \gamma^\mu u (p,s')=2 \delta_{s s'} p^\mu$ ,
\item $\overline v (p,s) u(p,s')=0$ ,
\item $\overline u (p,s) \gamma^\mu u(q,s')= \frac 1{2m} \overline u
(p) \left[ (p+q)^\mu+i \sigma^{\mu \nu} (p-q)_\nu \right] u (q)$
(Gordon decomposition) ,
\end{list}
where $s$ ($s'$) are the helicities of respective spinors.

Hint: See Halzen-Martin (ex. 6.1 and around), or Björken-Drell.

\end{list}

If you have not already done so, please fill in the student
survey. Forms can be found from the exercise box or from the web
page. 

Return the solved problems to the downstairs box in Laurikainen
building on Monday November 9th at 12 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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