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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 7}\\
%{\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 Derive eqs. (4.2.24--28) of Chaichian and Nelipa in detail. 


\item Derive eq. (4.2.33) of Chaichian and Nelipa in detail.

\item As in lectures, show that for the Lorentz gauge $\partial_\mu
A^\mu=0$ one has $\Delta_L(A)=\det M_L$. What is $\Delta(A)$ for the
axial gauge $n_\mu A^\mu=0$? \\ Hint: In addition to the lectures see
the literature (e.g. Bailin and Love section 10.4).

%\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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%\begin{list}%
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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 $a+b
%\rightarrow c+d+e$. \\ 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 The polarization vectors $\epsilon_\mu^{(\lambda)}(k)$,
($\lambda=1,2,3$) of a vector field with mass $M$ satisfy
\begin{eqnarray}
\epsilon_\mu^{(\lambda)}(k) k^\mu=0 \nonumber \\
\epsilon_\mu^{(\lambda)}(k) \epsilon^{\mu (\lambda ')}(k)
=-\delta^{\lambda \lambda '} . \nonumber
\end{eqnarray}
Show that
\begin{displaymath}
\sum_\lambda {\epsilon_\mu^{(\lambda)}(k)
\epsilon_\nu^{(\lambda)}(k)}=-\left(g_{\mu \nu}- \frac{k_\mu
k_\nu}{M^2} \right) .
\end{displaymath}
What is the respective relation for massless photon field? \\ Hint:
See e.g. Itzykson-Zuber, Halzen-Martin, ...

%\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}
%
%Hint: See Halzen-Martin, or Björken-Drell.

\end{list}


Return the solved problems to the downstairs box in Laurikainen
building on Monday November 2nd 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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