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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 4}\\

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\item 
Consider the fields
\begin{displaymath}
H= \left( \begin{array}{c} H_1 \\ H_2 \end{array} \right) ,
L= \left( \begin{array}{c} \nu_L \\ e_L \end{array} \right) , R=e_R ,
\end{displaymath}
of the standard model. Write down the transformation of these fields
under the electroweak symmetry group $SU(2)_L \times U(1)_Y$
explicitly. Then assume that field $H$ obtains a vacuum expectation
value, which can be chosen to be $\left< H \right>= (0, v)^T$. Show
that there is a residual symmetry $U(1)_Q$ that leaves the vacuum
state $\left< H \right>$ invariant. Find the generator of this
symmetry group and write down explicitly the transformation properties
of fields $H$, $L$ and $R$ under $U(1)_Q$ transformation. What are
the $U(1)_Q$ quantum numbers, i.e. the electric charges, of the
fields? \\
Hint: See Chaichian and Nelipa eqs. (6.1.5--6'), or e.g. Halzen and
Martin, chapter 15.

\item 
Study in detail spontaneous symmetry breaking in locally $U(1)$
invariant theory having Lagrangian
\begin{displaymath}
{\cal{L}} = {\overline{\psi}} i \gamma^\mu D_\mu \psi + (D^\mu
\phi)^* (D_\mu \phi) + m^2 \phi^* \phi-\frac 14 \lambda (\phi^*
\phi)^4 -\frac 14 F^{\mu \nu} F_{\mu \nu} ,
\end{displaymath}
where $D_\mu=\partial_\mu+igA_\mu$, $F_{\mu \nu}=\partial_\mu
A_\nu-\partial_\nu A_\mu$ and $\phi$, $\psi$ and $A_\mu$ are scalar,
spinor and vector fields. What happens to the Goldstone boson? Can
$A_\mu$ be a electromagnetic field? How could you make the fermion
massive? \\
Hint: See the lecture notes. Also Bailin and Love, section 13.5, might
prove useful.

\item 
Study the Lagrangian for the scalar and the gauge sector of the
standard model 
\begin{displaymath}
{\cal{L}}=(D_\mu H)^\dagger (D^\mu H) -V(H)- \frac 14 F_{\mu
\nu} F^{\mu \nu} -\frac 14 F^a_{\mu \nu} F_a^{\mu \nu} ,
\end{displaymath}
where
\begin{displaymath}
V(H)=-\mu^2 H^\dagger H+\frac 12 \lambda \left( H^\dagger
H \right) ^2 .
\end{displaymath}
Perform the spontaneous symmetry breaking. What happens to the
Goldstone bosons? Diagonalize the gauge boson mass matrix and find the
masses of the physical gauge bosons in terms of experimentally
measurable quantities ($e$, $G_F$ and $\theta_W$). Show that
$M_W/M_Z=\cos \theta_W$. \\ Hint: Halzen and Martin, sections 15.2--3,
might be helpful.

\item Show that that the conjugate Higgs doublet ${\tilde{H}}=-i
\tau_2 H^*$ transforms in (at least infinitesimal) $SU(2)$
transformations like Higgs doublet $H$. What are the electrical
charges of the components of the conjugate doublet ${\tilde{H}}$? Show
that the Yukawa terms
\begin{displaymath}
{\cal{L}}_{Yukawa} =-\lambda_d^{ij} {\overline{Q_i}} H
d_{Rj}-\lambda_u^{ij} {\overline{Q_i}} {\tilde{H}}
u_{Rj}-\lambda_e^{ij} {\overline{L_i}} H e_{Rj} +h.c. 
\end{displaymath}
are invariant under the electroweak group $SU(2)_L \times
U(1)_Y$. Given same VEV for the Higgs field as in problem 1, what are
the masses of the leptons and quarks expressed in terms of Yukawa
couplings and Higgs VEV? \\ Hint: Halzen and Martin, chapter 15, could
still be useful... 

\end{list}


Return the solved problems to the downstairs box in Laurikainen
building on Monday October 12th 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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