\input phyzzx
\def\dim{\mathop{\rm dim}\nolimits}
\hsize=6.65in \hoffset=-0.15in
\vsize=8.5in \voffset=0in
%\overfullrule=0pt
\noindent PHY--396~L.\qquad
Problem set \#25 (last!)\qquad
Due May 5, 2005.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip
%\setbox1=\vbox{\pointbegin\global\advance\listcount by -2}
\pointbegin
First of all, finish your previous reading assignments.
\medskip
\point
Next, another reading assignment:
Chapter 20 of {\sl Peskin \& Schroeder} about the Glashow--Weinberg--Salam
standard model of weak and EM interaction.
\subpar
I shall explain much of this material in class, but I want you to
read ahead, and I might skip some important details.
\medskip
\point
And now consider what happens when one adds extra Higgses to the GWS theory.
Specifically, let us add a real triplet of scalar fields $\varphi_a(x)$ in the
${\bf3}^0$ representation of the $SU(2)\times U(1)$ (\ie, isospin $I=1$,
hypercharge $Y=0$).
Suppose these scalars develop non-zero vacuum expectation values
$\vev{\varphi_a}\neq0$ but they are `aligned' with the VEV of
the standard Higgs scalars $\vev{H_i}$ such that the photon remains massless.
\spointbegin
Show that the triplet Higgs VEVs modify the $Z/W$ mass ratio (\ie,
$M_W\neq M_z\times\cos\theta_W$) but the couplings of the $W^\pm$ and
$Z^0$ bosons to the quarks and leptons remain exactly as in the standard model.
\spoint
At low energies $E\ll M_W,M_Z$, weak interactions of quarks and leptons
are governed by effective four-fermion couplings stemming from exchanges
of virtual $W^\pm$ and $Z^0$ particles.
In current-current form $J_\mu J^\mu$,
$$
{\cal L}_{\rm weak}\
=\ -{4G_F\over\sqrt{2}}\left[ J_L^+\cdot J_L^-\
+\ \rho\bigl(J_{L3}^{}\,-\,\sin^2\theta_W J_{EM}^{}\bigr)^2\right].
\eqno(1)
$$
In the standard model $\rho=1$ (make sure you understand why).
Show that adding triplet Higgs VEVs to the theory leads to $\rho>1$.
\bye