Chapter 15, The Weinberg-Salam Model and Beyond Video Solutions, Quarks And Leptons. An Introductory Course In Modern Particle Physics

01:47

Problem 1

As revision, derive (15.8) and (15.9) from the statements of the above paragraph.

Manik Pulyani

Numerade Educator

01:09

Show that in the Weinberg-Salam model
$$
\frac{1}{2 v^{2}}=\frac{g^{2}}{8 M_{W}^{2}}=\frac{G}{\sqrt{2}}
$$
and hence, using the empirical value of $G$ of Chapter 12 , verify that $v=246$ $\mathrm{GeV}$. Derive the mass relations
$$
M_{W}=\frac{37.3}{\sin \theta_{W}} \mathrm{GeV}, \quad M_{Z}=\frac{74.6}{\sin 2 \theta_{W}} \mathrm{GeV}
$$
and give the lower bounds for their masses. Predict $M_{W}$ and $M_{Z}$ using the experimental determination of $\sin ^{2} \theta_{W}$.
Very recently (1983) the W and Z bosons have been discovered at the CERN $\overline{\mathrm{p}} \mathrm{p}$ collider via the processes
$$
\begin{aligned}
&\overline{\mathrm{p} p} \rightarrow \mathrm{W}^{\pm} \mathrm{X} \rightarrow\left(\mathrm{e}^{\pm} \nu\right) \mathrm{X} \\
&\overline{\mathrm{p}} \mathrm{p} \rightarrow \mathrm{ZX} \rightarrow\left(\mathrm{e}^{+} \mathrm{e}^{-}\right) \mathrm{X}
\end{aligned}
$$
where $X$ denotes all the other particles produced in the high-energy head-on collision. By studying the momentum distribution of the emitted decay electrons and positrons, the masses are measured to be
$$
\begin{aligned}
M_{W} &=81 \pm 2 \mathrm{GeV} \\
M_{Z} &=93 \pm 2 \mathrm{GeV}
\end{aligned}
$$
which are in impressive agreement with the predictions of the standard electroweak model.

Keshav Singh

Numerade Educator

04:58

Problem 3

Suppose that the Higgs scalar field $\phi(x)$ has weak isospin $T=3$ and hypercharge $Y=-4$. If the neutral component $\phi^{0}$ (with $T^{3}=2$ ) develops a vacuum expectation value $v / \sqrt{2}$, show that
$$
\begin{aligned}
M_{W}^{2} &=\frac{g^{2}}{2} \phi^{\dagger}\left(T^{+} T^{-}+T^{-} T^{+}\right) \phi \\
&=4 g^{2} v^{2}
\end{aligned}
$$

Keshav Singh

Numerade Educator

01:44

Problem 4

Suppose that there exist several representations ( $i=$ $1, \ldots, N$ ) of Higgs scalars whose charge-zero members acquire vacuum expectation values $v_{i}$. Show that
$$
\rho \equiv\left(\frac{M_{W}}{M_{Z} \cos \theta_{W}}\right)^{2}=\frac{\sum v_{i}^{2}\left[T_{i}\left(T_{i}+1\right)-\frac{1}{4} Y_{i}^{2}\right]}{\sum \frac{1}{2} v_{i}^{2} Y_{i}^{2}}
$$
where $T_{i}$ and $Y_{i}$ are, respectively, the weak isospin and hypercharge of representation $i$. Show that $\rho=1$ if only Higgs doublets with $Y_{i}=+1$ exist.

Khoobchandra Agrawal

Numerade Educator

02:16

Problem 5

The Lagrangian for the scalar field, (15.13), contains trilinear $\mathrm{hW}^{+} \mathrm{W}^{-}$and quadrilinear $\mathrm{hh} \mathrm{W}^{+} \mathrm{W}^{-}$Higgs boson couplings. Use
$$
\phi=\sqrt{\frac{1}{2}}\left(\begin{array}{c}
0 \\
v+h(x)
\end{array}\right)
$$
[see (14.71)] to show that in the standard model the vertex factors are
$$
i g M_{W} \quad \text { and } \quad \frac{1}{4} i g^{2},
$$
respectively. Determine the $\mathrm{hZZ}$ and hhZZ vertex factors.