Show that in the Weinberg-Salam model (1)/(2 v^2)=(g^2)/(8 MW^2)=(G)/(√(2)) and hence, using the

step 1 For this problem on the topic of particle physics, we want to use the given data and place the p

Show that in the Weinberg-Salam model (1)/(2 v^2)=(g^2)/(8...

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 $M_{\mathrm{Z}}$ using the experimental determination of $\sin ^2 \theta_W$. Very recently (1983) the $\mathrm{W}$ and $\mathrm{Z}$ bosons have been discovered at the CERN $\overline{\mathrm{p}} \mathrm{p}$ collider via the processes $$ \begin{aligned} & \overline{\mathrm{p}} \mathrm{p} \rightarrow \mathrm{W}^{ \pm} \mathrm{X} \rightarrow\left(\mathrm{e}^{ \pm} \nu\right) \mathrm{X} \\ & \overline{\mathrm{p}} \mathrm{p} \rightarrow \mathrm{ZX} \rightarrow\left(\mathrm{e}^{+} \mathrm{e}^{-}\right) \mathrm{X}, \end{aligned} $$ where $\mathrm{X}$ 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.

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 $M_{\mathrm{Z}}$ using the experimental determination of $\sin ^2 \theta_W$.

Very recently (1983) the $\mathrm{W}$ and $\mathrm{Z}$ bosons have been discovered at the CERN $\overline{\mathrm{p}} \mathrm{p}$ collider via the processes
$$
\begin{aligned}
& \overline{\mathrm{p}} \mathrm{p} \rightarrow \mathrm{W}^{ \pm} \mathrm{X} \rightarrow\left(\mathrm{e}^{ \pm} \nu\right) \mathrm{X} \\
& \overline{\mathrm{p}} \mathrm{p} \rightarrow \mathrm{ZX} \rightarrow\left(\mathrm{e}^{+} \mathrm{e}^{-}\right) \mathrm{X},
\end{aligned}
$$
where $\mathrm{X}$ 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.

Key Concepts

Weak Mixing Angle

The weak mixing angle (?W) quantifies the mixing of the neutral gauge bosons corresponding to the two symmetry groups SU(2) and U(1) to produce the physical Z boson and the photon. It plays a central role in determining the properties and interactions of the electroweak gauge bosons, and the sin?W appears explicitly in the mass relations of the W and Z bosons.

Fermi Constant and Gauge Coupling Relations

The Fermi constant (G) is a measure of the strength of weak interactions, as measured in low-energy processes. In electroweak theory, it is related to the gauge coupling constant (g) and the vacuum expectation value (v) by a specific relation. This connection allows one to compute the masses of the W and Z bosons from the known value of G and the couplings, linking low-energy weak interaction phenomenology with the underlying gauge structure.

Mass Relations of Gauge Bosons

Mass relations in the electroweak theory emerge from the structure of the gauge couplings and the Higgs mechanism. The relations, such as M_W = ½vg and specific expressions involving sin?W for M_W and M_Z, provide a direct link between the theoretical parameters (like g, v, and ?W) and the measurable masses of the gauge bosons. These relations are essential for making testable predictions of the model and verifying them through experimental data.

Electroweak Unification (Weinberg-Salam Model)

This concept refers to the unified description of electromagnetic and weak interactions within a single gauge theory, where the gauge group is SU(2) × U(1). The Weinberg–Salam model explains how these forces emerge from the same underlying symmetry and how they break down into the distinct forces we observe at low energies, specifically through the introduction of the Higgs mechanism.

Vacuum Expectation Value (vev) and the Higgs Mechanism

The vacuum expectation value (v) is the nonzero value acquired by the Higgs field in its lowest energy state. This spontaneous symmetry breaking through the Higgs mechanism gives masses to the gauge bosons (W and Z bosons) and fermions while keeping the photon massless. The value of v is a crucial parameter in the theory and is determined by experimental inputs such as the Fermi constant.

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