Suppose that the Higgs scalar field ϕ(x) has weak isospin T=3 and hypercharge Y=-4. If the neutr

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Suppose that the Higgs scalar field ϕ(x) has weak isospin...

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

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

Key Concepts

Gauge Boson Mass Generation

This concept explains how gauge bosons acquire mass through their interactions with the Higgs field. By expanding the Higgs kinetic energy term in the Lagrangian and using the non-zero vev, one obtains mass terms for the W and Z bosons while preserving gauge invariance.

SU(2) Ladder Operators

These operators, often denoted as T? and T?, are the raising and lowering operators in the SU(2) Lie algebra. They are used to move between different components of an isospin multiplet. In mass calculations for gauge bosons, their quadratic combinations give contributions proportional to the squares of the vev.

Vacuum Expectation Value

The vacuum expectation value (vev) is the constant value that a scalar field adopts in its lowest energy state. In the context of the electroweak theory, the Higgs field’s non-zero vev leads to the spontaneous symmetry breaking that generates mass for gauge bosons and fermions.

Weak Isospin

Weak isospin is a quantum number associated with the SU(2) component of the electroweak gauge group. It organizes particles into multiplets and determines how fields transform under weak interactions. The eigenvalues of the third component (T^3) play an important role in defining the electric charge along with hypercharge.

Electroweak Symmetry Breaking

This concept refers to the process by which the underlying electroweak gauge symmetry is spontaneously broken due to the Higgs field acquiring a non-zero vacuum expectation value. It is central to the Standard Model in explaining how W and Z bosons gain mass while the photon remains massless.

Hypercharge

Hypercharge is the quantum number linked with the U(1) component of the electroweak gauge group. When combined with the weak isospin, it determines the electric charge of particles via the Gell-Mann–Nishijima formula, and plays a vital role in the gauge theory structure of the Standard Model.

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