Weinberg Angle and Electroweak Unification

Weinberg Angle and Electroweak Unification

The electroweak theory unifies electromagnetism and the weak nuclear force into a single framework. The key parameter that links the two interactions is the Weinberg angle (also called the weak mixing angle) $\theta_W$.

The Weinberg Angle

The measured value at the Z boson mass scale is:

$\sin^2\theta_W \approx 0.23122 \quad (Q = m_Z)$

This angle "mixes" the $SU(2)_L$ and $U(1)_Y$ gauge bosons to produce the photon and the Z boson. It relates the electromagnetic coupling $e$, the weak coupling $g$, and the hypercharge coupling $g'$:

$g \sin\theta_W = g' \cos\theta_W = e$

W and Z Boson Masses

At tree level, the electroweak theory predicts a precise relationship between the W and Z boson masses:

$m_W = m_Z \cos\theta_W = m_Z \sqrt{1 - \sin^2\theta_W}$

With $m_Z = 91.1876$ GeV and $\sin^2\theta_W = 0.23122$:

$m_W = 91.1876 \times \sqrt{1 - 0.23122} \approx 79.95 \text{ GeV}$

The measured value is $m_W \approx 80.377$ GeV — deviations are explained by radiative corrections.

Tree-Level W Mass from the Fermi Constant

The Fermi constant $G_F = 1.1663788 \times 10^{-5}$ GeV$^{-2}$ governs the strength of weak decays (like muon decay). At tree level, the W mass satisfies:

$m_W^2 \sin^2\theta_W = \frac{\pi \alpha}{\sqrt{2} \, G_F}$

$m_W = \frac{1}{\sin\theta_W} \sqrt{\frac{\pi \alpha}{\sqrt{2} \, G_F}}$

where $\alpha \approx 1/137.036$ is the fine-structure constant. This gives a slightly different value than the $m_Z \cos\theta_W$ relation because loop corrections enter differently.

Your Task

Implement three functions:

• w_mass_from_z(m_Z_GeV, sin2_theta_W) — W mass from the Z mass using $m_W = m_Z\sqrt{1-\sin^2\theta_W}$
• weinberg_angle_deg() — the Weinberg angle in degrees from $\sin^2\theta_W = 0.23122$
• w_mass_from_fermi(alpha, G_F_GeV2) — tree-level W mass from the Fermi constant

All physics constants must be defined inside each function body.