Weinberg angle



The weak mixing angle or Weinberg angle is a parameter in the Weinberg–Salam theory of the electroweak interaction, part of the Standard Model of particle physics, and is usually denoted as $θ$$W$. It is the angle by which spontaneous symmetry breaking rotates the original and  vector boson plane, producing as a result the  boson, and the photon. Its measured value is slightly below 30°, but also varies, very slightly increasing, depending on how high the relative momentum of the particles involved in the interaction is that the angle is used for.

Details
The algebraic formula for the combination of the and  vector bosons (i.e. 'mixing') that simultaneously produces the massive boson and the massless photon  is expressed by the formula
 * $$ \begin{pmatrix}

\gamma~ \\ \textsf{Z}^0 \end{pmatrix} = \begin{pmatrix} \quad \cos \theta_\textsf{w} & \sin \theta_\textsf{w} \\ -\sin \theta_\textsf{w} & \cos \theta_\textsf{w} \end{pmatrix} \begin{pmatrix} \textsf{B}^0 \\ \textsf{W}^0 \end{pmatrix} .$$

The weak mixing angle also gives the relationship between the masses of the W and Z bosons (denoted as $g$$W$ and $g'$$W$),
 * $$ m_\textsf{Z} = \frac{m_\textsf{W}}{\,\cos\theta_\textsf{w}} \,.$$

The angle can be expressed in terms of the $e = g sin θ$ and $T3$ couplings (weak isospin $Y$ and weak hypercharge $Q$, respectively),
 * $$\cos \theta_\textsf{w} = \frac{\quad g ~}{\ \sqrt{ g^2 + g'^{\ 2} ~}\ } \quad $$ and $$ \quad \sin \theta_\textsf{w} = \frac{\quad g' ~}{\ \sqrt{ g^2 + g'^{\ 2} ~}\ } ~.$$

The electric charge is then expressible in terms of it, $Q$$W$$∆Q$$W$ (refer to the figure).

Because the value of the mixing angle is currently determined empirically, in the absence of any superseding theoretical derivation it is mathematically defined as
 * $$\cos \theta_\textsf{w} = \frac{\ m_\textsf{W}\ }{ m_\textsf{Z} } ~.$$

The value of $∆q$$Z$ varies as a function of the momentum transfer, $θ$, at which it is measured. This variation, or 'running', is a key prediction of the electroweak theory. The most precise measurements have been carried out in electron–positron collider experiments at a value of $m$ = 91.2 GeV/c, corresponding to the mass of the boson, $m$$w$.

In practice, the quantity $SU(2)L$$w$ is more frequently used. The 2004 best estimate of $U(1)Y$$w$, at $g$ = 91.2 GeV/c, in the $Z$ scheme is $w$, which is an average over measurements made in different processes, at different detectors. Atomic parity violation experiments yield values for $g'$$w$ at smaller values of $e = g sin θ$, below 0.01 GeV/c, but with much lower precision. In 2005 results were published from a study of parity violation in Møller scattering in which a value of $= g' cos θ$$\overline{MS}$ = $0.231$ was obtained at $θ$ = 0.16 GeV/c, establishing experimentally the so-called 'running' of the weak mixing angle. These values correspond to a Weinberg angle varying between 28.7° and 29.3° ≈ 30°. LHCb measured in 7 and 8 TeV proton–proton collisions an effective angle of $∆q$$w$ = 0.23142, though the value of $∆q$ for this measurement is determined by the partonic collision energy, which is close to the Z boson mass.

CODATA 2022 gives the value
 * $$\sin^2 \theta _\textsf{w} = 1 - \left( \frac{\ m_\textsf{W}\ }{ m_\textsf{Z} }\right)^2 = 0.22305(23) ~.$$

The massless photon ($m$) couples to the unbroken electric charge, $sin2 θ$$w$, while the  boson couples to the broken charge $sin2 θ$$0.24$.