Weinberg Angle (relation Between Coupling Constants)

The weak mixing angle or Weinberg angle is a parameter in the Weinberg–Salam theory (by Steven Weinberg and Abdus Salam) of the electroweak interaction, part of the Standard Model of particle physics, and is usually denoted as . 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 and the massless photon () is expressed by the formula

$\begin \gamma~ \\ \textsf^0 \end = \begin \quad \cos \theta_\textsf & \sin \theta_\textsf \\ -\sin \theta_\textsf & \cos \theta_\textsf \end \begin \textsf^0 \\ \textsf^0 \end .$

The ''weak mixing angle'' also gives the relationship between the masses of the W and Z bosons (denoted as and ),

$m_\textsf = \frac \,.$

The angle can be expressed in terms of the and couplings (weak isospin and weak hypercharge , respectively),

$\cos \theta_\textsf = \frac \quad$ and $\quad \sin \theta_\textsf = \frac ~.$

The electric charge is then expressible in terms of it, (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

$\cos \theta_\textsf = \frac ~.$

The value of varies as a function of the momentum transfer, , 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 , corresponding to the mass of the  boson, .

In practice, the quantity is more frequently used. The 2004 best estimate of , at , in the scheme is , which is an average over measurements made in different processes, at different detectors. Atomic parity violation experiments yield values for at smaller values of , 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 was obtained at , establishing experimentally the so-called 'running' of the weak mixing angle. These values correspond to a Weinberg angle varying between 28.7° and . LHCb measured in 7 and 8 TeV proton–proton collisions an effective angle of ,

though the value of for this measurement is determined by the partonic collision energy, which is close to the Z boson mass.

CODATA 2022

gives the value

$\sin^2 \theta _\textsf = 1 - \left( \frac\right)^2 = 0.22305(23) ~.$

The massless photon () couples to the unbroken electric charge, , while the  boson couples to the broken charge .

Footnotes

References

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{{DEFAULTSORT:Weinberg Angle
Electroweak theory
Angle
Dimensionless numbers of physics
Physical constants
Steven Weinberg