Weinberg angle

The pattern of weak isospin, T₃, and weak hypercharge, Y_W, of the known elementary particles, showing electric charge, Q, along the Weinberg angle. The neutral Higgs field (circled) breaks the electroweak symmetry and interacts with other particles to give them mass. Three components of the Higgs field become part of the massive W and Z bosons.

The Weinberg angle or weak mixing 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 W0 and B⁰ vector boson plane, producing as a result the Z0 boson, and the photon.

$\begin{pmatrix} \gamma \\ Z^0 \end{pmatrix} = \begin{pmatrix} \cos \theta_W & \sin \theta_W \\ -\sin \theta_W & \cos \theta_W \end{pmatrix} \begin{pmatrix} B^0 \\ W^0 \end{pmatrix}$

It also gives the relationship between the masses of the W and Z bosons (denoted as m_W and m_Z):

$m_Z=\frac{m_W}{\cos\theta_W}$

The angle can be expressed in terms of the $SU(2)_L$ and $U(1)_Y$ coupling constants (g and g', respectively):

$\cos\theta_W = \frac{g}{\sqrt{g^2+g'^2}}$ and $\sin\theta_W = \frac{g'}{\sqrt{g^2+g'^2}}$

As the value of the mixing angle is currently determined empirically, it has been mathematically defined as:^[1]

$\cos\theta_W=\frac{m_W}{m_Z}$

The value of θ_W varies as a function of the momentum transfer, Q, 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 Q = 91.2 GeV/c, corresponding to the mass of the Z boson, m_Z.

In practice the quantity sin²θ_W is more frequently used. The 2004 best estimate of sin²θ_W, at Q = 91.2 GeV/c, in the MS scheme is 0.23120 ± 0.00015. Atomic parity violation experiments yield values for sin²θ_W at smaller values of Q, 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 sin²θ_W = 0.2397 ± 0.0013 was obtained at Q = 0.16 GeV/c, establishing experimentally the 'running' of the weak mixing angle. These values correspond to a Weinberg angle of ~30°. LHCb measured at Q = 7 and 8 TeV an effective angle of sin²(θ^eff_W) = 0.23142.

Note, however, that the specific value of the angle is not a prediction of the standard model: it is an open, unfixed parameter. At this time, there is no generally accepted theory that explains why the measured value is what it is.

See also

• Cabibbo angle

References

1. L. B. Okun (1982). Leptons and Quarks. North-Holland Physics Publishing. p. 214. ISBN 0-444-86924-7.

• C. Amsler et al. (Particle Data Group) (2008). "Review of Particle Physics – Electroweak model and constraints on new physics" (PDF). Physics Letters B 667 (1): 1. Bibcode:2008PhLB..667....1P. doi:10.1016/j.physletb.2008.07.018.
• E158: A Precision Measurement of the Weak Mixing Angle in Møller Scattering
• Q-weak: A Precision Test of the Standard Model and Determination of the Weak Charges of the Quarks through Parity-Violating Electron Scattering