Weinberg angle

Weinberg angle Template:MathTemplate:Sub, and relation between couplings Template:Math, Template:Math, and Template:MathTemplate:Sub. Adapted from Lee (1981).

The pattern of weak isospin, Template:Math, and weak hypercharge, Template:MathTemplate:Sub, of the known elementary particles, showing electric charge, Template:Math,Template:Efn along the Weinberg angle. The neutral Higgs field (upper left, 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 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 Template:MathTemplate:Sub. It is the angle by which spontaneous symmetry breaking rotates the original Template:SubatomicParticle and Template:SubatomicParticle vector boson plane, producing as a result the Template:SubatomicParticle 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 Template:SubatomicParticle and Template:SubatomicParticle vector bosons (i.e. 'mixing') that simultaneously produces the massive Template:Nowrap and the massless photon (Template:SubatomicParticle) is expressed by the formula

Template:In5$\left(\begin{array}{c}\gamma \\ {\mathsf{\text{Z}}}^0\end{array}\right)=\left(\begin{array}{cc}\cos {\theta }_{\mathsf{\text{w}}}& \sin {\theta }_{\mathsf{\text{w}}}\\ -\sin {\theta }_{\mathsf{\text{w}}}& \cos {\theta }_{\mathsf{\text{w}}}\end{array}\right)\left(\begin{array}{c}{\mathsf{\text{B}}}^0\\ {\mathsf{\text{W}}}^0\end{array}\right).$

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

Template:In5$m_{\mathsf{\text{Z}}}=\frac{m_{\mathsf{\text{W}}}}{\cos {\theta }_{\mathsf{\text{w}}}}.$

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

Template:In5$\cos {\theta }_{\mathsf{\text{w}}}=\frac{g}{\sqrt{g^2+g{\text{'}}^2}}$ and $\sin {\theta }_{\mathsf{\text{w}}}=\frac{g^{\prime }}{\sqrt{g^2+g{\text{'}}^2}}.$

The electric charge is then expressible in terms of it, Template:MathTemplate:SubTemplate:MathTemplate:Sub (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

Template:In5$\cos {\theta }_{\mathsf{\text{w}}}=\frac{m_{\mathsf{\text{W}}}}{m_{\mathsf{\text{Z}}}}.$

The value of Template:MathTemplate:Sub varies as a function of the momentum transfer, Template:Math, 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 Template:Nowrap, corresponding to the mass of the Template:SubatomicParticle boson, Template:MathTemplate:Sub.

In practice, the quantity Template:MathTemplate:Sub is more frequently used. The 2004 best estimate of Template:MathTemplate:Sub, at Template:Nowrap, in the [[minimal subtraction scheme|Template:Overline scheme]] is Template:Val, which is an average over measurements made in different processes, at different detectors. Atomic parity violation experiments yield values for Template:MathTemplate:Sub at smaller values of Template:Math, 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 Template:Nowrap was obtained at Template:Nowrap, establishing experimentally the so-called 'running' of the weak mixing angle. These values correspond to a Weinberg angle varying between 28.7° and Template:Nowrap. LHCb measured in 7 and 8 TeV proton–proton collisions an effective angle of Template:Nowrap, though the value of Template:Math for this measurement is determined by the partonic collision energy, which is close to the Z boson mass.

CODATA 2022 gives the value

Template:In5${\sin }^2{\theta }_{\mathsf{\text{w}}}=1-{\left(\frac{m_{\mathsf{\text{W}}}}{m_{\mathsf{\text{Z}}}}\right)}^2=0.22305(23).$Template:Efn

The massless photon (Template:Math) couples to the unbroken electric charge, Template:Nowrap, while the Template:SubatomicParticle boson couples to the broken charge Template:Nowrap.

References

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