Weinberg angle

The algebraic formula for the combination of the
W⁰
and
B⁰
vector bosons (i.e. 'mixing') that simultaneously produces the massive
Z⁰
 boson and the massless photon (
γ
) is expressed by the formula

${\displaystyle {\begin{pmatrix}\gamma \\Z^{0}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{\text{W}}&\sin \theta _{\text{W}}\\-\sin \theta _{\text{W}}&\cos \theta _{\text{W}}\end{pmatrix}}{\begin{pmatrix}B^{0}\\W^{0}\end{pmatrix}}.}$[3]

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

${\displaystyle m_{\text{Z}}={\frac {m_{\text{W}}}{\,\cos \theta _{\text{W}}\,}}.}$

The angle can be expressed in terms of the ${\displaystyle \mathrm {SU} (2)_{L}}$ and ${\displaystyle \mathrm {U} (1)_{Y}}$ couplings (weak isospin g and weak hypercharge g′, respectively),

${\displaystyle \cos \theta _{\text{W}}={\frac {g}{\,{\sqrt {g^{2}+g'^{2}\,}}\,}}\qquad }$ and ${\displaystyle \qquad \sin \theta _{\text{W}}={\frac {g'}{\,{\sqrt {g^{2}+g'^{2}\,}}\,}}.}$

The electric charge is then expressible in terms of it, e = g sin θ_W = g′ cos θ_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

${\displaystyle \cos \theta _{\text{W}}={\frac {\,m_{\text{W}}\,}{m_{\text{Z}}}}.}$[5]

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
Z⁰
 boson, m_Z.

In practice, the quantity sin² θ_W is more frequently used. The 2004 best estimate of sin² θ_W, at ∆q = 91.2 GeV/c, in the MS scheme is 0.23120 ± 0.00015, which is an average over measurements made in different processes, at different detectors. 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 sin² θ_W = 0.2397 ± 0.0013 was obtained at ∆q = 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 sin²( θ^eff
_W ) = 0.23142,[6] though the value of ∆q for this measurement is determined by the partonic collision energy, which is close to the Z boson mass.

CODATA 2018[4] gives the value

${\displaystyle \sin ^{2}\theta _{\text{W}}=1-\left(m_{\text{W}}/m_{\text{Z}}\right)^{2}=0.22290(30).}$[lower-alpha 2]

The massless photon (
γ
) couples to the unbroken electric charge, ${\displaystyle Q=T_{3}+Y_{W}/2}$, while the
Z⁰
 boson couples to the broken charge ${\displaystyle T_{3}-\sin ^{2}\theta _{W}~Q}$.

• Erler, J.; Freitas, A.; et al. (Particle Data Group (PDG)) (2019) [revised March 2018]. Review of the Standard Model (PDF) (Report).
• E158: A precision measurement of the weak mixing angle in Møller scattering. Stanford Linear Accelerator (SLAC) (Report). Stanford University.
• Q-weak: A precision test of the Standard Model and determination of the weak charges of the quarks through parity-violating electron scattering. Jefferson National Accelerator Lab. (Report). U.S. Department of Energy.