Weinberg angle

The weak mixing angle or Weinberg angle^[2] 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.^[3] 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.^[4]

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

     ${\displaystyle {\begin{pmatrix}\gamma ~\\{\textsf {Z}}^{0}\end{pmatrix}}={\begin{pmatrix}\quad \cos \theta _{\textsf {w}}&\sin \theta _{\textsf {w}}\\-\sin \theta _{\textsf {w}}&\cos \theta _{\textsf {w}}\end{pmatrix}}{\begin{pmatrix}{\textsf {B}}^{0}\\{\textsf {W}}^{0}\end{pmatrix}}.}$^[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),

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

The angle can be expressed in terms of the SU(2)_L and U(1)_Y couplings (weak isospin g and weak hypercharge g′, respectively),

     ${\displaystyle \cos \theta _{\textsf {w}}={\frac {\quad g~}{\ {\sqrt {g^{2}+g'^{\ 2}~}}\ }}\quad }$ and ${\displaystyle \quad \sin \theta _{\textsf {w}}={\frac {\quad g'~}{\ {\sqrt {g^{2}+g'^{\ 2}~}}\ }}~.}$

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

     ${\displaystyle \cos \theta _{\textsf {w}}={\frac {\ m_{\textsf {W}}\ }{m_{\textsf {Z}}}}~.}$^[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 sin² θ_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 2022^[4] gives the value

     ${\displaystyle \sin ^{2}\theta _{\textsf {w}}=1-\left({\frac {\ m_{\textsf {W}}\ }{m_{\textsf {Z}}}}\right)^{2}=0.22305(23)~.}$^[b]

The massless photon (
γ
) couples to the unbroken electric charge, Q = T₃ + ⁠ 1 / 2 ⁠ Y_w, while the
Z⁰
 boson couples to the broken charge T₃ − Q sin² θ_w.

• 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.