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]

Weinberg angle θ_W, and relation between couplings g, g′, and e = g sin θ_W. Adapted from Lee (1981).^[1]

The pattern of weak isospin, T₃, and weak hypercharge, Y_W, of the known elementary particles, showing electric charge, Q,^{[lower-alpha 1]} 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.

Details

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

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

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.

Footnotes

1. The electric charge Q is distinct from the similar-appearing symbol occasionally used for momentum-transfer ∆Q. This article uses ∆q, but upper case is common and may occur in some graphs.
2. Note that at present, there is no generally accepted theory that explains why the measured value θ_w ≈ 29° should be what it is. The specific value is not predicted by the Standard Model: The Weinberg angle θ_w is an open, free parameter, although it is constrained and predicted through other measurements of Standard Model quantities.

References

1. Lee, T.D. (1981). Particle Physics and Introduction to Field Theory.
2. Glashow, Sheldon (February 1961). "Partial-symmetries of weak interactions". Nuclear Physics. 22 (4): 579–588. Bibcode:1961NucPh..22..579G. doi:10.1016/0029-5582(61)90469-2.
3. Cheng, T.P.; Li, L.F. (2006). Gauge Theory of Elementary Particle Physics. Oxford University Press. pp. 349–355. ISBN 0-19-851961-3.
4. "Weak mixing angle". The NIST reference on constants, units, and uncertainty. 2022 CODATA value. National Institute of Standards and Technology. 30 May 2024. Retrieved 2024-05-30.
5. Okun, L.B. (1982). Leptons and Quarks. North-Holland Physics Publishing. p. 214. ISBN 0-444-86924-7.
6. Aaij, R.; Adeva, B.; Adinolfi, M.; Affolder, A.; Ajaltouni, Z.; Akar, S.; et al. (2015-11-27). "Measurement of the forward-backward asymmetry in Z/γ^∗ → μ⁺μ⁻ decays and determination of the effective weak mixing angle". Journal of High Energy Physics. 2015 (11): 190. doi:10.1007/JHEP11(2015)190. hdl:1721.1/116170. ISSN 1029-8479. S2CID 118478870.

• 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.