# Weinberg angle

Weinberg angle θW, and relation between couplings g, g′, and e = g sin θW . Adapted from Lee (1981).[1]
The pattern of weak isospin, T3, and weak hypercharge, YW, of the known elementary particles, showing electric charge, Q,[a] 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[2] is a parameter in the WeinbergSalam theory of the electroweak interaction, part of the Standard Model of particle physics, and is usually denoted as θW. 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 approximately 30°,[4] but varies slightly, depending on the relative momentum of the particles involved in the interactions the angle is used for.[4]

## 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

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

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

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

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 θW = g′ cos θW (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 θW 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, mZ.

In practice, the quantity sin2θW is more frequently used. The 2004 best estimate of sin2θW , at Q = 91.2 GeV/c , in the MS scheme is 0.23120 ± 0.00015 , which averages over measurements made in different processes and at different detectors. Atomic parity violation experiments yield values for sin2θW 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 sin2θW = 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 of ≈30°. LHCb measured in 7 and 8 TeV proton-proton collisions an effective angle of sin2( θeff
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

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

## Footnotes

1. ^ The electric charge Q is distinct from the similar-appearing symbol for momentum-transfer Q.
2. ^ Note that at present, there is no generally accepted theory that explains why the measured value θW ≈ 29° is what it is. The specific value is not predicted by the Standard Model: The Weinberg angle θW is an open, unfixed 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. doi:10.1016/0029-5582(61)90469-2.
3. ^ a b 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. ^ a b c "Weak mixing angle". The NIST Reference on Constants, Units, and Uncertainty. 2018 CODATA Value. National Institute of Standards and Technology. 20 May 2019. Retrieved 2019-05-20.
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.; Albrecht, J.; Alessio, F.; Alexander, M.; Ali, S.; Alkhazov, G. (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.