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]

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

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

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

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),

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

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

$\cos \theta _{\textsf {w}}={\frac {\ m_{\textsf {W}}\ }{m_{\textsf {Z}}}}~.$ ^[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 $sin 2 θ$ _w is more frequently used. The 2004 best estimate of $sin 2 θ$ _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 $sin 2 θ$ _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 $sin 2 θ$ _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 $sin 2 θ$ ^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

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

The massless photon ( $γ$ ) couples to the unbroken electric charge, $Q = T3 + .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠ 1 / 2 ⁠ Y$ _w, while the
Z⁰
boson couples to the broken charge $T 3 - Q sin 2 θ$ _w.

Footnotes

^ 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.
^ 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

^ Lee, T.D. (1981). Particle Physics and Introduction to Field Theory.
^ 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.
^ ^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.
^ ^a ^b "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.
^ Okun, L.B. (1982). Leptons and Quarks. North-Holland Physics Publishing. p. 214. ISBN 0-444-86924-7.
^ 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.

[2] 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.

[8] 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.

[1] Lee, T.D. (1981). Particle Physics and Introduction to Field Theory.

[3] 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.

[Cheng-Li-2006-4] Cheng, T.P.; Li, L.F. (2006). Gauge Theory of Elementary Particle Physics. Oxford University Press. pp. 349–355. ISBN 0-19-851961-3.

[wein-5] "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.

[6] Okun, L.B. (1982). Leptons and Quarks. North-Holland Physics Publishing. p. 214. ISBN 0-444-86924-7.

[7] 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.

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