# Weinberg angle

The weak mixing angle or Weinberg angle is a parameter in the Weinberg–Salam theory (by Steven Weinberg and Abdus Salam) 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 W$^0$ and B$^0$ vector boson plane, producing as a result the Z$^0$ boson, and the photon. 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.

## Details

The algebraic formula for the combination of the W$^0$ and B$^0$ vector bosons (i.e. 'mixing') that simultaneously produces the massive Z$^0$ boson and the massless photon ( γ ) is expressed by the formula
${\begin{pmatrix}\gamma ~\\{\textsf {Z}}^{0}\end{pmatrix}}={\begin{pmatrix}\quad \cos \theta _{\textsf {w}}&\sin \theta _{\textsf {w}}\\-\sin \theta _{\textsf {w}}&\cos \theta _{\textsf {w}}\end{pmatrix}}{\begin{pmatrix}{\textsf {B}}^{0}\\{\textsf {W}}^{0}\end{pmatrix}}.$
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
$\cos \theta _{\textsf {w}}={\frac {\ m_{\textsf {W}}\ }{m_{\textsf {Z}}}}~.$
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 Z$^0$ 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, 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 gives the value
$\sin ^{2}\theta _{\textsf {w}}=1-\left({\frac {\ m_{\textsf {W}}\ }{m_{\textsf {Z}}}}\right)^{2}=0.22305(23)~.$
The massless photon ( γ ) couples to the unbroken electric charge, Q = T$_3$ + ⁠ 1 / 2 ⁠ Y$_w$, while the Z$^0$ 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.