Weinbergwinkel

The Weinberg angle , according to Steven Weinberg , or electroweak mixing angle is a quantity in the theory of electroweak interaction that occurs in different contexts. It is one of the quantities that can not be predicted in the Standard Model , but has to be determined experimentally. ${\displaystyle \theta _{\text{W}}}$

The cosine of the Weinberg angle appears as the quotient of the masses of the W and Z bosons :

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

background

In the electroweak interaction, electromagnetic interactions and weak ones are united in that they are not separated by the massless photon or the physical massive bosons and mediated, but by four massless bosons and . The first couple with the strength to other particles, this couples with the strength , where the weak isospin and the weak is hypercharge . The electroweak interaction is broken spontaneously by the Higgs mechanism . The neutral ponds mix and to and fro : ${\displaystyle \gamma }$ ${\displaystyle W^{\pm }}$ ${\displaystyle Z}$ ${\displaystyle W^{1,2,3}}$ ${\displaystyle B}$ ${\displaystyle gT_{3}}$ ${\displaystyle B}$ ${\displaystyle g'Y_{\text{W}}}$ ${\displaystyle T_{3}}$ ${\displaystyle Y_{\text{W}}}$ ${\displaystyle W^{3}}$ ${\displaystyle B}$ ${\displaystyle Z}$ ${\displaystyle \gamma }$

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

The transformation matrix between these states can be understood as a rotation through an angle in two dimensions - the electroweak mixing angle.

Relationship between the various coupling constants and the electroweak mixing angle ${\displaystyle e,g,g'}$ ${\displaystyle \theta _{\text{W}}}$

The result of this mixture is that the photon couples to fermions with a strength , which is the electrical charge (in units of the elementary charge ). The Z boson couples to fermions with a strength of . It follows that ${\displaystyle gQ\sin \theta _{\text{W}}}$ ${\displaystyle Q}$ ${\displaystyle e}$ ${\displaystyle \textstyle {\frac {g}{\cos \theta _{\text{W}}}}\left(T_{3}-Q\sin ^{2}\theta _{\text{W}}\right)}$

${\displaystyle e=g\sin \theta _{\text{W}}}$

have to be. In contrast, the charged W bosons, since they are not affected by this mixture, continue to couple with a strength . ${\displaystyle gT_{3}}$

The different couplings to the Higgs field also mean that the bosons do not have the same mass. The photon is massless:

${\displaystyle m_{\gamma }=0}$,

and that's a factor heavier than that ${\displaystyle Z}$ ${\displaystyle \cos ^{-1}\theta _{\text{W}}}$ ${\displaystyle W^{\pm }}$

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

The weakness that the weak interaction at low energies shows compared to the electromagnetic one is not explained by a small coupling constant, as previously assumed, but by the propagator term , in which the large mass of the W or Z bosons enters the denominator as a square, while the mass of the photon is zero.

Experimental determination

The electroweak mixing angle cannot be measured directly, but it can be determined indirectly in various ways. Since it occurs in different contexts, the independent measurement of the Weinberg angle is an important precision test for the validity of the Standard Model.

One possibility is, for example, to measure the masses of the W and Z bosons and to calculate the mixing angle from this. Scattering experiments, on the other hand, are more precise, which make use of the mixture of the Z bosons and the photon and which measure an asymmetry in the differential cross section .

Since the coupling constants run , the Weinberg angle is also dependent on the energy scale considered. Furthermore, due to higher order effects in quantum field perturbation theory, the Weinberg angle is dependent on the renormalization scheme used .

The current value for the effective Weinberg angle is according to the Particle Data Group in the MS-bar scheme

${\displaystyle \sin ^{2}\theta _{\text{W}}(m_{Z})=0{,}23122(4)}$

and according to CODATA in the on-shell scheme

${\displaystyle \sin ^{2}\theta _{\text{W}}=0{,}2223(21)}$.

Individual evidence

1. ↑ Particle Data Group: Particle Physics Booklet . November 15, 2018, p. 7 .
2. ↑ Peter Mohr, Barry Taylor and David Newell: CODATA recommended values ​​of the fundamental physical constants: 2010 . In: Rev. Mod. Phys. tape 84 , no. 4 , 2012, p. 1587 .

literature

• Mattew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 (English).
• The ALEPH, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group and the SLD Electroweak and Heavy Flavor Groups: Precision Electroweak Measurements on the Z Resonance . In: Phys. Rept. tape 427 , no. 5 - 6 , 2006, pp. 257 - 451 , doi : 10.1016 / j.physrep.2005.12.006 , arxiv : hep-ex / 0509008 (English).