Weak isospin

In elementary particle physics , the weak isospin , also known as a weak charge , is a vector- valued quantity that is related to the weak interaction (just as the strong isospin is related to the strong interaction ). The amount of weak isospin is a quantum number and takes for fermions left-handed chirality the value ¹ / ₂ as well as right-handed chirality fermions for the value 0. The three components of the weak isospin are the charges of a - symmetry group that occurs in the Standard Model . ${\displaystyle T}$ ${\displaystyle SU(2)}$

As with the spin with its related name , the three components of the weak isospin cannot be measured at the same time , which is why, conventionally, in addition to the weak isospin, its projection onto the -axis or is specified (also referred to as the “third component”). This can assume the values . Since the magnitude of the weak isospin is always either ¹ ⁄ ₂ or 0, is . ${\displaystyle z}$ ${\displaystyle T_{z}}$ ${\displaystyle T_{3}}$ ${\displaystyle -T,-T+1,\dots ,T-1,T}$ ${\displaystyle T_{z}=\pm T}$

Details

For left-handed fermions

Due to the non-zero weak isospin, the left-handed fermions form weak isospin doublets , the entries of which are distinguished by the third component (the exact assignment of the particles to the third component of the weak isospin is convention; a general change in all signs would be physically irrelevant):

left handed fermions			black Isospin ${\displaystyle T_{z}}$
Leptons	Neutrinos	${\displaystyle \nu _{e},\nu _{\mu },\nu _{\tau }}$	+1/2
	charged leptons	${\displaystyle e^{-},\mu ^{-},\tau ^{-}}$	−1/2
Quarks	up-like	${\displaystyle u,c,t}$	+1/2
	down-like	${\displaystyle d',s',b'}$	−1/2

Here d ', s' and b' are the eigenstates of the quarks with regard to the weak interaction, which are linked by the CKM matrix with the eigenstates of the strong interaction .

The doublets for the three generations can be summarized as follows, with the index symbolizing left-handedness: ${\displaystyle L}$

${\displaystyle {\begin{alignedat}{2}{\begin{pmatrix}\nu _{e}\\e^{-}\end{pmatrix}}_{L}&,{\begin{pmatrix}\nu _{\mu }\\\mu ^{-}\end{pmatrix}}_{L}&&,{\begin{pmatrix}\nu _{\tau }\\\tau ^{-}\end{pmatrix}}_{L}\\{\begin{pmatrix}u\\d'\end{pmatrix}}_{L}&,{\begin{pmatrix}c\\s'\end{pmatrix}}_{L}&&,{\begin{pmatrix}t\\b'\end{pmatrix}}_{L}\end{alignedat}}}$

The charged currents of the weak interaction change the third component of the weak isospin, so that left-handed neutrinos and charged leptons ("upper" row of doublets) or left-handed quarks of the up and down types ("lower" row of doublets) each other can convert. In contrast, a neutral current of the weak interaction does not change the third component of the weak isospin.

For right-handed fermions

Make according to the right-handed fermion weak-isospin singlets :

${\displaystyle {\begin{alignedat}{2}e_{R}^{-}&,\mu _{R}^{-}&&,\tau _{R}^{-}\\u_{R}&,d_{R}&&,c_{R}\\s_{R}&,t_{R}&&,b_{R}\end{alignedat}}}$

Right-handed neutrinos are not mentioned in this list because their existence is debatable; in the standard model they do not take part in any interaction and are sterile particles (see sterile neutrino ).

Weak isospin in the standard model

In the standard model, weak interaction and electromagnetism are combined to form the electroweak interaction . The symmetry of the standard model is spontaneously broken and it is hidden. The result of the symmetry breaking shows that the third component of the weak isospin and the weak hypercharge are related to the electrical charge : ${\displaystyle SU(2)}$ ${\displaystyle Y_{W}}$ ${\displaystyle Q}$

${\displaystyle Q=T_{z}+{\tfrac {1}{2}}Y_{W}}$

Another convention normalizes the weak hypercharge differently:

${\displaystyle Q=T_{z}+Y'_{W}}$.

literature

• Bogdan Povh et al .: Particles and Cores . Springer, Berlin, Heidelberg 2006, ISBN 978-3-540-36685-0

Individual evidence

1. ↑ Jörn Bleck-Neuhaus: Elementary Particles. From the atoms to the standard model to the Higgs boson . 2nd Edition. Springer, Berlin Heidelberg 2013, ISBN 978-3-642-32578-6 .