Weak hypercharge

The weak Hyper charge ( W for engl. Weak , weak) is in the particle is a quantum number of elementary particles in connection with the electro-weak interaction . In contrast to the electrical charge , the weak hypercharge of a particle does not appear in everyday life. ${\displaystyle Y_{W}}$

The spontaneous symmetry breaking of the electroweak symmetry group by the Higgs mechanism relates the weak hypercharge to the third component of the weak isospin and the electrical charge. Because of this symmetry breaking, the weak hypercharge, like the weak isospin, is not a conserved quantity (see Fabri-Picasso theorem ).

background

The weak Hyper charge is the charge of the standard model occurring - symmetry group , the circuit group . The existence of this symmetry group means that the equations of motion for the wave function of the matter particles ( fermions ) must not change under the transformation with any function . The existence of bosonic gauge fields is always associated with such a symmetry . The calibration field belonging to the weak hypercharge is the physically unobservable boson. ${\displaystyle U(1)_{Y}}$ ${\displaystyle \psi (x)\to e^{\mathrm {i} Y\alpha (x)}\psi (x)}$ ${\displaystyle \alpha (x)}$ ${\displaystyle B}$

The Higgs mechanism breaks the symmetry of the standard model and the boson mixes with one of the gauge bosons of the symmetry to form the observable massive Z bosons and massless photons . The other two calibration bosons are not affected by this mixture. ${\displaystyle SU(2)_{L}\times U(1)_{Y}}$ ${\displaystyle B}$ ${\displaystyle SU(2)_{L}}$ ${\displaystyle Z^{0}}$ ${\displaystyle \gamma }$ ${\displaystyle SU(2)}$

values

The assignment of the weak hypercharges to the individual particles in the standard model results from the requirement for the standard model to be free from anomalies , so that the weak hypercharges are clearly defined except for a general normalization constant.

The weak hypercharge is

• for left-handed leptons: ${\displaystyle Y_{W}=-1}$
• for left-handed quarks: ${\displaystyle Y_{W}=+1/3}$
• for right-handed charged leptons: ${\displaystyle Y_{W}=-2}$
• for right-handed quarks of the up type: ${\displaystyle Y_{W}=+4/3}$
• for right-handed quarks of the down-type: . ${\displaystyle Y_{W}=-2/3}$

Right-handed uncharged leptons (neutrinos) do not exist according to the Standard Model; in some advanced theories they are Majorana fermions and carry as their own antiparticles . ${\displaystyle Y_{W}=0}$

Relation to the electric charge

The electric charge is defined by the weak hypercharge and the third component of the weak isospin by the relationship:

${\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{W}}$

This connection arises from the breaking of the electroweak symmetry group, after which this combination of charges continues to annihilate the quantum vacuum . Another normalization of the hypercharge selects this in such a way that then applies (and the above values ​​for the hypercharge must be halved). So in summary: ${\displaystyle Q=T_{3}+Y'_{W}}$

	Left handed	el. charge ${\displaystyle Q}$	black Isospin ${\displaystyle T_{z}}$	black Hyperldg. ${\displaystyle Y_{W}}$	Right handed	el. charge ${\displaystyle Q}$	black Isospin ${\displaystyle T_{z}}$	black Hyperldg. ${\displaystyle Y_{W}}$
Leptons	${\displaystyle \nu _{e},\nu _{\mu },\nu _{\tau }}$	0	+ ½	−1	-	-	-	-
	${\displaystyle e^{-},\mu ^{-},\tau ^{-}}$	−1	−½	−1	${\displaystyle e_{R}^{-},\mu _{R}^{-},\tau _{R}^{-}}$	−1	0	−2
Quarks	${\displaystyle u,c,t}$	+2/3	+ ½	+1/3	${\displaystyle u_{R},c_{R},t_{R}}$	+2/3	0	+4/3
	${\displaystyle d',s',b'}$	−1/3	−½	+1/3	${\displaystyle d_{R},s_{R},b_{R}}$	−1/3	0	−2/3

The down-like quarks in this table are in the eigen-states of the weak interaction. These are not the intrinsic mass states that can be measured by detectors , but a linear combination of them. The transition between the mass eigen basis and the eigen basis of the weak interaction is provided by the CKM matrix . ${\displaystyle d',s',b'}$ ${\displaystyle d,s,b}$

See also

• B - L , difference between baryons and lepton numbers, additional symmetry group of a large unified theory ${\displaystyle U(1)}$

Sources and footnotes

1. ↑ JA Minahan et al .: Comment on anomaly cancellation in the standard model . In: Phys. Rev. D. Volume 41 , no. 2 , 1990, p. 715-716 (English).
2. ↑ Mattew D. Schwartz: Quantum Field Theory and the standard model . 1st edition. Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 (English).