4.3 Electroweak theory 151gauge symmetry. The gauge coupling constants of theSU( 2 )andU( 1 )groups
should be taken as independent, and therefore theSU( 2 )×U( 1 )group cannot be
“unified” in a singleU( 2 )group.
4.3.1 Fermion content
In contrast to quarks, leptons are not involved in strong interactions, but both
leptonsandquarksparticipate in weak interactions. The three electrically charged
leptons, the electrone,the muonμ,and theτ-lepton,are partnered by neutrinos
νe,νμandντrespectively.
The neutrino masses are very small and we will first consider them as if they were
massless. The neutrino has spin 1/2 and, in principle, the normalized component
of spin in the direction of motion, called thehelicity, can take the value+1or− 1.
It has been found in experiments, however, that all neutrinos areleft-handed: they
have helicity−1, that is, their spins are always directed antiparallel to their veloc-
ity. All antineutrinos areright-handed. Hence, in weak interactions, the symmetry
between right- and left-handedness (parity-P) is broken and the corresponding the-
ory ischiral. Note that the notion of helicity is Lorentz-invariant only for massless
particles which move with the speed of light, otherwise one can always go to a
frame of reference moving faster than the particle and change its helicity.
The quarks and leptons are massive. However, because of the chiral nature of the
theory, mass terms cannot be introduced directly without spoiling gauge invariance.
In electroweak theory the masses arise as a result of interaction with a classical scalar
field. They will be considered later; until then, we will treatall fermionsas if they
were massless particles.
In weak interactions, the charged leptons can be converted into their correspond-
ing electrically neutral neutrinos. As a consequence the intermediate vector boson
must carry electric charge. Its antiparticle has the opposite charge and hence there
should be at least two gauge bosons responsible for weak interactions. The simplest
gauge group which can incorporate them is theSU( 2 )group.
Only the left-handed electroneLcan be converted into the left-handed neutrino
νe. They form anSU( 2 )doublet and transform as
ψeL≡(
νe
e)
L→UψeL, (4.37)whereUis a unitary 2×2 matrix with detU= 1 ,andνeandeLare Dirac spinors
describing the massless left-handed neutrino and electron. The right-handed elec-
tron is a singlet with respect to theSU( 2 )group:ψeR≡eR→ψeR.The concrete
form of Dirac spinors for chiral states depends on the Dirac matrix representa-
tion used. For instance, in the chiral representation the left-handed fermions are