Although weak interactions can be described by a non–Abelian group too, the phenomenology of Nature complicates the form of the corresponding Lagrangian.

The *CP* violation can be formalized assuming that weak interactions should act only on left–chirality states. Left–handed fermions therefore form a weak isospin doublet

where the left–handed states are

The electron neutrino, known to be nearly massless, is idealized to be exactly massless. As a consequence its right–handed state

would not exist and this leads to have only one right–handed fermion, which constitute the weak–isospin singlet

The group that describes the symmetry of weak isospin is therefore called , where stands for “left”. Anyway, neutral currents can couple even right–handed fermions, although in a different way.

The construction of a model which takes account of the peculiarities of the weak interaction was achieved by Glashow, Weinberg and Salam at the end of the 1960s. Weak and electromagnetic interactions were unified in the single symmetry group where , the *weak hypercharge*, is the generator for the group and it is obtained from the Gell-Mann — Nishijima relation .

The request of local gauge invariance leads to the introduction of four vector bosons.

Weak and electromagnetic interactions are considered unified in the single symmetry group where , the weak hypercharge, is the generator of the group that is given by the Gell-Mann–Nishijima relation between electrical charge and isospin .

The subscript is used to denote left-handed spinors recording the vector-axial nature of the charged currents and the subscript refers to the charge of strong interactions given by the *colour*.

The gauge fields associated with are , and , while is associated with the group. Charged bosons come from

The photon and the neutral boson can be obtained as a combination of the neutral fields

where the rotation angle takes the name of *electroweak mixing angle*, is the photon field, is the field associated to the boson.

###### References and further readings:

###### C. Quigg, *Gauge Theories of the Strong, Weak and Electromagnetic Interactions*, (Addison-Wesley Publishing Company), 1997.

###### F.Halzen and A.D. Martin, *Quarks and Leptons: an Introductory Course in Modern Particle Physics*, (John Wiley & Sons), 1984.

###### I. Aitchison, A. Hey, *Gauge theories in particle physics, a practical introduction – Volume. 1 – From Relativistic Quantum Mechanics to QED*, (IOP – Institute of Physics Publishing), 2003.

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