Weak Interactions

Although weak interactions can be described by a non–Abelian SU(2) 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

\nu_R=\frac{1}{2}(1+\gamma_5)\nu = 0

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

R\equiv e_R=\frac{1}{2}(1+\gamma_5)e.

The group that describes the symmetry of weak isospin is therefore called SU(2)_L, where L 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 SU(2)_L \otimes U(1)_Y symmetry group where Y, the weak hypercharge, is the generator for the group and it is obtained from the Gell-Mann — Nishijima relation Q=\frac{Y}{2}+I_3.
The request of local gauge invariance leads to the introduction of four vector bosons.

Weak and electromagnetic interactions are considered unified in the single U(1)_Y\otimes SU(2)_L symmetry group where Y, the weak hypercharge, is the generator of the group that is given by the Gell-Mann–Nishijima relation between electrical charge and isospin Q=\frac{Y}{2}+I_3.
The subscript L is used to denote left-handed spinors recording the vector-axial nature of the charged currents and the subscript C refers to the charge of strong interactions given by the colour.

The gauge fields associated with SU(2)_L are W^1, W^2 and W^0, while B is associated with the U(1)_Y group. Charged bosons W^\pm come from

 W^\pm_\mu=\frac{W_\mu^1\mp iW_\mu^2}{\sqrt 2}.

The photon and the neutral Z^0 boson can be obtained as a combination of the neutral fields

W^0_\mu= Z_\mu cos\theta_W + A_\mu sin\theta_W B^0_\mu=-Z_\mu sin\theta_W + A_\mu cos\theta_W 

where the rotation angle \theta_W takes the name of electroweak mixing angleA_\mu is the photon field, Z_\mu is the field associated to the Z^0 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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