A very natural way to generate the known neutrino masses is to minimally extend the SM including additional 2-spinors as RH neutrinos and at the same time extend the non-QCD electroweak SM gauge symmetry group to something like this:

The resulting model, initially proposed by Pati and Salam (** Phys. Rev. D.10. 275**) in 1973-1974. Mohapatra and Pati reviewed it in 1975, here

*. It is also reviewed in*

**Phys. Rev. D. 11. 2558***. This class of models were first proposed with the goal of seeking a spontaneous origin for parity (P) violations in weak interactions. CP and P are conserved at large energies but at low energies, however, the group breaks down spontaneouly at some scale . Any new physics correction to the SM would be of order*

**Unification and Supersymmetry: the frontiers of Quark-Lepton Physics. Springer-Verlag. N.Y.1986**

and where

If we choose the alternative , we obtain only small corrections, compatible with present known physics. We can explain in this case the small quantity of CP violation observed in current experiments and why the neutrino masses are so small, as we will see below a little bit.

The quarks and their fields, and the leptons and their fields , in the LR models transform as doublets under the group in a simple way. and . The gauge interactions are symmetric under left-handed and right-handed fermions. Thus, before spontaneous symmetry breaking, weak interactions, as any other interaction, would conserve parity symmetry and would become P-conserving at higher energies.

The breaking of the gauge symmetry is implemented by multiplets of LR symmetric Higgs fields. The concrete selection of these multiplets is NOT unique. It has been shown that in order to understand the smallness of the neutrino masses, it is convenient to choose respectively one doublet and two triplets in the following way:

The Yukawa couplings of these Higgs fields with the quarks and leptons are give by the lagrangian term

The gauge symmetry breaking in LR models happens in two steps:

1st. The is broken down to by the v.e.v. . It carries both and quantum numbers. It gives mass to charged and neutral RH gauge bosons, i.e.,

and

Furthermore, as consequence of the f-term in the lagrangian, above this stage of symmetry breaking also leads to a mass term for the right-handed neutrinos with order about .

2nd. As we break the SM symmetry by turning on the vev’s for the scalar fields

with

We give masses to the and bosons, as well as to quarks or leptons (). At the end of the process of spontaneous symmetry breaking (SSB), the two W bosons of the model will mix, the lowest physical mass eigenstate is identified as the observed W boson. Current experimental limits set the limit to . The LHC has also raised this bound the past year!

In the neutrino sector, the above Yukawa couplings after breaking by leads to the Dirac masses for the neutrino. The full process leads to the following mass matrix for the states in the general neutrino mass matrix

corresponding to the lighter and more massive neutrino states after the diagonalization procedure. In fact, the seesaw mechanism implies the eigenvalue

for the lowest mass, and the eigenvalue

for the (super)massive neutrino state. Several variants of the basic LR models include the option of having Dirac neutrinos at the expense of enlarging the particle content. The introduction of two new single fermions and a new set of carefully chosen Higgs bosons, allows us to write the mass matrix

This matrix leads to two different Dirac neutrinos, one heavy with mass and another lighter with mass . This light four component spinor has the correct weak interaction properties to be identified as the neutrino. A variant of this model can be constructed by addition of singlet quarks and leptons. We can arrange these new particles in order that the Dirac mass of the neutrino vanishes at tree level and/or arises at the one-loop level via boson mixing!

Left-Right symmetric(LR) models can be embedded in grand unification groups. The simplest GUT model that leads by successive stages of symmetry breaking to LR symmetric models at low energies is GUT-based models. An example of LR embedding GUT supersymmetric theory can be even discussed in the context of (super)string-inspired models.