LOG#119. Basic Neutrinology(IV).

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:

    \[ G(L,R)=SU(2)_L\times SU(2)_R\times U(1)_{B-L}\times P\]

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 Phys. Rev. D. 11. 2558. It is also reviewed in Unification and Supersymmetry: the frontiers of Quark-Lepton Physics. Springer-Verlag. N.Y.1986. 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 G(L,R) breaks down spontaneouly at some scale M_R. Any new physics correction to the SM would be of order

    \[ (M_L/M_R)^2\]

and where M\sim m_W

If we choose the alternative M_R>>M_L, 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 Q and their fields, and the leptons and their fields L, in the LR models transform as doublets under the group SU(2)_{L,R} in a simple way. (Q_L, L_L)\sim (2,1) and (Q_R,L_R)\sim (1,2). 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:

    \[ \phi\sim (2,2,0)\]

    \[ \Delta_L\sim (3,1,2)\]

    \[ \Delta_R\sim (1,3,2)\]

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

    \[ \mathcal{L}_Y=h_1\bar{L}_L\phi L_R+h_2\bar{L}_L\bar{\phi}L_R+h_1'\bar{Q}_L\phi Q_R+h'_2\bar{Q}_L\bar{\phi} Q_R+\]


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

1st. The SU(2)_R\times U(1)_{B-L} is broken down to U(1)_Y by the v.e.v. \langle \Delta_R^0\rangle=v_R\neq 0. It carries both SU(2)_R and U(1)_{B-L} quantum numbers. It gives mass to charged and neutral RH gauge bosons, i.e.,

M_{W_R}=gv_R and M_{Z'}=\sqrt{2}gv_R/\sqrt{1-\tan^2\theta_W}

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 \sim fv_R.

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

\langle \phi \rangle=\mbox{diag}(v_\kappa,v'_\kappa) with

v_R>>v'_\kappa>> v_\kappa

We give masses to the W_L and Z bosons, as well as to quarks or leptons (m_e\sim hv_\kappa). 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 M_{W_R}>550GeV. The LHC has also raised this bound the past year!

In the neutrino sector, the above Yukawa  couplings after SU(2)_L breaking by \langle \phi\rangle\neq 0 leads to the Dirac masses for the neutrino. The full process leads to the following mass matrix for the \nu, N states in the general neutrino mass matrix

\mathbb{M}_{\nu,N}=\begin{pmatrix}\sim 0 & hv_\kappa\\ hv_\kappa & fv_R\end{pmatrix}

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

m_{\nu_e}\approx (hv_\kappa)^2/fv_R

for the lowest mass, and the eigenvalue

m_N\approx fv_R

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 4\times 4 mass matrix

\mathbb{M}=\begin{pmatrix} 0 & m_D & 0 & 0\\ m_D & 0 & 0 & fv_R\\ 0 & 0 & 0 & \mu\\ 0 & fv_R & \mu & 0\end{pmatrix}

This matrix leads to two different Dirac neutrinos, one heavy with mass m_N\sim fv_R and another lighter with mass m_\nu\sim m_D\mu/fv_R. 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 W_L-W_R 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 SO(10) GUT-based models. An example of LR embedding GUT supersymmetric theory can be even discussed in the context of (super)string-inspired models.

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