LOG#122. Basic Neutrinology(VII).

The observed mass and mixing both in the neutrino and quark cases could be evidence for some interfamily hierarchy hinting that the lepton and quark sectors were, indeed, a result of the existence of a new quantum number related to “family”. We could name this family symmetry as U(1)_F. It was speculated by people like Froggatt long ago. The actual intrafamily hierarchy, i.e., the fact that m_u>>m_d in the quark sector, seem to require one of these symmetries to be anomalous.

A simple model with one family dependent anomalous U(1) beyond the SM was first proposed long ago to produce the given Yukawa coupling and their hierarchies, and the anomalies could be canceled by the Green-Schwarz mechanism which as by-product is able to fix the Weinberg angle as well. Several developments include the models inspired by the E_6\times E_8 GUT or the E_8\times E_8 heterotic superstring theory. The gauge structure of the model is that of the SM but enlarged by 3 abelian U(1) symmetries and their respective fields, sometimes denoted by X,Y^{1,2}. The first one is anomalous and family independent. Two of these fields, the non-anomalous, have specific dependencies on the 3 chiral families designed to reproduce the Yukawa hierarchies. There are right-handed neutrinos which “trigger” neutrino masses by some special types of seesaw mechanisms.

The 3 symmetries and their fields X,Y^{1,2} are usually spontaneously broken at some high energy scale M_X by stringy effects. It is assumed that 3 fields, \theta_i, with i=1,2,3, develop a non-null vev. These \theta_i fields are singlets under the SM gauge group but not under the abelian symmetries carried by X, Y^{1,2}. Thus, the Yukawa couplings appear as some effective operators after the U(1)_F spontaneous symmetry breaking. In the case of neutrinos, we have the mass lagrangian (at effective level):

\mathcal{L}_m\sim h_{ij}L_iH_uN_j^c\lambda^{q_i+n_j}+M_N\xi_{ij}N_i^cN_j^c\lambda^{n_i+n_j}

and where h_ {ij},\xi_{ij}\sim \mathcal{O}(1). The parameters \lambda determine the mass and mixing hierarchy with the aid of some simple relationships:

\lambda=\dfrac{\langle \theta\rangle}{M_X}\sim\sin\theta_c

and where \theta_c is the Cabibblo angle. The q_i,n_i are the U(1)_F charges assigned to the left handed leptons L and the right handed neutrinos N. These couplings generate the following mass matrices for neutrinos:

m_\nu^D=\mbox{diag}(\lambda^{q_1},\lambda^{q_2},\lambda^{q_3})\hat{h}\mbox{diag}(\lambda^{n_1},\lambda^{n_2},\lambda^{n_3})\langle H_u\rangle

M_\nu=\mbox{diag}(\lambda^{n_1},\lambda^{n_2},\lambda^{n_3})\hat{\xi}\mbox{diag}(\lambda^{n_1},\lambda^{n_2},\lambda^{n_3})M_N

From these matrices, the associated seesaw mechanism gives the formula for light neutrinos:

m_\nu\approx \dfrac{\langle H_u\rangle^2}{M_X}\mbox{diag}(\lambda^{q_1},\lambda^{q_2},\lambda^{q_3})\hat{h}\hat{\xi}^{-1}\hat{h}^T\mbox{diag}(\lambda^{q_1},\lambda^{q_2},\lambda^{q_3})

The neutrino mass mixing matrix depends only on the charges we assign to the LH neutrinos due to cancelation of RH neutrino charges and the seesaw mechanism. There is freedom in the assignment of the charges q_i. If the charges of the second and the third generation of leptos are equal (i.e., if q_2=q_3), then one is lead to a mass matrix with the following structure (or “texture”):

m_\nu\sim \begin{pmatrix}\lambda^6 & \lambda^3 & \lambda^3\\ \lambda^3 & a & b\\ \lambda^3 & b & c\end{pmatrix}

and where a,b,c\sim \mathcal{O}(1). This matrix can be diagonalized in a straightforward fashion by a large \nu_2-\nu_3 rotation. It is consistent (more or less), with a large \mu-\tau mixing. In this theory or model, the explanation of the large neutrino mixing angles is reduced to a theory of prefactors in front of powers of the parameters \lambda, related with the vev after the family group spontaneous symmetry breaking!

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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+

+f(L_LL_L\Delta_L+L_RL_R\Delta_R)+h.c.

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.