LOG#120. Basic Neutrinology(V).

Supersymmetry (SUSY) is one of the most discussed ideas in theoretical physics. I am not discussed its details here (yet, in this blog). However, in this thread, some general features are worth to be told about it. SUSY model generally include a symmetry called R-parity, and its breaking provide an interesting example of how we can generate neutrino masses WITHOUT using a right-handed neutrino at all. The price is simple: we have to add new particles and then we enlarge the Higgs sector. Of course, from a pure phenomenological point, the issue is to discover SUSY! On the theoretical aside, we can discuss any idea that experiments do not exclude. Today, after the last LHC run at 8TeV, we have not found SUSY particles, so the lower bounds of supersymmetric particles have been increased. Which path will Nature follow? SUSY, LR models -via GUTs or some preonic substructure, or something we can not even imagine right now? Only experiment will decide in the end…

In fact, in a generic SUSY model, dut to the Higgs and lepton doublet superfields, we have the same SU(3)_c\times SU(2)_L\times U(1)_Y quantum numbers. We also have in the so-called “superpotential” terms, bilinear or trilinear pieces in the superfields that violate the (global) baryon and lepton number explicitly. Thus, they lead to mas terms for the neutrino but also to proton decays with unacceptable high rates (below the actual lower limit of the proton lifetime, about 10^{33}  years!). To protect the proton experimental lifetime, we have to introduce BY HAND a new symmetry avoiding the terms that give that “too high” proton decay rate. In SUSY models, this new symmetry is generally played by the R-symmetry I mentioned above, and it is generally introduced in most of the simplest models including SUSY, like the Minimal Supersymmetric Standard Model (MSSM). A general SUSY superpotential can be written in this framework as

(1) \mathcal{W}'=\lambda{ijk}L_iL_jE_l^c+\lambda'_{ijk}L_iQ_jD_k^c+\lambda''_{ijk}D_i^cD_j^cU_k^c+\epsilon_iL_iH_2

A less radical solution is to allow for the existence in the superpotential of a bilinear term with structure \epsilon_3L_3H_2. This is the simplest way to realize the idea of generating the neutrino masses without spoiling the current limits of proton decay/lifetime. The bilinear violation of R-parity implied by the \epsilon_3 term leads by a minimization condition to a non-zero vacuum expectation value or vev, v_3. In such a model, the \tau neutrino acquire a mass due to the mixing between neutrinos and the neutralinos.The \nu_e, v_\mu neutrinos remain massless in this toy model, and it is supposed that they get masses from the scalar loop corrections. The model is phenomenologically equivalent to a “3 Higgs doublet” model where one of these doublets (the sneutrino) carry a lepton number which is broken spontaneously. The mass matrix for the neutralino-neutrino secto, in a “5×5” matrix display, is:

(2) \mathbb{M}=\begin{pmatrix}G_{2x2} & Q_{ab}^1 & Q_{ab}^2 & Q_{ab}^3\\ Q_{ab}^{1T} & 0 & -\mu & 0\\ Q_{ab}^{2T} & -\mu & 0 & \epsilon_3\\ Q_{ab}^{3T} & 0 & \epsilon_3 & 0\end{pmatrix}

and where the matrix G_{2x2}=\mbox{diag}(M_1, M_2) corresponds to the two “gauginos”. The matrix Q_{ab} is a 2×3 matrix and it contains the vevs of the two higgses H_1,H_2 plus the sneutrino, i.e., v_u, v_d, v_3 respectively. The remaining two rows are the Higgsinos and the tau neutrino. It is necessary to remember that gauginos and Higgsinos are the supersymmetric fermionic partners of the gauge fields and the Higgs fields, respectively.

I should explain a little more the supersymmetric terminology. The neutralino is a hypothetical particle predicted by supersymmetry. There are some neutralinos that are fermions and are electrically neutral, the lightest of which is typically stable. They can be seen as mixtures between binos and winos (the sparticles associated to the b quark and the W boson) and they are generally Majorana particles. Because these particles only interact with the weak vector bosons, they are not directly produced at hadron colliders in copious numbers. They primarily appear as particles in cascade decays of heavier particles (decays that happen in multiple steps) usually originating from colored  supersymmetric particles such as squarks or gluinos. In R-parity conserving models, the lightest neutralino is stable and all supersymmetric cascade-decays end up decaying into this particle which leaves the detector unseen and its existence can only be inferred by looking for unbalanced momentum (missing transverse energy) in a detector. As a heavy, stable particle, the lightest neutralino is an excellent candidate to comprise the universe’s cold dark matter. In many models the lightest neutralino can be produced thermally in the hot early Universe and leave approximately the right relic abundance to account for the observed dark matter. A lightest neutralino of roughly 10-10^4 GeV is the leading weakly interacting massive particle (WIMP) dark matter candidate.

Neutralino dark matter could be observed experimentally in nature either indirectly or directly. In the former case, gamma ray and neutrino telescopes look for evidence of neutralino annihilation in regions of high dark matter density such as the galactic or solar centre. In the latter case, special purpose experiments such as the (now running) Cryogenic Dark Matter Search (CDMS)  seek to detect the rare impacts of WIMPs in terrestrial detectors. These experiments have begun to probe interesting supersymmetric parameter space, excluding some models for neutralino dark matter, and upgraded experiments with greater sensitivity are under development.

If we return to the matrix (2) above, we observe that when we diagonalize it, a “seesaw”-like mechanism is again at mork. There, the role of M_D, M_R can be easily recognized. The \nu_\tau mass is provided by

m_{\nu_\tau}\propto \dfrac{(v_3')^2}{M}

where v_3'\equiv \epsilon_3v_d+\mu v_3 and M is the largest gaugino mass. However, an arbitrary SUSY model produces (unless M is “large” enough) still too large tau neutrino masses! To get a realistic and small (1777 GeV is “small”) tau neutrino mass, we have to assume some kind of “universality” between the “soft SUSY breaking” terms at the GUT scale. This solution is not “natural” but it does the work. In this case, the tau neutrino mass is predicted to be tiny due to cancellations between the two terms which makes negligible the vev v_3'. Thus, (2) can be also written as follows

(3) \begin{pmatrix}M_1 & 0 & -\frac{1}{2}g'v_d & \frac{1}{2}g'v_u & -\frac{1}{2}g'v_3\\ 0 & M_2 & \frac{1}{2}gv_d & -\frac{1}{2}gv_u & \frac{1}{2}gv_3\\ -\frac{1}{2}g'v_d & \frac{1}{2}gv_d & 0 & -\mu & 0\\ \frac{1}{2}g'v_u& -\frac{1}{2}gv_u& -\mu & 0 & \epsilon_3\\ -\frac{1}{2}g'v_3 & \frac{1}{2}gv_3 & 0 & \epsilon_3 & 0\end{pmatrix}

We can study now the elementary properties of neutrinos in some elementary superstring inspired models. In some of these models, the effective theory implies a supersymmetric (exceptional group) E_6 GUT with matter fields belong to the 27 dimensional representation of the exceptional group E_6 plus additional singlet fields. The model contains additional neutral leptons in each generation and the neutral E_6 singlets, the gauginos and the Higgsinos. As the previous model, but with a larger number of them, every neutral particle can “mix”, making the undestanding of the neutrino masses quite hard if no additional simplifications or assumptions are done into the theory. In fact, several of these mechanisms have been proposed in the literature to understand the neutrino masses. For instance, a huge neutral mixing mass matris is reduced drastically down to a “3×3” neutrino mass matrix result if we mix \nu and \nu^c with an additional neutral field T whose nature depends on the particular “model building” and “mechanism” we use. In some basis (\nu, \nu^c,T), the mass matrix can be rewritten

(4) M=\begin{pmatrix}0 & m_D & 0\\ m_D & 0 & \lambda_2v_R\\ 0 & \lambda_2v_R & \mu\end{pmatrix}

and where the \mu energy scale is (likely) close to zero. We distinguish two important cases:

1st. R-parity violation.

2nd. R-parity conservation and a “mixing” with the singlet.

In both cases, the sneutrinos, superpartners of \nu^c are assumed to acquire a v.e.v. with energy size v_R. In the first case, the T field corresponds to a gaugino with a Majorana mass \mu than can be produced at two-loops! Usually \mu\approx 100GeV, and if we assume \lambda v_R\approx 1 TeV, then additional dangerous mixing wiht the Higgsinos can be “neglected” and we are lead to a neutrino mass about m_\nu\sim 0.1eV, in agreement with current bounds. The important conclusion here is that we have obtained the smallness of the neutrino mass without any fine tuning of the parameters! Of course, this is quite subjective, but there is no doubt that this class of arguments are compelling to some SUSY defenders!

In the second case, the field T corresponds to one of the E_6 singlets. We have to rely on the symmetries that may arise in superstring theory on specific Calabi-Yau spaces to restric the Yukawa couplings till “reasonable” values. If we have \mu=0 in the matrix (4) above, we deduce that a massless neutrino and a massive Dirac neutrino can be generated from this structure. If we include a possible Majorana mass term of the sfermion at a scale \mu\approx 100GeV, we get similar values of the neutrino mass as the previous case.

Final remark: mass matrices, as we have studied here, have been proposed without embedding in a supersymmetric or any other deeper theoretical frameworks. In that case, small tree level neutrino masses can be obtained without the use of large scales. That is, the structure of the neutrino mass matrix is quite “model independent” (as the one in the CKM quark mixing) if we “measure it”. Models reducing to the neutrino or quark mass mixing matrices can be obtained with the use of large energy scales OR adding new (likely “dark”) particle species to the SM (not necessarily at very high energy scales!).

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


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.