# LOG#124. Basic Neutrinology(IX).

In supersymmetric LR models, inflation, baryogenesis (and/or leptogenesis) and neutrino oscillations can be closely related to each other. Baryosynthesis in GUTs is, in general, inconsistent with inflationary scenarios. The exponential expansion during the inflationary phase will wash out any baryon asymmetry generated previously by any GUT scale in your theory. One argument against this feature is the next idea: you can indeed generate the baryon or lepton asymmetry during the process of reheating at the end of inflation. This is a quite non-trivial mechanism. In this case, the physics of the “fundamental” scalar field that drives inflation, the so-called inflaton, would have to violate the CP symmetry, just as we know that weak interactions do! The challenge of any baryosynthesis model is to predict the observed asymmetry. It is generally written as a baryon to photon (in fact, a number of entropy) ratio. Tha baryon asymmetry is defined as

$\dfrac{n_B}{s}\equiv \dfrac{(n_b-n_{\bar{b}})}{s}$

At present time, there is only matter and only a very tiny (if any) amount of antimatter, and then $n_{\bar{b}}\sim 0$. The entropy density s is completely dominated by the contribution of relativistic particles so it is proportional to the photon number density. This number is calculated from CMBR measurements, and it shows to be about $s=7.05n_\gamma$. Thus,

$\dfrac{n_B}{s}\propto \dfrac{n_b}{n_\gamma}$

From BBN, we know that

$\dfrac{n_B}{n_\gamma}=(5.1\pm 0.3)\cdot 10^{-10}$

and

$\dfrac{n_B}{s}=(7.2\pm 0.4)\cdot 10^{-11}$

This value allows to obtain the observed lepton asymmetry ratio with analogue reasoning.

By the other hand, it has been shown that the “hybrid inflation” scenarios can be successfully realized in certain SUSY LR models with gauge groups

$G_{SUSY}\supset G_{PS}=SU(4)_c\times SU(2)_L\times SU(2)_R$

after SUSY symmetry breaking. This group is sometimes called the Pati-Salam group. The inflaton sector of this model is formed by two complex scalar fields $H,\theta$. At the end of the inflation do oscillate close to the SUSY minimum and respectively, they decay into a part of right-handed sneutrinos $\nu_i^c$ and neutrinos. Moreover, a primordial lepton asymmetry is generated by the decay of the superfield $\nu_2^c$ emerging as the decay product of the inflaton field. The superfield $\nu_2^c$ also decays into electroweak Higgs particles and (anti)lepton superfields. This lepton asymmetry is partially converted into baryon asymmetry by some non-perturbative sphalerons!

Remark: (Sphalerons). From the wikipedia entry we read that a sphaleron (Greek: σφαλερός “weak, dangerous”) is a static (time independent) solution to the electroweak field equations of the SM of particle physics, and it is involved in processes that violate baryon and lepton number.Such processes cannot be represented by Feynman graphs, and are therefore called non-perturbative effects in the electroweak theory (untested prediction right now). Geometrically, a sphaleron is simply a saddle point of the electroweak potential energy (in the infinite dimensional field space), much like the saddle point  of the surface $z(x,y)=x^2-y^2$ in three dimensional analytic geometry. In the standard model, processes violating baryon number convert three baryons to three antileptons, and related processes. This violates conservation of baryon number and lepton number, but the difference B-L is conserved. In fact, a sphaleron may convert baryons to anti-leptons and anti-baryons to leptons, and hence a quark may be converted to 2 anti-quarks and an anti-lepton, and an anti-quark may be converted to 2 quarks and a lepton. A sphaleron is similar to the midpoint($\tau=0$) of the instanton , so it is non-perturbative . This means that under normal conditions sphalerons are unobservably rare. However, they would have been more common at the higher temperatures of the early Universe.

The resulting lepton asymmetry can be written as a function of a number of parameters among them the neutrino masses and the mixing angles, and finally, this result can be compared with the observational constraints above in baryon asymmetry. However, this topic is highly non-trivial. It is not trivial that solutions satisfying the constraints above and other physical requirements can be found with natural values of the model parameters. In particular, it is shown that the value of the neutrino masses and the neutrino mixing angles which predict sensible values for the baryon or lepton asymmetry turn out to be also consistent with values require to solve the solar neutrino problem we have mentioned in this thread.

# 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

$(M_L/M_R)^2$

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.

# LOG#118. Basic Neutrinology(III).

## Mass terms

Phenomenologically, lagrangian mass terms can be understood as terms describing “transitions” between right (R) and left (L) handed states in the electroweak sector. For a given minimal, Lorentz invariant set of 4 fields ($\psi_L,\psi_R, \psi^c_L,\psi_R^c$), we have the components of a generic Dirac spinor. Thus, the most general mass part of a (likely extended) electroweak massive lagrangian can be written as follows:

$\mathcal{L}_m=m_D\bar{\psi}_L\psi_R+\dfrac{1}{2}m_T\left(\bar{\psi^c_L}\psi_L\right)+\dfrac{1}{2}m_S\left(\bar{\psi^c_R}\psi_R\right)+h.c.$

In terms of a “new” Majorana (real) field with $\nu^c=\nu$ and $N^c=N$, we have

$\nu=\dfrac{1}{\sqrt{2}}(\psi_L+\psi^c_L)$

$N=\dfrac{1}{\sqrt{2}}(\psi_R+\psi^c_R)$

and then, the massive lagrangian becomes

$\mathcal{L}_m=\begin{pmatrix}\bar{\nu} & \bar{N}\end{pmatrix}\mathbb{M}_{\nu,N}\begin{pmatrix}\nu\\ N\end{pmatrix}$

where the neutrino mass matrix is defined to be

$\mathbb{M}_{\nu,N}=\begin{pmatrix}m_T & m_D\\ m_D & m_S\end{pmatrix}$

We can diagonalize this mass matrix and then we will find the physical particle content! It is given (in general) by two Majorana mass eigenstates: the inclusion of the Majorana mass splits the 4 degenerate states of the Dirac field into two non-degenerate Majorana pairs. If we assume that the states $\nu, N$ are respectively “active” (i.e., they belong to some weak doublets) and sterile (weak singlets), then the terms corresponding to the Majorana masses $m_T, m_S$ transform as weak triplets and singlets respectively. While the term corresponding to $m_D$ is  an standard, weak singlet in most cases, Dirac mass term, its pressence shows to be essential in the next discussion. Indeed, this simple example can be easily generalized to three or more families, in which case the masses beocme matrices themselves. The complete full flavor mixing comes from any two different parts: the diagonalization of the charged lepton Yukawa couplings and that of the neutrino masses! Most of beyond Standard Model theories (specially those coming from GUTs) produce CKM-like leptonic mixing and this mixing is generally “arbitrary” with parameters only to be determined by the experiment. Only when you have an additional gauge symmetry (or some extra discrete symmetry), you can guess some of the mixing parameters from first principles. Therefore, the prediction of the neutrino oscillation/mixing parameters, as for the quark hierarchies and mixing, need further theoretical assumptions NOT included in the Standard Model. For instance, we could require that the $\nu_\mu-\nu_\tau$ mixing were “maximal” or to impose some “permutation symmetry” and derive the neutrino oscillation parameters from “tribimaximal” or “trimaximal” mixing. However, currently, the symmetry behind the neutrino mass matrix or the quark mixing matrix (the CKM mass matrix) are completely unknown. We can feel and “smell” there are some patterns there (something that suggests a “new” approximate broken symmetry related to flavor) but there is no current accepted working model for the neutrino mass matrix (or its quark analogue, the CKM mass matrix).

## The seesaw

When we diagonalize the above neutrino mass matrix, we can analyze different “limit” cases. In the case of a purely Dirac mass term, i.e., whenver $m_T=m_S=0$, then the $\nu, N$ states are degenerate with mass $m_D$ and a four component Dirac field can be “recovered” as $\nu'=\nu+N$, modulo some constant prefactor. It can be seen that, although violating individual lepton numbers, the Dirac mass term allows a conserve dlepton number $L=L_\nu+L_N$. This case in which the triplet and scalar masses are “tiny” or, equivalently, the case in which their Majorana mass “separation” is very small is sometimes called “pseudo-Dirac” case. In fact, it produces some interesting models both in Cosmology and particle physics. Inded, it could be possible that the 3 neutrino flavors we do know today were, in fact, neutrino (almost degenerated) triplets, i.e., every neutrino flavor could be formed by 3 very close Majorana states that we can not “resolve” using current technology.

In the general case, pure Majorana mass transition terms ($m_S, m_T$) arise in the lagrangian. Therefore, particle-antiparticle transitions violating the total lepton number by two units do appear ($\Delta L=\pm 2$). They can be understood as the creation or annihilation of two neutrinos, and thus, they allow the possibility of the existence of neutrinoless double beta decays! That is, only when the neutrino is a Majorana particle, the channel in which the total lepton number is violated opens.

When every mass term is allowed, there is an interesting case commonly referred as “the seesaw” limit. In this limit, taking the triplet mass to be zero and the singlet mass to be “huge” or “superheavy”, we deduce that

$m_T\sim 1/m_S\sim 0$ with $m_D< (the “seesaw” limit).

In this seesaw limit, the neutrino mass matrix can be diagonalized and it provides two eigenvalues:

$m_1\sim \dfrac{m_D^2}{m_S}<

$m_2\sim m_S$

Thus, the seesaw mechanism provide a way in which we obtain two VERY different mass eigenstates, i.e., two single particle states separated by a huge mass hierarchy! There is one (super)heavy neutrino (generally speaking, it corresponds to the right-handed neutrino) and a much lighter neutrino state, one that can be made relatively much lighter than a normal Dirac fermion mass. One fo the neutrino mass is “suppressed” and balanced up (hence the name “seesaw”) by the (super)heavy species. The seesaw mechanism is a “natural” way of generating two different (often VERY separated) mass scales!

The theory of the seesaw mechanism is very rich. I will not discuss its full potential here. There are 3 main types of seesaw mechanisms (generally named as type I, type II and type III) and some other less frequent variants and subvariants…It is an advanced topic for a whole future thread! 😉 However, I will draw you the 3 main Feynman graphs involved in these 3 main types of seesaw mechanisms:

## GUTs and neutrino mass models

Any fully satisfactory model that generates neutrino masses must contain a natural mechanism which allows us to explain their samll value, relative to that of their charged partners. Given the latest experimental hints and results, it would also be possible that it will include any comprehensive explanation for light sterile neutrinos and large, nearl maximal, mixing. This last idea is due to some “anomalies” coming from some neutrino experiments (specially those coming from reactors and the celebrated LSND experiment).

Different models can be distinguished according to the new particle content and spectrum, or according to the energy scale hierarchy they produce. With an extended particle content, different options open: if we want to brak the lepton number ant to generate neutrino masses without introducing new (unobserved) fermions in the SM, we must do it by adding to the SM Higgs sector fields carrying lepton numbers. Thus, one can arrange them to break the lepton number explicitly or spontaneously through interactions with these fields. If you want, this is another reason why the Higgs field matters: it allows to introduce fields carrying lepton numbers without adding any extra fermion field! Likely, the most straightforward approach to generate neutrino masses is to introduce for each neutrino an additional weak neutral single (that can be identified with the right-handed neutrino we can not observed due to be “very massive” and/or uncharged under the SM gauge group). This last fact strongly favors seesaw-like models!

For instance, the above features happen in the framework of LR (Left-Right) symmetric models in Grand Unified Theories (GUTs). There, the origin of the SM parity violation (explicit in the electroweak and weak sectors) is due to the spontaneous symmetry breaking of a baryon-lepton symmetry, and it yields a $B-L$ quantum number conservation/violation up to a degree that depends on the particular model. Thus, in $SO(10)$ GUT, the Majorana neutral particle N enters in a natural way in order to complete the matter multiplet. Therefore, N should be a $SU(3)\times SU(2)\times U(1)$ singlet, as we wished it to be.

If we use the energy scale as a guide where the new physics have relevant effects, unification (e.g., think about the previous SO(10) example) and the weak scale approach (radiative models and their effective theories) are usually distinguished and preferred form a pure QFT approach.

Despite the fact that the explanation of the known neutrino anomalies (the solar neutrino problem the first, but also the atmospheric neutrino flux and the reactor anomalies/neutrino beam anomalies) do not need or require the existence of an additional extra light/heavy sterile neutrino, some authors claim that they could exist after all. If every Marojana mass term is “small enough”, then active neutrinos can oscillate or mix into sterile (likely right-handed) fields/states. Light sterile neutrinos can appear in particularly special see-saw mechanisms if additional assumptions are considered (there, some models called “singular seesaw” models do exist as well). with some inevitable amount of “fine tuning”. The alternative to “fine tuning” would be seesaw-like suppression for sterile neutrinos involving new unknown (likely ultraweak or “dark”) interactions, i.e., family symmetries resulting in substantial field additions to the SM (some string theory models also suggest this possibility).

There is also weak scale models, i.e., radiative  generated mass models where the neutrino masses are zero at tree level and they constitute a very different type of models: they explain the smallness of the neutrino masses a priori for both active and sterile neutrinos. Loop corrections induce neutrino mass terms in these models. Thus, different mass scales are generated naturally by the different number of loops involved in the generation of each term. The actual implementation requires, however, the ad hoc (a posteriori) introduction of new Higgs particles with non-standard electroweak quantum numbers and lepton number violating couplings. This is the price we pay in an alternative approach.

The origin of the different Dirac and Majorana mass terms $m_S,m_T, m_D$ appearing in the neutrino (seesaw like) neutrino mass matrix is usually understood by a dynamical mechanism where at some energy scale it happens “naturally” and/or where some symmetry principle is spontaneously broke and invoked. Firstly, we face with the Dirac mass term. In one special case, $\nu_L$ and $\nu_R$ are SU(2) doublets and singlets respectively. The mass term describes a $\Delta I=1/2$ transition and it is generate from the SU(2) breaking via a Yukawa coupling:

$\mathcal{L}_{\mathcal{Y}}=h_i\begin{pmatrix} \bar{\nu}_i & \bar{l}_i\end{pmatrix}\begin{pmatrix}\phi^0\\ \phi^-\end{pmatrix}N_{R_i}+h.c.$

Here, $\phi^0, \phi^-$ are the components of certain Higgs doublet. The coefficients $h_i$ are the Yukawa couplings. After symmetry breaking, $m_D=h_iv/2$, where $v$ is the vacuum expectation value of the Higgs doublet. A Dirac mass term is qualitatively just like any other fermion mass, but that leads to the question of why it is so small in comparison with the rest of fermion masses: one would require $h(\nu_e)<10^{-10}$ in order to have $m(\nu_e)<10eV$. In other words, $h(\nu_e)/h(e)\sim 10^{-5}$ while for the hadronic sector we have $h(up)/h(down)\sim \mathcal{O}(1)$. In principle, it could be that there is no reason beyond the fine tuning of the Yukawa couplings (via Higgs vacuum expectation values to different fields) but, as much as large hierarchies or dimensionless ratios appear, it demands “an explanation”.

In the case of the Majorana mass term, the $m_S$ term will appear if $N$ is a gauge singlet on general grounds. In this case, a renormalizable mass term with structure

$L_N=m_SN^tN$

is allowed by the SM gauge group. However, it would bot be consistent in general with unified symmetries or general GUTs. That is, a full SO(10), for instance, and some complicated mechanisms should be used to describe and explain the presence of this term. The $m_S$ term is usually associated with the breaking of some larger symmetry group, and it is generally expected that its energy scale should be in a range covering from the few hundreds of $GeVs$ in LR models to GUTscale energies, or about $10^{15}-10^{17}$ GeV.

When the $m_T$ term is present, then $\nu_L$ are active. That is, whenever $\nu_L$ is active, there is a $m_T$ term. It belongs to some gauge doublet and it sometimes introduce non-renormalizable interactions. That is the reason why generally speaking models with $m_T=0$ are “preferred” over this alternative. In this case, we have $\Delta I=1$ and $m_T$ must be generated by either:

1) An elementary Higgs triplet.

2) An effective operator involving two Higgs doublets arranged to transform as a triplet.

In both cases, we can induce non-renormalizable interactions. In case 1), an elementary triplet $m_T\sim h_Tv_T$, where $h_T$ is a Yukawa coupling and $v_T$ the triplet v.e.v. The simplest realization is the so-called “old Gelmini-Roncadelli model”) and it is EXCLUDED by the LEP data on the Z-invisible width. This last result is due to the fact that the corresponding Majoron particle couples to the Z boson, and it increases significantly its width so we would have seen it at LEP. Some variants of this model involving the explicit lepton number violation or in which the Majoron is mainly a weak singlet (named invisible Majoron models) could still be possible, though, yet. In case 2), for an effective operator originated mass, one should expect $m_T\sim 1/M_{NP}$, where $M_{NP}$ is the scale of new physics wich generates the operator. Let me remark that both cases can trigger non-renormalizability in the extended gauge theory, a property which some people finds “disturbing”.

Final remarks: If $m_S\sim 1 TeV$ (typical in LR models), and with typical values of $m_D$, one would expect masses about $0.1eV, 10keV, 1MeV$ for the $\nu_{e,\mu,\tau}$ weak eigenstates, respectively. GUT theories motivates a bigger gap between the intermediate electroweak scale and the GUT scale. The gap can be as large as $10^{12}-10^{16}GeV$. In the lower end of this range, for $m_S\sim 10^{12}GeV$, we have some string-inspired models, GUT with multiple breaking stages and “mixed” models. At the upper end, for $m_S\sim 10^{16}$ (named GUT seesaw, with large Higgs representations), one typically finds smaller masses for the neutrinos, about $10^{-11}, 10^{-7}, 10^{-2}$ eV respectively for the 3 neutrino flavors (electron, muon and tau). Somehow, this radical approach is more difficult to fit into the present known experimental facts, that they suggest a milielectronvolt neutrino mass as the lighter neutrino mass, up to 1eV (if you consider some experiments as hinting a sterile neutrino as “yet possible”). Thus, neutrinos are hinting to the existence of some intermediate pre-GUT or GUT-like unification energy scale. Where is it? We don’t know! There are many possible models and theories GUT-like. For instance, the next scheme is possible

## Neutrinos and magnetic dipole moments

The magnetic dipole moment is another probe of possible new interactions and physics beyond the Standard Model. Majorana neutrinos have identically zero magnetic and electric dipole moments. Flavor transition magnetic moments are allowed however in general for both Dirac and Majorana neutrinos! Limits obtained from laboratory experiments (LEX) are of the order of a constant times $10^{-10}\mu_B$, where $\mu_B$ is the Bohr magneton. There are additional limits/bounds imposed by both stellar physics (or astrophysics) and cosmology in the range $10^{-11}-10^{-13}\mu_B$. In the SM, the electroweak sector can be extended to allow for Dirac neutrino masses, so that the nuetrino magnetid ipole moment is nonzero and given by

$\mu_\nu=\dfrac{3eG_Fm_\nu}{8\pi^2\sqrt{2}}=3\cdot 10^{-19}\left(\dfrac{m_\nu}{1eV}\right)\mu_B$

The proportionality of $\mu_\nu$ to the neutrino mass is due to the absence of an interaction with $\nu_R$ in this Dirac extended SM. Then, only its Yukawa coupling appears, and hence, the neutrino mass. In LR symmetric theories (like the mentioned SO(10) theory), the $\mu_\nu$ is proportional to the charged lepton mass. Based on general grounds, we find typical values about

$\mu_\nu\sim 10^{-13}-10^{-14}\mu_B$

These values are still too small to have odds of being measurable in current experiments or having practical astrophysical or cosmological consequences we could detect now. However, these magnetic dipole moments are important features of BSM models, so it is important to study them.

Magnetic moment interactions arise in ANY renormalizable gauge theory only as finite radiative corrections. The diagrams which generate a magnetic moment will also contribute to the neutrino mass once the external photon line is removed.In the absence of additional symmetries, a large magnetic moment is incompatible with a small neutrino mass. The way out to this NO-GO theorem suggested by Voloshin consists in defining a $SU(2)_\nu$ symmetry acting on the flavor space $(\nu, \nu^c)$, and then the magnetic moment term are singlets under this symmetry. In the limit of exact $SU(2)_\nu$ symmetry, the neutrino mass is forbidden BUT the magnetic moment $\mu_\nu$ is allowed. Diverse concrete models have been proposed where such extra symmetry is embedded into an extension of the SM (e.g., in LR models, with SUSY “horizontal” gauge symmetries, by Babu et al.).

What do you think? Some novel idea? Here you are a decision tree map (LOL):

However, we are far, far away to understand the neutrino hidden higher secrets! Here you are a basic “road map” towards superbeams and neutrino factories, yet an intermediate step before the mythical muon collider (yes, USA likely WANTS that muon collider, :P)…

May the neutrinos be with you!

PS: See you in my next neutrinology blog post!