Hi there! I am Amarashiki (a new “doctor” project). And I have a new upgraded The Spectrum Of Riemannium website!
I will keep this free site though, as backup material. But everything has been moved to the new URL: http://thespectrumofriemannium.com
ALERT: If you was reading my stuff via e-mail, RSS or any other magic device/too, upgrade my URL, please. I will NOT post here anymore. After all, I am paying for the new domain! And it brings new abilities or superpowers to my web! 🙂
I wish you will enjoy my new (improved) site, since even when this beginning trip has ended, a new one is coming! And I hope It will be satisfactory for all of us!
I will be happy to hear from any other suggestion or idea related to my site! Any desiderata? 😄
TSOR is just beginning!!!!!!!!
When neutrinos pass through matter or they propagate in a medium (not in the vacuum), a subtle and potentially important effect occurs. This is called the MSW effect (Mikheyev-Smirnov-Wolfenstein effect). It is pretty similar to a refraction of light in a medium, but now it happens that the particle (wave) propagating are not electromagnetic waves (photons) but neutrinos! In fact, the MSW effect consists in two different effects:
1st. A “resonance” enhancement of the neutrino oscillation pattern.
2nd. An adiabatic (i.e. slow) or partially adiabatic neutrino conversion (mixing).
In the presence of matter, the neutrino experiences scattering and absorption. This last phenomenon is always negligible (or almost in most cases). At very low energies, coherent elastic forward scattering is the most important process. Similarly to optics, the net effect is the appearance of a phase difference, a refractive index or, equivalently, a neutrino effective mass.
The neutrino effective mass can cause an important change in the neutrino oscillation pattern, depending on the densities and composition of the medium. It also depends on the nature of the neutrino (its energy, its type and its oscillation length). In the neutrino case, the medium is “flavor-dispersive”: the matter is usually non-symmetric with respect to the lepton numbers! Then, the effective neutrino mass is different for the different weak eigenstates!
I will try to explain it as simple as possible here. For instance, take the solar electron plasma. The electrons in the solar medium have charged current interactions with but not with . Thus, the resulting interaction energy is given by a interaction hamiltonian
where the numerical prefactor is conventional, is the Fermi constant and is the electron density. The corresponding neutral current interactions are identical fo al the neutrino species and, therefore, we have no net effect on their propagation. Hypothetical sterile neutrinos would have no interaction at all either. The effective global hamiltonian in flacor space is now the sum of two terms, the vacuum hamiltonian and the interaction part. We can write them together
The consequence of this new effective hamiltonian is that the oscillation probabilities of the neutrino in matter can be largely increased due to a resonance with matter. In matter, for the simplest case with 2 flavors and 2 dimensions, we can define an effective oscillation/mixing angle as
The presence of the term proportional to the electron density can produce “a resonance” nullifying the denominator. there is a critical density such as
for which the matter mixing angle becomes maximal and , irrespectively of the value of the mixing angle in vacuum . The probability that oscillates or mixes into a weak eigenstate after traveling a distance in this medium is give by the vacuum oscillation formula modified as follows:
2nd. The kinematical factor differs by the replacement of with . Hence, it follows that, at the critical density, we have the oscillation probability in matter (2 flavor and 2 dimensions):
This equation tells us that we can get a full conversion of electron neutrino weak eigenstates into muon weak eigenstates, provided that the length and energy of the neutrino satisfy the condition
There is a second interesting limit that is mentioned often. This limit happens whenever the electron density is so large such that , or equivalently, . In this (dense matter) limit, there are NO oscillation in matter (they are “density suppresed”) because vanishes and we have
Therefore, the lesson here is that a big density can spoil the phenomenon of neutrino oscillations!
In summary, we have learned here that:
1st. There are neutrino oscillations “triggered” by matter. Matter can enhance or enlarge neutrino mixing by “resonance”.
2nd. A high enough matter density can spoil the neutrino mixing (the complementary effect to the previous one).
The MSW effect is particularly important in the field of geoneutrinos and when the neutrinos pass through the Earth core or mantle, as much as it also matters inside the stars or in collapsing stars that will become into supernovae. The flavor of neutrino states follows changes in the matter density!
See you in my next neutrinological post!
Why is the case of massive neutrinos so relevant in contemporary physics? The full answer to this question would be very long. In fact, I am making this long thread about neutrinology in order you understand it a little bit. If neutrinos do have nonzero masses, then, due to the basic postulates of the quantum theory there will be in a “linear combination” or “mixing” among all the possible “states”. It also happens with quarks! This mixing will be observable even at macroscopic distances from the production point or source and it has very important practical consequences ONLY if the difference of the neutrino masses squared are very small. Mathematically speaking . Typically, , but some “subtle details” can increae this upper bound up to the keV scale (in the case of sterile or right-handed neutrinos, undetected till now).
In the presence of neutrino masses, the so-called “weak eigenstates” are different to “mass eigenstates”. There is a “transformation” or “mixing”/”oscillation” between them. This phenomenon is described by some unitary matrix U. The idea is:
If neutrinos can only be created and detected as a result of weak processes, at origin (or any arbitrary point) we have a weak eigenstate as a “rotation” of a mass eigenstate through the mixing matrix U:
In this post, I am only to introduce the elementary theory of neutrino oscillations (NO or NOCILLA)/neutrino mixing (NOMIX) from a purely heuristic viewpoint. I will be using natural units with .
If we ignore the effects of the neutrino spin, after some time the system will evolve into the next state (recall we use elementary hamiltonian evolution from quantum mechanics here):
and where is the free hamiltonian of the system, i.e., in vacuum. It will be characterized by certain eigenvalues
and here, using special relativity, we write
In most of the interesting cases (when and ), this relativistic dispersion relationship can be approximated by the next expression (it is the celebrated “ultra-relativistic” approximation):
The effective neutrino hamiltonian can be written as
In this last equation, we write
We can perform this derivation in a more rigorous mathematical structure, but I am not going to do it here today. The resulting theory of neutrino mixing and neutrino oscillations (NO) has a beautiful corresponded with Neutrino OScillation EXperiments (NOSEX). These experiments are usually analyzed under the simplest assumption of two flavor mixing, or equivalently, under the perspective of neutrino oscillations with 2 simple neutrino species we can understand this process better. In such a case, the neutrino mixing matrix U becomes a simple 2-dimensional orthogonal rotation matrix depending on a single parameter , the oscillation angle. If we repeat all the computations above in this simple case, we find that the probability that a weak interaction eigenstate neutrino has oscillated to other weak interaction eigenstate, say when the neutrino travels some distance (remember we are supposing the neutrino are “almost” massless, so they move very close to the speed of light) is, taking and ,
This important formula describes the probability of NO in the 2-flavor case. It is a very important and useful result! There, we have defined the oscillation length as
with . In practical units, we have
As you can observe, the probabilities depend on two factors: the mixing (oscillation) angle and the kinematical factor as a function of the traveled distance, the momentum of the neutrinos and the mass difference between the two species. If this mass difference were probed to be non-existent, the phenomenon of the neutrino oscillation would not be possible (it would have 0 probability!). To observe the neutrino oscillation, we have to make (observe) neutrinos in which some of this parameters are “big”, so the probability is significant. Interestingly, we can have different kind of neutrino oscillation experiments according to how large are these parameters. Namely:
–Long baseline experiments (LBE). This class of NOSEX happen whenever you have an oscillation length of order or bigger. Even, the neutrino oscillations of solar neutrinos (neutrinos emitted by the sun) and other astrophysical sources can also be understood as one of this. Neutrino beam experiments belong to this category as well.
-Short baseline experiments (SBE). This class of NOSEX happen whenever the distances than neutrino travel are lesser than hundreds of kilometers, perhaps some. Of course, the issue is conventional. Reactor experiments like KamLAND in Japan (Daya Bay in China, or RENO in South Korea) are experiments of this type.
Moreover, beyond reactor experiments, you also have neutrino beam experiments (T2K, , OPERA,…). Neutrino telescopes or detectors like IceCube are the next generation of neutrino “observers” after SuperKamiokande (SuperKamiokande will become HyperKamiokande in the near future, stay tuned!).
In summary, the phenomenon of neutrino mixing/neutrino oscillations/changing neutrino flavor transforms the neutrino in a very special particle under quantum and relativistic theories. Neutrinos are one of the best tools or probes to study matter since they only interact under weak interactions and gravity! Therefore, neutrinos are a powerful “laboratory” in which we can test or search for new physics (The fact that neutrinos are massive is, said this, a proof of new physics beyond the SM since the SM neutrinos are massless!). Indeed, the phenomenon is purely quantum and (special) relativist since the neutrinos are tiny particles and “very fast”. We have seen what are the main ideas behind this phenomenon and the main classes of neutrino experiments (long baseline and shortbaseline experiments). Moreover, we also have “passive” neutrino detectors like SuperKamiokande, IceCube and many others I will not quote here. They study the neutrino oscillations detecting atmospheric neutrinos (the result of cosmic rays hitting the atmosphere), solar neutrinos and other astrophysical sources of neutrinos (like supernovae!). I have talked you about cosmic relic neutrinos too in the previous post. Aren’t you convinced that neutrinos are cool? They are “metamorphic”, they have flavor, they are everywhere!
See you in my next neutrinological post!
The topic today is a fascinant subject in Neutrino Astronomy/Astrophysics/Cosmology. I have talked you in this thread about the cosmic neutrino background () and that the young neutrino Astronomy or neutrino telescopes will become more and more important in the future. The reasons are simple:
1st. If we want to study the early Universe, we need some “new” tool to overcome the last scattering surface as a consequence of the Cosmic Microwave Background (CMB). Neutrinos are such a new tool/probe! They only interact weakly with matter and we suspect that there are some important pieces of information related to the quark and lepton “complementarity” hidden in their mixing parameters.
2nd. Due to the GZK effect, we expect that the flux of cosmic rays will suffer a sudden cut-off at about , or about 8 joules. This Greisen–Zatsepin–Kuzmin limit (GZK limit) is a theoretical upper limit on the energy of cosmic rays, since at some high energy, that can be computed, they would interact with the CMB photons producing a delta particle () which would spoil the observed cosmic rays flux as its decays would not be detected after “a long trip”. Then, it can only be approached when the cosmic rays travel very long distances (hundreds of million light-years or more). Here you are a typical picture of SuperKamiokande cosmic ray detection:
The limit is at the same order of magnitude as the upper limit for energy at which cosmic rays have experimentally been detected. There are some current experiments that “claim” to have observed this GZK effect, but evidence is not conclusive yet as far as I know. Some experiments claim (circa 2013, July) to have observed it, other experiments claim to have observed events well above the GZK limit. The next generation of cosmic ray experiments will confirm this limit from SM physics or they will show us interesting new physics events!
Inspired by the GZK effect, some people have suggested an indirect way to detect the existence of the cosmic relic neutrinos. Remember, cosmic neutrinos have a temperature about if the SM is right, and their associated neutrino density now is about per cubic centimeter per species (neutrino plus antineutrino), or per cubic centimeter including the 3 flavors! Relic neutrinos are almost everywhere, but they are very, very feeble (neutral and weakly interacting) particles. While detecting the temperature is one of the most challenging tests of the standard cosmological model, we can try to detect the existence of these phantom neutrinos using a similar (quantum) trick than the one used in the GZK limit (there the delta particle resonance). If some ultra high energy cosmic ray (likely a neutrino coming from some astrophysical source) hits a “relic neutrino” with energy high enough to produce, say, a Z boson (neutral particle as the neutrino himself), then we should observe a “dip” in the cosmic ray spectrum corresponding to this “Z-burst” event! This mechanism is also called the ZeVatron or the Z-dip. It also shows the deep links between particle physics and Cosmology or Astrophysics. When an ultra-high energy cosmic neutrino collides with a relic anti-neutrino in our galaxy and annihilates to hadrons, this process proceeds via a (virtual) Z-boson:
The cross section for this process becomes large if the center of mass energy of the neutrino-antineutrino pair is equal to the Z-boson mass (such a peak in the cross section is what we call “resonance” in High Energy physics). Assuming that the relic anti-neutrino is at rest, the energy of the incident cosmic neutrino has to be the quantity:
In fact, this mechanism based on “neutral resonances” is completely “universal”! Nothing (except some hidden symmetry or similar) can allow the production of (neutral) particles using this cosmic method. For instance, if this argument is true, beyond the Z-burst, we should be able to detect Higgs-dips (Higgs-bursts) or H-dips, since, similarely we could have
or more generally, with some (likely) “dark” particle, we should also expect that
In the H-dip case, taking the measured Higgs mass from the last LHC run (about 126GeV), we get
In the arbitrary “dark” or “weakly interacting” particle, we have (in general, with ) the formulae:
Therefore, cosmic ray neutrino spectroscopy is a very interesting subject yet to come! It can provide:
1st. Evidences for relic neutrinos we expect from the standard cosmological model.
2nd. Evidence for the Higgs boson in astrophysical scenarios from cosmological neutrinos. Now, we know that the Higgs field and the Higgs particle do exist, so it is natural to seek out this H-dips as well!
3rd. Evidence for the additional neutral weakly interacting (and/or “dark”) particles from “unexpected” dips at ZeV (1ZeV=1Zetta electron-volt) or even higher energies! Of course, this is the most interesting part from the viewpoint of new physics searches!
Moreover, following one of the most interesting things in any research (expect the unexpected and try to explain it!) from the scientific viewpoint, I am quite sure the neutrino astronomy and its interplay with cosmic rays or this class of “neutrino spectroscopy” in the flux of cosmic rays open a very interesting window for the upcoming new physics. Are we ready for it? Maybe…After all, the neutrino mixing parameters are very different (“complementary”?) to the quark mixing parameters. You can observe it in this mass-flavor content plot:
Neutrino oscillations are a purely quantum effect, and thus, they open a really interesting “new channel” in which we can observe the whole Universe. Yes, neutrinos are cool!!! The coolest particles in all over the world! We can not imagine yet what neutrino will show and teach us about the current, past and future of the cosmological evolution.
Remark: When I saw the Fermi line and the claim of the Dark Matter particle “evidence” at about 130 GeV, I wondered if it could be, indeed, a hint of a similar “resonant” process in gamma rays, something like
since the line “peaked” close to the known Higgs-like particle mass (). Anyway, this line is controversial and its presence has yet to be proved with enough statistical confidence (5 sigmas are usually required in the particle physics community). Of course, the issue with this resonant hypothesis would be that we should expect that this particle would decay into hadrons leaving some indirect clues of those events. The Fermi line can indeed have more explanations and/or be a fluke in the data due to a bad modeling or a bad substraction of the background. Time will tell us if the Fermi line is really here as well.
Final (geek) remark: I wonder if the Doctor Who fans remember that the reality bomb of Davros and the Daleks used “Z-neutrinos“!!! I presently do not know if the people who wrote those scripts and imagined the Z-neutrino were aware of the Z-bursts…Or not… LOL The Z-neutrino powered crucible was really interesting…
And the reality bomb concept was really scaring…
However, neutrinos are pretty weakly interacting particles, at least when they have low energy, so we should have not fear them. After all, their future applications will surprise us much more. I am quite sure of it!
See you in my next neutrinological post!
May the Z(X)-burst induced superGZK neutrinos be with you!
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
At present time, there is only matter and only a very tiny (if any) amount of antimatter, and then . 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 . Thus,
From BBN, we know that
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
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 . At the end of the inflation do oscillate close to the SUSY minimum and respectively, they decay into a part of right-handed sneutrinos and neutrinos. Moreover, a primordial lepton asymmetry is generated by the decay of the superfield emerging as the decay product of the inflaton field. The superfield 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 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() 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.
There are some indirect constraints/bounds on neutrino masses provided by Cosmology. The most important is the one coming from the demand that the energy density of the neutrinos should not be too high, otherwise the Universe would collapse and it does not happen, apparently…
Firstly, stable neutrinos with low masses (about ) make a contribution to the total energy density of the Universe given by:
and where the total mass is defined to be the quantity
Here, the number of degrees of freedom for Dirac (Majorana) neutrinos in the framework of the Standard Model. The number density of the neutrino sea is revealed to be related to the photon number density by entropy conservation (entropy conservation is the key of this important cosmological result!) in the adiabatic expansion of the Universe:
From this, we can derive the relationship of the cosmic relic neutrino background (neutrinos coming from the Big Bang radiation when they lost the thermal equilibrium with photons!) or and the cosmic microwave background (CMB):
From the CMB radiation measurements we can obtain the value
for a perfect Planck blackbody spectrum with temperature
This CMB temperature implies that the temperature should be about
Remark: if you do change the number of neutrino degrees of freedom you also change the temperature of the and the quantity of neutrino “hot dark matter” present in the Universe!
Moreover, the neutrino density is related to the total neutrino density and the critical density as follows:
and where the critical density is about
When neutrinos “decouple” from the primordial plasma and they loose the thermal equilibrium, we have , and then we get
with the reduced Hubble constant. Recent analysis provide (PLANCK/WMAP).
There is another useful requirement for the neutrino density in Cosmology. It comes from the requirements of the BBN (Big Bang Nucleosynthesis). I talked about this in my Cosmology thread. Galactic structure and large scale observations also increase evidence that the matter density is:
These values are obtained through the use of the luminosity-density relations, galactic rotation curves and the observation of large scale flows. Here, the is the total mass density of the Universe as a fraction of the critical density . This includes radiation (photons), bayrons and non-baryonic “cold dark matter” (CDM) and “hot dark matter” (HDM). The two first components in the decomposition of
are rather well known. The photon density is
The deuterium abundance can be extracted from the BBN predictions and compared with the deuterium abundances in the stellar medium (i.e. at stars!). It shows that:
The HDM component is formed by relativistic long-lived particles with masses less than about . In the SM framework, the only HDM component are the neutrinos!
The simulations of structure formation made with (super)computers fit the observations ONLY when one has about 20% of HDM plus 80% of CDM. A stunning surprise certainly! Some of the best fits correspond to neutrinos with a total mass about 4.7eV, well above the current limit of neutrino mass bounds. We can evade this apparent contradiction if we suppose that there are some sterile neutrinos out there. However, the last cosmological data by PLANCK have decreased the enthusiasm by this alternative. The apparent conflict between theoretical cosmology and observational cosmology can be caused by both unprecise measurements or our misunderstanding of fundamental particle physics. Anyway observations of distant objects (with high redshift) favor a large cosmological constant instead of Hot Dark Matter hypothesis. Therefore, we are forced to conclude that the HDM of does not exceed even . Requiring that , we get that . Using the relationship with the total mass density, we can deduce that the total neutrino mass (or HDM in the SM) is about
Mass limits, in this case lower limits, for heavy or superheavy neutrinos ( or higher) can also be obtained along the same reasoning. The puzzle gets very different if the neutrinos were “unstable” particles. One gets then joint bounds on mass and timelife, and from them, we deduce limits that can overcome the previously seen limits (above).
There is another interesting limit to the density of neutrinos (or weakly interacting dark matter in general) that comes from the amount of accumulated “density” in the halos of astronomical objects. This is called the Tremaine-Gunn limit. Up to numerical prefactors, and with the simplest case where the halo is a singular isothermal sphere with , the reader can easily check that
Imposing the phase space bound at radius r then gives the lower bound
This bound yields . This is the Tremaine-Gunn bound. It is based on the idea that neutrinos form an important part of the galactic bulges and it uses the phase-space restriction from the Fermi-Dirac distribution to get the lower limit on the neutrino mass. I urge you to consult the literature or google to gather more information about this tool and its reliability.
Remark: The singular isothermal sphere is probably a good model where the rotation curve produced by the dark matter halo is flat, but certainly breaks down at small radius. Because the neutrino mass bound is stronger for smaller , the uncertainty in the halo core radius (interior to which the mass density saturates) limits the reliability of this neutrino mass bound. However, some authors take it seriously! As Feynman used to say, everything depends on the prejudges you have!
The abundance of additional weakly interacting light particles, such as a light sterile neutrino or additional relativistic degrees of freedom uncharged under the Standard Model can change the number of relativistic degrees of freedom . Sometimes you will hear about the number . Planck data, recently released, have decreased the hopes than we would be finding some additional relativistic degree of freedom that could mimic neutrinos. It is also constrained by the BBN and the deuterium abundances we measured from astrophysical objects. Any sterile neutrino or extra relativistic degree of freedom would enter into equilibrium with the active neutrinos via neutrino oscillations! A limit on the mass differences and mixing angle with another active neutrino of the type
should be accomplished in principle. From here, it can be deduced that the effective number of neutrino species allowed by neutrino oscillations is in fact a litle higher the the 3 light neutrinos we know from the Z-width bound:
PLANCK data suggest indeed that . However, systematical uncertainties in the derivation of the BBN make it too unreliable to be taken too seriously and it can eventually be avoided with care.
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 . It was speculated by people like Froggatt long ago. The actual intrafamily hierarchy, i.e., the fact that 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 GUT or the 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 . 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 are usually spontaneously broken at some high energy scale by stringy effects. It is assumed that 3 fields, , with , develop a non-null vev. These fields are singlets under the SM gauge group but not under the abelian symmetries carried by . Thus, the Yukawa couplings appear as some effective operators after the spontaneous symmetry breaking. In the case of neutrinos, we have the mass lagrangian (at effective level):
and where . The parameters determine the mass and mixing hierarchy with the aid of some simple relationships:
and where is the Cabibblo angle. The are the charges assigned to the left handed leptons L and the right handed neutrinos N. These couplings generate the following mass matrices for neutrinos:
From these matrices, the associated seesaw mechanism gives the formula for light neutrinos:
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 . If the charges of the second and the third generation of leptos are equal (i.e., if ), then one is lead to a mass matrix with the following structure (or “texture”):
and where . This matrix can be diagonalized in a straightforward fashion by a large rotation. It is consistent (more or less), with a large 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 , related with the vev after the family group spontaneous symmetry breaking!