# LOG#127. Basic Neutrinology(XII).

**Posted:**2013/07/22

**Filed under:**Basic Neutrinology, Physmatics |

**Tags:**electron density, mass eigenstates, matter density, MSW effect, neutrino mixing, neutrino oscillations, neutrino oscillations in matter, neutrino oscillatrions in vacuum, neutrino refraction, neutrinology, refraction, resonance, weak eigenstates Leave a comment

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

(1)

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

(2)

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

(3)

The presence of the term proportional to the electron density can produce “a resonance” nullifying the denominator. there is a critical density such as

(3)

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:

1st.

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):

(4)

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!

# LOG#126. Basic Neutrinology(XI).

**Posted:**2013/07/22

**Filed under:**Basic Neutrinology, Physmatics, The Standard Model: Basics |

**Tags:**IceCube, LBE, long baseline experiments, neutrino beam experiments, neutrino masses and lepton asymmetry, neutrino mixing, neutrino oscillation experiments, neutrino oscillations, neutrino oscillations in matter, neutrino oscillations in vacuum, neutrino telescopes, neutrinology, NOCILLA, NOSEX, reactor experiments, right-handed neutrinos, SBE, short baseline experiments, sterile neutrinos Leave a comment

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

and

In this last equation, we write

with

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 ,

(1)

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

(2)

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!

# LOG#120. Basic Neutrinology(V).

**Posted:**2013/07/15

**Filed under:**Basic Neutrinology, Physmatics, The Standard Model: Basics |

**Tags:**bino, dark matter, Dirac mass term, E(6) group, exceptional group GUT, gauginos, GUT, GUT scale, Higgsino, LR models, LSP, Majorana mass term, MSSM, neutralino, neutrino masses, neutrino mixing, proton decay, proton lifetime, R-parity, R-parity violations, seesaw, sfermion, singlets, sneutrino, soft SUSY breaking terms, string inspired models, superparticle, superpartner, superpotential, SUSY models of neutrino masses, vev, WIMPs, wino, Yukawa coupling, Zinos Leave a comment

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 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 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)

A less radical solution is to allow for the existence in the superpotential of a bilinear term with structure . 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 term leads by a minimization condition to a non-zero vacuum expectation value or vev, . In such a model, the neutrino acquire a mass due to the mixing between neutrinos and the neutralinos.The 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)

and where the matrix corresponds to the two “gauginos”. The matrix is a 2×3 matrix and it contains the vevs of the two higgses plus the sneutrino, i.e., 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 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 can be easily recognized. The mass is provided by

where and 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 . Thus, (2) can be also written as follows

(3)

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) GUT with matter fields belong to the 27 dimensional representation of the exceptional group plus additional singlet fields. The model contains additional neutral leptons in each generation and the neutral 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 and with an additional neutral field whose nature depends on the particular “model building” and “mechanism” we use. In some basis , the mass matrix can be rewritten

(4)

and where the 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 are assumed to acquire a v.e.v. with energy size . In the first case, the field corresponds to a gaugino with a Majorana mass than can be produced at two-loops! Usually , and if we assume , then additional dangerous mixing wiht the Higgsinos can be “neglected” and we are lead to a neutrino mass about , 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 corresponds to one of the 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 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 , 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!).