# LOG#126. Basic Neutrinology(XI).

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 $\Delta m_{ij}^2=m_i^2-m_j^2$. Typically, $\Delta m_{ij}\leq 1eV$, 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:

$\mbox{Neutrino masses}\neq 0\longrightarrow \mbox{Transitions between neutrino mass eigenstates}$

$\mbox{Transitions between mass eigenstates}\longrightarrow \mbox{Neutrino mixing matrix}$

$\mbox{Neutrino mixing matrix}\longrightarrow \mbox{Neutrino oscillations}$

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:

$\boxed{\vert \nu_w (0)\rangle =U\vert \nu_m (0)\rangle}$

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 $\hbar=c=1$.

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):

$\vert \nu_m (t)\rangle=\exp \left( -iHt\right)\vert \nu_m (t)\rangle$

and where $H$ is the free hamiltonian of the system, i.e., in vacuum. It will be characterized by certain eigenvalues

$H=\mbox{diag}(\ldots,E_i,\ldots)$

and here, using special relativity, we write $E_i^2=p_i^2+m_i^2$

In most of the interesting cases (when $E\sim MeV$ and $m\sim eV$), this relativistic dispersion relationship $E=E(p,m)$ can be approximated by the next expression (it is the celebrated “ultra-relativistic” approximation):

$p\simeq E$

$E\simeq p+\dfrac{m^2}{2p}$

The effective neutrino hamiltonian can be written as

$H_{eff}=\mbox{diag}(\ldots,m_i^2,\ldots)/2E$

and

$\vert \nu_m (t)\rangle=U\exp \left(-iH_{eff}t\right)U^+\vert \nu_w (0)\rangle=\exp \left(-iH_w^{eff}t\right)\vert \nu_m (0)\rangle$

In this last equation, we write

$H_w^{eff}\equiv \simeq \dfrac{M^2}{2E}$

with

$M\equiv U\mbox{diag}\left(\ldots,m_i^2,\ldots\right)U^+$

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 $\theta$, 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 $\vert \nu_w\rangle$ has oscillated to other weak interaction eigenstate, say $\vert \nu_w'\rangle$ when the neutrino travels some distance $l=ct$ (remember we are supposing the neutrino are “almost” massless, so they move very close to the speed of light) is, taking $\nu_m=\nu_e$ and $\nu_m'=\nu_\mu$,

(1) $\boxed{P(\nu_e\longrightarrow \nu_\mu;l)=\sin^22\theta\sin^2\left(\dfrac{l}{l_{osc}}\right)}$

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

$\dfrac{1}{l_{osc}}\equiv\dfrac{\Delta m^2 l}{4E}$

with $\Delta m^2=m_1^2-m_2^2$. In practical units, we have

(2) $\boxed{\dfrac{1}{l_{osc}}=\dfrac{\Delta m^2 l}{4E}\simeq 1.27\dfrac{\Delta m^2(eV^2)l(m)}{E(MeV)}=1.27\dfrac{\Delta m^2(eV^2)l(km)}{E(GeV)}}$

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 $l\sim 10^{2}km$ 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, $NO\nu A$, 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#116. Basic Neutrinology(I).

This new post ignites a new thread.

Subject: the Science of Neutrinos. Something I usually call Neutrinology.

I am sure you will enjoy it, since I will keep it elementary (even if I discuss some more advanced topics at some moments). Personally, I believe that the neutrinos are the coolest particles in the Standard Model, and their applications in Science (Physics and related areas) or even Technology in the future ( I will share my thoughts on this issue in a forthcoming post) will be even greater than those we have at current time.

Let me begin…

The existence of the phantasmagoric neutrinos ( light, electrically neutral and feebly -very weakly- interacting fermions) was first proposed by W. Pauli in 1930 to save the principle of energy conservation in the theory of nuclear beta decay. The idea was promptly adopted by the physics community but the detection of that particle remained elusive: how could we detect a particle that is electrically neutral and that interact very,very weakly with normal matter? In 1933, E. Fermi takes the neutrino hypothesis, gives the neutrino its name (meaning “little neutron”, since it was realized than neutrinos were not Chadwick’s neutrons) and builds his theory of beta decay and weak interactions. With respect to its mass, Pauli initially expected the mass of the neutrino to be small, but necessarily zero. Pauli believed (originally) that the neutrino should not be much more massive than the electron itself. In 1934, F. Perrin showed that its mass had to be less than that of the electron.

By the other hand, it was firstly proposed to detect neutrinos exploding nuclear bombs! However, it was only in 1956 that C. Cowan and F. Reines (in what today is known as the Reines-Cowan experiment) were able to detect and discover the neutrino (or more precisely, the antineutrino). In 1962, Leon M. Lederman, M. Schwartz, J. Steinberger and Danby et al. showed that more than one type of neutrino species $\nu_e,\nu_\mu$ should exist by first detecting interactions of the muon neutrino. They won the Nobel Prize in 1988.

When we discovered the third lepton, the tau particle (or tauon), in 1975 at the Stanford Linear Accelerator Center, it too was expected to have an associated neutrino particle. The first evidence for this 3rd neutrino “flavor” came from the observation of missing energy and momentum in tau decays. These decays were analogue to the beta decay behaviour leading to the discovery of the neutrino particle.

In 1989, the study of the Z boson lifetime allows us to show with great experimental confidence that only 3 light neutrino species (or flavors) do exist. In 2000, the first detection of tau neutrino ($\nu_\tau$ in addition to $\nu_e,\nu_\mu$) interactions was announced by the DONUT collaboration at Fermilab, making it the latest particle of the Standard Model to have been discovered until the recent Higgs particle discovery (circa 2012, about one year ago).

In 1998, research results at the Super-Kamiokande neutrino detector in Japan (and later, independently, from SNO, Canada) determined for the first time that neutrinos do indeed experiment “neutrino oscillations” (I usually call NOCILLA, or NO for short, this phenomenon), i.e., neutrinos flavor “oscillate” and change their flavor when they travel  “short/long” distances. SNO and Super-Kamiokande tested and confirmed this hypothesis using “solar neutrinos”. this (quantum) phenomenon implies that:

1st. Neutrinos do have a mass. If they were massless, they could not oscillate. Then, the old debate of massless vs. massive neutrinos was finally ended.

2nd. The solar neutrino problem is solved. Some solar neutrinos scape to the detection in Super-Kamiokande and SNO, since they could not detect all the neutrino species. It also solved the old issue of “solar neutrinos”. The flux of (detected) solar neutrinos was lesser than expected (generally speaking by a factor 2). The neutrino oscillation hypothesis solved it since it was imply the fact that some neutrinos have been “transformed” into a type we can not detect.

3rd. New physics does exist. There is new physics at some energy scale beyond the electroweak scale (the electroweak symmetry breaking and typical energy scale is about 100GeV). The SM is not complete. The SM does (indeed) “predict” that the neutrinos are massless. Or, at least, it can be made simpler if you make neutrinos to be massless neutrinos described by Weyl spinors. It shows that, after the discovery of neutrino oscillations, it is not the case. Neutrinos are massive particles. However, they could be Dirac spinors (as all the known spinors in the Standard Model, SM) or they could also be Majorana particles, neutral fermions described by “Majorana” spinors and that makes them to be their own antiparticles! Dirac particles are different to their antiparticles. Majorana particles ARE the same that their own antiparticles.

In the period 2001-2005, neutrino oscillations (NO)/neutrino mixing phenomena(NEMIX) were observed for the first time at a reactor experiment (this type of experiment are usually referred as short baseline experiment in the neutrino community) called KamLAND. They give a good estimate (by the first time) of the difference in the squares of the neutrino masses. In May 2010, it was reported that physicists from CERN and the Italian National Institute for Nuclear Physics, in Gran Sasso National Laboratory, had observed for the first time a transformation between neutrino flavors during an accelerator experiment (also called neutrino beam experiment, a class of neutrino experiment belonging to “long range” or “long” baseline experiments with neutrino particles). It was a new solid evidence that at least one neutrino species or flavor does have mass. In 2012, the Daya Bay Reactor experiment in China, and later RENO in South Korea measured the so called $\theta_{13}$ mixing angle, the last neutrino mixing angle remained to be measured from the neutrino mass matrix. It showed to be larger than expected and it was consistent with earlier, but less significant results by the experiments T2K (another neutrino beam experiment), MINOS (other neutrino beam experiment) and Double Chooz (a reactor neutrino experiment).

With the known value of $\theta_{13}$ there are some probabilities that the $NO\nu A$ experiment at USA can find the neutrino mass hierarchy. In fact, beyond to determine the spinorial character (Dirac or Majorana) of the neutrino particles, and to determine their masses (yeah, we have not been able to “weight” the neutrinos, but we are close to it: they are the only particle in the SM with no “precise” value of mass), the remaining problem with neutrinos is to determine what kind of spectrum they have and to measure the so called CP violating processes. There are generally 3 types of neutrino spectra usually discussed in the literature:

A) Normal Hierarchy (NH): $m_1<. This spectrum follows the same pattern in the observed charged leptons, i.e., $m(e)<. The electron is about 0.511MeV, muon is about 106 MeV and the tau particle is 1777MeV.

B) Inverted Hierarchy (IH): $m_1<. This spectrum follows a pattern similar to the electron shells in atoms. Every “new” shell is closer in energy (“mass”) to the previous “level”.

C) Quasidegenerated (or degenerated) hierarchy/spectrum (QD): $m_1\sim m_2\sim m_3$.

While the above experiments show that neutrinos do have mass, the absolute neutrino mass scale is still not known. There are reasons to believe that its mass scale is in the range of some milielectron-volts (meV) up to the electron-volt scale (eV) if some extra neutrino degree of freedom (sterile neutrinos) do appear. In fact, the Neutrino OScillation EXperiments (NOSEX) are sensitive only to the difference in the square of the neutrino masses. There are some strongest upper limits on the masses of neutrinos that come from Cosmology:

1) The Big Bang model states that there is a fixed ratio between the number of neutrino species and the number of photons in the cosmic microwave background (CMB). If the total energy of all the neutrino species exceeded an upper bound about

$m_\nu\leq 50eV$

per neutrino, then, there would be so much mass in the Universe that it would collapse. It does not (apparently) happen.

2) Cosmological data, such as the cosmic microwave background radiation, the galaxy surveys, or the technique of the Lyman-alpha forest indicate that the sum of the neutrino masses should be less than 0.3 eV (if we don’t include sterile neutrinos, new neutrino species uncharged under the SM gauge group, that could increase that upper bound a little bit).

3) Some early measurements coming from lensing data of a galaxy cluster were analyzed in 2009. They suggest that the neutrino mass upper bound is about 1.5eV. This result is compatible with all the above results.

Today, some measurements in controlled experiments have given us some data about the squared mass differences (from both, solar neutrinos, atmospheric neutrinos produced by cosmic rays and accelerator/reactor experiments):

1) From KamLAND (2005), we get

$\Delta m_{21}^2=0\mbox{.}000079eV^2$

2) From MINOS (2006), we get

$\Delta m_{32}^2=0\mbox{.}0027eV^2$

There are some increasing efforts to directly determine the absolute neutrino mass scale in different laboratory experiments (LEX), mainly:

1) Nuclear beta decay (KATRIN, MARE,…).

2) Neutrinoless double beta decay (e.g., GERDA; CUORE, Cuoricino, NEMO3,…). If the neutrino is a Majorana particle, a new kind of beta decay becomes possible: the double beta decay without neutrinos (i.e., two electrons emitted and no neutrino after this kind of decay).

Neutrinos have a unique place among all the SM elementary particles. Their role in the cosmic evolution and the fundamental asymmetries in the SM (like CP violating reactions, or the C, T, and P single violations) make them the most fascinating and interesting particle that we know today (well, maybe, today, the Higgs particle is also as mysterious as the neutrino itself). We believe that neutrinos play an important role in Beyond Standard Model (BSM) Physics. Specially, I would like to highlight two aspects:

1) Baryogenesis from leptogenesis. Neutrinos can allow us to understand how could the Universe end in such an state that it contains (essentially) baryons and no antibaryons (i.e., the apparent matter-antimatter asymmetry of the Universe can be “explained”, with some unsolved problems we have not completely understood, if massive neutrinos are present).

2) Asymmetric mass generation mechanisms or the seesaw. Neutrinos allow us to build an asymmetric mass mechanism known as “seesaw” that makes “some neutrino species/states” very light and other states become “superheavy”. This mechanism is unique and, from some  non-subjective viewpoint, “simple”.

After nearly a century, the question of the neutrino mass and its origin is still an open question and a hot topic in high energy physics, particle physics, astrophysics, cosmology and theoretical physics in general.

If we want to understand the fermion masses, a detailed determination of the neutrino mass is necessary. The question why the neutrino masses are much smaller than their charged partners could be important! The little hierarchy problem is the problem of why the neutrino mass scale is smaller than the other fermionic masses and the electroweak scale. Moreover, neutrinos are a powerful probe of new physics at scales larger than the electroweak scale. Why? It is simple. (Massive) Neutrinos only interact under weak interactions and gravity! At least from the SM perspective, neutrinos are uncharged under electromagnetism or the color group, so they can only interact via intermediate weak bosons AND gravity (via the undiscovered gravitons!).

If neutrino are massive particles, as they show to be with the neutrino oscillation phenomena, the superposition postulates of quantum theory state that neutrinos, particles with identical quantum numbers, could oscillate in flavor space since they are electrically neutral particles. If the absolute difference of masses among them is small, then these oscillations or neutrino (flavor) mixing could have important phenomenological consequences in Astrophysics or Cosmology. Furthermore, neutrinos are basic ingredients of these two fields (Astrophysics and Cosmology). There may be a hot dark matter component (HDM) in the Universe: simulations of structure formation fit the observations only when some significant quantity of HDM is included. If so, neutrinos would be there, at least by weight, and they would be one of the most important ingredients in the composition of the Universe.

Regardless the issue of mass and neutrino oscillations/mixing, astrophysical interests in the neutrino interactions and their properties arise from the fact that it is produced in high temperature/high density environment, such as collapsing stars and/or supernovae or related physical processes. Neutrino physics dominates the physics of those astrophysical objects. Indeed, the neutrino interactions with matter is so weak, that it passes generally unnoticed and travels freely through any ordinary matter existing in the Universe. Thus, neutrinos can travel millions of light years before they interact (in general) with some piece of matter! Neutrinos are a very efficient carrier of energy drain from optically thick objects and they can serve as very good probes for studying the interior of such objects. Neutrino astronomy is just being born in recent years. IceCube and future neutrino “telescopes” will be able to see the Universe in a range of wavelengths and frequencies we have not ever seen till now. Electromagnetic radiation becomes “opaque” at some very high energies that neutrinos are likely been able to explore! Isn’t it wonderful? Neutrinos are high energy “telescopes”!

By the other hand, the solar neutrino flux is, together with heliosysmology and the field of geoneutrinos (neutrinos coming from the inner shells of Earth), some of the known probes of solar core and the Earth core. A similar statement applies to objects like type-II supernovae. Indeed, the most interesting questions around supernovae and the explosion dynamics itself with the shock revival (and the synthesis of the heaviest elements by the so-called r-processes) could be positively affected by changes in the observed neutrino fluxes (via some processes called resonant conversion, and active-sterile conversions).

Finally, ultra high energy neutrinos are likely to be useful probes of diverse distant astrophysical objects. Active Galactic Nuclei (AGN) should be copious emitters of neutrinos, providing detectable point sources and and observable “diffuse” background which is larger in fact that the atmospheric neutrino background in the very high energy range. Relic cosmic neutrinos, their thermal background, known as the cosmic neutrino background, and their detection about 1.9K are one of the most important lacking missing pieces in the Standard Cosmological Model (LCDM).

Do you understand why neutrinos are my favorite particles? I will devote this basic thread to them. I will make some advanced topics in the future. I promise.

May the Neutrinos be with you!