LOG#127. Basic Neutrinology(XII).


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 \nu_e but not with \nu_\mu, \nu_\tau. Thus, the resulting interaction energy is given by a interaction hamiltonian

(1) H_{int}=\sqrt{2}G_FN_e

where the numerical prefactor is conventional, G_F is the Fermi constant and N_e 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) H_w^{eff}=H_w^{eff,vac}+H_{int}\begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}

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) \boxed{\sin\theta_M=\dfrac{\sin 2\theta/L_{osc}}{\left[\left(\cos 2\theta/L_{osc}-G_FN_e/\sqrt{2}\right)^2+\left(\sin 2\theta/L_{osc}\right)^2\right]^{1/2}}}

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

(3) \boxed{N_c^{osc}=\dfrac{\Delta m^2\cos 2\theta}{2\sqrt{2}EG_F}}

for which the matter mixing angle \theta_M becomes maximal and \sin 2\theta_M\longrightarrow 1, irrespectively of the value of the mixing angle in vacuum \theta. The probability that \nu_e oscillates or mixes into a \nu_\mu weak eigenstate after traveling a distance L in this medium is give by the vacuum oscillation formula modified as follows:

1st. \sin 2\theta\longrightarrow \sin 2\theta_M

2nd. The kinematical factor differs by the replacement of \Delta m^2 with \Delta m^2\sin 2\theta. Hence, it follows that, at the critical density, we have the oscillation probability in matter (2 flavor and 2 dimensions):

(4) \boxed{P_m (\nu_e\longrightarrow \nu_\mu;L)_{N_e=N_c^{osc}}=\sin^2\left(\sin 2\theta \dfrac{L}{L_{osc}}\right)}

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

\sin 2\theta \dfrac{L}{L_{osc}}=\dfrac{n\pi}{2} \forall n=1,2,3,\ldots,\infty

There is a second interesting limit that is mentioned often. This limit happens whenever the electron density N_e is so large such that \sin 2\theta_M\longrightarrow 0, or equivalently, \theta_M\longrightarrow \pi/2. In this (dense matter) limit, there are NO oscillation in matter (they are “density suppresed”) because \sin 2\theta_M vanishes and we have

P_m (\nu_e\longrightarrow \nu_\mu;L)_{\left(N_e>>\dfrac{\Delta m^2}{2\sqrt{2}EG_F}\right)}\longrightarrow 0

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).


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


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



\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}


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!