# 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#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.