# LOG#122. Basic Neutrinology(VII).

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 $U(1)_F$. It was speculated by people like Froggatt long ago. The actual intrafamily hierarchy, i.e., the fact that $m_u>>m_d$ 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 $E_6\times E_8$ GUT or the $E_8\times E_8$ 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 $X,Y^{1,2}$. 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 $X,Y^{1,2}$ are usually spontaneously broken at some high energy scale $M_X$ by stringy effects. It is assumed that 3 fields, $\theta_i$, with $i=1,2,3$, develop a non-null vev. These $\theta_i$ fields are singlets under the SM gauge group but not under the abelian symmetries carried by $X, Y^{1,2}$. Thus, the Yukawa couplings appear as some effective operators after the $U(1)_F$ spontaneous symmetry breaking. In the case of neutrinos, we have the mass lagrangian (at effective level):

$\mathcal{L}_m\sim h_{ij}L_iH_uN_j^c\lambda^{q_i+n_j}+M_N\xi_{ij}N_i^cN_j^c\lambda^{n_i+n_j}$

and where $h_ {ij},\xi_{ij}\sim \mathcal{O}(1)$. The parameters $\lambda$ determine the mass and mixing hierarchy with the aid of some simple relationships:

$\lambda=\dfrac{\langle \theta\rangle}{M_X}\sim\sin\theta_c$

and where $\theta_c$ is the Cabibblo angle. The $q_i,n_i$ are the $U(1)_F$ charges assigned to the left handed leptons L and the right handed neutrinos N. These couplings generate the following mass matrices for neutrinos:

$m_\nu^D=\mbox{diag}(\lambda^{q_1},\lambda^{q_2},\lambda^{q_3})\hat{h}\mbox{diag}(\lambda^{n_1},\lambda^{n_2},\lambda^{n_3})\langle H_u\rangle$

$M_\nu=\mbox{diag}(\lambda^{n_1},\lambda^{n_2},\lambda^{n_3})\hat{\xi}\mbox{diag}(\lambda^{n_1},\lambda^{n_2},\lambda^{n_3})M_N$

From these matrices, the associated seesaw mechanism gives the formula for light neutrinos:

$m_\nu\approx \dfrac{\langle H_u\rangle^2}{M_X}\mbox{diag}(\lambda^{q_1},\lambda^{q_2},\lambda^{q_3})\hat{h}\hat{\xi}^{-1}\hat{h}^T\mbox{diag}(\lambda^{q_1},\lambda^{q_2},\lambda^{q_3})$

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 $q_i$. If the charges of the second and the third generation of leptos are equal (i.e., if $q_2=q_3$), then one is lead to a mass matrix with the following structure (or “texture”):

$m_\nu\sim \begin{pmatrix}\lambda^6 & \lambda^3 & \lambda^3\\ \lambda^3 & a & b\\ \lambda^3 & b & c\end{pmatrix}$

and where $a,b,c\sim \mathcal{O}(1)$. This matrix can be diagonalized in a straightforward fashion by a large $\nu_2-\nu_3$ rotation. It is consistent (more or less), with a large $\mu-\tau$ 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 $\lambda$, related with the vev after the family group spontaneous symmetry breaking!

# LOG#118. Basic Neutrinology(III).

## Mass terms

Phenomenologically, lagrangian mass terms can be understood as terms describing “transitions” between right (R) and left (L) handed states in the electroweak sector. For a given minimal, Lorentz invariant set of 4 fields ($\psi_L,\psi_R, \psi^c_L,\psi_R^c$), we have the components of a generic Dirac spinor. Thus, the most general mass part of a (likely extended) electroweak massive lagrangian can be written as follows:

$\mathcal{L}_m=m_D\bar{\psi}_L\psi_R+\dfrac{1}{2}m_T\left(\bar{\psi^c_L}\psi_L\right)+\dfrac{1}{2}m_S\left(\bar{\psi^c_R}\psi_R\right)+h.c.$

In terms of a “new” Majorana (real) field with $\nu^c=\nu$ and $N^c=N$, we have

$\nu=\dfrac{1}{\sqrt{2}}(\psi_L+\psi^c_L)$

$N=\dfrac{1}{\sqrt{2}}(\psi_R+\psi^c_R)$

and then, the massive lagrangian becomes

$\mathcal{L}_m=\begin{pmatrix}\bar{\nu} & \bar{N}\end{pmatrix}\mathbb{M}_{\nu,N}\begin{pmatrix}\nu\\ N\end{pmatrix}$

where the neutrino mass matrix is defined to be

$\mathbb{M}_{\nu,N}=\begin{pmatrix}m_T & m_D\\ m_D & m_S\end{pmatrix}$

We can diagonalize this mass matrix and then we will find the physical particle content! It is given (in general) by two Majorana mass eigenstates: the inclusion of the Majorana mass splits the 4 degenerate states of the Dirac field into two non-degenerate Majorana pairs. If we assume that the states $\nu, N$ are respectively “active” (i.e., they belong to some weak doublets) and sterile (weak singlets), then the terms corresponding to the Majorana masses $m_T, m_S$ transform as weak triplets and singlets respectively. While the term corresponding to $m_D$ is  an standard, weak singlet in most cases, Dirac mass term, its pressence shows to be essential in the next discussion. Indeed, this simple example can be easily generalized to three or more families, in which case the masses beocme matrices themselves. The complete full flavor mixing comes from any two different parts: the diagonalization of the charged lepton Yukawa couplings and that of the neutrino masses! Most of beyond Standard Model theories (specially those coming from GUTs) produce CKM-like leptonic mixing and this mixing is generally “arbitrary” with parameters only to be determined by the experiment. Only when you have an additional gauge symmetry (or some extra discrete symmetry), you can guess some of the mixing parameters from first principles. Therefore, the prediction of the neutrino oscillation/mixing parameters, as for the quark hierarchies and mixing, need further theoretical assumptions NOT included in the Standard Model. For instance, we could require that the $\nu_\mu-\nu_\tau$ mixing were “maximal” or to impose some “permutation symmetry” and derive the neutrino oscillation parameters from “tribimaximal” or “trimaximal” mixing. However, currently, the symmetry behind the neutrino mass matrix or the quark mixing matrix (the CKM mass matrix) are completely unknown. We can feel and “smell” there are some patterns there (something that suggests a “new” approximate broken symmetry related to flavor) but there is no current accepted working model for the neutrino mass matrix (or its quark analogue, the CKM mass matrix).

## The seesaw

When we diagonalize the above neutrino mass matrix, we can analyze different “limit” cases. In the case of a purely Dirac mass term, i.e., whenver $m_T=m_S=0$, then the $\nu, N$ states are degenerate with mass $m_D$ and a four component Dirac field can be “recovered” as $\nu'=\nu+N$, modulo some constant prefactor. It can be seen that, although violating individual lepton numbers, the Dirac mass term allows a conserve dlepton number $L=L_\nu+L_N$. This case in which the triplet and scalar masses are “tiny” or, equivalently, the case in which their Majorana mass “separation” is very small is sometimes called “pseudo-Dirac” case. In fact, it produces some interesting models both in Cosmology and particle physics. Inded, it could be possible that the 3 neutrino flavors we do know today were, in fact, neutrino (almost degenerated) triplets, i.e., every neutrino flavor could be formed by 3 very close Majorana states that we can not “resolve” using current technology.

In the general case, pure Majorana mass transition terms ($m_S, m_T$) arise in the lagrangian. Therefore, particle-antiparticle transitions violating the total lepton number by two units do appear ($\Delta L=\pm 2$). They can be understood as the creation or annihilation of two neutrinos, and thus, they allow the possibility of the existence of neutrinoless double beta decays! That is, only when the neutrino is a Majorana particle, the channel in which the total lepton number is violated opens.

When every mass term is allowed, there is an interesting case commonly referred as “the seesaw” limit. In this limit, taking the triplet mass to be zero and the singlet mass to be “huge” or “superheavy”, we deduce that

$m_T\sim 1/m_S\sim 0$ with $m_D< (the “seesaw” limit).

In this seesaw limit, the neutrino mass matrix can be diagonalized and it provides two eigenvalues:

$m_1\sim \dfrac{m_D^2}{m_S}<

$m_2\sim m_S$

Thus, the seesaw mechanism provide a way in which we obtain two VERY different mass eigenstates, i.e., two single particle states separated by a huge mass hierarchy! There is one (super)heavy neutrino (generally speaking, it corresponds to the right-handed neutrino) and a much lighter neutrino state, one that can be made relatively much lighter than a normal Dirac fermion mass. One fo the neutrino mass is “suppressed” and balanced up (hence the name “seesaw”) by the (super)heavy species. The seesaw mechanism is a “natural” way of generating two different (often VERY separated) mass scales!

The theory of the seesaw mechanism is very rich. I will not discuss its full potential here. There are 3 main types of seesaw mechanisms (generally named as type I, type II and type III) and some other less frequent variants and subvariants…It is an advanced topic for a whole future thread! 😉 However, I will draw you the 3 main Feynman graphs involved in these 3 main types of seesaw mechanisms:

## GUTs and neutrino mass models

Any fully satisfactory model that generates neutrino masses must contain a natural mechanism which allows us to explain their samll value, relative to that of their charged partners. Given the latest experimental hints and results, it would also be possible that it will include any comprehensive explanation for light sterile neutrinos and large, nearl maximal, mixing. This last idea is due to some “anomalies” coming from some neutrino experiments (specially those coming from reactors and the celebrated LSND experiment).

Different models can be distinguished according to the new particle content and spectrum, or according to the energy scale hierarchy they produce. With an extended particle content, different options open: if we want to brak the lepton number ant to generate neutrino masses without introducing new (unobserved) fermions in the SM, we must do it by adding to the SM Higgs sector fields carrying lepton numbers. Thus, one can arrange them to break the lepton number explicitly or spontaneously through interactions with these fields. If you want, this is another reason why the Higgs field matters: it allows to introduce fields carrying lepton numbers without adding any extra fermion field! Likely, the most straightforward approach to generate neutrino masses is to introduce for each neutrino an additional weak neutral single (that can be identified with the right-handed neutrino we can not observed due to be “very massive” and/or uncharged under the SM gauge group). This last fact strongly favors seesaw-like models!

For instance, the above features happen in the framework of LR (Left-Right) symmetric models in Grand Unified Theories (GUTs). There, the origin of the SM parity violation (explicit in the electroweak and weak sectors) is due to the spontaneous symmetry breaking of a baryon-lepton symmetry, and it yields a $B-L$ quantum number conservation/violation up to a degree that depends on the particular model. Thus, in $SO(10)$ GUT, the Majorana neutral particle N enters in a natural way in order to complete the matter multiplet. Therefore, N should be a $SU(3)\times SU(2)\times U(1)$ singlet, as we wished it to be.

If we use the energy scale as a guide where the new physics have relevant effects, unification (e.g., think about the previous SO(10) example) and the weak scale approach (radiative models and their effective theories) are usually distinguished and preferred form a pure QFT approach.

Despite the fact that the explanation of the known neutrino anomalies (the solar neutrino problem the first, but also the atmospheric neutrino flux and the reactor anomalies/neutrino beam anomalies) do not need or require the existence of an additional extra light/heavy sterile neutrino, some authors claim that they could exist after all. If every Marojana mass term is “small enough”, then active neutrinos can oscillate or mix into sterile (likely right-handed) fields/states. Light sterile neutrinos can appear in particularly special see-saw mechanisms if additional assumptions are considered (there, some models called “singular seesaw” models do exist as well). with some inevitable amount of “fine tuning”. The alternative to “fine tuning” would be seesaw-like suppression for sterile neutrinos involving new unknown (likely ultraweak or “dark”) interactions, i.e., family symmetries resulting in substantial field additions to the SM (some string theory models also suggest this possibility).

There is also weak scale models, i.e., radiative  generated mass models where the neutrino masses are zero at tree level and they constitute a very different type of models: they explain the smallness of the neutrino masses a priori for both active and sterile neutrinos. Loop corrections induce neutrino mass terms in these models. Thus, different mass scales are generated naturally by the different number of loops involved in the generation of each term. The actual implementation requires, however, the ad hoc (a posteriori) introduction of new Higgs particles with non-standard electroweak quantum numbers and lepton number violating couplings. This is the price we pay in an alternative approach.

The origin of the different Dirac and Majorana mass terms $m_S,m_T, m_D$ appearing in the neutrino (seesaw like) neutrino mass matrix is usually understood by a dynamical mechanism where at some energy scale it happens “naturally” and/or where some symmetry principle is spontaneously broke and invoked. Firstly, we face with the Dirac mass term. In one special case, $\nu_L$ and $\nu_R$ are SU(2) doublets and singlets respectively. The mass term describes a $\Delta I=1/2$ transition and it is generate from the SU(2) breaking via a Yukawa coupling:

$\mathcal{L}_{\mathcal{Y}}=h_i\begin{pmatrix} \bar{\nu}_i & \bar{l}_i\end{pmatrix}\begin{pmatrix}\phi^0\\ \phi^-\end{pmatrix}N_{R_i}+h.c.$

Here, $\phi^0, \phi^-$ are the components of certain Higgs doublet. The coefficients $h_i$ are the Yukawa couplings. After symmetry breaking, $m_D=h_iv/2$, where $v$ is the vacuum expectation value of the Higgs doublet. A Dirac mass term is qualitatively just like any other fermion mass, but that leads to the question of why it is so small in comparison with the rest of fermion masses: one would require $h(\nu_e)<10^{-10}$ in order to have $m(\nu_e)<10eV$. In other words, $h(\nu_e)/h(e)\sim 10^{-5}$ while for the hadronic sector we have $h(up)/h(down)\sim \mathcal{O}(1)$. In principle, it could be that there is no reason beyond the fine tuning of the Yukawa couplings (via Higgs vacuum expectation values to different fields) but, as much as large hierarchies or dimensionless ratios appear, it demands “an explanation”.

In the case of the Majorana mass term, the $m_S$ term will appear if $N$ is a gauge singlet on general grounds. In this case, a renormalizable mass term with structure

$L_N=m_SN^tN$

is allowed by the SM gauge group. However, it would bot be consistent in general with unified symmetries or general GUTs. That is, a full SO(10), for instance, and some complicated mechanisms should be used to describe and explain the presence of this term. The $m_S$ term is usually associated with the breaking of some larger symmetry group, and it is generally expected that its energy scale should be in a range covering from the few hundreds of $GeVs$ in LR models to GUTscale energies, or about $10^{15}-10^{17}$ GeV.

When the $m_T$ term is present, then $\nu_L$ are active. That is, whenever $\nu_L$ is active, there is a $m_T$ term. It belongs to some gauge doublet and it sometimes introduce non-renormalizable interactions. That is the reason why generally speaking models with $m_T=0$ are “preferred” over this alternative. In this case, we have $\Delta I=1$ and $m_T$ must be generated by either:

1) An elementary Higgs triplet.

2) An effective operator involving two Higgs doublets arranged to transform as a triplet.

In both cases, we can induce non-renormalizable interactions. In case 1), an elementary triplet $m_T\sim h_Tv_T$, where $h_T$ is a Yukawa coupling and $v_T$ the triplet v.e.v. The simplest realization is the so-called “old Gelmini-Roncadelli model”) and it is EXCLUDED by the LEP data on the Z-invisible width. This last result is due to the fact that the corresponding Majoron particle couples to the Z boson, and it increases significantly its width so we would have seen it at LEP. Some variants of this model involving the explicit lepton number violation or in which the Majoron is mainly a weak singlet (named invisible Majoron models) could still be possible, though, yet. In case 2), for an effective operator originated mass, one should expect $m_T\sim 1/M_{NP}$, where $M_{NP}$ is the scale of new physics wich generates the operator. Let me remark that both cases can trigger non-renormalizability in the extended gauge theory, a property which some people finds “disturbing”.

Final remarks: If $m_S\sim 1 TeV$ (typical in LR models), and with typical values of $m_D$, one would expect masses about $0.1eV, 10keV, 1MeV$ for the $\nu_{e,\mu,\tau}$ weak eigenstates, respectively. GUT theories motivates a bigger gap between the intermediate electroweak scale and the GUT scale. The gap can be as large as $10^{12}-10^{16}GeV$. In the lower end of this range, for $m_S\sim 10^{12}GeV$, we have some string-inspired models, GUT with multiple breaking stages and “mixed” models. At the upper end, for $m_S\sim 10^{16}$ (named GUT seesaw, with large Higgs representations), one typically finds smaller masses for the neutrinos, about $10^{-11}, 10^{-7}, 10^{-2}$ eV respectively for the 3 neutrino flavors (electron, muon and tau). Somehow, this radical approach is more difficult to fit into the present known experimental facts, that they suggest a milielectronvolt neutrino mass as the lighter neutrino mass, up to 1eV (if you consider some experiments as hinting a sterile neutrino as “yet possible”). Thus, neutrinos are hinting to the existence of some intermediate pre-GUT or GUT-like unification energy scale. Where is it? We don’t know! There are many possible models and theories GUT-like. For instance, the next scheme is possible

## Neutrinos and magnetic dipole moments

The magnetic dipole moment is another probe of possible new interactions and physics beyond the Standard Model. Majorana neutrinos have identically zero magnetic and electric dipole moments. Flavor transition magnetic moments are allowed however in general for both Dirac and Majorana neutrinos! Limits obtained from laboratory experiments (LEX) are of the order of a constant times $10^{-10}\mu_B$, where $\mu_B$ is the Bohr magneton. There are additional limits/bounds imposed by both stellar physics (or astrophysics) and cosmology in the range $10^{-11}-10^{-13}\mu_B$. In the SM, the electroweak sector can be extended to allow for Dirac neutrino masses, so that the nuetrino magnetid ipole moment is nonzero and given by

$\mu_\nu=\dfrac{3eG_Fm_\nu}{8\pi^2\sqrt{2}}=3\cdot 10^{-19}\left(\dfrac{m_\nu}{1eV}\right)\mu_B$

The proportionality of $\mu_\nu$ to the neutrino mass is due to the absence of an interaction with $\nu_R$ in this Dirac extended SM. Then, only its Yukawa coupling appears, and hence, the neutrino mass. In LR symmetric theories (like the mentioned SO(10) theory), the $\mu_\nu$ is proportional to the charged lepton mass. Based on general grounds, we find typical values about

$\mu_\nu\sim 10^{-13}-10^{-14}\mu_B$

These values are still too small to have odds of being measurable in current experiments or having practical astrophysical or cosmological consequences we could detect now. However, these magnetic dipole moments are important features of BSM models, so it is important to study them.

Magnetic moment interactions arise in ANY renormalizable gauge theory only as finite radiative corrections. The diagrams which generate a magnetic moment will also contribute to the neutrino mass once the external photon line is removed.In the absence of additional symmetries, a large magnetic moment is incompatible with a small neutrino mass. The way out to this NO-GO theorem suggested by Voloshin consists in defining a $SU(2)_\nu$ symmetry acting on the flavor space $(\nu, \nu^c)$, and then the magnetic moment term are singlets under this symmetry. In the limit of exact $SU(2)_\nu$ symmetry, the neutrino mass is forbidden BUT the magnetic moment $\mu_\nu$ is allowed. Diverse concrete models have been proposed where such extra symmetry is embedded into an extension of the SM (e.g., in LR models, with SUSY “horizontal” gauge symmetries, by Babu et al.).

What do you think? Some novel idea? Here you are a decision tree map (LOL):

However, we are far, far away to understand the neutrino hidden higher secrets! Here you are a basic “road map” towards superbeams and neutrino factories, yet an intermediate step before the mythical muon collider (yes, USA likely WANTS that muon collider, :P)…

May the neutrinos be with you!

PS: See you in my next neutrinology blog post!

# LOG#117. Basic Neutrinology(II).

The current Standard Model of elementary particles and interactions supposes the existence of 3 neutrino species or flavors. They are neutral, upper components of “doublets” $L_i$ with respect to the $SU(2)_L$ group, the weak interaction group after the electroweak symmetry breaking, and we have:

$L_i\equiv \begin{pmatrix}\nu_i\\ l_i\end{pmatrix}$ $\forall i=(e,\mu,\tau)$

These doublets have the 3rd component of the weak isospin $I_{3W}=1/2$ and they are assigned an unit of the global $ith$ lepton number. Thus, we have electron, muon or tau lepton numbers. The 3 right-handed charged leptons have however no counterparts in the neutrino sector, and they transform as singlets with respect to the weak interaction. That is, there are no right-handed neutrinos in the SM, we have only left-handed neutrinos. Neutrinos are “vampires” and, at least at low energies (those we have explored till now), they have only one “mirror” face: the left-handed part of the helicities. No observed neutrino has shown to be right-handed.

Beyond mass and charge assignments and their oddities, in any other respect, neutrinos are very well behaved particles within the SM framework and some figures and facts are unambiguosly known about them. The LEP Z boson line-shape measurements imply tat there are only 3 ordinary/light (weakly interacting) neutrinos.

The Big Bang Nucleosynthesis (BBN) constrains the parameters of possible additional “sterile” neutrinos, non-weak interacting or those which interact and are produced only my mixing. All the existing data on the weak interaction processes and reactions in which neutrinos take part are perfectly described by the SM charged-current (CC) and neutral-current (NC) lagrangians:

$\displaystyle{\mathcal{L}_{I}(CC)=-\dfrac{1}{\sqrt{2}}\sum_{i=e,\mu,\tau}\bar{\nu}_{L,i}\gamma_\alpha l_{Li}W^\alpha+h.c.}$

$\displaystyle{\mathcal{L}_{I}(NC)=-\dfrac{1}{2\cos\theta_W}\sum_{i=e,\mu,\tau}\bar{\nu}_{L,i}\gamma_\alpha l_{Li}Z^\alpha+h.c.}$

and where $W^\alpha, Z^\alpha$ are the neutral and charged massive vector bosons of the weak interaction. The CC and NC interaction lagrangians conserve 3 total additive quantum numbers: the lepton numbers $L_{e}, L_\mu, L_\tau$, while the structure of the CC interactions is what determine the notion of flavor neutrinos $\nu_e, \nu_\mu, \nu_\tau$.

There are no hints (yet) in favor of the violation of the conservation of these (global) lepton numbers in weak interactions and this fact provides very strong bound on brancing ratios of rare, lepton number violating reactions. For instance (even when the next data is not completely updated), we generally have (up to a 90% of confidence level, C.L.):

1. $R(\mu\longrightarrow e\mu)<4.9\cdot 10^{-11}$

2. $R(\mu\longrightarrow 3e)<1.0\cdot 10^{-12}$

3. $R(\mu\longrightarrow e(2\gamma))<7.2\cdot 10^{-11}$

4. $R(\tau\longrightarrow e\gamma)<2.7\cdot 10^{-6}$

5. $R(\tau\longrightarrow \mu\gamma)<3.0\cdot 10^{-6}$

6. $R(\mu\longrightarrow 3e)< 2.9\cdot 10^{-6}$

As we can observe, these lepton number violating reactions, if they exist, are very weird. From the theoretical viewpoint, in the minimal extension of the SM where the right-handed neutrinos are introduced and the neutrino gets a mass, the branching ratio of the $\mu\longrightarrow e\gamma$ decay is given by (2 flavor mixing only is assumed):

$R(\mu\longrightarrow e\gamma)=G_F\left(\dfrac{\sin 2\theta \Delta m_{12}^2}{2M_W^2}\right)^2$

and where $m_{1,2}$ are the neutrino masses, $\Delta m_{12}^2$ their squared mass difference, $M_W$ is the W boson mass and $\theta$ is the mixing angle of their respective neutrino flavors in the lepton sector. Using the experimental upper bound on the heaviest neutrino (believed to be $\nu_\tau$ without loss of generality), we obtain that

$R^{theo}\sim 10^{-18}$

Thus, we get a value far from being measurable at present time as we can observe by direct comparison with the above experimental results!!!

In fact, the transition $\mu\longrightarrow e\gamma$ and similar reactions are very sensitive to new physics, and particularly, to new particles NOT contained in the current description of the Standard Model. However, the R value is quite “model-dependent” and it could change by several orders of magnitude if we modify the neutrino sector introducing some extra number of “heavy”/”superheavy” neutrinos.

See you in another Neutrinology post! May the neutrinos be with you until then!

# 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!

I discussed and reviewed the important Cherenkov effect and radiation in the previous post, here:

https://thespectrumofriemannium.wordpress.com/2012/10/16/log046-the-cherenkov-effect/

Today we are going to study a relatively new effect ( new experimentally speaking, because it was first detected when I was an undergraduate student, in 2000) but it is not so new from the theoretical aside (theoretically, it was predicted in 1962). This effect is closely related to the Cherenkov effect. It is named Askaryan effect or Askaryan radiation, see below after a brief recapitulation of the Cherenkov effect last post we are going to do in the next lines.

We do know that charged particles moving faster than light through the vacuum emit Cherenkov radiation. How can a particle move faster than light? The weak speed of a charged particle can exceed the speed of light. That is all. About some speculations about the so-called tachyonic gamma ray emissions, let me say that the existence of superluminal energy transfer has not been established so far, and one may ask why. There are two options:

1) The simplest solution is that superluminal quanta just do not exist, the vacuum speed of light being the definitive upper bound.

2) The second solution is that the interaction of superluminal radiation with matter is very small, the quotient of tachyonic and electric fine-structure constants being $q_{tach}^2/e^2<10^{-11}$. Therefore superluminal quanta and their substratum are hard to detect.

A related and very interesting question could be asked now related to the Cherenkov radiation we have studied here. What about neutral particles? Is there some analogue of Cherenkov radiation valid for chargeless or neutral particles? Because neutrinos are electrically neutral, conventional Cherenkov radiation of superluminal neutrinos does not arise or it is otherwise weakened. However neutrinos do carry electroweak charge and may emit certain Cherenkov-like radiation via weak interactions when traveling at superluminal speeds. The Askaryan effect/radiation is this Cherenkov-like effect for neutrinos, and we are going to enlighten your knowledge of this effect with this entry.

We are being bombarded by cosmic rays, and even more, we are being bombarded by neutrinos. Indeed, we expect that ultra-high energy (UHE) neutrinos or extreme ultra-high energy (EHE) neutrinos will hit us as too. When neutrinos interact wiht matter, they create some shower, specifically in dense media. Thus, we expect that the electrons and positrons which travel faster than the speed of light in these media or even in the air and they should emit (coherent) Cherenkov-like radiation.

Let me quote what wikipedia say about him: Gurgen Askaryan (December 14, 1928-1997) was a prominent Soviet (armenian) physicist, famous for his discovery of the self-focusing of light, pioneering studies of light-matter interactions, and the discovery and investigation of the interaction of high-energy particles with condensed matter. He published more than 200 papers about different topics in high-energy physics.

Other interesting ideas by Askaryan: the bubble chamber (he discovered the idea independently to Glaser, but he did not published it so he did not win the Nobel Prize), laser self-focussing (one of the main contributions of Askaryan to non-linear optics was the self-focusing of light), and the acoustic UHECR detection proposal. Askaryan was the first to note that the outer few metres of the Moon’s surface, known as the regolith, would be a sufficiently transparent medium for detecting microwaves from the charge excess in particle showers. The radio transparency of the regolith has since been confirmed by the Apollo missions.

## What is the Askaryan effect?

The next figure is from the Askaryan radiation detected by the ANITA experiment:

The Askaryan effect is the phenomenon whereby a particle traveling faster than the phase velocity of light in a dense dielectric medium (such as salt, ice or the lunar regolith) produces a shower of secondary charged particles which contain a charge anisotropy  and thus emits a cone of coherent radiation in the radio or microwave  part of the electromagnetic spectrum. It is similar, or more precisely it is based on the Cherenkov effect.

High energy processes such as Compton, Bhabha and Moller scattering along with positron annihilation  rapidly lead to about a 20%-30% negative charge asymmetry in the electron-photon part of a cascade. For instance, they can be initiated by UHE (higher than, e.g.,100 PeV) neutrinos.

1962, Askaryan first hypothesized this effect and suggested that it should lead to strong coherent radio and microwave Cherenkov emission for showers propagating within the dielectric. Since the dimensions of the clump of charged particles are small compared to the wavelength of the radio waves, the shower radiates coherent radio Cherenkov radiation whose power is proportional to the square of the net charge in the shower. The net charge in the shower is proportional to the primary energy so the radiated power scales quadratically with the shower energy, $P_{RF}\propto E^2$.

Indeed, these radio and coherent radiations are originated by the Cherenkov effect radiation. We do know that:

$\dfrac{P_{CR}}{d\nu}\propto \nu d\nu$

from the charged particle in a dense (refractive) medium experimenting Cherenkov radiation (CR). Every charge emittes a field $\vert E\vert\propto \exp (i\mathbf{k}\cdot\mathbf{r})$. Then, the power is proportional to $E^2$. In a dense medium:

$R_{M}\sim 10cm$

We have two different experimental and interesting cases:

A) The optical case, with $\lambda <. Then, we expect random phases and $P\propto N$.

B) The microwave case, with $\lambda>>R_M$. In this situation, we expect coherent radiation/waves with $P\propto N^2$.

We can exploit this effect in large natural volumes transparent to radio (dry): pure ice, salt formations, lunar regolith,…The peak of this coherent radiation for sand is produced at a frequency around $5GHz$, while the peak for ice is obtained around $2GHz$.

The first experimental confirmation of the Askaryan effect detection were the next two experiments:

1) 2000 Saltzberg et.al., SLAC. They used as target silica sand. The paper is this one http://arxiv.org/abs/hep-ex/0011001

2) 2002 Gorham et.al., SLAC. They used a synthetic salt target. The paper appeared in this place http://arxiv.org/abs/hep-ex/0108027

Indeed, in 1965, Askaryan himself proposes ice and salt as possible target media. The reasons are easy to understand:
1st. They provide high densities and then it means a higher probability for neutrino interaction.
2nd. They have a high refractive index. Therefore, the Cerenkov emission becomes important.
3rd. Salt and ice are radio transparent, and of course, they can be supplied in large volumes available throughout the world.

1) Low attenuation: clear signals from large detection volumes.
2) We can observe distant and inclined events.
3) It has a high duty cycle: good statistics in less time.
4) I has a relative low cost: large areas covered.
5) It is available for neutrinos and/or any other chargeless/neutral particle!

Problems with this Askaryan effect detection are, though: radio interference, correlation with shower parameters (still unclear), and that it is limited only to particles with very large energies, about $E>10^{17}eV$.

In summary:

Askaryan effect = coherent Cerenkov radiation from a charge excess induced by (likely) neutral/chargeless particles like (specially highly energetic) neutrinos passing through a dense medium.

## Why the Askaryan effect matters?

It matters since it allows for the detection of UHE neutrinos, and it is “universal” for chargeless/neutral particles like neutrinos, just in the same way that the Cherenkov effect is universal for charged particles. And tracking UHE neutrinos is important because they point out towards its source, and it is suspected they can help us to solve the riddle of the origin and composition of cosmic rays, the acceleration mechanism of cosmic radiation, the nuclear interactions of astrophysical objects, and tracking the highest energy emissions of the Universe we can observe at current time.

Is it real? Has it been detected? Yes, after 38 years, it has been detected. This effect was firstly demonstrated in sand (2000), rock salt (2004) and ice (2006), all done in a laboratory at SLAC and later it has been checked in several independent experiments around the world. Indeed, I remember to have heard about this effect during my darker years as undergraduate student. Fortunately or not, I forgot about it till now. In spite of the beauty of it!

Moreover, it has extra applications to neutrino detection using the Moon as target: GLUE (detectors are Goldstone RTs), NuMoon (Westerbork array; LOFAR), or RESUN (EVLA), or the LUNASKA project. Using ice as target, there has been other experiments checking the reality of this effect: FORTE (satellite observing Greenland ice sheet), RICE (co-deployed on AMANDA strings, viewing Antarctic ice), and the celebrated ANITA (balloon-borne over Antarctica, viewing Antarctic ice) experiment.

Furthermore, even some experiments have used the Moon (an it is likely some others will be built in the near future) as a neutrino detector using the Askaryan radiation (the analogue for neutral particles of the Cherenkov effect, don’t forget the spot!).

## Askaryan effect and the mysterious cosmic rays.

Askaryan radiation is important because is one of the portals of the UHE neutrino observation coming from cosmic rays. The mysteries of cosmic rays continue today. We have detected indeed extremely energetic cosmic rays beyond the $10^{20}eV$ scale. Their origin is yet unsolved. We hope that tracking neutrinos we will discover the sources of those rays and their nature/composition. We don’t understand or know any mechanism being able to accelerate particles up to those incredible particles. At current time, IceCube has not detected UHE neutrinos, and it is a serious issue for curren theories and models. It is a challenge if we don’t observe enough UHE neutrinos as the Standard Model would predict. Would it mean that cosmic rays are exclusively composed by heavy nuclei or protons? Are we making a bad modelling of the spectrum of the sources and the nuclear models of stars as it happened before the neutrino oscillations at SuperKamiokande and Kamikande were detected -e.g.:SN1987A? Is there some kind of new Physics living at those scales and avoiding the GZK limit we would naively expect from our current theories?