# LOG#121. Basic Neutrinology(VI).

**Posted:**2013/07/15

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

**Tags:**3-brane, bulk, bulk mode, bulk state, extra dimensions, extra space-like dimension, KK resonance, KK state, KK tower, left-handed neutrino, model building, neutrino mass matrix from extra dimensions, neutrinology, non-factorirable metric, Planck scale, Randall-Sundrum model, renormalizability, right-handed neutrino, SM brane, sterile neutrino, submicrometer extra dimension, submillimeter extra dimension Leave a comment

Models where the space-time is not 3+1 dimensional but higher dimensional (generally D=d+1=4+n dimensional, where n is the number of spacelike extra dimensions) are popular since the beginnings of the 20th century.

The fundamental scale of gravity need not to be the 4D “effective” Planck scale but a new scale (sometimes called ), and it could be as low as . The observed Planck scale (related to the Newton constant ) is then related to in dimensions by a relationship like the next equation:

Here, is the radius of the typical length of the extra dimensions. We can consider an hypertorus for simplicity (but other topologies are also studied in the literature). In fact, the coupling is if we choose . When we take more than one extra dimension, e.g., taking , the radius R of the extra dimension(s) can be as “large” as 1 millimeter! This fact can be understood as the “proof” that there could be hidden from us “large” extra dimensions. They could be only detected by many, extremely precise, measurements that exist at present or future experiments. However, it also provides a new test of new physics (perhaps fiction science for many physicists) and specially, we could explore the idea of hidden space dimensions and how or why is so feeble with respect to any other fundamental interaction.

According to the SM and the standard gravity framework (General Relativity), every group charged particle is localized on a 3-dimensional hypersurface that we could call “brane” (or SM brane). This brane is embedded in “the bulk” of the higher dimensional Universe (with extra space-like dimensions). All the particles can be separated into two categories: 1) those who live on the (SM) 3-brane, and 2) those who live “everywhere”, i.e., in “all the bulk” (including both the extra dimensions and our 3-brane where the SM fields only can propagate). The “bulk modes” are (generally speaking) quite “model dependent”, but any coupling between the brane where the SM lives and the bulk modes should be “suppressed” somehow. One alternative is provided by the geometrical factors of “extra dimensions” (like the one written above). Another option is to modify the metric where the fields propagate. This last recipe is the essence of non-factorizable models built by Randall, Sundrum, Shaposhnikov, Rubakov, Pavŝiĉ and many others as early as in the 80’s of the past century. Graviton and its “propagating degrees of freedom” or possible additional neutral states belongs to the second category. Indeed, the observed weakness of gravity in the 3-brane can be understood as a result of the “new space dimensions” in which gravity can live. However, there is no clear signal of extra dimensions until now (circa 2013, July).

The small coupling constant derived from the Planck mass above can also be used in order to explain the smallness of the neutrino masses! The left-handed neutrino having weak isospin and hypercharge is thought to reside in the SM brane in this picture. It can get a “naturally samll” Dirac mass through the mixing with some “bulk fermion” (e.g., the right-handed neutrino or any other neutral fermion under the SM gauge group) which can be interpreted as a right-handed neutrino :

Here, are the two Higgs doublet fields and the Yukawa coupling, respectively. After spontaneous symmetry breaking, this interaction will generate the Dirac mass term

The right-handed neutrino has a hole tower of Kaluza-Klein relatives . The masses of these states are given by

and the couples with all KK state having the same “mixing” mass. Thus, we can write the mass lagrangian as

with

Are you afraid of “infinite” neutrino flavors? The resulting neutrino mass matrix M is “an infinite array” with structure:

The eigenvalues of the matrix are given by a trascendental equation. In the limit where , or , the eigenvalues are , where and is a double eigenvalue (i.e., it is doubly degenerated). There are other examples with LR symmetry. For instance, right-handed neutrinos that, living on the SM brane, were additional neutrino species. In these models, it has been showed that the left-handed neutrino is exactly massless whereas the assumed bulk and “sterile” neutrino have a mass related to the size of the extra dimensions. These models produce masses that can be fitted to the expected values coming from estimations at hand with the neutrino oscillation data, but generally, this implies that there should be at least one extra dimension with size in the micrometer range or less!

The main issues that extra dimension models of neutrino masses do have is related to the question of the renormalizability of their interactions. With an infinite number of KK states and/or large extra dimensions, extreme care have to be taken in order to not spoil the SM renormalizability and, at some point, it implies that the KK tower must be truncated at some level. There is no general principle or symmetry that explain this cut-off to my knowledge.

**May the neutrinos and the extra dimensions be with you!**

**See you in my next neutrinological post!**

# LOG#120. Basic Neutrinology(V).

**Posted:**2013/07/15

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

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

Supersymmetry (SUSY) is one of the most discussed ideas in theoretical physics. I am not discussed its details here (yet, in this blog). However, in this thread, some general features are worth to be told about it. SUSY model generally include a symmetry called R-parity, and its breaking provide an interesting example of how we can generate neutrino masses WITHOUT using a right-handed neutrino at all. The price is simple: we have to add new particles and then we enlarge the Higgs sector. Of course, from a pure phenomenological point, the issue is to discover SUSY! On the theoretical aside, we can discuss any idea that experiments do not exclude. Today, after the last LHC run at 8TeV, we have not found SUSY particles, so the lower bounds of supersymmetric particles have been increased. Which path will Nature follow? SUSY, LR models -via GUTs or some preonic substructure, or something we can not even imagine right now? Only experiment will decide in the end…

In fact, in a generic SUSY model, dut to the Higgs and lepton doublet superfields, we have the same quantum numbers. We also have in the so-called “superpotential” terms, bilinear or trilinear pieces in the superfields that violate the (global) baryon and lepton number explicitly. Thus, they lead to mas terms for the neutrino but also to proton decays with unacceptable high rates (below the actual lower limit of the proton lifetime, about years!). To protect the proton experimental lifetime, we have to introduce BY HAND a new symmetry avoiding the terms that give that “too high” proton decay rate. In SUSY models, this new symmetry is generally played by the R-symmetry I mentioned above, and it is generally introduced in most of the simplest models including SUSY, like the Minimal Supersymmetric Standard Model (MSSM). A general SUSY superpotential can be written in this framework as

(1)

A less radical solution is to allow for the existence in the superpotential of a bilinear term with structure . This is the simplest way to realize the idea of generating the neutrino masses without spoiling the current limits of proton decay/lifetime. The bilinear violation of R-parity implied by the term leads by a minimization condition to a non-zero vacuum expectation value or vev, . In such a model, the neutrino acquire a mass due to the mixing between neutrinos and the neutralinos.The neutrinos remain massless in this toy model, and it is supposed that they get masses from the scalar loop corrections. The model is phenomenologically equivalent to a “3 Higgs doublet” model where one of these doublets (the sneutrino) carry a lepton number which is broken spontaneously. The mass matrix for the neutralino-neutrino secto, in a “5×5” matrix display, is:

(2)

and where the matrix corresponds to the two “gauginos”. The matrix is a 2×3 matrix and it contains the vevs of the two higgses plus the sneutrino, i.e., respectively. The remaining two rows are the Higgsinos and the tau neutrino. It is necessary to remember that gauginos and Higgsinos are the supersymmetric fermionic partners of the gauge fields and the Higgs fields, respectively.

I should explain a little more the supersymmetric terminology. The *neutralino* is a hypothetical particle predicted by supersymmetry. There are some neutralinos that are fermions and are electrically neutral, the lightest of which is typically stable. They can be seen as mixtures between binos and winos (the sparticles associated to the b quark and the W boson) and they are generally Majorana particles. Because these particles only interact with the weak vector bosons, they are not directly produced at hadron colliders in copious numbers. They primarily appear as particles in cascade decays of heavier particles (decays that happen in multiple steps) usually originating from colored supersymmetric particles such as squarks or gluinos. In R-parity conserving models, the lightest neutralino is stable and all supersymmetric cascade-decays end up decaying into this particle which leaves the detector unseen and its existence can only be inferred by looking for unbalanced momentum (missing transverse energy) in a detector. As a heavy, stable particle, the lightest neutralino is an excellent candidate to comprise the universe’s cold dark matter. In many models the lightest neutralino can be produced thermally in the hot early Universe and leave approximately the right relic abundance to account for the observed dark matter. A lightest neutralino of roughly GeV is the leading weakly interacting massive particle (WIMP) dark matter candidate.

**Neutralino dark matter** could be observed experimentally in nature either indirectly or directly. In the former case, gamma ray and neutrino telescopes look for evidence of neutralino annihilation in regions of high dark matter density such as the galactic or solar centre. In the latter case, special purpose experiments such as the (now running) Cryogenic Dark Matter Search (CDMS) seek to detect the rare impacts of WIMPs in terrestrial detectors. These experiments have begun to probe interesting supersymmetric parameter space, excluding some models for neutralino dark matter, and upgraded experiments with greater sensitivity are under development.

If we return to the matrix (2) above, we observe that when we diagonalize it, a “seesaw”-like mechanism is again at mork. There, the role of can be easily recognized. The mass is provided by

where and is the largest gaugino mass. However, an arbitrary SUSY model produces (unless M is “large” enough) still too large tau neutrino masses! To get a realistic and small (1777 GeV is “small”) tau neutrino mass, we have to assume some kind of “universality” between the “soft SUSY breaking” terms at the GUT scale. This solution is not “natural” but it does the work. In this case, the tau neutrino mass is predicted to be tiny due to cancellations between the two terms which makes negligible the vev . Thus, (2) can be also written as follows

(3)

We can study now the elementary properties of neutrinos in some elementary superstring inspired models. In some of these models, the effective theory implies a supersymmetric (exceptional group) GUT with matter fields belong to the 27 dimensional representation of the exceptional group plus additional singlet fields. The model contains additional neutral leptons in each generation and the neutral singlets, the gauginos and the Higgsinos. As the previous model, but with a larger number of them, every neutral particle can “mix”, making the undestanding of the neutrino masses quite hard if no additional simplifications or assumptions are done into the theory. In fact, several of these mechanisms have been proposed in the literature to understand the neutrino masses. For instance, a huge neutral mixing mass matris is reduced drastically down to a “3×3” neutrino mass matrix result if we mix and with an additional neutral field whose nature depends on the particular “model building” and “mechanism” we use. In some basis , the mass matrix can be rewritten

(4)

and where the energy scale is (likely) close to zero. We distinguish two important cases:

1st. R-parity violation.

2nd. R-parity conservation and a “mixing” with the singlet.

In both cases, the sneutrinos, superpartners of are assumed to acquire a v.e.v. with energy size . In the first case, the field corresponds to a gaugino with a Majorana mass than can be produced at two-loops! Usually , and if we assume , then additional dangerous mixing wiht the Higgsinos can be “neglected” and we are lead to a neutrino mass about , in agreement with current bounds. The important conclusion here is that we have obtained the smallness of the neutrino mass without any fine tuning of the parameters! Of course, this is quite subjective, but there is no doubt that this class of arguments are compelling to some SUSY defenders!

In the second case, the field corresponds to one of the singlets. We have to rely on the symmetries that may arise in superstring theory on specific Calabi-Yau spaces to restric the Yukawa couplings till “reasonable” values. If we have in the matrix (4) above, we deduce that a massless neutrino and a massive Dirac neutrino can be generated from this structure. If we include a possible Majorana mass term of the sfermion at a scale , we get similar values of the neutrino mass as the previous case.

**Final remark:** mass matrices, as we have studied here, have been proposed without embedding in a supersymmetric or any other deeper theoretical frameworks. In that case, small tree level neutrino masses can be obtained without the use of large scales. That is, the structure of the neutrino mass matrix is quite “model independent” (as the one in the CKM quark mixing) if we “measure it”. Models reducing to the neutrino or quark mass mixing matrices can be obtained with the use of large energy scales OR adding new (likely “dark”) particle species to the SM (not necessarily at very high energy scales!).

# LOG#119. Basic Neutrinology(IV).

**Posted:**2013/07/14

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

**Tags:**GUT, LR models, neutrinology, seesaw mechanism, SO(10) unification, SUSY unification Leave a comment

A very natural way to generate the known neutrino masses is to minimally extend the SM including additional 2-spinors as RH neutrinos and at the same time extend the non-QCD electroweak SM gauge symmetry group to something like this:

The resulting model, initially proposed by Pati and Salam (** Phys. Rev. D.10. 275**) in 1973-1974. Mohapatra and Pati reviewed it in 1975, here

*. It is also reviewed in*

**Phys. Rev. D. 11. 2558***. This class of models were first proposed with the goal of seeking a spontaneous origin for parity (P) violations in weak interactions. CP and P are conserved at large energies but at low energies, however, the group breaks down spontaneouly at some scale . Any new physics correction to the SM would be of order*

**Unification and Supersymmetry: the frontiers of Quark-Lepton Physics. Springer-Verlag. N.Y.1986**and where

If we choose the alternative , we obtain only small corrections, compatible with present known physics. We can explain in this case the small quantity of CP violation observed in current experiments and why the neutrino masses are so small, as we will see below a little bit.

The quarks and their fields, and the leptons and their fields , in the LR models transform as doublets under the group in a simple way. and . The gauge interactions are symmetric under left-handed and right-handed fermions. Thus, before spontaneous symmetry breaking, weak interactions, as any other interaction, would conserve parity symmetry and would become P-conserving at higher energies.

The breaking of the gauge symmetry is implemented by multiplets of LR symmetric Higgs fields. The concrete selection of these multiplets is NOT unique. It has been shown that in order to understand the smallness of the neutrino masses, it is convenient to choose respectively one doublet and two triplets in the following way:

The Yukawa couplings of these Higgs fields with the quarks and leptons are give by the lagrangian term

The gauge symmetry breaking in LR models happens in two steps:

1st. The is broken down to by the v.e.v. . It carries both and quantum numbers. It gives mass to charged and neutral RH gauge bosons, i.e.,

and

Furthermore, as consequence of the f-term in the lagrangian, above this stage of symmetry breaking also leads to a mass term for the right-handed neutrinos with order about .

2nd. As we break the SM symmetry by turning on the vev’s for the scalar fields

with

We give masses to the and bosons, as well as to quarks or leptons (). At the end of the process of spontaneous symmetry breaking (SSB), the two W bosons of the model will mix, the lowest physical mass eigenstate is identified as the observed W boson. Current experimental limits set the limit to . The LHC has also raised this bound the past year!

In the neutrino sector, the above Yukawa couplings after breaking by leads to the Dirac masses for the neutrino. The full process leads to the following mass matrix for the states in the general neutrino mass matrix

corresponding to the lighter and more massive neutrino states after the diagonalization procedure. In fact, the seesaw mechanism implies the eigenvalue

for the lowest mass, and the eigenvalue

for the (super)massive neutrino state. Several variants of the basic LR models include the option of having Dirac neutrinos at the expense of enlarging the particle content. The introduction of two new single fermions and a new set of carefully chosen Higgs bosons, allows us to write the mass matrix

This matrix leads to two different Dirac neutrinos, one heavy with mass and another lighter with mass . This light four component spinor has the correct weak interaction properties to be identified as the neutrino. A variant of this model can be constructed by addition of singlet quarks and leptons. We can arrange these new particles in order that the Dirac mass of the neutrino vanishes at tree level and/or arises at the one-loop level via boson mixing!

Left-Right symmetric(LR) models can be embedded in grand unification groups. The simplest GUT model that leads by successive stages of symmetry breaking to LR symmetric models at low energies is GUT-based models. An example of LR embedding GUT supersymmetric theory can be even discussed in the context of (super)string-inspired models.

# LOG#117. Basic Neutrinology(II).

**Posted:**2013/07/13

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

**Tags:**BSM, CC lagrangian, lepton number, lepton number violating reaction, lepton number violations, NC lagrangian, neutrinology, New Physics, Standard Model neutrinos Leave a comment

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” with respect to the group, the weak interaction group after the electroweak symmetry breaking, and we have:

These doublets have the 3rd component of the weak isospin and they are assigned an unit of the global 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:

and where 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 , while the structure of the CC interactions is what determine the notion of flavor neutrinos .

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.

2.

3.

4.

5.

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 decay is given by (2 flavor mixing only is assumed):

and where are the neutrino masses, their squared mass difference, is the W boson mass and 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 without loss of generality), we obtain that

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 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#115. Bohr’s legacy (III).

**Posted:**2013/07/10

**Filed under:**Chemistry, Gravitational theories, Physmatics, Quantum Gravity, Quantum Physics |

**Tags:**Bohr model, generalized uncertainty principle, gravatom, gravitational atom, MOG, MOND, Quantum Gravity, verlinde, verlinde's approach to gravity Leave a comment

# Dedicated to Niels Bohr

# and his atomic model

# (1913-2013)

# 3rd part:

# From gravatoms to dark matter

## Gravatoms

Imagine a proton an an electron were bound together in a hydrogen atom by gravitational forces and not by electric forces. We have two interesting problems to solve here:

1st. Find the formula for the spectrum (energy levels) of such a gravitational atom (or gravatom), and the radius of the ground state for the lowest level in this gravitational Bohr atom/gravatom.

2nd. Find the numerical value of the Bohr radius for the gravitational atom, the “rydberg”, and the “largest” energy separation between the energy levels found in the previous calculation.

We will take the values of the following fundamental constants:

, the reduced Planck constant.

, the proton mass.

, the electron mass.

, the gravitational Newton constant.

Let R be the radius of any electron orbit. The gravitational force between the electron and the proton is equal to:

(1)

The centripetal force is necessary to keep the electron in any circular orbit. According to the gravatom hypothesis, it yields the value of the gravitational force (the electric force is neglected):

(2)

(3)

Using the hypothesis of the Bohr atomic model in this point, i.e., that “the allowed orbits are those for whihc the electron’s orbital angular momentum about the nucleus is an integral multiple of “, we get

(4)

Then,

(5) and

From (3), we obtain

(6)

Comparing (5) with (6), we deduce that

(7)

and thus

(8)

This is the gravatom equivalent of Bohr radius in the common Bohr model for the hydrogen atom. To get the spectrum, we recall that total energy is the sum of kinetic and potential energy:

Using the value we obtained in (5), by direct substitution, we have

(9)

and then

(10)

and so the spectrum of this gravatom is given by

(11)

For n=1 (the ground state), we have the analogue of the Bohr radius in the gravatom to be:

For comparison, the radius of the known Universe is about . Therefore, !!!!!! is very huge because gravitational forces are much much weaker than electrostatic forces! Moreover, the energy in the ground state n=1 for this gravatom is:

The energy separation between this and the next gravitational level would be about this quantity in absolute value, i.e.,

This really tiny energy separation is beyond any current possible measurement. Therefore, we can not measure energy splittings in “gravatoms” with known techniques. Of course, gravatoms are a “toy-model” or hypothetical systems (bubble Universes?).

**Remark (I)**: The quantization of angular momentum provided the above gravatom spectrum. It is likely that a full Quantum Gravity theory provides additional corrections to the quantum potential, just in the same way that QED introduces logarithmic (vacuum polarization) corrections and others (due to relativity or additional quantum effects).

**Remark (II)**: Variations in the above quantization rules can modify the spectrum.

**Remark (III)**: In theories with extra dimensions, is changed by a higher value as a function of the compactification radius. So, the effect of large enough extra dimensions could be noticed as “dark matter” if it is “big enough”. Can you estimate how large could the compactification radius be in such a way that the separation between n=1 and n=2 for the gravatom could be measured with current technology?* Hint:* you need to know what is the tiniest energy separation we can measure with current experimental devices.

**Remark (IV)**: In Verlinde’s entropic approach to gravity, extra corrections arise due to the change of the functional entropy we choose. It can be due to extra dimensions and the (stringy) Generalized Uncertainty Principle as well.

## Gravatoms and Dark Matter: a missing link

I will end this thread of 3 posts devoted to Bohr’s centenary model to recall a connection between atomic physics and the famous Dark Matter problem! The calculations I performed above (and which anyone with a solid, yet elementary, ground knowledge in physics can do) reveals a surprising link between microscopic gravity and the dark matter problem. I mean, the problem of gravatoms can be matched to the problem of dark matter if we substitute the proton mass by the mass of a galaxy! It is not an unlikely option that the whole Dark Matter problem shows to be related to a right infrared/long scale modified gravitational theory induced by quantum gravity. Of course, this claim is quite an statement! I work on this path since months ago…Even when MOND (MOdified Newtonian Dynamics) or MOG (MOdified Gravity) have been seen as controversial since Milgrom’s and Moffat’s pioneer works, I believe it is yet to come its “to be or not to be” biggest test. Yes, even when some measurements like the Bullet Cluster observations and current simulations of galaxy formation requires a component of dark matter, I firmly believe (similarly, I think, to V. Rubin’s opinion) that if the current and the next generation of experiments trying to discover the “dark matter particle/family of particles” fails, we should take this option more seriously than some people are able to accept at current time.

**May the Bohr model and gravatoms be with you!**