Neutrino, Right-Handed (and Some SM Problems, 2021 Edition)

Contents

See-saw Mechanism

• Neutrino was proposed in 1930 as the missing part in beta decay.
• It was detected in 1942 by the beta capture experiment.

It was a cherished concept that theories in physics would be invariance if the left-handed space (x,y,z) is replaced by the right-handed version (-x,-y,-z) - a process called parity transformation. Parity violation in weak interaction was demonstrated in 1956 by beta decay of cobalt-60. The experiment shows that there is only left-handed neutrino and the right-handed version is absence (Figure 01,a).

Figure 01 See-saw Mechanism [view large image]

By 1975, all the elementary particles are codified in the Standard Model (SM). The mathematics is assembled into the Lagrangians as shown below just for the fermion mass generating term (see Figure 01,b for a pictorial representation) :

• According to SM, the mass term of the Lagrangian L_mass endows mass to the fermion, i.e.,
  L_mass = G_e(LHR + RHL) ---------- (1),
  where L is the left-handed fermion field, R the right-handed, H the Higgs field, G_e the Yukawa coupling constant (for each kind of fermion), and the little dagger "" denotes = (L)_ij = (L*)_ji.
• As shown in Figure 01,b top, the electron has both left-handed and right-handed components. Time development of the field alternates between the left/right handed version as it interacting with the Higgs. hield.
• The same diagram shows that the neutrino is massless as R 0 L_mass 0.
• The solar neutrino problem was observed in the mid-1960s. It concerned a large discrepancy between the flux of solar neutrinos as predicted from the Sun's luminosity. The problem was resolved around 2002 by detection of oscillation between the 3 flavors
  (,,). Such mixing in flight can happen only when the neutrinos have mass (see "Detection of Neutrino Mass").

The simplest way to introduce neutrino mass into the Standard Model is to add a mass term via the same Higgs mechanism for the electron and quark similar to Eq.(1) for each flavor. But this so-called "Dirac Mass" is calculated to be at least in the order of the electron's (too much) or the neutrino interactions with the Higgs boson at least 1012 times weaker than that of the top quark (too little for trying to reduce the mass ?).

Figure 02 Neutrino Mass [view large image]

Figure 02 shows the estimated mass of the neutrinos ~ 106 - 1012 times smaller than the other fermions.

• The Seesaw Mechanism - In this scheme, it is possible for right-handed neutrinos to have a mass of their own without relying on the Higgs boson. Unlike other quarks and leptons (including the electrons and left-handed neutrinos), the mass of the right-handed Majorana neutrino, M, is not tied to the mass scale of the Higgs boson, it can be much heavier than other particles. When a left-handed neutrino collides with the Higgs boson, it acquires a mass, m, which is comparable to the mass of other quarks and leptons. At the same time it transforms into a right-handed Majorana neutrino, which is much heavier than energy conservation would normally allow and to be the anti-neutrino itself. This kind of virtual particle can occur in a very short time interval as ascribed by the uncertainty principle. It has an effective mass of m²/M from these two alternative states (Figure 01,b).
  

  For m ~ 105 ev (~ mass of the electron), and M ~ 102 Gev (~ electro-weak symmetry breaking scale, see Figure 03), the neutrino mass would be of the order 0.1 ev. There is no need to reduce the strength for the Higgs coupling in this scenario. But the conservation of lepton number is violated. Such non-conserving process is still awaiting further experimental result (e.g., the neutrinoless double-beta-decay) to confirm. Although the seesaw model is extremely appealing in the sense that it gives a natural qualitative explanation of the smallness of neutrino masses, it still leaves too many possible reasonable combinations to use it for quantitative goals.

  Figure 03 Symmetry Breakings [view large image]

  To recap :

  • In SM, the fermion mass term (called Dirac mass) is given by L_Dirac = G_e(LHR + RHL).
  • In the See-Saw mechanism, since the Majorana neutrino is its own anti-neutrino, it is proposed that R (m/M)L, where m is the Dirac mass, and M is actually just a parameter but considered to be the mass of the Majorana neutrino.
  • Thus, L_Majo G_e(m/M)LHL.
  • After Symmetry Breaking, H settles down to a new "Vacuum Expectation Value" v = 246 Gev, L_Majo G_ev(m/M)LL.
  • Since the Dirac mass m = G_ev and LL = 1, L_Majo m²/M.
  • For electron, G_e = 2x10^-6 giving the Dirac mass m_e ~ 5 Mev. While for the top quark, G_e ~ 0.7 giving m_top ~ 172 Gev (see "Mass and coupling to the Higgs boson"). The problems with Dirac mass for neutrino are that it's too much if G_e is similar to the electron's, or G_e would be too feeble (?) ~ 10^-12 to get m_v ~ 0.1 ev (or it may be just the case ?). Hence the See-Saw mechanism is proposed to provide an alternate way. Validation of such scheme depends on the discovery of Majorana neutrino by the NDB experiment.

Leptogenesis

There are four different theories trying to explain the matter-antimatter asymmetry:


1. CP violation - The standard model of elementary particle suggests that when the universe was less than 10^-12 sec old, the condition was ripe for the production of more matter than antimatter with CP violation to provide the mechanism for different reaction rate (to produce matter and antimatter). The explanation sounds reasonable, however, theoretical calculation as well as experimental measurement shows an excess far too small to account for the observed degree of asymmetry.


2. Supersymmetry - Supersymmetry is one of the most promising extensions of the standard model of elementary particle. It encompasses many as-yet-unknown particles and perhaps new kind of interactions. It is suggested that new interaction outside the standard model might act differently on quarks and antiquarks, and produced the excess of quarks in our universe. Although there are inconclusive claims of discovery, so far (as of 2021) there is no hard evidence for supersymmetry from experiments.


3. Electric dipole moment - The presence of an electric dipole moment (EDM) in elementary particles would allow matter and antimatter to decay at different rates leading to a possible matter-antimatter asymmetry. But all experiments including the one by ACME in 2014 have failed to detect such EDM. See "Axion" for an attempt to explain the absence or near zero of the EDM.


4. Leptogenesis - This explanation also requires new physics beyond the standard model. It assumes the existence of a new type of very heavy neutrino in very early universe. According to the leptogenesis scenario, the heavy neutrinos would decay into either neutrinos or antineutrinos. Then the standard model predicts that certain reactions could occur in the very high-temperature conditions to convert antineutrinos into matter particles (and thus the lepton number is not conserved, e.g., from 0 to 2), eventually producing neutrons and protons leaving the universe devoid of antimatter (see Neutrinoless Double-Beta decay). So far there is only one controversial claim in 2001 to have observed such reaction.

baryogenesis) suggests that an exceptionally heavy but unstable breed of Majorana neutrino existed in the very early universe. Their subsequent decay generated more anti-leptons than leptons. A mechanism called "Sphaleron" then converted 1/3 of the excess anti-leptons into baryons leading to the imbalance between matter and antimatter at the dawn of time. Theoretical analysis shows that leptogenesis works best when the neutrino masses are in the range 0.1 ev - 1 mev; the mass of the Majorana neutrino and the reheating temperature must be larger than 109 Gev.

Figure 04 Leptogenesis [view large image]

Such theory is the most favored mechanism to explain the matter anti-matter asymmetry because there is good circumstantial evidence for existence of various ingredients (except the requirement of heavy Majorana neutrino at reheating when particles were massless according to SM). See detail :

Before going to the mathematical formalism, Figure 05 shows the time line leading to the domination of matter and beyond :



The Era of Inflation lasted from 10-35 to 10-33 sec. At the end of inflation, the inflaton field dumped its excessive energy to create particle/anti-particle pairs as it decayed to the true vacuum in a process called "reheating" (see also "Cosmic Inflation and Reheating"). It followecd a period with equal number of particles and anti-particles (all massless) until the "Spontaneous/Electroweak, Symmetry Breaking (SSB/ESB)" transition at 10-12 sec after BB. It it during this SSB/ESB process that particles became massive and leptogenesis transform the universe to matter only (see mathematical detail below).

Figure 05 Leptogenesis, Time Line [view large image]

CMB data reveal the extent of matter/anti-matter imbalance. It estimates 411 photons/cm3 as well as 4% of the total cosmic energy density (~ 10-29 gm/cm3) for baryons (mostly nucleons with mass ~ 1.6x10-24 gm), from which the ratio of matter/radiation = 6x10-10 is obtained.

Here's the mathematics :


• Leptogenesis is based on the following extension of the SM Lagrangian L₃ (see Figure 07). Beside the usual kinetic term, the interaction with the Higgs field is similar to the last term of L₃ with R (the right-handed lepton field) replaced by N (the right-handed heavy neutrino field) :
  

  Figure 06 Speed of Massless and Massive Particle [view large image]	(see Figure 06). The Lagrangian above is for generating the so-called "Dirac mass". There is another mass producing mechanism by Majorana neutrino, which is its own anti-self (see "See-Saw").

  
  
• In order to produce baryon asymmetry, it is necessary to satisfy the 3 conditions according to Andrei Sakharov :

  Baryon number violation - It could be fulfilled by the Sphaleron process which transform anti-leptons to baryons (Figure 04). It doesn't need energy to run in the symmetric phase before SSB/ESB. However, it requires lot of energy (~ 10 Tev) to operate afterward. Violation of C and CP invariance - It is expressed by : Thermal non-equilibrium - This condition is expressed by the wash-out indicator : Non-equilibrium is necessary to prevent excessive baryons to be wiped out by Sphaleron which can run both ways. This condition can be satified if the universe expands faster than the rate of reaction or decay leaving no chance for wash-out.

  Figure 07 Lepton Lagrangian [view large image]

  This condition determines the time when equilibrium is reached in relation to the processing of the following differential equations. The rate of cosmic expansion slows down over time, i.e., non-equilibrium previals mostly at earlier time.

• The asymmetry generation process is determined by the equations :
  
• Numerical Verification :
  • Development of matter/anti-matter asymmetry is assumed to occur at a short time interval about 10^-12 sec after BB around the processing of SSB/ESB (see Figure 08). The N neutrinos could be produced during reheating after inflation starting with an initial number density n₀. The solutions for n or n_B-L (in Eq.(2) or (4)) suggest that the N neutrino decay rate
    ~ 10¹² sec^-1 similar to many present-day meson decay.
  • The rate of cosmic expansion is expressed in term of the Hubble parameter H = (dR/dt)/R,

    where R is the scale factor for measuring the relative scale of the universe (R = 1, for the current epoch). The formula is : H = (8G/3)1/2 ~ 2.4x1012 sec-1 (for t ~ 10-12 sec), where ~ 1031 gm/cm3 (Figure 08) even through particles are restmass-less ( m0 = 0) before SSB/ESB, this data is still meaningful since E = m0c2/[1-(v/c)2]1/2, and the gravitational constant G = 6.67x10-8 cm3/sec2-gm. Thus, the wash-out indicator k ~ 0.4 indicating that the system is in non-equilibrium state around the time frame of 10-12 sec.

    Figure 08 Cosmic History [view large image]

  • According to the Uncertainty Principle Et = 10^-27 erg-sec, if the N neutrino were created soon after the epoch of

    Inflation at 10-33 sec (see Figure 09), then its mass M (in term of energy E) should not be lower than /10-33, i.e., M 106 erg = 0.6x109 Gev. Similar argument can also be applied to its decay product of anti-neutrino produced up to 10-12 sec around the SSB/ESB transition giving MB-L 0.6x10-3 ev (most probably ~ 0.1 ev), which is compatible with experimental measurement. This is used as evidence to support the leptogensis theory.

    Figure 09 Early Universe [view large image]

  • This step involves some arithmetic, and finally deduced a value for the CP violation efficiency .
    • The observed matter/anti-matter asymmetry is given by = matter/radiation = 6x10^-10. The present-day radiation number density = 411 cm^-3 is interpreted as the result of matter/anti-matter annihilation leaving behind the baryonic number density ~ 2.5x10^-7 cm^-3.
    • The leptogensis theory above expresses = (1/3)(n_B-L/411) which yields n_B-L = n₀ = 7.5x10^-7 cm^-3.
    • In the very early time of 10^-33 sec soon after inflation, the linear scale is 10^-25 cm less with a corresponding reduction in volume of 10^-75 cm³. Thus, n₀ = 7.5x10⁶⁸ cm^-3, and the radiation number density is 411x10⁷⁵ cm^-3.
    • Taking the mass (energy) of the N neutrino M = 10⁹ Gev as mentioned above, the matter (energy) density
      ₀ = 10⁵⁴ gm/cm³.
    • As shown in Figure 08 the matter density ₀ = 10⁶⁵ gm/cm³ at that time, thus = 10^-11 is required to account for the observed asymmetry by leptogenesis. The "B Meson Decay" experiments produce ~ 10^-19 revealing that such CP violation mechanism is not sufficient.

Ultra-High-Energy Cosmic Rays (UHECR)

		However, this kind of source seems to be unlikely since UHECRs are isotropic, and not affected by magnetic fields. The yellow line in Figure 10 is the observed cosmic ray energy spectrum. The three other lines show how suspected sources possibly contribute to the overall signature of cosmic rays. It is not known what causes the kinks at the "knee" and "ankle". Normally, cosmic rays bombard the Earth at a rate of about 1 particle/km2-sec. At ultra-high energies, above 1010 Gev, the rate falls to less than 1 particle/km2-year.
Figure 10 C-Rays Spectrum	Figure 11 Cosmic Rays, Origins of [view large image]	Figure 11 lists some of the sources such as Supernovae for High-Energy (up to 106 Gev), Black-holes for Medium Energy (up to 109 Gev). The UHECR (up to 1011 Gev) is the one difficult to explain. The following is an attemp to link such events to the right handed neutrinos in the very early universe.

• Most of the cosmic rays are thought to be the protons. Their energy are consumed by interaction with the CMB photons setting a limit of about a few million light-years (about the distance to Coma cluster) to be observable as high energy particles.
• However, such limit does not apply to neutrinos (and especially the right handed kind) as they only interact feebly with gravity. As shown in Figure 12, they can survive the journey to Earth all the way back to the end of inflation 10^-33 sec after BB.

As shown in Figure 04, the decay products from the right handed neutrino include quarks, leptons and neutrinos, which are those travelling unimpeded all the way to Earth and could be identified as UHECR with energy up to 1011 Gev (see Figure 12). High energy neutrino with 106 Gev has been captured by the IceCube experiment, a neutrino telescope made of a cubic kilometer of Antarctic ice (see "Multimessenger Search for the Sources of Cosmic Rays Using Cosmic Neutrinos" and "Neutrino Telescope"). However, they attribute the source to be a blazar, an energetic galaxy powered by a supermassive black hole.

Figure 12 Cosmic History [view large image]

Other explanation for UHECR involves Cosmic Defects, which are extremely high-energy phenomena occurred in very early universe. Cosmic defect could form at the boundaries of phase transition. It might decay and release burst of UHECR.

Dark Matter, Wimpzilla as

By observations over the years, the dark matter should have the characteristics as listed below (Figure 13) :



It interacts with other matter by gravity. It should be cold (i.e., speed << velocity of light), otherwise it could not clumped together to form structure such as the halo of galaxy. It would be even better if it is warm since it doesn't aggregate into smaller object like planet. As it does not absorb or emit electromagnetic radiation, it must be electrically neutral. It also does not interact with strong interaction as high energy cosmic rays fail to produce anything from the dark matter. Since there is no known mechanism to replenish dark matter, it must be stable on cosmic timescales. Very long life time is OK, the decay products may reveal some information.

Figure 13 Dark Matter Properties [view large image]

There should be enough mass and quantity of whatever species to account for the 27% cosmic matter density of present-day dark matter.

• WIMP - Many new particles with heavy mass appear in the Supersymmetry formulation. These are referred to as Weakly Interacting Massive Particles, or WIMPs. For example, the photino (the fermionic partner of photon) has a mass about 10 to 100

  times that of the proton. Most of these electrically neutral particles would, like neutrino, go straight through Earth. On rare occasion, however, one might interact with an atom in the material they pass through. So far, the only claimed detection of a dark matter particle (by an Italian team in 2000) has been strongly disputed. One particularly interesting WIMP is the lightest supersymmetric particle (LSP) - the neutralino, which could be a Majorana fermion (a particle which is its own antiparticle) and thus has more chance to produce the electron-positron pair. Figure 14 shows the region excluded in the dark-matter-mass vs WIMP-Nucleon-Cross-Section

  Figure 14 Dark Matter Constraints

  (via gravitational interaction) by various observations in the past or in the future (if nothing shows up).Anyway, Supersymmetry is becoming irrelevant as none of the super-partners show up in experiments (as of 2021).

• Axion -
  • It is a hypothetical particle originally proposed to resolve a problem within the SM formulation, which predicts CP violation in strong interaction. It implies a finite Electric Dipole Moment (EDM) for neutron, but no such thing has been found so far. Theorists modify the mathematical formula to resolve the problem.
  • It turns out that the manipulation is not simply inserting a term into the formula. An axion field A = f has to be introduced (see Figure 15,a and more Axion details). It evolved in the very early cosmic history starting soon after the inflation period, gradually generates an effect to cancel out the CP violation, and eventually acquires a steady mass (Figure 15,b).

  	Axion is a spin zero boson with no electrical charge. The range of possible axion mass : 5x10-5 ev < ma < 1.5x10-3 ev, and phase angle = /f = 2.155 for making the CP violation disappeared. Such low mass allows for long life (since there are no lighter particles to decay into). The number density Na ~ 14x106/cm3 would be in agreement with DM = 0.23, if ma = 10-4 ev.
  Figure 15 Axion [view large image]	Figure 15c lists the various dark matter candidates including the axion and other oddballs. Most of them has to be discovered except the neutrinos. It seems that the axion possesses all the ingredients as dark matter but it fails to show up in experiments (Figure 15c insert).

  
  
• Neutrino - It is a reasonable candidate for dark matter because of its unreactive nature. Theoretical calculations indicate that there should be as many as 100 million neutrinos for every atom in the universe. However, the recent estimates of neutrino mass is so slight that it could account for only about 0.1 - 7 per cent of the mass of the universe. Actually it is not a suitable candidate for moving too fast.



Wimpzilla - This type of neutrino is superheavy produced during the period of inflation (Figure 16). It could be the decay product of the "inflaton" driving the inflation. There are not many of such Wimpzilla around as a lot of mass-energy is packed into each one. Detection of Wimpzilla depends on whether they decay naturally or interact with non-dark matter occasionally. One possibility is that they are stable, i.e., neither decay nor interact - a scenario explaining all the futile searches of the dark matter.

Figure 16 Wimpzilla [view large image]

This one seems to have all the ingredients for dark matter too. It is just something like the neutrino except with heavy mass to keep it from moving too fast and to allow for sufficient amount similar to observation. However, it would not be the type proposed in leptogeneses, for which the life time is about 10-12 sec.

GUT in SM Extension

Although the Standard Model has been very successful in accounting for all experimental phenomena, it is not expected to be the ultimate theory because of the many problems (including neutrino mass now) it leaves unanswered. These objections suggest that there may be deeper symmetries underlying the standard model, leading perhaps to the unification of the strong and electroweak interactions into a single "Grand Unified theory (GUT)" at the dawn of time (see Figure 17). Such scheme is indeed possible if the internal rotation group is further generalized to SO(10). BTW, the SU(5) scheme is dead because its prediction of proton decay fails to materialized after more than 20 years of futile searches.

Figure 17 Cosmic Evolution [view large image]

The 10 dimensional SO(10) group introduces two additional color charges for weak interaction. It proposes :


• There would be 5 generalized color charges including the red (r), green (g), blue (b) in strong interaction, the yellow (y) and purple (p) for 3rd components of isospin t₃ in weak interaction (Figure 05b).


• The Table in Figure 18 lists all the fermions of the standard model in the last column. The generalized color charges are in the middle. The value of hypercharge Y for each fermion is calculated from the colored charges in solid circles (using the formula as shown on top of the table). The hollow circles would yield a value of Y for its opposite partner, either anti-fermion or fermion (anti-fermion has a "bar" sign on top).

The color charges in strong interaction and those charges for the weak interaction are now lumped together to form the generalized hypercharge Y = - ( r + g + b ) / 3 + ( y + p ) / 2. This is not exactly the electric charge, for example, the left-handed neutrino and electron carries 0 and -1 electric charge respectively yielding the average value -1/2 for Y. The hypercharge was invented to explain the state of the electron and neutrino prior to Spontaneous Symmetry Breaking (SSB), which endows mass and charge Q to these particles. After SSB, Q = Y + t3. For example, t3 = (1/2, -1/2) for (, e)L and (u, d)L while t3 = 0 for the right handed sector.

Figure 18 SO(10) Group in GUT [view large image]

The right-handed neutrino ()R is the odd-man/woman out in the model. It is there to provide a possible mechanism for the mass of the neutrinos (and more). As Y = t3 = Q = 0, it does not involve in any of the interactions. It just sits in the corner - never been detected (smelling like dark matter or something long gone ?).