Neutrino Mass

Contents

The Missing Neutrinos

		Back in the 1950s it was generally believed that neutrino has no mass and it exists only as a left-handed neutrino or right-handed anti-neutrino (see Figure 01, helicity is defined as the component of spin along the direction of motion, it is always perpendicular to the orbital angular momentum if there is any) participating in weak interaction. Later on it is found
Figure 01 Helicity [view large image]	Figure 02 Neutrino Oscillation [large image]	that there are three flavors of neutrino - the electron neutrino, muon neutrino, and tau neutrino. They are similar to each other except carrying different mass.

Detection of Neutrino Mass

This problem of missing neutrinos can be resolved if the neutrinos has mass. In this case, the existence of a neutrino can oscillate between the flavors as shown in Figure 02. According to quantum mechanics, the neutrino can be represented by two wave packets corresponding to the two flavors propagating at different speeds. Thus the interference pattern varies along the traveling path with different ratio (of the flavors) at any particular point.

		mixing as shown in Figure 04. The origin of mixing is not explained by the Standard Model (SM). Indeed, the massive neutrinos are the first experimental evidence for physics beyond SM, which is now regarded as an effective theory - a low energy approximation to a deeper, still unknown theory. Neutrino mixing is then considered as a correction within SM providing a window to the new discovery before formulating
Figure 03 CKM Matirx [view large image]	Figure 04 Mixing [view large image]	in detail the deeper theory.

		Figure 05 shows the agreement between the Super-K measure- ment and theory with neutrino oscillation. The neutrino in the upward direction would have to travel as long as 13,000 km, i.e., the diameter of the Earth. The horizontal direction would be about 500 km, i.e., the distance to the edge of the atmosphere (see Figure 06). The Sudbury Neutrino Observatory (SNO) in Ontario measured the total number of neutrinos from the Sun as well as the number of electron-neutrinos alone, and it shows that the total is much greater. The accounting seems to balance according to oscillation.
Figure 05 Measurements [view large image]	Figure 06 Super-K [view large image]

The probability of oscillation between 2 types of flavor neutrinos (i.e., , , and ) is given by the relation:



where _ij is the mixing angle, L is the distance traveled by the neutrino, E stands for the energy of the neutrino, and
_ji = m_j² - m_i² is the difference of the mass square. The mixing angles are determined from the amplitudes of the oscillation. The _jis can be calculated from the periods.

The solar neutrino measurements by SNO yields 12 ~ 30o, and 21 = 5x10-5 ev2. While those from Super-K gives 23 ~ 45o, and 32 = 3.5x10-3 ev2. The short-baseline (which implies larger mass difference) LSND experiment measured the oscillation of into . It yields ~ 1 ev2 and ~ 0o, which is very different from the other measurements. A sterile neutrino is required to reconcile all the data as shown in Figure 07a. Other experiments indicates 13 ~ 0o and the phase angle ~ 0o.

Figure 07a Mass Difference [view large image]

These data indicate large mixing between neutrinos (in comparison to quark) and small mass (at least million times smaller relative to electron's). But the data do not provide absolute mass measurement for the neutrinos. Direct measurements of the absolute neutrino mass impose the upper bounds: < 2.2 ev, < 190 kev, and < 18.2 Mev.

A 2007 report from the Fermilab experiment, known as MiniBooNE (for "Mini Booster Neutrino Experiment", see Figure 07b), found no evidence for the many of the muon neutrinos in the Fermilab beam oscillating into electron neutrinos (before reaching a detector 440 meters away). This study contradicts the LSND results and tends to refute the existence of the sterile neutrino. The news enables theoretical physicists to close an ugly chapter in the search for neutrino mass, because sterile neutrinos have no place in the standard model of particle physics. It would also have interfered with the growth of galaxies, changing the distribution of matter in the universe in a way that we do not observe, i.e., cosmologically, there should not be a sterile neutrino. However as the MiniBoone experiment

Figure 07b Miniboone [view large image]

has settled one problem, it reveals another anomaly of too many low-energy background electron neutrinos (Stay tune for further explanation).

Note that the MiniBooNE experiment has been constructed with the assistance from members of the LSND team. They would not be human if they didn't have a strong desire to see their signal confirmed and most of neutrino physics rewritten. And yet the setup is intentionally designed so that it would be almost impossible to bias the results one way or the other, and when it ruled against them, announced it openly to the world. That might not win them a Nobel prize, but it is still science at its best.

Figure 07c Neutrino Mass Spectrum [view large image]

The updated mass spectrum for three neutrinos is shown in Figure 07c, where it has been determined that 12 sun, and 23 atm. The fraction of each flavor state , ... is indicated by different pattern inside the bar.

Neutrino Mass Terms

• The Dirac Mass - The simplest way to introduce neutrino mass into the Standard Model is to add a neutrino mass term via the same Higgs mechanism for the electron and quark. The most general Lorentz invariant neutrino mass term that can be introduced into the Lagrangian density of the Standard Model (similar to the last term in Eq.(41c)) is:
  
  where m is a 3 X 3 matrix, which can be identified to the Higgs field coupling ₀G with and run over the three neutrino types , , , and _L, _R are left-handed and right-handed two-component spinor fields.
  Diagonalizing m with the help of unitary matrices (for i = 1, 2, 3):
  

  the mass term can be transformed to the standard Dirac form: The unitary transformation is expressed explicitly in Figure 03 and represented in pictorial form in Figure 04. It can be shown that for neutrino energies much greater than their mass, the right-handed field is much smaller than the left-handed field (Figure 08(b)). The lepton number remains a conserved quantity. But the predicted mass is 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, in an effort to reduce the mass).

  Figure 08 Neutrino Mass Models[view large image]

• Majorana Neutrino - As shown in Figure 08(a), the original Standard Model considers neutrino as massless and exists in the form of left-handed neutrino or right-handed anti-neutrino only. In addition, these two kinds are distinguished by the lepton number (for both electron and neutrino) L, which is +1 for neutrino and -1 for anti-neutrino. The lepton number is conserved in weak interaction according to the Standard Model. Since the neutrino has no charge, the lepton number is the only indictor to differentiate a neutrino from an anti-neutrino. When neutrino has mass, its speed would always be lower than the speed of light, theoretically an observer can move in a speed faster than the left-handed neutrino, overtakes this neutrino and sees a right-handed anti-neutrino (see Figure 09a) with corresponding change for the lepton number L from +1 to -1. In this case, the lepton number is not a valid indictor anymore or not conserved as it can vary from one moment to the next. Thus, there is no way to distinguish the neutrino and the anti-neutrino. This lepton number violating neutrino is its own anti-neutrino. Only the electrically neutral neutrino has this property among all the fermions. It is called Majorana neutrino in honor of Ettore Majorana, who first proposed this possibility.

  Mathematically, the left-handed and right-handed neutrino fields are related in the Majorana formulism: R = (i2)*L and L = -(i2)*R. It follows that the charge conjugate of a Majorana field is identical to itself, i.e., cL = L as intended. Following the same recipe of diagonalizing m by unitary matrices, the Lagrangian density of the mass term takes the form: . Within the framework of Standard Model, m can be identified to the Higgs field coupling as 20K. Here the coupling matrix K has dimension (mass)-1, which renders the theory unrenormalizable.

  Figure 09a Majorana Neutrino [view large image]

  This hypothetical particle can be confirmed by the "neutrinoless double decay process", which occurs with a very low probability and has not yet been detected.

  
  
• 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 (see Figure 08(c)), 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. This kind of virtual particle can occur in a very short time interval as ascribed by the uncertainty principle (Figure 09b). The neutrino has an effective mass of m²/M from these two alternative states. This formulism changes the relationship between the right-handed and left-handed neutrino fields R and *_L to:
  .
  Thus, R is (m/M) times small than *_L. The effective Lagrangian density then becomes:
  .
  

  For m ~ 100 Gev, and M ~ 1012 Gev, the neutrino mass would be of the order 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 09b Seesaw Mechanism [view large image]

Cosmic Neutrinos

• Neutrinos could constitute anything from 0.1 to 7 per cent of the mass of the universe (the dark matter). This range corresponds to the heaviest neutrino being in the mass range 0.05 to about 1 ev. Neutrinos any heavier than this would lead to galaxies being less clumped than actually observed by the recent 2dF Galaxy Redshift Survey.



• From the cosmic microwave background radiation (CMBR), astronomers can look back to an era about 380000 years after the Big Bang. A similar pattern created by relic neutrino allows examination of the universe at about 1 second after the cosmic genesis. It is estimated that the weak decoupling era produced about 300 neutrinos per cubic centimeter. The problem is that neutrinos are elusive by nature even under the best circumstances, and their interactivity decreases with the square of their energy. However, there is a special energy at which the probability of high energy cosmic-ray neutrinos interacting with a cold relic neutrino (or antinueutrino) goes way up. This occurs at the resonant energy of the

Z-particle, which is the end product of this neutrino-antineutrino collision with a certain signature as shown in Figure 10. There are two requirements for detection of the Z-burst. The first one, that neutrinos have mass, has been borne out in experiments. The second requirement, that neutrinos are somehow accelerated to tremendous energies in the ranges of 1022 - 1023 ev, is contingent on as-yet-unobserved physics (no cosmic-rays have been detected with energies much above 1020 ev). For the last 25 years the search for such event is unsuccessful in widely different locations such as the Moon, Greenland, and Antarctica. Confirmation of the Z-burst hypothesis would have several profound ramifications: First would be the clear-cut detection of the cosmic neutrino background. Second would be a determination of the neutrino mass. Third would be the observation of physics at the grand unification theory (GUT) scale, pointing to exotic GUT particles. The insert on the upper left corner of Figure 10 is a simulation

Figure 10 Relic Neutrino [view large image]

of the neutrino cosmic background. It contributes only a small net effect on the over all cosmic background.

	New evidence (2008) from the WMAP 5-year data shows that a sea of cosmic neutrinos permeates the universe. Cosmic neutrinos existed in such huge numbers they affected the universe’s early development. The data in Figure 11 show that, in the early epoch, neutrinos energy density made up 10% of the universe, while the current universe consists less than 1 % neutrinos as the energy density decreases
Figure 11 Composition from WMAP 5-Year Data [view large image]	with the cosmic expansion. This is in agreement with theories which based on the amount of helium seen today predict a sea of neutrinos should have been present when helium was made.