Physics 558 – Lecture 12

 

Neutrino masses, mixing and oscillations:

 

Our current understanding of neutrinos, the Standard Model and various experimental results suggest the following situation.  The Standard Model of the electroweak interactions requires a left-handed neutrino (and a right-handed antineutrino) for each generation with, as yet, no direct evidence of a non-zero mass.  Further there is considerable evidence suggesting that there are no right-handed neutrinos (or left-handed antineutrinos) contributing to current data.  Thus the Standard Model is typically characterized as containing exactly massless left-handed neutrinos.  This feature is indeed guaranteed, if the theory contains only the left-handed component of a Dirac neutrino.  Recall that a Dirac mass term in the Lagrangian requires both the left-handed and right-handed components,

 

                     

 

With

 

                                                  

 

we cannot build an appropriate Lorentz scalar term out of a single helicity component of a Dirac field, e.g.,

 

                                 

 

and similarly for the single right-handed component.  The (fatal) point here is the presence of the g0 in the definition of , which serves to change the sign in from of g5 when you pull out the projection operator.

The one theoretical flaw in this picture is that, since the neutrino has zero charge under the 2 fundamental conserved gauge interactions, the U(1) of E&M and the SU(3) of QCD (i.e., no electric charge and no color charge), it is quite possible that the neutrino is a Majorana rather than a Dirac fermion.  Recall that, while a massive Dirac fermion has 4 degrees of freedom (2 helicities each for the particle and antiparticle), a Majorana particle is self-conjugate with the particle and antiparticle being identical and the Majorana field represents just 2 degrees of freedom.  There are also several relevant experimental results (see the PDG web site for more details).

1.     Recall that, if the neutrino is a Majorana particle, then it is possible for the two neutrinos in double beta decays to annihilate each other.  Further, due to the chiral structure of the charged current coupling, one expects the annihilation process to be excluded unless the produced neutrinos are of mixed helicity, i.e., have a nonzero mass.   (In order for a left-handed neutrino to also act like a right-handed antineutrino, it must have both helicities present as in the decay on the pion.)  The results on neutrinoless double beta decay, while not yet confirming that the neutrino is a massive Majorana particle, also do not yet rule out this scenario (the expected rate is still below the observed limit).

2.     The results from the Super Kamiokande experiment on the leptons observed in the atmospheric showers of particles stimulated by cosmic rays incident on the top of the atmosphere seem to clearly indicate that the muon neutrino exhibits oscillatory behavior.  In particular, the flux of muon neutrinos in the showers is well below the expected flux, while the flux of electron neutrinos is consistent with expectations.  Likewise the results for solar neutrinos, e.g., the combined results from SNO and Super-K, suggest that the flux of low energy electron neutrinos from the “known” nuclear physics at the center of the sun is approximately 50% of that expected.  This situation is again most easily explained in a scenario where the electron neutrino oscillates into a different flavor with too little energy to interact on the earth via the charged current (i.e., too little energy to produce the corresponding charged lepton).  Both results suggest that at least two of the neutrinos have nonzero masses.  As we will see shortly the oscillation process is actually sensitive to the mass splitting between the neutrino mass eigenstates and these two measurements suggest two quite different scales for the two measured oscillation scenarios.  The atmospheric air shower data suggest that the muon neutrino oscillation (into something other than the electron neutrino) is characterized by .  The solar neutrino data, on the other hand, describe the oscillations of electron neutrinos (into something else) with a implied mass splitting of order .

3.     While most of the world’s short base line neutrino data (from both accelerators and reactors) are largely insensitive to the suggested parameter range for the oscillations suggested above, the latest results from both KamLAND (a detector in the Kamiokande facility that is sensitive to the  fluxes from 16 nuclear plants distributed around Japan at an average distance of about 180 km) and K2K (a  beam experiment also aimed at a detector at Kamiokande from 250 km away) find initial results consistent with (and constraining) the oscillation scenario of the previous paragraph.  Only the results from LSND (a short baseline low energy accelerator neutrino experiment looking primarily for   appearance in a  beam) suggest a scenario more complicated than the picture of 2 splitting scales outlined above.  In particular the LSND data suggest a mass splitting of order 1 eV2.  This is often interpreted as evidence for the participation of a 4th neutrino flavor, a sterile (not participating in the Standard Model interactions) neutrino.  We all await results from the MiniBooNE experiment to compare with the LSND data.

Not surprisingly (since I have chosen to introduce these issues), it is possible to use the theoretical conundrum at the beginning of this Lecture to address the experimental ones.  The scenario we are about to discuss arises “naturally” in theories with (broken) symmetries larger than those included in the Standard Model (and larger than a minimal SU(5) GUT), as we will eventually discuss.  However, it is important to note that the situation in neutrino physics is very fluid (i.e., exciting) and the model we will discuss is only one of many possible pictures.  Other scenarios involve nonstandard couplings and/or a role for extra dimensions.

 

Recall from Physics 557, Lecture 13 that we can define a conjugate spinor as

 

                                           

 

Here we want to apply this to a left-handed component to find a right handed one,

 

                                          

 

Again it is the g0 factor that is performing the magic of connecting a right-handed component to a left handed one.   Thus we want to contrast two possible spinor fields, a 4-component Dirac field

 

                                                    

 

and a 2-component Majorana field

 

                                 

constructed solely from the original left-handed component (of the Dirac field).  As required this Majorana field satisfies

 

                            

 

Thus the corresponding particle is its own antiparticle and must have zero charge for all relevant interactions.  We can now consider a Majorana mass term in our Lagrangian,

 

              

Unlike the similar expression for a regular Dirac field, these terms do not vanish identically (although 2 of the possible cross terms do).  Note that, since this particle is its own antiparticle, such a term can be interpreted not only as annihilating both a particle and its antiparticle or creating both a particle and its antiparticle or creating and annihilating a particle or an antiparticle (the Dirac interpretation) but also as creating or annihilating 2 particles (as we need for the neutrinoless double beta decay process).  Thus any quantum numbers carried by the particle will not be conserved.  Again we see that the Majorana particle must carry zero conserved charge.  However, there will be an issue with lepton number when we apply these techniques to neutrinos, as we will discuss below.  In particular, such a term is forbidden in the Standard Model with L (lepton number) conservation and in the minimal SU(5) Grand Unified Theory with B-L (baryon number minus lepton number) conservation.

 

If we start with both helicity components of a Dirac field, we can define the corresponding “right-handed” version of the Majorana field, and its mass term

 

              

 

We saw at the beginning of this lecture that the neutrino is a candidate to be a Majorana particle and that if, instead, we treated it as a usual, massive Dirac particle, we got into trouble with the required right-handed component, which we expect to be degenerate with the left and thus observed.  We will see now that, if we include both Dirac and Majorana mass terms, (an a priori crazy idea, perhaps), we can come up winners.

 

Consider a Lagrangian with the mass terms for both the left and right Majorana fields as above, but also with a Dirac term,

 

                               

 

In the basis of the Majorana fields we have the following matrix structure for the full mass term,

 

                                

 

Now imagine a scenario (still unmotivated) where  and .  We can diagonalize this system and expand in powers of the small parameter .  We find that the two eigenstates are

 

                    

 

Thus we have an essentially left-handed particle with vanishingly small mass (we’ll fix the minus sign below) and an essentially right-handed particle with a very large mass. By adding the Majorana mass terms (in effect, we added only the right-handed one but with a very large mass) we have gone from a Dirac particle with degenerate right- and left-handed components to two Majorana fields with very different masses.  This mechanism for obtaining necessarily one small and one large mass from the original off-diagonal matrix is often called the “see-saw” mechanism.

 

Now let us return to the Standard Model and neutrinos.  We have initially just the left-handed component in the SU(2)L


                                        

If we take nL to be the field yL used above, we can image awarding the neutrino a Majorana mass with the form

              

 

While this seems to allow a mass term without adding any further fields, it is really not acceptable.  Not only does such a term violate lepton conservation as noted above but we also see that, since nL is a member of an SU(2) doublet, so is nRC , and with the same quantum numbers.  Thus the terms in the Lagrangian above actually carry the quantum numbers of a triplet under SU(2)L, and violate that symmetry.  This is an explicit (not spontaneous) violation and contrary to the basic idea underlying the Standard Model.  We can turn this into a spontaneous breaking term if we add a second Higgs scalar field, H, which is a triplet under SU(2)L.  Thus the mass term looks like

 

                              

 

When H picks up an appropriate vev, the left-handed neutrino will get a mass.  However, this is really beyond the Standard Model and not typically viewed as a “natural” solution to the neutrino problem.  We would expect, as we shall discuss below, that we should associate any such new Higgs field, and its vev, as being part of the spontaneous breaking of some larger symmetry, e.g., some Grand Unified theory, and thus to occur at some scale well above 1 TeV, e.g., a GUT scale of  >1012 TeV.

 

A second way to proceed is to add a Dirac mass in analogy with the electron case.  Here we add a new right-handed neutral SU(2)L singlet,

 

                                                   

 

Note that this field is quite distinct (i.e., has different quantum numbers) from the field nRC defined above.  We can now add a Dirac mass without an extra Higgs field by defining a conjugate form of the existing complex scalar doublet,

 

                    

 

and writing

 

                     

 

The usual vev will put  in the upper component of jC and give the new 4-component neutrino a Dirac mass.   The problem here is that the natural result is Ge ~ Gn and thus me ~ mn.  This clearly is not a phenomenologically desirable result.

 

Instead we will make use of the scenario outlined earlier by allowing both a Majorana mass term and a Dirac mass term in the Lagrangian.  We assume that the Weinberg-Salam model is to be embedded in some larger symmetry, e.g., non-minimal SU(5) or SO(10), which allows us to motivate (i.e., to “understand”) the extra right-handed neutral field NR and some extra spontaneous symmetry breaking at a higher mass scale.  However, instead of awarding the Majorana mass to the left-handed neutrino, we give it to the right-handed (Weyl spinor) NR.  Now the corresponding Majorana mass term is a singlet under SU(2)L and can be easily, maybe even necessarily, associated with the symmetry breaking dynamics at the much higher mass scale, e.g., M ~ MGUT.  Thus, after both the high scale symmetry breaking and the electroweak symmetry breaking, the neutral fermion mass terms assume the form suggested earlier

 

                       

 

where we expect  and .   As above we can proceed to diagonalize this sector.

 

To prepare us for the interesting results below, let us first diagonalize the trivial case when there is no new dynamics and M = 0.  If we diagonalize in the usual way using a single unitary matrix, we find

 

                      

 

with eigenstates (and conjugates)

 

                         

 

In fact, we are interested in the eigenstates with positive eigenvalues and we are free to define a second matrix, UR, that, in this case, differs from UL by a diagonal phase matrix, UR = ULK.  With the choice

 

                                                   

 

we find

 

                                          

 

The corresponding eigenstates (and conjugates) are

 

                 

 

Thus we have 2 degenerate Majorana particles,

 

                                                 

 

that together constitute the expected single Dirac particle

 

                                     

 

Note that, as expected, (i.e., the Dirac particle and antiparticle are not identical).

 

Now we return to the case where .  Again we want to diagonalize as above (with the same K) but with a more general form for UL, which introduces a mixing angle q,

 

                                             

 

To leading order in the small ratio we find

 

                                         

 

The corresponding eigenstates are

 

    

 

So in this interesting case we have 2 non-degenerate Majorana states.  One is very light and essentially left-handed, while the other is very heavy and essentially right-handed.  The “see-saw” mechanism could be the explanation of the neutrinos we observe. 

 

Having established that we know how to award masses to the neutrinos (and may even be forced to by larger symmetries), and that these masses can be consistent with experimental observation (i.e., with the right-handed component being much heavier than the left), what is the easiest, or better the most exciting, way to see the effect of such masses?  As advertised, the phenomenon we want to discuss is neutrino oscillation.  This could explain both the Super-K observations concerning the reduction in the number of muon neutrinos in cosmic ray stimulated particle showers in the atmosphere and the reduction in the number of electron neutrinos observed to come from the sun.  To make the introduction to the mathematics simple we will consider the case of mixing of just two generations (at a time).  This would appear to be more than just an illustrative exercise as the Super-K results indicate rather clearly that the muon neutrinos oscillate but do not oscillate into electron neutrinos in a detectable fashion.  For the solar neutrino studies, it is the electron neutrinos that are oscillating but we have no indications yet as to which flavor neutrinos are being produced by electron neutrino oscillations.  However, the mass scales defining the oscillations in the two cases seem to be clearly distinct so both situations can be effectively described by a 2-generation formalism (i.e., the third mixing angle would seem to be quite small).  Having set up the problem, generalizations to 3 (or more) neutrino flavors are straightforward, if tedious. 

 

Consider 2 sets of basis states for 2 neutrinos.  One set is defined by the mass eigenstates, call them  and  with unequal masses m­1 and m2.  These are the states that propagate “trivially” in space-time.  The second set corresponds to the states coupled by the charged current to the charged leptons, for example call them  and .  In the most general case these two basis sets are not identical but rather one is a rotated version of the other, i.e., there is a unitary transformation from one to the other, which, in 2 D, is a real rotation.  We can write

 

                                       

 

The mixing angle q is a fundamental parameter of the theory, which, like the Cabibbo angle, is not determined by any of the physics in the Standard Model but can be represented in terms of the Yukawa couplings and vacuum expectation values we mentioned earlier.

 

Now we imagine that an electron neutrino is generated at time t = 0 by a nuclear reaction in the sun or in an air shower and focus on the subsequent evolution of states of definite momentum.  (To leading order in mn/E, which we assume is very small, this is equivalent to an analysis of states of definite energy.)  Such states will remain coherent in space and we can also project onto the basis of definite 3-momentum (the plane wave expansion).  For relativistic values of the 3-momentum p the two mass eigenstates will have energies

 

                                                   

 

For this relativistic case the distance traveled is the same as the time, (c = 1), so that the phase factor  is given, to a very good approximation, by  (with the same approximate form appearing for either definite p or definite E states).  Thus the initial electron neutrino will evolve as

 

                      

 

The physically interesting question is – what is the probability that the neutrino of interest is still an electron neutrino (i.e., interacts via the charge current to produce an electron) after time t?  This probability is given by

 

                       

 

We can rewrite this result in terms of the distance traveled as

 

                     

 

Thus, as long as neither the mixing angle q nor the mass splitting  vanishes, this probability will exhibit oscillatory behavior as a function of the distance (or time) traveled.  The quantity in the argument of the second sine function is just , where L0 is the vacuum oscillation length,

                                                    

 

(For once, the various factors, normally set = 1, are displayed.)  We can look for this effect either by measuring the disappearance of electron neutrinos or the appearance of muon neutrinos,

 

     

 

in this simple two component model.  Such questions will be more complex if all 3 generations participate.  It is also possible that there are “sterile” neutrinos that do not participate in the weak interactions but do contribute to oscillations.  If we think of writing the charged current or flavor basis, nl (l = e, m, t, …) in terms of the mass eigenstates, ni (i = 1, 2, 3, …), we have in the general case

 

                                                

 

with Uli the appropriate unitary matrix.  This matrix is the analogue of the CKM matrix for the quarks and is often labeled the MNS or Maki-Nakagawa-Sakata matrix.  The oscillation probability then becomes

 

                    

 

In current analyses, we invariably think in terms of the simple case of mixing a single pair and results are described in terms of values for the parameters sin22q, or tan2q, and Dm2. 

 

The case for the neutrinos observed (or not) from the sun has a long history starting in the 1970’s (actually the initial paper outlining the physics relevant to the SNO experiment was written in the late 1960’s by John Bahcall and an obscure graduate student named Steve Ellis).  We are interested in oscillations of electron neutrinos to some heavier flavor so that the corresponding charged leptons cannot be produced in the detectors on earth, since the neutrinos produced in the sun have fairly low energies.  Also, there are several distinct processes involved with the nuclear physics at the core of the sun contributing to the solar neutrino flux.  Hence the energy distribution of the neutrino flux has considerable structure, which plays a role in understanding the details of different experiments with differing energy sensitivities. 

 

If we assume that the only physics at work is oscillation in vacuum, then the way to observe a very reduced signal at a single energy is by assuming that we are near a minimum of the survival probability worked out above in the appropriate neutrino energy range and distance range (long!).  This scenario (some times called the “just so” scenario) corresponds to a large mixing angle (to get a big effect) and a very small mass difference, 10-10 < Dm2 < 10-9 eV2 to match the long travel distance. 

 

When we sum over the full energy spectrum and focus on the Super-K and SNO results, fits to both data sets prefer the large, but not maximal, mixing angle or LMA solution,  (unlike the quark case in the CKM matrix where the mixing angles are all small) and a mass splitting of order .  The various other scenarios, SMA, LOW, VAC, Just-So2, with other parameter ranges and/or a carefully tuned orbit for the earth seem now to be ruled out by the data  (see, for example, the recent review by S. Pakvasa and J.W.F. Valle, hep-ph/0301061). 

 

The Super-K results for atmospheric neutrinos (which revealed the first clear indications of neutrino oscillations) can be interpreted as corresponding largely to the oscillation  for neutrino energies of order 1 GeV (in the air shower) and a length scale of order 104 km (the radius of the earth).  (But note that other, more complicated scenarios are not ruled out and can be found in the literature.)  In particular, this experiment looks at the remnant neutrinos in the particle showers produced by cosmic ray particles colliding with nuclei in the upper atmosphere.  The showers contain copious quantities of pions, which decay primarily to muons and muon neutrinos.  The muons decay in turn into muon neutrinos, electrons and electron neutrinos.  At this naive level one expects to observe approximately twice as many muon neutrinos as electron neutrinos in such showers.  The experiment employs a large tank (~ 22.5 kilotons) of very pure water as both target and analyzer to look for the processes

 

                                               

 

What is detected is the Cherenkov light produced by the produced charged leptons.  The differences in the distribution of light within the Cherenkov cone (recall that Cherenkov is analogous to a shock wave – the charged lepton is traveling faster than the local speed of light, c/n, where n is the index of refraction of the water) allows for the separation of electron from muon events.  The initial focus was on the ratio of ratios,

 

                                                   

 

The denominator is a calculation by Monte Carlo simulation that includes our best understanding of the detailed structure of the evolution air showers and the details of the response of the detector (e.g., its ability to distinguish e’s and m’s).  Many systematic issues cancel in the ratio.  Further, the experiment can distinguish, on average, the direction of the incoming neutrino.  In particular, one can analyze separately down-going neutrinos, which travel approximately 15 km after being produced, and upward moving neutrinos, which travel some 13,000 km from birth to the detector.  One can even detect the dependence on the varying distance traveled as a function of the zenith angle, the angle from the local vertical, and thus of the continuously varying path length.  The observation that R is less than one is a signal that something interesting is occurring.   Looking separately at the muon and electron events as a function of L/En leads to the conclusion that it is the muon neutrinos, and not the electron neutrinos (i.e., the electron results match non-oscillatory expectations), which are oscillating.  Further, the details of the behavior of up-going versus down-going muon neutrinos suggest that there are oscillations over the long path length and not the short one, i.e., that , with sizeable mixing.  In the language of the mass eigenstates we can think of this as involving the mixing of state 2 with state 3, but, since, from above, states 1 and 2 are nearly degenerate on the mass difference scale relevant here, we really mean the mixing of 3 with some indistinguishable combination of 1 and 2.  In the 23 language the atmospheric data imply parameter values like  (i.e., a larger mixing angle than in the solar case) also expressed as , and .   The data further suggest that .

 

Thus the picture is that the three neutrino mass eigenstates split into 2 that are relatively degenerate and 1 that is well split.  What cannot be determined yet is whether the approximately degenerate states are higher in mass or lower in mass than the more widely separated state and just how the states are to be related to the flavor eigenstates.       

 

3                                                            2

                                                              1

 

 

                                    or

 

 

2

1                                                            3

 

 

In the detailed model outlined in the PDG report by Boris Kayser corresponding to the left-hand scenario, the lowest mass eigenstate (#1) is approximately 2/3 electron neutrino and 1/6 each of muon and tau neutrinos.  The next most massive state (#2) is approximately 1/3 of each flavor, while the most massive state (#3) is about ½ muon and ½ tau.

 

Note that, while the fits to the Super-K and SNO data plus the KamLAND results do not rule out extra sterile neutrinos, the data provide no evidence for their existence.

 

Another interesting feature is that the matter in the sun (or in the earth for the atmospheric neutrino analysis) can play a role.  For example, the evolution of the neutrinos while they are inside of the sun can be substantially altered by the matter, as noted by Mikheyev, Smirnov and Wolfenstein (the MSW effect).  The physics at work is the forward elastic scattering of the electron neutrinos from the electrons in the sun via the charged current interaction.  Since there is no corresponding scattering for the muon neutrinos (there are no muons in the sun), this will provide an extra contribution to the relative phase of the electron and muon neutrino wave functions.  Instead of an evolution equation that looks like

 

                          

 

where we have dropped the term proportional to the identity matrix corresponding to the shared overall phase, we find

 

                    

 

with

 

                                                    

 

The result of this effect can be studied in various ways and the detailed results depend on the variation of the electron density, ne, within the sun.  We will not study the details here except to note that it can produce a substantial enhancement of the oscillation within the sun and reduce the survival probability of the electron neutrinos at the earth.

 

With SNO now measuring the “total” solar neutrino flux via neutral current interactions, and various long baseline experiments involving neutrino beams of a variety of energies (K2K, KamLAND, Minos, etc.) coming online, we can hope for full clarification of the neutrino physics sector in a “timely” fashion.