MAJORANA NEUTRINOS

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MAJORANA NEUTRINOS

J. Bernabéu, F. Botella

To cite this version:

J. Bernabéu, F. Botella. MAJORANA NEUTRINOS. Journal de Physique Colloques, 1984, 45 (C3),

pp.C3-113-C3-119. �10.1051/jphyscol:1984322�. �jpa-00224036�

JOURNAL DE PHYSIQUE

Colloque C 3 , suppl6ment a u n03, Tome 45, mars 1984 page C3-113

MAJORANA NEUTRINOS

J. Bernabku and F.J. Botella

Department of T h e o r e t i c a l P h y s i c s , U n i v e r s i t y of V a l e n c i a , S p a i n

RCsumG - On prgsente de nouveaux phQnomSnes associ6s 3 la violation AL = 2 du nombre le~tonique, en relation avec la description de Majorana du champ du neutrino. Pour quelques g6n6rations, il apparait un nouveau type d'oscillation de neutrinos avec une probabilits proportionnelle aux masses des neutrinos. Le terme de masse de Majorana induit un mQlange atomique des Qlements Z et (2-2), qui conduit B la non stabilitg de l'atome 2.

Abstract - New phenomena associated with AL = 2 lepton number violation are discussed in connection with Majorana description of the neutrino field. For several generations, a new kind of neutrino oscillation appears with probabi- lity proportional to neutrino masses. Induced by the Majorana mass term, there is atom mixing of Z and (2-2) elements, leading to non stability of the otherwise stable 2-atom.

1. INTRODUCTION

It is apparent that neutrinos are different from other known fermions. First, the masses of the observed neutrinos are, if any, much smaller than those of the charged

leptons and quarks

Second, only left handed neutrinos have ever been observed. In fact, all present data on charged current weak interactions are consistent with (V-A) currents, so only the left handed components of neutrinos are produced in the processes. In contrast, quarks and charged leptons are known to have both right and left handed helicity states.

With these considerations in mind, we explore the possibility offered by Majorana fields to generate a mass term without the existence of right handed neutrino fields. Automatically, the Majorana mass term is lepton number violating AL = 2, leading to non vanishing probability for no neutrino double beta decay1). The propagators for Majoranas are discussed in Section 2. In Section 3 the implica- tions2) of the presence of several generations are studied. For Majorana neutrinos, CP violation is possible with only two generations. The phenomenon of "neutrino- antineutrino" oscillation appears in contrast with the Dirac case. Finally, in Section 4 we show the origin of atom mixing3) and the perspectives to observe no neutrino double electron capture.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984322

JOURNAL DE PHYSIQUE

2. MAJORANA FIELD

For a fermion field, in the representation in which y 5 is diagonal, it is possible to write a particular four-spinor solution with the following choice of components

where

The equation satisfied by the two-spinor is

its conjugate being the one corresponding to the other components i o2 <*(x).

In the case m = 0, Eq. (4) coincides with the Weyl equation. The solution (2) induced by ((x) is a Majorana field.

One can work with four-component spinors with an additional restriction imposed by

where C is the charge conjugation matrix

an6 @ is a phase. This leads to the following expansion

where u and uC are the four-spinors defined as for Dirac fermions.

For a Dirac fermion, +L(x) and VJR(x) are independent. With both, it is possible for the fermion to acqulre a Dirac mass. If, as assumed in spontaneous broken gauge theories, this mass is generated by the non vanishing vacuum expectation value of the Higgs field we can write a term in the Lagrangian

(8)

where the Higgs field ~ ( x ) has zero lepton number and a weak isospin 112. This is the scenario which is resented at the level of the standard SU(2) x ~ ( 1 ) gauge theory.

The question which arises for neutrinos is whether, with the absence of $R(~), it is possible to generate any mass at all. We have seen above that there are particular solutions, with only QL(x), which satisfy a Dirac equation with non vanishing mass. Therefore, we have to have an affirmative answer to our question.

In fact, $ (x) is a left-handed component, so that qR(x) could be replaced by

m). ^{~ i e} only difference between this last spinor and (~~(x) is in the Lorentz properties, but this can be arranged.

For charged leptons and quarks, which have non-zero electric charge and colour, a

mass term in the Lagrangian qLqL or lLIL is forbidden, because it is not an scalar

under SU(3)c x U(l)e.m.. These are local gauge symmetries supposed to be exact,

according with the present dogma. But neutrinos are colourless and have zero electric charge, so for them a mass term in the Lagrangian written as

seems quite legal. Eq. (9) constitutes a Majorana mass term. The presence of the Majorana mass term implies lepton number violation in two units. This is not in contradiction with the dogma.

To generate a Majorana mass by spontaneous symmetry breaking, we cannot stay at the level of the standard theory with a weak doublet of scalars. In fact,

needs a ~ ( x ) which is a member of a triplet in weak isospin. This appears natural- ly in grand unified theories as the ones based on the gauge group SO(10). At the level of the electro weak theory, one needs an extension of the Higgs sector.

With the expansion of the Majorana field as given by Eq. (7) we can build the following two point Green functions. The Dirac propagator corresponds to

as for Dirac fields. However, due to the simultaneous presence of both a(p,X) + and a+($,h) in the Majorana field $(x), the propagator

is non vanishing. Eq. (12) corresponds to the so-called "neutrino-antineutrino"

propagation, which is a AL = 2 term. When (12) is inserted between left-handed currents, the result is proportional to the neutrino mass.

3. NEUTRINO OSCILLATIONS

When several generations are considered, the mass parameter m in the Lagrangian is replaced by a matrix M. In the process of diagonalization, one gets from the weak current neutrino fields ( v ~ ) ~ the definite mass neutrino fields $(x) in the form

The non-trivial unitary transformation V induces the phenomenon of oscillations.

The definite mass field (13) is a Majorana field, because it satisfies

and V v"' is a diagonal matrix of phases. The matrix V is responsible, as for

the Dirac case, of neutrino oscillations due to flavour violation, governed by the first type propagation with time.

The existence of the non-vanishing second type propagation (12) induces a new kind

of neutrino oscillation. In fact, with double charge exchange, one can ask for the

propability amplitude that a charged lepton lj" becomes a charged lepton I;-. It

is given by

C3-116 JOURNAL DE PHYSIQUE

The result (15) is responsible of the set of new phenomena associated with AL = 2 processes. If two gen~rations are coupled, only a real parameter in V is necessary and a relative phase ela between the two neutrino fields. The amplitude for BB-decay without neutrinos would be proportional to

ml cos 2 B + eia m2 sin2 B

+ -

where m are the masses of neutrinos. For p + e conversion, however, one gets 1 9 2

an amplitude proportional to

(ml - e ia m2) sine (17)

Values of a # 0, .rr imply a signal of CP violation in the leptonic sector which, contrary to the quark sector, is possible even with only two generations. The value a = O(n) corresponds to equal (opposite) CP-eigenvalues of the two neutrino species.

The phenomenon of AL = 2 v-; oscillations4) can be discussed as follows. Assume that at time t = 0 an antineutrino (using conventional wisdom) is produced from the charged lepton ljC. We ask for the probability that at time t > 0 it behaves as a neutrino capable of producing a charged lepton li-. The time evolution is given by

where the CP-violating phase appears as a phase shift in the oscillatory term with time.

4. ATOM MIXING

We have seen that the AL = 2 neutrino mixing has the implication of non-vanishing no neutrino double beta decay of nuclei

/ ^e-

= + = =

* \ % r e -

Analogously, the no neutrino single electron capture (NONSEC) in atoms') becomes accessible

We are going to discuss with some detail the case of double electron capture ^{6 )}

(NONDEC) :

associated with a two nucleon mechanism e-e- pp

nn, H and H' being the vacancies in the daughter atom of the two captured :lectrons. From the point of view of the atom, the process (Z,A) + (2-2, A)H,H is a virtual mixing of the parent atom with the daughter atom with two elecfron holes. The process becomes real as the daughter atom de-excites (2-2, A ) ~ , ~ + (2-2, A)+ ---. To summarise, we have

6

z

(2-2)

which can be discussed as a two states problem, where 6 is the non diagonal term in the atomic mass matrix

This implies that the stationary states of matter, the ones with definite mass and life time, are not the atoms but linear combinations of Z and (2-2) atoms ! In the limit 6

A, T, the eigenvalues of the mass matrix M are given by

a s " r

A l " m + - - l - - A2+r2/4

so that the originally stable atom (in the absence of 6 ) has actually a NONDEC decay rate

The strategy suggested by Eq. (21) is the one saying that the most interesting cases are those for which A is closest to zero. The ideal situation would corres- pond to thecondition A

r , ^{but A} is a difference in nuclear energy levels, typically MeV, and r is the atomic X-ray decay rate, typically eV. So we are asking for a "monumental" coincidence7). From present knowledge of atomic masses, one finds some cases with A = 0 + few keV, but the errors are too big.

These limitations preclude the search for atom oscillations with time. Any

realistic time interval for an experiment satisfies t >> r-1

10-l5 sec, im lying

that the daughter atom disappears much before. Furthermore, t >>

10-If; sec,

so the oscillation is washed out. Only asymptotic times are available, for which

the Z atom has a (2-2) component with a probability fj2/(A2+r2/4). This leads to

the NONDEC decay rate T-' given in Eq. (21).

C3-118 JOURNAL DE PHYSIQUE

To estimate the non diagonal mass term 6, one has to proceed with the calculation of the diagram

.

This calculation has been made3' in some cases of good degeneracy, as it is the case for the mixing 112Cd (1.87)(1s)-2. One gets

5oSn * 48

which is about sixteen orders of magnitude smaller than the K L - K ~ mass difference.

The decay rate, in inverse years, is plotted in the Figure, for mass differences

between eV and several keV.

If, for the sake of the discussion, we assume A

100 eV, then the NONDEC and TWONDEC lifetimes are comparable, of the order years. The two decays have distinct signatures :

NONDEC

2 X-rays + y-rays cascade TWONDEC + 2 X-rays

and 50 decays of each type would take place per year and per kilogram of '12sn

(?. 100 Kg of natural tin).

5. CONCLUSION

We have discussed the richness of new phenomena if neutrinos are described by Majorana fields. There is a new kind of lepton number violating AL = 2 neutrino- antineutrino oscillation, with probabilities proportional to neutrino masses.

Because of the restriction imposed to Majorana fields, CP violation appears possible for the leptonic sector even with only two generations. In the AL = 2 neutrino oscillation, the CP violating phase manifests itself as a phase shift in the oscillatory term with time.

Induced by the second type propagation for Majorana neutrinos, there is a mixing of atoms, leading to non stability of otherwise non stable atoms. The decay rate for corresponding process of no neutrino double electron capture is, consequently, resonance enhanced. For the best found cases. oerhaos not the best existine cases.

one' finds a lifetime A 30)i ' r = years ( -

mvx 100

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5) VERGADOS J.D., CERN TH 3396 (1982)

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7) WINTER R.G., Phys. Rev. 100, 142 (1955)