KIAS-P99036 IFP-773-UNC HUTP-99/A033

Can the Zee ansatz for neutrino masses be correct?

Paul H. Frampton and Sheldon L. Glashow

Department of Physics and Astronomy, University of North Carolina

Chapel Hill, NC 27559

Lyman Laboratory, Harvard University

Cambridge, MA 02138

School of Physics, Korean Institute for Advanced Study

207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, Korea

^{†}

^{†}Research supported in part by the US Department of Energy under grant number DE-FG02-97ER-41036.

^{†}

^{†}Research supported in part by the National Science Foundation under grant number NSF-PHY-98-02709.

Working in the framework of three chiral neutrinos with Majorana masses, we investigate a scenario first realized in an explicit model by Zee: that the neutrino mass matrix is strictly off-diagonal in the flavor basis, with all its diagonal entries precisely zero. This CP-conserving ansatz leads to two relations among the three mixing angles and two squared mass differences. We impose the constraint to conform with experiment, which requires the to lie nearby one of four 1-parameter domains in -space. We exhibit the implications for solar and atmospheric neutrino oscillations in each of these cases. A unique version of the Zee ansatz survives confrontation with experimental data, one which necessarily involves maximal just-so vacuum oscillations of solar neutrinos.

06/99

The minimal standard model involves three chiral neutrino states, but it does not admit renormalizable interactions that can generate neutrino masses. Nevertheless, experimental evidence suggests that both solar and atmospheric neutrinos display flavor oscillations, and hence that neutrinos do have mass. Two very different neutrino squared-mass differences are required to fit the data:

where the neutrino masses are ordered such that:

and the subscripts and pertain to solar and atmospheric oscillations respectively. The large uncertainty in reflects the several potential explanations of the observed solar neutrino flux: in terms of vacuum oscillations or large-angle or small-angle MSW solutions, but in every case the two independent squared-mass differences must be widely spaced with

In a three-family scenario, four neutrino
mixing parameters suffice to describe neutrino oscillations,
akin to the four
Kobayashi-Maskawa parameters in the quark sector.^{†}^{†} Two
additional convention-independent
phases are measurable in principle, but they ordinarily do not affect
neutrino oscillations [1] [2].
Solar neutrinos
may exhibit an energy-independent time-averaged suppression due to
, as well as energy-dependent oscillations depending on
. Atmospheric neutrinos may exhibit oscillations due to
, but they
are almost entirely unaffected by . It is convenient
to define
neutrino mixing angles as follows:

with and standing for sines and cosines of . For neutrino masses satisfying (1), the vacuum survival probability of solar neutrinos is [3]:

whereas the transition probabilities of atmospheric neutrinos are:

None of these probabilities depend on , the measure of CP violation.

Let us turn to the origin of neutrino masses. Among the many renormalizable and gauge-invariant extensions of the standard model that can do the trick are:

The introduction of a complex triplet of mesons () coupled bilinearly to pairs of lepton doublets. They must also couple bilinearly to the Higgs doublet(s) so as to avoid spontaneous violation and the appearance of a massless and experimentally excluded majoron. This mechanism can generate an arbitrary complex symmetric Majorana mass matrix for neutrinos.

The introduction of singlet counterparts to the neutrinos with very large Majorana masses. The interplay between these mass terms and those generated by the Higgs boson—the so-called see-saw mechanism—yields an arbitrary but naturally small Majorana neutrino mass matrix.

The introduction of a charged singlet meson coupled antisymmetrically to pairs of lepton doublets, and a doubly-charged singlet meson coupled bilinearly both to pairs of lepton singlets and to pairs of -mesons. An arbitrary Majorana neutrino mass matrix is generated in two loops.

The introduction of a charged singlet meson coupled antisymmetrically to pairs of lepton doublets and (also antisymmetrically) to a pair of Higgs doublets. This simple mechanism was first proposed by Tony Zee [4] and results (at one loop) in a Majorana mass matrix in the flavor basis () of a special form:

We focus on the latter scenario. In particular, we adopt the Zee ansatz for without committing ourselves to the Zee mechanism for its origin. Related discussions of Eq. (7) appear elsewhere [1] [5]. The present work is essentially a continuation of ref. [1].

Because the diagonal entries of are zero, the amplitude for no-neutrino double beta decay vanishes at lowest order [4] and this process cannot proceed at an observable rate. Furthermore, the parameters and may be taken real and non-negative without loss of generality, whence becomes real as well as traceless and symmetric. With this convention, the analog to the Kobayashi-Maskawa matrix becomes orthogonal: is explicitly CP invariant and . But as we have noted, it is well known [3] that the mere existence of a squared-mass hierarchy virtually precludes any detectable manifestation of CP violation in the neutrino sector.

The sum of the neutrino masses (the eigenvalues of ) vanishes:

An important result emerges when the squared-mass hierarchy Eqs. (1) is taken into account along with Eq. (8). In the limit , two of the squared masses must be equal. There are two possibilities. In case A, we have and . This case arises iff at least one of the three parameters in vanishes. In case B, we have and . This case arises iff the three parameters in are equal to one another. Of course, is small but it does not vanish: in neither case can the above relations among neutrino masses be strictly satisfied. But they must be nearly satisfied. Consequently we may deduce certain approximate but useful restrictions on the permissable values of the neutrino mixing angles . Prior to examining these restrictions, we note that Eqs. (8) and (1) exclude the possibility that the three neutrinos are nearly degenerate in mass. If the Zee ansatz is even approximately realized in nature, no neutrino mass can exceed a small fraction of an elecron volt in magnitude and neutrinos are unlikely to contribute significantly to the dark matter of the universe.

We first consider case A. The relation may be obtained in three ways depending on which parameter in is set to zero. If , the quantum number is conserved. It follows that and . We see from Eq. (6) that atmospheric ’s oscillate exclusively into ’s and vice versa. This subcase (or any nearby assignment of mixing angles) is strongly disfavored by SuperKamiokande data [6].

Alternatively we may set to obtain conservation of . For this subcase, we obtain and . We see from Eq. (6) that atmospheric ’s do not oscillate at all. This subcase (or any nearby assignment of mixing angles) is also strongly disfavored by SuperKamiokande data [6].

The last (and best [1]) version of case A has and leads to conservation of . For this subcase, we obtain and . We see from Eq. (5) that solar neutrino oscillations are maximal:

Moreover, we see from Eq. (6) that atmospheric ’s oscillate exclusively into ’s with the unconstrained mixing angle :

This implementation of the Zee ansatz is compatible with experiment: It predicts maximal solar oscillations (without an energy-independent term) and it is consistent with the just-so vacuum oscillation hypothesis [7]. However, it is evidently not compatible with the small-angle MSW explanation of solar neutrino data. Neither is it compatible with the large-angle MSW solution, because it predicts virtually maximal solar neutrino oscillations. If is permitted to depart slightly from zero so as to generate a small finite value of , the coefficient of the oscillatory term in Eq. (9) will depart from unity by a term of order . The resulting solar-neutrino oscillations remain nearly maximal: they are energy independent and experimentally disfavored unless lies within the just-so domain.

We furthermore note that the version of the Zee ansatz admits atmospheric neutrino oscillations of the type with any value of the mixing angle . At the same time, it precludes all oscillations involving atmospheric . These results are quite in accord with SuperKamiokande data.

Others have noted [8] that the relation is preserved by radiative corrections for case A. This is not necessarily true for case B, where the relations satisfied by the neutrino masses, , are not a consequence of a symmetry principle. In any event, we argue that case B cannot fit the data. Ref. [1] shows that case B leads to the relation:

Along with Eqs. (6), this implies:

That is, we must have large (almost maximal) oscillations of atmospheric . This result is strongly disfavored by SuperKamiokande data [6], so that case B can be rejected without further ado.

Our conclusion is simple. We find that one (and only one!) of the realizations of the Zee ansatz incorporating a squared-mass hierarchy is compatible with both solar and atmospheric neutrino data. It corresponds to the assignments , and , with and . Near this domain, atmospheric electron neutrinos oscillate negligibly, while atmospheric muon neutrinos oscillate into tau neutrinos with the arbitrary mixing angle . Solar neutrino oscillations are very nearly maximal. They can be described by vacuum oscillations, but not by MSW oscillations. It is straightforward to implement the Zee model so as to conserve exactly so as to obtain to all orders [8]. Of course, this must not be done! Some unspecified new physics (beyond the introduction of Zee’s meson) is required to lift the degeneracy between and so as to yield an extreme hierarchy of neutrino squared-mass differences, with .

Acknowledgement

The authors thank the Korean Institute for Advanced Study for its gracious hospitality while this work was begun. Our research was supported in part by the U.S. Department of Energy under grant number DE-FG02-97ER-41036 and in part by the National Science Foundation under grant number NSF-PHY-98-02709.

References

[1][email protected] Jarlskog et al., Phys. Lett. B449 (1999) 240. [2][email protected] Glashow in Atti dei Convegni Lincei, Rome 133 (1997) 69. [3][email protected] Georgi and S.L. Glashow, hep-th/9808293. [4][email protected] Zee, Phys. Lett, 93B (1980) 389. [5][email protected] Yu. Smirnov and Zhijian Tao, Nucl. Phys. B426 (1994) 415; A. Yu. Smirnov and M. Tanimoto, Phys. Rev. D55 (1997) 1665. [6][email protected] Kajita for the SuperKamiokande collaboration, hep-ex/9807003. [7][email protected] Barger, K. Whisnant and R.J.N. Phillips, Phys. Rev. D24 (1981) 538; S.L. Glashow and L.M. Krauss, Phys. Lett. B190 (1987) 199. [8][email protected] Barbieri, G.G. Ross and A. Strumia, hep-ph/9906470.