

 18 May 95

IASSNS-HEP 95/37



May 1995 INDIRECT NEUTRINO OSCILLATIONS

K.S. BABUa; ?, JOGESH C. PATIb;c; y and FRANK WILCZEKb; z

aBartol Research Institute, University of Delaware

Newark, DE 19716

bSchool of Natural Sciences, Institute for Advanced Study

Olden Lane, Princeton, NJ 08540

cDepartment of Physics, University of Maryland

College Park, MD 20742

ABSTRACT We show how two different scales for oscillations between e and _ neutrinos, characterized by different mixing angles and effective mass scales, can arise in a simple and theoretically attractive framework. One scale characterizes direct oscillations, which can accommodate the MSW approach to the solar neutrino problem, whereas the other can be considered as arising indirectly, through virtual transitions involving the o/ neutrino with a mass , 1 eV. This indirect transition allows the possibility of observable _*_ $ _*e oscillations at accelerator and reactor energies. We discuss specifically the parameters suggested by a recent experiment at Los Alamos within this framework.

? Address b starting September 1995. Research supported in part by DOE Grant DE-FG02-

91ER406267, BABU@BARTOL.UDEL.EDU y Address c after July 1, 1995. Research supported in part by NSF Grant  and in part by sabbatical leave grant from the University of Maryland, PATI@UMDHEP.UMD.EDU z Research supported in part by DOE Grant DE-FG02-90ER40542, WILCZEK@SNS.IAS.EDU

The dearth of neutrinos observed to be emanating from the sun, compared to theoretical expectations, may be caused by the oscillation of these neutrinos, born as *e, into other types having smaller cross-sections at the detector.1 Both matterinduced (MSW)2 and vacuum ("just so")3 oscillations have been invoked in this regard. The range of (mass)2 differences of interest are of order , (10\Gamma 5 \Gamma 10\Gamma 4) eV2 or , (10\Gamma 11 \Gamma 10\Gamma 10) eV2 in the respective cases. These mass scales are considerably smaller than those of primary interest for accelerator oscillation experiments, which are of order , 1 eV2. Thus if *e $ *_ oscillations were to be observed in an accelerator experiment, it would appear at first sight as if one were confronted with some rather peculiar alternatives, e.g. that the relevant oscillation for the solar neutrino problem is *e $ *o/ , and the mass of *e is much closer to that of *o/ than to that of *_; or that some other hitherto undiscovered neutrino type is involved; or that the solar neutrino problem is solved in some other way than through neutrino oscillations.

In this brief note we shall discuss another alternative, slightly subtler but seemingly quite natural, which is fully consistent with the existence of *e $ *_ oscillations in both settings (solar and accelerator). It arises from a pattern of neutrino masses and mixings that has been suggested on independent theoretical grounds,4 as we shall recall below.

1. Indirect Mixing We assume that the electron neutrino *e can be expressed in terms of mass eigenstates *j, j = 1; 2; 3 in the form

*e =

3X

j=1

Uej*j (1:1)

and similarly for *_; *o/ . Possible mixing with heavier neutrinos, if any, will be assumed to be negligible so that the mixing matrix U is unitary. Thus a muon

2

antineutrino emitted at time zero evolves into the superposition

_*_(t) =

3X

j=1

U \Lambda _j exp(\Gamma iEjt)_*j (1:2)

at time t, where Ej = qm2j + p2 ss p + m2j =2p for the masses and momenta of interest. We wish to consider the possibility that m1 and m2 are very small relative to m3, such that we may ignore them. That is, the phase accumulations exp(\Gamma im2jL=2p) are supposed to differ very little from unity for j = 1; 2 and the lengths L and momenta p characteristic of accelerator experiments. This will embody our motivation above, by having m1 and m2 of the magnitude suggested by the solar neutrino problem (or smaller, in that we may have m1 o/ m2). Under these hypotheses, the probability for oscillations among various species, emitted at energy (or momentum) E to be observed at a distance L is given by

jh*e(0)j*o/ (L)ij2 = 4 sin2\Gamma m

2 3L

4E \Delta jUe

3Uo/3j

2

jh*_(0)j*o/ (L)ij2 = 4 sin2\Gamma m

2 3L

4E \Delta jU_

3Uo/3j

2

jh_*_(0)j_*e(L)ij2 = 4 sin2\Gamma m

2 3L

4E \Delta jUe

3U_3j

2 :

(1:3)

In the interesting case 1 ' jUo/3j2 AE jUe3j2; jU_3j2, it is natural to say that _*_ ! _*e oscillation proceeds indirectly, through a virtual _*o/ . The existence of such indirect mixing, of course, does not preclude the possibility of oscillations - conceivably much larger in amplitude - that become visible only at larger values of L=E, as for the case of solar neutrinos. It provides the slightly subtle alternative to which we previously alluded.

3

2. Numerical Parameters An appealing feature of (1.3) is that it ties together three different types of oscillations, that is _*_ ! _*e, _*e ! _*o/ , and *_ ! *o/ . Experimental constraints on the latter two processes can be combined to bound the first.

For concreteness, let us assume m23 = 2 eV2. From disappearance experiments at the Bugey nuclear reactor one has the bound5:

jUe3j2 ^ :02 : (2:1) From Fermilab experiment E531 and the CHARM-II experiment, one has the bound6:

jU_3j2 ^ :018 : (2:2)

By combining these, we find the upper bound for indirect mixing:

4jUe3j2jU_3j2 ^ 1:5 \Theta 10\Gamma 3 : (2:3)

Let us compare this with the results recently reported by Athanassopoulos et al.7 They indicate a _*_ ! _*e oscillation probability of (3:4\Sigma 1:8 (stat:)\Sigma 0:7)\Theta 10\Gamma 3, and the mixing parameter sin22` (deduced from their Fig. 3), which corresponds to the LHS of eq. (2.3), is nearly (1:2 \Gamma 2:5) \Theta 10\Gamma 3, for \Delta m2 ' 2 eV2. We see that these results are compatible with the indirect mixing hypothesis, though not by a wide margin. Clearly it would be absurd to claim this in any way as confirmation of the hypothesis; but we feel it does add some additional interest and plausibility to possible mixings of the order of magnitude being explored in the LAMPF experiment.

Bounds for other values of \Delta m2 are indicated in Table 1.

4

Table. 1: Limits on the mixing parameters jU_3j and jUe3j from ref. 6 and 5 respectively as a function of \Delta m2 ' m23. In deriving these limits, we have assumed jUo/3j ' 1 and used the unitarity relation jUo/3j2 + jU_3j2 + jUe3j2 = 1 to determine jUo/3j iteratively and in turn Ue3; U_3(via eq. (1.3)). The last column corresponds to the expected mixing probability for _*_ $ _*e oscillation at accelerators.

0B BBBB BBBB BBB@

\Delta m2(eV2) 4jU_3j2max 4jUe3j2max (4jU_3j2jUe3j2)max

0:5 0:25 0:04 2:5 \Theta 10\Gamma 3

1 0:09 0:06 1:4 \Theta 10\Gamma 3 2 0:07 0:08 1:5 \Theta 10\Gamma 3 4 0:05 0:15 2:0 \Theta 10\Gamma 3 6 0:03 0:17 1:1 \Theta 10\Gamma 3 8 0:018 0:17 0:77 \Theta 10\Gamma 3

1C CCCC CCCC CCCA

3. Theoretical Context Patterns of masses and mixing angles that allow accessible rates of indirect _*_ ! _*e oscillation as discussed above, and are compatible both with limits obtained from searches for direct _*e $ _*o/ and *_ $ *o/ oscillations5;6 and with the MSW solution to the solar neutrino problem are jUe3j ss (:10 \Gamma :16), jU_3j ss (:11 \Gamma :15) for m23 ss (:5 \Gamma 3) eV2; and m1 ! m2 , (2 \Gamma 3) \Theta 10\Gamma 3 eV, jU_1j , jUe2j ss (2:5 \Gamma 5) \Theta 10\Gamma 2 for the "small angle" MSW solution1, or m1 ! m2 , (2 \Gamma 10) \Theta 10\Gamma 3eV, jU_1j , jUe2j ss (:43 \Gamma :65) for the "large angle" MSW solution. It seems appropriate to mention now that this qualitative pattern of masses, and to a lesser extent of angles, has been suggested on quite independent grounds in the context of theoretical attempts to correlate quark and charged lepton, and predict neutrino, mass parameters.

Light neutrino masses with a hierarchy as exhibited above arise naturally in the context of a large class of unified gauge models - e.g. those with left-right symmetric gauge structures, which must include standard model singlet right-handed

5

neutrinos *iR, such as SO(10) or its subgroup SU (4) \Theta SU (2) \Theta SU (2). These models realize the famous see-saw mechanism, in which the *iR acquire large Majorana masses Mi. When these are combined with hierarchical Dirac masses mDi, one has the see-saw relation mi ss m2Di=Mi for the physical neutrino masses. If we assume that the Mi are all , 1012 GeV (within a factor of 10 (say)), then one finds the desired qualitative pattern of physical masses for mD1;2;3 ss 1 MeV; 300 MeV; 80 GeV, which are quite reasonable orders of magnitude to expect for the scale of Dirac masses in the corresponding families (compare with the masses of the u; c and t quarks). The mass scale , 1012 GeV for Mi is particularly intriguing because it has been suggested in other contexts, as the scale for Peccei-Quinn symmetry breaking, or for supersymmetry breaking in a hidden sector, or for preon-binding in a SUSY-composite model.

Theoretical ideas regarding mixing angles are even more tentative. One interesting idea8, that has had some success in providing a simple understanding of the inter-family mass hierarchy mu;d;e o/ mc;s;_ o/ mt;b;o/ and, with additional hypotheses, other important qualitative aspects of the quark and lepton mass matrices, deserves special mention. According to this idea the Dirac masses of the neutrinos as well as those of quarks and charged leptons arise indirectly through mixing with heavier vector-like families with masses , 1 TeV (generalized see-saw).9 If there is just one such vectorial family, which is a doublet of SU (2)L or SU (2)R, then only one light family receives a mass. With two vectorial families having the quantum numbers of a 16 and a 16 of SO(10), one obtains a hierarchical pattern of light masses for the three light families, and a parameter p ss 2qm0_=m0o/ ss (1=2 to 1=3), characterizing the _ \Gamma o/ mass hierarchy (at a high scale), appears.10 For details of a specific model of this type see ref. 4, especially case 2. This specific model suggests not only neutrino masses in the range mentioned above, but also sizable (, 5 \Gamma 15%) *_ \Gamma *o/ and *e \Gamma *o/ mixings, with the relation Ue2 ss U_1 ss (Ue3)( 2p ). Thus within this model only the large-angle MSW solution is compatible with the hypothesis of indirect _*_ $ _*e oscillation.

To summarize: the suggestion of indirect oscillation presented here raises the

6

interesting possibility that _*_ $ _*e oscillations in accelerator experiments, if observed, could reflect the mass of *o/ , (1 to few) eV, allowing *o/ to serve as a cosmologically significant hot component of dark matter; while the depletion of *e's from the sun would reflect direct mixing and \Delta m2 of approximately 10\Gamma 5 eV2 for the *e \Gamma *_ system.11 This scenario requires that not only _*_ $ _*e but also *_ $ *o/ and *e $ *o/ oscillations occur at levels accessible in the forseeable future.

4. References [1] Recently reviewed by P. Langacker, "Solar Neutrinos", Invited talk at the

Erice School, July (1994), .

[2] S.P. Mikheyev and A. Yu. Smirnov, Nuovo Cimento 9C, 17 (1986); L.

Wolfenstein, Phys. Rev. D17, 2369 (1978).

[3] S.L. Glashow and L. Krauss, Phys. Lett. B190, 199 (1988). [4] K.S. Babu, J.C. Pati and H. Stremnitzer, Phys. Lett. B264, 347 (1991). [5] B. Achkar et al., Nucl. Phys. B434, 503 (1995). [6] N. Ushida et al. (Fermilab E531 Collaboration), Phys. Rev. Lett. 57, 2897

(1986); M. Gruwe et al., (CHARM II Collaboration), Phys. Lett. B309, 463 (1993).

[7] C. Athanassopoulos et al., Preprint  April (1995). We use

numbers given in this paper. See also however J.E. Hill, University of Pennsylvania preprint, April (1995).

[8] J.C. Pati, Phys. Lett. B228, 228 (1989); K.S. Babu, J.C. Pati and H.

Stremnitzer, Phys. Lett. B256, 20 (1991); Phys. Rev. Lett. 67, 1688 (1991).

[9] Such mixings in the neutrino sector might lead to significant corrections in

eq. (1.3) and in the unitarity condition used in Table 1. In the model of [4] these corrections are not significant because UeN ^ 10\Gamma 2; U_N ^ 1=25 and Uo/N ^ 1=8, where N refers to the heavy neutral leptons.

7

[10] It needs to be mentioned that within supersymmetric composite models, the

coexistence of chiral and vector-like families seems to be a natural feature.8 Assuming supersymmetry, the number of such vectorial families with masses , 1 TeV cannot exceed two, regardless of their origin, because otherwise the QCD coupling will grow above 1 TeV and become confining well below 1016 GeV.

[11] For completeness, we mention that in addition to matter-enhanced direct

*e\Gamma *_ transition, the *e\Gamma *o/ oscillation suggested here would lead to a further depletion of *e for the solar neutrinos by about 12 \Gamma 4jUe3j2\Delta ss (4 \Gamma 5)%.

8

