On Neutrino Masses and Leptonic Mixing



Paul M. Fishbane
Physics Dept. and Institute for Nuclear and Particle Physics,
Univ. of Virginia, Charlottesville, VA 22901



Peter Kaus
Physics Dept., Univ. of California, Riverside, CA 92521



Abstract


Using recent data on neutrino oscillations, we argue that a
hierarchical solution for neutrino masses in a three family
context is possible, and that the masses of the and 
neutrinos are very nearly determined within that possibility.
We also examine the predictions of a model that determines
neutrino and charged lepton mass matrices as well as its
consistency with data.





1


1. Introduction


The purpose of this note is to recognize some simple features of neutrino masses within the
context of the most conservative assumptions, and to test these features within an
interesting predictive model of family symmetry. Our conception of conservative is the
following:
1. Neutrinos come in three families, with no additional species, sterile or otherwise.


2. Neutrinos come in right-handed singlets and left-handed doublets, consistent with the
other fermions of the standard model. They have mass either by a Dirac term alone in
the Lagrangian or by a combination of Dirac terms and Majorana terms that combine
through a seesaw mechanism.


3. Like the other sets of fermion families, the masses of neutrinos are hierarchical, with the
having the largest and the e having the smallest mass. (The data to be discussed below are
completely consistent with this assumption.)


4. The structure of the mass matrices of the charged leptons and the neutrinos determine
neutrino oscillations of the type that can explain recent experiments.


5. The overall scales of the masses in the different fermion sectors differ, and we have
nothing special to say about these scales (although we recall the grand unified theory

relation mb = m, a relation that follows in the model we discuss in Section 3). Instead,
we are interested only in mass ratios within a given family; the heaviest mass family of
a sector will be normalized throughout to unity.
These assumptions derive essentially from the idea that neutrinos are "normal," that
is, they have a hierarchical three-family structure similar to that of the other three sets of
fermions of the standard model, the up quarks, the down quarks, and the charged leptons1.
In each case, the masses of the fermions can be written as an overall scale times powers of
the sine of the Cabibbo angleequivalently the Wolfenstein expansion parameter. But
this fact is not very limitingwe have not yet specified the quantitative nature of the
hierarchy. We shall describe in Section 2 the freedom we think this gives us for the

neutrino sector, as well as the limits set by the data. The masses of the  and are very
nearly predicted.
A recent set of models [1,2] suggest within the framework of a low-energy
supersymmetric extension of the standard model that the existence of mass hierarchies
within fermionic sectors imply at least one additional U(1) family symmetry, one of which
must be anomalous, with a cancellation of its anomaly through the Green-Schwarz
mechanism then implying the presence of relations across fermionic sectors [3]. This has
the additional benefit of predicting itself. Effectively, the standard model is cut off at the
grand unified scale, and couplings are suppressed by powers of U(1) charges. These


1We should note here that in contrast to the other fermions, neutrinos can have Majorana mass terms and
hence a seesaw mechanism; this is an argument for neutrinos not being "normal." We simply assume that in
spite of what may be different dynamics neutrinos look like the other fermions.



2


suppressions appear in the mass matrices when symmetry is broken at the GUT scale, and
correspond to the powers of that appear in these matrices. Such models are especially
interesting because the additional U(1)'s are characteristic of superstrings, and moreover,
these symmetries are broken by effects associated with strings.
The mass matrices of both the quarks and the fermionsnot to mention the
relations across families and the value of the Cabibbo angle itselfare thereby predicted.
One has not only the charged and neutral leptons mass eigenvalues but the leptonic analog
of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix2 as well. With the generic form
predicted in this model the mass hierarchies predicted for the leptons cannot be
accommodated within the framework of the current data; however, with perturbative
extensions it is possible to do so. And while the mixing matrix itself cannot take a form

consistent with the maximal mixing between the  and sectors, it is possible to come
reasonably close to this limit. As we shall see, the leading mixing terms between the e and
the other neutrinos is then determined, suggesting ways to test the model further.


2. Remarks on Neutrino Masses


Recent experiments [4] have provided quantitative results on neutrino mass
differences.

 = m2 - m2 and  2
e = m2 - me . (2.1)

The remaining available atmospheric and long baseline data [5], if not the short
baseline accelerator data [6], is consistent (for recent analyses see [7] and [8]) with what we
refer to in Section 1 as the conservative point of view. This data is consistent with the small
angle MSW explanation [9] of the solar neutrino data [10] and the large - mixing
explanation of the atmospheric neutrino anomaly. For recent discussion of how one could
accommodate all the current data, including that of ref. [6] within a nonconservative
framework, we refer the reader elsewhere [11].
In order to ask what the quantitative nature of the hierarchy is, we first discuss the
charged leptons, for which we know the numbers. (While the mass scale at which we
discuss the quarks is important, for the leptons this is not an issue.) Are the charged leptons
like the quarks? The charged leptons seem to steer a course between the two quark families.
The up quarks have a mass hierarchy of the form 1:4:8 and the masses of the down
quarks have the ratio 1:2:4. The charged leptons have a mass hierarchy of the form
1:2:6. In other words, / is like s/b but e/ is like u/c. To find out what the data can tell
us we define the (for now) strictly phenomenological parameter R by

e/ = R / (2.2)

This parameter has the advantage of emphasizing the role of mass ratios, as is appropriate
for a hierarchy. The analog to this quantity for the down quarks is very close to unity, and if



2 Ramond et al refer to this as the MNS matrix after the early work of Z. Maki, M. Nakagawa, and S. Sakata,
Prog. Theo. Phys. 28, 247 (1962).



3


for the up quarks one replaces the masses by their squares, it is also close to unity.
Alternatively, the charged leptons are associated with an analog to R of O(2).
We can regard Eqs. (2.1) and (2.2) as three equations for the three neutrino masses

in terms of the three parameters  , e , and R. We take the first two parameters as given
- -
at the values  = 10 3 eV2 and  6
e = 5  10 eV2we include the effects of errors at the
end of this sectionand plot the results for the masses (or their ratios) as a function of R.
Because there are mass squares present, the resolution of the equations involves four
branches, two of which involve only sign difference and are of no importance. Two of the
branches give complex values for R is some critical range and the other two give complex
values for R outside a related critical range. The precise range limits depend on the 's and
have no special importance. Ignoring sign questions, we can reduce the solutions for the
masses to two branches, as shown in Figs. 1 to 3.





1. Solution of Eqs. (2.1) and (2.2) for the ratio m/m as a function of R. Two
branches are evident, labeled branch 1 and branch 2. For branch 2 the value of m/m
is insensitive to R.


We first remark that according to Fig. 1 there is no value for the ratio m /m as
small as O(4). Thus the data rule out the possibility that the neutrinos behave like the up-
quark sector.
From Fig. 1 we see the presence of two branches. For one branch, branch 1, the
masses are essentially large, although not outside the directly known bounds of neutrinos,
and not hierarchical. We show the masses themselves for this branch in Fig. 2.



4


2. Solution of Eqs. (2.1) and (2.2) for the logarithms of the masses me,, m
and m as functions of R for branch 1 (see Fig. 1).

At the minimum allowed R-value, which is 1, all three neutrinos masses begin essentially
degenerate, and as R increases from its minimum allowed value the two lightest neutrino

masses remain relatively close to one another. The ratio m/m decreases somewhat but
only attains a value something like 2 at the high end of the allowed R range. The second
branch, branch 2 (Fig. 3), shows two features interesting to us. Firstly, the masses are

arranged in a hierarchy, and secondly, the solutions for m and m are extremely stable as
R varies, with values around

-
m 3
 2.3  10 eV and m 0.032 eV. (2.3)


The value of the ratio m/m for branch 2 has a stable value around 0.071, which if, in
line with the down quarks and the charged leptons we interpret as 2, corresponds to a
value = 0.27. The fact that this is close to the measured value of the sine of the Cabibbo
angle, around 0.22, as well as remarks we make at the end of this section about the other
fermionic sectors, encourages us to consider branch 2 to be the relevant branch for neutrino
physics.





5





3. Solution of Eqs. (2.1) and (2.2) for the logarithms of the masses me, m and
m as functions of R for branch 2 (see Fig. 1).

We are also encouraged in this view by two papers we received after the completion of this
work, both of which extract the scale for the neutrino masses from the unification scale for
a supersymmetric grand unified theory, very much in the spirit of the model to be discussed
-
in the next section. Reference [12] gives m 10 2 eV, while ref. [13] estimates m 5 
-
10 2 eV.
While branch 2 very nearly predicts the  and neutrino masses independent of R,
the value of me does depends on R. In what follows we consider two possibilities. The case
R = 1 corresponds to the 1:2:4 pattern of the down quarks, while the case R = .08 (= 2)
corresponds to the 1:2:6 pattern of the charged leptons. We remark here that for the case
R = 1, the masses for branch 2, the one of interest, take the simple form

  e
 e

m = ;m = ;m
 = (2.4)
e
 - - -
e
  e
  e

While the case R = 1 seems to us more in line with the ideas discussed in this paper, the
current data do not allow us to choose between these cases. Below we describe how the two
patterns above fit into a model that generates neutrino masses. We shall treat the case
1:2:4 in some detail, then add briefly how the second case, the one with a 1:2:6 pattern,
affects our results. We shall see below that the degree to which the electron neutrino mixes
with the other two families will provide a test to distinguish between these possibilities.





6


It is worth reiterating here the significance of the appearance of in the neutrino
hierarchy. Once we believe in a particular set of hierarchies in a given fermionic sector, we
can extract the values of in each sector. In particular, we can take the mass ratios and
hierarchy form in the different sectors as follows:
 down quarks, normalized masses run to a scale of 1016-1017 GeV are [14] 1 to 0.034 to
-
1.610 3 and hierarchy 1: 2 4
d :d .
 up quarks, normalized masses run to a scale of 1016-1017 GeV are [14] 1 to 0.0036 to
-
1.310 5 and hierarchy 1: 4 8
u :u .
 -
charged leptons, normalized masses are 1 to 0.059 to 2.910 4 and hierarchy 1: 2 6
e :e .
 neutrinos, no direct data for me, but as described above we can use m and m with a
hierarchy 1:2.





4. The values of extracted by assuming that the mass ratios in a sector are exactly
equal to given powers of ; e.g. we get d from d = (ms/mb)1/2. = (0.034)1/2. Only
one point is shown for the up-quarks because the same value of is extracted from
the three mass ratios. For the neutrinos, only the ratio m/m with central values
for these masses (see the text discussion) is used.



Figure 4 show the values of extracted by these relations, assuming a coefficent of n in
the ratios exactly equal to one; e.g. we get d from d = (ms/mb)1/2. = (0.034)1/2. We can get
an idea of the possible range of the found from the neutrinos by including the errors in
- -
the measured values of  and  4 3
e , namely (in eV2)  = 5  10 to 6  10 (preferred
- - - -
value 10 3) and  6 5 6
e = 4  10 to 10 (preferred value 5  10 ). We now take the naive
limits andrecall the insensitivity of these masses to Rinsert these into the R = 1 result
[see Eq. (2.4)]



7



1/2

= e (2.5)
   

in order to find extreme values = 0.16 to 0.37 (preferred value 0.27). This value for is
certainly in the same range as the other values, around 0.25. Perhaps, this is more than a
numerical coincidence.
Finally the reader should keep in mind that even choosing a hierarchical branch, the
choice of R, and hence of the electron neutrino mass, is not dictated by anything other than
analogy to the other fermionic sectors. Here it is worthwhile to look at Figure 5. This graph
allows us to understand better the sense in which the neutrinos may, or may not, interpolate
the d-quark structure and the charged lepton structure.







5. The logarithms of the ratios m2/m3 and m1/m2, where 3 labels the heaviest of the
three families and 1 the lightest. The solid lines connect these ratios for the
three electrically charged fermionic sectors. The dotted line connects the
corresponding ratios for neutrinos for R = 1 and the dashed line connects the
corresponding ratios for neutrinos for R = O(2).

The graph contains the mass ratios m2/m1 and m3/m2, where in each case 1 refers to
the heaviest of the three families and 3 the lightest, for the three electrically charged
fermionic sectors. Within both and down- and up-quark sectors these two ratios are roughly
constant at 2 and 4 respectively. For the charged leptons, the first ratio is O(2) while the
second is O(4). The neutrinos resemble the down quarks if R = 1 and the charged leptons
if R = O(2).





8


3. Testing a Model of Family Symmetries


In recent work on supersymmetric models [1, 2], with additional string-inspired U(1)
symmetries including an anomalous U(1) added to the standard model, a mechanism for the

generation of charged lepton ( ) and neutrino () masses is described. A spontaneously






acquired vacuum expectation value of a supersymmetric chiral superfield gives hierarchical
masses to quarks and leptons by a Frogatt-Nielsen mechanism [15]. The hierarchy is
associated with charge-canceling powers in terms in the Lagrangian; connections to the
string appear here, through Green-Schwarz anomaly cancellations [3], hence to
determination of the charges associated with the new symmetries, and hence to the
hierarchical structure of masses and mixing angles.
Here we quote the results for the mass matrices of leptons and neutrinos in these
models and thereby test them. The model permits one to find, through the balancing of
U(1) charges, only the leading powers of in the entries of the mass matrices, with
unknown coefficients. In the constructive spirit of the model we assume only that these
coefficients, as well as others that arise below, are of order 1. Although it is certainly
possible that for unknown reasons these coefficients are as small as powers of or as large
as inverse powers of , such possibilities would ruin the predictive power and we explicitly
eschew them.


Charged leptons. The charged lepton mass matrix takes the form [1]







4 5 3

a a a
11 12 13

M 2
 = a a a
21 22 23 (3.1)






2
 a a a
31 32 33


While the coefficients aij can generally be complex, for our purposes it is enough to take
them to be real, and we do so henceforth both here and in the neutrino case. Since this
matrix is not hermitian, we study the hermitian form


H M M ; (3.2)






we study this form rather than its conjugate because the unitary matrix that diagonalizes
H will enter into the leptonic CKM matrix. We normalize so that the largest eigenvalue is






unity. A quick look at the three invariants of H reveals that without conditions on the a's






that lead to cancellations the masses squared appear in the ratio 1:2:10, not the desired
1:4:12. Moreover, terms in the mass matrix that are higher order in are necessary to get
the desired eigenvalues. The three eigenvalues of H will be to leading order denoted






m2 = 1, m2 = w 4 2 12
2 , and me = w3 , respectively. Higher order terms are of course
necessary to get the eigenvectors correctly.
Where do the higher order correction terms come from? Presumably they are
associated with quantum corrections. This is entirely reasonable in the context of the
models, where is proportional to the vacuum expectation value associated with a broken
symmetry. Our attitude towards these terms is to put in the minimum corrections necessary



9


to get the mass hierarchy right. While for the charged leptons we have not written the most
general terms that meet this criterion, corrections other than the ones we use will not
increase the size of leading contributions to the leptonic CKM matrix.
To illustrate this point, consider the corrected mass matrix







4 5 3
a a a !
11 12 13
!
M 2 3 2 (3.3)
=
a21 + b21 + c a a !
21 22 23
2 3 2
!

a31 + b31 + c a a
31 32 33



Calculation of the eigenvalues of H shows that one can set b21, b31, and c31 (or c21) to zero
"





and can still accommodate a nonzero value of w3, but that if c21 and c31 are both set to zero
one cannot accommodate w3. Thus we set b31 = c31 = 0; the parameter b21 will then have to
be zero to get the masses right. We normalize to a first eigenvalue m 2
= 1 to leading order
in by the condition

a 2 2
23 + a33 = 1; (3.4)


This suggests the trigonometric representation #

a23 = sin( ), a33 = cos( ), (3.5)
# #





a form that we'll use below when convenient. Finally we insist that the hierarchy be of the
desired form by setting a minimal number of these parameters in favor of w2 and w3; any
will do, and we choose to eliminate a21, a22, a13, and c21, leading to a mass matrix of the
form

$ '





4 5 11 33 3
& a a
a a )
11 12
& a )
31
& )





23 31 31 2 3 3 2 23 32 2 (3.6)
& a a a w w w + a a
M = + )
0 1 23
# a
& a a a a
33 33 11 32 - a a a )
12 31 33
& )
2

& a a a )
31 32 33
% (





Note the presence of the w3 factor in the term associated with c21, verifying that the
presence of such a correction is essential to the correct hierarchy.
2
The eigenvalues mi can now be calculated to higher orders by using the invariants;
in turn the eigenvectors can be calculated as a power series in . The matrix U that
#





diagonalizes H in the form
#





4 7





6 m2 0 0
e 9





U H U -1 6 2
= 0 m 9 (3.7)
2 0
# # #





6 9





5 0 0 m2 8
3





10


is composed of rows that are the eigenvectors of the respective eigenvalues. Here we quote
the correct form to O(5); the precise value of the coefficients is of no special
interest here and, except for the constant [O(0)] terms, we wait until we write the leptonic
CKM matrix before we insert the relevant ones.

A D





G 3H G 5H G 3H G 5
H
C 1 O + O O + O F
(3.8)
C F





G H I P G H I P G H
U 3 4 4
@ = cos sin
C O @ + O @
- + O F





C F





G 3H G H G H
I P 4 I P 4
C F
sin cos
B O @ + O @ + O E





The factor w3 appears only in the O(4) and O(5) coefficients.
Neutrinos. The leading order result for the mass matrix M comes from see-sawing
[16] Majorana and Dirac terms in the model of ref. [1]. Unlike the charged lepton form, it is
hermitian, which evidently simplifies the calculation. It takes the form

Q T





6 3 3
S n1 n
1 12 n13 V
S V
M = 3 (3.9)
S n12 n n
V
22 23
3
S V
R n13 n n U
23 33


We shall here make the initial assumption that the e mass is O(4) times the mass.
[Below we'll describe what happens when the ratio is O(6).] The three eigenvalues of M
should be to leading order m = 1, m = x 2 4
2 , and me = x3 , respectively, where x2 and
x3 are O(1). However, without conditions on coefficients, the masses that follow from Eq.
(3.9) are rather in the ratios 1:1:6, ratios that are inconsistent with the data as described in
Section 2. Again, both conditions on the nij and higher order terms in are necessary to
accommodate the preferred ratio(s) 1:2:4. The most general form that gives the ratio
1:2:4, correct to O(3) in its correctionswhich essentially means O(2) corrections in
the 22, 23, 32, and 33 elementsis:

A D





C F





6
C n M M
11 21 F
31
C F
n n (3.10)
13 22 2 3 3 2
C + -x x
M = n22 + p22 M F
32
C
n F
33
C p n p n x F
3 22 33 33 22 2 2
C n F
13 22 33 33
B n n + + - n
2 n n E
22 33



Here Mji stand for matrix elements which can be read off of the elements on the lower left
of this symmetric form. The pij are the coefficients of 2 in the higher order corrections to
the O(1) elements, and we have eliminated the coefficients n12, n23 and p23 in favor of the




11


leading mass coefficients x2 and x3. In addition, the normalization that gives m = 1
requires that


n22 + n33 = 1. (3.11)

Again, a trigonometric representation for n22 and n33 will be helpful, namely


n22 = sin2(), n33 = cos2(), (3.12)

The next step is to find the masses more precisely, then the corresponding
eigenvectors. Having done so, we can find the diagonalizing unitary matrix U for M,
whose rows are the eigenvectors:

W `





Y me 0 0 b
-
UMU 1 = Y 0 m 0 b . (3.13)
Y b





X a
0 0 m

To O(5), the form of U is

g r





h i h i
q 1 + O 2 + 4 + 3 + 5 + 3 + 5 t
cd ef Ocd ef O Ocd ef Ocd ef O Ocd ef Ocd ef
q t




(3.14)
h i h i h i
U = q O t
q + O 3 + 5 cos + 2 + 4 - sin + 2 + 4
c e c e c e c e c e c e
d f Od f Od f Od f Od f O
d f t
q t



h i h i
p O 3 + 5 sin + 2 + 4 cos + 2 + 4
s
cd ef Ocd ef Ocd ef Ocd ef Ocd ef O
cd ef


Again, we have been explicit only for the constant terms. We remark here that the e mass
parameter x3 already appears in terms that are first order in .
Leptonic CKM. The leptonic mixing of the weak interactions are summarized by V,
the leptonic analog to the CKM matrix, namely

-1
V = U U . (3.15)
u





If we choose the first quadrant for both u and , a choice that will not change the
conclusions below, then to leading order in each element V is



v y






x x
1 2 3 3 3 3

x + O - + O O + O

x2 (3.16)
x











x x
V = - - 3 cos - + O 3
cos - + O 2
-sin - + O
2

x x
2
x








3
3
2
2
x x

sin sin cos
w - - - + O - + O - + O
x
2





12


We have in this formula distinguished just for convenience the contribution to V from the
diagonalizing matrix for the neutrinos () and from the diagonalizing matrix for the
charged leptons (). The neutrino contributions dominate. We have also not bothered to
specify the O(3) contributions in the 13 element of V, since the coefficient depends on the
other free parameters n13, a11, a12, a31, and a32. All the other elements are determined by the
neutrino mass factors x2 and x3 and by the angles and .






Neutrino masses in a 1:
2:6 pattern. This case may be very simply abstracted
from the first by the simple expedient of setting x3 = 0 in Eq. (3.10). The e mass is then
given to leading order by


n n n2
m 11 33 13 6
, (3.17)
e
n33

with the other masses unchanged to leading order. One could, of course, replace the
parameters n13, say, in terms of this mass. The leptonic CKM matrix is equally given by Eq.
(3.14) with x3 = 0, i.e., it takes the modified form V,







d d d
2 3 3

1 +
O O O


d e f d e f d (3.18)
V = 3
g 2
g 2
cos sin
O - + O - - +
O







d d d
3 e f
g 2 e f
g 2

sin cos
O - + O - +
O


While the coefficients of the O(3) terms are not difficult to work out, they are of no special
interest here. The O(1) terms are unchanged, and of course this persists for any value of
me/m of order smaller than 2.

4. Discussion


Independent of model, but not of bias as to whether the neutrinos masses have a

hierarchical structure, the experiments that measure  and e seem to us to go a long
way towards measuring the masses m and m. Moreover, the hierarchical structure
suggested comes very close to the structure of the other three fermionic sectors, a fact
worthy of attention.
Next let us look in the context of the models studied at properties of the leptonic
CKM, particularly Eq. (3.14). We remark first that the leading order terms are determined
by the mass parameters and the two mixing angles. There are two issues that concern us,
the possibility of maximal mixing in the - sector, and the degree to which the e mixes
with the other neutrinos.
Maximal mixing in the 
- sector. Equation (3.14) in the crudest approximation
suggests a zeroth order mixing between the  and sectors and only small mixing of the e
sector with the  and sectors. This property, and its consistency with the Kamiokande
data, has been discussed already in Refs. [1] and [2], where reference is made to maximal
mixing. But is truly maximal mixing, in which all four elements in the lower right hand




13


corner of V are the same size, 1/2, possible? The spirit of the model requires us to start
with matrix elements of mass matrices that have a given leading power of with
coefficients of O(1). This means that the angles and cannot be near zero, nor near
h





/2. An example of "maximum" coefficients in the neutrino mass matrix is sin2 = 1/2 =
cos2; i.e., = /4. Similarly, we have "maximal" coefficients in the mass matrix for
charged leptons for = /4. But if we make this choice, then the - mixing disappears to
i





leading order in the leptonic CKM matrix.
Or we can turn this around and start with maximal mixing in the leptonic CKM

matrix, for which - = /4. We use this to set = - /4. The O(0) terms in the
i i





mass matrices are then cos , sin , cos2( - /4) and sin2( - /4). In Fig. 6 we plot
i i i i





these four functions; 0.25 is the largest value all four of these terms can exceed, indicating
that 1.3 may be the suitable region.
i





6. Maximal mixing in the leptonic CKM is assumed, meaning - = /4. We
i





use this to set = - /4. The O(0) terms in the mass matrices are then
i





cos , sin , cos2( - /4) and sin2( - /4), plotted here to show the
j j i i





possibilities for these terms.



Mixing of the electron sector. While the electron neutrino mixing is small, it does
not vanish. There is a first order term in the 12, 21, and 31 elements, proportional in each
case to (-x3/x2)1/2. In Section 1 we argued that for = 0.27, x2 = 1, and while we have no
direct way to measure x2, if R = 1, then x3 = 1 as well. This O() term is thus just itself,
namely 0.27. This is not only not an unmeasurably small amount of mixing, it comes close

to some of the "maximal" - mixing terms for realistic values of the angles and as
i





described above. It remains for a more careful analysis to decide whether this coupling is in



14


fact too large to be accommodated in a small angle MSW explanation of the solar neutrino
data, which may favor values of O(3).
In contrast, elements involving e of O() are absent in the alternative model where
neutrino masses are in a 1:2:6 pattern. Here, all terms in V [Eq. (3.16)] that couple the
electron neutrino to the - sector are O(3) 0.02. Thus at worst this becomes a way to
distinguish the two models described here, and at best it performs a testable prediction for
both versions of the model.




Acknowledgments


We thank P. Ramond for many useful conversations and the Aspen Center for Physics for
its hospitality. PMF is supported in part by the U.S. Department of Energy under grant
number DE-AS05-89ER40518.



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3. M. Green and J. Schwarz, Phys. Lett. B149, 117 (1984); M. Leurer, Y. Nir, and N.
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