a

r
X
i
v
:
h
e
p
-
p
h
/
9
2
0
9
2
9
9
v
1


3
0

S
e
p

1
9
9
2
McGill/92-22
hep-ph/9207207
rev. August 1992
Majorons without Majorana Masses
and Neutrinoless Double Beta Decay
C.P. Burgess and J.M. Cline

McGill University
3600 University Street
Montreal, Quebec, Canada H3A 2T8
We explain excess events near the endpoints of the double beta decay () spectra
of several elements, using the neutrinoless emission of massless Goldstone bosons. Models
with scalars carrying lepton number 2 are proposed for this purpose so that ordinary
neutrinoless is forbidden, and we can raise the scale of global symmetry breaking above
the 10 keV scale needed for observable emission of conventional Majorons in . The
electron spectrum has a dierent shape, and the rate depends on dierent nuclear matrix
elements, than for the emission of ordinary Majorons.

Present address: Theoretical Physics Institute, University of Minnesota, Minneapolis, MN

55455, USA
One of the deep questions of particle physics is whether there exist any fundamental
particles of spin zero. A candidate is the Majoron, the massless Goldstone boson that
would exist if lepton number (L) were spontaneously broken [1]. Majorons would be hard
to detect since they couple directly only to neutrinos, with a strength proportional to the
neutrino masses divided by the L-breaking scale v:
g

= m

/v. (1)
If g

were large enough, Majorons could be seen in double beta decay [2][3], a rare
process now observed in seven elements. Besides the usual neutrino-emitting mode
2
,
a Majoron could be emitted through annihilation of the virtual neutrinos,
M
. The
M
and
2
signals are distinguishable since for
M
the decay energy is shared among fewer
particles, skewing its electron spectrum toward higher energies.
In fact, a mysterious excess of high-energy electrons is seen in the spectra for
several elements. Such an observation was rst made in 1987 for
76
Ge
76
Se +2e

by
Avignone et al. [4], although they, and other groups, subsequently excluded a signal at the
original level [5]. Now the UC Irvine group also nds excess numbers of electrons near but
below the endpoints for
100
Mo,
82
Se and
150
Nd, with a statistical signicance of 5 [6].
Such events also persist in
76
Ge [7], at approximately a tenth of the original rate.
Formerly all these measurements were compared with the Gelmini-Roncadelli (GR)
model [8] [3], which however has been ruled out by LEPs bounds on the invisible Z width.
In this model lepton number was spontaneously broken by an electroweak-triplet Higgs
eld, resulting in both a Majorana mass and a direct coupling to the Majoron for
e
,
as in eq. (1). The presently observed excess events could then be explained by
M
if
g

e
1 10
4
, and the absence of
0
decay which would appear as a line at the
endpoint in the sum-energy spectrum requires m

e
<

1 eV. Eq. (1) then implies that

the triplet VEV must be unnaturally small,
v < 10 keV. (2)
(A similar bound follows from astrophysical considerations.)
With the demise of the GR model, it appeared that there existed no models capable
of predicting
M
at an observable rate. In this Letter we propose an alternative broad
class of Majoron models which might explain the excess events, while still preserving the
1
agreement between theory and geochemical experiments for the decay rate of
128
Te,
130
Te and
238
U [9].
We start by describing some generic features of
M
decays, and then turn to model-
specic issues. First, agreement with the Z-width measurement rules out a direct Majoron
coupling to the Z boson. Lepton number breaking, if it occurs, must come from an
electroweak-singlet eld. Majorons therefore cannot have renormalizable couplings to
e
;
instead,
M
proceeds through mixing of
e
with sterile neutrinos that couple to Majorons.
Next one must explain how
M
but not
0
events are seen, since lepton-breaking
generically gives both eects. The GR model did so by requiring a small lepton-number
breaking scale as in eq. (2), and the same expedient is used by alternate models where
electroweak-singlets rather than triplets do the lepton-breaking. But once such a small scale
is introduced by hand it is unnecessary for the scalar emitted in
M
to be a Goldstone
boson; it could just as well be given a mass of 10 keV. We refer to these as ordinary
Majorons (and denote their emission by
oM
) since they share many features (electron
spectra, etc.) of the GR model. These models all suer from a hierarchy problem since
the lepton-breaking VEV is not stable under renormalization and so must be articially
ne-tuned [10].
Here we introduce a class of models that do not require such a small scale, inspired
by realizing that the smallness of v in eq. (2) comes from the bound on
0
decay. This
restriction disappears if a lepton number, L, carried by electrons, remains unbroken, so
that
0
is completely forbidden. Then
M
occurs only if the scalar particle itself is
charged, i.e., it carries L() = 2, in units where L(e

) = +1. Fine-tuning of the scalar

mass is avoided if the scalar is a Goldstone boson, a charged Majoron. All of this can
be realized in a simple way, using for example a global SU(2)
s
U(1)
L
symmetry broken
down to U(1)
L
. (The U(1)
L
factor exists so that Majoron couplings to ordinary charged
leptons can be avoided.) Two of the resulting Goldstone bosons then carry the unbroken
charge, just as the would-be Goldstone bosons that make up the longitudinal W

s carry
electric charge in the Standard Model. We call all such models charged Majoron models
and denote the resulting process
cM
.
In what follows we compare charged and ordinary Majoron models with each another
and the data. We will: (i) show how the models can be experimentally distinguished by
the predicted shape of the sum electron spectrum, and (ii) nd the necessary conditions
for getting a large enough rate of
M
.
2
i. The electron spectrum. The decay rates for
2
,
oM
and
cM
all have the form
d() =
G
4
F
4
3
|A()|
2
d
n
, (3)
where G
F
is the Fermi constant, A is a matrix element (more about which later), and d
n
is the phase space, a function of the momenta and energies of the two outgoing electrons,
d
n
=
1
64
2
(Q
1

2
)
n
2

k=1
p
k

k
F(
k
)d
k
. (4)
F() is the Fermi function, equal to 1 when the nuclear charge vanishes, and Q

1
2 MeV is the endpoint energy for the nal-state electrons.
The integer n in eq. (4) depends on the decay channel: n = 5 for
2
, 3 for
cM
, and
1 for
oM
. n controls the shape of the electron energy spectrum, because for these decays
all lepton energies are p
F
100 MeV, the nuclear scale that determines the matrix
element A. Thus it is a good approximation relative error of order Q/p
F
1% to
neglect As dependence on the outgoing lepton energies. In particular, near the endpoint
the electron spectrum in
cM
goes like (Q E)
3
as E, the sum of electron energies,
approaches Q. This is intermediate between (Q E) for
oM
, and (Q E)
5
for
2
.
The spectra are plotted in Fig. 1. The distinction between
cM
and
oM
is one of our
main results.
Figure 1. The electron sum-energy spectrum for
2
(dashed),
cM
(solid) and

oM
(dot-dashed) decays.
3
We emphasize that the spectral shape dierence characterizes these two classes of
models, independently of any model-specic details. This traces to the vanishing of
cM
amplitudes at zero Majoron momentum, q = 0, due to the fact that Goldstone bosons are
always derivatively coupled. The puzzle is why
oM
amplitudes do not also vanish at
q = 0. There one nds that graphs with Majoron Brehmsstrahlung o the electron lines
survive as q 0 because the internal electron propagator diverges in this limit, leaving a
nite result. The same cannot happen for
cM
because the electron-Majoron coupling is
forbidden by the unbroken lepton symmetry in these models.
ii. The decay rate. Computing the absolute rate requires knowing the matrix element,
A, of eq. (3). Consider arbitrary Yukawa couplings between a set of neutrinos and a
complex scalar eld, ,
L
M
=
1
2

i
(a
ij
P
L
+b
ij
P
R
)
j

+c.c. (5)
Goldstone boson couplings are included here as a special case: if f is the decay constant,
X
ij
the conserved charge and m
ij
the left-handed neutrino mass matrix, then a
ij
takes
the form a = i(X
T
m + mX)/f, and similarly for b
ij
. Real scalar elds are also easily
incorporated by taking b = a

in all results. For example the GR Majoron coupling arising

from a triplet interaction L =
1
2
g L
e
P
L

a
L
e

a
+ c.c. would give b

ee
= a
ee
= ig.
To lowest (zeroeth) order in the scalar energy we get eq. (3) with n = 1 and
A(
oM
) =

ij
V
ei
V
ej
_
d
4
p
(2)
4
(4w
1
w
2
+p
2
w
5
) (a
ij
m

i
m

j
p
2
b
ij
)
(p
2
+m
2

i
i)(p
2
+m
2

j
i)
. (6)
V
ei
is the mixing matrix element for nding the mass eigenstate
i
in an electron charged-
current weak interaction. The form factors w
a
parametrize the matrix element of the
hadronic weak currents between initial and nal nuclei N and N

,
W

_
d
4
x N

|T

[J

(x)J

(0)] |N e
ipx
= w
1

+w
2
u

+w
3
(p

)+
+w
4

+w
5
p

,
(7)
and are functions of the invariants p
2
and up. u

is the four-velocity of the initial and nal

nuclei, which are equal since we neglect all dependence in W

on the nal-state momenta.

4
Although eq. (6) suces for
oM
it happens that for
cM
this lowest-order result
vanishes, making it necessary to work to next-higher order in the outgoing Majoron mo-
mentum, and leading to the dierence in electron spectra (above). The result has the form
of eq. (3), but with n = 3 and |A(
cM
)|
2
= |U(
cM
)|
2
+|V(
cM
)|
2
, where
U(
cM
) =

ij
V
ei
V
ej
b
ij
_
d
4
p
(2)
4
2 p
2
(w
3
2iw
4
)
(p
2
+m
2

i
i)(p
2
+m
2

j
i)
,
V(
cM
) =

ij
V
ei
V
ej
b
ij
_
d
4
p
(2)
4
p
0
(w
2
p
2
w
5
)
(p
2
+m
2

i
i)(p
2
+m
2

j
i)
.
(8)
Notice that whereas A(
oM
) has the same combination of form factors as
0
,

cM
depends on dierent nuclear matrix elements. (For
0
, A

i
V
2
ei
m

i
_
[d
4
p/(2)
4
] [(4w
1
w
2
+ p
2
w
5
)/(p
2
+ m
2

i
i)]; these are related to the Fermi and
Gamow-Tellertype form factors by w
1
=
1
3
w
GT
and w
2
= w
F
+
1
3
w
GT
.)
A striking feature of both eqs. (6) and (8) is that they vanish if all neutrinos share a
common mass, for in this limit the sum over states gives

ij
V
ei
V
ej
a
ij
= a
ee
. The latter is
zero since the Z
0
width constraint implies that the Majoron cannot directly couple to
e
.
As an important corollary, the amplitudes must also vanish if all of the
i
are negligibly
light compared to the Fermi momentum p
F
, above which the integrals in (6) and (8) are
eectively cut o. Thus in any model explaining the excess electron events by Majoron
couplings to a virtual neutrino, at least one of the
i
must have a mass m

i
>

100 MeV.
To make contact with the literature, we use the predictions of the GR model as a
benchmark. In this case
A(
M
)
GR
=
2

g
2
A
g
e
M, (9)
g
e
is the GR coupling of to
e
, g
V
and g
A
are the axial and vector couplings of the
nucleon weak currents, and M =

[
n

m
(g
V
/g
A
)
2
]h(r)
_
(see eq. (18)). We integrate
the phase space d
1
above a threshold E
th
where the anomalous events begin and the
contribution from ordinary decay should be small. Table 1 compares this with the
excess events seen by the Irvine group for the elements
82
Se,
100
Mo and
150
Nd [6], and in
the published spectrum of
76
Ge [11]. In all of these cases the excess events comprise R = 2
to 3 % of the total number observed. We also analyze the geochemically observed decays,
5
Table 1: parameters for emission of GR Majorons in double beta decay. T
1
1/2
is the inverse
half-life of the anomalous events; note that for the last three elements this is assumed to be the
entire rate; |M|
2
is the 0 matrix element extracted from Staudt et al.; and R is the ratio of
anomalous to all events hypothesized to be 1 for the geochemically observed decays.
1
m
7
e
is
the total phase space for Majoron events (see eq. (4)), while E
th
(MeV) denotes our choice for
the threshold value of the sum of the electron energies, above which essentially only excess events
appear.
1
m
7
e
is the part of phase space occurring above E
th
and g
e
= g
ee
is the coupling
needed to explain the rate using Majoron emission in the GR model.
Element T
1
1/2
(y
1
) |M|
2
m
2
e
R
1
E
th

1
g
e
76
Ge 2 10
23
1 10
5
0.02 2.0 1.5 0.9 1 10
4
82
Se 2 10
22
9 10
4
0.03 17 2.2 7.6 8 10
5
100
Mo 3 10
21
2 10
4
0.03 34 1.9 22 4 10
4
150
Nd 3 10
20
1 10
5
0.02 260 2.2 155 2 10
4
128
Te 5 10
25
7 10
4
1.00 0.23 0.0 0.23 4 10
5
130
Te 1 10
21
5 10
4
1.00 30 0.0 30 1 10
4
238
U 5 10
22
3 10
4
1.00 33 0.0 33 2 10
4
assuming all the events are due to
M
. Remarkably, g
e
lies in the range 4 10
5
to
4 10
4
for all seven elements.
The predictions for Te are of particular interest because of a recent measurement of the
ratio of decay rates (
130
Te)/(
128
Te)= (2.41 0.06) 10
3
[12]. If the same coupling
needed to account for the endpoint anomalies (g
e
1 10
4
) is used for these decays
then the GR and
oM
models predict too small a ratio: (
oM
) = (30.4/0.23)(5/7) = 93.
The same is not true for
cM
models because the rates scale with an extra factor of Q
2
relative to those for
oM
, and
128
Te has a small endpoint energy, Q 0.9 MeV. Thus we
nd (
cM
) = 770, in much better agreement with the experimental value.
To see whether the ordinary or charged Majoron models can predict a large enough
rate, we display the combinations of couplings, dened in (5), that are equivalent to the
GR model with coupling g
e
, so far as the total rate is concerned. For ordinary Majorons
g
e
=

ij
V
ei
V
ej
_
1
p
2
_
1
_
m

i
a
ij
m

j
p
2
b
ij
(p
2
+m
2

i
)(p
2
+m
2

j
)
_
. (10)
Here p is the four-momentum of the virtual neutrino, which as indicated by , is integrated
weighted by the nuclear form factors. An estimate for the size of g
e
is therefore obtained
6
by replacing p
2
by the characteristic momentum scale, p
2
F
(100 MeV)
2
. For charged
Majorons, in order of magnitude
g
e
=

ij
V
ei
V
ej
_
1
p
2
_
1
_
Qpb
ij
(p
2
+m
2

i
)(p
2
+m
2

j
)
_
X. (11)
The two qualitatively new features here are: (1) the additional suppression by the endpoint
energy, Q/p, due to the softer electron spectrum of these models, and (2) a ratio, X, of
the
cM
to
oM
nuclear matrix elements. Later we give a more quantitative comparison
of these matrix elements, in a specic model.
Consider now a representative model for each class. For
oM
, augment the Standard
Model with two Majorana-Dirac sterile neutrinos, s
+
and s

, and a complex singlet scalar,

. The model consists of interactions that preserve electron-type lepton number, assuming
L
e
(P
L
s

) = 1, and L
e
() = 2. To satisfy the
0
constraint we take = 0 and
ne-tune the mass to be
<

1 MeV as required by the

M
kinematics. The spectrum
contains one Dirac neutrino (
R
,
L
) with mass M,
R
= s
c

and
L
=
e
sin +s
+
cos .
The massless orthogonal combination

e
is predominantly
e
. The remaining two neutrinos

and

acquire Majorana masses at dimension ve.

In terms of the relevant Yukawa couplings to ,
L
y
=
1
2
g
+
( s
+
P
L
s
+
)
1
2
g

( s

P
L
s

+ c.c. (12)
the eective
M
coupling in this model becomes
g
e
=
_
1
p
2
_
1
_
(g

p
2
g
+
M
2
cos
2
)M
2
sin
2

p
2
(p
2
+M
2
)
2
_
. (13)
Since the light scalars couple to the massless neutrino

e
through mixing,
oM
may
proceed via the exchange of

e
, with the result g
e
is nonvanishing as M with
held xed. (Decoupling is not violated since 1/M if dimensionless couplings in the
lagrangian are held xed for M large.) Using

p
2
_
(100 MeV)
2
and 0.1 we nd
g
e
10
4
, as suggested by the anomalies. We choose M above the K

meson mass;
otherwise a heavy neutrino with such large mixing would have been seen as a peak in the
K e [13] or e [14] decay spectra.
For charged Majorons we suppose that the lepton symmetry group surviving at elec-
troweak energies is G
L
= SU(2)
s
U(1)
L
, and introduce the sterile Majorana neutrinos
N = (N
+
, N

), s
+
, and s

, whose left-handed components transform respectively as (2, 0),

7
(1, 1), and (1, 1) under G
L
. A sterile scalar doublet transforming as (2, 1) is also added,
whose VEV breaks G
L
U(1)
e
where L
e
= 2T
3
+L

. The renormalizable interactions pre-

dict three massless neutrinos

e
,

and

, and two heavy Dirac neutrinos of mass M and

, say.
e
is mostly

e
, but mixes with N
+
and s
+
similarly to the previous model. There
are three massless Goldstone boson states: a complex scalar having charge L
e
() = 2
and a real scalar with L
e
= 0.
For
cM
the unbroken lepton number not only forbids contributions from graphs
involving purely massless neutrinos, but it also causes the contribution from each heavy-
neutrino line to go like 1/M
2
rather than 1/M when M > 100 MeV. The equivalent GR
coupling of the Majoron in
cM
is roughly
g
e
sin
2

M
2v
Qp
F
M
2
(p
2
F
+M
2
)
2
X, (14)
if M is the heavy neutrino mass and v is the G
L
-breaking scale. (cf. eq. (11), which due
to the relation between b
ij
and the mass matrix does reproduce (14), despite appearances.)
For M v p
F
, 0.1 and X 1, g
e
is of order
2
Q/p
F
10
4
as desired.
To be more quantitative, we must evaluate the new nuclear matrix elements for
cM
,
which dier from the usual ones because the leptonic part of the amplitude is proportional
to the Majoron and virtual neutrino momenta through the factor [/ p, / q]. The p
0
piece (V
of eq. (8)) vanishes in the integral
_
dp
0
, up to small O(Q/p) corrections. To have a
0
+
0
+
nuclear transition, the p
i
piece (U in eq. (8)) must combine with odd-parity
nuclear operators, which come from the recoil corrections to the nucleon currents and p-
wave Coulomb corrections to the electron wave functions. Although they are formally of
higher order in or v/c of the nucleon than the usual Gamow-Teller and Fermi matrix
elements, explicit calculations by nuclear theorists show that they need not all be small,
as we will show [15].
For small mixing angles the matrix element becomes
|A(
cM
)|
2
=
2

2
(g
A
)
4
_
M
v
_
2

i=1,2
|M
i
|
2
, (15)
in which the nuclear matrix elements are
|M
1
|
2
=
7
9

_
(A
1
+iA
2
)

h
_

2
; |M
2
|
2
=
2
9

_
A
3

h
_

2
, (16)
8
and A
i
are operators
A
1
= r
_
(C
m

n
C
n

m
) + (g
V
/g
A
)
2
(D
n
D
m
)
+i(g
V
/g
A
)(D
n

m
+
n
D
m
)

;
A
2
=
Zr
2R
_
(g
V
/g
A
)
2
+
m

n
2
m
r
n
r

;
A
3
= r
_
(C
m

n
+C
n

m
) + (g
V
/g
A
)
2
(D
n
+D
m
)

.
(17)
We use the notation O
n
O
m
= 0
+
f
|

n,m
O
n
O
m

+
n

+
m
|0
+
i
, where
+
n
is the isospin raising
operator for the nth nucleon. Denitions of C and D can be found in the reviews by Doi
et al. [16] or Tomoda [17]. The neutrino potential

h depend on the internucleon separation
vector r = r
n
r
m
and the heavy neutrino mass M:

h =

M
2

r
(h(0) h(M)) ; h(M) =
_
d
3
p
2
2
e
ipr
1
( +)
, (18)
where = (p
2
+M
2
)
1/2
and 10 MeV is the average nuclear excitation energy.
In Table 2 we show the value of the nuclear matrix elements, (

i
|M
i
|
2
)
1/2
, needed for
charged Majoron emission to account for the anomalous events, assuming that
2
(M/v) =
3 10
2
(as would be the case for a neutrino with 17% mixing, the limit coming from
avor universality [18]). For comparison, we also show a similar matrix element that has
been calculated by Muto et al. [19], namely
M
R
= (g
V
/g
A
)

p
2
F
_
1
(D
n

m
+
n
D
m
) r h/r , (19)
Except for a minor dierence in the neutrino potential, the same matrix element is con-
tained in the A
1
term of eq. (16). This is only meant to be indicative because there are
further contributions to our

i
|M
i
|
2
which do not appear in the literature, and which
are conceivably bigger than M
R
.
From table 2 we see that charged Majoron emission explains the data remarkably
well, if the as-yet uncomputed matrix element for
238
U is similar to those of the other
nuclei. In particular the
cM
rates for the geochemically observed decays of
128
Te and
130
Te dominate over the contribution of
2
. As was described earlier, this is in better
agreement with the data than is the result for
oM
models.
Because of the extra suppression by (Q/p
F
)
2
in
cM
, we needed a heavy neutrino
mixing with
e
at the = 10% level, with a mass near 100 MeV. But there are strong
limits on for such a heavy neutrino. Of these, beam-dump experiments do not apply
9
Table 2: phase space and matrix elements for charged Majoron emission in double beta decay.
The phase space factors are dened analogously to those in table 1. We give the values of the
matrix elements which are needed to explain the data (assuming all events were
cM
for
238
U
and Te) for
2
M/v = 3 10
2
, and values of some representative matrix elements that have
been calculated; see eqs. (16) and (19), respectively.
Element
3
/m
9
e

3
/m
9
e
_
i
|M
i
|
2
_
1/2
needed |M
R
| (Muto et al.)
76
Ge 4.7 0.43 1.1 1.1
82
Se 78 7.9 0.99 0.95
100
Mo 160 42 1.4 1.1
150
Nd 1570 320 1.6 1.3
128
Te 0.13 0.13 0.32 0.92
130
Te 110 110 0.48 0.78
238
U 35 35 0.61 ?
to our model because decays invisibly into a light neutrino plus a Majoron. However,
searches for peaks in the spectra of , K e [20] rule out a long-lived heavy neutrino with
such large mixing, unless M > M
K
500 MeV. For such large M the
cM
rate becomes
suppressed, unless the unknown nuclear matrix elements turn out to be considerably larger
than those of ref. [19].
But there is a loophole in the peak search bounds: they do not apply if the 100 MeV
neutrino is a broad resonance. A width of 10 MeV is sucient to hide the
peak beneath backgounds [21], and the K decay searches do not extend below 140 MeV.
Such large widths occur in our model if the neutrino-Majoron coupling is strong, g 4,
in which case we must repeat the
M
analysis with new form factors parameterizing the
strongly-coupled neutrino sector. Then dimensional analysis again implies that the rate is
maximal if the scale for new neutrino physics is of order 100 MeV. It would be fascinating
if double beta decay experiments gave the rst hint of a strongly-coupled neutrino sector!
We note that v M (few hundred MeV) is also the scale at which the global
symmetry G
L
must break. While this is still annoyingly small compared to the weak
scale, it is a signicant improvement on the much smaller 10 keV scales needed in ordinary
Majoron models.
A nal challenge confronting any model is the strong limit, N

< 3.3, [22] on the num-

ber of neutrino species at nucleosynthesis. The large coupling implied by the anomalies
means that Majorons had an undiluted thermal energy density at MeV temperatures, a
10
potential disaster, since each Majoron counts as 4/7 of a neutrino. The three Majoron
states of the
cM
models therefore preclude more than two light neutrinos at nucleosyn-
thesis, which could happen if

(or

) decays before 0.8 MeV. One possibility is that

m

25 MeV (large enough to allow it to decouple before nucleosynthesis but still below
the laboratory mass bound), with a lifetime < 10
3
sec [23]. But even if

is light, it can
eectively contribute as negative number of neutrino species if it decays into
e
with life-
time 610
4
sec < < 210
2
sec [24], an even better prospect for our models. The rst
scenario, for example, occurs in our ordinary Majoron model via

+ , mediated
by a dimension-seven operator (L

H)P
L
(L

H)

. Similarly, we can use dimension-ve

and dimension-seven interactions to give majorana masses and lifetimes to

and

in
the charged Majoron model.
In summary, we have presented two classes of models for Majoron emission in double
beta decay which may be able to explain excess events near the endpoints of several
elements without seriously spoiling the agreement of geochemical observations with
theory. We propose a new class of models involving a charged Majoron that carries
an unbroken lepton number, which predict a softer sum energy spectrum than that of
ordinary Majorons, and thus make them experimentally distinguishable from the latter.
Charged-Majoron models are highly constrained, and require a larger, 100 MeV scale of
symmetry breaking than the unnaturally small 10 keV scale of ordinary Majoronsa 10
4
-
fold improvement. The detailed predictions for these models depend in part on the size of
certain nuclear matrix elements (eq. (16)) which dier from those appearing in the usual
amplitudes for
2
,
0
, or
oM
. All our models point toward a sterile neutrino with
a mass of a few hundred MeV, a large ( 0.1) angle for mixing with
e
, and possibly a
large ( 10 MeV) width. Nucleosynthesis requirements suggest that the neutrino which is
predominantly

should either have a mass that is not much below the laboratory lower
limit, or else a lifetime
<

10
3
sec.
We warmly thank F. Avignone, M. Moe and A. Turkevich for information about their
experiments, T. Kotani, R. Shrock and E. Takasugi for helpful correspondence, W. Haxton,
M. Luty, N. de Takacsy and P. Vogel for helpful discussions and R. Fernholz for his con-
tributions,including the gure. This work was supported in part by the Natural Sciences
and Engineering Research Council of Canada and les Fonds F.C.A.R. du Quebec.
11
References
[1] Y. Chikashige, R.N. Mohapatra and R.D. Peccei, Phys. Lett. 99B (1981) 411.
[2] M. Doi, T. Kotani and E. Takasugi, Phys. Rev. D37 (1988) 2572.
[3] H.M. Georgi, S.L. Glashow and S. Nussinov, Nucl. Phys. B193 (1981) 297.
[4] F.T. Avignone III et al., in Neutrino Masses and Neutrino Astrophysics, proceed-
ings of the IV Telemark Conference, Ashland, Wisconsin, 1987, edited by V. Barger,
F. Halzen, M. Marshak and K. Olive (World Scientic, Sinagpore, 1987), p. 248.
[5] P. Fisher et al., Phys. Lett. B192 (1987) 460; D.O. Caldwell et al., Phys. Rev. Lett. 59
(1987) 1649.
[6] M. Moe, M. Nelson, M. Vient and S. Elliott, preprint UCI-NEUTRINO 92-1.
[7] F.T. Avignone, private communication
[8] G.B. Gelmini and M. Roncadelli, Phys. Lett. 99B (1981) 411.
[9] We have been informed that the discrepancy between the
238
U observations of Turke-
vich, Economou and Cowan, (Phys. Rev. Lett. 67 (1991) 3211) and the calculations of
Staudt, Muto and Klapdor-Kleingrothaus (Europhys. Lett., 13 (1990) 31) have now
been resolved.
[10] It has recently been claimed by Z.G. Berezhiani, A.Yu. Smirnov and J.W.F. Valle,
(preprint FTUV/92-20, IFIC/92-21, LMU-05/92) that the required Majoron coupling,
g
M
10
4
, is suciently small to allow such a hierarchy below the weak scale. This
argument overlooks the fact that in these models the eective Majoron coupling mea-
sured in
M
decay is given by g
M
g
2
, where the small mixing angle, , is bounded
by oscillation and neutrino decay experiments to be very small. Since the coupling that
controls the hierarchy problem is g, and not g
M
, it is typically not small: g O(1).
[11] F.T. Avignone III et al., Phys. Lett. B256 (1991) 559.
[12] This result of Bernatowicz and Holenberg, together with its application to a
oM
description of the excess events may be found in the talk by W. Haxton, at Neutrino
92, Granada, Spain
[13] R.E. Shrock, Phys. Rev. D24 (1981) 1232; T. Yamazaki et al., Proceedings of the XIth
International Conference on Neutrino Physics and Astrophysics, eds. K. Kleinknecht
and E.A. Paschos (World Scientic, Singapore, 1984), p. 183.
[14] D.I. Britton et al., Phys. Rev. Lett. 68 (1992) 3000.
[15] The dierences between charged and ordinary Majorons can be seen in ref. [2], in what
they call the correction term; in contrast to their general analysis, our model has
part of this correction term being the leading contribution. Our results cannot simply
be extracted from ref. [2] because we dier on the form of the neutrino potential, and
in having the matrix element M
2
.
[16] M. Doi, T. Kotani and E. Takasugi, Prog. Theo Phys. Suppl. 83 (1985) 1.
[17] T. Tomoda, Rept. Prog. Phys. 54 (1991) 53.
12
[18] M. Gronau, C.N. Leung and J.L. Rosner, Phys. Rev. D 29, (1984) 2539.
[19] K. Muto, E. Bender and H.V. Klapdor, Z. Phys. A 334 (1989) 187, as tabulated in
[17].
[20] R.E. Shrock, Phys. Rev. D24 (1981) 1232; T. Yamazaki et al., Proceedings of the XIth
International Conference on Neutrino Physics and Astrophysics, eds. K. Kleinknecht
and E.A. Paschos (World Scientic, Singapore, 1984), p. 183.
[21] We thank D. Britton for helpful conversations on this point.
[22] T.P. Walker, G. Steigman, D.N. Schramm, K.A. Olive and H.S. Kang, Ap. J. 376
(1991) 51 and references therein.
[23] E.W. Kolb, M.S. Turner, A. Chakravorty and D.N. Schramm, Phys. Rev. Lett. 67
(1991) 533.
[24] K. Enqvist, K. Kainulainen and M. Thomson, Phys. Rev. Lett. 68 (1992) 744.
13