We propose a new scenario of neutrino masses with a Higgs triplet
$(\xi^{++},\xi^+,\xi^0)$ in a theory of large extra dimensions. Lepton number
violation in a distant brane acts as the source of a very small trilinear
coupling of $\xi$ to the standard Higgs doublet in our brane. Small realistic
Majorana neutrino masses are \underline{naturally} obtained with the
fundamental scale $M_* \sim {\cal O}(1)$ TeV, foretelling the possible
discovery of $\xi$ $(m_\xi\lsim M_*)$ at future colliders. Decays of $\xi^{++}$
into same-sign dileptons are fixed by the neutrino mass matrix. Observation of
$\mu-e$ conversion in nuclei is predicted.Comment: A comment on Tevatron reach and two references added. Discussion and
  conclusions unchange

A. Das

A. E. Faraggi

E. Ma

Ernest Ma

F. Cuypers

G. Dvali

H. C. Chiang

H. Plothow-Besch

I. Antoniadis

K. Huitu

K. R. Dienes

M. Raidal

Martti Raidal

N. Arkani-Hamed

R. N. Mohapatra

Utpal Sarkar

Y. Fukuda

Y. Suzuki

English

arXiv

We propose a new scenario of neutrino masses with a Higgs triplet (ξ++,ξ+,ξ 0) in a theory of large extra dimensions. Lepton number violation in a distant brane acts as the source of a very small trilinear coupling of ξ to the standard Higgs doublet in our brane. Small realistic Majorana neutrino masses are naturally obtained with the fundamental scale M∗ ~O(1) TeV, foretelling the possible discovery of ξ (mξ≲M∗) at future colliders. Decays of ξ++ into same-sign dileptons are fixed by the neutrino mass matrix. Observation of μ-e conversion in nuclei is predicted

Ma, Ernest

Raidal, Martti

Sarkar, Utpal

Publications of the IAS Fellows

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UCRHEP-T279
June 2000
Verifiable Model of Neutrino Masses
from Large Extra Dimensions
Ernest Ma1, Martti Raidal1,2, and Utpal Sarkar1,3
1 Department of Physics, University of California, Riverside, California 92521, USA
2 National Institute of Chemical and Biological Physics, Ra¨vala 10, 10143, Estonia
3 Physical Research Laboratory, Ahmedabad 380 009, India
Abstract
We propose a new scenario of neutrino masses with a Higgs triplet (ξ++, ξ+, ξ0) in
a theory of large extra dimensions. Lepton number violation in a distant brane acts
as the source of a very small trilinear coupling of ξ to the standard Higgs doublet in
our brane. Small realistic Majorana neutrino masses are naturally obtained with the
fundamental scale M∗ ∼ O(1) TeV, foretelling the possible discovery of ξ (mξ <∼ M∗)
at future colliders. Decays of ξ++ into same-sign dileptons are fixed by the neutrino
mass matrix. Observation of µ− e conversion in nuclei is predicted.
Recently it has been proposed [1] that the fundamental scale of quantum gravity could
be as low as a few TeV. In this theory, our four-dimensional world (called a 3-brane) is
localized to only one point in a space having n extra dimensions (the bulk). Assuming that
gravity is the only field which propagates in these extra dimensions, the fact that it is so
weak in our world may be understood. Since there are no scales larger than a few TeV in
this theory, the smallness of neutrino masses requires a different explanation than the usual
one that it is inversely proportional to some very large mass. There are already proposals
in this direction, where singlet fermions are introduced in the bulk [2, 3]. They become the
right-handed partners of the observed left-handed neutrinos, but with small Dirac masses
– the result of being suppressed by the volume of the extra dimensions. Note first that
this mechanism by itself does not generate a Majorana mass. Note also that such a model
of neutrino masses, like any other before it, has no distinct experimental signatures which
would set it apart from any other. Furthermore, the existence of singlet fermions in the bulk
is strongly constrained by supernova data to have an upper bound of 210/
√
n TeV on the
fundamental scale involved [4].
In this Letter we propose a simple, verifiable scenario in which the distant breaking
[5] of lepton number is used to generate small Majorana neutrino masses through a scalar
Higgs triplet [6] localized in our brane. The trilinear interaction of the triplet with the
standard Higgs doublet is induced by the “shining” of a scalar singlet which communicates
the violation of lepton number from a distant brane to our world through the large extra
space dimensions. Thus the amount of lepton number violation in our world is suppressed by
the distance between us and the other brane. Because the Higgs triplet should not be heavier
than the fundamental scale of the extra dimensions, this scenario provides the basis for a
rich and robust phenomenology of new observable processes at future high-energy colliders
as well as low-energy precision experiments.
2
The Higgs triplet may be produced at the upgraded Tevatron at Fermilab, the Large
Hadron Collider (LHC) at CERN, and perhaps a future linear collider (LC) or muon col-
lider (MC) or both. The decay branching fractions of the doubly charged member ξ++ of
the Higgs triplet into same-sign dileptons or the cross sections of the resonant processes
e−e−(µ−µ−) → l−i l−j may then be used to determine the magnitude of each element of
the neutrino mass matrix completely and uniquely up to an overall scale factor. Whereas
lepton-number violating processes are suppressed at low energies (because their amplitudes
are proportional to the neutrino mass matrix), large lepton-flavor violation through the Higgs
triplet is possible. Based on current neutrino-oscillation data, the rate of coherent µ−e con-
version in nuclei, with an amplitude enhanced by the large factor ln(m2ξ++/m
2
µ) over that of
µ → eγ, is predicted to be observable at the proposed Muon Electron Conversion (MECO)
experiment [7] at Brookhaven National Laboratory (BNL).
We extend the minimal standard model of particle interactions to include a heavy Higgs
triplet (ξ++, ξ+, ξ0) with the interaction
L = fij [ξ0νiνj + ξ+ (νilj + liνj)√
2
+ ξ++lilj ] + h.c., (1)
implying that ξ carries lepton number L = −2. The resulting Majorana neutrino mass
matrix is then given by
(Mν)ij = 2fij〈ξ0〉, (2)
where 〈ξ0〉 is the vacuum expectation value (VEV) of ξ0, hence lepton number must be
violated somehow. This may be achieved with the following addition to the Lagrangian:
L′ = hχ(ξ¯0φ0φ0 −
√
2ξ−φ+φ0 + ξ−−φ+φ+) + h.c., (3)
where χ is a complex neutral scalar singlet with L = −2. We then obtain [6]
〈ξ0〉 ≃ h〈χ〉〈φ
0〉2
m2ξ
. (4)
3
In other words, the breaking of L by 〈χ〉 is communicated to the lepton sector through 〈ξ〉.
As the neutrino masses are very small, we need mξ to be very large in Eq. (4). Suppose
mξ is only of order 1 TeV, then h〈χ〉 would have to be very small, i.e. h〈χ〉 ∼ 33 eV, if
〈ξ0〉 ∼ 1 eV, using 〈φ0〉 = 174 GeV. However, χ is a singlet, and as such, its VEV is expected
to be large, so this requires what we normally would consider as extremely unnatural fine
tuning. However, in a theory of large extra dimensions, a small 〈χ〉 may arise naturally, thus
allowing mξ to be of order 1 TeV and be directly observed experimentally.
The standard model fields together with ξ are localized in our world (a 3-brane P at
y = 0) and are blind to the extra space dimensions. Lepton number is assumed to be
conserved as far as these fields are concerned. The singlet χ is special; it propagates also in
the bulk carrying lepton number L = −2 and interacts in our brane according to Eq. (3).
The violation of lepton number happens at a distant 3-brane which then gets communicated
to our brane through 〈χ〉.
We assume the existence of a field η which is localized in a distant 3-brane (P ′) situated
at a point y = y∗ in the extra dimensions. It is a singlet under the standard model but has
L = 2 and couples to χ (with L = −2). When the field η acquires a VEV, lepton number is
broken maximally in the other brane. It will then act as a point source for L violation, and
the field χ is the messenger which carries it to our wall (the interface between our brane and
the bulk). The “shining” of χ at all points in our world is the mechanism [5] which breaks
lepton number and gives mass to the neutrinos.
At energies much below the fundamental scale M∗, the lepton-number violating effect
will be suppressed by the distance between the source brane at P ′ and our brane at P. We
assume that the source brane is situated at the farthest point in the extra dimensions so
that |y∗| = r is the radius of compactification and it is related to the fundamental scale M∗
4
and the reduced Planck scale (MP = 2.4× 1018 GeV) by the relation
rnMn+2∗ ∼M2P . (5)
This explains why lepton number is only violated weakly in our world.
We assume here that the source brane has the same dimensional structure as our world
and there are n extra dimensions. In our world (P) the field χ has only the lepton-number
conserving interaction
Sχ = h
∫
P
d4x ξ†(x)φ(x)φ(x)χ(x, y = 0). (6)
In the other brane (P ′) the field χ couples to the field η through the interaction
Sother =
∫
P ′
d4x′ µ2 η(x′)χ(x′, y = y∗), (7)
where µ is a mass parameter. Lepton number violation from 〈η〉 is carried by χ to our world
through its “shined” value 〈χ〉:
〈χ(x, y = 0)〉 = ∆n(r)〈η(x, y = y∗)〉, (8)
where 〈η〉 acts as a point source, and ∆n(r) is the Yukawa potential in n transverse dimen-
sions, i.e. [5]
∆n(r) =
1
(2pi)
n
2Mn−2∗
(
mχ
r
)n−2
2
Kn−2
2
(mχr) , (9)
K being the modified Bessel function. If the mass of the carrier field χ is large (mχr ≫ 1),
it has the profile
〈χ〉 ≈ m
n−3
2
χ
2(2pi)
n−1
2 Mn−3∗
e−mχr
r
n−1
2
. (10)
The suppression here is exponential, hence the amount of lepton-number violation in our
world is very small, but its precise value depends sensitively onmχ. An interesting alternative
is to have a light carrier field with a mass less than 1/r. However, it should be larger than
5
about (0.1 mm)−1, to be consistent with the present experimental data on gravitational
interactions.
If mχr ≪ 1, ∆n(r) is logarithmic for n = 2 and 〈χ〉 is not suppressed. For n > 2, the
asymptotic form of the profile of χ is
〈χ〉 ≈ Γ(
n−2
2
)
4pi
n
2
M∗
(M∗r)n−2
, (11)
which is suitably small for large r. Using Eqs. (2), (4), and (5), we then obtain
(Mν)ij ≈
Γ(n−2
2
)
2pi
n
2
h fij
〈φ〉2 M∗
m2ξ
(
M∗
MP
)2− 4
n
. (12)
For n = 3, M∗ ∼ 1 TeV, mξ <∼ 1 TeV and h ∼ f33 ∼ 0.25, we get (Mν)33 ∼ 0.03 eV, which
is of the right magnitude for understanding atmospheric neutrino oscillations. There is a
massless Goldstone boson (Majoron) corresponding to the spontaneous breaking of lepton
number in this scenario. It is a singlet and its coupling to the leptons in our world is
suppressed by 〈η〉−1. It is thus completely acceptable phenomenologically.
For n ≥ 4, this scenario requires a larger fundamental scale M∗ and is of less phenomeno-
logical interest. However, we have assumed here for simplicity the dimension of the source
brane to be the same as ours, which may not be the case. The distance between the branes
may also be smaller. On the other hand, such complications diminish the predictive power
of the proposed mechanism; they will not be considered any further.
The only pair production mechanism of ξ++ at the LHC is the Drell-Yan process mediated
by s-channel photon and Z-boson exchange [8]. Thus the production rate is enhanced by
the double charge of ξ++ and is uniquely determined by the gauge couplings. To obtain the
cross section of pp → ξ++ξ−− at the LHC, we have calculated the subprocesses involving
the u, u¯ and d, d¯ collisions and convoluted them over the parton distributions given by the
default set of the CERN library package PDFLIB [9]. The total cross section as a function
of the triplet mass mξ++ is plotted in Fig. 1.
6
Once produced, ξ++ will decay into same-sign dileptons. Because 〈ξ〉 is tiny, the decay
branching fraction of ξ++ → W+W+ is negligible. Thus the only possible decay channels in
our scenario are ξ++ → l+i l+j with the partial rates Γij = |fij |2mξ/4pi for i 6= j, and 1/2 smaller
for i = j. This same-sign dilepton signal at the invariant mass of ξ is very distinctive at the
LHC because it is completely background-free. Assuming the total integrated luminosity of
the LHC to be 1000 fb−1, the reconstruction efficiency of the event to be 10% (to be very
conservative) and that the predicted average of N = − ln(1− p) Poisson distributed events
would provide a discovery, the cross section in Fig. 1 implies at p = 95% confidence level that
mξ++ <∼ 1.2 TeV can be probed at the LHC [300 GeV at the Tevatron]. Its decay branching
fractions will then determine |fij|, i.e. the magnitude of each element of the neutrino mass
matrix up to an overall scale factor. This is the only model of neutrino masses which has
the promise of being verified from collider experiments.
Complementary measurement of |fij| is provided by the resonant processes e−e−(µ−µ−)→
l−i l
−
j at a future LC and/or MC. The ξ
++ mass reach in these colliders extends up to the col-
lision energies, which may be as high as 4 TeV. The sensitivity to |fij| depends on the beam
properties of the machines. The detailed estimate in Ref. [10] implies that |(f · f ∗)ij | >∼ 10−8
can be probed in these processes.
Consider now a phenomenological hierarchical neutrino mass matrix consistent with the
atmospheric and solar neutrino results [11]:
Mν = m


ax b bx
b x2 x
bx x 1

 , (13)
where m is the normalization mass and 0.67 < x < 1 determines the νµ → ντ mixing as
required by the atmospheric neutrinos. The three solutions of the solar neutrino problem
correspond to (i) large-angle matter-enhanced oscillations: a = 0.006, b = 0.04; (ii) small-
angle matter-enhanced oscillations: a = 4× 10−6; b = 0.0012; and (iii) vacuum oscillations:
7
a = 0.0002, b = 0.0012.
Given the pattern of fij via the neutrino mass matrix of Eq. (13), lepton-flavor violation
through ξ exchange may be observable at low energies. The amplitude of µ−e conversion in
nuclei is enhanced by ln(m2ξ++/m
2
µ) compared to that of µ→ eγ [12]. Planned experiments
will reach the sensitivity of 10−16 for µ− e conversion in aluminum [7] and 10−14 for µ→ eγ
[13]. The matrix element of photonic conversion is given by M = (4piα/q2)jµJµ, where q is
the momentum transfer with q2 ≈ −m2µ, J is the hadronic current, and
jλ = u¯(pe)
[
(fE0 + γ5fM0) γν
(
gλν − q
λqν
q2
)
+ (fM1 + γ5fE1) i σ
λν qν
mµ
]
u(pµ) (14)
is the leptonic current. The coherent µ− e conversion ratio in nuclei is given by
Rµe =
8α5 m5µ Z
4
eff Z |Fp(pe)|2
Γcapt
ξ20
q4
, (15)
where ξ20 = |fE0 + fM1|2 + |fE1 + fM0|2, and for 13Al, ZAleff = 11.62, FpAl(q) = 0.66, and
ΓAlcapt = 7.1× 105 s−1 [14]. We calculate the form factors induced by the one-loop diagrams
involving ξ++ and obtain
fE0 = fM0 =
∑
l
fµlf
∗
le
24pi2
[4sl + rF (sl)], (16)
fM1 = −fE1 =
∑
l
fµlf
∗
le
24pi2
sµ, (17)
where
F (sl) = ln sl +
(
1− 2sl
r
)√
1 +
4sl
r
ln
[√
r + 4sl +
√
r√
r + 4sl −
√
r
]
,
and r = −q2/m2ξ++, sl = m2l /m2ξ++, l = e, µ, τ . In the interesting limit sl → 0, we get
F (sl)→ ln r.
For numerical estimates we assume m = 0.03 eV in Eq. (13) and the large-angle matter-
enhanced oscillation solution to the solar neutrino problem. In Fig. 2 we plot the ratio of
8
µ− e conversion in aluminum as a function of the mass mξ++ and coupling h, assuming the
fundamental scale to be M∗ = 1 TeV. The behavior of this ratio can be understood from
Eq. (12): a fixed neutrino mass implies f ∝ m2ξ++ and f ∝ 1/h. Notice the complementarity
of collider and µ − e conversion experiments. For small mξ++, Rµe is suppressed while the
collider cross section is kinematically enhanced, and vice versa. Thus for M∗ = 1 TeV, the
MECO experiment should see a signal unless mξ++ <∼ 300 GeV. If mξ++ ∼M∗, which is the
most likely situation in our scenario, and h ∼ 1 which minimizes the signal, then MECO
will test our model up to the scale M∗ ∼ 7 TeV.
In conclusion, we have proposed a verifiable scenario of large extra dimensions in which
we invoke the distant breaking of lepton number to obtain small Majorana neutrino masses,
using a Higgs triplet ξ localized in our world. In this scenario, the fundamental scale M∗
may be a few TeV, thus mξ <∼ M∗ makes ξ kinematically accessible at the LHC and at
future lepton colliders. Our model has the unique feature that the decay branching fractions
of ξ++ into charged leptons l+i l
+
j determine directly |fij|, where fij is the neutrino mass
matrix up to an overall scale factor. Using present neutrino data, we also predict observable
µ−e conversion in nuclei. In particular, for hierarchical neutrino masses and the large-angle
matter-enhanced solution for the solar neutrino problem, the MECO experiment can test
our model up to mξ ∼M∗ ∼ 7 TeV.
Acknowledgement. This work was supported in part by the U. S. Department of Energy
under Grant No. DE-FG03-94ER40837.
References
[1] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B429, 263 (1998); I.
Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B436, 257
9
(1998); N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Rev.D59, 086004 (1999).
[2] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, and J. March-Russell, hep-ph/9811448; K.
R. Dienes, E. Dudas, and T. Gherghetta, Nucl. Phys. B557, 25 (1999).
[3] A. E. Faraggi and M. Pospelov, Phys. Lett. B458, 237 (1999); A. Das and O. C.
W. Kong, Phys. Lett. B470, 149 (1999); R. N. Mohapatra, S. Nandi, and A. Perez-
Lorenzana, Phys. Lett. B466, 115 (1999); G. Dvali and A. Yu. Smirnov, Nucl. Phys.
B563, 63 (1999); R. Barbieri, P. Creminelli, and A. Strumia, hep-ph/0002199.
[4] L. N. Chang, O. Lebedev, W. Loinaz, and T. Takeuchi, hep-ph/0005236.
[5] N. Arkani-Hamed and S. Dimopoulos, hep-ph/9811353; N. Arkani-Hamed, L. Hall, D.
Smith, and N. Weiner, Phys. Rev. D61, 116003 (2000); Y. Sakamura, hep-ph/9912511.
[6] E. Ma and U. Sarkar, Phys. Rev. Lett. 80, 5716 (1998).
[7] M. Bachmann et al., MECO Collaboration, Research Proposal E940 at BNL (1997).
[8] K. Huitu, J. Maalampi, A. Pietila, and M. Raidal, Nucl. Phys. B487, 27 (1997).
[9] H. Plothow-Besch, Comp. Phys. Comm. 75, 396 (1993).
[10] F. Cuypers and M. Raidal, Nucl. Phys. B501, 3 (1997); M. Raidal, Phys. Rev. D57,
2013 (1998).
[11] Y. Fukuda et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 81, 1158 (1998);
Phys. Rev. Lett. 81, 1562 (1998); Phys. Lett. B433, 9 (1998).
[12] M. Raidal and A. Santamaria, Phys. Lett. B421, 250 (1998).
[13] L. M. Barkov et al., Research Proposal at PSI (1999).
[14] H. C. Chiang et al., Nucl. Phys. A559, 526 (1993).
10
pp → x ++ x -- 
 
s[
fb
] 
m
x ++[GeV]
Figure 1: Cross section of ξ++ξ−− Drell-Yan pair production at LHC.
m  - e conversion in Al
h
m x +
+
[TeV
]
B
R
Figure 2: Rate of µ−e conversion in 13Al against the ξ++ mass and Higgs self coupling h for
M∗ = 1 TeV, assuming large-angle matter-enhanced solution to the solar neutrino problem.
11


Verifiable model of neutrino masses from large extra dimensions

Indian Academy of Sciences

ar
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-p
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06
04
6v
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un
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UCRHEP-T279
June 2000
Verifiable Model of Neutrino Masses
from Large Extra Dimensions
Ernest Ma1, Martti Raidal1,2, and Utpal Sarkar1,3
1 Department of Physics, University of California, Riverside, California 92521, USA
2 National Institute of Chemical and Biological Physics, Rävala 10, 10143, Estonia
3 Physical Research Laboratory, Ahmedabad 380 009, India
Abstract
We propose a new scenario of neutrino masses with a Higgs triplet (ξ++, ξ+, ξ0) in
a theory of large extra dimensions. Lepton number violation in a distant brane acts
as the source of a very small trilinear coupling of ξ to the standard Higgs doublet in
our brane. Small realistic Majorana neutrino masses are naturally obtained with the
fundamental scale M∗ ∼ O(1) TeV, foretelling the possible discovery of ξ (mξ <∼ M∗)
at future colliders. Decays of ξ++ into same-sign dileptons are fixed by the neutrino
mass matrix. Observation of µ − e conversion in nuclei is predicted.
Recently it has been proposed [1] that the fundamental scale of quantum gravity could
be as low as a few TeV. In this theory, our four-dimensional world (called a 3-brane) is
localized to only one point in a space having n extra dimensions (the bulk). Assuming that
gravity is the only field which propagates in these extra dimensions, the fact that it is so
weak in our world may be understood. Since there are no scales larger than a few TeV in
this theory, the smallness of neutrino masses requires a different explanation than the usual
one that it is inversely proportional to some very large mass. There are already proposals
in this direction, where singlet fermions are introduced in the bulk [2, 3]. They become the
right-handed partners of the observed left-handed neutrinos, but with small Dirac masses
– the result of being suppressed by the volume of the extra dimensions. Note first that
this mechanism by itself does not generate a Majorana mass. Note also that such a model
of neutrino masses, like any other before it, has no distinct experimental signatures which
would set it apart from any other. Furthermore, the existence of singlet fermions in the bulk
is strongly constrained by supernova data to have an upper bound of 210/
√
n TeV on the
fundamental scale involved [4].
In this Letter we propose a simple, verifiable scenario in which the distant breaking
[5] of lepton number is used to generate small Majorana neutrino masses through a scalar
Higgs triplet [6] localized in our brane. The trilinear interaction of the triplet with the
standard Higgs doublet is induced by the “shining” of a scalar singlet which communicates
the violation of lepton number from a distant brane to our world through the large extra
space dimensions. Thus the amount of lepton number violation in our world is suppressed by
the distance between us and the other brane. Because the Higgs triplet should not be heavier
than the fundamental scale of the extra dimensions, this scenario provides the basis for a
rich and robust phenomenology of new observable processes at future high-energy colliders
as well as low-energy precision experiments.
2
The Higgs triplet may be produced at the upgraded Tevatron at Fermilab, the Large
Hadron Collider (LHC) at CERN, and perhaps a future linear collider (LC) or muon col-
lider (MC) or both. The decay branching fractions of the doubly charged member ξ++ of
the Higgs triplet into same-sign dileptons or the cross sections of the resonant processes
e−e−(µ−µ−) → l−i l−j may then be used to determine the magnitude of each element of
the neutrino mass matrix completely and uniquely up to an overall scale factor. Whereas
lepton-number violating processes are suppressed at low energies (because their amplitudes
are proportional to the neutrino mass matrix), large lepton-flavor violation through the Higgs
triplet is possible. Based on current neutrino-oscillation data, the rate of coherent µ−e con-
version in nuclei, with an amplitude enhanced by the large factor ln(m2ξ++/m
2
µ) over that of
µ → eγ, is predicted to be observable at the proposed Muon Electron Conversion (MECO)
experiment [7] at Brookhaven National Laboratory (BNL).
We extend the minimal standard model of particle interactions to include a heavy Higgs
triplet (ξ++, ξ+, ξ0) with the interaction
L = fij [ξ0νiνj + ξ+
(νilj + liνj)√
2
+ ξ++lilj ] + h.c., (1)
implying that ξ carries lepton number L = −2. The resulting Majorana neutrino mass
matrix is then given by
(Mν)ij = 2fij〈ξ0〉, (2)
where 〈ξ0〉 is the vacuum expectation value (VEV) of ξ0, hence lepton number must be
violated somehow. This may be achieved with the following addition to the Lagrangian:
L′ = hχ(ξ̄0φ0φ0 −
√
2ξ−φ+φ0 + ξ−−φ+φ+) + h.c., (3)
where χ is a complex neutral scalar singlet with L = −2. We then obtain [6]
〈ξ0〉 ≃ h〈χ〉〈φ
0〉2
m2ξ
. (4)
3
In other words, the breaking of L by 〈χ〉 is communicated to the lepton sector through 〈ξ〉.
As the neutrino masses are very small, we need mξ to be very large in Eq. (4). Suppose
mξ is only of order 1 TeV, then h〈χ〉 would have to be very small, i.e. h〈χ〉 ∼ 33 eV, if
〈ξ0〉 ∼ 1 eV, using 〈φ0〉 = 174 GeV. However, χ is a singlet, and as such, its VEV is expected
to be large, so this requires what we normally would consider as extremely unnatural fine
tuning. However, in a theory of large extra dimensions, a small 〈χ〉 may arise naturally, thus
allowing mξ to be of order 1 TeV and be directly observed experimentally.
The standard model fields together with ξ are localized in our world (a 3-brane P at
y = 0) and are blind to the extra space dimensions. Lepton number is assumed to be
conserved as far as these fields are concerned. The singlet χ is special; it propagates also in
the bulk carrying lepton number L = −2 and interacts in our brane according to Eq. (3).
The violation of lepton number happens at a distant 3-brane which then gets communicated
to our brane through 〈χ〉.
We assume the existence of a field η which is localized in a distant 3-brane (P ′) situated
at a point y = y∗ in the extra dimensions. It is a singlet under the standard model but has
L = 2 and couples to χ (with L = −2). When the field η acquires a VEV, lepton number is
broken maximally in the other brane. It will then act as a point source for L violation, and
the field χ is the messenger which carries it to our wall (the interface between our brane and
the bulk). The “shining” of χ at all points in our world is the mechanism [5] which breaks
lepton number and gives mass to the neutrinos.
At energies much below the fundamental scale M∗, the lepton-number violating effect
will be suppressed by the distance between the source brane at P ′ and our brane at P. We
assume that the source brane is situated at the farthest point in the extra dimensions so
that |y∗| = r is the radius of compactification and it is related to the fundamental scale M∗
4
and the reduced Planck scale (MP = 2.4 × 1018 GeV) by the relation
rnMn+2∗ ∼ M2P . (5)
This explains why lepton number is only violated weakly in our world.
We assume here that the source brane has the same dimensional structure as our world
and there are n extra dimensions. In our world (P) the field χ has only the lepton-number
conserving interaction
Sχ = h
∫
P
d4x ξ†(x)φ(x)φ(x)χ(x, y = 0). (6)
In the other brane (P ′) the field χ couples to the field η through the interaction
Sother =
∫
P ′
d4x′ µ2 η(x′)χ(x′, y = y∗), (7)
where µ is a mass parameter. Lepton number violation from 〈η〉 is carried by χ to our world
through its “shined” value 〈χ〉:
〈χ(x, y = 0)〉 = ∆n(r)〈η(x, y = y∗)〉, (8)
where 〈η〉 acts as a point source, and ∆n(r) is the Yukawa potential in n transverse dimen-
sions, i.e. [5]
∆n(r) =
1
(2π)
n
2 Mn−2∗
(
mχ
r
)
n−2
2
Kn−2
2
(mχr) , (9)
K being the modified Bessel function. If the mass of the carrier field χ is large (mχr ≫ 1),
it has the profile
〈χ〉 ≈ m
n−3
2
χ
2(2π)
n−1
2 Mn−3∗
e−mχr
r
n−1
2
. (10)
The suppression here is exponential, hence the amount of lepton-number violation in our
world is very small, but its precise value depends sensitively on mχ. An interesting alternative
is to have a light carrier field with a mass less than 1/r. However, it should be larger than
5
about (0.1 mm)−1, to be consistent with the present experimental data on gravitational
interactions.
If mχr ≪ 1, ∆n(r) is logarithmic for n = 2 and 〈χ〉 is not suppressed. For n > 2, the
asymptotic form of the profile of χ is
〈χ〉 ≈ Γ(
n−2
2
)
4π
n
2
M∗
(M∗r)n−2
, (11)
which is suitably small for large r. Using Eqs. (2), (4), and (5), we then obtain
(Mν)ij ≈
Γ(n−2
2
)
2π
n
2
h fij
〈φ〉2 M∗
m2ξ
(
M∗
MP
)2− 4
n
. (12)
For n = 3, M∗ ∼ 1 TeV, mξ <∼ 1 TeV and h ∼ f33 ∼ 0.25, we get (Mν)33 ∼ 0.03 eV, which
is of the right magnitude for understanding atmospheric neutrino oscillations. There is a
massless Goldstone boson (Majoron) corresponding to the spontaneous breaking of lepton
number in this scenario. It is a singlet and its coupling to the leptons in our world is
suppressed by 〈η〉−1. It is thus completely acceptable phenomenologically.
For n ≥ 4, this scenario requires a larger fundamental scale M∗ and is of less phenomeno-
logical interest. However, we have assumed here for simplicity the dimension of the source
brane to be the same as ours, which may not be the case. The distance between the branes
may also be smaller. On the other hand, such complications diminish the predictive power
of the proposed mechanism; they will not be considered any further.
The only pair production mechanism of ξ++ at the LHC is the Drell-Yan process mediated
by s-channel photon and Z-boson exchange [8]. Thus the production rate is enhanced by
the double charge of ξ++ and is uniquely determined by the gauge couplings. To obtain the
cross section of pp → ξ++ξ−− at the LHC, we have calculated the subprocesses involving
the u, ū and d, d̄ collisions and convoluted them over the parton distributions given by the
default set of the CERN library package PDFLIB [9]. The total cross section as a function
of the triplet mass mξ++ is plotted in Fig. 1.
6
Once produced, ξ++ will decay into same-sign dileptons. Because 〈ξ〉 is tiny, the decay
branching fraction of ξ++ → W+W+ is negligible. Thus the only possible decay channels in
our scenario are ξ++ → l+i l+j with the partial rates Γij = |fij |2mξ/4π for i 6= j, and 1/2 smaller
for i = j. This same-sign dilepton signal at the invariant mass of ξ is very distinctive at the
LHC because it is completely background-free. Assuming the total integrated luminosity of
the LHC to be 1000 fb−1, the reconstruction efficiency of the event to be 10% (to be very
conservative) and that the predicted average of N = − ln(1 − p) Poisson distributed events
would provide a discovery, the cross section in Fig. 1 implies at p = 95% confidence level that
mξ++ <∼ 1.2 TeV can be probed at the LHC [300 GeV at the Tevatron]. Its decay branching
fractions will then determine |fij|, i.e. the magnitude of each element of the neutrino mass
matrix up to an overall scale factor. This is the only model of neutrino masses which has
the promise of being verified from collider experiments.
Complementary measurement of |fij| is provided by the resonant processes e−e−(µ−µ−) →
l−i l
−
j at a future LC and/or MC. The ξ
++ mass reach in these colliders extends up to the col-
lision energies, which may be as high as 4 TeV. The sensitivity to |fij| depends on the beam
properties of the machines. The detailed estimate in Ref. [10] implies that |(f · f ∗)ij | >∼ 10−8
can be probed in these processes.
Consider now a phenomenological hierarchical neutrino mass matrix consistent with the
atmospheric and solar neutrino results [11]:
Mν = m




ax b bx
b x2 x
bx x 1




, (13)
where m is the normalization mass and 0.67 < x < 1 determines the νµ → ντ mixing as
required by the atmospheric neutrinos. The three solutions of the solar neutrino problem
correspond to (i) large-angle matter-enhanced oscillations: a = 0.006, b = 0.04; (ii) small-
angle matter-enhanced oscillations: a = 4 × 10−6; b = 0.0012; and (iii) vacuum oscillations:
7
a = 0.0002, b = 0.0012.
Given the pattern of fij via the neutrino mass matrix of Eq. (13), lepton-flavor violation
through ξ exchange may be observable at low energies. The amplitude of µ−e conversion in
nuclei is enhanced by ln(m2ξ++/m
2
µ) compared to that of µ → eγ [12]. Planned experiments
will reach the sensitivity of 10−16 for µ− e conversion in aluminum [7] and 10−14 for µ → eγ
[13]. The matrix element of photonic conversion is given by M = (4πα/q2)jµJµ, where q is
the momentum transfer with q2 ≈ −m2µ, J is the hadronic current, and
jλ = ū(pe)
[
(fE0 + γ5fM0) γν
(
gλν − q
λqν
q2
)
+ (fM1 + γ5fE1) i σ
λν qν
mµ
]
u(pµ) (14)
is the leptonic current. The coherent µ − e conversion ratio in nuclei is given by
Rµe =
8α5 m5µ Z
4
eff Z |Fp(pe)|2
Γcapt
ξ20
q4
, (15)
where ξ20 = |fE0 + fM1|2 + |fE1 + fM0|2, and for 13Al, ZAleff = 11.62, Fp
Al
(q) = 0.66, and
ΓAlcapt = 7.1 × 105 s−1 [14]. We calculate the form factors induced by the one-loop diagrams
involving ξ++ and obtain
fE0 = fM0 =
∑
l
fµlf
∗
le
24π2
[4sl + rF (sl)], (16)
fM1 = −fE1 =
∑
l
fµlf
∗
le
24π2
sµ, (17)
where
F (sl) = ln sl +
(
1 − 2sl
r
)
√
1 +
4sl
r
ln
[√
r + 4sl +
√
r√
r + 4sl −
√
r
]
,
and r = −q2/m2ξ++, sl = m2l /m2ξ++, l = e, µ, τ . In the interesting limit sl → 0, we get
F (sl) → ln r.
For numerical estimates we assume m = 0.03 eV in Eq. (13) and the large-angle matter-
enhanced oscillation solution to the solar neutrino problem. In Fig. 2 we plot the ratio of
8
µ− e conversion in aluminum as a function of the mass mξ++ and coupling h, assuming the
fundamental scale to be M∗ = 1 TeV. The behavior of this ratio can be understood from
Eq. (12): a fixed neutrino mass implies f ∝ m2ξ++ and f ∝ 1/h. Notice the complementarity
of collider and µ − e conversion experiments. For small mξ++, Rµe is suppressed while the
collider cross section is kinematically enhanced, and vice versa. Thus for M∗ = 1 TeV, the
MECO experiment should see a signal unless mξ++ <∼ 300 GeV. If mξ++ ∼ M∗, which is the
most likely situation in our scenario, and h ∼ 1 which minimizes the signal, then MECO
will test our model up to the scale M∗ ∼ 7 TeV.
In conclusion, we have proposed a verifiable scenario of large extra dimensions in which
we invoke the distant breaking of lepton number to obtain small Majorana neutrino masses,
using a Higgs triplet ξ localized in our world. In this scenario, the fundamental scale M∗
may be a few TeV, thus mξ <∼ M∗ makes ξ kinematically accessible at the LHC and at
future lepton colliders. Our model has the unique feature that the decay branching fractions
of ξ++ into charged leptons l+i l
+
j determine directly |fij|, where fij is the neutrino mass
matrix up to an overall scale factor. Using present neutrino data, we also predict observable
µ−e conversion in nuclei. In particular, for hierarchical neutrino masses and the large-angle
matter-enhanced solution for the solar neutrino problem, the MECO experiment can test
our model up to mξ ∼ M∗ ∼ 7 TeV.
Acknowledgement. This work was supported in part by the U. S. Department of Energy
under Grant No. DE-FG03-94ER40837.
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10
pp → ξ++ξ-- 
 σ
[f
b]
 
mξ++[GeV]
Figure 1: Cross section of ξ++ξ−− Drell-Yan pair production at LHC.
µ - e conversion in Al
h
m ξ+
+
[Te
V]
B
R
Figure 2: Rate of µ−e conversion in 13Al against the ξ++ mass and Higgs self coupling h for
M∗ = 1 TeV, assuming large-angle matter-enhanced solution to the solar neutrino problem.
11


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Verifiable Model of Neutrino Masses from Large Extra Dimensions

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