Phenomenology of a TeV right-handed neutrino and the dark matter model

Phenomenology of a TeV right-handed neutrino and the dark matter model

Kingman Cheung

Department of Physics and NCTS, National Tsing Hua University, Hsinchu, Taiwan, Republic of China Osamu Seto

Institute of Physics, National Chiao Tung University, Hsinchu, Taiwan 300, Republic of China 共Received 8 March 2004; published 29 June 2004兲

In a model of a TeV right-handed共RH兲 neutrino by Krauss, Nasri, and Trodden, the sub-eV scale neutrino masses are generated via a three-loop diagram with the vanishing seesaw mass forbidden by a discrete sym-metry, and the TeV mass RH neutrino is simultaneously a novel candidate for cold dark matter. However, we show that with a single RH neutrino it is not possible to generate two mass-square differences as required by the oscillation data. We extend the model by introducing one more TeV RH neutrino and show that it is possible to satisfy the oscillation pattern within the modified model. After studying in detail the constraints coming from the dark matter, lepton flavor violation, the muon anomalous magnetic moment, and the neu-trinoless double beta decay, we explore the parameter space and derive predictions of the model. Finally, we study the production and decay signatures of the TeV RH neutrinos at TeV e⫹e⫺/␮⫹␮⫺colliders.

DOI: 10.1103/PhysRevD.69.113009 PACS number共s兲: 14.60.Pq, 12.60.Cn, 95.35.⫹d

I. INTRODUCTION

One of the most natural ways to generate a small neutrino mass is via the seesaw mechanism关1兴. There are very heavy right-handed neutrinos, which are gauge singlets of the stan-dard model 共SM兲, and so they could have a large Majorana mass MR. After electroweak symmetry breaking, a Dirac

mass term MDbetween the right-handed and the left-handed

neutrinos can be developed. Therefore, after diagonalizing the neutrino mass matrix, a small Majorana mass⬃m_D2/ M_R for the left-handed neutrino is obtained. This is a very natural mechanism, provided that MR⬃1011–1013 GeV. One

draw-back of this scheme is that these right-handed neutrinos are too heavy to be produced in any terrestrial experiments. Therefore, phenomenologically there are not many channels to test the mechanism. Although it could be possible to get some hints from the neutrino masses and mixing, it is rather difficult to reconstruct the parameters of the right-handed neutrinos using the low energy data 关2兴.

Another natural way to generate a small neutrino mass is via higher loop processes, e.g., the Zee model关3兴, with some lepton number violating couplings. However, these lepton number violating couplings are also subject to experimental constraints, e.g.,␮→e␥, ␶→e␥. In the Zee model, there are also extra scalars whose masses are of electroweak scale, and so can be observed at colliders关4兴.

On the other hand, recent cosmological observations have established the concordance cosmological model where the present energy density consists of about 73% of cosmologi-cal constant 共dark energy兲, 23% 共nonbaryonic兲 cold dark matter, and just 4% of baryons. To clarify the identity of the dark matter remains a prime open problem in cosmology and particle physics. Although quite a number of promising can-didates have been proposed and investigated in detail, other possibilities can never be neglected.

Recently, Krauss, Narsi, and Trodden 关5兴 considered an extension to the SM, similar to the Zee model, with two

additional charged scalar singlets and a TeV right-handed neutrino. They showed that with an additional discrete sym-metry the Dirac mass term between the left-handed and right-handed neutrinos are forbidden and thus avoiding the seesaw mass. Furthermore, the neutrino mass can only be generated at three-loop level, and sub-eV neutrino masses can be obtained with the masses of the charged scalars and the right-handed neutrino of order of TeV. Phenomenologi-cally, this model is interesting because the TeV right-handed neutrino can be produced at colliders and could be a dark matter candidate.

In this work, we explore in detail the phenomenology of the TeV right-handed 共RH兲 neutrinos. We shall extend the analysis to three families of left-handed neutrinos and ex-plore the region of the parameters that can accommodate the present oscillation data. In the course of our study, we found that the model in Ref.关5兴 with a single RH neutrino cannot explain the oscillation data, because it only gives one mass-square difference. We extend the model by adding another TeV RH neutrino, which is slightly heavier than the first one. We demonstrate that it is possible to accommodate the oscil-lation pattern. We also obtain the relic density of the RH neutrino, and discuss the possibility of detecting them if they form a substantial fraction of the dark matter. We also study the lepton number violating processes, the muon anomalous magnetic moment, and production at leptonic colliders. In particular, the pair production of N1N2,N2N2 at e⫹e⫺/␮⫹␮⫺ colliders gives rise to very interesting signa-tures. The N2 so produced will decay into N1 plus a pair of charged leptons inside the detector. Thus, the signature would be either one or two pairs of charged leptons plus a large missing energy.

The organization is as follows. We describe the model in the next section. In Sec. III, we explore all the phenomenol-ogy associated with the TeV RH neutrino. In Sec. IV, we discuss the signatures in collider experiments. Section V is devoted to a conclusion.

II. REVIEW OF THE MODEL

The model considered in Ref. 关5兴 has two extra charged scalar singlets S1,S2and a right-handed neutrino NR. A

dis-crete Z₂ symmetry is imposed on the particles, such that all SM particles and S1 are even under Z2 but S2,NR are odd

under Z2. Therefore, the Dirac mass term L¯␾NR is

forbid-den, where ␾ is the SM Higgs boson. The seesaw mass is avoided.

In the present work, we extend the model a bit further by adding the second TeV right-handed neutrino, which also has odd Z2parity. The reason for this is because with only 1 TeV RH neutrino, it is impossible to obtain two mass-square dif-ferences, as required by the oscillation data. However, with two TeV RH neutrinos it is possible to accommodate two mass-square differences with the corresponding large mixing angles. We will explicitly show this result in the next section. The most general form for the interaction Lagrangian is1

L⫽ f␣␤L␣TCi␶2L␤S1⫹⫹g1␣N1S2⫹ᐉ␣R⫹g2␣N2S2⫹ᐉ␣R ⫹V共S1,S2兲⫹H.c.⫹MN₁N1 T_CN 1⫹MN₂N2 T_CN 2, 共1兲 where ␣,␤ denote the family indices, C is the charge-conjugation operator, and V(S₁,S₂) contains a term ␭s(S1S2*)

2_{. Note that f}

␣␤ is antisymmetric under

inter-change of the family indices. Note that even with the pres-ence of the first term in the Lagrangian it cannot give rise to the one-loop Zee diagrams for neutrino mass generation, be-cause there is no mixing term between the Zee charged scalar S₁⫹ and the standard model Higgs doublet that can generate the charged lepton mass.

If the masses of N₁,N₂,S₁,S₂ are arranged such that MN₁⬍MN₂⬍MS₁⬍MS₂, N1would be stable if the Z2 parity is maintained. The N1could be a dark matter candidate pro-vided that its interaction is weak enough. Also, N₁,N₂must be pair produced or produced associated with S2 because of the Z2 parity. The N2 so produced would decay into N1 and a pair of charged leptons. The decay time may be long enough to produce a displaced vertex in the central detector. The S2, if produced, would also decay into N1, N2, and a charged lepton. We will discuss the phenomenology in de-tails in the next section.

III. PHENOMENOLOGY A. Neutrino masses and mixings

The goal here is to find the parameter space of the model in Eq.共1兲 such that the neutrino mass matrix so obtained can accommodate the maximal mixing for the atmospheric

neu-trino, the large mixing angle for the solar neuneu-trino, and the small mixing angle for␪13关6兴:

⌬matm⬇2.7⫻10⫺3 eV2, sin22␪atm⫽1.0, ⌬msol⬇7.1⫻10⫺5 eV2, tan2␪sol⫽0.45,

sin22␪13ⱗ0.1. 共2兲

The three-loop Feynman diagram that contributes to the neutrino mass matrix has been given in Ref. 关5兴. The neu-trino mass matrix ( M_␯)_␣␤is given by

共M␯兲␣␤⬃ 1 共4␲2_兲3 1 M_S 2 ␭sf␣␳mᐉ_␳g␳g␴mᐉ_␴f␴␤, 共3兲

where␣,␤denote the flavor of the neutrino. Note that in the Zee model, the neutrino mass matrix entries are proportional to f_␣␤ such that only off-diagonal matrix elements are non-zero. It is well known that the Zee model gives bimaximal mixings, which have some difficulties with the large-mixing angle solution of the solar neutrino关6兴. Here in Eq. 共3兲 we do not have the second Higgs doublet to give a mixing between the SM Higgs doublet and S₁⫹, and therefore the one-loop Zee-type diagrams are not possible. However, the mass ma-trix in Eq. 共3兲 allows for nonzero diagonal elements, which may allow the departure from the bimaximal mixings.

The mixing matrix between flavor eigenstates and mass eigenstates is given as U_␣i ⫽

冉

c13c12 s12c13 s13 ⫺s12c23⫺s23s13c12 c23c12⫺s23s13s12 s23c13 s23s12⫺s13c23c12 ⫺s23c12⫺s13s12c23 c23c13

冊

, 共4兲 where we have ignored the phases. The mass eigenvalues are given by

UTM U⫽Mdiag⫽diag共m1,m2,m3兲. 共5兲 The mass-square differences and mixing angles are related to oscillation data by ⌬msol 2 _⬅⌬m 21 2_⫽m 2 2_⫺m 1 2 ⌬matm 2 _⬅⌬m 32 2_⫽m 3 2_⫺m 2 2 ␪sol⬅␪12 ␪atm⬅␪23. 共6兲

From Eq.共3兲 the neutrino mass matrix is rewritten as

1_{In principle, there are terms sush as N}

1N2␾ and MN1N2. The latter explicitly gives a mixing between the two RH neutrinos, while the former also gives the mixing after the Higgs field devel-ops a vacuum expectation value共VEV兲. However, the mixing term can be rotated away by redefining the N1and N2fields. Effectively, the Lagrangian has the form given in Eq.共1兲.

共M␯兲␣␤⬃⫺ ␭s 共4␲2_兲3_M S₂

冉

共 f mg兲e 2 _{共 f mg兲} e共 f mg兲␮ 共 f mg兲e共 f mg兲␶ 共 f mg兲e共 f mg兲␮ 共 f mg兲_␮ 2 共 f mg兲␮共 f mg兲␶ 共 f mg兲e共 f mg兲␶ 共 f mg兲␮共 f mg兲␶ 共 f mg兲␶ 2

冊

, 共7兲 where ( f mg)_␣⫽兺_␳f_␣␳m_ᐉ

␳g␳, the mass eigenvalues are

given by m1⫽m2⫽0, 共8兲 m3⬃⫺ ␭s 共4␲2_兲3_M S₂ 关共 f mg兲e 2 ⫹共 f mg兲␮2⫹共 f mg兲␶2兴. 共9兲

This model obviously cannot explain the neutrino oscillation data because of the vanishing ⌬m₂₁2 .

Hereafter we would like to discuss a possibility to im-prove this shortcoming. The reason that this model predicts two vanishing mass eigenvalues is the proportionality rela-tion in the mass matrix共7兲. Therefore it is necessary to break the proportionality relation. Although one way to improve the mass matrix might be to add small perturbations to the original mass matrix, we, however, found that this approach cannot resolve the difficulty. Instead, we consider a modifi-cation of the right-handed neutrino sector. As mentioned be-fore, we employ two TeV RH neutrinos, the mass matrix共7兲 is replaced by 共M␯兲␣␤⬃ 1 共4␲2_兲3 1 M_S 2 ␭s

兺

I_⫽1,2共 f mgI兲␣共gI m f兲_␤, 共10兲

where I denotes the two RH neutrinos.

If we assume ( f mg2)␮Ⰶ( f mg1)e, Eq.共10兲 is rewritten as

共M␯兲␣␤⬃⫺ ␭s共 f mg1兲e 2 共4␲2_兲3_M S₂

冉

1⫹c2 w t⫹cd w w2 wt t⫹cd wt t2⫹d2

冊

, 共11兲 w⫽共 f mg1兲␮/共 f mg1兲e, t⫽共 f mg1兲␶/共 f mg1兲e, c⫽共 f mg2兲e/共 f mg1兲e, 共12兲 d⫽共 f mg2兲␶/共 f mg1兲e,

and has one zero and two nonzero eigenvalues:

m_⫾⬃⫺␭s共 f mg1兲e 2 共4␲2_兲3_M S₂ ␭_⫾, 共13兲 where 2␭_⫾⫽1⫹w2⫹t2⫹c2⫹d2⫾

冑

共1⫹w2⫹t2⫹c2⫹d2兲2⫺4共d2⫹c2w2⫹d2w2⫺2cdt⫹c2t2兲, 共14兲

and each of the mixing angles is given by

t23⫽ w共␭_⫹⫺c2⫺d2兲 t共␭_⫹⫺c2兲⫹cd, 共15兲 s₁₃⫽ ␭⫹⫺d 2_⫺tcd

冑

共␭⫹⫺d2⫺tcd兲2⫹共1⫹t23 2 _兲w2_共␭ ⫹⫺c2⫺d2兲2 , 共16兲 c₁₂⫽ 1 c13 dw

冑

共c2_⫹d2_兲w2_{⫹共ct⫺d兲}2, 共17兲 where we adopt the normal mass hierarchy. Indeed, we found that the correct mixing angles could not be realized if we assumed the inverted mass hierarchy here. Here t23⯝1, s13 2 Ⰶ1 imply w⯝t, ␭⫹Ⰷc2,d2 and w2Ⰷ1. This means t2

⯝w2_Ⰷ1,c2_,d2_{. Definitely, from} sin22␪13⯝ 2 w2

冉

1⫺ tcd ␭⫹

冊

2

冉

2 1⫹t₂₃2

冊

ⱗ0.1, 共18兲 we obtain w2ⲏ20. Since Eq. 共17兲 is rewritten as

t₁₂2 ⯝c

2_w2_{⫹共cw⫺d兲}2

d2w2 , 共19兲

where c13⯝1 and w⯝t are used, we obtain c2

d2⬃ 1

4, 共20兲

by comparing with Eq. 共2兲. From the mass-square differ-ences,

⌬msol 2 ⌬matm 2 ⯝

冉

␭⫺ ␭⫹

冊

2 ⯝

冉

2c 2_w2_⫹d2_w2_⫺2cdw 4w4

冊

2 ⯝

冉

3c 2 2w2

冊

2 ⬃10⫺2_{, 共21兲} we find c2⬃⫾4 3

冉

w2 20

冊

. 共22兲

Finally, ⌬m_atm2 ⯝m₃2⫽m_⫹2 is rewritten as 2.7⫻10⫺3 eV2⯝

冉

⫺40␭s共 f mg1兲e 2 共4␲2_兲3_M S₂

冊

2

冉

w2 20

冊

2 , 共23兲 where we used␭_⫹⯝2w2_{. In Sec. III E, we find some} param-eter space that leads to correct mixing angles and mass-square differences, after considering also the constraints from the dark matter relic density and lepton flavor violation.

B. Neutrinoless double beta decay

A novel feature of the Majorana neutrino is the existence of neutrinoless double beta decay, which essentially requires a nonzero entry ( M_␯)eeof the neutrino mass matrix. Its

non-observation has put an upper bound on the size of ( M_␯)ee

ⱗ1 eV 关7兴.

In the model with two RH neutrinos, ( M_␯)ee is estimated

to be 共M␯兲ee⬃⫺ ␭s 共4␲2_兲3_M S₂ 关共 f mg1兲e 2 ⫹共 f mg2兲e 2_兴 ⬃3⫻10⫺3

冉

1⫾3 4共20/w 2_兲 2

冊

eV 共24兲

by using Eqs.共22兲 and 共23兲. Thus, we find that this model is consistent with the current experimental bound. Such a small ( M_␯)ee may still be within reach of the GENIUS

neutrino-less double beta decay experiment 关8兴.

C. Dark matter: Density and detection

The lightest RH neutrino is stable because of the assumed discrete symmetry. Here we consider the relic density of the lightest RH neutrino, and the relic density must be less than the critical density of the Universe. First of all, we verify that the second lightest RH neutrino is of no relevance here be-cause of the short decay time. The heavier RH neutrino will decay into the lighter one and two right-handed charged lep-tons, N2→N1ᐉ_␣⫺ᐉ_␤⫹ (␣,␤ denote flavors兲, and its decay width is given by ⌫N₂⫽ MN₂ 512␲3兩g1␤g2␣兩 2 1 2␮_s2

冋

2共1⫺␮s兲共␮1⫺␮s兲共␮1⫹␮s ⫹␮1␮s⫺3␮s 2_兲log

冉

␮s⫺␮1 ␮s⫺1

冊

⫹共1⫺␮1兲␮s共2␮1⫺5␮s ⫺5␮1␮s⫹6␮s 2_兲⫺2␮ 1 2 log␮1

册

, 共25兲 where ␮1⫽MN₁ 2 / M_N 2 2 ,␮s⫽MS₂ 2 / M_N 2 2

. In the worst case when M_N

2 is very close to MN1, say, they are both of order

1 TeV but differ by 1 GeV only, and we set gi⬃0.1. In this

case, the decay width is then of order 104⫺105 s⫺1, i.e., the decay time is still many orders smaller than the age of the present Universe. Therefore, the presence of N2 will not af-fect the relic density of N1.

The relevant interactions for the annihilation is N1N1 →ᐉ␣R⫹ ᐉ␤R⫺ through charged scalar S2⫹ exchange. The corre-sponding invariant matrix element is given by

兩M兩2_⫽兩g1␣g1␤兩 2 4

冋

共2q1•p1兲2q2•p2 共t⫺MS₂ 2 _兲2 ⫹ 共2q2•p1兲2q1•p2 共u⫺MS₂ 2 _兲2 ⫺ 2 MN1 2 2 p1•p2 共t⫺MS₂ 2 _兲共u⫺M S₂ 2 _兲

册

, 共26兲

where q_i and p_i are four-momenta of the incoming N₁ par-ticles and the outgoing leptons, respectively. Then, we obtain

2q₁02q₂0␴v⫽ d 3_p 1 共2␲兲2_{2 p} 1 0 d3p2 共2␲兲2_{2 p} 2 0共2␲兲 2_兩M兩2␦(4)_共q 1⫹q2⫺p1⫺p2兲 共27兲 ⫽ 1 8␲ 兩g1␣g1␤兩2 共MS₂ 2 _⫹s/2⫺M N₁ 2 _兲2

冋

m_l2_␣⫹m_l2_␤ 2

冉

s 2⫺MN1 2

冊

⫹8₃ 共MS2 2 _⫺M N₁ 2 _兲2_{⫹共s/2兲共M} S₂ 2 _⫺M N₁ 2 _兲⫹s2_/8 共MS₂ 2 _⫹s/2⫺M N₁ 2 _兲2 s 4

冉

s 4⫺MN₁ 2

冊

册

, 共28兲

where ml␣ is the lepton mass. We expanded兩M兩2in powers

of the three-momenta of these particles and integrated over the scattering angle in the second line. Following Ref. 关9兴, the thermal averaged annihilation rate is estimated to be

具

␴v

典

⫽

冋

M_N 1 2 T 2␲2 K2

冉

MN₁ T

冊

册

⫺2 T 4共2␲兲4

冕

4 M_N 1 2 ⬁ ds ⫻

冑

s⫺4M_N 1 2 _K 1共

冑

s/T兲共2q1 0_2q 2 0_␴ v兲 ⯝

兺

f 兩g1␣g1␤兩2 32␲ M_S 2 4 _⫹M N₁ 4 共MS₂ 2 _⫹M N₁ 2 _兲44 MN1 2

冉

T MN₁

冊

⬅␴0

冉

T MN₁

冊

, 共29兲

where兺_fdenotes the summation over lepton flavors, and we have omitted the contributions from the S-wave annihilation terms, which are suppressed by the masses of the final state leptons. The relic mass density is given by

⍀N₁h2⫽1.1⫻109 2共MN₁/T兲

冑

g *Mp

具

␴v

典

冏

_T d GeV⫺1, 共30兲 where Td is the decoupling temperature, which is determined

as M_N 1 Td ⯝ln

冋

0.152

冑

g *共Td兲 M_p␴₀M_N 1

册

⫺3 2ln ln

冋

0.152

冑

g *共Td兲 M_p␴₀M_N 1

册

, 共31兲 and g

*is the total number of relativistic degrees of freedom

in the thermal bath 关10兴.

By comparing with the recent data from the Wilkinson Microwave Anisotropy Probe共WMAP兲 关11兴, we find

⍀D Mh2⫽0.113⫽2.2⫻1012

冉

MN₁ 103 GeV

冊

共MN₁/Td兲2

冑

g *Mp␴0MN₁ . 共32兲 We can calculate␴₀ from Eqs.共31兲 and 共32兲, and we obtain

␴0⯝1.4⫻10⫺7

冉

102 g *共Td兲

冊

1/2

再

1⫹0.07 ln

冋冉

MN₁ 103 GeV

冊

⫻

冉

10 2 g *共Td兲

冊

册冎

GeV⫺2, 共33兲

if we ignore the second term in Eq. 共31兲. Indeed, we can confirm the validity of this assumption within about 10% error by using Eq.共33兲. Actually, Eq. 共31兲 is evaluated to be

MN₁ T_d ⯝ln共2.5⫻10 13_兲⫺3 2ln ln共2.5⫻10 13_兲 ⫽31⫺5.1⫽26. 共34兲

Our result of

具

␴v

典

is consistent with a previous estimation 关12兴. Equations 共29兲 and 共33兲 read

兺

f 兩g1␣ g1␤兩2 ⯝1

冉

MN1 1.3⫻102 GeV

冊

2

冉

1⫹M_S 2 2 _{/ M} N₁ 2 1⫹2

冊

4

冉

1⫹22 1⫹M_S 2 4 / M_N 1 4

冊

. 共35兲 It is obvious that the RH neutrino must be as light as ⬃102 _{GeV and at least one of g}

1␣ should be of order of unity, such that the relic density is consistent with the dark matter measurement.2As the mass difference between MS₂

and N1 becomes larger, the upper bound on MN₁ becomes

smaller provided that we keep gⱗ1.

The detection of the RH neutrinos as a dark matter can-didate depends on its annihilation cross section and its scat-tering cross section with nucleons. Conventional search of dark matter employs an elastic scattering signal of the dark matter with the nucleons. We do not expect that the NR dark

matter would be easily identified by this method, given its very mild interaction. In addition, because of the Majorana nature the annihilation into a pair charged lepton at the present velocity (vrel⬃0) is also highly suppressed by the small lepton mass, even in the case of the tau lepton. How-ever, one possibility was pointed out by Baltz and Bergstrom 关12兴 that the annihilation N1N1→ᐉ⫹ᐉ⫺␥ would not suffer from helicity suppression. The rate of this process is approxi-mately␣/␲ times the annihilation rate at the freeze-out. As will be indicated later, the dominant mode would be

␮⫹_␮⫺_{␥. There is a slight chance to observe the excess in}

positron, but, however, the energy spectrum is softened be-cause of the cascade from the muon decay. However, the chance of observing the photon spectrum is somewhat better 关12兴.

D. Lepton flavor changing processes and gÀ2 There are two sources of lepton flavor violation in Eq.共1兲. The first one is from the interaction f_␣␤L_␣TCi␶2L␤S1⫹. This one is similar to the Zee model.共However, the present model would not give rise to neutrino mass terms in one loop be-cause of the absence of the S₁⫹-␾mixing.兲 The flavor violat-ing amplitude of ᐉ_␣→ᐉ_␳ via an intermediate ␯_␤ would be proportional to兩 f_␣␤f_␤␳兩. The second source is from the term g_I␣N_IS₂⫹ᐉ_␣Rin the Lagrangian共1兲. The flavor violating

am-2_{Krauss et al.关5兴 claimed that M}

N_R⬃1 TeV and g2⬃0.1 is

con-sistent with the dark matter constraint, but in their rough estimation a numerical factor of (TD/ MN)/8⬃200 is missing from the

plitude of ᐉ_␣→ᐉ_␤ via an intermediate NI would be

propor-tional to兩gI␣gI␤兩. We apply these two sources to the

radia-tive decays ofᐉ_␣→ᐉ_␤␥ and the muon anomalous magnetic moment.

The new contribution to the muon anomalous magnetic moment can be expressed as

⌬a␮⫽ m_␮2 96␲2

冉

兩 f␮␶兩2⫹兩f␮e兩2 M_S 1 2 ⫹ 6兩g1␮兩2 M_S 2 2 F2共MN₁ 2 _{/ M} S₂ 2 _兲 ⫹6兩g2␮兩 2 M_S 2 2 F2共MN2 2 / MS₂ 2 _兲

冊

, 共36兲 where F2(x)⫽(1⫺6x⫹3x2⫹2x3⫺6x2ln x)/6(1⫺x)4. The function F2(x)→1/6 for x→0, and F2(0.25)⬇0.125. We naively set F2(x)⫽1/6 for a simple estimate. Therefore, we obtain ⌬a␮⫽3⫻10⫺10

冋

共兩 f␮␶兩2⫹兩f␮e兩2兲

冉

2⫻102 GeV MS₁

冊

2 ⫹共兩g1␮兩2⫹兩g2␮兩2兲

冉

2⫻102 GeV MS₂

冊

2

册

ⱗ10⫺9_{, 共37兲}

which implies that f23, f21,g1␮,g2␮can be as large as O(1) for O(200 GeV) S₁⫹,S₂⫹without contributing in a significant level to⌬a_␮.

Among the radiative decays ␮→e␥ is the most con-strained experimentally, B(␮→e␥)⬍1.2⫻10⫺11 关13兴. The contribution of the our model is

B共␮→e␥兲⫽ ␣v 4 384␲

冋

兩 f␮␶f␶e兩2 MS₁ 4 ⫹ 36兩g_1eg_1␮兩2 MS₂ 4 F2 2_共M N₁ 2 / M_S 2 2 _兲 ⫹36兩g2eg2␮兩2 M_S 2 4 F2 2_共M N₂ 2 _{/ M} S₂ 2 _兲

册

_{, 共38兲} where v⫽246 GeV. Again we take F2(x)⫽1/6 and O(200 GeV) mass for S₁⫹,S₂⫹for a simple estimate:

B共␮→e␥兲⫽1.4⫻10⫺5

冋

共兩 f_␮␶f_␶e兩2兲

冉

2⫻10 2 _GeV MS₁

冊

4 ⫹兩g1eg1␮兩2

冉

2⫻102 _GeV MS₂

冊

4 ⫹兩g2eg2␮兩2

冉

2⫻102 GeV M_S 2

冊

4

册

⬍1.2⫻10⫺11_, 共39兲 which implies that

兩 fe␶f␶␮兩⬍1⫻10⫺3,

兩g1eg1␮兩⬍1⫻10⫺3,

兩g2eg2␮兩⬍1⫻10⫺3. 共40兲 This is in contrast to a work by Dicus et al. 关14兴. In their model, the couplings g_i’s are much larger than f_{i j}’s.

E. An example of consistent model parameters Here we summarize the constraints from previous subsec-tions, and illustrate some allowed parameter space. The prime constraints come from neutrino oscillations. The maxi-mal mixing and the mass-square difference required in the atmospheric neutrino and the small ␪13 read

f_␶␮m_␮g1␮⯝ f␮␶m␶g1␶Ⰷ fe␮m␮g1␮⫹ fe␶m␶g1␶ ⬃

冑

_␭1 s

冉

M_S 2 102 GeV

冊

MeV, 共41兲 where the terms f_␶emeg1e and f␮emeg1e have been omitted because these terms are suppressed by electron mass. The large mixing angle and the mass-square difference required in the solar neutrino are given by

f_␶emeg2e⫹ f␶␮m␮g2␮⯝2共 fe␮m␮g2␮⫹ fe␶m␶g2␶兲 Ⰷ f␮emeg2e⫹ f␮␶m␶g2␶, 共42兲

冉

fe␮m␮g2␮⫹ fe␶m␶g2␶ f_␶␮m_␮g1␮

冊

2 ⯝2 3⫻10 ⫺1_{. 共43兲}

On the other hand, the dark matter constraint requires at least one of the g1e,g1␮,g1␶ to be of order of unity. While the muon anomalous magnetic moment does not impose any strong constraints, lepton flavor violating processes, espe-cially B(␮→e␥), give the following strong constraints:

兩 f␮␶f␶e兩ⱗ1⫻10⫺3, 共44兲

兩g1eg1␮兩,兩g2eg2␮兩ⱗ1⫻10⫺3. 共45兲 Now, let us look for an example of consistent parameters. From Eq.共41兲, we obtain 兩m_␮g1␮兩⯝兩m␶g1␶兩, in other words 兩g1␮兩Ⰷ兩g1␶兩, and

f_␶␮Ⰷ fe␮⫹ f␶e. 共46兲

Since either g_1␮ or g_1e must be of order of unity from the dark matter constraint, we take g1␮⯝1. From Eqs. 共42兲 and 共43兲 with g2␶⯝0, we obtain

f_␶␮⯝2 f_e␮,兩m_␮g_2␮兩Ⰷ兩m_eg_2e兩 共47兲 and

g₂2_␮⯝8/3⫻10⫺1g2_1␮⯝0.27共g1␮/1兲2. 共48兲 Equations共46兲 and 共47兲 can be rewritten as

1Ⰷ1 2⫹

f_␶e

where we find that a mild cancellation between fe␮and f␶eis

necessary. For instance, f_␶e/ f_␶␮⫽⫺1/3. The strong cancel-lation corresponds to the small␪13. However, a cancellation with too high accuracy would require a␭s, which is too big

by Eq.共41兲. Therefore, one can say that this model predicts a relatively large mixing in ␪13. Now we obtain an example set of parameters that makes this model workable and it is

兩g1e兩ⱗ1⫻10⫺3, 兩g1␮兩⯝1, 兩g1␶兩⯝0.06, 兩g2e兩ⱗ2⫻10⫺3, 兩g2␮兩⯝0.5, 兩g2␶兩⬍10⫺2,

共50兲 fe␮⯝1⫻10⫺2, f␶␮⯝2⫻10⫺2, fe␶⬃⫺ fe␮.

IV. PRODUCTION AT e¿eÀ,µ¿µÀCOLLIDERS

The decay of N2may have an interesting signature, a displaced vertex, in colliders. Depending on the parameters, N2could be able to travel a typical distance, e.g., mm, in the detector without depositing any kinetic energy, and suddenly decay into N1 and two charged leptons. The signature is very striking.

The N1N1, N2N2, and N1N2 pairs can be directly produced at e⫹e⫺ colliders. The differential cross section for e⫹e⫺ →NINI, I⫽1,2, is given by d␴ d cos␪共e ⫹_e⫺_→N INI兲⫽ g_Ie4 256␲ ␤I s

冋

共t⫺MN_I 2 _兲2 共t⫺MS₂ 2 _兲2⫹ 共u⫺MN_I 2 _兲2 共u⫺MS₂ 2 _兲2⫺ 2 M_N I 2 s 共t⫺MS₂ 2 _兲共u⫺M S₂ 2 _兲

册

, 共51兲 where ␤I⫽

冑

1⫺4MN_I 2 /s, t⫽M_N I 2 _{⫺(s/2)(1⫺} ␤Icos␪), u⫽MN_I 2 _{⫺(s/2)(1⫹}

␤Icos␪). The total cross section is obtained by

integrating over the angle␪:

␴共e⫹_e⫺_→N INI兲⫽ gIe 4 64␲s 2共xI⫺xs兲2⫹xs ⫺2xI 3_⫹x I 2_共6x s⫹1兲⫺2xIxs共3xs⫹2兲⫹xs共1⫹xs兲共1⫹2xs兲

冋

␤I共⫺2xI⫹2xs⫹1兲 ⫹2„共xI⫺xs兲2⫹xs…log

冉

2xI⫺2xs⫹␤I⫺1 2x_I⫺2x_s⫺␤_I⫺1

冊册

, 共52兲 where xI⫽MN_I 2 /s and xs⫽MS₂ 2

/s. For N1N2 production the differential cross section is given by

d␴ d cos␪共e ⫹_e⫺_→N 1N2兲⫽ 兩g1eg2e兩2 128␲ ␤12 s

冋

共t⫺MN₁ 2 _兲共t⫺M N₂ 2 _兲 共t⫺MS₂ 2 _兲2 ⫹ 共u⫺MN₁ 2 _兲共u⫺M N₂ 2 _兲 共u⫺MS₂ 2 _兲2 ⫺ 2 MN₁MN₂s 共t⫺MS₂ 2 _兲共u⫺M S₂ 2 _兲

册

, 共53兲 and the integrated cross section is

␴共e⫹_e⫺_→N 1N2兲⫽ 兩g1eg2e兩2 128␲ ␤12 s 4 ␤12s共⫺1⫹x1⫹x2⫺2xs兲共⫺1⫹␤12⫹x1⫹x2⫺2xs兲共1⫹␤12⫺x1⫺x2⫹2xs兲 ⫻

再

␤12s共⫺1⫹x1⫹x2⫺2xs兲共⫺1⫹␤12 2_⫹2x 1⫺x1 2_⫹2x 2⫺6x1x2⫺x2 2_⫺4x s⫹8x1xs⫹8x2xs⫺8xs 2_兲 ⫹s关2

冑

x1x2⫹共x1⫹x2兲共x1⫹x2⫺4xs⫺1兲⫹2xs共2xs⫹1兲兴关␤12 2_{⫺共⫺1⫹x} 1⫹x2⫺2xs兲2兴 ⫻log

冉

⫺1⫺␤12⫹x1⫹x2⫺2xs ⫺1⫹␤12⫹x1⫹x2⫺2xs

冊冎

, 共54兲

where ␤12⫽

冑

(1⫺x1⫺x2)2⫺4x1x2. The above cross sec-tion formulas are equally valid for ␮⫹␮⫺ collisions. Since the constraints from the last section restrict g1eand g2eto be hopelessly small, we shall concentrate on using g1␮ and g2␮.

The production cross sections for the N₂N₂ and N₁N₂ pairs are given in Figs. 1共a兲 and 1共b兲, respectively, for

冑

s ⫽0.5,1,1.5 TeV and for MN₂ from 150 to 800 GeV, and we

have set g1␮⫽1, g2␮⫽0.5 关see Eq. 共50兲兴. In the curve for N₁N₂, we set M_N

in-terested in the N1N2,N2N2 production, because of its inter-esting signature.

As we have calculated the decay width of N2 in Eq.共25兲, the N2 can decay into N1 plus two charged leptons, either promptly or after traveling a visible distance from the inter-action point. It depends on the parameters involved, mainly the largest of g1␤g2␣. As seen in Eq. 共50兲 the largest is 兩g1␮g2␮兩⬃0.5, and so the decay of N2is prompt. Therefore, in the case of N1N2production, the signature would be a pair of charged leptons plus missing energies, because the N₁’s would escape the detection. The charged lepton pair is likely to be on one side of the event. In case of N2N2 production,

the signature would be two pairs of charged leptons with a large missing energy. Note that in the case of N1N1 produc-tion, there is nothing in the final state that can be detected. From Fig. 1 the production cross sections are of order O(10⫺100 fb), which implies plenty of events with O(100 fb⫺1) luminosity.

One may also consider S₂⫹S₂⫺ pair production. The S2 so produced will decay into S₂⫾→N1ᐉ_␣R⫾ or N2ᐉ_␣R⫾ , where ᐉ␣ ⫽e,␮,␶. However, the constraints on the parameter space require the mass of MS₂ substantially heavier than N1 and N2, and therefore the S2⫹S2⫺ pair production cross section is relatively much smaller.

V. CONCLUSIONS

In this paper, we have discussed a model that explains the small neutrino mass and dark matter in the Universe at the same time. Such a model was proposed by Krauss et al. as a modification of Zee model. However, our study revealed that their original model is unfortunately not capable of explain-ing the neutrino oscillation pattern.

We have extended the model by introducing another right-handed neutrino. We succeed in showing that such an exten-sion is possible to achieve the correct neutrino mixing pat-tern. A prediction of this model is the normal mass hierarchy. In addition, the undiscovered mixing angle ␪13 is relatively large, because of the requirement of a mild cancellation be-tween the parameters for a small␪₁₃and a sensible coupling of the charged scalar,␭s.

The relic density of the lightest right-handed neutrino has also been revisited. Under the constraint by WMAP we found that the mass of the right-handed neutrino cannot be as large as TeV but only of order 1⫻102 GeV, after a careful treatment of the calculation. In addition, other constraints including the muon anomalous magnetic moment, radiative decay of muon, and neutrinoless double beta decay have also been studied. With all the constraints we are still able to find a sensible region of parameter space.

Finally, our improved model has an interesting signature at leptonic colliders via pair production of right-handed neu-trinos, in particular N1N2 and N2N2. The N2 so produced will decay into N₁ plus two charged leptons. Thus, the sig-nature is either one or two pairs of charged leptons with a large missing energy. Hence, this model can be tested not only by neutrino experiments but also by collider experi-ments.

ACKNOWLEDGMENTS

This research of K.C was supported in part by the Na-tional Science Council of Taiwan R.O.C. under grant no. NSC 92-2112-M-007-053. O.S is supported by the National Science Council of Taiwan under the grant no. NSC 92-2811-M-009-018.

FIG. 1. Production cross sections for 共a兲 N2N2 and 共b兲 N1N2 pairs for

冑

s⫽0.5,1.0,1.5 TeV at l⫹l⫺ collisions. We have set g1␮

⫽1, g2␮⫽0.5, as suggested by Eq. 共50兲, MS₂⫽500 GeV, and

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