Predictive model for dark matter, dark energy, neutrino masses, and leptogenesis at the TeV scale

Narendra Sahu and Utpal SarkarTheory Division, Physical Research Laboratory, Navarangpura, Ahmedabad 380 009, India

(Received 18 January 2007; published 30 August 2007)

We propose a new mechanism of TeV scale leptogenesis where the chemical potential of right-handedelectrons is passed on to the B� L asymmetry of the Universe in the presence of sphalerons. The modelhas the virtue that the origin of neutrino masses is independent of the scale of leptogenesis. As a result, themodel could be extended to explain dark matter, dark energy, neutrino masses, and leptogenesis at the TeVscale. The most attractive feature of this model is that it predicts a few hundred GeV triplet Higgs scalarthat can be tested at LHC or the International Linear Collider.

DOI: 10.1103/PhysRevD.76.045014 PACS numbers: 12.60.Fr, 14.60.St, 95.35.+d, 98.80.Cq

I. INTRODUCTION

In the canonical seesaw models [1] the physical neutrinomasses are largely suppressed by the scale of the leptonnumber violation, which is also the scale of leptogenesis.The observed baryon asymmetry and the low energy neu-trino oscillation data then give a lower bound on the scaleof leptogenesis as�109 GeV [2]. Alternately, in the tripletseesaw models [3], it is equally difficult to generate Lasymmetry at the TeV scale because the interaction ofSU�2�L triplets with the gauge bosons keeps them inequilibrium up to a very high scale �1010 GeV [4].However, in models of extra dimensions [5] and modelsof dark energy [6], the masses of the triplet Higgs scalarscould be low enough for them to be accessible in LHC orthe International Linear Collider (ILC), but in those modelsleptogenesis is difficult. Even in the left-right symmetricmodels in which there are both right-handed neutrinos andtriplet Higgs scalars contributing to the neutrino masses, itis difficult to have triplet Higgs scalars in the range of LHCor ILC [7]. It may be possible to have resonant leptogenesis[8] with light triplet Higgs scalars [9], but the resonantcondition requires a very high degree of fine tuning.

In this paper we introduce a new mechanism of lepto-genesis at the TeV scale. We ensure that the lepton numberviolation required for the neutrino masses does not conflictwith the lepton number violation required for leptogenesis.This led us to propose a model which is capable of explain-ing dark matter, dark energy, neutrino masses and lepto-genesis at the TeV scale. Moreover, the model predicts afew hundred GeV triplet Higgs whose decay through thesame sign dilepton signal could be tested either through thee�e� collision at linear collider or through the pp collisionat LHC.

II. THE MODEL

In addition to the quarks, leptons, and the usual Higgsdoublet � � �1; 2; 1�, we introduce two triplet Higgs sca-lars � � �1; 3; 2� and � � �1; 3; 2�, two singlet scalars�� � �1; 1;�2� and T0 � �1; 1; 0�, and a doublet Higgs� � �1; 2; 1�. The transformations of the fields are given

under the standard model (SM) gauge group SU�3�c �SU�2�L �U�1�Y . There are also three heavy singlet fermi-ons Sa � �1; 1; 0�, a 1, 2, 3. A global symmetry U�1�Xallows us to distinguish between the L-number violationfor neutrino masses and the L-number violation for lepto-genesis. Under U�1�X, the fields ‘TiL � ��; e�iL ��1; 2;�1�, eiR � �1; 1;�2�, ��, and T0 carry a quantumnumber 1; �, Sa, a 1, 2, 3, and � carry a quantumnumber zero; and � and � carry quantum numbers �2and 2, respectively. We assume that M� M�, while both� and � contribute equally to the effective neutrino masses.Moreover, if neutrino mass varies on the cosmological timescale, then it behaves as a negative pressure fluid and henceexplains the accelerating expansion of the present Universe[10].1 With a survival Z2 symmetry, the neutral componentof � represents the candidate of dark matter [12].

Taking into account the above defined quantum numbersof the fields, we now write down the Lagrangian symmetricunder U�1�X. The terms in the Lagrangian relevant to therest of our discussion are given by

�L � fij�‘iL‘jL ���A��y���M2��y��M2

��y�

� hia �eiRSa�� �MsabSaSb � yij� �‘iLejR

�M2TTyT � �TjTj4 � ��jTj2j�j2 � ��jTj2j�j2

� fT��yTT � ���j��j2j�j2 � ���j�

�j2j�j2

� V�� � H:c:; (1)

where V�� constitutes all possible quadratic and quarticterms symmetric under U�1�X. The typical dimension fullcoupling ��A� �A, A being the acceleron field2 that isresponsible for the accelerating expansion of the Universe.We introduce the U�1�X symmetry breaking soft terms

�Lsoft m2TTT �m������ H:c: (2)

1The connection between neutrino mass and dark energy,which is required for accelerating expansion of the presentUniverse in large extradimension scenario, is discussed inRef. [11].

2The origin of this acceleron field is beyond the scope of thispaper. See, for example, Ref. [13].

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If T carries the L number by one unit, then the first termexplicitly breaks the L number in the scalar sector. Thesecond term on the other hand conserves the L number if�� and � possess an equal and opposite L number.3 Thisleads the interactions of the fields Sa, i 1, 2, 3 to beL-number conserving. As we shall discuss later, this cangenerate the L asymmetry of the Universe, while theneutrino masses come from the L-number conserving in-teraction term �y�TT after the field T acquires a vacuumexpectation value (VEV).

III. NEUTRINO MASSES

The Higgs field � acquires a very small VEV

h�i ���A�v2

M2�

; (3)

where v h�i, � is the SM Higgs doublet. However, wenote that the field � does not acquire a VEV at the treelevel.

The scalar field T acquires a VEV at a few TeV, whichthen induces a small VEV to the scalar field �. TheGoldstone boson corresponding to the L-number violation,the would-be Majoron, and the Goldstone boson corre-sponding to U�1�X symmetry will have a mass of the orderof a few TeV and will not contribute to the Z decay width.The VEV of the field � would give a small Majorana massto the neutrinos.

The VEV of the singlet field T gives rise to a mixingbetween � and � through the effective mass term

�L�� m2s�y�; (4)

where the mass parameter ms ��������������fThTi

2p

is of the order ofTeV, similar to the mass scale of T. The effective couplingsof the different triplet Higgs scalars, which give theL-number violating interactions in the left-handed sector,are then given by

�L�-mass fij�‘i‘j ���A�m2s

M2�

�y��� fijm2s

M2�

�‘i‘j

���A��y��� H:c: (5)

The field � then acquires an induced VEV,

h�i ���A�v2m2

s

M2�M

2�

: (6)

The VEVs of both the fields � and � will contribute toneutrino masses by equal amount and thus the neutrinomasses are given by

�m��ij �fij��A�v2m2

s

M2�M

2�

: (7)

Since the absorptive part of the off-diagonal one-loopself-energy terms in the decay of triplets � and � is zero,their decay cannot produce any L asymmetry even thoughtheir decay violates the L number. However, the possibilityof erasing any preexisting L asymmetry through the �L 2 processes mediated by � and � should not be avoidedunless their masses are very large and hence suppressed incomparison to the electroweak breaking scale. In particu-lar, the important erasure processes are

‘‘$ �$ �� and ‘‘$ �$ ��: (8)

If m2s M2

�, then the L-number violating processes me-diated through � and � are suppressed by �m2

s=M2�M

2��,

and hence practically do not contribute to the above erasureprocesses. Thus, a fresh L asymmetry can be produced atthe TeV scale.

IV. LEPTOGENESIS

We introduce the following two cases for generating Lasymmetry, which is then transferred to the required Basymmetry of the Universe.

Case I: The explicit L-number violation.—First we con-sider the case where the L number is explicitly broken inthe singlet sector. This is possible if ��, and hence � doesnot possess any L number. Therefore, the decays of thesinglet fermions Sa, a 1, 2, 3 can generate a net Lasymmetry of the Universe through

Sa ! e�iR � �� ! e�iR � �

�:

We work on the basis that Msab is diagonal and M3 >M2 >M1, where Ma Msaa. Similar to the usual right-handed neutrino decays generating L asymmetry [14],there are now one-loop, self-energy, and vertex-type dia-grams that can interfere with the tree-level decays to gen-erate a CP asymmetry. The decay of the field S1 can nowgenerate a CP asymmetry

" �Xi

���S1 ! e�iR�

�� � ��S1 ! e�iR���

�tot�S1�

�

’1

8�M1

M2

Im �hhy�i1�hhy�i1�P

ajha1j

2 : (9)

Thus, an excess of eiR over eciR is produced in the thermalplasma. This will be converted to an excess of eiL over eciLthrough the t-channel scattering process eiReciR $ �0 $eiLe

ciL. This can be understood as follows. Let us define the

chemical potential associated with the eR field as �eR �0 ��BL, where �BL is the chemical potential contribut-ing to B� L asymmetry and �0 is independent of B� L.Thus, at equilibrium we have

3If �� does not possess any L number, then the interaction ofSa explicitly breaks the L number, and hence the decay of thelightest Sa gives rise to a net L asymmetry, as in the case of right-handed neutrino decay [14].

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�eL �eR ��� �BL ��0 ���: (10)

We see that �eL is also associated with the same chemicalpotential �BL. Hence, the B� L asymmetry produced inthe right-handed sector will be transferred to the left-handed sector. A net baryon asymmetry of the Universeis then produced through the sphaleron transitions, whichconserve B� L but violate B� L. Since the source of theL-number violation for this asymmetry is different fromthe neutrino masses, there is no bound on the mass scale ofS1 from the low energy neutrino oscillation data.Therefore, the mass scale of S1 can be as low as a fewTeV. Note that the mechanism for L asymmetry proposedhere is different from an earlier proposal of right-handedsector leptogenesis [15]. The survival asymmetry in the �fields is then transferred to � fields through the trilinearsoft term introduced in Eq. (2).

Case II:: Conserved L number.—We now consider thecase where the L number is conserved in the singlet sector.This is possible if ������ possesses an L number exactlyopposite to that of e�R �e

�R �. Therefore, the decays of the

singlet fermions Sa, a 1, 2, 3 cannot generate any Lasymmetry. However, they produce an equal and oppositeasymmetry between ������ and e�R �e

�R � fields as given by

Eq. (9). If these two asymmetries cancel each other, thenthere is no remaining L asymmetry. However, we can seefrom the Lagrangians (1) and (2) that none of the inter-actions can transfer the L asymmetry from �� to the leptondoublets while eR is transferring the L asymmetry from thesinglet sector to the usual lepton doublets through � �‘LeRcoupling. Note that the coupling through which the asym-metry between �� and e�R is produced is already gone outof the thermal equilibrium. So, it will no longer allow thetwo asymmetries to cancel with each other. The asymmetryin the � fields is finally transferred to the � fields throughthe trilinear soft term introduced in Eq. (2).

V. DARK MATTER

As the Universe expands, the temperature of the thermalbath falls. As a result the heavy fields �� and T0 areannihilated to the lighter fields� and � as they are allowedby the Lagrangians (1) and (2). Notice that there is a Z2

symmetry of the Lagrangians (1) and (2) under which Sa,a 1, 2, 3, ��, and � are odd while all other fields areeven. Since the neutral component of � is the lightest one,it can be stable because of Z2 symmetry. Therefore, theneutral component of � behaves as dark matter.

After T gets a VEV, the effective potential describing theinteractions of � and � can be given by

V�� ��m2

� ���fTm2s

�j�j2 �

�m2� �

��fTm2s

�j�j2

� �1j�j4 � �2j�j

4 � �3j�j2j�j2 � �4j�

y�j2;

(11)

where we have made use of the fact thatms ��������������fThTi2

pand

��, �� are the quartic couplings of T with � and �,

respectively. For m2� > �

��fT�m2

s > 0 and m2�, ���fT �m

2s > 0,

the minimum of the potential is given by

h�i 0v

� �and h�i

00

� �: (12)

The VEVof � gives masses to the SM fermions and gaugebosons. The physical mass of the SM Higgs is then given

by mh �������������4�1v2

p. The physical masses of the real and

imaginary parts of the neutral component of the � fieldare almost the same and are given by

m2�0R;I m2

� ���fTm2s � ��3 � �4�v

2: (13)

Since � is odd under the surviving Z2 symmetry it cannotdecay to any of the conventional SM fields, and hence theneutral component of � constitutes the dark matter com-ponent of the Universe. �0

R;I are in thermal equilibriumthrough the interactions �2�

0R;I and ��3 � �4��

0R;Ih

2.Assuming that m�0

R;I< mw, mh, the direct annihilation of

a pair of �0R;I below their mass scale, to SM Higgs and W�

W�, is kinematically forbidden. However, a pair of �0R;I

can be annihilated to the SM fields, f �f; , W�W�; ZZ; gg;hh; etc., through the exchange of neutral Higgs h. Thecorresponding scattering cross section in the limit m�0

R;I<

mW , mh is given by [16]

hjvj ’�2m2

�0R;I

m4h

; (14)

where � ��3 � �4�.We assume that at a temperature TD, �ann=H�TD� ’ 1,

where TD is the temperature of the thermal bath when �0R;I

got decoupled and

H�TD� 1:67g1=2� �T2

D=Mpl� (15)

is the corresponding Hubble expansion parameter withg� ’ 100 being the effective number of relativistic degreesof freedom. Using Eq. (14) the rate of annihilation of �0

R;I

to the SM fields can be given by �ann n�0hhjvji, wheren�0 is the density of �0

R;I at the decoupled epoch. Using thefact that �ann=H�TD� ’ 1 one can get [17]

zD �m�0

R;I

TD’ ln

� Nann�2m3

�0R;IMpl

1:67g1=2� �2��3=2m4

h

�; (16)

where Nann is the number of annihilation channels, whichwe have taken roughly to be 10. Since �0

R;I is stable in thecosmological time scale, we have to make sure that itshould not overclose the Universe. Therefore, we calculatethe energy density of �0

R;I at the present epoch. The numberdensity of �0

R;I at the present epoch is given by

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n�0R;I�T0� �T0=TD�

3n�0R;I�TD�; (17)

where T0 2:75 K, the temperature of present cosmicmicrowave background radiation. We then calculate theenergy density at the present epoch,

�0R;I�T0� ’

�0:98� 10�4 eV

cm3

�1

Nann�2

�mh=GeV�4

�m�0R;I=GeV�2

� 1� ��; (18)

where � 1. The critical energy density of the presentUniverse is

c 3H20=8�GN � 104h2 eV=cm3: (19)

At present the contribution of dark matter to the criticalenergy density of the Universe is precisely given by�DMh

2 0:111� 006 [18]. Assuming that �0R;I is a can-

didate of dark matter, we have shown in Fig. 1 the allowedmasses of �0

R;I up to 80 GeV for a wide spectrum of SMHiggs masses.

VI. DARK ENERGY AND NEUTRINOS

It has been observed that the present Universe is ex-panding in an accelerating rate. This can be attributed tothe dynamical scalar field A [19], which evolves on thecosmological time scale. If the neutrino mass arises froman interaction with the acceleron field, whose effectivepotential changes as a function of the background neutrinodensity, then the observed neutrino masses can be linked tothe observed acceleration of the Universe [10].

Since the neutrino mass depends on A, it varies on thecosmological time scale such that the effective neutrinomass is given by the Lagrangian

�L

�fij��A�

v2m2s

M2�M

2�

�i�j � H:c:�� V0; (20)

where V0 is the acceleron potential. A typical form of thepotential is given by [6]

V0 �4 ln�1� j ��j��A�j�: (21)

The two terms in the above Lagrangian (20) act in oppositedirections such that the effective potential

V�m�� m�n� � V0�m�� (22)

today settles at a nonzero positive value. From the aboveeffective potential we can calculate the equation of state

w �1� ��=��� ��A��; (23)

where w is defined by V / R�3�1�w�. At present the con-tribution of light neutrinos having masses varying from5� 10�4 eV to 1 MeV to the critical energy density of theUniverse is �� � 0:0076=h2 [18]. Hence, one effectivelygets w ’ �1. For naturalness we chose ��A�m2

s

M2�

� 1 eV such

that M� can be a few hundred GeV to explain the sub-eVneutrino masses, and �� 10�3 eV such that the varyingneutrino mass can be linked to the dark energy componentof the Universe.

VII. COLLIDER SIGNATURE OF DOUBLYCHARGED PARTICLES

The doubly charged component of the light triplet Higgs� can be observed through its decay into same sign dilep-tons [20]. Since M� � M�, the production of � particlesin comparison to � is highly suppressed. From Eq. (5) onecan see that the decay ��� ! ���� is suppressed sincethe decay rate involves the factor ��A�m

2s

M2�

� 1 eV. While the

decay mode ��� ! h�W� is phase space suppressed, thedecay mode ��� ! W�W� is suppressed because theVEV of � is small, which is required for sub-eV neutrinomasses as well as to maintain the parameter of SM asunity. Therefore, once it is produced, � mostly decaysthrough the same sign dileptons: ��� ! ‘�‘�. Note that

100 200 300 400 500 600mh GeV

10

20

30

40

50

60

70

80

m0

GeV

100 200 300 400 500 600mh GeV

10

20

30

40

50

60

70

80

m0

GeV

FIG. 1 (color online). The allowed region of dark matter at the 1 C.L. is shown in the plane of mh versus m�0 with �2 0:5 (left-hand side) and �2 0:1 (right-hand side).

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the doubly charged particles cannot couple to quarks, andtherefore the SM background of the process ��� ! ‘�‘�

is quite clean and the detection will be unmistakable.Hence, it is worth looking for the signature of ��� eitherat LHC or ILC.

From Eq. (5) the decay rate of the process ��� ! ‘�‘�

is given by

�ii jfiij2

8�M��� and �ij

jfijj2

4�M��� ; (24)

where fij is highly constrained from the lepton flavorviolating decays. From the observed neutrino masses wehave fijx� 10�12, where x �h�i=v�. If fij * x, thenfrom the lepton flavor violating decay ��� ! ‘�i ‘

�j one

can study the pattern of neutrino masses and mixing [21].

VIII. CONCLUSIONS

We introduced a new mechanism of leptogenesis in thesinglet sector that allowed us to extend the model toexplain dark matter, dark energy, neutrino masses, andleptogenesis at the TeV scale. This scenario predicts afew hundred GeV triplet scalar that contributes to theneutrino masses. This makes the model predictable and itwill be possible to verify the model at ILC or LHC throughthe same sign dilepton decay of the doubly charged parti-cles. This also opens an window for studying the neutrinomass spectrum in future colliders (LHC or ILC). Since thelepton number violations required for lepton asymmetryand neutrino masses are different, the leptogenesis scalecan be lowered to as low as a few TeV.

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