WIMP dark matter as radiative neutrino mass messenger
The minimal seesaw extension of the Standard Model requires two electroweak singlet fermions in order to accommodate the neutrino oscillation parameters at tree level. Here we consider a next to minimal extension where light neutrino masses are generated radiatively by two electroweak fermions: one singlet and one triplet under SU(2). These should be odd under a parity symmetry and their mixing gives rise to a stable weakly interactive massive particle (WIMP) dark matter candidate. For mass in the GeV–TeV range, it reproduces the correct relic density, and provides an observable signal in nuclear recoil direct detection experiments. The fermion triplet component of the dark matter has gauge interactions, making it also detectable at present and near future collider experiments.
Edificio Institutos de Paterna, Apt 22085, E–46071 Valencia, Spain AHEP Group, Institut de Física Corpuscular – C.S.I.C./Universitat de València
Edificio Institutos de Paterna, Apt 22085, E–46071 Valencia, SpainInstitut für Theoretische Physik und Astrophysik, Universität Würzburg,
97074 Würzburg, Germany. AHEP Group, Institut de Física Corpuscular – C.S.I.C./Universitat de València
Edificio Institutos de Paterna, Apt 22085, E–46071 Valencia, SpainPontificia Universidad Católica de Chile, Facultad de Física. Av. Vicuña Mackenna 4860. Macul. Santiago de Chile, Chile. AHEP Group, Institut de Física Corpuscular – C.S.I.C./Universitat de València
Edificio Institutos de Paterna, Apt 22085, E–46071 Valencia, Spain\arxivnumber
Despite the successful discovery of the Higgs boson, so far the Large Hadron Collider (LHC) has not discovered any new physics, so neutrino physics remains, together with dark matter, as the main motivation to go beyond the Standard Model (SM). Neutrino oscillation experiments indicate two different neutrino mass squared differences Schwetz:2011zk (); Tortola:2012te (). As a result at least two of the three active neutrino must be massive, though the oscillation interpretation is compatible with one of the neutrinos being massless. In the Standard Model neutrinos have no mass at the renormalizable level. However they can get a Majorana mass by means of the dimension-5 Weinberg operator,
where is an effective scale, a dimensionless coefficient and and denote the lepton and Higgs isodoublets, respecively. This operator should be understood as encoding new physics associated to heavy “messenger” states whose fundamental renormalizable interactions should be prescribed. The smallness of neutrino masses compared to the other fermion masses, suggests that the messenger scale must is much higher than the electroweak scale if the coefficient in equation 1 is of . For example, the scale should be close to the Grand Unification scale if is generated at tree level. One popular mechanism to generate the dimension-5 operator is the so–called seesaw mechanism. Its most general realization is the so called “1-2-3” seesaw scheme Schechter:1981cv () with singlet, doublet and triplet scalar fields with vevs respectively , and . Assuming extra singlet fermions (right-handed neutrinos), the “1-2-3” scheme is described by the matrix
The vevs obey the seesaw relation
giving two contributions to the light neutrino masses , called respectively type-II and type-I seesaw. Assuming , namely no Higgs triplet 111Note that in pure type-II seesaw, only one extra scalar field is required, in contrast with type-I, where at least two fermion singlets must be assumed., the light neutrino masses arise only from the type-I seesaw contribution. In this case it is well known that in order to accommodate the neutrino oscillation parameters, at least two right-handed neutrinos are required, namely . We call the case minimal. Note that in this case one neutrino mass is zero and so the absolute neutrino mass scale is fixed. Typically the next to minimal case is to assume three sequential right-handed neutrinos, that is . An alternative seesaw mechanism is the so called type-III in which the heavy the “right-handed” neutrino “messenger” states are replaced by SU(2) triplet fermions Foot:1988aq (). As for the type-I seesaw case, one must assume at least two fermion triplets (if only fermion triplets are present) in order to accommodate current neutrino oscillation data.
There is an interesting way to induce the dimension-5 operator by mimicking the seesaw mechanism at the radiative level. This requires the fermion messengers to be odd under an ad-hoc symmetry in order to accommodate a stable dark matter (DM) candidate. In this case one can have “scotogenic” Ma:2006km () neutrino masses, induced by dark matter exchange. This trick can be realized either in type-I or type-III seesaw schemes Ma:2006km (); Ma:2008cu (). To induce Yukawa couplings between the extra fermions and the Standard Model leptons, one must include additional scalar doublets, odd under the assumed symmetry, and without vacuum expectation value. In order to complete the saga in this paper we propose a hybrid scotogenic construction which consists in having just one singlet fermion () but adding one triplet fermion as well.
This also gives rise to light neutrino masses, calculable at the one
loop level, as illustrated in
figure 1 222Note the scalar contributions
come from the scalar and pseudoscalar pieces of the field ..
However, due to triplet–singlet mixing, the lightest combimation of
the neutral component of the fermion triplet and the singlet will be
stable and can play the role of WIMP dark matter. We show that it
provides a phenomenologically interesting alternative to all previous
“scotogenic” proposals since here the dark matter can have sizeable
gauge interactions. As a result, in addition to direct and indirect
detection signatures, it can also be kinematically accessible to
present colliders such as the LHC.
Existing collider searches at LEP Ellis:1988zy (); L3:2001PhLB () and LHC CMS:2012PhLB (), set a nominal lower bound of 100 GeV for the masses of new charged particles. However, coannihilations present in the early universe, between the neutral and charged components, set the dark matter mass to be of the order of Ma:2008cu ()
in order to explain the observed abundance planck:2013 ():
Radiative neutrino masses generated by at least two generations of fermion singlets or triplets have been studied in Ref. Kubo:2006yx (). Here we focus on the radiative neutrino mass generation with one singlet and one triplet fermion which has interesting phenomenological consequences compared to the cases aforementioned cases. In our scenario, the dark matter candidate can indeed be observed not only in indirect but can also be kinematically accessible to current collider searches, and need not obey Eq. (4). Moreover, we will show that, in contrast to the proposed schemes in Refs. Ma:2006km (); Ma:2008cu () in our framework amplitudes leading naturally to direct detection processes appear at the tree level, thanks to singlet-triplet mixing effects.
The rest of this paper is organized as follows: in section 2 we introduce the new fields and interactions present in the model, making emphasis upon the mixing matrices and the radiative neutrino mass generation mechanism. Section 3 is devoted to numerical results on the phenomenology of dark matter in this model. An interesting feature of the model is the wide range of possible dark matter masses, ranging from 1 GeV to a few TeV. We also briefly discuss some the implications for LHC physics. In Section 4 we give our conclusions.
2 The model
2.1 The Model and the Particle Content
The new fields with respect to the Standard Model include one Majorana fermion
triplet and a Majorana fermion singlet both with zero
hypercharge and both odd under an ad-hoc symmetry . We also
include a scalar doublet with same quantum numbers as the Higgs
doublet, but odd under . In addition, we require that not
to acquire a vev.
As a result, neutrino masses are not generated at tree level by a
type-I/III seesaw mechanism. Instead they are one-loop calculable,
from diagrams in Fig. 1.
Furthermore, this symmetry forbids the decays of the lightest
odd particle into Standard Model particles, which is a mixture of the neutral
component of and . As a result this becomes a viable dark
matter candidate. Note also that our proposed model does not modify
quark dynamics, since neither of the new fields couples to quarks.
The fermion triplet, can be expanded as follows ( are the Pauli matrices):
The is exactly conserved in the Lagrangian, moreover, it allows interactions between dark matter and leptons, in fact, this is the origin of radiative neutrino masses. The Yukawa couplings between the triplet and leptons play an important role in the dark matter production. Finally a triplet scalar is introduced in order to mix the neutral part of the fermion triplet and the fermion singlet . This triplet scalar field also has zero hypercharge and is even under the symmetry, thus, its neutral component can acquire a nonzero vev.
2.2 Yukawa Interactions and Fermion Masses
The most general and Lorentz invariant Lagrangian is given as
The symbol stands for the Lorentz charge conjugation matrix and .
The Yukawa term is the SM Yukawa interaction for leptons, taken as diagonal matrix in the flavor basis333We can always go to this basis with a unitary transformation.. On the other hand the Yukawa coupling mixes the and fields and when the neutral part of the field acquire a vev , the dark matter particle can be identified to the lightest mass eigenstate of the mass matrix,
in the basis . As a result one gets the following tree level fermion masses
where is the mixing angle between and .
Here and characterize the Majorana mass terms for the
triplet and the singlet, respectively. The term is also
the mass of the charged component of the field, this issue is
important because the mass splitting between and the
dark matter candidate will play a role in the calculation of its relic
density. As we will see later, the splitting induced by
allows us to relax the constraints on the dark matter coming from the
existence of .
2.3 Scalar potential and spectrum
The most general scalar potential, even under , including the fields , and and allowing for spontaneous symmetry breaking, may be written as:
where the fields , and , can be written as follows:
where and are the vevs of and fields respectively. We have three charged fields one of which is absorbed by the boson, three CP-even physical neutral fields, and two CP-odd neutral fields one of which is absorbed by the boson 444Remember that the neutral part of field is real, so it does not contribute to the CP-odd sector..
Let us first consider the charged scalar sector.
The charged Goldstone boson is a linear combination of the
and the , changing the definition for the
boson mass from that in the Standard Model : .
Note that this places a constraint on the vev of from
electroweak precision tests Gunion:1989ci (); Gunion:1989we (), one
can expect roughly this vev to be less than 7 GeV, in order to keep
the experimental range, and alter the value inside the
experimental error band.
Apart from the boson, the two charged scalars have mass:
Notice that the nonzero vacuum expectation value will play an important role in generating the novel phenomenological effects of interest to us (see below). Now let us consider the neutral part: the minimization conditions of the Higgs potential allow vevs for the neutral part of the usual field as well as for the neutral part of the field. The mass matrix for neutral scalar eigenstates in the basis is:
where and are the tadpoles for and and are described in Appendix A.2. The presence of the vev induces the mixing between and . The corresponding eigenvalues give us the masses of the Standard Model Higgs doublet and the second neutral scalar both labelled as .
On the other hand, the field does not acquire vev, therefore, the mass eigenvalues of the neutral , charged and pseudoscalar are decoupled. The spectrum for and fields is:
2.4 Radiative Neutrino Masses
In this model, neutrino masses are generated at one loop. The dark matter candidate particle acts as a messenger for the masses. The relevant interactions for radiative neutrino mass generation arise from from Eqs. (10) and (16) and can be written in terms of the tree level mass eigenstates. Symbolically, one can rewrite the relevant terms for this purpose as:
Here the field are the mass eigenstate of the matrix (11) and is a matrix and is given by
where is the orthogonal matrix that diagonalizes the matrix in equation (11). There are two contributions to the neutrino masses from the loops in figure 1, where the and fields are involved in the loop. With the above ingredients, from the diagram in Fig. 1 one finds that the neutrino mass matrix is given by:
The functions correspond essentially to a differences of the Veltman functions Passarino:1978jh (), when evaluated at different scalar masses, note they have mass dimensions. The index runs over the mass eigenvalues, i.e. . Note that these masses are independent of the renormalization scale. In the equation below, each stands for the mass values of the fields.
It is useful to rewrite the equation 25 in a compact way as follows
which is formally equivalent to the standard type-I seesaw relation with Schechter:1980gr (). This is a diagonal matrix while plays the role of the Dirac mass matrix, in our case it is a matrix. It is not difficult to see that we can fit the required neutrino oscillation parameters Schwetz:2011zk (); Tortola:2012te (), for example, by means of the Casas Ibarra parametrization Casas:2001sr ().
In order to get an idea about the order of magnitude of the parameters required for producing the correct neutrino masses, one can consider a special limit in equation 25. For example, in cases where both are lighter than the other fields, we have from 25:
Here is the coupling introduced in equation 16. The are the masses of the neutral fermion fields . The mass term comes from writing the masses of the , and in the following way: , see appendix A.1 for more details. In particular we are interested in the magnitude of the Yukawa couplings required in order to have neutrino with masses of the order of eV. For masses of of order of 10 GeV and of order of 1000 GeV, and couplings not too small, namely of order of , one finds that the values for are in the order of the bottom Yukawa coupling . Hence it is not necessary to have a tiny Yukawa for obtaining the correct neutrino masses.
3 Fermion Dark Matter
As previously described the model contains two classes of potential dark matter candidates. One class are the odd scalars: and , when any of them is the lightest odd particle. Their phenomenology is very close to the inert doublet dark matter model LopezHonorez:2006gr () or discrete dark matter models Hirsch:2010ru (); Boucenna:2011tj (). For this reason here we focus our analysis on the other candidates which are the fermion states . In this case, the dark matter candidate is a mixed state between and . This interplay brings an enriched dark matter phenomenology with respect to models with only singlets or triplets.
For models with only fermion triplets as dark matter, equivalent in our model to taking , the main constraints come from the observed relic abundance (equation 5). Coannihilations between and are efficient processes due to the mass degeneracy between them, controlling the relic abundance. These processes force the dark matter mass to be 2.7 TeV. In addition, direct detection occurs only at the one loop level Cirelli:2005uq (), see Fig. 4.
Most of the corresponding features have been already studied in Ma:2008cu (); Chao:2012sz (). In figure 2, we show the coannihilation channels present in our model in terms of gauge eigenstates, except for the even scalars. The dark matter mass can be much smaller for singlets fulfulling the contraint. However, processes related to direct detection are absent at tree level Schmidt:2012yg () for singlets too.
The presence of the scalar triplet and its nonzero vev induces a mixing between and , implying coannihilations that can be important when the dark matter has a large component of . This mixing also breaks the degeneracy between the mass eigenstate fermions and . However, in this case, the mass degeneracy with the charged fermion is increased and forces the dark matter to be . Other coannihilation processes occur when is also degenerate with . For the opposite case, when is mainly , the model reproduces the phenomenology of the fermion singlet dark matter where the main signature is the annihilation into neutrinos and charged leptons (as in leptophilic dark matter) without any direct detection prospective Schmidt:2012yg (). The potential scenarios present in the model have the best of singlet-only or triplets-only scenarios and more. In addition, the dark matter phenomenology includes new annihilation and coannihilation channels when kinematically accessible.
The presence of the scalar triplet also induces an interaction between dark matter and quarks (direct detection) via the exchange of neutral scalar , as illustrated in In Fig 3, we show the main diagrams of the model related to indirect and direct searches. The model can potentially produce the typical annihilation channels appearing in generic weakly interactive massive particle dark matter models. Indeed, our dark matter candidate mimicks the Lightest Supersymmetric Particle (neutralino) present in supergravity-like versions the Minimal Supersymmetric Standard Model with R-parity conservation. The latter would correspond here to our assumed symmetry.
In order to study the dark matter phenomenology, we have implemented
the lagrangian (equation 10) using the standard codes
LanHEP lanhep:1996 (); lanhep:2009 (); lanhep:2010 () and
Micromegas micromegas:2013 ().
We scan the parameter space of the model within the ranges indicated
in Tab. 2. We also take into account the following
constraints: perturbatibity and a Higgs–like
scalar at 125 GeV.
Also we take into account the constraints from the relic
abundance planck:2013 () as well as the lower bound on the masses
of new non-colored charged particles coming from
LEP L3:2001PhLB () and LHC CMS:2012PhLB () collider searches,
roughly translated to .
We calculate the thermally averaged annihilation cross section
, and the spin independent cross section
In figure 5, we present the results of the scan in terms of the annihilation cross section versus the dark matter mass. Moreover, we show in color scale the quantity:
which estimates how degenerate is the dark matter mass with respect to . Small values of imply dark matter with a large component of and large value implies a large component of . This quantity has implications for coannihilation processes discussed previously. We notice that regions with low dark matter masses ( GeV) are less degenerate mainly because . In this region the dark matter contains a large component of . As expected, the TeV region is dominated by dark matter with large component of . The mass range 100–800 GeV is particularly interesting because any of the new charged particles are accessible at LHC. Moreover, when the mixing is non-zero and , the annihilation channels into quarks and leptons are naturally enhanced due to the -channel resonance in the process:
This is translated into higher expected fluxes of gamma–rays and
cosmic–rays for indirect searches as well as higher spin independent
Now, turning to the direct detection perspectives, the plot of the spin–independent cross section versus the dark matter mass is shown in figure 6. The scattering with quarks is described only with one diagram (the exchange of scalars ), also shown in figure 4. The size of the interaction will depend directly on the mixing . For masses larger than 100 GeV, we observe an increase of because maximal mixing can be obtained for and for . This does not occur for masses much lower to 100 GeV since the dark matter becomes mainly a pure . Moreover, the model produces large enough to be observed in direct detection experiments such XENON100 XENON:2012Ph () (yellow line).
Finally, we note that the new particles introduced in our model can be kinematically accessible at the LHC. Here we briefly comment on relevant production cross sections for the LHC. Both, ATLAS ATLAS-CONF-2013-019 () and CMS CMS:2012PhLB () have searched for pair production of heavy triplet fermions: , deriving lower limits on of the order of GeV CMS:2012PhLB () and GeV ATLAS-CONF-2013-019 (), respectively. However, these bounds do not apply to our model, because the final state topologies used in these searches, tri-leptons in case of CMS CMS:2012PhLB () and four charged leptons in ATLAS ATLAS-CONF-2013-019 (), are based on the assumption that decays to the final states . As a result of the symmetry present in our model, however, the lightest fermion or scalar is stable and all heavier -odd states will decay to this lightest state. Thus, the intermediate states and , which have the largest production cross sections of all new particles in our model, will not give rise to three and four charged lepton signals.
Instead, the phenomenology of and depends on the unknown mass ordering of fermions and scalars. Since we have assumed in this paper that the lighter of the fermions is the dark matter, we will discuss only this case here. Then, the phenomenology depends on whether the lightest of the neutral fermions, , is mostly singlet or mostly triplet. Consider first the case . Then, from the pair , only decays via , where the can be on-shell or off-shell. Thus, the final state consists mostly one charged lepton plus missing energy. The other possibility is pair production of via photon exchange, which leads to plus missing energy. In both cases, standard model backgrounds will be large and the LHC data probably does not give any competitive limits yet. We expect that LHC data at 14 TeV with increased statistics may constrain part of the parameter space. A quantitative study would require a MonteCarlo analisys which is beyond the scope of this work.
Conversely, for the case , the will decay to plus either one on-shell or off-shell Higgs state, depending on kinematics. In this case the final state will be one charged lepton plus up to four b-jets plus missing momentum. This topology is not covered by any searches at the LHC so far, as far as we are aware.
Also, the new neutral and charged scalars can be searched for at the LHC. All possible signals have, however, rather small production cross sections. Neither nor have couplings to quarks and only (both charged and neutral) can be produced at the LHC due to its mixing with the Standard Model Higgs field . Final states will be very much SM-Higgs like, but the event numbers will depend quadratically on this mixing, which supposedly is a small number, since the observed state with a mass of roughly GeV behaves rather closely like A Standard Model Higgs. Searches for a heavier state with Standard Model like Higgs properties Chatrchyan:2013yoa () exclude scalars with standard coupling strength now up to roughly 700 GeV. However, upper limits on in the mass range GeV are currently only of the order . The next run at the LHC, with its projected luminosity of order fb, should allow to probe much smaller mixing angles.
We have presented a next-to minimal extension of the Standard Model including new -odd majorana fermions, one singlet and one triplet under weak SU(2), as well as a -odd scalar doublet . We also include a -even triplet scalar in order induce the mixing in the fermionic sector –. The solar and atmospheric neutrino mass scales are then generated at one-loop level, with the lightest neutrino remaining massless. This way our model combines the ingredients present in Refs. Ma:2006km (); Ma:2008cu () with a richer phenomenology.
The unbroken symmetry implies that the lightest -odd particle is stable and may play the role of dark matter. We analyze the viability of the model using state-of-art codes for dark matter phenomenology. We focus our attention to the fermionic dark matter case. The mixing between and the neutral component of relaxes the effects of coannihilations between the dark matter candidate and the charged component of . In the pure triplet case, the dark matter mass is forced to be 2.7 TeV in order to reproduce the observed dark matter abundance value. However, in the presence of mixing the effect of coannihilations is weaker, allowing for a reduced dark matter mass down to the GeV range. Thanks to that, the charged can be much lighter than in the pure triplet case, openning the possibility of new signatures at colliders such as the LHC. In addition, the dark matter candidate can interact with quarks at tree level and then produce direct detection signal that may be observed or constrained in current direct searches experiments such XENON100.
This work was supported by the Spanish MINECO under grants FPA2011-22975 and MULTIDARK CSD2009-00064 (Consolider-Ingenio 2010 Programme), by Prometeo/2009/091 (Generalitat Valenciana), and by the EU ITN UNILHC PITN-GA-2009-237920. S.M. thanks to DFG grant WI 2639/4-1 for financial support. N.R. thanks to CONICYT doctoral grant, Marco A. Díaz for useful discussions and comments, the EPLANET grant for funding the stay in Valencia, and the IFIC–AHEP group in Valencia for the hospitality. R.L. also thanks to V. Ţăranu for her support.
Appendix A Appendix
a.1 Approximations for Neutrino Masses.
Starting from the equation 25, one can perform some approximations
to examine neutrino masses for cases of interest, for example, cases with one of
the masses being the lightest between ,
, and .
One wants to establish the relation between neutrino masses and the other parameters in the lagrangian in a suitable form. In principle, neutrino masses depend on the masses of neutral fields and the masses of the , but the dependence of the parameters of the scalar sector is more complicated, given the structure of the masses of the fields (see equations 21 and 22). One can take these equations and write them in the following way:
Where is a complicated function of the parameters of the scalar potential. One can write the equation 26 as follows:
One can identify two interesting limit cases. When then the function can be written as:
Therefore, the neutrino mass matrix in this approximation is given by:
The other case is given by , the procedure is not difficult, the result is:
In this case, the neutrino mass matrix is given by:
a.2 Minimization conditions
The tadpole equations were computed in order to find the minimum of the scalar potential, thus, the linear terms of the scalar potential at tree level can be written as:
Where the tadpoles are:
In order to have an invariant vacuum, the vev has
to vanish, which is extracted from the equation 40. For
the vev , one can choose the value to be nonzero solving the equation
in the parenthesis, in equal manner, one obtain the vev , in
terms of the other parameters of the potential.
The numerical values of the vevs and are restricted to reproduce the measured values of gauge boson masses, this allows to have the value for GeV, and GeV, as one can see in the section 2.3.
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