# Radiative neutrino mass generation from WIMP dark matter

Roberto A. Lineros Instituto de Física Corpuscular – CSIC/U. Valencia,
Parc Científic, calle Catedrático José Beltrán 2, E-46980 Paterna, Spain
<EMAIL_ADDRESS>
###### Abstract

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 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 potentially detectable at present and near
future collider experiments.

##  1 Introduction

It is well established that most of the matter content of the Universe is in
the form of Dark Matter (DM). The latest observations of the Cosmic Microwave
Background anisotropies [1] implies that the DM relic abundance in the
$\Lambda$CDM model is

$\Omega_{\rm DM}h^{2}=0.1198\pm 0.0015\,,$ (1)

where $h=0.678\pm 0.009$ is the scale factor for Hubble expansion rate [1].
The value of the relic abundance can be explained in terms of Weakly
Interactive Massive Particles (WIMPs). This class of particles are not part of
the Standard Model (SM) indicating the necessity to go beyond. The existence
of DM motivates, in part, searches of new particles in particle accelerator
like the Large Hadron Collider (LHC).

In general words, WIMP-DM particles can easily reproduce the observed value of
the relic abundance thanks to the _freeze-out mechanism_. This mechanism comes
from the combined effect of the expansion of the Universe and the interaction
rate among DM and the primordial plasma during the early Universe.

For WIMP particles in GeV–TeV range, the relic abundance is

$\Omega_{\rm WIMP}h^{2}\simeq 0.1\frac{3\times 10^{-26}\,{\rm cm}^{3}/{\rm
s}}{\langle\sigma v\rangle}\,,$ (2)

where $\langle\sigma v\rangle$ is the DM thermally averaged cross section. The
correct value of $\Omega_{\rm DM}h^{2}$ is then obtained when $\langle\sigma
v\rangle$ is of the order of weak interaction cross sections i.e. $3\times
10^{-26}\,{\rm cm}^{3}/{\rm s}$. In minimal scenarios, DM models are
constructed on top of the SM providing also new mediators to act as portal
between the dark and visible sectors. However, it seems reasonable that DM is
part of dark sector that includes new particles and interactions like in
Supersymmetric, extra dimensions [2], or dark gauge group models [3].

On the other hand, neutrinos also give an indication that the SM is not
complete. In the SM, neutrinos are massless but the observation of neutrino
oscillations indicate the contrary, i.e. they are massive and with the largest
mass squared differences of $\sim 10^{-3}\,{\rm eV}^{2}$ [4, 5]. The most
known mechanism of neutrino mass generation are the so called _see-saw_ models
(see for instance [6] and references within), where the smallness of neutrino
masses is generated at tree level by extremely heavy states such in the case
of the Type-I see-saw. However, it is also possible to generate the neutrino
masses via loop processes. One realization are the _scotogenic_ models [7, 8].
In one of its versions, the SM is enlarged with an extra SU$(2)_{L}$ doublet
scalar and a majorana fermion singlet. In another version, it gets an extra
SU$(2)_{L}$ doublet scalar but a SU$(2)_{L}$ fermion triplet with zero
hypercharge. In this type of models, an ad-hoc symmetry is imposed and
therefore neutrino mass generation at tree level is forbidden. This scheme is
very tantalizing due to that the ad-hod symmetry allows to have a DM
candidate.

Our model [9] is based on a mixed realization of the scotogenic models. We
show that in this scheme the phenomenology is richer and allows to have a
WIMP-DM candidate with a sizeable interaction with quarks and a mass ranging
from GeV to TeV.

##  2 The model

As previously stated, the model is a mix realization of two scotogenic models.
Besides the particle content of the SM, we have included a majorana fermion
singlet $N$, a SU$(2)_{L}$ fermion triplet $\Sigma$, a SU$(2)_{L}$ scalar
triplet $\Omega$; the latter three are without hypercharge, and finally a
SU$(2)_{L}$ scalar doublet $\eta$. The new fields, except $\Omega$, are
charged with a $Z_{2}$ symmetry that ensures the DM stability. Due to $\Omega$
is not protected by the symmetry, it will participate in the spontaneous
symmetry breaking of the SU$(2)_{L}\times$ U$(1)_{Y}$ SM group. All the matter
content of the model is summarized in Tab. 1.

| Standard Model | Fermions | Scalars
---|---|---|---
| $L$ | $e$ | $\phi$ | $\Sigma$ | N | $\eta$ | $\Omega$
SU$(2)_{L}$ | 2 | 1 | 2 | 3 | 1 | 2 | 3
U$(1)_{Y}$ | -1 | -2 | 1 | 0 | 0 | 1 | 0
$Z_{2}$ | $+$ | $+$ | $+$ | $-$ | $-$ | $-$ | $+$
Table 1: Matter content.

Using the symmetries described in the matter content, we write the lagrangian
of the model

$\displaystyle\mathcal{L}$ $\displaystyle\supset$ $\displaystyle-
Y_{\alpha\beta}\,\overline{L}_{\alpha}e_{\beta}\phi-
Y_{\Sigma_{\alpha}}\overline{L}_{\alpha}C\Sigma^{\dagger}\tilde{\eta}-\frac{1}{4}M_{\Sigma}\mbox{Tr}\left[\overline{\Sigma}^{c}\Sigma\right]+$
(3) $\displaystyle-
Y_{\Omega}\mbox{Tr}\left[\overline{\Sigma}\Omega\right]N-Y_{N_{\alpha}}\overline{L}_{\alpha}\tilde{\eta}N-\frac{1}{2}M_{N}\overline{N}^{c}N+h.c.\,-V_{\rm
scal},$

where $Y_{X}$ stand for yukawa couplings, $M_{X}$ are mass terms of $\Sigma$
and $N$, and $V_{\rm scal}$ is the scalar potential. The details about the
scalar potential are in Ref. [9].

The neutrino masses arises at one loop mainly due to the terms $L\eta N$ and
$L\Sigma\eta$ present in the lagrangian. We obtain that the neutrino mass
matrix in the flavor basis is

$M_{\alpha\beta}^{\nu}=\sum_{\sigma=1,2}\frac{h_{\alpha\sigma}h_{\beta\sigma}}{8\pi^{2}}\frac{\lambda_{5}v_{h}^{2}}{m_{0}^{2}}M_{k}\,,$
(4)

where $h_{\alpha\sigma}$ is a $3\times 2$ matrix that encodes the values of
the yukawa terms $Y_{\Sigma}$ and $Y_{N}$. $\lambda_{5}$ and $m_{0}$ are terms
coming from the scalar sector. $v_{h}$ is the vacuum expectation value of the
doublet $\phi$, and $M_{k}$ the masses of the mass eigenstates from the mix of
$N$ and $\Sigma^{0}$ (neutral component of the triplet $\Sigma$).

The model contains two possible DM candidates. One is the neutral component of
the scalar doublet $\eta$ which has a similar phenomenology to the inert higgs
model; and the second is the fermion which is a combination among $N$ and
$\Sigma^{0}$. In our analysis, we focus on the fermion DM candidate. The two
neutral fermions mass eigenstate are combination of $N$ and $\Sigma^{0}$ and
we will denote as $\chi^{0}_{1,2}$. This class of mixed state, between singlet
and triplet, was not possible in the traditional versions of the scotogenic
model. In our model, this is possible thanks to $\Omega$ that allows the term
$\Omega\Sigma N$ in the lagrangian. This term also generates a mass splitting
between the neutral and charged component of the triplet $\Sigma$ which later
it will reveal to be essential for the DM phenomenology.

Figure 1: Annihilation cross section versus DM mass. More details in Ref. [9]

At this point, it is important to describe the mass spectrum of the fermion
states:

$\displaystyle m_{\chi^{\pm}}$ $\displaystyle=$ $\displaystyle M_{\Sigma}\,,$
(5) $\displaystyle m_{\chi^{0}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(M_{\Sigma}+M_{N}-\sqrt{\left(M_{\Sigma}-M_{N}\right)^{2}+4\left(2Y_{\Omega}v_{\Omega}\right)^{2}}\right)\,,$
(6) $\displaystyle m_{\chi^{0}_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(M_{\Sigma}+M_{N}+\sqrt{\left(M_{\Sigma}-M_{N}\right)^{2}+4\left(2Y_{\Omega}v_{\Omega}\right)^{2}}\right)\,,$
(7) $\displaystyle\tan(2\alpha)$ $\displaystyle=$
$\displaystyle\frac{4Y_{\Omega}v_{\Omega}}{M_{\Sigma}-M_{N}}\,,$ (8)

where $v_{\Omega}$ is the vacuum expectation value of the neutral component of
$\Omega$ and $\alpha$ is the mixing angle between $N$ and $\Sigma^{0}$. Here
it is easy to appreciate the role of $v_{\Omega}$ breaking the degeneracy
between $\chi^{\pm}$ and the $\chi^{0}_{i}$ with larger component of
$\Sigma^{0}$.

We use the codes LanHEP [10, 11, 12] and MicrOmegas [13] to scan on the
parameter space (See details in Ref. [9]) including constraints from neutrino
masses, perturvativity limits of the adimensional parameters, and LEP [14] and
CMS [15] searches on new charged particles. Also, we require that
$m_{\chi^{0}_{1}}<m_{\chi^{\pm}},m_{\eta}$. We obtained the parameter space
compatible with current relic abundance value (Eq. 1). The allowed region is
compatible with DM masses in the range of $10$ to $10^{4}$ GeV.

In Fig. 1, we present the result of the scan in terms of the annihilation
cross section (relevant for indirect detection searches) versus the DM mass.
The color code corresponds to the value of

$\xi=\frac{M_{\Sigma}-m_{\rm DM}}{m_{\rm DM}}\,,$ (9)

which is related to the amount of $\Sigma^{0}$ present in DM candidate. This
quantity reveals three regimes. The low DM mass corresponds to a DM which is
mainly singlet $N$. This is the consequence of the LEP constraints on charge
particles. At the TeV masses, the DM candidate is mostly triplet. This is
produced due to annihilation channels to gauge bosons, coannihilation with
$\chi^{\pm}$, and $\chi^{\pm}$ annihilations. In the range between 100 GeV to
1 TeV, the DM is more a mixed state $N$-$\Sigma^{0}$. This regime is very
interesting because annihilations into quarks become relevant. This occurs
because $\Omega\Sigma N$ allows a higgs portal.

Same three regimes can be observed in Fig. 2 where we show the spin-
independent cross section versus DM mass. Let us highlight that in traditional
scotogenic models the expected spin-independent cross section is largely
suppressed because DM can only interact with quarks via loops. However, in our
model, the direct detection signal is of the order of the reach of current
experiments thanks to the higgs portal produced by the term $\Omega\Sigma N$.
In Fig. 2, we also show latest exclusion limits of XENON100 [16] as reference.

Figure 2: Spin-independent cross section versus DM mass. More details in Ref.
[9]

##  3 Conclusions

The nature behind DM is still a mystery. WIMPs are the most attractive
candidates due to many search strategies. We have proposed a new scotogenic
model that provide a neutrino mass mechanism at one loop and a fermion WIMP-DM
candidate. The fermion DM in this model provides features that are not present
in the traditional scotogenic models. One feature is the "tree level" spin
independent cross section that is at the reach of current and future direct
detection experiment. Also, the DM mass can go from 10 GeV to 10 TeV which is
broader with respect to previous models.

## Acknowledgments

We would like to thank the organizers of the TAUP 2015 for an interesting
conference. This work was supported by the Spanish MINECO under grants
FPA2014-58183-P, and MULTIDARK CSD2009-00064 (Consolider-Ingenio 2010
Programme); by Generalitat Valenciana grant PROMETEOII/2014/084, and Centro de
Excelencia Severo Ochoa SEV-2014-0398. R. L. is supported by a Juan de la
Cierva contract JCI-2012-12901 (MINECO).

## References

  * [1] Planck Collaboration 2015 ArXiv e-prints (Preprint 1502.01589)
  * [2] Lineros R A and Pereira dos Santos F A 2014 J. Cosmology Astropart. Phys. 10 059 (Preprint 1407.5999)
  * [3] França U, Lineros R A, Palacio J and Pastor S 2013 Phys. Rev. D 87 123521 (Preprint 1303.1776)
  * [4] Schwetz T, Tortola M and Valle J 2011 New J.Phys. 13 109401 (Preprint 1108.1376)
  * [5] Forero D, Tortola M and Valle J 2012 Phys.Rev. D86 073012 (Preprint 1205.4018)
  * [6] Lattanzi M, Lineros R A and Taoso M 2014 New Journal of Physics 16 125012 (Preprint 1406.0004)
  * [7] Ma E 2006 Phys.Rev. D73 077301 (Preprint hep-ph/0601225)
  * [8] Ma E and Suematsu D 2009 Mod.Phys.Lett. A24 583–589 (Preprint 0809.0942)
  * [9] Hirsch M, Lineros R A, Morisi S, Palacio J, Rojas N and Valle J W F 2013 Journal of High Energy Physics 10 149 (Preprint 1307.8134)
  * [10] Semenov A V 1996 ArXiv High Energy Physics - Phenomenology e-prints (Preprint arXiv:hep-ph/9608488)
  * [11] Semenov A V 2009 Computer Physics Communications 180 431–454 (Preprint 0805.0555)
  * [12] Semenov A 2010 ArXiv e-prints (Preprint 1005.1909)
  * [13] Belanger G, Boudjema F, Pukhov A and Semenov A 2013 ArXiv e-prints (Preprint 1305.0237)
  * [14] L3 Collaboration 2001 Physics Letters B 517 75–85 (Preprint arXiv:hep-ex/0107015)
  * [15] CMS Collaboration 2012 Physics Letters B 718 348–368 (Preprint 1210.1797)
  * [16] Aprile E et al. 2012 Physical Review Letters 109 181301 (Preprint 1207.5988)