# Are small neutrino masses unveiling the missing mass problem of the Universe?

###### Abstract

We present a scenario in which a remarkably simple relation linking
dark matter properties and neutrino masses naturally emerges. This
framework points towards a low energy theory where the neutrino mass
originates from the existence of a light scalar dark matter particle
in the MeV mass range. A very surprising aspect of this scenario is
that the required MeV dark matter is one of the favored candidates
to explain the mysterious emission of 511 keV photons from the
center of our galaxy. A possible interpretation of these findings is
that dark matter is the stepping stone of a theory beyond the
standard model instead of being an embarrassing relic whose energy
density must be accounted for in any successful model building.

Lapth-1169/06; IPM/P-2006/077; IPPP/06/84; DCPT/06/168.

## I Introduction

The discovery of non-zero neutrino masses in neutrino oscillation experiments nuosci and the increasing evidence for about 23 of the content of the Universe being in the form of dark matter DMhint are the two main indications for physics beyond the Standard Model. These two issues, the origin of neutrino masses and the nature of dark matter, have been long standing problems in particle physics. Yet, in general, they are considered as two different topics and current explanations rely on completely different mechanisms, involving unrelated particles and scales. Although there have been some proposals to establish a link nuDM1 ; nuDM2 ; nuDM3 ; Ma:2006km ; Ma06a ; Ma06b ; Ma06c ; nuDM4 ; nuDM5 , a simple and natural picture in which the neutrino mass scale, the dark matter properties and dark matter abundance would be quantitatively related is still missing. In particular, to the best of our knowledge, there is no model in the literature which uniquely determines the dark matter scale and predicts, at the same time, a direct connection between the smallness of neutrino masses and the observed dark matter relic density.

In this letter, we present a scenario where such a prediction exists and therefore establish a quantitative link between these two fields. Our framework is strongly inspired by the class of models proposed independently in Ref. Palomares-Ruiz:2005vf to explain the signal observed in the LSND neutrino experiment LSND , in Refs. bens ; bf to illustrate that annihilating dark matter can be as light as a few MeV and in Ref. 511 ; ascasibar to explain the mysterious presence of low energy positrons in the center of our galaxy SPI . It is based on the following Lagrangian:

(1) |

where is a coupling constant, is a Majorana neutrino (with a mass ), is a neutral scalar (singlet of ) which plays the role of dark matter (hereafter referred to as the SLIM particle for Scalar as LIght as MeV), and is the standard left-handed neutrino. Since the mass of the particle is of Majorana type, lepton number is not conserved. As one can notice, the Lagrangian above contains only one interaction term. Since it breaks the electroweak symmetry for the case that we detail, it has to be regarded as a low energy effective Lagrangian.

Here we show that, with such a Lagrangian, a remarkably simple relationship between the dark matter cross section and the neutrino mass scale naturally emerges. Moreover the requirement of sub-eV neutrino masses, as imposed by experimental constraints, points towards light dark matter particles (with a mass of a few MeV). Our expression therefore suggests that the issues regarding the dark matter and neutrino masses are not only closely related but they also share the same low energy origin. A possible interpretation of these findings is that dark matter is fundamental. It may be the first step towards a low energy theory beyond the standard model.

## Ii Linking dark matter and neutrino mass

In the Lagrangian given in Eq. (1), is a scalar particle which may either be real or complex, and is a heavy neutrino with a Majorana mass . The particle is stable (it cannot decay into ) and constitutes our dark matter candidate. In contrast decays into and with a decay rate .

One consequence of the Lagrangian given in Eq. 1 is that left-handed neutrinos acquire a mass term via the one-loop correction depicted in Fig.1 Ma:1998dn . This mechanism is the same as in Refs. Ma:1998dn ; Ma:2006km ; Ma06a ; Ma06b ; Ma06c except that, in our scenario, is a singlet under the electroweak symmetry. Like in Refs. Ma:2006km ; Ma06a ; Ma06b ; Ma06c , we assume that does not have a vacuum expectation value, so Eq. 1 does not induce any tree level contribution to the left-handed neutrino mass which could dominate over the contribution discussed in this letter.

In this scenario, light neutrinos are predicted to be Majorana particles. This prediction is important because it can be tested in neutrinoless double beta decay experiments doublebeta1 ; doublebeta2 .

A real scalar field gives a contribution to which is given by:

(2) |

In this expression and (the ultraviolet cut-off of the effective theory) are free parameters.

From Eq. 1, one can draw the three diagrams shown in Fig. 2 which demonstrate that SLIM particles annihilate into two neutrinos (or two antineutrinos) as well as neutrino-antineutrino pairs.

Owing to these annihilations, the SLIM number density decreases with time. The rate at which the SLIM particles disappear is controlled by the total SLIM pair annihilation cross section. The three diagrams in Fig. 2 are the only annihilation channels available at tree-level. Hence the sum of these three contributions sets the annihilation rate and therefore determines the SLIM relic density.

The cross sections associated with the SLIM pair annihilations into either neutrino or antineutrino pairs (see the first two diagrams of Fig. 2), times –the relative velocity of the initial state particles–, are given by:

(3) |

where the notation denotes the thermal average of the quantity in the brackets.

In contrast, the cross section associated with the SLIM pair annihilations into neutrino-antineutrino pairs (see last diagram of Fig. 2) is suppressed by the ratio (or in the case of complex particles, where is the dark matter momentum). This cross section is therefore negligible with respect to the two others.

Hence (the total annihilation cross section times ) approximately corresponds to . Any relationship which involves this quantity is necessarily related to the dark matter abundance.

The remarkable point is that, for , the second term in Eq. (2) can be neglected. Then, using Eq. (3), Eq. (2) can be very simply rewritten as:

(4) |

This relation shows that, in our scenario, the left-handed neutrino masses and the dark matter abundance are very strongly related.

The simplicity of Eq. (4) implies that one can make firm predictions. The order of magnitude of can be obtained from measurements in neutrino experiments; The value of is set by the requirement that the SLIM relic density must correspond to the measured dark matter abundance. Thus, the only free parameters in Eq. 4 are , and . Since the dependence of Eq. (4) on is only logarithmic and since varying the ratio between 0 and 1 does not modify the result of Eq. (4) by more than a factor 2, is the only important free parameter.

In order to explain the observed dark matter abundance, the total SLIM pair annihilation cross section must be (see Refs. leeweinberg ; bens ) of the order of :

(5) |

It is worth noticing that this value is, in first approximation, independent of the dark matter mass and corresponds to a coupling

(6) |

If we now insert Eq. 5 into Eq. 4 and take, for instance, and consider , we obtain that can only vary within the range :

(7) |

This range will be narrowed down by improving the bounds on neutrino masses or, possibly, by directly measuring them. To be accurate, one should take into account flavor effects, i.e. one should specify the combination of neutrino flavors to which is coupled, keeping in mind that at least a second (heavier) is necessary to lead to at least two massive neutrinos. This would be done in a forthcoming paper.

Since , we can therefore conclude from Eq. 7 that MeV. The exact value of depends on the actual cut-off of the theory but, as already mentioned, this dependence is only logarithmic. Note that the above range implies that is an electroweak singlet or has very weak couplings to the the Standard Model boson.

Let us now discuss the case of a complex scalar field, where and are real fields with masses and . The various equations obtained for real are modified but the overall picture remains the same. In particular, Eq. 2 becomes

(9) |

and Eq. 4 becomes

(10) |

where we have neglected the terms suppressed by . Note that the cut-off dependence drops out in Eq. 9. In Eq. 10, the neutrino mass is determined by the quantity while, in Eq. 4, it was determined by . Hence, instead of Eq. 7, we now obtain:

(11) |

For definiteness, we have assumed that the ratio was ranging from 10 to . In Eq. 10, the mass is a free parameter which can be much larger than the mass of the boson. Hence, in the complex case (unlike the real case), the particle can have electroweak couplings.

If, for example, one expects that the unstable particle, , decays into plus a pair of neutrino and antineutrino. The particle, being stable, would be our dark matter candidate. Note also that if the mass splitting between and is small, one has to take into account the coannihilations between and for the calculation of the dark matter relic density. This may slightly change Eqs. 10 and 11.

In summary, if is a real field, the natural scale for the dark matter mass is the MeV range or below. As discussed in Section III, a dark matter mass much smaller than a few MeV poses some conflict with observations. Thus a dark matter mass in the MeV range is the preferred solution in the real case. If is a complex field, a suitable scale is also the MeV range although Eq. 11 does not uniquely predict that the dark matter mass must lie in the low energy range.

Obtaining the MeV scale is quite an amazing finding since this corresponds to the dark matter mass range which is required to explain the 511 keV emission line from the center of our galaxy bens ; bf ; 511 .

Note that if is mixed with light neutrinos and has a mass MeV, it might be responsible for the LSND signal Palomares-Ruiz:2005vf .

## Iii Constraints

The scenario that we discussed in the present letter satisfies the constraints from direct and indirect dark matter detection experiments. It also satisfies cosmological constraints. Large scale structure arguments force the SLIM particle to have a mass greater than a few keV, which is consistent with the present scenario. SLIM particles are also consistent with the constraints obtained in Refs. structure1 ; structure2 .

In supernovae, the strong interactions between the SLIM particles and neutrinos would maintain them in equilibrium. However, owing to the weakness of the interactions (if any) between the SLIMs and the rest of the Standard Model particles, neutrinos are emitted at approximately the same temperature as in the standard scenario without SLIM interactions. Thus, considering the present observational, as well as theoretical uncertainties, no bound can be obtained. However, in the case of future supernova neutrino observations, one may be able to test this scenario by studying the neutrino energy spectra.

SLIM particles should not affect primordial nucleosynthesis. For masses above MeV, no new light degrees of freedom (dof) are present at big bang nucleosynthesis epoch. For masses below MeV, each scalar would contribute 4/7 dof and each fermion 1 dof. Analysis of cosmological data, for instance the combination of the CMB determination of the baryon-to-photon ratio with primordial light-element abundances observations or with large scale structure data, lead to an upper bound on the number of extra dof of 1.5 (at CL) BBN1 ; BBN2 ; BBN3 .

As far as laboratory constraints are concerned, light scalar emission has not been observed in pion and kaon decays. For kinematically allowed decays a very conservative bound can be obtained, which constrains the coupling in Eq.(1) to be piondecays1 ; piondecays2 ; piondecays3 . Improving the present experimental bounds seems nevertheless feasible. For real , the upper bound on and the relatively large value of the coupling (see Eq. 8) promise observable effects in Kaon and pion decay experiments. This would make this scenario even more appealing as it could be tested soon. Many other constraints were discussed in Ref. bf with the conclusions that this scenario is perfectly viable.

There are certainly many ways to obtain the effective low energy
( breaking) term of Eq.(1) from
an underlying theory. If the particle is a singlet,
this interaction term is necessarily effective. It can be obtained,
in particular, from the exchange of an additional scalar
doublet^{1}^{1}1A model where such particle content has been
considered in a different mass range can be found in
Ref.Ma4 ., an extra vector like fermion singlet, or an extra
vector-like fermion doublet. The extra particles then have to be
well above the MeV scale. If is not a singlet, there
are various possibilities to obtain the same interaction term as in
Eq.(1). This “fundamental” Lagrangian was in fact
first proposed in bf , with a mirror fermion (doublet of
). In Ref. bf , was not a Majorana
particle, so it could not lead to the mechanism described in this
letter. However, one can consider a more sophisticated model where
is still a mirror particle (it would belong, together with a
charged lepton , to a right-handed doublet) but with
an “effective” mass that is induced from symmetry
breaking. This Majorana mass can be obtained easily from a
“mirror” seesaw mechanism between and an extra left-handed
singlet fermion (to be added to the lagrangian of
Ref. bf ). If the mass of the is not far above the
electroweak scale, and if the allowed ‘’ Yukawa coupling is
large enough, this seesaw can lead to a mass well above GeV, which can satisfy the constraint on the invisible decay
width of the -boson. This would require a complex , as
explained above. In this model, the lightest component would
be stable for example if, like and , it is odd under a
symmetry (similarly to models considered in
Ref. Ma:1998dn ; Ma:2006km ). This solution may be interesting
since, in Ref. ascasibar , it is shown that mirror fermions
can lead to the 511 keV line. This interaction term might also
reflect more exotic possibilities. A systematic study of all these
possibilities and related constraints is beyond the scope of this
letter.

## Iv Conclusion

In this letter, we propose a simple scenario, based on a single interaction term (Eq. 1), where our dark matter candidate is an electroweak singlet scalar, which interacts with a Majorana fermion and a left-handed neutrino.

This term generates left-handed neutrino masses through a one-loop diagram which can be directly related to the SLIM pair annihilation cross section into neutrinos. This leads to very simple relationships, Eqs. 4 and 10, which constrain the SLIM particle (our dark matter candidate) to be light. The natural scale which arises from these equations is the MeV scale (see Eqs. 7 and 11), providing a quantitative link between the dark matter characteristics and the neutrino masses.

A very exciting point is that a dark matter particle with a mass of only a few MeV also explains the morphology of the 511 keV-line emission in our galaxy. In addition, for lighter masses, it could also offer a possible explanation for the LSND signal.

If this picture is experimentally confirmed, our vision of dark matter in the universe has to be modified. It may play a fundamental role and even be an active component of the universe (whose presence is crucial) instead of being a simple relic. Sub-eV neutrino masses could be the experimental manifestation of MeV particles, possibly indicating the existence of a low energy theory difficult to access in collider/accelerator experiments due to the lack of luminosity at these energies.

Accurate measurements of left-handed neutrino masses (and the study of neutrino properties in general) could finally open up new possibilities to answer the question of the origin of the low energy positrons in our galaxy.

## Acknowledgment

We would like to thank F. Boudjema, P. Gagnon, A. Kusenko, E. Ma, C. Nicholls, S. T. Petcov, M. Raidal, L. Violini for useful discussions. The authors would like to thank the Theory Division at CERN for hospitality. SPR is partially supported by the Spanish grant FPA2005-01678 of the MCT.

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