Dark Matter and The Seesaw Scale

We discuss the possibility to find an upper bound on the seesaw scale using the cosmological bound on the cold dark matter relic density. We investigate a simple relation between the origin of neutrino masses and the properties of a dark matter candidate in a simple theory where the new symmetry breaking scale defines the seesaw scale. Imposing the cosmological bounds, we find an upper bound of order multi-TeV on the lepton number violation scale. We investigate the predictions for direct and indirect detection dark matter experiments, and the possible signatures at the Large Hadron Collider.


INTRODUCTION
The origin of neutrino masses and the nature of the cold dark matter in the Universe are two of the most exciting open problems in particle physics and cosmology. We know today about several mechanisms to generate neutrino masses, see for example Ref. [1], but the so-called seesaw mechanism [2] is considered the most appealing and simple mechanism for Majorana neutrino masses. Unfortunately, we only know that the upper bound on the seesaw scale is about 10 14 GeV, which is an energy scale very far from any future collider experiment. Therefore, it is not clear we could test the mechanism behind neutrino mass. There are also many possible candidates to describe the cold dark matter in the Universe, see for example Ref. [3]. The weakly interacting massive particles (WIMPs) have been popular dark matter candidates in the last decade but the recent experimental results tell us that maybe one should think about other possibilities. However, it is fair to say that the idea of describing the dark matter with WIMPs is so appealing that it is better to understand and revise all constraints and the different models before abandoning this idea.
The discovery of lepton number violating signatures in low energy experiments or at colliders will be striking signals for new physics beyond the Standard Model. In low energy experiments we could discover neutrinoless double beta decay, for a review see Ref. [4], and at colliders different signatures with same-sign leptons could be seen [5]. These discoveries will be crucial to establish the origin of neutrino masses.
We understand the origin of charged fermion masses in the Standard Model through the spontaneous breaking of the electroweak symmetry. In the same way, we could understand the origin of the seesaw scale if B −L is a local symmetry spontaneously broken through the Higgs mechanism. Unfortunately, as in canonical seesaw, the upper bound on the B − L is typically very large, M B−L 10 14 GeV. There are two known ways to establish a much smaller upper bound on the B − L breaking scale: a) In the context of the minimal supersymmetric U (1) B−L theory [6] the B − L breaking scale is defined by the supersymmetry breaking scale. Then, if low energy supersymmetry is realized at the multi-TeV scale, we could discover lepton number violating signatures at colliders. b) The second possibility is to use the cosmological bounds on the dark matter relic density to impose an upper bound on the B − L breaking scale in the case where the dark matter is charged under the same gauge symmetry.
In this article, we focus on the second possibility mentioned above in order to find an upper bound on the B − L seesaw scale. In this theory, the dark matter candidate is a vector-like fermion which is a SM singlet but charged under the B − L gauge symmetry. We find that, using the constraints on the cold dark matter relic density, the upper bound on the B − L is in the multi-TeV region. Therefore, one can expect exotic signatures at colliders with same-sign multi-leptons and displaced vertices. This connection between the cosmological dark matter bounds and exotic signatures at colliders is very unique and one could hope to test the origin of neutrino masses at colliders. See Fig. 1 for a simple way to illustrate this correlation.

NEUTRINO MASSES AND THE B − L SCALE
The simplest gauge theory where one can understand dynamically the origin of neutrino masses is based on the B − L gauge symmetry. In this context, we add three copies of righthanded neutrinos to define an anomaly free theory, and one can easily implement the seesaw mechanism [2] for Majorana arXiv:1803.07462v2 [hep-ph] 10 Aug 2018 neutrino masses. The relevant Lagrangian for the generation of neutrino mass is given by where ν R ∼ (1, 1, 0, −1) are the right-handed neutrinos, H ∼ (1, 2, 1/2, 0) is the Standard Model Higgs, and S BL ∼ (1, 1, 0, 2) is the new Higgs responsible for the spontaneous breaking of B − L. Using the above interactions one can generate masses for the Standard Model neutrinos through the well-known Type I seesaw mechanism [2], which are given by Here v 0 / √ 2 is the vacuum expectation value of the Standard Model Higgs. We note that, in this case, the masses of the right-handed neutrinos, M R = √ 2λ R v BL , are defined by the B − L breaking scale. The seesaw scale, in general, is unknown; the only thing we know is that the upper bound should be the canonical seesaw scale 10 14 GeV. It is important to mention that if M R is at the TeV scale one can generate masses for Standard Model neutrinos in agreement with the experiments if m D < 10 −3 GeV, and one can produce the right-handed neutrinos at the LHC through the B − L neutral gauge boson, pp → Z * BL → N i N i [7][8][9][10][11], giving rise to striking lepton number violating signatures with same-sign leptons and multijets. Now, since the observation of lepton number violation is crucial to learn about the origin of neutrino masses, it is important to understand the possibility to find an upper bound on the B − L breaking scale which is much smaller than the canonical seesaw scale. Then, we could hope to test the origin of neutrino masses at current or future experiments. We know about two different class of theories where it is possible to find an upper bound on the B − L breaking scale: • In Ref. [6], one of us (P.F.P.) and collaborators pointed out that in the minimal supersymmetric B − L model the gauge symmetry must be broken by the vacuum expectation value of the 'right-handed' sneutrinos. Then one predicts that R−parity must be spontaneously broken, and one expects the existence of lepton number violation. In this context, the R−parity and lepton number violation scales are defined by the supersymmetry breaking scale. Then, if one has low energy supersymmetry at the multi-TeV scale, and this theory is true, one should discover lepton number violation at current or future colliders. For detailed studies see Refs. [12][13][14][15].
• The second possibility is discussed in details in this article. We will show that if one has a fermionic cold dark matter candidate which is charged under the B −L gauge symmetry, it is possible to find an upper bound on the B − L breaking scale in the multi-TeV region using the cosmological bounds on the dark matter relic density. Therefore, this theory provides a simple scenario which motivates the search for lepton number violation at colliders.
These two scenarios provide two major examples of theories where one could expect the discovery of lepton number violating processes at the multi-TeV scale. We will focus on the second example and investigate the impact of all dark matter bounds.

DARK MATTER AND THE B − L SCALE
One can write a very simple model to generate Majorana neutrino masses and to explain the presence of cold dark matter in the Universe based on the spontaneous breaking of the U (1) B−L gauge symmetry. The relevant Lagrangian for our discussion is given by where Here, |n| = 1, 3 in order to avoid the decay of χ = χ L + χ R , which must be stable, and can be a good cold dark matter candidate. In the case the proposed theory is sensitive to UV physics, we note that mixing among neutrinos and the dark matter candidate could be originated by non-renormalizable operators only for a choice of n odd, whereas even and fractionally charges would be safe. The kinetic mixing between the B − L gauge boson and the hypercharge gauge boson is neglected for simplicity. A similar model has been partially investigated before in Ref. [16], where the main emphasis was the study of the gamma lines from dark matter annihilation. Our main goal here is to investigate in detail the connection between the cosmological bounds and the lepton number violation scale, and understand the implications for the search for lepton number violation at colliders.
Higgs Sector: The Higgs sector of this theory is composed of the SM Higgs H ∼ (1, 2, 1/2, 0) and S BL ∼ (1, 1, 0, 2) and the scalar potential is given by where In this case the physical states are: where After symmetry breaking one finds that the mass for the B −L gauge boson is given by The dark matter candidate χ = χ L + χ R is a Dirac fermion with mass M χ . We focus on the case where the dark matter candidate is a Dirac fermion because in the other cases, scalar or Majorana fermion, the annihilation cross section through the B − L gauge boson is suppressed. This simple dark matter model has the following relevant parameters for the dark matter study: As we have mentioned above the properties of the DM candidate χ are very simple since it is a vector-like fermion, χ = χ L + χ R , and it has interactions only with the B − L gauge boson. The two body annihilation channels arē where the first one is the dominant channel when 2M χ < M Z BL , and the second channels are possible when • Relic Density In order to compute the relic density, we use the analytic approximation [17] where M Pl = 1.22 × 10 19 GeV is the Planck scale and  the function J(x f ) reads as where g * is the total number of effective relativistic degrees of freedom at the time of freeze-out. The thermally averaged annihilation cross section times velocity σv is a function of x = M χ /T , and is given by where K 1 (x) and K 2 (x) are the modified Bessel functions. The freeze-out parameter x f can be computed using where g is the number of degrees of freedom of the dark matter particle.
In Fig. 2, we show the numerical results for the branch- However, as we will discuss later, perturbativity bounds constrain this channel in such a way that the annihilation into fermions will always dominate over annihilation into two gauge bosons, regardless of the choice of the B − L charge of the dark matter candidate. Furthermore, as we will see, it does not make sense to consider larger values of n because the direct detection bounds are much stronger and one will only find consistent solutions when the gauge boson is very heavy.
In Fig. 3, we show the predictions for the relic density when M Z BL = 3.5 TeV, g BL = 0.5 and different B − L charges for the dark matter candidate. The charges n = 1/3, n = 2/3, and n = 2 are represented by a solid and dashed lines, respectively. One can see that when one has large values of the dark matter B − L charge one can achieve the relic density in agreement with cosmology even if we are far from the resonance M Z BL ≈ 2M χ because in this case the annihilation into two gauge bosons has a larger contribution. In these studies we consider only the main annihilation channels in the numerical studies, and we will focus on n = 1/3, and n = 2 as illustrative examples.

• Direct Detection
The elastic spin-independent nucleon-dark matter cross section is given by where M N is the nucleon mass. We note that σ SI χN is independent of the matrix elements. The cross section can be rewritten as for a given value of n. In Fig. 4, we show the predictions for the direct detection cross-section σ SI χN for points with n = 1/3 satisfying the relic density and LEP bounds. We show the Xenon-1T [20,21] and Xenon-nT [22] bounds to understand the available region of the parameter space which is still in agreement with the direct detection and the expected region which could be tested in the near future. The different colored points correspond to the predictions when we use different values for the gauge coupling, g BL = 0 − 0.25 (blue), g BL = 0.25 − 0.5 (gold), g BL = 0.5 − 0.75 (green) and g BL = 0.75−1.0 (orange). Clearly, in these scenarios the dark matter mass should be above 1 TeV to be in agreement with the direct detection bounds. In this model we can have two gamma lines from dark matter annihilation,χχ → Zγ, andχχ → hγ. However, due to the fact that the cross section for the final state radiation processesχχ →f f γ are much larger, one cannot identify the gamma line from the continuum spectrum. The numerical results for these gamma lines were studied in Ref. [16]. In Fig. 5, we show the allowed parameter space for thermal averaged dark matter annihilation cross section into two bottom quarks (upper-panel), and two tau leptons (lowerpanel), compatible with the relic density constraint. The gray shaded area shows the parameter space excluded by the experimental bounds from FermiLAT [23]. As one can see, for choices of low n, the allowed parameter space is compatible with these bounds.

• Upper bound on the Symmetry Breaking Scale
In Figs. 6, and 7, we show the allowed region in the M Z BL − M χ plane when Ω DM h 2 ≤ 0.1199 ± 0.027, in agreement with the LEP bounds. In Fig. 6, we show the allowed solutions when the dark matter B − L charge is 1/3, and the allowed region for n = 2 is shown in Fig. 7.
As we can see, the maximum allowed value for M Z BL is around 25 TeV when n = 1/3, while the upper bound on M Z BL when n = 2 is around 130 TeV. Clearly, each Here we use the perturbative bound gBL < 2 √ π.
choice of n, and g BL define a theory which is bounded from above. However, regardless of the value of the B − L charge, and the choice of g BL , we note that there is an absolute upper bound at the multi-TeV scale for the seesaw scale. This statement may not be trivially seen in Fig. 6 and 7, because it seems that the larger the B−L charge, and the coupling g BL , the larger the upper bound on the B − L breaking scale. However, the coupling g BL is bounded by perturbativity, and in the limit of large n, the annihilation into two fermions, which defines the upper bound on the resonance, becomes insensitive to the B − L charge for large values thereof. We note that the annihilation channel into to new gauge bosons is irrelevant for defining the upper bound since it is bounded by perturbativity of the B − L coupling. These results are crucial to understand the testability of this theory at colliders. Clearly, we could test at the Large Hadron Collider only one fraction of the parameter space.
For completeness of the discussion on the upper bound for the seesaw scale, we would like to mention that the upper bound coming from the cosmological bound on the relic density is in agreement with partial-wave unitarity of the S-matrix. It is well known that, from a naive model-independent study, partial wave unitarity requires that M χ < 340 TeV [24]. However, in this model, the partial wave expansion only becomes relevant in regions of the parameter space which are not allowed by cosmology. Therefore, unitarity of the Smatrix does not make any influence on the upper bound for the seesaw scale.
The main implication of these results is that there is a hope to test the existence of lepton number violation since the upper bound on the B − L breaking scale is much smaller than the canonical seesaw scale. Here we use the perturbative bound gBL < √ π.

LEPTON NUMBER VIOLATION AT THE LHC
In the previous section we have shown the possibility to find an upper bound on the B − L symmetry breaking scale using the constraints from the dark matter relic density. Therefore, one can hope to test the existence of a new force associated to B − L, and observe lepton number violation at the LHC. Unfortunately, the upper bound is large if n = 2, but still we can hope to test this theory if the symmetry is broken much below the upper bound.
The right-handed neutrinos can be produced at the LHC through the neutral gauge boson Z BL , i.e. pp → Z * BL → N i N i or through the Higgses present in the theory [7][8][9][10][11]. Since the production mechanisms through the Higgses suffer from the dependence on the mixing angle in the Higgs sector, we focus our discussion on the production through the B − L gauge boson. The right-handed neutrinos could have the following decays: Therefore, the lepton number violating signatures can be observed when the right-handed neutrinos decay into charged leptons, and one has the following channels with the samesign leptons The number of these events is given by where L is the integrated luminosity. Here, the factor two is included to discuss the channels with same-sign leptons without distinguishing the electric charge of the leptons in the final state.
In Fig. 8 (upper-panel), we show the predictions for the cross section when √ s = 13 TeV, and for different scenarios consistent with relic density bounds. Here we choose n = 1/3, and numerical values for M χ , M Z BL , and g BL which satisfy the cosmological bounds. In the lower-panel, we show the number of events assuming a luminosity L = 50 fb −1 .
We have reviewed the current LHC bounds and unfortunately they cannot exclude the region of the parameter space studied here. A more detailed experimental study will help to understand the testability of this theory with more luminosity. Similar results are shown in Fig. 9, when √ s = 14 TeV. However, in this case one expects a large number of events for the same-sign leptons for right-handed neutrino masses below 1 TeV.
In Fig. 10, we show the predictions for the next generation of proton-proton collider at 100 TeV. In this case one should be able to test most of the parameter space even when the right-handed neutrinos are in the multi-TeV region. See Ref. [25] for the discovery potential of the 100 TeV collider. We would like to mention that, in a large part of the parameter space, the right handed neutrino decays can give rise to displaced vertices [11]. Since the seesaw scale has to be at most at the multi-TeV scale ∼ O(10 2 TeV), the Yukawa Dirac Y ν has to be small in order to reproduce the measured active neutrino masses. A small Y ν enhances the lifetime of the right-handed neutrinos, which then become long-lived particles. Therefore, as a consequence of having a low seesaw scale, displaced vertices arise as an exotic signatures predicted by the model. For instance, when the right-handed neutrino mass is about 400 GeV, the decay length can be L = (10 −3 − 10 −1 )mm.
This simple study motivates the search of lepton number violating signatures at the LHC or at future colliders.

FINAL DISCUSSION
We have discussed the simple seesaw mechanism for neutrino masses where the seesaw scale is defined by the B − L symmetry breaking scale and the scenarios where it is possible to find an upper bound on the seesaw scale using theoretical arguments or cosmological bounds. We have investigated the relation between the origin of neutrino masses and the properties of a simple cold dark matter candidate in the context of a theory based on the B − L gauge symmetry. In this context the upper bound on the seesaw scale is in the multi-TeV region and one predicts the existence of exotic lepton number violating signatures at colliders. We use all cosmological constraints and investigate The parton level cross section for this process is where t = (p q − p N ) 2 and the color factor for quarks is N c = 3. The integrated expression reads as