Bound-state dark matter with Majorana neutrinos

We propose a simple scenario in which dark matter (DM) emerges as a stable neutral hadronic thermal relics, its stability following from an exact $\operatorname{U}(1)_D$ symmetry. Neutrinos pick up radiatively induced Majorana masses from the exchange of colored DM constituents. There is a common origin for both dark matter and neutrino mass, with a lower bound for neutrinoless double beta decay. Direct DM searches at nuclear recoil experiments will test the proposal, which may also lead to other phenomenological signals at future hadron collider and lepton flavour violation experiments.

I. The heavy messenger Q is charged under U(1) D and can act as constituent dark matter, made stable thanks to a U(1) D symmetry.  Note that all dark sector particles carry colour. For definiteness, we assign them to the octet SU(3) c representation. This ensures the viability of the bound-state DM scenario [1]. Notice, however, that our neutrino mass discussion also holds if they had different SU(3) c transformation properties. The U(1) D symmetry implies that Q has only a Dirac-type mass term. Hence the new terms in the Lagragian are the following where summation is implied over repeated indices, and trace over N c = 8 is implicit. The Higgs potential contains the following terms Note that terms like η † a H 2 are also forbidden and, as a consequence, the real and imaginary parts of the scalars with nonzero dark charges are degenerate. Moreover, the CP-even and CP-odd scalars do not mix, thanks to our CP conservation assumption.
The Higgs boson is the same as that in the standard model. The neutral scalar components η 0 a of the dark charge carrying scalars η a , with cuadratic mass coefficients µ a in Fig. 1, mix into two complex mass eigenstates S a . Since the tree-level mass term is forbidden by symmetry, neutrino masses are calculable at one-loop order, by the Feynman diagram displayed in Fig. 1. One finds the following effective mass matrix ) and the SU(3) c color factor N c is assumed to be 8, since the new particles running in the loop transform as octets. Note that the effective one-loop induced neutrino mass matrix has in general rank two, implying two non-vanishing neutrino masses (similar general structure occurs also in non-colored models [12][13][14]), as required to account for neutrino oscillation data.
As a simple numerical estimate, assuming µ 2 η1 M 2 Q , we consider the case µ 2 η1 = µ 2 η2 λ η1η2H v 2 and λ 3ηH , λ 4ηH 1. Taking m 2 S2 − m 2 S1 = λ η1η2H v 2 and m 2 Small neutrino masses are protected by the small parameter λ η1η2H in whose absence the theory acquires a larger symmetry. As a result, adequate neutrino masses arise naturally for reasonable values of the Yukawa couplings and scalar potential parameters. Due to the flavor structure of the neutrino mass matrix (Eq. (3)), one can express five of the six Yukawa-couplings h i and y i in terms of the neutrino observables [15]. Choosing h 1 as a free parameter we have where Note that, in the diagonalization condition U T M ν U = diag(m 1 , m 2 , m 3 ) we have used the PDG form [16] of the lepton mixing matrix [17], namely U = V P and P = diag(1, e iα/2 , 1).

III. STABILITY OF BOUND-STATE DARK MATTER AND NEUTRINO MASSES
As suggested in [18] the radiative nature of neutrino mass generation is used to ensure dark matter stability. Moreover, this idea is combined with the proposal that dark matter emerges in the form of stable neutral hadronic thermal relics, as a neutral bound-state of colored constituents, such as QQ, where Q is a vector-like color octet isosinglet fermion (for a general discussion of the cosmology of a stable colored relic see [19]). A necessary and sufficient condition for dark matter stability in this case is the presence of a global U (1) D dark baryon number 1 , under which the Q is charged [1]. In our present model construction such symmetry also gives rise to radiative neutrino masses. In fact, both dark matter stability, and the Dirac nature of the exotic fermion Q are equivalent, resulting from dark charge conservation.
Note that, by construction, the new Yukawa interactions in Eq. (1) do not affect the stability of our bound state dark matter, since the colored scalars are heavier than Q, and QQ annihilation processes mediated by η are forbidden by the conserved dark symmetry. An adequate thermal relic density of bound-state dark matter requires the lightest constituent vector-like color octet Dirac fermion, Q, to have a mass ≈ 12.5 TeV, so that the QQ hadron weighs approximately 25 TeV [1].
Bound-state dark matter made up by our Dirac octets will be seen in direct searches for nuclear recoil at underground dark matter experiments. The relevant spin-independent cross-section is given as and depends rather sensitively on the dark matter mass M QQ = 2M Q , as shown in the red line in Fig. 2. Note that the star in the figure assumes that the bound-state DM makes up 100% of the cosmological dark matter. If an additional dark matter component is present, e.g. made of axions, then bound-state dark matter masses below 25 TeV become allowed, as indicated by the red line. In this case their spin-independent cross section would be larger, though their share in the relic density will be lower. The blue line denotes the current Xenon1T limit after 1.0 t×yr exposure [22]. This should be compared with the future sensitivities expected at XENON1T [23], LZ [24] and DARWIN [25] indicated by the black (dashed, and dot-dashed) lines. We also note that, within the standard thermal cosmological scenario, DM masses above 25 TeV are ruled out by current observations of the Planck collaboration [26] (gray band). Notice also that the current LHC limit of 2 TeV [27,28], implies that the cross section is always small enough so as to have the bound-state dark matter candidate reaching underground detectors.

IV. NEUTRINOLESS DOUBLE BETA DECAY
In contrast to the proposal in Ref. [8], here total lepton number is a broken symmetry, hence we expect neutrinoless double beta decay to occur. The effective mass parameter characterizing the amplitude for neutrinoless double beta decay is given by [29] m ee = j U 2 ej m j = c 2 12 c 2 13 m 1 + s 2 12 c 2 13 m 2 e 2iφ12 + s 2 13 m 3 e 2iφ13 , where m i are the neutrino masses, c 12 and s 13 correspond to the angles measured from oscillations and φ 12 , φ 13 are the Majorana phases (here we use the symmetrical parametrization of the lepton mixing matrix [17]).
Since our model predicts the lightest neutrino to be massless, m 1 = 0, it follows that there is one single physical (relative) Majorana phase φ ≡ φ 12 − φ 13 . Furthermore, one can also write the three physical masses directly in terms of the squared mass splittings measured in neutrino oscillation experiments. Depending on the ordering these masses read NO : m 2 = ∆m 2 21 , m 3 = ∆m 2 31 , IO : m 1 = ∆m 2 13 , m 2 = ∆m 2 13 + ∆m 2 12 . (yellow band) and inverted (green band) mass orderings, varying the neutrino oscillation parameters within 3σ [10,11] of their best fit values. One sees that, in contrast to the general three-neutrino scenario, here the 0νββ amplitude never vanishes, even when the neutrino mass ordering is of the normal type (models with this feature typically require the existence of specific flavor symmetries [30][31][32]). The top four horizontal bands represent the 90% C.L. upper limits from CUORE [33], EXO-200 Phase II [34], GERDA Phase II [35] and KamLAND-Zen [36] experiments. The sensitivity bands for the upcoming nEXO experiment after 10 years of data taking [37] as well as for the SNO+ Phase II [38] and LEGEND [39] experiments are indicated by the horizontal red bands.

V. COUPLING EVOLUTION
Given that we have colored multiplets it is interesting to illustrate their effect in the running of coupling constants. At one loop level the evolution of gauge couplings is governed by where the b i coefficients are determined by For our model, they are given as where n η is the number of colored scalar fields η ∼ (8, 2, 1/2). One can easily see that asymptotic freedom can be lost due to the presence of the octets. However, in our case n η = 2, hence no Landau pole appears up to the Planck scale, as long as the scalars are heavier than the fermion octets Q ∼ (8, 1, 0, 1). This is illustrated in Fig. 4. Notice that the situation displayed in Fig. 4 seems to threaten the idea of unification of couplings. This, however, is not the full story. In a Grand Unified Theory (GUT) all the fields come in representations of the GUT group. Complete representations of the GUT group do not spoil unification, but change the value of the coupling constant at M GU T . A particular GUT embedding of the color octets Q ∼ (8, 1, 0), however, lies beyond the scope of this work.

VI. PHENOMENOLOGY
The cosmological relic abundance requires M Q ≈ 12.5 TeV, which lies beyond the kinematical reach of the LHC. A future hadron collider with a center of mass energy of 100 TeV, would probe masses up to M Q 15 TeV [1,40]. In contrast with bound-state dark matter with radiative Dirac neutrino masses induced by color octets circulating the loop [8], in the Majorana case the color octet scalars can lie at the same scale as the color octet fermion. In fact, if the lightest color octet scalar, η 1 , transforming as a weak doublet, has a mass close to that of Q, they could be pair-produced with similar cross sections 2 . The pair produced scalars further decay into a QQ pair and charged leptons or neutrinos. This gives rise to similar signals with long-lived bound states but with extra charged leptons or missing transverse energy.
Note also that our bound-state dark matter constituents can mediate charged lepton flavor violation effects, whose rates will depend on the scalar masses and may reach detectability levels.

VII. SUMMARY AND OUTLOOK
We have proposed a simple and viable theory in which dark matter emerges as a stable neutral hadronic thermal relics, whose stability results from an exact U(1) D symmetry. Neutrinos pick up radiatively induced Majorana masses from the exchange of colored dark matter constituents, giving a common origin for both dark matter and neutrino mass, with a lower bound for neutrinoless double beta decay and direct tests at direct DM searches at nuclear recoil experiments. The scheme provides a consistent ultraviolet complete setup, free of Landau poles all the way up to the Planck scale, provided the scalars are heavy enough. The new states may also lead to other phenomenological signals at future hadron collider as well as lepton flavour violation experiments.