Absolute neutrino mass as the missing link to the dark sector

With the KATRIN experiment, the determination of the absolute neutrino mass scale down to cosmologically favored values has come into reach. We show that this measurement provides the missing link between the Standard Model and the dark sector in scotogenic models, where the suppression of the neutrino masses is economically explained by their only indirect coupling to the Higgs field. We determine the linear relation between the electron neutrino mass and the scalar coupling $\lambda_5$ associated with the dark neutral scalar mass splitting to be $\lambda_5=3.1\times10^{-9}\ m_{\nu_e}/$eV. This relation then induces correlations among the DM and new scalar masses and their Yukawa couplings. Together, KATRIN and future lepton flavor violation experiments can then probe the fermion DM parameter space, irrespective of the neutrino mass hierarchy and CP phase.


MS-TP-20-28
The identification of cold Dark Matter (DM) -a mysterious particle that according to most cosmological models is five times more abundant in the Universe than ordinary matter -is one of the most urgent challenges in modern physics. Neutrinos as only weakly interacting massive particles in the Standard Model (SM) have the right characteristics of a DM candidate, but are neither cold nor can they, due to their tiny masses, contribute more than a small fraction (between 0.5 and 1.6%) to the measured total DM relic density [1,2]. Nevertheless the idea that neutrinos and DM might be related is intriguing and has led to an enormous theoretical activity on so-called radiative seesaw models, where the suppression of the SM neutrino masses is due to their only indirect interaction (via DM) with the SM Higgs field [3][4][5].
While the fact that at least two of the three neutrino flavors are massive has been deduced about 20 years ago from atmospheric [6] and solar [7,8] neutrino oscillations, their absolute masses are still unknown. The KATRIN experiment has recently improved the upper limit on the electron (anti)neutrino mass to 1.1 eV [9] and ultimately aims at a sensitivity of 0.2 eV [10]. This value would rival the cosmological constraint on the sum of the SM neutrino masses of i m νi < 0.12 eV, assuming the ΛCDM model and normal hierarchy (NH), the minimal value allowed by oscillation experiments being 0.06 eV [11]. An inverted hierarchy (IH) is still a possiblity, although the long-baseline experiments T2K and NOνA and further evidence from reactor and atmospheric neutrinos favor NH. For the CP phase, T2K and to a lesser degree also NOνA data seem to favor 3π/2 (π/2) for NH (IH) [12].
In radiative seesaw models, the SM neutrinos ν, even under a discrete Z 2 symmetry, interact with the (also Z 2even) SM Higgs field φ and obtain their masses via a dark (Z 2 -odd) sector, which contains only a small number of new multiplets (typically up to four new scalar/fermion singlets, doublets or triplets under SU(2) L ) [4]. In Ma √ m ν /m N . In the inverse seesaw, ν − N mixing is even smaller, i.e. m ν /m 2 , but ν − S mixing is much larger, i.e. m 2 /m 1 , which is only bounded at present by about 0.03 [15]. In the double seesaw, the effective mass of N is m 2 In the original scotogenic model [7], neutrino mass is radiatively induced by heavy neutral Majorana singlet fermions N 1,2,3 as shown in Fig. 2. However, they may be replaced by Dirac fermions.
In that case, a U (1) D symmetry may be defined [16], under which η 1,2 transform oppositely. If Z 2 symmetry is retained, then a radiative inverse seesaw neutrino mass is also possible [17,18]. We discuss here instead the new mechanism of Fig. 1, based on the third one-loop realization of neutrino mass first presented in Ref. [2]. The smallness of m N , i.e. the Majorana mass of N L , may be naturally connected to the violation of lepton number by two units, as in the original inverse seesaw proposal using Eq. (1). It may also be a two-loop effect as first proposed in Ref. [19], with a number of subsequent papers by other authors, including Refs. [20][21][22].
In our model, lepton number is carried by (E 0 , E − ) L,R as well as N L . This means that the Yukawa term N L (E 0 m E comes from the invariant mass term (Ē 0 This means that the one-loop integral of Fig. 1 is well approximated by This expression is saw.
The radiative m a discrete flavor sy where I is given then the charged-l Higgs doublet of t discovery [23,24] o trino mass matrix, mass-squared mat where A and B by a symmetry fi µ − τ exchange in Refs. [27,28]. A mal ν µ − ν τ mixi exp(−iδ) = ±i, w scheme is thus a from Eq. (7), it is c degenerate neutrin Neutrino mass generation in scotogenic models.
famous scotogenic model (see Fig. 1), only one additional scalar doublet η and (for three massive SM neutrinos) three generations of fermion singlets N i (sterile neutrinos with i = 1, 2, 3) are required [3]. The parameter space is therefore much smaller than, e.g., the one of supersymmetry and can be better constrained with neutrino oscillation data via the Casas-Ibarra method [13], limits on lepton flavor violation (LFV) [14], and measurements of the DM relic density [15]. Nevertheless, these previous analyses found that the dark scalar/fermion masses as well as their scalar and Yukawa couplings could still vary over several orders of magnitude.
In this Letter, we demonstrate that a determination of the absolute electron neutrino mass, which has now come into reach, will provide additional stringent constraints on the dark sector of the scotogenic model in a way that is almost independent of the neutrino hierarchy and CP phase. In particular, we determine the linear relation between the absolute electron neutrino mass and the scalar coupling associated with the mass splitting of the dark neutral scalars. This linear dependence induces correlations among the other parameters of the model, i.e. the DM and scalar masses and their Yukawa couplings, which we can also quantify. Together, current neutrino mass and future LFV experiments can then probe almost the entire fermion DM parameter space. In the original scotogenic model, the SM is enlarged with a dark sector containing only two new types of fields, a complex Higgs doublet η and three generations of fermion singlets N i [3]. In the Lagrangian of this model we define fermions in terms of Weyl spinors and denote the three generations of left-handed SM lepton doublets with L α (α = 1, 2, 3). The fermion singlet with the smallest mass m Ni is assumed to be the DM candidate. SM neutrinos ν couple to the SM Higgs field φ and obtain their mass only at one loop (see Fig. 1) via the 3 × 3 matrices of Yukawa couplings y iα and the scalar potential The parameters m φ and λ 1 are fixed by the known SM Higgs vacuum expectation value (VEV) φ 0 = 246 GeV/ √ 2 [1] and the LHC measurement of the (squared) SM Higgs boson mass m 2 h = 2λ 1 φ 0 2 = −2m 2 φ = (125 GeV) 2 [16]. To ensure that the scalar potential is bounded from below and the vacuum is stable, we must have while perturbativity imposes |λ 2,3,4,5 | < 4π. The inert doublet η does not acquire a VEV, so that λ 2 induces only self-interactions and decouples from the phenomenology. We set λ 2 = 0.5 without loss of generality. The masses of the charged scalar component η + and the real and imaginary parts of the neutral component η 0 = (η R + iη I )/ √ 2 are then Note that it is natural for λ 5 and the mass difference m 2 R − m 2 I = 2λ 5 φ 0 2 to be small, since if λ 5 was exactly zero, it would induce a conserved lepton number and massless neutrinos [17]. Following previous work [15], we scan over the range 10 −12 < |λ 5 | < 10 −8 . For vanishing λ 3 , the LEP limit on charged particles [18] implies a lower limit on the scalar mass range m η ∈ [0.1; 10] TeV, which we also employ for the sterile neutrino masses m Ni . As m 2 η dominates over φ 0 2 in much of the parameter space, the scalar couplings λ 3 and λ 4 will play a subdominant role, and η + will be close in mass to both η R and η I .
The branching ratios (BRs) and conversion rates (CRs) depend on the charged scalar and sterile neutrino masses and their Yukawa couplings through dipole/non-dipole form factors and box diagrams [14] and are calculated with SPheno 4.0.3 [26]. We also restrict the DM relic density with mi-crOMEGAs 5.0.8 [27] to the central value Ωh 2 = 0.12 measured by Planck [11], allowing for a theoretical uncertainty of 0.02 [28,29]. In the standard freeze-out scenario, the relic density results from DM annihilation processes in the early Universe, here of the lightest sterile neutrinos into leptonic final states via charged and neutral scalars in the t-channel. Coannihilation processes, which may occur in fine-tuned scenarios with nearly mass-degenerate scalars and fermions [30], are required to contribute less than 1%. Direct DM detection is theoretically possible at one loop, but is currently beyond the experimental reach [31].

NUMERICAL RESULTS
For electron neutrino masses of 1.1 eV to 0.2 eV as currently explored by KATRIN [9,10], i.e. larger than the minimal i m νi > 0.06 eV, but approaching the cosmological upper limit of 0.12 eV [11], the mass differences, PMNS matrix U and rotation angles θ i play a subdominant role, and the eigenvalues of the Yukawas matrices y α take similar values. This is demonstrated in Fig. 2 (grey points), where the ratio |y 2 /y 1 | varies over its full range at low m ν1 , but only by about a factor of two at large m ν1 . In addition, the LFV processes l α → l β γ and l α → 3l β impose upper limits on both y α and y β , limiting their ratio further (blue). Conversely, to obtain the correct DM relic density, the Yukawas must not be too small (green), so that the combination of all constraints leads indeed to |y 2 /y 1 | ∼ 1 (red points). We have verified that this result is independent of the neutrino mass hierarchy and holds also for the ratios |y 3 /y 1 | and |y 3 /y 2 |.
The linear dependence of the neutrino mass matrix (m ν ) αβ on the dark sector-Higgs boson coupling |λ 5 | in Eq. (8) can then be made explicit by studying the dependence of |λ 5 | on the lightest eigenvalue m ν1 . It emerges in Fig. 3 after imposing LFV (blue), relic density (green) and all (red points) constraints and can be fitted at 90% C.L. to becomes independent of m ν1 . The dark sector-Higgs boson coupling λ 5 can therefore be predicted, once the absolute neutrino mass scale is known. Furthermore, with m ν1 /|λ 5 | fixed, the Yukawas in Eq. (8) become correlated with the DM and scalar masses. Since the ratio of the latter is in addition constrained by the relic density (m R,I /m N1 ∼ 1.5), the leading term in Eq. (8) becomes proportional to |y 1 | 2 /m N1 , which allows us to fit this dependence in Fig. 4 at 90% C.L. as   the other fermions N 2,3 being significantly heavier. This implies, that if the DM mass is known, we can predict its Yukawa coupling to the SM leptons. Our findings imply that the fermion DM parameter space of the scotogenic model can be almost completely tested with LFV experiments and a measurement or limit on the direct neutrino mass, as can be seen from Fig. 5. The current limit on BR(µ → eγ) (blue) [20] imposes stronger bounds than the one for BR(µ → 3e) (red) [22], but this is expected to change soon [21,23]. Independently of the neutrino mass hierarchy, the models that survive even these future tight constraints can be probed in an orthogonal way by new limits or measurements of the absolute neutrino mass, if they reach indeed the region of cosmologically favored values [9,10].

SUMMARY AND OUTLOOK
As we have demonstrated in this Letter, the parameter space of fermion DM in Ma's scotogenic model is now severely constrained. In particular, an electron neutrino mass measurement would allow to directly predict the dark sector-Higgs boson coupling λ 5 and to test the complete parameter space of the model in an orthogonal way to LFV, while a DM mass measurement would result in a prediction for its Yukawa coupling to the SM leptons. This is due to the strong mutual constraints inherent in the one-loop diagram (Fig. 1) for neutrino mass generation, which is topologically similar to a penguin diagram mediating LFV and (when cut on the internal fermion line) to DM annihilation. The correlations are absent for scalar DM, i.e. when the diagram is cut on the internal scalar lines, since the scalar doublets can annihilate into weak gauge bosons. The case of scalar-fermion coannihilation has been studied elsewhere [30].
Our observations generalize to other scotogenic models such as those with triplet fermions [32] and/or singletdoublet scalars [33], where the neutrino mass matrices take forms similar to Eq. (8). However, since the neutral components of electroweak triplets can annihilate into gauge bosons, the Yukawas and LFV processes are generally smaller, so that collider constraints must also be considered [34,35].