Dark matter origins of neutrino masses
DDark matter origins of neutrino masses
Wei-Chih Huang ∗ and Frank F. Deppisch † Department of Physics and Astronomy, University College London,London WC1E 6BT, United Kingdom
We propose a simple scenario that directly connects the dark matter (DM) and neutrino massscales. Based on an interaction between the DM particle χ and the neutrino ν of the form χχνν/ Λ ,the DM annihilation cross section into the neutrino is determined and a neutrino mass is radiativelyinduced. Using the observed neutrino mass scale and the DM relic density, the DM mass andthe effective scale Λ are found to be of the order MeV and GeV, respectively. We construct anultraviolet-complete toy model based on the inverse seesaw mechanism which realizes this potentialconnection between DM and neutrino physics. INTRODUCTION
The Standard Model (SM) is not able to explain theexistence of Dark Matter (DM) in the universe as well asthe finite masses of neutrinos. Experimentally, both phe-nomena are firmly established. The two neutrino mass-squared differences are very well measured in neutrino os-cillation experiments [1]. Together with the upper limiton the sum of the neutrino masses, (cid:80) m ν (cid:46) .
66 eV,derived from cosmological observations [2], they implythat the heaviest active neutrino has a mass of 0.05 to0.22 eV. While the DM mass is largely unconstrained,the crucially important DM relic abundance is very wellmeasured at Ω h = 0 .
12 [2].Connections between DM physics and the origin andsize of the neutrino masses have been proposed in theliterature in the context of radiative neutrino mass mod-els, for example in Refs. [3–5], where the neutrino massis induced radiatively with DM particles and heavy neu-trinos in the loop. In these models, the neutrino massscale depends on the DM and heavy neutrino masses aswell as various coupling constants. This implies that theDM mass can not be uniquely determined given the ob-servations, unless other model parameters are fixed. Analternative scenario was proposed in Refs. [6, 7]. Simi-lar to our case, it connects neutrino physics with an MeVscale DM particle, although the underlying model is quitedifferent.In this work, we propose a simple scenario that con-nects the DM particle and neutrino mass scales. Startingwith an effective 6-dimensional operator χχνν Λ , (1)where χ refers to a gauge singlet Majorana DM parti-cle, while ν is the SM neutrino . Here and in the fol-lowing, we use the two-component Weyl spinor nota-tion for all fermionic fields. We implicitly assume that We neglect the flavour structure of the three neutrinos and workwith one Majorana neutrino field with mass scale m ν ≈ . χ is odd under a Z symmetry to ensure its stability.Assuming that this operator is the only one couplingDM to SM particles, the DM annihilation cross sectiontimes the DM relative velocity v rel is approximated by σv rel ≈ m χ / ( π Λ ). This implies a DM relic abundanceof Ω h ≈ . × − GeV − / ( σv rel ). On the other hand,the neutrino receives a radiative mass by contracting two χ fields in the interaction operator, m ν ≈ m χ / ( π Λ ).Using the experimental data on the DM relic abundanceand the light neutrino mass scale, the DM mass m χ andthe scale Λ of the interaction operator can be determinedeasily, m χ ≈ . (cid:16) m ν . (cid:17) / (cid:18) Ω h . (cid:19) / , (2)Λ ≈ . (cid:16) m ν . (cid:17) / (cid:18) Ω h . (cid:19) / . (3)Naturally, m χ and Λ are of the order MeV and GeV, re-spectively. The effective operator scale Λ is far below theelectroweak (EW) scale, which is why the operator is notinvariant under the SM gauge group. It also naturallyimplies the existence of at least one more particle lighterthan the EW scale in order to obtain the interaction op-erator χχνν/ Λ . We proceed by constructing a possiblemodel that realizes the previous operator in two steps:firstly by discussing an effective Lagrangian, and then apossible fully ultraviolet (UV)-complete toy model. EFFECTIVE LAGRANGIAN
The natural scale of the operator (1) is GeV and inorder to discuss a possible SM effective model we haveto introduce another light particle that connects the DMsector with the SM. In addition, we assume that the onlysource of lepton number violation (LNV), that generatesthe DM Majorana mass, is situated in the hidden sec-tor. We do not specify this source of LNV but it couldfor example result from a seesaw-like mechanism in thehidden sector. Note that one has to make sure that inthe UV-complete theory, the hidden sector does not cou-ple to the SM directly, i.e. it has to go through the DM a r X i v : . [ h e p - ph ] M a y χχ νν −→ p ×× χχ νν −→ p FIG. 1: Loop diagrams generating a Majorana neutrino mass.The arrow represents chirality; if the arrow direction is thesame as that of the momentum, it represents the left-handedchirality. One of the contributions (right panel) involves achirality flip. particle χ . Therefore, any other effective operators haveto conserve lepton number which for example forbids theWeinberg operator LHLH .We introduce a complex scalar Φ with two units oflepton number, L (Φ) = 2 which connects the DM andSM sectors, L ⊃ c Φ χχ + Φ ∗ LHLH Λ ∗ + h.c.. (4)Here, L and H are the SM lepton and Higgs boson dou-blets, respectively. Choosing L ( χ ) = −
1, the Lagrangian(4) conserves lepton number. After integrating out Φ andEW breaking, H = (0 , v ) T , one obtains L ⊃ χχνν Λ + h.c., (5)where Λ = Λ ∗ m Φ / ( √ c v ).Due to the Majorana nature of the DM particle χ ,the light neutrino ν will obtain a loop-induced Majoranamass as shown in Fig. 1. Treating Λ as a dimensionfulcoupling constant instead of a cut-off scale, the neutrinomass becomes m ν = m χ π Λ (cid:18) m χ µ − (cid:19) , (6)using the dimensional regularization scheme with mod-ified minimal subtraction , renormalized at the scale µ .We take µ to be the neutrino mass m ν , with the incomingmomentum p set to zero.On the other hand, the relic abundance of χ is deter-mined by the same effective operator . The DM annihi-lation cross section reads, up to v , σv rel = m χ π Λ (cid:18) v (cid:19) . (7) As we shall see later, this is well justified in the UV-completemodel which has exactly same loop structure. One has to include the contribution from χ † χ † ν † ν † / Λ , whichinvolve a different chirality. The interference between the twodifferent chirality contributions is tiny, being proportional to thevery small neutrino mass m ν . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:76) (cid:72) GeV (cid:76) m Χ (cid:72) G e V (cid:76) (cid:87) h (cid:61) (cid:87) h (cid:62) (cid:87) h (cid:60) m Ν (cid:61) m Ν (cid:61) FIG. 2: Neutrino mass m ν and DM relic abundance Ω h as afunction of the DM mass m χ and the effective DM-neutrinointeraction scale Λ. The red curve corresponds to the cor-rect DM density, while the blue (purple) line refers to theupper (lower) limit on the heaviest active neutrino mass, asdenoted. At the intersection, m χ ≈ . ≈ We base the computation of the DM relic density onthe thermally averaged annihilation cross section (cid:104) σv rel (cid:105) ,where we include the fact that the number of relativis-tic degrees of freedom is much smaller for MeV DM asopposed to GeV DM, as described in Ref. [8]Given the observed neutrino mass scale and the DMrelic density, the DM mass m χ is around the sub-MeVscale while Λ ≈ m χ , Λ and the observables m ν , Ω h is shown inFig. 2. The red curve denotes the observed relic abun-dance Ω h = 0 .
12 while the blue (purple) line corre-sponds to the upper (lower) limit on the heaviest activeneutrino mass. The fact that Λ is much smaller than theEW scale justifies the explicit EW symmetry breaking ofEq. (5) and it implies the existence of the light particleΦ in this scenario.
UV-COMPLETE TOY MODEL
As a final step, we construct a UV-complete toy modelthat in turn generates the effective Lagrangian and thelow energy DM-neutrino interaction, as shown in Fig. 3.The corresponding Lagrangian reads
L ⊃ c χ + (cid:104) Φ χ (cid:105) ) χχ + c Φ χχ + c Φ ∗ ξξ + yLHN − m Φ χ Φ χ Φ ∗ χ − m Φ ΦΦ ∗ − m N N ξ + h.c., (8)where Φ χ and Φ are scalar fields with lepton number L = 2, N and ξ are heavy Dirac neutrinos with op-posite L . The vacuum expectation value (VEV) (cid:104) Φ χ (cid:105) Field
L H N χ ξ Φ χ Φ[ SU (2) L ] Y − / / L Z + + + – + + +TABLE I: Particle content and corresponding quantum num-bers in the toy model. of Φ χ generates the DM mass m χ = c (cid:104) Φ χ (cid:105) . In prin-ciple, Φ χ could be very heavy compared to (cid:104) Φ χ (cid:105) . Forinstance, Φ χ may couple to another scalar φ such that (cid:104) Φ χ (cid:105) = (cid:104) φ (cid:105) /m Φ χ (cid:28) (cid:104) φ (cid:105) , m Φ χ , similar to the type-II see-saw mechanism. In other words, the small VEV (cid:104) Φ χ (cid:105) canin this case be triggered by the VEV of φ , suppressed bythe heavy Φ χ mass, m Φ χ . Moreover, the massless Ma-joron from φ could be removed by gauging B − L . Thequantum numbers of the various fields are listed in Ta-ble I. The Z symmetry is imposed to guarantee the sta-bility of DM and forbid the mixing between DM and theSM neutrino. Lepton number is spontaneously brokenafter Φ χ obtains a VEV, giving a Majorana mass to χ .Moreover, χ induces the mixing between Φ χ and Φ suchthat LNV is transferred to the heavy neutrino N andfinally to ν via the heavy-light neutrino mixing. Inte-grating out the heavy particles, we obtain the effectiveDM-neutrino interaction χχνν/ Λ , withΛ = 1 √ c c m N m Φ yv . (9)Alternatively, the χ -induced mixing between Φ χ and Φgives rise to a linear term in Φ once Φ χ acquires a VEV.This linear term will induce a small VEV of Φ as (cid:104) Φ (cid:105) = c χχ/m Φ , (10)where χχ represents the χ -loop of mass dimension three.It in turn gives a small Majorana mass term to ξ , c (cid:104) Φ (cid:105) ξξ .The full neutrino mass matrix in the basis ( ν , N , ξ ) reads yv yv m N m N c c χχm , (11)which is exactly the inverse seesaw [9]. The resultinglight neutrino mass will be m ν = 2 c c y v χχ/ ( m Φ m N ),which implies Eq. (9) from Eq. (5). In addition, for Λ ≈ m N m Φ ≈
100 GeV if all couplings are of O (1). This also means that m N is bounded from above, m N (cid:46)
100 TeV since m Φ is required to be larger than m χ ,otherwise Φ can not be regarded to be heavy to generatethe effective operator χχνν/ Λ . On the other hand, m N is also bounded from below m N (cid:38)
100 GeV for y = O (1)due to constraints from EW precision and flavor changingneutral currents data [10–12]. Φ χχ ννξ Nξ N ⟨ H ⟩⟨ H ⟩ ++ FIG. 3: Diagram generating the effective DM-neutrino inter-action in the UV-complete toy model. For illustration, weonly show the diagram with a right-handed χ . The UV-complete toy model satisfies two very impor-tant requirements necessary for this mechanism to work:Firstly, LNV arises in the hidden sector, and it is medi-ated to the SM sector (including right-handed neutrinos)only by the DM particle χ . Secondly, the heavy parti-cles that are being integrated do not enter the χ -loop,which radiatively induces the light neutrino mass. It isa distinctive feature of the model, setting it apart fromthe existing literature, for instance Refs. [13–15], whereheavy particles exist inside loops that give rise to radia-tive neutrino masses. It is this feature that renders ourmodel more predictive. CONCLUSIONS
In this letter, we propose a simple scenario that di-rectly connects the physics of DM and neutrino masses.The introduced operator χχνν/ Λ induces a radiativeMajorana neutrino mass as well as leads to the DM an-nihilation. Given the observed DM density and the neu-trino mass scale, the DM mass m χ and the operator scaleΛ are uniquely fixed to be of order MeV and GeV, re-spectively. In a UV-complete toy model, we postulatethe breaking of lepton number in a hidden sector that ismediated via a DM loop to the visible sector and thusto the light neutrinos, generating the effective Weinbergoperator. To our knowledge this has not been explored inthe literature but we find this possibility rather sugges-tive and intriguing; it would for example motivate whylepton number is only slightly broken in the visible sec-tor and the DM loop mediation is quite natural with thepresence of a Z symmetry to ensure the DM stability,only allowing the DM particle to interact in pairs.In our letter we focus purely on the relation betweenthe neutrino mass generation and the DM annihilation.As an outlook, we would like to comment on other po-tential signatures of the model. The DM annihilationcross section is S -wave dominated without velocity sup-pression. This implies a neutrino flux due to ongoing DMannihilation, for example, from the Galactic center. TheDM mass and hence the energy scale of the neutrino fluxis of order MeV, in the vicinity of the energy thresholdof neutrino experiments such as Super-Kamiokande [16],KamLAND [17, 18], SNO [19] and Borexino [20]. Weestimate the expected monochromatic neutrino flux asΦ ν ≈
300 (MeV /m χ ) cm − s − using the calculation ofRef. [21]. Such a flux would give rise to a few events foran exposure of a Mton · yr.The fact that the effective scale Λ is naturally of orderGeV implies the existence of light exotic states. Withregard to direct searches at colliders, it is difficult tomake a general statement without fully specifying a UV-complete model. For a TeV scale neutrino N , m Φ ≈ GeV.The scalar Φ only couples indirectly through N and ishardly constrained by collider searches. If Φ couples tothe SM Higgs via Φ ∗ Φ H ( H ), invisible Higgs decays with-out phase space suppression would be generated. Themass range of the heavy quasi-Dirac neutrino N (and ξ )is confined to be 100 GeV (cid:46) m N (cid:46)
100 TeV with a rela-tively large heavy-light neutrino mixing. Therefore, theheavy neutrino production cross section could be sizeableat the LHC.Finally, we would like to point out that the MeV scaleDM particle can contribute to the entropy of the universeduring the time of Big Bang Nucleosynthesis (BBN). Inour scenario, DM is still in thermal equilibrium with neu-trinos after they decouple from the thermal bath aroundthe temperature T = 2 . N ν during the time of last scattering producingthe cosmic microwave background (CMB). MeV scaleMajorana DM coupling to the neutrinos will result in N BBN ν = 4 and N CMB ν = 4 . . < N BBN ν < . N CMB ν = 3 . ± .
34 fromthe CMB data alone [2]. Potentially more severe isthe constraint from the determination of the primordialDeuterium abundance. The observationally determinedvalue, expressed relative to the Hydrogen abundance, is
D/H = (2 . ± . × − [1]. On the other hand, aMajorana DM particle with mass m χ ≈ . D/H ≈ × − [24]. Compatibility with ob-servation in this case requires a DM mass m χ (cid:38) N, ξ ) are needed. In this case, one can have a larger DM mass (cid:38) m ν small, such that the CMB and BBNconstraints do not apply. The corresponding neutrinomass with m χ (cid:38) m ν ≈ . χ mass, effectively turning χ intoa quasi-Dirac particle. This would increase the DM masswhile the Majorana neutrino mass, proportional to theMajorana DM mass component, could be kept constant.The two solutions, however, will modify the DM and neu-trino mass link but they could be implemented in a con-trollable way via the help of a symmetry such that theconnect between DM and neutrino physics still exists. Acknowledgments
The authors are especially thankful to John Ellis andJose Valle for useful comments, and Andr´e de Gouvˆeafor very detailed and useful comments on the draft. Wethank Pedro Schwaller for pointing out the tension withthe CMB measurement of the number of relativistic de-grees of freedom. W.-C. H. is grateful for the hospitalityof the CERN theory group and the AHEP group at IFIC,where part of this work was performed. This work issupported by the London Centre for Terauniverse Stud-ies (LCTS), using funding from the European ResearchCouncil via the Advanced Investigator Grant 267352. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] Particle Data Group, K. Olive et al., Chin.Phys.
C38 ,090001 (2014).[2] Planck Collaboration, P. Ade et al., Astron.Astrophys.(2014), 1303.5076.[3] L. M. Krauss, S. Nasri, and M. Trodden, Phys.Rev.
D67 ,085002 (2003), hep-ph/0210389.[4] E. Ma, Phys.Rev.
D73 , 077301 (2006), hep-ph/0601225.[5] F. Deppisch and W.-C. Huang, JHEP , 066 (2015),1411.2922.[6] C. Boehm, Y. Farzan, T. Hambye, S. Palomares-Ruiz,and S. Pascoli, Phys.Rev.
D77 , 043516 (2008), hep-ph/0612228.[7] Y. Farzan, Phys.Rev.
D80 , 073009 (2009), 0908.3729.[8] K. Griest and D. Seckel, Phys.Rev.
D43 , 3191 (1991).[9] R. Mohapatra and J. Valle, Phys.Rev.
D34 , 1642 (1986).[10] F. del Aguila, J. de Blas, and M. Perez-Victoria,Phys.Rev.
D78 , 013010 (2008), 0803.4008.[11] A. Atre, T. Han, S. Pascoli, and B. Zhang, JHEP ,030 (2009), 0901.3589.[12] F. F. Deppisch, P. S. B. Dev, and A. Pilaftsis, (2015),1502.06541.[13] K. Babu and C. N. Leung, Nucl.Phys.
B619 , 667 (2001),hep-ph/0106054. [14] A. de Gouvea and J. Jenkins, Phys.Rev.
D77 , 013008(2008), 0708.1344.[15] P. W. Angel, N. L. Rodd, and R. R. Volkas, Phys.Rev.
D87 , 073007 (2013), 1212.6111.[16] Super-Kamiokande Collaboration, A. Renshaw, (2014),1403.4575.[17] KamLAND Collaboration, S. Abe et al., Phys.Rev.
C84 ,035804 (2011), 1106.0861.[18] KamLAND Collaboration, A. Gando et al., (2014),1405.6190.[19] SNO Collaboration, B. Aharmim et al., Phys.Rev.
C81 ,055504 (2010), 0910.2984. [20] Borexino Collaboration, G. Bellini et al., Phys.Rev.
D89 ,112007 (2014), 1308.0443.[21] S. Palomares-Ruiz and S. Pascoli, Phys.Rev.
D77 ,025025 (2008), 0710.5420.[22] K. Enqvist, K. Kainulainen, and V. Semikoz, Nucl.Phys.
B374 , 392 (1992).[23] C. Boehm, M. J. Dolan, and C. McCabe, JCAP ,027 (2012), 1207.0497.[24] K. M. Nollett and G. Steigman, (2014), 1411.6005.[25] R. H. Cyburt, B. D. Fields, K. A. Olive, and E. Skillman,Astropart.Phys.23