Physics Letters B 726 (2013) 744–746

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Physics Letters B

www.elsevier.com/locate/physletb

New scotogenic model of neutrino mass with U (1)D gauge interaction

Ernest Ma a, Ivica Picek b,∗, Branimir Radovcic b

a Department of Physics and Astronomy, University of California, Riverside, CA 92521, USAb Department of Physics, Faculty of Science, University of Zagreb, P.O.B. 331, HR-10002 Zagreb, Croatia

a r t i c l e i n f o a b s t r a c t

Article history:Received 31 August 2013Accepted 24 September 2013Available online 30 September 2013Editor: J. Hisano

We propose a new realization of the one-loop radiative model of neutrino mass generated by dark matter(scotogenic), where the particles in the loop have an additional U (1)D gauge symmetry, which may beexact or broken to Z2. This model is relevant to a number of astrophysical observations, including AMS-02and the dark-matter distribution in dwarf galactic halos.

© 2013 Elsevier B.V. All rights reserved.

The notion that dark matter (DM) is the origin of neutrinomass (scotogenic) is by now a common theme among many stud-ies. The first one-loop realization [1], as shown in Fig. 1, re-mains the simplest such example. The standard model (SM) ofquark and lepton interactions is augmented by three neutral singletMajorana fermions N1,2,3 and a second scalar doublet (η+, η0).A new discrete Z2 symmetry is imposed so that the new parti-cles are odd and all the SM particles even. The complex scalarη0 = (ηR + iηI )/

√2 is split by the allowed (λ5/2)(Φ†η)2 + H.c.

term in the Higgs potential so that mR �= mI and the scotogenicneutrino mass is given by [1]

(Mν)i j =∑

k

hikh jk Mk

16π2

[m2

R

m2R − M2

k

lnm2

R

M2k

− m2I

m2I − M2

k

lnm2

I

M2k

].

(1)

The DM candidate is either ηR (assuming of course that mR < mI )or N1 (assuming of course M1 < M2,3). Many studies and varia-tions of this original model are now available in the literature. Oneimportant extension is the promotion of the stabilizing discrete Z2symmetry to a U (1)D gauge symmetry [2,3], which gets broken toZ2 through an additional scalar field. This has two effects: (1) thestability of dark matter is now protected against possible violationof the Z2 symmetry from higher-dimensional operators includingthose of quantum gravity, (2) the force carriers (both vector andscalar) between DM particles may be relevant in explaining a num-ber of astrophysical observations.

In this Letter, we propose a new scotogenic model with a U (1)D

gauge symmetry which may be exact or broken to Z2. The newparticles are two scalar doublets (η+

1 , η01) ∼ 1 and (η+

2 , η02) ∼ −1

under U (1)D , and three neutral singlet Dirac fermions N1,2,3 ∼ 1under U (1)D . The allowed couplings completing the loop, as

* Corresponding author.

0370-2693/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physletb.2013.09.049

Fig. 1. One-loop generation of neutrino mass with Z2 symmetry.

Fig. 2. One-loop generation of neutrino mass with U (1)D symmetry.

shown in Fig. 2, are h1 NRνLη01, h2NLνLη

02, and (Φ†η1)(Φ

†η2)

which mixes η01 and η0

2. Let(η0

1

η02

)=

(cos θ sin θ

− sin θ cos θ

)(χ1χ2

), (2)

where χ1,2 are mass eigenstates, then the analog of Eq. (1) be-comes

(Mν)i j = sin θ cos θ∑

k

[(h1)ki(h2)kj + (h2)ki(h1)kj]Mk

8π2

×[

m21

m2 − M2ln

m21

M2− m2

2

m2 − M2ln

m22

M2

], (3)

1 k k 2 k k

E. Ma et al. / Physics Letters B 726 (2013) 744–746 745

Fig. 3. Values of DM couplings αD (left) and yL , yR (right) as a function of DM mass required to obtain observed relic abundance of DM in the Universe. For simplicity wehave chosen yL = yR .

where m1,2 are the masses of χ1,2 and Mk the mass of Nk . Notethat in contrast to Fig. 1, Majorana neutrino masses are obtainedin Fig. 2 even though only Dirac masses appear in the loop. Atthis point, the U (1)D gauge symmetry may remain exact, in whichcase there is a massless dark photon. However, we can also breakthe U (1)D gauge symmetry to Z2 by a complex singlet scalar fieldζ ∼ 2, in which case there is a massive dark photon γD as well asa dark Higgs boson, both of which may be relevant in astrophysicsas force carriers between DM particles.

If U (1)D is unbroken, only N1 is a DM candidate because η01,2

are not split in their real and imaginary parts, which means thattheir interaction with nuclei through Z exchange cannot be sup-pressed and thus ruled out by direct-search data as a possibleDM candidate. In the presence of U (1)D breaking with the al-lowed yLζ

†NL NL and yRζ †NR NR couplings, N is no longer a Diracfermion, but if these new terms are small, it may still be a pseudo-Dirac particle. At the same time, the ζη

†1η2 coupling allows split-

ting of the real and imaginary parts of η1,2.There is yet another scenario, where the gauge U (1)D symme-

try becomes an exact global U (1)D symmetry. This is accomplishedif ζ is forbidden to couple to N or η1,2, by choosing for exampleζ ∼ 3. The spontaneous breaking of the gauge U (1)D symmetrynow results in a global U (1)D symmetry, under which only N andη1,2 transform. This means that dark Higgs is no longer a force car-rier for the dark matter N , but the vector force carrier γD remainsand is no longer massless.

In the following we choose our DM candidate to be the light-est Dirac (or pseudo-Dirac) N and investigate how it fits intothe standard thermal WIMP (Weakly Interacting Massive Particle)paradigm. The dark photon γD may be massless [4] in which case arealistic scenario would require N to be heavier than about 1 TeV.If U (1)D is broken by ζ = (u + ρ + iσ)/

√2, where u = √

2〈ζ 〉,then γD is massive together with ρ . In the following we will as-sume u to be small compared to the decoupling temperature ofN , in which case its relic abundance is determined by the un-broken theory, whereas at present, its interaction with ordinarymatter is determined by the broken theory. In the early Universe,N N would annihilate to γDγD and ζ ζ ∗ . Since the dark scalar sin-glet ζ must mix with the SM Higgs doublet Φ in the most gen-eral scalar potential containing both, and the dark photon γD mayhave kinetic mixing [5] with the SM photon, these processes willallow N to have the correct thermal relic abundance to be a suit-able DM candidate. Furthermore, for γD and ρ lighter than about0.1 GeV, a number of astrophysical observations at present may beexplained.

Our DM scenario assumes N to be much heavier than theU (1)D breaking scale. Thus N is in general pseudo-Dirac. As faras relic abundance is concerned, it behaves as a Dirac fermion [6].Further, since it can annihilate into scalars (ζ ζ ∗) or vectors (γDγD)

instead of just SM quarks and leptons, its cross section is notsuppressed by fermion mass. Its thermally averaged s-wave anni-hilation cross sections to γDγD and ζ ζ ∗ are given by

⟨σ(N N → γDγD)v

⟩ = πα2D

M21

, (4)

⟨σ

(N N → ζ ζ ∗)v

⟩ = (|yL |2 + |yR |2)2 − (yL y∗R − y∗

L yR)2

16π M21

, (5)

where αD = g2D/4π is the dark fine structure constant and we

have neglected the masses of γD and ζ .In Fig. 3 we display the values of DM couplings required to ob-

tain the observed value for the dark-matter relic density of theUniverse, ΩDMh2 = 0.1187(17) [7]. For example, if M1 = 1 TeV,then we need either αD = 0.04 or yL = yR = 0.48.

As U (1)D is broken, the Dirac DM fermion N splits up intotwo Majorana fermions of about equal mass. The heavier stateΣ2 will decay into the lighter state Σ1 and a force carrier(Σ2 → Σ1γD ,Σ1ρ) if kinematically allowed. If the mass split-ting is smaller than the mass of the force carriers, Σ2 will decaythrough an off-shell force carrier or η1,2 to Σ1 and a pair of SMleptons.

There are two important phenomenological implications of ourU (1)D DM scenario. First, the large positron excess observed byPAMELA [8,9] requires an enhancement of the DM annihilationcross section at present compared to what it was at the time offreeze-out. This may be accomplished [10] by the inclusion of anew force in the dark sector, resulting in a Sommerfeld enhance-ment of the cross section from multiple exchange of the light forcecarrier. Recent AMS-02 results [11] may also be explained [12] ina similar way. In our case, since ρ mixes with h and γD mixeswith γ , their decays to μ−μ+ and e−e+ are ideal for such a pur-pose.

Second, DM self-interactions change its density profile from theusual collisionless WIMP scenario. To reconcile the theoretical pre-diction with the present astronomical observation of the halosof dwarf galaxies, a rather large cross section per unit DM mass∼1 cm2/g is required, and may be achieved [13,14] with ratherlight force mediators, such as M1 = 1 TeV and mρ,γD ∼ 4 MeV, orM1 = 100 GeV and mρ,γD ∼ 20 MeV.

746 E. Ma et al. / Physics Letters B 726 (2013) 744–746

Finally, additional insight into DM candidates in our sce-nario may come from direct detection experiments. The currentXENON100 limits [15] are already sensitive to very small cou-plings corresponding to the mixing of the dark-force carriers withthe appropriate SM bosons. For a benchmark value 10−10 for thecoupling involved in the kinetic mixing of the dark photon withthe SM photon, and for a 10–100 MeV dark force-carrier mass,XENON100 excludes self-interacting DM with a mass larger than∼300 GeV [14].

Acknowledgements

The work of E.M. is supported in part by the US Department ofEnergy under Grant No. DE-FG03-94ER40837. I.P. and B.R. are sup-ported by the Croatian Ministry of Science, Education and Sportsunder Contract No. 119-0982930-1016.

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