Dark Matter and the elusive \mathbf{Z'} in a dynamical Inverse Seesaw scenario
Valentina De Romeri, Enrique Fernandez-Martinez, Julia Gehrlein, Pedro A. N. Machado, Viviana Niro
FFERMILAB-PUB-17-285-TFTUAM-17-14IFT-UAM/CSIC-17-070
Prepared for submission to JHEP
Dark Matter and the elusive Z (cid:48) in a dynamical Inverse Seesaw scenario Valentina De Romeri, a Enrique Fernandez-Martinez, b,c
Julia Gehrlein, b,c
Pedro A. N. Machado d and Viviana Niro e a AHEP Group, Instituto de F´ısica Corpuscular, C.S.I.C./Universitat de Val`encia,Calle Catedr´atico Jos´e Beltr´an, 2 E-46980 Paterna, Spain b Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid,Cantoblanco E-28049 Madrid, Spain c Instituto de F´ısica Te´orica UAM/CSIC,Calle Nicol´as Cabrera 13-15, Cantoblanco E-28049 Madrid, Spain d Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, IL, 60510, USA e Institut f¨ur Theoretische Physik, Ruprecht-Karls-Universit¨at Heidelberg, Philosophenweg 16, 69120Heidelberg, Germany
E-mail: [email protected], [email protected],[email protected], [email protected],[email protected]
Abstract:
The Inverse Seesaw naturally explains the smallness of neutrino masses viaan approximate B − L symmetry broken only by a correspondingly small parameter. Inthis work the possible dynamical generation of the Inverse Seesaw neutrino mass mecha-nism from the spontaneous breaking of a gauged U (1) B − L symmetry is investigated.Interestingly, the Inverse Seesaw pattern requires a chiral content such that anomaly can-cellation predicts the existence of extra fermions belonging to a dark sector with large,non-trivial, charges under the U (1) B − L . We investigate the phenomenology associatedto these new states and find that one of them is a viable dark matter candidate with massaround the TeV scale, whose interaction with the Standard Model is mediated by the Z (cid:48) boson associated to the gauged U (1) B − L symmetry. Given the large charges required foranomaly cancellation in the dark sector, the B − L Z (cid:48) interacts preferentially with this darksector rather than with the Standard Model. This suppresses the rate at direct detectionsearches and thus alleviates the constraints on Z (cid:48) -mediated dark matter relic abundance.The collider phenomenology of this elusive Z (cid:48) is also discussed. Keywords:
Neutrino Physics, Dark Matter a r X i v : . [ h e p - ph ] N ov ontents N eff The simplest and most popular mechanism to accommodate the evidence for neutrinomasses and mixings [1–6] and to naturally explain their extreme smallness, calls upon theintroduction of right-handed neutrinos through the celebrated Seesaw mechanism [7–12].Its appeal stems from the simplicity of its particle content, consisting only of the right-handed neutrinos otherwise conspicuously missing from the Standard Model (SM) ingredi-ents. In the Seesaw mechanism, the smallness of neutrino masses is explained through theratio of their Dirac masses and the Majorana mass term of the extra fermion singlets. Un-fortunately, this very same ratio suppresses any phenomenological probe of the existence ofthis mechanism. Indeed, either the right-handed neutrino masses would be too large to bereached by our highest energy colliders, or the Dirac masses, and hence the Yukawa interac-tions that mediate the right-handed neutrino phenomenology, would be too small for evenour more accurate precision probes through flavour and precision electroweak observables.However, a large hierarchy of scales is not the only possibility to naturally explainthe smallness of neutrino masses. Indeed, neutrino masses are protected by the B − L (Baryon minus Lepton number) global symmetry, otherwise exact in the SM. Thus, ifthis symmetry is only mildly broken, neutrino masses will be necessarily suppressed bythe small B − L -breaking parameters. Conversely, the production and detection of theextra right-handed neutrinos at colliders as well as their indirect effects in flavour andprecision electroweak observables are not protected by the B − L symmetry and therefore– 1 –ot necessarily suppressed, leading to a much richer and interesting phenomenology. Thisis the rationale behind the popular Inverse Seesaw Mechanism [13] (ISS) as well as theLinear [14, 15] and Double Seesaw [13, 16–18] variants.In the presence of right-handed neutrinos, B − L is the only flavour-universal SMquantum number that is not anomalous, besides hypercharge. Therefore, just like theaddition of right-handed neutrinos, a very natural plausible SM extension is the gaugingof this symmetry. In this work these two elements are combined to explore a possibledynamical origin of the ISS pattern from the spontaneous breaking of the gauged B − L symmetry.Previous models in the literature have been constructed using the ISS idea or gauging B − L to explain the smallness of the neutrino masses, see e.g. [19–24]. A minimal modelin which the ISS is realised dynamically and where the smallness of the Lepton NumberViolating (LNV) term is generated at the two-loop level was studied in [25]. Concerning U (1) B − L extensions of the SM with an ISS generation of neutrino masses, several modelshave been investigated [26–29]. A common origin of both sterile neutrinos and Dark Matter(DM) has been proposed in [30, 31]. An ISS model which incorporates a keV sterile neutrinoas a DM candidate was constructed in e.g. [32]. Neutrino masses break B − L , if thissymmetry is not gauged and dynamically broken, a massless Goldstone boson, the Majoron,appears in the spectrum. Such models have been investigated for example in [30, 33].Interestingly, since the ISS mechanism requires a chiral pattern in the neutrino sector,the gauging of B − L predicts the existence of extra fermion singlets with non-trivial chargesso as to cancel the anomalies. We find that these extra states may play the role of DMcandidates as thermally produced Weakly Interacting Massive Particles (WIMPs) (see forinstance [34, 35] for a review).Indeed, the extra states would form a dark sector , only connected to the SM via the Z (cid:48) gauge boson associated to the B − L symmetry and, more indirectly, through the mixingof the scalar responsible for the spontaneous symmetry breaking of B − L with the Higgsboson. For the simplest charge assignment, this dark sector would be constituted by oneheavy Dirac and one massless Weyl fermion with large B − L charges. These large chargesmake the Z (cid:48) couple preferentially to the dark sector rather than to the SM, making itparticularly elusive . In this work the phenomenology associated with this dark sector andthe elusive Z (cid:48) is investigated. We find that the heavy Dirac fermion of the dark sector canbe a viable DM candidate with its relic abundance mediated by the elusive Z (cid:48) . Conversely,the massless Weyl fermion can be probed through measurements of the relativistic degreesof freedom in the early Universe. The collider phenomenology of the elusive Z (cid:48) is alsoinvestigated and the LHC bounds are derived.The paper is structured as follows. In Sec. 2 we describe the features of the model,namely its Lagrangian and particle content. In Sec. 3 we analyse the phenomenology of theDM candidate and its viability. The collider phenomenology of the Z (cid:48) boson is discussedin Sec. 4. Finally, in Secs. 5 and 6 we summarise our results and conclude.– 2 – The model
The usual ISS model consists of the addition of a pair of right-handed SM singlet fermions(right-handed neutrinos) for each massive active neutrino [13, 36–38]. These extra fermioncopies, say N R and N (cid:48) R , carry a global Lepton Number (LN) of +1 and −
1, respectively,and this leads to the following mass Lagrangian − L
ISS = ¯ LY ν (cid:101) HN R + N cR M N N (cid:48) R + N (cid:48) cR µ N (cid:48) R + h . c ., (2.1)where Y ν is the neutrino Yukawa coupling matrix, (cid:101) H = iσ H ∗ ( H being the SM Higgsdoublet) and L is the SM lepton doublet. Moreover, M N is a LN conserving matrix, whilethe mass matrix µ breaks LN explicitly by 2 units.The right-handed neutrinos can be integrated out, leading to the Weinberg opera-tor [39] which generates masses for the light, active neutrinos of the form: m ν ∼ v Y ν M − N µ ( M TN ) − Y Tν . (2.2)Having TeV-scale right-handed neutrinos (e.g. motivated by naturalness [40, 41]) and O (1) Yukawa couplings would require µ ∼ O (keV). In the original ISS formulation [13], thesmallness of this LNV parameter arises from a superstring inspired E6 scenario. Alternativeexplanations call upon other extensions of the SM such as Supersymmetry and GrandUnified Theories (see for instance [15, 42]). Here a dynamical origin for µ will be insteadexplored. The µ parameter is technically natural: since it is the only parameter that breaksLN, its running is multiplicative and thus once chosen to be small, it will remain small atall energy scales.To promote the LN breaking parameter µ in the ISS scenario to a dynamical quantity,we choose to gauge the B − L number [43]. The spontaneous breaking of this symmetrywill convey LN breaking, generate neutrino masses via a scalar vev, and give rise to amassive vector boson, dubbed here Z (cid:48) . B − L is an accidental symmetry of the SM, andit is well motivated in theories in which quarks and leptons are unified [44–47]. In unifiedtheories, the chiral anomalies cancel within each family, provided that SM fermion singletswith charge +1 are included. In the usual ISS framework, this is not the case due to thepresence of right-handed neutrinos with charges +1 and −
1. The triangle anomalies thatdo not cancel are those involving three U (1) B − L vertices, as well as one U (1) B − L vertexand gravity. Therefore, to achieve anomaly cancellation for gauged B − L we have toinclude additional chiral content to the model with charges that satisfy (cid:88) Q i = 0 ⇒ (cid:88) Q iL − (cid:88) Q iR = 0 , (2.3) (cid:88) Q i = 0 ⇒ (cid:88) Q iL − (cid:88) Q iR = 0 , (2.4)where the first and second equation refer to the mixed gravity- U (1) B − L and U (1) B − L anomalies, respectively. The index i runs through all fermions of the model.In the following subsections we will discuss the fermion and the scalar sectors of themodel in more detail. – 3 –article φ φ ν L N R N (cid:48) R χ R χ L ωU (1) B − L charge +1 +2 − − Table 1:
Neutral fermions and singlet scalars with their U (1) B − L charge and their mul-tiplicity. φ , are SM singlet scalars while N R , N (cid:48) R and χ R are right-handed and χ L and ω are left-handed SM singlet fermions respectively. Besides the anomaly constraint, the ISS mechanism can only work with a certain number of N R and N (cid:48) R fields (see, e.g., Ref. [48]). We find a phenomenologically interesting and viablescenario which consists of the following copies of SM fermion singlets and their respective B − L charges: 3 N R with charge −
1; 3 N (cid:48) R with charge +1; 1 χ R with charge +5; 1 χ L withcharge +4 and 1 ω with charge +4 Some of these right-handed neutrinos allow for a massterm, namely, M N N cR N (cid:48) R , but to lift the mass of the other sterile fermions and to generateSM neutrino masses, two extra scalars are introduced. Thus, besides the Higgs doublet H ,the scalar fields φ with B − L charge +1 and φ with charge +2 are considered. The SMleptons have B − L charge −
1, while the quarks have charge 1 /
3. The scalar and fermioncontent of the model, related to neutrino mass generation, is summarised in Table 1. Themost general Lagrangian in the neutrino sector is then given by −L ν = ¯ LY ν (cid:101) HN R + N cR M N N (cid:48) R + φ N cR Y N N R + φ ∗ ( N (cid:48) R ) c Y (cid:48) N N (cid:48) R + φ ∗ χ L Y χ χ R + h . c ., (2.5)where the capitalised variables are to be understood as matrices (the indices were omitted).The singlet fermion spectrum splits into two parts, an ISS sector composed by ν L , N R ,and N (cid:48) R , and a dark sector with χ L and χ R , as can be seen in the following mass matrixwritten in the basis ( ν cL , N R , N (cid:48) R , χ cL , χ R ): M = Y ν (cid:101) H Y Tν (cid:101) H † Y N φ M N M TN Y (cid:48) N φ ∗ Y χ φ ∗ Y Tχ φ . (2.6)The dynamical equivalent of the µ parameter can be identified with Y (cid:48) N φ ∗ . After φ develops a vacuum expectation value (vev) a Dirac fermion χ = ( χ L , χ R ) and a massless Introducing 2 N R and 3 N (cid:48) R as for example in [32] leads to a keV sterile neutrino as a potentiallyinteresting warm DM candidate [49] in the spectrum due to the mismatch between the number of N R and N (cid:48) R . However, the relic abundance of this sterile neutrino, if thermally produced via freeze out, is an orderof magnitude too large. Thus, in order to avoid its thermalisation, very small Yukawa couplings and mixingsmust be adopted instead. Notice that a coupling φ ∗ ωY ω χ R , while allowed, can always be reabsorbed into φ ∗ χ L Y χ χ R through arotation between ω and χ L . The analogous term Y N φ - also dynamically generated - contributes to neutrino masses only at theone-loop level and is therefore typically sub-leading. – 4 –ermion ω are formed in the dark sector. Although the cosmological impact of this extrarelativistic degree of freedom may seem worrisome at first, we will show later that thecontribution to N eff is suppressed as this sector is well secluded from the SM.To recover a TeV-scale ISS scenario with the correct neutrino masses and O (1) Yukawacouplings, v ≡ (cid:104) φ (cid:105) ∼ keV (cid:28) v (where v = (cid:104) H (cid:105) = 246 GeV is the electroweak vev) and M R ∼ TeV are needed. Moreover, the mass of the B − L gauge boson will be linked tothe vevs of φ and φ , and hence to lift its mass above the electroweak scale will require v ≡ (cid:104) φ (cid:105) (cid:38) TeV. In particular, we will show that a triple scalar coupling ηφ φ ∗ can inducea small v even when v is large, similar to what occurs in the type-II seesaw [12, 52–55].After the spontaneous symmetry breaking, the particle spectrum would then consist of a B − L gauge boson, 3 pseudo-Dirac neutrino pairs and a Dirac dark fermion at the TeVscale, as well as a massless dark fermion. The SM neutrinos would in turn develop smallmasses via the ISS in the usual way. Interestingly, both dark fermions only interact withthe SM via the new gauge boson Z (cid:48) and via the suppressed mixing of φ with the Higgs.They are also stable and thus the heavy dark fermion is a natural WIMP DM candidate.Since all new fermions carry B − L charge, they all couple to the Z (cid:48) , but specially the onesin the dark sector which have larger B − L charge. The scalar potential of the model can be written as V = m H H † H + λ H H † H ) + m φ ∗ φ + m φ ∗ φ + λ φ ∗ φ ) + λ φ ∗ φ ) (2.7)+ λ φ ∗ φ )( φ ∗ φ ) + λ H φ ∗ φ )( H † H ) + λ H φ ∗ φ )( H † H ) − η ( φ φ ∗ + φ ∗ φ ) . Both m H and m are negative, but m is positive and large. Then, for suitable values ofthe quartic couplings, the vev of φ , v , is only induced by the vev of φ , v , through η andthus it can be made small. With the convention φ j = ( v j + ϕ j + i a j ) / √ H = ( v + h + iG Z ) / √ G Z is theGoldstone associated with the Z boson mass), the minimisation of the potential yields m H = − (cid:0) λ H v + λ H v + 2 λ H v (cid:1) (cid:39) − (cid:0) λ H v + 2 λ H v (cid:1) , (2.8) m = − (cid:16) λ v + λ H v − √ ηv + λ v (cid:17) (cid:39) − (cid:0) λ v + λ H v (cid:1) , (2.9) m = (cid:32) √ ηv − λ (cid:33) v − λ v − λ H v (cid:39) √ ηv v , (2.10)or, equivalently, v (cid:39) √ ηv m . (2.11)Clearly, when η → m → ∞ , the vev of φ goes to zero. For example, to obtain v ∼ O (keV), one could have m ∼
10 TeV, v ∼
10 TeV, and η ∼ − GeV. The neutral– 5 –calar mass matrix is then given by M (cid:39) λ H v λ H v v/ λ H v v/ λ v −√ ηv −√ ηv ηv / √ v . (2.12)Higgs data constrain the mixing angle between Re( H ) and Re( φ ) to be below ∼
30% [56].Moreover, since η (cid:28) m , v , the mixing between the new scalars is also small. Thus, themasses of the physical scalars h , ϕ and ϕ are approximately m h = λ H v , m ϕ = λ v , and m ϕ = m / , (2.13)while the mixing angles α and α between h − ϕ and ϕ − ϕ , respectively, aretan α (cid:39) λ H λ v v , and tan α (cid:39) v v . (2.14)If v ∼ TeV and the quartics λ and λ H are O (1), the mixing α is expected to be smallbut non-negligible. A mixing between the Higgs doublet and a scalar singlet can onlydiminish the Higgs couplings to SM particles. Concretely, the couplings of the Higgs togauge bosons and fermions, relative to the SM couplings, are κ F = κ V = cos α , (2.15)which is constrained to be cos α > .
92 (or equivalently sin α < .
39) [57]. Since themassless fermion does not couple to any scalar, and all other extra particles in the modelare heavy, the modifications to the SM Higgs couplings are the only phenomenologicalimpact of the model on Higgs physics. The other mixing angle, α , is very small since it isproportional to the LN breaking vev and thus is related to neutrino masses. Its presencewill induce a mixing between the Higgs and ϕ , but for the parameters of interest here itis unobservable.Besides Higgs physics, the direct production of ϕ at LHC via its mixing with the Higgswould be possible if it is light enough. Otherwise, loop effects that would change the W mass bound can also test this scenario imposing sin α (cid:46) . m ϕ = 800 GeV [56].Apart from that, the only physical pseudoscalar degree of freedom is A = 1 (cid:112) v + 4 v [2 v a − v a ] (2.16)and its mass is degenerate with the heavy scalar mass, m A (cid:39) m ϕ .We have built this model in SARAH
SPheno
CalcHep [64] which are then used to study the DMphenomenology with
Micromegas χ Z ′ ff χχ ϕ Z ′ Z ′ χχ χ Z ′ Z ′ χχ χ Z ′ Z ′ Figure 1:
DM annihilation channels χ ¯ χ → f ¯ f via the Z (cid:48) boson and χ ¯ χ → Z (cid:48) Z (cid:48) . The χ ¯ χ → Z (cid:48) Z (cid:48) channel opens up when M Z (cid:48) < m χ . Since the process χ ¯ χ → ϕ → Z (cid:48) Z (cid:48) isvelocity suppressed this diagram is typically subleading. As discussed in the previous section, in this dynamical realisation of the ISS mechanismwe have two stable fermions. One of them is a Dirac fermion, χ = ( χ L , χ R ), which acquiresa mass from φ , and therefore is manifest at the TeV scale. The other, ω , is massless andwill contribute to the number of relativistic species in the early Universe. First we analyseif χ can yield the observed DM abundance of the Universe. In the early Universe, χ is in thermal equilibrium with the plasma due to its gauge inter-action with Z (cid:48) . The relevant part of the Lagrangian is L DM = − g BL ¯ χγ µ (5 P R + 4 P L ) χZ (cid:48) µ + 12 M Z (cid:48) Z (cid:48) µ Z (cid:48) µ − m χ ¯ χχ, (3.1)where M Z (cid:48) = g BL (cid:113) v + 4 v (cid:39) g BL v , and m χ = Y χ v / √ , (3.2)and P R,L are the chirality projectors.The main annihilation channels of χ are χ ¯ χ → f ¯ f via the Z (cid:48) boson exchange and χ ¯ χ → Z (cid:48) Z (cid:48) - if kinematically allowed (see fig. 1).The annihilation cross section to a fermion species f , at leading order in v , reads: (cid:104) σ v (cid:105) ff (cid:39) n c ( q χ L + q χ R ) q f L + q f R π g m χ (4 m χ − M Z (cid:48) ) + Γ Z (cid:48) M Z (cid:48) + O (cid:0) v (cid:1) , (3.3)see e.g. [66, 67], where n c is the color factor of the final state fermion (=1 for leptons), q χ L = 4 and q χ R = 5 and q f L,R are the B − L charges of the left- and right-handedcomponents of the DM candidate χ and of the fermion f , respectively. Moreover, thepartial decay width of the Z (cid:48) into a pair of fermions (including the DM, for which f = χ )is given by – 7 – ffZ (cid:48) = n c g (cid:16) q f L q f R m f + (cid:16) q f L + q f R (cid:17) (cid:16) M Z (cid:48) − m f (cid:17)(cid:17) (cid:113) M Z (cid:48) − m f πM Z (cid:48) . (3.4)When M Z (cid:48) < m χ , the annihilation channel χ ¯ χ → Z (cid:48) Z (cid:48) is also available. The crosssection for this process (lower diagrams in fig. 1) is given by (to leading order in the relativevelocity) [66] (cid:104) σ v (cid:105) Z (cid:48) Z (cid:48) (cid:39) πm χ M Z (cid:48) (cid:18) − M Z (cid:48) m χ (cid:19) / (cid:18) − M Z (cid:48) m χ (cid:19) − (cid:0) g ( q χ R + q χ L ) ( q χ R − q χ L ) m χ + (cid:0) ( q χ R − q χ L ) + ( q χ R + q χ L ) − q χ R − q χ L ) ( q χ R + q χ L ) (cid:1) g M Z (cid:48) (cid:1) , (3.5)The χ ¯ χ → ϕ → Z (cid:48) Z (cid:48) (upper right diagram in fig. 1) channel is velocity suppressed andhence typically subleading. Further decay channels like χ ¯ χ → ϕ ϕ and χ ¯ χ → Z (cid:48) ϕ openwhen 2 m χ > m ϕ + m ϕ ( m ϕ + m Z (cid:48) ). With m χ = Y χ / √ v , m ϕ = √ λ v , m Z (cid:48) = g BL v and the additional constraint from perturbativity Y χ ≤ χ ¯ χ → Z (cid:48) h is also subleading due to the mixing angle α between ϕ − h which is small although non-negligible (cf. Eq. (2.14)).The relic density of χ has been computed numerically with Micromegas obtainingalso, for several points of the parameter space, the DM freeze-out temperature at whichthe annihilation rate becomes smaller than the Hubble rate (cid:104) σ v (cid:105) n χ (cid:46) H . Given the freeze-out temperature and the annihilation cross sections of Eqs. (3.3) and (3.5), the DM relicdensity can thus be estimated by [68]:Ω χ h = 2 . · m χ T f . o .χ M P l √ g (cid:63) (cid:104) σ v (cid:105) , (3.6)where g (cid:63) is the number of degrees of freedom in radiation at the temperature of freeze-out ofthe DM ( T f . o .χ ), (cid:104) σ v (cid:105) is its thermally averaged annihilation cross section and M P l = 1 . · GeV is the Planck mass. In Sec. 5 we will use this estimation of Ω χ h together with itsconstraint Ω χ h (cid:39) . ± . The same Z (cid:48) couplings that contribute to the relic abundance can give rise to signals in DMdirect detection experiments. The DM-SM interactions in the model via the Z (cid:48) are eithervector-vector or axial-vector interactions. Indeed, the Z (cid:48) - SM interactions are vectorial(with the exception of the couplings to neutrinos) while χ has different left- and right-handed charges. The axial-vector interaction does not lead to a signal in direct detectionand the vector-vector interaction leads to a spin-independent cross section [71].The cross section for coherent elastic scattering on a nucleon is σ DD χ = µ χ N π (cid:18) g M Z (cid:48) (cid:19) (3.7)– 8 –here µ χ N is the reduced mass of the DM-nucleon system. The strongest bounds on thespin-independent scattering cross section come from LUX [72] and XENON1T [73]. Theconstraint on the DM-nucleon scattering cross section is σ DD χ < − pb for m χ = 1 TeVand σ DD χ < − pb for m χ = 10 TeV. The experimental bound on the spin-independentcross section (Eq. (3.7)) allows to derive a lower bound on the vev of φ : v [GeV] > (cid:18) . · σ DD χ [pb] (cid:19) / . (3.8)This bound pushes the DM mass to be m χ (cid:38) TeV. For instance, for g BL = 0 .
25 and m Z (cid:48) = 10 TeV, a DM mass m χ = 3 . σ DD χ ∼ × − pb. In turn,this bound translates into a lower limit on the vev of φ : v (cid:38)
40 TeV (with Y χ (cid:38) . In full generality, the annihilation of χ today could lead also to indirect detection signatures,in the form of charged cosmic rays, neutrinos and gamma rays. However, since the mainannihilation channel of χ is via the Z (cid:48) which couples dominantly to the dark sector, thebounds from indirect detection searches turn out to be subdominant.The strongest experimental bounds come from gamma rays produced through directemission from the annihilation of χ into τ + τ − . Both the constraints from the Fermi-LATSpace Telescope (6-year observation of gamma rays from dwarf spheroidal galaxies) [76]and H.E.S.S. (10-year observation of gamma rays from the Galactic Center) [77] are notvery stringent for the range of DM masses considered here. Indeed, the current experimen-tal bounds on the velocity-weighted annihilation cross section < σv > ( χ ¯ χ → τ + τ − ) rangefrom 10 − cm s − to 10 − cm s − for DM masses between 1 and 10 TeV. These valuesare more than two orders of magnitude above the values obtained for the regions of theparameter space in which we obtain the correct relic abundance (notice that the branchingratio of the DM annihilation to χ into τ + τ − is only about 5%). Future experiments likeCTA [78] could be suited to sensitively address DM masses in the range of interest of thismodel ( m χ (cid:38) N eff The presence of the massless fermion ω implies a contribution to the number of relativisticdegrees of freedom in the early Universe. In the following, we discuss its contribution to theeffective number of neutrino species, N eff , which has been measured to be N exp eff = 3 . ± . ω only interacts with the SM via the Z (cid:48) , its contribution to N eff willbe washed out through entropy injection to the thermal bath by the number of relativistic– 9 –egrees of freedom g (cid:63) ( T ) at the time of its decoupling:∆ N eff = (cid:18) T f . o .ω T ν (cid:19) = (cid:18) g (cid:63) ( T f . o .ω ) (cid:19) / , (3.9)where T f . o .ω is the freeze-out temperature of ω and T ν is the temperature of the neutrinobackground. The freeze-out temperature can be estimated when the Hubble expansionrate of the Universe H = 1 . √ g (cid:63) T /M P l overcomes the ω interaction rate Γ = < σv > n ω leading to: ( T f . o .ω ) ∼ . √ g (cid:63) M Z (cid:48) M P l g (cid:80) f ( q f L + q f R ) . (3.10)With the typical values that satisfy the correct DM relic abundance: m Z (cid:48) ∼ O (10 TeV)and g BL ∼ O (0.1) ω would therefore freeze out at T f . o .ω ∼ ω decouples and the contributionof the latter to the number of degrees of freedom in radiation will be suppressed:∆ N eff ≈ .
026 (3.11)which is one order of magnitude smaller than the current uncertainty on N eff . For gaugeboson masses between 1-50 TeV and gauge couplings between 0.01 and 0.5, ∆ N eff ∈ [0 . , . N eff matches the sensitivity expected from aEUCLID-like survey [79, 80] and would be an interesting probe of the model in the future. The new gauge boson can lead to resonant signals at the LHC. Dissimilarly from the widelystudied case of a sequential Z (cid:48) boson, where the new boson decays dominantly to dijets,the elusive Z (cid:48) couples more strongly to leptons than to quarks (due to the B − L number).Furthermore, it has large couplings to the SM singlets, specially χ and ω which carry large B − L charges. Thus, typical branching ratios are ∼
70% invisible (i.e. into SM neutrinosand ω ), ∼
12% to quarks and ∼
18% to charged leptons. LHC Z (cid:48) → e + e − , µ + µ − resonantsearches [81, 82] can be easily recast into constraints on the elusive Z (cid:48) . The productioncross section times branching ratio to dileptons is given by σ ( pp → Z (cid:48) → (cid:96) ¯ (cid:96) ) = (cid:88) q C qq sM Z (cid:48) Γ( Z (cid:48) → q ¯ q )BR( Z (cid:48) → (cid:96) ¯ (cid:96) ) , (4.1)where s is the center of mass energy, Γ( Z (cid:48) → q ¯ q ) is the partial width to q ¯ q pair given byEq. (3.4), and C qq is the q ¯ q luminosity function obtained here using the parton distributionfunction MSTW2008NLO [83]. To have some insight on what to expect, we compare our Z (cid:48) with the usual sequential standard model (SSM) Z (cid:48) , in which all couplings to fermions If the decay channels to the other SM singlets are kinematically accessible, specially into χ and intothe N R , N (cid:48) R pseudo-Dirac pairs, the invisible branching ratio can go up to ∼ Z (cid:48) even moreelusive and rendering these collider constraints irrelevant with respect to direct DM searches. – 10 – m Z ' [ TeV ] g B L m χ = L HC c on s t r a i n t s non - perturbative region Ω m > L U X X E N O N T ( y r ) L Z ( ) m Z ' [ TeV ] g B L m χ = L HC c on s t r a i n t s non - perturbative region Ω m > L U X X E N O N T ( y r ) L Z ( ) m Z ' [ TeV ] g B L m χ =
10 TeV L HC c on s t r a i n t s non - perturbative region Ω m > L U X XE N O N T ( y r ) L Z ( )
10 20 30 40 500.00.10.20.30.40.50.60.7 m Z ' [ TeV ] g B L m χ =
20 TeV L HC c on s t r a i n t s non - perturbative region Ω m > L U X XE N O N T ( y r ) L Z ( ) Figure 2:
Summary plots of our results. The red region to the left is excluded by LHCconstraints on the Z (cid:48) (see text for details), the region above g BL > . g BL · q max ≤ √ π . In the blue shaded region DM is overabundant. The orangecoloured region is already excluded by direct detection constraints from LUX [72], theshort-dashed line indicates the future constraints from XENON1T [74] (projected sensi-tivity assuming 2 t · y ), the long-dashed line the future constraints from LZ [75] (projectedsensitivity for 1000d of data taking). are equal to the Z couplings. The dominant production mode is again q ¯ q → Z (cid:48) though thecoupling in our case is mostly vectorial. The main dissimilarity arrives from the branchingratio to dileptons, as there are many additional fermions charged under the new gaugegroup. In summary, only O (1) differences in the gauge coupling bounds are expected,between the SSM Z (cid:48) and our elusive Z (cid:48) . – 11 – Results
We now combine in fig. 2 the constraints coming from DM relic abundance, DM directdetection experiments and collider searches. We can clearly see the synergy between thesedifferent observables. Since the DM candidate in our model is a thermal WIMP, the relicabundance constraint puts a lower bound on the gauge coupling, excluding the blue shadedregion in the panels of fig. 2. On the other hand, LHC resonant searches essentially puta lower bound on the mass of the Z (cid:48) (red shaded region), while the LUX direct detectionexperiment constrains the product g BL · M Z (cid:48) from above (orange shaded region). Forreference, we also show the prospects for future direct detection experiments, namely,XENON1T (orange short-dashed line, projected sensitivity assuming 2 t · y ) and LZ (orangelong-dashed line, projected sensitivity for 1000d of data taking). Finally, if the gaugecoupling is too large, perturbativity will be lost. To estimate this region we adopt theconstraint g BL · q max ≤ √ π and being the largest B − L charge q max = 5, we obtain g BL > . m χ . First, we see that for DM masses at 1 TeV (upper left panel), there is only a tinyallowed region in which the relic abundance is set via resonant χ ¯ χ → Z (cid:48) → f ¯ f annihilation.For larger masses, the allowed region grows but some amount of enhancement is in any caseneeded so that the Z (cid:48) mass needs to be around twice the DM mass in order to obtain thecorrect relic abundance. For m χ above 20 TeV (lower right panel), the allowed parameterspace cannot be fully probed even with generation-2 DM direct detection experiments.On top of the DM and collider phenomenology discussed here, this model allows fora rich phenomenology in other sectors. In full analogy to the standard ISS model, thedynamical ISS mechanism here considered is also capable of generating a large CP asym-metry in the lepton sector at the TeV scale, thus allowing for a possible explanation of thebaryon asymmetry of the Universe via leptogenesis [84–87].Moreover, the heavy sterile states typically introduced in ISS scenarios, namely the threepseudo-Dirac pairs from the states N R and N (cid:48) R can lead to new contributions to a widearray of observables [12, 88–111] such as weak universality, lepton flavour violating or pre-cision electroweak observables, which allow to constrain the mixing of the SM neutrinoswith the extra heavy pseudo-Dirac pairs to the level of 10 − or even better for some ele-ments [112, 113]. The simplest extension to the SM particle content so as to accommodate the experimentalevidence for neutrino masses and mixings is the addition of right-handed neutrinos, makingthe neutrino sector more symmetric to its charged lepton and quark counterparts. In thiscontext, the popular Seesaw mechanism also gives a rationale for the extreme smallnessof these neutrino masses as compared to the rest of the SM fermions through a hierarchybetween two different energy scales: the electroweak scale – at which Dirac neutrino masses– 12 –re induced – and a much larger energy scale tantalizingly close to the Grand Unificationscale at which Lepton Number is explicitly broken by the Majorana mass of the right-handed neutrinos. On the other hand, this very natural option to explain the smallness ofneutrino masses automatically makes the mass of the Higgs extremely unnatural, given thehierarchy problem that is hence introduced between the electroweak scale and the heavySeesaw scale.The ISS mechanism provides an elegant solution to this tension by lowering the Seesawscale close to the electroweak scale, thus avoiding the Higgs hierarchy problem altogether.In the ISS the smallness of neutrino masses is thus not explained by a strong hierarchybetween these scales but rather by a symmetry argument. Since neutrino masses areprotected by the Lepton Number symmetry, or rather B − L in its non-anomalous version,if this symmetry is only mildly broken, neutrino masses will be naturally suppressed bythe small parameters breaking this symmetry. In this work, the possibility of breaking thisgauged symmetry dynamically has been explored.Since the ISS mechanism requires a chiral structure of the extra right-handed neutrinosunder the B − L symmetry, some extra states are predicted for this symmetry to be gaugeddue to anomaly cancellation. The minimal such extension requires the addition of threenew fields with large non-trivial B − L charges. Upon the spontaneous breaking of the B − L symmetry, two of these extra fields become a massive heavy fermion around theTeV scale while the third remains massless. Given their large charges, the Z (cid:48) gauge bosonmediating the B − L symmetry couples preferentially to this new dark sector and muchmore weakly to the SM leptons and particularly to quarks, making it rather elusive .The phenomenology of this new dark sector and the elusive Z (cid:48) has been investigated.We find that the heavy Dirac fermion is a viable DM candidate in some regions of the pa-rameter space. While the elusive nature of the heavy Z (cid:48) makes its search rather challengingat the LHC, it would also mediate spin-independent direct detection cross sections for theDM candidate, which place very stringent constraints in the scenario. Given its preferenceto couple to the dark sector and its suppressed couplings to quarks, the strong tensionbetween direct detection searches and the correct relic abundance for Z (cid:48) mediated DMis mildly alleviated and some parts of the parameter space, not far from the resonance,survive present constraints. Future DM searches by XENON1T and LZ will be able toconstrain this possibility even further. Finally, the massless dark fermion will contribute tothe amount of relativistic degrees of freedom in the early Universe. While its contributionto the effective number of neutrinos is too small to be constrained with present data, futureEUCLID-like surveys could reach a sensitivity close to their expected contribution, makingthis alternative probe a promising complementary way to test this scenario. Acknowledgements
VDR would like to thank A. Vicente for valuable assistance on SARAH and SPheno. JGwould like to thank Fermilab for kind hospitality during the final stages of this project.This work is supported in part by the EU grants H2020-MSCA-ITN-2015/674896-Elusivesand H2020-MSCA-2015-690575-InvisiblesPlus. VDR acknowledges support by the Span-– 13 –sh grant SEV-2014-0398 (MINECO) and partial support by the Spanish grants FPA2014-58183-P, Multidark CSD2009-00064 and PROMETEOII/2014/084 (Generalitat Valenciana).EFM acknowledges support from the EU FP7 Marie Curie Actions CIG NeuProbes (PCIG11-GA-2012-321582), ”Spanish Agencia Estatal de Investigaci´on” (AEI) and the EU ”FondoEuropeo de Desarrollo Regional” (FEDER) through the project FPA2016-78645-P andthe Spanish MINECO through the “Ram´on y Cajal” programme (RYC2011-07710) andthrough the Centro de Excelencia Severo Ochoa Program under grant SEV-2012-0249 andthe HPC-Hydra cluster at IFT. The work of VN was supported by the SFB-TransregioTR33 “The Dark Universe”. This manuscript has been authored by Fermi Research Al-liance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of En-ergy, Office of Science, Office of High Energy Physics. The United States Governmentretains and the publisher, by accepting the article for publication, acknowledges that theUnited States Government retains a non-exclusive, paid-up, irrevocable, world-wide licenseto publish or reproduce the published form of this manuscript, or allow others to do so, forUnited States Government purposes.
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