The scale-invariant scotogenic model

Journal of High Energy Physics, Jun 2016

We investigate a minimal scale-invariant implementation of the scotogenic model and show that viable electroweak symmetry breaking can occur while simultaneously generating one-loop neutrino masses and the dark matter relic abundance. The model predicts the existence of a singlet scalar (dilaton) that plays the dual roles of triggering electroweak symmetry breaking and sourcing lepton number violation. Important constraints are studied, including those from lepton flavor violating effects and dark matter direct-detection experiments. The latter turn out to be somewhat severe, already excluding large regions of parameter space. None the less, viable regions of parameter space are found, corresponding to dark matter masses below (roughly) 10 GeV and above 200 GeV.

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The scale-invariant scotogenic model

Received: May The scale-invariant scotogenic model Amine Ahriche 0 1 3 4 5 6 Kristian L. McDonald 0 1 3 6 Salah Nasri 0 1 2 3 6 0 School of Physics, The University of Sydney 1 Strada Costiera 11 , I-34014, Trieste , Italy 2 Physics Department, UAE University 3 PB 98 Ouled Aissa , DZ-18000 Jijel , Algeria 4 The Abdus Salam International Centre for Theoretical Physics 5 Department of Physics, University of Jijel 6 POB 17551 , Al Ain , United Arab Emirates We investigate a minimal scale-invariant implementation of the scotogenic model and show that viable electroweak symmetry breaking can occur while simultaneously generating one-loop neutrino masses and the dark matter relic abundance. The model predicts the existence of a singlet scalar (dilaton) that plays the dual roles of triggering electroweak symmetry breaking and sourcing lepton number violation. Important constraints are studied, including those from lepton avor violating e ects and dark matter direct-detection experiments. The latter turn out to be somewhat severe, already excluding large regions of parameter space. None the less, viable regions of parameter space are found, corresponding to dark matter masses below (roughly) 10 GeV and above 200 GeV. Beyond Standard Model; Neutrino Physics - The scale-invariant scotogenic model 2 3 5 6 7 8 1 1 Introduction 2.1 2.2 Symmetry breaking The scalar spectrum Neutrino mass 4 Invisible Higgs decays Lepton avor violating decays Dark matter The discovery of the Higgs boson provides an explanation for the origin of mass in the origin of the O(100) GeV mass-parameter that determines the weak scale in the SM also remains a mystery. Thus, with regard to the mechanisms of mass in the universe, there remains much to be discovered. The scotogenic model is a simple framework that aims to address some of these shortcomings [1]. It o ers an explanation for the origin of neutrino mass and the nature of DM by proposing a common or uni ed solution to these puzzles. In this approach, neutrinos acquire mass as a radiative e ect, at the one-loop level, due to interactions with a Z2-odd sector that includes DM candidates. The resulting theory gives a simple model for neutrino mass and DM, and has been well-studied in the literature [2{8]. Motivated by the simplicity of the scotogenic model, and our inadequate understanding O(TeV), enhancing the prospects for testing the model. The resulting theory provides a common framework for the aforementioned problems relating to mass | namely the origin of neutrino mass, the origin of the weak scale, and the nature of DM. We investigate the SI scotogenic model in detail, demonstrating that viable electroweak symmetry breaking can be achieved, while simultaneously generating neutrino masses and the DM relic abundance. The model predicts a singlet scalar (dilaton) that plays two important roles | it triggers electroweak symmetry breaking and sources the lepton number violation that allows radiative neutrino mass. Important constraints are studied, including those from lepton avor violating e ects, DM direct-detection experiments, and the Higgs sector, such as the invisible Higgs decay width and Higgs-dilaton mixing. Direct-detection constraints turn out to be rather severe and we nd that large regions of parameter space are already excluded. None the less, viable parameter space is found with a DM mass below (roughly) 10 GeV or above 200 GeV. The model can be experimentally probed in ! e + a number of ways, including: searches, future direct-detection experiments, precision studies of the Higgs decays h ! and h ! Z, and collider searches for an inert doublet. Before proceeding we note that a number of earlier papers have studied relationships between neutrino mass and DM; see e.g. refs. [10{64], and also ref. [65], in which DM stability follows from an accidental symmetry. Earlier works investigating SI extensions of the SM appear in refs. [66{81] and, in particular, studies of SI models for neutrino mass can be found in refs. [82{92]. The structure of this paper is as follows. In section 2 we introduce the model and detail the symmetry breaking sector. We turn our attention to the origin of neutrino mass in section 3 and discuss various constraints in sections 4 and 5. Dark matter is discussed in section 6 and our main analysis and results appear in section 7. Conclusions are drawn in section 8. 2 The scale-invariant scotogenic model The minimal SI implementation of the scotogenic model is obtained by extending the SM to include three generations of gauge-singlet fermions, NiR (1; 1; 0), where i = 1; 2; 3; labels generations, a second SM-like scalar doublet, S (1; 2; 1), and a singlet scalar (1; 1; 0). A Z2 symmetry with action fNR; Sg ! fNR; Sg is imposed on the model.1 The scalar , as well as the SM elds, transform trivially under this symmetry. The lightest Z2-odd particle is stable and may be a DM candidate; this should be taken as either the lightest singlet fermion N1 or a neutral component of the the doublet S, as discussed below. The scalar plays the dual roles of sourcing lepton number violation, to allow neutrino mass, and triggering electroweak symmetry breaking. 1This model was also mentioned in refs. [93, 94]. { 2 { With this eld content, the most-general Lagrangian consistent with both the SI and Z2 symmetries contains the terms L iNR yi NicR NiR gi NiRL S V ( ; S; H); (2.1) where L letters, ; (1; 2; 1) denotes the SM lepton doublets, with generations labeled by Greek = e; ; . We denote the SM scalar doublet as H (1; 2; 1) and V ( ; S; H) is the most-general scalar potential consistent with the symmetries. The SI symmetry precludes any dimensionful parameters in the model, including bare Majorana mass terms In the absence of dimensionful parameters, the scalar potential contains only quartic in(2.2) (2.4) HJEP06(21)8 for the fermions Ni. 2.1 Symmetry breaking teractions: V ( ; S; H) = G (X) = X2 5 can be taken real without loss of generality. The desired VEV pattern has hSi = 0, to preserve the Z2 symmetry, with hHi 6= 0 and h i 6= 0, to break both the SI and electroweak symmetries. In addition to the doublet scalar S, we shall see that the spectrum contains an SM-like scalar h1 and a dilaton h2. Radiative corrections play an important role in triggering the desired symmetry breaking pattern. A full analysis of the potential requires the inclusion of leading-order loop corrections; however, in general, the full one-loop corrected potential is not analytically tractable. None the less, as discussed in ref. [92] (and guided by ref. [95]), simple analytic expressions can be obtained by noting the following. Loop corrections involving SM elds are dominated by top-quark loops, due to the large Yukawa coupling. To allow viable electroweak symmetry breaking and give a positively-valued dilaton mass, these corrections must be dominated by loop corrections from a beyond-SM scalar, namely S. Thus, loop corrections from S and t are expected to dominate and, to reasonable approximation, one can neglect loop corrections involving the light scalars (namely the SM-like Higgs and the H = (0; h=p2), the one-loop corrected potential for h and is dilaton). More precisely, this gives an approximation to the potential up to corrections of O(Mh41 =MS4) [92], which is reasonable provided one restricts attention to MS & 200 GeV. Adopting this approximation, and writing the SM scalar in unitary gauge as V1 l (h; ) = 4 H h4 + 4 H 2h2 + 4 4 + X i=all elds ni G Mi2 (h; ) ; (2.3) where ni is a multiplicity factor, is the renormalization scale, and the sum is over all elds barring the light scalars (h and ) and the light SM fermions (all but the top-quark). The function G is given by In the absence of bare dimensionful parameters, the eld-dependent masses can be written as where i and i are constants. Symmetry breaking is triggered via dimensional transmutation, introducing a dimensionful parameter into the theory in exchange for one of the dimensionless couplings (which is now xed in terms of the other parameters). Analyzing the potential reveals a minimum with both h i x 6= 0 and hhi v 6= 0 for H < 0. If one considers the tree-level poten tial, the desired VEV pattern is triggered at the scale where the running couplings obey H( ) = 0. Including loop corrections, subject to our approximation, is also satis ed. Absent ne-tuning, we observe that with H; H = O(1) one obtains v x and the exotic scale is expected near the TeV scale. Eqs. (2.6) and (2.7) ensure that the tadpoles vanish. One-loop vacuum stability requires that the couplings obey: 1H l; 1 l; 1Hl + 2 q 1 l 1 l > 0; H where the one-loop couplings are de ned as 1 l = H 1 l = 1Hl = : Eq. (2.8) guarantees that the masses for the neutral scalars h and are strictly positive, forcing one of the beyond-SM scalars in the doublet S to be the heaviest particle in the spectrum, to overcome top-quark contributions to the dilaton mass. Demanding also ensures that the vacuum with v 6= 0 and x 6= 0 is preferred over the vacuum with a 1Hl < 0 single nonzero VEV. 2.2 The scalar spectrum Writing the inert-doublet as S = (S+; (S0 + iA)=p2)T , the components have masses MS2+ = MS20;A = 2 2 S x2 + 2 two mass eigenstates, which we denote by h1;2, 5-term splits the neutral scalar masses MS0 and MA, with the splitting becoming 1.2 After symmetry breaking, the scalars h and mix to give h1 = h cos h sin h ; h2 = h sin h + cos h : Due to the Z2 symmetry, the neutral components of S do not mix with these elds. At tree-level the mixing angle is determined by the VEVS, x px2 + v2 ; ch cos h = sh sin h = v px2 + v2 ; and the SM-like scalar mass is given by massless at tree-level, though radiative corrections induce Mh2 6= 0. A useful approximation for Mh2 is [95] Mh22 ' 8 2(x2 + v2) 1 ( Mh41 + 6M W4 + 3MZ4 12Mt4 + 2MS4+ + MA4 + MS40 (2.11) (2.12) (2.13) 3 2 X i=1 ) Mi4 : (2.14) Here the singlet fermion masses are given by Mi = yi x, and are ordered as M1 < M2 < M3. Eq. (2.14) shows that viable symmetry breaking requires one of the scalars S+, S0 or A to be the heaviest particle in the spectrum, to overcome negative loop contributions to Mh2 from the top quark and the fermions Ni. Tree-level expressions for Mh1 and h are presented above for convenience, however, in our numerical analysis (detailed below), we use the mass eigenvalues Mh1;2 and the mixing angle h obtained by diagonalizing the one-loop corrected potential. We note that the SI symmetry imposes non-trivial constraints on the model, with and H xed by eqs. (2.6) and (2.7), and the Higgs mass Mh1 ' 125 GeV further xes H . 3 Neutrino mass The combined terms in eqs. (2.1) and (2.2) explicitly break lepton number symmetry, giving rise to radiative neutrino mass at the one-loop level, as shown in gure 1. Observe that plays a key role in allowing the neutrino mass diagram, without which neutrinos would remain massless.3 Calculating the mass diagram gives (M ) = i X gi gi Mi MA2 MA2 : (3.1) the limit 5 ! 0. are possible [96]. 2Note that the limit 5 1 is technically natural due to the restoration of lepton number symmetry in 3The Feynman diagram in gure 1 is an example of the SI type T3 one-loop topology. Related variants { 5 { (M ) ' X gi gi 5v2 i 1 i : (3.2) Note that the Z2 symmetry prevents mixing between SM neutrinos and the exotics Ni. One can relate the entries in the neutrino mass matrix to the elements of the Pontecorvo-Maki-Nakawaga-Sakata (PMNS) mixing matrix [97, 98] elements. We parameterize the latter as 0 (3.3) j m213j = 2:55+0:06 0:09 with d being the Dirac phase and Um = diag(1; ei =2; ei =2) giving the dependence on the Majorana phases ; . We use the shorthand sij sin ij and cij cos ij to refer to s23 = 0:43+00::0033, and the mass-squared di erences: 2 m221 = 7:62+0:19 0:19 the mixing angles. In our numerical scans of the parameter space in the model, we t to the best- t experimental values for the mixing angles: s213 = 0:025+00::000033, s212 = 0:320+00::001167, 10 5 eV2 and To determine the parameter space that generates viable neutrino masses, we use the (M ) = X gi gi i = g T g ; i i = Mi MA2 MA2 M 2 ln i MA2 : According to the Casas-Ibarra parameterization, the coupling g can be written as g = Dp 1 RDpm U y; where Dp 1 = diag q 1 ; 1 q 2 ; 1 q 3 1 , Dpm = diag pm1; pm2; pm3 , and R is an orthogonal rotation matrix (m1;2;3 are the neutrino eigen-masses). (3.4) (3.5) (3.6) { 6 { Invisible Higgs decays The model is subject to constraints on the branching fraction for invisible Higgs decays, B(h ! inv) < 17% [101]. One should use inv available, with corresponding decay widths given by fh2h2g; fNDMNDMg, when kinematically 122 is de ned in eq. (6.11) below. As a result of the SI symmetry, the coupling 122 vanishes at tree-level, and the non-zero loop-level coupling is su ciently small to ensure that decay to h2 pairs is highly suppressed.4 5 avor violating decays The new elds give rise to one-loop contributions to Br( ! e e), the corresponding branching fraction is ! e + . Normalized relative to where AD is the dipole form factor: with the loop function given by Br( Br( ! e ) ! e e) = 4G2F 3(4 )3 em jADj2 ; AD = X geigi 1 i A simple change of labels allows one to use the above formulae for the related decay + . In our analysis we also include the constraint from neutrino-less double Note that, in general, the scotogenic model is subject to strong LFV constraints, relating to the fact that the DM annihilates via the same Yukawa couplings that mediate LFV processes. Consequently one cannot decouple the two e ects and there can be tension between the demands of suppressed LFV processes and the attainment of a viable DM abundance (actually, in the scotogenic model, constraints from other LFV processes, like -e conversion, can be more severe than the above LFV decays; see the 3rd and 4th 4Note that h2 decays to SM states, similar to a light SM Higgs boson but with suppression by the mixing angle, s2 . However, dedicated ATLAS or CMS searches for such light scalars, in the channels 2b, 2 h or 2 , do not currently exist, so we classify the decay h1 ! h2h2 as invisible. In practice, however, the suppression of (h1 ! h2h2) due to SI symmetry renders this point moot. (5.1) (5.2) (5.3) { 7 { papers in refs. [2{8]). However, we shall see that the situation di ers in the SI model, due to additional annihilation processes mediated by the dilaton. This provides a degree of decoupling between the LFV processes and DM annihilations, such that LFV bounds are more readily satis ed. Thus, for our purposes, it is su cient to consider the above LFV decays (we shall see that the viable parameter space includes regions well-below the LFV bounds, so slightly stronger bounds do not have a large e ect). We note that the correlation between ! e and the DM relic abundance, for the case of fermionic DM in the scotogenic model, was rst noted in ref. [102], while ref. [103] noted that models with a singlet scalar allow one to decouple these issues. 6 6.1 Dark matter Relic density As the universe cools, the temperature eventually drops below the DM mass. Consequently the DM number density becomes Boltzmann suppressed and the DM annihilation rate can become comparable to the Hubble parameter. At a certain temperature the DM particles freeze out of equilibrium, such that the DM number density in a comoving volume henceforth remains constant. The cold DM relic abundance therefore depends on the total thermally averaged annihilation cross section HJEP06(21)8 h (NDM NDM)vri = h (NDM NDM ! X)vri = X X X X Z 1 4MD2M ds N DM NDM!X (s) s where vr is the relative velocity, s is the Mandelstam variable, K1;2 are the modi ed Bessel functions and at the CM energy ps. At freeze-out, the thermal relic density can be given in terms of the NDM NDM!X (s) is the annihilation cross due to the channel NDM NDM ! X, thermally averaged annihilation cross section by DMh2 ' pg Mpl(GeV) h (NDM NDM)vri ; (1:07 109)xF where Mpl is the Plank mass and g counts the e ective degrees of freedom of the relativistic elds in equilibrium. The inverse freeze-out temperature, xF = MDM=TF , can be determined iteratively from the equation xF = log 8 3pg xF r 45 MDMMpl h (NDMNDM)vri ! : In the present model, the classes of DM annihilation channels are shown in gure 2. The DM can annihilate into: (1) charged leptons and neutrinos, ` ` + and , including LFV nal states with 6 = , (2) SM fermions and gauge bosons bb, tt, W +W , ZZ and the scalars SS, and (3) nal states comprised of the Higgs and/or dilaton, hihk. The rst class of channels are h1;2-mediated s-channel processes, the second class are S-mediated t-channel processes while the third class contains both s- and t-channels processes mediated by h1;2. p s T (6.1) ; (6.2) (6.3) { 8 { N1 N1 N1 (a) hj (d) S Lβ hi hk N1 N1 N1 (b) (e) S Lβ hi hk N1 N1 N1 hj (c) (f) hi hk XSM,S The cross section for the annihilation channel into charged leptons5 is given by [104] HJEP06(21)8 (6.4) : (6.5) (6.6) p s ; (6.7) (NDMNDM ! ` `+)vr = 1 8 s(MS2+ + s 2 3 s 4 jg1 g1 j 2 MD2M MD2M + 2s )2 " m`2 + m2 ` 2 s 2 MD2M (MS2+ MD2M)2 + 2s (MS2+ MD2M) + s82 # (MS2+ MD2M + 2s )2 The cross section for annihilation into neutrinos can be obtained from eq. (6.4) by replacing MS2+ ! MS20 and sending the charged lepton masses to zero, i.e., (NDMNDM ! )vr = jg1 g1 j 2 12 s 4 MD2M (MS20 MD2M)2 + 2s (MS20 (MS20 MD2M + 2s )4 2 MD2M) + s8 : (2) s-channel processes. The processes NDMNDM ! bb, tt, W +W and ZZ can occur as shown in gure 2-c. The corresponding amplitude can be written as M = ichshy1u (k2) u (k1) s i Mh21 i Mh22 ! Mh!SM mh ! p s ; with Mh!SM (mh ! Higgs mass replaced as mh ! ps. This leads to the cross section ps) being the amplitude of the Higgs decay h ! XSMXSM, with the (NDMNDM ! XSMXSM) r = 8pss2hc2hy12 s s 1 Mh22 2 h!XSMXSM mh ! where h!XSMXSM (mh ! ps) is the total decay width, with mh ! ps. 5For same- avor charged leptons ( = ), there are also s-channel processes mediated by h1;2. However, these are proportional to their Yukawa couplings and may therefore be ignored. s 1 Mh21 { 9 { (3) Higgs channel. The DM can self-annihilate into hihk, as seen in gure 2-d, -e and -f. The amplitude squared is given by jMj2 = 2y~D2Ms " ch 1ik s Mh21 + s sh 2ik Mh22 " ch 1ik s #2 + 4cicky~D3MMDM + (t 2ci2c2ky~D4M MD2M) 2 + a2 2ci2c2ky~D4M + a (u (t MD2M)2 2ci2c2ky~D4M MD2M) (u MD2M) Mh21 + s sh 2ik Mh22 # s Mh2i + Mh2k + a MD2M s + Mh2i u 4MD2MMh2k + MD2M + Mh2i MD2M + Mh2i u Mh2k ! MD2M sMh2i 4MD2MMh2i + MD2M + Mh2k u MD2M + Mh2k t sMh2k MD2M + Mh2i t MD2M + Mh2k t Similarly, the SS annihilation cross section can written as (NDMNDM ! SS)vr = S h h 1 s where S0 = A = 1; S+ = 2, and 1SS and 2SS are the triple couplings of a scalar h1;2 with two S elds, given by 1S+S = 3chv Sshx; 2S+S = 3shv + Schx; 1 2 1 2 1S0S0;1AA = 2S0S0;2AA = (e ) LN1 q = aq qq NDcMNDM; + MD2M +Mh2k u MD2M +Mh2i u s s with s, t and u being the Mandelstam variables, and the Yukawa couplings are de ned as y~DM y1, c1 ch and c2 sh. Here, we integrate the phase space numerically to obtain the cross section for a given value of s. At tree-level the e ective cubic scalar couplings ( 1ik and 2ik) are given by 3 111 = 6 H chv 3 Hc2hshv + 3 Hchs2hx 6 s3hx; though for completeness we employ the one-loop results, obtained from the loop-corrected potential following ref. [105]. We note that the (leading order) absence of the cubic interactions h1h22 and h32, is a general feature of SI models. 6.3 Direct detection With regard to direct-detection experiments, interactions between the DM and quarks are described by an e ective low-energy Lagrangian: with where Consequently, the e ective nucleon-DM interaction is written as aq = shchMqMDM 2 h i hH0i " 1 B the baryon mass in the chiral limit [106]. This leads to the following nucleon-DM elastic cross section in the chiral limit det = s4hM 2 N MN The analysis below will show that the upper bound reported by LUX experiment [109, 110] provides a stringent constraint on det. 7 Analysis and results Next we turn to our numerical analysis and results. We perform a numerical scan of the parameter space to determine whether radiative electroweak symmetry breaking is compatible with one-loop radiative neutrino mass and singlet neutrino DM. In the scans, we enforce the minimization conditions, eqs. (2.6) and (2.7), vacuum stability via eq. (2.8), and demand that the SM-like Higgs mass is in the experimentally allowed range, Mh1 = 125:09 0:21 GeV. Compatibility with constraints from LEP (OPAL) on a light Higgs [107] are enforced, and we consider the constraint from the Higgs invisible decay, (however, we only nd viable benchmark points for h i & 150 GeV).6 B(h ! inv) < 17%, [101]. Dimensionless couplings are restricted to the perturbative range throughout, and we consider values of 100 GeV < h i < 5 TeV for the beyond-SM VEV The scan reveals a spread of viable values for the dilaton mass Mh2 , consistent with OPAL, as plotted in gure 3. In the scan we tend to nd Mh2 in the range O(1) GeV . Mh2 . 90 GeV. Lighter values of Mh2 seemingly require an amount of engineered cancellation among the radiative mass-corrections from fermions and bosons, or larger values for h i; see eq. (2.14). We noticed that regions with h i & 500 GeV tend to be preferred. We further scan for parameter space giving viable neutrino masses and mixing, subject to the LFV and muon anomalous magnetic moment constraints, while simultaneously generating a viable DM relic density. Figure 4 shows viable benchmark sets for the Yukawa couplings gi , along with the corresponding LFV branching ratios and a contributions. The couplings gi are typically well-below the perturbative bound. Note that the range for 6In principle, one can consider larger values for h i. However, these require hierarchically small couplings in the scalar potential [108], which we do not consider here. M(e10-3 S dashed line denotes the degenerate case, i.e, min jgj = max jgj. Right: the LFV branching ratios versus the muon anomalous magnetic moment, both scaled by the experimental bounds. the Yukawa couplings varies over several orders of magnitude. This re ects the freedom to take the lepton-number violating quartic coupling 5 to be small, and accordingly transfer some of the neutrino mass suppression between the Yukawa and quartic coupling sectors. The capacity to obtain viable neutrino masses, with Yukawa couplings that vary over a considerable range, in uences the strength of the signal from LFV decays. Figure 4 shows that the bound from ! e gives important constraints in parameter space with larger gi , while smaller values of gi allow the model to easily evade the bound. Constraints from the weaker ! bound are readily satis ed. Also, we veri ed that constraints from neutrino-less double-beta decay searches are satis ed by the benchmark points. With regards to the DM relic density, recall that there are multiple classes of annihilation channels, namely NDMNDM ! X (X = ` ` , , bb, tt, W W , ZZ, SS, h1;2h1;2). 200 100 0 1 10-2 10-4 lepton pairs, gauge bosons, heavy quarks and scalars. Right: the charged scalar masses MS+ versus the DM mass. The palette shows the DM Yukawa coupling yDM y1. Depending on the speci c value of the DM mass, a given channel may be signi cant or suppressed. To probe the role of the distinct channels, in gure 5-left we plot the contribution of each channel relative to the total cross section at freeze-out, X = tot, versus the DM mass. Annihilations into lepton pairs typically play a subdominant role. These are mediated by the couplings gi , whose values should be su ciently small to ensure viable neutrino masses and consistency with LFV constraints. For lighter values of MDM . 75 GeV, the cross section tends to be dominated by annihilations into b quarks, while annihilations into Z2-even neutral scalar nal states (X = hh with h h1;2) are dominant for heavier values of MDM & 125 GeV. In the intermediate range, annihilations into gauge bosons can also be important. For completeness, we include the nal states X = 2S in the plot, for components of the doublet S. Although the doublet scalars are typically heavier than the DM, thermal uctuations can allow a contribution from these modes (though the e ect is clearly subdominant, as seen in the gure). Figure 5-right shows the mass of the charged scalar, MS+ , versus the DM mass. In the lighter DM mass range, MDM . O(100) GeV, one notices that the charged scalar mass should not exceed 450 GeV, while for larger values of MDM one can have MS+ at the TeV scale. Such light charged scalars may be of phenomenological interest as they can be within reach of collider experiments. We note that gure 5-right contains disconnected regions for viable DM, with the region 31 GeV . MDM . 48 GeV not returning viable benchmark points. This \missing region" results from an over-abundance of DM, due to an insu ciently large, thermallyaveraged annihilation cross section. In the small MDM region, the annihilation cross section is dominated by bb nal states, with an important sub-contribution from annihilations into dilatons. However, below MDM 48 GeV, we nd that the dilaton contribution is too small to allow the observed relic abundance. The allowed island at MDM . 31 GeV corresponds to parameter space that approaches the h2 resonance, such that 2MDM is around, or just below, the dilaton mass, namely MDM . Mh2 =2 (the dilaton mass is shown in gure 6). This enhances annihilations into SM nal states. The corresponding enhancement to the s-channel process NDMNDM ! h2h2, via an intermediate h2, is not su cient to overcome 10-40 10-41 10-42 90 80 70 60 50 40 30 20 10 0 LUX(2015) 1 10 100 1000 MDM (GeV) recent constraints from LUX, while the palette gives the mass for the neutral beyond-SM scalar (dilaton), Mh2 , in units of GeV. the small cubic coupling 222, as shown in eq. (6.11). Note also that points in the region MDM . Mh1 =2 60 GeV experience some enhancement from the h1 resonance. Such enhancements do not occur in heavier MDM regions, as both the dilaton and Higgs are much lighter than the DM. Throughout the lighter MDM regions, the Higgs may decay into NDM and h2 nal states, though the bound on invisible Higgs decays is readily satis ed. The decay h1 ! NDMNDM is su ciently small due to Yukawa suppression (in addition to small h mixing), as seen from the palette in gure 5-right, while the decay h1 ! h2h2 is suppressed by the small cubic scalar coupling 122. Next we consider the constraints from direct-detection experiments. We plot the directdetection cross section versus the DM mass for the benchmark parameter sets in gure 6. The mass of the dilaton, Mh2 , in units of GeV, is shown in the corresponding palette. One immediately observes that direct-detection limits from LUX [109, 110] impose very serious constraints on the model, with a large number of benchmark sets already excluded. The plot shows that the surviving benchmark points mostly occur for MDM . 10 GeV, with a smaller number of viable points found for MDM & 200 GeV. Benchmarks with intermediate MDM values are excluded. The viable parameter space typically requires a lighter dilaton mass, Mh2 . 10 GeV, as all benchmarks with Mh2 & 50 GeV are excluded. It is clear from the gure that the surviving benchmark sets can be probed in forthcoming directdetection experiments. In gure 7 we consider the oblique parameters. The variation with respect to the mixing parameter sin2 h is shown in the left panel. One notices that the sin2 h dependence is not the dominant source of variation. There is some sensitivity to sin2 h, primarily in S. However, for a given xed value of sin2 h, benchmark points occur along the majority of the V-shaped curve traced out in the plot. Thus, the sin2 h dependence is not driving the variation. The dependence of the oblique parameters on the dimensionless mass-di erence for components of S, namely MS0 ) =2MS+ , is shown in the right panel 0.4 0, as seen in the plot, while larger mass-splittings can con ict with ellipsoids show the 68%, 95% and 99% CL., respectively. In the Left frame, the palette shows the mixing sin2 h between the Higgs and the dilaton; in the Right frame it shows the relative mass splitting, of gure 7. The plot shows that the majority of the variation in T is due to the mass splitting encoded in . This is expected. The T parameter is sensitive to isospin violation and thus constrains the splitting for SU(2)L multiplets. Viable benchmark points occur in The benchmark points include a range of values for the mass-splitting parameter giving rise to the variation in gure 7. However, in general, one can take the couplings 4;5 in the scalar potential su ciently small to ensure the mass-splitting for S+, S0 and A is consistent with oblique constraints. From the (technical) naturalness perspective, arbitrarily small values of 5 are allowed, due to the enhanced lepton number symmetry for 5 ! 0.7 Natural values of 4 are bounded from below by one-loop gauge contributions to the operator jHySj2. Consequently the mass splitting for components of S is not expected to be smaller than the one-loop induced splitting, which is safely within the bounds. Thus, although the oblique parameters can exclude some regions of parameter space, the constraints are readily evaded. The exotics in the model can also give new contributions to the Higgs decays h ! Z . The ratio of the corresponding widths, relative to the SM values, is plotted in gure 8. One sees that the overwhelming majority of the benchmark points are consistent with constraints from ATLAS and CMS. Importantly, more-precise measurements by ATLAS and CMS during Run II of the LHC will provide further probes of the model. Before concluding, we note that our analysis reveals considerable di erences between the SI scotogenic model and the standard (non-SI) scotogenic model. These relate primarily to the presence of the dilaton. The coupling between and the DM provides new annihilation channels for the sterile neutrino DM. This alleviates the need for larger Yukawa couplings gi , normally required in the scotogenic model to generate the relic density, and 7In practice, the demand of viable neutrino masses gives a Yukawa coupling-dependent lower bound on 5. 1.5 1.4 reduces the tension with LFV constraints. However, the dilaton also permits new channels at direct-detection experiments making these constraints more severe for the SI model. As a rough guide, one expects stronger LFV signals for the scotogenic model, and stronger direct-detection signals for the SI scotogenic model. 8 In this work, we performed a detailed study of the minimal SI scotogenic model. Our analysis demonstrates the existence of viable parameter space in which one obtains radiative electroweak symmetry breaking, one-loop neutrino masses and a good DM candidate. The model predicts a new scalar with O(GeV) mass. This eld plays the dual roles of triggering electroweak symmetry breaking and sourcing lepton number symmetry violation. The model can give observable signals in LFV searches, direct-detection experiments, and precision searches for the Higgs decays h ! and h ! Z. It also predicts a scalar doublet S, whose mass is expected to be . TeV, within reach of collider experiments. The model is subject to strong constraints from direct-detection experiments; viable parameter space was found for MDM . 10 GeV and MDM & 200 GeV, while intermediate values for MDM appear excluded. Acknowledgments AA is supported by the Algerian Ministry of Higher Education and Scienti c Research under the CNEPRU Project No D01720130042. KM is supported by the Australian Research Council. 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Amine Ahriche, Kristian L. McDonald, Salah Nasri. The scale-invariant scotogenic model, Journal of High Energy Physics, 2016, 182, DOI: 10.1007/JHEP06(2016)182