Dark matter and LHC phenomenology of a scale invariant scotogenic model
DDark matter and LHC phenomenology of a scale-invariant scotogenic model
Chao Guo , ∗ Shu-Yuan Guo , , † and Yi Liao , ‡ School of Physics, Nankai University, Tianjin 300071, China Center for High Energy Physics, Peking University, Beijing 100871, China (Dated: October 2, 2019)We study the phenomenology of a model that addresses the neutrino mass, darkmatter, and generation of the electroweak scale in a single framework. Electroweaksymmetry breaking is realized via the Coleman-Weinberg mechanism in a classicallyscale invariant theory, while the neutrino mass is generated radiatively through in-teractions with dark matter in a typically scotogenic manner. The model introducesa scalar triplet and singlet and a vector-like fermion doublet that carry an odd parityof Z , and an even parity scalar singlet that helps preserve classical scale invariance.We sample over the parameter space by taking into account various experimentalconstraints from the dark matter relic density and direct detection, direct scalarsearches, neutrino mass, and charged lepton flavor violating decays. We then exam-ine by detailed simulations possible signatures at the LHC to find some benchmarkpoints of the free parameters. We find that the future high-luminosity LHC will havea significant potential in detecting new physics signals in the dilepton channel. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] O c t I. INTRODUCTION
The existence of tiny neutrino mass and dark matter (DM) provides two pieces of evidence beyond thestandard model (SM). Moreover, the SM is afflicted with some theoretical flaws such as the naturalness of theelectroweak scale around hundreds of GeV. Hence, it would be interesting to investigate whether it is possibleto address these issues within a single framework.Electroweak symmetry breakdown could occur via the Coleman-Weinberg mechanism [1], in which theelectroweak scale is induced in a scale-invariant theory through radiative effects instead of being put in byhand through a quadratic term of the wrong sign. In recent years, there have been multiple attempts toincorporate DM in the setting of the neutrino mass model, at one-loop [2], two-loop [3] and higher-looplevels [4]. The idea is to induce a tiny radiative neutrino mass through interactions with new heavy particlesthat are protected by a global symmetry, so that the lightest of the new particles could serve as a DMcandidate. In the minimal model, for instance, a scalar doublet and fermion singlets are introduced. Thiswas generalized in Ref. [5] by restricting the SM representations of new particles to be no larger than theadjoint representation. Recently, the authors of Ref. [6] proposed to marry the minimal scotogenic modelwith the idea of scale invariance by assuming a new scalar singlet. This scalar singlet plays an importantrole in triggering the radiative breakdown of scale invariance while the lightest fermion singlet serves as DM.The parameter space was found to be severely constrained by direct detection experiments of DM, and viableregions of parameters exist for a DM mass either smaller than ∼
10 GeV or larger than ∼
200 GeV. Otherusages of scale invariance in the context of neutrino mass and dark matter and related extensions of the SMare provided in Refs. [7–9] and the references therein.From a practical point of view scale invariance reduces the number of free parameters in a model withmultiple scalars. In this work, we incorporate scale invariance into a non-minimal scotogenic model suggestedin Ref. [5]. We add a new scalar singlet φ to preserve scale invariance on top of the new fields alreadyintroduced, i.e., a scalar triplet ∆ and singlet η plus a vector-like fermion doublet F , all of which are protectedby a Z symmetry. Such a model appeared as one of the various realizations of scale invariant scotogenicmodels in Ref. [10]. As in the general case with multiple scalars [11], the SM Higgs doublet and the singlet φ can develop vacuum expectation values (VEVs) radiatively, thus spontaneously breaking scale invarianceand electroweak symmetry and generating masses for all particles including neutrinos and DM in particular.The lighter of the singlet η and the neutral component of the triplet ∆ could serve as DM. We study variousconstraints on the new particles and interactions, and investigate the feasibility of detecting the triplet scalarsand vector-like fermions through multi-lepton signatures at the LHC.The paper is organized as follows. In section II, we introduce the model and discuss radiative breakdown ofscale invariance and electroweak symmetry. The radiative neutrino mass and constraints from lepton-flavor-violating (LFV) processes are briefly addressed. In section III, we study the parameter space that survivesthe most stringent constraints coming from relic density and direct detection of DM. The feasibility to detectnew particles at LHC is simulated in section IV using the multi-lepton signatures. We summarize our mainresults in section V. II. MODEL AND CONSTRAINTS
The scotogenic model, on which our work is based was proposed in Ref. [5]. It extends the content of theSM fields by an SU (2) L triplet ∆, a scalar singlet η and a vector-like fermion doublet F . An exact Z parity isassigned to these new fields to stabilize the lightest neutral particle as DM. We introduce a new scalar singlet φ that helps preserve scale invariance at the classical level, but spontaneously breaks it at the quantum level.The quantum numbers of these fields together with the SM Higgs doublet Φ and left-handed lepton doublet L L are collected in table I. TABLE I. Relevant fields and their quantum numbers under the SM group and a new Z group. L L Φ F η ∆ φSU (2) L U (1) Y −
12 12 −
12 0 1 0 Z + + − − − + The general Yukawa interactions involving the new vector-like fermion are given by − L
Yuk. ⊃ y η F R L L η + y ∆ L cL (cid:15) ∆ F L + y φ F L F R φ + h.c. , (1)where L cL is the charge conjugation of L L and (cid:15) is the antisymmetric tensor so that (cid:15)L cL transforms as an SU (2) L doublet. The Yukawa couplings y η, ∆ ,φ are complex matrices with respect to the lepton flavor and newfermion indices. The F and ∆ fields are cast in the form F = NE − , ∆ = ∆ + / √ ++ ∆ − ∆ + / √ . (2)A mass term for F is forbidden by scale invariance, and it does not mix with the SM leptons due to the Z symmetry. Similarly, Φ and φ on the one side do not mix with η and ∆ on the other side. Since η is purereal, F would have to carry one unit of the lepton number to keep it conserved in the y η term. This wouldin turn require ∆ to have two negative units of lepton number in the y ∆ term. However, then a term linearin ∆ (the λ η ∆Φ term) in the scalar potential would still break the lepton number. As shown in the following,the Majorana neutrino mass generated at one loop is indeed proportional to all of the couplings mentionedabove.The complete scale and Z invariant scalar potential is generally given as: V = λ Φ | Φ | + λ ∆1 (cid:2) Tr (cid:0) ∆ † ∆ (cid:1)(cid:3) + λ ∆2 Tr (cid:0) ∆ † ∆∆ † ∆ (cid:1) + 14 λ η η + 14 λ φ φ + λ Φ∆1 | Φ | Tr (cid:0) ∆ † ∆ (cid:1) + λ Φ∆2 Φ † ∆ † ∆Φ + 12 λ Φ η | Φ | η + 12 λ Φ φ | Φ | φ + 12 λ η ∆ η Tr (cid:0) ∆ † ∆ (cid:1) + 12 λ φ ∆ φ Tr (cid:0) ∆ † ∆ (cid:1) + 14 λ ηφ η φ + λ η ∆Φ η (cid:16) ˜Φ † ∆ † Φ + h.c. (cid:17) , (3)with the trace taken in weak isospin space. We follow the approach in Ref. [11] that generalizes the study ofRef. [1] to the case with multiple scalars. Only the Z even scalar fields Φ and φ can develop a nonvanishingVEV. Requiring that the VEVs (cid:104) Φ (cid:105) = v/ √ (cid:104) φ (cid:105) = u occur in the flat direction, where the above treelevel potential vanishes, one must have λ φ = 4 λ Φ λ φ with λ Φ , λ φ > > λ Φ φ . The ratio of the VEVs is givenby v u = − λ Φ φ λ Φ = − λ φ λ Φ φ , (4)while their absolute values are determined by higher order terms in the effective potential. Note that u doesnot contribute to the masses of the weak gauge bosons, since φ is a neutral singlet, and v ≈
246 GeV.The scalar doublet Φ contains three would-be Goldstone bosons that become the longitudinal componentsof the weak gauge bosons. Its remaining degree of freedom mixes with the singlet φ into a physical neutralscalar H of mass m H = v (cid:112) λ Φ − λ Φ φ , which is identified with the discovered 125 GeV scalar, and a physicalscalar h , which will only gain a radiative mass. Employing the result from Ref. [11], we obtain m h = 8 B (cid:104) ϕ (cid:105) , (5)where (cid:104) ϕ (cid:105) = v + u = − m H /λ Φ φ , and B = 164 π (cid:104) ϕ (cid:105) (cid:2) Tr M S + 3Tr M V − M F (cid:3) , (6)which sums over the masses of all scalars, gauge bosons and fermions. The new scalars must be sufficientlyheavy to make B >
0. The scalar singlet η mixes with the neutral component ∆ of the triplet scalar throughthe λ η ∆Φ coupling. Since the coupling also enters the radiative neutrino mass as shown in Fig. 1, it should benaturally small. The mass splitting among the components ∆ ++ , + , is controlled by the λ Φ∆2 coupling. As wedo not discuss cascade decays between those components in this work, we simply assume it vanishes. Thus, m = 12 (cid:0) λ Φ∆1 v + λ φ ∆ u (cid:1) , m η = 12 (cid:0) λ Φ η v + λ ηφ u (cid:1) . (7)The vacuum stability demands the couplings to satisfy the conditions λ − loopΦ > , λ − loop φ > , λ − loopΦ φ + 2 (cid:113) λ − loopΦ λ − loop φ > , (8)and the existence of two nonvanishing VEVs requires λ − loopΦ φ <
0. Here the λ − loop couplings have includedthe one-loop corrections, and are defined by partial derivatives of the effective potential in the same form asappearing in, e.g., Ref. [6]. In Fig. 2, they are shown as functions of | λ Φ φ | at m η = 200 GeV. It can be seenthat λ Φ φ must lie in the range ( − . , − . M αβν = √ π ) λ η ∆Φ v m ∆ + m η (cid:88) i (cid:16) y αi ∆ y iβη + y βi ∆ y iαη (cid:17) R iη R i ∆ R iη − R i ∆ (cid:18) ln R iη − ( R iη ) − ln R i ∆ − ( R i ∆ ) (cid:19) , (9) ⌫ L F R ⌘ F L ⌫ L FIG. 1. Feynman graph for one-loop radiative neutrino mass. where R i ∆ ,η = m N i /m ∆ ,η . In the basis where the charged leptons are already diagonalized, diagonalization U T PMNS M ν U PMNS = m ν will yield the neutrino masses m ν i and the PMNS matrix U PMNS . Although the de-sired order of magnitude for neutrino masses can be always realized by adjusting jointly the new couplingsand masses, we are interested in the region of parameter space where some new particles would in prin-ciple be reachable at the LHC, i.e., with a mass not exceeding few TeV; e.g., assuming ( m ∆ , m η , m N ) ∼ (10 , , y ∆ ∼ y η ∼ − , λ η ∆Φ ∼ − , will yield O (cid:0) . (cid:1) . With generally complex 3 × n F matrix y ∆ and n F × y η and a diagonal n F × n F real matrix formed with R i ∆ ,η , where n F is the numberof new doublet fermions, it is also easy to accommodate the measured PMNS matrix with free Dirac and Ma-jorana CP violation phases. In our phenomenological analysis, we assume n F = 3 almost degenerate doubletfermions, although it would be enough to generate two non-vanishing neutrino masses with two fermions.The lepton-flavor-violating (LFV) processes generically take place in neutrino mass models. Currently,the most stringent experimental bounds are set in the µ − e sector, with Br( µ → eγ ) < . × − [12] andBr( µ → eee ) < . × − [13] for the decays and Br( µ Au → e Au) < . × − [14] for µ − e conversion innuclei. For the model under consideration, these processes appear at the one-loop level. We have calculatedthe branching ratios, and found that those bounds can be readily avoided, e.g., at the benchmark point ofmasses of this study, namely m ∆ ∼ , m η ∼
200 GeV , m F ∼
650 GeV, assuming no special flavorstructure for the Yukawa matrices y ∆ ∼ y η ∼ y , we found that a loose bound y (cid:46) O (0 .
1) can satisfy all of the ●●● ● ●●●● ●● ● ●● ●●● ● ●●● ●● ● ●●●● ● ●●● ●● ●● ●● ● ●●● ●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ●● ● ●● ●●●●●● ●●●●●●●●●●● ●● ●● ●●●● ●● ●●● ●●●● ●●● ●●●●●●● ●●●● ● ●●● ● ● ● ●●● ●● ● ●●● ●● ●● ● ●● ● ●● ● ●● ● ●●●● ●● ●● ● ● ●●●● ●● ●●● ●● ●● ●● ●●● ●●●● ●●● ●●●● ●● ● ●● ●●● ●● ●●● ●●● ● ●●●● ●● ●●●●● ●● ● ●● ●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●● ● ●●●●●●● ●●●● ●●● ●●●●●●●●●● ●●●●● ●●●● ●● ●●●●● ●●● ●●●●●● ●●●●●●●●● ●●● ●●● ●● ●●● ● ●●●● ●● ● ●● ●●● ● ●●● ●● ● ●●●● ● ●●● ●● ●● ●● ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●● ●●●●●●●●●●● ●● ●● ●●●● ●● ●●● ●●●● ●●● ●●●●●●● ●●●● ● ●●●●●●●●●●● ● ●● ●● ● ●●●● ●● ●● ●● ●● ● ●●●● ●●● ● ●●●● ●● ●●● ●● ●● ●● ●●● ●●●● ●●● ●●●● ●● ● ●● ●●● ●● ●●● ●●● ● ●●●● ●● ●●●●● ●● ● ●● ●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ●●●● ●● ● ●● ●●● ● ●● ●● ● ●●● ● ●●● ●● ●● ●● ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●● ●●●●●● ● ● ●● ● ●●● ● ●● ● ●●●●●● ●●●● ●● ● ●● ●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ●●●● ●● ● ●● ●●● ● ●●● ●● ● ●●●● ● ●●● ●● ●● ●● ● ●●● ●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ●● ● ● ●● ●●●●●● ●●●●●●●●●●● ●● ●● ●●●● ●● ●●● ●●●● ●●● ●●●●●●● ●●●● ● ●●●●●●●● ●●● ● ● ● ●●● ●● ● ●●●● ●● ●● ● ●● ● ●● ● ●●● ●●● ● ●●●● ●● ●● ●● ●● ●● ●●● ●●●● ●●● ●●●● ●● ● ●● ●●● ●● ●●● ●●● ● ●●●● ●● ●●●●● ●● ● ●● ●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●● ● ●●●●● ●●●● ●●●● ●●● ●●●● ●●●●●●● ●●●●●● ●●●● ●● ●●●●●●● ●●● ●●●●●● ●●●●●●●●● ●●● ●●● ●● ����� ����� ����� ����� ������������������������� - λ Φϕ λ � - ���� λ Φ � - ���� λ ϕ � - ���� λ Φϕ � - ���� + � λ Φ � - ���� λ ϕ � - ���� - λ Φϕ � - ���� FIG. 2. One-loop corrected couplings are shown as functions of | λ Φ φ | at m η = 200 GeV. above LFV constraints. III. DARK MATTER DIRECT DETECTION
The lightest Z -odd neutral particle could in principle serve as a DM particle. However, to avoid strongconstraints from direct detection, it should preferably not couple to the Z boson. As a matter of fact, such aDM candidate would otherwise be heavier than 2.5 TeV [5, 15, 16], making detection of all Z -odd particlesessentially impossible at the LHC. We thus choose to work with the scalar singlet η as DM and expect tohave viable parameter space for a DM mass at the electroweak scale.The DM annihilation proceeds predominantly through the s -channel H/h exchange and contact interactionsinvolving
H/h as in a Higgs-portal DM model. The contribution from the t − channel exchange of the fermion F , which results in the (cid:96)(cid:96) (cid:48) , νν (cid:48) final states, is negligible as it is suppressed by the small Yukawa couplingsentering the neutrino mass. The set of relevant couplings is thus ( λ Φ , λ φ , λ Φ φ , λ Φ η , λ ηφ ). The fixed mass of m H = 125 GeV and the flat direction condition provide two constraints on λ Φ , λ φ and λ Φ φ . The mixing of φ and Φ into H and h is determined by the angle θ with sin θ = λ Φ φ / ( λ Φ φ − λ Φ ) = − λ Φ φ v /m H . The angle θ enters the η pair interactions with the H and h fields through λ Φ η and λ ηφ terms. As we will see shortly, theinterference effects between H and h are already rich enough with the θ angle and one of the two couplings λ Φ η , λ ηφ . We therefore technically switch off one of them, say λ Φ η = 0, to reduce the number of parameters.This leaves us with two free parameters, which we choose to be ( λ Φ φ , λ ηφ ), or equivalently ( λ Φ φ , m η ).In our numerical analysis, we use the package micrOMEGAs [17] to calculate the DM relic density and crosssection for DM-nucleon scattering. We scan the two free parameters in the following ranges | λ Φ φ | ∈ (0 . , . , m η ∈ (45 , , (10)while the other new particle masses are fixed as m ∆ = 1000 GeV , m F = 650 GeV . (11)Our numerical results are shown as a function of the DM mass m η in Fig. 3 for the spin-independent crosssection σ SI in DM direct detection and in Fig. 4 for the coupling − λ Φ φ (left longitudinal axis) and m h (rightlongitudinal axis). In these figures, all displayed points pass the DM relic density requirement. The redpoints are excluded by the PandaX-II experiment [18], the black ones are further excluded by the Xenon1Tresult [19], and the green ones are still allowed by all current experimental data.Two competing physical effects are responsible for the shapes of the points in Figs. 3 and 4 and in particularfor the jitter behaviour in the low DM mass region. The first is the destructive interference between the scalars H and h in the DM annihilation (via s -channel exchanges) and direct detection ( t -channel) amplitudes. Theycontribute a similar term but differ in sign, because the relevant couplings are proportional to ± sin θ cos θ .This effect is strong when the mass m h of h approaches m H of the SM-like Higgs H . The second effect is the ● ●● ● ●● ● ●● ●●●●●● ●●● ●● ● ●● ● ●● ●●● ●● ●● ●●● ●● ●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●● ●● ●●● ●●● ●●● ●●●●●● ●●● ●●●●● ●●●● ●●●●●● ●●● ● ●●●●● ●● ●● ●●●● ●● ●●●●● ●●●● ●●●●●●●●● ●●●● ●●●●●●● ●● ●● ●● ●●●● ●●● ●●●●●● ● ●●●● ●● ●● ●●● ● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ●● ●●●●●●●●●●●●●●● ● ● ●●●● ● ●● ●●● ●● ● ●●● ● ●●●● ● ● ●●●● ●● ●● ●● ●●● ● ●● ●●●●● ●●●●●●● ●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●● ●●● ●●●●●● ● ●●●●●●●●● ●●●●●●● ●●● ●●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● �� �� ��� ��� ��� ��� ����� - �� �� - �� �� - �� �� - �� �� - � �� - � � η ( ��� ) σ �� ( �� ) ������ - ������� �� FIG. 3. Sampled spin-independent cross section for DM scattering off the nucleon as a function of DM mass. Thetwo curves are the upper bounds by the PandaX-II [18] and Xenon1T [19] experiments. on-resonance enhancement of DM annihilation when the DM mass m η approaches half of the scalar mass m H or m h . The first effect takes place without requiring the couplings themselves to be necessarily small, and itis evident due to the presence of a dip in σ SI at m η (cid:46)
400 GeV, where m h crosses m H , but in the absenceof a corresponding dip in − λ Φ φ . The second effect would appear as a sudden and deep dip at m η ≈ m H / H . This then requires the relevant couplings to bevery small to avoid over-annihilation of DM. However, the presence of a second scalar h makes the situationhighly involved, especially in the region m η ∼ −
70 GeV, where m h varies rapidly and moves across thevalue of m H . In this region, the destructive interference and successive appearance and disappearance of oneor two resonances overlap to varying extents, and also explain the unusual rising behaviour as m η increases0 ● ●●● ●● ● ●● ●●●●●● ●●● ●● ● ●●● ●● ●●●●● ●● ●●● ●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●● ●● ●●● ●●● ●●● ●●●●●●●●● ●●●●● ●●●●●●●●●● ●●●●●●●●●●● ●● ●●●● ●● ●●●●● ●●●● ●●●●●●●●● ●●●● ●●●●●●● ●● ●● ●● ●●●● ●●● ●●●●●● ● ●●●● ●● ●● ●●● ● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●● ●●●●●●●● ●●●●● ●● ●●●●●● ●●●●●● ●●● ●●● ●●●●●● ●●●●●●●●●●● ●●●●● ●●●●●●● ●● ●●●●●● ●●●●●●● ●● ●●● ●●●● ●●●● ●● ●● ●●●●●●●●●●●●●●●● ●●●● ● ● ●●●● ● ●● ●●● ●● ● ●●● ● ●●●●● ● ● ●●●● ●● ●● ●● ●●● ● ●● ●●●●● ●●●●●●●● ●●●● ●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●●●● ●●● ●●● ●●●●●● ● ●●●●●●●●● ●●●●●●● ●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ●● ●●●●● ●●●● ●●●●●●● ●●●●● ●●●●● ●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● �� �� ��� ��� ��� ��� ��������������������������������� �������������� � η ( ��� ) - λ Φϕ � � ( ��� ) | ��� θ |= ������ ��� = ���� FIG. 4. Scanning results from DM simulation projected on the coupling − λ Φ φ and m h against DM mass m η . Thehorizontal dashed line indicates the upper bound on | sin θ | , and the dashed curve shows the branching ratio upperbound on invisible Higgs decay. from ∼
70 GeV to ∼
100 GeV.As mentioned above, the s -channel annihilation through the exchange of H/h is part of the dominantDM annihilation channels. In the low mass region DM annihilates mainly into b ¯ b and W + W − final statesthrough s -channel exchange, while in the high mass region it annihilates dominantly into the HH, hh, Hh particles. The transition point appears around m η ∈ (160 , λ Φ φ as thecontact annihilation channels are opened and start to contribute. Comparing to Ref. [6], which excluded themass window of a singlet fermion DM from 10 GeV to 200 GeV, here we find that a significant portion of the1mass range is still viable for a singlet scalar DM.When DM is light enough, m η < m H /
2, it contributes to the invisible decay of the Higgs boson. The decaywidth is given by Γ( H → ηη ) = λ φ v π ( m H + λ Φ φ v ) m η m H (cid:115) − m η m H . (12)The latest searches for invisible Higgs decays by the CMS based on the 5 . , . , . − data collected at7 , ,
13 TeV, respectively, set a combined bound on the invisible branching ratio Br inv < .
24 [20] in theproduction modes of gg F, VBF, ZH , and W H . At the ATLAS, the most stringent bound comes from thestudy on the 8 TeV 20 . − data through the VBF production, Br inv < .
28 [21]. It is easy to check thatinvisible decays are currently less restrictive than direct detection. Finally, the mixing between the Φ and φ fields suppresses all the couplings between the Higgs boson H and SM particles. Using the data on directsearches of the Higgs boson, an upper bound | sin θ | < .
37 has been achieved at 95% C.L. [22]. We can seefrom Fig. 4 that our survived sample points safely avoid this constraint.
IV. LHC PHENOMENOLOGY
As discussed in section II, the scalar particles must be sufficiently heavy to guarantee the expected sponta-neous symmetry breakdown. In contrast, we have chosen the singlet scalar as DM to avoid strong constraintsfrom direct detection. Thus, a natural order of masses for the Z odd particles suggests itself: m ∆ > m F > m η (13)The Z symmetry implies that a heavier Z -odd particle decays into a lighter one plus SM particles. Since wehave assumed a degenerate spectrum for the triplet scalars, the permitted decays are ∆ ++ → E + (cid:96) + , ∆ + → ( E + ν, N (cid:96) + ) , E + → η(cid:96) + plus their conjugates, where (cid:96) is a charged lepton. These decays generate eventswith multiple charged leptons at LHC, while the pure neutral decays ∆ → N ν and N → νη will appear asmissing energy.Now, we focus on searching for the LHC signatures of the new particles and interactions. We followthe standard procedure for event simulation and analysis using a series of software programs. We utilize2 FeynRules [23] to obtain the
UFO [24] model file, which is input into
MadGraph aMC@NLO [25] to generateparton level events. The
NNPDF2.3 [26] LO parton distribution function set passing through
Pythia 6 [27] isused to generate showering and hadronization, successively. The events then pass
Dephes3 [28] for detectorsimulation. After simulation, we use
MadAnalysis5 [29] to obtain various final-state distributions for analysis.Finally,
CheckMATE [30] is used to examine whether the benchmark points we choose are excluded or not at95% C.L. E (cid:43) E (cid:45) (cid:68) (cid:177)(cid:177) (cid:68) (cid:161) (cid:68) (cid:43)(cid:43) (cid:68) (cid:45)(cid:45) (cid:68) (cid:43) (cid:68) (cid:45)
500 1000 1500 20000.0010.010.1110 M (cid:68) , E (cid:72) GeV (cid:76) Σ (cid:72) f b (cid:76) FIG. 5. Cross sections for pair and associated production of ∆ ++ ∆ −− , ∆ ±± ∆ ∓ , ∆ + ∆ − and E + E − at 13 TeV LHCas a function of their masses m ∆ and m E ± . Let us first take a look at the production of heavy particles through electroweak interactions. The crosssections for the dominant pair and associated production of the scalar triplet and fermion doublet are shownin Fig. 5 at 13 TeV LHC. The production of E + E − , ∆ ±± ∆ ∓ , and ∆ ++ ∆ −− will lead respectively to signaturesof dilepton, trilepton, and four-lepton with a large missing transverse energy /E T carried away by the DM3particle η . We discard the dilepton events produced via ∆ + ∆ − production, because its cross section is toosmall compared with E + E − . At the same time, we combine the trilepton and four-lepton signatures (termedmulti-lepton below) to enhance the significance. To summarize, the following two signatures will be studiedin detail: • (cid:96) ± + /E T from E + E − production, • (cid:96) ± + /E T and 4 (cid:96) ± + /E T from ∆ ±± ∆ ∓ and ∆ ++ ∆ −− production,where (cid:96) = e, µ in our definition of a lepton for LHC signatures.The search for dilepton and multi-lepton plus /E T signatures has recently been performed at 13 TeV LHCby ATLAS [31] based on simplified SUSY models. Owing to model dependence, its exclusion line has tobe recast for the model under consideration. Fixing the DM mass at 200 GeV and according to entire cutsin [31], we have used CheckMATE to perform this recasting job, and have confirmed that heavy leptons areexcluded at a mass of 600 GeV. Thus, we choose m E = 650 GeV. Considering the order of masses in eq. (13)and the experimental lower bound on m ∆ [32], we work with the following benchmark point to illustrate thetestability of signatures at LHC: m ∆ = 1000 GeV , m E = 650 GeV , m η = 200 GeV . (14)Since the current 36.1/fb data at 13 TeV LHC has a small significance for the signals under consideration, wepresent our results for the future HL-LHC with an integrated luminosity of 3000/fb at 13 TeV. A. Dilepton Signature
The dilepton signature from the pair production of E ± is as follows: pp → E + E − → η(cid:96) + η(cid:96) − → (cid:96) + (cid:96) − + /E T , (15)where (cid:96) = e, µ for collider simulations. Requiring clean backgrounds, we concentrate on the final states asATLAS did for the dilepton signature. The dominant sources of background are di-bosons ( W Z, ZZ, W W ),4tri-bosons (
V V V with V = W, Z ), top pairs ( t ¯ t ), and top plus boson (mainly from t ¯ tV ) with leptonic decaysof W, Z , all of which we have included at the leading order, i.e., without a K factor. We adopt the sameselection criteria as ATLAS for a more reasonable analysis.We start with some basic cuts: p (cid:96) T >
25 GeV , p (cid:96) T >
20 GeV , m (cid:96)(cid:96) >
40 GeV , | η | < . , (16)where p (cid:96) ( (cid:96) ) T denotes the transverse momentum of the more (less) energetic one in the two charged leptons, m (cid:96)(cid:96) is the dilepton invariant mass, and η is the pseudorapidity. In Fig. 6, the distributions of the numbers ofleptons N ( (cid:96) ) and bottom-quark jets N ( b ) at 13 TeV are shown. In order to make backgrounds cleaner, weapply the following cuts: N ( (cid:96) + ) = 1 and N ( (cid:96) − ) = 1; N ( b ) = 0 , (17)which require exactly a pair of oppositely charged leptons and no appearance of a b -jet to cut the backgroundscoming from t ¯ t and t ¯ tV production. Using Madanalysis5 , we obtain other important distributions shown inFig. 7. As the large DM and fermion masses make the signal deviate significantly from the backgrounds inthe p (cid:96) T and /E T distributions, we impose further cuts on them: p (cid:96) T >
250 GeV , /E T > . (18)In table II we show the cut-flow for the dilepton signature at our main benchmark point in eq. (14)and for dominant backgrounds. Also included are two more benchmark points with slightly larger masses( m ∆ = 1200 GeV and m E = 700 ,
750 GeV), for the purpose of comparison. Although the triplet scalardoes not directly affect the dilepton signal a larger mass helps stabilize the vacuum when the fermion massincreases. The efficiency of the p (cid:96) T and /E T cuts is clearly appreciated, while the t ¯ t and t ¯ tV backgrounds aresignificantly reduced by demanding N ( b ) = 0. As expected, a larger fermion mass m E results in smaller crosssection and signal events. After all the cuts, we have about 1601 .
1, 1367 .
7, and 1020 . FIG. 6. Distributions of N ( (cid:96) ) and N ( b ) at 13 TeV LHC for dilepton signature.Channels No cuts Basic cuts N ( (cid:96) ) = 2 N ( b ) = 0 p (cid:96) T >
250 GeV /E T >
200 GeV1000-650-200 4653 3132.9 3103.1 3015.6 2531.7 1601.11200-700-200 3294 2073.9 2055.8 1994.2 1688.2 1367.71200-750-200 2300 1446.3 1432.0 1386.7 1211.0 1020.3
W Z
W W ZZ V V V t ¯ t t ¯ tV − . Each set of numbers in the first column shows the values of m ∆ ,E,η in GeV ateach benchmark point. FIG. 7. Distributions of p (cid:96) T , m (cid:96)(cid:96) , E T , and /E T at 13 TeV LHC for dilepton signature. B. Multi-lepton Signature
The multi-lepton signature originates from the production of ∆ ++ ∆ −− and ∆ ±± ∆ ∓ and their sequentialdecays: pp → ∆ ++ ∆ −− → E + (cid:96) + E − (cid:96) − → η(cid:96) + (cid:96) + η(cid:96) − (cid:96) − → (cid:96) + (cid:96) − + /E T ,pp → ∆ ±± ∆ ∓ → E ± (cid:96) ± E ∓ ν/N (cid:96) ∓ → η(cid:96) ± (cid:96) ± η(cid:96) ∓ ν/ην(cid:96) ∓ → (cid:96) ± (cid:96) ∓ + /E T . (19)We start again with the basic cuts in Eq. (16), then we select the trilepton and four-lepton events by imposingthe following criteria: N ( (cid:96) ± ) = 2 and N ( (cid:96) ∓ ) = 1 , or N ( (cid:96) + ) = 2 and N ( (cid:96) − ) = 2; N ( b ) = 0 . (20)We demand that the trilepton events contain no like-sign trileptons, and that in the four-lepton events thereare exactly two positively and two negatively charged leptons. From the possibly relevant distributions shownin Fig. 8, we propose the stricter cuts: p (cid:96) T >
350 GeV , /E T >
250 GeV , M ( (cid:96) + (cid:96) + (cid:96) − ) >
800 GeV . (21)Table III shows the cut-flow for the multi-lepton signature at the benchmark point eq. (14), and anothertwo with slightly heavier triplet scalars and for the main backgrounds. The production cross section and thenumber of signal events decrease as m ∆ increases. Because of the relatively small cross sections for the tri-and four-lepton signal events, we provide only the results for the HL-LHC mode. The cuts we employed hereare efficient enough in preserving signals while suppressing backgrounds. The p (cid:96) T cut is still the golden cutfor the multi-lepton case for reducing a large portion of the backgrounds. After the /E T cut, only the W Z events in the backgrounds are significant and are further diminished by a strong cut on the invariant massfor the trilepton system M ( (cid:96) + (cid:96) + (cid:96) − ). The backgrounds in the multi-lepton case are much cleaner than in thedilepton case, however at the cost of less signal events: after all the cuts, there remain only about 16, 7.9,and 5.95 signal events, respectively.We summarize in Table IV our signal and background events together with their significance after all thecuts for both di- and multi-lepton cases. The multi-lepton signal is much cleaner than the dilepton one in8 FIG. 8. Distributions at 13 TeV LHC for multi-lepton signature of N ( (cid:96) ), N ( b ), /E T , M T ( (cid:96) + (cid:96) + (cid:96) − ), M ( (cid:96) + (cid:96) + (cid:96) − ), and p (cid:96) T . Channels No cuts Basic cuts N ( (cid:96) ) cuts N ( b ) = 0 p (cid:96) T >
250 GeV /E T >
200 GeV M ( (cid:96) + (cid:96) + (cid:96) − ) >
800 GeV1000-650-200 56.13 55.67 35.66 34.57 30.55 22.97 16.051050-650-200 26.30 22.0 17.01 16.35 15.10 11.22 7.901100-650-200 18.8 15.76 12.19 11.71 11.03 8.27 5.95
W Z
W W ZZ V V V t ¯ t t ¯ tV − . Each set of numbers in the first column shows the values of m ∆ ,E,η in GeV ateach benchmark point. both aspects of signals and backgrounds, however its significance is much less than the latter. This leaves thedilepton signal as a better search channel for the future HL-LHC. V. CONCLUSION
We have studied the dark matter and LHC phenomenology in a scale-invariant scotogenic model, whichaddresses three issues beyond the standard model in one framework, i.e., neutrino mass, dark matter, andgeneration of the electroweak scale. We incorporated the constraints coming from dark matter relic densityand direct detection and bounds from direct searches and invisible decays of the Higgs boson, and searched forviable parameter space for the most relevant parameters in the scalar potential λ Φ φ , m η . To test the modelfurther at high energy colliders, we proposed to employ the dilepton and multi-lepton signatures and made adetailed simulation at 13 TeV LHC with an integrated luminosity of 3000 fb − . We found that the dilepton0 Benchmark points S B S/ √ S + B dilepton case 1000-650-200 1601.1 2624.25 24.61200-700-200 1367.7 2624.2 21.61200-750-200 1020.3 2624.2 16.9multi-lepton case 1000-650-200 16.05 6.84 3.361050-650-200 7.90 6.83 2.061100-650-200 5.95 6.84 1.66TABLE IV. Summary of numbers of signal and background events after all cuts and significance for dilepton andmulti-lepton cases. channel, mainly due to its large cross section, is the most promising to probe in the future high-luminosityLHC run. ACKNOWLEDGEMENT
CG and SYG are very grateful to Junjie Cao, Liangliang Shang and Yuanfang Yue for their help inconfiguring the package
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