Collider phenomenology of new neutral scalars in a flavored multi-Higgs model

Collider phenomenology of new neutral scalars in a flavored multi-Higgs model P. M. Ferreira,1,2,* João Gonçalves ,3,†Antonio P. Morais ,3,4,‡António Onofre,5,§ Roman Pasechnik ,6,∥and Vasileios Vatellis 3,¶ 1Instituto Superior de Engenharia de Lisboa—ISEL, 1959-007 Lisboa, Portugal 2Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal 3Departamento de Física, Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal 4Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland 5Departamento de Física, Universidade do Minho, 4710-057 Braga, Portugal 6Department of Astronomy and Theoretical Physics, Lund University, SE-223 62 Lund, Sweden (Received 31 January 2023; accepted 14 May 2023; published 30 May 2023) In this work, we propose and explore for the first time a new collider signature of heavy neutral scalars typically found in many distinct classes of multi-Higgs models. This signature, particularly relevant in the context of the Large Hadron Collider (LHC) measurements, is based on a topology with two charged leptons and four jets arising from first and second generation quarks. As an important benchmark scenario of the multi-Higgs models, we focus on a recently proposed Branco-Grimus-Lavoura (BGL) type model enhanced with an Abelian U(1) flavor symmetry and featuring an additional sector of right-handed neutrinos. We discuss how kinematics of the scalar fields in this model can be used to efficiently separate the signal from the dominant backgrounds and explore the discovery potential of the new heavy scalars in the forthcoming LHC runs. The proposed method can be applied for analysis of the statistical significance of heavy scalars’production at the LHC and future colliders in any multi-Higgs model. DOI: 10.1103/PhysRevD.107.095041 I. INTRODUCTION The current basis for our understanding of particle physics is leaning on the theoretical framework of the Standard Model (SM), which was ultimately confirmed by the discovery of the Higgs boson [1,2], whose properties closely match the SM expectations. Despite this, an explanation for neutrino masses, for the observed hierarchies in the fermionic sector and for the existence of dark matter cannot be accommodated in the SM. Possible solutions to the aforementioned questions require a new physics (NP) framework that typically incorporates new scalar fields, including both electrically charged and neutral Higgs states with distinct CP properties. The presence of multiple beyond the SM (BSM) particles can lead to interesting phenomenology at collider experiments, with a multitude of possible different final states. Indeed, over the years, various searches have been conducted by the experimental community, including decays that involve vector bosons [3–5], multijets [6,7], and charged leptons [8–10]. However, the presence of new scalars that interact and/or mix with one another lead to additional topologies, which have not been covered by experiments. In particular, decay chains featuring various BSM fields in internal propagators offer a solid physics case to test generic multi-Higgs models at the LHC. In this regard, while there have been a number of recent searches [11,12],theyhave been limited in scope, especially when compared to topologies where a BSM field decays immediately into a pair of SM particles. Therefore, the proposal in this article aims at filling such a gap and enlarging the explored parameter space in view of the LHC run-III, which is already collecting data, and of the upcoming High-Luminosity (HL) upgrade. In this regard, we consider a particular example of a multi-Higgs model featuring the Branco-Grimus-Lavoura (BGL) flavor structure [13] as a benchmark for our *[email protected] †[email protected] ‡[email protected] §[email protected] ∥[email protected] ¶vasileios.v[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 107, 095041 (2023) 2470-0010=2023=107(9)=095041(21) 095041-1 Published by the American Physical Society phenomenological analysis. Besides the two Higgs doublets and a complex singlet scalar charged under a global flavor U(1) symmetry, the model also incorporates an additional sector with three generations of right-handed neutrinos and a type-I seesaw mechanism for the generation of light neutrino masses, which was recently explored by some of the authors in [14]. In this work, a comprehensive phenomenological analysis of this model has been performed, where electroweak (EW) precision, Higgs, flavor, and the existing collider observables were scrutinized. A selection of the phenomenologically validated points are then used as benchmark scenarios for exploring the discovery potential of the heavy neutral BSM scalars in the considered NP framework. This article is organized as follows. In Sec. II,we briefly discuss a particular benchmark NP model that is chosen for our numerical analysis. In Sec. III,we give a general overview of the current experimental status regarding the search for heavy BSM scalar states at collider experiments. In Sec. IV,wediscussanumber of processes involving such states that have been so far a subject of little or no attention in the literature. Here, we propose a new promising signature featuring a final state with four jets and a pair of charged leptons and with multiple neutral BSM scalars in internal propagators. In Sec. V, we present the details of the newly developed methodology, while in Sec. VI, we calculate the statistical significance for the discovery/exclusion of new Higgs states. Finally, in Sec. VII, we conclude and summarize our results. II. BENCHMARK MODEL The recently proposed BGL-like next-to-minimal two Higgs doublet model (BGL-NTHDM) has been thoroughly discussed both from the theoretical and phenomenological points of view in [14]. Here, for completeness of information, we briefly present its key features. The BGL structure results from a global flavor U(1) symmetry, broken once the Higgs doublets and the scalar singlet develop vacuum expectation values (VEVs) [13]. The allowed Yukawa interactions read as −LYukawa ¼q0 LΓaΦad0 Rþq0 LΔa ˜ Φau0 Rþl0 LΠaΦae0 R þl0 LΣa ˜ ΦaνRþ1 2νc RðAþBSþCSÞνRþH:c:; ð2:1Þ where the index a¼1, 2 runs over the two doublets Φaand all the matrices are written in the flavor basis, with Γa,Δa being the Yukawa matrices for down-quarks, up-quarks, and Πa,Σaare the charged lepton and neutrino Yukawa matrices, respectively. Besides, here Band Care the Majorana-like Yukawa couplings with a complex SU(2) singlet scalar field S, whereas Ais a Majorana mass term for the right-handed neutrinos. We adopt the notation where ˜ Φ≡iσ2Φand the superscript 0 indicates that the fields are written in the gauge basis. The textures of the quark-sector Yukawa matrices are given as Γ1∶0 B @ ××× ××× 000 1 C A;Γ2∶0 B @ 000 000 ××× 1 C A; Δ1∶0 B @ ××0 ××0 000 1 C A;Δ2∶0 B @ 000 000 00× 1 C A;ð2:2Þ while those of the charged and neutral leptons read as Π1;Σ1;B¼0 B @ ××0 ××0 000 1 C A; Π2;Σ2¼0 B @ 000 000 00× 1 C AA¼0; C¼0 B @ 00× 00× ××0 1 C A:ð2:3Þ In the gauge basis, the fermion mass matrices can be cast as M0 u≡1 ffiffiffi 2 pðv1Δ1þv2Δ2Þ; M0 d¼1 ffiffiffi 2 pðv1Γ1þv2Γ2Þ; M0 e¼1 ffiffiffi 2 pðv1Π1þv2Π2Þ;ð2:4Þ where u,d,edenote up, down quarks, and charged leptons, while v1and v2are the VEVs arising from the first and second Higgs doublets, respectively. Such mass matrices can be rotated to the physical basis via bi-unitary transformations. Generically, each fermion f¼u,d,ecan be diagonalized as follows: Nf¼U† fLM0 fUfR;ð2:5Þ where Uare the unitary matrices, with the subscripts L(R) denoting left (right) chirality, and Nfare the diagonal fermion mass matrices. It follows from the textures in Eqs. (2.2) and (2.3) that both the charged lepton and up-quark Yukawa matrices can be simultaneously P. M. FERREIRA et al. PHYS. REV. D 107, 095041 (2023) 095041-2 diagonalized, and therefore, there are no tree-level FCNCs for both sectors. For down-quarks, Γ1and Γ2can not be simultaneously diagonalized, and therefore, FCNCs will be present readily at tree level, mediated by new BSM scalars. As in the standard BGL construction [13], the effect of the global flavor symmetry results in a suppression of FCNCs by the off diagonal elements of the Cabibbo-KobayashiMaskawa (CKM) matrix. While not particularly relevant for the current study, the model also features an additional sector of massive righthanded neutrinos, which, for completeness, is briefly outlined below (for more details, see [14]). Defining the neutrino fields as n0 L≡ν0 L νc R;ð2:6Þ the mass matrix can be cast in the standard seesaw format as M≡0mD mT DMR;ð2:7Þ where one defines mD≡1 ffiffiffi 2 pðv1Σ1þv2Σ2Þ; MR≡AþvS ffiffiffi 2 pðBþCÞ;ð2:8Þ and vSis the VEV in the real component of the complex scalar singlet. As it was noted in our previous work [14], the model possesses enough freedom to simultaneously fit the Pontecorvo-Maki-NakagawaSakata (PMNS) neutrino mixing matrix and the active neutrino mass differences. The scalar potential is defined as V¼V0þV1, with V0¼μ2 ajΦaj2þλajΦaj4þλ3jΦ1j2jΦ2j2þλ4jΦ† 1Φ2j2 þμS 2jSj2þλ0 1jSj4þλ0 2jΦ1j2jSj2þλ0 3jΦ2j2jSj2; V1¼μ2 3Φ† 2Φ1þ1 2μ2 bS2þa1Φ† 1Φ2Sþa2Φ† 1Φ2S† þa3Φ† 1Φ2S2þH:c:ð2:9Þ While V0contains the quartic couplings (λ1;2;3;4;λ0 1;2;3) and the quadratic mass terms (μ2 1;2;S), V1contains the soft U(1)-breaking interactions (μ3;μb;a 1;2), as well as the quartic coupling between the singlet and the two Higgs doublets (a3) invariant under the U(1) flavor symmetry. Once the scalars develop VEVs, six physical states emerge, including three CP-even neutral scalars H1,H2, and H3, with H1corresponding to the SM-like Higgs boson, two CP-odd neutral scalars A2and A3, and a charged scalar H. III. HEAVY HIGGS PARTNERS AT THE LHC Multi-Higgs models provide a plethora of new scalar states that can be either charged or neutral and possess CP-even or CP-odd properties, leading to characteristic signatures in collider measurements. A summary of the most recent searches, performed in years of 2020 and 2021 at both the CMS and ATLAS experiments, is shown in Table I. Various combinations of final states have been searched for, particularly, for the neutral fields commonly referred to as Higgs partners, Hand A, including final states with light jets, bjets and charged leptons. It is also interesting to note that searches involving decays into other BSM fields are also included, such as A→HZ0 in [12] and H→AA in [10]. These channels are of particular relevance for the Higgs partners’search since interactions between different BSM scalars are common to most (if not all) multi-Higgs extensions of the SM. We do note that in [10], the search focused in the low-mass regime for the CP-odd scalar, complementing the high-mass regime in ATLAS for the processes A→τþτ−and A→τþτ−bb [8] and A→γγ [3], has been performed. Note, the CP-odd (A) and CP-even (H) scalars often share the same final states as highlighted in Table I. Regarding the charged Higgs bosons, some searches have also been reported recently, with a focus on the tbHþ vertex, either through decay into tb [17,18] or considering a charged Higgs state produced via a top/antibottom pair [7]. An additional search focusing on the vector-boson fusion (VBF) mechanism was also performed by CMS [19]. The top/antibottom channel appears to be the preferred channel in the search for charged Higgs bosons, corroborated by previous searches in [20–26]. Indeed, these studies indicate that for a charged Higgs state with a mass below that of the top quark, it will be predominantly produced via top quark decays, whereas heavy charged scalars would be typically produced in association with a top quark [27,28]. For masses below the kinematic threshold for the production of a top quark, the decay modes H→τντbecome dominant [27,28]. Besides the searches in [3,10] (see also Table I), the preference in the literature has been given to the BSM scenarios where new scalars purely decay into SM states. However, multi-Higgs models enable interactions among different BSM scalars such that a richer set of final states involving multiple scalars is possible and must be considered as discussed below. IV. LHC PHENOMENOLOGY OF BSM SCALARS The model considered in this article was previously validated in [14], where several points consistent with EW, Higgs, collider, and flavor physics observables were COLLIDER PHENOMENOLOGY OF NEW NEUTRAL SCALARS IN …PHYS. REV. D 107, 095041 (2023) 095041-3 found.1In particular, it was shown that scenarios with light scalars, i.e., being not too far above the Higgs boson mass, are still allowed and can be potentially probed at the LHC run-III or its high-luminosity (HL) phase. Testing these possibilities is, therefore, of phenomenological interest. Our focus here is on either single or double-production channels for the new physics states present in the model, in particular, charged-, (H), neutral- (H2and H3), and pseudoscalars (A2 and A3). While a multitude of topologies for collider searches of these states can be proposed, our current analysis is dedicated to a single production process featuring the final states of four jets and two charged leptons and leaving other possible topologies for a future work. A. CP-even neutral scalars The model considered in this work features two CP-even scalars H2;3with different couplings, masses (satisfying mH3>m H2) and branching ratios (BRs) for separate decay modes. In what follows, both H2;3are assumed to be heavier than the Higgs boson (denoted as H1). Here, we are only interested in channels with sizeable BRs by requiring them to be greater than 20%. As it was previously shown in [14], the decay channels with small BRs typically result in small cross sections, even below the allowed sensitivity of the HL phase of the LHC. With this in mind, we have TABLE I. A summary of the most recent searches conducted between the years of 2020 and 2021 by the ATLAS and CMS experiments at the LHC. Here, H1is the SM-like Higgs boson, Arepresents a neutral pseudoscalar, His a CP-even neutral scalar, and His a charged scalar. Here, “BR”stands for the corresponding branching ratio. As for the production mechanisms of the Higgs states, “ggF”indicates the gluon-gluon fusion, while “VBF”corresponds to the vector boson fusion. Scalar field Decay channel Mass limits (GeV) Comments References AA→τþτ−[200, 2500] Limits given in terms of σ×BR [8] A→τþτ−b¯ b[200, 2500] Limits given in terms of σ×BR [8] H1→AZ0[0.5, 4.0] Hadronic decays with BRðA→ggÞ¼1or BRðA→s¯ sÞ¼1[11] AA →b¯ bb¯ b[20, 60] Limits given in terms of σ×BR [15] Associated Z0production A→HZ0 Limits mHvs mA[12] Multiple channels 2l2b,2l4j,2l4b A→γγ [160, 2800] Limits given in terms of σ×BR [3] HH→τþτ−[200, 2500] Limits given in terms of σ×BR [8] H→τþτ−b¯ b[200, 2500] Limits given in terms of σ×BR [8] HH →b¯ bb¯ b[260, 1000] Vector-boson fusion [6] Coupling constraints H→VV [300, 3200] ggF First two for Kaluza-Klein (KK) massive gravitons, third for radion. [4] [300, 760] VBF Vindicates vector boson [300, 2000] ggF H→Z0Z0[400, 2000] Various widths assumptions [5] VBF and gluon fusion Fully and semileptonic H→γγ [160, 2800] Limits given in terms of σ×BR [3] HðH1Þ→AA [16, 62] H1→AA →b¯ bμþμ−[16] [15, 60] H1→AA →lþl−lþl−[9] [3.6, 21] HðH1Þ→AA →μþμ−τþτ−[10] Hpp →tbHþ[200, 2000] In both: Hþ→tb [17] [200, 3000] Constraints of m Hvs tan β(both) [18] Limits as σ× BR (both) H→WZ0[200, 1500] Considers VBF production [19] Limits as σ×BR H→cs [80, 160] Assumes BRðH→csÞ¼1[7] Limits as BRðt→HþbÞvs mHþ 1Some of the data and the UFO model (both P ython2 and P ython3 versions) are publicly available in one of the author’s GitHub page (see https://github.com/Mrazi09/BGL-ML-project). The numerical values for the various couplings/masses is also shown in Appendix B. P. M. FERREIRA et al. PHYS. REV. D 107, 095041 (2023) 095041-4 selected six preferred scenarios among the valid parameter-space points of [14], for which the characteristics of the next-tolightest Higgs state H2—the mass and the dominant BR—read as Benchmark H1∶MH2¼599.09 GeV;BRðH2→A2Z0Þ¼90.7%; Benchmark H2∶MH2¼286.92 GeV;BRðH2→H1H1Þ¼66.7%; Benchmark H3∶MH2¼527.55 GeV;BRðH2→A2Z0Þ¼52.1%; Benchmark H4∶MH2¼397.45 GeV;BRðH2→A2Z0Þ¼41.7%; Benchmark H5∶MH2¼215.56 GeV;BRðH2→c¯ cÞ¼56.0%; Benchmark H6∶MH2¼402.81 GeV;BRðH2→A2Z0Þ¼89.6%; that are worth of a dedicated phenomenological analysis. Note, however, that single H2production processes with the dominant decay channels corresponding to those of the benchmarks H1, H2, H3, H4, and H6 have already been studied at the LHC. In particular, the most recent H2 searches in the H1, H3, H4, and H6 channels are already indicated in Table I. Besides, final states with at least two b jets were considered for the benchmark H2 [29–34]. Notice that the benchmark H5, whose BR into a pair of charm quarks is 56%, represents a rather attractive physics case motivating the searches involving light jets in the final state. Of course, a fully hadronic signal is not optimal in the context of the LHC since the SM multijet background is expected to dominate over the signal. While fully hadronic final states have been searched for at the ATLAS experiment (see, e.g., [6,35–37]), such a signature can become a lot cleaner, e.g., at future lepton colliders. With this in mind, one may consider two possible cases—double and single H2production—separately. For the latter case, the focus is typically on associated production with a Z0boson (see Fig. 1) subsequently decaying into a pair of leptons, thus minimizing the relevant QCD backgrounds. For the former case, all leading-order (LO) diagrams are shown in Fig. 2,where both sand tchannels contribute to the matrix element. Additionally, the gluon-gluon fusion mode is relevant and needs also to be included in the calculation. The major ingredients of the background are represented by diboson, V þjets and t¯ tproduction modes that will be carefully considered in what follows. FIG. 1. The LO Feynman diagrams for production of the single H2CP-even scalar in the schannel (top two diagrams) and the t channel (bottom diagram) processes. Here, qdenotes the SM quarks, gcorresponds to a gluon, and Ai¼2;3—to CP-odd scalars, while j stands for a jet originating from the physical quarks of first and second generations u,s,d,c, and their antiparticles. COLLIDER PHENOMENOLOGY OF NEW NEUTRAL SCALARS IN …PHYS. REV. D 107, 095041 (2023) 095041-5 We can now focus our attention on the heaviest scalar state, H3. Again, we are interested in points with BRs higher than 20%. Considering the same benchmark scenarios as above, Benchmark H1∶MH3¼907.61 GeV;BRðH3→A2A2Þ¼79.0%; Benchmark H2∶MH3¼741.53 GeV;BRðH3→HþW−Þ¼22.3%; Benchmark H3∶MH3¼724.46 GeV;BRðH3→A2A2Þ¼35.6%; Benchmark H4∶MH3¼705.62 GeV;BRðH3→A2Z0Þ¼33.9%; Benchmark H5∶MH3¼626.92 GeV;BRðH3→HþH−Þ¼50.3%; Benchmark H6∶MH3¼439.59 GeV;BRðH3→A2Z0Þ¼66.9%; one can deduce topologies that are distinct from those of the H2scalar. While a number of interesting channels can be derived from here, one may disregard the benchmarks H1 and H3 as the corresponding decay channel H3→A2A2 has already been studied in dedicated experimental analyses [9,10,16]. Additionally, we note that for the benchmarks H4 and H6, there have already been searches performed by ATLAS [12] focusing on the inverse channel, A→HZ0. Our current work represents a novel analysis of H3production in the benchmarks scenarios H2 and H5 whose final-state topologies, to the best of our knowledge, have not yet been investigated by LHC experiments. Indeed, the corresponding decays are of great interest as they allow us to simultaneously probe the masses and/or couplings of both the neutral and charged scalars, with the latter decaying before reaching the detector. For this purpose, one should consider the largest corresponding BRs which, for the benchmark H2, read as BRðHþ→A2WþÞ¼72.6%;BRðA2→c¯ cÞ¼97.3%: Combining this information, the proposed topology is shown in Fig. 3. As one can notice, this results in a rich final state featuring four jets originating from quarks, one muon and a neutrino that acts as missing transverse energy (MET) in the detector. Besides, note that three of the new scalars contribute in the internal propagators as virtual states within the same topology, thus, opening up a new possibility to constrain the charged, the CP-even and the CP-odd sectors with a single process. FIG. 2. The LO Feynman diagrams for pair-production of the H2CP-even scalars in the schannel (top two diagrams) and the t channel (bottom diagram) processes. Here, qdenotes the SM quarks, gcorresponds to a gluon, and Hi¼1;2;3—to CP-even scalars, while jstands for a jet originating from the physical quarks of first and second generations u,s,d,c, and their antiparticles. P. M. FERREIRA et al. PHYS. REV. D 107, 095041 (2023) 095041-6 In order to ensure that H3and Hare produced on shell, one requires the following mass hierarchy: MH3>M HþþMW;M Hþ>M A2þMW:ð4:1Þ For the benchmark scenario H2, the condition (4.1) is satisfied, with MHþ¼448.52 GeV and MA2¼188.01 GeV. Based on the ATLAS analysis [38], for this type of topology, the main backgrounds include t¯ tþjets, Vþjets (with V¼W;Z0), and single top production. An additional sizeable contribution from fakes (events with either jet or photon that are mistagged as a lepton) and nonprompt leptons (leptons that originate from decays from the hadronized quarks in the jets) is also relevant. For the benchmark H5, the neutral scalar would decay into a pair of charged Higgs bosons. However, based on the available branching ratios, the dominant decays would proceed into a c¯ spair, with BRðHþ→c¯ sÞ¼92.8%, resulting in an undesirable fully hadronic final state. B. CP-odd neutral scalars In addition to the CP-even states discussed above, our framework features two heavy CP-odd fields, A2and A3 (with the mass hierarchy of mA2<m A3) whose signal and background topologies will also be analyzed in this work. Again, here we focus on specific topologies suggested by A2;3decay modes with the largest BRs. Starting with the lightest one, A2, the corresponding masses and the dominant BRs for the benchmark scenarios from H1 to H6 read Benchmark H1∶MA2¼314.99 GeV;BRðA2→c¯ cÞ¼86.5%; Benchmark H2∶MA2¼159.20 GeV;BRðA2→c¯ cÞ¼97.3%; Benchmark H3∶MA2¼205.41 GeV;BRðA2→c¯ cÞ¼97.3%; Benchmark H4∶MA2¼188.01 GeV;BRðA2→c¯ cÞ¼97.3%; Benchmark H5∶MA2¼177.49 GeV;BRðA2→c¯ cÞ¼96.9%; Benchmark H6∶MA2¼217.36 GeV;BRðA2→c¯ cÞ¼96.7%: We notice that for these scenarios that we have picked, A2 almost always decays into c¯ cpair. So far, most searches at the LHC have focused on the lepton channel (see Table I and references therein) and mostly on the low-mass domain [9,16],mA<62 GeV, i.e., well below the A2 masses in the above benchmarks. Please do note that the pattern of high BRs to a pair of cquarks should not be interpreted as a feature of the model. Indeed, there are valid parameter points where such decays are not dominant. In what follows, we would like to consider the topologies involving A2decays into light jets. In particular, we consider very similar topologies to those already shown in Figs. 1and 2, with the replacement H2→A2. Various LO diagrams that contribute to such a signal are shown in Figs. 4and 5. Since the final states are identical to those of H2, the main irreducible backgrounds remain the same as for the H2topologies considered above. For the heavier CP-odd state, A3, the dominant BRs read as FIG. 3. The LO Feynman diagrams for production of the single H3CP-even scalar. Here, qdenotes the SM quarks and jstands for a jet originating from the physical quarks of first and second generations u,s,d,c, and their antiparticles. While not explicitly shown here, the gluon-gluon fusion contribution may play an important role in the cross section and hence has been included in the numerical analysis. COLLIDER PHENOMENOLOGY OF NEW NEUTRAL SCALARS IN …PHYS. REV. D 107, 095041 (2023) 095041-7 FIG. 5. The LO Feynman diagrams for pair production of the A2CP-even scalars in the schannel (top two diagrams) and the tchannel (bottom diagram) processes. Here, qdenotes the SM quarks, gcorresponds to a gluon, and Hi¼1;2;3—to CP-even scalars, while jstands for a jet originating from the physical quarks of first and second generations u,s,d,cand their antiparticles. FIG. 4. The LO Feynman diagrams for production of the single A2CP-odd scalar in the schannel (top two diagrams) and the tchannel (bottom diagram) processes. Here, qdenotes the SM quarks, gcorresponds to a gluon, and Hi¼1;2;3—to CP-even scalars, while jstands for a jet originating from the physical quarks of first and second generations u,s,d,c(and their antiparticles) and l¼e,μdenotes the first and second generation charged leptons (and their antiparticles). P. M. FERREIRA et al. PHYS. REV. D 107, 095041 (2023) 095041-8 Benchmark H1∶MA3¼955.09 GeV;BRðA3→A2H2Þ¼24.5%; Benchmark H2∶MA3¼566.75 GeV;BRðA3→H2Z0Þ¼44.4%; Benchmark H3∶MA3¼707.60 GeV;BRðA3→HþW−Þ¼36.1%; Benchmark H4∶MA3¼611.07 GeV;BRðA3→H2Z0Þ¼38.78%; Benchmark H5∶MA3¼683.60 GeV;BRðA3→HþW−Þ¼27.1%; Benchmark H6∶MA3¼772.32 GeV;BRðA3→A2H2Þ¼59.6%: Combining this information with that of Table I, we note that the preferred channels in the search for A3would rely on A3→A2H2(in the benchmark scenarios H1 and H6) or A3→HþW−(in the benchmark scenarios H3 and H5) decays. The remaining benchmarks H2 and H4 result in topologies that have been probed already by ATLAS Collaboration in [12]. Both dominant A3decay modes in H1/H6 and H3/H5 are interesting since the vast majority of analyses performed at the LHC so far typically do not consider decay channels that involve multiple BSM particles in the internal propagators (the existing examples are listed in Table I), which results in considerably weaker constraints on these decay modes. In our analysis, we first focus on the decay A3→A2H2 which is dominant in the benchmark scenarios H1 and H6. Then, combining with the information on the BRs for H2 and A2shown above, we deduce that the optimal final state in this case would be 4jþ2l, where we assume a Z0boson decaying into a pair of leptons. The LO Feynman diagram for such a topology is shown in Fig. 6, with the background being dominated by Z0þjets, t¯ tand diboson production processes. C. Charged scalars For each of the six considered benchmark points, the masses and the largest BRs for the charged scalar Hread as Benchmark H1∶MH¼566.40 GeV;BRðHþ→A2WþÞ¼93.6%; Benchmark H2∶MH¼411.94 GeV;BRðHþ→A2WþÞ¼72.6%; Benchmark H3∶MH¼524.70 GeV;BRðHþ→A2WþÞ¼91.0%; Benchmark H4∶MH¼448.52 GeV;BRðHþ→A2WþÞ¼88.8%; Benchmark H5∶MH¼156.89 GeV;BRðHþ→c¯ sÞ¼92.8%; Benchmark H6∶MH¼330.88 GeV;BRðHþ→A2WþÞ¼63.8%: Here, one notices that there are two interesting distinct scenarios that can be probed at the LHC. In particular, the benchmarks scenarios H1 to H4 and H6 result in the following process: pp →Hþ→A2Wþ→jjlþνl.Note that the majority of the experimental studies focusing on the charged Higgs boson search assume its production via the tbHþcoupling. The more interesting and novel channels that can be considered are those where Hþis produced via lighter quarks as shown by a LO diagram in Fig. 7. Additionally, for the benchmark H5, the Hþcoupling to light quarks, c¯ s, is dominant. Indeed, as noted throughout this work, a substantial part of the preferred decay channels of the new scalars tend to be those involving decays to light quarks, which are not well constrained by collider measurements so far. The main irreducible background is expected to be dominated by W þjets, diboson, and t¯ tproduction processes. D. Production cross sections For the topologies introduced above, we have computed the corresponding production cross sections for protonproton collisions at the centre-of-mass energy of ffiffiffi s p¼ 14 TeV using M ad G raph. These results are listed in Table II. One can readily observe that every single topology for the selected benchmarks shown in Table II can potentially be probed both at the LHC Run-III and at the HL-LHC. Note that all considered signal processes are within the reach of the current data for the run-II luminosity of L¼139 fb−1. It is then possible to look for these signals without having to wait for new batches of data. In fact, we can deduce from such a simple calculation that the single A3production process is the most challenging for future searches, with the smallest amount of expected events, i.e., N¼17 at the LHC run-III and N¼178 at the HL-LHC. On the other hand, production COLLIDER PHENOMENOLOGY OF NEW NEUTRAL SCALARS IN …PHYS. REV. D 107, 095041 (2023) 095041-9 relaxing such selection criteria from MðjÞ>15 GeV to MðjÞ>10 GeV and from ΔM<25 GeV to ΔM<35 GeV, we notice from Table Vthat both benchmark scenarios can now be potentially probed at both run-III and HL-LHC. Although, note that for benchmark H6, only one event is expected at run-III. Of course, these conclusions are taken from a limited sample of benchmark points, and it is possible that other coupling/mass combinations may lead to different conclusions. At any rate, the results from the considered benchmarks already indicate a nonzero number of expected events, which can enable one to set exclusion limits for the production of these scalars in different regions of the phase space. VI. STATISTICAL SIGNIFICANCE WITH DEEP LEARNING The determination of the statistical significance of the considered benchmark points is performed using evolutionary algorithms based on deep learning (DL) methods. The employed methodology is the same as in previous works by some of the authors [51–53], and thus in what follows, we only present the key features. As detailed in [52], the inputs of the DL evolutionary method consist of normalized kinematic and angular distributions, which in the current analysis are shown in Fig. 9. Additionally, due to the unbalanced nature of the various classes, i.e., some having more data entries than others, we use a SMOTE algorithm [54] to balance our dataset and use it as input data for the neural networks (NNs). The latter are constructed using T ensor F low [55]. Last but not least, the calculation of the statistical significance is performed using as metric the Asimov significance, defined as ZA¼2ðsþbÞlnðsþbÞðbþσ2 bÞ b2þðsþbÞσ2 b −b2 σ2 b ln1þσ2 bs bðbþσbÞ1=2 ;ð6:1Þ where sis the number of signal events, bis the number of background events and σbis the variance of the background events, with systematic uncertainties of 1%.3 FIG. 12. Statistical significance with the mass cuts MðjÞ>15 GeV and ΔM<25 GeV. Results are shown for an integrated luminosity of 3000 fb−1and for the NN architecture indicated in Table VIII of Appendix A. FIG. 13. Statistical significance with the mass cuts MðjÞ>10 GeV and ΔM<35 GeV. Results are shown for an integrated luminosity of 3000 fb−1and for the NN architecture indicated in Table VII of Appendix A. 3Based on the European strategy for particle physics (see [56]), both ATLAS and CMS, a guideline of 1%–1.5% systematic was set. Additionally, a recent update [57] improved this estimate to 0.83% based on the collected data from run-II, making a global 1% a realistic benchmark scenario for both run-III and the HL phase. We varied by a factor 2 around 1% systematics and have not found any significant deviations from the calculated values reported in the text. P. M. FERREIRA et al. PHYS. REV. D 107, 095041 (2023) 095041-16 In what follows, we determine the statistical significance for our benchmark scenarios, considering first the scenario H1 and the HL phase of the LHC. We show the result in terms of the cut on the classifier score in Figs. 12 and 13. The numerical values for the employed cuts are determined by the NN and can be interpreted as a label for identifying whether a set of features can be classified either as signal or as a background event. From here, we also indicate the best Asimov significance obtained for each of the metrics, whose values for both of the employed cuts read as ðMðjÞ>15=ΔM<35 GeVÞ∶ZA¼19.89σ ðMðjÞ>10=ΔM<25 GeVÞ∶ZA¼10.36σ:ð6:2Þ Notice that, for both cases, we obtain a statistical significance well beyond 5σ, suggesting that the benchmark H1 is a good candidate to be tested at the LHC. One can also confirm that more lenient constraints lead to a greater significance. Indeed, while more restrictive constraints on the phase space result in further reduced backgrounds, the expected number of events for the signal becomes larger when more lenient cuts are considered. Combining this with the good separation obtained between the background and the signal, see Fig. 11,theoverall significance then becomes larger. For the case of the benchmark H6, at the HL phase, the statistical significance reads as ðMðjÞ>15=ΔM<35 GeVÞ∶ZA¼5.12σ ðMðjÞ>10=ΔM<25 GeVÞ∶ZA¼2.02σ;ð6:3Þ from where we conclude that for the more restrictive cuts, we can only probe this point with a significance of 2.02σ, whereas for more lenient ones, we obtain a statistical significance of 5.12σ. We have also found that this method works more efficiently for larger masses of the scalar fields inside the decay chain, as is the case of H1 in comparison to H6. This results from the fact that, for the lighter scalars’ scenarios, the cut on the invariant masses of the light jets can severely reduce the cross section. For completeness, we also extended this analysis into luminosities of 300 fb−1and 1000 fb−1presenting the results in Table VI. In particular, we have found that the benchmark H1 can already be tested at run-III, with a significance of up to 5.48σor 1.96σ, whether we consider the less restrictive or the more lenient selection criteria, respectively. For benchmark H6, on the other hand, it can only be tested at the HL phase of the LHC. VII. CONCLUSION In this article, we have conducted a collider phenomenological analysis using as an example model a BGL version of the NTHDM. We have focused on a potential signal topology characterized by two charged leptons and four jets in its final state. Based on Monte Carlo generated datasets, we have demonstrated that one can use the massdifference information between two pairs of the jets in order to efficiently reconstruct all intermediate particles, while simultaneously damping the main irreducible backgrounds, such as Z0þjets. We have used the TMVA framework to identify the ten best observables that offer the greatest discriminating power between signal and background distributions. These were subsequently used as features in a deep-learning evolution algorithm engineered to find the neural network that best optimizes the statistical significance of a hypothetical discovery. We have employed such methods on preselected benchmark points, which are consistent with electroweak and flavor observables’constraints, and whose cross sections are high enough for it to be accessible at future collider experiments. In this regard, we found that one can potentially test the considered model with a statistical significance greater than 5 standard deviations, for both the LHC run-III and its future HL upgrade, depending on the benchmark and employed selection criteria. More importantly, it is worth mentioning that the main take-away message of this study is that both the methods TABLE VI. The statistical significance for each of the benchmarks/cuts mentioned in the text. Benchmark points that are not shown indicate that the signal cross section is too low for events to be produced at any given luminosity. Benchmarks H1 MðjÞ>15 GeV=ΔM<25 GeV H6 MðjÞ>15 GeV=ΔM<25 GeV 300 fb−1ZA1.96σ0.27σ 1000 fb−1ZA4.81σ0.80σ 3000 fb−1ZA10.36σ2.02σ Benchmarks H1 MðjÞ>10 GeV=ΔM<35 GeV H6 MðjÞ>10 GeV=ΔM<35 GeV 300 fb−1ZA5.48σ1.29σ 1000 fb−1ZA11.49σ2.95σ 3000 fb−1ZA19.89σ5.12σ COLLIDER PHENOMENOLOGY OF NEW NEUTRAL SCALARS IN …PHYS. REV. D 107, 095041 (2023) 095041-17 and the considered signal topology are model independent and can be applied to any multiscalar scenario featuring, at least, three new physical neutral Higgs bosons allowed to couple in a triple vertex interaction. ACKNOWLEDGMENTS We thank Felipe F. Freitas for the fruitful discussions during the initial stages of this project. J. G., A. P. M., and V. V. are supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT—Fundação para a Ciência e a Tecnologia), Grants Reference No. UIDB/04106/2020 and No. UIDP/04106/2020. A. P. M., J. G. and V. V. are supported by the Projects No. PTDC/FIS-PAR/31000/2017 and No. CERN/FIS-PAR/0021/2021. A. P. M. is also supported by national funds (OE), through FCT, I. P., in the scope of the framework contract foreseen in the numbers 4, 5, and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. J.G is also directly funded by FCT through a doctoral program grant with the Grant Reference No. 2021.04527.B.D. R. P. is supported in part by the Swedish Research Council grant, Contract No. 2016-05996, as well as by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 668679). P. F. is supported by Centro de Física Teórica e Computacional - Universidade de Lisboa (CFTC-UL) under FCT Contracts No. UIDB/00618/2020, No. UIDP/00618/2020, and by the Projects No. CERN/FISPAR/0002/2017 and No. CERN/FIS-PAR/0014/2019. A. O. is supported by the FCT Project No. CERN/FIS-PAR/ 0029/2019. APPENDIX A: NEURAL NETWORK ARCHITECTURES APPENDIX B: NUMERICAL BENCHMARKS Here, we write down the numerical values for the benchmark points that were studied in this work. The values indicated here are not exact, but rounded to 3 significant figures. For exact values, as well as some complementary information, see https://github.com/Mrazi09/BGL-ML-project. Quadratic mass terms are in units of GeV2(μ2 1,μ2 2,μ2 3,μ2 b, and μ2 S) and masses are in units GeV (MH2,MH3,MA2,MA3,MH). The VEV of the singlet field Sis in unit of GeV (vS), (i) H1 λ1¼2.413;λ2¼0.104;λ3¼1.515;λ4¼4.224;λ0 1¼0.0257 λ0 2¼−2.721;λ0 3¼−0.0397;μ2 1¼1.973 ×107;μ2 2¼2.799 ×105 μ2 3¼−3.132 ×106;μ2 S¼−8.430 ×106;μ2 b¼−2.837 ×105 a3¼0.433;v S¼3781.31;M H2¼599.085;M H3¼907.607 MA2¼315.991;M A3¼955.091;M H¼566.363.ðB1Þ TABLE VIII. NN architectures that were found by the evolutionary algorithm for the benchmarks H1/H6. for the selection criteria MðjÞ>10 GeV and ΔM<35 GeV. Architecture Neurons: 256 for input and hidden layers and 6 for output; Number of layers: 1 input þ3hidden þ1output; Regularizer: L1L2; Initializer: For input and hidden layers, VarianceScaling with normal distribution and in fan in mode. Output layer, VarianceScaling with uniform distribution and in fan average mode; Activation functions: tanh for input/hidden layers. Sigmoid for output layer Optimizer: Adam TABLE VII. NN architectures that were found by the evolutionary algorithm for the benchmarks H1/H6. for the selection criteria MðjÞ>15 GeV and ΔM<25 GeV. Architecture Neurons: 1024 for input and hidden layers and 6 for output; Number of layers: 1 input þ2hidden þ1output; Regularizer: L1L2; Initializer: For input and hidden layers, VarianceScaling with uniform distribution and in fan in mode. Output layer, VarianceScaling with uniform distribution and in fan average mode; Activation functions: Sigmoid Optimizer: Adam P. M. FERREIRA et al. PHYS. REV. D 107, 095041 (2023) 095041-18 (ii) H2 λ1¼4.123;λ2¼0.133;λ3¼−1.229;λ4¼1.996;λ0 1¼0.330 λ0 2¼4.099;λ0 3¼−0.248;μ2 1¼−1.030 ×106;μ2 2¼6.889 ×104 μ2 3¼−2.392 ×105;μ2 S¼−1.339 ×105;μ2 b¼−6.275 ×104 a3¼0.716;v S¼244.648;M H2¼286.917;M H3¼741.153 MA2¼159.196;M A3¼566.752;M H¼411.940.ðB2Þ (iii) H3 λ1¼1.689;λ2¼0.124;λ3¼1.414;λ4¼2.782;λ0 1¼0.195 λ0 2¼−3.492;λ0 3¼−0.224;μ2 1¼2.211 ×106;μ2 2¼1.223 ×105 μ2 3¼−4.958 ×105;μ2 S¼−1.216 ×105;μ2 b¼−9.680 ×104 a3¼0.808;v S¼1065.478;M H2¼527.554;M H3¼724.461 MA2¼205.411;M A3¼707.602;M H¼524.699.ðB3Þ (iv) H4 λ1¼0.914;λ2¼0.129;λ3¼−0.0672;λ4¼2.251;λ0 1¼0.137 λ0 2¼0.0990;λ0 3¼−0.166;μ2 1¼1.309 ×105;μ2 2¼1.107 ×105 μ2 3¼−4.037 ×105;μ2 S¼−1.181 ×105;μ2 b¼−7.311 ×104 a3¼0.536;v S¼1184.279;M H2¼397.452;M H3¼705.621 MA2¼188.008;M A3¼611.067;M H¼448.519.ðB4Þ (v) H5 λ1¼2.699;λ2¼0.138;λ3¼−0.497;λ4¼0.756;λ0 1¼0.0401 λ0 2¼−0.442;λ0 3¼−0.0673;μ2 1¼1.096 ×106;μ2 2¼1.540 ×105 μ2 3¼−3.714 ×105;μ2 S¼3.554 ×104;μ2 b¼−2.270 ×105 a3¼0.152;v S¼2190.371;M H2¼215.563;M H3¼626.921 MA2¼177.492;M A3¼683.604;M H¼156.890.ðB5Þ (vi) H6 λ1¼−3.480;λ2¼0.110;λ3¼−0.489;λ4¼2.766;λ0 1¼0.00574 λ0 2¼−1.839;λ0 3¼−0.0510;μ2 1¼1.336 ×107;μ2 2¼3.634 ×105 μ2 3¼−1.899 ×106;μ2 S¼1.454 ×105;μ2 b¼−2.275 ×105 a3¼0.260;v S¼3794.448;M H2¼402.807;M H3¼429.589 MA2¼217.359;M A3¼772.323;M H¼330.882.ðB6Þ COLLIDER PHENOMENOLOGY OF NEW NEUTRAL SCALARS IN …PHYS. 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