Jhep01(2016)160

JHEP01(2016)160 Springer January 15, 2016 January 27, 2016 December 9, 2015 : : : 10.1007/JHEP01(2016)160 Received Accepted Published doi: Published for SISSA by b [email protected] , and O. Fischer a . 3 1512.00869 The Authors. c Beyond Standard Model, Higgs Physics

Non-Abelian family symmetries offer a very promising explanation for the , [email protected] One of our central results is the possibility to have dark matter fermions and at least Highfield SO17 1BJ, Southampton, UnitedDepartment Kingdom of Physics, UniversityKlingelbergstr. of 82, Basel, CH-4056 Basel, Switzerland E-mail: School of Physics and Astronomy, University of Southampton, b a Open Access Article funded by SCOAP testing this class of models. Keywords: ArXiv ePrint: masses and mixing of the Standard Model fermions. one familon with massesparticular on we discuss and the characteristic even signaturesfields below in collider with the experiments a from experimentally non-Abelian light accessible familon family TeV symmetry, and scale. we show In that run I of the LHC is already Abstract: flavour structure in the Standard Modeldark and matter its consists extensions. in Wevisible explore fermions and the that possibility dark transform that sector under areresponsible for a linked spontaneously by family breaking the the symmetry, family familonsmodels such symmetry. - with that We study Standard non-Abelian the three Model family representative gauge symmetries singlet that scalars, have been shown capable to explain the I. de Medeiros Varzielas Non-Abelian family symmetries asmatter portals to dark JHEP01(2016)160 ]. 2 ] with an argument that 16 1 16 19 6 9 20 10 13 4 – 1 – 11 18 5 10 17 6 11 3 1 21 23 lepton model with leptophilic DM with neutrino-familon interactions and DM 4 4 2.2.2 SM-familon2.2.3 interactions: neutrinos The2.2.4 dark sector Dark matter relic density 2.1.1 The2.1.2 scalar sector SM-familon2.1.3 interactions: charged leptons TheA dark sector 2.2.1 The scalar sector A rules 4 The experimental status of DM signals are not conclusive: there are some hints for 3.5 Familon strahlung and single top production A 3.1 Effective framework3.2 for non-Abelian flavour Production symmetries 3.3 mechanisms Higgs production3.4 and decay Lepton flavour violation and lepton non-universality 2.3 GUT inspired SU(3) family symmetry 2.2 2.1 zling pattern of hierarchical massesto and the small fermion large mixing mixing inthe in the SM quark the does sector, lepton not opposed the sector, offer observed is a relic beyond density suitable that present candidate is understanding. that about five can Furthermore, times take more theDM abundant signals than rôleof baryonic in the matter indirect DM [ detection to experiments, make but up there is no unambiguous signal in direct observations that the SM doesSM not (BSM). address Two and prominent whichand examples motivate the the of observation extensions of unresolved Beyond collisionless, observations the cold arearranged Dark in the Matter three flavour (DM). generations problem, has The beenconnects fact addressed that the the recently SM by fermions gauge ref. are [ group with the number of fermion generations. However, the puz- 1 Introduction The Standard Model (SM) isthe a very experimental efficient observations and well in tested elementary theory particle that explains physics. almost all However, there are several 4 Conclusions A 3 Collider phenomenology of non-Abelian Familon fields Contents 1 Introduction 2 Specific models JHEP01(2016)160 ]. ]. ], 9 23 19 – , 21 18 symmetry as L − B ]. The generic proper- 15 – 11 parity, extra dimensions and Grand R – 2 – ]. The authors found that the DM and FS mass ]. From a theoretical point of view, a new unbroken 6 17 – 3 ] for considerations of relating DM parity with lepton 10 ]), or in situations where all fermion masses arise radiatively at 9 ], as the FS will control the flavour problem, while the theory also ad- 8 ]. TeV to match the observational constraints. Furthermore, in refs. [ ∼ 20 ], where it was shown that, compared to the Higgs portal, the DM in the Familon ] for a recent review. 7 In this paper, we study the phenomenological implications of effective models with non- Other approaches to connect the FS to DM involve charging Higgs doublets under the Recently, the possibility of a deeper connection between the DM and flavour were stud- More often than not, the introduced DM candidate is an additional particle (scalar FSs are particularly appealing in BSM theories with additional particles, such as addi- 16 symmetry is evoked, because the DM phenomenology does not depend very much on 2 FS (this is thethe case loop in level [ [ Abelian FS and flavoured DM.mixing We consider shows non-Abelian FSs hints because that the observed those leptonic are preferred when fitting the observed patterns [ or scalar) DM andpromoted the into SM physical to scalarscale be fields should given be [ by aminimal gauged flavor violation FS, scenarios where werebetween the compared SM with Yukawa and matrices an DM SU(3) are is FS, where mediated the by interaction a FS singlet. to connect the visible sectorties to of a flavoured scalar DM charged — underFS the the [ FS Flavon [ or Familonportal portal setup allows — for were smaller investigated DMtheir masses for predictivity. since an the A Abelian direct very detection detailed constraints study lose investigated some the of connection of flavoured (fermionic sector is the so-calledHiggs “Higgs boson, and portal”, which leads where toall the the the SM dark kinematically final states and available from SM visible DM particle annihilation sector content to interact according be spread to viaied over their the in Yukawa different couplings. frameworks by independent groups. Many of them explore the Higgs portal lightest particle charged under it.we investigate Other in possibilities this include paper,parity. using The see U(1) protective [ symmetry for theheart DM of leads the to theory: having two the different non-Abelianin symmetries (gauge) at the symmetry the in DM the SM sector. sector, and The the symmetry most often studied interaction between the dark and the visible dresses other shortcomings of the SM, for instance by providing suitableor DM candidates fermion) [ charged underZ an Abelianthe symmetry. explicit Usually choice the for simplest the possibility stabilising symmetry, of as a long as it prevents the decay of the most promising BSM approach toexhibited deal with by the the flavour masses problem,often and i.e. referred the mixings to peculiar of as structure the aref. Family fermions. [ Symmetry (FS), In as this it approach, relates the the symmetry generations is tional of Higgs fermions, fields see [ symmetry can beinto used SM to particles. explainis the So-called the DM “top-down” case properties approaches e.g.Unified often by for Theories generate preventing (GUTs). Supersymmetry decays a (SUSY) Inis of “dark with the introduced, sector”, the above which mentioned DM as distinguishes examples, at a least new one sector new from symmetry the SM. Symmetries are also the detection experiments, see e.g. refs. [ JHEP01(2016)160 we introduce three effective 2 in the literature. flavons – 3 – , we discuss the production of familons at lepton and hadron colliders, 3 , which are also often referred to as we summarise our results and, after a brief discussion, present our conclusions. 4 The structure of our paper is as follows. In section We note that an interesting and new feature of the Familon portal that we explore in The FS models are expected to be representative for the class of models where the The models considered here share the generic feature of a dark sector, that is charged familons 2 Specific models In FS models, theFS, SM and extra fermions scalar are SM gaugeVEVs usually singlet with embedded fields a — in the specific a familonsmixings alignment, — specific can are such be representation introduced, that explained. that of the acquire the FS is broken and the fermion masses and same section, the relic abundance constraintstudied. as In well section as relevant otherand constraints are the briefly constraints resultingsection from a selection of present experimental observations. In GUTs), as opposed tomuch the simpler familon models. coupling Regarding exclusively the tocoupling latter, the either we leptons, to further which the distinguish are charged, the typically or cases the of neutral the familon leptonmodels sector. with fermionic DM candidates and a different non-Abelian Familon portal. In the this article is the possibilitythat that may the allow underlying to FS distinguishFS can one give is FS rise from non-Abelian). to another observationalIn one features, We terms (and therefore in of aim particular discriminating determinedifferent power, to if we models, highlight the investigate e.g. generic observables whether that features may the of help familon this to is type distinguish coupling of both DM. to quarks and leptons (as in portal — much in thein same terms way that of more thethe generic SM dark SUSY phenomenology models sector in make compared the similarfeatures fermion to predictions of sector, the the but non-Abelian MSSM. rather familon to portal capture They to the are the phenomenological not here considered intended dark to sector. explain specific signatures and predictionspresent for experimental collider (non-)observation phenomenology ofputting are deviations bounds possible, from on and the combinations that of SM the the predictions model is parameters. already interactions between dark and SM sector are exclusively given by the non-Abelian familon under non-Abelian FS, togetheristic with UV-complete FS scalar models familonare in fields. simplified a such They SUSY thattional (and are the constraints. in derived number one from of In case free real- thethe a model following electroweak GUT) we parameters scale, framework, show, can theon and that be flavour observed they with related changing relic familons to neutral and abundance observa- currents DM as can masses well be around as accommodated. experimental Furthermore constraints we show, that expectation value (VEV), which canDM explain particles that the are observed singlets flavour underof the structure. SM the gauge For FS, group fermionic and the residethe interaction in non-trivial between multiplets visible and dark sector can be mediated exclusively by Typically the FS is broken by familon fields (SM gauge singlets) developing a vacuum JHEP01(2016)160 ] . 1 27 , (2.1) 26 -singlet 4 A . The field , we further c L ν − B ]. Therein, the SM 25 [ and the three 13 θ χ , and exclusive couplings to ν φ , charges, is summarised in table and φ L ` L we limit the discussion to the charged − . φ 4 B + A SM χ L L -singlets, which allows to fit their masses and contains the terms of the scalar potential that + with the right-handed neutrinos -triplet fermion field symmetry, the group multiplication of which is 4 4 c i – 4 – φ symmetry, and add to the field content an extra 4 A χ A SM 4 L A -triplets and the right-handed leptons are assigned L A symmetry and the experimentally observed leptonic 4 4 A -triplet scalars = 4 A A L within the SUSY framework. In this type of model it is possible 3, all of which are singlets with respect to the SM gauge group. ] that account for the observed value of , . 2 symmetry. The left-handed fields among the charged leptons of the , 24 13 θ 4 , for the chosen basis. The models are based on the UV complete models A = 1 i to address also the quark sector. However, for the sake of simplicity, we A , 4 c i — the familon — that is a SM gauge singlet, and transforms non-trivially includes the kinetic terms for the dark fermions as well as the Yukawa- A χ ` contains the usual SM field content (including their interactions with the χ ], which is based on SO(10) GUT and a continuous SU(3) FS which necessarily φ L lepton model with leptophilic DM 22 F-term alignment SM 4 L A The DM in this model consists in the The third model of this section considers a FS embedding in a SUSY GUT framework Two of the models are based on the In this section, we present three different flavour models based on UV-complete theo- familon), like interactions with the familon,include while the familon field.and their We implications now separately, turn althoughlepton to for interactions the with discussion the of familon the fields. three contributing terms content of the model,The along Lagrangian with density the of FS the and considered U(1) model is given by where into the three different singlet representations of fermion fields In order to prevent a mixingenforce between the the baryon-minus-lepton number to be a global symmetry: U(1) As mentioned above, we positscalar a field global with respect to the SM field content are embedded into addresses both quarkswherein and discrete leptons. subgroups werereactor Similar used mixing frameworks angle in were order explored to in generate refs. different paths [ 2.1 to a non-zero to extend the choose to keep themixings quark as fields in as the trivial SM. as in ref. [ listed in appendix in a SUSY framework [ leptons are embedded into amixing global is generated throughthe two charged and neutral leptons,so-called respectively. Their specific VEVs are aligned through the ries, that can lead toThe a realistic models description share of the theintroduced pattern as general of DM fermions’ feature candidates masses ofit and and additional mixings. exclusively arranged couples SM as to a gauge the multiplet singlet familon of fields. fermions, the respective that FS, are such that JHEP01(2016)160 , ), h ` φ,s c φ s (2.3) (2.4) (2.5) (2.2) ] ` φ 22 GeV, ` . were not H, φ [ † a ] ∝ H ` ] φ † ` = 246 ` φ ` φ ` φ [ EW , which is strongly v H ` 1 1 0 0 H φ /u φ c EW 0 2 for the specific familon c − v 1 / τ 00 1 ] -product rules, the antisym- and the linear combinations u . 2 EW † ` c 4 00 v 1 φ µ ` A ` φ,s H c φ ` φ 0, a hierarchy exists between the [ φ 0 c √ c 1 = . ' ] † ` = mixes with the real Higgs scalar ) 0 1 φ 1 , c φ ` ` c 1 1 1 1 φ φ φ L e 0) [ − EW , 2 + 4 v c + 4 0 c − 3 3 1 1 1 u, φ,s ν . New fields are gauge singlets with respect to the φ,s φH c 1 c c L ] – 5 – † ` − = ( √ 2( 0 c 3 u , m φ B i 1 ` χ l p φ φ φ,s [ h c 1 = . In the following, we assume that this hierarchy is U(1) ] c 2 00 2 ` † u χ u × φ p ` 4 φ A c 1 [ = 1 introduces the mass-term c 0 φ EW − H v m ` 3 1 1 0 0 0 0 χ χ -fields vanish identically and terms involving [ s ` φ ] : c ` † is given by φ u φ L u ` − φ [ 4 B s . ] A ` † u φ ` , with the masses . For the coupling parameter being smaller than zero, the familon field itself U(1) 3 φ is the Higgs doublet. Note, that due to the 1 [ ` ` φ φ H φ,s c 2 − . Field assignment within ` φ = After symmetry breaking, the familon field When the electroweak symmetry gets broken by the Higgs-VEV, φ = L φ It is worth noting,scale, that defined the by three resulting physical familons all reside on the same mass acquires a vacuum expectation value (VEV) in a specific direction in flavour space: where the magnitude present, since this is required in the UV-complete theorywith this the model mixing is based beingsuppressed on. proportional due to to the theto here ratio considered be of VEV non-zero, the hierarchy. the Considering VEVs, mass only eigenstates the are coupling given by metric contractions of shown above. the coupling parameter component In the case ofelectroweak an approximate VEV cancellation, and mass-terms). We consider then the following non-SM part of the scalar potential: where the 2.1.1 The scalar sector In the following we do not considerwhich tri-linear is combinations motivated of the because familon they fieldscomplete are ( forbidden theory. by additional We symmetries also present in disregard the UV for now bi-linear combinations of familons (explicit Table 1 SM gauge group. The SM gauge charges for the SM fields are unchanged. JHEP01(2016)160 ) 1 ` φ (2.8) (2.9) (2.6) (2.7) around -doublets are purely H L 3 , 2 ) an effective ` (which consist and φ 2.6 φ φ -field masses. The , 3, the dark sector is . , χ c 3 2 c fields emerges: , χ τ 00 00 χ ] ] i = 1 χ L ` ` φ , i [ φ . c i [ 3 ) τ χ 3 χ y χ y prohibits direct couplings between . + , y − ) 1 c 2 τ c 2 χ µ 0 χ , y ] 0 , y i µ allow for the construction of an effective, ] 1 L χ χ 1 ` , y ` y e φ φ [ y [ µ ]. When the heavy FN messenger fields are 2 y χ ]. – 6 – symmetries, when the VEVs of the scalar fields y Diag( 25 + 3 allow for three different , (which consists exclusively of the component -singlet fermions, , Diag( 4 c − 29 4 × 0 2 , e 24 A ] c 1 , 9 φ ∝ A i u χ L L ] ` = χ = 1 φ M ` [ ] for further details. Expanding both, e φ [ DM y , i projects out the charged leptons from the SU(2) 1 i 25 M χ , χ y y Λ H H 24 ⊃ ⊃ − ), that couple exclusively off-diagonally to the dark fermions. 3 , and the three , χ =5 2 χ d eff ` L φ O -doublet L ], and the specific renormalisable interactions between the charged leptons develops its VEV, the following mass matrix of the 28 -triplet, 4 ` are inserted: φ A fields is also diagonal, as opposed to those of the lighter fields -symmetry assignment to the fermions in table H L χ − B The three parameters and and Λ encodes the mass scale and coupling of the FN messenger particle from the i i ` of the components U(1) the dark-sector fermions and SMforbidden, leptons, which such in that decays turn of stablises the the former lightest into dark the fermion latter are on cosmological time scales. After the The couplings of the physicalto familon the mimicking the charged-lepton sector. Omittingthe the following kinetic terms: terms, the dark sector contains off-diagonal, and can lead tofor interesting cases lepton with flavour multiple changing Higgs phenomena bosons as [ was shown 2.1.3 The dark sector With the after breaking of theφ electroweak and We emphasize, that Yukawa couplings of the physical charged leptons to the L UV-complete theory, see [ their respective VEVs yieldsYukawa-like interaction from between the the invariant familonsthe effective and Higgs operators the boson eq. charged and ( leptons, the and charged also leptons. between The mass matrix of the charged leptons arises charged leptons, the Higgs boson, and the familon where the SU(2) The hierarchy of themechanism [ charged leptonand masses the is familons explained wereintegrated by out, considered the the charge in assignments Froggatt-Nielsen from [ table (FN) non-renormalizable dimension five operator, that introduces an interaction between the SM 2.1.2 SM-familon interactions: charged leptons JHEP01(2016)160 ]. can 32 )[ (2.10) , which φH c χ ] into the contributes 30 [ a T3PS 1.0 1 -exchange into two χ in the left-hand plot of 2 h 2 together with a Higgs boson. , on the other hand, is very Feynrules 2.3 1 b ` ≤ φ 1 φH , with the final state given by Higgs- 10 TeV a , c ≤ 3 χ 3 y annihilation via s-channel familon-exchange χ = annihilation via t-channel a: , m 2 1 TeV 2 . χ 3 ` χ c: , y 2 φ ≤ to the total annihilation cross section becomes the same as φ . The amplitude from diagram – 7 – 3 , m = , c φ 1 2 m 1 are non-zero. However, due to the small Yukawa b: φ χ 2 m χ 1 i √ y m m ]. All the parameter space scans in the following have χ y ≤ ≤ ≤ = = Λ = 10 TeV 31 and [ 1 u Before they freeze out from the primordial plasma, the φ χ 01 . m 0 m 10 GeV 100 GeV fields, is kept in thermal equilibrium by the processes as depicted χ micrOMEGAs 4.1.8 . The details, i.e. which of the processes is dominant, are depending on the . Feynman diagrams depicting the annihilation channels of the fermionic DM, 1 . Therein, we vary the parameters in the following ranges: 2 The model as defined above has been implemented via numerical tool been carried out using the ToolWe for show Parallel a Processing in general Parameterfigure overview Scans ( over the produced relic density Ω amplitude that corresponds to diagram dominant. Due to thisthe process observed relic being density a possibleof t-channel resonant for interaction, annihilation, larger this which DM makes is masses the discussed (not matching below), considering of compared the to special the case other two processes. lead to the SM final statesfamilon fields of (if two kinematically Higgs available), bosons or (also evenThis counting a potentially the familon large Goldstone contribution bosons) to oron the two the total annihilation mass cross relations section ofcase is the of very the familons, dependent familon the fields DM being fermions, (much) lighter and than the the Higgs DM fields, boson the mass. contribution from In the the to the total annihilationparameters, cross provided the section couplings ofcouplings the of the DM SM candidates leptons,relic this for amplitude abundance. all alone is values The insufficientpowerful of process to in give the corresponding creating rise model to to a the large diagram observed annihilation cross section; a non-zero scalar coupling Dark Matter relicDM, density. given by the in figure explicit relation of the masses Figure 1 couples only to the (non-Abelian)into familon portal two SM fermionsscalars (here, or leptons familon only). fields, iffamilon kinematically fields. allowed. Note, thatthe familon due is to considered the part possible of decays the of SM the field familon content. into two (or more) SM fields, JHEP01(2016)160 . i i χ χ 000 y 10 (2.11) or and/or i φH χ c y , a suitable 1 χ 1. The shown . m 0 2 D , . 5 1 . is smaller than the 1000 0 TeV > @ , χ Χ 3 0 , . 2 m m χ is the most relevant, i.e. = 1 m , respectively, always allows 1 1 χ φH c m minimal relic densiy, imposing the for resonant s-channel annihilation. = 100 i 0 and χ φ y φ , that is compatible with the observed m 4 6 1 . χ - - 5 m Right: .

m 0.01 10 10

W m

). 2 ' 5 . χ 0 2.10 m – 8 – ∼ 5000 χ 600 GeV, if the DM mass m ∼ = 10 TeV. The resulting relic density yields the maximum 3 , 1000 2 shows the lower contour of the relic density with resonant χ 2 m ], is shown in the ﬁgure by the black line. It shows that the 500 2 D and 1 [ . GeV @ 1 08 TeV. For DM masses below this bound, the observed value for the 0 Χ . = 1 and the relation χ m 1 m ∼ , for the three sets of parameters: ∼ φH 100 . c 3 1 := φ overview of the produced relic density. The model parameters were marginalised is excluded in this parameter scan. The observational value for the DM = χ 50 m 1 m χi φ , such that the observation can be matched. y Left: χ m 5 . m . 0 Now we will study the connection between DM and familon masses that is imposed Resonant s-channel annihilation occurs when the resonance condition is met: ' 10 . It is possible to identify which of the diagrams in ﬁgure 8 6 4 4 1 -

10 10 10 100

0.01 10 χ

W φH 2 SM final state particles. the remaining parameters.smaller For than masses 0.1, which insidefamilon can the VEV, be smaller contours, remedied mass the for splittingsc abundance between instance the by tends DM allowing to fields,which for or be parameters larger a have values reduction of the of the most effect on the relic density, and which are the dominant combination of the familon and thefor DM masses, the i.e. correct relic density.is The shown in reflection figure of thislines constraint of on different the mass colorparameters parameter each from space denotes this the set, contour where line the for relic a abundance of specific 0.1 value can for be the produced coupling by adjusting The right-hand plot of figure annihilation, and thus yields therelic upper density, bound to be on relic density can be matched by adjusting (i.e. reducing) the coupling parameters from demanding to match observed relic density. For the case of This case, which canone, be as considered it leads as to fine the tuning maximal DM of masses the still model compatible with parameters, the is observed relic a density. limiting m relic density, being observation can be matchedfield for masses, DM between masses, 100familon given GeV mass by and the minimum among the three within the numericalparameters domains defined inFurthermore, eq. ( value for We note, that the possibility of resonant s-channel annihilation, i.e. the fine tuning of Figure 2 JHEP01(2016)160 , χ 1 φ , the 2 -triplet 4 A , and a global 4 = Λ = 1000 GeV 1. . A u 0 500 , 5 2 ν . 0 − , 0 . . Fixed φ 1 3 0 H φ m = 1 2 , with couplings to the right- ν 0 c √ 400 φH 1 φ τ c = = 1 c 00 charge assignments in table φ i µ 1.0 χ m L . On the right-hand branch of the y − b c D B 1 1 1 1 1 e and . This implies, that in order to have a DM 1 GeV 3 @ , 2 300 1 1 ), yields a relevant contribution to the total 2 Χ L . New fields are gauge singlets with respect to the φ m − L 2.6 – 9 – − B c ), in order not to result in overabundant DM. 3 3 1 ν 3, from mixing with the neutrinos. The non-Abelian , U(1) 2.11 c -triplet SM singlet fermion field. The familon is acting 2 4 × , 0.5 200 4 A -triplet SM singlet scalar 4 A is the mass of = 1 3 1 0 2 χ χ A 1 TeV, the mass of the DM and the familon fields have to be φ , i m c i ∼ L χ − χ

0.1

4 B

m D @ A 100 . The numbers in the plot refer to , that are an ]. As in the previous section, the introduced FS is 1 U(1) c 800 600 400 200

25 Φ GeV m -singlets m 4 , the green line converges onto the case of resonant s-channel annihilation, × A 3 with neutrino-familon interactions and DM = 10 -symmetry is enforced to prevent the dark fermions, given by the . Relation between familon and DM masses from the observed relic density, for the familon 3 4 . Field assignment within , L 2 − A χ B In figure On the upper branch of each contour line, the main annihilation products are pairs m Figure 3 fields with fixed massand ratios: and the three familon portal is given byhanded the neutrinos as a Majoron, and inand developing a breaks VEV, the it gives lepton a number. mass term to Due the to right-handed neutrinos the U(1) 2.2 In this section, we consider adiscussed second familon in model, that ref. derives [ fromU(1) the SUSY frameworks annihilation rate. which is shown by the right-handcandidate plot with in a figure mass related by the resonance condition ( of Higgs scalars,contour and lines, the the relevant four-pointwhich corresponds diagram interaction to between is the the operator DM, in the eq. Higgs ( boson and the Table 2 SM gauge group. The SM gauge charges for the SM fields are unchanged. JHEP01(2016)160 ν φ . For (2.16) (2.14) (2.15) (2.12) (2.13) and the EW w with mass . . 3 c ν ν 2 φ w > v 2 EW w v + = 780 GeV, where 2 H ν ν w φ φ , c +    1, the lightest eigenstate ) = +2, in order to allow 1 δ c = 2 ν χ φ 2 , c + ε 2 + c/ , s + . ] χ 0 c/ c − for , δ b φH < − ε Lν ) = 0. Note that these assignments [ ,s H , c + χ ν ν 1 ˜ δ b 1 φ . H 3 c φ c c ) c H 2 − y c a 16 2 + c/ , and between the DM and = − + = c/ − s 2 b ] c ), with a (possible) hierarchy LH c − w, w, w ν and a c – 10 – ν = ( 2.12 δ b [ 2 2 , c i ν + ν 2 ]. For the sake of simplicity, we consider the scalar c/ c/ φ , c : φ c h φ 1 − − ) = +2, while B-L( c y + 25 φH c , b b c − χ a . In the following, we consider one physical familon field + 14 24 1 w    ) is relaxed to = ⊃ − c ] whence this model is derived, requires the familon field 2 2 being linear combinations of the VEV-magnitude ν and c w L ,s 2.13 = 8 yields the magnitude of the familon VEV is 25 ν , = ) is given by the linear combination φ 3 , which is suppressed compared to the mass scale of the other two 2 ν − 24 / 2 ) to apply also for the model of this section, with the substitution , c φ i 1 2.14 a, b, c, δ ) c M δ 2.2 , b are possible for smaller values of = 10 2 + c w ε φH c + 8 (36 1 = c × 1 c w . We find, that the condition ν = 20 φ a Without considering bi-linear and tri-linear terms in the familon fields as discussed pre- → ` 2.2.2 SM-familon interactions:The neutrinos part of the Lagrangianneutrinos, density is of given the by model contains Yukawa-like terms including the of the mass matrixgiven by ( familon fields, that iswith given a by mass on thethe electroweak TeV scale, scale. and two degenerate familon fields with masses above coupling parameters When the condition in eq. ( with the parameters allows for the VEVinstance, alignment from eq. ( larger values for viously, mass-terms for the familons emerge after symmetry breaking with the mass matrix tential terms through F-termspotential [ from eq. ( φ The SUSY framework [ to develop a VEV in the particular direction in flavour space: In this type of model the VEVs are usually obtained from the SUSY preserving superpo- for the familon portal betweenthe SM RH neutrinos, and we DM. opt for Asprevent B-L( the any DM invariants between fields the must DM be and distinguished from 2.2.1 The scalar sector candidate DM fields need to have the symmetry charge B-L( JHEP01(2016)160 . φH and c ]. We χ (2.19) (2.18) (2.17) triplet, m 34 4 , φν A to perform 33 y ν φ 001 [ , T3PS . c χ 2 = 0 and χ 2 = 1 and the s-channel 3 | ν θ | φ φν χ y y = + χ c y χ 3 microMEGAs symmetries, and insertion of the χ acquires its VEV: 2 L = 10 TeV ν ν φ 2 φ χ y heavy ν ≤ φ + w . m c 1 TeV φH χ , which is shown by the red line in the left- y χ 1 , that is a SM gauge singlet and an , respectively. and SU(2) 1 , c ≤ Φ 10 TeV χ . It shows, that for DM masses between 60 GeV = χ 4 φν , and the hieararchy of the couplings φν 1 – 11 – m 4 y χ ≤ ν A N light ν φ , y χ φ m ∼ χ χ y m m y χ and > m m χ = ≤ ≤ ≤ = 100 GeV, χ y c N m χ 01 ) in figure ] ) allows for to the correct abundance with the appropriate . m 0 χ ν to implement the model into 2.19 φ 2.19 10 GeV [ , that is a singlet under the SM gauge group as well as under c 100 GeV χ y χ = , being on or above the TeV scale, the heavy ones will decay rapidly . The red area above indicates, that the correct relic density can be N χχ 5 . The relic density is then minimised for ν . The breaking of the Feynrules m φ a ? fields have Yukawa-type interactions with the familon fields L H 1 2 χ iτ = ˜ in the limits of eq. ( H 4 TeV the observation can be matched. In general, the abundance constraint imposes . The 4 We first consider the case of The mass eigenstates have to include the observed three light (active) neutrino fields. ∼ A light ν φ In this case, thegiven Feynman by diagram figure that dominatesdiagram the being total on annihilation resonance,hand cross i.e. plot section 2 of is figure obtained by tuning the couplings in the ranges defined in eq.and ( no relation between DM mass andm familon mass in this model;choice each of combination the of other parameters. while we use for the active-sterile (ordisplay light-heavy) the neutrino resulting mixing: range for the relic density, marginalised over the remaining parameters As before, we use the parameter space scans. We confine the model parameters to the following ranges: with the ensuing masses being degenerate, once the 2.2.4 Dark matter relic density The dark sector consists inand a chiral one fermion conjugate field the handed neutrinos are notheavy part (sterile) of neutrinos the isheavy neutrinos, DM; not when exactly the zero,into the mixing and light between for ones. light the (active) here and considered2.2.3 masses of The the dark sector respective scalar VEVs, results in Majorana and Dirac masses for theFurther neutrino mass heavy matrix. (mostly sterile)heavy neutrinos to are escape present, experimentalfollowing, that detection we up have consider to the to heavy now be neutrinos or either to to sufficiently have be masses only on slightly the mixed. TeV scale. In the The right- where JHEP01(2016)160 Χ ), m Χ ν < TeV 3 Ν m φ TeV Φ 10 m 0.6 10 , = ). = = N N N m , m m GeV , , ), yields w/m 2.19 ( 0.01 0.1 1.0 100 = = = = H H H N O Φ Φ Φ c c c m heavy ν 2 → ν φ 2000 . We observe, that, while into Higgs and Goldstone 1000 5 D ν 5000 φ GeV . We note that, for the above 500 @ ν Χ φ m 01. All Yukawa couplings equal one, 200 . D 1000 100 = 0 GeV @ φH = 1. In this scenario, the Yukawa coupling 500 , that allows the production of the observed 5 4 1 c Χ i - - 0.1

0.01 χ 10 10 m

0.001 χH

h W c 2 – 12 – 1 0 = = and H H Φ Φ c c χ 100 > m 50 = 1, Blue line: 1, respectively. In this case, the scalar parameters are dominating 1000 . N 0 , 5 4 m φH -

c 0.1 01 10 100

. 800 10

h W 2 the blue line is the lower limit on the abundance. It is given by resonant = 0 D 600 φH GeV @ c = 1. This case is shown — with the heavy neutrinos masses being ﬁxed: red line: Χ Right: m 400 φν y Left: = , the minimal relic density matches that of the other case, where only annihila- 200 . . Overview of the produced relic density with the non-Abelian familon portal given by N φH c 0 4 5 1 - - = 100 GeV — by the blue curve in the left-hand plot in ﬁgure 10 The maximal mass for the DM particles The second relevant case to consider, is an intermediate one, where the familon cou- = 100 GeV. becomes irrelevant for the numerical evaluation. Alternatively, the resonant s-channel 0.1 < m 0.01 10 10

0.001

. The model parameters were marginalised within the numerical domains deﬁned in (

N χ N

ν W φν 2 also allows for large DM masses. The possible mass range for this resonant annihilation is bosons is enhanced depending onwhich the we familon will VEV discuss and in its more mass, detail with below. a factor relic density, is given when y tion into right-handed neutrinosreduces is the dominant. annihilation Passing crossthe the section partial kinematic compared decay threshold, to width however, a the of significant red the contribution line. familon to into thesetting This two total of comes heavy parameters, decay about neutrinos, the width Γ( partial since of decay the width of the familon plings m m Figure 5 m s-channel production of twocontrolling heavy the cross neutrinos section (beingthe are lighter same, the just than respective with the Yukawa couplings.in DM). importance The over two Relevant the other parameters neutrino colored ones. lines are φ Figure 4 JHEP01(2016)160 , c i w × ψ L leads . The (2.20) i . This w χ W SU(2) M × . ]. At the Pati- ], such that an χ field has no zero 22 35 [ m 123 ν ¯ φ φ ] and include subscript m 22 ' strong scalar interactions, field has a zero in the 1st . This implies for the DM ν ]. In the following, we posit a 23 φ ¯ φ 22 : in this figure, the intermediate HH ), and the )[ 5 b without on the weak scale, when the familon − → . , that is enlarged by decays into two ν ), the ν ν φ c, c, c π φ , b, Π , a φ m 0 r , ν φ m . c , which is also motivated from the GUT model. ≤ – 13 – c ψ w and the conjugate of the right-handed SM fields as b φH and c i ψ ψ a . The case with the familon being lighter than the 1 and kinematically available heavy neutrinos in the annihi- . φH . This case is shown by the dark red line in the right plot of c c = 0 1 and φH c 1 c and then to the SM gauge group, combined with an SU(3) FS [ is the generation index, that is in here also the SU(3) FS index. As the conjugate . Notably, this process does not increase the allowed mass range for the DM. i R 5 The model further contains three familon scalar fields, singlets under the gauge groups A theoretically limiting case is given by the parameter settings that are denoted by field has a zero in the 1st and 2nd entry (0 3 ¯ in the name that isφ a reminder of theentry specific and VEV the direction other forentries two and each with all of three equal the entries magnitudes familons: withhierarchy equal of (0 the magnitudes the ( familon VEVs, of right-handed spinors transformmass under terms the can Lorentz be group made between as left-handed spinors,and Dirac anti-triplets under theWe SU(3) assume FS, that which the three allowswith familons to different develop form magnitudes. non-zero invariants For VEVs with clarity, in we the different follow fermions. FS the alignments notation and of [ This model is based onSU(2) an SO(10) SUSY GUT brokenSalam to stage the the Pati-Salam gauge SU(4) group isall Left-Right the symmetric, left-handed and SM we fermions denote as where the field that contains are not compatible withbeyond the the TeV scale abundance is constraint. notinteracts very with compatible the Equivalently with Higgs one boson. can state,2.3 that GUT inspired SU(3) family symmetry Here Π isrelic the density, phase that space parametera factor sets perturbed that for sense, are the which, violating however, decay the has above no bound implication are for meaningless our in previous study, since they comes from the total decayHiggs width bosons, of the partial the decay familon,perturbative width Γ of treatment which of is proportional theupper to bound model the on familon breaks the VEV down, model when parameters emerges: Γ the scalar couplings to the dominant contribution toprocess the annihilation shown cross in section figure beingfigure given by the t-channel the green line, which always stays above the observation, except for scenario is shown for lation cross section, by thescalar red case, line. emphasising The the massstrating, range power is of that slightly the the smaller, scalar comparedwhich relic with couplings, is the density important but to can at preserve the be the same matched hierarchy time of also demon- the VEVs, which requires small values for shown by the three curves in the right-hand plot of figure JHEP01(2016)160 that (2.23) (2.21) (2.22) (2.24) 45 ): H . . 2 2.21 H Λ ) 1 ) c c j ): χ k 3 ψ 2 ¯ φ j 23 χ j 3 ¯ φ ¯ φ − i 3 )( ¯ i 2 φ ), the term giving rise c ( ψ χ , a 1 field component to be an ijk 0 i 123 χ , 3 ¯ φ ( ( φ 3 3 ¯ φ 123 = (0 c i ) + 3 . + 3 ¯ . c φ 2 h χ )    H Λ 1 v ) 2 χ c j + √ ψ − ] for further details. We shall not consider 1 0 0 h 0 1 0 0 0 0 1 ( − c j 123 ) 22 i ¯ χ φ    3 ψ 2 )( i 3 as a triplet, in which case they could have mass χ i – 14 – ¯ ( φ c i m ψ 2 3 ( and results in a mass matrix for the dark fermions χ c ¯ 3 φ i 23 k 3 = ψ ¯ ¯ φ φ 3 2 ( , see ref. [ 2 ¯ φ a acquires its VEV cj ) + Λ 2 45 χ 3 3 123 c M i to couple to the fermion current and the Higgs boson. c Λ φ χ χ 3 HH 3 As in the previously discussed examples, we first consider + ) ¯ χ φ ijk c j 2 ψ − H Λ 3 ) j 23 c c j ¯ φ χ ψ ). Explicitly, the dark fermions couple directly to the familon fields 2 )( j 3 i ¯ χ φ as anti-triplet and ( ψ 1 3 )( i 2.23 i i 23 ¯ φ χ ¯ φ ψ Given that the SM fermions originate from the SO(10) GUT fields, the i 3 3 ¯ φ y ( 3 − c = ⊃ , and residing on the electroweak scale. i χχ † 3 3 ¯ φ φ ψφH i 3 Among the invariants that can be constructed with the field content introduced above, L ¯ φ L 2 right-handed dark fermions toto transform the as simple anti-triplets invariant term underof SU(3) the FS, form which of eq. canvia ( lead the following Yukawa terms: left-handed and right-handed components havethe the SU(3) FS. same The transformation same propertieswould argument be under need to not have be trueterms for by the themselves. dark fermions:sector This a to simple is interact option with not the the familon case portal. we We are therefore interested choose in, both as the left-handed we and require the dark If no other massuse terms this as are a included,independent motivation the to parameter, lightest consider arising familon the e.g.m remains mass from of exactly small the massless. soft lightest We SUSY breakingDark mass sector. terms such as following SU(3) invariant tri-linear with the Levi-Civita tensor This term allows the familon Familon mass structure. the case that theexclusively familon masses the arise familon from fields the and terms in no the explicit scalar mass potential terms. that contain Then we can write down the breaking, i.e. when theto familon the mass of the 3rd generation fermions is the most relevant one from eq. ( The parameter Λ is again the mass scale of the FN messengers that were integrated out. The equality of the coefficientsstructures of symmetric. the last two Furthermore, termsgives a different is term Clebsch-Gordan due coefficients to exists to SO(10), with2nd the making generation, a charged the ( leptons Georgi-Jarlskog Yukawa and field downthis quarks contribution for in the the following, for the sake of presentational simplicity. After symmetry one can find termsgive rise that to are the connectingdiscussion masses the are of fermions the charged and fermions. scalars The of most the relevant model, of these and terms can for our JHEP01(2016)160 33 . 123 3 η ¯ y φ θ (2.27) (2.25) (2.26) to the 23 s ¯ φ i 123 have to be 10 TeV, for ¯ is shown in φ h and the DM ∼ 3 3 ¯ φ and i 2, with the mixing lose their validity. 23 θ/ i, j, k, l, m ¯ φ 33 h η , = MicrOMEGAs 2 cm θ , ), the physical DM field χ and l 3 with a mass proportional to = c as well as the heavy familon and 3 3 χ χ ¯ 1 φ 21 2.26 . 3 θ 21 η k 123 χ 33 ¯ φ θ η , η T3PS 2 c j 23 12 Λ ) can be realised. and ¯ 10 TeV, φ + η i has the same overall charge as † 3 1 12 , ≤ c EW ¯ 3 φ η v 1 , 3 3 χ ¯ . φ ( 2 2 φ 3 1 0 ξ χ ¯ O φ 2 is a part of the label. Thus, after symmetry ≤ ≤ + θ , m a linear combination of the flavour eigenstates s k 3 3 33 H ij ¯ y φ – 15 – 3 η 33 + θ φ cj η 2 m s c shows the masses of the familon c χ i χ ≤ ≤ ≤ 6 χ (1) coefficients, and the indices 1 ), we consider the additional terms χ fixed, corresponds to a specific fermion mass eigenstate, O , with the mixing in eq. ( 01 ijk 1 with one of the others, and the remaining contract with . 1 θ 001 c . 0 s i 1 i, j 0 χ 2.21 ξ 2 = 10 GeV χ 1, such that we approximate ⊃ 33 η , with and χχφ . Therein, the relevant model parameters have been varied in the above are 2 is lifted and a mass of , cj 2 1 i L 6 c χ θ ξ i 33 s χ χ 1 η , where the superscript χ have been neglected, considering their masses to be above ij η 2 , 1 3 )). The = 10 TeV and the other DM particles ¯ φ emerge, together with the massless fermion a : a 2.21 cj being a function of the VEV-ratios that mixes the DM fields. Since light familon couples to χ 3 3 θ tensor). Including these contributions, it is easy to find parameter settings such that ¯ φ The relic density produced in a scatter scan with It is important to notice, that the contributions from the VEVs Further invariants with the familons and the dark fermions can be constructed, which that are compatible with the abundance constraint. Larger values for the masses are and i 33 simplicity. The rightη panel of figure possible, when the resonantblue s-channel dots annihilation in is thepossible, same considered, but figure. which the simplifications is In due shown this to by case, the the masslessness masses of for DM and familon up to 10 TeV are while Λ = components the left panel offollowing figure ranges: with the coefficients angle field also interacts with the light familon field, with a coupling strength proportional to one the masslessness of mass matrix render the massχ eigenstate tion is suppressed by the VEV hierarchytheory. and the The FN messenger presence scale, of fromoriginal the the model, UV-complete given second that term the(see is combination eq. necessarily of ( allowed bycontracted the in symmetries all of possible the ways ( model that are implicit in eq. ( which contribute to the fermion masses after symmetry breaking, although their contribu- which we label breaking, two degenerate Dirac massthe VEV eigenstates add to the masses of the dark sector. Keeping in mind the charges of the original GUT The combination of JHEP01(2016)160 0 qq f 1000 (3.1) (3.2) ). The . -triplets 4 ) 2.27 500 a ] considers A ] 37 φq [ D s ] 200 φq GeV [ @ 5 c 33 Λ, with Λ absorbing Η / relation between the DM + m 100 a ] EW v φq [ a 50 Right: = ] φq φq [

c D @ + 20 , s ] 0 20 50 500 200 100

φq

q q 1000

[

Φ s GeV

m φ ] 3 0 qq not necessarily of the same flavour, and φq [ f 0 3 q c – 16 – ⊃ + 5000 00 1 φqq and ] L q φq [ 0 1 ] 1000 φq [ , for instance when the quarks are embedded into D 2 ). They can also arise from adding quark-familon interactions c , with the model parameters in the domain defined in eq. ( ]. The recent and thorough collider analysis in ref. [ 500 3 2.2 + GeV ¯ φ 36 1 @ 2.21 ] Χ m φq [ and the quarks 1 ] φ . 100 φq [ χ overview of the produced relic density with the non-Abelian familon portal given 1 c m 50 ( 2 φq ∼ Left: f φ . ⊃ m 5 7 5 - 10 0.1 10 10 φq The above effective quark-familon interactions take place for instance in the GUT 10

1000

0.001

L h W 2 like the leptons: We have introduced thethe effective coupling “messenger and coupling” mass of the messenger field from the UV complete model that has been with the familon being an effective coupling constant. inspired model, see eq. ( to the model from section 3.1 Effective frameworkAn for effective non-Abelian interaction term flavour between symmetries thecan quark be current generated and after the electroweak lightest symmetry familon field, breaking, that has the form on the mixing betweenUV-cutoff the (lightest) to familon be and aboveFS the the VEVs Higgs TeV boson. to scale, suppress and Here,that the the we other scalar consider observables resulting become mixing the hierarchy relevant. between ofobservables, In with the the the a Higgs electroweak following, focus boson and we on will and high-energy investigate the colliders. some familons, of those such 3 Collider phenomenology ofWe non-Abelian discuss implications Familon and fields signalsplete at investigations, see high [ energy collidersa in low this energy section, theory for from more a com- specific UV-complete flavour model, with particular emphasis by the third component of black horizontal line denotes the valuecandidate of the mass observed and relic density. theis lightest matched. familon The from blueimplies the dots requirement denote that the the fine-tuned scenario observed with relic resonantly abundance annihilating DM, which Figure 6 JHEP01(2016)160 , 7 (3.3) = 150 GeV, q φ φ is the VEV of m w ) is given by: Familon strahlung. ), and 3.1 ). 3.2 ∗ (10 pb) for q /w Right: O EW v ( O (1), the production cross section for from eq. ( i O c w , = q g φq 0 f : gluon fusion. 0 qq f qq 0 c Left = 1 TeV. ]. Familon strahlung can be the most efficient – 17 – q q = φ 36 0 P m qq f ]. , and flavour violating familon-quark couplings, is given t ), the effective vertex from eq. ( φ = 7 TeV can be as large as 38 s 3.2 < m √ 01 pb) for φ . , in models with a non-Abelian FS it is possible to have light m (0 2 O ψ ψ . The production mechanisms. ψ . Therefore, provided that denotes the coupling constant(s) between the familon and the quarks under 0 g g 0 2 qq f qq ), we find that the dominant contribution to familon production stems from the 0 c Figure 7 in the following and assume to reside above the TeV scale. The VEV hieararchy results q q 3.2 Another production mechanism for familons is familon strahlung, which occurs when The effective interaction of familons with the quark currents allows the production As shown in section coupling. w P a quark changes flavour whilemass radiating off is a larger familon, in thanproduction particular the mechanism when for the familon initial the mass quark’s eq. familons. [ ( From the familon-quarkφut interaction as defined in of the familonexpression fields is proportional via to the gluon correspondingof one fusion, in the analogously SM, with tofamilons a via proportionality the factor gluon Higgs fusionwhile boson. at it drops down The to resulting where consideration, given by thethe sum familon over field. the relevant where the gray blobswhich denote are the generated effective when verticesout. the between In FN the the familon messenger framework and from of the the eq. ( fermions, UV-complete theory is integrated by the decay ofinvestigated for the instance top in quark ref. to [ a charm3.2 and a light Production familon, mechanisms whichThe has different processes already from been which a (light) familon can be produced are shown in figure in the quark-familon interactions to be suppressed by familon fields (with masses onTeV the scale electroweak scale), and together a withfamilon successful cutoff masses, scales DM meaning above candidate. the A promising discovery channel for very light integrated out. After symmetry breakingby the familon develops a VEV, which we will denote JHEP01(2016)160 (3.4) , with /s 07 MeV, . , adds to 2 q φ 2 is less than m = 4 → φ , is shown in = 150 GeV, a h 2 m ut φ SM f , ,Γ m φ tot its mass. q 2, the observed relic / 1000 m H = 1 and < m ut f φ 800 m . pb, respectively, are possible. , which results in a factor D q 2 ut f TeV 600 GeV × @ 13 100 GeV Φ m TeV – 18 – 7 w < 9.7 MeV, which in turn limits the product of the imply, that, for < φH 400 pb and 87 c φ 2.2 2 2 ]. It appears pointless, however, to search for familons ut † f → ]. Compared to the SM prediction of Γ = 7 TeV and 13 TeV, with

× φφ s 39 D @ [ √ H 200 † H MeV [ 1 5 2

7 9

. . 50 20 10

ut 2

pb f

Σ +7 − 2 1 . = 6 tot . Production cross section for familon strahlung at the LHC from partonic initial states, . We note, that for 8 The production cross section for familon strahlung, normalised by There is one more contribution to familon production, that comes from the mixing term being the square of the center of mass energy of the parton, and This constraint on the familon-Higgsabundance coupling observation together in with thedensity section constraints can from only the be DM the accounted familon-neutrino for interactions when are the very DM strong. is heavier than the sterile neutrinos and the Higgs total decay width.results The in determination Γ of the Higgsthis total yields decay width the at thefamilon upper LHC Higgs bound coupling Γ and the familon VEV: 3.3 Higgs productionThe and interactions decay between the Higgs bosonfields and to the decay familon into allow the thehalf heavier lighter the among one. the Higgs two When mass, the the mass partial of decay the width lightest of familon the Higgs boson into 2 cross sections. It turns out,bosons, that i.e. the production the of longitudinalcompared familons to modes in the the of fusion two thethe of aforementioned two Goldstones weak production Goldstone are gauge mechanisms. coupling bosons,s to This are the is parton completely as current negligible expected, since figure production cross section of 25 in the scalar potential, produced from Higgs bosons, because the latter have themselves rather small production Figure 8 normalised by the effective familon-quarktions coupling. have been In used. this figure, the CTEQ parton density func- JHEP01(2016)160 , 1 ≥ − . w (3.6) (3.7) (3.5) H m 20 fb 2 ' < . Assuming ]. The non- φ int . In order to , which means L m 40 H the discovery is µ, τ SM H 2 SM 2 240 GeV. We can 1 N = − N ∼ ` q ) being the branching (10) LFV events for a ≤ for O 01), a discovery of LFV . HH H φ e` 2 (0 f , we use the production cross ] or the International Linear N → O = 150 GeV = 1 TeV, H = 1 TeV. φ 2 φ 42 φ φ φ = 150 GeV = 1 TeV, N m m 3 TeV. If both effective couplings m φ φ , allows the familon to decay into two ≤ m m s H φ 4 φ ) for ) for 2 ) 6 4 m m 2 eτ ) for ) for π m f 5 3 (10 (10 > − − eµ O O 16 φ f and the integrated luminosity ( ( m (10 (10 must be small compared to (10) TeV for 8 – 19 – × O O = H 0 φ 2 ( 2 ≥ O qq Yukawa coupling, i.e. N f LVF ≤ w ) τ φ, ) σ would lead to flavour violating final state fermions and an HH are order one, and HH eτ → 2.3 → f φ φ and Br( = 7 TeV from figure 0 = Br( s eµ qq = 150 GeV, and f f 200, requires that H √ φ φ ], this final state could be studied with great precision, for instance when 2 ∼ m N 43 ) at H φ SM 2 m N ( φ σ ) implies that, if we assume all coefficients to be equal to one, we get with Λ = ], the Circular Electron Positron Collider (CEPC) [ 3.3 41 The SM cross section for di-Higgs production (at next-to-next-to leading order) is The complementary situation, given by (100) TeV for with the effective lowno energy background LFV and electron-lepton adiscovery. couplings perfect At detector the environment, FCC-ee,feasible we with when demand a both total integratedare luminosity of of the 10 same ab magnitude as the Collider (ILC) [ investigating the Higgs boson propertiesapproximate at the center-of-mass LFV energies violating of cross section with another very promising signalinstance for the model non-Abelian in FS section associated and Higgs the boson associated in the familoncolliders final fields. such state, as which the For does planned notee) electron-positron have mode [ a SM of background. the Future At Circular lepton Collider (FCC- which can be usedeq. to ( constrain the parametersO of the UV complete model.3.4 For instance Lepton flavourThe violation violation and of lepton lepton non-universality flavour (LFV) in the final states at high-energy colliders provide statistical one sigma fluctuationinto of the the bound SM-predicted number of events, which translates about 10 fb atobservation 8 of TeV center-of-mass discrepancies energy, inthe and the LHC, about SM predicted 40 fb numberquantify at what of we 14 di-Higgs TeV mean [ events bythat “small”, in the we resulting run impose number I the of condition at familon-produced di-Higgs events has to be smaller than the for two exemplary values ofratio the familon of mass, the and familon with into Br( two Higgs bosons, one of which may be virtual, if Higgs bosons, which addsmeasurement to of the the number triple-Higgs of couplingestimate di-Higgs the at events, number the of which familon-produced LHC provide di-Higgs andsection the events, all important future colliders.which In yields order to JHEP01(2016)160 ) 6 and from (3.9) (10 (3.10) τ O 13 − 10 × 7 . ], the introduction 44 ) of 5 eγ ]. Since also these decays → 44 [ µ 2 ( , − ] ), the following interaction terms Br ) ) 036 46 . c c 1 2 0 q q 2.22 ], or other viable data-driven patterns )( )( 3 3 q q 47 090 074 . . ]. , ) GeV (3.8) 0 0 4 ) 44 c 3 ) + ( ) + ( q c c 3 3 and eq. ( (10 q q )( ). The FS and familon VEVs considered here – 20 – 3 745 )( )( . q 1 2 2.3 q q ≥ O Kee = 0 Λ :( : 2( :( ] → 1 3 2 3 3 3 24 ¯ ¯ ¯ φ φ φ 300 GeV. As soon as at least one of the effective familon B LHCb K ( ≤ R B / φ ) ]. m 45 Kµµ , that receives contributions from loops of familon and FN messen- eγ → B → ], a very relevant constraint comes from the experimental non-observation ( µ 24 ≡ B K R , LFV bounds, and rare quark decays [ K Furthermore, the measurement of lepton universality provides a generic means of as- Another observable that is sensitive to LFV is given by the charged lepton flavour vi- Non-Abelian FSs often give rise to LFV decays of the Higgs boson at the loop level, R decays themselves. The here considered non-Abelian FS models by themselves do not From the GUT inspired modelbetween in section the familon fields and the quark currents are obtained for the leptoquark Yukawa couplings.patterns It match so is well interesting the thatby data-driven the patterns theory-driven, imposed FS-inspired on leptoquark Yukawa couplings 3.5 Familon strahlung and single top production where very naturally justify thethe textures lepton flavour of isolation the textures leptoquark proposed in Yukawa couplings [ that account for lead to a sizeable modificationcouplings of being lepton universality on observables, due thesuppressed to order the by of familon-lepton the the messenger Yukawaof couplings, scale. leptoquarks and to the However, FS interactions asuniversal being models was flavour further makes shown decays it from in possible the ref. LHCb to [ measurement explain [ the recent hints for lepton non- the MEG collaboration [ sessing new physics couplingsinteractions of to the the leptons leptonµ in sector. the decay It modes amounts of to various measuring mesons, the but weak also of the from the non-Abelian FSmass to scale the of cLHV the decay FN rate messengers yields [ an estimated lower bound on the which does not change much with the recent update on Higgs-boson events, a detection of those decays might indeedolating be (cLFV) possible. decays, to whichcussed non-Abelian in FSs ref. contribute [ atof the one-loop the level. process Asgers, dis- whenever the familon couples off-diagonally to charged leptons. The contribution couplings is as small aslighter the than muon 10 Yukawa GeV coupling in or below, order the to familon produce would the havethe required to branchings number be for of which signal are expectedare events. to free be of smaller SM than background, 10 and the FCC-ee (and CEPC) are estimated to produce would still be possible for JHEP01(2016)160 ], ], 53 22 2 pb , with c 3 , where q χ 3 q 115 , which is an < m 8 ∼ φ m ) gives rise to a light = 150 GeV and 1.0 TeV, 1. In order to assess the 3.10 ). Moreover, in the quark φ c 1 q m suggest that )( 3 VEVs of the other familons [ ) signal events might have been 6 q 3 b, c (10 01. We note, that this still allows for ) + ( . O c 3 0 , and two heavy familons that consist q 3 3 ¯ ∼ φ )( 1 ) or 2 q 4 , we assume that the masses of the lighter ut f (10 events. Since the statistical fluctuations of this bc/a – 21 – O 6 ∼ 10 ut f × . The light familon eigenstate will therefore couple ]), which is compatible with the SM prediction [ 2 3 2 ¯ φ 52 ' – , the breaking of the FS, eq. ( t n . 50 and 2 , 2.3 c 1 1 3 q ¯ φ and )) that consists mostly in 2 , 1 q 2.23 ), with a small coupling to ( c 3 q ] and ATLAS [ )( 3 q 49 , 48 In this paper we have studied three examples from the class of models with non-Abelian In comparison, the measured single top production cross section of The familon-quark interactions therefore give rise to an excellent search channel at the We try to estimate the prospects of detecting such a resonance by comparing event The constraints from the DM analysis shown in figure These mixing effects imply that the off-diagonal interaction of the lightest familon As discussed in section represent the potential family of dark fermions, which implies that the decays of the Model, and they canthe easily observed provide dark a matter connection relic density to through viable the dark familon sectorsfamily portal. symmetries, in with order the dark generate sectorthe consisting Standard in Model fermions that gauge are group, gauge and singlets under transform non-trivially under the family symmetry, 4 Conclusions Family symmetries are wellpatterns motivated through of the fermionic possibility masses of and explaining mixing, the the observed so-called flavour problem of the Standard number are of the samemight order be of possible magnitude — as withthe the a number systematic resonance of analysis already of hypothetical from the signalat the kinematic events, present distributions 13 it TeV, — data. the to Since extract chancesalso the of by production its finding cross luminosity light sections upgrade. familons increase are enhanced at run II of the LHC, and produced, that consist inrespectively. a single top plus a familon(CMS with [ yields a total event sample of LHC, namely the search forfamilon, a for resonance instance in in the a invariant mass di-jet, of that the is decay associated productscounts. with of During a the run single top. I at the LHC, up to order-of-magnitude prediction rather than an upper bound, due toχ the assumptions made. light familon lead to theHiggs detectable bosons,) final states if being kinematically SM allowed, fermions a of fraction the of third which family, (and can be flavour violating. mass eigenstate with theeffective quark familon-up-top mass coupling eigenstates constant quark has eigenstates and to the quark be mixingswhich stem leads from to the the roughthe estimate sizeable of cross-section for familon strahlung of order femtobarn, see figure familon (see eq. ( dominantly of themostly fields to ( mass basis we find thesmall heaviest fractions quark of mass eigenstate to consist dominantly of JHEP01(2016)160 10 TeV. In particular, the grand ∼ – 22 – , which amounts to a light familon field with potentially 2 , which enhances the predictivity of the respective model. We have shown (10) GeV, the model with familon-charged-lepton interactions requires dark 50 GeV, such that the observed abundance can be matched. On the other 3 O ∼ Since the family symmetries are introduced to explain the observed pattern of masses In particular we find that, on the one hand, the bounds from the colliders effectively An intriguing possibility is given by the combination of the constraints on the mass and Concerning collider phenomenology at the LHC, we have investigated interesting sig- We have shown that regions in the parameter space of those models exist that al- respect to flavour observables.ogy investigated It here would allows begiven to interesting by distinguish to the see, Abelian betweenof the whether and model the the two specific classes phenomenol- non-Abelian predictionsencouraging. from of case. family family The symmetries symmetry for prospects collider for phenomenology future are investigations very collider data. Onconstraints tells the us, other that we hand, canprovided the expect the to fact effective start couplings that seeingnot are excesses the too that order effective might far one hint parameters above and at the familons, the are electroweak flavour-symmetry-breaking subject scale. scale to is and mixings of the Standard Model fermions, they tend to lead to similar predictions with collider is compatible withconstraints the from constraints the on collider the experiments dark and sector, precision and experiments it such ispush as already the MEG. subject familon to vacuumorder expectation to values accommodate of the the non-observation of effective the models considered to familon-induced higher excesses values, in in coupling parameters from section flavour violating couplings tofrom the section fermions, with thethat results the from potential collider phenomenology of causing flavour violating signals at the LHC or at a future lepton corresponding to less complicatedflavour violating family leptonic symmetry final models,the states it tri-linear at Higgs is future coupling possibleeffects lepton or from to its colliders, the test UV decay along complete width. lepton with modelslepton with flavour differences It a violating is in non-Abelian decays also family and symmetry, possible the such as to violation charged test of lepton the universality low-energy in meson decays. is considered. natures, such as flavoursingle violation in tops, fermionic which final allowand states, to the additional directly quarks. di-Higgs test In events the the and effective case coupling of the between familon the fields familon coupling fields exclusively to the lepton sector, unification inspired model allowsas for small masses as ofmatter familon masses fields above and 100 GeV darkmasses and matter the of candidate model with familon-neutrinohand, interactions dark allows for matter massesmodels beyond considered here, a unless few the TeV fine-tuned lead case to of a overabundant resonant dark annihilation matter cross for section the dark sector. low for matching theresonant observed annihilation, dark and matter for reliccessible dark density, electroweak matter even scale, and without familon with the masses the fine-tuning on UV of the cutoff experimentally above ac- which gives rise to (non-Abelian) familon portal interactions between the visible and the JHEP01(2016)160 4 , ), 3 A , b (A.4) (A.1) (A.2) (A.3) 2 ) (same , b 1). The (2011) 3 1 b ≡ 43 3 . , ωa = ( ω 2 a ].    products: B 2 2 1 3 4 ) triplet: b b b ), , ω 3 2 1 a A 3 1 ]. a a a , with a 3 SPIRE , a − − − 2 IN π/ 3 2 1 2 = ( i , a ][ b b b SPIRE 1 e 2 1 3 a Gen. Rel. Grav. a a a IN [ ≡ , TA . ]. ,    , = ( 0 00 ω ( 1 2 1 1 1 , and 1 respectively. The group A 00 ]. ω ∼ ∼ ∼ = , 1 SPIRE ) ) ) 0 a ]. ] 2 3 1 IN b b b Young Pulsars and the Galactic Center and 3 3 3 ][ SPIRE 2 a a a AB [ ω IN arXiv:1303.5076 + + + SPIRE [ ][ 3 1 2 IN , [ b b b ) and an anti-symmetric ( 2 2 2 s arXiv:1502.02805    a a a Planck 2013 results. XVI. Cosmological , 2 1 3 + + + – 23 – b b b 3 2 1 1 2 3 a a a b b b The GeV Excess Shining Through: Background 1 1 1 ] with square brackets indicating (2014) A16 − − − a a a ) transforming as 1 ]. 3 2 1 arXiv:1410.2376 00 25 b b b , 1 = ( = ( 2 1 3 571 0 ] = ( a a a × 00 24 ] ), which permits any use, distribution and reproduction in arXiv:1404.2318 ] 0 [ − − − SPIRE AB 1 3 2 AB IN [ arXiv:1504.02477 AB b b b [ ), (1 [ , 3 matrices. For specific triplets 1 3 2 Millisecond pulsar interpretation of the Galactic center gamma-ray ][ 00 (2014) 76 [ a a a 1 × 2 2 2 Are we Really Seeing Dark Matter Signals from the Milk Way Center? × (2014) 1 58    CC-BY 4.0 1 3 Gravitational anomaly and fundamental forces 3-4 ), (1 , (diagonal in the basis we are considering), 0 This article is distributed under the terms of the Creative Commons = 1 T s Astron. Astrophys. ] × , (non-trivial) are conjugate to one another, transforming under a specific collaboration, P.A.R. Ade et al., 00 , by getting multiplied respectively by AB JHEAp [ T arXiv:1001.3236 , rules [ 4 and 1 0 ). The conventions follow [ A Frascati Phys. Ser. Systematics for the Inner Galaxy Analysis GeV Gamma-ray Excess Planck parameters excess 2499 has 4 irreducible representations, 1 triplet and three singlets. 1 is the trivial singlet, F. Calore, I. Cholis and C. Weniger, R.M. O’Leary, M.D. Kistler, M. Kerr and J. Dexter, Q. Yuan and B. Zhang, G.A. Gómez-Vargas, J.J. van der Bij, B 4 [5] [6] [2] [3] [4] [1] any medium, provided the original author(s) and source areReferences credited. Open Access. Attribution License ( It is also possible to construct a symmetric ( products are (1 acts on triplets through 3 under generator for A A and 1 generator, This project has received funding fromhas the Swiss received National funding Science Foundation. fromsearch, This the technological project development European and demonstration Union’s under2012-327195 Seventh grant SIFT. agreement Framework no Programme PIEF-GA- for re- Acknowledgments JHEP01(2016)160 ]. , 02 , JHEP , D 82 (2015) JHEP 13 SPIRE , θ D 92 JHEP D 85 and Large Nucl. Phys. IN Phys. Lett. , , , ][ 115 13 New J. Phys. θ , ]. ]. Phys. Rev. , ]. Phys. Rev. Phys. Rev. SPIRE SPIRE , , IN IN ][ ]. SPIRE ][ IN Dark Matter and Gauged Phys. Rev. Lett. Neutrino Mass and Mixing: ][ arXiv:1402.4271 ]. Leptonic mixing, family , [ ]. (2011) 033007 SPIRE ]. IN symmetry at colliders and in the ][ SPIRE 4 SPIRE D 83 IN SPIRE A IN Discrete dark matter ][ IN ][ ][ Neutrino Model for Nonzero arXiv:1311.3213 ]. ]. family symmetry and neutrino bi-tri-maximal arXiv:1206.1570 4 [ (2014) 045018 [ A Non-Abelian Discrete Dark Matter ]. ]. ]. ]. ]. 16 arXiv:1505.03862 Symmetryless Dark Matter Phys. Rev. Beyond Minimal Lepton Flavored Dark Matter Flavored dark matter beyond Minimal Flavor SPIRE SPIRE [ ]. Flavor portal to dark matter , – 24 – ]. IN IN Ultraviolet Completion of Flavour Models hep-ph/0507176 SPIRE UV completions of flavour models and large ]. ][ ][ SPIRE SPIRE [ SPIRE SPIRE IN IN IN IN IN SPIRE ][ Scotogenic SPIRE ][ ][ ][ ][ (2014) 091801 IN arXiv:1504.03955 IN arXiv:1405.6709 SPIRE arXiv:1104.2601 ][ (2012) 1250134 (2015) 130 [ [ ][ [ IN T(13) Flavor Symmetry and Decaying Dark Matter ]. ]. New J. Phys. 112 (2006) 31 ][ , 12 Non-Abelian family symmetries in Pati-Salam unification Family symmetries and alignment in multi-Higgs doublet models A 27 SPIRE SPIRE (2014) 72 IN IN B 733 JHEP (2015) 080 [ arXiv:1011.5753 arXiv:1103.3053 (2011) 597 , ][ [ [ 10 08 arXiv:1111.1270 arXiv:1501.07268 arXiv:1007.0871 arXiv:1211.5370 arXiv:1111.3952 hep-ph/0405152 [ [ [ [ [ [ arXiv:1011.6662 B 701 Phys. Rev. Lett. JHEP [ , JHEP , , arXiv:1502.02200 Nucl. Phys. Radiative Origin of All Quark and Lepton Masses through DarkNon-Abelian discrete Matter symmetries with and Flavor neutrino masses: Two examples Derivation of Dark Matter Parity from Lepton Parity (2011) 303 (2011) 207 [ , Int. J. Mod. Phys. , (2013) 065 (2012) 097 (2004) 104 CP arXiv:1101.0602 (2011) 062 03 01 mixing symmetries and neutrino phenomenology [ Symmetry 6 Flavor Symmetries Violation arXiv:1510.04694 δ (2015) 016004 B 848 B 700 (2012) 033008 011801 (2010) 116003 from Theory to Experiment Phys. Lett. universe I. de Medeiros Varzielas, I. de Medeiros Varzielas, O. Fischer and V. Maurer, S.F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, I. de Medeiros Varzielas and D. Pidt, I. de Medeiros Varzielas, I. de Medeiros Varzielas, R. Gonzalez Felipe and H. Serodio, I. de Medeiros Varzielas and L. Merlo, E. Ma, E. Ma, I. de Medeiros Varzielas and G.G. Ross, SU(3) P. Agrawal, M. Blanke and K. Gemmler, M.-C. Chen, J. Huang and V. Takhistov, E. Ma, A. Natale and A. Rashed, L. Calibbi, A. Crivellin and B. Zald´ıvar, F. Bishara, A. Greljo, J.F. Kamenik, E. Stamou and J. Zupan, A. Adulpravitchai, B. Batell and J. Pradler, Y. Kajiyama, K. Kannike and M. Raidal, E. Ma, M. Hirsch, S. Morisi, E. Peinado and J.W.F. Valle, Y. Kajiyama and H. Okada, [8] [9] [7] [25] [26] [23] [24] [20] [21] [22] [18] [19] [15] [16] [17] [13] [14] [10] [11] [12] JHEP01(2016)160 , , ] decay γ ]. + ]. (2014) e Phys. Rev. → Phys. Rev. , SPIRE , + IN SPIRE D 90 µ [ ]. ]. ]. IN ]. First Look at the Comput. Phys. ][ , in Abelian and ]. arXiv:1412.3671 SPIRE µτ SPIRE SPIRE [ (2014) 2250 SPIRE IN ]. IN IN [ → IN FeynRules 2.0 – A complete ]. Phys. Rev. ][ ][ SPIRE ][ , 185 IN SPIRE ][ Higgs ]. IN SPIRE (2015) 281 arXiv:1411.5606 IN MicrOMEGAs4.1: two dark matter ][ ]. , ][ arXiv:1407.6129 (1990) 1215 ]. ]. [ Higgs-flavon mixing and LHC SPIRE ]. IN 64 SPIRE B 896 ]. ][ IN SPIRE SPIRE arXiv:1308.6176 ][ IN IN arXiv:1502.05915 SPIRE [ arXiv:1505.07122 [ [ IN SPIRE ][ collaboration, M. Bicer et al., [ (2015) 322 Perturbative Unitarity Revisited: A New Upper arXiv:1407.6607 IN ][ Discrete family symmetry, Higgs mediators and Total Width of 125 GeV Higgs Boson – 25 – [ hep-ph/9907213 ][ Clues for flavor from rare lepton and quark decays circular colliders [ 192 Nucl. Phys. Comput. Phys. Commun. New constraint on the existence of the arXiv:1303.0754 − ]. ]. , Phenomenology of flavon fields at the LHC , e [ (2014) 164 + Phys. Rev. Lett. (1979) 277 (2015) 053 Hierarchy of Quark Masses, Cabibbo Angles and e , (2015) 053 Higgs pair production at next-to-next-to-leading logarithmic 01 arXiv:1203.6636 SPIRE SPIRE 05 (2014) 094 [ 09 ]. IN IN Non-unitarity of the leptonic mixing matrix: Present bounds and Testing sterile neutrino extensions of the Standard Model at [ arXiv:1503.01073 10 B 147 ][ [ Natural fermion mass hierarchy and new signals for the Higgs arXiv:1203.3456 (2000) 033002 JHEP [ JHEP , SPIRE arXiv:1503.01084 JHEP , arXiv:0911.2149 (2013) 201801 [ , IN [ JHEP (2012) 041 ][ D 62 , 110 12 (2016) 195 The International Linear Collider Technical Design Report – Volume 2: Nucl. Phys. , T3PS v1.0: Tool for Parallel Processing in Parameter Scans Comput. Phys. Commun. 198 (2015) 072 , Higgs measurement at (2012) 261801 arXiv:1306.6352 ]. JHEP arXiv:1406.6054 , , 06 Phys. Rev. [ collaboration, J. Adam et al., (2010) 036012 , 108 13 SPIRE IN arXiv:1310.1921 Physics JHEP MEG Phys. Rev. Lett. accuracy at the LHC TLEP Design Study WorkingPhysics Group Case of TLEP boson Lett. D 81 phenomenology in a simplified076004 model of broken flavor symmetry future lepton colliders Bound on the Higgs Boson Mass candidates Commun. future sensitivities [ toolbox for tree-level phenomenology [ theta CP-violation non-Abelian flavor symmetry models I. de Medeiros Varzielas and G. Hiller, M. Ruan, H. Baer et al., V. Barger, M. Ishida and W.-Y. Keung, D. de Florian and J. Mazzitelli, E.L. Berger, S.B. Giddings, H. Wang and H. Zhang, K.S. Babu and S. Nandi, S. Antusch and O. Fischer, L. Durand, J.M. Johnson and J.L. Lopez, K. Tsumura and L. Velasco-Sevilla, V. Maurer, S. Antusch and O. Fischer, A. Alloul, N.D. Christensen, C. Degrande, C. Duhr and B. Fuks, G. Bélanger,F. Boudjema, A. Pukhov and A. Semenov, C.D. Froggatt and H.B. Nielsen, J. Heeck, M. Holthausen, W. Rodejohann and Y. Shimizu, I. de Medeiros Varzielas and G.G. Ross, [44] [45] [41] [42] [43] [39] [40] [37] [38] [34] [35] [36] [32] [33] [30] [31] [28] [29] [27] JHEP01(2016)160 ]. ]. (2012) pp ] SPIRE IN Phys. Rev. B 717 , The Large ]. ][ ]. , in -channel in s decays boson and a top quark SPIRE SPIRE arXiv:1506.02800 IN − IN ` W ][ arXiv:1209.3489 + ][ Phys. Lett. ` [ , + K -channel cross section in → arXiv:1205.5764 t [ + B ]. physics beyond the standard model (2013) 022003 arXiv:1209.4533 s`` arXiv:1408.1627 SPIRE [ (2012) 142 [ (2015), pg. 259–300, [ 110 IN → -channel single top-quark production cross t ][ b Top-Quark Physics at the LHC – 26 – B 716 ]. Springer (2012) 035 , TeV with the ATLAS detector using the Matrix Element TeV with the ATLAS detector ]. 12 and future (2014) 054014 SPIRE = 8 = 7 K IN s Phys. Rev. Lett. [ s R , Phys. Lett. √ SPIRE √ JHEP D 90 , arXiv:1406.6482 IN , [ Evidence for single top-quark production in the Measurement of the Evidence for the associated production of a TeV ][ TeV Test of lepton universality using Measurement of the single-top-quark Evidence for associated production of a single top quark and W boson in TeV = 7 s = 7 = 7 s √ s Phys. Rev. √ collisions at , √ pp (2014) 151601 collaboration, collaboration, collaboration, ]. arXiv:1511.05980 , collaboration, collaboration, 113 arXiv:1205.3130 [ collisions at SPIRE IN Hadron Collider – Harvest of Run 1 ATLAS in ATLAS at ATLAS 330 proton-proton collisions at Method CMS pp [ ATLAS section in Lett. opportunities CMS collisions at K. Kröninger,A. B. Meyer and P. Uwer, G. Hiller and M. Schmaltz, LHCb collaboration, [51] [53] [52] [49] [50] [47] [48] [46]