JHEP12(2019)112 Springer July 10, 2019 : December 3, 2019 December 16, 2019 November 26, 2019 : : : Received Revised Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP12(2019)112 -physics anomalies from B [email protected] , -mesons. It seems interesting to assume that this leptoquark B . 3 1906.11666 The Authors. c Beyond , GUT, and Composite Models

A vector leptoquark at the TeV scale, mostly coupled to the of the , [email protected] -physics, as well as the one-loop potential for the pseudo Nambu-Goldstone . The -anomalies, predicting very small couplings to the Right-handed currents without any Av. Bustillo 9500, 8400, S.C. deE-mail: Bariloche, Argentina Centro Bariloche, At´omico Instituto Balseiro and CONICET, Open Access Article funded by SCOAP ArXiv ePrint: B theory has a rich phenomenology and a candidate forKeywords: dark . Nambu-Goldstone . WeModel show that fermions can anarchic partial accommodateB compositeness the couplings of of theadditional Left-handed Standard hypothesis. currents required By by makingnamics, the use in of terms an of effective weakly theory coupled description resonances, we of are the able strong to dy- compute the corrections to violation in the decaysbelongs of to the samelem, beyond since the the third StandardHiggs generation Model potential. of sector fermions We thatTeV play present solves that the a the leading contains composite hierarchy role the Grand prob- in required Unified the Theory vector instability with leptoquark of resonances the and at develops the the Higgs as a pseudo Abstract: third generation, is the preferred option to explain the hints of flavor universality Leandro Da Rold and Federico Lamagna A vector leptoquark fora the composite GUT JHEP12(2019)112 , . 23 30 ]. Although there is no 9 – 6 [ 19 ¯ ` s` couplings → 1 b U 7 8 3 -, both in the charged current process 24 B – 1 – great physicist and greater person, who left us too soon We dedicate this work to the memory of Eduardo Pont´on -anomalies 25 B 28 31 ) ∗ ( D for each kind of interaction, leading to one of the most interesting 16 11 -symmetry R 4 σ P 19 9 29 1 5 14 26 22 10 ], and in the neutral current process 5 – -anomalies 1 [ B c`ν 4.2 Bounds 4.3 Predictions for 4.4 Phenomenology of4.5 the pNGB scalars Case without 2.7 Effective theory 2.8 Potential 4.1 EW symmetry breaking and spectrum of resonances 2.2 Fermions 2.3 Partial compositeness2.4 and flavor structure and2.5 conservation 2.6 Bounds 2.1 Coset structure → b conclusive evidence of new physicsModel yet, the (SM) are deviations above from 3 thechallenges predictions of of the flavor Standard physics.but There is at no theoretical evidence level of it a is common very origin interesting for both to deviations, explore this possibility. These anomalies 1 Introduction In the lastuniversality years (LFU) different in semileptonic experiments decays of have reported hints of violation of lepton flavor D A model in 5 dimensions 5 Conclusions A Representations of SO(12) B Mass matrices C Numerical estimates of the Right-handed 3 A three-site model 4 Phenomenology Contents 1 Introduction 2 A composite GUT for the JHEP12(2019)112 , ) 1 ∗ 1 ( U D R ], transforming , requires a well 10 1 [ U 1 . 2 U R ] have made a detailed ). In these scenarios the 30 , SM 28 -physics can be explained by , B 17 ]. Refs. [ -anomalies, besides a 35 B – 11 – 2 – of H. Composite GUTs can also solve the hierarchy SM -physics puzzle with a single new at the TeV scale. B ] for a solution with a single scalar leptoquark 36 As usual in composite Higgs models, since the elementary fields do 2 ]. 37 under the SM gauge group, with a mass of order few TeV and interactions 3 / 2 ) 1 , In the present work we do not pursue precise gauge coupling unification. Instead we Grand Unified Theories (GUT) naturally predict the presence of leptoquarks, but As is well known new dynamics at the TeV scale, mainly coupled to the third genera- Many references have shown that the deviations in 3 See also, for example,In ref. this [ kind of composite GUTs, the global subgroup H must also contain the custodial symmetry of 1 2 -anomalies, the Right-handed ones are fixed to obtain the Yukawa couplings. As all the resonances, onceB the Left-handed mixings of the are fixed to explainthe Higgs the sector to have a chance to pass the electroweak precision tests (EWPT). physics in composite modelsthe is elementary anarchic fermions partial withknown, compositeness, the in SCFT where this are the case dominated interactionsat a by low of hierarchy energies linear of by interactions. the elementary-compositeand running As linear mixing of is angles mixings the of well linear can the couplings, beprocesses. CKM leading matrix, generated to and the Since simultaneously hierarchical suppressing the spectrum the same flavor violating elementary-composite mixings enter in the interactions with are guided by the lowcontains energy phenomenology, and one we resonances demandas G with to the well be proper as such that quantum adefined the numbers SCFT NGB flavor to be Higgs. structure identified of with A couplings. solution to One the of the most interesting approaches to flavor emerge as aexample composite in Nambu-Goldstone ref. Boson [ not (NGB) furnish state full in representationsinduce G/H, a of as potential the for discussed global theelectroweak for NGBs. symmetry symmetry Under breaking of some (EWSB) the suitable dynamically. conditions SCFT, this at potential can loop trigger level they composite GUTs usually thecontaining grand as unified a group subgroupgauge H the and is gauge a symmetry fieldsSCFT, of global of and the the symmetry gauging SM the SM of subgroup are (G theproblem: G taken SCFT, as if elementary the fields, SCFT weakly has coupled a to the larger group G, spontaneously broken to H, the Higgs can it would be very interesting to find a commonusually origin with for masses both at the phenomena. a scale new of grand strongly unification. coupledbe However, in taken field composite at theory GUTs, the where (SCFT) TeV is scale, leading introduced, to the leptoquark compositeness resonances scale with can masses of few TeV. In analysis of flavor observables,and showing simultaneously what satisfy the kind bounds of fromas other couplings the flavor best can observables. one explain to This explain scenario the the remains anomalies tion, is also neededtion to stabilize between the these Higgs deviations potential. and Although the there hierarchy is problem, no from obvious a connec- theoretical perspective, a non-trivial flavor structure, mainly coupled to the fermionsadding of to the the third SM generation. as a spin ( one leptoquark,with known Left-handed in currents the of literature SM as fermions [ could be explained by introducing new physics at a few TeV scale, with interactions having JHEP12(2019)112 . In . Fi- ) 5 ∗ ( D R , leading to some freedom a priori ], although in a different context. . In the same section we present ) 38 ∗ ( we show a composite GUT contain- we describe the phenomenology of the K 4 2 R with Left-handed currents and negligible 1 and U ) ∗ – 3 – ( D . Thus anarchic partial compositeness gives, as -anomalies R 1 U B we present an effective description of the resonances of 3 , the resulting Right-handed mixings are suppressed by the L τ -physics, as B , the third generation of Left-handed leptons play an important role L τ that can create spin 1/2 resonances, transforming with linear irreducible SCFT O -anomalies. We describe the coset structure of the SCFT, the content of NGBs, Our paper is organized as follows: in section An effective weakly coupled description of the above dynamics can be obtained by B , are taken of order few TeV, whereas the interactions between them are characterized ∗ -anomalies and stabilize the Higgs potential. Besides, due to the large degree of compos- with the adjoint representation ofoperators G. Besides, we alsorepresentations assume of that G. the These there representationsfor are are fermionic not model fixed building.m The masses of the first level of resonances, collectively denoted as We consider a theory withsector two different called sectors: elementary an that SCFT ishave or weakly a composite coupled simple sector with global and the symmetry another a SCFT. group subgroup G, The H. spontaneously SCFT This broken isof breaking by assumed generates the the to a strong coset set dynamics ofconserved G/H. to NGBs Noether associated currents Some to of the of the broken SCFT these generators can NGBs create will resonances be of spin identified one, as transforming a composite Higgs. The boson (pNGB) states.an We appendix present we our present conclusions a and 5D some model discussions that in can also section 2 be used to A describe composite the SCFT. GUT for the the SCFT in terms of atheory, three-site we model. scan the In parameter section spacetrum finding of regions new with . EWSBthem and We also we with compute calculate the the the present spec- corrections bounds,nally, we to as comment several well very observables, briefly as comparing on the the corrections phenomenology to of the flavor new quantities pseudo as Nambu Goldstone the fermionic representations andestimates flavor structure, associated as to well asthe some effective important low bounds energythe and SCFT, physics whose obtained structure afterpotential depends of integration only the of on NGBs. the the In pattern section massive of resonances symmetries, of and the one-loop itenes required for in the potential. This situation was considered in ref.ing [ the usual ingredientsthe of composite Higgs models, as well as a vector leptoquark for working with a theorysite of describing resonances. the elementary WeSCFT. sector will and consider In the a other this three-siteexplicitly, two case theory, as sites with the well describing the one-loop as resonancesfined first of the potential predictions. the spectrum of of the WeB new NGBs will states is and show finite their that and couplings, composite can leading be GUTs to calculated can well de- simultaneously explain the ratio of charged leptonsmall mass over Right the handed Higgsa couplings vacuum very expectation with good value (vev), approximation,interactions giving with interactions very Right of handed currents, without any additional hypothesis. requires a large mixing for JHEP12(2019)112 is 1 (2.2) (2.3) (2.4) (2.1) U , and 2 , 1 − U m min leptoquark resonances, H 1 0 ≡ U Z decomposes as: X ., c 11 . and c 0 U(1) ⊕ W × 6 R √ / 3 1 ]. − 40 ) 1 SU(2) , 1 × , However, since in this case L ¯ 3 ., SU(2). Besides, we identify hyper- . ( c 4 . × the representation , thus for simplicity we assume that , that are proportional to X ) ⊕ π ∗ T ( 4 0 ], the last one, as far as we know, has not been + h SU(2) ) 6 min D 4 2 37 × R SU(2) √  , c 2 ∼ ∗ + , SCFT ψ g 1 R – 4 – O and ( 3 ¯ ψ  ) SU(3) T ⊕ ∗ ψ ( 0 ω ≡ K ) → SM 1 R g , Y ] for leptoquarks). 1 L ⊃ , 10 1 SO(4) ( can be associated with an unbroken generator. Let us discuss ] and eventually the first generation [ U(1) and SO(4) × ∼ 1 39 × . Under SO(10) and H U 10 11 , taken as: ⊕ ∗ SO(6) SU(3) g 1 → ⊃ ∼ 11 SU(4) SO(10) ∼ , of order TeV. ∗ → being the coupling at the high ultraviolet (UV) scale Λ at which this Lagrangian explicitly breaks the global symmetry of the SCFT. The fermions of the SM have /g ψ ∗ ω m SM The set of broken generators in the coset SO(12)/SO(11) transform, under SO(11), We start by describing some features of the unbroken group. SO(11) contains SO(10) The gauge fields and fermions of the SM are external to the SCFT, they are taken as We assume that bilinear interactions are suppressed, havingThe no first impact two in properties the were already phenomenology, except shown in ref. [ 3 4 = SO(11) possibly for the sector [ considered before. with the representation with SO(6) charge with the following combination: SO(12)/SO(11), such that the coset structure in some detail. that, as is well knownextension from of the the study SM ofproperty gauge GUTs, is: group. can accommodate A a possible Left-Right pattern symmetric of subgroups that allows to see this one state that has(we the follow proper the quantum notation numbersassociated of to to be ref. a identified [ brokenresulting with generator, in the it a is suppressed heavierthus than, can effect not for in be example, accommodated. For this reason, we will consider instead a larger coset: 2.1 Coset structure SO(11)/SO(10) is thecontains minimal the coset SM of gauge symmetryas simple group a groups as pNGB with well and, as the custodial after following symmetry, proper it properties: identification delivers a of it Higgs hypercharge, it contains a composite spin linear interactions with the SCFT, that also break G explicitly: with is defined. all the couplings betweenf resonances are of the same order. Theelementary NGB fields. decay Demanding constant Gof is G to contain the SM gauge symmetry group, the gauging by a single coupling JHEP12(2019)112 , . ), B 6, 10 SM √ con- 2.3 (2.6) (2.5) / . 6 X c . √ T / c ) can be will lead 1 2 ) can induce ⊕ P − 2.6 2 2 6 , R ˜ V 3 √ 2 / , T 1 3 ) ( is associated to a 1 ≡ , solution. ⊕ , the representation 1 Y 11 , as well as two new 6 1 , 6 11 √ U 3 / / 1 2 . ) − 2 ) , 1 3 120 , ( 1 ⊂ , ∼ leptoquarks. The leptoquark in , and the second one of the 3 , that under SO(11) decomposes ( 2 1 ˜ V 45 this parity can be written in terms U ⊕ 66 0 ) , the present model has a global U(1) 12 1 , ), the multiplets ( 1 2.4 , q, d, ` , 2.3 . However in this case the NGB leptoquark 1 . In generic leptoquark models . ( 3 ) the decomposition of c 45 . / . As we will show, the presence of c 4 ⊕ – 5 – ⊂ 12 2.4 − ., 0 j ) ⊕ c ) as: δ . 1 6 3 c ) shows the transformation properties of the NGBs, , 12 , √ i 3 / 1 ⊕ δ min 2 , 2.4 2 6 − -physics that can destabilize the 1 ) √ ( − / B , q, e 1 and ( 1 can be identified with , ⊕ ij ) 6 1 δ 0 10 1 / , whereas the ones associated to the color triplet lead to a -symmetry, that corresponds to a parity and enlarges SO(12) , ) in the literature. , 7 2 11 , for the representation ) 3 1 = 1 H ⊂ 1 Z , 2 , ¯ S , ij 3 A 3 , in ( ( 3 P and u, ` that interact linearly with the SM fermions can be decomposed 1 1 ( ⊕ U 0 55 ⊕ ) 0 2 ) SCFT of . We have shown in eq. ( , 1 10 2 O , , there is another spin one leptoquark: , 11 1 10 1 ⊕ 1 , ( U as sums of irreducible representations. To avoid explicit breaking of G ⊕ 8 45 ( ⊕ 12 matrix as: 55 c.c. means that, for the complex representations as the color triplet, one has to SM ∼ ∼ , giving contributions to × ∼ 1 and the Higgs field G ⊕ S 55 ϕ We will add a discrete It is also possible to choose other identifications of hypercharge, as Besides The currents of the SCFT associated to the global symmetry SO(12) can create spin one 66 decomposes under SO(10) and H is associated to an unbroken generator, whereas the leptoquark in partners of the SM fermions can be found in the following representations of SO(10): A Right-handed neutrino can beSO(10). embedded Larger in representations a are singlet also or in possible. the The adjoint representations representation of of eq. ( 2.2 Fermions The operators under these decompositions must contain the representations of the SM fermions. Given eq. ( to O(12).generators We are, are respectively, interested odd andbasis in even defined the under in this transformation appendix parity,of under leading a to which 12 odd broken NGBs.to and In several the unbroken simplifications as well as a candidate for dark matter. that forbids decay. TheSM other fermions. two states do not have dimension-four operators with that allow to embed is an 55 broken one, thus the former results lighter than thestates latter. transforming as: ( baryon decay, however, as we will show in section where the first lineWith contains the the identification decomposition of oftained hypercharge the of in eq. the ( states that transform withas: the adjoint representation 55 add the charge conjugate one.those Eq. associated ( to thecall colorless generators lead toleptoquark two usually multiplets: called a SM singlet that we where JHEP12(2019)112 ) ` 2.7 SCFT 66 up to and ), and O interact u min ` 2.3 ) and ( H and 2.6 can distinguish u , P q × × × 66 66 66 220 220 ) (2.7) 0 can interact with SO(12) . Each elementary fermion q 10 , ) , we will briefly comment on ⊕ 1 P SCFT 220 . The parity ⊕ 45 55 55 11 11 11 11 O 165 165 ( 55 SO(11) . Besides, from eqs. ( 10 , we assume that ⊕ . ( . In the following we will assign the ⊂ ) 1 and 0 can interact with several components ⊕ 0 4.5 ) is odd, thus if we assign a well defined SCFT 220 0 ` 45 0 10 SCFT 1 66 O 45 45 10 10 10 45 ⊕ SCFT 66 10 120 10 O SO(10) O and ⊕ u 120 – 6 – ( 6 45 6 6 6 ( √ √ √ √ 0 0 0 0 ∼ / / / / ) ) ) ) 1 is conserved and the elementary fermions ∼ 1 1 1 2 3 2 1 − ) ) ) or with the ) , , , , 55 P 2 1 3 min 1 2 1 2 1 11 , , , interact with is even and , , , , , ⊕ H 2 1 1 11 e 1 1 1 1 1 ⊕ 0 , , , ( ( ( ( , 3 3 3 ⊂ ¯ 3 ( ( ( 10 55 165 ( and : ∼ ∼ 10 d 66 + + − − − − − − P 66 220 1 ` . In particular, as shown in table e q d u ϕ ¯ H S field to the elementary fermions, and we will mix them with the components s inside 1 SCFT 10 O only, whereas , however we will assume that the SCFT operators have different anomalous di- : either with the . Embedding of the composite partners of the elementary fermions, from SCFT 66 SCFT 220 O SCFT 66 O It is also interesting to consider the scenarioLet us without now describe the interactions between the elementary fermions and the Higgs. In order to obtain interactions between all the SM fermions and the SCFT, we will con- O of the SCFT operators shown in that table. the consequences of this assumption in the section Since bilinear interactions with the Higgs have been assumed to be suppressed, the interac- one can see thatof the elementary fermions between both parity to the elementaryinteract fermions, only with oneparities multiplet of of table SO(11) in and mensions, such that one ofas the a interactions simplification dominates of over this thejust situation other with we one (see will section consider thatwith each elementary fermion interacts marks to distinguish SO(10) representationsrepresentations that of arise SO(11). from the decomposition of different sider that the following operators arecan present: interact with more than one SCFT operator, for example embedded in representations of SO(12), we are interested in the following: where we have shown the decompositions under SO(11) and SO(10). We have used the Table 1 SO(12). In the last threerepresentation lines of of SO(11). the table we show the NGBs that transform with the fundamental JHEP12(2019)112 , ], ∗ SM 42 m [ ] We (2.8) ψ   43 ≡ .[ , whereas Ψ ]. It is also jk ∗ g -odd, but for 429 37 f/m P ψ ⊕ λ and = , thus our choice is of SO(11), however (1), as well as their 165 ψ . When considering ). The resonances of ψj . Thus for Λ θ . The mixings of the R ... 2 , both λ ⊕ 3 / ψ 32 u 2.1 u 5 × O  ⊕  tan ∗ − 2 it is not. In this way a ∗ ∗ 11 g ψ / g g 11 L 5 3 ]. ∆ and ∼ is in a representation of G q ψ   ∼ ` 41 ' Λ) ψ 11 / ∼ ψ ∼ ∗ ) leads to Yukawa interactions of t × 165 ψ m y -symmetry is respected by Yukawa ( y 2.1 × 55 , P ' ∼ : Ψ 55 , as well as a massive resonance with a ψ 1 ψ ψ λ  ¯ , and ψP . f 6 , that contains 11 / ψ 1 ) λ – 7 – 12 2 , 3 ⊃ ( Ψ eff ¯ q L ≡ must be understood. Since Ψ q is suppressed, whereas for ∆ qP ψ λ q, u, d, `, e = ψ , that when acting on Ψ selects the component with the same quantum ψ (1). The hierarchy of SM fermionic masses is driven by hierarchical mixings, P , for example: ¯ ψ ×O 2 the coupling ∗ / 5 m > The interactions between the elementary fermions and the Higgs require insertions of The mass eigenstates can be obtained after a rotation of angle: Assuming that the running of the couplings is driven by the dimension of operator: Usually the SM fermions are embedded in the representation ψ , the coupling can be estimated to scale as: , thus the Yukawa couplings can be estimated as: ψ ψ couplings between fermionic resonancesmasses: are of the samelight order: fermions require at leastrequires one sizable of mixing the for chiralquarks mixings both can being chiralities, small, be since: whereas related the 1 with top mass the CKM angles and the masses, assuming that each λ generations, the mixings andwill couplings assume acquire generation that indices: SFCT the there flavor are structure no preferred of directions the in flavor SCFT space, is therefore anarchic, all the this coefficients means of the that in the if ∆ hierarchy of mixings can be obtained for different fermionsrealizing [ partial compositeness. Thistially rotation composite, leads with to degree achiral of component chiral compositeness that massless is state partially that elementary. is par- and Ψ is inthe a projector representation of G,numbers as the GUT symmetry, strictly speaking one has∆ to add Ψ (we use capital lettersinteractions for lead resonances, and to small mixing: letters for elementary fields). The linear where a sum over simplicity we will not consider them. 2.3 Partial compositeness andAt flavor low energies structure one can consider an effective description of the SCFT in terms of resonances for the down-type andcompatible the with charged leptons: the Higgsinteractions. embedded in an in order to do thatpossible one to has take to other take representations, a as different identification of hypercharge [ the SCFT interact with thethe composite elementary fermions. Higgs The and SCFT eq.to has ( obtain a the global proper unbroken Yukawa interactions symmetry of SO(11),containing the thus SM, the in the order partners interactions between of the resonances the the up-type SM quarks, fermions from and the the embeddings Higgs of must table be SO(11)-invariant. For tions with the Higgs are mediated by the linear interactions of eq. ( JHEP12(2019)112 2 `  ] (2.9) , the (2.12) (2.11) (2.10)  ej 45  , 2 e (1).  44 for , , ∼ O 3 3 In the present q q jk 2.5   5 c ∗ ∗ . SM SM d s g g y y 3 C 2 C SM τ λ λ . We will also show GeV. y 3 ` ∼ ∼  16 , 22 is the Cabibbo angle. ∼ . , 1 2 3 0 d d 10 ek e k R    e ∗ ' ∗ ) ∼ g 2 -anomalies can be fitted with g n µ ` C ( 3 dj 1  ] for some examples. is universal, due to the global B `  λ U ∗ 2 is understood. The couplings can for their numerical estimates. µ jk 43 g  , γ c √ n C and j R 1 ` ¯ d ∼ 40 ,   , f ) ,,   ) n 39 ( Rjk n 3 3 SM µ ( Rjk q q g , y   , and besides, at least for the second and , the mixings of the Right-handed charged ∗ ∗ 3 3 + `i SM SM q e u c ∼ g g    y y SM b 2 3 C 2 C ∗ k L , as well as the ones of section e y – 8 – λ λ g `  , g dj )  ∗  n µ g ( ∼ ∼ ∼ `k 1 2 2  e ` 1 2 3 U ∗  d u u µ g states, a sum over γ ) for qj 1 j  L  ¯ U q 2 2.9 1 ) jk ,  e n c √  ( Ljk , in the present scenario the g SM e ∼ y 2.5 ) leptoquarks are of special interest for our analysis: n ∼ L ⊃ ( Ljk 1 ,  ,  ,  1 g e 3 3 U 3  q q 2 arises from the SO(11) generators, and the factor q   ∗  SM t g √ ∗ 2 C 3 C y numerates the 1 / g λ λ `  ) can be safely ignored. See appendix n become very suppressed, and the Right-handed interactions of the second ∼ ∼ ∼ 2 3 1 . Since these hierarchical mixings lead to small mixing angles in the matrices ) q q u 2.11 3 n    ` ( Rjk is the SM Yukawa coupling of the quark are not fixed by these equations.  g ∗ g  SM f y 3 e  Using the estimates of eq. ( The interactions with the spin one resonances have a flavor structure similar to the The lepton sector depends on the nature of the neutrino and the realization of their and In fact this scale depends on the nature of the leptoquark, as well as on the size of its couplings to the 5 3 q 2.4 Baryon andLeptoquarks lepton can number mediate conservation baryonable, decay unless making the they theory have phenomenologically very unaccept- large masses,SM typically fermions. of order where the factor 1 couplings term of eq. ( where the index be estimated as: elaborate more on the neutrino masses, see refs. [ Yukawa couplings, except thatsymmetry in of this the case SCFT,The thus the interactions generically factor with they are misaligned with the Yukawa couplings. leptons are also hierarchical: third generations, they areand smaller than thediagonalizing corresponding the charged Left-handed mass ones: matrix,be the generated large in mixing the angles of neutrino the sector. PMNS We matrix will must assume this to be the case, and we will not As we will show ina section hierarchical mixing ofin that the section Left-handed that, leptons: for the given values of where  masses. The masses of the charged leptons require: chiral SM multiplet interacts predominantly with just one resonance, one gets: [ JHEP12(2019)112 1 L U re- 4.3 ) ∗ ( (2.14) (2.15) (2.13) (2.16) D R = 0. , including the SCFT q ) ∗ is given by: ( O , ). This symmetry D , resulting in a too L )] ∗ R ijkl l L 2.2 ` , . C , ≡ /m µ and 2 ` γ )  2 ∗ k L ( 2 , only the effect of the 031 005 SM ¯ . f ` , can be approximated by: . . v τ` D -physics, and in section 0 0 B )( C `l R SCFT ` j B L  2.3    q O `k + 2 µ 23 33  L L γ = R 334 258 g g qj i L 3 . .  q ∗ ∗ tb T ts qi defined in eq. ( V V  , with masses of order few TeV, that = 0 = 0 = SCFT ` 2 ∗ jk X ∗ ) + (¯ ˜ − ∗ V c O Y l L D SM D 1 il ` L c a  σ ∼ µ γ (1) arising from the dependence of the Wilson 3233 )2 – 9 – 1 k L ,R,R n ¯ C ` ( U 2 SM )( 2, and we have assumed that the contribution from that, to linear order in v ∼ O m j 015 003 L ) 2 √ . . ∗ q ∗ ( 23 0 0 1 + 2 a ) ]. As discussed in section c D f/ n σ (   Ljk ∗ ' µ 17 R g g γ ) ) ) ∗ ∗ ) for the quark degree of compositeness, a fit of n i L ( ( ' ( Lil 299 297 ] and the experimental value of q . . SM g D D 1 [(¯ 2.9 U 49 R R ], the Weinberg dimension five operator can be induced, with a = ], are: – = 0 = 0 m the generator of the U(1) ijkl 2 SM 9 37 TeV, with 46 , v D -physics it is convenient to work with the effective theory resulting / C ijkl 5 X SM D 1 R B C T R ⊃ ∼ 1 eff U L , with /m being generation indices. The dimensionless coefficient X 33 T L 3 g ) gives a contribution to / -anomalies 23 2 c B p i, j, k, l 2.13 The SM prediction [ As discussed in ref. [ = Using the estimates ofquires eq. ( recent results of Belle [ Eq. ( where we have used that: the lightest resonance dominatesmodel. the Below sum, we estimate aswe the we show contributions the will of numerical our show predictions model that in to happens a three-site in model. a three-site with leptoquarks on Left-handed currentseffective is Lagrangian important in our model. We obtain the following 2.5 In order to study the from the tree-level integrationclosely of follow the the resonances. analysis of Except ref. where [ explicitly stated, we will number conservation. Thus in the present model Wilson coefficient that can belarge generically estimated contribution to to be neutrino ofglobal masses. order symmetry To to avoid these themixing contributions with composite one the sector, can elementary add assigning fields, a for the U(1) example: usual numbers to the operators in principle could couplegroup to contains diquarks a inducingB generator baryon that decay. can Howeverassigns be the the identified expected SO(11) with baryon sub- elementary an number states, to operator forbidding the the of resonances, coupling of baryon and leptoquarks acts number: to in diquarks the and usual ensuring baryon way on the model there are, for example, vector leptoquarks JHEP12(2019)112 , ∗ τ × /f /g and 3 2 ` − few  (2.18) (2.19) (2.17) `` 9 3 q C 10 ∼ , leading  L 2 × Z b ) ), leads to ∼ -anomalies, g ∗ 7 . B 0 2.9 g/g ( ' 3 2 ` 3 e . ξ  TeV, the Left-handed ∼ 12 . mixes with a resonance ∼ -quark). Since it is not 0 symmetry that protects L Z L τ b f τ  /g . For negligible coupling to LR L 40 Z τ . . s`` 0 2 ] δg ]. We will show the numerical ` −  . → 57 , that is in agreement with the SM . 54 . TeV. This bound can be compared b , ) one obtains: = τ symmetry also protects TeV. Although this puzzle seems to / / 8 17 . Making use of eq. ( . 0014 1 3333 4.2 3 . 0 2.10 2322 LR − 0 C ∼ . C are fixed, the Right-handed ones can be 10 1  08 ] the violation of LFU can be parametrized ∗ leptoquark arises from LFU violation in 1 . ts U τ 0 U V 1 53 can induce large deviations in the couplings ' tb U − π /m 3 1 /m 0000 V we will describe the numerical predictions in a ` 4 . – 10 – 33 2 U and 33 L em , realizing a discrete . Given our choice for the embedding of the reso- L = 1 g coupling of the µ 4.2 2 α 2–3 is enough to satisfy both requirements. g = 1 L /m

) b TeV. This implies that, for ¯ 23

, the discrete 22 R c ∼ / L = W W W τ µ L ∗ L T W W : τ µ g g b g ) 23

g g Zb (1) µµ ∗ 10 c

( 32 C . In our model this ratio can be estimated as: = ( L D to pass the constraints [ g ∆ 23 R 2 ∼ O 2 point to LFU violation in ) L ) − . In section ) L g 3 ∗ ), we obtain /f ( /g T = − 3 ∗ ` K , the does not play any important role in the . g modifies the  2 ∗ µµ R at the same time, this coupling gets a larger modification, requiring 9 3 ` 2.19 1 defined in eq. (24). However, in our model q  C /g  U Z ν and ( ξ 3 g ∆  − R ] for an explicit calculation in the case of the 3 1 , that are in agreement with the SM at the per mil level also. In generic 10 ` ) we obtain: ] ) and ( 56 T TeV, that fixes the order of magnitude of  ¯ ν / × , with 52 1 2 ` = . – 4 Zν 2.14 . 2.18 0 ξ 0 L 50 3 ∼ is not fixed by them if the latter limit is satisfied, as we will assume from now on. ∼ ' T and ]. In this case the leading tree-level corrections are 1 2 is: [ Z ` ` (see ref. [ /f e L  55  2 A similar situation holds for The large degree of compositeness of Once the Left-handed mixings of As long as The deviations in ¯ τ [ `` /g ` 3 10  L − Z ` must have a large degree of compositeness. C L 3 Z τ q three-site model. extra tuning of orderresults performing ( a tree-level calculation in section nance mixing with the elementary to corrections of order 10 δg having g 10 possible to protect with the value required tointroduce fit some tension, a factor Zτ models with partial compositeness the correction to these couplings can be estimated as: One-loop radiative corrections can be estimated as: [ From eqs. ( One of the mostdecays. stringent At constraints one-loop on a prediction at the per milin level. the Following ref. ratio: [ thus adjusted to obtain theand proper masses. Using eq. ( 2.6 Bounds Using eq. (  therefore we obtain: τ , the preferred contribution from∆ new physics to the Wilson coefficients ∆ coefficient on the coupling JHEP12(2019)112 ] , ) ], ∗ 3 ( 64 59 CP and D R (2.21) (2.20) 12 ], the existence 2 40 ) ˆ a -anomalies, ref. [ B (Π ˆ a SO(11) being a function X = H ∈ 2 6–7 TeV. An interesting alter- in the -system that gives , (20–40) TeV. There have been a ˆ a & K O  T , or if it is broken by the couplings ∗ , ρ ˆ a 3 & 2 m , showing that both scenarios can ex- f Π ], as well as the presence of flavor sym- `j 1 SO(12) and  mixing combined with a solution to − 61 Π = Π ∼ , B ) G ∈ 2 require `i 60 − 10 TeV. Concerning the ], slightly increasing the amount of tuning of  ρ 2 ], where the lepton doublet and singlet of the ¯ ρ/f (5) TeV and B – 11 – ) , 66 . j L O 40 d /f , with cos( the broken generators of the coset SO(12)/SO(11). † µ Π , and & i γ symmetry in the composite “leptonic” sector. Either ˆ a H e ei i L . Although the composite fermion multiplets contain f  ¯ T 3 U d = Π + 75 TeV [ G ∼ ) . CP are independent as required in our set-up (and similar for U 0 `i ], where the authors consider a composite sector with `i ×  and ( ρ/f  ρ & 3 64 2 ], bounds from are relaxed to f ) sin( ∗ L 65 and give even stronger bounds: d U(1) m + R qi × ]. In the lepton sector the electron dipole moment and the flavor s  I eγ interact with SCFT operators that have different charges under U(1) can be written as: 58 i = Π). In the fundamental and adjoint representations of SO(12), transforms as: → ` , U U G µ U ]. The proposal of ref. [ ( 7 TeV. This condition introduces a tension with EW precision tests, that . and H 0 63 i , q . the real NGB d.o.f., and 62 (10) TeV [ f ˆ a O and Π: As discussed in ref. [ As in any other model with anarchic partial compositeness from linear interactions, & G , the constraints on , respectively, ∗ ψ Under SO(12), of 66 based only on symmetryNGB principles. unitary matrix: One of the main objects for this constructionwith is Π the the model. 2.7 Effective theory An effective low energy theory,and obtained containing the after elementary integration fermions of and the gauge resonances fields of as well the as SCFT, the NGBs, can be built thus in this scenario the Right-handed fermions). lead to usually require, at least: analyses two different cases: plain the anomalies.by This extending proposal G can to bestates G implemented with straightforwardly lepton in andsense our that baryon model, number, the elementary fermions are not unified, in the from operators like (¯ native is proposed insymmetry, as ref. well [ as aif the flavor U(1) elementary-composite interactionsλ respect U(1) the presence of naturally tinyof bilinear extended interactions color in symmetries anarchicmetries in scenarios [ the [ SCFT [ first generation are elementary, asquarks, well as can the be first implemented generationdangerous straightforward Right-handed contributions in up- to and the the down- present aformentioned processes model. are In suppressed, though this transitions case, the most there are several quantitiestron that dipole push moment the thatm compositeness requires scale to largerviolating values: decay the neu- few proposals for these problems, as the presence different scales for different flavors [ JHEP12(2019)112 as r ψ (2.25) (2.22) (2.23) (2.24) we will 6 3 r ) 0 f ψ , in the following we will † p 6 U ( . 0 r ff  v f Π r )  2 U f ¯ . sin ψ ( ) generates a mass term for the elec- ˆ a 2 . µ f r d X 2 ˆ a µ = 2.22 ], one can build SO(12)-invariants by dress- 2 0 SM is proportional to d f v 2 2 0 4 f,f X f 69 , r ff ≡ – 12 – + ξ 68 , we embed the elementary fermions in the repre- f , eq. ( v L ⊃ . is the usual covariant derivative containing the SM pψ 6 1 2.2 µ f ¯ a representation of SO(11). To quadratic order in the ψ D f = (246GeV) r ] Z , these invariants are actually invariants of SO(12). 67 are the form factors that codify the information arising from 2 U SM f v 0 X q, u, d, `, e ff has a vev: , where ⊃ = , with a r 0 , and then forming with them SO(11)-invariants. Thanks to the T eff H † a µ to L , that transforms with a representation of SO(12), we define e f, f U , factorizing that power of momentum from the form factors. stands for the elementary kinetic term, it will be taken to unity in 0 ψ ψ + f r ff ˆ a Z have the same chirality, Π Π 0 T p ˆ a µ f d → 6 0 = and r ff f U µ We also embed the elementary gauge fields in the adjoint representation of SO(12), by If Given a field As usual in the CCWZ formalism [ Let us focus now on the fermions. Although the elementary fields do not fill complete The kinetic term of the NGBs can be written in terms of the Maurer-Cartan form: D As is well known, in the case of quadratic invariants depending on two fields that are in the same 6 † trade Π adding non-dynamical degrees of freedom. The effective Lagrangian atrepresentation quadratic of G, order there in is a linear combination of invariants that is independent of the NGBs. microscopic parameters of theform SCFT, requires but a they modeldiscrete not version for depend of the on an SCFT, theshow extra an NGBs. as dimension explicit for with realization Their a in example precise terms finite an of number extra-dimensional a of three-site theory, sites. model. or In a section The coefficient numerical calculations. Π the integration of the resonances of the SCFT, they depend on momentum and on the the projection of elementary fermions, the effective Lagrangian can be written as: Following the discussions ofsentations section of SO(12) shown in table ing the fieldstransformation with properties of representations of SO(12), we findby it adding useful non-dynamical to fields, embed that them must in be full put SO(12) to representations zero in the end of the calculations. coset SO(5)/SO(4), namely: As usual in CHM we define: [ iU gauge fields: Assuming that only troweak (EW) gauge bosons, that is identical to the case of the MCHM based on the JHEP12(2019)112 ): (2.28) (2.29) (2.30) (2.26) (2.27) ) in the 2.27 ) for each 2.27   , respectively, X 2.30 f u, d , e, ν . Y ) X = = . We only present f ) and ( 2 , + Π # 2.25 r and U(1) X ) , f , f f ν L . r q r ` a Z † ( Π Π p 6 U L L ( = 0 r f r f X i i r g ¯ | f 2 f Π , SU(2) r r r c X X ) M L,R U = X = = µ − | X a L L ) ( , f f being the form factors that codify the . R j µ f Π Π r g r r g a X Π ) u,d,e,ν X r a ), we show them in table a i = + + Π f , ν , R = v/f f a – 13 – r f r e + + Π and Π 2 a . Z 2 Π Π p c 2 0 Π p . R )( µ R cos( a r f r e a L /g i i f g Z ≡ + h Z − r r v ( R X X , and keeping just the dynamical fields corresponding to = 1 − c can be expressed in terms of trigonometric functions of j µ + Π f v can be computed by matching eqs. ( " a is: f with the NGBs evaluated in their vevs. Assuming that only g = = r L f f has not been written, it could be present or not, depending j X Z µν µ R R M j f Z eff e f a P X R L , ( ) and ¯ L 2 f ν 2 Π 1 2 Π p and   /p ⊃ g,w,b v/f ν r a and Π X p = , i , eff a µ f , = 0), can be obtained by computing the poles of eq. ( r f p sin( L u,d,e,ν 7 i 2 1 X M r qf r `e f = − ≡ f Π Π Z + the elementary gauge couplings of SU(3) r r f f v . The field j j ⊃ y s µν 2 0 a η − g 0 r r eff g X X = L = = = and e a µν f w P Z 0 M M The spectrum of the resonances that are not excited by the SM fermions, as the exotic The spectrum of fermions of a given species, including the resonances that can be ex- The functions For the bosonic sector: The functions It is useful to consider g . Defining In some case there are also non-exotic resonances that are decoupled from the elementary SM fermions, , 7 s 0 ones (that have species of fermions. thus their spectrum is given by the poles of the correlators. compute once the corresponding representations are built. cited by the corresponding elementary fermion, canof be motion obtained of by computing the the dynamical equations elementary fermions present in the Lagrangian of eq. ( v/f the invariants involving fields withrole large in degree the of potential compositeness, that that determines play the an vev, important the other invariants are straightforward to g and: on the realization of neutrino masses. background of the Higgs vev, they are given by: where the sum of the second line is over the dynamical gauge fields of the SM. We will call the NGBs. the Higgs has athe non-trivial SM, vev we obtain: with SCFT dynamics after the integration of the spin one resonances, that are independent of the elementary gauge fields JHEP12(2019)112 1 ¯ S 2 / (2.31) (2.32) 2 2 v , with and / s k w i 2 v g s − H : 1 , c ] ϕ 70 g . 1 0 i is a complicated ) i V 2 , w 2 , k √ g U √ / , / v u v s j = ( s v ! in powers of v t 0 a a ic ic V K K and the scalar leptoquark are − , a . We only show the invariants of ) ϕ R L 1 0 e v/f 3. We have neither shown spinor, i , u , can be calculated by making use of L 2 , + log det 2 , e K / = cos L 0 f f 2 2 v L v / = 1 s K K , ν ν c 2 v i i L s − , the effect of the other fields is sub-leading , d 1 and V L u log det – 14 – R 2 v 2 v on , with v/f u − i c s i i = ( ): w

t w s only to the potential of the Higgs. 4 : of the third generation, we will also consider the effect w ) L p L = sin 2.25 1 0 d 4 L π i ` v is a color singlet, at one-loop level the contribute d s (2 H 2 Z a , f / and of the kinetic and mass terms, in the background of the Higgs vev, a 2 2 v 1 2 L and the / r s f R K u v 2 j and i 1 u a s = − ¯ , with an infinite series of terms. In order to analyse the stability S , 1 2 L 1 1 L we have included the elementary gauge fields of SU(3) ¯ V S and q + ). The specific form of the fermionic contribution depends on the em- a r f f i are the fermionic and bosonic matrices in the quadratic effective La- f and g . We have used and the weak fields K 2.25 1 ¯ 55 11 K ¯ k f S , for H g SO(11) , V ⊃ ϕ and eff 8, and the ones of SU(2) ) and ( f L . Invariants K we have included only the fermions of the third generation giving the largest con- 66 ,... 2.21 Since the dependence on the pNGBs is contained in the matrix At one-loop level, the Coleman-Weinberg potential can be written as: [ f is a matrix of dimension eleven. The matrices SO(12) is a singlet of SU(2) = 1 f 1 ¯ S only to the potential of function of of the potential, we find it useful to perform an expansion of nor vector, indices. The superindexdenominators 0 in means the that argument the of NGBsdent the are of determinants evaluated the just to subtract NGBs. zero, aK thus Notice divergent the that, term since indepen- the quarkseqs. have ( color indices, forbedding our of approximation the elementary fermions into SO(12), leaving freedom for model building. Since For tribution to k of the gluons and it will not be taken into account. where grangian of elementary fields, eq. ( true NGBs. Since thesymmetry interactions of with the thethus SCFT elementary these sector down scalars explicitly to become break pNGBs.elementary the the The fields SM global potential that group, is have dominatedthat at the by role loop the largest is contributions played couplings level by of with a the the potential SCFT. is In generated, the present model, with no vev for the fields that have a non-negligible degree of compositeness. 2.8 Potential If the SCFT is considered in isolation, the Higgs, the singlet Table 2 JHEP12(2019)112 . ϕ (2.33) (2.34) (2.35) , ), as well , , , )] ..., L  ] ν ! ! 2.29 + 2 | 2 | ). The absence 45 g 55 g u 55 u 1 4 45 g 11 u 55 g + Π ¯ S f Π M | ` Π Π 2 2 + Π + Π Z π − ) and ( − ( + Π ϕ − | 2 2 − 1 2 p p ) u ¯ S 11 u p g (16 10 g w R 11 g is minimized by a non- Z ϕ 2.28 / Π u Z Z Π λ Π 4 ∗ c − V − + -symmetry. Notice that in m N 2 log[ 2 2 f + Π P 4 9 3  ϕ 16 − R 2 + 4 | we will show numerical results u − ∼ )] − Z ) H L 11 q Φ | e 55 q )( 4.1 λ 55 u Π 11 u 11 u L Hϕ Π u 2 − | λ + Π + Π − + Π ` q + Π + 11 u 55 q + Π u can be present, triggering a vev for 2 ) and Z Z u | 11 u ( Π 2 M Z 1 , as usual in composite Higgs models. L 2 Z c ϕ f Π ¯ u ξ S p )( 2 | − N Z 2 π ( | 55 q 2 55 u , – 15 – p + 10 + 2 H (16 | 2 log[ M 1 / + Π | 55 u ¯ 4 ∗ S 11 q 11 ` 11 u − log[ q 55 q 55 ` . As is well known, EWPT demands a separation Π H ) Π Π m c Z f λ 2 f w ( − N  − − + Π c + Π + Π 2 . Generically, the quadratic and quartic coefficients can ∼ ]+ and zero leptoquark and singlet vev. Using the estimates , the quadratic coefficients are: u ` N q 11 u r g ∼ 2 55 ` 55 q + Π − ) Z 1 v Z Z H Π 2 2 Φ Π Π 2 | 0 and suitable quartic couplings, c +2 )] p 2 4 m /λ L Φ N | w d >

4 ( 2 H and Π . We will also show the predictions for the masses of the pNGBs. Z 4 4 4 Φ 2 ϕ 0 m ) ) ) p p p − λ em 4 4 4 + Π π π π − r ff d d d + , m q (2 (2 (2 1 2 = log( | Z ¯ 2 S ( U(1) 2 Φ 9 2 . The dominant contributions in our model are given by: 2 Z Z Z | m v × p ϕ 2 Φ  c , leading to a tuning of order − − − 0, 4 f m ) [ p = = = log[ 4 < π c ,ϕ 1 2 ϕ d 1 2 H ¯ 2 S ¯ (2 N and S 2 H m . 2 m m , nor of v H, ) can be expressed as momentum integrals of combinations of the fermionic and X m 1 Z − 2 ¯ S Φ= ' 2.33 In the next section we will show an effective description of the resonances of the We have also computed the quartic couplings, but weWe do consider not show now them the because potential they that determines the Higgs vev, in the case of neither For the embedding of table For ' V V as table SCFT that allows to modelfor the regions form of factors. the parameter Inpreserving space section SU(3) where the one-loop potential breaks the EW symmetry, where the form factors can be obtained by making use of the eqs. ( coefficients. lead to too long expressions. vev of the last termsbedding. show the Having gauge explicit contributions, expressions of that the are form independent factors of it the is fermion possible em- to compute these of the previousparameter paragraph space in one the obtains: between solution for the Higgs vev, for generic regions of the be estimated to be ofof order: terms with anthe odd absence number of of this symmetry, fields a is term guaranteed linear by in trivial the Higgs vev: where the dots standeq. ( for higher orderbosonic terms. form factors Π The quadratic and quartic coefficients of to fourth order, obtaining: JHEP12(2019)112 , jµ +1 H j A j (3.1) , the G Ω × i , whose j 0 − g j =SO(11) is Ω 2 µ 1 , q, u, d, `, e − . Matching the j # π , fixing = ) 4 2 2 iA | f − 2 j + g  -models based on the Ω j σ 2 µ + Ω 2 µ D , g | − 1 1 ∂ g g tr( = + 2 j j  4 2 f Ω − 0 µ g + SM g D = aµν j 2 transforming linearly under G F − SM . In fact a subindex in the coupling at g j , and ]. We show a moose diagram of such a a j , with jµν SM /f j j to the embedding of the gauge fields into g F 71 Π + 1 are connected by 2 j i µ g 1 e j  a 4 2 = – 16 – − , g j =SO(12), however only a subgroup H " and 1 2 2 g , j =1 X j for a model in 5-dimensions). Site-0 contains the el- for + D . SM , we find it useful to add non-dynamical elementary degrees . Moose diagram of the three-site theory. g 1 aµν f ∼ 2.7 a 0 µν g , with a field Ω f , must be understood, distinguishing the different factors of the 2 0 g 1 +1) . Site-2 has massive Dirac fermions transforming with irreducible Figure 1 SM 4 j 1 g − +( the field strength on site-0 and j (see appendix = to the embeddings of the elementary fermions, with G are the gauge couplings on site- 1 / b f jµν j =SO(12), as well as massive Dirac fermions transforming with irreducible L ψ F g +1 1 j -model fields provide 132 NGBs, in the unitary gauge 121 NGBs become the G ). σ µ and × 0 j A µν f The The Lagrangian of the bosonic sector is: As described in section ≡ µ a site-0, as well asSM in gauge symmetry. longitudinal degrees of freedom of theat 66 spin site-2, one that resonances at become site-1 massive. and the 11 55 resonances NGBs remain in the spectrum, they correspond to with ( coupling of the unbroken diagonalsize group can leads be to: estimated as SO(12), and later ones are defined in table and parametrizing the coset. of freedom that allowthese to embeddings embed it the is straightforward elementaryin to fields write the in invariants, end the full spurion of multiplets fields the of are set calculation. SO(12). to zero Using We will call will be denoted withsymmetry capital G letters and subindicesrepresentations indicating of the G site.representations of Site-1 a has global a symmetrygauged local G on this site. Twocoset nearest G sites minimal set-up that allowsnite to one-loop obtain potential the istheory proper a in leptoquark three-site figure interactions, theory asementary [ well fermions as and a gaugewhereas fi- the fields other of two sites the contain the SM, lightest that set of will resonances be of the denoted SCFT, with whose fields small letters, 3 A three-site model We consider an effectiveelementary description SM of fermions the and SCFT gauge dynamics and bosons its in interactions terms with of the a discrete composite model. The JHEP12(2019)112 . ) ) e ` λ 220 1 2.7 3.3 (3.4) (3.2) (3.3) being and . and r 66 2 u d B , Ψ λ q 2 and one in r Ω , R 2 66 1 Thus eq. ( 66 Ψ mix with Ψ ¯ r Ψ , 8 R 2 e . 2 ¯ f Ψ 2 r , , 220 j 1 R 2 and λ m d + r r , X 66 2 under SO(11), see eq. ( ., + , we obtained that Ψ c 2 r . , R 2 66 2 labels the SO(12) representation Ψ 2.5 ) are shown in appendix ¯ Ψ , whereas + h R r ) contains the kinetic terms of the 6 , D 2 3.4 66 1 R 2 R 2 3.3 .. m Ψ . ¯ Ψ c 2 i . 2 2 11 1 Ω , f + through a mass term that is SO(11)-invariant: R 1 55 + h ) and ( X + R 1 ¯ Ψ =  66 2 r 2 1 f 2 1 3.1 )Ψ f f ) for the other fermions of the third generation. , one would have to include two massive Dirac + mix with Ψ 2 R 1 1 – 17 – , 6 g Dψ R 1 = u m ( 66 2 f λ 2.2 ¯ 2 are dimensionless numbers that, as discussed below ψ O Ψ 1 − i f + f 6 D and 6 D + λ R 1 i 2 66 ` ( Ψ ¯ 66 1 , , and Ψ 1 R 1 , and we will neglect the effects of Ψ i q R can mix with Ψ Ψ ¯ Ω , distorting the spectrum of fermions. We are neglecting these effects Ψ τ 66 j 1 f + 55 ¯ Ω + ψ 220 2 f L, , such that fermionic and bosonic resonances have masses of the 1 66 f 220 2 ¯ 2 and ψ R f ) and the discussions of section Ψ f λ )Ψ 2 R 6 f 1 Dψ 55 g b , as well as the kinetic and mass terms of the fermions on sites 1 and f λ f 2.10 R, m ¯ f 66 2 X ψ ∼ = 0, Ψ i . These terms, that give different masses to each SO(11) multiplet of ¯ 1 Ψ − r 0 2 , e f q,`,u 2 f λ X m 6 = D X q, u, d, `, e , + i 220 f m ( + ˆ ) and ( = 1 = = f 1 66 or f 55 d couplings with leptoquarks, from now on we will consider that there is just f 2.9 ¯ λ and Ψ L + R, 220 2 ` 1 , as: 66 . By going to the unitary gauge, one can obtain the decay constant of the physical = Ψ f f ϕ ), can span a hierarchy of values, being very small for the fermions of first and second ∼ 1 f 55 g are very suppressed, to simplify our analysis of the potential, as well as the analysis 2 L and L, 2.8 The mass matrices obtained from eqs. ( Following the discussion of section The Lagrangian of the fermionic sector is: 66 2 . The elementary fermions ∼ Even if and q ¯ 8 Ψ 1 220 1 2 1 ¯ ˆ m also. one resonance in eachsimplifies site: to: Ψ From eqs. ( of the third generation havemust a be large suppressed degree of to compositeness, reproduceΨ whereas the the bottom mixing and masses.of Given that the mixings with eq. ( generations, as well as fermions per generation in220 each composite site, one in the representation fermions, are allowed becausethe on mixing site-2 between only fermions SO(11)thanks to is located the gauged. on transformation properties different Them of sites, second the line matrices these of contains terms NGBs.same are size. The masses gauge The are mixing invariant taken parameters as of the corresponding fermion.elementary fermions The first line2. of eq. The ( last termthe of representations the obtained first afterfor line contains the Ψ masses decomposition for of the Ψ fermions on site-2, with where a generation index is understood. The superindex NGBs: S JHEP12(2019)112 . 1 f 1 , the (3.6) (3.7) (3.5) g j f ' j = 0, the g f λ . 2 and / 11 1 2, that does not , , f / 4 1 ) 55 g 2 2  , f i 0 = 2 ' . For the fermions we g g ) + r 2 2  2 1 f 2.7 2 f that split in two levels, as , O ( 2 1 λ + , p 55 1 } +  ), after diagonalization of this ] g 2 2 2 2 is the projection of the massless 1 g f 55 3.1 , 2 el 2 2 , 2 1 + x 55 m λ , 2 , 1 ( 2 2 1 g 2 q ) λ m + 4  11 , 2 2 0 + 2 + ) g r 2 2 2 , , where m ) 2 f ' el 2 2 2 , f 2 g, m → 2 2 2 1 – 18 – 2 x , g λ 2 1 − 2 1 55 , − λ g 1 2 as shown in eq. ( 1 m − + . 2 , m [( 2 2 q 2 2 ) f u ` 55 g g , 2 2  0 1 λ λ the mass eigenstates have masses: 2 = that split in two levels, with masses: , and all the fermionic mass parameters of order g g r 2 , 2 g m g 2 → → g  1 + 11 q q m 2 1 ∼ m = λ λ g . There are two multiplets in the ( 1 ( ( 2 2 + 0 1 2 q , g q q g g 1 1 λ   λ ∼ m , is defined as − = = = g { 2 , `  q and u 1 2 1    λ of SO(11), located on site-1, with mass 2 = 1 = f 2 g )  the elementary and composite states are mixed. After this mixing, and 11 =  ( r 0 1 g f m can be defined as g  In the next sections we will show numerical predictions for the spectrum and couplings. Integrating the resonances at sites 2 and 3, it is possible to obtain explicit expressions One can also perform a biunitary diagonalization of the fermionic mass matrices before For the fermions one can proceed in an analogous way. Taking the limit of The states at sites 2 and 3 give an effective description of the lightest level of resonances of SO(11), that are mixed by We will take into accountthe all Higgs the vev. mixings for these predictions, including those inducedfor by the form factors of the low energy effective theory of section The mixing with the elementarywhat states is also usually shifts known the as masses light of custodians the when resonances, the leading mixing to is large. same size as the masses of the spin oneEWSB. resonances. Considering just one generation, we define the degree of compositeness as: Since we will take masses of the fermionic resonances, before mixing with the elementary states, are of the eigenstate onto the elementary one. masses of the fermionicmixings resonances between sites: depend onwell the as masses two at multiplets in each the site, as well as on the it can beEWSB. obtained by calculation of the massless eigenstate of the mass matrix before 55 mixing, for Turning-on before EWSB, one obtainsto a one set of correspondence massless withthe and the massless partially gauge states: composite bosons fields of that the are SM. in one The degree of compositeness of of the SCFT. To gainthe some insight spectrum into neglecting the the dynamics of mixingfrom the with the SCFT, the it Higgs elementary is fields useful vev.representation to as study well Let first as us the startmix contributions with with the any spin other one state states, in there this is limit. one multiplet There in are the two multiplets in the representation JHEP12(2019)112 1 ¯ S , (3.8) (3.9) ϕ (3.10) . )] -symmetry, r , 2 LR P m + ). The scan was coupling saturates p , 3.1 )( ) p , W τ 2 1 ) f + r , 2 1 2 1 g , , it is possible to be within ) m L + r , m τ , 2 u 2 p + ( )] 2 , λ 2 2 2 2 , m /d , q ] f − g u λ r ( 2 , , ( λ 2 2 2 + 2 1 ( . After that we study the corrections f 0 λ 2 1 S M m 2 2 1. After that, we study the corrections g 2 . . At sites 1 and 2, gauge couplings were 2 Z g − ( 0 n = = f 2 2 2 )][ , f r ∼ 2 1 πf r qu , ) + r uu . 2 2 λ 2 /d , ξ − Π Π ) ) f m 2 2 r − 2 , in the cases with EWSB. We discarded all the p ) f 2 2 + – 19 – 1 2 − 1 [2 v 2 g 1 m + p 2 m , 1 , 2 f 2 ) f 2 ) ( ) 1 )( 1% for f r − 2 , . r r 2 1 f , p , 2 2 p 2 0 ( g 2 2 2 − 2 2 1 f − m 2 × g p 2 , 1 p , m , m 2 2 at tree level, showing that, thanks to the 2 1 + q ` λ m (2 2 λ λ L − − few 2 1 ( ( p b 2 4 − f S S + ( p p − ∼ 2 4 4 , as well as the lightest 2 [( p 2 = = 1 ( 2 and 1 f 2 1 2 = = U f r `` , r qq 0 f L 2 1 2 τ Π Π λ 11 g 55 g λ λλ Π Π = [ ) = ) = d r r , , defined as: 2 2 M , m 0 λ, m couplings of ( and show that, for large degree of compositeness of S and ) λ, λ Z ∗ ( ( S D M of the experimental average value. In this case the correction to R σ degrees of compositenesswould larger be at than larger 0.5, mixings.in For as each the point the we Higgs calculated points vev, thepoints one-loop obtaining consistent potential with the at with maximal value all breaking, orders phenomenology of asobtained well the as Higgs mass those by without calculating breaking. the For curvature each around point the kept, minimum. we The top mass the pNGB potential andrandomly, the with Higgs a vev. numbersite of For dimensionful this, constrains. couplings should we not As letchosen exceed we most 4 in wish of the to the intervalto maintain parameters [2.1,3.6], perturbativity, match to and at vary the the each also couplings value optimized at of to the select the elementary the SM site elementary-composite couplings, couplings where such adjusted as as to discussed have below the fermionic eq. ( 1 the bounds. Finally, we discuss the phenomenology of4.1 the pNGBs. EW symmetryWe breaking have and spectrum performed of a resonances scan of the parameter space of the three-site model, calculating the SM states have theand proper the masses, spin and one showing theto states predictions the for the massesthe of corrections are of order to 4 Phenomenology In this section weof study the resonances phenomenology effectively of describedcompute the first by composite the GUT, the spectrum with three-site the of model lowest pNGBs, level of looking the for previous regions section. of the We parameter space where For the gauge fields we get: with obtain: JHEP12(2019)112 ). 2 2.34 7 TeV were . 0 ≥ = 246 GeV, as f ξf √ one by custodial symmetry. In figure W 150] GeV, a smaller window requires more , slightly larger than the experimental value. t , however, since the blue circles do not show ϕ [100 /m – 20 – ∈ h h m > m obtained with the random scan, selecting points with ). m 1 ¯ h S 15] TeV. m m . 2.23 0 , 1 . [0 versus ∈ t mass is related with the h m we show both masses plotted one against the other, along with 180] GeV and Z m , 3 , we only show the results with both masses positive. Red triangles 1 [120 ¯ S ∈ , than the other way around. For masses larger than 1 TeV it seems that m t ϕ m = 18]TeV and . ϕ 0 > m , . Scatter plot for top mass versus Higgs mass. Only points with m 1 mass was fixed to its experimental value by the rescaling described in the previous ¯ 12 S . 7 TeV, . m [0 W 700 GeV. The random points contain the phenomenologically interesting region of the 0 We have also calculated the masses of the other scalar states by taking into account We defined a benchmark window, selecting phenomenologically viable points, as ∈ ≥ t there are more redthat triangles pattern, with we canof not be points sure lying if inthese this states the a are benchmark fluctuation in due region,100–700 GeV. the to or the range if somewhat 0.1–2.5 TeV, it small with number is a a larger reliable density tendency. of points The in masses the of interval The gauge contribution is always positive,or whereas negative. the fermionic In onethe can figure be line either positive correspond to points within thelie benchmark outside region, that whereas region. blue Justwith circles by are counting for red triangles points we that obtain that there are more points f > SM, although the model prefers a ratio both: the gauge and the fermion contribution to the potential, as shown in eq. ( f time of CPU running, withThe almost no impact in theparagraph, phenomenology whereas that the we want towe study. show a scatter plot of is calculated as well, byof biunitary vev. diagonalization of Last, the by upindicated rescaling quark in all mass the matrix the matching in of dimensionful presence eq. parameters, ( we fixed: Figure 2 selected for thism plot. In grey we highlight the limits for the benchmark window, defined by JHEP12(2019)112 0 . L Z τ and 1–4 TeV, states, we L b ∼ we present 1 1 4 U U m . In figure boson, the next 5 states ) ∗ ( Z D R collisions is highly suppressed, as the pp can be understood from the discussion of f with couplings mostly to the third generation 0 – 21 – Z we calculated its mass and its coupling to 1 U particle mass as a function of the mass of the scalar LQ. Plotted in state, as a function of the decay constant of the pNGBs, for ). We obtain that, for the benchmark region, ϕ implies that one of these states does not couple to the physical 1 3.6 U P 700 GeV and in red the ones inside the benchmark window. Also included > . 1 ¯ S m is zero. As such, their generation in , while the 5th one is around 5% heavier, and does mix with the elementary = 0 . Z ϕ ) Z n m ( 1 . Scatter plot of the U /m By calculating the eigenvalues of the mass matrix of the spin one neutral states, we For the vector leptoquark ) n ( 1 . As the first generation of quarks and leptons have a very low degree of compositeness, U in the approximation incomposite which they areparton distribution fully functions elementary, of the theirA 2nd process couplings and that 3rd to can generations theseis put in four fully bounds the tops, proton on are which these suppressed. constrains the size of 4 quark operators that can be generated as a find the spectrum of neutralare vector of resonances. two Besides kinds. theelementary The SM lightest 4 are actuallyZ degenerate in mass, and do not mix with the fermions, and as such itthe does mass not of contribute to thethe the lightest benchmark couplings region. Thethe dependence spectrum with above eq. ( with a larger densityg of points near 1–2 TeV. In the next section we will show the ratio This was also done bythe diagonalizing Lagrangian the in fermion terms andobtained of boson three physical mass masses degrees matrices, and and ofhave three by freedom. chosen, different writing the couplings. As parity there For the are fermion three embeddings that we Figure 3 blue are points with f is the line JHEP12(2019)112 .                  ▲  ξ     ▲    ▲                                  ▲          ▲   ▲     ▲  ▲     ▲    ]. In       ▲   ▲   ▲ ▲ 0.12              ▲  , as a          ▲   ▲                ▲ ▲ 1    ▲    78 ▲     ▲          ▲    ▲  ▲ ▲    ▲   ▲  ▲     ▲  U –   ▲  ▲      ▲    ]. In the    ▲  ▲   ▲  ▲   ▲   ▲  ▲         ▲ ▲    ▲ ▲                   73   ▲ ▲      ▲  ▲    ▲   ▲   57        ▲ 0.10         ▲ ▲ ▲  ▲ ▲      ▲      ▲    ▲         ▲            ▲  ▲   ▲   ▲      ▲  ▲   ▲   ▲ ▲  ▲ ▲             ▲  ▲     ▲   ▲    resonance, as a             ▲               ▲     ▲      ▲ ▲    0 ▲  ▲  ▲ ▲ ▲  ▲  ▲ ▲  ▲  ▲     ▲  ▲ ▲  ▲ ▲ ▲    ▲ ▲  ▲     ▲ ▲  Z 0.08 ▲   ▲     ▲    ▲    ▲   ▲     ▲  ▲  02, increasing the ▲ ▲   ▲  ▲   ▲  ▲ ▲ ▲  ▲  ▲ ▲    .  ▲ ▲  ▲ ▲    ▲      ▲ ▲ ▲ ▲ ▲  ▲ ▲ ▲   ▲   ▲ ▲ 0  ▲   ▲ ▲     ▲ ▲ ▲ ▲              ▲       ▲  ▲   ▲ ▲   ▲ ▲ ▲    ▲ ▲   ▲  ▲ ▲   ▲ ▲ ▲ ▲ ▲ ▲ ▲    ▲  ▲ ▲ ▲ , the dispersion arising ▲     ▲  .       ▲  ▲    ▲ ▲ ▲ ▲ ▲ ▲  ▲ ▲ ▲ ▲ 0.06  ξ   ▲ ▲ ▲  ▲ ▲ ▲   ▲   ▲     ▲   ▲  ▲  ▲ ▲ ξ ▲     ▲ ▲  ▲ ▲  ▲ ▲ ▲  ▲     ▲ ▲ ▲ ▲    ▲ 1 onwards. The coupling ▲ ▲ ▲  ▲ ▲  ▲ . ▲ ▲     ▲ ▲  ▲ ▲ ▲ ▲ ▲   ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲  ▲ 0 1. The distinction between ▲ ▲  ▲ ▲  coupling, as a function of ▲ ▲ ▲  ▲  ▲ ▲ ▲  ▲  ▲ . ▲ ▲ ▲  ▲ ▲   ▲ ▲ ▲  ▲ ▲  ▲ ▲  ▲ ▲ ▲  ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0  ▲ 0.04 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲  L ▲ ▲ . ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲  ▲ ▲ ▲  ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ Z ▲ ▲ ▲ ▲ τ ▲   ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲    ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ g ▲  ▲ ▲ ▲ ξ  . ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲  ▲ ▲  ▲ ▲ ▲ ▲  ▲ ▲  ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ξ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ larger than for the charged lepton. ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0.02 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 2 g ▲ ▲ ▲ in our three-site model. By rewriting ▲ ▲ ▲ ▲ ], but in this case, since there is no ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ to , since the values and ▲ / L ), one can understand the dependence . We find that the relative corrections 57 b g Z  0.00 2 1 0 6 5 4 3 2.23 and and , it scales linearly with , the bounds require ν 2.2 , ` 5 – 22 –  L , for points inside the phenomenological window. 2.6 τ ξ 2.0 to 2 TeV for values of , for the benchmark region. We observe the predicted linear Z f & 0 1.8 detection and those that pass the experimental constraints Z 0 , as well as eq. ( m f Z . Z 1.6 b g as a function of 1.4 we present the relative difference of 0 Z and has been measured with an accuracy of order few per mil [ , for points on the benchmark window. Red triangles pass the experimental 2 5 m Z Z ν τ that does couple to first generation, following experimental bounds on g g /f . Also pictured is the line at 1.5 TeV, often cited as a bound for its mass in direct 0 scales with , 1.2 f 2 SM Z Z τ 0 v . We see that g Z ]. However, all points in the benchmark window pass these constraints, as ξ = m 1.0 ]. On the right panel we show a scatter plot of mass for the first 72 ξ . On the left panel we show a scatter plot of mass of the vector leptoquark 79 with we present 0.8 0 4 Z We omit the same graph for the coupling of The coupling has also been measured at the per mil level [ m 0 2 1 4 3 Z ν symmetry protection, the corrections areAs a can factor be 1 seenamount in of tuning. the left panel of figure distribution of this coupling are similar. right panel of figure As expected from the estimates offrom section the dependence ofto the the coupling coupling on cang reach values below 2–5 per mil from 4.2 Bounds We have computed the coupling of the three-site interaction Lagrangian inthe terms couplings of the physical states, one finds the value of Using that of points that are ruled outcan by be seen bythose the that marker, do where not. red triangles are points that pass, and blue circles are exchange [ they are not tooto restrictive. the In lightest that case,these points heavy were vector bosons only into acceptedfigure either or leptons, rejected light jets, according bottom and top quarks [ function of the Higgs decaydependence constant with searches [ function of constraints, whereas blue circles do not. Figure 4 JHEP12(2019)112 . ) ), n ( U 02, 06. . . 3333 C 4.1 0 (4.1) (4.2) 0.12 C ≤ = 0 ) ) n ), that is: n ( 0.10 U ( U C C 4.1 n n ) is dominated 0.08 P P on the right panel, 4.2 L ¯ . In it, we show the τ L ). Making use of the D 0.06 , ) on eq. ( R Zτ .  33 , whereas the dispersion 2.16 ) L 2 vs 23 33 n g 0.04 c c ( U ∗ ! ∗ ts D ) ) C 1 33 n n R ( U ( /V L 1 + can be estimated as: 0.02 23 m we present the coefficients  vg . L ) 6 g

∗ 23 33 ( , as a function of the bottom quark ∗ couplings give constraints on tb 3233 c c 1 2 τ D 0.00 V 2 C C R W ≡ λ 0.010 0.008 0.006 0.004 0.002 0.000 − is given in eq. ( ) and n ) (1 ( U on the left panel, and ∗ 3333 b ( leads to larger 33 ¯ ν C D . In figure ) q /c R n  ∼ Zν ( 1 – 23 – 32 U c  ,C 0.12 ) we present a plot of we sum both contributions, and multiply by a ran- 23 33 /m n ( L L ) U 7 ) ∗ 33 g g n ( C ( L ∗ ∗ 0.10 D tb ts . g ` n V V decays, and the rest of the points up to R  X , LFU violation in τ − ) = 0.08 ∗ 1 leptoquarks to ( 2.6 (1), arising from the last two factors of the r.h.s. of eq. (  1 D 3. In figure , this coefficient can actually reach values of order 0.1, we do not 3333 O U R 6 < 0.06 C 3233 | by knowing c C | 0.04 3233 , for points in the benchmark region. . We present our prediction in two groups, one such that C ξ σ to account for the factors ), with 0.02 c 1 states having nonzero couplings to (2) U 1 . We find that the coefficient of the lightest state is approximately an order of C . Relative correction of the couplings q U  0.00 + 0.06 0.02 0.03 0.04 0.05 0.01 0.00 1 ------To obtain the coefficient of As described in section (1) U C ( c SM prediction, the worldfor average 1, for 2 experimental and 3 values,to along be with consistent confidence withAs LFU ellipses can in be seen in figure by the lightest state.of As points expected, is generated larger similar by dependence the is random found variation for of the otherdom parameters factor of the model. A Thus, up to awe factor can of obtain of both mixing magnitude higher than theto coefficient the difference of in the masses heaviest between state, both the states. suppression Thus mainly the due sum of eq. ( with 4.3 Predictions for The contribution of the flavor structure arising from partial compositeness, We have computed thiswe Wilson will coefficient distinguish for the the pointsthat benchmark that a points. pass large these portion In constraints of the form the next those former section that can do also not, explain showing Figure 5 as a function of JHEP12(2019)112 ). σ region is very σ Z ν g . q  from ξ are the same as for the ], and references therein. 2 81 , 80 , 38 for points inside the benchmark window, as a for our model. In red are plotted those points 2 06. Also plotted is the experimental point along ∗ . ) ) D 1 n ( U R = 0 – 24 – ) /m n ( U ) and 33 n . On the other hand, the bound on C ( L D 1]. One can see that many points lie in the 1 n W τ . vg R g 0 P , 2( / 06 . = 1 [0 ) ∈ n 02, consistent with current bounds on LFU violation in tau decays. In ( . U ) 0 C n ( U ≤ ellipse from below. C ) n n ( σ U C P n P . Scatter plot of . Theoretical predictions for Let us start with theMCHM pNGB with Higgs. fermions Since embeddedphenomenology the in is invariants of similar the table to fundamentalsively that representation discussed case. of in We SO(5), the will literature, the not see Higgs describe for it instance here, refs. as [ it has been exten- without violating the bounds from stringent, we have checked thatthe only border 6% of of the the 1 points satisfy this bound, almost reaching 4.4 Phenomenology of the pNGB scalars blue, we plot the restwith of the points, confidence up ellipses to for coverage probabilities of 68.27%, 95.45% and 99.73% (1show to points 3 with Figure 7 that agree with Figure 6 function of the degree of compositeness of the quark doublet of the third generation: JHEP12(2019)112 ’, R is ψ ν and 10 55 L , or in . ` 2 emitted (with P µ and ϕ ∗ γ . This ∗ 1 µν 1 Z pψ ¯ ¯ 6 S 10 S G ¯ ψ L ¯ q µν ϕ is odd and ϕ < m G µ ∂ ϕ . It is also possible 11 , they can mediate 1 m P ¯ = 1 TeV, however these S is odd under decays also, mediated m R ϕ ∼ q` O , being a color triplet, can Hν 1 ). Since and L ¯ S ¯ ` , with the virtual ϕ 2.7 ¯ ` -even, with SM fields and just of the third generation, we have m , with the singlets giving missing off-shell for of order P . The presence of the operators of L 1 1 L ` ¯ ¯ qϕϕ` ` S S ¯ and with a dark matter candidate). `ϕϕ † . This is the main creation channel at . In this case ) m at LHC. ¯ q` ∗ 1 ∗ , they are: H → ( q and ¯ tν . The first case leads to a stable particle 1 ∗ S 1 ϕ with even parity (+1). In the first case, ¯ K 1 ¯ S interact simultaneously with S ¯ R q` ¯ S ϕ` R → , with R R u ϕ ∗ 1 L ν u ∗ allows to include a dimension four operator: 1 ¯ ` ¯ and q` S -symmetry is violated by the interactions with → ¯ S and → 1 P P tν ¯ – 25 – 1 S → gg , as shown in eq. ( and ¯ ϕ ¯ and , since the operator S ∗ ` → L 66 R , and the dimension five operators: is stable and can be a good dark matter candidate 2 p` u ¯ qZ 6 q` decay: | of ∗ ∗ 1 1 1 ϕ 1 ¯ ¯ ¯ S S S ¯ , and is not phenomenologically viable. The second case S | 55 L 1 q ϕ ¯ S → even under . The lowest dimensional operators with these fields have ϕ and R , one could get a final state with a pair of quarks and leptons of ϕ ν and : at LHC give bounds on -symmetry has important consequences for the phenomenology ϕ , or the decay 1 2 11 ¯ P or | 1 S ¯ q` -symmetry ¯ S ∗ 1 1 H P ¯ | S ¯ S ϕ , for example in , → 1 . Depending on the relation between ¯ ϕ S R ϕ ] for another scenario solving that can mediate 82 He , it is not possible to assign a well defined parity to the elementary fermions R ∗ 1 ν . Let us consider two different cases: first the situation with no elementary P ¯ ∗ S 1 ϕ we have shown the spectrum of these states, showing that both situations are could be created in pairs through a dimension six operator as ¯ L S ¯ q 3 R ϕ ϕ u and = = ¯ In this case there are many new operators, as for example dimension one and dimension Direct searches of Let us discuss briefly the creation of Adding an elementary The presence of the 1 ¯ S , and second the case with an elementary uν qe R the SCFT is much smallernot than studied the them. mixing of three operators: being any elementary fermion), ¯ the elementary sector. There arecase several where possibilities for the this elementary violation,respectively fermions we contained have in studied the even under such that the elementary-composite interactionsto are break invariant this under symmetry with the other elementary fermions, but since their mixing with the decay channelsdifferent are cases. model dependent, a4.5 dedicated analysis Case must without beAnother possible done scenario is the for case where the energy. association with from the initial proton. A detailed study of these processes isbounds beyond the depend scope on of this which work. are the dominant decay channels. Since in the present model be created in pairs byLHC. QCD As interactions: discussed ininteresting the case previous of a paragraphs, stable thethe final third state generation is as model well as dependent. two In scalar the singlets: possible in the model. O by its interactions with can not have Yukawa interactions with O either the decay with and color: is much more interesting(see because also ref. [ In figure of ν it is not possibleone power to of writedimension either a gauge six invariant and operator, contain both fields, JHEP12(2019)112 1 . ψ 1 U λ ¯ can S SCFT ψ R can be O u and ), under R e at the UV ϕ are for the leptoquark and 2.8 ’, as well as ` ` 1 is generated, ˆ L λ ¯ λ S ). In the limit ` 10 and ϕ L , = ` a large degree of L R ` q d ) and λ ) is defined ∗ with ( u D ’ for λ R 2.8 R u and 10 u ˆ λ , whereas and = u u 66 λ at the EW scale. Therefore the for ψ respectively ( ˆ λ ` , that are in agreement with the SM 10 ˆ λ L must be small, leading to a suppression . As discussed below eq. ( b f and R τ and ψ and , it opens new decay channels for λ u h ˆ i  λ L and -symmetry, confirming that, in the absence of τ . ϕ 1 – 26 – h R P b U is determined by the anomalous dimension of the and . We have computed the one-loop potential to all f ψ . At the UV scale where eq. ( ϕ . The elementary-composite interactions, dominated λ requires some extra tuning. SCFT ψ 220 P O couplings of -anomalies. We have shown that an anarchic flavor structure of the third generation is required. This configuration could Z B L q , that we will call 10 symmetry, by properly choosing the representations of the fermionic violation can be controlled by the mixing of with the Right handed currents, and realizing the scenario in which and 1 in the cases without P , and it contains massive spin one states that can be identified with L LR U ` ϕ with H L . This vev has a number of new effects, as: it gives large contributions to the ` f and ϕ i ∼ SO(11) has the following properties: it contains the SM gauge symmetry and the = 0 the symmetry is recovered, while taking these mixings small the violation ϕ ` → , however, since in our scenario the EW scale is taken only one order of magnitude h ˆ λ ψ is suppressed, and one can obtain . In particular, we have shown that to reproduce the shift in ˆ λ We will not elaborate more on this interesting case, and leave it for future work. The amount of 1 = P U u ˆ by the third generation, generateger a EWSB potential and for give the masses NGBsthe to at bottom the one-loop quark extra and level, NGBs. tau, thatin the To can obtain mixing the trig- the of coupling suppression of inonly the Left masses handed of currents interact with induce large corrections to at the per milwith level, a however well we known have shownoperators that under the it global is symmetry possiblebe of the to embedded SCFT. protect in We those the haveembedded couplings adjoint shown in that representation the of representation SO(12), the leptoquarks addressing the of the SCFT, togetherSM with spectrum partial and compositeness CKM, fromof simultaneously linear leading mixing, to can acompositeness reproduce suitable of the flavor pattern of couplings We have consideredgroup a SO(12) strongly spontaneouslySO(12) broken coupled to field SO(11) theorycustodial by symmetry, with the it strong a developsand dynamics. a unified a set SM global of The singlet NGBs breaking symmetry that include the Higgs, an scenario with small violation of 5 Conclusions transforms linearly with ascale. representation of At low SO(12), IR thus scalesand SO(12) is spontaneously broken, allowingsmaller a different than evolution that of infrared (IR)natural scale, to the expect window a for running large is hierarchy rather between small and it is not the mixing of mixings used in the previousλ sections of the article: of some suitable conditions thecorresponding size SCFT of operator that can be estimatedorders to in be of order tuning EW scale, it induces mixing between dimension one and three change drastically the potential, since a vev for JHEP12(2019)112 . - 1. . P C 0 and . In , the ) ) L ∗ . ∗ ( q ( ξ D coupling D R R W of the third with mass of 1 R 1% for . u U and at LHC could allow to leptoquark. However, L ` 1 ϕ are given in appendix , U L 1 q U and are of order 0 1 L ¯ ¯ S τ give an extra suppression to the L 1 U , bounds from this coupling require Zτ m Z ν region, such that: an improvement of g σ 2–2 TeV, and a lightest and showed that it is possible to obtain and . 1 L b ¯ U L . EWPT give lower bounds on the later of Zb – 27 – W of order 0 ϕ and as a NGB and generate a Z H and 1 ¯ S , without conflict with other observables as the can be solved by properly choosing a small mixing for ) if measurements of this coupling increase their precision. ) ∗ ( ∗ 4–6 TeV. These values of ( D W τ 02 do not enter into de 1 K . g & R 0 R 1 , with the same central value, could not be explained in the present U . ) is associated to broken generators, its mass results heavier than the , that results approximately a factor 4 smaller than what is needed to ∗ m ξ ) ( 1 ∗ ( D almost saturate the bounds, thus in the present model one can expect -anomalies must be done, particularly at LHC. We find this avenue very U D R B W τ R g 02, increasing the amount of tuning and introducing some tension with . . The anomalies in 0 Finally, we stress that the coset SO(11)/SO(10) is big enough to contain the SM and We have discussed very briefly the phenomenology of the new scalar states at LHC, We have shown an explicit realization of the SCFT dynamics in terms of a weakly cou- L of the second generation, estimates of the couplings with τ . L Acknowledgments We thank Eduardo Andr´esfor helpCONICET with Argentina group for theory, financial Bartosz support Fornal for with discussions PIP-0299. and lightest resonances associatedorder to 2–3 TeV, thus contribution to fit the anomalies. It couldthan be SO(12) interesting that to could study do the the possibility job. of finding a group smaller related with the interesting and leave its study for future work. the custodial symmetry, develop since in this case symmetry, as well as a lightor elementary not, Right-handed a neutrino detailed with studydistinguish parity of +1, the the are different production present realizations and of detectionthe of the couplings model. are fixed, Besides, giving thethe a size scope rather and predictive of flavor scenario. structure this Although of work, such study a is careful beyond analysis of direct signals of new physics that could be model. For the pointscorrections of to the parameterto space measure that deviations induce in the proper shift on finding a very rich set of signals. Since the phenomenology depends on whether the We have also shown that theOn corrections the to other hand,ξ since there is nofact protection the for points with the precission of region, obtaining a largestates. set We obtained of masses points of order with 1–3 EWSB, TeV. We as computed wellthe the as proper couplings correction the of to rightto spectrum of SM ` pled theory of resonances. Forthe that lowest we level have of built resonances anto by effective compute making low the energy use spectrum theory of and containing athat couplings three-site of is theory. the finite. This resonances, description asgeneration, allows well Choosing we as have a the scanned one-loop large the potential parameter degree space of of compositeness the three-site for theory within a natural JHEP12(2019)112 , the (A.2) (A.3) (A.1) , with , , , 12 } e ) ) ) 4 4 12 , , 11 3 3 , = ˆ T 10 0 ,... n T ) + − T 2 2 2 + , , , 1 1 = 2 2 9 , T T , 8 ( ( , , 1 T 1 2 1 2 ( ) ) , + ) 11 10 − − ⊕ , , 8 7 9 7 , , 0 , ` < m 7 = = to point in a direction of T 6 ) T 1 L R `m 3 n 3 + − T 2 , − T T 1 − { SO(6), where we will embed ) with indices different from 10 11 9 , ( , , , 8 6 6 8 × 3 ( T T T 1 A.1 ( ( ( √ ⊕ 2 1 2 1 2 2 1 ,T,T 6 , l < m . ) ) ., = = = = √ 4 4 ) , , / c , . 2 2 2 ., `k , c T − c δ 55 . ) ) by using the structure constants, or, by c ⊕ 11 − T + SU(3) SU(3) SU(3) SU(3) 1 2 4 6 8 mj ⊕ , 0 3 3 66 δ ⊕ , , ) ⊕ 1 1 1 to identify which representations contain SM 1 0 , − 1 11 T T ) , 3 ( ( – 28 – 1 ( 1 1 2 1 2 ∼ ∼ , , min mk 1 ⊕ 1 δ . ( , = = 6 12 66 `j ) 1 inside SO(4) can be defined by: δ √ L R ( ⊕ 2 2 ( 11 / , i 0 1 R ⊕ ) ) . The SO(4) algebra is defined by allowing indices to run 10 ,T = ) 2 1 T X ) , , 3 ,T,T , 2 1 jk + ) ) 11 ,T , ) , , 1 SU(2) 9 ) 10 11 , , ,T 1 3 10 , , U(1) 9 8 ( ( × , 9 7 ) 1 T `,m ,T T 7 ( 3 × L T , T ) ⊕ ⊕ T 2 ( C + 3 + ⊕ , − T + 6 6 T 9 2 + U(1) inside SO(6) can be defined by: , 7 0 √ √ , 8 ) / / 8 11 10 + 6 × , , , 0 1 1 1 T 6 8 6 4 ) ) ) − T , , SU(3) T T −T T 1 1 1 2 4 3 4( ( ( ( ( , , , , T × , 1 2 2 2 2 1 1 1 1 ( − 1 1 2 1 R T , , , 1 2 ( ( = = = = = 1 3 3 1 2 − ( ( ( ⊕ = = ∼ ∼ ∼ SU(2) U(1) L SU(3) SU(3) SU(3) SU(3) R 1 1 5 7 1 3 1 × T 11 55 T T T T T T L Decomposing them further under H We construct the adjoint representation ( The smallest representations of SO(12), and their decompositions under SO(10): The algebra of SU(2) An SO(11) subgroup can be defined by choosing a vector ˆ the complex conjugate representationsto must the be original added one. only As when we they need are to not consider a equivalent parity transformation in order to forbid odd fermions, we get: using the generators of the algebra as a basis of this vector space. An algebra of SU(3) from 1 to 4, while the SO(6) algebra by those indices between 6 and 11. the 12-dimensional space.algebra For of instance, SO(11) selecting“12”. is the defined Inside twelfth by coordinate, SO(11),SU(2) generators we ˆ as can in define eq. the ( subgroup SO(4) In this appendix we give arepresentations, brief of description of the the algebra,fundamental group as representation well SO(12). is as given the by lowest Acoefficients: dimensional the simple set of basis generators for the algebra of SO(12) in the A Representations of SO(12) JHEP12(2019)112 (B.1) (B.2) (A.4) (A.5) (A.6) (A.7)                  2 11 , , 1 0 0 2 λ m 2 55 , , 1 0 0 0 0 2 λ m 2 55 , , 1 0 0 0 0 0 0 2 λ m partners, since their mixings R 2 55 e . , , 1 0 0 0 0 0 00 0 0 0 2 R λ d m    v , nor 2 1 2 , c 55 , representations, we can find how this , R 1 0 0 0 0 0 0 u 1 2 m d λ λ λ m 1 12 11 2 2 11 − 1 , 55 − √ 1 2 / 1 m = = , λ = = v 1 0 0 0 0 0 0 m s 1 q λ 0 0 0 0 – 29 – u 11 λ 55 11 P iλ    P P P 2 = √ d representation within the adjoint, so they will be odd that contains the adequate quantum numbers. In the / 2 1 v , M 1 0 00 0 0 0 s m 11 u 66 λ iλ − 2 1 , 1 0 0 0 m λ 2 0 0 0 √ 2 2 as: / v 2 v/ 2 v/ 00 0 0 000 0 0 0 0 0 0 s c s 12 q q q λ λ iλ − can be built with the product of two                  66 = u For the down type quarks, the matrix is a smaller three by three matrix as there is For the up type quarks we get a nine by nine matrix, as there are four elements inside M basis where we order the three elements as site 0, 1 and 2 respectively, we get: There is a row of zeros because there are no mixing for only one representation inside the adjoint representation withbasis the where same we SM firstand quantum put then numbers the those as of elementary this site fermion, 2, fermion. then we the get: In four the representations of site 1, B Mass matrices In this appendix weconsider show just the one generation mass and matricesare we in do small. not the include three-site model, for the fermions we And the pNGB areunder inside this this parity. The way to representdiagonal this matrix parity with transformation its first intation 11 the entries basis +1, and hereparity the defined acts last would entry on -1. be the adjoint As as by the a decomposing adjoint represen- the product of representations: this transformation has a determinantunder equal this to parity. -1. One We wayrepresentation wish to to achieve make this the is pNGB to states make odd this parity act over the fundamental terms in the pNGB potential, we will have to extend the group from SO(12) to O(12), as JHEP12(2019)112 2 ). ∼ / 1 ) 3 ∗ ` (B.3) (B.4) (C.2) (C.1) 2.12  /vg e m is not fixed ( 1 ` ∼  1 . , e                 1  U 2 11 scenario), we obtain , , 2.5 ∼ 1 m TeV 0 0 2    1 λ `  m =  2 2 f 2 55     , , 1 0 0 0 0 3 3 3 2 + ` ` 1 ` 1 1 couplings λ 0 0    state, we get a three by three 2 1 m τ τ τ 2 2 f 1 1 , x v v 2 m m m 2 U 55 b v s U d , 3 C 2 C , 2 1 one coupling 1 0 0 0 0 0 0 λ λ    2 m g m m λ x x x m 2 2 2 2 6 5 5 ` g can be reproduced by taking ` 1 ` 1 1    2 − − − 2 1 2 55 g , ) g , µ µ µ 2 2 ∗ 1 0 0 0 0 0 0 2 2 2 10 10 10 2 2 ( v v 2 m m m λ f f v b s d 3 C 2 C K m × × × λ λ − m m m R 2 5 3 6 1 ,  1 0 0 0 0 0 0 1 1 1 2 ` ` 1 x x x ` 1 1 m 2 λ    6 6 5 – 30 – f 2 . Taking for simplicity e e e 2 − − − 2 2 g 2 and 1 , ` v v 1 + m 2 m m 1 0 0 0 0 0 0  b s d g v ) 3 C 2 C m 10 10 10 λ 2 0 0 1 ∗ 2 λ λ 2 ( m m m f 2 ), and assuming that all the couplings between the f × × × 1  , D 1 0 0 0 0 0 0     2 9 2 − 2 1 R m λ 3 ` 1 6 u  2.10 g ) takes the values: 2  1 − , 6 5 2. As discussed at the end of section 1    . m − − ∼ λ 10 0 C.1 1 2 ) ` 10 10 × n λ ∼ ( R 00 0 0 00 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 6 leptoquarks to elementary fermions were estimated in eq. ( g ) and ( − 3 1 ` 1 U    , eq. (                 2.9 ∗ U / M g ∼ 2 = ` ) √  e n / ( R 1 M g ∼ 3 q resonances we get a 14-by-14 matrix, as there are 8 elements in the algebra asso-  -anomalies, as long as TeV and Z ∼ B / 3 1 u U  The experimental values of For We do the same for the bosonic resonances. For the For the charged lepton we also get a nine-by-nine mass matrix: m 3 u by the and  Making use ofresonances eqs. are ( of thefor same the order Right-handed (this couplings: is known as the which are not present inWe site do 2. not Adding show a the source matrix in the because site it 0, is we tooC get 14 large. degrees of freedom. Numerical estimates ofThe the couplings Right-handed of the ciated with the Z quantum numbers, but three of those belong to the unbroken generators, matrix, as there is,in in the site site 1 2 bothsite only broken 1 an and and unbroken one. 2, unbroken followed generators We by order identified the the with broken basis it, one as but in the site two 1. unbroken generators We get in JHEP12(2019)112 : 55 SM . To (D.1) (D.2) L and , . , U       ,    # # Q # ) ) ) ) − − , survives and leads +) 5 −− −− − ( (+ (+ ( A ( (++) q q u u ` ` r R r R ˜ r R r R ˜ r R r R ˜ L L Q Q L L ) ., .. " " " +) − c c . . − ) for the decomposition of c c = = = ( (++) (+ , ⊕ ⊕ 2 2.5 55 11 55 R 11 R R R 55 R 11 R / can be obtained by working in a 6 3 1 Q Q L L U U / / 1 − 2 3 ) ) )          , that can be expressed straightfor- 2 2 1 , , , = = = SM 3 1 3 R R R = ( = ( = ( ` q u r r r ,U – 31 – ,Q ,L          , , , # of the third generation, that are embedded in the # # u q ` r ˜ r ) ) ˜ ` r ˜ +) +) − ⊕ ⊕ ⊕ − − −− ( (++) q ` u (++) ( ( (+ and consider just the interactions with the Left-handed and u u q q r r r ` ` r r r L r L ˜ ˜ L L L r L r ˜ u ∼ ∼ ∼ R U U Q Q L L ) ) g , " " " − +) q 55 55 11 −− = − = = (+ ( ( 55 11 55 L 11 L L L 11 L 55 L L U U Q Q L of SO(12). We will use large letters for the 5D fields:          = = = 66 are reducible representations of G geometry. The extra dimension has two boundaries, respectively called UV- L L L u 5 L U r ˜ Q . min and ` The boundary conditions of the 5D fermions can be taken as: There is a 5D fermion for each SM fermion multiplet of the SM. We will describe There is an SO(12) gauge symmetry in the bulk, broken to SO(11) in the IR by r ˜ , q Notice that, since SO(12)tions is of broken different to SO(11) SO(11) components in of the the 5D IR fermions boundary, can the be boundary different condi- in the IR. The under H r wardly in terms of irreducible ones, see for example eq. ( only those associatedrepresentation to simplify the explanation of thedefine boundary the conditions following of notation the for the 5D decomposition fermions, of we representations find of it SO(11) useful under G to boundary conditions, and toconditions. the SM A zero gauge mode symmetryto of group the the in NGBs, fifth the that component UVsimilar of transform also to the by in the gauge boundary MCHM the field, arising fundamental from representation SO(5)/SO(4). of SO(11). This is very theory with extra dimensions. We considerand a 5D an spacetime, AdS with a compactboundary extra dimension and IR-boundary, leadingscale to is the taken well of known order Randall-Sundrum TeV, whereas model. the The UV one IR is of the order of the Planck scale. accuracy one cancurrents. neglect D A model inAn 5 holographic dimensions dual of the SCFT described in section The Right-handed couplings are much smaller than the Left-handed ones, thus with good ˜ JHEP12(2019)112 , ]. Phys. (D.3) , 08 SPIRE ., ]. (2018) IN c decays . ][ Erratum ibid. Phys. Rev. − , transforming JHEP [ , ` . L , + h in the decay + SPIRE D 97 χ ` ) 55 IN ∗ 11 R + ][ D Q decays K ( decays Decays and Implications 11 L R − → ` − ¯ L τ inside ` + ¯ 2 ν + ` 3 + − ` / B m Phys. Rev. + (2015) 111803 τ 2 0 , ) ) ∗ K ∗ + arXiv:1303.0571 1 ( ]. K , [ D → 115 55 R 3 with a semileptonic tagging method → L + decays → ) 0 ∗ B 55 L SPIRE ]. τ ¯ B B ¯ D Q IN arXiv:1612.00529 ( 1 [ ]. ][ R m SPIRE + IN and (2013) 072012 lepton polarization and SPIRE ][ ) 55 R τ IN L ]. D Phys. Rev. Lett. – 32 – ( L ][ ), ¯ D 88 χ R ), there remain three massless chiral fields that can µ ]. ` ¯ ν (2017) 211801 m − SPIRE in terms of Left-handed sources localized on the UV, D.3 µ IN + + 118 L SPIRE ∗ ][ of the SM. arXiv:1903.09252 55 ), which permits any use, distribution and reproduction in R IN D [ ]. L U ` ][ Phys. Rev. and L → , arXiv:1406.6482 ¯ χ 0 and eq. ( [ u Q ¯ Measurement of an Excess of B and SPIRE L ( Measurement of Measurement of the m Test of lepton universality using Test of lepton universality with Search for lepton-universality violation in Measurement of the ratio of branching fractions Test of Lepton Flavor Universality by the measurement of the χ arXiv:1506.08614 IN B R [ / + ˜ u ) CC-BY 4.0 leptoquarks are given by the lightest Kaluza-Klein states of the 5D , τ branching fraction using three-prong ¯ 55 ν L R 1 (2019) 191801 Phys. Rev. Lett. q τ This article is distributed under the terms of the Creative Commons − U U ν , τ ]. τ + arXiv:1705.05802 55 L + ¯ ν τ ∗ [ 122 ¯ (2014) 151601 Q − 67 ]: describing D , u collaboration, τ ∗− arXiv:1711.02505 collaboration, collaboration, ∗ [ 41 D m 41 → D 113 (2015) 159901] [ 0 = → ¯ , as well as the following mass terms localized on the IR: in terms of Right-handed ones, only a set of UV-sources that can be matched with → B 0 ( IR ¯ 55 Lett. (2017) 055 Rev. Lett. B 072013 Belle arXiv:1904.08794 B 115 Belle B BaBar for Charged Higgs Bosons U We will not elaborate more on this description. As usual in this kind of theories, the degree of compositeness of the 0-modes isThe con- lightest The fermions with (++) boundary conditions lead to chiral 0-modes. After taking into We include also a 4D chiral fermion localized in the IR boundary: L LHCb collaboration, LHCb collaboration, LHCb collaboration, LHCb collaboration, LHCb collaboration, [7] [8] [4] [5] [6] [2] [3] [1] any medium, provided the original author(s) and source areReferences credited. gauge field associated to the generators transforming as ( Open Access. Attribution License ( trolled by theintegration bulk of mass the of bulkfunctions degrees the [ of 5D freedom, fermions. and they can The be form expressed factors in terms can of be Bessel calculated by that are compatible with the SO(11) symmetry of theaccount the IR presence boundary. of be identified with approach [ and the SM ones are dynamical, i.e.: they have (+)as UV a boundary conditions. boundary conditions on the UV can be understood easily by making use of the holographic JHEP12(2019)112 ] 3 ]. B ] , PS ] ] , (2015) (2018) B SPIRE Phys. ] , IN ] flavour 115 , and ][ ) C 78 0 ( Phys. Lett. -physics K , B s`` U(2) ]. → ]. ]. arXiv:1603.04993 Simultaneous decays at Belle b (2017) 015 [ ]. puzzles arXiv:1712.01368 − [ ) ` SPIRE ∗ 01 arXiv:1806.02312 ( arXiv:1801.07256 + SPIRE SPIRE IN [ D [ ` Eur. Phys. J. IN IN ∗ Phys. Rev. Lett. 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