Published Version

PUBLISHED VERSION

James Barnard, Daniel Murnane, Martin White, Anthony G. Williams Constraining fine tuning in composite Higgs models with partially composite leptons Journal of High Energy Physics, 2017; 2017(9):049-1-049-31

Originally published at: http://doi.org/10.1007/JHEP09(2017)049

PERMISSIONS http://creativecommons.org/licenses/by/4.0/

7 November 2017

http://hdl.handle.net/2440/109233 JHEP09(2017)049 has ν b − − t τ , and so , of SO(5) − − Springer q l τ b July 19, 2017 parameters, 14 April 11, 2017 : p : August 13, 2017 n : September 12, 2017 : Revised Received SO(4) MCHMs with all Accepted , where MCHM → Published 5-5-5 14-1-10 and Anthony G. Williams Published for SISSA by b https://doi.org/10.1007/JHEP09(2017)049 [email protected] , tau partner in representation l , and MCHM Martin White 5-5-5 14-14-10 b , MCHM observables. We then use a novel scanning procedure to perform enjoys a relatively low increase in ﬁne tuning even for a future lower . o 5-5-5 5-5-5 3 n Daniel Murnane, 1703.07653 a 5-5-5 14-1-10 The Authors. c Technicolor and Composite Models, Beyond Standard Model, Eﬀective Field

Minimal Composite Higgs Models (MCHM) have long provided a solution to , [email protected] E-mail: [email protected] ARC Centre of Excellence forUniversity Particle of Physics Melbourne, Victoria at 3000, theARC Australia Terascale, Centre School of of Excellence Physics, forUniversity Particle of Physics Adelaide, South at Australia the 5005, Terascale, Department Australia of Physics, b a Open Access Article funded by SCOAP Higgs-gluon coupling deviation of 2%). Keywords: Theories ArXiv ePrint: one properly accounts forfine-tuning all of sources each scenario, ofbehave and fine differently the tuning. with future respect prospects. Wethat to study Noting the future both MCHM that improvements the the in different currentbound collider scenarios on relative measurements, the we top find partner masses of 3.4 TeV (or equivalently a maximum Higgs-fermion or These are the MCHM the lepton doubleton. partner in We representation findmoderately that reduces embedding the tuning at for leastresults), the case one but of massive that low lepton top this partner in masses is the (in balanced line symmetric with against previous the increased complexity of the model when the LHC. We develop ato new fine fine tuning tuning (single, measurethat double, that must triple, accurately reproduce counts etc) each thata contribution can comprehensive occur study in of athird three theory generation different fermions with two-site, included, 4D, distinguished by SO(5) the choice of the lepton embeddings. Abstract: the hierarchy problem of the Standardthat Model, are yet becoming suffer from increasingly various problematic sources of with fine the tuning lack of new physics observations at James Barnard, Constraining fine tuning inwith composite partially Higgs composite models leptons JHEP09(2017)049 – 1 24 26 27 10 10 11 16 24 – 1 – 13 3 9 20 8 20 ]. An important innovation in more recent work is the 17 5 , 4 fine-tuning 3 fine-tuning 5-5-5 5-5-5 5-5-5 14-14-10 5-5-5 14-1-10 fine-tuning 1 11 21 5-5-5 14-14-10 5-5-5 14-1-10 5-5-5 5-5-5 5-5-5 14-14-10 5-5-5 14-1-10 5-5-5 5-5-5 2.3.1 MCHM 2.3.2 MCHM 2.3.3 MCHM A.1 MCHM A.2 MCHM A.3 MCHM 5.1 MCHM 5.2 MCHM 5.3 MCHM 2.1 Model2.2 overview Details of2.3 the gauge sector Details of the fermion sector ]. As seen in the case of the pion mass in QCD, such a theory naturally contains a hierarchy example a strongly-coupledcongregate, new and sector), an near anomalouslythe light which fermion object. heavy mass One hierarchies resonancesnotion [ can are of also expected “partial use compositeness”, to compositenessof in to elementary which explain fields SM and fermions the and new gauge fields composite are states. a mixture Whilst the precise choice of global The hierarchy problem ofexistence the of new Standard physics at Model the (SM) TeVinvolves scale. the has A replacement long well-known of and the been compellingas extension elementary used of a Higgs the to boson pseudo-Nambu-Goldstone SM boson by motivate a of the 3 composite a state new, that spontaneously emerges brokenbetween global a symmetry characteristic [ mass scale associated with some new fundamental physics (for a 1 Introduction A Fermion representation expressions 6 Conclusions 3 Scan details 4 Fine-tuning in many observables 5 Results Contents 1 Introduction 2 The minimal composite Higgs JHEP09(2017)049 → X U(1) ]. Naively, × 21 representation, but the 14 ], making light top partners 20 – representation provides a partial 16 , 14 12 – 10 – 2 – ] for examples), the model based on SO(5) ]. This symmetry breaking pattern fixes the embedding 13 – 6 15 , 14 representations of SO(5) leads to a well-known “double tuning” effect, representations, one obtains two functional forms in the Higgs potential 10 for the leptons, with a suppressed contribution from the quarks. The two is known as the Minimal Composite Higgs Model (MCHM), since this is or 10 X 5 or 5 U(1) × Embedding at least one quark chirality in the In MCHM theories without partially composite leptons, one observes that embedding The current lack of evidence for BSM physics at the LHC is telling us that the com- the lack of a colourquarks factor. in However, if one putsat the leading leptons order in a terms in the Higgsan potential additional naturally ad-hoc come tuning out compared to to be the of case similar of order, the with initial quark less approach. need for is to include partially compositeany leptons complete in description the in model,it any since they might case, must and be be theythe accounted expected can for model. that in introduce At the extra anycontribution effects leptons given (assuming [ order similar play embeddings in to only the thethe quarks) a Higgs quark will potential small be contribution calculation, considerably due role smaller for to than in example, the the much the lepton smaller phenomenology Yukawa couplings of and, to a lesser extent, solution to this problem, sincepotential one at now leading obtains order. two differentreduce In functional the practise, Higgs forms mass, however, in a which thescale turns large Higgs out in ad-hoc to these tuning be models. generically is much then higher It required than has to the recently electroweak been shown, however, that a more elegant solution contribution to the Higgs potential,has since a the minimum resulting at functional thea form origin. (in second The the term inclusion Higgs of which field) parameters formally that can subleading enter give contributions the produces rise coefficientfrom of to their the electroweak natural leading term size, symmetry can independently breaking be of provided tuned the sufficiently that initial far tuning. the down the quarks in which can ultimately befirst traced tune back the parameters to to the obtainthis calculation a does Higgs of not VEV lead the below to the Higgs viable compositeness potential. electroweak scale. symmetry One Even breaking then, if must one includes only the leading the Higgs VEV, thecontribute Higgs an independent mass source and ofmass the is fine specifically fermion tuning. correlated masses, with It thethe each is mass embedding of of known, of the for which lightest the example, can top compositeessential partner that in for states for the certain naturalness in principle Higgs choices in SO(5) of these [ models. positeness scale in anythan realised the composite electroweak scale, Higgstuning in scenario in which is case these probably it theories. significantlybeing is higher that Fine worth the critically tuning Higgs examining vacuum comesness expectation the from scale. value level a (VEV) of One must variety fine be must of kept also sources, below make the the the composite- most theory obvious reproduce a variety of observations including SO(4) the smallest symmetry group consistent withNambu-Goldstone custodial bosons symmetry [ that leads toof exactly the four SM gauge sectorSM fermion but sector, leaves considerable in freedom which case of it choice is in most the usual embedding to of explore the several possible options. symmetry remains open (see [ JHEP09(2017)049 5 . These . 6 respectively, in observables, and 14-1-10 N 5-5-5 implementation of the quarks for good measure. b lepton and top quark masses, in ], comprehensive scans of MCHM MultiNest τ 22 quark, b ] which leads to interesting conclusions. We 21 observables, and in doing so we count the total – 3 – , or in the fully-composite tau N 14-14-10 , and review the standard derivations of the relevant fermion 2 ] to cope with we outline how to deal with the tuning of , before we present our final conclusions in section 4 5 22 . In section 3 ]) and associate the Higgs with a pseudo Nambu-Goldstone boson (pNGB) that 32 representations for the quarks), and scan over three different lepton representations. – The rest of our paper is structured as follows. We provide a short overview of the two- In previous work by a subset of the current authors [ 23 10 2.1 Model overview A particularly elegant mechanism forposite realising sector a that compositein Higgs emerges [ boson from is a to take confining a gauge com- theory (e.g. the theories considered observables as functions ofin the section model parameters.present a Our computationally scanning effectiveare technique method presented is of in producing described section a total fine tuning.2 Our results The minimal composite Higgs composite Higgs models,between all parameters of to produce which known can phenomenology. be expectedsite to 4D rely MCHM on inexpressions delicate section from cancellations the composite fermion Lagrangian. This includes formulae for the SM fine tuning in a moresolve accurate the way second than problem in bynested [ using sampling a algorithm, combination plus of adesired the second regions stage in of our MCMChard candidate sampling, to MCHM to parameter find efficiently find spaces. by the random Such regions means, are and punishingly our approach will be useful in the study of other addition to the Higgs massrequires and a VEV. sophisticated Such treatment, athe both large method in number used the of to observables definition findgive and the of rise parameters small the to regions fine of the tuning thefine-tuning correct considerable measure, measure SM parameter and of behaviour. space [ volume We that approach the first problem by generalising the or These are the compositethe lepton symmetric-antisymmetric doublet, taumodels and have between tau 19 neutrinoparameter and in space 25 that input correctly parameters, reproduce and the we must find regions of this large boson resonance masses, Higgspaper, coupling we deviations revisit and thisleptons, the work also with compositeness introducing the scale. the addition possibilityAs such, In of of the partially this the matter composite content partially ofquantum composite numbers the of models third the considered generation heaviest includes flavourrepresentation composite of (since fermions SM the quarks with details and the of leptons. the tuning We consider will one be quark relatively insensitive to the choice of scenarios without partially composite leptonsof the were SM performed, top with quark threeof embedding different the (and parameter choices lighter space fermions consistent neglected). withwere the identified In Higgs each and VEV, case, top used thefunction quark to regions mass of and obtain the existing current Higgs and and mass projected hypothetical limits constraints on on fine the tuning top as partner a masses, charged vector- JHEP09(2017)049 , ]. X × = X L 1 35 (2.1) , G ). The . This U(1) 0 0 } 1). One 34 ˆ a , , × G 0 T 1 18 , { 0 , 0 , with a U(1) , 1 ], and we refer the = SO(4) = (0 1 35 i H , 1 generators, Φ 22 1 h , ) ˆ a T ˆ a is then spontaneously broken down to Π i 1 turns out to be necessary to provide the , by introducing spurious, non-dynamical G X X × exp( i 0 0 1 spontaneously breaks to G – 4 – U(1) Φ 1 can be parameterised by a field Φ h . . Site 1 is populated by composite partners to × 1 G 1 0 G 1 H × → (Π) = 0 0 1 1 , and the NGBs associated with this breaking are eaten G Φ G G . Site 0 and Site 1 are connected by a link field Ω that will = SO(5) 1 ], and produces four pNGBs which is exactly the number re- 0 0 Φ G 1 33 g (leaving the SM fermions in incomplete representations of → 0 0 1 down to the SM electroweak gauge group, but the explicit breaking is . At the same time, G G G , where the extra factor U(1) X SO(4) preserves precision electroweak measurements through compatibility = 0 and a non-zero vacuum expectation value of contains the four NGBs for the four broken SO(5) X gauge fields to produce massive, vector bosons transforming in the adjoint → ˆ a Q U(1) 1 T ˆ a G × subgroup of 1 Y The spontaneous breaking One can now “turn off” the spurious fields on Site 0, and gauge only the SM SU(2) The models that we consider consist of an elementary site (Site 0) and a composite can write: where Π field transforms as Φ U(1) effect is to break weak due to the factthan that their the SM couplings counterparts. and Hence, masses the on Higgs the remains compositecharge light site due of are to much its larger pNGB nature. representation of producing 4 NGBs that providenature a are fully linear composite combinations Higgs of“partially boson. the composite”. elementary The and SM composite fields source observed fields, in i.e. they are site are non-dynamical. Thisto is construct done the purely Lagrangian for ofunder mathematical the the low-energy convenience, global effective as theory product it bythe group allows writing global, us terms diagonal symmetric subgroup by the correct hypercharge forwith the the fermions. same (gauged) symmetry Siteexcluding group the 0 and the Higgs is same doublet.an fermion populated representations exact It by as global is the fermion symmetry SM, usefulgauge and to and gauge promote fermion fields the fields, elementary and symmetry by group temporarily to assuming that all fields on the elementary site (Site 1),the as elementary summaried fields, innew to figure strongly-coupled be sector. thoughtSO(5) It of contains as fields the invariant under first a set global of symmetry resonances arising from the the precise embeddings ofpaper, the we focus elementary on fermionsThe the in mechanics two-site of representations 4D the of models collectivestudy breaking SO(5). that uses will were be well-established previously In broadly bosonic described summarised this reader expressions, below. in to e.g., [ these However, derived this for in athe [ pedagogical fermion guide. sector under We will the delve breaking. into more detail of the behaviour of tern SO(5) with custodial symmetry [ quired to form aenergy Higgs theorems, and doublet. hencetions the The only between interactions remaining the of task SM the is fermions pNGBs to and specify are the the determined composite form by of sector, low the which interac- boils down to choosing derives from the spontaneous breaking of a global symmetry. The symmetry breaking pat- JHEP09(2017)049 . 1 ,G (2.5) (2.6) (2.2) (2.3) (2.4) being 0 0 q G , , and its ∈ T ) G 1 2 factor, but the c ξ c , g ν b 0 X → 0 0 − g / / D , 0 1 D i † 1 i c G c g p ¯ ν b ¯ Ω 2 2 × 0 field strength tensors, , ξ, g 1 0 0 V , B for the U(1) G → 0 m , one can show that Λ m , Λ , X ν b } U(1) T 1 d d × = (0 0 /H ˆ h f 1 i + + c Φ c G ! . Terms involving lighter fermions t h cot τ 0 0 h 0 0 ∈ ˆ h SO(5) / D

/ D ˆ , a D i 4 i ∈ c = c ¯ t . ,T ¯ τ , h Y 1 2 and 1 ! 3 2 µν 2 4 H is the NGB decay constant. h h , h B T 2 ∈ T U(1) T f + + µν m ) Λ – 5 – a t , h m Λ h × 1 3 B 1 d τ T 0 Φ = ΩΦ L { d , c h , ih ih and 2 1 h 1

g . Therefore we match fields and see v f + 4 , s ˆ h f T + 0 = q ) , l basis 0 − 0 0 sin / , v H 1 D , / 1 D µν = sin 1 ˆ h ¯ qi li ¯ ξ 2 -model, and which transforms as Ω W 2 = (0 , σ h f i µν representing the elementary right-handed top-like quark, Φ = Q Φ = (0 L c H W t h m Λ 0 m Λ , ]. q l 2 2 1 d d = sin g 35 4 ⇒ ..., h c = . We identify the usual Higgs doublet as representing the SU(2) − + + + ˆ , a } h f h = ˆ a µν 0 h h L √ ,B = = sin = ˆ h µν h ˆ s h W = { 4 The Lagrangian for our models can be constructed by combining separate contributions For brevity, we have dropped from this discussion an extra link field Ω h 1 a left-handed doublet forinvolving the the third elementary quark gauge generation,are fields and neglected is so denoted (but on. by noted The for covariant derivative completeness by thedetails dots), can and be we found in also [ note that the quark with and the matter content: where that describe theelementary elementary site and contribution composite from Site sites, 0 and is given the by mixing between them. The with the usual vacuum where In this gauge, the would-beare NGBs set that to are zero. eaten Using to the generate SO(5) the composite vector masses Ω parameterises theaddition NGBs allows the from realisation the offinal partially spontaneous Lagrangian composite of breaking fermions bilinears through of thatphysical the involve content of a presence the fermion in theory on the becomeswhich Site apparent can 0 when one be and transforms a accomplished to fermion via: the on unitary Site gauge, 1. The be described by a separate JHEP09(2017)049 , 1 (2.7) by Φ 1 . Three } ) SO(4) → c 1 are the composite B , off-diagonal mass , Q, T, B 1 Ψ } q, t, b, l, τ, ν / c D (Φ { m i gauge field; Φ contains V c Y by removing non-dynamic , 1 ¯ c 0 B − x G T c + , c B ¯ U(1) by gauging the SM group. 1 QB ,L c × / B T Y D ) that couple the fermions to the 0 Y V ¯ ,B Φ) m , Bi c µ 1 U(1) T − + D ,T SU(2) × c c c T Q → Φ)( 1 ¯ { QT weak / ,µ D T 1 ..., i Y 0+1 c D ¯ m ( T – 6 – 2 1 SU(2) 2 , Q, T, B, L, − + f c terms are not present despite being allowed by all → T (Φ + 1 1 + h.c. + T Y x ¯ / c BB D µν X B ρ Q ¯ T i m U(1) + → V} × X,µν − c ρ 1 c Q ,B 2 by Ω, explicit SO(5) X 1 1 g ¯ is the covariant derivative involving the composite gauge fields. TT / D 4 i T 1 SU(2) 0+1 c → T − m ¯ D × Q 0 − µν + x and their charge conjugates ρ c SO(5) L, T Q are the field strength tensors for the composite, µν 1 → ¯ ρ U(1) → V} QQ / X 1 D 2 , ρ Q ρ × g 1 Q T ¯ 4 , and Yukawa-like terms, The group structure of the two-site model considered here, with coloured-in regions 0 Qi m { Ψ Y − + − + and SO(5) m = × ρ 0 1 The composite site Lagrangian contains the terms: L Q, T, B, L, types of term can beterms, written for the fermions:Higgs. diagonal mass These terms, terms areHiggs given below potential for finite each means fermionsymmetries that representation. of The the need model. to keep the where the Higgs fields, and { Dirac fermions that mix, respectively, with the elementary fields explained shortly. denoting gauged groups. ThereSO(5) are four sources of breaking:fields, SU(2) spontaneous SO(5) kinetic terms do not yet follow canonical normalisation. The normalisation factor will be Figure 1. JHEP09(2017)049 , = T ψ (2.9) (2.8) θ ) can (2.11) (2.10) V and , c term, and t T c c , t , B c ν / pb c L c b Q B b ¯ 2 3 , µ | ) ψ B (Ω) W , h b ) and 2 , M for the representations that p R T , h ( b q T 2 c 1 ..., − | , b p m ) symmetry. This mixing G ( t Q c + Λ . d 1 ψ , , + Π Ψ T G ρ WB q c c m t × / pt ψ + h.c. + c 0 d + Π ¯ t (Ω) ) V G t )(1 + Π ν c → ψ R B ν , h for the t µ 2 ψ 2 2 p Q v B f ( is a covariant derivative that contains (Ω) ) c m + Λ t u ..., q ≈ c n (1 + Π , h d 2 2 0+1 R p p is the transverse projection operator. Once + Π qQ ν ( D v f c T ) is the fermion contribution to the potential, B / pb bb (Ω) P b ¯ log – 7 – ) + Λ 2 q ) 2 + h.c. + β + Π 2.10 γ , h R p T 2 , h } 2 q c ν 2 2 2 ν p τ π p ( (Ω), that correspond to the p = sin = W ( b d w b µ R + Λ ξ → 16 (Ω) M † τ W ∞ ) + R 0 + Π τ, b Ω) log Π τ c Z , h 2 / pt 9 2 tt ψ → ¯ t µ 0+1 p ) ) ( . + Λ D Nc , h 2 l, t c 4 h W , h 2 p 2 p Π 2 lL p Ω)( βs ( 2 → ( t π p t ,µ t,b,τ,ν q + X T d µν (Ω) M { 16 = l 2 h P ψ 0+1 . The potential is expanded up to quartic order in the Higgs fields, to are the form factors and + Π + + R 1 2 ∞ 2 ψ l γs D 0 ( Z M − = 2 Ω 4 + Λ f ≡ − eff ) = L and = h are embedded in. Since the elementary fermions are not canonically normalised, ( i 1]. After we have our effective theory, it turns out to be convenient to redefine the . m , V L [0 etc The one-loop Higgs potential can be shown to be At low energies, the composite site degrees of freedom ( Finally, we can write the mixing Lagrangian as: , ∈ c ψ make connection with the usual SM Higgs potential. The Higgs VEV is then given by: The second term in theand first will line be of discussed eq. ineach ( the fermion next Nc section. It includes a factor for the number of colours of where Π a choice has beenexpressions for made the for form factors the can precise be embedding obtained. of the elementary fermions, explicit encode the details of the composite site: so on. Thatd is, we parameterise thescale of elementary-composite each mixing bare by mass an to canonical angle normalisation: tan be Λ integrated out to obtain an effective theory, where momentum-dependent form factors both elementary andand composite so fields. on, inis The a accomplished form remaining that using terms is projections, q, mix consistent t with the original the actual couplings on the mixing terms go like where Ω is the link field defined earlier, and JHEP09(2017)049 ]. χ - χ 35 , (2.14) (2.16) (2.12) (2.17) (2.13) 22 to give the f , # . ) ) 2 ρ 2 Ω m f − + 2 θ 2 2 t c found is consistent with p ) f ]. In practise, the Higgs 2 ρ ( )( f (both in the gauge and the 10] TeV (2.15) 2 ρ 2 a 21 m , g 5 m 1 2 . − πf via the mixing angle and masses . [0 2 a − = ) Ω 2 m ∈ 2 a f p ( , v a 2 ρ . )( . 2 θ (0 ) m t c , m ) ξ ) and 2 ρ ψ ) c , v − f m , )Π , hχχ (0 (1 + (1 ( ρ 2 2 ψ hχχ , v ξ SM g ( g c ) can be calculated from the form factors in 2 c β M and hence we can simply rescale (0 SM " , m – 8 – f 8 c ψ 2 2 h 2 c s Π f = = 2 1 4 /f ρ t, b, τ 2 p 2 h g χ v r 1 2 + m = = coupling ) ) = 2 ρ 2 ρ ψ ψ χ 2 ρ - m m boson is m ) χ 2 θ − t W 2 1] p , )( . We vary these parameters in the intervals: (1 + [0 2 θ a t ∈ − m 2 θ , assumed to be small, quantifies the amount of mixing between the t , m p θ (1 + ( e 2 2 2 , ρ and 2 p g g . For each point, we check that the value of tan g ρ ρ − m ≡ = = θ with the SM Higgs- θ t t > m c W ): a Π m 2.10 The form factor for the Form factors in the gauge sector depend only on the symmetry breaking pattern so are In the following, we will explore three different theories that are distinguished by with all dimensionful parameters havingfermion magnitudes sectors). less than 4 the same in all models studied here. We vary An angle elementary and composite sectors, whilstare given the by masses of the two lightest vector resonances Comparison of these predictionslimits with on current the and fine anticipated tuning collider of results each2.2 will theory. give us Details ofThe the gauge gauge sector sector is common to each of our theories, and is unchanged from [ and calculate the spectrumSM of Higgs predicted couplings. resonances The and lattercoupling the are parameterised expected as deviations a from fraction the of the composite Higgs- lepton. The tauthe neutrino lepton will be composite treatedVEV partners as only can massless, ever realise appears howevercorrect a certain in Higgs the representations see-saw VEV ratio of model insteadthis [ of rescaling, treating we it take as the an points extra that input give parameter. correct After values performing for the remaining observables the choice of embeddingparameter for space the to leptons. findare points For the that each Higgs reproduce model, VEV measured we and observables. scan mass, the These and composite observables the sector masses of the top quark, bottom quark and tau The mass ofeq. each ( SM fermion ( and the (composite) Higgs mass by JHEP09(2017)049 . , , i ]: τ 14 , i . 37

, 5-5-5 5-5-5 (2.21) (2.18) (2.19) (2.20) . The τ 36 r θ 022 . ,τ 0 2 / cos ,τ with spin 1 − 2 / / A ρ χ 1 coupling if one A m + 024 . b 0 + r = htt ,b ,b 3 2 2 ρ ) depend on the way / ) and performing the / ≈ − 1 1 couplings that we will m # SO(4) composite Higgs A ,τ A 2.9 2 ρ 2 1 3 1 3 2.10 / m → 1 hττ ) + + 2 a 2 θ t t ,t boson), by r m 2 ,t / and i, A 2 1 / W and the symmetric traceless , tau partner in representation 1 l A ]: (1 + with mass group. That is, there is more than 4 3 095 A coupling, which is given by [ " or . 1 hbb 0 37 10 4 3 0 + , ln Z 3 1 + ) − is the SO(5) hγγ ξ . 2 ρ htt A V ν b r is a m 2 θ − − − 1 ) 072 t and t . τ 2 θ ) 1 A V t − 0 2 ρ ρ −

q l is the loop function for particle p part of the Higgs potential of: m m – 9 – = = 2 h i,χ ≈ − −

(1 + = s A 2 a V χ ,b 1 2 r r − ρ / m 2 a coupling is the same as that of the ,χ 1 ( ,χ 2 m 2 4 ρ . MCHM / m / 1 ( 1 gauge coupling. Plugging into ( m , the antisymmetric 2 9 hgg A A ,A 5 π L 2 χ 2 χ Q 64 Q coupling (where 375 5-5-5 14-1-10 c c χ . χ − 1 . These are approximately [ N N c χ = with mass ≈ . The quantum numbers and masses are given, to a very good N hV V g 1 ,t t,b,τ Y t,b,τ γ 2 . = are the modiﬁcations to the = / θ ξ ξ 1 1 t τ r P U(1) P and MCHM + ,A + × ], we will focus on three interesting ensembles of embedding: MCHM 1 V and L , the fundamental r A 324 35 b 1 1 . , r 8 is the observed SU(2) A , 21

t 2 14-14-10 5-5-5 r g = ≈ − There is also a correction to the loop-induced The composite sector features several massive vector-boson resonances that are charged γ 1 r A model with the lepton doubletand partner so in on. representation Wein hold the the definition quark of sectorqualitatively fixed each similar but model. results vary with the Weof respect composite expect our to lepton that lepton the representations embedding alternate relative choices quark differences (although in embeddings the fine would absolute tuning give scale for may each differ). that each composite fermion isone embedded way in of the SO(5) representingunder the an fermion SO(5) multiplet rotation. inthe We the are trivial Lagrangian interested such in that allAs of it per the is [ lowest invariant dimensionMCHM representations, Finally, the modification to the neglects the contribution of lighter states. 2.3 Details ofAs the noted fermion above, sector the specific fermion form factors that enter eq. ( The lighter fermion contributions areincluded negligible here compared for to completeness, the though heavier are terms. not They included are in the fine tunings given below. where describe in the followingand sections, number and of colours under SU(2) approximation, by effect is to modify the integral results in a contribution to the at leading order in where JHEP09(2017)049 ), ). 3 / 2 2.7 1 (2.24) (2.22) (2.23) ∈ 10-10-10 R . t ) c , and 0 V † 5 SU(2) only allows V ∼ × Φ)(Φ 5-5-5 ν ¯ )) to provide cancella- L ( 4 ψ ), we add terms such as ν d ( Y SU(2) , 2.8 1 ' − ) + ∼ O c 5 T † T ∼ , ν ) and one SM singlet (e.g. Φ)(Φ 3 τ are the composite partners for the left- g, t, b, τ . / ,L ¯ 2 L τ ν ) we must introduce a second fiveplet to ) present, we have extra quadratic invariants ( = 3 τ 2 / ,L , Y 1 τ , ϕ 2 − ( ; and ξ ξ 1 6 ,L – 10 – . ) + 2 / 14-14 b ∈ c 6 − 1 ∈ ,L / 2 7 − 1 B L 3 ,Q 2 † q / R m 1 t √ 1 b m Q − = 5 ; 3 Φ)(Φ ϕ / b ∼ ]. With a r 2 ), that provide quadratic and quartic Higgs fields without ¯ 1 Q ( 18 b m ,B , , and the same in the leptonic sector. b Y 2.26 c b 17 , and so on. Similarly, for equation ( models, we require two partners for each doublet. This is required ) with mass Q c notation, the first layer of multiplets containing top-like massive 3 b ) with mass ) + q / T ]. We thus need to add the appropriate terms to equation ( c 3 t 1 Y / ∆ ¯ T − 2 34 Q q † T 5-5-5 ,Q Y + ¯ ,B ,B with mass 3 U(1) are the composite partners of the right-handed elementary top, bottom, 14-14-10 5-5-5 5-5-5 5-5-5 3 3 c t / m / / 3 2 Φ)(Φ 2 5 / × V Q 5 t 2 t , T T q → L ¯ T Q T ∼ ( c ∆ . t = = ( = ( ¯ q Y 3 6 6 ,T ¯ QT t / / / → = 2 1 7 T A.1 T,B, Q c Y 1 2 2 y In SU(2) The Yukawa couplings in the composite sector are: Q L m • • • q ∆ ¯ representation. This parametrically enlarged tuningrepresentation emerges structures, from the requiring subleadingtions of terms Higgs (i.e. mass terms [ in the Yukawa sector, eq.requiring ( significant subleading terms for cancellation. These can be seen explicitly in the pendix 2.3.2 MCHM Our second case embeds thetations. leptonic These sector give in us the symmetric the and freedom antisymmetric to represen- avoid the double-tuning present in the fundamental More details for the model (including expressions for the form factors) are given in ap- resonances are The remaining modifications to the Higgs couplings are now: the coupling of oneTo SM couple doublet another (e.g. SMpreserve singlet symmetry (e.g. [ i.e. q where tau and tau neutrinohanded respectively. states in the thirdIt is generation a quark quirk and that in leptonsince doublets the of decomposition the of elementary a sector. composite fiveplet under SO(4) We begin with thedamental case representation. of the new Ingeneration composite fermion, this sector and case, particles two for each we each embedded have left-handed in doublet: a the partner fun- for each right-handed, third 2.3.1 MCHM JHEP09(2017)049 2 ), Φ n ) at c 4 L V (2.29) (2.30) (2.31) (2.25) (2.26) (2.27) (2.28) 5-5-5 14-1-10 ¯ , s L . These 2 † , m τ s a Φ ( L m ν . O Y 0 m . ( 1]; . 0 L , 10 Φ [0 10 c Φ) + , m c V ∈ B V ∼ 10] TeV; ¯ T L θ we have † , † t V ∼ 5 , m . Φ T ν a , [0 Y 1 y and tau lepton Φ)(Φ , m ∈ − b ¯ ) + L d 1 a a † , c Q m y + 4)˜ 1 3 T 14-14-10 (Φ , m ξ − , m τ ρ − T ∼ ˜ u Φ) Y 23 14 m Q a ¯ L y † − m 2 4)˜ Φ + , for the tuning reason above. However, (Φ ξ c T ∼ 1 τ g, t, b, τ . − T Y − 14 ξ ¯ L = 2(20 † 14 ) + , bottom quark . Φ − c t , ϕ (2(5 τ ∼ a ξ ξ Y B ξ m y † – 11 – , 2 , L, A.3 − − 3) ξ 3 ξ − 1 ) + 1 / 2 c 1 − − 1 . √ √ ,L Φ)(Φ − B ξ − b 1 3 † 5 / ¯ = 1 Q (6 √ 1 5-5-5 5-5-5 ( ∼ ϕ − b r = = 5 , top quark Y Φ)(Φ g τ b ,B H ∼ r d ¯ Q , r m ) + ( b c , that we use to derive four observables measurable at the LHC: b ,B T , r Y d t † r . 5-5-5 5-5-5 ) + c ,Q Φ)(Φ T 3 t A.2 † ,Q / 5-5-5 14-1-10 ¯ 2 Q 3 ( / 5 t 2 Y are the proto-Yukawa couplings for the composite tau partner, as described 5 ∼ Φ)(Φ t ). a = ∼ ¯ ˜ Q 2 ψ y ,T y ( , d t u ( a L ,T Y y Q u O The bare masses of the lightest scalarThe resonances angle of composite-elementary mixing in theThe on-diagonal gauge bare sector masses of the top partners = Q The free parameters are: We now have one partner for each SM lepton (since the lepton embedding no longer These are required only for MCHM • • • y 2 L The models described above have betweenand 25 27 independent in parameters in the the MCHM the MCHM masses of the SM Higgs four observables determine the likelihood of a particular parameter point. The top partners are as above. The modification to the Higgs couplings isFurther now: details can be found in appendix 3 Scan details The Yukawa couplings in the composite sector are given in appendix 2.3.3 MCHM Our final model embeds thewe are lepton now doublet interested in in a to seeing a the effect partner of in a the fully singlet composite representation:, tau. That is, the tau couples where in the next section. Further details on this model, includuing form factor expressions, are The remaining modifications to the Higgs couplings are now order follows the 5-5-5 pattern), and the same as above for the quarkThe sector, Yukawa couplings in the composite sector are source term form factors listed in the appendix, where for JHEP09(2017)049 , b , d t (3.2) (3.3) (3.1) , d d 10] TeV; q , 5 , d parameters . u q p [0 d N ∈ n y , m τ y , , m d y ! 10] TeV; and 2 , ] , m . u 10 a y exp x − 2 . [ p O m ) ) N a ∈ − d x σ ) ( ) ν p x 2( y x ) ( ( 3 Z p x a | ) ), O x τ [ O implementation of the nested sampling tech- y ( th observable with experimentally measured )( (˜ i − p , x |

τ – 12 – O , y ) = ( b p exp , we wish to obtain the region of the parameter space O . | , y x t a Z , and the product runs over all observables. For our x y Y ( Multinest p = a exp O 5-5-5 14-10-10 Z ) = , is the Bayesian evidence 1], where the extrema are respectively fully elementary or the likelihood of any particular model with via Bayes’ theorem x , ), of the distribution of model parameters we can determine Z | } x x [0 τ ( O ( p ∈ p , m b ν 10] TeV where the indices are described in the previous section; , d , m , τ t 5 d . 2 , m [0 is the error in ], where a random sampling approach was used), we scan the full dimen- h ), ∈ n a l 21 m characterises how close we want the masses to be to their observed values. ]. This has proven very successful in exploring complicated, multidimensional σ V ) is the predicted value of the , d a , x 40 a ( σ l ≡ { , m – a d a exp T ( l O 38 O O fully composite . The off-diagonal bare masses of the topThe partners proto-Yukawa couplings The extent tod which the measured SM particles are composite m Given a set of input parameters, Rather than make simplifying assumptions to reduce the complexity of the parameter This is required only for MCHM • • • 3 is long convergence times due toexpected a in rapidly a falling largeon acceptance volume delicate rate, where cancellations something correctly between which reproducing termsThe is the goal in to required of complicated be our observables functions study, depends however, of is merely the to input find parameters. large samples of points with the correct SM The nested sampling algorithmfirst evaluates transforming the the evidence multidimensional bybe integral Monte evaluated into Carlo numerically). a integration one-dimensional (after product, Correctly and integral it that weighted is can posterior thesetuning samples samples results. that are Even we with use obtained the in as nested the sampling following a sections technique, by- we to find determine that our the fine scans have very The normalisation constant, value purposes Given a prior knowledge, the posterior probability of Given x where nique [ functions encountered in aapply range it of here, cosmology we and first particle formulate physics the examples. scan In asin order a which to Bayesian the inference masses problem of as the follows. SM fermions included in our study match the observed values. space (as in [ sionality of each model using the JHEP09(2017)049 O in a (4.1) (4.2) (3.4) O ∇ = 125 h m 5GeV; where . = 0 . 1 } 3 O 8 . . 2 , 2 = 1 . 3 τ , m , 170 to produce the observable = ), one might simply take the , i ~x 4 exp ( x O O O 130 = O

, ∆ i O O      } ≤ { p 5GeV; τ ∂ ∂x 1 2 . i 0 used in this scan were O x , m i BG, BG,

... b BG,n ], with step sizes for each parameter given 7 O ∆ ∆ . 4 ∆ 42 , m ) = t      i – 13 – = 2 x = ( , m b a H m O BG O m = ∇ 3 ]. = ∆ O 41 } ≤ { 3 . O BG,i 1 , ∆ 2 . Alternatively, one may define a vector of BG measures . 2 15GeV; , BG,i 140 , = 155 120 t { m = 2 O This gives us a few hundred viable points for each model. We use each of these as Approximately 80 million points are sampled for each model, with around 40,000 The particular values for the observables The values are not precisely the experimentally determined values — they have strong and electroweak 4 RGE running applied, as outlined in [ where we use the definitionsfor from each the parameter. previous section. Tomaximum find This of the gives all a total the measure fine-tuning ∆ ofthe in fine intuitive tuning way, ∆. The amount of finehas tuning historically in been any that particular introduced parameter by Barbieri and Giudice, Including a compositenatural Higgs new sector physics is aboveand 1TeV. a leptons goes Partial well-established further compositeness method to of raisedeal of this with the scale fine-tuning raising heaviest without comparisons unsatisfying flavour between the fine of models,extends tuning. scale the we quarks usual would To of concept like consistently of to tuning explore to a one measure that that consistently considers every source of tuning by 0.01 times theMetropolis-Hastings output current that value pass of the mass the cuts. parameter.4 Our final plots Fine-tuning use in points many from observables the the starting point for afor Markov each Chain model, Monte Carlo givingWe sampling us use of a the the same more Metropolis-Hastings parameter thorough algorithm space exploration [ of each possible preferred region. the true observable valuesSM are values assumed with standard to deviations be as normally given. distributed aroundpassing the initial EWSB predicted conditions. Wethe choose correct to SM study behaviour the by subset applying that mass are cuts in as the follows: vicinity of on the scans, but merelynot run seek for to long enough make toparameters. statistical obtain inferences hundreds from of our suitable final points. results, We do and we5GeV; use flat priors on all behaviour in order to analyse their behaviour. We thus do not impose strict convergence JHEP09(2017)049 ) ) → 4.6 4.5 (4.7) (4.6) (4.3) (4.4) (4.5) ac 2 ) being 4.1 ], it is often 0 and ∆ 22 → bc 2 . For aligned tunings b O ∇ . , then ∆ a b O . 0. Noting that equation ( exp | O a O κ . → O . = to be independent. Configuration ) = 1 2 → ∇ O ab 2 ab 2

|∇ , a quantity displaying these crite- b ca 2 c b b } . ab 2 ∆ b O O O = a O , 2 O and + ∆ b ) b

4.5 c ~x ∆ ( ∆ should then fulfil the criteria that (i) for all observables = → O · ∇ 2 b a ab 2 ∆ O . observables O observables, ∆ . Thus, ∆ o 2 ∇ κ o · ∇ n n a O ∇ , then b O ∇ λ = Configuration 1 algebraically satisfies criterion (iii) in a simple extension of eq. ( The total fine tuning ∆ For any two particular tuning vectors . This comes from both eq. ( a ab 2 O to unordered pairs over insensitive to aobservables, scaling the third criterionthere is are not two unique. configurations for, In1 e.g. particular, has observables for all four dependency and onvariables, one shown five observable, in observables, configuration figure 2 has the dependency shared across measure. For three observables, the measure satisfying these is One can see that∆ for observable ∇ is the area spanned by any two tuning vectors,independent this it behaviour be should a be maximum,for intuitive. (ii) the for limiting only case one of independent two observable independent it observables, vanish, it and simply (iii) be the single double-tuning For orthogonal tunings, tuning — a highertuning order should tuning. be If simply they thethe are product higher completely of tuning orthogonal, each should single then disappear. tuning. the higher If order they areria completely is parallel, This has been the state-of-thethe art until case recently. that However, these as pointed fine-tuningfrom out vectors more by are than [ not one aligned. source That and is, the the fine-tuning fine-tuning measure may should come reflect this special double Similarly, to extend to and take the magnitude of this vector over JHEP09(2017)049 ), = 4.7 (4.8) (4.9) N (4.10) (4.11) (4.12) ∇ . d 10% typically. For O . 2 < κ exp = O . c = O 1 2 O

ab... N c c c . ∆ O particular observables O O and abc 3 b · ∇ · ∇ · ∇ N O ∆ b c 1 2 a 1 | ) κ O O O ...

N 4.7 = = ∆ b O abc 3 1 κ ∆ -th order of tuning of a set of = N a O We have one set of dependen- ) is given by 3 above configuration 1. This is the limit of inaccuracy in equation ( / (a) cies The configurations available for one source of double-tuning amongst four observables. ,... -th higher order tuning over b N O ∇ , The generalisation of equation ( a O ∇ Figure 2. This is the measureproving by the naturalness which of we the evaluate MCHM. Before the stating success and comparing of our each results leptonic with those embedding in im- and the Finally, we simply sum each order of tuning for a measure of higher order tuning: In general, the ( Being a volume, thissum follows various the triple same tunings behaviour with as the the extension double tuning derived above. We where we take the volume spanned by three particular tuning vectors: a factor of 4 but for randomly distributedthe purposes observables the of correction fine-tuningmeasure being drops as a an to good order approximation. of magnitude calculation, we will accept this However, calculating eq. ( JHEP09(2017)049 1 ≈ 2 3) (4.16) (4.17) (4.14) (4.15) (4.18) (4.13) / , o n ∼ o , the Higgs coupling ratios goes as . 2 n T . O m = p . Of course there is a further will not affect the measure as 2 n = 120 ), ∆ 2 o N 2)! N/ p ∼ n 2) 2 b − 4.3 , n / 1 ∼ o ∇ (5 − 2 a n ! ! . N o , o ∇ p 9 n o n 2 27 n q n 1)2!( ∼ √

· ∼ − 1 2 ∼ o 1) . / – 16 – n 1 − 2 − 1 O

( / b (5 o b 3 p ∆ n from equation ( n = · ∇ 5 3 · ∇ p ∝ ∝ ! b a n o 3 3 2 ∇ ∇ n ≈ ∆ ∆ b a

1 · ∇ · ∇ − 1 a a ], it must be noted that our new measure will give larger abso- factor o 5 ∇ ∇ )): n

follow this pattern of 22 , 4.6 ∝ ∝ N 21 2 2 , the lightest top partner resonance mass ∆ ∆ ρ m ) and ( 4.5 ] to this paper of 22 Considering all of these artefacts of the tuning measure, we arrive at a generic increase Higher orders ∆ At order one of tuning, the number of observables We remind the reader that Barnard, White have three observables and nine parameters. 5 8, we also add in the possibility of order-four tuning, which is generically a factor of ∆ . Below, we present the scan results inpoint. terms of the The fine-tuning found tuning atresonance each of mass viable parameter each lepton embedding is shown against the lightest vector-boson from [ 5 Results scaling of the measure whenthree considering to higher four numbers observables, of not1 observables. only do When we going increase from greater the than fine order-three. tuning out-of-hand by (4 assuming mostly orthogonallinearly observables. with That number of is,goes both as at parameters second and order, observables. the At measure third scales order, the measure (equations ( they are averaged out. In terms of At order two of higher order tuning — that is, double fine tuning — the measure goes as lute numbers for theof fine parameters, tuning. arbitrary increase(i.e. To of see observables, compare and why, the genuinely consider generalnew more three double sensitive observables). factors: expressions tuning For random of arbitrarydependencies. fine-tuning Higgs increase vectors, mass/vev we to would the expect double the tunings following general of the of previous studies [ JHEP09(2017)049 ), 5-5-5 5-5-5 4.18 model ], i.e. a 21 . The full 5-5-5 5-5-5 4 (20) [ , which is to be ], which uses the /ξ ≡ O 22 37TeV. However, this . = 1 with ∆ 6 / 7 2 m lepton double tuning required to 5-5-5 14-1-10 and . Our measure gives higher values for fine ξ . A convex hull is provided to understand the 2 – 17 – ] (1) to raise the top partner masses that can be /f 1TeV partners. 2 21 or the naive fine-tuning measure 1 v 1 1TeV. This is in agreement with [ ≈ O ≡ ≥ ξ , which shows the fine tuning for the MCHM 4 ]. If we were interested in comparing with lepton-insensitive 22 fine-tuning ]. Our present case, however, is complicated by the fact that the inclusion 5-5-5 5-5-5 22 (100) and the least finely tuned was MCHM , along with the unnormalised results. For the rest of this section, we stick 3 ≡ O A general note on the results described below is in order. The argument for including A comparison of our new tuning with less sophisticated tuning measures can be seen in and the vacuum misalignment χ model is particularly badlyachieve tuned, EWSB. due Our to previous the studyest quark showed a top sharply partner linear mass relationship andlepton between the sector the fine [ light- tuning ofof the the lepton point sector for introduces a both model extra parameters that and did extra not potential include sources of the tuning. as colliders are able to exclude the 5.1 MCHM Here we present the resultstuning for is the quite fundamental severe, representation, partlya found due in minimum to figure tuning the of generic ∆ = fine 1082 tuning at reasons a explained top above, with partner mass of non-significant effect. Our results reflectof this, two with between even model smaller differences tunings. ofwhen This up tuning to dependencies can a are be factor fully considered considered,lepton a there result embeddings is of no in tuning-based the the preference newnot between tuning lepton-inclusive equally measure: MCHM. scale However, as tunings top partner between masses models grow, do so this may be a point of model distinction same sophisticated scanning technique.order-of-magnitude measure. Regarding That the is, second, weof consider ten we differences as take in being tuning the not of significant.in tuning less In considering than as this leptons a sense, an as factor there in iswith previous already ∆ some papers, question where of the the usefulness most finely tuned was MCHM composite lepton partners isfound two-fold in [ the parameter space,judicious irrespective choices of of tuning, lepton and embedding. (2)that Regarding to the lower allow first, the for we tuning find top by large making partner parameter masses volumes as a function oftuning relative the to the vacuum single misalignment expected. tuning ∆ In this case, withtations the of leptons SO(5), and the quarksof all lepton the embedded sector in model, is fundamental which not represen- suffers contributing from much the at double all tuning to effect the highlighted previously. phenomenology in figure to using the newcomparison measure of without our lepton additional embeddingsthe normalisation, (since difference which we between use will the the permit number same a of observables in parameters relative each is case, notthe and significant). bottom right panel of figure general limits of minimalnot fine always tuning appear (note that tothat given the be the fine-tuning convex). is logarithmic generally scale, two Wethan orders the observe, the of hull magnitude top-only in may higher case line inmodels, of this with lepton-sensitive for [ case the example, prediction we above could ( normalise by this factor. Such a normalised plot is given r JHEP09(2017)049 multiplet is the 6 / 7 2 does not mix directly with the 6 / 7 2 – 18 – A comparison of non-normalised (upper) and normalised (lower) fine tunings in the . This can be understood from the fact that the 3 / 2 scales, and greater divergenceevidence from to Standard suggest Model that Higgsif the coupling fine this predictions. tuning mass There rises exceedslightest more is top 3 steeply TeV. partner with are the We significantly1 lightest also less partner finely see mass tuned that than points points for where it which tends the to be the These sources include themasses single of tuning the Higgs associated andcombinations with SM of reproducing fermions, these and the observables. the Higgs newwith It VEV lower possibilities is masses and for for still multiple new true, tunings particles, across however, a that smaller hierarchy the between fine elementary and tuning composite decreases Figure 3. mass of top partners. JHEP09(2017)049 6 TeV. Currently, however, Run . – 19 – model as a function of Higgs coupling ratios, lightest scalar 5-5-5 5-5-5 . A precision of less than 3% on the Higgs couplings to gluons or fermions 3 / 2 1 Tuning in the MCHM elementary top quark, and hencethat its of mass the is lesswould constrained lead and to easier a to dramaticthe keep increase same light in tuning than the limits fineI as tuning constraints excluding of still top the allow partners model. even up the This to least precision 2 finely provides tuned configurations. Figure 4. resonance mass, top partner masses, and vacuum misalignment. JHEP09(2017)049 . 7 model for 3 TeV, which . is the lightest 5-5-5 5-5-5 6 , Higgs coupling / 7 = 3 2 6 / 7 2 case, tuning increases 5-5-5 14-1-10 m -specific tuning required 5-5-5 5-5-5 5-5-5 14-14-10 2% (in the MCHM ≈ ψ r – 20 – 4 TeV. Note that the Higgs-tau coupling modifi- . . 5 . We find a lower measure of tuning in this case than for 6 . A minimum fine tuning was found to be ∆ = 637 at a top 37 TeV. As such, by the order-of-magnitude argument, the . 34 TeV. The fine tuning again decreases with lower masses for 2.3 . = 1 fine-tuning = 1 fine-tuning 6 6 / / 7 7 2 2 m m 5-5-5 14-1-10 5-5-5 14-14-10 A measurement of Higgs-top coupling up to 3% would provide the same tuning con- There is evidence to suggest that, unlike in the MCHM a natural symmetric representationpartner shows mass a exclusion sharp limits rise andmodel in more remains the precise relatively fine Higgs untuned couplingcorresponds tuning even measurements, to with at a the better coupling top present top ratio partnermodifications precision have masses of identical of fine-tunings regardless of the species of particle being coupled 5.3 MCHM Finally, we show the resultsThe for tuning the is case of similarpartner a to fully mass the composite of previous tau case,tuning lepton, found with at in these a figure low minimum masses tuning does of ∆ not = prefer 594 this at to a the top previous models. However, where straint as excluding top partnerscation up has a to different 3 structurecation from is the much other more models forgivingat considered. — low there In tuning. exists this parameter This case, space is the with modifi- shown very in little figure modification term from the leptons,introduce additional and complexity to a do sub-leading so,representation. thus contribution lessening Due the from attractiveness to the ofcurrent the this, quarks, symmetric embedding we one to consider be has the not had tuning significant. to difference between the previous and both the cancellation ofto double achieve tuning, low and Higgs,partner masses. the top Our MCHM and tuningas measure tau also a masses counts negative the that feature. increasethrough in may Thus, organising the be number although to of more one have parameters can significant a alleviate at leading the higher order double top contribution tuning in to this the model quartic Higgs potential extreme difficulty of findingnear viable the points convex in hull. thislow model top The leads partner tuning to masses, isbe a but somewhat attributed may poor below be to sampling the that comparable density tuning tuning at of measure in higher the used. masses. a MCHM Where particular Again, previous the parameter, works reason we consider can consider only a the worst cumulative measure that is sensitive to new particles, a smallerHiggs hierarchy coupling in predictions. scales, and Wetop greater see partner, divergence again we from generally that Standard find in Model a the lower cases tuning. wheremore the quickly for top partner masses greater than 1 TeV. We caution, however, that the Here we present the results for theand case the of tau symmetric lepton, representations found for the inthe leptonic figure doublet fundamental, which cantuning be described partly in attributed section topartner the mass convenient of cancellation of double 5.2 MCHM JHEP09(2017)049 5-5-5 14-1-10 scenarios are 5-5-5 14-1-10 ). In this case, the current 4.16 and MCHM 5-5-5 14-14-10 – 21 – , MCHM 5-5-5 5-5-5 representation of SO(5) is less dramatic than previous and a minimum tuning of ∆ = 594 for the MCHM 14 The tuning of Higgs-tau coupling modifications. 5-5-5 14-14-10 Figure 5. To deal with the significantly large parameter spaces encountered in MCHM models be compared to tunings of other models that have also been appropriately normalised. model. The absolute value ofvalues due these to fine the tunings new is measure,each significantly but order worse one of than may tuning, previous choose as quoted tominimum given normalise fine in out tunings the the for complexity equation the cost MCHM followingapproximately at eq. 10%, ( 20% and 20% respectively. Note that these optimistic tunings should studies due to theeach complexity of cost the built thirdtrino in generation in to lepton fundamental representations the doublet, of new right-handed SO(5)which leads is tau fine to expected and tuning a to minimum right-handed measure. increase finemeasurements tau tuning with of Embedding neu- of top ∆ the partner = mass Higgs 1082, ∆ limits couplings. = at 637 the This for LHC, can the and be MCHM better collider compared with a minimum tuning of proves extremely difficult to find viableweeks points of for study cluster in running thesetop models, in and typically each requiring right-handed case. bottomsentation The are of third all SO(5). generation assumed Contrary quark toat to doublet, be least previous right-handed embedded one work, in of we the the find fundamental leptons that repre- in the a advantage of embedding find that the resultingany measure given scales problem, with and thus the naturally number penalises of additional observables model andwith complexity. parameters leptons in included, webination developed of a nested sophisticated sampling sampling and approach Markov Chain based Monte on Carlo a sampling. com- Even with this, it We have performed comprehensive scansposite to Higgs study scenarios the withembeddings. fine realistic tuning In lepton of doing sectors, three so,the distinguished Minimal we expanded by Com- have range the had of choice to single of develop and lepton a multiple new tunings fine that tuning can measure occur that in counts these scenarios. We with). This leaves the fully compositeonce tau further scenario Run as the II likely data most-natural is representation released. 6 Conclusions JHEP09(2017)049 – 22 – model as a function of Higgs coupling ratios, lightest 5-5-5 14-14-10 Tuning in the MCHM scalar resonance mass, top partner masses, and vacuum misalignment. Figure 6. JHEP09(2017)049 – 23 – model as a function of Higgs coupling ratios, lightest scalar 5-5-5 14-1-10 Tuning in the MCHM resonance mass, top partner masses, and vacuum misalignment. Figure 7. JHEP09(2017)049 , 2 4 22 m (A.1) 2 3 m 4 p + 2 5 ) + m 2 4 2 2 m ) can be written m + scenario, although + ). Each representa- 2 3 2.10 2 3 scenario, meanwhile, 6 m m p t A.3 2 2 (4) q + − ˆ m Π 2 2 ), ( ) and ( 5-5-5 14-14-10 t (4) 2 5 m + − 4 5-5-5 14-1-10 ˆ Π 2 3 p m 2.9 t + A.2 (1) q ]. They are included here for − m 2 1 2 3 + ˆ 2 1 Π ) + 35 ), ( 2 4 m m 2 4 t (1) m ( ) 2 2 2 ˆ m 2 h Π 2 m 2 p + s A.1 m p + 2 2 (4) + ) t − + + 2 3 − h 2 2 3 ˆ m 2 3 s 2 4 M 2 3 2 1 b m m (4) q m m m − − , will look similarly unnatural once Higgs m 2 2 ˆ + ( 2 1 Π + 4 2 2 m 2 2 2 m (1) + t m p m ˆ m 5-5-5 5-5-5 + 2 + (1 4 TeV. The MCHM + t M – 24 – . (4) q − 2 2 + 2 2 m + ˆ 2 3 t (4) Π 1 2 1 m 2 h 2 1 m ˆ 2 1 Π s m 2 1 m m + m 2 2 2 m ( m + − 2 4 2 m ∆ ) 1 2 t p p 2 1 2 1 2 ) Q t m q T + − 2 q d 2 t h t d ∆ s ∆ Q T 2 ) = ∆ ) = 3 TeV. Higgs coupling limits and top partner limits provide 1 m m 2 5 . ( ( √ ∆) = ∆ ∆) = ∆ ∆ , , 4 , 4 , m = = = 1 4 t t c t ∆ , m , m Π , 3 3 M , m Π 4 3 , m , m , m 2 2 , m 5-5-5 5-5-5 3 2 , m , m , m 1 1 , m 2 1 m m ( ( m , m ( L R 1 B A A m Finally, it is interesting to note that our explored models behave differently with respect ( ], so we present them in full for each representation. The expressions for the SO(4) M A completeness. A.1 MCHM Top quark: The precise expressions for the source terms35 in this study are slightly differentdecomposed from both form [ factors are to be found originally in [ A Fermion representation expressions The source term form factors implicitlyin defined terms in of equations ( thetion’s decomposed form form factors factor generally expressions depend ( on the four following functions: Australian Research Council Future Fellowshipported FT140100244. by DTM the and ARC(CE110001104) AGW. Centre and are of the sup- Excellence Centresupported for for by the Particle an Subatomic Physics Australian Structure at Government Research of the Training Matter Terascale Program (CSSM). (CoEPP) (RTP) Scholarship. DTM is or top partnercomplementary limits probes of for the 3 naturalness of composite Higgs scenarios inAcknowledgments the next decade. We thank Peter Stangl for helpful comments. The work of MJW is supported by the to future improvements in collidercurrently measurements. less The fine-tuned MCHM thancoupling the measurements MCHM ofpartner the exclusion fermion limits reach decay aenjoys channels mass a of reach relatively 3 a low precision increase of in 3%, fine tuning, or even top up to Higgs coupling limits of 2%, JHEP09(2017)049 → q (A.2) , ) t/b b τ ν (4) l l (4) (4) q Λ , ˆ ˆ ˆ Π Π Π 0) (4) ν (4) τ (4) b t/b , − − ˆ − q ) Π ˆ ˆ Π Π , , Λ b τ ν ) t/b − ) , (1) l l (1) (1) q T/B − − 0) 0) Λ ˆ ˆ ˆ Y Π , Π t/b , Π t/b , q (1) ν (1) τ (1) b 0) T/B Λ + ˆ Λ t,b Π , ˆ ˆ , 2 h 2 h Y Π 2 h Π q , T/B 2 2 T/B 2 s s 0 s 0 Y Λ Y , ) + ) , ) ) ) T /B , (4) (4) T /B (4) ν τ 2 h b + + 0) 0) + 2 h Y 2 h Y + + s ˆ ˆ ˆ , t/b , s s t/b M M M q T/B b ν τ T/B T /B T /B (4) l l (4) (4) q − Λ , m , m Λ Y Y Y − − Y − − , − ˆ ˆ T /B T /B ˆ T /B T /B 0 , 0 Π Π Π 0 Y Y , Y Y , 0 + + , , m , m , + + (1) (1) + (1) ν τ b 0 0 + (1 , m , m + (1 , m , m + (1 ˆ ˆ ˆ , , t T/B τ ν T/B M M – 25 – M 0 0 T /B T /B 0 0 (4) l l (4) (4) q T /B T /B , , Y Y , , (4) ν Y Y ˆ ˆ ˆ (4) τ (4) b Π Π Π , m ˆ 2 h , m 2 h 2 h T/B T/B Π ˆ ˆ Π Π s s s , m , m + + + , m , m T/B T/B T/B T/B t/b t/b + 0 0 0 0 + + , m , m 2 2 − 2 − − Q , , Q , , ) ) 2 ) 2 2 , m , m 1 1 , m , m ) 1 b ν τ t/b t/b m ) m ) t/b t/b L L ( Q ( b ν τ Q Q N T Q Q t/b t/b q B q t/b t/b 2 l 2 l q 2 q T/B T/B 2 ν d d d 2 τ 2 b d B d h h B d h b Q Q ν τ Q Q m m m m ∆ ∆ ∆ s ∆ s m m s ( ( ∆ ∆ L N L T B Q ( ( ( ( m m m m 2 2 2 ( ( R R ( ( M M L L 1 1 m m m m 1 m m ( ( √ √ ( ( ( √ ( A A B B B A A A A B ======c c b b c ν ν τ τ b ν τ Π Π Π t/b t/b (1) (4) (1) t/b (4) t/b M t/b t/b M M Π Π Π (1) q (4) q ˆ ˆ ˆ ˆ Π Π ˆ ˆ Π Π M M . ν → τ, b The same expressions apply for the leptonic form factors, with the substitutions → l, t with the SO(4) decomposed form factors given by Tau neutrino lepton: Tau lepton: Bottom quark: JHEP09(2017)049 (9) τ ˆ M + 3 (4) τ ˆ M 8 − (9) l (1) τ ) ˆ ) Π l τ ˆ , 0) , 0) , ν (6) M ) , ) , Λ Λ − ) 5 ˆ ν 5 l Π , τ , 5 2) 2) 2) / 0 0 / / Λ / Λ Λ / (4) l ) , , − , , , V V V )4 2 h )4 5 5 ˆ 2 2 2 Π s / / T / T / / ,Y ,Y ν (4) ,Y 2 h 5 , ˜ ˜ Y Y V 2 V T 2 ) 2 )4 s )4 ˆ ) Π / / 2 h Y / − τ 3 4 T T + + ,Y ,Y s T T ˜ ˜ T 2 h Y Λ Y 2 2 h 2 2 + Y T Y (4 s , T Y − / s / − l 0) Y + + 5 , Y T , , 2 1 T 4 5 T + + Λ + Y ) − i ) √ T T , Y T Y l τ 0) T 0) T 1 (9) l Y T (9) τ 8 + + ( + ( Y Y , T , Y , Λ Y Λ (4) Y + (1 ν , m + + ˆ Y ˆ ) Π Π , , T T T T 2 ˆ + ν ν (6) 0 0) , m 0 Y Y T Y Y T M / + ( – 26 – + ( (9) τ , m , m (9) l , , , , m 0 Λ Y 2 h Y ˆ , m 2 h Π V + 4 V 0 , ˆ + 3 V , s ˆ V T T Π T T s 0 Π (4) l , m , m , , m , m 0 Y Y Y Y , T + ˆ ,Y 0 0 , m 0 , m , Π 0 0 − − − + (4) l (4) τ T , m , , m , , , , m , 0 V V T 2 1 ˆ , m , m ˆ 1 , m , , m ) T V − 2 , m Π T T T Π T T T ) 0 0 (4) τ 0 0 0 8 8 L , m N q , , m , , , m , , , m q L ˆ 2 ν Π L 2 l (4) l d h , m , m , m , m , m h , m , m , m L L L L L m d − T T T − s ( ˆ s ∆ L L L L L Π L L L L N ∆ m m m m m m m m m ( 2 1 B ( ( ( ( ( (1) l m m ( ( ( (1) τ m m m m m m m m ( ( ( ( ( ( ( ( − ( ( √ R R R M L L L R R ˆ ˆ + 2 Π Π (9) + τ B B B B B 5 5 B B B A A A A A A A A A ˆ = = = M (9) τ (9) l ) c 2 ν ν ======ˆ Π ˆ ν Π ) − 2 h Π 2 h s M Π + (1) τ (4) ν (6) ν (9) (9) l (4) l (1) l (9) τ (4) τ s + τ (4) τ 5 − 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Π Π Π Π Π Π Π Π 2 ) ˆ M ) M − (1 T L τ 2 l 2 h d 5-5-5 14-14-10 d s (4 5 ∆ L T ∆ i 1 4 1 5 √ 3 m m ( ( 2 + + = = = c τ τ τ Π M Π with the SO(4) decomposed form factors given by Tau neutrino lepton: The quark expressions are as above. Tau lepton: A.2 MCHM JHEP09(2017)049 (A.4) (A.3) , ) l 2) Λ / , V , 2 ) / ,Y ν V 0) 0 , Λ , 0 , ,Y V , 0 0 0 , , , , 0 ) , m V , V τ 0 0 , Λ , , m 2 h L , 2 , m s l 0) (9) l (0 , (0 , m Λ (0 ) ( ˆ L R 5 Π − , τ 2) / B 5 B A A 1 (9) l / 2) ) − (6) ν Λ / )4 ν ˆ , / V Π ˆ l Π = = )4 T V Λ (4) l Λ ˜ ,Y , Y T − l ˆ , Π 2 ,Y + 4 ˜ (4) l (6) ν Y (4) l 2 Λ / 2 + ˆ ˆ / Π Π ˆ (4) ν , / Π ) T + (4) l T 2 T ˆ 2 h T Π Y ˆ / Y − Π s T Y Y 8 V Y + 2 h − + + (4) l + ( s − ,Y T , ˆ T Π + ( 1 2 Y T ) T T Y Y τ Y (1) l T (4) + + (1 ν Λ Y , m ˆ ) Π , m + , m , ˆ , m ν V , m l 5 0) V M – 27 – V (6) ν (9) l V 0 , Λ , m 2 h (1) τ Λ (9) l , 5 ˆ , ˆ s Π 0 , Π ) , , m ˆ l 2) , m Π ˆ / , m , ) T Π 5 2 h ) , m / T l 4 L Λ τ − s + / + 0) 0) T T T + (1) V , τ + , 4 , Λ , Λ 1 m 2 2 , m 2 p , 5 − ˆ ) 5 , 2 , m ( ) ) 2 / ,Y L M , m , m ) 0 / ν p T / 0 , m ) 2) L q V L 0 h V , 4 L L , M 2 l 2 ν 4 (1 T / T Λ L h L m d d , s 5 ), which permits any use, distribution and reproduction in ν 2 l 5 ,Y m 2 h ( , d s d V A L V ∆ V ∆ m m / ,Y 5 ( p / p 0 s m 0 ,Y i ∆ ( ( 2 T L ∆ , 4 0 B ( , 4 , T 1 4 0 T m m 4 B ,Y √ , ) − 2 ( ( √ M M , T l m m , m i B 0 0) p V p ( − ( / ,Y 0 , ,Y + T , Λ A A − T , T V 0 0 = = = V , 0 , m , , L = = = , m , 0 c ν ν = = = L ,Y , m ,Y ,Y , T 0 T L ν c τ τ m 0 , m 0 0 L , 0 Π τ m ( M , 0 Π , , , CC-BY 4.0 m Π 0 ( (4) (4) (1) M ν τ τ , m Π , m ( , m , 0 T L L B ( ˆ ˆ ˆ B L , L L L M M M M This article is distributed under the terms of the Creative Commons m m m M ( ( ( m (0 m m m A ( ( ( ( 5-5-5 14-1-10 A L R R R i B B B B A A A A − − ======(4) τ (4) ν (1) (9) l (1) l (1) ν τ ˆ ˆ ˆ ˆ ˆ ˆ Π Π Π Π M M Open Access. 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