The Strongly-Interacting Light Higgs

The strongly-interacting light Higgs

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Please note that terms and conditions apply. JHEP06(2007)045 , the . We ρ ρ m m , hep062007045.pdf May 28, 2007 June 13, 2007 April 23, 2007 osite Models. symmetry break- Accepted: low the scale Received: Published: and Riccardo ergy longitudinal vector c s only. Phenomenological [email protected] , terize these models: e a composite object of the alous Higgs couplings. We fi- om a strongly-interacting sector -Yvette, France sensitive to the new strong force 1015 Lausanne, Switzerland a, Barcelona, Spain Alex Pomarol ab http://jhep.sissa.it/archive/papers/jhep062007045 /j , their coupling. An effective low-energy Lagrangian ρ g Christophe.Grojean@cern.ch , Published by Institute of Physics Publishing for SISSA Christophe Grojean, a Beyond Standard Model, Higgs Physics, Technicolor and Comp We develop a simple description of models where electroweak [email protected] ad [email protected] Institut Physiques, de des EPFL, Théorie Phénomènes CH- E-mail: IFAE, Universitat de Autònoma Barcelona, 08193 Bellaterr CERN, Theory Division, CH–1211Service Geneva de 23, Physique Switzerland CEA Théorique, Saclay, F91191 Gif-sur SISSA 2007 c b d a c ° approach proves to be useful for LHC and ILC phenomenology be Abstract: identify two classes of operators:and those those that are that genuinely areprospects sensitive for to the LHC theboson and spectrum scattering, the of strong ILC double-Higgs thenally include production resonance discuss the and study the anom of possibilitystrong high-en that sector. the top quark could alsoKeywords: b ing is triggered by a lightas composite Higgs, a which pseudo-Goldstone emerges boson. fr Two parameters broadly charac mass scale of the new resonances and Rattazzi Gian Francesco Giudice, The strongly-interacting light Higgs JHEP06(2007)045 2 4 8 1 2 16 18 30 22 31 34 11 29 boson, ruling out, for Z eak symmetry breaking. ictions showed that the ections, received a further EP1 has provided us with hether the dynamics respon- ng to a weakly-coupled Higgs f the ere viewed as the prototypes of s. Indeed, the good agreement best agreement between experi- d to be the most realistic realiza- R t – 1 – 2.1 Definition of2.2 SILH Constructing the2.3 effective action The SILH effective lagrangian 3.1 Holographic composite3.2 Higgs model Littlest Higgs3.3 model Little Higgs model with custodial symmetry 20 5.1 Phenomenology of a strongly-interacting instance, the simplest forms ofa technicolor strongly-interacting models, electroweak which w sector.ments and Moreover, theory the was obtainedself-interaction. for a Finally, light supersymmetry, Higgs, which correspondition appeare of a light Higgs with mass stabilized under quantum corr 1. Introduction The main goal of theA LHC crucial is issue to that unveilsible experiments the should for mechanism be symmetry of able breaking electrow convincing to is indications settle is in weakly w favor orof of strongly weakly-coupled precision coupled. dynamic measurements L withnew dynamics the cannot Standard significantly influence Model the (SM) properties pred o 2. The structure of SILH Contents 1. Introduction 3. Relating the SILH to explicit models 5. Strongly-interacting top quark 4. Phenomenology of SILH 6. Conclusions A. Integrating out vectors and scalars B. Effective Lagrangian in the canonicalC. basis Loop functions for the Higgs radiative decays 36 37 JHEP06(2007)045 ρ m the SM gauge and a scale ρ henomenology of SM g g k of discovery of a symmetry into fine- scale. There has been marize our results and SILH and construct the eaking [1 – 6]. Still, the nsible for EW symmetry namics at a higher scale, found to be in accord with M properties in Higgs and by the connection between g from new dynamics [7, 8] interactions will help, in case Higgs. Then we describe in . In section 5, we extend our cular model realization. We rovided by direct production ak data. A more promising tive particle emerging from a then our tests can be used to c models previously proposed geometries, have led the way eaking. If no new states are , a coupling wing. In addition to the vector eracting dynamics seemed hard belong to a strongly-interacting indicating by ely missed by direct searches at ight Higgs (SILH). Of course, in ttle Higgs [2], Holographic Higgs our model-independent approach vation. Moreover, new theoreti- s with the SM fields. In section 3, – 2 – In this paper we want to study the general properties and the p The situation has swayed back after the LEP2 results. The lac This paper is organized as follows. In section 2, we define the Higgs boson belowtuning 114 GeV territory, or partially undermining ofcal its any developments, mostly original new influenced moti statesstrongly-interacting by gauge has extra theories dimensions forced and and to super gravity the on construction warped complete of replacement new of the models Higgsto of sector implement, with electroweak mostly strongly-int symmetry becauseapproach br is of to constraints keep from thewhich Higgs electrowe becomes boson strong as atvarious an attempts a effective to scale field realize arisin not suchas scenario, much Goldstone including larger bosons the than Li [5, the 6] Fermi or not [9], and other variations. supersymmetric unification. scenarios in whichfocusing a on light features Higgs thatwill is are refer associated to quite this with independent scenariomany of strong specific as the to models, dy parti the the Strongly-Interactingof best L experimental new signals states, will whilelongitudinal be here gauge p boson we processes. concentrate on Still,is useful. we deviations believe The from that tests S weof propose here new on discoveries, Higgs to and establish gauge-boson sector if ultimately the new responsible particlesobserved, for indeed or electroweak if the symmetry resonancesinvestigate br are whether too the broad Higgs is tostrongly-interacting weakly be sector, coupled identified, whose or is discoverythe an has LHC. effec been bar low-energy effective theory that describeswe its discuss interaction how this effectivein Lagrangian the is literature, related like tosection 4, the specifi how Holographic the Higgs SILHanalysis can and to be the the tested Little in casedraw collider of our experiments conclusions a in composite section top 6. quark and2. finally we The sum structure of SILH 2.1 Definition of SILH The structure of the theoriesbosons we want and to consider fermions is of thebreaking, the follo which is SM, broadly there characterized exists by a two parameters new sector respo boost by the LEP1 measurements of gauge coupling constants, describing the mass of heavy physical states. Collectively JHEP06(2007)045 . ρ 2 ρ × m 6= 0 m . In ) (2.1) ρ ρ π g 4 SU(2) /m / precisely × ρ SM = 0, (1) and the g α U SM × by the equation g coset space of a ρ in the fundamental nd by their quartic m H / and the mass scale of limit SU(2) G . In that case eq. (2.1) amely interactions that f / and π ring represent a “weakly 4 ρ 3) imal possibilities in which g ng sector. The SM vector ably reveal new physics in the top quark Yukawa), we ∼ ffective) action that are not 1, we are interested in the tant possibility is represented by hierarchy problem, it should ρ will be the key signature of of a heavy Higgs boson, using an ed to scale like ( he new sector as “the strong Higgs complex doublet spans M Yukawas. ses involving the Higgs boson y means of the SU(3) , the pure low-energy effective g that is to say below 1 ρ NDA) [10]. Because of the first π 4 to m < ρ corresponds to a maximally strongly- g . However, once the SM couplings are 4 is related to π , which is parametrically lower than the 4 Φ ρ f 2 ρ f . ensures that the loop expansion parameter g ∼ m ρ ρ g ρ + non-linear theory. In this case the role of g g 2 -model can become before it is replaced by a = – 3 – σ Φ H ρ 2 ρ / G m m − and the non-zero Yukawas explicitly break the Gold- -model scale = σ Y V on the coset space. In particular a mass term for the Higgs (1) 1 SO(4). U G / × -model would become strongly coupled. The coupling is an exact Goldstone boson, living in the . The upper bound on σ π 4 H < ∼ ρ g < ∼ is less than unity, while the limit where the 2 -model UV completion of the ) SM σ g π is played respectively by the mass of the heavy scalar modes a SO(4), a simple UV completion could consist of a real scalar Φ πf gauge coupling and by means of proto-Yukawa interactions, n 4 / 2 ρ / Y g ρ A second crucial assumption we are going to make is that in the The gauging of SU(2) As we shall explain below, the Along the same lines we could even describe the SM in the limit g 1 ( (1) -model description breaks down above a scale ∼ and Yukawa couplings (basically the weak gauge coupling and assume inequality, by a slightsector”. abuse of The language, Higgs webosons multiplet shall and is fermions refer are assumed weakly to coupled to to t the belong strong sector to b the stro coupled theory in the spirit of naive dimensional analysis ( U in the low-energy effective field theory will give rise to the S the Higgs doublet spontaneously broken symmetry ofthe the complex strong Higgs sector. doublet Two spans min the whole coset space are SU( custodially symmetric SO(5) the various realizations ofmodel-independent effects, SILH. which Here, couldand/or as be longitudinal stated visible gauge in in bosons,the proces section electroweak and breaking which sector. would unmistak coupling. For instance, inSO(5) the interesting case in which the Observation at the LHC of the new states with mass σ and stone symmetry of the strong sector leading to terms in the (e invariant under the action of of SO(5) and quartic potential, interesting models, the Higgs mass parameter is thus expect measures how strong themore coupling fundamental of description. the Thea simplest linear example of this is generated at 1-loop.soften If the the new sensitivity dynamics of is the addressing Higgs the mass to short distances, Fully strongly interacting theories, like QCD, correspond scale 4 the QCD states. Oncoupled” the deformation other of this hand, QCD-like the pattern. theories For we are conside expresses the usual NDA relation between the pion decay cons JHEP06(2007)045 (2.2) (2.3) ow the In these . We have (the quartic 2 . Moreover . R λ ρ 5 √ m ). Nonetheless = SM ρ g g one parameter, the , i ≫ H h eakly-coupled boundary slice of AdS = adratic corrections to the ds. Since we are assuming ggses [1]. There, the scale r is composite so that the 3 upled bulk dynamics. Ex- ctroweak vector bosons and f ” mass scale is more accurately described -breaking sector corresponds ρ e Higgs mass. In Little-Higgs we now want to derive the form g ul. d coupling play respectively the . ρ ion to ultra-high scales. The same low- ) to strong ( m SM N . g ) is close in spirit to recent studies of two (and es, our paper focuses on the low-energy effects e of ρ ace by turning on the suitable boundary terms, per is however complementary to that refs. [12]: ≡ ∼ , he Higgsless limit). ,g ρ N π πR m 4 √ = – 4 – = Λ . To be explicit, consider a 5-dimensional (5D) ρ 2 g N ¶ ρ gauge theory, we also expect the hadrons to interact π g 4 . These relations then imply 2 5 N µ . This is also basically the picture that holds in extra- compactified on a circle or orbifold of radius /g 5 N 2 g ) -model, where weakly-coupled states appear below the naive ). On the other hand, according to 5D NDA, the physical π σ (4 πR ( ∼ / 2 5 g ≡ , while the number of weakly-coupled Kaluza-Klein modes bel 2 ρ ρ g g of the new states, characterizes this “deformation”. and and . It is useful to focus on the simplest possibility where just -model involving only 3 (and not 4) Goldstone bosons, with ρ ρ g σ πf /R m A more interesting possibility arises when the strong secto Other models that basically fall into our class are Little Hi Summarizing, in several models of interest the electroweak = 1 is represented by the masses of the partners of top quark, ele Notice that the AdS geometry only matters for the extrapolat Our simplified approach based on two parameters ( 2 3 ρ ρ m cut-off of the model is Λ gauge theory with 5D coupling cut-off can be basically interpreted as extra-dimensional realizations, the Kaluza-Kleinrole mass of an dynamics while the Higgsamples sector of is this part type of are a the more Holographic strongly-co Goldstones [6] over a dimensional constructions where the SM is represented by a w energy dynamics of ref. [6]along could the also lines be of realized ref. over flat [11] sp (although ref. [11] focussed on t which becomes weaker at large analogous turned on, such a limited UV completion fails to screen the qu Higgs mass. corrections to the Higgs mass are screened above the “hadron Higgs, the states thatmodels soften there the is more quadratic parameter correction freedom, to and th the coupling m Higgs coupling), and the physical Higgs mass playing the rol while those studies focus onin the Higgs physics and of the vector new boson heavy interactions. stat three) site models [12]. The phenomenological goal of our pa if the underlying theory iswith a a coupling large- by a set of couplings that can range from weak ( we shall still find our simplified characterization very usef to a “deformation” of acut-off pure 4 coupling 2.2 Constructing the effective action Under the assumptions of the class ofof theories the defined most above, general effective Lagrangian for the SM + Higgs fiel JHEP06(2007)045 , ρ ≡ ρ /R m . and 1 Π m (2.7) (2.5) (2.6) (2.4) # = A µ ile the ≡ s ∂ ←→ T . . . ρ M m e covariant )+ ρ = 0, and to turn A and Π . , we parametrize ,∂/m G ¶ . SM Φ g ¸ H , in principle there will be µ U, ( · · · H . If the structure of the roup ←→ D d then integrate out the ρ † he leading two-derivative (2) general form of the action s with mass of order m must be he identification rate on this case. Later we H at the structure in eq. (2.5) as well as the leading self- L 4 ρ der terms in the low-energy mions. Indicating by Π) + ) ated in” for purposes that will 4 ρ rms of all orders in derivatives µ π ¶µ . g bove structure). Moreover this ∂ /m ←→ cal Lagrangian involves at most A (4 H ng leading (dimension-6) interac- cription of the string. For instance µ T A ←→ D )+ † Π)(Π Π ρ µ H ≡ ∂ ←→ µ , then the presence or absence of the Φ’s Π 2 , we can make the identifications: (Π ,∂/m T f s (0) c 3 1 2 Φ L · · · /M – 5 – + U, 1 ( Π+ ´ )+ µ ∼ (1) µ H ∂ † L Π D Π i 2 µ H e µ ) 2 ρ ∂ ³ π D g = · µ (4 ∂ U Tr ( Tr of radius transform as a reducible representation of ´ 2 2 6 A )+ f f H ρ T T † ≡ = H 0 ³ L µ ,∂/m 4 ρ ∂ 2 ρ Φ 2 g m H f ) (provided the power divergent loops are computed by NDA, wh U, c 2 ( . πR (0) π Π)Π. Once we interpret Π as the Higgs doublet and include gaug ( 2 L µ / / 4 ∂ 2 " 5 for each quadratic invariant. 2 ρ ( g g 4 ρ is the Goldstone combination defined in eq. (A.3) of appendix f 2 ρ , it makes sense to focus first on the strong sector in the limit − g = m µ = SM 2 ρ D s Π) = g g In order to get the truly low-energy effective action we shoul µ respectively the broken and the unbroken generators of the g When the coset generators > g 4 ∂ L a ρ shall discuss the moretwo realistic derivatives: situation where in theaction that crucially classi case depends the onterm structure the defines quantum of numbers relation the of (2.1)interactions higher-or the for Φ’s. the T Goldstone decay constant has no impact on the low-energy theory. We shall first concent terms in eq. (2.5) isappear the already most in general one, the in classical particular Lagrangian if te and Φ’s and also include the quantum fluctuations at scales below In the action we have keptbecome massive degrees momentarily of more freedom clear. “integr Oneis can obtained for in instance the check compactification th of a 5D gauge theory with t g log-divergent and finite piecessame automatically structure characterizes satisfy the effective the field-theoryin a des type I compactified on a and on later the couplings of this sector to the SM vectors and fer Here a different Π( derivatives, we obtain that eq.tions (2.6) describes the followi T the Goldstone field by the matrix In addition we assume the strong sector features a set of field which we collectively indicateincluding by quantum Φ. fluctuations from By scales our shorter assumptions, than the 1 JHEP06(2007)045 = an λ = 0, (2.8) f the a (2.10) H /c and T c (with = 1. /v ) (2.9) f . 2 H is present, and µν m /f µν ρ -model must be /c 2 B α B σ T and quartic Higgs ν m √ ) c H ∂ 3 both in the cases f ) ( H / y = ν H ¶ responding renormaliz- † ] is weighted by a factor f D gs . H ν y ( = 4 coefficients of the higher- of the H 3 † µ unting for Higgs field and volving Higgs bosons and ( er than ) ´ SO(4) one finds gses and a pair of photons a , D H ←→ D † H µ / the various possibilities. For H uging of the SM group: /c µ † + wa terms and Higgs potential. . Using the CCWZ construc- y rules 1 and 2. Basically, the . For instance the addition of D 6 H xpects these breaking terms to in terms of the two appearing [ c D an H α i µν µ (1) one finds ructure of dimension-6 operators ( /f 2 ³ i i . | − G H -model structure, up to the overall U 2 . When the SM subgroup is weakly λ 2 σ H = = ρ µν = 6 × f µ c → B is in order; this same rule implies that 3 and discussed before induces the operators D µν /m H/f / | HB − O † α HG µ . For SO(5) F 2 1 † . O f H D ¶ H H 2 − † . SU(2) H /f c α . ≡ / H /f = i = 2 ) µ H v ) g H – 6 – +h µν iA O i H µν R /c (1), with the definitions † W + y are fixed by the ν W c U H µ ) Hf ( D T ∂ and ( × L c a H ¯ f ν ¶ → + µν H D µ H † α B and ( i µ ∂ H H σ µν SU(2) H is broken at tree level by the weak gauging of the SM group ←→ D † f c i / ) 2 y → G σ y f † HB H c α † µ H µ H H D µ ( i i = and of the flavour symmetry of the SM control the overall size o = = G BB ). W λ O HW O O -model contributions give . SO(4) and SU(3) σ 2 ρ ), valid up to corrections of order / 2 /m v coupling suppressed by the same (weak)able coupling interaction associated in to the the SM cor Lagrangian (e.g., Yukawa couplin 1 two Higgs doublet legs involves the factor each extra insertion of a gauge field strength gauged, the replacement (2 The global symmetry From eqs. (2.5)–(2.6) we can deduce the rules to estimate the Of special phenomenological relevance are the operators in / 3. Higher-dimensional operators that violate the symmetry 2. Each extra derivative is weighted by a factor 1 1. Each extra Goldstone leg is weighted by a factor 1 2 H For instance, the shift symmetry breaking terms, whilederivative rules insertions. 1 We and can 2 thus determine formulate the rule co 3: simple expressions of the Goldstonesatisfy field the are same involved, one field e selection and derivative rules expansions of expressed b In section 3, weour shall present present goal a we more just detailed need to analysis remark of that all if no new scale oth and by the weak interactions that underlie the origin of Yuka in eq. (2.7). The coefficients because of custodial symmetry, and for SU(3) normalization which depends on the definition of dimensional operators in the low-energy effective Lagrangi The pure Here we have made a Higgs field redefinition appropriate constant) to write the operator of SO(5) m gauge fields, and inor particular gluons those like involving one or two Hig tion [13], we derive, in appendixinvolving A, Higgs the and following general gauge st field strengths, arising by weak ga JHEP06(2007)045 . . ) i B is 5 σ W O and BB , and µν , i µν O B W µν W EM O µν α G ≡ µν (like under which HB µν † γγ G W HG H † l photon to two of → and a milder growth H = ρ h h extra suppression. T = m BB 2 ρ g , where 1, and all vectors are O O rms, the couplings that e classical action. Minimal /g H hese theories correspond ≤ can instead be generated ) not interact at tree level 2 SM vative operators like those µν B g in ives. Holographic Goldstone e of ect we remark an interesting s like er suppressed by a Goldstone B O e equal to 2. In these theories d upon the simple gauging of mal” theories, the classical ac- + f heavy vector fields. We show is both charge and color neutral, , it is also manifest that none of or In the case of minimally-coupled vanishes at leading order in rs in eqs. (2.9)–(2.10) by µν (1) generator h 6 . Notice that, using integration by 2 W µν W U 2.9)–(2.10) can appear in the classi- ssive vector fields through the requirement F ( O 2 0 HB π µν e. For instance the gauge symmetry breaking or those leading to of ghosts at the scale O . . According to counting rule 2, all these (with real photons). On the other hand, q g W m † γγ and also through the chiral anomaly. On the other O γγ q and H → m . Two linearly independent combinations of and → and bear an extra one-loop suppression. By = h , as they formally involve two extra covariant – 7 – 2 ρ HW h HB . EM (1) ρ O ZW must intervene, so that their coefficient must be α /m L ). Normally one gets a , O m O ρ h . The operators T ρ /g does not break the HW from should also arise at one-loop and moreover, because of and /g O Q SM ) and to a correction to the gyromagnetic ratio of the only if there exists a field Φ with the appropriate quantum g Zγ H SM BB (1) Z ) ρ contribute to g and O U → m µν × h B B ZW c + and O , O g µν W O nor W O ( µν ZB contribute respectively to a vertex that couples an on-shel B O exists at zeroth order in both † counts like a four derivative object and is discarded from th HB must therefore appear in H 2 µν 2 O F ) = − µ HB V π µ + and ZB ∂ π , O O According to the general expression in eq. (2.5), four-deri The latter property sets the rule to count derivatives for ma A similar result holds in low-energy QCD. The coupling 6 5 HW HW of the amplitudes at energies above the scale coupling along with the gauge principle ensures the absence term ( gets generated at subleading order in the quark mass that the action for the eaten Goldstones be a 2-derivative on hand, there are contributions to While, expectedly, operators involving gluons do not arise these operators contributes to the process parts, we could equivalently parametrize the set of operato and Again, neither theories, higher-derivative operators like thosecal in eqs. low-energy action ( below O associated to (spontaneously-broken) gauge symmetries. On the other hand,with neutral in states minimally-coupled and all theoriesO gyromagnetic photons ratios are do fixed to b numbers to mediate the correspondingdifference operator. between In this resp of operators affecting the coupling between Higgs and photon operators have a coefficient of order 1 derivatives with respect to a Higgs kinetic term. The absenc neutral states (a Higgs and a due to the Goldstone symmetry. Since the neutral Higgs explicitly break this shiftthe symmetry SM and cannot group be describedbreak generate by the rule symmetry 2. generated In by order to generate these te suppressed by extra powers of ( in eqs. (2.9)–(2.10) cantion arise including at the tree heavy level. fieldsmodels However Φ in and involves “nor Little at Higgs mostto two are minimally-coupled derivat of field this theories type. where To the be states more have specific, sp t the gauging of just SU(3) the same argument the previous symmetry-based argument, theysymmetry-breaking should power be of furth in minimally-coupled theories by the tree-level exchange o the physical Higgs boson shifts. Operators like JHEP06(2007)045 . 2 ) ρ (2.11) (2.14) (2.12) (2.13) a ) πm Of course ρν minimally- (4 7 G / ρ 1 D ∼ ( a doublet. By the ) L µν . At first glace we would G ρ µ rs in eq. (2.13) can be in a minimally coupled m D , [14] of all the high-spin states nimally coupled classical w-energy dimension-6 ef- ≡ rs involving only covariant ator involving two Higgses g 2 n SU(2) groups and the Holographic i = ( α cients are therefore in general ors transforming respectively minimal coupling. are minimally coupled, in the odel with a heavy vector that her-spin (higher-derivative) theory , as opposed to the genuinely KK g ) ct corrections from the strong 2 π R π m 4 O f 4 ∼ L . < < s F ) ( ρ . ρ gnetic ratio M c ρµ ) f g ρν g y t G oton can scatter off a dilaton, a neutral scalar, ded it exists, a realization of the Holographic H B 2 ρ + b νρ ∂ D α G )( ν )( † a µ µν HH G H B – 8 – † 2 = 2, indicating a close similarity to a minimally-coupled µ abc ∂ D g f ( λH 2 ρ = = ( 1 + contributes to the magnetic dipole and to the electric m g α B 3 2 W H O 3 O 2 H O , for which all loops are equally important in the spirit of π m i 4 ) h . 2 ρ ρν ρ ∼ 1 k ρµ m m W ρ ρ g W . They are thus generally expected with a coefficient D j νρ ( i . The two operators in eq. (2.14) cannot arise at tree level in W ) W 2 ν ) µν i ρ µ W m W µ ρ g ijk D ( ǫ / = ( = W W At the dimension-6 level, there is one last independent oper For completenes we should also list the dimension-6 operato 2 3 One may wonder how our results would change in a genuinely hig 7 O O strongly coupled case As we show in thetheory appendix by A integrating this out can a be massive generated at scalar tree transforming level as a coupled theories. For instance like string theory. To be specific, one could consider, provi Goldstone in weakly-coupled string theory and take the limi NDA. and four covariant derivatives As we show ingenerated the at appendix tree level A,as through see a the weak eq. exchange triplet, (A.11), of as aof massive the singlet order vect and three 1 as operato a color octet. Their coeffi resembles both the Little HiggsGoldstone. models Given with that product all of the gauge known examples with this explicitly in appendix A, in the context of a simplified m corresponding to effects that aresector. all subleading to more dire derivatives and field strengths expect a drastic change.at For instance, tree it level. is On obvious the that other a hand, ph a specific study of the gyroma following of this paper, we will work under the assumption of this makes a difference when we consider models at equations of motion this term can, however, be rewritten as quadrupole of the 2.3 The SILH effective lagrangian We now basically havefective all Lagrangian. the We ingredients willLagrangian to work at write under the down scale the the assumption lo of a mi of the open string remarkably gives the result theory. JHEP06(2007)045 n- (2.15) (2.17) (2.16) g must at 95% , as we to zero. cations, 3 H T c − c ) at 95% CL). 10 µν µν 3 × ¶ B B − ) ν 3 H . ∂ violates custodial 10 H µ 1 ( . ν T × ¶ D D ←→ c aµν ( 9 † cally normalized, and ξ < H . . † ). Let us start with the G 1 ) µ T , or, which is equivalent, H acceptable. In the next rangian with coefficients 2 ←→ D H a µν with coefficient rgy effective action for the † µ < c /f receives a model dependent ve the vector bosons. The ξ < ¶µ ¶ α H . 3 D H in Higgs and vector boson T T HG c ( H y discussed, by using the Fierz − ional to . µ † c H he form ′ 2 this operator and set µ endent operators involving four ′ ) ggs scenario this corresponds to g f 2 ρ 10 H s. Also, we recall the definition < c g ←→ D = 246 GeV 2 H 2 +h 2 t ρ † B 3 m † × π 2 g y HB 2 − / R ic H 1 2 H 1 ic 16 . µ f 10 − 2 1 S + 2 Hf g 2 +( ´ + i − × g π L T ) f α F ¯ c c f i 2 7 µν . 16 G µν H parameter H 1 2 is turned on, W + † W + ρ ) in various models. − , can be written in terms of a combination of √ → ν ´ H 2 ³ | H T – 9 – SM α µν f D . This is because this term purely originates from H c ν g 2 ( y † H 2 ρ = B H y f D µ ¶ c ( H /g µν to the i D 2 ³ f µ | H σ y b µ T † µ ξ, H ) + ∂ HB T † ←→ D † . 3 i c H ´ H ´ σ µ H , v H † = H ) 2 2 ρ 2 2 † H D † g g † b H ( v f T H 2 2 H SO(4). When g µ H 2 ³ f f µ ′ ≡ ≡ / ³ g suggests that new physics relevant for electroweak breakin 2 2 2 ρ µ g D ξ ρ π π γ HW λ ∂ is universal at leading order in the Yukawa couplings, and no ( 2 W T m c 6 2 c f 2 ∆ ic are pure numbers of order unity. For phenomenological appli 16 16 c ic y H − f c i c 2 + − + + c to vary one finds instead H µ b = S D † by a Higgs field redefinition H SILH y . The third operator L ≡ , c T 6 c H , c µ T In what follows, we will comment on the operators in eq. (2.15 The coefficient Using the rules described in section 2.2, we obtain a low-ene D and ←→ fields and two covariant derivatives. Two are shown in our Lag , c † H H Indeed, the bound on contribution, which should besection, small we will enough briefly to discuss make the the size model of CL (letting also Because of this strong limit, we will neglect new effects from We will later discusscoupling the constants Lagrangian terms that purely invol be approximately custodial-invariant. In ourassuming Goldstone the Hi coset SO(5) operators involving more than two Higgsidentities fields. for the As previousl Pauli matrices,H one can write three indep From the SM fit of electroweak data [16], we find universal effects will appear at order we have switched togauge couplings a explicitly notation appear inH in which covariant derivative gauge fields are canoni will show in sectionscattering 4, plays at a high-energy crucial colliders. role in testing The the operator SIL proport c leading dimension-6 operators involving the Higgs field of t symmetry and gives a contribution c by using the leading order equations of motion. The operator JHEP06(2007)045 . , ) p 2 ) (1) ggs nge ρ and /f L 2 upled 2 /g (2.19) (2.18) ′ , these v 6 TeV). g ( 2 ρ . SM 1 O g /g 2 2 happens to )( / π 2 1 B ) . Notice that c 16 /f B 2 c + SM ∼ v . Moreover, while g + contributes to the W N c are suppressed by an W B HB ∼ c c c d quarks by ppear only for sizable g ], we obtain the bound are like a Higgs mass ( ρ c ). Therefore, in weakly- g g > ∼ H t and O and ρ ction. Although they are and are subleading with respect /c , unless ower counting rules we then 2-to-2 scattering amplitudes m γ W is of order ( ρ c c HW g and g breaking without excessive fine c W,B ned in the previous section (see W,B c c c theory where )( BB 2 times the trivial factors of ρ . give only subleading effects. Since, , O N and 2 4 /g 2 ρ 2 W ´ ρ γ 2 originate from the 1-loop action c m ρ g . We have here assumed this simplest π W,B m . Notice that according to this counting g 2 ρ 4 c /m ) ³ HB appearing in eq. (2.15). More precisely, for is ( m 2 H B /N are generated respectively by tree-level ex- might be a promissing direction [15]. c ) c 2 ρ 2 B m H c B π c N + /g c 5 . + and . 16 – 10 – 1 W y 2 SM / c c g W and c 2 SM HW g = ( c W ( < ∼ c b S ξ ∼ 1 coincides with the usual technicolor result. Recently, it -model matrices. Indeed, the field redefinition mentioned 2 H ∼ σ ξ m defined in eq. (2.17), this bound becomes ξ ), the two contributions are comparable but, in strongly-co , they cannot be enhanced above their 1-loop size by the excha of the g B with respect to those proportional to 5 TeV at 95% CL. (this bound corresponds to assuming a light Hi p ∼ , c . H ) , in most cases the contribution from 2 ρ ρ . W g c 2 H /g / ), which for ), the operators proportional to > g c 1 2 g ) ρ π SM g B g = 0; by relaxing this request the bound becomes c ≫ (16 b ρ T . Then eq. (2.19) requires a rather large value of indirectly correct the physical Higgs coupling to gluons an + g ξ y is defined in ref. [16]. Using the SM fit of electroweak data [16 ≡ W c c Nξ/ terms, like b S ρ ( 2 4 g The operators with coefficients The operators proportional to As discussed in section 2.2, the operators proportional to A linear combination of the operators with coefficients > ∼ and D parameter of electroweak precision data: . In the simplest models ∼ 2 ρ 2 3 H b b be accidentally small. the non linearity in As we show invalues section of 4, new effects in Higgs physics at the LHC a where above precisely generates this universal from the point of view of the Goldstone symmetry, terms are down with respect to the others by 1 Their effect is then important only in the weakly coupled limi extra power ( under our assumptionH of minimal coupling for the classical a of any spin 0 or 1 massive field. In the case of a large c m and ∆ In terms of the parameter with respect to the SM, the direct contribution of term with extra fieldexpect strength their insertions. coefficient According to to roughly our scale p like g possibility, which accounts for the extra S has been pointed out that walking at small change of a massive triplet and singlet vector field as explai also eq. (A.11) inwith appendix respect A). to Their the operator relative proportional importance to in coupled theories ( S theories ( as we will show intuning section prefer 3, realistic models of electroweak to the one from JHEP06(2007)045 ss a ) ρν (2.21) (2.20) ct (or, G ρ D contribute ( a ) B 2 c µν ). The Georgi- ρ G . µ m B and c D ( W 2 ρ + ension-6 CP-invariant 2 3 e-loop effects involving 2 e to the effective La- g c m . W g 2 ρ c 2 viewing these models we g normalized fields, and in- ts uark and its explicit form ection 2.2. The operators c , 2 ρ 2 W ian for the vectors: erator involving Higgs and 2 ese operators can be rewrit- m m s, while Little Higgses belong Given that the experimental zation of the terms that ex- − 2 ρ 2 is fully saturated by quantum c ρµ ) ′ s. We can broadly distinguish g g G ρν es higher than B b νρ B 2 ρ c G ∂ ν = a µ ) for the operator involving photons )( 2 t [16] G µν y ( as the coupling of the largest contribution B Y , in the moderately strong coupling regime abc we refer to strong-sector states with mass µ 2 f b S g ∂ 2 ρ SM ( 3 3 g L,R and 2 ρ 2 m g ′ = 2 g – 11 – g . We have already discussed in some detail the m 3 π 2 ρ W G B c , Y g 2 are weaker than those on 2 SM 16 c 2 g 2 ρ 2 W B + − 2 m i . This result, albeit in a different basis, agrees with m c 2 ρ ) 2 y g arise from virtual tree-level exchange of massive vectors, c g k ρµ ρν g 2 W and W W c 2 c ρ j νρ W D 2 = ( W c and i ν ) i µ W B , while in the second the quartic coupling is a tree-level effe µν 2 induced by the weak gauging of the SM group. Here we will discu ρ W c W G are comparable to that on , m µ ijk ǫ W Y D 2 2 ρ ( c 3 the SM fermions while by Ψ 2 ρ 2 g m 2 g and m W π 2 ρ 3 W L,R In this class of models, the full Higgs potential arises by on c g 2 f W 16 c 2 + − , the constaints on = . ρ The Lagrangian terms in eqs. (2.15) and (2.20) include 14 dim Finally, for completeness, we give the dimension-6 Lagrang > g m vect ρ L SM particles. The dominant contribution is given by the top q Class 1. Kaplan model and Holographic Goldstones areto in the the second. first clas dicate In by what follows we∼ shall work with canonically equivalently, it arises from quantum corrections at energi the various possibilities fortwo the classes Yukawa and of Higgs models: coupling effects in at the first the class, scale the Higgs potential explicit breaking of operators involving only Higgsfermions, and gauge the fields one plus associated a 15th to op g to the electroweak precision parameters By using the equations often motion for as the contact SM interaction fieldbound among strengths, on th the electroweak currents. grangian (2.15) below thewould masses like to of give theplicitly a new break synthetic the states. but Goldstone comprehensive Before symmetry characteri re ref. [17] for the same class of operators. 3. Relating the SILH to explicitIn models this section, we consider a few explicit models that reduc phenomenological purposes, we have chosen in the corresponding SM loop, i.e., where the coefficients areproportional dictated to by the arguments given in s (gluons), respectively. from the second term in eq. (A.11) of appendix A. The coefficien JHEP06(2007)045 ons (3.2) (3.3) (3.4) (3.5) (3.1) global 5 1) while the quantum num- ∼ lings of eq. (3.5) e Higgs potential ξ is needed in order ally distinguish two ξ have definite quantum (i.e., is thus predicted for the pect to the quartic, thus This case is realized when f ve the SM fermions couple L,R t , in this type of models, it O m t is the one implemented in breaking of the SU(3) . st one corresponds to models i ∼ nt form than the one from the e to the interaction ) qualitatively corresponds to the are the SM Yukawa matrices. In H . , h f b c S . H/f y , ( universality will come only at higher ) ) determines the universal coefficient ˆ V y 2 +h c , H × L ( ξ , H/f S somewhat less than maximal we must tune 2 O ( O 1 has the size of an electroweak loop. (By 2 , the more tuning on t ρ O π y R 2 ρ ¯ ρ g y 2 SM f R g g g ∼ 16 f R L y H P ξ ¯ – 12 – × ∼ f breaking couplings, possibly involving only the R f + 4 ρ f b 2 S ρ . These models must therefore largely satisfy our y G L g R b ¯ m S and f O f π L ∼ y ¯ 4 f ) L ). An upper bound around 2 ∼ y H ( π ρ g in order to have a Higgs mass above the experimental bound. V 16 1. In order to achieve that, we clearly need extra contributi / , the generic prediction for 2 ρ 2 SM < ) g g ρ ( 2 2 t ) y ≫ π . On the other hand, for π/g 4 ρ / ∼ g SM (4 ρ g g , and therefore eq. (3.5) formally determines the spurionic λ ∼ G ≫ . ρ N ξ . The possibility of generating Yukawas from the linear coup g are matrices in flavor space. In the simplest cases, . By simple power counting, the top-quark contribution to th technicolors.) Then the smaller is some operator of the strong sector, while is a polynomial whose expansion at f L,R y S y N L,R y P O y The second possibility to generate Yukawa couplings is to ha of the effective Lagrangian (2.15). Violations of by an amount ( y to satify the experimental bound on assumption ξ to the potential, associated to other order in Higgs quartic is c where Then the generic prediction of this class of models is case of Higgs mass, in someis analogy mandatory with that supersymmetry. Notice that has the form interpreting depends on the origin of the top Yukawa coupling. We can basic possibilities to generate thewith minimal Yukawa couplings. flavor violation [18] The in simple which the only source of which for maximal strength Notice also that numbers under heavy states. Provided thetop extra quark, contributions we have differe maysuppressing tune a little bit the quadratic term with res linearly to fermionic operators of the strong sector: where was first suggested in ref. [19] for Technicolor models, and i where bers of flavor symmetry of the SMthe are the fermions Yukawa matrices couple themselves. to the strong sector bilinearly the low-energy effective theory, the above term will give ris JHEP06(2007)045 , R rs y and (3.8) (3.6) (3.7) (3.9) L y eq. (3.6). , while . The top b T y dependent) c eak symmetry . Obviously, 2 ρ For the same 10 H /g 9 2 t , and therefore a y t . rsal ( ¸ πy R 4 3 x transforming as an object Ψ ) L . By going to canonically π a couplings are generated 2 ρ have the form ¯ 4 Ψ let under the global group coefficient at leading order n of the physical states of / sh. The allowed values for instead of /g ρ y 2) under the custodial group ed by some little tuning of g ρ . c )+ , ( on to terms of the form (3.3), ) 2 L,R should be considered as part of /g ξ , fic model at hand. Depending on the y are expected to be t ibution eq. (3.3), which has a minimum ∼ 2 ρ which is left massless by the second and y R 4 g b R H/f T = (1 f H/f R 2 ( λ y ( c π R The result however is that the Higgs mass is R ˆ or V N assless composite. P O 16 and . L ential terms eq. (3.8) align to the wrong vacuum L × and Ψ f ) non-universal corrections to Ψ R = bb 2 ¯ R 2 ρ 1), ρ y R would be f π , ¯ b Z g T f L 2 L,R /g g y y 16 ρ R O g y 2 L,R ∼ – 13 – × = (2 y f ( 4 ρ 2 ξ , ρ L y that will scale like )+ g O 2 ρ 2 L m , being an isospin singlet, will not contribute to O g y L respectively , which is a coupling of intermediate strength. If the y 300 GeV) can in principle be obtained. ∼ L,R ρ H/f ρ ∼ f g ) ( g f ∼ b L b y H P g ∼ ( δg √ R V R Ψ y ∼ , the corrections to L ¯ f are flavor universal so will be the the linear combination of R or , the potential they generate may or may not trigger electrow ρ L ρ L,R ρ g R g y , will have to enter at least at fourth order. y L,R g = 2. Therefore, P · R y ∼ strongly depend on the quantum numbers of the mixing operato c ρ This remark explains the normalization of the mixing term in I ∼ SU(2) R y m 8 L × and L,R y corresponds to a contribution to the vector boson mass matri L y L this leads to a Higgs quartic coupling b T one has P . t R R . Nevertheless, the exchange of Ψ will give rise to non-unive y m y , must therefore be equally important. This is easily achiev f < ∼ ∼ L,R ∼ y L L i ∼ . In the simplest case in which y y = 0. In that model the formally subdominant genuine top contr Whether this can be achieved in practice depends on the speci For instance, for Notice that H thanks to an accidental numerical suppression of eq. (3.8). 9 8 i h quantum numbers of 10 L,R , its contribution to the Higgs potential eq. (3.8) will vani R H normalized fermions one induces then moderately heavy Higgs bosonreason ( the suppression of the coefficients of the couplings corrections to the kinetic terms of For in the particular caseG in which the right-handed top is a sing in the third terms in eq. (3.6) has the naturalG interpretation of a m polynomials O Holographic Higgs models [6]. Writing eq. (3.5) as a functio the strong sector Ψ, onethrough can a see sort that of in universal these see-saw models the Yukaw Notice that for contribution to the Higgscorrections potential scaling receives like now, in additi For SO(4) = SU(2) the strong sector. The effective SM Yukawa couplings after integrating out the Ψ breaking. For instance in the model in ref. [6], the Higgs pot h at y with custodial isospin bounded being a doublet of SU(2) JHEP06(2007)045 b b ρ b R T 11 . m /g /g t and b b 3 and m / (3.10) (3.11) in the L and is δg δg 1 is com- , and a SU(2) , which ∼ − 2 L b 1), other πf × y π , = 4 demands a /g L b cust 16 Q b ∼ / m δg 2 for the special /g = (1 g b b δg R r with the relation = /g is to introduce into tion by which only tribution to b O , but in this case , in contrast to the 2 ρ b b T . A larger ρ ) to be compared to δg t g 2 /g 2), /m π m b e , ith 2 W will have extra supression δg < ∼ r previous warnings about 2) under SU(2) (16 will control the breaking of , ntum corrections at the m b exchanging the SU(2) / T 2 ρ = (2 g o . without being in stark conflict ∼ is a singlet under the custodial LR 4 cust lectromagnetic charge ξ . L ρ implies that the Higgs mass can o the larger scale Λ b P ∼ R S 2 ρ g ), the bound from m H ns that are easily accessible at the O t y 4 L g ρ 2 y 2 y g 2 SM 2 SM c π √ g . We then have g /f is less severe. In this case, however, ∼ N 2 16 ∼ + v ∼ b 2 SM T R . This can occur, for example, for the assignment g 2 L y b ∼ λ will generically have problems with ( y 1). Now H /g ∼ b transforms as (1 T , b 2 ρ R ) in order to reduce flavor violating effects are of the order of the t L 2 δg m L,R y ρ f 2 L – 14 – O g 2 ). A possible way to reduce = (1 y π below the experimental bound for 2 2 SM π ξ , R g π ∼ 16 b 2 ρ 2 L T O g (16 y R ∼ / (16 y 2 ) ∼ 2), ξ/ Λ , 2 b ρ H b , while the two couplings are constrained to have a sizeable ( g g and 2 SM c δg R V g y t N = (2 y 05. The reason of this tight bound is that , while the one from . ∼ L ∼ b 0 S ∼ λ O b . A less constrained, and thus less tuned scenario, can arise T t L y y 2). . For this value of ξ < , that can reduce b T 2 ) = (1 ρ R imply O /m ρ , corresponding to a composite demands a small These are models realizing the clever Little Higgs construc t cust /g y b , since this would induce large effects on = 3 is the number of colors. The experimental bounds, togethe T 1 for models of Class 1. The relation R L m group within SO(4). In that case, the leading tree-level con , c b y 2) and ∼ L , L ∼ N . We can take y R y L b 2 T Apart from the above exceptional case in which y One must, however, check that these extra states do not have e = (3 ∼ /f 11 L 2 t parable to the one from small can be naturally set to zero, and one can take less minimal case where product to reproduce the mass of the Higgs is again bounded to be parametrically where with the bound on y possibly heavier Higgs, can remarkably be made compatible w SU(2) For mix with experimental bounds, as we will see in section 5. Of course ou obtaining proper electroweak breaking still apply here. O The minimization yelds the parametric relation group and drops out of eq. (3.9). However case in which the theory possesses an additional parity alternatives for the quantum numbers of v and and therefore comes always too large, the theory custodial partners for the top whose masses the custodial symmetry [20]. In this case the contribution t now be above the experimental bound without requiring a larg factors ( models of Class 1. In a related way we have roughly estimated to be of order This possibility implies the presenceLHC of or extra light even fermio at Tevatron. Class 2. the quadratic term in the Higgs potential is saturated by qua scale. The Higgs quartic term, in particular, is sensitive t JHEP06(2007)045 . n is ρ g top l, in b S (3.12) of the , while -model ). This f f 4 σ ng some ρ ρ g T the use of g associated 2 g H π ρ / ∼ in the Little g , and like the ∼ G ρ ρ (16 b S T truly represents m / m ρ , 2 m T g g T 4 t y m ∼ ) effects including those in Higgs physics. Notice b 2 can be achieved by using T , the terms of dimension T els with the above charac- the Goldstone suppression t be those dictated by the T y b /g S c g c osite Higgs model of [6, 20]. y coupled the new sector is, ution, the expression for e scales c . 2 SM rection, which is typically only l of electroweak triplet scalars g nguish a coupling cts in Higgs physics are of order ading contribution to the mass g and , which can be numerically acceptable. modify at tree level the structure or parameters: in the new gauge sector. 2 is somewhat favored. Then in the (1) violations of the , t H π t ρ y c O 2 ρ g y 2 T 16 g g term changes sign thus restoring electroweak / . Now, the result for ) ∼ 2 2 ρ t requiring a severe tuning of parameters 2 π g T /y 2 g g 4 16 π that are potentially important at small ) (and also associated with the mass g 2 ρ ( , induce b 16 T T ∼ ρ ∼ /g / g g 2 – 15 – T 2 ρ 2 W g 2 SM ∼ m g m with respect to its typical one-loop size, making it -model does not posses a custodial symmetry, one ( ∼ σ b S O ∼ ) SM down to 2 g is controlled by the vector boson mass b b S S , then, by the isosping argument illustrated before, we the vector boson loops dominate the mass term in the potentia b /f ρ S 2 g in eq. (2.15), leading roughly to v /g t ( . For instance, in addition to 4 y ρ ) T ρ g m Therefore this class of models prefers a weak coupling in the /g ∼ O > ∼ even more. But even when the model is custodially symmetric i t y ρ 12 b T ( T g . A more detailed scrutiny of these effect requires consideri . g t ∼ ,y , if the underlying T v/f , which is parametrically like a SM loop effects. Expectedly, T c 2 b T ) π = 0, there are corrections to > g 4 ρ / g (1), and this operator is as important as T SM g g O ( is ∼ structure. In realistic LH models a weak 2 2 T Concerning Notice that the general class of In the following, we will give three explicit examples of mod Notice that if we take H /g /f 12 / 2 2 t partners of the top quark. Since the scalar potential tends to be dominated by the top contrib Higgs is parametrically the samemarginally of acceptable. a 1-loop Better electroweak cor agreement with the bound on more appropriately the flexibility the Litlle Higgs possesses in the strong sect numerically acceptable. a spectrum of different couplings. In particular we can disti showing that there is space to relax also does not manifestly push towards a large resonance sector but still a somewhat large coupling to the extra gauge factors and a coupling which case either thesymmetry. relaxation Thus effect we saturates can or naturally the relax mass will have a significant and disfavoring large the limit of the strong sector(these Lagrangian, effects unlike in the particular Higgs the mass are potentia not screened above th This is because SM custodial breaking couplings like generally expect Higgs quartic they formallyof have the tree-level triplets size). is If controlled the by le explicit model, which is beyond the scope of our brief survey also favors we have just discussed, would, for our effective Lagrangian isthat more is motivated, the the bigger more strongl teristics. We will first concentrate on the Holographic Comp however that in the weakly-coupled limit,v all anomalous effe structure below the scale y higher than 6 inG the two derivative Higgs Lagrangian would no contribution of the top partners to the Higgs-gluon couplin JHEP06(2007)045 d X ge. on in (3.14) (3.13) sets the where 1 R 3 , we obtain /L T P + , labels the extra /M (4). X ! ass mixing terms. W z O auge fields in UV- 0 = H e the red-shift factor m nder the weak SU(2) ∈ undary fields can be tle Higgs model with Y T R 4 he mass of the lightest xing terms between the H ese mixing couplings are he correct SM spectrum. 0 corresponds to an explicit bulk field. As in eq. (3.6), dstones, the bulk contains − (where k , onal theory in AdS space- assive Kaluza-Klein states 2 5 ll follow ref. [20] and embed reaking pattern of the bulk espond to the new “strong” 0 g Ã L q e UV boundary, while the other , is assumed to be compactified π ). This is the correct separation = 8 Π= 3 1 TeV). The bulk gauge symmetry /k ,x z 0 ∼ ≃ L 1 ρ , f 6= m /L 1) 1 z , = ( 0 ∼ ρ A , g 0 , 0 – 16 – , (4), we expect four Goldstone bosons parametrized and O = (0 is used to explain the hierarchy i → 1 , is also needed in these models to guarantee a Higgs mass Σ 1 h ρ /L π L g 0 3 4 L is the 3rd-component isospin of SU(2) ) and bulk fields ≃ , 16 implying that the coupling among resonances is always lar R 3 ,x ρ T 0 /f representation of SO(5). We can again separate each 5D fermi ∼ and it is called the IR boundary. The energy scale 1 m Π L ) 5 e 0 0 i = L Σ SU(3) on the IR. The hypercharge is defined by /L h z 1 ( SU(3) is broken down to the SM gauge group on the UV boundary an ≫ × L A 1 × X SO(4) coset [6]: Σ= L ln( X charge and / 2 (1) = g X (1) U z is the bulk SO(5) gauge coupling. In the particular case wher is a real 4-component vector, which transforms as a doublet u > ∼ U × (1) 5 k × g H U 2 5 g (4) We will follow the Holographic approach and separate the 5D g O mass gap of the model (the Kaluza-Klein mass where between the two boundaries dimension in conformal coordinates) andone it is is referred at as th custodial symmetry [21]. 3.1 Holographic composite Higgs model The Holographic Higgs modeltime. [6, This 20] space-time, is of basedby constant on radius two a of 4D five-dimensi curvature boundaries. 1 One boundary is located at Then we move to the Littlest Higgs model [2] and finally to a Lit SO(5) to a UV-boundary field, to beYukawa associated couplings to are the SM generated fermion,SM in plus fermions a this and the modeldetermined heavy through by fermionic the mass bulk 5D modes. mi fermion The masses and size can of be th chosen to give t that is the above the experimental bound.and The the interaction fields between theThese on terms m the only respect UV-boundary thebreaking (the SM of gauge SM the symmetry SO(5) fields) and symmetry. therefore the is For SM the fermions due fermion in the sector, only we to wi m As we explained before, a large boundary fields to make contactassociated with to the the theory SMsector. gauge defined bosons, Let in while us the section analyzeand bulk 2; IR-boundary this states is new corr given the by sector. SO(5) UV-bo Since the symmetry-b by the SO(5) where group and can be associateda with massive the Higgs. tower Apart ofstate from 4D is the Gol given states, by the gauge Kaluza-Klein modes. T JHEP06(2007)045 (3) O (3.16) (3.17) (3.15) (3.18) (3.21) (3.23) (3.24) (3.19) (3.22) (3.20) coupling. For . , ξ ξ 2 2 corrections) with 3 3 , hff 2 ρ , − ¸ − ξ 2 µ 1 1 /m Z − ents of the effective op- , ≃ 2 W µ ≃ 1 Z ) h ) ) m f 2 are constants), we have ian eq. (2.15). This second ≃ W /f /f /f ) θ i i i i β 2 sin s as eq. (3.16), is performed in h gs potential can approximately h f h h 1 h /f h h h ( i . 2 M h and h T 2 cos . α 2 sin( tan(2 ), eq. (3.15) gives )= Σ) + coupling obtained from the Higgs po- 2sin = cos h f cos( µ h = 0. This is due to the custodial i µ ( = − h D f T h/f h i can be obtained from the kinetic term of sin 1 W c . h . . = hhh h µ h H 2 Σ)( ¯ ¯ ¯ ¯ = = cos gf c W µ h ] (where ) , i = 1 ¯ ¯ ¯ ¯ = 1 = 0 · 0 h h D ) ( h , ) 6 y ( can be unambiguously computed by comparing H – 17 – h 0 c )= = c h h/f c 2 and ( 2 W ( h ∂h 2 h f 2 f H ¯ ¯ ¯ ¯ ¯ ( ∂h T ff, m c 2 W ) ) c ∂m W h = h/f, cos h 3 ∂m m ( ) ( m ) β h f ) V + ∂h kin ( h 3 h ( − m h ( L 1 ∂ W f µ − W α 6 1 [ ) ) gm M h∂ = gm we must match the h h = 0 or, equivalently, 2 µ ( ( h/f 6 ∂ = ρ c V 2 yuk 1 W 2 = 2 h L m W = sin 4 g∂ hff f hWW ≃ g gm kin = g can be similarly deduced from the calculation of the ) W L h gm y ( c m hhh 2 V 2 H g W coupling for canonical fields in eq. (3.16) (neglecting gm is a constant. We then obtain m 4 M Let us calculate the contribution of this model to the coeffici hW W the Holographic Higgs model of ref. [20] we have be written as tential to eq. (B.5). In the model of ref. [20] in which the Hig where Eq. (3.16) tells us that ∆ In the unitary gauge where Σ = (sin and, using eq. (B.3), we find erators of eq. (2.15). The coefficients invariance of eq. (3.15). The value of that, comparing it with eq. (B.4), leads to From eq. (3.19), (3.23) and (B.5) we get the same quantity deducedstep, from which our requires general writing effective eq.appendix Lagrang (2.15) B. in From the eq. same (3.16) field we basi have The coefficient To obtain the coefficient where the Goldstone bosons: the JHEP06(2007)045 ′ g (3.27) (3.29) (3.25) (3.28) (3.26) SU(2) and / R g ) are only , Y L g depend only on and 6 c try down to SO(5). W , 2 1 , , and efficients are therefore b L S y ry and we consider the ¢ c 2 2 W in the spinorial represen- r are, as we said, completely (4) symmetry of the model m ile below the scale of the heavy 5) symmetric representation T O ¶ l be discussed in section 5. tive to the details of the 5D c grangian with a SU(3) 1+ e ¡ X 1 2 5 at leading order can be trusted g W and . . 6 2 , although generated at tree-level, c c 0 + y . H . 2 ρ 2 W c will not be presented here. They are c 1 2 5 = 1 (and therefore    m g m ≃ 2 B and γ,g 2 µ B 2 1 c 2 2 π y 2 ′ c π subgroup of SU(5) is gauged ( 1 4 128 W, g 27 2 , 256 2 Y 27 c and T 1 = ≃ c , c = (1) – 18 – ,    ξ 9 2 5 . 2 U and B can be obtained respectively from the parameters H g 1 g = c c = 1. 8 × 3 B i ≃ 6 2 HW,HB = R c Σ c c W,B 2 = , Y h c 2 1 W ¶ b S c 2 L and , at tree level, is given by [6] π 64 SU(2) 2 W we obtain b 9 S gauge coupling. Using eq. (2.21), we get W m × µ 2 Y = 0 and 2 5 X L c 2 g ≃ y g , 4 c and W = 2 W,B c c W W . The coefficients , the predictions for , we obtain π X 4 5 B c /g . The parameter < 5 = is the bulk U(1) g Y ρ g W X = c 5 r g and The coefficients It is assumed that a UV dynamics breaks the global SU(5) symme Although the calculations for W , b S tation of SO(5), we find In the model of ref. [6], in which the SM fermions are embedded where This breaking is conveniently parametrizedacquiring in a terms vev of of the a form SU( valid for are the respective gauge couplings).new particles, As this we model are can goingcoset be to described structure. show, by our effective La even in the non-perturbative regime. The coefficients Using eq. (2.18) and eq. (3.25) together with the fact that th determined by the symmetry breaking pattern of the model, wh depends on the particular structure of the top sector and wil where the way we embed theindependent SM of fermions the into five-dimensional the dynamics. SO(5) group. These3.2 co Littlest Higgs model The Littlest Higgs modelversion where [2] only is a SU(2) based on a global SU(5) symmet For the parameters generated at the one-loopmodel. level and Similarly, the are non-universal therefore contribution very to sensi implies JHEP06(2007)045 . . 2 = f f . ged R 2 Y ) ρ g . 2 R /g (3.31) (3.30) (3.35) (3.34) (3.32) (3.33) m g T (1) f ˜ 2 H U tion 2 of + √ ˜ H 2 L × , = g ) . R ⋆ Y φ    = ( T H , is ˜ H SU(2) 2 W Σ+Σ ⋆    he gauge symmetry m is × ˜ Y 2 ˜ 4 H , H φ ( / L µ R − − gauge coupling is turned a ⋆ B † ⋆ ′ σ L ˜ , gauge sector. According to (the subscripts denote the φ SU(2) due to the fact that H ig is 2 − o the constraint is the decay constant of the SU(2) | / − 0 es its usual expression 1 − Y SU(2) 3 rom the kinetic term of the Σ − s f t acquires a mass of order Σ )    × (1) f SU(2). µ T ∂ L U R µ    † 3 , and A × Σ Π= Σ | ⋆ i a R µ µ L 2 A H Tr D 2 2 √ † and the mass of the axial vector is = f Σ+Σ Σ iσ 2 1 . µ R µ R R µ g = A D ¶ ( ]+ 2 1 ≡ ˜ R 3 Tr – 19 – H , , A ρ with 2 2 Σ ig 1 2 2 g f µ ,    − ∂ 0 i † 3 and a neutral singlet , = ) SU(2) coset. This totally fixes the relative coefficient 2 1 Σ Σ T 1 L h µ / L µ φ − 2 ∂ /f , A / a Π 2 1 i Tr[ , as it is explained formally in appendix A, as well as the σ 2 2 − e respects a SU(3) global symmetry and the doublet and the R f    µ A 2 = Σ+Σ R the low-energy effective Lagrangian can be mapped onto the L µ aL µ /f A A 13 T ( Π L , the Littlest Higgs model has the structure described in sec i = = diag e ig f down to the diagonal subgroup SU(2) and we are left with a char i L µ R -model weakly coupled to a SU(2) Y − g Σ A R σ h Σ order, the integration of the scalar triplet is equivalent t µ , a charged triplet vev gives a mass to the axial part of SU(2) /f 2 2 ∂ . By the construction of the model, when the SU(2) Π i / i 2 symmetric complex matrix, 2 /f R 1 parametrizes the SU(3) SU(2) e Σ SU(2) h H × 3 /g × / Σ= SU(2) Lagrangian charges of the fields). These Goldstones are parametrized by µ L / Σ= + 1 Y D The After integrating out At the 1 2 L is a 2 13 (1) /g coset model. The interactionsfield of the Goldstones originate f where the covariant derivative accommodates the gauging of φ Among the 14 Goldstone bosons, 3 are eaten in the breaking of t SU(2) where Σ between the two independent invariants that exist in SU(3) this coset decomposes into 2 irreducible representations o with While the gauge coupling of the unbroken vectorial SU(2) tak singlet are exact Goldstone bosons while the charged triple SU(3) a SU(3) 1 Below the scale doublet off, the gauging of SU(2) U charged scalar triplet, our description we should then identify JHEP06(2007)045 × × of he are L 1 (3.38) (3.39) (3.36) (3.40) (3.37) (3.41) g . In that (SO(5) / L and g 2 ∼ , g (the subscripts R R g 0 . are important. , g s ) , L 6 R g /f /g = 0 ry breakings SU(2) 6 -model structure below L nveniently parametrized v g σ B ( 2 t m O ∼ coupling at zero momentum: ρ SO(4) structure. , c = meters are found to be, in the / ll be described, below the mass ed on the coset SO(9) /g , = 1 SM , Y g hW W ¶ . , 2 ρ W 2 W ξ 8 2    m 5 m 16 − 4 and we are left with a charged doublet and a neutral singlet 2 L 2 , c R 1 1 . g = g Y 1 µ 1 (1) subgroup gauged ( 1 4 1 16 φ H = U 4 W SU(2), breaks custodial symmetry and hence (1) − = 1 / × U , c – 20 – = H gm    c , W T → = = = 0 2 ρ i 2 W T SU(2) m , c Σ (1) c m 4 h × U hWW − g = 0 R × = coefficient. The source of this custodial breaking is the vev B R -model, SU(3) b T T σ c , c , SU(2) , a charged triplet 0 2 1 = 0, since the only gauge fields integrated out form an adjoint 2 ρ × φ can be easily deduced from the 2 W of the corresponding SILH Lagrangian can be computed along t m B = L i m c 2 H c c W and SU(2) = c b S W . It can always be fine-tuned away, for instance by taking φ is now lost, since the corrections of order , ρ g SU(2) m ≫ → ρ g , a neutral triplet The coefficients The global symmetry breaking of this Little Higgs model is co 2 / 1 case SU(2). The value of we obtain a non-vanishing that, together with eq. (B.3), leads to of the triplet Notice that the low-energy From this and eqs. (2.16), (2.18), (2.21) we deduce that while the other coefficients are unaffected. However, the exac where we have taken the scale 3.3 Little Higgs model withThe custodial littlest symmetry Higgs model with custodial symmetry [21] is bas lines outlined in thelimit previous subsection. The oblique para the respective gauge couplings). Thisof Little the Higgs new model resonances, wi by a SILH Lagrangian with a SO(5) by a symmetric representation of SO(9) taking a vev of the for Among the 20 Goldstone bosons, 6 are eaten in the gauge symmet SU(2) H SO(4)) with an SU(2) JHEP06(2007)045 t . (3.43) (3.45) (3.42) (3.46) (3.47) (3.49) (3.44) (3.48) singlet    is a real 2 2 φ √ 4 coupling at / φ/ e Goldstones T − H 2 0 . for concreteness). 2 hW W 2 √ ρ √ f g 2 R , H/ g H/ ≡ . ts. The kinetic term of 2 ρ − 2 W = m R 2 0 cay constants in the two m . g ± H en by the usual formulae , obtained after integrating = 1 straint ne doublet, while √ 2 2 ρ ρ ′ 2 A 2 2 / 1 rangian computed in the Holographic g 4 B g 1 = g (1) gauge couplings are turned m T 2 0 φ/ c H U L = , m H c + g 2 − f 2 R 1 . ) g    , Y 2 R Σ g µ 2 ρ = 2 W = 1 and D . + 2 m † ′ m Π= 1 . 2 T 1 i g 2 2 Σ ρ W g 2 1 f 2 g g µ 2 D = HH = ( , c = Tr H – 21 – H = c 2 , 2 B = 0 4 respects a SO(5) global symmetry whose breaking φ f with 2 2 2 , W 1 T g R = , the heavy vector fields transform as a neutral triplet, , m (we have considered 2 i , c + Y = 0 L f Σ f ) 2 L h b 1 ρ T (1) g 2 L = 1 g /f g U , B Π = i ≡ c + × 2 ρ 2 2 W 2 2 e and SU(2) 2 ρ -model structure below the scale 1 m + gauge fields that do not couple to fermions and integrating ou g W m g σ L m = R W = c = ( doublet as exact Goldstone bosons while the triplets and the /f b S T H H Π i 2 W SO(4) SU(2) charges of the fields). A convenient parametrization of thes e / can be computed, exactly as before, by looking at the m i × Y Σ H L h c (1) /f U Π i e By the construction of the model, when the SU(2) and 4 symmetric matrix and it contains the singlet and the triple Σ= = 0. We obtain is the real 4-component vector corresponding to the Goldsto × 2 Under the unbroken SU(2) denote the H the Σ field generates the interactions among the Goldstones a neutral singlet and a charged singlet whose masses are is The oblique corrections are found to be which allow us to identify the coefficients of the effective Lag The factor 2 of disagreement with the value, eq. (3.19), of 4 The gauge couplings of the unbroken gauge symmetries are giv off, the gauging ofto SU(2) SO(4) leaves the acquire a mass of order Higgs model simplymodels. fixes the relative normalization of the de The value of Hence, the SO(5) p out the SU(2) as well the heavy triplet and scalar, which amounts to the con JHEP06(2007)045 . ¸ i ν ν A [23]. A 2 (4.3) (4.5) (4.7) (4.4) (4.1) (4.2) (4.6) i ¶ = 0). . . . nal to γ ¶ i M + k . . . h W h ¶ W θ = Z + θ ˆ c µν 2 2 µi v Z tan h 2 and the gauge A 3 µν − and the photon h µν F + + tan µ W D h θ pped the effects of γZ Z Z W l masses and gauge c k µ c 2ˆ W ¯ = he gauge kinetic terms θ ff sin 2 ¢ 2 f Z 1 µ v g otons. Notice also that the + m µν + osed by the usual SM part (shifted such that y ν W c 2sin h s and study how they can be Z ary gauge and write the SILH ν ew interaction terms involving − Z d + − ˆ c . µν · ¶ γ W relation Z κ µν = 246 GeV). Similarly, we redefine − . . . D W µν v θ µ − Z + µν 2 Z 4 2 ρ W 2 (and similarly for the 2 h g HB ν g tan c 3 − 2 ν + ± to the SM input parameters (fermion masses µ − W + W W 6 θ )+ W ), and new interactions involving only gauge Z . 1 ¡ c + 3 µ 2 ν h c g – 22 – . µ 4 ∂ L W ¸¾ vh Z = ) µ − and Z a µν +h 1 Z ¸ +tan 2 W g µν f 2 H ± + G ν ν ˆ c v c A W 2 m HB W γ W θ 2 ρ 2 Z HW 6 aµν c µ κ c c µν m 2 ∂ m G , κ W cos ´ + g − D ) . Notice that for on-shell gauge bosons θ 2 c ρ = = ρ π − − h µ 2 t ig µν g µν 4 µ y Z HB 1 πg ± g µν W s ³ W c − 4 ¤ α h∂ + sin W W µν + + µ we have kept only the first powers in the Higgs field µν c + − (ˆ ∂ + generate a Higgs coupling to gauge bosons which is proportio B h µν B 2 2 , g ρ c ν 2 , and vacuum expectation value L W Z g ) B · ¶ HW µν HW ∂ g (3) 2 µν c c c Z H µ B F v h ( c κ W 2 ∂ 2 m 2 HW θ ¶· µν . In ´ W c b ´ + 2 2 SW ρ = 2 π F and ˆ θ W 1+ W HW v ρ π g µ 4 θ h 3 c γ W 2 W g 2 ρ 4 µ 2 c c ³ θ ) which, at leading order, are given by W O µν ) 2 ( ³ − 2 2 c + H 2 , V πg (cos π D c g + tan + 4 v h 2 ρ 2 ρ B L αg 2 W 2 W tan (4 2 ig 4 sin 2 ( we have included only trilinear terms in gauge bosons and dro HB µ W W m m + c m m O − ξ c − c − + ), written in terms of the usual SM input parameters (physica , V = = = = = ˆ = = , Higgs mass L ) and W SM h γ b f Z V S 2 µ W ˆ c κ γZ L L ˆ c L the gauge fields andcanonical. the In gauge this coupling way,( constants the and SILH we effective make Lagrangian t is comp couplings), by new Higgs interactions ( We reabsorb the contributions from c bosons ( tested at future colliders.the Let physical us Higgs boson. start by Foreffective considering simplicity, Lagrangian we the in work eq. n in (2.15) the only unit for the real Higgs fiel 4. Phenomenology of SILH In this section we analyze the effects of the SILH interaction A m In fields. We have defined O Therefore ˆ mass, as in the SM, andcorrections do to not generate trilinear any vector Higgs boson coupling to vertices ph satisfy the JHEP06(2007)045 as and y (4.9) c (4.12) (4.11) (4.10) (4.13) . HW ¶¸ c and is then to ¶¸ Z is universal hat removes J γ H so shifts the H γ γZ c J c c H + c 2 c 4 + 2 ρ Z g g I γ 4 in many different I + h + nly for rather heavy Z ay widths in units of W from the three coeffi- γ ˆ c W ¶¸ ¶¸ BR ˆ c 2 /J 2 ρ g g g /J 2 2 ρ e effect of Z g W c × g g γ I I ˆ c easure of the Higgs decay 2 t − I bution. The loop functions 2 ρ h 2 2 y model-independent oper- ρ y − ¶¸ g s (for a review of the Higgs H σ g g 4 e the spirit of our analysis is g the Higgs field according to 1+ c H Z t the dominant effects should fore relatively light. However, 1+ ggs branching ratios, but only c ˆ c o understand the nature of the − + + generates an extra contribution 2 2 ρ + g g H Z H γ H c H c c H )] (4.8) c − /I µ c /I + H Z + γ the subleading effects from ξ H c + J y y J c y c c − + c µ 2 2 from precise measurements of the Higgs 2 1+ 1 ). The contribution from y 1+ ξ µ SILH h · c y µ ) c µ − (2 Re SM 1 Re BR ξ ξ Re · ´ ξ ξ , ZZ and – 23 – − − − − − ) − SM ∗ 1 1 1 [1 ( H , the coefficient · ´ · W c · h ) ∗ + W ( SM SM SM SM + ¢ , since their SM contribution occurs only at loop level. ) ) ) W ¯ f 2, with respect to their SM value, the couplings of the has less phenomenological relevance since it affects only ZZ W f ). / gg γZ γγ hgg → f , see eq. (4.1), modify the SM predictions for Higgs produc- H → → h γZ → → m h → → c ξc L h h h h h h and ³ ¡ ³ or − v hγγ = Γ( = Γ = Γ( = Γ ( = Γ = Γ ) and branching ratios ( . The rules of SILH select the operators proportional to h , although (see appendix B for an alternative redefinition of the Higgs t σ SILH SILH SILH SILH SILH SILH to all other fields. Notice that the Higgs field redefinition al ) ) ) ) ) ¢ γZ ¯ γZ H h − f c gg γγ f γZ ξc ZZ (but not of , → W y → + → c → H h → → 1+ h , h h are given in appendix C. h m W h ¡ H √ c Γ ( J Γ ( Γ ( Γ → Γ ( h/ The new interactions in We can express the modified Higgs couplings in terms of the dec The leading effects on Higgs physics, relative to the SM, come , which are parametrically smaller than a SM one-loop contri h → and Γ( HB the derivative terms of the Higgs — first term of eq. (4.1)). Th renormalize by a factor 1 tion and decay. At quadratic order in to the Higgs kinetic term.h This can be reabsorbed by redefinin Here we have neglected in Γ( canonical field value of the SM prediction, expressedproperties in in terms the of SM, physical see pole [22]), masse I cients c the most important onesator for analyses LHC [24 studies, – 26] as whichappear opposed often in to lead the totall to vertices the conclusion tha the decay production rate ( Therefore, we believe that anHiggs important experimental boson task will t be the extraction of for all Higgs couplings andthe therefore total it does decay not width affectwidth and the at the Hi production the cross LHCHiggs section. bosons, is well very The above difficult m theto and two consider gauge the it Higgs boson as can threshold, afor be whil pseudo-Goldstone a boson, reasonably light and done Higgs, there LHC o experiments can measure the product JHEP06(2007)045 , , . ξ b 1 → − = 1 ) h γ ) and H σ 1. The /I /c γ − y J c SM ) (1 + y )] = 0, ∆[Γ( 4 and , it is possible to / − ξc 1 σ BR 2 ( − ross sections ( W / − = 1 + ξ ). The SILH Lagrangian W SILH ios between the rates of H from the SM expectation ) c )] = tth certainties drop out. Our ike the SLHC. At a linear linear collider can test the → e till be limited by statistics, − erent decay modes. In mea- h arious channels with 20–40 % W σ BR nd top-strahlung; decay into Γ( + / annels at the LHC with production ) W ) = ( ZZ → . h → coupling is quite challenging [28]. This up to 0.2–0.4. ξ σ BR | Γ( b h H ( ξ can reach the percent level [29], providing c / / y ) ) c − h | ), and topstrahlung ( γγ h – 24 – BR )= = 0, the deviation is universal in every production σ BR → and × | H h = 1 and we have included also the terms quadratic in ξ h σ /c H σ BR H y c ( | c / /c ) y c , ∆[Γ( y 4, ξc / 2 σ BR ) defined as ∆( − = 1 ) prediction is that ∆[Γ( ξ BR H SM c )] = g − ≫ W ρ + g W The deviations from the SM predictions of Higgs production c → h and (virtual) weak gauge bosons. At the LHC with about 300 fb Γ( In figure 1, we show our prediction for the relative deviation Cleaner experimental information can be extracted from rat / γ ) ¯ f , Figure 1: not explicitly shown in eqs. (4.8)–(4.13). channels: production through gluon, gauge-boson fusion, a parameters are set by predictions are shown forvia some of vector-boson the fusion main (VBF), Higgs gluon discovery fusion ch ( in the main channels for Higgs discovery at the LHC, in the cas will translate into a sensitivity on decay branching ratios ( τ measure Higgs production rate timesprecision branching ratio [27], in although the a v determination of the (as in the Holographic Higgs). For channel and is given by ∆( processes with the same Higgssurements production of mechanism, these but diff ratiosleading-order ( of decay rates, many systematic un f a very sensitive probe on the new-physics scale. Moreover, a However, the Higgs coupling determinationsand at therefore the they LHC can willcollider, benefit s like from the ILC, a precisions luminosity on upgrading, l JHEP06(2007)045 der .16) ittle = 0, WW . The (4.17) (4.15) (4.14) (4.16) , ρ SM s 2 g m H f c , associated h = racy of about T ¢ portional to the ± L W , ± ) L in the limit , t ecay rates, could be a tence of a new strong ξ W 6 H + [30]. ¢ 2 4]. This corresponds to a s f 1 → corresponding interaction X g coupling is reached at a ( e theory, namely ′ − L follows from the non-linear up to the cut-off scale Λ. In ± H elastic amplitude is softened L ight, we obtain strong V c e bosons with large invariant to be found in the very high- s ξ M, of the terms growing with L efore, in theories with a SILH, W it of the scattering amplitudes ew light particles are discovered V = e of the light Higgs, longitudinal ± L ¢ → W y at the cutoff scale. Indeed, the − L ¡ W pp ¡ + L −A σ W 2 ) in eq. (2.15), using the equivalence the- = ) ξ ¢ H → , and to all orders in 0 † L H in eq. (4.1) prevents Higgs exchange diagrams 2 c − Z L H ξ 0 L ( – 25 – H W s/f µ Z = ( shifts. Therefore we expect that corrections can c ∂ + L ) H v → c W H ¢ ¡ † − L X A of integrated luminosity, it should be possible to identify H ′ W L ( 1 V µ + L − L ∂ , W V t ¡ 2 ≡ H . f → c A 5–0.7. H is the cross section in the SM without Higgs, at the leading or . 0 O = = = 0 pp 6 H ¡ ) ). The growth with energy of the amplitudes in eqs. (4.14)–(4 ¢ ¢ ¢ ≃ depends on the specific model realization. In the case of the L 2 ρ σ 0 0 L L X − ξ L -model, corresponding to the action of the generator ρ ′ L Z Z σ H W m 0 V L ± c . Notice that the result in eqs. (4.14)–(4.16) is exactly pro s/m + L L Z ( ρ V W , since the triple Higgs coupling can be measured with an accu O W /g 6 → ρ c = 500 GeV and an integrated luminosity of 1000 fb → -model is exact. The absence of corrections in → . With about 200 fb 0 L → s σ 2 0 L ) Z πm 0 pp L √ Z 0 ( L 4 Z πv ± σ Z 0 L ∼ ¡ (4 From the operator Deviations from the SM predictions of Higgs production and d Z W ¡ ¡ A s/ A A orem [31], it isfor easy longitudinal to gauge derive bosons the following high-energy lim symmetry of the scattering at high energies. with the neutral Higgs, under which arise only at is strictly valid only up to the maximum energy of our effectiv Higgs, we expect that the amplitudes continue to grow with 5D models, like the Holographicby Goldstone, KK the exchange, growth of but the scale the inelastic channel dominate and stron behaviour above scattering amplitudes obtained in athe Higgsless cross SM section [31]. at Ther themasses LHC can for be producing written longitudinal as gaug when the from accomplishing the exactenergy cancellation, in present the in amplitudes. the S Therefore, although the Higgs is l where in hint towards models with strong dynamics,at especially if the no n LHC. However,interaction. they The do most characteristic notenergy signals regime. unambiguously of Indeed, imply a a the peculiarity SILHgauge-boson of exis have SILH scattering is that, amplitudes inbecomes spit grow strong, with eventually energy violatingextra and Higgs tree-level the kinetic unitarit term proportional to sensitivity up to the signal of a Higgsless SM with about 30–50% accuracy [32 – 3 existence of This result is correct to leading order in 10% for JHEP06(2007)045 . g − L H O c ) W [35], + L (4.19) (4.18) X 0 and L bγγ ¯ Z ,W b 0 0 L L BB Z Z indicated by , and with a O 0 L jets are often , → M background. b < δ ’s appears more | W,B pp litudes necessarily hh, Z ( η W O which are, at most, ∆ | → 2 ρ δ,M eal . σ pair production rate at one boson and therefore (4) symmetry of the op- s gg son, the ¶ 2 s/m O H δ f 2 g coefficient. Because of this c c 2 n with energy. Although we uplings is crucial for testing ergies. 2 ρ he LHC is not so prominent. n of SILH is that the strong with sufficient luminosity, the = of Higgs pairs. Indeed we find h two like-sign leptons, where (4.14), SM ) ¢ ten) Goldstones, corresponding antity) of /m studied in the channel tanh gauge bosons and a single Higgs 1 hh hh − multiplet, the amplitudes for Higgs ∝ 9 → → µ H − 6 L 1 gg and thus indirectly test the non-linearity ( W + 2 ρ + L H c . The sum rule in eq. (4.19) is a characteristic /m ≃A ) W 2 2 ρ ¡ g X . Indeed the contribution of , their effects in Higgs decay rates are subleading – 26 – makes the SILH rate unobservable. However, in y A , on the other hand, only depend on the scale of − L c g parity embedded in the SILH SM ) = W O , even though they arise from the interplay between 4 2 γγ g s > M 6 ¢ + contributes to processes like L Z hh c and ≫ g → W hh and H ρ O → h c g and → → times the SM contribution. While at electroweak energies (4) symmetry of the gg y BB 0 L 2 ρ O ( are computed with a cut on the pseudorapidity separation c pp O Z ( A 0 , L s/m Z δ,M δ,M ¡ σ σ W,B A is purely generated by the strongly-interacting sector, as O = H H c O ) jets, studied in refs. [35, 25]. with high invariant masses could be distinguished from the S coefficient. Also hhX b ¯ νν 2 ρ bb , under which each Goldstone change sign. Non-vanishing amp ¯ ± → b ℓ H /m ± 2 ρ O pp ℓ ( g The operator As an example, the operator In the SILH framework, the Higgs is viewed as a pseudo-Goldst Using eqs. (4.14), (4.15) and (4.18), we can relate the Higgs → ∼ δ,M σ but the small branching ratio of SILH, one can take advantage of the growth of the cross sectio do not perform here a detailedsignal study, of it may be possible that, Notice however that, because of the high boost of the Higgs bo of SILH. However, the signalIt from was suggested Higgs-pair that, production for at a light t Higgs, this process is best promising for detection. Thehh cleanest channel is the one wit not well separated. The case in which the Higgs decays to two r cut on the two-particle invariant mass ˆ its with respect to those induced by weak and strong couplings, are sensitive to of the Higgs sector.SILH. Therefore The probing operators the effects of these co fact, in the strong coupling limit new physics, not on its strength, as indicated by their to amplitudes scales like this effect is very small, it can become a sizeable at higher en Here all cross sections with scattering amplitudes between the two final-state particles (a boost-invariant qu its properties are directlyto related the to those longitudinal of gaugegauge the bosons. boson exact scattering (ea is Thus, accompanied athat, by as strong generic production a predictio consequence of the pair-production grow with the center-of-mass energy as eq. Notice that scattering amplitudesvanish. involving This longitudinal is a consequence of the involve an even number of each species of Goldstones. the LHC to the longitudinal gauge boson cross sections 2 erator JHEP06(2007)045 , . ρ → ± H ρ m ′ e to that (4.21) (4.20) qq proved , where W ass. For 2 ρ . Indeed, ρ m /g and coupling ≃ 2 ρ g can contribute g s se quantities are . Therefore, it is ρ with coefficients of W,B m ergy O re important because . le s. Indeed, they induce W,B is O + H 5 fb determination at LEP and should be tested in vector . , ρ irect resonance production. one charged resonances ble-Higgs production at the 0 osons. Indeed, by using the b 2 ρ ons) equal to S s kind of new physics at the SM is quantitatively similar n be rewritten as the product s in eq. (4.5) can be probed at m 6 HB s to the technirho, in composite = bosons) equal to ¶ c ˆ s g electroweak-breaking sector in ¯ ¯ ¯ ¯ ρ L m and ˆ sdτ τd 3 TeV 2 ρ 4 µ g HW 2 πg c 12 ¶ and can give new-physics effects in ¯ can be tested through precise measurements . ρ π = in Little-Higgs models. They have mass g 4 H ¢ W,B ± µ – 27 – H c X W = HW,HB + ¢ O + H , and to the magnetic moment anomaly of the X ρ + W,B , they give rise to the operators → Zγ ρ ± H O ρ m pp → ¡ , in spite of their overall 1-loop suppression, are sensitiv . → h σ ∓ L pp up to 6–8 TeV. W ¡ [36], but at a sub-TeV linear collider the precision can be im smaller) to that of ρ ± L σ HW,HB 2 2 m − O π ,W is the parton luminosity at an energy equal to the resonance m [37, 38]. This is highly competitive with the ± 16 L –10 / 4 1 4 TeV, we find was recently studied in ref. [25]. Also the operators 2 ρ W − − g 0 L g /dτ < ∼ ) relative to the SM contribution. In practice, however, the . The cross section for the resonant production of L 2 O 2 ρ ρ –10 , Z 3 ˆ m sd ± /f L − /m 2 2 < ∼ τ/ v , indicating that in principle they probe the strong dynamic g ( 2 ρ As discussed in section 1, the signals studied in this paper a The operators The effect of the operators , hW O 0 L /m 2 ρ of the same order of magnitude of the SM one, for the maximal en in the effective theory below the corresponding resonances canLHC be from directly produced. Dou where 2 TeV to the weak sector (quarks, leptons and transverse gauge bos boson production, where(indeed their even contribution relative to the coupling to the strong sector (Higgs and longitudinal gauge are they are model-independent testspresence of of a a strongly-interactin lightLHC Higgs. may However, come the from first production evidence of for the thi resonances at the mass sca can probe values of experimentally not well accessible. Therefore to high-energy production ofequations Higgs of and motion for longitudinal theof gauge gauge b a bosons, these fermionic operators currenthZ ca times a bilinear in g corrections to the process of triple gauge vertices. Atthe the level LHC, of the 10 anomalous coupling These particles can be interpreted asmodels, bound or states analogou as the heavy gauge bosons useful to compare theAs indirect an effects illustrative studied example, here we with consider the the d case of new spin- order up to 10 JHEP06(2007)045 , ρ m πf and – 4 (4.22) (4.23) H scales. πf c < ρ parameter, m b S . , 1 ates at the LHC 1 omes increasingly − − ¶ lored regions of the ¶ up to about 30 TeV. 4 4 4 g ρ 4 g ρ g annels (in some specific ding of the LHC. Higgs- g 24 πf — see section 3). Notice 24 gauge and Higgs bosons is ρ and probe processes highly ecise measurements of triple ng triple gauge vertices or, 1+ , resonance production at the /g 1+ s production at the LHC can ctly into leptons or top quarks. ρ µ eresting region of the 4 H 2 e boson and Higgs production, ls becomes more important in t also because the decay width µ c g ty on 4 1 2 m 4 4 ) resonance searches become less ρ th a light Higgs boson. g ρ g = 2 g , these decay modes are suppressed ¢ -model would become fully strongly- ρ − = , the scale at which new states appear. σ g ρ ¢ W ¯ t m 0 b (large Z → πf → can be probed at colliders by studying pair- − H and the indirect signal gains importance. While ρ ρ – 28 – − H ¡ ρ ρ up to 6-8 TeV. With complementary information m ¡ m ρ BR up to 5–7 TeV, mostly testing the existence of m 3 1 BR , we have described the SILH in terms of the two pa- i πf = = c ¢ ¢ ¯ ν − up to about 4 TeV. These studies are complementary to Higgs µ comes from the experimental constraint on the to the top can be larger than , the scale at which the ρ hW is obtained from the theoretical NDA requirement → πf H πf m ρ ρ − H → ). However, at low 4 ρ ρ m . ¡ g − H ρ . An alternative description cane be done in terms of two mass ρ g ρ ¡ g BR BR (small and branching ratios are grows, the experimental identification of the resonance bec ρ ± H ρ πf m space. Precise measurements of Higgs production and decay r ρ g ρ m Searches at the LHC, and possibly at the ILC, will probe unexp On the other side, the parameter For order-unity coefficients The – . These measurements can be improved with a luminosity upgra πf y will be able to explore values of 4 while a lower bound on 4 see eq. (2.18). interacting in the absence of new resonances, and Analyses of strong gauge-bosonbe scattering sensitive to and values double-Higg of 4 c physics studies at a linear collider could reach a sensitivi An upper bound on precision measurements, as they test only the coefficient rameters They can be chosen as 4 characteristic of a strong electroweak-breaking sector wi hard, not only because thebecomes leptonic large. signal is Detection suppressed,experimentally of bu challenging a and broad the resonancethe study decaying region of of into large indirect signa that, as and gauge and Higgsmodels bosons the then coupling provide of the dominant decay ch production of longitudinalmore gauge directly, bosons by producing and the new Higgs, resonances. by For fixed testi The resonances are most easily detected whenHowever, they as decay dire shown in eqs. (4.22)–(4.23), for large the search for new resonancesgauge is most vertices favorable at at the the LHC, ILC pr can test LHC will overwhelm the indirectat signal of large longitudinal 4 gaug from collider data, we will explore a large portion of the int effective in constraining the parameter plane, testing the composite nature of the Higgs. JHEP06(2007)045 ing (5.3) (5.1) (5.2) is the metry, cust . m leg added to ) L R R q t t µ µ γ γ L where R ¯ q ¯ t H )( is to reduce the scale since their effects are µ R cust s, the right-handed top in eq. (5.3) contributes 2 ρ t R D c µ m ence. In models in which we can satisfy the bound q † ogical implications of this 4 γ . /m ρ are strongly coupled. We c H ) ∼ R 1 ¯ t L m ( 2 of the SILH, the top quark is on 2 for the case of the Higgs q (1) L ξ , y effective Lagrangian can be troweak-breaking sector. Here H t transforms non-trivially under 2 f ticing that each µ 4 2 ic ≪ f γ c R op quark and therefore it violates ¶ t L + q f + Λ and cust )(¯ R µ L t L m t h.c. ). We find three dimension-6 operators q 2 R µ 2 c µ γ / + transforms as a singlet. This guarantees γ 1 ρ R ¯ 02 t L R . has, however, severe constraints from flavor R at the one-loop level q t (¯ H fm ˜ L b ( Ht q µ 2 T can be viewed as originating from an insertion q 4 = 0 / L f c D ¯ y q – 29 – † 2 . The first term of eq. (5.1) was already included c ρ 4 H + Λ . Assuming H g † − 2 f R L t 2 v 2 R t q H c : i f π 2 R t ic c R 2 σ y 002 translates, via eq. (5.2), into a severe upper bound c t . 16 q µ f + 0 c N γ L < ∼ + ¯ q ∼ h.c. can be easily satisfied in models in which the strong sector b dimension-6 operators in the low-energy Lagrangian involv R T H b T + µ 2 R Hb c R D i /f L ¯ q σ ˜ line. The second term of eq. (5.1) violates the custodial sym Ht † H L . The difference 1. † ¯ and involving H R q y t c 2 ∼ H 2 H . For example, the operator proportional to (3) L † b f L R 2 /f y ic b c H q f t c + 2 y t on the ): f c L 2 the 95% CL bound , b even if L . This bound on ρ t ξ b H/f T † m 2 R Let us first consider the case in which, in addition to the Higg Similarly, we can consider the case in which c = 0 at tree-level. Another possibility to evade the bound on = ( H ∼ L R leading interactions carries an extra factor 1 The possibility of having a strongly-coupled 5. Strongly-interacting top quark In section 3 we have seen that, in some explicit realizations required to be strongly coupledwe to want the to resonances study, of instrongly-coupled the a top elec model-independent quark, way, much the inboson. phenomenol the same spirit of secti also belongs to thewritten by strongly-coupled generalizing sector. the rules 1, The 2 low-energ and 3 of section 2.2, no suppressed by 1 on We are not considering dimension-6 operators suppressed by where Λ is the scaleΛ that cuts off the one-loop momentum diverg preserves a custodial symmetry under which c physics due to smaller than those in eq. (5.1) for large Λ in eq. (5.2). This can be achieved in models in which of in eq. (2.15). Nevertheless, herethe it universality is of only present for the t and therefore it generates a contribution to the custodial group as discussed in section 3. In this case Λ mass of the custodial partners of the from have now the following 1 q JHEP06(2007)045 , al ¯ νν and s with (5.7) (5.5) (5.6) (5.4) (5.8) (5.9) . An X ¯ tt L h b γγ e generic → → B , − , the bound in , generates the ℓ = 800 GeV and ub Z + s ℓ V ¯ tth, h s √ X → ngle . → at the couplings gg models whose low energy ¶ mass eigenstate. From the B ], within the framework of ξ , o estimate the new-physics ed by the d ng an alignment of R f , ifications to the couplings of ) c . í L t W ρ P H 2 θ , . , f c m m 2 3 t ρ T 3 , one can reach a sensitivity up to 2 bd − + 1 can also be tested in flavor-violating into the θ ln m m R − − y t 2 2 10 2 t B L R 8sin c 2 W v f t b W m × ) ln θ R + m − B does not exceed 20% of the experimental tj 2 2 t 1 V W t W m c Zt B ∗ q θ < x µ ³ ti sin = 1 processes like 4 ( m 2 V 2 ξ = 300 fb f ξc W can only be measured with accuracy at future mesons F F ¶ ξf – 30 – W Q θ − sin 2 3 L ( θ R 2 R ub bd B 1 π c µ t . Thus, we will not further consider this possibility. in units of the SM contribution θ h = V 2 ξαG L R sin ¯ = q fγ t µ √ R ij B W g 3 cos c c 4 Zt q j L 2 m 4 ij m d gm SM ij ˜ c 2 = − ˜ c ∆ µ ξc can be measured in the process γ , etc. The typical experimental sensitivities or theoretic ij = = i L ˜ c ¯ /ǫ d ′ htt R ǫ ij t g htt ˜ c , R g = 500 GeV and Zt ¯ νν g s + √ π parametrizes the projection of are the electric charge and the third isospin component of th up to 5% can be reached at a linear collider with → bd f θ [29]. The coupling T + is the left chiral projector, and htt 1 g K L − , P and − , ℓ f 04 [29]. Deviations on the SM vertex f , the mass difference of neutral . + are given by Q B 0 ℓ colliders. For R m Flavor constraints have been studied in detail, e.g. see [39 ∼ t − → R e = 1000 fb R ¯ s t + effective operator value, we obtain fermion where processes. For example, one-loop penguin diagrams, mediat requirement that the new contribution to ∆ B accuracy on ξc Z Therefore, unless there is a flavor-symmetry reason for havi L At the LHC the coupling to ∆ The presence of the operators (5.1) gives non-universal mod the top to the Higgs and gauge bosons. In particular, we find th where the angle uncertainties of these processescontribution, it is is no useful better to than express ˜ 10–20%. T This operator contributes to many rare ∆ e warped extra dimensions andeffective they description apply is described to by holographic our Higgs Lagrangian5.1 (2.15) Phenomenology of a strongly-interacting a mass eigenstate more accurate than the corresponding CKM a eq. (5.5) disfavors a strongly-coupled JHEP06(2007)045 . , ¯ t ) t 1 t 2 D as 4 − c ti and, /f θ 2 (5.13) (5.12) (5.11) (5.10) ξ v a little R )( 2 c . On the 2 where a b S π /f ¯ t 2 16 ¯ tt v with 100 fb t / . 2 . 4 g → − 34 . 0 , T C 10 pp MeV MeV. 2 N ≃ × ) ( or proportional to 11 11 ] gives also contributions Zt − − ∼ x t . e obtaining W t nts, this prediction is not ontribution to 4 scattering that grows with b 10 10 S x etermined by extra param- c ) of 2 2 − ¯ t section tter in electroweak data for imple technicolor is however t × × stone bosons, a light pseudo- ¶ a mild constraint on cZ larger than the corresponding Zt ) ln 4 2 )(1 cb tc with respect to Standard Model e of the absence of a light Higgs x haps most dramatic such possi- θ tc V n in some new strong sector then 2 → Zt < W t θ can potentially lead to observable x µ , the mass difference of neutral t ¶ x D ( 2 D R ) ub tu c in rare top decays. By defining m θ m V R BR R µ 2 ξc (1 + 2 c (3 + 2 ) ( 2 2 4 ¶ W t W t ¶ − x x cb tc W θ 10 V tc − θ − µ × – 31 – gθ state and the mass eigenstates, we find t (1 R 2 W t 4 R x t ξc ξc 16 cos ) = 2 = + 6 µ t cZ 2 tu π W t θ 2 m x 2 → tc 7 θ t ( 2 2 )= D scattering, this operator induces a − v f cZ BR D 2[1 m and → WW t 4 2 t . This gives a branching ratio t )= 2 t ξc for Γ( . /m 3 2 W t is possible, but requires a mixing angle /m H cb x 2 W c ( 2 Z = R V c f m m D = = m ∆ W t Zt x x Let us finally comment on possible implications of the operat signals. It is also interesting to consider effects of the mixing angle between the current For mixing angles of thefar order from of the the present corresponding experimental bound, CKM which eleme is ∆ where with improved experimental accuracy, the coefficient CKM element Since the LHC is expected to reach a sensitivity on where Analogously to 6. Conclusions If the weak scale originates fromthe dimensional physics transmutatio of the Higgs willexpectations. manifest important Technicolor deviations represents thebility: simplest no and narrow per state canat be odds identified with as electroweak precision theresonance. tests Higgs and boson. Models largely where, S becaus inGoldstone addition to Higgs the appears three in eatentwo the Gold reasons. low-energy theory On can one fare hand, be a light Higgs screens the infrared c This shows that present limits from flavour physics give only a signal from energy. At the LHC this will give an enhancement of the cross- to flavor processes. For example it contributes to ∆ mesons: bit below 1, which is enough to suppress the UV contribution pair is produced by the new 4-top interaction. The coefficient other hand, the vacuum dynamicseters of (SM the couplings pseudo-Goldstone among is d them) and therefore one can imagin JHEP06(2007)045 are g c hile in and , is dictated γ ρ c g is large. , indicating the 2 ρ ) g ρ past on the effective his paper, these new /m ory [7]. A crucial test ρ . The rest of the terms g represent genuine “form gluon and photon fields. One can distinguish two ( aracterize these theories alizable operators in the these are the masses and /f dea of a pseudo-Goldstone g c pactifications (Holographic imary goal of this paper. ∼ ich allow for a quantitative provided sensitive to the new strong e favored to be large, others the construction of such an from the SM in the physics thus encompasses all models ch for new resonances at the tone Higgs. This is of course CD-like proposals. For what = 1 should nicely complement the ly-coupled nature of the Higgs. he Higgs mass. In the limit proach is best motivated in the ρ and dressings of the quadratic (free) to the spectrum. The “new cou- . This second factor, which deter- γ ), we have derived the form of the 2 c ρ /m olor, less effort has been devoted to ρ , where the resonances are narrow. ρ , ) that can be seen as arising from the ρ g /g 2 ρ g B ,m c ρ 2 SM , g /m . In 5D models, these are respectively g ρ W g c )(1 2 π – 32 – 16 / , and suppresses these effects at large where the resonances become heavier and broader. , are determined by an expansion in the Higgs field, 2 SM f coefficient. Remarkably, the coefficients y ρ g ρ c ( 2 ρ g g ) factor times 2 ∼ = /m f and ρ 2 and a coupling are special in that they have a structure similar to the form 6 π m c ρ , (16 HB m T / c O , H c and is large. Models based on 5D are favored to be in this regime, w HW ρ O g , these models coincide with a generic strongly-coupled the π 4 After its proposal [7] and some work in the eighties [8], the i Using our simplified description in terms of ( → ρ the Kaluza-Klein massself-couplings and coupling. that regulate Ing the Little-Higgs quadratic models divergence of t Indeed, the leading dimension-6 operators have a coefficient of these theories willLHC. undoubtedly This proceed is through certainly the more sear true at small enough below the experimental bound. While work has been done in the low-energy description of Higgsless theoriesthe like technic construction of anpartly effective justified theory by for the thedescription pseudo-Golds actual of existence the ofeffective resonance specific theory sector. models is an wh important Still, task, we and this believe has that been theHiggs pr was recently revivedGoldstones) by and its in realizationsmodels Little in represent Higgs warped weakly-coupled com models. variantspertains of low-energy the As phenomenology, original wein Q we emphasized terms found in of it a t mass useful scale to ch relevance of these effects even when the resonances are heavy However, these theoriesof also the predict Higgs. importantlow-energy deviations description. These The deviations studydirect are of searches, these especially associated indirect for to effects large non-renorm leading dimension-6 effective Lagrangian.with Goldstone Higgs, Our although description theregime effective-Lagrangian ap where Little Higgs models therecan is more be freedom weak). (someclasses couplings Our ar effective of Lagrangian effects, is theforce, shown and “new in the couplings”, “form eq. factors” (2.15). whichplings”, which described are are by basically genuinely sensitive factor” effects since they have a 1 product of a strong loop 1 associated to a suppression factor and test its strong self interaction, characterized by by the Goldstone symmetryThe and operators by its preservation by both the in eq. (2.15), can beHiggs action: basically as viewed such as they higher-derivative More do not precisely, test the equally well operators the proportional strong to mines the dependence on just JHEP06(2007)045 . n of 2 SM 2 g , the 1 with a /f − ≫ 2 smaller) n to the v 2 ρ 2 hh g π on → ght Higgs are 16 / ector tends to L 2 ρ , in the absence s form factors. V 2 g L /f V largely improve the 2 v g issue which may be elative size scales with wth and the amplitude erence in these searches. l couplings of the Higgs, t the idea of a composite , given that they arise at ned by observing the self e corrections is the Higgs ess theories: through the gs. A direct assessment of decay channels at the LHC nd. Of course the upgraded le gauge vertices at the ILC igh-energy scattering among alars, would be an indirect anomalous couplings between zed vector bosons. The way dinal vector boson grows like up to 30 TeV. If, at this level, Higgs production. Given the the LHC. With 300 fb from elementary fermions. In ) and in particular to the top y han the “new couplings” at mea- , like for the “new couplings”. In ilar (indeed even πf c ρ 2 TeV), or what can be achieved at g 3 TeV). > ∼ ∼ ρ ρ m m ( b S – 33 – should be tested in vector boson production, where ) that depends on HB 2 ρ . On the other hand the effects of the “new couplings” c 01, corresponding to 4 2 . m 0 2 /f π and 2 is between 0.5 and 0.7. Our study further motivates analyses ∼ v -jets from Higgs decay and the presence of a rapidity gap, it ) 2 (16 b 2 ρ / HW /f up to 6–8 TeV. This sensitivity is far superior to what has bee 2 ρ c /g 2 g ρ v 2 SM m g down to of the ) and Higgs coupling to fermions ( . As the Higgs plays the role of a fourth Goldstone, in additio and thus dominate for a strongly-coupled Higgs sector with 2 . In particular their effects on the on-shell couplings of a li T 2 H = ( 2 ρ 2 p c /f should allow a test of these interactions with a sensitivity 2 ρ /f 2 channels we also have strong double-Higgs production . In this respect these effects can be practically classified a /f 2 /m v 1 2 2 L . Therefore E /m v − ρ V E W,B 2 W L c V /m m scattering, even in presence of a light Higgs. An interestin ρ 2. The detection of a deviation from the SM in this range of . g . In our case, the light Higgs fails to fully moderate this gro → 2 One aspect of theories with a composite Higgs is that the top s Through precise measurements of Higgs physics, the ILC will The form factors lead to corrections to SM amplitudes whose r L /v WW V 2 L the LHC, through direct resonance production ( reached at LEP, through the measurement of practice, however, they are experimentallysuring less relevant t no deviation from thelight SM Higgs is is detected, ruledcan it out. test will Measurements the be of value fair the of anomalous to trip say tha their contribution relative to the SM is quantitatively sim luminosity of the second phase of LHC would makesensitivity a on crucial diff may be possible to selects these events over the QCD backgrou factors but a coefficient expected sensitivity on comparable cross section. Itlongitudinal is well vector known bosons that is the not study of a h straightforward task at to that of of studied is the possibilityhigh energy to and detect the reaction of double- energy like of order are of order order 0 V behaves like of new lightbut states, clear and signature in ofthe particular new strongly-coupled strong of nature dynamics additionalinteractions of involving light among the the sc the Hig Higgs Higgsto would and only study the be these longitudinallyscattering obtai interactions polari of is vector the bosonsthe same that Higgsless as were case, in collinearly theE radiated ordinary scattering Higgsl amplitude among longitu quark. The measurement ofwith all possible 300 Higgs production fb and Our conclusions differ from theHiggs, widely photons held expectation and that gluonsone-loop should level be in the theincluding most SM. important those effect We to do photonsself-interaction find and ( corrections gluons, to but all the on-shel origin of thes JHEP06(2007)045 H at ets ne ρ (A.5) (A.3) (A.4) (A.1) m elax some . and assuming † ) v (Π) one has then , g g ∼ (Π f → h µ tudy of this and other ∂ ) A2005-02211 and DURSI work. T-2004-503369. The work , g rite terms in the Goldstone transform under a local ix, we predict flavor effects µ ele, G. Polesello, M. Porrati ) goes all the way to being E is associated to the reaction R ing out fields of mass µ bly, for i (Π t ) (A.2) . Although model dependent, E ons. This work has been partly + R ih , g s to generate Yukawa couplings, t or µ , − ) † L (Π and D ) † t i a ) , g ξ µ , g ≡ , g ) possibly within future experimental D (Π ≡ a (Π h a (Π h T ξ h a µ , ... E to the strong sector. It is then natural to (Π)) i D (Π) g (Π) µ t ( m + µ y D – 34 – E U ρ ) . Under the group action Π A ) vertex. In section 5, we have therefore extended g , ∆ T H , g ≡ , g √ A µ bb ¯ a cZ (Π T D Z (Π ∼ a i h (Π) h iξ → = e gU t U = , µ h ∂ G on the Goldstone operator Π, defined in eq. (2.4), is given by (Π)) = transforms like the associated gauge field. Massive multipl and the νν † (Π)) = g g are the broken and unbroken generators respectively, we defi + ( µ U ( µ π E δρ a µ ∈ G T g → → E → D + and also has phenomenological advantages, in that it allows to r K A (Π) (Π) , T µ µ R − t E D ℓ + . If ℓ † s X gUh → → B a mixing pattern that follows( the size entries of the CKM matr couple with intermediate strength reach. However, the leadingof signature four of top-quark topimplications production. compositeness of composite We Higgs and plan top to at perform the LHC a inAcknowledgments detailed a future s We would like to thank I. Antoniadis,and A. especially Ceccucci, R. T. Contino Han, and B. M I.supported Low by the for European valuable conversati Commission underof contract MRTN-C A.P. was partly supported byResearch the Project FEDER SGR2005-00916. Research Project FP A. Integrating out vectors and scalars Here we describe the low-energy action obtained by integrat consider the possibility that one of the two helicities ( composite. Moreover, in mosta of the composite realistic construction our effective Lagrangian to the case of a fully composite there are important implications in flavor physics. Remarka significant bounds from is an element of the unbroken subgroup U where Notice that for space dependent Π configurations, with transformations under tree level. We shall needaction. the standard The CCWZ action notation of [13] to w symmetry, in particular JHEP06(2007)045 - ). ≡ H x . It can ( 2 U h (A.8) (A.9) (A.6) ) G O (A.10) µ defines follows, ≡ µ iA g E otons and + µ ∂ o that one may ( † , U )] emerge from ¯ , E transforms as the ) ee level in minimally ( 5 , a µ ) (A.7) 2 HB µν ¯ ) 4 E is a symmetric space.) (Π d to couplings involving cussing on the case of a µ O F ν O (Π ctures H re known to give rise to a ¯ ast four Goldstones. D / O e non-linearly realized −E ates. This is indeed the case + µ and G µ ¸ ¯ ve two gluons and two Higgses. derivatives. On the other hand D . Thus V Higgses. One distinctive feature Π l in a minimally coupled theory ( Π+ . The action of the global group µ 2 ρ HW µ SM . These are however shown to give H m O ν D ←→ G = Tr[ D 2 1 ¯ ←→ D . In the ungauged limit we have the 2 Π . ν Π , + ¯ H i . Then the most general two derivative 2 D Π gauge group [13]. In particular 2 )Φ H⊃ G µ ) · − ¯ , while at lowest order in Π , g D 1 6 H V 1 µν µ µ ¯ F D O ¯ (Π − A ( D h 4 1 – 35 – under Π or µ -covariant derivative, one could write down other and , O → )= µ − ]. Substituting eqs. (A.7)–(A.8) into eq. (A.9) we ν ν D E )] ν ¯ H µ E A ¯ Φ ¯ ¯ D E E ( , ν on the massive fields Φ. D , D tranformations µ )= ¯ A -tranformations and under the genuinely local µ D µ ¯ µν E ν (Π H G [ µ E D is obtained by changing eq. (A.3) to i F covariant derivative. (In the last second equality of both i µ 2 ρ ¯ ) D E , D ¯ the full through the exchange of heavy vector states. Let us briefly E + µ + m H ( ¯ + SM (Π µ µ D µ B = ¯ ¯ µ ∂ E µ µν G E , c i ν 0 ν F D A ∂ ¯ + L D + 4 ρ is the ν = = emerge by expanding − µ W ¯ m µ to transform like D c µ µ ∂ ν ). Indeed since we treat the gauge fields as spectators in what B = Tr[ is that they give rise to interactions involving on-shell ph ¯ ¯ ν E ¯ E D µ transforming in the adjoint of iA O 1 ¯ µ ∝ ≡ D V ∂ O µ b π, A + µ S HB V ¯ ( D ¯ µ D µ O and do not lead to any extra interactions for on-shell photons, s ¯ ∂ E ) = i ¯ E W B = ( O and O )+ µ µν F D -covariant derivative and HW π, A We want to classify the 4-derivative structures that can lea The question remains onto which effects can be generated at tr The weak gauging of H ( O µ is realized through the “local” W -invariant action is given by ¯ D where two Goldstones and two gauge fields. There are 2 relevant stru where equations we have specified to the interesting case in which find that fill reducible representations Φ of the unbroken group be written, just by using the rules for a “local” structures like coupled theories, such as Holographicof Goldstones or Little either the same effects at dimension 6 or terms involving at le Using the last 3 equations the most general Lagrangian for th gauge field both under the global electrically neutral states. Thiswhere photon cannot interactions occur are at purelyO tree dictated by leve covariant is also evident that operatorsIndicating of the by above form cannot invol expect them to arisefor at tree both level Holographic bycontribution integrating Goldstones out to and heavy Little st Higgses, which a the option to choose outline how this effects come about within our formalism by fo massive vector G we can, without loss of generality, gauge the full It is useful to have the expressions of G i JHEP06(2007)045 . ρ m (B.1) ∼ (A.13) (A.12) (A.11) we find . Notice , the one ′ V A . . The second ) B ρν O have more than B ρ . . . µ ∂ . (A.10) mimics the are irrelevant to our and )( )+ −E ¯ µν E W ( µ B O ν V ρ ) precisely like ars. At the two derivative other possible contraction µ strength and with at least x F ow energy, diagonal SU(2), ∂ ( m is generated in general ρ Higgs model with a product ( ≡ h 2 ρ s derivative terms that makes . D matically eliminates the term lve then at least 4 derivatives. n are now given by µ ise to 2 ρ , where we have suppressed the = m ˆ gs: V ¶ 1 2 µ ρ ¸ pure gauge kinetic terms m g 2 g 3 D v 1 2 h + µ 3 . . . , 2 ) D = + + ∂ H ) B 2 4 v 2 ¯ 2 v E h 4 h ( O D ¶ and Φ , we can eliminate these derivative terms by + µν )( ξ † H µ F h c µ 3 H D µ in the above equation. According to this change 2 – 36 – ), cannot arise from scalar exhange as it involves D 25 Φ ξ ¯ 2 ρ E µ D H 2 H ( -quantum numbers of Φ and the contractions ensure g 1 − µ c D 2 2 ρ ) 6 ]. Notice that terms involving covariant derivatives H 1 D c ν m − ν − 6 E → iρν ) h , V µ D ¯ µ W E )( ( ρ + V → † [ i 3 D µν h H h )( transform thus like gauge fields. By integrating out F ν + ) ¶ i ¯ D µν µ E V H ( µ V playing the role of the vector boson of the second SU(2) 2 c W ν 3 D µ µν ∂ V which would imply the presence of a scalar ghost with mass F and D − 2 ρ − ( 2 6 g 1 A 2 ρ then amounts to ν ]) c 4 µ V m 1 H − µ 2 ρ ˆ ·µ V g ∂ , 2 = 2 H v µ 2 = m L V = [ i ∆ V µν − W − F 2 ½ ξ µ O Let us consider now the effect of integrating out massive scal = 2 part. The more interesting terms arise at dimension 8, and ˆ V = will also have to transform under the genuinely local µ J -invariance. Integrating out Φ, the leading operators invo h ∂ L performing the following non-linear redefinition of the Hig term, instead, gives rise to four derivative corrections to analysis. B. Effective Lagrangian in the canonicalThe basis first term of thedifficult effective to Lagrangian read (4.1) off involves physical Higg effects. At order a where the dots indicate4-derivatives. terms As more we than already quadratic explained in the the first term field gives r At dimension 6, one is then easily convinced that only one ter which can be induced byat the dimension exchange 6 of order, massive ( doublets. The The gauging of H V that the structure we havegroup thus structure outlined with is the same of Little where acting on the homogeneously transforming combination level we can have mixings of the type After this redefinition, the corrections to the SM Lagrangia indices; we assume, of course that the two derivatives. A( limitation to two derivatives, also auto under which the SMeffects of fermions the are Goldstone uncharged. whichunder breaks which The the both gauge mass group term to in the, l eq the following correction to the low-energy effective action JHEP06(2007)045 2 ρ ¸ gs ν A ξ/g . . . (B.3) (B.5) (B.2) (B.4) 2 ¶ g + W ¶ θ Z e operator ˆ c µν Z tan µν − F W θ γZ W c c 2ˆ µ W sin 2 Z o compare it with the θ µ g the Higgs equation of 2 µ Z + . ¶ ν 2sin , Z xtract the predictions of these · · · ve decays in eqs. (4.11)–(4.13), ¶¸ − Z ˆ c í H + µν · y 2 c c 2 Z 3 2 µν h v + µν 2 D − , µ Z 6 H + 2 i c Z c ξ v h 2 ρ µ ³ 2 HB g H 2 µ g c ξ ξ c 2 ¸ 2 Z W − − θ m )+ 2 1+ . 1 . . . . 2 1 – 37 – h · H c coupling, we have h 4 . + c f 3 2 2 H ¸¾ v W W W − h h 2 +h µ +tan m m a µν ) gm + gm 2 4 ν gm − G H c and W HW = W = = c aµν and at zero momentum (or, equivalently, neglecting + + µν µ y G + ξ c D W g hhh hff hff − − 2 c µν ρ g g µ . For this purpose it will be useful the give the Higgs couplin , ¶ hWW 2 i t g c W y +(3 W πg s 4 · · · h µν W α c ¶ + + (ˆ + hW W 2 H 2 2 ρ 2 2 W h c g v µν ) corrections coming from matching the SM contribution to th g 2 2 F S + 2 + y α HW ¶· µν c ( c v h 2 2 F O µ v µ ·µ h γ 2 2 ρ 2 c 2 W ) ¯ 2 2 ff + πg π m g f 4 v h αg v (4 H m µ c + − + + − Here we give the loop functions describing the Higgs radiati C. Loop functions for the Higgs radiative decays including the Eq. (B.2) could alsomotion. have been The obtained Lagrangianeffective Lagrangian from eq. arising from eq. (B.2) specific (4.1)models SILH is models, for by in and the usin e coefficients a more suitable basis t corrected by eq. (B.2) at order corrections). For the JHEP06(2007)045 B ) B B , (C.8) (C.7) (C.6) (C.3) (C.2) (C.4) WZ (2002) ) (C.1) New J. ,x ]. 07 , WH x WH ( ]. x 1 Nucl. Phys. ( Nucl. Phys. Phys. Lett. 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