DDFPD-2015/TH/11
Beyond the Minimal Top Partner Decay
Javi Serra
Dipartimento di Fisica e Astronomia, Universit`a di Padova & INFN, Sezione di Padova,Via Marzolo 8, I-35131 Padova, Italy [email protected]
Abstract
Light top partners are the prime sign of naturalness in composite Higgs models.We explore here the possibility of non-standard top partner phenomenology. We showthat even in the simplest extension of the minimal composite Higgs model, featuring anextra singlet pseudo Nambu-Goldstone boson, the branching ratios of the top partnersinto standard channels can be significantly altered, with no substantial change in thegenerated Higgs potential. Together with the variety of possible final states from thedecay of the pseudo-scalar singlet, this motivates more extensive analyses in the searchfor the top partners. a r X i v : . [ h e p - ph ] J a n Introduction
Composite Higgs models aim at solving the electroweak hierarchy problem by postulating anew strongly interacting sector that dynamically generates the Higgs field. This emerges asa pseudo Nambu-Goldstone boson (pNGB), which explains why it is parametrically lighterthan any typical composite resonance. Ultimately, the Higgs is screened from high energyscales on account of its composite nature [1–4].In this class of scenarios, a set of vector-like composite fermions linked to the top quarkis responsible for keeping the Higgs potential under control [5, 6]. As long as the mass of thetop partners is below the TeV, the electroweak scale and the Higgs mass can be reproducedwithout significant fine tuning. This follows from a simple estimate, based on power countingand selection rules, of the size of the Higgs potential V (cid:39) − µ | H | + λ | H | generated byloops of the top and its partners,∆ µ ∼ y t π m T ≈ (90 GeV) (cid:16) m T
500 GeV (cid:17) , ∆ λ ∼ y t π g T ≈ . (cid:16) g T (cid:17) . (1)The top Yukawa coupling y t is the largest coupling in the Standard Model (SM) that explicitlybreaks the global shift symmetry protecting the Higgs. The top partner mass m T controls thesize of the potential, while the top partner coupling, defined as g T ≡ m T /f , where f is thecompositeness scale of the Higgs, determines the physical Higgs mass once the electroweaksymmetry is broken. These estimates, verified in explicit constructions, point towards lightand weakly coupled top partners saturating the radiatively generated Higgs potential [7–10].The absence so far of any evidence of Higgs compositeness, in electroweak precision testsor Higgs couplings measurements, has pushed the scale f to somewhat unnatural values f (cid:38)
600 GeV [11, 12], and driven with it these models into the (cid:46)
10% fine-tuned territory.Besides, the ATLAS and CMS collaborations have also directly searched, without success,for the top partners potentially produced during the first LHC run [13, 14]. The lowerbounds placed on their masses, m T (cid:38)
800 GeV, have started to build up the tension withnaturalness. With the increase in energy and luminosity that will come with the second runof the LHC, the mass reach of direct searches will be substantially higher. Such an upgradewill provide an excellent opportunity for uncovering the symmetry mechanism protecting theHiggs potential and the agents implementing it, but it will also become a crucial test of theidea, given its present degree of tuning. In this regard, it is very important to understandthe level of model dependence involved in the actual experimental searches of top partners.Such searches are mainly based on pair production through QCD interactions (and seldomon single production via electroweak interactions), and decays to W ± , Z , or h , plus a topor a bottom quark. However, there exist models, implementing the twin Higgs mechanism,in which the Higgs potential is controlled by top partners that are neutral under the SMgauge group, in particular under SU(3) C color [15]. This possibility, although theoreticallychallenging, provides a proof of principle for natural theories with no direct signals at the1HC, at least of the standard kind. Another, more modest, approach towards unusualphenomenology for the top partners regards non-standard decay channels [16, 17]. Thesecould proceed via new light states, a natural option being other pNGBs. In fact, givenour ignorance about the UV degrees of freedom participating of the strong dynamics, theappearance of extra light scalars in the IR is a well-motivated possibility. The mass of theseextra scalars could receive contributions, along with the Higgs, from top loops, in which case∆ m ∼ (cid:15) y t π m T . (2)This kind of contributions are generically below the top partner masses m T , given the implicitassumption that the couplings (in the case above y t ) that explicitly break the correspondingshift symmetry, are small perturbations. Besides, extra parameters, denoted above by (cid:15) ,must always be kept in mind, to account for the different selection rules associated with theextra shift symmetry. These could actually render the entire top contribution vanishing, andthe extra scalars naturally very light, as much as allowed by experimental searches.In this work we study the feasibility of non-minimal top partner decays within a compositeHiggs model featuring a single extra pNGB, in addition to the Higgs complex doublet. TheNext to Minimal Composite Higgs model (NMCHM) is based on a global SO(6) symmetryspontaneously broken to SO(5) [18]. The extra light scalar η is a singlet under the SM gaugesymmetries, and we further take it to be a CP-odd state. We will show that in this scenarioa subset of the top partners can have a significant branching ratio into the pseudo-scalarsinglet and a top quark, becoming even the dominant one under some circumstances. Thiscomes about without affecting the level of tuning required to reproduce the Higgs potential.By focussing on two specific examples we will show how this can be possible. On the onehand, new sources of explicit breaking of the global symmetries can be introduced that, whilegiving rise to a dominant coupling of the top and its partner to η , do not directly break theshift symmetry protecting the Higgs. In the case these extra interactions do contribute to theHiggs potential, they do it in such a way as to reduce the overall contribution. On the otherhand, the extended global symmetry structure predicts additional top partners that decayexclusively to the singlet. The phenomenology of η is mainly dictated, as that of the Higgs,by considerations regarding the symmetries of the low energy effective theory. Dependingon those symmetries and their breaking by the interactions with the SM fields, the singletcan present a varied pattern of decay channels, and therefore also the final products of thedecays of the top partners can be variable. Moreover, given that the compositeness scale of η is the same as that of the Higgs, the phenomenology of the singlet is mainly controlledby dimension five operators suppressed by f , which are not usually considered in colliderstudies of this type of scalars. The extra decay channel of the top partners to η , along withthe diversity of decays of such a scalar, motivates extended searches for both particles. This model can be realized as a theory with four flavors of strongly interacting technifermions in apseudo-real representation of the confining gauge group [19, 20].
The Higgs complex doublet and an extra singlet η arise as the NGBs of the spontaneoussymmetry breaking SO(6) / SO(5) ∼ = SU(4) / Sp(4) [18]. This coset indeed contains five scalardegrees of freedom, transforming in the representation of SO(5). This decomposes as a + = ( , ) + ( , ) of the custodial symmetry SO(4) ∼ = SU(2) L × SU(2) R . The associatedGoldstone matrix, U (Π) = exp (cid:0) i √ i ( x ) T i (cid:1) , can be conveniently written as U (Π) = × − h √ − h − η − hη √ − h − η h − hη √ − h − η − η √ − h − η η − h − η (cid:112) − h − η , (3)where we have eliminated the three NGBs eventually eaten by the W ± and the Z . TheGoldstone matrix transforms as U (Π) → gU (Π)ˆ h † (Π , g ), with g a global SO(6) transfor-mation and ˆ h a local (dependent on Π( x )) SO(5) transformation. When constructing theeffective Lagrangians for the NGBs, we will often make use of a projector into the bro-ken directions Σ = (cid:0) (cid:1) T . With it we can define the Goldstone multipletΣ = U (Π)Σ = (cid:0) h η (cid:112) − h − η (cid:1) T .The kinetic term for the NGBs is given by the leading invariant term in derivates, O ( ∂ ), f d iµ d µi = f D µ Σ) † ( D µ Σ) = 12 ( ∂ µ h ) + 12 ( ∂ µ η ) + 12 ( h∂ µ h + η∂ µ η ) f − h − η + g h (cid:18) W + µ W µ − + 12 cos θ W Z µ Z µ (cid:19) , (4)where we have given dimensions to the NGBs: h → h/f , η → η/f . The object d µ is defined as d iµ ≡ − i Tr[ T i U † D µ U ], and it is one of the basic building blocks of our effective Lagrangians(see appendix A for more details). As in the SM, once h gets a vacuum expectation value(VEV), (cid:104) h (cid:105) = v ≈
246 GeV, the weak gauge bosons W ± and Z become massive. It is ofphenomenological relevance that h has extra derivative self-interactions and interactions with η . The first implies that after electroweak symmetry breaking (EWSB), the kinetic term of3he Higgs receives an extra positive contribution of order v /f , which has the net effect ofsuppressing all of the Higgs interactions. The second gives rise, if kinematically allowed, toa non-standard Higgs decay to two η ’s controlled by 1 /f .As far as the interactions in Eq.(4) are concerned, the scalar singlet can either be CP-evenor -odd. Actually, the Lagrangian Eq. (4) is invariant under a set of discrete Z transforma-tions that act individually on each of the NGBs as Π i → − Π i , as well as under the spacetimeparity P : x → − x , t → t , Π i → Π i . We will be particularly interested in the combination CP = C P P , which defines the CP symmetry of the NGBs in SO(6) / SO(5): h → h and η → − η . The automorphism C is identified with charge conjugation, while P correspondsto the grading of the algebra, under which all the unbroken generators remain unchanged T a → + T a ∀ a , while the broken generators change sign, T i → − T i ∀ i . In this work we willassume that the strong sector respects CP, and that it remains unbroken to a high degreeof approximation by the interactions with the SM fields (keeping in mind the amount of CPviolation needed to reproduce the SM). This assumption is in fact necessary to avoid toolarge contributions to CP-violating observables. Furthermore, the SO(6) / SO(5) coset admitsa Wess-Zumino-Witten (WZW) term [21], arising at the next to leading order in derivatives, O ( ∂ ), that respects CP. This term could play an important role in the phenomenology of η , since it gives rise, at leading order in f , to the interactions: ηf (cid:15) µνρσ π (cid:88) a = a C ,a L ,Y n a g a F aµν F aρσ , (5)where F aµν are the field strengths of the SM SU(3) C × SU(2) L × U(1) Y gauge group, g a thecorresponding gauge couplings, and n a the anomaly coefficients, which carry informationabout the underlying UV structure of the theory. In particular, given the SU(4) × SU(3) C global symmetry structure of the strong sector under consideration, n g = 0 and n W = − n B . The potential for the pNGBs depends on how the associated shift symmetries are explicitlybroken. The SM already contains relevant symmetry breaking parameters: the top Yukawacoupling, and the SU(2) L gauge coupling. In order to understand how the global symmetriesare broken, we need to specify how the top quark, q L and t R , and the gauge bosons W a are coupled to the strong sector. The latter is fixed by gauge invariance, that is gaugefields couple to the strong sector’s associated conserved currents L ⊃ gW aµ J µ a . The former In the basis we have used to write the Goldstone matrix in Eq. (3), these discrete transformations aregiven by C = diag( − , , − , , − , −
1) and P = diag(1 , , , , , − P is actuallyan outer automorphism, not contained in SO(6), and generically it should not be respected by higher orderterms in the Lagrangian expansion in derivatives. L ⊃ λ L ¯ q L O q + λ R ¯ O t t R + h.c. [22]. Given that both the operators O q,t andthe current J µ a are part of entire representations of SO(6) (the first to be specified, and thesecond in the adjoint ), while the SM fields do not fill complete SO(6) multiplets, theseinteractions break explicitly the global symmetries. Notice that in partial compositeness thebreaking introduced by the top Yukawa is a consequence of the combined breaking introducedby λ L and λ R , since y t ∝ λ L λ R . We should observe as well that, in order to reproduce thecorrect hypercharges of the O q,t components mixing with q L and t R , an extra unbroken U(1) X global symmetry must be introduced. The hypercharge of the states of the strong sector isthen given by Y = T R + X , where T R is the U(1) generator inside SU(2) R . While the pNGBsdo not carry X charge, we will assume that the fermionic operators O q,t (and their associatedresonances, i.e. the top partners) have X = 2 / L gauge bosons can be derived once we prop-erly identify which generators of SO(6) are associated to the SU(2) L current (see appendix Afor an explicit expression). Such generators, times the weak gauge coupling g , can then beviewed as spurionic fields, from which SO(6) invariants can be constructed. At leading order, O ( g ), they lead to the potential: V g = c g f (cid:88) a = a L Σ T ( gT aL )( gT aL )Σ = c g g h , (6)where we recall that Σ = U Σ . Notice that only a mass term for the Higgs is generated, butthere is no potential for the η . This is a consequence of the fact that η is a singlet underSU(2) L , thus the gauging of SU(2) L does not break the U(1) η shift symmetry protecting thesinglet. We can estimate the size of the coefficient as c g ∼ m ρ / π , where m ρ is the massof the vector resonances cutting off the loop of W ’s. For m ρ = 2 . ≈
20% on the Higgs mass term.In a similar fashion we can identify the contributions to the potential from loops of q L and/or t R . For this we need to specify the transformation properties of the operators O q,t the top couples to, and then the actual interactions λ ¯ t O will only be restricted by therequirement that they should respect the SM gauge symmetries. It will be convenient to usethe embedding fields Q L = b L υ b L + t L υ t L and T R = t R υ t R , in order to write the couplingsof the top to the composite fermionic operators as L ⊃ λ L ( ¯ Q L ) I O Iq + λ R ( ¯ O t ) I ( T R ) I , wherethe index I runs over SO(6) components. Then λ L υ b L , λ L υ t L , and λ R υ t R can be treated asspurions from which we can compute the potential.We will be considering two different sets of representations for the operators O q,t . Forthe first we will assume that both O q,t transform in vector representation of SO(6), with5 = 2 / In that case the embeddings of q L and t R are given by L : υ b L = 1 √ (cid:0) i +1 0 0 0 0 (cid:1) T , υ t L = 1 √ (cid:0) i − (cid:1) T . (7) R : υ t R = (cid:0) i γ (cid:1) T . (8)Notice that the embedding in Eq. (8) implies that t R couples to two different components of O t , with relative strengths set by γ . This parameter can be taken to be real and positivewithout loss of generality. The couplings of q L and t R specified by the above embeddings leadto the potential, at leading order in the symmetry breaking couplings, O ( λ L ) and O ( λ R ), V λ L = c L f (cid:88) α = t L ,b L (Σ T λ L υ α )( λ L υ † α Σ) = c L λ L h . (9) V λ R = c R f (Σ T λ R υ t R )( λ R υ † t R Σ) = c R λ R (cid:2) f − h + ( γ − η (cid:3) . (10)We would like to point out several important aspects of Eqs. (9) and (10). First, there isno contribution to any h dependent term from γλ R , given that this coupling does not breakthe Goldstone-Higgs shift symmetry. Second, there is no contribution to any η dependentterm when γ →
1, given that in this limit there is no breaking of the Goldstone-singlet shiftsymmetry. In particular, the interaction of q L with the strong sector does not break theU(1) η , only those of t R do for γ (cid:54) = 1. This is understood by observing that one can formallyassign to q L a definite U(1) η charge, T η Q L = 0 Q L , and likewise for t R , T η T R = − / √ T R ,provided γ = 1. A simple estimate leads to the coefficients c L,R ∼ m / π , where m Ψ isthe mass scale at which the top loop is cut off. We will compute below in a specific examplethe actual dependence of c L,R on the top partners masses. From the embeddings of q L and t R in Eqs. (7) and (8) we can also derive the top Yukawa coupling: y t f ( ¯ Q L Σ)(Σ T T R ) + h.c. = − y t √ t L ht R (cid:32)(cid:115) − h f − η f + iγ ηf (cid:33) + h.c. . (11)Finally, we must notice that a Higgs quartic term is not generated at leading order in λ L or λ R . This must therefore come from subleading O ( λ ) terms. One of such terms (the onethat actually descends from the top Yukawa coupling above), reads, V λ L λ R = c LR f (cid:88) α = t L ,b L | (Σ T λ R υ t R )( λ L υ † α Σ) | = c LR λ L λ R h (cid:2) f − h + ( γ − η (cid:3) , (12)with the estimate c LR ∼ / π . The fact that any term in the potential involving thesinglet vanishes in the limit γ → λ R . This is the extension to SO(6) / SO(5) of the minimal SO(5) / SO(4) model with the L - and R -handed topembedded in the vector representation of SO(5) [23].
6e will also be considering the alternative option in which the operator O q transformsin the (cid:48) (symmetric and traceless component of × ) representation of SO(6), while O t is just a singlet , both with X = 2 /
3. Since in this case t R has a trivial embedding, λ R does not give rise to any explicit breaking. We only need to specify the embedding of q L , ina symmetric traceless tensor: (cid:48) L : ˆ υ b L = 1 √ × − i ˆ γυ b L υ b L − i ˆ γυ Tb L υ Tb L , ˆ υ t L = 1 √ × − i ˆ γυ t L υ t L − i ˆ γυ Tt L υ Tt L , (13)where υ b L ,t L have been given in Eq. (7), and ˆ γ ∈ R + , consistently with CP symmetry. Thisembedding of q L gives rise, at leading order in λ L , to two different invariants in the potential: V (1) λ L = c (1) L f (cid:88) α = t L ,b L Σ T ( λ L ˆ υ α )( λ L ˆ υ α ) † Σ = c (1) L λ L (cid:20) γ − h γ − η (cid:21) , (14) V (2) λ L = c (2) L f (cid:88) α = t L ,b L (Σ T λ L ˆ υ α Σ)(Σ T λ L ˆ υ α Σ) † = c (2) L λ L h (cid:2) − h + (ˆ γ − η (cid:3) . (15)The interactions of q L now break the U(1) η shift symmetry, whenever ˆ γ (cid:54) = 1. Conversely, thesinglet becomes massless in the limit ˆ γ →
1. The Higgs potential receives contributions from λ L and also from ˆ γλ L . However, moderate values of ˆ γ tend to reduce the Higgs mass term.This will become clear in the specific example presented below. Contrary to the previousembeddings, a Higgs quartic is generated at leading order in the breakings. Therefore it isnot necessary to involve subleading terms to reproduce the Higgs potential. Finally, the topYukawa coupling is given by the SO(6) invariant y t √ f (Σ T ¯ Q (cid:48) L Σ) t R + h.c. = − y t √ t L ht R (cid:32)(cid:115) − h f − η f + i ˆ γ ηf (cid:33) + h.c. . (16)In the next sections we will present simple realizations of the two cases considered above,where the fermionic operators O q,t are interpreted as light composite resonances, below themass of the cutoff Λ (cid:46) πf . We will assume that these top partners saturate the Higgspotential, in order to gain a qualitative and somewhat quantitative understanding of theirrole in reproducing the electroweak VEV and the Higgs mass.We should also keep in mind that other possible sources of explicit breaking beyond thoseassociated to the SM could be present. For instance, in the present scenario a plausible sourceof breaking could be given by − c M M T Σ = − c M m (cid:112) − h − η , (17)where M ≡ m Σ and c M ∼ πf . If the mass m is a relevant perturbation at the compos-iteness scale, the term above could have a significant impact in the pNGB potential [19]. This contribution could originate from a non-vanishing mass term for the technifermions, see foonote 1.In the estimate of c M we have assumed the free field scaling for the technifermion bilinear. R -handedbottom to the strong sector. If we assume that b R is embedded in a of SO(6), the subsequentcontributions to the pNGB potential will be similar to those of t R in Eq. (10), with λ R → λ bR and γ → γ b . Therefore, if γ b (cid:29) y b ∼ λ bR λ bL ,and without contributing significantly to the Higgs potential (notice in particular that thespurion γ b λ bR has different CP quantum numbers than y b ). This is just one possibility thatreflects the fact that the potential for η is subject to more model dependencies than that ofthe Higgs. L + R With a simple effective Lagrangian containing the top partners, we can understand how theirmasses fix the coefficients c L,R and c LR in Eqs. (9), (10), and (12). To this aim, we introducea complete multiplet of massive top partners in the vector representation of SO(6),Ψ L,R = (cid:18) Ψ Ψ (cid:19) L,R , Ψ L,R = 1 √ i ( B − X / ) B + X / i ( X / + T ) X / − T √ i T (cid:48) L,R . (18)As the notation suggests, Ψ decomposes under SO(5) as a 5-plet Ψ and a singlet Ψ . TheLagrangian for the top sector then reads, − L Ψ = λ Ψ f ¯Ψ L Ψ R − y Ψ f ( ¯Ψ L Σ)(Σ T Ψ R ) + λ L f ¯ Q L Ψ R + λ R f ¯Ψ L T R + h.c. , (19)where the embeddings Q L and T R have been identified in Eqs. (7) and (8). Such a Lagrangianis often found in 2-site descriptions of composite Higgs models [7]. Its symmetry featuresare clear once we perform a SO(6) rotation on Ψ L,R that eliminates the NGB dependence inthe y Ψ term, moving it to the mixing terms λ L,R , − ˜ L Ψ = M ¯Ψ L Ψ R + M ¯Ψ L Ψ R + λ L f ¯ Q L U Ψ R + λ R f ¯Ψ L U † T R + h.c. , (20)where M = λ Ψ f and M = ( λ Ψ − y Ψ ) f (notice that the masses of the 5-plet and the singletare independent). The collective pattern of SO(6) symmetry breaking is now apparent. Both y Ψ = ( M − M ) /f and λ L or λ R are needed in order to generate a non-trivial potential forthe NGBs. This also implies that any one-loop contribution to the potential will be at mostlogarithmically divergent within this simple model.8he top partners masses, at leading order in λ L,R and neglecting EWSB effects, whichare suppressed by v /f , are given by m X / = m X / = M , m B (cid:39) m T (cid:39) (cid:113) M + ( λ L f ) ,m T (cid:48) (cid:39) (cid:113) M + ( γλ R f ) , m Ψ (cid:39) (cid:113) M + ( λ R f ) . (21)The top Yukawa coupling in Eq. (11), arising through the mass-mixing of the elementarystates q L and t R and the composite resonances in Ψ, is given by y t = y Ψ λ L fm T λ R fm Ψ m X / m T (cid:48) . (22)From the Lagrangian Eq. (19) it is clear why in order to generate a top Yukawa the couplings y Ψ , λ L , and λ R are needed. The first is the Yukawa-type coupling for the Ψ fields, whilethe last two give rise to the necessary mixing angles λ L f /m T and λ R f /m Ψ for q L and t R respectively. Besides, the factor of m X / /m T (cid:48) arises from the extra mixing of t R with T (cid:48) R .This extra factor favors large values of M in order to reproduce the large top Yukawa.A standard computation of the Coleman-Weinberg potential yields the following resultfor the coefficients c L and c R in Eqs. (9) and (10): c L = c R = 38 π (cid:20) M log (cid:18) Λ M (cid:19) − M log (cid:18) Λ M (cid:19)(cid:21) . (23)As expected, these are logarithmically divergent, the scale Λ to be interpreted as the massof a second layer of heavier fermionic resonances. Furthermore, c L and c R vanish in the limit y Ψ = M − M →
0, as we advanced after inspecting the symmetry properties of the topLagrangian. The coefficient c LR in Eq. (12) is instead finite at one loop, since it requires fourinsertions of the symmetry breaking couplings λ L,R , c LR = 34 π M − M (cid:20) M − M + M M log (cid:18) M M (cid:19)(cid:21) . (24)Similar expressions are obtained for the terms arising at order λ L and λ R . To understandunder which conditions and with how much tuning the Higgs potential can be reproducedin this simple model, we must take into account that the top Yukawa coupling Eq. (22)establishes a relation between the couplings λ L,R and the top partners masses. It followsthen that the leading contributions to the Higgs mass term, expressed in terms of the massof the top partners Ψ and X / , are ζ (∆ µ ) L (cid:39) − ζ (∆ µ ) R (cid:39) ∓ y t π m X / m Ψ f | m Ψ ± m X / | (cid:34) m X / log (cid:32) Λ m X / (cid:33) − m log (cid:18) Λ m (cid:19)(cid:35) , (25)9here we have defined ζ ≡ λ R /λ L . These contributions scale as ∆ µ ∼ ( m Ψ /f ) f , thatis with the third power of the top partner’s mass. The two contributions are equal in sizebut opposite in sign when ζ = 1 / √
2. This can be traced back to the fact that both q L and t R have been embedded in the same representation of SO(6). At O ( λ L,R ) there isno effect of a non-vanishing γ . This arises at the next to leading order, primarily fromthe dependence introduced through the Yukawa of the top, and it is then suppressed by( γλ R f /M ) . Whenever this ratio is small, we can approximate(∆ µ ) γ (cid:39) γ y t ζ f min( m X / , m Ψ )2 m X / (∆ µ ) R . (26)This contribution increases the Higgs mass term in the region of small m X / . We shouldnotice though that for γ (cid:54) = 0, there is a lower theoretical bound on m X / , ( m X / ) min = γy t f ,which arises from the requirement to reproduce the large top Yukawa. This is the main effectof a non-vanishing γ in what regards the Higgs potential.Given the current bounds on f and the masses of the top partners, the contributions inEq. (25) must be finely cancelled in order to reproduce the correct Higgs mass term. Sincein this simple model c L = c R , the cancellation can be achieved by adjusting ζ (cid:39) / √ λ R (cid:39) λ L / √
2, how important depending on how heavy the composite vector resonancesare. Likewise for an extra contribution from Eq. (17) (also positive). In this regard, noticethat this latter term can only play a role in the Higgs potential (given c M ∼ πf ) if m/f (cid:38) y t ( m Ψ /f ) / (4 π ) ≈ − for m Ψ /f = 2. In this simple model however there cannotbe a large departure from ζ = 1 / √
2, because in that case the Higgs quartic is not reproduced,see the discussion after Eq. (27). One other possibility to tune down µ is to consider M < m (cid:39) m X / , whilestill reproducing the top Yukawa (recall that y t ∼ | M − M | ). On top of this, the effect of anon-vanishing γ in Eq. (26) is to disfavor the regions where m X / (cid:28) m Ψ , basically becauseof ( m X / ) min ∝ γ . On the other hand, for m X / (cid:29) m Ψ , the dependence on γ becomessmall. Therefore γ (cid:54) = 0 favors a light singlet top partner in this simple model. We will showin section 4 that for γ (cid:38) decays predominantly to η t .The masses of the top partners determine also the Higgs quartic coupling. The leadingcontribution, which arises at O ( λ L,R ), takes a simple form in the limit ζ → / √ λ (cid:39) y t π m X / m f ( m X / − m ) log (cid:32) m X / m (cid:33) . (27)This scales with the second power of the top partner’s mass λ ∼ ( m Ψ /f ) . It follows thenthat reproducing the lightness of the Higgs requires that one of the top partners, either the10 .0 0.5 1.0 1.5 2.0 2.50.00.51.01.52.02.5 m X (cid:144) (cid:72) TeV (cid:76) m (cid:89) (cid:72) T e V (cid:76) L (cid:43) R f (cid:61) . T e V , (cid:200) Λ R (cid:200) (cid:61) (cid:200) Λ L (cid:200) (cid:144) , Γ (cid:61) (cid:37) (cid:37) (cid:37) Figure 1: Contour lines of tuning µ / (∆ µ ) R (solid black) and regions with Higgs quartic0 . (cid:54) λ (cid:54) .
14 (blue), in the plane of the top partners masses m X / and m Ψ , for γ = 2, ζ = 1 / √
2, and f = 0 . γ = 0 (dotted black) are also shownfor comparison. The red lines delimitate the region (upper-right) where the top Yukawa canbe reproduced. For γ = 0 the lower bound on m X / goes to zero. Notice that for large m X / there is little difference between the solid and dashed lines. We have taken Λ = 4 √ m X / m Ψ .singlet or the 5-plet, is weakly coupled, g Ψ = ( m Ψ /f ) (cid:46)
2. Departures from the relation λ R = λ L / √ g Ψ . Taking into account in addition the theoretical lowerbound on m X / (for γ (cid:54) = 0), and the one present also for the mass of Ψ , ( m Ψ ) min = ζy t f ,one finds that ζ ∈ (1 / , m X , m Ψ ), and for a representative set of parameters. Let us stress that the numbers inthis plot have been obtained under the approximations explained above, but its qualitativefeatures properly reflect the effects of the top partners on the Higgs potential. We simplydefined the tuning as the ratio of the largest contribution to µ , which for the parameterstaken in the plot corresponds to (∆ µ ) R , over its correct value µ ≈ (90 GeV) .The contribution of the top sector to the mass of the singlet is correlated with the degreeof tuning to be enforced on the Higgs mass term,(∆ m η ) R = ( γ − µ ) R ≈ ± (320 GeV) (cid:18) | γ − | (cid:19) (cid:18) µ / (∆ µ ) R (cid:19) . (28)11t is important to recall that in the limit γ → µ ) L , then (∆ µ ) R is negative, and m η is positive only for γ < γ > m η >
0, if some other contribution overcompensates Eq. (28). Ifsuch a contribution also adds to the Higgs potential, like Eq. (17) [19], then it must be aleading one. This in turn requires γ ∼ µ ) R >
0, then m η > γ >
1, while for γ < η from getting a VEV. Notice that we are focussing on (cid:104) η (cid:105) = 0 to keep the Higgs from inheriting the properties of a pseudo-scalar singlet. In anycase we should keep in mind that m η is exposed to large model dependencies. (cid:48) L + R Here a (cid:48) multiplet of Dirac fermions Ψ is coupled to q L , breaking explicitly the SO(6) sym-metry, while t R couples to a singlet of SO(6). Given that this latter mixing does not intro-duce any explicit breaking, we can actually dispense with the composite singlet and directlyintroduce a coupling of t R to the SO(5) singlet component of the (cid:48) . Then the effective La-grangian, in the field basis where the NGB dependence comes with the elementary-compositecoupling λ L , reads − ˜ L Ψ = M ¯Ψ L Ψ R + M ¯Ψ L Ψ R + M ¯Ψ L Ψ R + λ L f Tr[ ¯ Q (cid:48) L U Ψ R U T ] + λ R f ¯Ψ L t R + h.c. , (29)where the embedding Q (cid:48) L has been given in Eq. (13), and an explicit matrix form for Ψ isgiven in appendix A. Notice that Ψ decomposes as a + + of SO(5). From Eq. (29)one can understand that λ L is needed to generate a potential for the Higgs and η . Afterperforming a SO(6) NGB-dependent rotation on Ψ R , it becomes explicit that either M , M , or M , are also needed.After mixing (at zeroth order in h and η ) the q L and t R states with the resonances in Ψwith the proper gauge quantum numbers, the top Yukawa coupling in Eq. (16) is generated, y t = (cid:112) / λ L λ R f M M (cid:112) M + (ˆ γλ L f ) (cid:112) M + ( λ L f ) (cid:112) M + ( λ R f ) . (30)The presence of two different couplings for q L , one of them proportional to ˆ γ , introduces twomixing angles for t L . Recalling that λ R does not introduce any explicit breaking of SO(6),already from Eq. (30) one can see that the regime M (cid:28) λ R f will be preferred. This allows λ L , which controls the size of the top sector contribution to the pNGB potential, to be thesmallest possible compatible with the large top Yukawa, λ L (cid:39) y t (cid:112) /
12. From now on wewill take M = 0. The mass of the singlet top partner is then m Ψ = λ R f , and given that itis not associated to any breaking, it drops out completely from the potential.12ndeed, the computation of the Coleman-Weinberg potential gives rise to the followingcoefficients c (1) L and c (2) L in Eqs. (14) and (15): c (1) L = 34 π (cid:16) ˜ M − ˜ M (cid:17) , c (2) L = 320 π (cid:16) M − M (cid:17) , (31)where we have defined ˜ M , ≡ M , log(Λ /M , ). Taking into account Eq. (30), andkeeping the leading order terms O ( λ L ) only, this model predicts,(∆ µ ) L (cid:39) y t π (cid:104) M (7 − ˆ γ ) − ˜ M (23 − γ ) (cid:105) , (32) λ (cid:39) y t π M − M f , (33)for the Higgs mass term and quartic. Both arise at leading order, and they are sensitive toa second level of resonances through Λ, which have been reabsorbed in the effective masses˜ M , . The mass of the singlet is given by(∆ m η ) L (cid:39) (ˆ γ −
1) 15 y t π (cid:16) ˜ M − ˜ M (cid:17) . (34)As we advanced, in the limit ˆ γ → γ , η becomes massless in limit M → M ( c (1) L → M and ˜ M , which is satisfied without fine tuning aslong as | ˜ M /f | (cid:46) . | ˜ M /f | (cid:46) .
5. Therefore a light Higgs requires weakly coupled toppartners. The relation between ˜ M and ˜ M enforced by λ fixes also the level of tuning inthe Higgs mass term, as well as the mass of the singlet, as a function of a single top partner’smass parameter, µ (cid:39) (3 − ˆ γ ) 3 y t ˜ M π + λ − ˆ γ ) f + (∆ µ ) g , m η (cid:39) (ˆ γ −
1) 4 π λf − y t ˜ M π . (35)This is neglecting other contributions, such as Eq. (17), as well O ( y t ) terms. It is importantfor the phenomenology of the top partners (see section 4) to discuss the role of ˆ γ . For ˆ γ < M such that m η is positive (which corresponds to ˜ M (cid:62) ˜ M ).For ˆ γ >
1, this becomes an upper bound. Naively ˆ γ > µ , for fixed ˜ M , and thus reduces the level of fine tuning. However, forˆ γ (cid:54) = 0 there is a theoretical lower bound on M from the requirement to reproduce the topYukawa, ( M ) min = ˆ γ (cid:112) / y t f (as well as ( M ) min = (cid:112) / y t f ), which forces large valuesof M , increasing the tuning for both the Higgs mass term and quartic. This lower boundon M has to be compared with the upper bound on ˜ M enforcing m η >
0. However, thiscomparison relies on the scale Λ, and therefore we cannot directly establish the consistency13 .0 0.5 1.0 1.5 2.0 2.50.00.51.01.52.02.5 M (cid:142) (cid:72) TeV (cid:76) M (cid:142) (cid:72) T e V (cid:76) L (cid:43) R f (cid:61) . T e V , Γ (cid:96) (cid:61) (cid:37) (cid:37) (cid:37) Figure 2: Contour lines of tuning as defined in the text (solid black) and regions with Higgsquartic 0 . (cid:54) λ (cid:54) .
14 (blue), in the plane of the top partners mass parameters ˜ M and˜ M , with f = 0 . γ = 4. Contour lines of tuning for ˆ γ = 0 (dotted black) arealso shown for comparison. The red lines delimitate the region (upper-right) where the topYukawa can be reproduced, and we have simply taken Λ = 2 . γ = 0 the lowerbound on M goes to zero.of these two limits without an explicit model that eliminates this lack of calculability (suchas a 3-site model). We can nevertheless say that ˆ γ (cid:38)
1, but not excessively large, is preferredin this model. We can also give a simple estimate of the top sector contribution to the massof η by setting ˜ M = 0 in Eq. (35),∆ m η ∼ (ˆ γ − m h f v (cid:39) (230 GeV) (cid:18) | ˆ γ − | (cid:19) (cid:18) v /f (cid:19) . (36)To illustrate the points discussed above, we show in Figure 2 contour lines for the tun-ing in this model, as well as the region where the Higgs quartic is reproduced, in theplane ( ˜ M , ˜ M ), and for a representative set of parameters. We defined the tuning asmax[( ∂ log µ /∂ log ˜ M i )( ∂ log λ/∂ log ˜ M i )], for ˜ M i = ˜ M , ˜ M , thus treating as separate con-tributions to the potential the terms proportional to ˜ M and those proportional to ˜ M .14 Non-Minimal Top Partner Phenomenology
We have explicitly shown in the previous section that in the NMCHM the masses of the toppartners control the size of the Higgs potential. The mass of the extra singlet η also getscontributions from the top partners, such that ∆ m η ∼ ( γ − m . The extra symmetrybreaking coupling associated to γ (or ˆ γ depending on the embedding of the top) does notmodify significantly the predictions for the Higgs potential, and in particular γ = O (1)does not give rise to a larger level of tuning. Interestingly, we will show in this sectionthat the decay channel Ψ → ηt becomes important when γ (cid:54) = 0, where Ψ is the toppartner in the singlet representation of SO(5). We will also show that one of the toppartners belonging to the multiplet of SO(5) decays exclusively to ηt . In order to arriveat such results, we will derive, for a single SO(5) multiplet of top partners at a time, itsinteractions with the NGBs and the SM fields. We will make use of effective Lagrangiansthat implicitly assume that other composite resonances, in particular other multiplets oftop partners, are heavy and lie at or beyond the cutoff. Such type of Lagrangians mustbe invariant under local SO(5) transformations, thus reproducing the non-linearly realizedSO(6) symmetry of the strong sector. Its building blocks are i) the top partners, belongingto a given SO(5) multiplet (and with a definite X charge, for the cases considered hereequal to 2/3), ii) the derivatives of the NGBs, introduced through d iµ = − i Tr[ T i U † D µ U ]and the SO(5) gauge connection e aµ = − i Tr[ T a U † D µ U ], with T i and T a the broken andunbroken generators of SO(6) respectively (see Appendix A for details), and iii) the SMstates, specifically the SU(3) C × SU(2) L × U(1) Y gauge fields and the top quark. Regardingthe latter, recall that we specified its embedding in SO(6) representations, Eqs. (7) and (8)or Eq. (13). In order to include them in our effective Lagrangian, we will use the NGBmatrix U to form the dressed fields U iI ( Q L ) I and U I ( Q L ) I transforming as a and a under SO(5) respectively, and likewise for T R and Q (cid:48) L . This approach, also followed in [24],is very efficient in systematically identifying the leading interactions of the top partners, inan expansion in derivatives and symmetry breaking couplings. Let us focus first on the phenomenology of the top partner singlet of SO(5), Ψ , for thecase where both the q L and t R are embedded in the / of SO(6) × U(1) X . The effective Other than the extra decay to ηt , Ψ has the same main characteristics as the top partner singlet ofSO(4) in the SO(5) / SO(4) model, denoted by ˜ T in [24]. This is also the case for some of the top partners in the , although we will not discuss them here. y L,R , is L L + R Ψ = i ¯ q L /Dq L + i ¯ t R /Dt R + i ¯Ψ /D Ψ − M Ψ ¯Ψ Ψ − y L f ( ¯ Q L ) I U I Ψ R − y R f ¯Ψ L U I ( T R ) I + h.c. . (37)The parameters of this Lagrangian can be taken to be real without any loss of generality andconsistently with CP conservation. The covariant derivatives acting on q L and t R encodethe usual SM gauge interactions, and given that Ψ has hypercharge Y = X = 2 / t R ), its covariant derivativecontains also the corresponding gauge connections. Importantly, only the last two terms inEq. (37) depend on the NGBs, and in a non-derivative way. Both of them induce a mixingbetween the Ψ and the SM top, but only the term proportional to y R does it at leadingorder in v/f (recall (cid:104) h (cid:105) = v ). Then the masses of the top and the top partner (we use thesame notation before and after rotation to the mass basis) read m t (cid:39) y L y R (cid:112) g + y R v √ , m Ψ (cid:39) f (cid:113) g + y R (38)where we defined g Ψ ≡ M Ψ /f , and neglected subleading O ( y L,R v /g f ) terms.The most relevant interactions for what regards the decays of Ψ (and its single pro-duction) come from the trilinear couplings between a top partner, a third generation SMquark, and a single NGB, either the physical Higgs scalar h , the pseudo-scalar η , or thelongitudinal components of the W ± and the Z . The latter are, by the equivalence theorem,well approximated by the Goldstone degrees of freedom in the Higgs field, φ ± and φ , re-spectively. We will consider only the leading couplings arising at zeroth order in v/f . Wewill therefore neglect the interactions with the transverse components of the W ± and the Z , given that these are diagonal in flavor space, and only after EWSB they give rise to acoupling of a SM quark and a top partner. Besides, the electroweak gauge couplings g or g (cid:48) are smaller than the Yukawa-type couplings, proportional to y L , y R (cid:38) y t . Under theseapproximations, the linear couplings of the top partner Ψ are g Ψ y L (cid:112) g + y R (cid:20) √ h − iφ )¯ t L Ψ R − φ − ¯ b L Ψ R (cid:21) − i g Ψ y R γ (cid:112) g + y R η ¯Ψ L t R + h.c. . (39)These then imply the following approximate relations between the branching ratios of Ψ :BR(Ψ → ηt )BR(Ψ → ht ) (cid:39) BR(Ψ → ηt )BR(Ψ → Zt ) (cid:39) → ηt )BR(Ψ → W + b ) (cid:39) y R γ y L , (40) We are neglecting terms at the same order in y L and y R but with one extra derivative, given that theseare effectively suppressed by g Ψ /g ρ , where g ρ is a strong coupling associated to heavy composite states, inthe sense g ρ (cid:29) g Ψ ≡ M Ψ /f . With a slight abuse of notation, we will denote with h also the scalar fluctuation around the electroweakVEV, that is h → v + h . m Ψ (cid:29) m η + m t . Therefore, the decay channel Ψ → ηt becomes important with γ . As an example, we canmatch the parameters in the effective Lagrangian Eq. (37) to the model presented in section3.1, after integrating out the 5-plet Ψ . One then obtains g Ψ = λ Ψ − y Ψ = M /f , y R = λ R , y L = λ L , (41)and consequently, BR(Ψ → ηt ) (cid:39) −
11 + ( γζ ) / , (42)where ζ = λ R /λ L . Recalling that from fine-tuning considerations the regime ζ (cid:39) / √ → ηt ) (cid:39)
33% for γ = 2, a branching ratioas large as that into W + b (which is the dominant channel in the SO(5) / SO(4) model). InFigure 3 we show the branching ratios of Ψ as a function of its mass, for y R = y L / √ γ = 2 or 4. We have fixed in both cases m η = 300 GeV, to illustrate the fact that if thereare no kinematical suppressions the decay Ψ → ηt can dominate. Notice however that if γ becomes very large, such a decay is only kinematically allowed for a heavy Ψ , given thatthe mass of η grows with γ (and we do not wish to tune down m η ). Furthermore, recallthat for increasing γ the theoretical lower bound on the mass of the 5-plet of top partnersalso grows. In summary, the non-standard ηt decay can dominate, but not at the level ofmaking the other decays negligible.The phenomenology of Ψ for the case where the q L and t R are embedded respectively inthe (cid:48) / and / can be described a similar way. The leading terms in the correspondingeffective Lagrangian are L (cid:48) L + R Ψ = i ¯ q L /Dq L + i ¯ t R /Dt R + i ¯Ψ /D Ψ − M Ψ ¯Ψ Ψ + y L f U I ( ¯ Q (cid:48) L ) IJ U J Ψ R + y L c t f U I ( ¯ Q (cid:48) L ) IJ U J t R + h.c. . (43)The Yukawa coupling of t R does not need to be suppressed with respect to that of Ψ R ,therefore c t = O (1). Only the terms in the second line depend on the NGBs: at leadingorder in v/f , the last gives rise to the top mass, m t (cid:39) y L c t v , while the first gives rise to theleading non-diagonal interactions of Ψ with q L . The mass of Ψ at this order is simply M Ψ .Here we will once again neglect the couplings to transverse gauge bosons, subleading in the v/f expansion and also because g, g (cid:48) < y L (cid:39) y t . Then the relevant interactions are − y L (cid:20) √ h − iφ )¯ t L Ψ R − φ − ¯ b L Ψ R + i ˆ γ vf η ¯ t L Ψ R (cid:21) + h.c. . (44)Notice that because of SU(2) L quantum numbers, the leading coupling of Ψ to η arises atorder v/f , but it is enhanced by ˆ γ . The branching ratios of Ψ are then:BR(Ψ → ηt )BR(Ψ → ht ) (cid:39) BR(Ψ → ηt )BR(Ψ → Zt ) (cid:39) → ηt )BR(Ψ → W + b ) (cid:39) ˆ γ v f . (45) We have also neglected in Eq. (40) the rescaling of all the couplings of h from the correction to its kineticterm in Eq. (4). .6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.00.10.20.30.40.50.60.7 m (cid:89) (cid:72) TeV (cid:76) BR (cid:72) (cid:89) (cid:76) L (cid:43) R f (cid:61) . T e V , (cid:200) y R (cid:200) (cid:61) (cid:200) y L (cid:200) (cid:144) , Γ (cid:61) ht Zt Wb Η t m (cid:89) (cid:72) TeV (cid:76) BR (cid:72) (cid:89) (cid:76) L (cid:43) R f (cid:61) . T e V , (cid:200) y R (cid:200) (cid:61) (cid:200) y L (cid:200) (cid:144) , Γ (cid:61) ht Zt Wb Η t Figure 3: Branching ratios of Ψ to ht (dotted black), Zt (dot-dashed green), W b (dashedblue), and ηt (solid red), in the L + R model. The left panel correspond to γ = 2 and theright one to γ = 4. The singlet mass has been fixed in both cases to m η = 300 GeV, while y R = y L / √ γ , toovercome the v /f suppression. For instance, given ˆ γ = 4 and f = 600 GeV, one findsBR(Ψ → ηt ) (cid:39) M Ψ = (cid:113) M + ( λ R f ) , y L = − (cid:114) λ L M (cid:112) M + ( λ R f ) , c t = − λ R fM . (46)Recall in particular that the regime M /f (cid:28) λ R was preferred for fine-tuning considerations,in which case M was playing little role in the Higgs potential. Besides, even though largevalues of ˆ γ were preferable, the theoretical lower limit on M scaled also with ˆ γ , possiblybecoming the leading source of tuning for ˆ γ (cid:29)
1. Therefore, we can conclude that a non-standard branching ratio of Ψ to ηt can become comparable to those into the standardchannels, but not dominant. We discuss next the phenomenology of the top partner 5-plet of SO(5), Ψ . For the casewhere both the q L and t R are embedded in the / of SO(6) × U(1) X , the leading effective18agrangian reads L L + R Ψ = i ¯ q L /Dq L + i ¯ t R /Dt R + i ¯Ψ ( /D + ie µ γ µ )Ψ − M Ψ ¯Ψ Ψ − y L f ( ¯ Q L ) I U Ii (Ψ R ) i − y R f ( ¯Ψ L ) i U iI ( T R ) I + h.c. . (47)Its parameters can be taken to be real, consistently with CP conservation and without loss ofgenerality. The covariant derivative acting on Ψ contains the gauge connection associatedwith the X = 2 / D Ψ = ( ∂ − ig (cid:48) (2 / B − ig s G )Ψ . The kinetic term for Ψ contains also the object e µ , required by the local SO(5) symmetry. Such a term encondes the proper electroweakgauge interactions of the components in the 5-plet: ( T, B ) = / , ( X / , X / ) = / and T (cid:48) = / under SU(2) L × U(1) Y , where we recall that Y = T R + X . The masses of thesetop partners, after accounting for the mixing with q L and t R , are m X / = m X / = M Ψ ≡ g Ψ f , m B (cid:39) m T (cid:39) f (cid:113) g + y L , m T (cid:48) (cid:39) (cid:113) g + ( γy R ) , (48)while the top mass is m t (cid:39) y L y R g Ψ (cid:112) g + y L (cid:112) g + ( γy R ) v √ . (49)In these expressions we neglected EWSB corrections, effectively suppressed by y L,R v /g f .The decays of the top partners in the 5-plet are mostly determined by the trilinearinteractions from the second line in Eq. (47), which involve a NGB and a third generationSM quark. The interactions with the transverse gauge bosons, from the first line in Eq. (47),are effectively suppressed by v/f and g/y L,R , and we will neglect them in what follows (this isin analogy with the interactions of the singlet Ψ in section 4.1). Under these approximations,the relevant couplings of the top partners in Ψ are y R c Ψ R (cid:20) c Ψ L √ h − iφ ) ¯ T L − √ h + iφ ) ¯ X / L − c Ψ L φ − ¯ B L − φ + ¯ X / L + iη ¯ T (cid:48) L (cid:21) t R + h.c. , (50)where c Ψ R ≡ g Ψ / (cid:112) g + γ y R and c Ψ L ≡ g Ψ / (cid:112) g + y L . Notice first that all these interac-tions are proportional to y R . The Yukawa-type invariant proportional to y L in Eq. (47) doesnot give rise to couplings of the SU(2) L doublets in Ψ with q L unless the electroweak sym-metry is broken (thus they arise at order v/f ), and likewise for the singlet T (cid:48) . Furthermore,the trilinear couplings φ ¯ Bb , φ + ¯ X / b are absent, in analogy with the same couplings in theSO(5) / SO(4) model (see [24] for details). Finally, the interactions φ + ¯ T (cid:48) b , ( h − iφ ) ¯ T (cid:48) t , η ¯ T t ,and η ¯ X / t are also vanishing. This is due to a parity symmetry P η : H → H, η → − η and T, B, X / , X / → T, B, X / , X / , T (cid:48) → − T (cid:48) , which is preserved by the leading interac-tions in the effective Lagrangian Eq. (47). Summarizing, the interactions in Eq. (50) implythe following branching ratios for the Ψ components:BR( T, X / → ht ) (cid:39) BR(
T, X / → Zt ) (cid:39) , BR( B → W − t ) (cid:39) BR( X / → W + t ) (cid:39) BR( T (cid:48) → ηt ) (cid:39) , (51)19eglecting kinematical factors. Interestingly, the extra top partner T (cid:48) , associated to thelarger SO(6) symmetry of the NMHCM (compared with the SO(5) of the minimal model),decays exclusively to the extra NGB η . The signatures at colliders from the T (cid:48) could provideimportant indications towards such an extended symmetry structure.Let us comment on one more possibility regarding the decays of the 5-plet, still for the q L and t R being embedded in the . One of the conclusions extracted from the model presentedin section 3.1 was that for γ (cid:54) = 0 Ψ must be generically heavier than the singlet Ψ . Thatbeing the case, we may wonder how the decays of the 5-plet would change by including Ψ in the effective Lagrangian. Given that both of them are composite states, their interactionswould be stronger than those with the elementary q L or t R . Indeed, at leading order inderivatives we can add to the effective Lagrangian the term i c L ( ¯Ψ L ) i d iµ γ µ Ψ L + h.c. + L ↔ R , (52)with c L , c R = O (1) and real. Within the assumption that the 5-plet is heavier than thesinglet, we will keep only the leading order interactions in g Ψ = M Ψ /f . This implies inparticular that we will neglect the mixings of q L and t R with Ψ . We must also keep in mindthat the interactions in Eq. (52) involve derivatives of the NGBs, which after integrating byparts give rise to couplings proportional to the masses of the top partners. Then the relevanttrilinear interactions, coming from Eq. (52), are c R g Ψ y R (cid:112) g + y R ¯ T L ( h − iφ ) t R + h.c. , (53) (cid:32) c L (cid:113) g + y R − c R g Ψ g Ψ (cid:112) g + y R (cid:33) ¯ T L ( h − iφ )Ψ R + ( c R g Ψ − c L g Ψ ) ¯ T R ( h − iφ )Ψ L + h.c. , (54)for the T top partner, and where y R is the Yukawa coupling of t R with Ψ in Eq. (37). Thecouplings of the rest of top partners in Ψ can be easily obtained by making the substitutions T → X / along with ( h − iφ ) → − ( h + iφ ), T → T (cid:48) with ( h − iφ ) → √ iη , T → X / with ( h − iφ ) → √ φ + , and T → B with ( h − iφ ) → −√ φ − . Even thought the decay to t R is still relevant, as long as g Ψ > y R the decays of the 5-plet to Ψ easily dominate:BR(Ψ → Ψ Π)BR(Ψ → t Π) (cid:39) c L ( g + y R ) + c R g c R y R , (55)under the assumption that the decay is kinematically allowed. Finally, notice that onceagain T (cid:48) decays exclusively to η .The phenomenology of Ψ when the q L and t R are embedded respectively in the (cid:48) / and / is described by the effective Lagrangian L (cid:48) L + R Ψ = i ¯ q L /Dq L + i ¯ t R /Dt R + i ¯Ψ ( /D + ie µ γ µ )Ψ − M Ψ ¯Ψ Ψ (56)+ i c R ( ¯Ψ R ) i d iµ γ µ t R + 2 y L f U I ( ¯ Q (cid:48) L ) IJ U Ji (Ψ R ) i + y L c t f U I ( ¯ Q (cid:48) L ) IJ U J t R + h.c. , t R instead of Ψ R . Given that the R -handedtop interacts like a singlet of SO(6), it may couple strongly to Ψ without inducing a largeHiggs potential. Therefore, the trilinear interactions from the first term in the second lineof Eq. (56), with c R = O (1), will generically dominate over the interactions from the secondterm. Under this assumption, we find that the relevant couplings of Ψ are c R (cid:113) g + y L (cid:104) − ( h − iφ ) ¯ T L + √ φ − ¯ B L (cid:105) t R + c R g Ψ (cid:104) ( h + iφ ) ¯ X / L + √ φ + ¯ X / L − i √ η ¯ T (cid:48) L (cid:105) t R + h.c. , (57)where g Ψ ≡ M Ψ /f , and the masses of the top partners are m X / (cid:39) m X / (cid:39) m T (cid:48) (cid:39) g Ψ f ,and m B (cid:39) m T (cid:39) f (cid:112) g + y L , while the mass of the top is m t (cid:39) y L c t v . Therefore we findthe branching ratios: BR( T, X / → ht ) (cid:39) BR(
T, X / → Zt ) (cid:39) , BR( B → W − t ) (cid:39) BR( X / → W + t ) (cid:39) BR( T (cid:48) → ηt ) (cid:39) . (58)These are the same branching ratios as in the case with q L and t R embedded in the ofSO(6), Eq. (51), even though in this case they arise from Lagrangian terms with derivativesacting on the NGBs. This means that while in the previous case the presence of a lightsinglet Ψ to which Ψ could decay to would easily dominate the branching ratios, in thepresent case t R could be as strongly coupled as a hypothetically light Ψ , and the decaychannels Ψ → t Π and Ψ → Ψ Π would generically be comparable.
The non-standard phenomenology of the top partners in the NMCHM relies on the extraGoldstone mode η . Specifically, the final state particles in the production and subsequentdecay of Ψ and T (cid:48) will be ultimately determined by the decay products of η . Besides,understanding the phenomenology at colliders of the pseudo-scalar singlet is important perse. This is the aim of this section.Before we do so, let us briefly comment on the couplings of the Higgs. These are modifiedwith respect to the SM ones mainly because of the non-linearities associated with its NGBnature. From Eq. (4), the kinetic term of the Higgs gets shifted after EWSB, which hasthe net effect of suppressing all of the Higgs interactions. On top of this, there are furthercorrections to the couplings to fermions, due to the non-standard Higgs dependence of theirYukawa couplings. Following the usual parametrization L h = hv (cid:0) a (cid:2) m W W + µ W − µ + m Z Z µ Z µ (cid:3) − c hψ m ψ ¯ ψψ (cid:1) , (59)21ne finds a = √ − ξ , where ξ = v /f , and c hψ = (1 − ξ ) / √ − ξ , for both embeddingsof q L and t R considered in this work, L + R and (cid:48) L + R . Notice however that theseembeddings have only been identified for the top, and they need not be the same for thelight quarks or the leptons. We have not included in Eq. (59) the couplings of the Higgsto photons or gluons, since they are not significantly modified, beyond the rescaling of thetop and W loops induced by a, c ψ (cid:54) = 1. In particular, light top partners do not give largecontributions to such couplings in the models considered here [25, 26]. They do affect theYukawa coupling of the top, by an amount of order λ L,R v /m , which we will neglect tofirst approximation. Finally, the absence of mixing between the Higgs and the η implies nofurther modifications of the Higgs couplings. We can parametrize the linear couplings of η in a similar fashion as those of the Higgs, L η = − i ηv (cid:0) c ηt m t ¯ tγ t + c ηb m b ¯ bγ b + c ηt m τ ¯ τ γ τ + c ηc m c ¯ cγ c (cid:1) + ηv (cid:16) c ηg α s π G µν ˜ G µν + c ηγ α π A µν ˜ A µν (cid:17) + ηv (cid:16) c ηγZ α π A µν ˜ Z µν + c ηZ α π Z µν ˜ Z µν + c ηW α π W + µν ˜ W − µν (cid:17) , (60)where we have now included couplings to the SM field strengths. These are important fortwo reasons: first, given that η is neutral under the electroweak interactions and it does notmix with the Higgs, the couplings to W + µ W − µ and Z µ Z µ vanish. Therefore the couplings to F µν ˜ F µν are the leading ones to any of the SM gauge vectors. And second, there can be directcontributions to this kind of couplings from UV physics, as explained in section 2. Indeedwe find from the anomalous term in Eq. (5), c ηg = n g (2 / (cid:112) ξ = 0 , c ηγ = ( n W + n B )(2 / (cid:112) ξ = 0 , c ηW = ( n W / sin θ W )(2 / (cid:112) ξ , (61)given n g = 0 and n W = − n B . The rest of the couplings in Eq. (60) are fixed by the relations c ηγZ = ( c ηW − c ηγ ) tan θ W and c ηZ = c ηW − ( c ηW − c ηγ ) tan θ W . Of course, given a non-vanishingcoupling of η to SM fermions, these can also contribute to the effective couplings to gluonsand electroweak gauge bosons, much in the same way as they do for the Higgs (see forinstance [27]). In order to fix the coefficients c ηψ in Eq. (60), we must specify the embeddingsof ψ = t, b, τ, c . Let us assume that for the bottom, tau, and charm, these are the same thanfor the top, that is either L + R or (cid:48) L + R (we rename ˆ γ ≡ γ for notational simplicityin this section). In both cases we find c ηi = γ i (cid:112) ξ , i = t, b, τ, c , (62)where we recall ξ = v /f . For what regards the values of the different γ ’s, let us recall thatin the limit γ i → η symmetry is unbroken and the singlet does not get a potential This is true at least for what regards the scalar potential generated by the third generation quarks andthe SU(2) L × U(1) Y gauge bosons, see section 3.
00 200 300 400 5000.0010.0050.0100.0500.1000.5001.000 m Η (cid:72) GeV (cid:76) BR (cid:72) Η (cid:76) Γ t (cid:61) Γ b (cid:61) Γ Τ (cid:61) Γ c (cid:61) n W (cid:61) t t (cid:42) b bg g Τ Τ c cW (cid:43) W (cid:45) Γ Γ
Z Z Γ Z
100 200 300 400 5000.0010.0050.0100.0500.1000.5001.000 m Η (cid:72) GeV (cid:76) BR (cid:72) Η (cid:76) Γ t (cid:61) Γ Τ (cid:61) Γ c (cid:61) Γ b (cid:61) n W (cid:61) t t (cid:42) b bg g Τ Τ c cW (cid:43) W (cid:45) Γ Γ
Z Z Γ Z Figure 4: Branching ratios of η to t ¯ t ∗ (black), b ¯ b (blue), gg (red), τ ¯ τ (green), c ¯ c (purple), W + W − (orange), γγ (turquoise), ZZ (brown), and γZ (magenta). The left panel correspondto universal couplings to SM fermions γ t = γ b = γ τ = γ c = 1, while in the right pane wesuppressed the coupling to bottoms, γ b = 0. The anomaly coefficient has been fixed in bothcases to n W = 2.from loops of the fermion i . Let us also notice that the limit γ i → P η parity symmetry under which the fermion i is even and η is odd. This just means that thereare selection rules which we can use to naturally take either γ i = O (1) or γ i (cid:28) η couplings carry a √ ξ suppressing factor. Consequently, for γ i = 1 the singleproduction cross-sections of the singlet are the same as those of a SM Higgs of mass m η ,times a ξ factor. This of course excludes all processes involving the electroweak gauge bosons,i.e. vector boson fusion and Higgs-strahlung. The suppression due to ξ (cid:46) . η ’s. For what regards the branching ratios, the factor of ξ drops out (they do not depend on f ). Therefore, for γ i = 1 and keeping in mind that η doesnot couple linearly to the longitudinal W ± and Z , the BR’s of the singlet should follow thesame pattern as those of the SM Higgs. This is explicitly shown in the left panel of Figure 4.There we neglected the contributions from loops of SM fermions to the couplings with γZ , ZZ , and W + W − . These are generically subleading, and moreover they are the only onesthat receive a contribution directly from the UV anomalies, for which we took n W = 2. Sincethe couplings to tops is much larger than the rest, we also included off-shell top effects in thedecay to t ¯ t (see for instance [28]). From the left panel of Figure 4 we can then conclude thatfor O (1) couplings to SM fermions, η mostly decays to bottom pairs below the t ¯ t threshold,while above it decays to top pairs. The situation changes significantly if γ b (cid:28) t ¯ t threshold η mostly decays to gluons inthat case. This is shown in the right panel of Figure 4. There are several other situations23hat we could consider, exposing the variability of the phenomenology of the pseudo-scalarsinglet. When γ t >
1, for which the decay Ψ → ηt is enhanced, the decay of η to gluons isenhanced because of the larger contribution from the top loop, and dominates over b ¯ b at lowmasses (low in the sense of below the t ¯ t threshold). If γ b (cid:28) γ τ (cid:29)
1, the BR( η → τ ¯ τ )is enhanced and dominates over that to gluons, thus the singlet becomes a τ ¯ τ resonance atlow masses. And if both γ b , γ τ (cid:28) γ c (cid:29)
1, then η becomes a c ¯ c resonance, or in otherwords it decays mostly to jets. Finally, when γ t = 0 and the rest of Yukawa couplings areorder one, the BR( η → b ¯ b ) dominates over the whole mass range.Notice that in the discussion above we have assumed that the couplings of η to fermionsrespected CP, that is γ i ∈ R . If that was not the case, a tadpole term would be induced for η , which nevertheless would be proportional to y i (cid:61) [ γ i ] (cid:60) [ γ i ], thus small and under control.Furthermore, let us recall that the predictions for m η in section 3 were close to the t ¯ t threshold, implying that both possibilities m η ≶ m t should be equally considered. However,we do not contemplate here the case in which the singlet is light enough for the Higgs todecay to ηη (we refer to [11] where this possibility is partly discussed in the light of theHiggs discovery). Finally, let us notice that given the prospect of top partners decayingsignificantly to ηt , their production could become an important source of η ’s. Another extraproduction mechanism for the pseudo-scalar could proceed via a large coupling to bottomquarks, γ b (cid:29)
1, boosting production through or in association with bottom quarks.
Top partners are expected to be the first sign of new physics associated to the naturalnessproblem of the electroweak scale, both in composite Higgs models and in supersymmetricextensions of the SM. In this work we have investigated the role of the top partners inthe Next to Minimal Composite Higgs Model. These fermionic resonances, related to thetop quark, control the size of the Higgs potential by effectively cutting off the radiativecontributions associated to the top Yukawa coupling. We have explicitly shown that in theNMCHM, keeping fine tuning to the minimum and reproducing the Higgs mass requires thetop partners to be light and weakly coupled, aspect shared with most models.One of the characteristic features of the NMCHM is the presence in the spectrum of a lightpseudo-scalar η , singlet under the SM gauge symmetries. This arises as a Nambu-Goldstoneboson along with the Higgs from the spontaneous breaking of a global SO(6) symmetry downto SO(5). Interestingly, the decay patterns of the top partners can be significantly affected bythis extra state. We have identified under which conditions the decays of Ψ , a top partnersinglet of SO(5), are dominated by the ηt channel. We have also shown that certain exotictop partners in the 5-plet of SO(5), which arise from the extended symmetry structure of theNMCHM, decay to ηt only. Motivated by the preference, in the simple models studied here,24or a singlet top partner lighter than the 5-plet, we have discussed as well the feasibility ofthe decays Ψ → Ψ Π, Π = W ± , Z, h, η . In addition, we have explicitly verified with severalexamples the viability of all such non-standard decays with respect to the generation of theHiggs potential. It is worth noting that while the NMCHM is the simplest extension of theminimal composite Higgs model with custodial protection [6], there is a plethora of otherpossibilities for the quantum numbers of non-minimal NGBs, which could play a similar rolein top partner decays [4].One question we have left unanswered in this work is how much the experimental boundson the top partners change given BR(Ψ → ηt ) (cid:54) = 0. If we simply take the extra decaychannel as a reduction of the standard branching ratios ( ht , W ± b , Zt ), then we roughlyestimate that the bounds could go down as much as ∼
100 GeV for the singlet Ψ , whilethey would be absent for those top partners that decay exclusively to ηt . However, theexperimental searches could be recasted or adjusted to look for the different pattern of finalstate particles from the production and decay of these top partners. We expect that thecorresponding analyses could reach comparable sensitivities as the current ones (see [30, 31]where this subject is addressed). Nevertheless, the search for non-minimal top partner decayscould also provide compelling information about the underlying symmetry structure of theelectroweak scale. From another point of view, analyses incorporating inclusive decays suchas Ψ → t + X would certainly contribute to cover most of the ground regarding detection oftop partners at colliders, much in the same way as the study of non-standard Higgs decayshas been carried out [27].The mass of the pseudo-scalar singlet is predicted to be a factor ∼ f /v larger than thatof the Higgs. Above the t ¯ t threshold η decays almost with branching ratio one to top pairs,while for lower masses its decays are more model dependent. When the coupling to bottomsis unsuppressed, η → b ¯ b dominates. Instead, if the singlet does not couple to bottoms,detection at colliders becomes challenging, since it mostly decays to pairs of jets. Still, thecoupling of η to taus could be enhanced, in which case η → τ ¯ τ would become the dominantdecay channel, and likewise for η → c ¯ c . The singlet is mostly produced through gluonfusion, although with a cross section suppressed by v /f . It is important to remark that η generically presents a phenomenology substantially different than that of an elementary(pseudo-)scalar singlet.In conclusion, the NMCHM is a simple non-minimal composite Higgs model whichpresents a top partner phenomenology that is non-standard, while retaining experimentalconsistency with little tuning. 25 cknowledgements I would like to thank Clara Peset, Riccardo Torre and Andrea Wulzer for helpful discussionsand valuable comments on the manuscript. This work was supported in part by the MIUR-FIRB Grant RBFR12H1MW.
A Explicit representations
In the vectorial representation of SO(6), we have chosen the generators as( T αL ) IJ = − i (cid:20) (cid:15) abc (cid:0) δ bI δ cJ − δ bJ δ cI (cid:1) + (cid:0) δ aI δ J − δ aJ δ I (cid:1)(cid:21) , α = 1 , , , ( T αR ) IJ = − i (cid:20) (cid:15) abc (cid:0) δ bI δ cJ − δ bJ δ cI (cid:1) − (cid:0) δ aI δ J − δ aJ δ I (cid:1)(cid:21) , α = 1 , , , ( T β ) IJ = − i √ (cid:0) δ iI δ J − δ iJ δ I (cid:1) , β = 1 , . . . , , ( T β ) IJ = − i √ (cid:0) δ iI δ J − δ iJ δ I (cid:1) , β = 1 , . . . , , ( T η ) IJ = − i √ (cid:0) δ I δ J − δ J δ I (cid:1) , (63)where I, J = 1 , . . . ,
6. The SO(5) unbroken generators are identified with T a = { T αL , T αR , T β } ,while the SO(6) / SO(5) broken generators are T i = { T β , T η } . The generators T αR,L spanthe custodial SO(4) ∼ = SU(2) L × SU(2) R subgroup of SO(5), while T η is the extra Cartangenerator, corresponding to the U(1) η abelian symmetry. The SM electroweak symmetrygroup is identified with the generators of SO(6) as T aL = T αL and Y = T R .From the Goldstone matrix U (Π( x )) we can construct the d and e symbols [29], − iU † D µ U = d iµ T i + e aµ T a ≡ d µ + e µ , (64)which transform as d µ → ˆ h (Π , g ) d µ ˆ h † (Π , g ) , (65) e µ → ˆ h (Π , g ) e µ ˆ h † (Π , g ) − i ˆ h (Π , g ) ∂ µ ˆ h † (Π , g ) , (66)with g a global SO(6) transformation and ˆ h a local (dependent on Π( x )) SO(5) transfor-mation. Given that the SM subgroup of SO(6) is gauged, we must also consider local26 transformations. These are incorporated through D µ = ∂ µ − iA µ in Eq. (64), where A µ = A aµ T a = gW aµ T aL + g (cid:48) B µ Y . At lowest order in the NGBs, the d and e symbols then read d iµ = √ f D µ Π i + · · · , e aµ = − A aµ + · · · . (67)The SO(5) multiplets of top partners introduced in sections 3 and 4 transform asΨ → Ψ , Ψ → ˆ h (Π , g )Ψ , Ψ → ˆ h (Π , g )Ψ ˆ h T (Π , g ) . (68)Finally, an explicit representation for the top partners Ψ in the (cid:48) of SO(6) is given byΨ = (cid:18) Ψ − × Ψ / √
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