Odd Top Partners at the LHC
Archana Anandakrishnan, Jack H. Collins, Marco Farina, Eric Kuflik, Maxim Perelstein
PPrepared for submission to JHEP
Odd Top Partners at the LHC
Archana Anandakrishnan, Jack H Collins, Marco Farina, Eric Kuflik, and MaximPerelstein
Department of Physics, LEPP, Cornell University, Ithaca, NY 14853, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
LHC searches for fermionic top partners T focus on three decay topologies: T → bW , T → tZ , and T → th . However, top partners may carry new conserved quan-tum numbers that forbid these decays. The simplest possibility is a conserved parity, underwhich the top partner is odd and all SM states are even. In this case, decays of top partnersmay involve new particle-odd scalars, leading to signal topologies more commonly associatedwith supersymmetry, either with or without R-parity conservation. We study a simplifiedmodel in which this possibility is realized, and estimate the bounds on the top partner massin this model implied by LHC searches for supersymmetry. We find that the bounds can besignificantly weaker than in the conventional top partner decay scenario. For example, if thenew parity is exact, a 500 GeV top partner is allowed as long as the lightest parity-odd scalarmass is between 325 and 500 GeV. The lower allowed top partner mass reduces the need forfine-tuning in the Higgs mass parameter, compared to the conventional decay scenario. Wealso present an explicit model, the Oddest Little Higgs, which exhibits this phenomenology. a r X i v : . [ h e p - ph ] D ec ontents T ¯ T → t ¯ tηη T ¯ T → b ¯ bω + ω − Many well-motivated extensions of the Standard Model (SM) at the weak scale contain “toppartners”, particles that cancel the quadratic divergence in the top loop contribution tothe Higgs mass parameter. Quantum numbers of the top partners are somewhat model-dependent. In a large class of SM extensions, including Little Higgs models [1] and five-dimensional “Pseudo-Goldstone Higgs” models [2] (see [3–6] for reviews), the top partner is fermionic (spin-1/2), colored (fundamental representation of the SM SU (3) C ), has an electriccharge of +2 /
3, and is mostly an SU (2) W singlet. This particular species of top partner willbe the focus of this paper.Collider phenomenology of the top partner is largely determined by its mass and itsquantum numbers. A fermionic top partner T in the ( , ) +2 / representation of the SMgauge group is expected to be pair-produced at the LHC through QCD interactions, and– 1 –ecay to tZ , th , and bW , with branching ratios of 25%, 25%, and 50%, respectively, fixed bythe Goldstone boson equivalence theorem [7, 8]. LHC experiments have pursued dedicatedsearches for these processes, and their non-observation places a strong lower bound on thetop partner mass: roughly, m T > ∼
800 GeV from a recent ATLAS search based on 19.5 fb − of 8 TeV data [9] (see also [10] from CMS). These bounds, together with the discovery of a125 GeV Higgs boson, rule out the most natural parameter region of the model. The requiredfine-tuning can be estimated as (see, e.g. , [11])∆ ≈ λ t m T π m h log Λ m T > ∼ , (1.1)where m T is the top partner mass, and Λ ∼
10 TeV is the cutoff scale of the model. It seemsthat top partners of this kind are increasingly endangered, at least if naturalness is to betaken seriously as a guide to the new physics landscape.This conclusion may need to be modified, however, if top partners do not decay accordingto the pattern assumed in the LHC searches. This is the possibility that we investigate here.Deviating from the standard top partner decay pattern requires two ingredients. Firstly, T needs alternate particles to decay to. Secondly, the couplings leading to the standard decaysneed to be suppressed. The first objective can be achieved by using global symmetry breakingpatterns which contain more pseudo-Nambu-Goldstone bosons (pNGBs) than just the Higgs.This opens up the possibility that the top partner decays into a top and a neutral pNGB,or bottom and charged pNGB. The second objective can be achieved by implementing anapproximate parity symmetry, under which all SM particles are even, and the top partnerand the new pNGBs are odd. The possibility that the top partner which cancels the quadraticdivergences coming from top loops is odd under such a parity was first considered in [15], inthe context of Little Higgs models with T-parity [16, 17]. In the case of an exact symmetry,heavy odd states will decay into light odd states and SM particles, and the lightest odd statewould be stable. In the presence of small parity breaking, the lightest odd state will decay.We therefore consider Lagrangians with the generic form: L = L even + (cid:15) L odd , (1.2)where L even contains all of the parity preserving interactions, including all of the SM couplingsand those required for the cancelation of quadratic divergences from top loops. L odd containsall parity breaking interactions, which will be responsible for the decay of the lightest oddparticle. The spurion (cid:15) schematically represents the size of the parity violation. Since thereis an enhanced symmetry in the limit (cid:15) →
0, it is technically natural for these couplings tobe very small. Models with non-standard top partner decays have been previously considered, for example, in Refs. [12–14]; however, those models did not include a parity symmetry, so that both standard and non-standard T decays were allowed. In contrast, here we will study models in which T → tZ , th , and bW decays areforbidden by symmetry. – 2 – TeVfew TeV200 GeV T ω, ηφT (cid:48) , Q (cid:48) W (cid:48) , Z (cid:48) Figure 1 : The type of spectrum considered in this paper. Solid, orange lines representparity-odd particles, while parity-even states are represented by dashed blue lines. We assumethat LHC phenomenology is dominated by a set of parity-odd states below the TeV scale,including a single top partner responsible for the cancelation of quadratic divergences, anda set of scalars that allow it to have interesting phenomenology. There may be additionalfermionic and bosonic states at a multi-TeV scale associated with a UV completion of thispicture.In this paper we do not consider explicit extensions of the gauge sector which cancel thequadratic divergences coming from gauge boson loops. In the absence of new states associatedwith the gauge sector below a few TeV, there remains a residual little hierarchy problem inthe gauge sector. This possibility was explored in [18], if the cutoff is not low. Alternatively,the cancelation of divergences in the gauge sector can be decoupled from that in the fermionsector by having two symmetry breaking scales [19], or by introducing supersymmetry at anintermediate scale (in which case the cancelation is achieved as in the MSSM by gauginos). Inboth of these cases, the new states can have masses of order a few TeV, without introducingsignificant fine tuning in the Higgs potential. Our choice of focusing only on the top andscalar sector is motivated by simplicity in the effective theory, but our models could derivefrom a UV completion in any of these categories.The rest of the paper is organized as follows. A simplified model that encompasses theLHC phenomenology of interest to us, and the three particular top partner decay scenariosthat occur naturally in this model, are presented in Section 2. The LHC constraints on thissimplified model, within each of the T decay scenarios, are studied in detail in Section 3. InSection 4, we describe the “Oddest Little Higgs”, a complete non-linear sigma model (NLSM)that gives rise to the simplified model, and hence the LHC phenomenology, considered in thefirst half of the paper. We summarize our conclusions in Section 5. We suppose that the Higgs is a pNGB of a spontaneously broken approximate global sym-metry, and extend the SM top sector so that the top Yukawa couplings only break the globalsymmetry collectively, eliminating the one-loop quadratic divergence in the Higgs mass pa-– 3 –ameter. We further assume that the non-linear sigma model (NLSM) that encodes the globalsymmetry breaking, as well as the extended top sector, are invariant under a parity such thatall SM particles are even and any new particles with masses (cid:46) T and T c ; and additional scalars which are pNGBs of the global symmetry, η and ω . Theirquantum numbers of these states under the SM SU(2) L × U(1) Y gauge symmetry are asfollows: T, T c : ± / , η : , ω : . (2.1)We assume that one of the electrically neutral scalars, either η or ω , is the lightest t-oddparticle (LtP); otherwise, strong LHC limits on stable charged particles [20] would apply ifthe LtP were both charged and long lived.The LHC phenomenology is described by the following simplified Lagrangian: L = L SM + L Kin + 12 m η + m ω Tr (cid:2) ω (cid:3) + m T T T c + y η T t c η + y ω f ( Q L ωH ) T c + 1 f (cid:16) Q L (cid:15) ( u ) η u c Hη + Q L (cid:15) ( d ) η d c H ∗ η + Q L (cid:15) ( u ) ω u c ωH + Q L (cid:15) ( d ) ω d c ωH ∗ (cid:17) . (2.2)Here, f is the mass scale at which the non-renormalizable interactions of the model aregenerated; in the OLH model, it is identified with the “pion decay constant” of the NLSM.The t-parity preserving couplings in the first line arise in the OLH model from the sameoperators responsible for the top Yukawa, and generically y η , y ω ∼ O (1). We assume thatthe similar parity-preserving couplings involving the light quarks are Yukawa suppressed andnegligible. The couplings in the second line of Eq. (2.2) encode the possibility of small t-parity violation; in the presence of these couplings, the LtP can decay to SM quarks, leadingto interesting LHC signatures. The (cid:15) couplings are matrices in flavor space and are notrelated to the SM Yukawas, and therefore have much more freedom in their flavor structure.The most flavor-safe structures would be minimally flavor-violating (MFV) or universal, butanarchic and inverted structures are also possible so long as the overall scale of these spurionsis sufficiently small to avoid flavor constraints. This is technically natural due to the enhancedsymmetry when all of these couplings are set to zero. The LHC constraints will generally beweakest when the decay products are light jets, and for simplicity we will assume that theLtP either decays exclusively to first generation quarks, or is stable on detector time scalesand neutral.At the LHC, the t-odd top partners will be pair produced with a QCD production crosssection [21]. Unlike the traditional top partners, the single production of such partners is– 4 – cenario 1 m T m T − m t m η T → tηm ω Scenario 2 m T m T − m t m η T → bω + m ω + Cascade m T m T − m t m η T → bω + ω + → η ¯ qq m ω + Figure 2 : Decay scenarios depending on the mass hierarchies. The decay T → tη willtypically dominate if it is kinematically allowed (scenario 1). If m η > m T − m t , then thedecay T → bω + will dominate if it is allowed (scenario 2). If m T > m ω > m η > m T − m t ,then cascade decays may be typical.forbidden by t-parity. (T-violating interactions may induce single production cross sectionof order (cid:15) ; we assume that this is too small to play a role in the LHC phenomenology.)The experimental signatures of the t-odd top partner are model-dependent, since a varietyof decay patterns are possible. Three phenomenologically distinct, simple scenarios can berealized by the Lagrangian of Eq. (2.2): Scenario 1: Singlet LtP
In the OLH model, it is natural for η to be the LtP, since ω receives quadratically divergentcontributions to its mass from gauge loops, while η does not. If this is the case, the decay T → tη will typically dominate. (Even if decays to ω are kinematically accessible, thecorresponding couplings are suppressed by a factor of ( v/f ).) If t-parity is exact, η is a stable,weakly-interacting particle, leading to a SUSY-like signature t ¯ t + E/ T , see Section 3.1.1. Ift-parity is approximate so that the decay η → jj is allowed, the final state is instead t ¯ tjjjj ,with two jet pairs forming resonances with the same mass, m η . The η decays may be eitherprompt or displaced, depending on the value of (cid:15) . Hadronic decays of the top can resultin final states with 10 hard jets (including two b ’s), potentially more with additional hardgluon emissions. This scenario will therefore be strongly constrained by multi-jet R-ParityViolating (RPV) gluino searches, as we discuss in Section 3.1.2. Scenario 2: Triplet LtP
Since the size of the UV contributions to the scalar masses is not calculable, we should alsoconsider the possibility that ω is the LtP. In this case, if T → tη is not kinematically available,the top partner will decay via T → tω or T → bω + . The first of these decays leads to thesame phenomenology as scenario 1. However, if m T − m b > m ω > m T − m t , the decay– 5 – → bω + dominates. Radiative corrections and non-renormalizable operators in the OLHmodel inevitably induce a small splitting, typically O (10 MeV), between the ω states. Weassume that ω is the LtP, in which case ω ± will decay to q ¯ q (cid:48) ω or (cid:96) ± νω ; however, the jetsand leptons produced in these decays are too soft to be detected. If t-parity is exact, thisscenario results in a signature b ¯ b + E/ T , covered by SUSY searches, see Section 3.2.1. If t-parityis approximate, the b ¯ bjjjj final state is produced, and constraints from multi-jet searches willapply. However if the T - ω mass splitting is small, the b jets will typically be soft, relaxingthe constraints from such searches. This will be discussed in Section 3.2.2. Scenario 3: Cascade Decays
Finally, it is also possible that both η and ω are light enough to participate in the decaysof the top partner, leading to cascade decays and complex, high-multiplicity signatures. Forexample, the chain T → bω + , ω + → q ¯ q (cid:48) η , may produce a b ¯ b + 4 j + E/ T final state, if thet-parity is exact, or a b ¯ b + 8 j final state, if it is approximate. Some of the jets may be softdepending on the T - ω and ω - η mass splittings. Electroweak precision data place significant constraints on the parameter space of modelswith fermionic top partners, which need to be taken into account in any discussion of directsearches. For example, in Littlest Higgs models, based on the same coset as our OLH model,potentially large tree-level contributions to electroweak precision observables arise from thevacuum expectation value (vev) of the triplet scalar, and from Z (cid:48) exchange diagrams [22].Neither of these effects is present in our model: triplet vev is forbidden by t-parity, while Z (cid:48) bosons do not appear at the scale f . Moreover, the leading one-loop contributions to theelectroweak precision observables that dominate the constraints in the Littlest Higgs modelwith T-parity [23] are also absent, since those loops involve parity-even top partners absentin our model [15]. Thus, we expect the precision electroweak constraints on our model to bequite weak.Here, we consider the contributions to precision electroweak observables produced by theparticles and interactions of the simplified model, Eq. (2.2). These are in a sense “irreducible”,since they follow directly from the structure that gives rise to the LHC signatures of interestto us. It turns out that these contributions are in fact quite small, allowing the t-odd toppartners to be as light as 300 GeV. Of course, a more complete description of the physicsthat gives rise to Eq. (2.2) will generally introduce additional, model-dependent contributionsto precision electroweak observables; we leave an analysis of those contributions in the OLHmodel for future work.Starting with Eq. (2.2) and integrating out the heavy top partner and ω triplet leads toone-loop corrections to the top Z ¯ b L b L vertex. Following the conventions of [24], the correctionsto coupling are δg bL (cid:39) | y ω | π v f (cid:20)(cid:18) −
12 + s w (cid:19) log Λ m T + (cid:18) −
12 + 43 s w (cid:19) log Λ m ω (cid:21) + finite . (2.3)– 6 –ere the coupling δg bL is defined by effective Lagrangian L eff = es w c w Z µ ( g bL + δg bL )¯ b L γ µ b L and g bL = + s w is the SM coupling. The divergence indicates that there is a countertermsomewhere in the full theory, that can contribute to δg bL but is incalculable within the chiralLagrangian. We can still get an estimate on the constraint, requiring that the above contri-bution not be too large for Λ = 4 πf ∼ √ πm T , where in the last step we used the relation m T ≈ f / √ g bL (SM) = − . +0 . − . , g bL (LEP) = − . +0 . − . . The one-loop contribution can only worsen the fit. Requiring that the top partner does notcontribute another 2 σ deviation from the SM prediction constrains y ω vf (cid:46) . . (2.4)Given that generically y ω ∼
1, this bound is satisfied for f > ∼
500 GeV, or (again using m T ≈ f / √
2) for m T > ∼
300 GeV.The light scalar triplet, ω can contribute logarithmically divergent contributions to the W boson mass, if the masses of charged and neutral component are split. The correspondingcontribution to the Peskin-Takeuchi oblique T parameter [26, 27] is δT = 12 πs w c w δm ω m Z log Λ m ω + , (2.5)where δm ω ≡ m ω + − m ω . The current bounds on T constrain δm ω (cid:46)
200 GeV . A generalUV-completion can be expected to generate mass-splitting δm ω ∼ a × v f m ω , where a is amodel-dependent numerical factor. In the OLH model presented in Section 4, we find a = at leading order. Assuming m ω ∼ v , as will be typical for the phenomenological scenariosconsidered here, we find a bound f > ∼
500 GeV, corresponding to m T > ∼
300 GeV in the OLHmodel.
In this section, we estimate the current LHC bounds on the different topologies describedabove. To do so we recast searches performed by both ATLAS and CMS, mainly in thecontext of supersymmetric models. In all cases the pp → T ¯ T process has been simulated with MadGraph5 aMC@NLO 2.2.3 [28], using CTEQ6L parton distribution functions [29], followedby decaying, showering and detector simulation performed through
Pythia 6.4 [30, 31] and
PGS4 [32]. After all cuts the LO cross sections have been rescaled by a K-factor extractedfrom [21], which amounts to a factor ∼ . T → tη or T → bω + , and by the properties ofthe scalar involved in the decay chain, in particular whether it is stable or decays promptly. We– 7 –
00 400 500 600 700 800 900100200300400500600700800 m T ( GeV ) m η ( G e V ) T → t η ( missing E T ) m T - m η = m t m T = m η
200 300 400 500 600 700 800 9000200400600800 m T (cid:72) GeV (cid:76) m Η (cid:72) G e V (cid:76) T (cid:174) t (cid:72) Η(cid:174) jj (cid:76) m T (cid:45) m Η (cid:61) m t m T (cid:61) m Η Figure 3 : LHC bounds for scenario 1, T → tη . Left panel: Exact t-parity case. Theblue/orange shaded areas are excluded by the CMS [34]/ATLAS [35] searches for isolatedlepton, jets, and missing transverse momentum, assuming the same acceptance and cut effi-ciency for spin-1/2 and spin-0 signal models. The dashed line indicates the bound from theCMS cut-and-count search in the same channel [36], including the difference in the cut effi-ciencies. The purple area is excluded by the mono-jet search [37]. Right panel: Approximatet-parity case, η → jj . The blue shaded area is excluded by the ATLAS multijet analysis [38].In both panels, below the horizontal gray line the Higgs decay h → ηη is kinematicallyaccessible.do not consider the case where LtP lifetime corresponds to displaced decays inside a detector,since displaced decays into jet pairs are very strongly constrained at the LHC independent ofthe details of the event [33]. In all scenarios we assume 100% branching ratio in the channelsof interest for both T and the scalars. T ¯ T → t ¯ tηη If the singlet η is the LtP, the decay T → tη dominates. We consider two cases: exact t-parity(stable LtP) and broken t-parity (unstable LtP). The signal topology in this case is identical to that of stop squark (˜ t ) pair-production, wherethe stop decays via ˜ t → t ˜ N and ˜ N is a stable neutralino. Many searches for this SUSY processhave been performed at the LHC. In the region of the parameter space where a two-bodydecay to t ˜ N is kinematically allowed, the strongest bounds can be derived from the ATLASand CMS searches for isolated lepton, jets, and missing transverse momentum (MET) [34–36].The ATLAS collaboration supplies acceptances and efficiencies to pass the selection cuts as afunction of m ˜ t and m ˜ N for m ˜ t <
800 GeV. We assume that these acceptances and efficiencies– 8 –pply to the fermionic top partners as well, with m T = m ˜ t and m η = m ˜ N . This assumptionignores the differences in the kinematic distributions of the fermionic and scalar top partners;we will comment on this effect below. We then use the calculated T pair-production crosssection and the 95% C.L. bounds reported by ATLAS to place constraints on the m T - m η plane,shown in the left panel of Fig. 3 (solid orange line). Likewise, the CMS collaboration providesa 95% C.L. upper bound on the pp → ˜ t ˜ t ∗ cross-section, in m ˜ t − m ˜ N plane, for m ˜ t <
900 GeVand m ˜ t − m ˜ N >
100 GeV. Neglecting the differences in kinematic distributions, we use thecalculated T pair-production cross section to obtain the bound shown in Fig. 3 (solid blueline).To test the effect of the differences in kinematic distributions of spin-1/2 top partner andstop signals, we compared the efficiency of the cut-and-count search presented in Ref. [36]for the cases of the T → tη and ˜ t → t ˜ N signal models, for a grid of points in the parameterspace. We find that across the parameter space, the efficiency is significantly lower in thecase of the T → tη signal, compared to the ˜ t → t ˜ N signal with the same mother and daughtermasses. The reason is that the spin-1/2 top partners on average have smaller production-frame velocity compared to stops of the same mass, due to a steeper rise of the cross sectionat the kinematic threshold in the spin-1/2 case. This translates into lower MET and lower p T of the visible decay products. The bound from the cut-and-count search [36] on ourmodel, including the effect of kinematic distributions, is shown by the dashed blue line inFig. 3. Unfortunately, we were not able to evaluate the effect of kinematic distributions onthe other relevant searches in this channel, since they involve advanced multivariate statisticaltechniques such as boosted decision trees. However, we note that for the case of stops, thebounds imposed by the cut-and-count search [36] are only slightly weaker than those fromthe more complex searches. We expect the same to be true for the spin-1/2 top partner,meaning that the true bound is somewhat, but not dramatically, stronger than indicated bythe dashed line. In any case, this analysis strongly suggests that the solid blue and orangelines in Fig. 3 represent a very conservative interpretation of the data, and the true boundsare likely to be significantly weaker. We conclude that for a light LtP these searches canprobe fermionic top partners up to 650 GeV (cid:46) m T (cid:46) m T − m η <
175 GeV their sensitivity is substantially degraded, leaving a window thatis unconstrained.In this compressed region, constraints from the mono-jet search [37] become important.In this case, we use
CheckMate [39], based on the fast detector simulation
DELPHES 3 [40] torecast the bounds in terms of our model. This procedure automatically takes into accountthe differences in kinematic distributions between our model and the case of stops. Theexcluded region is also shown in the left panel of Fig. 3 (purple line). This search rules outvery degenerate spectra below m T ≈
400 GeV (which compares to the reach of ≈
300 GeVfor stops), and does not impose any constraint for heavier top partners. The CMS search forsoft leptons in association with initial-state radiation (ISR) jet and MET [41], may also berelevant in the compressed region. This search has a similar reach for stops as the ATLASmonojet search, and we expect the same is true for fermionic top partners. The compressed– 9 –egion is also probed by the ATLAS search in the W + W − topology [42] This analysis issensitive for stops only in the region m ˜ t < ∼
200 GeV, and while the top partner bound isprobably somewhat stronger due to higher production cross section, the rapid decrease of thecross section with mass implies that this search does not constrain the masses of interest tous. Therefore we do not explicitly recast it in this work. We conclude that top partners withmass m T (cid:38)
400 GeV are not yet constrained by searches in this compressed regime, whichcompares to ∼
300 GeV for stops.
The decay chain of interest in this case is T → t ( η → jj ). Most searches with tops in thefinal states rely on the presence of extra leptons, as in the case of standard fermionic toppartners decays in tZ or bW , or rely on same-sign dileptons as typical in supersymmetricmodels involving stops. As such they do not apply to our case. We thus require the topsto decay hadronically and we recast an ATLAS analysis for massive particles decaying tomultiple jets, designed to search for RPV gluinos [38].The analysis requires ≥ ≥ p T and | η | < .
8. Different searchregions are categorized by different p T cut and number of minimum required b-tagged jets.In particular our signal at parton level is comprised of 2 b’s and 8 jets. Given the presence ofb’s and the fact that intermediate state particles are on-shell, we find the most constrainingsearch category to be the one requiring a minimum of 2 b-tags and p T >
80 GeV for all ≥ ±
190 events, while 1560 events have been observedduring data taking, corresponding to 20 . − of collected luminosity at 8 TeV.First we compute the expected number of signal events for each point in parameter space.The signal likelihood is then estimated through the standard CL s technique, where we fix theexpected background to its central value. The 95% C.L. excluded area is shown in Figure 3,right panel. The upper bound on the top partner mass is at most m T (cid:38)
850 GeV, anddegrades to approximately 700 GeV in the light-LtP region m η = 0, and to as low as 500GeV in the quasi-degenerate region m η ≈ m T . In the former region, η is produced witha large boost, so that the two jets stemming from its decay are often merged. This effectreduces the total number of jets of the final state, making it less likely to pass the ≥ η ’s are produced almost at rest in the lab frame,and thus their decays produce softer jets which often fail to pass the p T >
80 GeV cut.Let us briefly comment on possible constraints in similar scenarios with different η decays,namely into third generation quarks. If η → b ¯ b we can expect the bounds to be somewhatstronger than in the light generation case, since the higher number of b’s in the final stateincreases the probability of passing the b-tag cut, while the kinematics is nearly identical.If η → t ¯ t , an interesting six tops final state appears which is not directly addressed by anysearch at present. However, a recent recast [43] points to bounds of the order m T (cid:38)
700 GeVfor most η masses.If the η → jj decay is long lived on detector scales, much stronger constraints comingfrom the CMS displaced dijet search [33] apply for lifetimes between 1 mm and 1 km. For the– 10 –
00 400 500 600 700 800 900 1000200300400500600700 m T ( GeV ) m ω ( G e V ) T → b ω + ( missing E T ) m T - m ω = m t m T = m ω
200 400 600 800100200300400500600700 m T (cid:72) GeV (cid:76) m Ω (cid:72) G e V (cid:76) T (cid:174) b (cid:72) Ω (cid:43) (cid:174) jj (cid:76) m T (cid:45) m Ω (cid:61) m t m T (cid:61) m Ω Figure 4 : LHC bounds for Scenario 2, T → bω + . Left panel: Exact t-parity case. The blueshaded area is excluded by the ATLAS search for 2 b-jets and E/ T [33]. The purple area isexcluded by the mono-jet search [37]. Right panel: Approximate t-parity case, η → jj . Theblue shaded area is excluded by the ATLAS multijet analysis [38], while the red shaded areais excluded by the CMS dijet resonances search [48].case where m η > m T , the topology is very similar to the displaced gaugino decay, ˜ g/ ˜ N → jjj ,studied in [44–47]. T ¯ T → b ¯ bω + ω − We next consider the scenario where ω is the LtP, and ω + and ω nearly degenerate. Inthe case of exact t-parity, the ω + decays to ω and soft leptons or jets, which are too softto be detected. In the case of approximate t-parity, the direct decay ω + → q ¯ q is permittedalong with the decay via an intermediate ω . Both of these channels are phenomenologicallyequivalent, appearing as ω + → jj . We assume that m η > m T , so that η plays no role inthe top partner decays. We focus on the decay T → bω + , which we assume is the dominanttop partner decay. This assumption is a good approximation for m T > m ω > ∼ m T − m t . If m ω < m T − m t , the top partner would decay in both bω + and tω channels. The latterchannel produces signals identical to the ones considered in Sec. 3.1 above. Since we will findthat the mass bounds on the bω + and tω channels are quite similar, we do not attempt adetailed combination of the two; either one can be taken as a good estimate of the bound onthis scenario in the region m ω < m T − m t . In this case, ω escapes the detector undetected, resulting in a 2 b + E/ T signature. The signaltopology is identical to sbottom squark (˜ b ) pair-production, where the sbottom decays via˜ b → b ˜ N and ˜ N is a stable neutralino The strongest bounds can be derived from the ATLASsearch for two b-jets and missing transverse momentum [33]. We recast this search in terms– 11 –f our signal model using CheckMate . The 95% C.L. constraints on the m T - m η plane areshown in the left panel of Fig. 4. For light ω , the top partner masses up to at least 800 GeVare ruled out; the true bound is probably higher, but no information on cross section boundsbeyond 800 GeV was provided in [33]. Again, the bound is weakened significantly if T and ω are quasi-degenerate, even for a rather modest degree of degeneracy: for example, m T = 500GeV is allowed if m T − m ω < ∼
100 GeV.In the compressed region, we again evaluate constraints from the mono-jet search [37],recasting it using
CheckMate . The excluded region is shown in the left panel of Fig. 4 (purpleline). We conclude that top partners as light as 400 GeV are allowed, as long as T and ω aredegenerate at a O (10%) level. The decay chain of interest here is T → b ( ω + → jj ). Notice that the T pair productionsignature here closely resembles the gluino pair production signal, with R-parity violatingdecay ˜ g → bjj . Thus the search recast in Section 3.1.2 is relevant also in this case. Weproceed as before using the same search category. The results are shown in the right panelof Fig. 4. For generic spectra, the top partner mass below 700 −
750 GeV is excluded. Thenear-degenerate region, m T ≈ m ω , is not constrained by this search: the two b’s present inthe final state are required to pass a p T >
80 GeV cut and will often fail in this region. Inthis case, the signal topology is similar to a pair of massive particles decaying into two jetseach. With this in mind we recast a dedicated CMS search looking for pair-produced dijetresonances [48]. This search is also motivated by RPV supersymmetry and is specificallyintended for stop pair production with RPV decays in two (light) quarks. Events with atleast 4 jets with p T >
80 GeV or p T >
120 GeV and | η | < . m T and m η , simulate the original ˜ t ˜ t signal for corresponding m ˜ t and m ˜ N andcompute the cut efficiencies, and repeat the procedure for the T ¯ T signal. Finally we rescalethe T pair production cross section by the ratio of the efficiencies, and extract a limit on m T corresponding to the 95% C.L. upper bound of the CMS analysis. The result is shown inFigure 4, as the red shaded area on the right panel. The results are consistent with CMSbounds once the difference in the pair production cross section between fermionic and scalartop partners is taken into account, which amounts to a factor of ∼
6. We conclude that toppartner masses below about 550 GeV are excluded for any value of m ω .We conclude this section by noticing that CMS has performed a similar search for pairproduction of 3-jet resonances [49]. This search places gluino mass bounds very similar tothose of the ATLAS multi jet search recast above, and the limitations of the two searches,such as the jet p T cuts that degrade the efficiency in the m ω ≈ m T region, are also similar.Thus, we do not expect the CMS search to add significantly to the recast bounds from theATLAS search shown in Figs. 3 and 4. – 12 – .3 Scenario 3: Cascade Top Partner Decays In this scenario, the parameter space is mode complicated than in the other two, since m T , m η and m ω all play a role. However, the signal topologies are quite similar. The typicalfinal states are 2 b + 4 j + E/ T (exact t-parity) or 2 b + 8 j (approximate t-parity), the same asin the scenario 1, T → tη , with hadronic top decays. There may be slight differences in thekinematic distributions since no on-shell tops/ W s are present in the cascade scenario, but wedo not expect them to have a significant effect on the mass bounds. The only possibility tosignificantly relax the top partner mass bound seems to be to assume an approximate triplemass degeneracy of m T , m η and m ω , and exact t-parity. This case is very similar to the decay T → tη with an off-shell top, which was already considered in Scenario 1. In this section we finally describe a non-linear sigma model which can reproduce the simplifiedmodel used in earlier sections in certain regions of its parameter space. We use the LittlestHiggs coset [1], SU(5)/SO(5), which preserves custodial symmetry and provides a collectivetree level quartic for the Higgs. The Goldstones are parameterized by the field ΣΣ = e i Π odd /f e i Π h /f Σ e i Π Th /f e i Π T odd /f = e i Π odd /f e i Π h /f e i Π odd /f Σ , (4.1)with Σ = , (4.2)and we have chosen to separate the Goldstone fields as follows:Π odd = ω − η/ √
20 0 φ (cid:112) / η φ † ω T − η/ √ , Π h = H ∗ / √ H T / √ H † / √ H/ √ . (4.3)As is typical in Little Higgs models based on this coset, the pNGB φ will get a quadraticallydivergent contribution to its mass and is generically expected to be heavier than the otherscalars. We impose a t-parity symmetry which has the following action on the scalar sectorΣ → Σ t ≡ Ω Σ Σ † Ω Σ , Ω Σ = − . (4.4)On the Goldstone fields, this has the action: H → H, η → − η, ω → − ω, φ → − φ. (4.5)– 13 –n contrast to the original Littlest Higgs construction, we gauge only the SM SU(2) L andU(1) Y subgroups of SU(5), with generators: Q a = σ a / − σ a ∗ / , Y = diag(1 , , , − , − / . (4.6)The t-parity acts trivially on the gauge fields. The H field has the quantum numbers ofthe SM Higgs, while the Goldstone fields η , ω , φ have quantum numbers , , underSU(2) L × U(1) Y . The global symmetry is broken explicitly by the gauge couplings, and alsoby the Yukawa couplings described below. Quantum effects will then generate a potentialfor the Goldstone fields. This potential is discussed in detail in Section 4.1. For reasonablechoices of model parameters, a tachyonic mass term is generated for H , triggering EWSB,while all other Goldstones acquire positive mass. It can be easily shown that e i Π h /f = + (cid:113) − s h i √ s h − + (cid:113) − s h i √ s h (cid:113) − s h i √ s h − + (cid:113) − s h i √ s h + (cid:113) − s h , s h ≡ sin (cid:32) √ hf (cid:33) , (4.7)where H = ( π + , ( h + iπ ) / √ T and we dropped the π fields that are eaten in the EWSB.Reproducing the W mass requires (cid:104) s h (cid:105) = 2 v f (cid:18) − v f (cid:19) , (4.8)with v = ( √ G F ) − / ≈
246 GeV. After EWSB, the t-odd pseudo-Goldstones decompose as ω = ω a σ a / (cid:32) ω / ω + / √ ω − / √ − ω / (cid:33) , (4.9) φ = φ a σ a = (cid:32) φ ++ φ + / √ φ + / √ − φ + iφ P ) / √ (cid:33) . (4.10)In order to build a Lagrangian of the form of Eq. (1.2), a candidate operator O can beadded in the following way: L ⊃ (cid:0) O + O t (cid:1) + (cid:15) (cid:0) O − O t (cid:1) , (4.11)where O t is the t-image of the operator O , and (cid:15) is a small parameter. The top sector of ourmodel consists of a triplet χ , and two singlets u c , u c (where the superscript c indicates thefield is a color antifundamental, and all fermion fields are two-component left-handed Weylspinors). The action of t-parity on these fermions is: u c t ←→ u c , χ t ←→ Ω χ χ, Ω χ = diag (1 , , − . (4.12)– 14 –he third, odd component of χ will marry the odd linear combination of u c , u c , gaining alarge Dirac mass and leaving the SM third generation quarks massless before EWSB. Thet-preserving top Yukawas are given by: L Yuk = − y t f (cid:0) χ i O i u c + χ ti O ti u c (cid:1) + h.c. , (4.13)where O i = (cid:15) ijk (cid:15) xy Σ jx Σ ky ; (4.14) O ti = (cid:15) ijk (cid:15) xy Σ tjx Σ tky . Here all repeated indices are summed over: i, j, k = 1 , , x, y = 4 ,
5. The Higgs isprotected by two SU(3) subgroups of the full SU(5), and these are interchanged by t-parity.Each term in this Lagrangian breaks one SU(3) while preserving the other, so the full globalsymmetry protecting the Higgs is completely broken only non-locally in theory space. Thisguarantees the absence of quadratic divergences in the Higgs mass at one-loop, amelioratingthe little hierarchy problem [1].In the top sector, the mass eigenbasis before EWSB is obtained by the following fieldredefinitions: t c = 1 √ u c + u c ) , (4.15) T c = 1 √ u c − u c ) . (4.16) χ = (cid:16) σ · Q, T (cid:17) . (4.17)Expanding out the Σ field to quadratic order in H , the Lagrangian reads: L Higgs = − y t √ f T T c + y t HQt c + y t √ | H | f T T c + h.c. (4.18)It can be easily seen that the quadratic divergence from the T -loop cancels that of the t -loop by noticing that the trace of the Higgs dependent masses, Tr M ( h ) = m T ( h ) + m t ( h ),vanishes at order h . Before electroweak symmetry breaking the odd top partner T gets amass ˆ m T ≡ y t f / √
2, and the top quark is massless. After EWSB, the leading couplings ofthe 3rd generation quarks to the Goldstones is given by:
L ⊃ y t f s h t L t c + (cid:114) iy t T t c η + iy t √ s h b L T c (cid:0) ω − − φ − (cid:1) (4.19) iy t √ s h t L T c (cid:18) √ η + 1 √ ω − φ − iφ P (cid:19) + h.c. , where s h is defined in Eqs. (4.7), (4.8). These are exactly the t-preserving couplings ofEq. (2.2). It can be seen that the leading decay for the top partner will be T → tη if thischannel is kinematically available, as the decays to φ and ω involve couplings suppressed by– 15 – s h (cid:105) ∼ v/f . However, if the mass splitting between T and η is sufficiently small so that thisdecay cannot proceed on-shell then the decays T → bω + may dominate if either of these arekinematically available. We note that while the doubly charged scalar φ ±± could result insome striking signatures, it is unlikely to play an important role in the phenomenology of thetop partner due to its quadradically divergent mass and due to the fact that its couplings tothe top partner only arise at higher order in the v/f expansion.In the phenomenological analysis of Sections 2 and 3, we also considered a scenario withapproximate t-parity, where the pseudo-Goldstones η and ω may decay to quark pairs. Toincorporate this possibility in the OLH model, we can introduce couplings of the form L odd ⊃ Q ˆ ia (cid:15) ( u ) ab u cb (cid:16) O ˆ i − O t ˆ i (cid:17) + Q ˆ ia (cid:15) ( d ) ab d cb (cid:16) O ∗ ˆ i − O ∗ t ˆ i (cid:17) , (4.20)where all repeated indices are summed over: ˆ i = 1 , a, b run over three generations.The flavor structure of the (cid:15) couplings will determine the decays of the lightest t-odd state. In this section we describe qualitatively the contributions to the Goldstone potential, leav-ing the lengthy explicit formula to the Appendix. We introduce a tree level mass for theGoldstones by including the following explicit global symmetry breaking (but custodial andt-parity invariant) term in to the scalar potential: V ⊃ f Tr [ M Σ] + h.c. (4.21)where: M = 132 m m − m m . (4.22)This particular normalization is chosen for convenience after expanding out the Σ field interms of the Goldstones. When expanded in terms of the Goldstone fields, it introduces amass for η of m , and a mass contribution for ω and φ of m . In order to reproduce thecompressed spectrum of section 3.2.2, we will need to make m η > m T − m t . This will requireus to explore the region of parameter space where m , and possibly also m are not negligiblysmall compared to f . A precise study would require considering all operators that can beconstructed, consistent with the symmetries, in powers of M/f . However for the purposes ofthis work, we only introduce the additional operators that will add qualitatively new featuresto the potential, setting the other coefficients to zero for simplicity.Quadratically divergent fermion loops involving the couplings in Eqs. (4.13), require theintroduction of a counterterm: L ⊃ y t f c T (cid:15) ijk (cid:15) klm (cid:15) xy (cid:15) wz Σ ix Σ jy Σ ∗ lw Σ ∗ mz + t-image , (4.23)– 16 –here c T is an O (1) number determined by UV physics. This contributes the ordinary treelevel collective quartic for the Higgs, as well as a large contribution to the φ mass. We alsoinclude the operator: L ⊃ y t f c T M (cid:15) ijk (cid:15) klm (cid:15) xy (cid:15) wz M ix Σ jy Σ ∗ lw Σ ∗ mz + t-image + h.c. . (4.24)which also typically has an O (1) coefficient c T M and a parametric suppression
M/f . Thisoperator contributes to the masses of the all of the Goldstones.The additional fermion loop contributions to the Goldstone potential are calculated usingthe Coleman-Weinberg (CW) potential [50] V CW = − N c π (cid:88) i M i (cid:18) log (cid:18) M i Λ (cid:19) − (cid:19) (4.25)where the sum is over the eigenvalues of the fermion mass matrix. There is a log-divergentpiece which contributes to the Higgs quartic and φ mass which is degenerate with the quadraticdivergence in Eq. (4.23) and can therefore be absorbed by a redefinition of c T . Remaininglog divergences are cut off at a scale Λ = y f , with y ∼ O (2). This may be the scale of newfermion resonances, an example of which is given in Appendix (A). This log divergent andadditional finite parts contribute to both the Higgs mass and quartic, but only contributesto the masses of the other Goldstones after EWSB and so this effect is suppressed by v /f .Quadratically divergent gauge boson loops require counterterms of the form L ⊃ c g f Tr [ Q Σ Q ∗ Σ ∗ ] + c Y g Y f Tr [ Y Σ Y Σ ∗ ] . (4.26)These operators provide tree level contributions to m H , m φ , m ω , and the Higgs quartic, andsub-leading corrections and mixings after EWSB. Additional terms obtained by includinginsertions of the mass matrix M are degenerate with a redefinition of the mass matrix.For obtaining the correct Higgs potential in the compressed scenario, we also introducethe following term which explicitly breaks all of the symmetries protecting the Higgs L ⊃ f m (cid:0) Σ + h.c. (cid:1) . (4.27)This operator provides positive masses for η and h , but does not contribute to the Higgsquartic. The role of this term will be discussed in more detail in Appendix (B). The top partner has mass O ( f / √ φ and ω get quadratically divergent contri-butions to the masses, typically raising them significantly above the Higgs mass unless there issome additional tuning. On the other hand, the loop generated mass for η is of order v/ √ π and so unless there are large tree level contributions to its mass it tends to be somewhatlighter than the Higgs. – 17 –ase f / GeV ( m / GeV) ( m / GeV) ( m / GeV) y c T c T M c Y c A1 1320 200 − − − Table 1 : Input Lagrangian parameters for sample spectra.Case m T / GeV (cid:8) m φ , m ω , m η (cid:9) / GeV (cid:8) m φ ± , m ω ± (cid:9) / GeVA1 900 { } { } A2 810 { } { } B 600 { } { } C 600 { } { } Table 2 : Sample mass hierarchies.In tables (1), (2), we show sample parameter space points of the Oddest Littlest Higgswhich reproduce the simplified phenomenological models of section (2). Case A1 is typical ifthe tree level breakings of the global symmetry are small, with η being the lightest pNGB.The decay T → tη dominates and so this reproduces scenario 1 of Sec. 2, with the η decayinginto two hard jets. Case A2 has a very light η such that it will be highly boosted whenproduced from decays of T , and so its decay products will be observed as a single jet. Thisplaces it in the narrow window of Fig. 3 for light η where the exclusion limits are weaker,but the model parameters are tuned to avoid a large branching fraction h → ηη . Case B hasa compressed spectrum, with the top partner decaying via T → bω + as the decay T → tη isnot kinematically available. Raising the η mass is achieved via large tree level contributionsfrom m and m . In this scenario, the dominant contributions to the tuning in the Higgsmass parameter are actually coming from m and c Y , with top loops being subleading. Anaive estimate of the tuning in the Higgs mass parameter coming from these contributions is O (5%), as discussed in Appendix (B). In case C, the mass hierarchy will lead to a cascadedecay of the form T → bω + → bq ¯ qη . This possibility was mentioned in Sec. 2, although wehave not discussed it in detail. Fermionic top partners are well motivated theoretically, and form an important component ofnew physics search program at the LHC. Currently, the experimental searches focus on threedecay topologies: T → bW , T → tZ , and T → th . However, top partners may carry newconserved quantum numbers that forbid these decays. The simplest possibility is a conservedparity, under which the top partner is odd and all SM states are even. In this case, decaysof top partners may involve new particle-odd scalars, leading to non-standard experimental– 18 –ignatures. If the parity is exact, the lightest particle-odd scalar is stable, and assuming thatit is weakly interacting, the scenario is characterized by missing transverse energy signatures,with signal topologies identical to stops in R-parity conserving supersymmetry. If, on theother hand, the parity is only approximate, the lightest parity-odd scalar may decay, forexample, into jets, resulting in multi-jet or tops+jets final states similar to those producedby gluinos and stops in R-parity violating supersymmetry. In either case, we found that thecurrent LHC lower bounds on the top partner mass are similar to those in the conventionaldecay scenario, m T > ∼ −
900 GeV, if the mass of the lightest t-odd scalar is well below m T .If, on the other hand, the top partner and the lightest t-odd scalar are somewhat degeneratein mass, the bounds can be relaxed significantly. For example, in the case of exact t-parityand decays into a gauge-singlet scalar η , a 500 GeV top partner is allowed as long as m η isbetween 325 and 500 GeV. The low allowed top partner mass reduces the need for fine-tuningin the Higgs mass parameter, compared to the conventional decay scenario, making this classof models a theoretically attractive possibility. In the OLH model considered in Sec. 4, thiscan only be achieved at the expense of introducing new tunings of tree-level parametersassociated with raising the mass of η . It remains an interesting open question whether asimilar model can be constructed in which a compressed spectrum can be arranged withoutdirectly impacting the tuning of the Higgs mass parameter.An interesting issue not investigated here is the possibility that the t-parity is anoma-lous [51, 52]. Whether or not such an anomaly is present depends entirely on the UV comple-tion of the TeV-scale NLSM [53, 54], and it is certainly consistent to assume that the anomalyis absent. If it were present, it would give rise to a phenomenologically interesting possibilityof the lightest t-odd scalar decaying to two SM massive vector bosons, for example η → ZZ .Depending on the size of the explicit t-parity violating couplings, these decays may becomedominant. Hadronic Z decays would give rise to signatures similar to the ones consideredin the approximate t-parity scenarios we studied, but with higher jet multiplicity and softerjets. Leptonic Z decays may also be exploited in this case. We leave a detailed analysis ofthis possibility for future work.A natural by-product of our scenario is that, if the t-parity is exact and non-anomalous,the lightest t-odd particle can be a dark matter candidate. Unlike the LHT models, wherethe stable dark matter candidate is usually a spin-1 T-odd partner of the hyper charge gaugeboson [55], in this case the dark matter particle would be a scalar. It would be interesting tounderstand if the correct relic abundance can be obtained in viable and phenomenologicallyinteresting regions of the model parameter space. Note Added:
While we were completing this manuscript, we became aware of Ref. [56] where similar ideaswere pursued in the context of holographic Composite Higgs models.– 19 – cknowledgements
We are grateful to James Alexander, Gustaaf Brooijmans, Jeff Dror, Nathan Mirman, JavierSerra, and Ennio Salvioni for useful discussions related to this work. This research is sup-ported by the U.S. National Science Foundation through grant PHY-1316222. We also ac-knowledge the support of the Bethe Postdoctoral Fellowship (EK) and the John and DavidBoochever Prize Fellowship in Fundamental Theoretical Physics (JC).
A An Extended Fermion Sector for the Oddest Littlest Higgs
In this section we describe an extended fermion sector for the OLH model which cuts off logdivergences in the Higgs potential that are not degenerate with the quadratically divergentpiece responsible for the collective quartic. The top sector of this model consists of threetriplets χ , χ , χ c , and two singlets u c , u c . The action of t-parity on these fermions is: χ t ←→ χ , u c t ←→ u c , χ c t ←→ Ω χ χ c , Ω χ = diag ( − , − , . (A.1)We will see that these fields decompose in to a t-even SM left handed doublet and righthanded singlet (the top quark and left-handed bottom), a light t-odd singlet top partnerwhich cancels the quadratic divergence of the top, and a heavy triplet of fermions – an odddoublet, and an even singlet. The t-preserving top Yukawas are given by: L Yuk = − y f (cid:2) χ i O i u c + χ i O ti u c (cid:3) + y √ f [( χ · χ c + χ · Ω χ · χ c )] + h.c. , i = 1 , , , (A.2)where O i and O ti are given as in Eq. (4.14). These Yukawas are very similar to those in [15],except that because t-parity acts trivially on the gauge sector, we assume that the fermionmultiplets transform as incomplete linear representations of the SU(5) global symmetry groupand we do not require that they have non-linear transformations under SO(5) .The triplets decompose in the following way: χ = 1 √ (cid:32) Q + Q (cid:48) T + T (cid:48) (cid:33) , χ = 1 √ (cid:32) Q − Q (cid:48) − T + T (cid:48) (cid:33) , χ c = (cid:32) Q (cid:48) c u c (cid:33) (A.3)and then we make the following field redefinitions: t c = 1 (cid:112) y + y (cid:18) y u c − y √ u c + u c ) (cid:19) , (A.4) T (cid:48) c = 1 (cid:112) y + y (cid:18) y u c + y √ u c + u c ) (cid:19) , (A.5) T c = 1 √ u c − u c ) . (A.6) An extension of the gauge sector at ∼ (few TeV) would require that t-parity act non-trivially on the fullgauge group, necessitating the introduction of complete multiplets or non-linear symmetry transformations onthe fermions. – 20 –xpanding out the Σ field to leading order in H , the Lagrangian takes a particularly simpleform in this new basis: L leading = − y y t (cid:112) y − y t f T T c − √ y (cid:112) y − y t f T (cid:48) T c (cid:48) + f y Q (cid:48) Q c (cid:48) (A.7)+ y t ( QH ) t c + y t (cid:112) y − y t ( QH ) T c (cid:48) + √ y y t (cid:112) y − y t ( Q (cid:48) H ) T c + h.c. , where we have replaced y by y t : y t = 2 y y y + y . (A.8)In the limit y (cid:29) y t , Eq. (A.7) reduces to: L ⊃ − y t √ f T T c − y f (cid:0) T (cid:48) T c (cid:48) + Q (cid:48) Q c (cid:48) (cid:1) + y t ( QH ) t c + y t √ y ( QH ) T c (cid:48) + y ( Q (cid:48) H ) T c . (A.9)This limit is a decoupling limit, in which the primed fields form a heavy and nearly degeneratetriplet, leaving just the physical top quark and the odd top partner in the low energy spectrum.The parities of the various fields are: + − Q, t c , T ( c ) (cid:48) T ( c ) , Q ( c ) (cid:48) .Before electroweak symmetry breaking, the primed fields acquire a mass ˆ m (cid:48) ≡ y f , theodd top partner T gets a mass ˆ m T ≡ y t f / √
2, and the top quark is massless. Integratingout the primed fields at tree level will generate custodial symmetry violating couplings forthe light top partner, which will generate corrections to the T parameter at one-loop. Thesefields also serve to cut off the logarithmic divergences in the loop generated Higgs potentialof the OLH. In the limit y (cid:29) y t , this is the only role they play and can otherwise be ignoredin the collider phenomenology of the model. B Oddest Littlest Higgs Potential
The Higgs potential is given by: V higgs = 14 m f s h + 116 λf s h − π m t (cid:0) s h (cid:1) (cid:18) log (cid:18) m t ( s h ) µ (cid:19) − (cid:19) + O (cid:0) s h (cid:1) (B.1)where m t (cid:0) s h (cid:1) is the Higgs-dependent top mass: m t = 14 y t f s h + O (cid:0) s h (cid:1) . (B.2)The scale µ in Eq. (B.1) will be set to the top mass so that the log vanishes at the potentialminimum, though the term still plays a role in setting the minimum of the potential. Thepotential is minimized with: m = − λf s h + 332 π y t f s h (B.3)– 21 – y / y t G ( y / y t ) y / y t F ( y / y t ) Figure 5 : G and F functions which contribute to the Higgs mass parameter and quartic.resulting in a physical Higgs mass: m h = (125 GeV) = λf s h (cid:0) − s h (cid:1) (B.4)= 2 λv (cid:18) − v f (cid:19) . The Goldstones in the Oddest Littlest Higgs model of section (4.1) have masses given by(in the limit y (cid:29) y t ): m η = m + m + 110 y t c T M m + O (cid:0) s h (cid:1) , (B.5) m ω = m (cid:18) y t c T M (cid:19) + 8 c g f + O (cid:0) s h (cid:1) , (B.6) m φ = m (cid:18) y t c T M (cid:19) + 8 c g f + 4 c Y g Y f + 4 y t c T f + O (cid:0) s h (cid:1) . (B.7)The mass parameter and quartic of the Higgs potential are given by: m = 38 m (cid:18) y t c T M (cid:19) + 58 m + 516 m + 3 c g f + c Y g Y f − y t f G (cid:18) y y t (cid:19) (B.8) λ = 38 m f (cid:18) y t c T M (cid:19) + 58 m f + 3 c g + c Y g Y + y t c T + 3 y t π log m T m t + y t F (cid:18) y y t (cid:19) (B.9)where F and G are contributions generated by fermion loops between the scales of the lightand heavy top partners, shown below and plotted in Fig (5).– 22 – ( x ) = 38 π x x − x + 1 (cid:20) log (cid:0) x (cid:1) + (2 x −
1) log (cid:18) − x (cid:19)(cid:21) (B.10) F ( x ) = 316 π (cid:34) − − x ( x − + log (cid:0) x (cid:1) − (cid:18) x (2 x − (cid:19) log (cid:18) − x (cid:19) (B.11)+ (cid:18) x ( x − − x (2 x − (cid:19) log (cid:0) x − (cid:1) (cid:35) The expression for the quartic, Eq. (B.9), can be rewritten in terms of the Goldstone massesas follows λ = 18 f (cid:0) m φ + m ω + 5 m η (cid:1) + 3 y t π log m T m t + y t F (cid:18) y y t (cid:19) − m f . (B.12)In order to arrange for the compressed spectrum of Section (?), it is required that all of thescalars have masses (cid:38) f / √
2. It is clear from this expression that a small quartic can onlybe achieved in this case if m is large. In case B of the sample spectra, we have obtaineda tachyonic Higgs mass parameter and a small quartic using negative c Y and tachyonic m ,which provide important negative contributions in expressions (B.8), (B.9) to balance thepositive contributions that are needed for heavy scalars, while the large η mass (requiredto make the T → tη decays kinematically forbidden) is obtained with a large m . Thecancelation between the contributions from m and c Y is the dominant contribution to thetuning of the Higgs mass parameter in this case. A naive estimate of the tuning in the Higgsmass parameter is given by: ∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Max (cid:2) δm i (cid:3) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (B.13)where δm i are the individual contributions to the mass parameter in Eq. (B.8). By thismeasure, case B has a tuning ∆ − = 4%. A model which can reproduce the compressedscenario without additional tuning in the Higgs potential would be interesting, and it wouldrequire either that the parity-preserving couplings of η to the top partner are suppressed, themass of η can be raised without large contributions to the Higgs potential, or that the statedoes not exist in the first place. References [1] N. Arkani-Hamed, A. Cohen, E. Katz, and A. Nelson,
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