Probing top-partners in Higgs + jets

Fermionic top-partners arise in models such as Composite Higgs and Little Higgs. They modify Higgs properties, in particular how the Higgs couples to top quarks. Alas, there is a low-energy cancellation acting in the coupling of the Higgs boson to gluons and photons. As a result of this cancellation, no information about the spectrum and couplings of the top-partners can be obtained in gluon fusion to Higgs, just the overall new physics scale. In this paper we show that this is not the case when hard radiation is taken into account. Indeed, differential distributions in Higgs plus jets are sensitive to the top-partner mass and coupling to the Higgs. We exploit the transverse momentum distribution of the hard jet to obtain limits on the top-partners in the 14 TeV LHC run, finding that 300 ifb of data of 14 TeV LHC are sufficient to rule out top-sector mixing angles $\sin^2 (\theta_R)$>0.05 for top-partners with masses from 300 GeV to above 2 TeV.


Introduction
The idea that the Higgs is a composite resonance, manifestation of the breaking at high energies of a global symmetry is an old one [1]. This idea has been thoroughly explored and explicit realizations are built as Little Higgs (LH) [2], Composite Higgs (CH) [3] and Partial compositeness [4] models. In these models, the pseudo-Goldstone nature of the Higgs explains why other resonances of the new sector have not been seen yet, but introduces the hurdle of how to generate a potential for the Higgs, a mass and self-interactions. Successful electroweak symmetry breaking requires new states to generate a sizable potential, and those are typically top-partners. Top-partners are heavy resonances with the same quantum numbers as the top and couple strongly to the Higgs. Their contribution is essential to raise the Higgs mass to acceptable levels.
Top-partners are then a key piece to understand electroweak symmetry breaking, but searching for them is more complicated that one would expect. Although they contribute to the hgg coupling, there is a cancellation at low energies which renders this coupling insensitive to the mass and coupling of the top-partner [5,6]. Instead, the coupling is only sensitive to v 2 /f 2 , where f is the scale of breaking of the global symmetry leading to the pseudo-Goldstone sector. As a result, fits on the rates of Higgs production and decay into various final states are only sensitive to this parameter [7,8], and not to the individual coupling and mass of the top-partners. Double Higgs production pp → hh [9] is one obvious place to look for signs of top-partners. However, this process has a small cross section that is largely insensitive to finite top-partner masses.
Top-partners can be searched for directly, both produced in pairs pp → TT and singly produced pp → T + X. However, direct production carries more model dependence since the search strategies and limits depend on how the top-partner decays -an aspect of the model usually unrelated to electroweak symmetry breaking. Most experimental searches focus on the decay modes T → W + b, T → Zt, and/or T → ht. The bounds, assuming these three modes are the only ones available are roughly 700 − 800 GeV [10]. However if there are other decays possible, such as to exotic pseudo-Goldstone states [11,12], the T width will increase and the bounds will weaken.
Associated top-partner plus Higgs production pp → TT h is one way to directly test the hT T coupling. However, the cross section for this process falls steeply as the top-partners become heavy. Additionally, this method requires reconstructing the T produced with the Higgs, which is sensitive to the model-dependent details of how T decays.
In this paper we show that, unlike pp → h, the process pp → Higgs plus high-p T jet is sensitive to the individual coupling and mass of the top-partner. The reason for this can by seen by inspecting the diagrams that contribute to pp → h + j. One contribution comes from box diagrams, shown on the left in Fig. 1. As the additional gluon probes the fermion loop, it is not surprising that these diagrams carry a dependence on the fermion mass. The second contribution to pp → h + j comes from familiar hgg triangle diagrams stitched on to additional partons. Because of their similarity to gg → h diagrams, one may think these diagrams are not sensitive to the internal fermion mass. This is not true; to make a final state with high-p T , the intermediate gluon in diagrams such as the right side of Fig. 1 must have high virtuality. The high virtuality of the incoming gluon means the fermion triangle is resolved at a different, shorter scale compared to gg → h production, and the process becomes sensitive to the fermion mass. Therefore, by studying pp → h+j and comparing to SM rates, one can bound the top-partner and its Higgs coupling independent of the details of the T decay.
The setup for the remainder of this paper is as follows: In Sec. 2, we describe the lowenergy cancellation at the level of dimension-six operators, and how the extra jet would come from an effective theory including dimension-eight operators. We also set the notation and translation between our parametrization and common models in the literature. In Sec. 4, we show that the sensitivity to mass and couplings arise as double logarithmic terms in the matrix element in the high-p T limit. We then numerically study h + j production, both at parton level and after including parton distribution functions. In Sec. 5 we discuss the stability of our results at leading order when higher order corrections and experimental uncertainties would be included. Finally, in Sec. 6, we use the differential distributions to set limits on the top-partner masses and couplings in the 14 TeV LHC run. Figure 1. Typical diagrams contributing to pp → h + j.

Top-partners
In this section we describe the effects on Higgs production due to a new colored fermion that mixes with the top quark, which we will call the top-partner. To set limits, we present a explicit choice of mass mixing, which can be mapped into from several CH and LH models. Our study will focus on the Higgs production in association with jets, and, in the next section, we explain why one requires extra hard radiation.

Mass matrix
In this section we parametrize the top-partner sector as a Dirac fermion T = (T L , T R ), with mass M and a mixing with the SM top given by ∆. Without loss of generality, one can then write the mass matrix between the top t L,R and top-partner T L,R as This mass matrix can be diagonalized by a bi-unitary transformation, on the left with a mixing angle of θ L and on the right with a mixing angle of θ R . Identifying ytv √ 2 = m, we can trade the three parameters m, ∆, M for the two mass eigenstates m t , M T and one of the mixing angles θ R . Expanding the Higgs about its vacuum expectation value, we can then find the couplings of the mass eigenstate top-quark/top-partner to the Higgs boson. The diagonal Higgs couplings in terms of m t , M T , θ R are: where and η = ∆/M .
From the coupling expressions in Eq. (2.2), we can quickly see that the gg → h amplitude is insensitive to the mixing angle θ R . The gg → h amplitude from a single fermion loop can be written as [13]: and C 0 is the three-point Passarino-Veltman function, see Ref. [13] for conventions and the explicit form of C 0 . When the fermion running in the loop is heavy (τ → 0), A i asymptotes to a constant value, A(0) = −4/3 and the amplitude is independent of the fermion mass. Therefore, if we combine two gg → h amplitudes, both coming from fermions far heavier than the Higgs mass, the net amplitude also insensitive to the individual fermion masses and only depends on the strength of the Higgs-fermion couplings:  . A crucial requirement for this insensitivity is that both fermions are heavy compared to the external momenta. This requirement teaches us two things. First, it tells us that this insensitivity of gg → h to SM fermion -new fermion mixing is only possible for the top sector; all other quarks are light compared to m 2 H so Eq. (2.5) no longer holds 1 . Second, being a 2 → 1 process, the total invariant mass entering the loop (ŝ) in gg → h is fixed to m 2 H . However, this is not true for more general processes, such as when the Higgs recoils agains other final state particles; there,ŝ m 2 H is possible. We emphasize that, to guaranteeŝ m 2 H , one must focus on Higgs production with lots of recoil. Once higherorder corrections to gg → h [15][16][17][18] are taken into account, the Higgs will acquire some recoil. However, inclusive pp → h + X is dominated by p T m H , which is insufficient to unveil the properties of the internal fermion loop. We must instead look to Higgses produced in association with one or more high-p T objects.

Low energy Higgs theorems and the insensitivity of the hgg coupling
In this section we describe how a low energy theorem is responsible for the insensitivity of the dimension-five coupling hgg. Consider a colored fermionic particle which transforms under the fundamental of SU (3) c , and whose mass comes at least partially from electroweak symmetry breaking, M = M (H). In this case, it is well known [19,20] that the effect of this particle in the hgg coupling at low energies (E M ) would be described by where the coefficients A n can be written as

7)
H is the Higgs doublet, and M is the heavy fermion mass matrix.
In Composite Higgs and in Little Higgs models, the Higgs is a pseudo-Goldstone boson. This property restricts the coupling of the Higgs to fermions, hence the form of M. In these models, the form of the mass matrix factorizes as follows where f is the scale at which the global symmetry is broken, resulting in the appearance of the pseudo-Goldstone boson sector. For example, in the minimal Composite Higgs (i.e. coset SO(5)/SO(4)), ρ = sin 2 (2H/f ). This is similar to the fact that the pion non-derivative interactions appear as a function of the spurion π/f π . As a result of this restriction, when one evaluates the effect of the fermion sector on the hgg coupling, the dependence in the coupling and mass (i.e. the dependence in the piece ρ in Eq. 2.8) factors out and one is left with which is just a function of the parameter All dependence on the coupling and mass of the top-partners disappears in the low energy limit. This statement goes beyond the leading m → ∞ terms calculated in in Eq. (2.5). It tells us that even when higher order 1/m terms are taken into account, the hgg coupling in CH and LH models will only depend on ξ.
From the point of view of the effective theory, the inclusion of a hard jet in the final state corresponds to adding higher dimension operators. At the level of processes with one extra gluon, one needs to consider three dimension-seven operators [21] (i.e. dimension-eight operators with one v-insertion), which have the form As we will see in the next Sec. 4, the effect of top-partners in the processes involving those operators does carry information about the coupling and masses of the top-partners.

Top-partners in pseudo-Goldstone Higgs models
In models where the Higgs is a pseudo-Goldstone boson and assuming only one top-partner, the coupling of the top (t L,R ) and top-partner (T L,R ) mass eigenstates to the Higgs can be written in terms of field-dependent masses: where, at lowest order in the strong scale f , m t (h) and M T (h) can be parametrized as [22] where c σ is a contribution coming from the non-linearity of the Higgs in pseudo-Goldstone models; this piece is model dependent, but O( v 2 f 2 ). Expanding h → v + h and continuing to work to lowest order in ξ = v 2 /f 2 , the Higgs couplings in Eq. (3.2) can be massaged into the same form as Eq. (2.2): Therefore, we can identify where the O(ξ 2 ) correction includes the non-linear piece c σ . Despite the fact that CH and LH models come in many varieties and have various field content and underlying symmetry, the mass matrices for the top-partner sector -at least for several well-studied models -can all be cast in the form Eq. (2.1) up to terms of order O(1/f 2 ). This mapping is shown explicitly in Appendix A. Following the steps in Eq. (3.2 -3.4) for a given CH or LH model, we find where c is an order one coefficient arising from the linear coupling of elementary and composite fermions. Different CH, LH models yield different c. For example, c = 1 in the littlest Higgs model. In Fig. 2 we show the relation between the mixing angle and the parameters in the parametrization in Eq. (3.2). Large values of sin 2 (θ R ) imply low values of the scale of breaking of the global symmetry or large coupling a T , i.e. λ T y t .

Current bounds
The scale of breaking f depends on the UV completion of the theory. This scale is subject to electroweak precision tests [23] and flavor constraints, which depend on the assumptions on the symmetry structure and spectrum of the theory. For example, one could imagine that the UV completion preserves custodial symmetry [24], or that there is a spectrum designed to minimize the S parameter [25]. One could also assume there is a specific flavor structure [26] in the model at the scale f which keeps the flavor constraints under control. Regardless of these UV-sensitive issues, we expect modifications on the way the Higgs realizes electroweak symmetry breaking, hence modifications on the Higgs couplings to SM fields. Keeping an open mind about the UV structure of the top-partner theories, we will consider ξ 0.3, the current bounds from Higgs signal-strength fits [27] (though the actual bound on ξ depends on the specific model). In practice, the parameter sin 2 (θ R ) is more convenient to use than a T and ξ. Motivated by the bounds on ξ and the expression for a T (Eq. (3.5)), we consider sin 2 (θ R ) ≤ 0.4 in all numerical studies. Searches for top-partners in pair production through color processes, i.e. pp → TT compete with the search we propose here, but the comparison would depend on the electroweak quantum numbers, e.g. the left and right handed composition [10] and what they decay to. The phenomenology could be driven by leptonic channels [28] or more complicared multijet or boosted signatures [29,30]. Similarly, single production of the toppartner depends on the flavor structure of the model and how electroweak precision is addressed [29,31].

The process pp → H + j
Having mapped the top-mixing sector of CH and LH models into our parameterization, we are ready to explore the effects of top-partners on Higgs plus jet production. We start by looking at some limiting cases, then give numerical results both at parton level and after including the parton distribution functions (PDFs) 2 .

Generalities
When the Higgs is produced in association with a jet, the assumptions of the low-energy theorem no longer holds. Specifically, for a given p T,h = p T,j , there is a bound onŝ, For sufficient p T , thisŝ is no longer small compared to the mass of the fermion (top, or top-partner) running around the loop, we can no longer take the simpleŝ → 0 limit and must retain the full dependence of the loop functions onŝ/m 2 f . To get some idea of how the h + j cross section changes withŝ/m 2 f , we can look at the limiting cases: i.) high-p T and ii.) low-p T .
There are four partonic subprocess that contribute to pp → h + j, The actual breakdown of the subprocesses depends on p T , the scale choice, and the PDFs, but gluon-gluon initiated subprocesses typically dominate, so we focus on gg → h+g for now. The gg → h + g cross section can be decomposed as a sum over the various gluon helicity configurations [33,34] and the different fermions running in the loop: where β H is the final state velocity, λ i = ± are the helicities of the 3 gluons 3 , and f i indicates the different fermion species running in the loop. For simplicity, when looking at the limiting cases, we will focus on one helicity configuration, M +++ . We will also consider only one fermion species (mass m, Yukawa coupling y = m v κ) running around the loop and take the center-of-mass rapidity (y * ) to be zero.
• In the high-p T limit p T m, m H , M +++ contains single-and double-logarithms of the form [33,34] M +++ • For low p T , there is no dependence on the fermion mass since we are back in the gg → h limit of Sec. 2.2. Instead: Having shown the two kinematic limits, let's now consider the form of M +++ when there are two contributions, one from a lighter (EW-scale) fermion (i.e. the top quark, with mass m t , coupling κ tt ) and one from a heavier, TeV-scale fermion (the top-partner, mass M T , coupling κ T T ). When the final state has low-p T , the Higgs is approximately at rest, and the the low-energy theorem applies. Raising the p T , we enter an intermediate regime where the p T O(m t ) but p T M T . Approximating the top and top-partner contributions with the high-p T and low-p T limits, respectively, the matrix element in this regime is (schematically, and up to higher order corrections): We see that the top-partner leads to a term in the amplitude proportional to p T . This linear term will lead to a slower dropoff in the cross section as we push to higher p T . The matrix element in this kinematic region is sensitive to the top mass and Yukawa, and the top-partner Yukawa. There is no dependence on the top-partner mass until we go to an even higher p T regime, p T m t , m H , M T . There, (4.6)

Matrix Element level
We now turn to numerics to study how the matrix elements change in a top-partner setup as the final state p T is increased. Since gg is the dominant contribution to the total cross section, let us continue to focus on gg → h + g. A useful variable is the ratio of partonic matrix elements squared: ( 4.7) The Mandelstam variables depend on m H , the p T of the Higgs (or the recoiling jet) and the rapidity of the center-of-mass frame, y * . For a given p T , the minimumŝ occurs when y * = 0. Asŝ = 2 p T ( p 2 T + m 2 H + p T ) + m 2 H ,t =û = (m 2 H −ŝ)/2, in this kinematic region Eq. (4.7) is a function of p T , the heavy fermion mass M T , and the mixing angle θ R . Fixing M T to three different values, the ratio of partonic matrix-elements squared is shown in Fig. 3 as a function of sin 2 (θ R ) and p T . The shapes of the contours in Fig. 3 can be understood by the different functional forms of Eq. (4.5) and Eq. (4.6): for p T M T (below the red dashed line) the ratios have a similar shape for all three M T values, while for p T M T the contours change shape and their values depend on the M T assumed. Large ratios ∼ O(5) are possible, however the largest differences come at high-p T where the cross section is smallest. To gauge the effect on the full cross section we need to fold in parton distribution functions.

Including the effect of PDFs and running
We now move onto the effect of including scale and PDF effects. This has been done by adapting Herwig [35] amplitudes to include contributions from a top-partner. The modified matrix elements were then interfaced with HOPPET [36] and LHAPDF [37] to generate the distributions. We also implemented the top-partner in MCFM [38] to check our results 5 . For the SM, our calculation includes the effects of both the bottom and top quarks; for the toppartner scenarios we include the top, top-partner (with θ R dependent Yukawa couplings),  and bottom quark contributions. The differential p T distribution is shown below in Fig. 4 for the SM and six top-partner scenarios -three different M T values and two different sin 2 (θ R ) values. This plot exhibits the same features we saw at the partonic level, though diluted by the PDFs. First, as dictated by the low-energy theorem, all top-partner scenarios converge to the SM result at low-p T . Second, as suggested by the analytic results in Sec. 4.1, the p T -spectra in top-partner scenarios is harder than the SM. Finally, the spectra for a given mixing angle are not sensitive to the top-partner mass until the final state p T ∼ M T .
The difference in the p T spectrum between the SM and a theory with a top-partner is our main result. The full p T spectrum is, however, an experimentally difficult quantity to measure since the higher p T bins will suffer from low statistics. A similar, though perhaps experimentally more tractable observable is the net Higgs plus jet cross section for all events that satisfy a given p T cut, i.e.
Using σ(p T > p cut T ), we define a new variable δ, . (4.9) which encapsulates the effect of a top-partner in the cross section. Here, σ t+T is the cross-section in a theory with a top-partner of a given mass and mixing angle, while σ t is the cross-section for the SM. In Fig. 5 we show the value of δ as a function of p cut T for different values of M T and the mixing angle. Obviously, the effect increases with the topmixing angle. As in the differential distributions, heavier top-partners lead to a harder p T spectrum, but the effect δ is negligible until p T > M T . To generate this plot, we have taken TeV. While gluon-initiated subprocess dominate pp → h+j for low p T , it is interesting to see how the breakdown of the cross section into partonic subprocesses changes as we increase the p T . In Fig. 6 we plot the ratio dσ i dp T / dσ tot dp T , i = gg, gq +qg, orqq (4.10) in the SM and in the theory with a 1 TeV top-partner (here, dσtot dp T is the differential distribution including all channels in the respective theory). The dominant cross section  Figure 6. Breakdown of the differential cross section dσ/dp T into different initial state channels, dσ i /dp T , where i = gg (solid), qg +qg (dashed) and qq (dotted). The blue (red) lines correspond to the SM (top-partner) theory. The top-partner in this plot corresponds to M T = 1 TeV and sin 2 (θ R ) = 0.4. The contribution from gq + gq is not shown (thus the sum does not equal 1.0) since it is identical to qg +qg.
corresponds to gg for jet p T 800 GeV, after which qg becomes the dominant subprocess. The crossover is delayed in the case of a theory with a top-partner respect to the SM, as the former exhibits a harder spectrum. Note also that the qg and qq initial states do depend on the quark mass. For example, in the right-hand diagram in Fig. 1, the dependence on the quark in the loop can be understood as the t-channel gluon virtuality enhancing the double-logarithmic structure in the matrix element. The sharp features in theqq subprocess at p T ∼ m T and p T ∼ M T come from a resonant enhancement in the loop functions near s ∼ 4 p 2 T ∼ 4m 2 t orŝ ∼ 4 M 2 T respectively. Had we plotted to p T > 1 TeV, theqq fraction in the top-partner scenario would shrink again.
Although we have calculated loops of top quarks and top-partners, our pp → h + j calculation is still a lowest-order calculation. Being a lowest order (LO) result -especially given that the cross section depends on α 3 s -one immediate worry is that our result may be highly dependent on the scale choice and the choice of PDF. However, provided we look at a ratio of cross sections, such as δ(p cut T ), one might expect most dependence on these input choices should drop out. We have confirmed this intuition with cross-checks. First, calculating δ(p cut T ) for three different values of the factorization and renormalization scheme, µ R = µ F = µ = p T + p 2 T + m 2 h /2, p 2 T + m 2 h and m H , we find the difference in the ratio between the three schemes, i.e. δ(p cut T , µ)/δ(p cut T , µ ) is below the percent level. Next, we verified the stability of δ(p cut T ) under changes in the PDF schemes by comparing δ(p cut T ) calculated with two different PDF sets. Using top-partner parameters M T = 1 TeV, sin 2 (θ R ) = 0.4, the ratio of δ(p cut T ) calculated with MSTW2008nlo68cl PDFs [42] to δ(p cut T ) calculated using cteq6mE [43] is shown below in Fig. 7. The effect is less than 2% in the range of p T we will consider. . The ratio of δ(p cut T ) calculated with MSTW2008nlo68cl parton distribution functions to δ(p T ) calculated with cteq6mE. The top-partner used for calculating δ(p T ) has mass 1 TeV and mixing angle sin 2 (θ R ) = 0.4. All distributions were generated using 8 TeV LHC parameters.
We move on to study the effect of the collider energy by comparing the results for √ s = 8 TeV and √ s = 14 TeV. The quantity δ(p cut T ) is shown in Fig. 8. Comparing with the same quantity at √ s = 8 TeV Fig. 5, one can see that the ratio does not depend strongly on the energy of the collider.
Finally, a comment on the dependence of the result on the rapidity acceptance for the jet. The topology we are looking at, with a Higgs recoiling against a high-p T jet tends to produce very central events. This is just because at high p T there is not enough phase space to produce high rapidity jets. Indeed, our Herwig implementation, in which we have integrated over all rapidities, is in agreement with MCFM with a cut |η| < 5. We have checked in MCFM that moving the cut on jet rapidity from |η| < 5 to |η| < 2.5, which corresponds to the acceptance of the CMS and ATLAS central trackers, does not alter our results.

Stability against higher order corrections and experimental uncertainties
In Sec. 4 we discussed the stability of the results when changing the renormalization scale and PDF sets, finding that the effect is at the percent level. In this section we will focus on the effect of adding higher order corrections and experimental uncertainties. Currently, there is no available computation of Higgs plus jet at next-to-leading order (NLO) including finite mass effects 6 . This calculation is beyond the scope of this paper, but given its importance for constraining new physics, one would hope that it becomes available in the near future. Given this situation, the best one can do is to evaluate the NLO effects, differentially, in the infinite top mass limit. We have evaluated the K-factor, the LO and NLO Higgs plus jets using MCFM [38] in the infinite top mass limit in the differential distribution dσ/dp T . The K-factor is rather flat (roughly O(2)) as a function of p T for µ = p 2 T + m 2 H , but has a slope for µ = m H . We expect the higher order corrections to produce changes in shape once the finite mass effects are taken into account. Nevertheless, as our observable is an integrated cross section, dominated by the region near the p T cut, we expect higher-order corrections to just amount to an overall K-factor, although this expectation should be corroborated with a explicit calculation. Moreover, one should aim to obtain as much information as possible from the differential cross section, whereas in this paper we have to limit ourselves to an integrated cross section with a p T cut. With a NLO calculation with finite mass effects, the differential distribution would become a more powerful tool to disentangle new physics.
The most important experimental uncertainty for our observable would be energy and momentum smearing of the Higgs or the recoil jet. As we are using integrated cross sections as in Eq. (4.8), the effect of smearing would affect the region near the cut. Similarly, the effect of the underlying event would also produce some momentum smearing, although we expect it would be negligible at p T > 100 GeV [45]. Therefore, at least for p T cuts greater than ∼ 200 GeV, we believe experimental effects should be small and will affect the SM and top-partner scenarios in a similar way.

Mass limits on top-partners
In this section we study the sensitivity of the 14 TeV LHC to the top-partner mass and coupling. The events we are focusing on are characterized by a high-p T jet plus a Higgs boson.
Given a particular Higgs plus jet final state and some amount of luminosity, we can set limits on the top partners by comparing two hypothesis: SM Higgs plus jet production vs. Higgs plus jet production in a top partner scenario, where the latter hypothesis is a function of M T and sin 2 (θ R ). For simplicity, and since there is no dedicated CMS/ATLAS search in Higgs plus hard jet to work off of, we will quantify the difference between the two hypothesis with the variable where S is the signal and S 0 is the SM piece, S 0 = σ t (p T > p cut T ) × L. We claim sensitivity to rule out a top-partner at the 95 % confidence level if S/ √ S 0 at luminosity L is bigger than 2.0. This test statistics is only approximate as it assumes that the SM background can be completely removed. This is a reasonable assumption in the clean leptonic and photon final states. For the higher rate, hadronic Higgs decay modes the SM background is more problematic, though the requirement of a hard jet in the event is a useful handle for suppressing background. Dedicated studies of the backgrounds in all Higgs final states for Higgs plus hard jet events are well motivated, but beyond the scope of this paper.
Our test statistics also assumes that the cut efficiency for the SM and new physics Higgs plus jet events is the same, and that the Higgs branching ratios are not modified by new physics 7 . A final caveat in our significance measure is that we use LO cross sections only. As we mentioned in Sec. 5, the complete, mass-dependent higher order corrections are not known yet and may carry some non-trivial fermion mass and p T dependence.
In Fig. 9, we show the significance as a function of the mixing angle for a standard luminosity fo 300 fb −1 . With mixing angles sin 2 (θ R ) 0.05, one would have sensitivity in a range from around 300 GeV to above 2 TeV. Recall that in Fig. 2  between the mixing angle and the top-partner parametrizations in the literature. A limit of sin 2 (θ R ) at 0.05, is equivalent to a limit on the scale of breaking f for a fixed value of a T . For example, In Sec. 4.3, we showed that δ(p cut T ) is very stable against changes in definitions of renormalization scale and PDF sets. We have checked that the quantity S/ √ S 0 is also rather stable. To do so, we define where ω i is a label for the choice of running parameters. The value of ∆ for the same two choices of PDF schemes mentioned in Sec. 4.3 is shown below in Fig. 10. As before, the effect at the sub-percent level. We have also checked agains changes in renormalization scales and PDF sets within a PDF scheme. From Fig. 9, we see that the sensitivity curves are fairly flat, indicating that the S/ √ S 0 is mainly sensitive to the coupling. To see the difference between higher top-partner masses, we would need to look at higher-p T , where there is simply not enough rate at √ s = 14 TeV.
This fact makes the Higgs plus jet search quite complementary to traditional pp → TT top-partner searches, where the production the rate is set by M T alone. The decay of top-partners is more model dependent. However, at least in simple setups, the decay is completely governed by "Goldstone-equivalence" and is thus independent of the TT h coupling. As the sensitivity is rather flat with M T for M T 600 GeV, one can plot the luminosity required to set a exclusion as a function of the top-mixing angle alone. This is shown in the left panel of Fig. 11, where we have chosen a cut on p T of 200 GeV. In the right panel of Fig. 11 we show the effect of changing this cut for sin 2 (θ R ) = 0.2. As the cut increases, the sensitivity does increases until at about p T 400 GeV, the cut is too hard and the sensitivity starts decreasing.

Conclusions
In this paper we have presented a first step to search for top-partners in events where the Higgs is produced in association with hard jets. This topology avoids the well-known lowenergy cancellation acting the hgg coupling when the Higgs is a pseudo-Goldstone boson that renders the gg → h process insensitive to the mass and coupling of the top-partner.
Our analysis is motivated by these type of models, but it just relies on the presence of a top-partner with couplings to the Higgs coming from electroweak symmetry breaking 8 . We have worked out the dependence of the spectrum on the top-partners using variables with are not directly the differential distribution, but integrated distributions with a cut on p T . We checked that the results at leading order are stable against choices of renormalization scales and PDF sets. We discussed what would be the effect of including NLO corrections. Unfortunately, no NLO computation is available in the finite mass limit. We did check that in the infinite mass limit the K-factor on the differential distribution is flat for appropriate choices of the renormalization scale.
Finally, we have assigned a significance for a top-partner signal, finding that 200 fb −1 of data may access very low mixing angles sin 2 (θ R ) <0.05, for a large range of top-partner masses. These results are certainly encouraging and warrant more dedicated study.
Furthermore, more information could be obtained by looking at the differential distribution, as opposed to the integrated one. This study would require an excellent understanding of the NLO corrections of this distribution, a calculation we hope will become available in the near future.  Table 1. Translation between our parametrization and the choices in the model space.