Digging for Top Squarks from Higgs data: from signal strengths to differential distributions

One way to hunt for top squarks is to look for deviations from the Standard Model in loop level processes involving Higgses. This method is indirect, but complementary to direct searches as it does not rely on specific top squark decays. Studying inclusive Higgs production $pp \to h$ alone is insufficient, since there are parameter regions where the effects of the two top squarks approximately cancel. This degeneracy can be broken by looking at the rate for highly boosted Higgses recoiling against a jet, $pp \to h + \text{jet}$. In this paper we perform a detailed study of the complementarity of the inclusive and highly boosted processes at the LHC, both in existing Run 1 and Run 2 data, and looking forward to high luminosity. To break the degeneracy, our calculation must maintain the full mass dependence in the loop functions and therefore cannot be recast in an effective field theory framework. We quantify the dependence of both topologies in the top squark parameter space, and outline which levels of experimental and theoretical understanding would be needed for boosted Higgses to be competitive with inclusive Higgs production.


Introduction
The second phase of the LHC is well underway and will usher in the era of precision Higgs physics, hunting for any sign of a deviation from the Standard Model (SM) expectations. Being a hadron collider, collisions at the LHC are inherently chaotic and complicated by low-energy QCD effects. This ultimately limits the accuracy at which particle properties can be measured. Future lepton colliders, such as CEPC, FCC-ee, and ILC, CLIC and TLEP [1] offer a cleaner environment and improved precision, but none of the currently discussed possibilities possess the energy of the LHC.
In this paper we study the interplay between the two approaches, precision and energy. As our testing ground, we will use the top squark sector of the minimal supersymmetric standard mode (MSSM). Top squarks modify Higgs properties at loop level [2][3][4][5][6], and the size and nature of the effects therefore depend on the energies involved. At low energies (on-shell Higgs), top squark loops will modify both Higgs production and decay, which we encapsulate using signal strengths. For our highenergy observable we will look at Higgs plus jet production.
While top squarks can be searched for directly [7], direct searches always rely on assumptions about the decay products, branching ratios, and spectrum. Indirect probes of the top squark sector, such as Higgs coupling measurements, require far fewer assumptions and in most circumstances only depend on the masses and mixings of the top squarks. Effects of top quarks on Higgs production, on the other hand, are independent of the rest of the spectrum and fixed to a large extent by supersymmetry, although related to the sbottom mass spectrum and direct searches [8].
Top squark effects in Higgs plus jet production have been explored previously in Ref. [9], where the authors focused on the region where the stop effects cancel in inclusive production. In Ref. [10], the differential rates of Higgs plus jets where discussed in the context of the MSSM, but no explicit expressions or analytical discussion the differential rates was given. Nowadays, a tool called SusHi [11] allows the user to compute differential rates and interface with Monte Carlo generators such as aMC@NLO [12]. Compared to that work, the present study is more comprehensive, covering a wider array of top squark scenarios and comparing the top squark sensitivity of inclusive Higgs cross section measurements with that of highly boosted Higgses. We also provide a more detailed discussion of the calculation of Higgs plus jet with scalar contributions, with cross checks in the soft and collinear limits. Our differential rates are computed using a fast Gaussian integrator, allowing the user to obtain the differential rates in a fraction of seconds, hence it is useful when dealing with spectrum parameter scans. There are also several studies of generic new physics loops affecting Higgs production within the framework of effective field theory [13][14][15][16][17][18] both at leading and next-to-leading (NLO) accuracy, where by definition the mass of the new particles is assumed to be large compared to the momentum of the process. Our study includes the full superpartner mass dependence. As we will show, this additional information allows us to differentiate between supersymmetric spectra that are degenerate when √ŝ m N P . The utility of Higgs plus jet has been explored in the past in a similar fashion to find fermionic top-partners [9,[19][20][21].
The setup of this paper is as follows: in Sec. 2 we define the top squark parameters and explore their contribution to inclusive Higgs production. Next, in Sec. 2.2, we introduce Higgs plus jet production and present our analytic results. The full form of the scalar loop contribution to Higgs plus jet is postponed to appendix A along with several cross-checks in special kinematic regions. In Sec. 3 and Sec. 4, we introduce the simulation tools and numerically explore the complementarity between inclusive Higgs production and Higgs plus jet for exposing top squark signals. Finally, in Sec. 5, we conclude.

Top squarks and the Higgs
While the minimal supersymmetric Standard Model (MSSM) is a vast framework with rich phenomenology, for the purposes of this work the only aspects of the MSSM that we care about is the top squark couplings to the Higgs and their masses. As such, we are not interested in features that require knowledge of the complete spectrum, such as how the measured Higgs mass is achieved or how/whether there is a viable dark matter candidate. The top squark sector of the MSSM consists of two complex scalar fields,t L ,t R , both of which receive the bulk of their mass from supersymmetry breaking. In addition to |t L | 2 , |t R | 2 type masses, the two scalars can mix through interaction with one of the MSSM Higgses. Rather than working with the entries of the top squark mass matrix, we will parameterize the stop sector by: where the lightest top squark mass is denoted by mt 1 , the mixing angle by θ which lies in the interval [−π/2, π, 2], and ∆m is the separation with the next state, ∆m 2 = m 2 t 2 − m 2 t 1 . With this parametrization, one can write the coupling of the lightest Higgs boson to the mass eigenstate top squarks as where and v = 2m W /g 246 GeV. Here we use the shorthand notation s β ≡ sin β, c β ≡ cos β, s W ≡ sin θ W , etc., where tan β is the ratio of the two Higgs vevs and α is the mixing angle rotating the CP-even neutral components of the two Higgses to the mass eigenstates, h and H 1 . In the decoupling limit [22], valid as long as the mass of the pseudoscalar A is large compared to the weak scale (m A m Z ), the angles α and β are related by α = β − π 2 and the couplings simplify to where the coefficients α 1,2 reduce to We will assume m A m Z throughout this paper. Having defined how the top squarks couple to the lightest Higgs boson, we next study their impact on inclusive Higgs production and Higgs plus jet production. In both cases, the top squarks enter at loop level, induced by gluons and/or quarks. As the stop-gluon coupling is fixed by SU (3) invariance, the top squark contribution to Higgs (and Higgs + jet) production for a givenŝ is a function of the stop masses and mixing alone.

Top squark contributions to gg → h
Focusing first on Higgs production via gluon fusion, the ratio of cross section in the MSSM to the cross section in the SM is given by [23][24][25] (2.10) The function F g depend on the masses of particles in the loop and their couplings to the Higgs. For the SM, the only important contribution is the top quark, while in the MSSM loops of both types of top squarks will contribute 2 . Both the MSSM and SM contributions receive higher order QCD corrections. As these corrections are not the same, there is some residual effect after taking the ratio which we encapsulate into the factor C g . Expanding out the mass and coupling dependence of F g for the cases of interest: (2.12) For a given particle i running around the gluon fusion loop, the functions F 0 and F 1/2 depend on the dimensionless variable τ i = m 2 h /(4m 2 i ) and can be further decomposed as: To gain some insight into Eq. (2.11), it is useful to take some limits. If both the top squarks are heavy, mt 1,2 m h , the top squark contribution can be matched onto local operators in the context of an EFT analyses [26], and the functions F g , C g simplify. Neglecting the effects of gluinos, squark mixing, and any running that would sum the large logs that would appear if the matching was done in two stages (e.g. one top squark eigenstate at a time), the perturbative matching correction is [27]: (2.14) Carrying out mt 1,2 m h in F g (mt 1 , mt 2 , θt), we find (2.15) up to corrections of O(g 2 ). Clearly, Eq. (2.15) is the sum of two types of terms, a positive-definite contribution from both mass eigenstates, and another dependent on the mixing angle. As such, the size -and even overall sign -of F g (mt 1 , mt 2 , θt) depends on the details of the mixing in the stop sector. The stop contribution depends on the mass and chirality of the eigenstates. If the overall contribution is dominated by a light eigenstate of pure handedness (θ → 0) the sum will be negative, whereas if the mixing term proportional to ∆m 2 is dominant then the sum will be positive. Stated another way, in the case of zero mixing we expect there to be an enhancement of σ(gg → h), but in the case where there is sizable mixing and the ∆m 2 term dominates, the separation between the two eigenstates will govern the suppression. Dialling the mixing between the θ → 0 limit and the limit of large ∆m 2 , it is clear that there are slices of parameter space where the contributions from the two states partially cancel each other and F g (mt 1 , mt 2 , θt) ∼ 0. For these parameter regions, gg → h will have reduced sensitivity of the top squark sector. We emphasize that, while the possibility of cancellation in F g when there is large mixing among the top squarks is easiest to see analytically in the mt 1 , mt 2 m t limit, it is not restricted to that parameter region.
As we will show below, by adding an extra jet to the final state pp → h + jet and focusing on regions where the Higgs has high p T , we can disrupt the cancellation among top squark loops. Whereas these cancellations may be accidental, and hence unmotivated from a model building point of view, adding differential information from pp → h + jet can only increase the amount of information we have on the stop sector. Moreover, future measurements of deviations in both total rates and differential rates may allow us to distinguish among different types of stop sectors, pointing out to specific UV realizations of supersymmetry.

Top squark contributions to pp → h + jet
Higgs plus jet production in the SM is a one-loop process induced by gg, qg,qg orqq partons. The top squark contribution has been calculated previously in Ref. [9][10][11]; however this is the first instance where the analytic form of the amplitude is given. The analytic form is useful as it allows us to understand how the contributions behave in different kinematic limits. In addition to the analytic expression for the top squark contributions, we also show their behaviour in the soft and collinear limits in appendix B. While not strictly required for our numerical analysis, the soft and collinear limits serve as a valuable cross-check.
The largest component of pp → h + j comes from the gg initial state. The gg → hg amplitude can be expressed in terms of eight primitive helicity amplitudes M h 1 h 2 h 3 corresponding to the possible choices for each gluon helicity h i = ±. We use the convention that the momenta of gluons p 1 and p 2 are incoming, and that of gluon p 3 is outgoing, so that the Mandelstam variables are defined as (2.16) The helicity amplitudes are then related to the full, un-averaged amplitude squared via (2.17) The index i here refers to the particle running in the loop needed to couple the gluons to the Higgs. After applying parity and crossing symmetry, only two of the helicity amplitudes are independent, which we take to be M i +++ and M i ++− . The amplitudes for fermions in the loops (needed for i = t, b) can be found in appendix A of Ref. [30].
The contributions to the helicity amplitudes due to loops containing a top squark with mass m = mt i and coupling to the Higgs g ht iti , are: and In these expressions we define (2.20) The functions B 1 , C 1 , D 0 are 1-loop basic scalar integrals. They are functions of s, t, u, the mass of the particle in the loop, and the Higgs mass; their definitions can be found in [31]. The function E 0 introduced in [30] is an auxiliary function defined as where C 0 is again a 1-loop scalar integral defined in [31]. The other pp → hj subprocesses (qq → hg, qg → hq,qg → hq) are controlled by a third function, the un-averaged amplitude squared The SM amplitudes can be found are in Ref. [30], while for scalars running in the loop we have: We can get the amplitudes for the subprocesses qg → hq and gq → hq from the above by swapping the Mandelstam variable s and u, and s and t respectively. Before we investigate the numerical impact of these corrections 3 to the p Tspectrum of a Higgs boson recoiling against one jet, let us discuss the qualitative feature of the above amplitudes. For the sake of illustration, we consider only M +++ , but similar results hold for other amplitudes as well and are presented in appendix A. We are particularly interested in how the SM (found in eq. (A.15) of ref. [30]) and top squark contributions behave when the momentum flowing through the loop is either much smaller or much larger than the loop particle masses. We denote m as the scale of the superpartner masses, m ∼ mt 1,2 , and distinguish two regimes: Using the results in eq. (A.2), we find that the scalar contribution reduces to  .3). In the SM case, the triangle loops contribute a factor of 1/m 2 t but have two extra powers of m t in the numerator -one from the Yukawa coupling of the top, the other from a helicity flip imposed by the interaction with the Higgs. As a result: 25) with no dependence on the top mass, as in the total cross section gg → h. Note that this low p T limit holds with a very good approximation also in the region p T ∼ m, m t .
• High-p T limit p T m, m t , m H . In this limit, the amplitudes for both squarks and fermion loops reduce to single and double logarithms of p T /m, do not depend on the mass of the particle in the loops, but only on kinematic invariants. Also, A i are the same for all processes involving any fermion in the loops coupling in the same way a top does (e.g. a top partner), whileÃ i are the same for all process with scalars in the loops.
Having seen the behavior of different components of the Higgs plus jet amplitude in the low and high-p T regime, we can combine things to get a sense of the behavior of the amplitude as a whole. In the limit where the p T of the Higgs is much less than m t or either of the top squark masses, the combination of Eq. (2.25), (2.24) and (2.15) yields: (2.27) In this limit, the top squark contributions are combined into F g , the same function appearing in inclusive production. In fact, in the m t → ∞, mt 1 , mt 2 m t limit, the exact combination appearing in Eq. (2.27). As such, the pattern of deviations in the low-p T regime of h + jet production will mirror those of pp → h. In particular, the parameter regions where the two stops cancel (e.g. when F g (mt 1 , mt 2 , θt) ∼ 0), both inclusive pp → h and pp → h + jet processes will appear SM-like.
At intermediate p T , m t < p T < mt i we can approximate the amplitude as the sum of a high-p T piece (Eq. (2.26a)) for the top plus a decoupled piece (Eq. (2.24)) for the top squarks: The contribution of the top quarks is again proportional to F g -so if there is a cancellation between two top squark contributions in the total cross section it will persist in this regime.
To break the cancellation, we must go to p T higher than the mass of the lighter stop. Here, we can approximate the amplitude as a high-p T contribution from the top loop and lightest top squark loop, and a decoupled piece for the heavier top squark: The couplings g ht iti multiply different kinematic functions rather than combining into F g , so the amplitude is sensitive to the top squarks even when parameters conspire to make F g ∼ 0. Note that any cancellation between the contributions of the stops is also broken at very high p T , i.e. p T mt 1 , mt 2 . To summarise, we have shown that by looking at the high-p T behavior of h + j once can break model degeneracies in the top squark sector, opening up sensitivity to parameter space that more inclusive searches cannot probe. However, the added information in h + j comes only when we consider p T higher than the mass of the lightest top squark. Whether or not one can expect to reach such kinematic p T regime at the LHC depends on the physical masses mt 1,2 . To get a more quantitative idea of the size of the deviation top squarks can cause, we turn to numerics.

Numerical Results
Using the squared matrix elements in Eqs. (2.17) and (2.22), we compute the p T spectrum of the Higgs dσ/dp T . The actual calculation of the p T spectra results from interfacing a modified version of HERWIG [33] with the parton density toolkit HOPPET [34]. Our results have been validated against the existing program SusHi [11], using the MSSM input card, with very large masses for the bottom squarks and the gluinos. The main difference between our implementation and SusHi is that we compute the full p T spectrum with a Gaussian integrator in a single run, whereas SusHi uses a Monte-Carlo integrator to provide a single p T -bin for each run. In terms of performance, with a single current CPU, with our implementation one can obtain the entire p T -spectrum for a mass point in less than a second, whereas to run SusHi in one single p T -bin would take about a minute.
In Fig. 1, we show numerical results for dσ/dp T for mt 1 = 600 GeV and four different values of the mass difference ∆m, obtained with the MSTW2008NLO parton distribution set [35], for tan β = 10 and maximal mixing, i.e. θ = π/4, as well as the corresponding prediction in the SM. All distributions have been obtained by setting both renormalisation scale µ R and factorisation scale µ F equal to (p T + p 2 T + m 2 H )/2. We note immediately that, for the chosen parameters, the difference between the SM spectrum and that with additional top squarks in the loops is not huge, at most 30% in the highest p T bins. The smallness of the effect is expected from the analytical results in Sec. 2.2, Eqs. (2.24-2.29).  In order to better assess deviations from the SM behaviour in the spectra, we construct the cumulative distribution σ(p cut T ), defined by and we consider the deviation δ(p cut T ) of the cumulative cross section σ(p cut T ) from its expected value in the Standard Model as follows: . (3. 2) The δ values corresponding to the parameters used in Fig. 1 are shown in Fig. 2. Note that each prediction consists only of a single curve obtained by fixing µ R = µ F = (p T + p 2 T + m 2 H )/2 in each p T spectrum. We have checked that a simultaneous variation µ R and µ F in the range 1/2 ≤ µ R /µ F ≤ 2 in the numerator and denominator in Eq. (3.2) has a negligible impact on δ(p cut T ). Therefore, no band associated with such scale variations is shown in Fig 2. Let us now comment on the robustness of the variable δ respect to the introduction of NLO effects. Recently, a complete calculation of the differential Higgs+jet process at NLO with fulll top-mass effects has been performed [36], see also [37,38] for recent predictions with an approximate treatment of the top-quark mass dependence. The authors in Ref. [36] find that the K-factor (the ratio NLO/LO) for this process is roughly 2 and quite flat with p T . Hence, one could simply re-scale the SM contribution in Eq. (3.2) by this K-factor, improving the SM prediction to NLO. But  the equivalent calculation for the BSM contribution is not known. Nevertheless, one can expect that the K-factor of the BSM contribution will be of similar size as the SM one. In fact, on one hand, one can argue that for large stop masses, the BSM contributions behaves in a similar way as the heavy-top effective theory (m t → ∞), which has a K-factor of about 2. On the other hand, Ref. [36] also shows that for finite masses the K-factor one obtains is of the same order as that of the heavy-top effective theory. Therefore, one would then expect a similar K-factor (flat with p T and of order 2) in the stop contribution to σ, and hence our calculation of the quantity δ and LO should be similar to the NLO. This suggests that this variable may be robust against higher-order contributions, although this statement will have to be validated when NLO corrections to the BSM p T spectrum will become available. In the next section we show plots with δ at LO, and the reader should keep in mind the caveats mentioned before.

Comparing inclusive Higgs and high-p T Higgs sensitivities
Having reviewed the top squark contributions to inclusive Higgs production and Higgs plus jet, we now compare the LHC sensitivity in the two modes. For our comparison, we will fix θ and tan β, and plot our results as contours in the mt 1 , ∆m plane. For θ, we pick two benchmarks, θ = 0, θ = π/4; these correspond to the extremes of no mixing and maximal mixing among the different top squarks -other choices of mixing angle would fall between the two.
We now discuss the range of parameters that can be probed through the variable δ, by showing different contours for δ(p cut T ) (translated into a percent deviation) as a function of the mass of the lightest top squark mt 1 and the top squark mass difference ∆m. The filled contours correspond to δ(p cut T ), as defined in Eq. (3.2), whereas the dashed lines represent analogous contours for the total cross section, obtained from eq. (2.10). All the results we show correspond to tan β = 10. We have checked that other values of tan β lead to similar results. Furthermore, all the parameters we have considered are not excluded by present data of the Higgs p T spectrum [39].
For inclusive Higgs production, each value of θ, mt 1 , ∆m can be mapped to κ g following Eq.  [41]) data. The blue, red, and black dashed lines show future sensitivity, quoted in terms of the percent deviation in the inclusive cross section coming from the top squarks (for the future bounds, the region to the left is excluded). As the difference between the two frames in Figs. 4 and 5 applies only to the Higgs plus jet mode, the dashed lines are the same in both panels. When the two top squarks are highly mixed we can easily spot the parameter region where a cancellation occurs between them: for θ = π/4, the cancellation occurs along a line between (mt 1 , ∆m) = (200 GeV, 400 GeV) and extending to (1000 GeV, 700 GeV). When θ = 0, there can be no cancellation between top squarks and the bounds in (mt 1 , ∆m) space look qualitatively different.
The inclusive Higgs production contours are overlayed on top of Higgs plus jet δ(p cut T ) contours, with δ(p cut T ) defined in Eq. (3.2). In the left panel, we show δ(200 GeV) while the right panel we show δ(600 GeV); in both frames we show δ deviations of 5, 10, 15 and 20%. Focusing first on the maximal mixing case (Fig. 4) and comparing the two panels, we can see the impact of the Higgs p T cut. For p cut T = 200 GeV, we have p cut T mt 1 so the the top squarks can be considered as decoupled in the bins of the Higgs p T spectrum that contribute most to δ(p cut T ). As shown in Eq. (2.29), the top squark contribution in this regime is proportional to the same term F g appearing in eq. (2.15). Therefore, whenever there is a cancellation in the total Higgs cross section, there will also be a cancellation in δ(p cut T ). This cancellation can also be appreciated by looking at the curves in Fig. 2. Picking to ∆m = 600 GeV as an example top squark mass, we see that we need to increase p cut T to more than 600 GeV to see an appreciable deviation of δ(p cut T ) from one. In fact, for p cut T > 600 GeV we start to open at least the loop containing the lighter top squark. This slightly larger sensitivity then is reflected in the right panel of Fig. 4, as the e.g. δ = 10% contour cuts out more parameter space than the 10% contour for inclusive Higgs production (dashed blue) for mt 1 400 GeV.
In Fig. 5, we show the analogous plots for the case in which no mixing occurs, i.e. θ = 0. As in the maximal mixing scenario, when p cut T = 200 GeV both the Higgs cross section and the Higgs p T spectrum have very similar sensitivities. This picture changes when increasing p cut T to 600 GeV, where the sensitivity of the Higgs p T spectrum is essentially doubled with respect to that of the total cross section. This increase in mass sensitivity as one increases the cut is expected from the analytical analysis and can be seen in Fig. 3.
As there is no possible cancellation among top squarks when θ = 0, F g < 0 and the amplitudes for pp → h and pp → h + jet are always increased by new physics. In this way, the θ = 0 case is the scalar analogue of the contribution of a fermionic partner of the top presented in [19]. However, the sensitivity of the p T spectrum to contributions from fermionic top partners found in [19] is much larger than the sensitivity to scalars we find here. This difference is due to the specific interplay between a top and a top-partner in composite Higgs models for different values of the Higgs transverse momentum. In the case of composite Higgs models, there is a cancellation between the top contribution and the top partner contribution which occurs whenever both states are decoupled. As the p T of the Higgs increases above m t , a heavy top-partner stays decoupled, while the top quark behaves as a light particle, breaking the cancellation between contributions. This is not the case for top squarks, where there is a region of parameter space where the contribution of the two top squarks cancels when they are both decoupled, while the top quark contribution remains SM-like. Therefore, in order to break the cancellation, one needs to reach transverse momenta that exceed the mass of the lighter top squark.
Furthermore, due to the fact that a top partner is a chiral fermion while a top squark is a scalar, the contribution to pp → h + jet from a heavy top squark is suppressed by one extra power of the top squark mass. Finally, note that in this paper we are using Higgs data to indirectly probe top squarks. Currently, indirect top squark bounds are in the range of mt ∼ 300 GeV [29,42]. The most recent direct searches for stops reach the TeV region, however this is not a fair comparison as direct searches are based on the assumption that stops decay into specific final states, largely involving missing energy signatures [7].

Discussion and Outlook
In this paper we studied how new colored scalars, top squarks, could affect the production of the Higgs boson. We were particularly interested in the interplay between inclusive gluon-fusion and differential Higgs production, h+jet.
At first glance, one would think the inclusive production g g → h should be the dominant handle on possible new coloured particles. This production enjoys rich statistics and a solid experimental and theoretical understanding. The effect of new physics, however, could be reduced due to symmetries (such as in many models in Composite Higgs) or simply due to accidental cancellations (as in classes of SUSY sectors). Under these circumstances, the study of differential rates of the Higgs production in association with a high-p T jet is then the best handle to uncover new physics.
In prior work, Ref. [19], we studied the sensitivity of the h+jet channel in the quite dramatic case of fermionic top-partners in composite Higgs models, where low-energy theorems tend to protect the inclusive production from any variations from the SM; see also Refs. [20,21].
In this paper we have focused on scalar top-partners, and in particular on supersymmetric top squarks, where no such low-energy theorems are present. Nevertheless, we found that there is an interesting interplay between the information contained in the h+jet differential production and the inclusive production. We presented analytical expressions for the corresponding amplitudes including the dependence on the stop spectrum, as opposed to previous studies [9][10][11].
Obviously, if an accidental cancellation occurs in the stop sector leading to a reduced gluon fusion rate [9], the differential production could become the best handle to discover new physics. Even if no dramatic cancellation occurs, the h+jet rate still adds value to the search for new physics. Searches in gluon-fusion inclusive and the boosted Higgs topologies face very different background challenges. Indeed, as we have seen in searches by ATLAS and CMS, some decays of the Higgs may be more accessible in the boosted regime than in the inclusive case. Hence, a combined analysis of the two topologies would provide us with the best handle to dig top squarks from the LHC data.
We have obtained analytical expressions for the h+jet amplitude in various p T regimes and found that the information one could gain on stops using differential rates is, unsurprisingly, concentrated in the regime p T mt. We then performed a numerical study to evaluate these relative effects. As a simple measure of the differential rate, we have defined a cumulative variable, namely the excess of events above a certain bin in p T with respect to the SM, δ(p T ). We argued that δ may be more robust against theoretical and experimental uncertainties than a fully differential study. With this naive measure of new physics in the high-p T region, we chose two benchmark values p T > 200 and 600 GeV and compared future prospects for inclusive and differential information under some assumptions regarding the level of uncertainties for each topology. As expected, larger cuts on p T can lead to increased sensitivity, but the gain has to be weighted against the loss of statistics. This is a similar situation encountered when using the missing-energy distribution in searches for SUSY Dark Matter [43] and a similar detailed analysis should be done. Such an analysis in the case of Higgs+jet topologies is feasible (at LO) with current Monte-Carlo event generators, through aMCSusHi, the interface of the fixed-order program SusHi to aMC@NLO [12].
In order to completely assess which values of δ can be actually probed by experiments we need to be able to accurately determine the SM contribution. This requires considering all possible backgrounds to Higgs production in the selected decay channels, e.g. h → γγ or to four-leptons, estimating the associated systematic uncertainties and performing a suitable statistical data analysis. Such an study is beyond the scope of this paper, but one can examine the results in Ref. [44], where the authors considered the effect of κ g as an effective operator (see also Ref. [45] for an NLO analysis). Specifically, using the transverse momentum spectrum of a Higgs decaying into τ τ and W W * , recoiling against a jet with p t,j > 200 GeV, and 3 ab −1 of luminosity at the HL-LHC, Ref. [44] claims that it is possible to exclude at 95% confidence level values of κ g in the range κ g < −0.4 and κ g > 0.3, with the additional constraint that no deviation is seen in the Higgs total cross section, and assuming that experimental systematic uncertainties are at most 10% 4 . For p t,j > 200 GeV, these values of κ g correspond to roughly a 6-7% deviation from the SM. This means that obtaining a similar sensitivity in the present case is not unreasonable.
However, the analysis in Ref. [44] relies on the dramatic growth with energy of the higher-dimensional operator, which in turns results in deviations from the SM on the Higgs transverse momentum spectrum which become as big as 80% for p T > 600 GeV. This is quite different from our situation, where increasing the cut on the jet transverse momentum does not lead to huge deviations from the SM. Indeed, let us compare the prospects drawn in Ref. [44] with a simple case where one single stop dominates the phenomenology. Namely, let us take Eqs. (2.28) and (2.15) in the case mt 1 mt 2 and θ=0, leading to κ g 1 + . A 10% reach in κ g would then mean a limit mt 1 > 1 TeV, which is clearly beyond the sensitivity we expect when looking at the full stop contributions.
In conclusion, we hope this paper serves to motivate the experimental collaborations to perform a combined analysis of gluon-fusion and differential information to search for new physics. We have provided an analytical understanding of the differential rates in various regimes of p T and defined a useful, but rather simplistic, variable δ to encompass some of the differential information. We also hope to encourage theorists to perform the calculation of the differential distribution for stops at NLO QCD, which would be ultimately needed to sensibly compare with the SM predictions in the high-p T tails.
First, we give the expansion of the scalar integrals appearing in the amplitudes: This gives Similarly, B Higgs plus one jet in the soft and collinear limit.
In this appendix we report the soft and collinear limits of the amplitudes and matrix elements for Higgs plus one-jet production computed in section 2.2. We believe this information might be useful for future studies aiming at exploiting the analytical properties of the matrix elements.
The main property of soft and collinear limits of matrix elements is that they factorize into the product of the tree-level matrix element and universal functions. Therefore, we first need the expression of the Born matrix element. Due to conservation of angular momentum, the amplitude for the process gg → h is non-zero only if the two gluons have the same helicity, say both positive. The un-averaged matrix element squared for this process is The top squark contribution to the above equation is with g ht iti is either of the couplings defined in equations (2.2), (2.3).

B.1 Soft limit
The soft limit p 3 → 0 corresponds to Keeping the most relevant terms in this limit, (2.18) gives u)) .

(B.7)
Evaluating again all scalar integrals in the soft limit we get (B.9) 5 The √ 2 factors comes from the differing normalisation factors for gauge group generators tr[T a T b ] = δ ab in the spinor helicity formalism, compared to the usual tr[T a T b ] = 1 2 δ ab . This is compensated by a relative √ 2 factor associated to the gauge coupling.
Since we have not used the spinor-helicity formalism, it is not immediate to rephrase our expressions in terms of helicity products. However, for real momenta, spinor products are simply equal to the square root of the relevant momentum invariant, up to a phase. The universal soft factor has an implicit helicity set by the helicity of the soft gluon, and so the choice of translating to angle or square bracket spinor products is fixed by this. We then obtain from (B.6) and (B.8) that Mt i +++ and Mt i ++− have the correct behavior (B.9) in the soft limit, modulo an overall phase that depends on the gluon helicity. This phase is the same as for the standard model case, and therefore can be factored out of each helicity amplitude and will not contribute to the amplitude squared.
Similarly, for the other helicity configuration we obtain Substituting this expression in (B.15) leads to M −++ 0 in the collinear limit u → 0.

Collecting all results we have
To check the correctness of the above limits, we have to translate our conventions for helicity and splitting fraction into the ones available in the literature, in which all momenta are considered to be outgoing. First, we need to swap the helicity of each incoming particle. Additionally, the relation of z to the momenta is different when the collinear gluons are outgoing. One can switch between the two cases by making the replacement z → 1 z . Adopting the usual convention of associating negative momentum signs to angle spinors we expect the behavior [47,48]  We must now translate (B.17) to helicity language. The translation from Mandelstam variables to spinor invariants is similar to the soft case, although the helicity consideration is slightly subtler. As the three legs of the splitting amplitude are collinear, we no longer have information about the contribution from each individual leg, as the helicity spinors become proportional. Instead what matters is the overall (outgoing) helicity of the three, which governs whether it is appropriate to translate to angle or square brackets, and with this consideration we indeed find the correct momentum dependence. However, this is not relevant in the end because, up to an overall phase [p 1 p 3 ] ∼ p 1 p 3 ∼ √ −u.