Measurement of the Triple Higgs Coupling at a HE-LHC

The currently unmeasured triple Higgs coupling is one of the strong motivations for future physics programs at the LHC and beyond. A sufficiently precise measurement can lead to qualitative changes in our understanding of electroweak symmetry breaking and the cosmological history of the Higgs potential. As such, the quantitative measurement of this coupling is now one of the benchmark measurements for any proposed collider. We study the capability of a potential $27\,\mathrm{TeV}$ HE-LHC upgrade in measuring the Higgs trilinear coupling via the di-Higgs production process in the $b\bar{b}\gamma\gamma$ channel. We emphasize that a key background from single Higgs production via gluon fusion has been underestimated and underappreciated in prior studies. We perform a detailed study taking into account two different potential detector scenarios, and validate against HL-LHC projections from ATLAS. We find that the di-Higgs production process can be observed at $\geq 4.5 \sigma$, corresponding to a $\sim 40\%$ measurement of the Higgs self-coupling, with $15\,\mathrm{ab}^{-1}$ of data at the HE-LHC.


Introduction
Since the discovery of the Higgs boson [1,2], one of the primary goals in high energy physics has been to understand its properties better. While the Higgs looks very similar to the Standard Model (SM) Higgs, many of its properties have yet to been measured, or measured sufficiently well to test for many possibilities of beyond the SM (BSM) physics. One of the most exciting possibilities is reaching the level of precision where quantitative measurements can lead to qualitative changes in our understanding of electroweak symmetry breaking (EWSB).
While the quantitative improvement in Higgs couplings to other particles has been studied in many contexts at future colliders, e.g. naturalness [3,4], the most exciting possibility is the measurement of the Higgs self interaction. Such an interaction has never been observed in nature and it is the only direct window to further information about the Higgs potential itself. Currently we only have experimentally determined the location of a minimum of the Higgs potential and the value of the second derivative at that minimum. To learn more about the nature of the Higgs and its cosmological history requires the measurement of the triple Higgs coupling and beyond. Unfortunately, measuring the Higgs self-coupling at the 14 TeV LHC appears exceedingly difficult unless its value deviates substantially from the Standard Model prediction [5,6], although there is potential room for incremental improvement [7]. Unfortunately, simply measuring the coupling is not enough to necessarily change our qualitative understanding of the Higgs potential or its cosmological evolution. For example, the strength of a potential EW phase transition (EWPT) depends critically on the value of the effective triple Higgs coupling, and must be measured to the O(10%) level or better to distinguish the order of the phase transition in some cases [8,9]. Furthermore it has been recently shown that there might not even be an EW phase transition at typical EW scales, or even possibly at any scale [10]. In these cases the triple Higgs coupling can distinguish the qualitatively different scenario, but an even more precise measurement is needed than to distinguish a first-order from second-order EWPT. Therefore the actual quantitative precision a future collider program can reach in this coupling is of paramount importance.
With planning for the High-Luminosity phase of the LHC underway, there is growing interest in the possibility of an upgrade to the center of mass energy of the LHC. Such an upgrade, known as High-Energy (HE) LHC, would reach a center of mass energy of 27 TeV, and may be the next opportunity to explore physics at the energy frontier. While the measurement of the self-coupling has been well studied at a potential 100 TeV collider [11][12][13][14][15][16][17] and proposed lepton colliders [18][19][20], there are far fewer studies for the HE-LHC [21]. In particular, despite the extensive work for 100 TeV hadron colliders, the relevant backgrounds are still being understood and better estimated. In this work we carefully analyze the HE-LHC measurement of the triple Higgs coupling from di-Higgs production in the bbγγ channel. We implement this study using a Delphes 3 [22] based detector simulation and two possible parametrizations for ECAL resolution. We validate our study against existing ATLAS results as described in the Appendix and provide 14 TeV projections alongside those for HE-LHC. Contrary to some existing studies for future hadron colliders, we find that the largest background, after typical analysis cuts to avoid the tth background, comes from single Higgs production via gluon fusion in association with additional jets. We study this background from a number of different perspectives and emphasize that this background increases in importance as one moves to higher energy colliders. In our study we do not optimize for observation of di-Higgs production alone, as could be done, as sensitivity to anomalous couplings is the primary focus for elucidating properties of the Higgs potential.
The rest of the paper is structured as follows, in Section 2, we discuss the details of di-Higgs production via gluon fusion at a hadron collider, focusing in particular on the decay channel hh → bbγγ and the simulation of the signal and backgrounds. Section 3 contains more information on the HE-LHC scenario and the detector simulation used for our study. In Section 4 we describe our cut-and-count analysis and the significance and precision attainable on the di-Higgs production cross section and trilinear coupling, as well as several benchmark figures on the capabilities of an HE-LHC compared to the High Luminosity (HL-LHC) scenario currently planned. We summarize our results in Section 5. In the Appendix we demonstrate the validation of our methods through a comparison with the ATLAS projection for this channel at the HL-LHC [6].

Di-Higgs Production
The most direct way to measure the Higgs trilinear coupling, λ 3 , at a hadron collider is via the Higgs pair production process, which arises primarily from gg → hh. Our conventions are chosen such that after EWSB the interactions from the Higgs potential are given by so that λ 3 = λ 4 = 1 in the SM and m h = √ 2λv 2 is the physical Higgs boson mass. The lowest order diagrams contributing to di-Higgs production for gg → hh in the SM, shown in Fig. 1, arise from the "triangle" diagram, as in single-Higgs production with an additional hhh vertex, and from the "box" diagram, which is independent of λ 3 . To leading order, these amplitudes scale as [12]: Due to the additional fermion line in the box diagram, the two amplitudes interfere destructively, leading to a ∼ 50% reduction in the total cross section in the Standard Model, with a maximum cancellation near λ 3 ∼ 2. In this study, we focus exclusively on the gluon-gluon fusion channel of di-Higgs production, which dominates at all relevant energies [23]. Only the leading-order (LO) cross section was known exactly for many years [24][25][26], with a number of approximations to account for higher-order corrections. These include next-to-leading order (NLO) [27] and NNLO [28,29] corrections in QCD, using the infinite top quark mass limit, as well as estimates of threshold resummation effects at NNLL accuracy [30,31]. Recently, the cross section was computed exactly at NLO, including all top-quark mass effects [32,33] and matched to parton showers [34]. A better approximation can thus be obtained by shifting the NNLO+NNLL values in [35] to account for the finite-mass effects, leading to the values shown in Table 1 Table 1: The Standard Model cross-section for di-Higgs production via gluon-gluon fusion at several different center of mass energies, based on the "Born-projected approximation" in [36]. Scale uncertainties are shown as superscripts/subscripts.

Signal and Primary Backgrounds
There are a number of decay channels available for hh production, many of which can be exploited by a future collider when the production rate is relatively large. These include bbγγ, bbbb, bbτ + τ − , bb4 , and bb + − , where the leptons can arise either directly from the Higgs decay, or from an intermediate W/Z (see Table 2 for a table of some common hh final state branching ratios). Here, we focus on the hh → bbγγ decay channel (BR = 0.264% in the SM), which allows reasonable control of the backgrounds while maintaining a high enough rate from the large h → bb branching ratio. This channel has been shown to give the highest significance on the signal at future colliders [12], although the hh → bbbb prospects are competitive at 14 TeV [37] and significant improvements have been made in hh → bbτ τ as well [38].
0.011% 4.8 × 10 −5 2.6 × 10 −6 γγ 5.2 × 10 −6 5.6 × 10 −7 ZZ(4 ) 1.5 × 10 −8 Backgrounds to the hh → bbγγ mode arise from the single higgs production modes, ggF (γγ) (where the bb pair is produced by extra radiation), Zh(γγ), and tth(γγ). There are also a number of non-resonant backgrounds arising from processes such as bbγγ, jjγγ, bbjγ, ccjγ, bbjj, and Z(bb)γγ, where j refers to a jet arising from light (u, d, s, c) quarks or gluons. Finally, there are also backgrounds due to top-pair production, e.g. tt and ttγ, where the additional photons can arise from either misidentified light jets or electrons. We neglect backgrounds where more than two misidentified particles are required, such as ttγγ and bjjj, as their contributions are expected to be negligible.  Table 3: List of signal and background processes and the event generator used to simulate their matrix elements and parton showering. Also shown are the cross sections of each process and the corresponding order in QCD at which the cross section used to normalize the expected event rate was computed. Note that we use the same naming scheme for backgrounds as ATLAS, but we do not list bbh as it is included in ggF as described in the text. * The hh signal is scaled to NLO in QCD with the full m t dependence, but also includes NNLO corrections in the m t → ∞ limit as well as threshold resummation effects to NNLL. † The ggF background was produced at leading order and in the m t → ∞ limit, but with two extra real emissions included. See text for details.

Signal and Background Simulations
For the signal simulation, the leading order loop level process is simulated directly using MadGraph5 aMC@NLO [39,40] using the NNPDF2.3LO PDF set [41] including all finite top mass effects. The MadSpin package [42] was used for the Higgs boson decays and Pythia 8 [43] for the showering and hadronization of events. The LO signal is normalized to match the NLO+NNLO/NNLL cross sections listed in Table 1. Signal events were also generated with the trilinear coupling λ 3 (defined as in Eq. (2.1)) modified to values ranging from −1 to 10 times the SM value. Some kinematics features of the di-Higgs signal at parton level are illustrated in Fig. 2 for different values of λ 3 . Several of these features can be understood from the naive scaling in Eq. (2.2); in particular, for large values of λ 3 , the process becomes dominated by the "triangle" diagram, which falls off for largeŝ as clearly demonstrated in Fig. 2.
The background from single-Higgs production via gluon fusion, ggF (γγ), was ignored in a number of previous projections, but, as demonstrated in Section 4, it is one of the dominant backgrounds to di-Higgs production. As such, we'll treat the generation of this background with special care in Section 2.3.1. Events from other single Higgs production modes -Zh(γγ) and tth(γγ) 1 -were generated directly in Pythia 8 at LO using the CTEQ6L1 PDF sets [44]. To account for higher order effects, the 14 TeV samples were normalized to the NLO cross sections based on the LHC Higgs Cross Section working group recommendations [35]. These recommendations were also used to compute k-factors at 7, 8, 13 and 14 TeV. These k-factors were found to vary modestly as a function of energy, so we extrapolated to 27 TeV based on a second-order polynomial in √ E. The extrapolated k-factors are found to vary only by 4% and 6% for tth and Zh respectively. Since these differences are relatively mild, we use these k-factors to normalize the LO samples, resulting in the cross sections presented in Table 3.
Non-resonant background events were generated with MadGraph5 aMC@NLO [39] using the CTEQ6L1 PDF set and interfaced with Pythia 8 [43] for showering and hadronization. For all of the non-resonant QCD backgrounds, one additional jet in the matrix element was allowed, and MLM matching [45,46] was implemented with the parameter xqcut set to 30 GeV to prevent over-counting in phase space. For all of the non-resonant backgrounds, a common set of generator cuts was used: Jets produced in the hard process were required to have p T,j > 20 GeV and |η j | < 4.0 while photons were required to have p T,γ > 25 GeV and |η γ | < 2.7. For backgrounds where a photon pair is produced in the hard process, the invariant mass of the photon pair was limited to 90 < m γγ < 180 GeV. Finally, the invariant mass of a b-quark pair was required to be greater than 45 GeV, and all pairs of light jets (including g, u, d, s, and c) were required to have m jj > 25 GeV. To avoid overlaps between the different samples, events were vetoed in the ccγγ, jjγγ and ccjγ samples if they contained two b-quarks, and from the jjγγ samples if they contained two c-quarks.
Finally, the tt and ttγ processes were produced directly at LO in MadGraph interfaced to Pythia 8 with the CTEQ6L1 PDF set. The tt total cross section at NNLO in QCD was recomputed at both 14 and 27 TeV, with soft-gluon resummation effects to NNLL, using Top++2.0 [47]. The NNLO+NNLL cross section is then used to normalize the events. Events in the tt sample were vetoed if they contained any true photons. For ttγ, the NLO cross section is available at 14 TeV [48]. We use this cross section to normalize our 14 TeV sample, and compute a k-factor, k = 1.48, which we take to be the same at 27 TeV.

Details of the ggF Background
The background from single-Higgs production via gluon fusion turns out to be a dominant contribution to the background for hh production, as the true h → γγ process is irreducible, and QCD interactions lead to a significant continuum of bb, some of which will be indistinguishable from the h → bb signal regardless of cuts. However, because the lowest-order process is lacking the two partons necessary to fake a bb signal, the background is dependent on different modeling choices. To account for all potential production mechanisms of the additional partons, we generate the sample inclusively in the heavy top-mass limit, allowing up to two extra partons in the matrix element, where the partons can be any light jets or a c-or b-quark. As with the signal process, the Higgs in the sample is decayed to two photons using MadSpin before passing events to Pythia 8 and Delphes 3 with MLM matching between the parton-level events and the parton shower. Note that this is different than how this process was simulated in [17] where this background was generated with Powheg-Box, interfaced to Pythia 6. In [17] this was done to mimic the ATLAS background generation of this process for the HL-LHC [6], which we also compare to for validation in Appendix A. We have found roughly similar results with Powheg-Box and MadGraph however, the Powheg-Box codes generate separately at NLO the h, h + j, and h + jj processes, and in [17] only the h code was used, which in turn means the additional b-jets are generated only from the parton shower, or splitting from the single emission interfaced to the parton shower. The double real emission graph is not included in the method of ATLAS [6] and [17], nor is the hard gluon splitting diagram. We find it is much simpler to generate the inclusive matched sample in MadGraph in the infinite top mass limit (which Powheg-Box also takes) with all the relevant processes which lead to hard b's. We comment on these differences further in both section Section 4.2.1 and Appendix A, and demonstrate the robustness of our estimate.
Because our final selection will require two additional jets with respect to the LO process, a significant portion of the selected ggF sample will be from the h+2j portion of the sample, so using the inclusive k-factor (known to N 3 LO [49,50]) would be an overestimate. The kfactor for exclusive h+2j production has been computed at NLO [51][52][53], and was found to be a mild ∼ 15% effect. While the m t → ∞ limit is known to be inaccurate for energies 2m t , the bulk of the background comes from events with p T,h ∼ m h , where finite top-mass effects are expected to be small, as has been shown explicitly for exclusive h + j production [40,54]. Note also that our choice of p T cuts in Section 4 also eliminates effects from the interference of top-and bottom-quark loops in the h+j sample 2 . Additional cross-checks of our modeling of the ggF background are described in Section 4.2.1.

HE-LHC measurement
An exciting possibility for the LHC is an upgrade to center of mass energies of √ s = 27 TeV.
This could be achieved by upgrading the 8.33 T field dipole magnets currently installed at the LHC with 16 T dipole fields that are being developed for FCC-hh studies [55]. Such an upgrade would substantially extend the reach of the LHC for new physics.
For the remainder of this study, we consider two benchmark scenarios: HL-LHC with √ s = 14 TeV and an integrated luminosity L = 3 ab −1 (the same as considered in [6]) and HE-LHC with √ s = 27 TeV and L = 15 ab −1 , which represents the full improvement possible with an upgraded collider. An upgrade to 27 TeV will present a number of challenges to future detectors, particularly in understanding the effects of pile-up interactions. For simplicity, we assume that the projected performance of the ATLAS detector at the HL-LHC [56,57] can be replicated. We also consider the effects of improved electromagnetic calorimeter resolution, which would help discriminate true h → γγ decays from background. These two detector performance scenarios are discussed more in the following section.

Detector Assumptions and Event Reconstruction
To approximate the HE-LHC detector scenario, we use Delphes 3 [22] to simulate the performance of an upgraded ATLAS-like detector. A custom input card based off the current ATLAS card available in Delphes v3.4.1 is used, with slight modifications to better match the projected performance of the ATLAS and CMS detectors for HL-LHC based on the current scoping documents [56,58] and other projections [57,59,60]. These modifications were validated by comparing to the most recent projection for measuring λ 3 at 14 TeV [6] (see Appendix A for details). The custom card also allows us to directly simulate the b-tagging performance, as well as the rates for jets or electrons to fake a photon in the detector, discussed in more detail below. The efficiency of identifying photons is taken to match the results of the "Tight-ID" requirement in [57], which reaches approximately 85%, but falls off for photons with p T 120 GeV. The probability of an electron being misidentified as a photon is taken to be 2% for |η| < 1.7 and 5% otherwise [6].
Following showering and hadronization, jets are clustered in Delphes 3 with the FastJet package [61] using the anti-k t algorithm [62] with a distance parameter of R = 0.4. The btagging efficiencies and mis-tagging rates are taken as functions of p T as follows: where here only j indicates a jet that contains no b or c quarks. These probabilities correspond to roughly 70%, 20%, and 1% respectively. The probability for a light jet to fake a photon in the detector is also taken as a function of p T : which peaks at 5 × 10 −4 for p T,j ∼ 60 GeV before dropping asymptotically to 1.1 × 10 −4 . Aside from the usual jet energy scale correction applied in Delphes 3 at simulation level, an additional correction is made to the momentum of b-tagged jets at analysis level by scaling their four-momentum by a function of the jet p T that starts at 1.08 for low p T and falls off smoothly to 1.0 around p T,j ∼ 100 GeV. This is to help correct for energy losses during reconstruction, and is a rough approximation to the ptcorr correction used by ATLAS [63]. The correction was found to give a m bb peak centered at m h = 125 GeV, and slightly improve the resolution on the bb invariant mass (See Appendix A for more details).

E-Cal Resolution
To simulate the resolution of photons in the Electromagnetic Calorimeter, two parameterizations were used. The first parameterization (hereon referred to as "Regular") was fixed to best fit the performance expected at HL-LHC based on [57]. The second parameterization is taken from the "Medium" benchmark from 100 TeV studies of double Higgs production [12], E ⊕ 1.59 (3.8) The "Regular" parameterization is used to validate our setup with previous ATLAS studies at 14 TeV, as described in Appendix A.

Event Selection
For tabulating the signal and backgrounds, we consider isolated photons and jets with p T > 30 GeV and |η| < 2.5. A reweighting procedure is then implemented to account for the probabilities associated with b-tagging and light jets faking photons listed above. We select events containing at least two photon and b-jet candidates to be consistent with the signal. The weighted events are then subjected to selection criteria chosen to optimize the SM di-Higgs signal strength.
The leading (sub-leading) photon and b-jet are required to have p T > 60 (35) GeV. The diphoton invariant mass is required to satisfy 122 < m γγ < 128 GeV, (122.8 < m γγ < 127.2 GeV) in the "Regular" ("Improved") scenario, while the invariant mass of the b-quark pair is constrained to 100 < m bb < 150 GeV in both cases.
Compared to the ATLAS 14 TeV study [6], we impose somewhat stricter cuts on the p T of each reconstructed Higgs, demanding that p T,h > 125 GeV. Our requirements for the final state particle p T are also somewhat more stringent. The cuts on the invariant radial distance between each jet or photon in a reconstructed Higgs are instead looser, requiring only ∆R bb , ∆R γγ < 3.5. The cuts on p T and ∆R of the reconstructed Higgs are tightly correlated with cuts on the invariant mass of the Higgs pair, m hh , and these values correspond roughly to requiring m hh 350 GeV (this is similar to the cut considered in [16]). We also impose an additional cut on the decay angle of the Higgs boson pair evaluated in the lab frame, requiring | cos θ hh | < 0.8. This was seen in refs. [11,13] to significantly reduce backgrounds from QCD processes as well as tth production As in ref. [6], events with any isolated leptons and more than 5 jets are rejected to reduce backgrounds from top quark decays.
The full list of analysis cuts is summarized below: • At least 2 isolated photons and b-tagged jets with leading p T > 60 GeV and subleading p T > 35 GeV, all with |η γ,b | < 2.5.
• No isolated leptons with p T > 25 GeV.
To optimize the cut on m γγ , the expected significance (S/ √ B) was computed for each set of samples with the width of the m γγ window around m h = 125 GeV varied up to 8 GeV. The results are plotted in Fig. 4.
The selection efficiency for the di-Higgs signal is 2.56% for both the Reg. and Imp. scenarios. Note that, as can be seen in Figs. 2 and 3, the kinematics of the hh → bbγγ signal change for non-SM values of λ 3 , and the optimal selection criteria for the SM signal are not necessarily the same as for other values. For our purposes, we assume that any deviation from λ 3 = 1 is small, and leave an analysis optimized for excluding significant departures from the SM to future work.

Expected Event Yields
The expected number of events from each signal and background channel based on our simulations is shown in Table 4   Events / arb. unit  The expected significance at 14 TeV (computed as S/ √ B) is found to be 1.05 ± 0.01, comparable to previous projections [6]. At 27 TeV, with 15 ab −1 , a significance of 4.51 ± 0.04 (4.70 ± 0.04) is attainable in the Regular (Improved) scenario.
As evident in Table 4, the most significant backgrounds arise from ggF (γγ), tth(γγ), bbγγ, and bbjγ. The ggF and tth backgrounds are difficult to suppress, as they include a true h → γγ decay that is indistinguishable from that of the hh system, as demonstrated in Fig. 5. The bbγγ background similarly can only be reduced by improving the calorimeter resolution and tightening the cut on the diphoton invariant mass. The bbjγ background, on the other hand, is strongly dependent on the jet-faking-photon probability, and could be significantly reduced if true photons can be more reliably distinguished from jets.

The ggF Background
As shown in Table 4, we find the background from single-Higgs production via gluon fusion to be very significant, with an event rate of the same order as the signal. This result differs from some previous estimates [5,6] which found the gluon fusion background to be small compared to the signal, as well as other studies which considered it to be negligible compared to the other single Higgs and irreducible backgrounds [11][12][13][14][15][16]. Ref. [17] does include the gluonfusion background, in a similar setup as the ATLAS studies, and finds it to be ∼ 2 times larger than previous estimates, but still smaller than our result shown above.
To understand the source of this large background, we perform a further analysis of the simulated events in our ggF (γγ) sample described above. While there are contributions  Table 4: The expected number of events from each signal/background process at 14 and 27 TeV based on MC simulation, using the cross sections shown in Table 3. Also shown are the total background and significance (computed as S/ √ B) at each energy.
from mistagged jets, we find that ∼ 75% of the events passing all selection criteria arise from two true b-jets. While the showering of events can lead to additional b-jets in our event samples, it's apparent that the production of a hard bb pair accompanying Higgs production via gluon-fusion is a non-negligible component of the background. To understand this more fully, and as an extra check on the size of the ggF (γγ) background estimate, we compute the parton-level cross section for gg → hbb production in the m t → ∞ limit, with y b set to zero to ensure we are including only the effects of a hard gluon splitting to bb in association with gg → h. This is done with a set of generator level cuts intended to mimic the analysis cuts described above. In particular, we demand: • p T,b1 > 60 GeV, p T,b2 > 35 GeV, • m bb > 100 GeV, • p T,h > 125 GeV 1 .  Figure 5: Stacked histograms showing the contribution of events from the signal (red) and each background category (single Higgs production in blue, tt, ttγ and Zγγ in yellow, and QCD backgrounds in grey) passing every cut except the cut on m γγ (top) and m bb (bottom) in the "Improved" detector scenario. The single Higgs background shows an additional peak at m bb < 120 GeV due to the contribution from the Z(bb)h(γγ) background.
The remaining cuts apply only to the true h → γγ decay, and are expected to have an efficiency close to unity. After folding in the h → γγ branching ratio, the resulting cross section is 4.9 × 10 −2 fb at √ s = 27 TeV, or ∼ 740 events at 15 ab −1 . While the requirement of two b-tags and other detector efficiencies will reduce this number, this represents only the leading process in ggF (γγ) and its contribution is clearly non-negligible, supporting the findings in Table 4.

Limit Setting on the Higgs Self-Coupling
To understand the attainable precision on λ 3 , we assume a hypothetical observation of S + B events after all selection cuts, with S and B as given in Table 4. This allows us to derive 68 and 95% confidence intervals on the expected number of signal events using a likelihood scan, including only the MC and statistical uncertainties. The expected number of signal events with 15 ab −1 integrated luminosity is plotted in Figs. 6, 7, along with the 1σ (2σ) regions in green (yellow). We can also compute the expected number of events at a given integrated luminosity as a function of λ 3 , taking into account both the varying σ hh cross section and the modified acceptance due to changes in the signal kinematics. For L = 15 ab −1 this is shown in red in Figs. Figs. 6, 7. The intersection of this curve with the 1 and 2σ regions indicate the expected precision on λ 3 (ignoring systematic uncertainties). We find Note that, as a result of the destructive interference discussed in Section 1, there is a degeneracy in the expected number of events around λ 3 ∼ 5. However, the kinematic structure of the hh signal is very different at large values of λ 3 , and such values could be easily rejected using differential measurements (e.g, with m hh = m bbγγ or p T,hh ), so the degeneracy can be safely ignored for the purposes of this work.  Table 4 at L = 15 ab −1 for HE-LHC with the regular detector performance assumption. The black dashed line indicates the expected number of events from signal while the green (yellow) regions show the 1σ (2σ) uncertainty regions arising from a likelihood scan with the statistical and MC uncertainties on the signal and background counts. The red curve shows the expected number of events from signal in a background free measurement as a function of λ 3 , accounting for the changes in the signal acceptance due to kinematic differences at different λ 3 .

Conclusions
We have presented a detailed study of the potential for a √ s = 27 TeV hadron collider such as HE-LHC to search for di-Higgs production in the bbγγ final state. In addition to the signal, all important backgrounds are simulated using Monte Carlo, with parton-showering and hadronization. We consider detector scenarios based on the projected performance of the ATLAS detector at HL-LHC, and estimate the effects of the detector performance using the  Figure 7: The same as Fig. 6 but for the "Improved" detector scenario.
Delphes 3 package. Backgrounds arising from the mistagging of a light flavor jet as a b-jet and from jets faking photons in the calorimeter have also been included using a reweighting procedure. Compared to the study of [16], which also considered measuring λ 3 at a 27 TeV collider, we find that the most important background comes from gluon fusion production of a single Higgs in association with additional b-jets. This work complements the results of [16] by considering the additional backgrounds arising from the aforementioned ggF (γγ), as well as Zγγ, tt, ttγ, and QCD backgrounds requiring more than one particle to be misidentified. In our cut-based analysis, we find an expected significance of 4.51 ± 0.04 σ (4.70 ± 0.04 σ) in the Regular (Improved) calorimeter resolution scenarios, and at the 68% C.L., this sensitivity corresponds to a precision of 46% (43%) on the Higgs trilinear coupling, neglecting systematic uncertainties. This is consistent with [16] given the difference in backgrounds and analysis. Further improvements could be made to this result if one was interested in improving the significance at λ 3 = 1, however, it is not clear how much further the precision of the coupling can be improved in this channel given the irreducible backgrounds. Recent results from ATLAS and CMS with 36 fb −1 integrated luminosity indicate that while the bbγγ channel is the cleanest, competitive limits can be set in the bbbb, bbτ τ and bbV V channels as well [38,64,65]. Thus, assuming similar gains for these channels at √ s = 27 TeV, it's possible that a combined O(10 − 20%) measurement of λ 3 may be possible at HE-LHC, making such a collider a powerful probe of electroweak symmetry breaking and BSM physics.

A Validation with ATLAS Study
In order to validate our performance assumptions for a detector at HE-LHC, and to keep consistent with the current projections for HL-LHC, we've made an effort where possible to align our definitions with the projections available from ATLAS [6]. This allows us to compare the results of our MC samples directly with the results presented in [6], using the same cuts described in their paper. In this appendix we describe the results of this comparison, highlighting some of the discrepancies and improvements made in our study. The full set of event selection criteria implemented in the ATLAS study are as follows: • ≥ 2 isolated photons with p T > 30 GeV, |η| < 1.37 or 1.52 < |η| < 2.37, • ≥ 2 jets identified as b-jets with leading/subleading p T > 40/30 GeV, |η| < 2.4, • < 6 jets with p T > 30 GeV, |η| < 2.5, • no isolated 1 leptons with p T > 25 GeV, |η| < 2.5 • 0.4 < ∆R bb , ∆R γγ < 2.0, and 0.4 < ∆R γjet , • 122 < m γγ < 128 GeV, 100 < m bb < 150 GeV, • and p T,γγ , p T,bb > 80 GeV.  Table 5: Cut flow for the signal process hh → bbγγ using the ATLAS event selection criteria. Shown are the results taken from [6] and the results obtained with the MC samples generated as described in Section 2.3 with the detector parameterization described in Section 3.1, using the HL-LHC benchmarks. Rows where the selection efficiency is not directly comparable between this work and [6] due to differences in the reweighting procedure are omitted.
The MC samples produced in our study (described in Section 2.3) were subjected to the cuts summarized above. The expected number of events from each signal and background channel, at 3 ab −1 are listed in Table 6, alongside the results of [6], combining the "barrelbarrel" and "other" categorizations therein.

Process
Expected Events (3 ab −1 ) ATLAS [6] This 1.00 ± 0.02 0.91 ± 0.01 Table 6: Comparison of the expected number of events from each signal/background process at 14 TeV with 3 ab −1 integrated luminosity from the ATLAS HL-LHC projections in [6] and this work, after imposing the selection criteria summarized in Appendix A. For the ATLAS expectations, we've combined the events categorized as "barrel-barrel" and "other", which is consistent with our criteria. Also shown are the total background and significance (computed as S/ √ B) from each study.
In Table 5, we compare the selection efficiency for the signal process (hh → bbγγ) after each successive selection criterion from the ATLAS study is applied. There are two small differences, in the isolated lepton veto efficiency, and the efficiency of the m bb cut. After correspondence with the authors of [6], it appears the discrepancy in the isolated lepton veto arises from muons in b-jets being erroneously labelled as isolated in the ATLAS study. Including the events which were discarded based on this isolation cut appears to resolve the discrepancy in the signal efficiency, however, it's unclear to what extent the additional isolated leptons suppressed the expected background numbers, making a precise comparison to our results difficult.
The difference in the m bb cut efficiency can be attributed to differences in corrections made to b-jet four momenta in our study compared with the ATLAS study. The ATLAS work includes corrections such as the muon-in-jet and PtReco corrections, which have been shown to increase the resolution of the h → bb mass peak in previous ATLAS studies [63]. In contrast, we've approximated these corrections with only an additional energy scaling for b-jets, which corrects the m bb peak location, but does not increase the resolution.
There are several differences between our setup and that used in the ATLAS study that should be noted when comparing the results. The ggF (γγ) and tt backgrounds in the ATLAS study were generated using Powheg-Box, interfaced to Pythia 6. In our study, these were instead produced in MadGraph5 aMC@NLOinterfaced to Pythia 8, with up to two extra partons (including b-jets) at the matrix element level. Additionally, the ATLAS study required the tt and ttγ samples to have at least one lepton in the final state at truth level, while we made no such requirement in our samples. The tt and ttγ samples were normalized using the same method in both studies. Finally, while the bbh(γγ) background is included as part of the inclusive ggF (γγ) sample in our analysis, we also simulate it separately using the same methods used for tth and Zh to facilitate comparison with the ATLAS results.
The most significant differences between the two columns in Table 6 are in the tth(γγ), ggF (γγ), and ttγ backgrounds. As discussed above, the ttγ backgrounds in our study were not required to have a lepton, which likely explains the discrepancy. The ggF backgrounds were also simulated differently in the two studies, which may explain in part the difference in the two projections. Finally, the tth background in our sample is slightly higher, which is likely due to the presence of jets containing muons that were errantly vetoed in the ATLAS study. This is in fact a general trend, as nearly all of our signal and background estimates are somewhat higher on average than those in the ATLAS study, but agree with the estimates of [17], which used the same cuts.
With the caveats discussed above, the results shown in Tabs. 5 and 6 indicate that our Monte Carlo setup interfaced with Delphes 3 accurately describes a detector performance similar to the expectations at HL-LHC. We take this as confirmation that our setup can also give a reasonable projection to the 27 TeV HE-LHC, with the HL-LHC performance as a useful benchmark