Di-Higgs final states augMT2ed -- selecting $hh$ events at the high luminosity LHC

Higgs boson self-interactions can be investigated via di-Higgs ($pp\to hh+X$) production at the LHC. With a small ${\cal{O}}(30)$ fb Standard Model production cross section, and a large $t\bar t$ background, this measurement has been considered challenging, even at a luminosity-upgraded LHC. We demonstrate that by using simple kinematic bounding variables, of the sort already employed in existing LHC searches, the dominant $t\bar t$ background can be largely eliminated. Simulations of the signal and the dominant background demonstrate the prospect for measurement of the di-Higgs production cross section at the 30% level using 3/ab of integrated luminosity at a high-luminosity LHC. This corresponds to a Higgs self-coupling determination with 60% accuracy in the $b\bar b \tau^+\tau^-$ mode, with potential for further improvements from e.g. subjet technologies and from additional di-Higgs decay channels.


INTRODUCTION
After a particle consistent with the Standard Model (SM) Higgs boson has been discovered at the LHC [1, 2], we have the final irrefutable experimental evidence of the realisation of a Higgs mechanism in nature [3][4][5][6]. This discovery alone, however, does not provide us the full details of this symmetry breaking sector. In particular, we do not have any additional information other than the existence of a (local) symmetry-breaking minimum and the Higgs potential's curvature at this point in field space. These are rather generic properties of symmetry breaking potentials which can easily be reconciled with more complex scenarios of electroweak symmetry breaking. These typically exhibit a significantly different form of the Higgs self-interaction from the SM * , and to obtain a better understanding of how electroweak symmetry breaking comes about, we need to find a way to discriminate between these different realisations.
The only direct way to provide a satisfying discrimination between the SM symmetry breaking sector and more complicated realisations is probing higher order terms of the Higgs potential directly. In practice this means studying multi-Higgs final states and inferring the relevant couplings from data. The size of the cross sections at the LHC and future colliders effectively limits such a program to the investigation of the trilinear Higgs coupling λ [8]. In the SM, λ is a function of the Higgs mass * For example in scenarios in which the electroweak symmetry is broken radiatively, we typically encounter Coleman-Weinberg type potentials [7] which exhibit an infinite power series in the Higgs field with model-dependent expansion coefficients. m h and the quartic Higgs interaction η, where we have expanded the potential around the nonzero Higgs vacuum expectation value in unitary gauge in the second line which yields λ SM = m h η/2. The effort of phenomenologically reconstructing the trilinear Higgs coupling is based on di-Higgs production pp → hh + X [9][10][11][12][13][14] and dates back more than a decade [15], but in the light of the recent Higgs discovery it has gained new momentum [16][17][18][19][20][21][22][23][24][25]. Probably the most promising approach to infer the trilinear coupling which has been proposed so far is via the hh → bbτ + τ − channel at the LHC, using boosted techniques [26,27] as reported first in the hadron-level analysis of Ref. [16]. That analysis was conservative in the sense that it did not employ selection criteria based on missing transverse momentum, which have the potential to reduce the most challenging tt backgrounds.
In the present letter we complement the analysis of Ref. [16] along these lines and also address the question of the extent to which a successful analysis of the di-Higgs final state will depend on the overall Higgs boost. We concentrate on the bbτ + τ − mode, for which the tt background process dominates. We use kinematical properties of the decay of Eq. (2) to greatly reduce the tt background.
While we focus on the bbτ + τ − mode in this letter, we note that variants of the technique would be applicable to a broader range of di-Higgs decay modes, particularly others also involving the h → bb and h → W + W − decays, which have the largest branching ratios for a 125 GeV Standard Model Higgs boson.

KINEMATIC BOUNDING VARIABLES
The dominant tt background can be reduced by using the m T2 variable, sometimes called the 'stransverse mass' [28,29]. This mass-bound variable was designed for the case where a pair of equal-mass particles decay, and where one daughter from each parent, B or B , is a visible particle, and the other, C or C is not observed. Since the Cs are invisible their individual four-momenta are not known. However the vector sum p Σ T of the transverse momentum components of C and C can be determined from momentum conservation in the plane perpendicular to the beam.
For any given event m T2 is defined to be the maximal possible mass of the parent particle A consistent with the constraints; that is m T2 provides the greatest lower bound on m A = m A given the experimental observables [30].
In the context of the di-Higgs decay (1) the dominant background process (2) satisfies the assumptions under which m T2 is useful: the dileptonic (di-tau) tt background involves the pair-production of identical-mass parents; and each of which decays to a final state which contains visible particles (the b jets, and visible τ decay products) and invisible particles (the neutrinos both from the W decays and from the leptonic or hadronic τ decays). We can therefore build a kinematical variable from the observed final state particles which is bounded above by the top quark mass for the tt background, but remains unbounded above for the di-Higgs signal process.
The m T2 variable can be explicitly constructed [28] as where m T is the transverse mass constructed from m B , m C , b T and c T , while m T is the transverse mass constructed from m B , m C , b T and c T , and where the minimisation is over all hypothesised transverse momenta c T and c T for the invisible particles which sum to the constraint p Σ T , which is usually the observed missing transverse momentum / p T . The transverse mass m T is itself defined by where the 'transverse energy' e for each particle is defined by Variants † of m T2 address cases where some or all of the A, B, A or B particles are composed of four-vector sums. Such variants are designed for more complicated n-body decays with n > 2 or for the case of sequential decays with on-shell intermediates. While these mass-bounding variables were originally proposed to gain sensitivity to the masses of new particles at hadron colliders, they have also proved effective in searches [33][34][35][36][37].
For the hh → bbτ + τ − case, an appropriate variable is constructed as follows. The b jets resulting from each of the two top quark decays enter (3) as the visible particles B and B . The components C and C in (3) which form the transverse momentum constraint should then be the sum of the decay products of the W bosons. The appropriate vector sum p Σ T for the constraint in (3) contains both visible and invisible components, where the first line sums the missing transverse momentum / p T (from all neutrinos from the leptonic W decays, including subsequent leptonic or hadronic τ decays), and the visible transverse momentum from each of the two reconstructed τ candidates.
The resulting variable is by construction bounded above by m t for the tt background process (in the narrow width approximation, and in the absence of detector resolution effects). By contrast, for the hh signal the m T2 distribution can reach very large values, in principle up to √ s/2.

Detector simulation
We model the effects of detector resolution and efficiency using a custom detector simulation based closely on the ATLAS 'Kraków' parameterisation [38]. The The bbW + W − column considers only the decay of W bosons to τ leptons, and already includes the corresponding branching ratios. The final column shows the signal to background ratio. The numbers in brackets follow from a more conservative τ τ mass reconstruction, described in the text. The last rows correspond to exemplary cuts p T,bb ≥ 175 GeV followed by mT2 ≥ 125 GeV.
parameters employed provide conservative estimates of the ATLAS detector performance for the phase-II highluminosity LHC machine (HL-LHC), which is expected to deliver an integrated luminosity of 3 ab −1 to each of the two general-purpose experiments. In particular we model pile-up (at µ = 80) and ΣE T dependent resolutions for jets and for / p T . Jets are reconstructed with the anti-k t jet clustering algorithm [39,40] with radius parameter 0.6. Tau lepton reconstruction efficiencies and fake rates are included, based on Ref. [38], as are jet resolutions, and b-jet efficiencies and fake rates.

Event generation
To generate the signal and background events we closely follow Ref. [16] (details of the comparison of the signal Monte Carlo that underlies this study and comparisons against earlier results can be found therein). Signal events p(g)p(g) → hh + X (which dominate the inclusive hh cross section) are generated with a combination of the Vbfnlo [41] and FeynArts/FormCalc/LoopTools [42,43] frameworks. We generate events in the Les Houches standard [44] which we pass to Herwig++ [45] for showering and hadronisation of the selected h → bb, τ + τ − final states. We use a flat NLO QCD factor to account for higher order perturbative corrections by effectively normalizing to an inclusive cross section of σ = 33.89 fb [19,46].
The QCD and electroweak bbτ + τ − backgrounds are generated with Sherpa [47] and the tt background of Eq. (2) is generated with MadEvent 5 [48]. The bbW + W − NLO cross sections have been computed in Ref. [49] (we use K 1.5 and specify W → τ ν τ in Herwig++ during showering and hadronisation to increase the efficiency for the cut selection), for the mixed QCD/electroweak and the purely electroweak contributions we use the corrections to Zbb (K 1.4) and ZZ (K 1.6) production using Mcfm [50][51][52].

Event selection
Events are assumed to pass the trigger if there are at least two τ s with visible p T > 40 GeV or at least one τ with visible p T > 60 GeV. Both leptonic and hadronic decays of τ s are included. Selected events are required to have exactly two reconstructed τ s (leptonic or hadronic) and exactly two reconstructed and b-tagged jets.
The reconstruction of the di-tau mass is important in discriminating the h → τ + τ − from the Z → τ + τ − background. The LHC experiments typically employ sophisticated mass-reconstruction methods which include kinematic constraints but also likelihood functions or multivariate techniques trained to mitigate against detector resolution [53,54]. We use a simpler, purely kinematic reconstruction of the di-tau mass, which is not expected to perform as well as the techniques used by the experiments in the presence of detector smearing. To estimate the systematic impact of the τ reconstruction on h → τ τ selection, we perform the same m > τ τ reconstruction with and without simulation of the / p T resolution. The more sophisticated techniques used by the experiments which mitigate against detector resolution can be expected to lie between our two estimates.
In each case we construct a τ + τ − invariant mass bound m > τ τ using the greatest lower bound m Higgs−bound τ τ on m h given the visible momenta, / p T and m τ constraints [55]. When detector smearing leads to events where m Higgs−bound τ τ does not exist, the τ mass constraints are dropped, and the resulting transverse mass m T is used as the greatest lower bound m > τ τ on m h . In each case we require that m > τ τ lie within a 50 GeV window. In the analysis without / p T smearing we choose 100 GeV < m τ τ < 150 GeV, while we select 80 GeV < m τ τ < 130 GeV when smearing is included. Note that in the latter case Z → τ + τ − is a large contamination of the signal region defined by the invariant mass windows. By calibrating the Higgs mass reconstruction from h → τ + τ − as already presently performed in the Z → τ + τ − case [56,57], this contamination could be reduced.
The bb invariant mass is calculated from the four-vector sum of the two b-tagged jets. Events are selected if they satisfy 100 GeV < m bb < 150 GeV.

RESULTS
The numbers of events passing each of the selection criteria are tabulated in Tab. I. We find that the transverse momentum and m T2 observables are necessary for background suppression, and, hence, for a potentially successful measurement of the di-Higgs final state in a hadronically busy environment. The normalized m T2 and p T,bb distributions after the selection shown in Tab. I are plotted in Fig. 1. It can be seen that each of the two variables offers good signal versus background discrimination at the large integrated luminosities anticipated at the high luminosity LHC. We also observe that, m T2 and p T,bb encode orthogonal information and they can be combined towards an optimised search strategy.
We find it is straightforward to obtain signal-tobackground (S/B) ratios of ∼ 1/5 while retaining acceptably large signal cross section. These ratios are reexpressed in Fig. 1 which depicts the luminosity contours that are necessary to claim a 5σ discovery of di-Higgs production on the basis of a simple 'cut and count' experiment that makes the rectangular cut requirements that both p T,bb > p T,bb (cut) and m T2 > m T2 (cut). Both axes stop at rather low values of p T,bb , m T2 since a tighter selection would be dependent on the tail of the tt distribution where S/ √ B does not provide an appropriate indicator of sensitivity. We find that the HL-LHC has good sensitivity to the hh production at high luminosity. For an example selection we obtain a cross section FIG. 2: Luminosity in fb −1 required to reach S/ √ B = 5 for di-Higgs production based on simple rectangular cuts on p T,bb and mT2. Numbers in red show luminosities that would require a combination of the ATLAS and CMS data sets from a 3 ab −1 high luminosity LHC. measurement in the 30% range (including the statistical background uncertainty).
The sensitivity to the Higgs trilinear coupling follows from destructive interference with other SM diagrams (see Ref. [16]), such that Using the full parton-level p(g)p(g) → hh + X calculation [16] we find that the quoted 30% cross section uncertainty translates into 60% level sensitivity to the Higgs trilinear coupling in the part of the p T,bb distribution which is relevant for this analysis, p T,bb 180 GeV.
As an alternative to a 'cut and count' analysis we construct a two dimensional likelihood from (m T2 , p T,bb ) to obtain an estimate of the maximal sensitivity that is encoded in these observables, including their correlation [58]. Figures 1 and 2 show that the best sensitivity will result from energetic events either with large m T2 or large p T,bb or both. Using the likelihood method we find fractional uncertainty in the cross section of [σ/σ(hh) SM ] excl 0.37 [1.00] for 3 ab −1 at 95% confidence level (7) using the CL(s) method [59]. The sensitivity to the pp → hh + X cross section as captured in Eq. (7) can be rephrased into an expected upper 95% CL bound on the Higgs self-interaction in the bbτ + τ − channel via Eq. (6). For a background-only hypothesis with no true hh production we would find a limit on the self-coupling of where it should be noted that a 95% CL of 3 λ SM is more stringent than the case of 1 λ SM due to the destructive interference (6).
A measurement of λ at this level would be sufficient constrain a wide range of scenarios of electroweak symmetry breaking, such as composite-Higgs models and pseudo-dilaton models which can lead to large increases in the Higgs self-coupling. While the limit might be somewhat degraded by additional systematic uncertainties in background determination, it also has the potential to be improved by using a subjet analysis [16], and/or by using the more sophisticated di-tau mass reconstruction techniques already employed by the LHC experiments.

SUMMARY AND CONCLUSIONS
Following the discovery of a Higgs boson, one of the top priorities at the LHC is to address the mechanism of electroweak symmetry breaking at a more fundamental level.
In this work we have shown that the 3 ab −1 high luminosity LHC will have sensitivity to the Higgs self-coupling in the favoured bbτ + τ − channel using two simple kinematic variables m T2 and p T,bb each of which independently suppresses the dominant tt background.
We have used parameterised detector simulations of the ATLAS detector as expected for a high-luminosity environment throughout.
Using a two dimensional log-likelihood approach, the null hypothesis of σ(hh) = 0 would constrain the Higgs trilinear coupling to λ 3.0 [1.0] λ SM at the 95% confidence level. An exemplary cross section measurement with 30% precision translates into a measurement of λ at the 60% level.
However further improvements to the presented analysis are possible: 1. Jet substructure techniques allow one to narrow the invariant bb mass window [16], thus leading to a larger rejection of the bbW + W − and bbτ + τ − backgrounds.
3. We have used a definition of m T2 which does not include any information about the τ lepton momenta other than the sum of their p T in (4). Using calibrated taggers, further kinematic information is available by modifying Eq. (5) through pairing τ and b objects, and exploiting the Jacobian peak of m bτ in the top decay [60]. One can pair the hardest τ with that particular b jet that yields an m bτ value that is closer to 140 GeV (the maximum of the Jacobian peak). The leftover b jet is paired with the softer τ , and the following substitutions used in Eq. (5) m vis → m vis bτ , where the latter line indicates that the visible τ decay products are included in the invariant visible mass definition.
A combination of such techniques can be used by the LHC experiments to gain improved sensitivity to the Higgs self-coupling -and hence to the nature of electroweak symmetry breaking.