M. Gouzevitch, C .Vernieri
It has been shown (see [428–431] and references therein) that the resolution of the measured objects in the final state of p − p collisions can be improved by forcing well-defined kinematic hypotheses through an event-by-event least square fitting technique. The resulting chi-square of the fit can be interpreted as the probability of the proposed kinematic hypotheses to be true for the observed event.
In the searches for resonances decaying into H H , the Higgs boson mass, as measured by both ATLAS and CMS experiments [432], could be used as a kinematic constraint in the event recon- struction. The kinematic fit procedure is extremely effective for improving the four body invari- ant mass. Such kinematic constraints are widely used for measurements where a decay proceeds through some known intermediate state. For example, in the case of H H → b ¯bb ¯b the kinematic fit technique aims to fit the measured quantities, i.e. the four b-jet four vectors, to certain hypotheses within their uncertainty, as described in [433]. On an event-by-event basis, it builds aχ2 func- tion using the four-vectors of the final state objects and their resolutions. Theχ2is minimised by correcting the measured quantities within their resolutions, fulfilling the kinematic constraints by using Lagrangian multipliers. In this case, the number of degrees of freedom allowed in the fit is
ten, as there are four jets (three degrees of freedom for each jet) and the two constraints from each di-jet invariant mass. The outcome of the kinematic fit is a set of corrections for each of the mea- sured quantities, which translate into in an improved four-body invariant mass resolution. The information provided by the minimisedχ2is the measure of the probability for the observed event to be compatible with the proposed kinematic. The correction factors are then applied to each jet to improve the four-body invariant mass reconstruction. This procedure usesη, φ and pTinforma- tion for each jet and their related uncertainty. As the jet angles are measured with a better relative resolution than the jet-pT, the corrections mainly affect the jet transverse momentum.
The improvement in resolution for the reconstructed signal resonance ranges from 20 to 40% depending on the mass hypothesis for the CMS H H → b ¯bb ¯b resonant search [402], resulting in an improvement of the sensitivity of 10–20%. Similar improvements are also observed in CMS
H H → b ¯bτ+τ− searches at 13 TeV [434] or H H → b ¯bγγ at 8 TeV [435]. The asymmetry of the
corrections, due to the jet momentum resolution across the pT range considered, results in a lin- ear mass shift as function of the resonant mass. The relative improvement is large for the low- est mass resonant hypotheses, since by construction once the two Higgs boson masses are con- strained to the nominal value of the Higgs boson mass, the resolution of the four-body invariant mass ∼ 2mH+∆(EH 1, pH 1, EH 2, pH 2) is dominated by the precision of the 2mH∼ 250 GeV term [436]. The application of the kinematic fit could potentially be extended to other final states involving
b-jets, to further improve the resolution of the mH H invariant mass on top of the dedicated b-jet
specific corrections, as the two methods exploit orthogonal information. Indeed the sensitivity of the CMS search for H H → b ¯bτ+τ−is enhanced by the use of the kinematic fit, which exploits the four-momenta of both theτ and b-jets and the pmissT vector in the event, and is performed under the hypothesis of two 125 GeV Higgs bosons decaying into a bottom quark pair and aτ lepton pair. The use of the kinematic fit improves the resolution on mH H by about a factor of two compared to the four-body invariant mass of the reconstructed leptons and jets [434]. The decay products of
theτ leptons are assumed to be collinear in the fit, since they are highly boosted as they originate
from an object that is heavy when compared to their own mass. In the decay of the twoτ leptons, at least two neutrinos are involved and there is no precise measurement of their original energies. For this reason, theτ lepton energies are constrained from the balance of the fitted H boson transverse momentum and the reconstructed transverse recoil, pmissT , as detailed in Ref. [437].
A simplified version of the kinematic fit is used by the ATLAS H H → b ¯bγγ and H H → b ¯bb ¯b searches [148,149], where mb ¯b is constrained by a simple multiplicative factor mH= 125/mb ¯b be- fore reconstructing mH H. This improves the mH H resolution, on average, by 30–60% across the resonance mass range of interest as shown in Figure4.6and sculpts the non-resonant background in the low mH Hrange.
The CMS H H → b ¯bγγ [438] search applies two different scaling factors for the mγγand mb ¯b, and approximates the kinematic fit procedure by defining a modified mH Hestimator. The so called “reduced” mH H mass [439] is shown in the following equation:
f
MX= mj jγγ− (mb ¯b− mH) − (mγγ− mH). (4.1) This estimator subtracts the out-of-cone and resolution effects that impact the mb ¯b mass more than the jet pT. While the kinematic fit scales the jet momentum, this method attempts to directly
correct the mb ¯bmass. The mj jγγis also corrected for the reconstructed mγγvalue, even if its res- olution is much better compared to mb ¯b. The use ofMfX instead mj jγγimproves the mH H recon- struction by 25 to 30 GeV in absolute, that have the most visible effect at mass resonant hypotheses, as shown in Fig.4.6. For resonant mass of 300 GeV the resolution reduces from roughly 50 to 20
4.7. H H specific analysis techniques 113
Figure 4.6: Reconstructed mH H with (solid lines) and without (dashed lines) the dijet mass con- straint, for a subset of the mass points used for the resonant H H → b ¯bγγ searches of ATLAS [149] (left) and CMS (right) [438].
GeV. CMS also uses anMfX estimator for the boosted H H → b ¯bb ¯b searches [440,441], reporting an improvement of about 10% for the dijet mass resolution.