Measurement of the top quark mass in the dileptonic t t-bar decay channel using the mass observables M[bl], M[T2], and M[blnu] in pp collisions at sqrt(s) = 8 TeV

TL;DR

This paper presents a precise measurement of the top quark mass in dileptonic decay channels using innovative kinematic observables and Gaussian process modeling, based on 8 TeV proton-proton collision data from the CMS experiment.

Contribution

The study introduces a combined analysis of multiple mass observables with Gaussian process regression to improve top quark mass measurement accuracy.

Findings

Measured top quark mass as 172.22 GeV with high precision.

Utilized novel kinematic observables M[bl], M[T2], and M[blnu].

Achieved detailed modeling of observable shapes and sensitivities.

Abstract

A measurement of the top quark mass (M[t]) in the dileptonic t t-bar decay channel is performed using data from proton-proton collisions at a center-of-mass energy of 8 TeV. The data was recorded by the CMS experiment at the LHC and corresponding to an integrated luminosity of 19.7 +/- 0.5 inverse femtobarns. Events are selected with two oppositely charged leptons (l = e, mu) and two jets identified as originating from b quarks. The analysis is based on three kinematic observables whose distributions are sensitive to the value of M[t]. An invariant mass observable, M[bl], and a `stransverse mass' observable, M[T2], are employed in a simultaneous fit to determine the value of M[t] and an overall jet energy scale factor (JSF). A complementary approach is used to construct an invariant mass observable, M[blnu], that is combined with M[T2] to measure M[t]. The shapes of the observables, along with their evolutions in M[t] and JSF, are modeled by a nonparametric Gaussian process regression technique. The sensitivity of the observables to the value of M[t] is investigated using a Fisher information density method. The top quark mass is measured to be 172.22 +/- 0.18 (stat) +0.89/-0.93 (syst) GeV.

Tables

Table 1. Figure 7: Likelihood fit results using 1000 1000 1000 bootstrap pseudo-experiments for the (top) 2D fit, (center left) 1D fit, and (center right) MAOS fit. (Bottom) hybrid fit results given by the linear combination in Eq. ( 9 ) of the 1D and 2D fits. The error bars represent the statistical uncertainty corresponding to the number of pseudo-experiments in each bin.

$M_{\text{PQt}}^{\text{hyb}}=172.22\pm 0.18{\text{stat}}_{-0.93}^{+0.89}\text{syst}\text{GeV}.$

Equations

$$M_{\text{PQb}\ell }=\sqrt{(p_{\text{PQb}}+p_{\ell }{)}^2},$$

$$(M_{\text{PQb}\ell }{)}_{\text{max}}=\frac{\sqrt{(M_{\text{PQt}}^2-M_{\text{PW}}^2)(M_{\text{PW}}^2-M_{\nu }^2)}}{M_{\text{PW}}}.$$

$$M_{\mathrm{T}}=\sqrt{m_{\ell }^2+m_{\nu }^2+2(E_{\mathrm{T}\ell }{E}_{\mathrm{T}\nu }-{\vec{p}}_{\mathrm{T}\ell }\cdot {\vec{p}}_{\mathrm{T}\nu })},$$

$$M_{\mathrm{T}2}=\underset{{\text{ptvec}}^{\mathrm{a}}+{\text{ptvec}}^{\mathrm{b}}=\text{ptvecmiss}}{\min }[\max \{M_{\mathrm{T}}^{\mathrm{a}},M_{\mathrm{T}}^{\mathrm{b}}\}],$$

$$M({\ell }^{+}\nu )=M({\ell }^{-}\overline{\nu })=M_{\text{PW}}=80.4\text{GeV}.$$

$$p_{z\nu }=\frac{1}{E_{\mathrm{T}\ell }^2}\left[p_{z\ell }A\pm \sqrt{p_{z\ell }^2+E_{\mathrm{T}\ell }^2}\sqrt{A^2-(E_{\mathrm{T}\ell }{E}_{\mathrm{T}\nu }{)}^2}\right],$$

$${\mathcal{L}}^x(M_{\text{PQt}},\mathrm{JSF})=\prod _{i}f(x_i|M_{\text{PQt}},\mathrm{JSF}).$$

$$\mathcal{L}(M_{\text{PQt}},\mathrm{JSF})$$

$$=\prod _{i}f(x_{1i}|M_{\text{PQt}},\mathrm{JSF})f(x_{2i}|M_{\text{PQt}},\mathrm{JSF}).$$

$$M_{\text{PQt}}^{\text{hyb}}=w_{\text{hyb}}{M}_{\text{PQt}}^{\text{1D}}+(1-w_{\text{hyb}})M_{\text{PQt}}^{\text{2D}},$$

$$\log \mathcal{L}(m)=\sum _i^N\log f(x_i|m),$$

$$\mathcal{I}(m)=\int {\left(\frac{\partial }{\partial m}\log f(x|m)\right)}^{2}f(x|m)\text{rd}x.$$

$${\sigma }_m^2\ge \frac{1}{N\mathcal{I}(m)},$$

$$s(x|m)\equiv \frac{1}{f(x|m)}{\left[\frac{\partial f(x|m)}{\partial m}\right]}^2.$$

$${\mathbf{u}}_i\equiv (x_i,M_{\text{PQt}}{}_i,{\mathrm{JSF}}_i).$$

$$\mathbf{C}=\left[\begin{array}[]{cc}\sigma_{\text{train}}^{2}&\rho\sigma_{\text{train}}\sigma_{\text{test}}\\ \rho\sigma_{\text{train}}\sigma_{\text{test}}&\sigma_{\text{test}}^{2}\end{array}\right],$$

$${\mu }_{N+1}$$

$${\sigma }_{N+1}^2$$

$$\begin{gathered} \text{cov}(f({\mathbf{u}}_i),f({\mathbf{u}}_j))=N_1[N_2\exp \left\{-\frac{1}{2{\theta }_1^2}(x_i-x_j{)}^2\right\} \\ \exp \{-\frac{1}{2(1-{\rho }^2)}(\frac{1}{{\theta }_2^2}({M_{\text{PQt}}}_i-{M_{\text{PQt}}}_j{)}^2+\frac{1}{{\theta }_3^2}({\mathrm{JSF}}_i-{\mathrm{JSF}}_j{)}^2 \\ -\frac{2\rho }{{\theta }_2{\theta }_3}({M_{\text{PQt}}}_i-{M_{\text{PQt}}}_j)({\mathrm{JSF}}_i-{\mathrm{JSF}}_j))\}+{\sigma }_i^2{\delta }_{ij}]. \end{gathered}$$

Full text

0.1 Introduction

The top quark mass is a fundamental parameter of the standard model (SM), and an important component in global electroweak fits evaluating the self-consistency of the SM [1]. In addition, the value of $M_{\PQt}$ has implications for the stability of the SM electroweak vacuum due to the role of the top quark in the quartic term of the Higgs potential [2]. Measurements of $M_{\PQt}$ have been conducted by the CDF and \DZEROexperiments at the Tevatron, and by the ATLAS and CMS experiments at the CERN LHC. These measurements are typically calibrated against the top quark mass parameter in Monte Carlo (MC) simulation. Studies suggest that this parameter can be related to the top quark mass in a theoretically well-defined scheme with a precision of about 1\GeV[3]. A combination of measurements including all four experiments and \ttbar decay channels with zero, one, or two high-\ptelectrons or muons (all-hadronic, semileptonic, and dileptonic, respectively) gives a value of $173.34\pm 0.36\stat\pm 0.67\syst\GeV$ [4] for the top quark mass. Currently, the most precise experimental determination of $M_{\PQt}$ is provided by CMS using a combination of measurements in all \ttbar decay channels, yielding a value of $172.44\pm 0.13\stat\pm 0.47\syst\GeV$ [5]. In the dileptonic \ttbar decay channel, the ATLAS [6] and CMS [5] Collaborations have recently determined $M_{\PQt}$ to be $172.99\pm 0.41\stat\pm 0.74\syst\GeV$ and $172.82\pm 0.19\stat\pm 1.22\syst\GeV$, respectively. This paper presents a reanalysis of the dileptonic \ttbardata set recorded in 2012, with a primary motivation of reducing the systematic uncertainties in $M_{\PQt}$ determination.

The dileptonic top quark pair (\ttbar) decay topology, $\ttbar\to(\PQb\ell^{+}\nu)(\PAQb\ell^{-}\overline{\nu})$, with $\ell=(\Pe,\mu)$, presents a challenge in mass measurement arising primarily from the presence of two neutrinos in the final state. While the undetected \ptvec of a single final-state neutrino in a semileptonic \ttbar decay can be inferred from the momentum imbalance in the event, the allocation of momentum imbalance between the two neutrinos in a dileptonic \ttbar decay is unknown a priori. For this reason, the dileptonic \ttbar system is kinematically underconstrained, and mass determination cannot be easily conducted on an event-by-event basis. Instead, the mass of the parent top quarks in the dileptonic \ttbarsystem can be extracted from kinematic features over an ensemble of events, with the help of appropriate observables and reconstruction techniques.

The measurement reported in this paper is based on a set of observables that have been proposed specifically for mass reconstruction in underconstrained decay topologies. These observables include the invariant mass, $M_{\PQb\ell}$, of a $\PQb\ell$ system, a ‘stransverse mass’ variable, $M_{\mathrm{T}2}^{\PQb\PQb}$, constructed with the $\PQb$ and $\PAQb$ daughters of the \ttbar system [7, 8, 9], and the invariant mass of a $\PQb\ell\nu$ system, $M_{\PQb\ell\nu}$, where the neutrino momentum is estimated by the $M_{\mathrm{T}2}$-assisted on-shell (MAOS) reconstruction technique [10]. The MAOS reconstruction technique builds on $M_{\mathrm{T}2}$ by exploiting the neutrino momenta estimates that are by-products of the $M_{\mathrm{T}2}$ algorithm. The sensitivity of the $M_{\PQb\ell}$, $M_{\mathrm{T}2}^{\PQb\PQb}$, and $M_{\PQb\ell\nu}$ observables to the value of $M_{\PQt}$ is investigated using a Fisher information density method. Distributions of $M_{\PQb\ell}$ and $M_{\mathrm{T}2}^{\PQb\PQb}$ in dileptonic events contain a sharp edge descending to a kinematic endpoint, the location of which is sensitive to the value of $M_{\PQt}$. Recently, masses of the top quark, W boson ($M_{\PW}$), and neutrino ($M_{\nu}$) were extracted in a simultaneous fit using the endpoints of these distributions in dileptonic \ttbar events [11]. The $M_{\PQb\ell}$, $M_{\mathrm{T}2}^{\PQb\PQb}$, and MAOS $M_{\PQb\ell\nu}$ observables are described in more detail in Section 0.4.

One of the dominant sources of systematic uncertainty limiting the precision of this measurement comes from the overall uncertainty in jet energy scale (JES). To address the JES uncertainty, we introduce a technique that uses the $M_{\PQb\ell}$ and $M_{\mathrm{T}2}^{\PQb\PQb}$ observables to determine an overall jet energy scale factor (JSF) simultaneously with the top quark mass, where the JSF is defined as a multiplicative factor scaling the four-vectors of all jets in the event. Similar techniques have been developed for the all-hadronic and semileptonic \ttbar channels, where the jet pair originating from a \PW boson decay is used to determine the JSF [5]. Because light-quark jets from the \PW boson decay are used to calibrate the energy scale of \PQbjets arising from the \cPqt and $\overline{\cPqt}$ decays, these methods are sensitive to flavor-dependent uncertainties that emerge from differences in the response of \PQbjets and light-quark jets. In the method featured here, the JSF is determined in the dileptonic \ttbar channel without relying on a \PW boson decaying to jets. Instead, it achieves sensitivity to the JSF through the kinematic differences between \PQbjets, which are subject to JSF scaling, and leptons, which are not. Because it does not use light quarks from a hadronic W boson decay, this approach is insensitive to flavor-dependent JES uncertainties.

To model the $M_{\PQb\ell}$, $M_{\mathrm{T}2}^{\PQb\PQb}$, and MAOS $M_{\PQb\ell\nu}$ distribution shapes, we use a Gaussian process (GP) regression technique [12, 13]. This technique is nonparametric, and thus largely model-independent. It is effective in modeling distribution shapes when no theoretical guidance is available to specify a functional form. The distribution shapes can conveniently be modeled as functions of multiple variables. In this analysis, three variables are used: the value of the relevant observable ($M_{\PQb\ell}$, $M_{\mathrm{T}2}^{\PQb\PQb}$, or $M_{\PQb\ell\nu}$), $M_{\PQt}$, and the JSF. The shapes are determined using simulated events generated with seven different values of $M_{\PQt}$ ranging from $166.5$ to $178.5\GeV$, and with five values of JSF, ranging from $0.97$ to $1.03$, applied to the jets in each event. Each shape ultimately models the distributions of the observables together with their evolution in $M_{\PQt}$ and in JSF.

0.2 The CMS detector

The central feature of the CMS apparatus is a superconducting solenoid of 6\unitm internal diameter, providing a magnetic field of 3.8\unitT. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. The tracker has a track-finding efficiency of more than 99% for muons with transverse momentum $\pt>1\GeV$ and pseudorapidity $\abs{\eta}<2.4$. The ECAL is a fine-grained hermetic calorimeter with quasi-projective geometry, and is distributed in the barrel region of $\abs{\eta}<1.48$ and in two endcaps that extend up to $\abs{\eta}<3.0$. The HCAL barrel and endcaps similarly cover the region $\abs{\eta}<3.0$. In addition to the barrel and endcap detectors, CMS has extensive forward calorimetry. Muons are measured in gas-ionization detectors, which are embedded in the steel flux-return yoke outside of the solenoid. The silicon tracker and muon systems play a crucial role in the identification of jets originating from the hadronization of \PQbquarks [14]. Events of interest are selected using a two-tiered trigger system [15]. The first level, composed of custom hardware processors, uses information from the calorimeters and muon detectors to select events at a rate of around 100\unitkHz within a time interval of less than 4\mus. The second level, known as the high-level trigger, consists of a farm of processors running a version of the full event reconstruction software optimized for fast processing, and reduces the event rate to less than 1\unitkHz before data storage. A more detailed description of the CMS detector, together with a definition of the coordinate system used, can be found in Ref. [16].

0.3 Data sets and event selection

We select dileptonic \ttbar events from a data set recorded at $\sqrt{s}=8$\TeVduring 2012 corresponding to an integrated luminosity of $19.7\pm 0.5\fbinv$ [17]. Events are required to pass one of several triggers that require at least two leptons, $\Pe\Pe$, $\Pe\mu$, or $\mu\mu$, where the leading (higher-\pt) lepton satisfies $\pt>17\GeV$ and the subleading lepton satisfies $\pt>8\GeV$.

A particle-flow (PF) algorithm [18, 19] is used to reconstruct and identify each individual particle in an event by combining information from various subdetectors of CMS. Each event is required to have at least one reconstructed collision vertex, with the primary vertex selected as the one containing the largest $\sum\pt^{2}$ of associated tracks. Electron candidates are reconstructed by matching a cluster of energy deposits in the ECAL to a reconstructed track [20]. They are required to satisfy $\pt>20\GeV$ and $\abs{\eta}<2.5$. Muon candidates are reconstructed in a global fit that combines information from the silicon tracker and muon system [21], and must have $\pt>20\GeV$ and $\abs{\eta}<2.4$. A requirement on the relative isolation is imposed inside a cone $\Delta R=\sqrt{\smash[b]{(\Delta\eta)^{2}+(\Delta\phi)^{2}}}$ around each lepton candidate, where $\phi$ is the azimuthal angle in radians. A parameter $I_{\text{rel}}=\sum{{\pt}_{i}}/\pt^{\ell}$ is defined, where the sum includes all reconstructed PF candidates inside the cone (excluding the lepton itself), and $\pt^{\ell}$ is the lepton \pt. Electron (muon) candidates are required to have $I_{\text{rel}}<0.15$ $(0.2)$ with $\Delta R<0.3$ $(0.4)$. Events selected offline are required to contain exactly two such leptons, $\Pe\Pe$, $\Pe\mu$, or $\mu\mu$, with opposite charge. For events containing an \EE or \MM pair, contributions from low-mass resonances are suppressed by requiring an invariant mass of the lepton pair $M_{\ell\ell}>20\GeV$, while contributions from \PZ boson decays are suppressed by requiring that $\abs{M_{\PZ}-M_{\ell\ell}}>15\GeV$, where $M_{\PZ}=91.2\GeV$ [22].

Hadronic jets are clustered from PF candidates with the infrared and collinear safe anti-\ktalgorithm [23], with a distance parameter $R$ of 0.5, as implemented in the \FASTJETpackage [24]. The jet momentum is determined as the vectorial sum of all particle momenta in this jet. Corrections to the JES and jet energy resolution (JER) are derived using MC simulation, and are confirmed with measurements of the energy balance in quantum chromodynamics (QCD) dijet, QCD multijet, photon+jet, and Z+jet events [25]. Muons, electrons, and charged hadrons originating from multiple collisions within the same or nearby bunch crossings (pileup), are not included in the jet reconstruction. Contributions from neutral hadrons originating from pileup are estimated and subtracted from the JES. Jets originating from the hadronization of \PQbquarks are identified with a combined secondary vertex (CSV) \PQbtagging algorithm [14], combining information from the jet secondary vertex with the impact parameter significances of its constituent tracks. The algorithm yields a tagging efficiency of approximately $85\%$ and a misidentification rate of $10\%$. Events are required to contain at least two jets that pass the \PQbtagging algorithm and satisfy $\pt>30\GeV$ and $\abs{\eta}<2.5$. In this analysis, the two jets satisfying these requirements that have the highest CSV discriminator values are referred to as \PQbjets.

The missing transverse momentum vector is defined as $\ptvecmiss=-\sum{\ptvec{}_{i}}$, where the sum includes all reconstructed PF candidates in an event [26]. Its magnitude is referred to as \ptmiss. Corrections to the JES and JER are propagated into \ptmiss, as well as an offset correction that accounts for pileup interactions. An additional correction mitigates a mild azimuthal dependence, arising from imperfect detector alignment and other effects, which is observed in the reconstructed \ptmiss. To further suppress contributions from Drell–Yan processes, events containing an \EE or \MM pair are required to have $\ptmiss>40\GeV$.

Simulated \ttbar signal events are generated with the \MADGRAPH5.1.5.11 matrix-element generator [27], combined with MadSpin to include spin correlations of the top quark decay products [28], \PYTHIA6.426 with the $\mathrm{Z}2^{*}$ tune for parton showering [29], and \TAUOLAfor the decay of $\tau$ leptons [30]. Parton distribution functions (PDFs) are described by the CTEQ6L1 set [31]. The \ttbar signal events are generated with seven different values of $M_{\PQt}$ ranging from $166.5$ to $178.5\GeV$. The contribution from the \PW associated single top quark production (t\PW) is simulated with \POWHEG1.380 [32, 33, 34, 35], where the value of $M_{\PQt}$ is assumed to be $172.5\GeV$. Background events from \PW+jets and Z+jets production are generated with \MADGRAPH 5.1.3.30, and contributions from \PW\PW, \PW\PZ, $\Z\Z$ processes are simulated with \PYTHIA. The CMS detector response to the simulated events is modelled with \GEANTfour[36]. All background processes are normalized to their predicted cross sections [37, 38, 39, 40, 41].

With the requirements outlined previously, $41\,640$ \ttbar candidate events are selected in data. The sample composition is estimated in simulation to be 95% dileptonic \ttbar, 4% single top quark, and 1% other processes including diboson, \PW+jets, and Drell–Yan production, as well as semileptonic and all-hadronic \ttbar.

0.4 Observables

The observables featured in this study have been developed for physics scenarios where undetected particles, such as neutrinos, carry away a portion of the kinematic information necessary for full event reconstruction. In the dileptonic \ttbar system, distributions in these observables contain endpoints, edges, and peak regions that are sensitive to the top quark mass. The observables are described in more detail below.

0.4.1 The $M_{\PQb\ell}$ observable

The $M_{\PQb\ell}$ observable is defined as

[TABLE]

where $p_{\PQb}$ and $p_{\ell}$ are four-vectors corresponding to a \PQbjet and lepton, respectively. The $\PQb\ell$ pairs underlying each value of $M_{\PQb\ell}$ are chosen out of four possible combinations by an algorithm described below. The $M_{\PQb\ell}$ observable contains a kinematic endpoint that occurs when the \PQbjet and lepton are directly back-to-back in the top quark rest frame. The location of this endpoint, $(M_{\PQb\ell})_{\text{max}}$, is a function of the masses involved in the decay:

[TABLE]

With $M_{\PQt}=172.5$\GeV, $M_{\PW}=80.4$\GeV[22], and $M_{\nu}=0$, we have $(M_{\PQb\ell})_{\text{max}}=152.6$\GeV. Although this endpoint is a theoretical maximum on the value of $M_{\PQb\ell}$ at leading order, events are still observed beyond this value due to background contamination, resolution effects, and nonzero particle widths.

The $M_{\PQb\ell}$ distribution is shown in data and MC simulation in Fig. 1 (left), with a breakdown of signal and background events shown in the simulation. The ‘signal’ category includes \ttbar dilepton decays where both \PQbjets are correctly identified by the \PQbtagging algorithm. The background categories include: ‘mistag’ dilepton decays where a light quark or gluon jet is incorrectly selected by the \PQbtagging algorithm; ‘$\tau$ decays’ where dilepton events include at least one $\tau$ lepton in the final state subsequently decaying leptonically; and ‘hadronic decays’ that include events where at least one of the top quarks decays hadronically. The ‘non-\ttbar bkg’ category consists of single top quark, diboson, \PW+jets, and Drell–Yan processes. Events in which a top quark decays through a $\tau$ lepton contain extra neutrinos stemming from the leptonic $\tau$ decay. Although the extra neutrinos cause a small distortion to the kinematic distributions, these events still contribute to the sensitivity of the measurement.

The sensitivity of the $M_{\PQb\ell}$ observable to the value of $M_{\PQt}$ is demonstrated in Fig. 1 (right), where $M_{\PQb\ell}$ shapes corresponding to three values of the top quark mass in MC simulation ($M_{\PQt}^{\text{MC}}$) are shown. The variation between these shapes reveals regions of the $M_{\PQb\ell}$ distribution that are sensitive to the value of $M_{\PQt}$, such as the edges to the left and right of the $M_{\PQb\ell}$ peak, and regions that are not sensitive, such as the stationary point where the three shapes intersect. To provide a quantitative description of these effects, we introduce a ‘local shape sensitivity’ function, also known as the Fisher information density, shown in Figs. 1, 3, and 4. This function conveys the sensitivity of an observable at a specific point on its shape. For the $M_{\PQb\ell}$ observable, the local shape sensitivity function peaks near the kinematic endpoint ($M_{\PQb\ell}\sim 150\GeV$), and has a zero value at the stationary point ($M_{\PQb\ell}\sim 105\GeV$). The integral of this function over its range is proportional to $1/\sigma_{M_{\PQt}}^{2}$, where $\sigma_{M_{\PQt}}$ is the statistical uncertainty on a measurement of $M_{\PQt}$. A full description of the local shape sensitivity function is given in Appendix .11.

\PQb jet and lepton combinatorics

The two \PQbjets and two leptons stemming from each \ttbar decay give rise to a two-fold matching ambiguity, with two correct and two incorrect $\PQb\ell$ pairings possible in each event. Pairings in which the \PQbjet and lepton emerge from different top quarks do not necessarily obey the upper bound described in Eq. (2), and thus do not have a clean kinematic endpoint in $M_{\PQb\ell}$. Although a priori it is experimentally difficult to distinguish between correct and incorrect pairings, one possible approach is to select the smallest two $M_{\PQb\ell}$ values in each event. This way, the kinematic endpoint of the distribution is preserved – even if the smallest two $M_{\PQb\ell}$ values do not correspond to the correct pairings, they are guaranteed to fall below the correct pairings, which do respect the endpoint. In this analysis, we employ a slightly more sophisticated matching technique, introduced in Ref. [11], where either two or three $\PQb\ell$ pairs are selected in each event.

By selecting either two or three $\PQb\ell$ pairs in each event, the technique employed in this analysis has the benefit of increased statistical power, while preserving the kinematic endpoint of $M_{\PQb\ell}$. Although they are not necessarily the correct pairs, the corresponding $M_{\PQb\ell}$ values are guaranteed by construction to be less than or equal to those of the correct pairs. The matching technique is based on the following prescription:

match each \PQbjet with the lepton that produces the lower $M_{\PQb\ell}$ value; 2. 2.

match each lepton with the \PQbjet that produces the lower $M_{\PQb\ell}$ value.

This recipe produces either two or three values of $M_{\PQb\ell}$. In the latter case, two different leptons may be successfully paired with the same \PQbjet, and vice versa. Such a configuration highlights the difference between this recipe and the simpler approach of choosing the smallest two values of $M_{\PQb\ell}$, which do not necessarily incorporate both \PQbjets and both leptons in the event. For example, this could occur if both \PQbjets are matched to a single lepton. In these cases, the next largest $M_{\PQb\ell}$ value is also needed to ensure both \PQbjets and both leptons from the event are used.

0.4.2 The $M_{\mathrm{T}2}$ observable

The $M_{\mathrm{T}2}$ ‘stransverse mass’ observable [7, 8] is based on the transverse mass, $M_{\mathrm{T}}$. The transverse mass of the W boson in a $\PW\to\ell\nu$ decay is given by

[TABLE]

where $E_{\mathrm{T}x}^{2}=m_{x}^{2}+\ptvec^{\,2}$ for $x\in\{\ell,\nu\}$, $m_{x}$ is the particle mass, and $\vec{p}_{\mathrm{T}x}$ is the particle momentum projected onto the plane perpendicular to the beams. This quantity exhibits a kinematic endpoint at the parent mass, $M_{\PW}$, which occurs in configurations when both the lepton and neutrino momenta lie entirely in the transverse plane (up to a common longitudinal boost).

The dileptonic \ttbar system has two layers of decays, with $\cPqt\to\PW\PQb$ in the first step followed by $\PW\to\ell\nu$ in the second. The result is an event topology with two identical branches, $\cPqt\to\PQb\ell^{+}\nu$ and $\cPaqt\to\PAQb\ell^{-}\PAGn$, each with a visible ($\PQb\ell$) and invisible ($\nu$) component. In this case, one value of $M_{\mathrm{T}}$ can be computed for each branch. The invisible particle momentum associated with each branch, however, is not known. While for a semileptonic \ttbar decay, with only one $\PW\to\ell\nu$ decay, the neutrino \ptvec is estimated from the \ptvecmiss in the event, a dileptonic \ttbar decay includes two neutrinos, for which the allocation of \ptvecmiss between them is unknown.

The $M_{\mathrm{T}2}$ observable is an extension of $M_{\mathrm{T}}$ for a system with two identical decay branches, ‘a’ and ‘b’, such as those in the dileptonic \ttbar system. Here, the invisible particle momenta, $\ptvec^{\,\mathrm{a}}$ and $\ptvec^{\,\mathrm{b}}$, must add up to the total \ptvecmiss. The strategy of $M_{\mathrm{T}2}$ is to impose this constraint on the invisible particle momenta, while also performing a minimization in order to preserve the kinematic endpoint of $M_{\mathrm{T}}$. For a general event with a symmetric decay topology, $M_{\mathrm{T}2}$ is defined as

[TABLE]

where $M_{\mathrm{T}}^{\mathrm{a}}$ and $M_{\mathrm{T}}^{\mathrm{b}}$ correspond to the two decay branches. If the invisible particle mass is known, it can be incorporated into the $M_{\mathrm{T}2}$ calculation as well, yielding an endpoint at the parent particle mass. Although the final values of $\ptvec^{\,\mathrm{a}}$ and $\ptvec^{\,\mathrm{b}}$ are typically treated as intermediate quantities in the $M_{\mathrm{T}2}$ algorithm, they are employed as neutrino \ptvec estimates in the MAOS reconstruction technique described in Section 0.4.3.

The $M_{\mathrm{T}2}$ subsystems

In the \ttbar system, there are several ways in which $M_{\mathrm{T}2}$ can be computed, depending on how the decay products are grouped together. The $M_{\mathrm{T}2}$ algorithm classifies them into three categories: upstream, visible, and child particles [42]. The child particles are those at the end of the decay chain that are unobservable or simply treated as unobservable. In the latter case, the child particle momenta are added to the \ptvecmiss vector. The visible particles are those whose \ptvec values are measured and used in the calculations; and the upstream particles are those from further up in the decay chain, including any initial-state radiation (ISR) accompanying the hard collision.

In general, the child, visible, and upstream particles may actually be collections of objects, creating three possible subsystems in the dileptonic \ttbar event topology. These subsystems are illustrated in Fig. 2. For simplicity, we refer to the corresponding $M_{\mathrm{T}2}$ observables as $M_{\mathrm{T}2}^{\PQb\PQb}$, $M_{\mathrm{T}2}^{\ell\ell}$, and $M_{\mathrm{T}2}^{\PQb\ell}$, where:

• •

The $\boldsymbol{M}_{\text{{T2}}}^{\boldsymbol{\ell\ell}}$ observable uses the two leptons as visible particles, treating the neutrinos as invisible child particles, and combining the \PQbjets with all other upstream particles in the event.

• •

The $\boldsymbol{M}_{\text{{T2}}}^{\mathbf{bb}}$ observable uses the \PQbjets as visible particles, and treats the W bosons as child particles, ignoring the fact that their charged daughter leptons are indeed observable. It considers only ISR jets as generators of upstream momentum.

• •

The $\boldsymbol{M}_{\text{{T2}}}^{\mathbf{b}\boldsymbol{\ell}}$ observable combines the \PQbjet and the lepton to form a single visible system, and takes the neutrinos as the invisible particles. A two-fold matching ambiguity results from the matching of \PQbjets to leptons in each event. In order to preserve the kinematic endpoint of the $M_{\mathrm{T}2}^{\PQb\ell}$ distribution, the $\PQb\ell$ pair with the smallest value of $M_{\mathrm{T}2}^{\PQb\ell}$ is used in each event.

These observables are identical, respectively, to $M_{\mathrm{T}2}^{(2,2,1)}$, $M_{\mathrm{T}2}^{(2,1,0)}$, $M_{\mathrm{T}2}^{(2,2,0)}$ of Ref. [42], and $\mu_{\mathrm{b}\mathrm{b}}$, $\mu_{\ell\ell}$, $\mu_{\mathrm{b}\ell}$ of Ref. [11].

The subsystem observable $M_{\mathrm{T}2}^{\PQb\PQb}$ is employed in this study to complement the observable $M_{\PQb\ell}$. The $M_{\mathrm{T}2}^{\PQb\PQb}$ observable contains an endpoint at the value of $M_{\PQt}$, and can be combined with $M_{\PQb\ell}$ to mitigate uncertainties due to the JES. This feature is discussed further in Section 0.5. The distribution of $M_{\mathrm{T}2}^{\PQb\PQb}$ and its sensitivity to the value of $M_{\PQt}$ are shown in Fig. 3. Although $M_{\mathrm{T}2}^{\ell\ell}$ is not directly sensitive to $M_{\PQt}$, the neutrino \ptvec estimates that are a by-product of its computation are used as an input into the MAOS $M_{\PQb\ell\nu}$ reconstruction technique described in Section 0.4.3.

The $M_{\mathrm{T}2}^{\PQb\PQb}$ distribution employed in this analysis includes a kinematic requirement on the upstream momentum, defined as $\ptvec^{\text{ upst}}=\sum_{\rm reco}{\ptvec{}_{i}}-\sum_{\text{b jets}}{\ptvec{}_{i}}-\sum_{\text{leptons}}{\ptvec{}_{i}}$, where the sums are conducted over all reconstructed PF candidates, \PQbjets, and leptons in each event, respectively. The direction of $\ptvec^{\text{ upst}}$ is required to lie outside the opening angle between the two \PQbjet \ptvec vectors in the event. This requirement primarily impacts events at low values of $M_{\mathrm{T}2}^{\PQb\PQb}$, and its effect on the statistical sensitivity of the observable is small.

0.4.3 The MAOS $M_{\PQb\ell\nu}$ observable

The MAOS reconstruction technique employed in this analysis is based on the subsystem observable $M_{\mathrm{T}2}^{\ell\ell}$. In the $M_{\mathrm{T}2}^{\ell\ell}$ algorithm, an $M_{\mathrm{T}}$ variable, defined in Eq. (3), is constructed from the $\ell^{+}\nu$ and $\ell^{-}\overline{\nu}$ pairs corresponding to each of the \ttbar decay branches. Because the values of neutrino \ptvec are unknown, a minimization is conducted in Eq. (4) over possible values consistent with the measured \ptvecmiss in each event.

The MAOS technique employs the neutrino \ptvecvalues that are determined by the $M_{\mathrm{T}2}^{\ell\ell}$ minimization to construct full $\PQb\ell\nu$ invariant mass estimates corresponding to each of the \ttbardecay branches. Given the neutrino \ptvec values, the remaining $z$-components of their momenta are obtained by enforcing the W mass on-shell requirement [22]

[TABLE]

This yields a longitudinal momentum for each neutrino given by

[TABLE]

where $A=\frac{1}{2}(M_{\PW}^{2}+M_{\nu}^{2}+M_{\ell}^{2})+\ptvec{}_{\ell}\cdot\ptvec{}_{\nu}$ [10]. Given these estimates for the neutrino three-momenta together with $M_{\nu}=0$, we have the required four vectors to construct an $M_{\PQb\ell\nu}$ invariant mass corresponding to the decay products of each top quark.

The quadratic equations in Eq. (6) underlying the W mass on-shell requirement provide up to two solutions for each value of $p_{z\nu}$, yielding a two-fold ambiguity for each neutrino momentum. In addition, there is a two-fold ambiguity resulting from the matching of \PQbjets to $\ell\nu$ pairs in the construction of $\PQb\ell\nu$ invariant masses. No matching ambiguity exists between leptons and neutrinos, since the $\ell^{+}\nu$ and $\ell^{-}\overline{\nu}$ pairs have been fixed by the $M_{\mathrm{T}2}^{\ell\ell}$ algorithm. The combined four-fold ambiguity, along with the two top quark decays in each event, gives up to eight possible values of $M_{\PQb\ell\nu}$. In the measurement, all of the available values are used: for each $\ell\nu$ pair, this includes up to two neutrino $p_{z\nu}$ solutions, and two $\PQb$-$\ell\nu$ matches. The distribution of MAOS $M_{\PQb\ell\nu}$ and its sensitivity to the value of $M_{\PQt}$ are shown in Fig. 4.

0.5 Simultaneous determination of $M_{\PQt}$ and JSF

To mitigate the impact of JES uncertainties on the precision of this measurement, we introduce a technique that allows a JSF parameter to be fit simultaneously with $M_{\PQt}$. The JSF is a constant multiplicative factor that calibrates the overall energy scale of reconstructed jets. It is applied in addition to the standard JES calibration, which corrects the jet response as a function of \ptand $\eta$. The dominant component of uncertainty in the JES calibration can be attributed to a global factor in jet response, which is captured in the JSF.

The challenge in determining the JSF simultaneously with $M_{\PQt}$ stems from the large degree of correlation between these parameters. In the top quark decay, $\PQt\to\PQb\ell\nu$, the JSF directly affects the momentum of the \PQbjet, and indirectly, the inferred momentum of the neutrino, by scaling all jets entering the \ptmiss sum. The $M_{\PQt}$ parameter affects the momenta of these two particles in addition to the lepton produced in the top quark decay. In the context of observables and distribution shapes, variations in the $M_{\PQt}$ and JSF parameters cause shape changes that are difficult to distinguish. For this reason, a shape-based analysis using a single observable can be implemented to determine either $M_{\PQt}$ or JSF, but not both simultaneously.

To determine the $M_{\PQt}$ and JSF parameters simultaneously, we construct a likelihood function that contains two distributions corresponding to the $M_{\PQb\ell}$ and $M_{\mathrm{T}2}^{\PQb\PQb}$ observables. In this configuration, variations in the parameters produce shifts in each individual distribution. They also create a relative shift between the distributions that provides the additional constraint needed for a simultaneous fit of $M_{\PQt}$ and JSF. The dependence of the $M_{\PQb\ell}$ and $M_{\mathrm{T}2}^{\PQb\PQb}$ distribution shapes on $M_{\PQt}$ is shown in Figs. 1 and 3, and their dependence on the JSF is shown in Fig. 5. The difference in response between the $M_{\PQb\ell}$ and $M_{\mathrm{T}2}^{\PQb\PQb}$ shapes to the JSF parameter is rooted in the reconstructed objects underlying the $M_{\PQb\ell}$ and $M_{\mathrm{T}2}^{\PQb\PQb}$ observables – while each value of $M_{\PQb\ell}$ uses one \PQbjet and one lepton, each value of $M_{\mathrm{T}2}^{\PQb\PQb}$ uses two \PQbjets and no leptons for the visible system. Thus, $M_{\mathrm{T}2}^{\PQb\PQb}$ exhibits a stronger dependence on the JSF. The likelihood fit used in this measurement is described in more detail in Section 0.7.

0.6 Gaussian processes for shape estimation

In this analysis, the $M_{\PQb\ell}$, $M_{\mathrm{T}2}^{\PQb\PQb}$, and $M_{\PQb\ell\nu}$ distribution shapes are modeled with a GP regression technique that has two main advantages over other commonly-used shape estimate methods. First, the GP shape is nonparametric, determined only by a set of training points and hyperparameters that regulate smoothing; and second, it can be easily trained as a function of several variables simultaneously. The latter feature allows one to capture the smooth evolution of the distribution shapes as the $M_{\PQt}$ and JSF parameters are varied. A detailed introduction to GPs can be found in Refs. [12, 13]. Here, we give a brief overview of the GP regression technique, with further discussion provided in Appendix .12.

The likelihood fit described in Section 0.7 uses distribution shapes of the form $f(x|M_{\PQt},\mathrm{JSF})$, where $x$ is the value of an observable ($M_{\PQb\ell}$, $M_{\mathrm{T}2}^{\PQb\PQb}$, or $M_{\PQb\ell\nu}$), and $M_{\PQt}$ and $\mathrm{JSF}$ are free parameters in the fit. The shapes $f$ are shown in Figs. 1, 3, and 4 for each observable, where the free parameters are set to $M_{\PQt}=166.5,172.5,$ or $178.5\GeV$ and $\mathrm{JSF}=1$. In Fig. 5, shapes corresponding to the $M_{\PQb\ell}$ and $M_{\mathrm{T}2}^{\PQb\PQb}$ observables are shown with the free parameters set to $M_{\PQt}=172.5\GeV$ and $\mathrm{JSF}=0.97,1.00,$ or $1.03$. In the figures, these shapes are represented as functions of a single variable (the observable $x$) with $M_{\PQt}$ and $\mathrm{JSF}$ fixed. In GP regression, however, each shape is treated as a function of all three quantities ($x$, $M_{\PQt}$, and $\mathrm{JSF}$), and can be described as a probability density in three dimensions.

Each GP shape is trained using binned distributions of the observable $x$ in MC simulation. For each observable, $35$ binned distributions are used, corresponding to seven values of $M_{\PQt}^{\text{MC}}$ ranging from $166.5$ to $178.5$\GeVand five values of $\mathrm{JSF}$ ranging from $0.97$ to $1.03$. Each distribution has 75 bins in $x$, yielding a total of $2625$ training points at which the value of $f$ is known and used as an input into the GP regression process. Each training point is specified by its values of $x$, $M_{\PQt}$, and $\mathrm{JSF}$. The GP regression technique interpolates between the discrete values of $x$, $M_{\PQt}$, and $\mathrm{JSF}$ covered by these training points to provide a shape that is smooth over its range. The smoothness properties of each shape are determined by a kernel function that is set by the analyzer. The GP shapes in this analysis correspond to the kernel function given in Eq. 18 of Appendix .12.

The binned distributions used to construct each GP shape are normalized to unity. However, the normalization of the GP shape itself may deviate slightly from unity due to minor imperfections in shape modeling. To mitigate this effect, the GP shape normalization is recomputed for each value of $M_{\PQt}$ and JSF at which the shape is evaluated. In a likelihood fit, the normalization is recomputed for every variation of the fit parameters.

0.7 Fit strategy

This measurement employs an unbinned maximum-likelihood fit using the $M_{\PQb\ell}$, $M_{\mathrm{T}2}^{\PQb\PQb}$, and MAOS $M_{\PQb\ell\nu}$ observables described in Section 0.4, along with the GP shape estimate technique described in Section 0.6. The MC samples used to train the GP shapes include the \ttbar signal and background processes described in Section 0.3.

The likelihood constructed from a single observable, $x$, is given by:

[TABLE]

Here, the distribution shape $f$ depends on the value of the free parameters $M_{\PQt}$ and $\mathrm{JSF}$, and expresses the likelihood of drawing some event $i$ where the value of the observable is $x_{i}$. It is normalized to unity over its range for all values of $M_{\PQt}$ and $\mathrm{JSF}$. The parameters $M_{\PQt}$ and $\mathrm{JSF}$ are varied in the fit to maximize the value of the likelihood.

A likelihood containing two observables, $x_{1}$ and $x_{2}$, is constructed as a product of individual likelihoods:

[TABLE]

This analysis employs three different versions of the likelihood fit:

the 1D fit uses the $M_{\PQb\ell}$ and $M_{\mathrm{T}2}^{\PQb\PQb}$ observables to determine $M_{\PQt}$, and JSF is constrained to be unity; 2. 2.

the 2D fit also uses $M_{\PQb\ell}$ and $M_{\mathrm{T}2}^{\PQb\PQb}$  but imposes no constraint on the JSF and determines $M_{\PQt}$ and JSF simultaneously; 3. 3.

the MAOS fit uses the $M_{\mathrm{T}2}^{\PQb\PQb}$ and $M_{\PQb\ell\nu}$ observables to determine $M_{\PQt}$, and JSF is constrained to be unity.

Among these versions, the 1D fit provides the best precision on the value of $M_{\PQt}$. The 2D fit mitigates the JES uncertainties, which are the largest source of systematic error in the 1D approach. The MAOS fit is expected to yield results similar to the 1D fit, and is presented as a viable alternative that substitutes the $M_{\PQb\ell}$ observable for MAOS $M_{\PQb\ell\nu}$. The best overall precision on $M_{\PQt}$ is given by a combination of the 1D and 2D fits, which is discussed below. The fit results are discussed in Section 0.9.

The central value and statistical uncertainty on $M_{\PQt}$ and JSF are determined using the bootstrapping technique [43]. This method is based on pseudo-experiments rather than the shape of the total likelihood defined in Eq. (8) near its maximum, and thus mitigates the effects of correlation between the two observables, $x_{1}$ and $x_{2}$, in the likelihood. The technique also mitigates possible correlations within the $M_{\PQb\ell}$ and $M_{\PQb\ell\nu}$ observables when multiple values of the observable occur in a single event. The bootstrapping technique is primarily relevant for statistical uncertainty determination, which may otherwise be affected by correlations in the likelihood. The technique has a negligible impact on the central values of $M_{\PQt}$ and JSF. The bootstrap pseudo-experiments are constructed by resampling the full data set with replacement, where the size of each pseudo-experiment is fixed to have the number of events in data ($41\,640$ events). Events are selected at random from the full data set, so that a particular event has the same probability of being chosen at any stage during the sampling process. In this procedure, a single event may be selected more than once for any given pseudo-experiment. In data, all events have an equal probability to be selected. In simulation, the probability of selecting a particular event is proportional to its weight, containing the relevant cross sections, as well as corrections for MC modeling and object reconstruction efficiencies.

The performance of the likelihood fitting approach described above is evaluated using events in simulation, where the true values of $M_{\PQt}$ and JSF are known. The fit is conducted using seven different values of $M_{\PQt}^{\text{MC}}$ ranging from $166.5$ to $178.5$\GeVfor each version of the likelihood fit. The results of this performance study are shown in Fig. 6. The likelihood fits are consistent with zero bias, showing that the GP shape modeling technique accurately captures the distribution shapes and their evolution over several values of $M_{\PQt}^{\text{MC}}$. For this reason, no calibration of the fit is necessary for an unbiased determination of the $M_{\PQt}$ and JSF parameters.

Combination of 1D and 2D fits

The 1D and 2D fits discussed above have differing sensitivities to various sources of systematic uncertainty in this measurement. Although the 2D fit successfully mitigates the JES uncertainties, which dominate in the 1D fit, other uncertainties in the 2D method are larger and cause the total precision to worsen (Section 0.8). The best overall precision on the value of $M_{\PQt}$ is provided by a hybrid fit, defined as a linear combination of the 1D and 2D fits. The measured value of $M_{\PQt}$ in the hybrid fit is given by:

[TABLE]

where the parameter $w_{\text{hyb}}$ determines the relative weight between the 1D and 2D fits in the combination. The value of $M_{\PQt}^{\text{hyb}}$ and its statistical uncertainty are extracted using bootstrap pseudo-experiments, as described above. In each pseudo-experiment, the measured value of $M_{\PQt}^{\text{hyb}}$ is given by the linear combination in Eq. (9) of the measured $M_{\PQt}^{\text{1D}}$ and $M_{\PQt}^{\text{2D}}$ values. A value of $w_{\text{hyb}}=0.8$ is found to achieve the best precision on $M_{\PQt}$ when both statistical and systematic uncertainties are taken into account. The performance of the hybrid fit, evaluated using MC samples corresponding to seven values of $M_{\PQt}^{\text{MC}}$, is shown in Fig. 6.

0.8 Systematic uncertainties

The systematic uncertainties evaluated in this measurement are given in Table 0.8. The uncertainties include experimental effects from detector calibration and object reconstruction, and modeling effects mostly arising from the simulation of QCD processes. All uncertainties are determined by conducting the likelihood fit using events from MC simulation with the relevant parameters varied by ${\pm}1\Delta$, where $\Delta$ is the uncertainty on a particular parameter. The difference in the measured top quark mass ($\delta M_{\PQt}$) or JSF ($\delta\mathrm{JSF}$) is taken to be the corresponding systematic uncertainty. For uncertainties that are evaluated by comparing two or more independent MC samples, the values of $\delta M_{\PQt}$ and $\delta\mathrm{JSF}$ may be subject to statistical fluctuations. For this reason, if the value of $\delta M_{\PQt}$ or $\delta\mathrm{JSF}$ is smaller than its statistical uncertainty in a particular systematic variation, the statistical uncertainty is quoted as the systematic uncertainty. Finally, if a systematic uncertainty is one-sided, where both $+\Delta$ and $-\Delta$ variations produce $\delta M_{\PQt}$ or $\delta\mathrm{JSF}$ shifts of the same sign, the larger shift is taken as the symmetric systematic uncertainty.

In the hybrid fit, the systematic uncertainties are evaluated according to the linear combination in Eq. (9). For each systematic variation, this gives $\delta M_{\PQt}^{\text{hyb}}=w_{\text{hyb}}\delta M_{\PQt}^{\text{1D}}+(1-w_{\text{hyb}})\delta M_{\PQt}^{\text{2D}}$. This approach provides the smallest overall uncertainty, with the largest contributions stemming from the JES, \PQbquark fragmentation modeling, and hard scattering scale. The next most precise result is given by the 1D fit, also dominated by the same sources of uncertainty. The JES uncertainties are successfully mitigated in the 2D fit. The 2D fit, however, is more sensitive to the uncertainties in the top quark \ptspectrum, matching scale, and underlying event tune, so the total systematic uncertainty for the 2D fit is larger than that of the 1D fit. The MAOS fit has a larger total systematic uncertainty than the 1D fit due to its sensitivity to the JES, top quark \ptspectrum, and \PQbquark fragmentation modeling uncertainties. Further details on each source of systematic uncertainty are given below.