Extraction of the Top-Quark Mass

Abstract

The choice of the top-quark mass value in a certain scheme affects the predicted production cross sections for ${\text {t}\bar{\text {t}}}$ pairs as well as the kinematics of their decay products. Section 6.1 is dedicated to the extraction of the top-quark pole mass $m_t^{\text {pole}}$ from $\sigma _{{\text {t}\bar{\text {t}}}}$. A determination of $m_t^{\text {MC}}$ and studies to extract a well-defined value for $m_t^{\text {pole}}$ from the kinematics of decay products are presented in Sect. 6.2.

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The choice of the top-quark mass value in a certain scheme affects the predicted production cross sections for ${\text {t}\bar{\text {t}}}$ pairs as well as the kinematics of their decay products. Section 6.1 is dedicated to the extraction of the top-quark pole mass $m_t^{\text {pole}}$ from $\sigma _{{\text {t}\bar{\text {t}}}}$. A determination of $m_t^{\text {MC}}$ and studies to extract a well-defined value for $m_t^{\text {pole}}$ from the kinematics of decay products are presented in Sect. 6.2.

6.1 Determination of ${m}_{{t}}$ from $\sigma _{{\text {t}\bar{\text {t}}}}$

The inclusive ${\text {t}\bar{\text {t}}}$ production cross sections at $\sqrt{s}=7\text { and }8\,\text {TeV}$, precisely determined in Chap. 5 are employed to extract $m_t^{\text {pole}}$. The extraction is performed by a joint-likelihood approach confronting the measured cross sections with their predicted values for each $\sqrt{s}$ and is described in Sect. 6.1.1. The results for $m_t^{\text {pole}}$ at $\sqrt{s}=7\,\text {TeV}$ and $\sqrt{s}=8\,\text {TeV}$ are combined in Sect. 6.1.2.

6.1.1 Extraction Technique

The predicted ${\text {t}\bar{\text {t}}}$ production cross section, $\sigma _{{\text {t}\bar{\text {t}}}} ^{\text {pred}}$, and the measured cross section, $\sigma _{{\text {t}\bar{\text {t}}}}$, depend on the choice of the top-quark mass value. For the prediction, the dependence is more pronounced and affects the production rate directly, whereas for the measurement, it enters mainly through small acceptance effects.

The mass dependence of the measured value of $\sigma _{{\text {t}\bar{\text {t}}}}$ is evaluated by employing two signal MC samples with $m_t^{\text {MC}} =166.5$ GeV and $m_t^{\text {MC}} =178.5$ GeV in addition to the nominal simulation with $m_t^{\text {MC}} =172.5$ GeV. The ${\text {t}\bar{\text {t}}}$ cross-section measurement, as described in Chap. 5, is repeated for each additional mass hypothesis. For each of them, variations in the distributions employed for the fit due to all detector related uncertainties, uncertainties from hadronization modeling, and PDF are re-evaluated. The simulations required to assess the remaining modeling uncertainties are only generated for the nominal $m_t^{\text {MC}}$. In these cases ($Q^2$ and ME-PS matching scale, ME generator, CR and UE tunes), the relative uncertainty estimated for the nominal mass is propagated to the other mass points. The uncertainty due to the top-$p_{\mathrm {T}}$ modeling is based on a measurement performed for $m_t^{\text {MC}} =172.5$ GeV. Therefore, the reweighting procedure applied to the simulation to match the measured spectrum is not valid for other $m_t^{\text {MC}}$ values. To account for this, the relative variations are extrapolated form the nominal mass point, but the corresponding nuisance parameter is left free in the fit. This is expressed in terms of a floating prior as defined in Sect. 5.2.

Fig. 6.1

Predicted dependence of the generated lepton $p_{\mathrm {T}}$ (left) and $\eta$ (right) on the $m_t^{\text {MC}}$ hypothesis. All distributions are normalized and compared to the prediction for $m_t^{\text {MC}} =172.5$ GeV. Leptons $p_{\mathrm {T}}$ or $\eta$ larger (lower) than the displayed range are included in the last (first) bins

The three cross-section values obtained for $m_t^{\text {MC}}$ = 165.5, 172.5, and 178.5 GeV are fitted with an exponential function to obtain a continuous dependence of $\sigma _{{\text {t}\bar{\text {t}}}}$ on $m_t^{\text {MC}}$ as:

$$\begin{aligned} \sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}(8\,\text {TeV},m_t^{\text {MC}})= & {} \exp {\left( -0.267617\cdot (m_t^{\text {MC}}/\,\text {GeV}-176.729 ) \right) }+242.6 {\,\text {pb}} \qquad \quad \end{aligned}$$

(6.1)

$$\begin{aligned} \sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}(7\,\text {TeV},m_t^{\text {MC}})= & {} \exp {\left( -0.130183\cdot (m_t^{\text {MC}}/\,\text {GeV}-184.100 ) \right) }+169.9 {\,\text {pb}} \text {.} \end{aligned}$$

(6.2)

An exponential dependence¹ is chosen since the acceptance $A_{\text {e}\mu }$ is expected to saturate for large $m_t^{\text {MC}}$ values with respect to the beam energy, at which the ${\text {t}\bar{\text {t}}}$ pair is produced without additional momentum. For lower $m_t^{\text {MC}}$ or higher beam energies, the top quarks acquire more momentum in z direction and are produced with larger rapidities. These momenta are propagated to the decay products of the top quarks. The $p_{\mathrm {T}}$ of the leptons from the ${\text {t}\bar{\text {t}}}$ decay decreases with $m_t^{\text {MC}}$, as shown in Fig. 6.1, while the leptons are produced with larger $\eta$. Both effects lead to a decrease of the acceptance. A corresponding increase of $A_{\text {e}\mu }$ can be observed for larger $m_t^{\text {MC}}$ values. In consequence, the dependence of $\sigma _{{\text {t}\bar{\text {t}}}}$ on the choice of $m_t^{\text {MC}}$ is more pronounced the smaller the ratio ${m_t^{\text {MC}}}{/}{\sqrt{s}}$ becomes.

The measured dependence of $\sigma _{{\text {t}\bar{\text {t}}}}$ on $m_t^{\text {MC}}$ is expressed in terms of a likelihood constructed from $\sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}(m_t^{\text {MC}})$ and its uncertainty. The relative total uncertainties increasing or decreasing the cross section $\sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}$, $\Delta _{\text {meas},\pm }$, are almost constant for all mass hypotheses. In order to express the likelihood constructed from $\sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}(m_t^{\text {MC}})$ in terms of $m_t^{\text {pole}}$, an additional relative uncertainty $\Delta _\text {def}(m_t^{\text {MC}})$ is assigned accounting for the difference between $m_t^{\text {pole}}$ and $m_t^{\text {MC}}$, estimated to be about 1 GeV. It is calculated from the fitted dependence as:

$$\begin{aligned} \Delta _{\text {def},\pm }(m_t^{\text {MC}}) = \frac{|\sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}(m_t^{\text {MC}} \mp 1\,\text {GeV}) - \sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}(m_t^{\text {MC}})|}{\sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}(m_t^{\text {MC}})} \end{aligned}$$

(6.3)

and added in quadrature to $\Delta _{\text {meas},\pm }$. Then the final uncertainties on the measured dependence result

$$\begin{aligned} \tilde{\Delta }^2_{\text {meas},\pm } = \Delta ^2_{\text {meas},\pm } + \Delta ^2_{\text {def},\pm } \text {.} \end{aligned}$$

(6.4)

The asymmetric uncertainties on $\sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}$ are expressed in terms of an asymmetric Gaussian $G_a(x,y,w_+,w_-)$:

$$\begin{aligned} G_a(x,y,w_+,w_-) = \frac{ (x-y)^2 }{2c\cdot x} \text { with } c = \left\{ \begin{array}{ll} w_+\ , &{} x-y > 0 \\ w_-\ , &{} x-y \le 0 \end{array}\right. \end{aligned}$$

(6.5)

and the final likelihood for the measured dependence $L_{\text {meas}}(m_t^{\text {pole}},\sigma _{{\text {t}\bar{\text {t}}}})$ becomes

$$\begin{aligned} L_{\text {meas}}(m_t^{\text {pole}},\sigma _{{\text {t}\bar{\text {t}}}}) = exp\left[ -0.5 \cdot G_a\left( \sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}(m_t^{\text {MC}} =m_t^{\text {pole}}), \sigma _{{\text {t}\bar{\text {t}}}},\tilde{\Delta }_{\text {meas},+}, \tilde{\Delta }_{\text {meas},-} \right) \right] \text {.}\qquad \end{aligned}$$

(6.6)

The predicted dependence of $\sigma _{{\text {t}\bar{\text {t}}}}$ on $m_t^{\text {pole}}$ at NNLO$+$NNLL is determined with top++ employing four PDF sets and setting $\alpha _S (M_Z)=0.118\pm 0.001$: NNPDF3.0 [1], CT14 [2], and MMHT2014 [3], with $M_Z$ being the Z-boson mass. For the ABM12 [4] PDF set, $\alpha _S$ is set to the value given the PDF set. The value of $m_t^{\text {pole}}$ in the calculation is varied in 1 GeV steps between 166.5 and 178.5 GeV. The resulting 13 central values are fitted with a sixth-order polynomial to obtain a continuous dependence on $m_t^{\text {pole}}$.

The relative uncertainties are constant with respect to different mass hypotheses. In addition to variations due to the choice of renormalization and factorization scales, PDF, and $\alpha _S$, an uncertainty of 1.79 % (7$\,\text {TeV}$) and 1.72 % (8 TeV) is assigned to the predicted cross section values to account for the uncertainty on the LHC beam energy [5]. Analogue to Eq. 6.6, a likelihood $\hat{L}_{\text {pred}}(m_t^{\text {pole}},\sigma _{{\text {t}\bar{\text {t}}}})$ is defined, including the beam-energy uncertainty, PDF, and $\alpha _S$ variations summed in quadrature to a relative uncertainty, $\Delta _{p,\pm }$:

$$\begin{aligned} \hat{L}_{\text {pred}}(m_t^{\text {pole}},\sigma _{{\text {t}\bar{\text {t}}}}) = exp\left[ -0.5 \cdot G_a\left( \sigma _{{\text {t}\bar{\text {t}}}} ^\text {pred}(m_t^{\text {pole}}), \sigma _{{\text {t}\bar{\text {t}}}},\Delta _{\text {p},+}, \Delta _{\text {p},-} \right) \right] \end{aligned}$$

(6.7)

Given that no particular probability distribution is known that is adequate to model the confidence interval obtained from variations of renormalization and factorization scales, the corresponding uncertainty on the prediction is approximated using a box prior. Following [6], this prior is convoluted with $\hat{L}_{\text {pred}}$ as

$$\begin{aligned} L_{\text {pred}}(m_t,\sigma _{{\text {t}\bar{\text {t}}}}) = \frac{1}{C(m_t^{\text {pole}})} \left( \text {erf} \left[ \frac{\sigma _{{\text {t}\bar{\text {t}}}}^{(h)}(m_t^{\text {pole}})-\sigma _{{\text {t}\bar{\text {t}}}}}{\sqrt{2} \Delta _{p,+}} \right] - \text {erf} \left[ \frac{\sigma _{{\text {t}\bar{\text {t}}}}^{(l)}(m_t^{\text {pole}})-\sigma _{{\text {t}\bar{\text {t}}}}}{\sqrt{2} \Delta _{p,-}} \right] \right) \text {.} \end{aligned}$$

(6.8)

Here, $\sigma _{{\text {t}\bar{\text {t}}}}^{(h)}(m_t^{\text {pole}})$ and $\sigma _{{\text {t}\bar{\text {t}}}}^{(l)}(m_t^{\text {pole}})$ denote the upper and lower predicted cross section values, respectively, from independent variations of renormalization and factorization scales by a factor of 2. The normalization factor $C(m_t^{\text {pole}})$ is given by the maximum value of $L_{\text {pred}}(m_t^{\text {pole}},\sigma _{{\text {t}\bar{\text {t}}}})$ for a free $\sigma _{{\text {t}\bar{\text {t}}}}$ and a fixed $m_t^{\text {pole}}$. It only differs from 1 if the contributions of PDF, $\alpha _S$, and the beam energy to the total uncertainty dominate significantly over the uncertainties due to variations of the renormalization and factorization scales.

The value of $m_t^{\text {pole}}$ is extracted from the product of the likelihood for the measured and predicted dependence $L_{joint} = L_{\text {pred}} \cdot L_{\text {meas}}$. Its maximum corresponds to the most probable $m_t^{\text {pole}}$. The total uncertainty on $m_t^{\text {pole}}$ is determined from the maximum spread of the $L_{joint} = exp(-0.5)$ contour in $m_t^{\text {pole}}$.

The measured $m_t^{\text {pole}}$ and the likelihoods for the measured and predicted dependence of $\sigma _{{\text {t}\bar{\text {t}}}}$ on $m_t^{\text {pole}}$ are shown in Fig. 6.2 for the NNPDF3.0 PDF set. The dependence of the measured cross section on $m_t^{\text {pole}}$ is mild, but more pronounced for $\sqrt{s}=8\,\text {TeV}$.

Fig. 6.2

Likelihood for the predicted and measured dependence of the ${\text {t}\bar{\text {t}}}$ production cross section, $\sigma _{{\text {t}\bar{\text {t}}}}$, on the top-quark pole mass, $m_t^{\text {pole}}$, for $\sqrt{s}=7\,\text {TeV}$ (bottom) and 8 TeV (top). The prediction is calculated with top++ employing the NNPDF3.0 PDF set. The measured dependencies with their 1$\sigma$-uncertainties are represented by the dashed lines. The extracted pole mass values are indicated by black symbols, their total 1$\sigma$-uncertainty by a black contour, corresponding to $-2 \ln { L_{joint}}= 1$

The procedure is repeated for each PDF set. Uncertainties are evaluated at 68 % CL. For CT14, the total PDF uncertainty is provided at 90 % CL and scaled accordingly. The resulting values for $m_t^{\text {pole}}$ are listed in Table 6.1. A fully consistent extraction using ABM12 is not possible since the resulting $m_t^{\text {pole}}$ of 165.5 GeV is smaller than the probed range, 166.5–178.5 GeV, and is in the steeply falling part of $\sigma _{{\text {t}\bar{\text {t}}}} ^\text {meas}(m_t^{\text {pole}})$ at $\sqrt{s}=8\,\text {TeV}$.

Table 6.1 Top quark pole mass at NNLO$+$NNLL extracted by confronting the measured ${\text {t}\bar{\text {t}}}$ production cross section at $\sqrt{s}$ = 7 and 8 TeV with predictions employing different PDF sets

6.1.2 Combination of $\mathrm {m}_\mathrm {t}$ at $\sqrt{s}$ = 7 and 8 TeV

The results for $m_t^{\text {pole}}$ obtained at $\sqrt{s}=7\text { and }8\,\text {TeV}$ are combined for each PDF set using a weighted mean defined as:

$$\begin{aligned} \langle m_t^{\text {pole}} \rangle = \left( \Delta _{u,7}^{-2} + \Delta _{u,8}^{-2} \right) ^{-1} \cdot \left( \frac{m_t^{\text {pole}} (7\,\text {TeV})}{\Delta _{u,7}^2} + \frac{m_t^{\text {pole}} (8\,\text {TeV})}{\Delta _{u,8}^2} \right) \end{aligned}$$

(6.9)

with $\Delta _{u,7}$ ($\Delta _{u,8}$) being the uncorrelated parts of the total uncertainty on $m_t^{\text {pole}}$ at $\sqrt{s}=7$ (8) TeV, determined as follows.

The uncertainties on the measured $\sigma _{{\text {t}\bar{\text {t}}}}$ comprise the uncertainty on $\sigma _{{\text {t}\bar{\text {t}}},\text {vis}}$ in the visible kinematic range $\Delta ^\text {vis}$, for which the correlation coefficient $\rho _{7,8}$ is obtained from the fit, and the fully correlated extrapolation uncertainties. With $\rho _{7,8}=0.30$, the uncorrelated part of $\Delta ^\text {vis}$ is removed by scaling $\Delta ^\text {vis} \rightarrow \rho _{7,8}\Delta ^\text {vis}$. The extrapolation of $\sigma _{{\text {t}\bar{\text {t}}},\text {vis}}$ to $\sigma _{{\text {t}\bar{\text {t}}}}$ is performed by adding the corresponding uncertainties in quadrature and the extraction of $m_t^{\text {pole}}$ is repeated. The resulting uncertainty on $m_t^{\text {pole}}$, ${\Delta _c}$, only includes fully correlated uncertainties, since the uncertainties on the prediction are also assumed to be fully correlated. The uncorrelated part $\Delta _{u}$ of the total uncertainty, $\Delta _\text {tot}$, can therefore be determined as:

$$\begin{aligned} \Delta _{u}^2 = \Delta ^2_\text {tot} - {\Delta _c}^2 \end{aligned}$$

(6.10)

The weighted mean of $m_t^{\text {pole}}$ $\langle m_t^{\text {pole}} \rangle$ is calculated using Eq. 6.9. The contribution of $\Delta _{u,7}$ and $\Delta _{u,8}$ to its uncertainty is determined as:

$$\begin{aligned} \Delta _{u,\text {comb}}^2 = \left( \Delta _{u,7}^{-2} + \Delta _{u,8}^{-2} \right) ^{-1} \text {.} \end{aligned}$$

(6.11)

The fully correlated contribution is determined by varying $m_t^{\text {pole}} (7\,\text {TeV})$ and $m_t^{\text {pole}} (8\,\text {TeV})$ simultaneously within the fully correlated uncertainty. The resulting combined values for the top quark pole mass are listed in Table 6.2. The combined $m_t^{\text {pole}}$ agree well for different PDF sets. Their precision of 1–1.3 % is similar to a recent determination from normalized differential ${\text {t}\bar{\text {t}}}$ production cross sections predicted at NLO accuracy (1.3 %) [7], and supersedes the precision achieved for the extraction from the inclusive ${\text {t}\bar{\text {t}}}$ production cross section (1.5 %) [5] using calculations at NNLO.

Table 6.2 Combined top quark pole mass at NNLO$+$NNLL extracted by confronting the measured ${\text {t}\bar{\text {t}}}$ production cross section with predictions employing different PDF sets

6.2 Determination of ${m}_{{t}}$ from the Lepton-b-Jet Invariant Mass Distribution

Besides the total ${\text {t}\bar{\text {t}}}$ production cross section, additional information contained in differential cross sections can be used to determine ${m}_{{t}}$ if an observable is chosen that is particularly sensitive to ${m}_{{t}}$ and, in the best case, insensitive to certain systematic uncertainties.

Such an observable is the invariant mass distribution of the lepton and the $\text {b}\,\text {jet}$ ($\text {m}_{\text {lb}}$) in dileptonic ${\text {t}\bar{\text {t}}}$ events [8]. Calculations at NLO for ${\text {t}\bar{\text {t}}}$ production and decay are available [9, 10]. The shape of the $\text {m}_{\text {lb}}$ distribution is affected by the choice of ${m}_{{t}}$. It is in principle under good theoretical control over the entire range that is relevant for measurements of ${m}_{{t}}$, but the way higher-order effects are included in the measurement could be crucial [11, 12].

In the following, an analysis of the $\text {m}_{\text {lb}}$ observable and a determination of ${m}_{{t}}$ from its shape is presented, using ${\text {t}\bar{\text {t}}}$ candidate events in the $e \mu$ channel from the data taken at $\sqrt{s}=8\,\text {TeV}$. The event selection is described in Sect. 6.2.1. The observable $\text {m}_{\text {lb}}$ is defined in Sect. 6.2.2. A direct measurement of $m_t^{\text {MC}}$ is performed by confronting the measured shape to predictions using MC simulation as described in Sect. 6.2.4. In Sect. 6.2.5, a generic approach for an alternative top-quark mass measurement is presented, comparing the measured $\text {m}_{\text {lb}}$ distribution to fixed-order QCD calculations at LO and at NLO.

6.2.1 Event Selection

The extraction of the top-quark mass from the $\text {m}_{\text {lb}}$ shape requires a very clean signal with minimal contribution from background processes, since those contributions decrease the sensitivity of the distribution to the top-quark mass. Therefore, in addition to the trigger criteria and the dilepton selection described in Chap. 4, at least two jets, and one $\text {b}\,\text {jet}$ are required. The loose working point for the b-tagging algorithm is employed, see Sect. 4.3.2. The additional criteria reduce the total predicted contribution from background processes to 7 %, while keeping about 29,300 ${\text {t}\bar{\text {t}}}$ signal events for analysis. As shown in Fig. 6.3, the kinematics of the selected lepton candidates as well as of the leading $\text {b}\,\text {jet}$ are well described by the simulation.

Fig. 6.3

Transverse momentum (left column) and pseudorapidity (right column) of the leading $\text {b}\,\text {jet}$ (first row), the leading lepton (second row), and the second leading lepton (third row). The hatched bands correspond to statistical and systematic uncertainties added in quadrature. The lower panels depict the ratio of observed and predicted yields. Here, the small contribution to the uncertainty from MC statistics is indicated by a gray shaded band

6.2.2 Definition of the $\text {m}_{\text {lb}}$ Observable

The top-quark decay chain considered in this analysis is $t \rightarrow W b$ followed by $W \rightarrow l \nu$. At LO and neglecting lepton and b-quark masses, one finds

$$\begin{aligned} \text {m}_{\text {lb}} ^2 = \frac{m_t^2 - m_W^2}{2} \left( 1 - \cos \theta _{lb} \right) , \end{aligned}$$

(6.12)

where $m_W$ is the mass of the $\text {W}$ boson and $\theta _{lb}$ is the opening angle between the lepton and the b-quark in the $\text {W}$-boson rest frame. This relation already illustrates that the $\text {m}_{\text {lb}}$ distribution has an endpoint at

$$\begin{aligned} \max (\text {m}_{\text {lb}}) \approx \sqrt{m_t^2 - m_W^2}\text {.} \end{aligned}$$

(6.13)

For a top-quark mass of 173 GeV, $\max (\text {m}_{\text {lb}})$ is around 153 GeV. The LO distribution is diluted by higher-order effects, but remains sensitive to ${m}_{{t}}$.

Experimental effects such as the limited detector acceptance and the finite resolution in the reconstruction of the lepton and jet four-momenta further decrease this sensitivity. In addition, the reconstructed jets cannot be associated to a particular top quark without dedicated reconstruction algorithms. For this analysis, a simple algorithmic approach for reconstructing $\text {m}_{\text {lb}}$ is sufficient. The permutation $\text {m}_{\text {lb}}^{\text {min}}$ is chosen that minimizes the value of $\text {m}_{\text {lb}}$ in each event when pairing the leading $\text {b}\,\text {jet}$ with the leading or second-leading lepton ($\text {e}$ or $\mu$) candidate.² The resulting distribution provides a good sensitivity to the choice of $m_t^{\text {MC}}$, especially for $\text {m}_{\text {lb}}^{\text {min}}\approx 150$ GeV, as shown in Fig. 6.4. For $m_t^{\text {MC}}$ =172.5 GeV, expected and observed event yields agree well.

In addition, a predicted quantity $\text {m}_{\text {lb,pred}}^{\text {min}}$ is defined based on generator information. The leading $\text {b}$ quark and both leptons ($\text {e}$, $\mu$ or $\tau$) from the $\text {W}$-boson decay are required to be in the visible phase space, defined as $p_{\mathrm {T}} >20$ GeV (leptons) or $p_{\mathrm {T}} >30$ GeV ($\text {b}$ quark) and $|\eta |<2.4$. Leptons and the leading $\text {b}$ quark are paired according to the same algorithm used for $\text {m}_{\text {lb}}^{\text {min}}$. The fraction of correct pairings of $\text {b}$ quark and lepton to the corresponding top quark is studied using MadGraph+pythia and is found to be 72 %.

Fig. 6.4

Left Reconstructed $\text {m}_{\text {lb}}^{\text {min}}$ for $m_t^{\text {MC}} =172.5$ GeV. The hatched band corresponds to the statistical and systematic uncertainties added in quadrature. The lower panel depicts the ratio of observed and predicted yields. Here, the uncertainty from MC statistics is indicated by a gray shaded band. Right Dependence of the normalized $\text {m}_{\text {lb}}^{\text {min}}$ distribution on the choice of $m_t^{\text {MC}}$ normalized to the total number of selected events. The lower panel shows the relative difference of the shape for each $m_t^{\text {MC}}$ with respect to $m_t^{\text {MC}} =172.5$ GeV

6.2.3 Extraction Technique and Systematic Uncertainties

The value of $m_t^{\text {MC}}$ is determined by comparison of the measured and the expected normalized $\text {m}_{\text {lb}}^{\text {min}}$ distributions, including contributions from the signal and background processes. The normalization factor $n_{\text {pred}}$ ($n_{\text {obs}}$) is derived from the integral of the expected (observed) $\text {m}_{\text {lb}}^{\text {min}}$ distribution. The observed yields, $N_{\text {obs},i}$, are confronted with their expectation $N_{\text {pred},i}$ in bin i of the distribution for different values of $m_t^{\text {MC}}$. For this purpose, an estimator $\chi ^2(m_t^{\text {MC}},i)$ is defined as:

$$\begin{aligned} \chi ^2(m_t^{\text {MC}},i) = \frac{(N_{\text {pred},i}(m_t^{\text {MC}})/n_\text {pred}-N_{\text {obs},i}/n_\text {obs})^2}{(\Delta _{\text {pred},i}/n_\text {pred})^2+(\Delta _{\text {obs},i}/n_\text {obs})^2} \text {,} \end{aligned}$$

(6.14)

with $\Delta _{\text {pred},i}$ and $\Delta _{\text {obs},i}$ being the statistical uncertainties of the expected and observed yields, respectively.

The yields $N_{\text {pred},i}(m_t^{\text {MC}})$ are evaluated for $m_t^{\text {MC}}$ = 166.5, 169.5, 171.5, 172.5, 173.5, 175.5, and 178.5 GeV using dedicated simulations. In order to derive a continuous dependence on $m_t^{\text {MC}}$, the resulting yields are fitted with second-order polynomials, which describe this dependence well. The fitted curves for each bin are shown in Appendix D. A global estimator $\chi ^2(m_t^{\text {MC}})$ is derived by summing $\chi ^2(m_t^{\text {MC}},i)$ over all bins i. The top-quark mass is determined from its minimum $\chi ^2_{\text {min}}$. The statistical uncertainty is obtained by applying the criterion $\chi ^2(m_t^{\text {MC}}) = \chi ^2_{\text {min}} + 1$.

The same sources of systematic uncertainties discussed in Sect. 5.2 are considered in this analysis. The impact of each source on $m_t^{\text {MC}}$ is evaluated by varying the corresponding parameter, and determining the expected event yield as a function of $\text {m}_{\text {lb}}^{\text {min}}$. The corresponding value of $m_t^{\text {MC}}$ is extracted and the difference to the nominal result is taken as systematic uncertainty. For variations of the UE modeling and the ME-PS matching scale, these deviations are smaller than the statistical uncertainty on $m_t^{\text {MC}}$ due to fluctuations in the simulation. Therefore, these statistical uncertainties are taken as systematic uncertainty, instead. The variations of $m_t^{\text {MC}}$ with respect to individual components of the JES: flavor group describing gluon, $\text {c}$-, $\text {b}$-, and light-quark response are added linearly to account for the correlation among them. The remaining contributions to the total uncertainty on $m_t^{\text {MC}}$ are added in quadrature.

In addition, systematic uncertainties related to the extraction procedure are studied and discussed in the following. These are assumptions on the statistical model used to define $\chi ^2(m_t^{\text {MC}})$, the ${m}_{{t}}$-dependence of the contribution from $\text {tW}$ processes, and the parameterization of the predicted dependence as a function of $m_t^{\text {MC}}$.

Fig. 6.5

Pull distribution for the extraction of $m_t^{\text {MC}}$ from the $\text {m}_{\text {lb}}^{\text {min}}$ distribution evaluated for 3 different hypothesis of $m_t^{\text {MC}}$. The difference between hypothesis $m_t^\text {in}$ and extracted value $m_t^\text {out}$ is divided by the statistical uncertainty of $m_t^\text {out}$, $\Delta _{\text {stat}}$

6.2.3.1 Assumptions on the Statistical Model

For the definition of $\chi ^2(m_t^{\text {MC}})$ two assumptions are made: (a) the statistical uncertainties have Gaussian form and (b) all bins are uncorrelated. The assumption (a) might bias $m_t^{\text {MC}}$ through bins with low statistics. The assumption (b) is true for all bins except one, which is correlated with the remaining ones through the normalization requirement. The effect of these assumptions can be quantified with pseudo-experiments. These are performed for three initial mass hypotheses $m_t^{in}=169.5$, 172.5, and 175.5 GeV. Poisson-distributed pseudo-data are generated in each bin using $N_{\text {pred},i}(m_t^{in})$ as the central value. Fluctuations of the predicted yields are simulated for background and signal contributions independently, following the approach described in Sect. 5.7.1. For each of in total 3 $\times$ 10,000 pseudo-experiments, the pull is calculated. The resulting pull distributions are fitted with a Gaussian, as shown in Fig. 6.5. For all $m_t^{in}$, the peak positions are consistent with 0 indicating a bias-free measurement. Also the statistical uncertainty extracted by the criterion $\chi ^2_{\text {min}} + 1$ is well modeled since the pull widths are compatible with 1. Therefore, the simplifications in the definition of $\chi ^2(m_t^{\text {MC}})$ do not affect the extracted top-quark mass value.

Fig. 6.6

Relative variation of the $\text {m}_{\text {lb}}^{\text {min}}$ shape for different top-mass hypothesis with respect to a hypothesis of $m_t=172.5$ GeV. The value of $m_t$ is varied independently for the ${\text {t}\bar{\text {t}}}$ signal and the $\text {tW}$ predictions

6.2.3.2 Contribution from $\text {tW}$ Processes

The expected contribution from $\text {tW}$ processes also depends on the top-quark mass. However, only variations of the ${\text {t}\bar{\text {t}}}$ signal contribution are considered in the $m_t^{\text {MC}}$-dependence of the $\text {m}_{\text {lb}}^{\text {min}}$ shape. The simulation of $\text {tW}$ events is not available for all seven $m_t^{\text {MC}}$ hypotheses and therefore only the simulated MC sample with $m_t^{\text {MC}}$ fixed to 172.5 GeV is employed. The effect of this approximation on the final result is studied by comparing 5 scenarios using simulated ${\text {t}\bar{\text {t}}}$ and $\text {tW}$ events generated with $m_t^{\text {MC}} =166.5$, 172.5, and 178.5 GeV. In each case, the $\text {m}_{\text {lb}}^{\text {min}}$ shape is evaluated for a different choice of $m_t^{\text {MC}}$ in the simulation of ${\text {t}\bar{\text {t}}}$ signal and $\text {tW}$ processes, indicated as ${\text {t}\bar{\text {t}}} (m_t^{\text {MC}})$ or $\text {tW} (m_t^{\text {MC}})$, respectively. The following scenarios are considered:

• 0 ${\text {t}\bar{\text {t}}}$(172.5 GeV) and $\text {tW}$(172.5 GeV)

• 1 ${\text {t}\bar{\text {t}}} (166.5\,\text {GeV}$) and $\text {tW} (166.5\,\text {GeV})$

• 2 ${\text {t}\bar{\text {t}}} (166.5\,\text {GeV}$) and $\text {tW} (172.5\,\text {GeV})$

• 3 ${\text {t}\bar{\text {t}}} (178.5\,\text {GeV})$ and $\text {tW} (172.5\,\text {GeV})$

• 4 ${\text {t}\bar{\text {t}}} (178.5\,\text {GeV})$ and $\text {tW} (178.5\,\text {GeV})$

The resulting normalized $\text {m}_{\text {lb}}^{\text {min}}$ distributions are compared to scenario 0, as presented in Fig. 6.6. Consistent variations of $m_t^{\text {MC}}$ in the $\text {tW}$ and ${\text {t}\bar{\text {t}}}$ simulation (1 and 4) lead to a slightly increased sensitivity of the distribution to $m_t^{\text {MC}}$ and would thus increase the statistical precision of the extracted ${m}_{{t}}$. An upper limit on a possible bias can be estimated by comparing the maximum relative deviations of consistent and inconsistent variations. These are below 10 % in the sensitive region with $\text {m}_{\text {lb}}^{\text {min}}\approx 150$ GeV and hence correspond to a maximum bias of 0.1 GeV per 1 GeV difference to $m_t^{\text {MC}} =172.5$ GeV which is assigned as an additional systematic uncertainty ($\mathbf{tW }({m}_{{t}})$).

6.2.3.3 Parametrization of the $m_t^{\text {MC}}$ Dependence

Second-order polynomials describe the dependence of the predicted yields in each bin of the $\text {m}_{\text {lb}}^{\text {min}}$ distribution well. However, a possible impact of the choice of the functional form is studied. The bin-wise fits are performed for third-order polynomials up to fifth-order polynomials. For each, the extraction procedure is repeated and the maximum deviation from the nominal result for $m_t^{\text {MC}}$ of 70 MeV is taken as parametrization uncertainty.

In addition, the dependence of the extracted $m_t^{\text {MC}}$ on the number of $m_t^{\text {MC}}$ hypothesis used to derive the parameterization is studied. For this purpose, 3 mass points are used instead of 7 to derive the parameterization: a central point with $m_t^{\text {MC}} =172.5$ GeV, and the most significant variations $m_t^{\text {MC}} =166.5$ GeV and $m_t^{\text {MC}} =178.5$ GeV. Only a small shift of $+60\,\text {MeV}$ of the extracted $m_t^{\text {MC}}$ value is observed. In the following, 7 $m_t^{\text {MC}}$ hypotheses are employed for the ${\text {t}\bar{\text {t}}}$ signal simulation.

6.2.4 Determination of ${\mathrm {m_t^{ MC}}}$ from the $\text {m}_{\text {lb}}^{\text {min}}$ Shape

The measured shape of the $\text {m}_{\text {lb}}^{\text {min}}$ distribution is compared to the prediction, comprising contributions from background processes and the ${\text {t}\bar{\text {t}}}$ signal modeled by MadGraph+pythia, as presented in Fig. 6.7. The data show the most compatibility with the expected shape for $m_t^{\text {MC}} =172.5$ GeV, while deviations from this value of $\pm 6$ GeV are disfavored. The minimization of the global $\chi ^2(m_t^{\text {MC}})$ results in a top-quark MC mass value of

$$\begin{aligned} m_t^{\text {MC}} =172.8\,^{+1.3}_{-1.0} \,\text {GeV} \text {,} \end{aligned}$$

(6.15)

consistent with the world average [13]. The total uncertainty is larger than the one in measurements based on semileptonic ${\text {t}\bar{\text {t}}}$ decays, where an in-situ JES calibration is performed [14]. However, this analysis has partially complementary uncertainties and provides a result more precise than other measurements in the dilepton channel [15, 16]. All contributions to the total uncertainty are listed in Table 6.3. The dominant uncertainties arise from the JES and the hadronization modeling. Furthermore, variations of top $p_{\mathrm {T}}$ and $Q^{2}$ scale have a large effect on the total uncertainty. Negligible contributions come from variations affecting the normalization of the prediction, e.g. from the luminosity or trigger uncertainties.

Fig. 6.7

Normalized selected event yields presented as a function of $\text {m}_{\text {lb}}^{\text {min}}$. The closed symbols represent data points and the error bars their statistical uncertainties. The predicted yields are obtained with a top-quark MC mass hypothesis of ${m}_{{t}} =178.5$ GeV (red band), ${m}_{{t}} =172.5$ GeV (green band), and ${m}_{{t}} =166.5$ GeV (blue band). The width of the bands indicate the statistical uncertainties on the prediction. The inset shows the $\chi ^2$ distribution as a function of ${m}_{{t}}$ as determined from the comparison of data and predictions

Table 6.3 Breakdown of systematic uncertainties on the top-quark MC mass value, obtained by confronting the shape of the $\text {m}_{\text {lb}}^{\text {min}}$ distribution to predictions by MadGraph+pythia

6.2.5 Folding: Comparison to Fixed-Order Calculations

In the following, a technique is introduced which allows to use fixed-order calculations (in particular mcfm) to determine the top-quark mass, by comparison with experimentally measured distributions. In general, these calculations provide the possibility to extract the top quark mass in a well-defined scheme. However, the predicted distributions can not be compared directly to the reconstructed quantities. The folding approach presented in this thesis allows to fold a predicted observable to its reconstructed counterpart, e.g. the ${\text {t}\bar{\text {t}}}$ production cross section as a function of $\text {m}_{\text {lb,pred}}^{\text {min}}$ to the event yields as a function of $\text {m}_{\text {lb}}^{\text {min}}$. The folded prediction can then be compared to the data without the need of a full detector simulation.

For this purpose, a response matrix M is defined as

$$\begin{aligned} \vec {N}_{reco} = \mathcal {L} \cdot M \vec {\sigma } \text {.} \end{aligned}$$

(6.16)

The event yields or differential cross sections in each bin of $\text {m}_{\text {lb}}^{\text {min}}$ or $\text {m}_{\text {lb,pred}}^{\text {min}}$ are represented by entries in $\vec {N}_{reco}$ and $\vec {\sigma }$, respectively. The response matrix comprises resolution effects through non-diagonal entries. Bin-wise selection efficiencies and acceptance corrections are included in the normalization of each column. The matrix is determined from the MadGraph+pythia signal simulation. For each simulated event, $\text {m}_{\text {lb,pred}}^{\text {min}}$ is calculated and associated to a certain bin i. If the event passes all reconstruction requirements, $\text {m}_{\text {lb}}^{\text {min}}$ is determined to be within bin j. The response matrix is calculated as

$$\begin{aligned} M_{ij} = \frac{1}{c_i + \tilde{\epsilon }_i} \sum _k{w_{i,j,k}} \text {,} \end{aligned}$$

(6.17)

with $w_{i,j,k}$ being the weight assigned to event k due to correction factors discussed in Chap. 4. The matrix is normalized with $c_i=\sum _{j,k}{w_{i,j,k}}$ and $\tilde{\epsilon }_i$. The latter term implements the reconstruction efficiency and acceptance effects by summing all weights of events generated in bin i, that do not pass the selection requirements. The resulting response matrix for $\text {m}_{\text {lb}}^{\text {min}}$ is shown in Fig. 6.8. Dominant diagonal elements indicate a strong correlation between the generated and reconstructed observables. Thus, the sensitivity of $\text {m}_{\text {lb}}^{\text {min}}$ to the top quark mass is not significantly decreased by the limited detector resolution. The intermediate leptonic $\tau$ decays from $\text {W} \rightarrow \tau + \nu \rightarrow \text {e}/\mu + 2\nu$ are considered signal for the reconstruction of $\text {m}_{\text {lb}}^{\text {min}}$, while $\text {m}_{\text {lb,pred}}^{\text {min}}$ is defined for prompt leptons from the $\text {W} \rightarrow \text {e}/\mu /\tau$ decay. The dominant diagonal elements in M demonstrate that this fact does not lead to a significantly softer reconstructed $\text {m}_{\text {lb}}^{\text {min}}$ distribution. Nevertheless, even a pronounced bias would be corrected for by the folding technique.

Fig. 6.8

Response matrix for $\text {m}_{\text {lb}}^{\text {min}}$, quantifying detector resolution and event reconstruction effects as defined in Eq. 6.17. The matrix relates the predicted event rate as a function of $\text {m}_{\text {lb,pred}}^{\text {min}}$ to the reconstructed event rate as a function of $\text {m}_{\text {lb}}^{\text {min}}$. Bins without entries are left white

The limited number of generated events leads to statistical uncertainties on each element $M_{ij}$. These are estimated using a binomial approximation, since the calculation of $M_{ij}$ employs statistically correlated terms as:

$$\begin{aligned} {(\Delta M_{ij})}_\text {stat} = \sqrt{\frac{M_{ij}(1-M_{ij})}{\sum _k{w_{i,j,k}}}} \text {.} \end{aligned}$$

(6.18)

These statistical uncertainties decrease with increasing simulated event rates (cross sections) as a function of $\text {m}_{\text {lb}}^{\text {min}}$ ($\text {m}_{\text {lb,pred}}^{\text {min}}$) and are propagated to the folded distribution.

Systematic uncertainties related to the detector response and the signal modeling by MadGraph+pythia enter the response matrix. For each uncertainty source and choice of $m_t^{\text {MC}}$, a new response matrix is derived. In cases where systematic variations can not be performed due to missing simulation for $m_t^{\text {MC}}$ values other than 172.5 GeV, relative uncertainties are propagated from the response matrix for $m_t^{\text {MC}} =172.5$ GeV. This applies to variations of the $Q^2$ and matching scales, the ME generator, CR and UE tunes.

The resulting set of response matrices allows to fold a predicted ${\text {t}\bar{\text {t}}}$ production cross section as a function of $\text {m}_{\text {lb,pred}}^{\text {min}}$ and compare the resulting shape directly to the data, once contributions from background processes are added. The requirements on the visible phase space in the definition of $\text {m}_{\text {lb,pred}}^{\text {min}}$ (see Sect. 6.2.2) reduce the impact of acceptance corrections.

Alternatively, an unfolding of the measured event yields to measured differential cross-sections can be performed. Within the unfolding, the response matrix M is inverted. This inversion can be ill-posed. Even though, ill-posed problems can be solved with regularization techniques [17], the statistical fluctuations in $M^{-1}$ typically demand a coarser binning of the unfolded distributions [18] in comparison to the one used for $\text {m}_{\text {lb}}$ here, which leads to a decrease in sensitivity to ${m}_{{t}}$. Furthermore, unfolding introduces statistical correlations between the bins of the unfolded distribution. These correlations must be taken into account in the estimator used to perform the extraction, since statistical uncertainties on the measurement are typically not negligible. Thus, the folding technique represents a robust, precise, and statistically well-defined method to extract ${m}_{{t}}$.

However, the definition of $\text {m}_{\text {lb,pred}}^{\text {min}}$ used in MadGraph+pythia to derive the response matrices and in the prediction, in this case mcfm, must coincide. In addition, the $m_t^{\text {MC}}$-dependence of the response matrices might lead to a bias when extracting ${m}_{{t}}$ in a well-defined scheme. These issues would also apply to the case of an unfolding approach and are discussed in the following.

Fig. 6.9

Feynman diagrams for the top quark decay at LO (a), an example of a virtual (b) and a real (c) correction to the decay at NLO. Figure d Shows two real emissions. The corrections (a)–(c) are implemented in mcfm. MadGraph includes (a), while (c) and (d) are modeled by the parton shower in pythia

6.2.5.1 Calculation of $\text {m}_{\text {lb,pred}}^{\text {min}}$ in mcfm

The response matrices, as described above, are derived using MadGraph+pythia. The decay of the ${\text {t}\bar{\text {t}}}$ pairs is simulated with MadSpin, which implements LO predictions. In a second step, real emissions are modeled with pythia. Therefore, the $\text {b}$ quark and leptons can be considered either before or after this step (before or after radiation). In mcfm, the production and decay of ${\text {t}\bar{\text {t}}}$ pairs can be predicted with LO or NLO accuracy. The NWA allows to separate both amplitudes, such that calculations at NLO (LO) for the production can be combined with calculations at LO (NLO) for the decay. In particular, the $\text {b}$ quark momentum is affected by different choices for the decay.

Diagrams for the top-quark decay that are implemented in MadGraph+pythia and mcfm are schematically compared in Fig. 6.9. The most consistent definition of $\text {m}_{\text {lb,pred}}^{\text {min}}$ is achieved by considering the leptons and quarks given by calculations at LO for the decay in mcfm and their counterparts in MadGraph+pythia before radiation (Fig. 6.9a). The corresponding mcfm routines are modified accordingly and adapted to apply the visible phase space requirements given in Sect. 6.2.2. For each top-quark mass hypothesis considered in MadGraph+pythia, the ${\text {t}\bar{\text {t}}}$ production cross section as a function of $\text {m}_{\text {lb,pred}}^{\text {min}}$ is calculated. Concerning the production, LO and NLO calculations are employed to extract the top-quark mass ${m}_{{t}} ^\text {LO}$ and ${m}_{{t}} ^\text {NLO}$, respectively. The predictions are obtained using the MSTW2008 [19] PDF set at LO (NLO), with $\alpha _S (M_Z)=0.1394$ (0.1202), and setting the $\text {b}$-quark mass to 4.75 GeV. Renormalization and factorization scales are set to $m_t^{\text {pole}}$. The full configuration of mcfm is listed in Appendix D.

6.2.5.2 Dependence of the Response Matrix on $m_t^{\text {MC}}$

For each $m_t^{\text {pole}}$ hypothesis used in mcfm, the corresponding response matrix derived for $m_t^{\text {MC}}$, $M(m_t^{\text {MC}} =m_t^{\text {pole}})$, is used for the folding. However, $m_t^{\text {pole}}$ and $m_t^{\text {MC}}$ are not equal. Therefore, this procedure can introduce a bias if the response matrix depends strongly on $m_t^{\text {MC}}$. This possible bias is studied with MadGraph+pythia by comparing the initial and extracted values of $m_t^{\text {MC}}$. An artificial mismatch is introduced between $m_t^{\text {MC}}$ used for the predicted $\text {m}_{\text {lb,pred}}^{\text {min}}$ shape and the folding matrix. The nominal $\text {m}_{\text {lb,pred}}^{\text {min}}$ distribution predicted with $m_t^{\text {MC}} =172.5$ GeV is folded with response matrices $M(m_t^{\text {MC}})$ corresponding to $m_t^{\text {MC}} =166.5$, 172.5, and 178.5 GeV. The resulting normalized event yields as a function of $\text {m}_{\text {lb}}^{\text {min}}$ are presented in Fig. 6.10.

Fig. 6.10

Folded and normalized $\text {m}_{\text {lb,pred}}^{\text {min}}$ distribution as predicted by MadGraph+pythia for $m_t^{\text {MC}} =172.5$ GeV. The folding is performed with the response matrix M corresponding to $m_t^{\text {MC}} =172.5$ GeV (nominal), $m_t^{\text {MC}} =178.5$ GeV, or $m_t^{\text {MC}} =166.5$ GeV. The statistical uncertainties are indicated with error bars. The lower panel shows the ratio of all distributions with respect to the nominal one. For the latter, statistical uncertainties are indicated with a shaded band

The mismatch of $\pm 6$ GeV leads to small variations in the folded distributions. Their effect on the extracted top-quark mass is quantified by using these distributions instead of the one measured in data and performing the extraction of $m_t^{\text {MC}}$ as described in Sect. 6.2.3. A bias of 200$\,\text {MeV}$ per 1 GeV mismatch is observed. Consequently, an additional systematic uncertainty ($\varvec{m_t}$ definition) of 200$\,\text {MeV}$ is assigned to the folding procedure, since the difference between $m_t^{\text {pole}}$ and $m_t^{\text {MC}}$ is estimated to be of the order of 1 GeV.

6.2.5.3 Extracted Top-Quark Mass and Systematic Uncertainties

The total uncertainty of the top-quark mass extracted from mcfm is composed of experimental uncertainties that affect the response matrix, systematic uncertainties related to the folding and extraction procedure, and theoretical uncertainties affecting the mcfm calculation. The latter comprise the PDF uncertainty (PDF $_\mathbf{mcfm }$) calculated at 68 % CL according to the prescriptions of the MSTW2008 group. The value of $\alpha _S$ is varied using dedicated PDF sets [20] and differences to the central result are considered as additional uncertainty ($\varvec{\alpha _S}$). The renormalization and factorization scales are varied independently by a factor of 2 up and down. The maximum deviation in each bin of $\text {m}_{\text {lb,pred}}^{\text {min}}$ is taken as a systematic uncertainty (scale $_\mathbf{mcfm }$). The mass of the $\text {b}$ quark is varied by $\pm 0.25$ GeV, resulting in an uncertainty (b-quark mass) on the extracted top-quark mass. Due to lacking statistics in the corresponding simulation, the response matrices for variations of CR and UE models suffer large fluctuations. Thus, these uncertainties are taken from the corresponding values obtained for $m_t^{\text {MC}}$ measured using MadGraph+pythia.

The resulting top-quark masses extracted from the mcfm prediction employing calculations at (N)LO for the ${\text {t}\bar{\text {t}}}$ production and at LO for the top-quark decay are

$$\begin{aligned} {m}_{{t}} ^\text {LO}= & {} 171.8 ^{+1.1}_{-1.0}\,\text {GeV} \text { and}\end{aligned}$$

(6.19)

$$\begin{aligned} {m}_{{t}} ^\text {NLO}= & {} 171.5 ^{+1.1}_{-1.0}\,\text {GeV} \text {.} \end{aligned}$$

(6.20)

Table 6.4 Breakdown of systematic uncertainties on the top-quark mass values extracted using mcfm

A breakdown of the systematic uncertainties is provided in Table 6.4. The impact of top-$p_{\mathrm {T}}$ and PDF uncertainties on the mcfm-based result is slightly reduced in comparison to the mass determination based on MadGraph+pythia, since both affect only the simulation of the detector response, and not the shape of the predicted cross section as a function of $\text {m}_{\text {lb,pred}}^{\text {min}}$. Both, ${m}_{{t}} ^\text {LO}$ and ${m}_{{t}} ^\text {NLO}$ agree within a few 100$\,\text {MeV}$. The ${m}_{{t}}$ extracted from the $\text {m}_{\text {lb}}^{\text {min}}$ shape is mostly independent of the production mechanism. Hence, an extension of the studies presented here regarding the treatment of the ${\text {t}\bar{\text {t}}}$ decay would be beneficial. This could be achieved by defining $\text {m}_{\text {lb,pred}}^{\text {min}}$ in terms of a $\text {b}\,\text {jet}$ instead of a $\text {b}$ quark. In consequence, the method could be applied to predictions that do not separate the amplitudes for production and decay and take into account effects of a finite top-quark width. The corresponding calculations have been performed including the subsequent $\text {W}$-boson decays to leptons and studies show that uncertainties due to variations of renormalization and factorization scale seem to be underestimated when using the NWA [12]. Therefore, the uncertainties on ${m}_{{t}} ^\text {LO}$ and ${m}_{{t}} ^\text {NLO}$ should be interpreted as a lower limit. Nevertheless, these predictions are not publicly available in a form that is applicable to an experimental analysis.