Phenomenology of tt¯ j + X production at the LHC

JHEP05(2022)146 Published for SISSA by Springer Received:March 4, 2022 Accepted:April 25, 2022 Published:May 23, 2022 Phenomenology of t¯ tj +Xproduction at the LHC Simone Alioli,aJuan Fuster,bMaria Vittoria Garzelli,cAlessandro Gavardi,a Adrian Irles,bDavide Melini,dSven-Olaf Moch,cPeter Uwereand Katharina Voßf,c aDipartimento di Fisica “G. Occhialini”, Università degli Studi di Milano-Bicocca and INFN, Sezione di Milano Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy bIFIC, Universitat de València and CSIC, Catedrático Jose Beltrán 2, E-46980 Paterna, Spain cII. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany dPhysics Department, Technion-Institute of Technology, Haifa 3200003, Israel eInstitut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany fCenter For Particle Physics Siegen, Department Physik, Universität Siegen, Walter Flex Str. 3, D-57068 Siegen, Germany E-mail: [email protected],[email protected], [email protected],[email protected], [email protected],[email protected], [email protected],[email protected], [email protected] Abstract: We present phenomenological results for t¯ tj +Xproduction at the Large Hadron Collider, of interest for designing forthcoming experimental analyses of this process. We focus on those cases where the t¯ tj +Xprocess is considered as a signal. We discuss present theoretical uncertainties and the dependence on relevant input parameters entering the computation. For the Rdistribution, which depends on the invariant mass of the t¯ tjsystem, we present reference predictions in the on-shell, MS and MSR top-quark mass renormalization schemes, applying the latter scheme to this process for the first time. Our conclusions are particularly interesting for those analyses aiming at extracting the topquark mass from cross-section measurements. Keywords: Specific QCD Phenomenology, Top Quark ArXiv ePrint: 2202.07975 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP05(2022)146 JHEP05(2022)146 Contents 1 Introduction and motivations for this work 1 2 Computational framework 3 3 Phenomenology of t¯ tj production at the LHC 6 3.1 Options for scale choice and scale uncertainty evaluation 7 3.2 Comparison of NLO and LO differential cross sections 14 3.3 Effects of PDF + αs(Mz)variation 23 3.4 Effects of variation of the Rparameter used in jet reconstruction 29 4 Theoretical predictions for top-quark mass measurements using the R distribution 32 4.1 Setup of the calculation 32 4.2 Conversion to the MS and MSR mass scheme 33 4.3 Exemplary results for the Rdistribution using different parton distribution function sets and different renormalization schemes 35 4.4 Off-shell effects and non-resonant/non-factorizable contributions 38 5 Conclusions 45 A Numerical results for cross sections for different scales, top-quark masses and analysis cuts 47 1 Introduction and motivations for this work The t¯ tj+Xproduction process at the Large Hadron Collider (LHC) is very interesting both as a signal and as a background in numerous experimental analyses. When considered as a signal, it can be used to extract precise values of the top-quark mass from measurements of differential cross sections particularly sensitive to the value of this Standard Model (SM) parameter. In particular, in ref. [1] it has been argued that the additional jet activity in the t¯ tj +Xprocess due to gluon radiation can lead to an enhanced mass sensitivity of the respective inclusive and differential cross sections, compared to the corresponding inclusive and differential cross sections for the t¯ t+Xprocess. This may allow for a more precise measurement, while keeping the advantage of uniquely fixing the renormalization scheme of the extracted mass value through higher-order theoretical predictions. More precisely, ref. [1] introduced the following observable R(mR t, ρs) = 1 σt¯ t+1-jet dσt¯ t+1-jet dρs (mR t, ρs),(1.1) – 1 – JHEP05(2022)146 with ρsgiven by ρs=2m0 √st¯ tj .(1.2) In the above definition, √st¯ tj is the invariant mass of the t¯ tj system, built from the topquark pair and the hardest jet satisfying typical transverse momentum and pseudorapidity cuts depending on the experimental analysis, and mR tis used to denote the top-quark mass in the renormalization scheme R. We have not explicitly specified the renormalization scheme for the top-quark mass, as different schemes can be and have been employed. The ‘mass’ m0occurring in the definition of ρsis an arbitrary scale to make ρsdimensionless. In practical applications it is fixed to a value of the order of the top-quark mass, e.g. m0= 170 GeV that we will use in the following. As a proof of concept, the measurement of Rwas used by the ATLAS collaboration to determine the top-quark mass in the onshell scheme using LHC data collected at 7 TeV [2]. As a follow-up, the aforementioned published experimental results were used in ref. [3] to determine the top-quark mass in the MS renormalization scheme. Later, also LHC data collected at 8 TeV were used by both the ATLAS and CMS collaborations [4,5] to extract the top-quark mass, whereas new analyses exploiting the large statistics accumulated at √S= 13 TeV are in preparation. Motivated by the experimental needs for the ongoing and forthcoming analyses, we present in this work theoretical predictions for the t¯ tj +Xprocess, discuss their main theoretical uncertainties and study their dependence on the choice of the input parameters and on some of the experimental cuts for the jet reconstruction procedure. We also investigate the role of different top-quark mass renormalization schemes. Besides the on-shell and the MS schemes, already used in previous papers, in this work we apply, for the first time, the MSR top-quark mass renormalization scheme [6,7] in deriving predictions for this process. We limit our discussion to fixed-order predictions in the stable top-quark case, considering the sophisticated techniques that have been developed by the experimental collaborations to reconstruct top quarks from their decay products and to unfold their results from the particle to the parton level (see for example refs. [5,8]). We provide reference predictions at next-to-leading order (NLO) accuracy, for different top-quark mass renormalization schemes and state-of-the-art choices for other input quantities, with the intent of providing useful information and concrete numerical results to the experimental collaborations in view of their ongoing t¯ tj +Xstudies and top-quark mass extractions. In fact, to determine the top-quark mass in the experimental analyses in preparation, as well as in those already published so far, the unfolded measurements are compared with theoretical predicitions with this accuracy in QCD. The manuscript is organized as follows. In section 2we briefly describe the theoretical framework. Theoretical predictions for fiducial inclusive and differential cross sections and their uncertainties are discussed in section 3, where we show their dependence on various inputs, including renormalization and factorization scale choices, parton distribution functions (PDFs) and the jet algorithm Rparameter. In section 4we focus on the R distribution, providing detailed predictions useful to the experimental collaborations to extract the top-quark mass value in different mass renormalization schemes. Furthermore we investigate the impact of off-shell effects and non-resonant/non-factorizable contributions. – 2 – JHEP05(2022)146 We conclude in section 5. Further material and tables with our reference numerical cross sections are collected in appendix A. 2 Computational framework NLO QCD corrections for t¯ tj +Xhadroproduction at the LHC have been computed and discussed in a number of papers, using multiple methods. In particular, refs. [9,10] consider the top quark as stable, refs. [11] and [12] complement NLO QCD production with top-quark decays at LO and NLO, respectively, by working in the narrow width approximation (NWA) retaining spin correlations,1whereas refs. [13,14] include full off-shellness and resonance effects for the cases where the top and antitop quarks decay leptonically. One-loop amplitudes for the parton-parton →t¯ tj process were first obtained analytically in refs. [9,10]. Refs. [11,12] used instead generalized D-dimensional unitarity [15] in a numerical implementation, whereas refs. [13,14] are based on numerical results from Helac-nlo [16], which provides one-loop amplitudes through a numerical implementation of the OPP method [17] complemented by effective Feynman rules for the calculation of the contribution due to the R2rational terms [18]. Nowadays, the same amplitudes can also be easily obtained, still numerically, by making use of various other automated tools (the so-called one-loop providers), such as MadLoop [19], GoSam [20,21], OpenLoops [22,23] and Recola [24,25]. Real emission amplitudes can also be easily obtained numerically, and multiple automatic frameworks for the cancellation of the infrared singularities when combining real and virtual contributions at NLO have also been developed, working according to either the Catani-Seymour (CS) dipole subtraction method for calculations with massive partons [26], or the Frixione-Kunst-Signer (FKS) method [27], or the Nagy-Soper subtraction formalism [28,29]. These developments have allowed for the automatic computation of fixed-order NLO cross sections with a variety of different tools, leading to predictions fully consistent among each other.2Additionally, one-loop providers have been interfaced to computational frameworks providing NLO QCD matching to parton shower (PS) approaches, considering both the POWHEG [32,33] and the MC@NLO [34] matching methods. In particular, first predictions with NLO QCD + PS accuracy for t¯ tj production were obtained in the PowHel framework [35], using numerical matrix elements from Helac-nlo as input to the Powheg-Box implementation [36], interfaced to Shower Monte Carlo (SMC) codes. Top-quark decays as well as PS emissions besides the first one and hadronization effects were taken care of by the interface of the produced events at the first radiation emission level to the Pythia [37] and Herwig [38] SMC. Subsequently, the analytical amplitudes of refs. [9,10] were also used as input to the Powheg-Box, producing, in case of stable top quarks, predictions [39] consistent with the previous ones. As an alternative to the top-quark decay by the SMC code, an in-house implementation of 1In ref. [12] jet radiation from the top-quark decay products is also included. 2The fully local schemes for the subtraction of infrared divergences employed at NLO are a necessary prerequisite for the consistency of the predictions. This is not guaranteed in different regularization frameworks, which use non-local methods for the subtraction, as pointed out recently in case of the Drell-Yan process at NNLO in refs. [30,31]. For the t¯ tj process NNLO predictions are not yet available. – 3 – JHEP05(2022)146 top-quark decays including spin correlations in the NWA was considered in ref. [39]. On the other hand, one-loop amplitudes as automatically obtained by MadLoop are at the core of the t¯ tj NLO QCD + PS computation in the Madgraph5_aMC@NLO framework [40], which provides PS-dependent matching terms for various versions of the Pythia and Herwig SMC approaches [37,38,41–43]. The MC@NLO matching method has also been implemented in the t¯ tj NLO QCD + PS implementation of ref. [44], using amplitudes from Helac-nlo, in association with the Deductor PS approach [45,46]. Furthermore, NLO electroweak (EW) corrections have been recently computed in ref. [47], and embedded in the multi-jet merging technique MEPS@NLO [48], giving rise to merged predictions for t¯ tand t¯ tj including NLO QCD + EW radiative corrections3together with PS and hadronization effects as available in the Sherpa SMC [49]. In this work we have used various frameworks to compute predictions for the t¯ tj process at NLO QCD accuracy: i) the implementation as presented in refs. [9,10] and ii) the Powheg-Box, which makes use of the FKS method for the infrared subtraction as numerically implemented inside the code, together with either the analytical amplitudes of refs. [9,10], implemented in a C++ library, or the numerical amplitudes from OpenLoops 2. In particular, the first ones have been interfaced in the Powheg-Box-v1 framework, whereas the second ones have been interfaced to the Powheg-Box-v2, respectively. We have extensively cross-checked these frameworks by comparing predictions for many differential distributions among each other, using also different systems of cuts. In all cases we have obtained perfect agreement. An illustration of these cross-checks can be found in figure 1, using as an example the ρsdistribution introduced in section 1. The computations of t¯ tj +Xhadroproduction have all been performed considering five active flavors at the scales relevant for this process, an assumption commonly adopted for most of the top quark related production processes.4Various modern PDF sets have been taken as input. In particular, in the present work we make use of the ABMP16 [52], CT18 [53], MMHT2014 [54], MSHT20 [55] and NNPDF3.1 [56] NLO PDF sets, together with their associated αs(MZ)value, and αsevolution at two-loops according to the Lhapdf interface [57] for the phenomenological predictions presented in section 3. 3In this implementation the dominant virtual electroweak corrections are incorporated exactly, whereas the NLO QED bremsstrahlung is accounted for in an approximate way. 4Processes involving bottom-quarks as final states, like e.g. single-top and t¯ tb¯ bproduction, besides in schemes with five active flavors, have also been computed in the decoupling factorization and renormalization scheme with four active flavors, i.e. considering massive b-quarks (see e.g. refs. [50,51]). For t¯ tb¯ bthe two descriptions have been shown to be compatible within uncertainties [51], at least for the experimental set of analysis cuts considered in that work. In t¯ tj production at hadron colliders, the final-state jet jcan also be a b-jet. This case is however suppressed with respect to the cases where jis a light jet, also due to the smallness of the b-quark PDF with respect to the ones for other partons. Therefore, given that b-initiated contributions (absent when bis considered as a massive quark) are indeed small when bis taken to be massless, and that u,d,c,squarks are massless in both cases, we do not expect a large difference in the description of the t¯ tj production using either four or five active flavors. On the other hand, when considering scales well above mt, as appropriate e.g. for the high-pt Tand high-mt¯ tj tails, it might be worth to investigate the t¯ tj process even in scheme with six active flavors. This may be done by using appropriate PDFs with six active flavors above the top-quark threshold. Most of the PDF fits available nowadays are however limited to a maximum of five active flavors. – 4 – JHEP05(2022)146 0 100 200 300 400 500 600 700 800 900 dσ/dρs[pb] µ0=mt= 172 GeV, pj T>30 GeV,|ηj|<2.5, R = 0.4, Nj≥1 DUW V1 V2 0.0 0.2 0.4 0.6 0.8 1.0 ρs 0.96 0.98 1.00 1.02 1.04 ratio w.r.t.V2 0 100 200 300 400 500 600 700 800 900 dσ/dρs[pb] µ0=mt= 172 GeV, pj T>50 GeV,|ηj|<2.5, R = 0.4, Nj≥1 DUW V1 V2 0.0 0.2 0.4 0.6 0.8 1.0 ρs 0.96 0.98 1.00 1.02 1.04 ratio w.r.t.V2 Figure 1. Comparison of predictions of the ρsdistribution for t¯ tj +Xproduction in pp collisions at √S= 13 TeV using different frameworks described in the text: Powheg-Box-v2 (black) vs. Powheg-Box-v1 (green) vs. the in-house NLO implementation of Dittmaier-Uwer-Weinzierl (blue) in an analysis setup where at least one light jet reconstructed with the anti-kTalgorithm with R= 0.4is required, with |ηj|<2.5and a pj T-cut of at least 30 GeV (left panel) and 50 GeV (right panel). The renormalization and factorization scales are fixed to the value of the top-quark mass mt= 172 GeV. PDF uncertainties are computed according to the specific prescriptions associated to each set. As we will show in the following, for the distributions and the kinematical regions in which we are most interested, it turns out that PDF uncertainties obtained in NLO hadronic computations involving NLO PDFs (+ αs) and NLO partonic cross sections have approximately the same relative size as PDF uncertainties obtained in LO hadronic computations involving NLO PDFs (+ αs) and LO partonic cross sections. Therefore, considering that the second approach is computationally much faster than the first one, we make use of it in section 3in order to compute the NLO PDF uncertainty bands for a number of different PDF fits. We still make use of the first approach only when computing PDF uncertainties in association with our nominal PDF set (CT18 NLO). Additionally, we present some considerations concerning the effect of the simultaneous variation of αs(MZ) and PDFs, using as a basis a set of ABMP16 NLO PDF + αS(MZ)fits, which fully preserve the correlations between these quantities. The predictions presented in section 3refer to the case when the top-quark mass is renormalized in the on-shell scheme, i.e. mt=mpole t. On the other hand, in section 4we – 5 – JHEP05(2022)146 discuss how to obtain predictions with the top-quark mass renormalized in the MSR and MS schemes, while we always used the latter scheme for αsrenormalization, and present the corresponding results for selected distributions. As further input, in section 3we consider various choices for the renormalization and factorization scales µRand µF. To that end, we define the quantities HB Tand mB t¯ tj as HB T= rpt, B T 2+m2 t+rp¯ t, B T 2+m2 t+pj, B T!,(2.1) mB t¯ tj =q(pB t+pB ¯ t+pB j)2,(2.2) which, as emphasized by the “B” superscript, are computed using the four-momenta of the outgoing particles of the underlying Born phase-space configuration in the Powheg-Box framework. The quantities HTand mt¯ tj are defined in analogy to eqs. (2.1) and (2.2) using instead the real emission kinematics at NLO for the four-momenta of the outgoing particles, i.e. pt,p¯ tand pj. Central predictions are obtained by fixing µR=µF=µ0, with µ0being one of the following options: 1) µ0=mt,2) µ0=HB T/2,3) µ0=HB T/4,4) µ0=mB t¯ tj/2.(2.3) The scale uncertainties are obtained by the usual seven-point prescription, i.e. by varying independently µRand µFby factors of [1/2, 2] around their central value, excluding the extreme configurations (µR,µF) = (2, 0.5) µ0and (0.5, 2) µ0. We reconstruct jets following the E-recombination scheme, according to which the energy and the momentum of a jet are defined as the sums of the energies and the momenta of its constituents, by using the anti-kTjet clustering algorithm [58], with two different values for the jet radius parameter, R= 0.4and R= 0.8. The first value is the present default value adopted by both the ATLAS and CMS collaborations in their analyses. We consider the second value as a possible alternative, in order to study the sensitivity of the predictions and their uncertainties in dependence on R. A general discussion of the effects of these inputs on predictions for total and differential cross sections is included in section 3. 3 Phenomenology of t¯ tj production at the LHC If not specified otherwise, the predictions presented in the following refer to our default configuration. This corresponds to a center-of-mass energy of √S= 13 TeV. The topquark mass renormalized in the on-shell scheme is set to mpole t= 172 GeV. In our fixed-order computation the top quarks are considered as stable and the CT18 NLO PDF set is used as default. At the phenomenological analysis level, at least one jet is required with a transverse momentum pj T>30 GeV and an absolute pseudorapidity |ηj|<2.4. Jets are reconstructed using the anti-kTjet clustering algorithm from FastJet [59] (version 3.3.4) with R= 0.4 and the E-recombination scheme. This system of analysis cuts closely resembles typical systems of cuts used by the ATLAS and CMS experimental collaborations. – 6 – JHEP05(2022)146 10−5 10−4 10−3 10−2 10−1 100 dσ/dx [pb/GeV] pj T>30 GeV,|ηj|<2.4,R= 0.4,Nj≥1,µ0=mt x=HT x=HB T x=mt¯ tj x=mB t¯ tj 0 1 2HT/HB T mt¯ tj/mB t¯ tj 400 600 800 1000 1200 1400 1600 1800 2000 x[GeV] 0 1 2 HT/mt¯ tj HB T/mB t¯ tj Figure 2. Differential cross section as a function of HB T(blue solid), HT(blue dashed), mB t¯ tj (black solid) and mt¯ tj (black dashed), calculated for the pp →t¯ tj +Xprocess at a center of mass energy of √S= 13 TeV, using as input a central scale µ0set to µ0=mt. 3.1 Options for scale choice and scale uncertainty evaluation Before studying uncertainties related to PDF and Rvariation, as explained in subsection 3.3 and 3.4, respectively, we investigated the influence of different central scale definitions on differential cross sections at NLO, considering many different observables. As mentioned in eq. (2.3), we have considered four central scale choices, a static scale and three dynamical scales. The investigation of different µ0choices was motivated by observations presented in ref. [14], where a more stable behavior of the scale variation uncertainty was found when using a dynamical scale with respect to the static scale case. However, while in the present manuscript the top quarks are considered as stable and the effects of relatively loose systems of cuts are investigated, in line with the procedures applied in ongoing experimental t¯ tj+X analyses for the determination of the top-quark mass, in ref. [14] the t¯ tj +Xprocess was studied in presence of fully leptonic top-quark decays and the analyzed events fulfilled more specific analysis cuts, which differ substantially from the more inclusive cuts used here. This difference motivates the work presented in this subsection. In figure 2the NLO distributions in the variables HB T(blue solid), mB t¯ tj (black solid), HT(blue dashed) and mt¯ tj (black dashed) as defined in eqs. (2.1) and (2.2) are shown for comparison. All these differential cross sections were obtained by setting the central scale to the fixed value µ0=mt. In the lower panels ratios between various pairs of these – 7 – JHEP05(2022)146 µ0σLO [pb] δscale [pb] σNLO [pb] δscale [pb] K=σNLO/σLO mt294.3 +138.0 (+47%) −87.5 (−30%) 359.4 +15.3 (+4%) −41.5 (−12%) 1.22 HB T/2231.08 +100.30 (+43%) −65.42 (−28%) 331.9 +34.9 (+11%) −47.2 (−14%) 1.44 HB T/4331.4 +159.5(+48%) −100.3(−30%) 366.8 +3.3(+1%) −34.9(−10%) 1.11 mB t¯ tj/2202.17 +83.91 (+42%) −55.61 (−28%) 302.5 +35 (+12%) −43.2 (−14%) 1.50 Table 1. Integrated cross section at LO and NLO of the process pp →t¯ tj +Xwith the analysis cuts Nj≥1,pj T>30 GeV and |ηj|<2.4, where the anti-kTjet clustering algorithm with R= 0.4 and the CT18 NLO PDF set were used. The scale variation uncertainty δscale is obtained according to the seven-point scale variation procedure described in section 2. In each δscale column, the first number refers to the absolute uncertainty due to scale variation, whereas the second number, indicated in parentheses, gives the same uncertainty in a percentage format. distributions are shown. It is evident that the mB t¯ tj and mt¯ tj spectra are harder than the HB Tand HTones. This leads to reduced integrated and differential cross sections when using the scales mB t¯ tjor mt¯ tj-based on the invariant-mass of the t¯ tj-system compared to the HB Tor HT-based scales as the coupling αs(µR)decreases for increasing scales µR. The same considerations also apply when comparing predictions obtained with µ0=HB T/2and µ0=HB T/4, as we will discuss next. The dynamical central scale choices µ0=HB T/2 and µ0=mB t¯ tj/2lead in general to larger numerical values for the renormalization and factorization scales, than the static scale choice µ0=mt. They are exactly equal to the latter only at threshold. Comparing the dashed and solid histograms in figure 2, we also expect that the slightly different options HTand mt¯ tj for the dynamical scales, i.e. building them from the real emission kinematics instead of from the underlying Born configuration, will lead to slightly larger cross-section values. The values of the integrated cross sections at LO and NLO under our default system of cuts described at the beginning of this section are presented in table 1for the four central scale choices in eq. (2.3). Thereby the seven-point scale variation was used to estimate the scale uncertainty δscale. The resulting K-factors, given by the ratio of the NLO and LO integrated cross sections evaluated with each central scale, are presented in the last column. Both for the NLO and LO computations the CT18 NLO PDF set was used, accompanied by the associated default αNLO s(MZ)value, equal to 0.118. Typical values of the strong coupling accompanying LO PDF sets are in general larger, amounting to αLO s(MZ)≃0.13.5 Using the static scale choice leads to larger values of the integrated cross section than the corresponding calculation with the dynamical scales µ0=HB T/2and µ0=mB t¯ tj/2. On the other hand, the cross section obtained with the static scale is smaller than that obtained with the dynamical scale µ0=HB T/4. In the threshold region, µ0=HB T/4is close to mt/2, 5On the basis of this consideration, the LO cross sections, obtained using as input NLO (PDFs + αs(MZ)) values and two-loop αsevolution, have generally smaller values than those obtained with a LO (PDF + αs(MZ)) set. – 8 – JHEP05(2022)146 perturbative expansion. This applies for those cases where the same initial-state partonic channels are present at both orders.6 To this end we compare the LO and NLO predictions obtained with the static and the dynamical scale choices.7In figure 7the LO and NLO differential cross sections in ρs are compared among each other, using the scale choices in eq. (2.3). In the middle ratio plot, below the ρsdistribution, the scale variation bands, normalized to the result obtained with KR=KF= 1 at either LO (black) or NLO (blue), are shown. The red line in the ratio plot indicates the differential K-factor (dσNLO/dρs)(dσLO/dρs). In contrast, in the lowest ratio plot the scale variation bands are both rescaled by the LO result. This way it can be easily checked if the NLO and LO scale variation bands overlap. As expected, the NLO scale variation band is generally smaller compared to the LO one. It turns out that, for ρs<0.9, the NLO predictions obtained with the dynamical scales µ0=HB T/2and HB T/4are shifted to higher values with respect to the LO predictions, by an almost uniform differential K-factor of ∼1.5and 1.1, respectively, as visible in figure 7. The scale uncertainty band is thereby reduced when going from LO to NLO and shows a nearly uniform width over the whole range of the ρsdistribution, amounting to about ±10 % for µ0=HB T/2and to about ( +1 %,−10 %) for µ0=HB T/4. On the other hand, the same distribution evaluated with the scale µ0=mB t¯ tj/2, as shown in the second panel of figure 7, is characterized by a larger differential K-factor for low values of ρs, where (dσNLO/dρs)/(dσLO/dρs)∼1.9, than for high values of ρs, where the differential K-factor is found to be ∼1.4(excluding the very last bin near threshold). Applying the scale definition µ0=mB t¯ tj/2, the LO and NLO scale variation bands depart from each other in the low ρsregion, signaling that this region is not well described. The aforementioned behaviour can be understood from the fact that the process is a multi-scale problem. While for inclusive quantities, the invariant mass of the final states at the underlying Born level mB t¯ tj/2gives an appropriate scale setting, the scale relevant for the emission of the additional jet can be much lower. As a consequence, one expects that observables relying significantly on the additional emission are not well described using the mB t¯ tj/2scale. This implies a large K-factor using this setting as at NLO accuracy the scale dependence should be reduced with respect to LO and differences in the predictions on the basis of different choices should be attenuated as well. In case of the static scale µ0=mt, the central values of the NLO and LO distributions are strongly distorted one with respect to each other, as shown in the first panel of figure 7. A crossing of the NLO and LO distributions evaluated with KR=KF= 1 occurs in the same region of the ρsdistribution, namely at ρs∼0.2, where the NLO predictions, evaluated at the scales that allow to build the seven-point scale variation band, cross among each others (see the left panel of figure 4). Additionally, with the static scale definition, the NLO and LO scale variation bands are only marginally overlapping in the high ρsregion. 6Differently from the case of t¯ tproduction, for t¯ tj production the qg,gq,¯qg and g¯qchannels contribute already at LO, together with the gg and q¯qchannels. 7Both NLO and LO predictions are obtained by using the CT18 NLO PDF set with its own αS(MZ)=0.118 value and two-loop αsevolution, see also sections 3.1 and 3.3. – 15 – JHEP05(2022)146 0 500 1000 dσ/dρs[pb] µ0=mpole tµ0=mB ttj/2µ0=HB T/2µ0=HB T/4 LO NLO NLO/LO 0 1 2 scalevar/ central 0.0 0.2 0.4 0.6 0.8 1.0 ρs 0 1 2 NLO/LO 0.2 0.4 0.6 0.8 1.0 ρs 0.2 0.4 0.6 0.8 1.0 ρs 0.2 0.4 0.6 0.8 1.0 ρs pj T>30 GeV,|ηj|<2.4, R = 0.4, Nj≥1 Figure 7. Central predictions for the ρsdistribution at NLO (blue) and LO (black), together the corresponding NLO and LO seven-point scale variation uncertainty bands, using as a central scale µ0=mt,mB t¯ tj/2,HB T/2and HB T/4(from left to right). In the middle ratio plot the scale variation uncertainty at either NLO or LO is rescaled by the central scale prediction at NLO or LO respectively. To visualize the difference between the NLO and LO differential cross section the ratio of the NLO and LO central scale predictions is shown in red. In the bottom ratio insets both the NLO and LO scale variation uncertainty bands are rescaled by the LO central scale prediction. As follows from the relation between ρsand mt¯ tj in the definition of ρs, the features of the ρsdistribution at low ρscalculated with the different scales are of course reflected in the mt¯ tj distributions for large values of mt¯ tj, shown in figure 8. Another distribution, in which the more stable behavior in the high-energy tail with the dynamical scale choices is visible, is the invariant mass of the top-quark pair mt¯ t= p(pt+p¯ t)2, shown in figure 9. In case of the static scale assignment (left panel), for large values of the invariant mass of the top-quark pair, the scale variation band at NLO departs from the LO one. This signals again an instability in the high energy tail of the distribution. The width of the NLO scale uncertainty band is smaller with respect to the LO one at small values of mt¯ t, whereas in the high energy tail the NLO scale variation band shows a strong increase, being of similar size as the LO band for mt¯ t= 1.5 TeV. Also in this distribution a distortion of the central NLO prediction with respect to the corresponding LO one is visible, the NLO values going from +40% of the LO values in the region of small mt¯ t(large values of ρs) to −50% in the region of high mt¯ t. The predictions obtained with the dynamical scale show a much smaller distortion of the central values when going from LO to NLO, while the width of the scale variation bands are quite stable over the whole range of explored mt¯ tvalues. Similar features as for the behavior of the NLO uncertainty bands can also be observed in e.g. the transverse momentum distribution of the top-quark pt T, shown in figure 10, – 16 – JHEP05(2022)146 0.0 0.5 1.0 dσ/dmt¯ tj [pb/GeV] µ0=mpole tµ0=mB ttj/2µ0=HB T/2µ0=HB T/4 LO NLO NLO/LO 0 1 2 scalevar/ central 500 1000 1500 2000 mt¯ tj [GeV] 0 1 2 NLO/LO 500 1000 1500 2000 mt¯ tj [GeV] 500 1000 1500 2000 mt¯ tj [GeV] 500 1000 1500 2000 mt¯ tj [GeV] pj T>30 GeV,|ηj|<2.4, R = 0.4, Nj≥1 Figure 8. Same as in figure 7, but for the mt¯ tj distribution. 10−3 10−2 10−1 100 dσ/dmt¯ t[pb/GeV] µ0=mpole tµ0=mB ttj/2µ0=HB T/2µ0=HB T/4 LO NLO NLO/LO 0 1 2 scalevar/ central 500 1000 1500 2000 mt¯ t[GeV] 0 1 2 NLO/ LO 500 1000 1500 2000 mt¯ t[GeV] 500 1000 1500 2000 mt¯ t[GeV] 500 1000 1500 2000 mt¯ t[GeV] pj T>30 GeV,|ηj|<2.4, R = 0.4, Nj≥1 Figure 9. Same as in figure 7, but for the mt¯ tdistribution. – 17 – JHEP05(2022)146 10−1 100 dσ/dpt T[pb/GeV] µ0=mpole tµ0=mB ttj/2µ0=HB T/2µ0=HB T/4 LO NLO NLO/LO 0 1 2 scalevar/ central 0 100 200 300 400 500 pt T[GeV] 0 1 2 NLO/ LO 100 200 300 400 500 pt T[GeV] 100 200 300 400 500 pt T[GeV] 100 200 300 400 500 pt T[GeV] pj T>30 GeV,|ηj|<2.4, R = 0.4, Nj≥1 Figure 10. Same as in figure 7, but for the pt Tdistribution. 0 50 100 150 dσ/dyt[pb] µ0=mpole tµ0=mB ttj/2µ0=HB T/2µ0=HB T/4 LO NLO NLO/LO 0 1 2 scalevar/ central −4−2 0 2 4 yt 0 1 2 NLO/LO −2 0 2 4 yt −2 0 2 4 yt −2 0 2 4 yt pj T>30 GeV,|ηj|<2.4, R = 0.4, Nj≥1 Figure 11. Same as in figure 7, but for the ytdistribution. – 18 – JHEP05(2022)146 0 50 100 dσ/dyt[pb] µ0=mpole tµ0=HB T/4µ0=HB T/2µ0=mB ttj/2 pj T>30 GeV pj T>50 GeV pj T>50 GeV/ pj T>30 GeV 0.5 1.0 1.5 scalevar/ central −4−2 0 2 4 yt 0.5 1.0 1.5 pj T>50 GeV/ pj T>30 GeV −2 0 2 4 yt −2 0 2 4 yt −2 0 2 4 yt |ηj|<2.4, R = 0.4, Nj≥1 Figure 12. NLO central predictions for the ytdistribution with a pj T-analysis cut of 30 GeV (black) and 50 GeV (blue), accompanied by the corresponding seven-point scale variation uncertainty bands, obtained using as a central scale µ0=mt,HB T/4,HB T/2and mB t¯ tj/2(from left to right). In the middle ratio plot the scale variation uncertainty band with either pj T-cut is rescaled by the central scale prediction with the same analysis cut. To visualize the difference between the differential cross section obtained with the different pj T-cuts the ratio of the central scale predictions is shown in red. In the bottom ratio plot the scale variation uncertainty bands using either analysis cut are rescaled by the central scale result using the requirement pj T>30 GeV. 0 50 100 150 dσ/dyt¯ t[pb] µ0=mpole tµ0=mB ttj/2µ0=HB T/2µ0=HB T/4 LO NLO NLO/LO 0 1 2 scalevar/ central −4−2 0 2 4 yt¯ t 0 1 2 NLO/LO −2 0 2 4 yt¯ t −2 0 2 4 yt¯ t −2 0 2 4 yt¯ t pj T>30 GeV,|ηj|<2.4, R = 0.4, Nj≥1 Figure 13. Same as in figure 7, but for the yt¯ tdistribution. – 19 – JHEP05(2022)146 10−2 10−1 100 101 dσ/dpj1 T[pb/GeV] µ0=mpole tµ0=mB ttj/2µ0=HB T/2µ0=HB T/4 LO NLO NLO/LO 0 1 2 scalevar/ central 100 200 300 400 500 pj1 T[GeV] 0 1 2 NLO/ LO 100 200 300 400 500 pj1 T[GeV] 100 200 300 400 500 pj1 T[GeV] 100 200 300 400 500 pj1 T[GeV] pj T>30 GeV,|ηj|<2.4, R = 0.4, Nj≥1 Figure 14. Same as in figure 7, but for the pj1 Tdistribution. 0 50 100 dσ/dηj1[pb] µ0=mpole tµ0=mB ttj/2µ0=HB T/2µ0=HB T/4 LO NLO NLO/LO 0 1 2 scalevar/ central −2−1 0 1 2 ηj1 0 1 2 NLO/LO −2−1 0 1 2 ηj1 −2−1 0 1 2 ηj1 −2−1 0 1 2 ηj1 pj T>30 GeV,|ηj|<2.4, R = 0.4, Nj≥1 Figure 15. Same as in figure 7, but for the ηj1distribution. – 20 – JHEP05(2022)146 0 10 20 30 40 50 dσ/dηj2[pb] µ0=mpole tµ0=mB ttj/2µ0=HB T/2µ0=HB T/4 LO −2−1 0 1 2 ηj2 0 1 2 scalevar/ central −2−1 0 1 2 ηj2 −2−1 0 1 2 ηj2 −2−1 0 1 2 ηj2 pj T>30 GeV,|ηj|<2.4, R = 0.4, Nj≥1 Figure 16. Same as in figure 7, but for the ηj2distribution. The content of the panels reflect the fact that the second jet appears for the first time in NLO calculations and, therefore, the accuracy of the predictions for it is limited to LO. although in that case a distortion of NLO predictions with respect to LO ones is present even when using the dynamical scales. In figure 11 the differential cross section is shown as a function of the rapidity of the top-quark yt. The scale variation bands in the static and dynamical scale cases have a more similar width. Using µ0=HB T/4leads to the smallest size of the scale uncertainty band. For the ytdistribution the dynamical scale choice does not lead to an improvement in the stability of the prediction. In fact, as also observed in ref. [14], the observable ytreceives contributions from all phase-space regions, with most of them from the bulk, where the differences between dynamical and static scales are reduced. The shape of the ytdistribution is almost independent on the pj Tcut applied in the reconstruction of the hardest jet. This is shown in figure 12, where this observable is plotted for two different pj Tcuts, namely pj T>30 GeV and pj T>50 GeV. The integrated cross section becomes lower when applying a stronger pj T-cut, but the ratio of the distributions dσ/dyt(pj T>50 GeV)/dσ/dyt(pj T>30 GeV)is almost uniform, showing that no region in the ytdistribution has a much stronger dependence on this analysis cut compared to the rest of the distribution. As far as the differential contributions are concerned, the dynamical scale µ0=HB T/4 gives in most cases a flat K-factor close to 1, suggesting that potentially large logarithms are effectively absorbed when adopting this particular choice. While the two other dynamical scales also show a flat K-factor for most distributions, the K-factor value itself is much larger. Uncalculated higher orders could thus be important. This interpretation is supported by the larger scale uncertainty observed for these two – 21 – JHEP05(2022)146 dynamical scales. For most distributions using the static scale leads to a K-factor which depends significantly on the phase space. This is in particular true for the distributions where the individual bins set different energy scales. This is not surprising as a static treatment cannot absorb corresponding logarithms. However, as long as the tails of the distributions are avoided, the static scale choice leads to a smaller K-factor then the two dynamical scales mB t¯ tj/2and HB T/2. We can thus conclude that the static scale is still a reasonable choice for many distributions, at least for the time being. An improved understanding of the perturbative series for the t¯ tj process will be possible when higherorder calculations beyond NLO will start to appear. Since the integrated cross section is dominated by the threshold region, in which the static and dynamical scales µ0=HB T/2and µ0=mB t¯ tj/2give similar predictions, no large differences in the size of the scale variation bands of the ytpredictions when using either one or the other scale are expected, as observed in figure 11. It can also be seen in figure 11 that the LO result varies much stronger going from the static to the dynamical scales, than the NLO result. In the region of large absolute rapidity of the top quark the NLO scale variation band starts to depart from the LO one. This is observed using all four scale choices. The t-channel diagrams contributing in this region lead to final state particles closely following the direction of the initial state particles, such that these final state particles have a large rapidity. Removing the ηj-cut would result in a t-channel singularity explaining the aforementioned behavior observed at large |yt|. In case of t¯ t production, and more generally heavy-quark production, this was already investigated in ref. [60] and [61]. We also observe that similar trends as for the ytdistribution occur even for the yt¯ tdistribution, plotted for various scale choices in figure 13. Additionally, we looked at the differential distributions of the hardest and second hardest jet. The transverse momentum of the hardest jet is shown in figure 14, where the high-energy tail seems again to be better described using a dynamical scale definition, than in case of a static scale. Looking at the pseudorapidity of the hardest jet ηj1and second hardest jet ηj2, a relatively flat differential K-factor is found using either central scale, comparing figures 15 and 16, respectively. Predictions for the second hardest jet are only available in the NLO t¯ tj calculation, since at LO only one light parton is present in the final state. As expected from accuracy considerations, these predictions are characterized by larger scale uncertainty bands than those for the first hardest jet. The size of the scale uncertainty bands for the ηj2distribution in figure 16 is even larger than the size of the LO scale uncertainty band for ηj1in figure 15. As visible in each panel of both figures, for all central scale choices, the size of scale uncertainties accompanying both the ηj1and ηj2distributions does not depend on the pseudorapidity value, i.e. it is uniform, differently from the size of scale uncertainties accompanying the top-quark ytdistribution, which increase at large absolute values of rapidity, as discussed above. Some of the described differential cross sections were also discussed in ref. [14], where similar trends were observed, although using more exclusive cuts (and different scales) as mentioned at the beginning of this section. – 22 – JHEP05(2022)146 0 2 4 dσ/dpj1 T[pb/GeV] LO NLO 0.8 1.0 1.2 PDF/central 100 200 300 400 500 pj1 T[GeV] 1 2 NLO/LO CT18NLO, µ0= HB T/2 0 25 50 75 dσ/dyj1[pb] LO NLO 0.8 1.0 1.2 PDF/central −2−1 0 1 2 yj1 1 2 NLO/LO CT18NLO, µ0= HB T/2 Figure 17. Central predictions for the pj1 T(left) and yj1(right) distributions in a full NLO computation (blue) and in a LO computation (black), using as input in both cases the CT18 NLO PDFs + αs(MZ)value, two-loop evolution of αsand the scale µ0=HB T/2. PDF uncertainty bands computed according to the CT18 prescription and rescaled to the 68% C.L. are also shown. 0.0 0.5 1.0 dσ/dmt¯ t[pb/GeV] LO NLO 0.8 1.0 1.2 PDF/central 250 500 750 1000 1250 1500 1750 2000 mt¯ t[GeV] 1 2 NLO/LO CT18NLO, µ0= HB T/2 0 50 100 dσ/dyt¯ t[pb] LO NLO 0.8 1.0 1.2 PDF/central −4−2 0 2 4 yt¯ t 1 2 NLO/LO CT18NLO, µ0= HB T/2 Figure 18. Same as figure 17, but for the mt¯ t(left) and the yt¯ t(right) distributions. 3.3 Effects of PDF + αs(Mz)variation In this subsection we discuss the effect of PDF + αs(MZ)variations. We take as default the CT18 NLO PDF set, with their associated αs(MZ)value, the latter being subject to two-loop evolution in QCD as provided by the Lhapdf interface. We compute central predictions using the set 0 (best fit) and we determine the PDF uncertainty band from the 58 available eigenvectors, according to the prescription provided by the CT18 collaboration.8 8The PDF uncertainties quoted by CT18 denote the 90% confidence level (C.L.) and have to be reduced by a factor of 1.645 for comparison with the 68% C.L. uncertainties quoted by other groups. – 23 – JHEP05(2022)146 0.0 0.2 0.4 0.6 dσ/dmt¯ tj [pb/GeV] LO NLO 0.8 1.0 1.2 PDF/central 500 1000 1500 2000 mt¯ tj [GeV] 1 2 NLO/LO CT18NLO, µ0= HB T/2 0 250 500 750 dσ/dρs[pb] LO NLO 0.8 1.0 1.2 PDF/central 0.0 0.2 0.4 0.6 0.8 1.0 ρs 1 2 NLO/LO CT18NLO, µ0= HB T/2 Figure 19. Same as figure 17, but for the mt¯ tj (left) and the ρs(right) distributions. 0.0 0.2 0.4 0.6 dσ/dmt¯ tj [pb/GeV] µ0=HB T/2 µ0=HB T/4 0.8 1.0 1.2 PDF/central 500 1000 1500 2000 mt¯ tj [GeV] 0 1 2 µHB T/4/µHB T/2 CT18NLO,LO simulation 0 250 500 750 dσ/dρs[pb] µ0=HB T/2 µ0=HB T/4 0.8 1.0 1.2 PDF/central 0.0 0.2 0.4 0.6 0.8 1.0 ρs 0 1 2 µHB T/4/µHB T/2 CT18NLO,LO simulation Figure 20. Central predictions for the mt¯ tj (left) and ρs(right) distributions in a LO computation with the scales µ0=HB T/2(black) and µ0=HB T/4(blue), using as input in both cases the CT18 NLO PDFs + αs(MZ)value and two-loop evolution of αs. PDF uncertainty bands computed according to the CT18 prescription and rescaled to 68% C.L. are also shown. First of all we observe that the relative size of the PDF uncertainty band for many distributions remains approximately the same when making a LO computation with NLO PDFs instead of a full NLO computation. This is shown in figure 17 and 18 in case of various representative distributions, i.e. pj1 T,yj1,mt¯ tand yt¯ t, by using the central scale µ0=HB T/2for all predictions. This is also true for the mt¯ tj and ρsdistributions for ρsvalues not too large (corresponding to large enough mt¯ tj values), as shown in figure 19. On the other hand, for very large ρsvalues (ρs>0.82), corresponding to small enough mt¯ tj (approximately, mt¯ tj <400 GeV), the PDF uncertainty band obtained in the LO computation is smaller – 24 – JHEP05(2022)146 0 2 4 dσ/dpj1 T[pb/GeV] µ0=mpole tµ0=HB T/2µ0=HB T/4 R= 0.4 R= 0.8 R= 0.8/ R= 0.4 0.75 1.00 1.25 100 200 300 400 500 pj1 T[GeV] 0.75 1.00 1.25 R= 0.8/R= 0.4 100 200 300 400 500 pj1 T[GeV] 100 200 300 400 500 pj1 T[GeV] pj T>30 GeV,|ηj|<2.4, Nj≥1 Figure 29. Same as figure 27, but for the pj1 Tdistribution. 0.0 0.5 1.0 1.5 dσ/dmt¯ t[pb/GeV] µ0=mpole tµ0=HB T/2µ0=HB T/4 R= 0.4 R= 0.8 R= 0.8/ R= 0.4 0.75 1.00 1.25 250 500 750 1000 1250 1500 1750 2000 mt¯ t[GeV] 0.75 1.00 1.25 R= 0.8/R= 0.4 500 750 1000 1250 1500 1750 2000 mt¯ t[GeV] 500 750 1000 1250 1500 1750 2000 mt¯ t[GeV] pj T>30 GeV,|ηj|<2.4, Nj≥1 Figure 30. Same as figure 27, but for the mt¯ tdistribution. – 31 – JHEP05(2022)146 showing the mt¯ tj and the pj1 Tdistributions. Our conclusion is that the size of the scale uncertainty bands is almost insensitive to the Rvariation in the range [0.4 - 0.8]. On the other hand, the effect of a change in the Rvalue on observables which only depend on the top quarks just amounts to a rescaling of the cross section, with no impact on the shape of the corresponding distributions, as shown e.g. in figure 30 for the case of the mt¯ tdistribution. Reassured by the fact that no significant shape distortion caused by the different value of the jet radius Rappears in the NLO distributions we are interested in, we want however to remind the reader that some caution is required when extracting from fixed-order calculations the final physical dependence of the cross section and of inclusive distributions on R. This is due to the fact that the R-dependence of the cross section is not influenced by virtual corrections, but only by the additional real radiation. We thus expect it to display, at the NLO level, an unphysical behavior, in the form of a logarithmic divergence at small R. Only after adding shower and hadronization effects one obtains a more physical description, usually resulting in an increase of the slope of the Rdependence of the dσ/dR cross section, compared to fixed-order calculations. This is a known effect, since hadron formation further randomizes the particles’ momenta, driving even more energy out of the cone. It is also known that, on the other hand, the underlying event counteracts the effect of hadronization, since it generates soft hadrons that bring more energy into the jet cone, with a probability proportional to its area. 4 Theoretical predictions for top-quark mass measurements using the R distribution Some results for the Rdistribution have already been presented in section 3. However, for systematic extractions of the top-quark mass using as a basis the t¯ tj +Xsamples of events collected at the LHC at √S=13 TeV, or future samples at higher center-of-mass energies like e.g. √S=14 TeV, many more predictions are required, using different PDF sets and covering a large top-quark mass range in small mass steps. The purpose of this section is to provide extensive results which can be used by the ATLAS and CMS experimental collaborations to infer the top-quark mass values in different mass renormalization schemes and allow also to study various systematic uncertainties. In the experimental analyses the binning of the Rdistribution has been optimized to minimize systematic and statistical uncertainties. For the exemplary results shown in the following, we use the current binning and analysis setups as provided by ATLAS and CMS. As the analysis setup differs slightly from the one adopted in section 3, we collect the details in the next subsection. 4.1 Setup of the calculation The results presented in section 4.3 have NLO QCD accuracy, refer to stable top quarks and were obtained by properly extending the framework of refs. [9,10], after having verified that, for the case of top-quark mass renormalized in the on-shell scheme, it gives predictions compatible with the Powheg-Box implementation used in section 3, as already discussed there. As in section 3, the anti-kTjet algorithm [58], as implemented in FastJet [59], is used with Rset to 0.4 and employing the E-scheme for parton recombination into jets. – 32 – JHEP05(2022)146 Additional cuts are applied to the jets as returned by FastJet. First of all, a minimal transverse momentum pmin Tis required for jet detection. We have produced results for different choices of pmin T:pmin T∈{30 GeV, 50 GeV, 75 GeV, 100 GeV}. This set of pmin T values extends the one considered in section 3. In addition, in order to be consistent with the most recent experimental setups, we have computed predictions for two different choices of the maximal value of the pseudorapidity ηj: results using |ηj|<2.5as well as results using |ηj|<2.4have been produced (in section 3only |ηj|<2.4was used). If more than one jet satisfies these cuts (the top quarks are assumed to be always detected and not passed to the jet-algorithm), the highest energetic jet is used to calculate ρs. For the binning in ρswe have investigated different choices, as provided by the experiments. In the following, predictions are presented using {0., 0.18,0.22,0.27,0.32,0.38,0.45,0.53,0.62,0.71,1.} for the bin boundaries. For the hadronic center of mass energies we used both √S= 13 TeV and √S= 14 TeV, in view of Run 3 and the High Luminosity phase at the LHC. Further predictions for other center-of-mass energies can be provided if required by the experiments. Despite the slightly enhanced mass sensitivity of the Rdistribution compared to the inclusive cross section, the mass effects are still rather small and high precision in the numerical Monte Carlo integration is required to make them visible. For the virtual corrections to the partonic process gg →t¯ tg additional runs were required to achieve a precision at the sub-percent level in the combined result. To allow interpolations in the mass, results for different top-quark mass values were produced using a step size of 1GeV. For the numerical evaluation of the derivative of the leading order cross section required in the conversion to the MS or MSR heavy-quark mass renormalization scheme (see section 4.2) a step size of 0.5GeV is used. Employing different methods to calculate the derivative, we have checked that the step size of 0.5GeV leads to negligible uncertainties for the derivative. As the pole mass mtand the MS mass m(m)differ by about 10 GeV (doing the conversion at four loops), we produced results for mass values between 158 and 178 GeV, allowing to cover not only the relevant range in the pole mass but also in the MS and MSR mass. The MSR mass depends on the choice of the Rscale. For typical Rvalues between O(GeV) and the MS mass m(m), the value of the MSR mass lies between the MS mass m(m)(R=m(m)) and the pole mass (R∼1GeV). We used the (in alphabetic order) ABMP16_5_nlo [52], CT18NLO [64], MMHT2014nlo68cl [54], as well as MSHT20nlo_as118 [55], and NNPDF31_nlo_as_0118 [56] PDF sets. The latter two fix αs(MZ)to 0.118 and in the evaluation of the NLO predictions we have always used the αs(MZ)value as provided by the corresponding PDF set. In addition, we also produced results always using a same fixed αs(MZ)value, following the prescription applied in refs. [9,10]. 4.2 Conversion to the MS and MSR mass scheme As mentioned before, the Rdistribution can also be expressed using renormalization schemes different from the pole mass scheme usually applied. In the following, we briefly – 33 – JHEP05(2022)146 describe how the predictions obtained in the pole mass scheme can be translated to the MS or to the MSR mass scheme. We follow the method outlined in ref. [65] and present only the main steps as the details can be found in ref. [65]. We start with the discussion of the MS mass. In perturbation theory the pole mass and MS mass are related through a finite renormalization and the mass given in one scheme can be expressed in terms of the mass given in a second scheme. For the concrete case the relation between the top-quark pole mass mtand MS mass m(µ)reads: mt=m(µ)1−c1(LM)a(nf)(µ) + c1(LM)2−c2(LM)a(nf)2(µ) + . . . .(4.1) with a(nf)(µ) = α(nf) s(µ) πand LM= ln µ2 m2 t!.(4.2) The coefficients ciare known up to four-loop order [66] and can be found conveniently collected for example in refs. [67,68]. At NLO accuracy only c1is required which reads: c1(L) = −4 3−L. (4.3) At the order we are working, LMcan be replaced with LM= ln µ2 m(µ)2. The difference between using m(µ)or mtis of higher order in the coupling constant and does not contribute at the order we are working here. Using in addition m(m)instead of m(µ)leads to LM= 0. In refs. [9,10] the top-quark loops in the gluon self energy are subtracted at finite momentum. At NLO accuracy the strong coupling defined in this way is equivalent to α(nf−1) s(µ), i.e. αsrenormalized in the MS scheme with nf−1active flavors. Note that all quarks except the top one are treated as massless. As the difference between α(nf) s(µ) and α(nf−1) s(µ)is of order (α(nf−1) s(µ))2, one finally obtains mt=m(m)1−c1(0)a(nf−1)(m(m)) + . . . =m(m)1−c1(0)a(nf−1)(µ) + . . . .(4.4) Starting with the perturbative expansion of the cross section using the pole mass σ= (a(5)(µ))3σ(0)(mt)+(a(5)(µ))4σ(1)(mt) + . . . , (4.5) where we have set nf= 6, and using the above relation between mtand m(m), one obtains after expanding in αs σ=(a(5)(µ))3σ(0)(m(m))+(a(5)(µ))4 σ(1)(m(m))−c1(0) dσ(0)(mt) dmtmt=m(m) m(m) +.... (4.6) We note that this formula can be applied to the inclusive cross section but also to individual bins of differential distributions. The definition of the MSR mass [6,7] employs the fact that the MS mass is free of renormalon ambiguities and that the renormalon ambiguity does not depend on the mass of the heavy quark. Accordingly the MSR mass is defined through mMSR =mt−Rd1(0)a(nf)(R) + d2(0)(a(nf)(R))2+. . . ,(4.7) – 34 – JHEP05(2022)146 PDF sets •ABMP16 •CT18NLO •MMHT14 •MSHT20 •NNPDF3.1 Mass renormalization •pole mass scheme •MS mass scheme •MSRp mass scheme pcut ⊥ •pcut ⊥= 30 GeV •pcut ⊥= 50 GeV •pcut ⊥= 75 GeV •pcut ⊥= 100 GeV Pseudo rapidiy cut • |η|<2.4 • |η|<2.5 Figure 31. Overview of the different settings used as input in the calculation. with the expansion coefficients diimplicitly given by mt=m(m)1 + d1(0)a(nf)(m) + d2(0)(a(nf)(m))2+. . . . Note that the coefficients dican be expressed in terms of the ciintroduced above, e.g. d1(0) = −c1(0). Using again a(nf−1)(R)instead of a(nf)(R)and expressing the cross section in terms of the MSR mass leads to σ=(a(5)(µ))3σ(0)(mMSRp)+(a(5)(µ))4 σ(1)(mMSRp)−c1(0) dσ(0)(mt) dmtmt=mMSRp R +.... (4.8) We note that there are two variants of the MSR scheme, leading respectively to the natural MSR mass and the practical MSR mass, differing according to the way the transition from nfactive flavors to nf−1active flavors is done. Expressing α(nf) s(µ)in terms of α(nf−1) s(µ) corresponds to the practical MSR mass scheme. This is indicated by the additional “p” in the subscript of the MSR mass mMSRp. For the present application, the difference between the natural and the practical scheme is negligible, as has been shown in refs. [7,69,70]. Using eqs. (4.6) and (4.8) it is straightforward to convert the results using the pole mass scheme into the MS or MSR scheme. 4.3 Exemplary results for the Rdistribution using different parton distribution function sets and different renormalization schemes In this section we provide theoretical predictions using the setup described above. The results allow for top-quark mass determinations from the measured Rdistribution. Figure 31 gives an overview of the different settings used in the calculation. In addition, we have considered different bin choices. Here we only show selected results. Some reference cross sections are collected in the appendix Aand the complete set of predictions is available online at [71]. The first three panels of figure 32 show the Rdistribution using different schemes for renormalizing the top-quark mass: the on-shell scheme, the MS scheme, as well as the MSR scheme. In each panel, the Rdistribution is depicted for a central mass value together with the variation by ±5 GeV. We fix the renormalization and factorization scale to µF=µR=m, where mis the top-quark mass in the considered scheme. The central – 35 – JHEP05(2022)146 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0 0.5 1 1.5 2 2.5 3 3.5 ) t =m 0 µ, t (mR =1 R =K F =167 GeV, K t m =1 R =K F =172 GeV, K t m =1 R =K F =177 GeV, K t m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0 0.5 1 1.5 2 2.5 3 3.5 =m(m)) 0 µ(m(m)),R =1 R =K F m(m)=158 GeV, K =1 R =K F m(m)=163 GeV, K =1 R =K F m(m)=168 GeV, K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0 0.5 1 1.5 2 2.5 3 3.5 ) MSRp =m 0 µ, MSRp (mR =1 R =K F =167 GeV, K MSRp m =1 R =K F =172 GeV, K MSRp m =1 R =K F =177 GeV, K MSRp m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 ref R / R =172 GeV t m =172 GeV MSRp m m(m) =163 GeV Figure 32.Rdistribution using different renormalization schemes for the top-quark mass, i.e. the pole mass mt, the MS mass m(m)and the MSR mass mMSRp. Different lines correspond to different input masses, implying different choices for the renormalization and the factorization scales (assumed to be equal). In the last panel, ratios of predictions with the m(m)and mMSRp masses with respect to predictions with the on-shell mass are shown. The central CT18NLO PDF set is used as input in all panels. mass values in the three different schemes are chosen such that they roughly correspond to each other after doing the conversion at 4-loop accuracy. As we used a step size of 1GeV for the masses in the calculation, this correspondence is obviously not exact. Although small, the effect of mass variation on the Rdistribution is clearly visible within each of the three considered schemes. In the fourth plot in figure 32, we show the predictions using the MS and MSR masses divided by the predictions obtained in the pole mass scheme. The MSR masses were computed at a scale R= 3 GeV. Note that because of the aforementioned step size of 1GeV we do not expect perfect agreement. The predictions using either the pole mass or the MSR mass agree with each other. However, a sizeable difference is observed – 36 – JHEP05(2022)146 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 ref R / R =172 GeV t m =172 GeV MSRp m m(m) =165 GeV Figure 33. Ratio of Rin the MS and MSR scheme to Rref in the pole scheme, similar to the fourth plot in figure 32, however using m(m) = 165 GeV instead of 163 GeV. using the MS mass. This is a consequence of the large impact of the NNLO and N3LO corrections in the conversion formulas. Using an MS mass value of 165 GeV, which is close to the value obtained using the one-loop conversion formula, the agreement is much better, as expected. The corresponding ratio plot is shown in figure 33. In figure 34 we show the Rdistribution for different PDF choices divided by the results for the CT18NLO set. Apart from the first bin, where the differences are larger, the CT18NLO, MMHT14-nlo68cl, MSHT20nlo_as118 and NNPDF3.1_nlo_as_0118 central PDF sets give similar results at the level of a few percent. We note that in all results shown in figure 34 the αs(MZ)value and evolution is used as provided by the PDF set. The differences in the first bin which corresponds to high energetic events and is thus sensitive to the large xvalues of the PDFs is irrelevant for practical applications due to the small number of high energetic events. The ABMP16_5_nlo PDF set shows slightly larger deviations reaching up to 5% if one excludes the three highest energetic bins (ρs<0.3). In figure 35 we show the scale dependence of the Rdistribution using different renormalization schemes for the top-quark mass. We used µF=µR=µ0together with µ0=m,m/2,2m, where mdenotes the respective mass. For the pole mass and the MSR mass the behavior is very similar. A major difference is only observed in the highest energetic bin, where the MSR mass leads to a larger scale dependence of the Rdistribution. Again this is most likely of no practical relevance. In case of the MSR mass one could use a larger value for the R-scale bringing the MSR mass closer to the MS mass. However, the corresponding plot for the MS mass suggests that in this case the scale variation would become even larger. In the range 0.2< ρs<0.7the scale variation band using the MS mass is similar to the one obtained by working in the poleand the MSR-scheme. However, in the bin close to the threshold one obtains very large scale variation effects using the MS mass. The origin of this effect is the strong mass dependence in the threshold region which leads to large contribution when using eq. (4.6) to convert from the pole mass to the MS mass. Figure 36 is similar to figure 32 using however a step – 37 – JHEP05(2022)146 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.8 0.85 0.9 0.95 1 1.05 1.1 ref R / R CT18NLO ABMP16 MMHT14 NNPDF3.1 MSHT20 =1 R =K F , K t =m 0 µ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.8 0.85 0.9 0.95 1 1.05 1.1 ref R / R CT18NLO ABMP16 MMHT14 NNPDF3.1 MSHT20 =1 R =K F =m(m), K 0 µ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.8 0.85 0.9 0.95 1 1.05 1.1 ref R / R CT18NLO ABMP16 MMHT14 NNPDF3.1 MSHT20 =1 R =K F , K MSRp =m 0 µ Figure 34.Rdistribution using different PDF sets. We chose again µF=µR=µ0where µ0is set equal to the respective mass value. size of 1GeV instead of 5GeV. From the approximately constant step size one can see that in all bins the mass dependence is to good approximation linear, allowing a linear interpolation when performing a χ2fit to determine the top-quark mass from the measured distributions. 4.4 Off-shell effects and non-resonant/non-factorizable contributions As mentioned already, the results presented in previous sections can only be compared with data which have been unfolded to the parton level. The unfolding procedure accounts for effects due to the top-quark decay, additional gluon radiation from top-quark decay products as well as hadronization and experimental detector effects. Unfolding is a well established method and has been applied in the past to a variety of different observables. Obviously the unfolding procedure introduces additional uncertainties which need to be taken into account. In ref. [72] an alternative approach has been investigated. Instead of unfolding the measured data to the level of the intermediate top quarks, the observable is computed using – 38 – JHEP05(2022)146 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 ref R / R =1 R =K F , K t =m 0 µ =2 R =K F , K t =m 0 µ =0.5 R =K F , K t =m 0 µ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 ref R / R =1 R =K F =m(m), K 0 µ =2 R =K F =m(m), K 0 µ =0.5 R =K F =m(m), K 0 µ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 ref R / R =1 R =K F , K MSRp =m 0 µ =2 R =K F , K MSRp =m 0 µ =0.5 R =K F , K MSRp =m 0 µ Figure 35. Ratios of Rdistributions using different scale choices, i.e. µR=KRµ0,µF=KFµ0 with µ0=mand KF=KR= 0.5, 1, 2. Different panels refer to different renormalization schemes for the top-quark mass, i.e. the pole scheme (upper left), the MS scheme (upper right) and the MSR one (lower). the momenta of the final state particles in pp →e+νeµ−¯νµb¯ bj+Xevents, which is a typical final state for t¯ tj +Xassuming that both the top and antitop quarks decay leptonically. In principle, this allows to incorporate some of the aforementioned effects like the top-quark decay or radiation from top-quark decay products into the theoretical prediction instead of modeling them with Monte Carlo (MC) programs. In particular, spin effects like for example spin correlations are automatically taken into account and passed on to the top-quark decay products. To compare with the results of ref. [72] we consider parton level with decay of the top quarks instead of using the intermediate top quarks to define the final state. An expected advantage of using the results of ref. [72] is that unfolding uncertainties should be reduced as less modeling is introduced in the partonic description. The calculation in ref. [72] was performed at NLO accuracy in QCD keeping not only finite width – 39 – JHEP05(2022)146 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.7 0.8 0.9 1 1.1 1.2 1.3 ref R / R ∆ −=172 t m ∆ +=172 t m = 5 GeV∆ = 4 GeV∆ = 3 GeV∆ = 2 GeV∆ = 1 GeV∆ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.7 0.8 0.9 1 1.1 1.2 1.3 ref R / R ∆ −m(m)=163 ∆ +m(m)=163 = 5 GeV∆ = 4 GeV∆ = 3 GeV∆ = 2 GeV∆ = 1 GeV∆ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ρ 0.7 0.8 0.9 1 1.1 1.2 1.3 ref R / R ∆ −=172 MSRp m ∆ +=172 MSRp m = 5 GeV∆ = 4 GeV∆ = 3 GeV∆ = 2 GeV∆ = 1 GeV∆ Figure 36. Ratios of Rdistributions using as input different top-quark mass values and the Rdistribution obtained with a fixed central top-quark mass value (assumed to be equal to 173 GeV in all renormalization schemes). Different panels refer to different renormalization schemes for the top-quark mass. effects for the top quark but also the corresponding effects for the intermediate W-bosons. Furthermore, non-resonant/non-factorizable contributions were also included in the prediction. The theoretical description thus includes many physical effects. In the following we will refer to it as full calculation (Full NLO). However, this calculation has not yet been matched to a parton shower, therefore shower effects are not presently accounted for, and hadronization and further soft physics effects are also absent. In addition, two further approximations to model off-shell effects are studied in ref. [72]. First, the approximation in which the two decay chains pp →t¯ tj +X→e+νeµ−¯νµb¯ bj +X, pp →t¯ t→e+νeµ−¯νµb¯ bj +X are calculated in the narrow width approximation, called NWA in ref. [72]. The production – 40 – JHEP05(2022)146 extractions of the top-quark mass parameter using LHC Run 2 and future Run 3 data, while hoping for future theoretical and experimental developments which might further increase the accuracy of these extractions and decrease the associated systematic uncertainties, during the HL-LHC phase. Acknowledgments We would like to thank Malgorzata Worek for providing us with numerical input for the study of off-shell effects. We are grateful to Giuseppe Bevilacqua for further clarifications and to André Hoang, Katerina Lipka, Marcel Vos, and Sebastian Wuchterl for useful discussions. The work of S.A. and A.G. is supported by the ERC Starting Grant REINVENT714788. S.A. also acknowledges funding from Fondazione Cariplo and Regione Lombardia, grant 2017-2070 and from MIUR through the FARE grant R18ZRBEAFC. J.F. and A.I. acknowledge support from projects PGC2018-094856-B-100 (MICINN/FEDER), PROMETEO-2018/060 and CIDEGENT/2020/21 (Generalitat Valenciana), and the iLINK grant LINKB20065 (CSIC). The work of M.V.G., S.M. and P.U. was supported in part by the Bundesministerium für Bildung and Forschung under contracts 05H21GUCCA and 05H18KHCA1. A Numerical results for cross sections for different scales, top-quark masses and analysis cuts In the following we present predictions for cross sections for t¯ tj production in pp collisions at √S= 13 TeV at the LHC for future reference. In tables 3,4,5and 6, using as input mpole t= 172 GeV and the CT18 NLO PDF central set, we provide predictions for the ρs distribution under the default system of cuts described at the beginning of section 3, that is Nj≥1,pj T>30 GeV, |ηj|<2.4for the different central scale choices: µR=µF=µ0=mt, HB T/2,HB T/4and mB t¯ tj, as defined in section 2. Scale variation effects, computed by varying simultaneously µRand µFby a factor of two around the central value are also shown. Covering a range of top-quark mass values, a series of slightly different cuts pj T>30, 50,75 and 100 GeV along with |ηj|<2.4, and using as input the ABMP16_5_nlo PDF set, tables 7–14 provide further examples for reference cross sections, the complete set of which is collected in the online repository [71]. – 47 – JHEP05(2022)146 xlxrKR=KF= 0.5KR=KF= 1 KR=KF= 2 0.0 0.18 12.6±0.2 22.0±0.1 21.70±0.07 0.18 0.22 136±1 166.7±0.6 154.7±0.4 0.22 0.27 269±1 300.1±0.9 271.5±0.9 0.27 0.32 437±1 457.3±0.9 409.8±0.6 0.32 0.38 609±2 616±1 546.6±0.8 0.38 0.45 765±2 752±1 663.6±0.6 0.45 0.53 845±1 814±1 717±2 0.53 0.62 800±1 759.8±0.9 667.8±0.6 0.62 0.71 615±1 576.2±0.9 505.6±0.6 0.71 1.0 133.1±0.3 123.8±0.2 108.5±0.1 Table 3. NLO predictions for the ρsdistribution [pb] for t¯ tj +Xhadroproduction at √S=13 TeV using as input the CT18 NLO central PDF set, mt= 172 GeV and µ0=mt, for Nj≥1,pj T> 30 GeV, |ηj|<2.4. See section 3for more detail. xlxrKR=KF= 0.5KR=KF= 1 KR=KF= 2 0.0 0.18 23.91±0.09 21.10±0.06 17.38±0.05 0.18 0.22 171.9±0.8 152.4±0.5 126.7±0.3 0.22 0.27 306±1 269.9±0.8 226.6±0.4 0.27 0.32 464±1 413.9±0.7 348.6±0.5 0.32 0.38 626±2 558.7±0.9 473.5±0.6 0.38 0.45 760±1 684.9±0.8 584.5±0.6 0.45 0.53 828±2 749.5±0.8 643.4±0.9 0.53 0.62 7745±1 706.1±0.7 609.6±0.6 0.62 0.71 592±1 541.1±0.6 470.1±0.5 0.71 1.0 127.9±0.3 117.6±0.2 102.5±0.2 Table 4. Same as table 3, but for µ0=HB T/2. – 48 – JHEP05(2022)146 xlxrKR=KF= 0.5KR=KF= 1 KR=KF= 2 0.0 0.18 22.23±0.1 23.9±0.1 21.1±0.1 0.18 0.22 164±1 171.9±0.8 152.4±0.5 0.22 0.27 294±1 306±1 269.9±0.8 0.27 0.32 446±2 464±1 413.9±0.7 0.32 0.38 601±2 626±2 558.7±0.9 0.38 0.45 732±2 760±1 684.9±0.8 0.45 0.53 797±2 828±2 749.5±0.8 0.53 0.62 743±2 775±1 706.1±0.7 0.62 0.71 567±2 592±1 541.1±0.6 0.71 1.0 123.4±0.8 127.9±0.3 117.6±0.2 Table 5. Same as table 3, but for µ0=HB T/4. xlxrKR=KF= 0.5KR=KF= 1 KR=KF= 2 0.0 0.18 19.59±0.06 16.26±0.04 13.17±0.03 0.18 0.22 146.2±0.4 123.1±0.3 100.8±0.3 0.22 0.27 265.2±0.6 226.0±0.5 186.5±0.4 0.27 0.32 412.5±0.7 355.0±0.5 296.2±0.5 0.32 0.38 565.3±0.8 492.5±0.6 413.9±0.5 0.38 0.45 698±1 618.1±0.6 523.4±0.5 0.45 0.53 768±1 689.2±0.6 590.5±0.7 0.53 0.62 725.9±0.9 659.1±0.6 570.4±0.5 0.62 0.71 554.0±0.9 510.3±0.8 446.1±0.4 0.71 1.0 119.9±0.4 111.9±0.2 98.6±0.1 Table 6. Same as table 3, but for µ0=mB t¯ tj/2. – 49 – JHEP05(2022)146 dσt¯ tj+X dρsdistribution with Nj≥1, pT,j >30 GeV, |ηj|<2.4,KR=KF= 1 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 19.49(6) 19.32(6) 19.23(6) 18.91(6) 18.79(6) 18.70(6) 18.48(6) 18.16(5) 18.13(5) 17.97(5) 17.72(5) 17.64(5) 17.49(5) 0.18 0.22 159.1(6) 158.6(5) 156.4(5) 153.7(5) 152.7(5) 151.2(5) 149.6(5) 146.6(5) 146.0(5) 143.9(5) 142.1(4) 142.3(4) 139.6(4) 0.22 0.27 299.0(8) 295.4(7) 291.3(7) 287.4(7) 285.6(7) 281.2(7) 278.4(7) 274.0(7) 271.8(6) 269.0(6) 264.4(6) 262.4(6) 259.4(6) 0.27 0.32 475.0(1) 469.0(1) 461.0(1) 457.0(1) 452.0(1) 443.2(1) 439.6(1) 432.2(9) 426.1(9) 421.2(9) 414.9(9) 409.4(9) 401.9(9) 0.32 0.38 658.0(1) 650.0(1) 639.0(1) 632.0(1) 619.0(1) 612.0(1) 604.0(1) 596.0(1) 585.0(1) 576.0(1) 568.8(1) 559.7(1) 553.0(1) 0.38 0.45 830.0(1) 813.0(1) 803.0(1) 788.0(1) 776.0(1) 762.0(1) 749.0(1) 739.0(1) 728.0(1) 716.0(1) 703.0(1) 692.0(1) 679.0(1) 0.45 0.53 931.0(1) 914.0(1) 894.0(1) 878.0(1) 861.0(1) 843.0(1) 829.0(1) 811.0(1) 796.0(1) 781.0(1) 768.0(1) 750.0(1) 736.0(1) 0.53 0.62 899.0(1) 883.0(1) 858.0(1) 840.0(1) 823.0(1) 806.0(1) 783.0(1) 763.0(1) 746.0(1) 731.0(1) 711.0(1) 695.0(1) 678.0(1) 0.62 0.71 732.0(1) 709.0(1) 690.0(1) 667.0(1) 644.0(1) 625.0(1) 605.0(1) 586.0(1) 566.1(1) 546.6(1) 529.1(9) 511.9(9) 493.7(9) 0.71 1.00 200.0(3) 188.2(3) 176.5(3) 165.3(3) 155.2(3) 144.9(3) 135.4(3) 126.6(2) 118.3(2) 109.8(2) 103.0(2) 95.4(2) 88.8(2) dσt¯ tj+X dρsdistribution with Nj≥1, pT,j >30 GeV, |ηj|<2.4,KR=KF= 2 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 19.30(4) 19.24(4) 18.88(4) 18.67(4) 18.52(4) 18.36(4) 18.19(3) 18.01(3) 17.80(3) 17.58(3) 17.41(3) 17.32(3) 17.13(3) 0.18 0.22 149.4(3) 146.4(3) 145.4(3) 143.8(3) 142.0(3) 140.0(3) 138.3(3) 136.6(3) 135.2(3) 133.7(3) 131.6(3) 130.5(3) 128.6(3) 0.22 0.27 271.3(5) 267.4(5) 264.4(5) 260.4(5) 258.0(4) 254.3(4) 251.1(4) 248.5(4) 245.8(4) 242.4(4) 240.0(4) 235.7(4) 232.8(4) 0.27 0.32 424.4(7) 418.4(7) 411.7(7) 407.7(7) 401.0(6) 395.1(7) 389.8(6) 383.6(6) 380.1(6) 374.7(6) 369.3(6) 364.2(6) 359.7(6) 0.32 0.38 584.2(8) 575.0(8) 564.6(8) 556.4(8) 548.0(7) 540.0(7) 532.2(7) 524.3(7) 517.6(7) 510.0(7) 501.3(7) 494.4(6) 487.6(6) 0.38 0.45 729.0(8) 717.1(8) 703.8(8) 693.8(8) 680.9(8) 669.6(8) 659.3(8) 649.2(8) 637.6(7) 627.1(7) 615.7(7) 606.6(7) 596.3(7) 0.45 0.53 812.8(9) 796.3(9) 782.0(9) 767.1(9) 752.1(8) 740.2(8) 723.9(8) 710.6(8) 696.5(8) 683.7(8) 670.1(7) 657.6(7) 645.1(7) 0.53 0.62 789.9(9) 770.3(9) 752.3(9) 735.4(9) 717.9(8) 700.5(8) 683.0(8) 669.9(8) 652.9(7) 637.1(7) 622.8(7) 606.5(7) 592.1(7) 0.62 0.71 636.7(8) 618.8(8) 598.9(8) 581.7(8) 561.3(7) 546.1(7) 527.9(7) 508.9(7) 492.6(7) 477.1(6) 460.4(6) 445.8(6) 430.7(6) 0.71 1.00 174.0(2) 163.0(2) 153.6(2) 143.8(2) 134.8(2) 125.7(2) 117.8(2) 110.2(2) 103.1(2) 95.9(2) 89.8(1) 83.0(1) 77.1(1) dσt¯ tj+X dρsdistribution with Nj≥1, pT,j >30 GeV, |ηj|<2.4,KR=KF= 0.5 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 10.5(1) 10.9(1) 10.7(1) 10.5(1) 10.4(1) 10.7(1) 10.7(1) 10.6(1) 10.58(9) 10.59(9) 10.59(9) 10.53(9) 10.38(9) 0.18 0.22 125.9(9) 124.6(9) 124.1(9) 122.9(8) 123.3(9) 120.7(9) 121.3(8) 117.1(9) 118.1(8) 115.9(8) 116.0(7) 115.3(8) 114.8(8) 0.22 0.27 265.0(1) 260.0(1) 259.0(1) 258.0(1) 255.0(1) 252.0(1) 246.0(1) 246.0(1) 242.0(1) 240.0(1) 237.0(1) 236.0(1) 232.6(1) 0.27 0.32 446.0(2) 443.0(2) 439.0(2) 434.0(2) 423.0(2) 422.0(2) 418.0(2) 410.0(2) 405.0(2) 402.0(2) 397.0(2) 388.0(1) 387.0(1) 0.32 0.38 646.0(2) 643.0(2) 636.0(2) 623.0(2) 615.0(2) 610.0(2) 596.0(2) 588.0(2) 584.0(2) 569.0(2) 565.0(2) 554.0(2) 548.0(2) 0.38 0.45 846.0(2) 829.0(2) 815.0(2) 805.0(2) 789.0(2) 779.0(2) 764.0(2) 755.0(2) 738.0(2) 727.0(2) 715.0(2) 707.0(2) 691.0(2) 0.45 0.53 966.0(2) 946.0(2) 933.0(2) 913.0(2) 896.0(2) 875.0(2) 862.0(2) 843.0(2) 828.0(2) 811.0(2) 799.0(2) 782.0(2) 766.0(2) 0.53 0.62 955.0(2) 933.0(2) 910.0(2) 889.0(2) 866.0(2) 844.0(2) 826.0(2) 809.0(2) 789.0(2) 770.0(2) 751.0(2) 735.0(2) 719.0(2) 0.62 0.71 780.0(2) 757.0(2) 733.0(2) 710.0(2) 688.0(2) 664.0(2) 643.0(2) 626.0(2) 604.0(2) 586.0(2) 564.0(2) 544.0(1) 528.0(1) 0.71 1.00 215.2(6) 203.0(5) 192.2(5) 178.3(5) 167.9(5) 157.8(5) 146.3(4) 137.0(4) 128.3(4) 119.2(4) 111.4(4) 103.0(3) 96.3(3) Table 7. NLO predictions for the ρsdistribution in the pole mass scheme for √S= 13 TeV using as input the static scale µ0=mtand the ABMP16 NLO PDF set. At least one jet with |ηj|<2.4and pj T>30 GeV is required. – 50 – JHEP05(2022)146 dσt¯ tj+X dρsdistribution with Nj≥1, pj T>50 GeV, |ηj|<2.4,KR=KF= 1 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 19.23(6) 19.09(6) 18.94(5) 18.63(6) 18.52(6) 18.42(5) 18.19(5) 17.89(5) 17.86(5) 17.67(5) 17.46(5) 17.38(5) 17.22(5) 0.18 0.22 152.3(5) 152.1(5) 149.7(4) 147.3(5) 146.3(5) 145.0(4) 143.4(4) 139.9(4) 139.6(4) 137.5(4) 136.7(4) 135.9(4) 133.6(4) 0.22 0.27 276.7(7) 273.0(6) 268.6(6) 265.6(6) 263.1(6) 259.1(6) 255.7(6) 253.3(6) 249.9(5) 247.4(5) 244.0(5) 240.7(5) 237.7(5) 0.27 0.32 420.1(9) 412.0(9) 405.6(8) 401.0(9) 396.9(8) 388.2(8) 386.0(8) 378.3(8) 373.8(7) 368.8(7) 362.2(7) 357.7(7) 350.9(7) 0.32 0.38 547.6(1) 541.6(1) 532.0(9) 525.8(9) 514.7(9) 508.5(9) 499.1(8) 494.1(8) 483.8(8) 474.8(8) 469.0(8) 462.2(8) 456.1(7) 0.38 0.45 647.5(1) 634.7(1) 623.2(9) 612.8(9) 601.0(9) 591.3(9) 579.8(8) 570.5(8) 560.7(8) 551.9(8) 541.3(7) 531.9(8) 522.6(8) 0.45 0.53 666.3(9) 653.1(9) 638.4(9) 624.0(9) 612.5(9) 597.9(8) 586.8(8) 571.3(8) 561.9(8) 550.7(8) 538.9(7) 527.1(7) 516.6(7) 0.53 0.62 573.9(9) 561.5(9) 546.1(8) 531.4(8) 519.5(7) 504.3(8) 491.6(7) 477.7(7) 465.2(7) 454.9(7) 440.1(6) 428.3(7) 415.8(6) 0.62 0.71 393.1(7) 379.4(7) 366.0(7) 353.0(7) 337.4(6) 325.8(6) 312.2(6) 301.3(6) 290.5(6) 276.9(6) 265.5(5) 255.4(5) 244.4(5) 0.71 1.00 66.2(2) 61.5(2) 56.8(1) 52.4(1) 48.7(1) 44.7(1) 41.3(1) 37.7(1) 34.7(1) 31.7(1) 29.28(9) 26.71(9) 24.27(8) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>50 GeV, |ηj|<2.4,KR=KF= 2 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 18.83(4) 18.77(4) 18.41(4) 18.22(4) 18.09(3) 17.93(3) 17.74(3) 17.53(3) 17.36(3) 17.16(3) 16.96(3) 16.89(3) 16.66(3) 0.18 0.22 142.0(3) 139.0(3) 138.0(3) 136.7(3) 134.6(3) 133.1(3) 131.0(3) 129.3(3) 128.2(3) 126.8(3) 124.7(3) 123.6(3) 122.0(3) 0.22 0.27 248.4(4) 245.0(4) 242.1(4) 238.3(4) 235.9(4) 233.1(4) 229.9(4) 227.0(4) 223.9(3) 221.0(4) 218.0(4) 215.1(4) 212.2(3) 0.27 0.32 370.6(6) 365.5(6) 359.0(6) 355.4(6) 348.9(5) 343.8(5) 339.0(5) 333.9(5) 329.8(5) 325.0(5) 320.4(5) 316.2(5) 312.0(4) 0.32 0.38 482.6(6) 473.6(6) 466.6(6) 459.3(6) 451.7(5) 444.4(5) 438.5(5) 431.2(5) 424.8(5) 418.3(5) 411.8(5) 405.3(5) 397.9(5) 0.38 0.45 562.1(6) 553.4(6) 544.0(6) 534.0(6) 523.5(6) 514.5(5) 505.7(6) 497.0(5) 487.3(5) 479.2(5) 469.5(5) 461.7(5) 453.5(5) 0.45 0.53 577.4(6) 565.8(6) 552.3(6) 541.8(6) 530.8(5) 519.7(5) 508.1(5) 497.9(5) 487.6(5) 476.7(5) 467.5(5) 457.0(5) 447.2(5) 0.53 0.62 499.3(6) 485.6(5) 473.4(5) 461.5(5) 449.0(5) 436.3(5) 423.8(5) 414.4(5) 401.7(5) 391.3(4) 380.0(4) 370.7(4) 360.3(4) 0.62 0.71 340.3(5) 327.7(5) 316.0(5) 303.2(4) 291.5(4) 281.0(4) 269.9(4) 259.1(4) 249.1(4) 239.5(4) 229.2(3) 219.8(3) 211.8(3) 0.71 1.00 57.0(1) 52.7(1) 48.8(1) 45.12(9) 41.78(8) 38.57(8) 35.36(8) 32.54(7) 29.92(7) 27.41(7) 25.07(6) 22.92(6) 20.79(6) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>50 GeV, |ηj|<2.4,KR=KF= 0.5 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 11.1(1) 11.3(1) 11.2(1) 10.99(9) 10.98(9) 11.24(9) 11.03(9) 10.96(9) 11.02(9) 10.99(8) 11.04(8) 10.89(8) 10.85(8) 0.18 0.22 124.1(9) 123.6(8) 122.9(8) 121.5(8) 121.1(8) 119.9(8) 119.7(7) 115.6(8) 116.9(7) 114.7(7) 114.4(7) 113.8(7) 113.2(7) 0.22 0.27 252.0(1) 247.0(1) 244.0(1) 244.3(1) 240.0(1) 238.0(1) 232.7(1) 233.0(1) 229.1(1) 227.0(9) 223.6(9) 222.8(9) 220.3(8) 0.27 0.32 403.0(1) 396.0(2) 395.0(1) 390.0(1) 381.0(1) 377.0(1) 376.0(1) 367.0(1) 363.0(1) 360.0(1) 355.0(1) 347.0(1) 347.0(1) 0.32 0.38 553.0(2) 546.0(2) 538.0(2) 526.0(1) 522.0(2) 515.0(1) 501.0(1) 499.0(1) 493.0(1) 481.0(1) 477.0(1) 470.0(1) 460.0(1) 0.38 0.45 670.0(2) 656.0(2) 644.0(2) 636.0(1) 623.0(2) 616.0(1) 604.0(1) 593.0(1) 581.0(1) 575.0(1) 561.0(1) 554.0(1) 542.0(1) 0.45 0.53 705.0(2) 691.0(2) 677.0(2) 662.0(1) 645.0(1) 634.0(1) 621.0(1) 607.0(1) 596.0(1) 584.0(1) 574.0(1) 560.0(1) 549.0(1) 0.53 0.62 623.0(1) 607.0(1) 588.0(1) 572.0(1) 560.0(1) 542.0(1) 528.0(1) 515.0(1) 498.0(1) 489.0(1) 476.0(1) 463.0(1) 452.0(1) 0.62 0.71 430.0(1) 413.0(1) 400.0(1) 385.0(1) 367.0(1) 356.0(1) 341.9(1) 329.3(1) 315.1(1) 305.9(9) 289.2(9) 279.8(8) 268.2(8) 0.71 1.00 73.1(3) 67.7(2) 63.0(2) 58.1(2) 54.0(2) 49.7(2) 45.8(2) 42.2(2) 38.9(2) 35.5(2) 32.7(1) 29.7(1) 27.1(1) Table 8. Same as 7but for pj T>50 GeV. – 51 – JHEP05(2022)146 dσt¯ tj+X dρsdistribution with Nj≥1, pj T>75 GeV, |ηj|<2.4,KR=KF= 1 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 18.69(6) 18.48(6) 18.39(5) 18.13(5) 17.98(5) 17.87(5) 17.65(5) 17.38(4) 17.37(5) 17.16(5) 16.98(4) 16.87(4) 16.74(5) 0.18 0.22 142.0(5) 141.3(4) 139.4(4) 137.4(4) 135.5(4) 134.7(4) 133.1(4) 130.4(4) 129.7(4) 127.5(4) 126.8(4) 125.7(4) 123.7(4) 0.22 0.27 245.7(5) 242.3(5) 239.0(5) 234.9(5) 232.5(5) 229.4(5) 226.3(5) 223.9(5) 220.7(5) 218.4(4) 215.0(4) 212.5(4) 209.7(5) 0.27 0.32 351.3(7) 345.6(7) 341.1(6) 336.5(7) 332.9(7) 325.9(6) 323.1(6) 317.5(6) 312.9(6) 308.2(6) 304.3(6) 299.2(6) 294.2(6) 0.32 0.38 435.7(8) 429.4(7) 422.4(7) 417.0(7) 407.6(7) 402.3(6) 396.1(6) 390.1(6) 382.2(6) 376.9(6) 369.6(6) 364.4(6) 359.4(6) 0.38 0.45 478.6(7) 469.1(7) 459.0(7) 452.0(7) 443.3(7) 434.6(6) 427.2(6) 419.6(6) 411.7(6) 403.4(6) 396.2(6) 388.7(6) 382.2(6) 0.45 0.53 449.1(7) 438.3(7) 428.6(6) 419.4(7) 410.1(6) 399.4(6) 392.0(6) 381.6(5) 373.3(5) 364.6(5) 356.7(5) 347.7(5) 339.7(5) 0.53 0.62 337.1(6) 328.1(6) 316.9(5) 307.5(5) 299.4(5) 289.6(5) 280.9(5) 271.6(4) 262.7(4) 255.1(4) 246.8(4) 239.6(4) 231.1(4) 0.62 0.71 179.7(4) 171.5(4) 163.0(4) 155.5(4) 147.8(4) 140.6(3) 133.5(3) 127.0(3) 121.0(3) 114.1(3) 107.6(3) 102.3(3) 96.6(3) 0.71 1.00 14.44(6) 12.98(6) 11.64(5) 10.37(5) 9.28(5) 8.29(4) 7.39(4) 6.58(4) 5.83(4) 5.10(3) 4.46(3) 3.87(3) 3.34(3) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>75 GeV, |ηj|<2.4,KR=KF= 2 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 18.22(4) 18.14(3) 17.79(3) 17.64(3) 17.49(3) 17.27(3) 17.12(3) 16.94(3) 16.75(3) 16.58(3) 16.39(3) 16.29(3) 16.09(3) 0.18 0.22 131.7(3) 128.8(3) 128.0(3) 126.8(3) 125.0(3) 123.1(2) 121.3(2) 119.5(2) 118.7(2) 117.7(2) 115.2(2) 114.2(2) 113.1(2) 0.22 0.27 219.7(3) 217.0(3) 213.9(3) 210.7(3) 208.2(3) 205.6(3) 203.0(3) 200.1(3) 197.1(3) 195.2(3) 192.5(3) 189.6(3) 187.3(3) 0.27 0.32 311.7(5) 306.2(5) 301.8(4) 298.3(5) 293.0(4) 288.7(4) 284.2(4) 279.6(4) 275.9(4) 271.6(4) 267.4(4) 264.0(4) 260.4(4) 0.32 0.38 382.7(5) 376.5(5) 370.1(5) 363.6(5) 357.6(4) 351.7(4) 346.0(4) 340.4(4) 335.3(4) 329.6(4) 324.3(4) 318.6(4) 312.9(4) 0.38 0.45 415.6(5) 409.0(4) 400.0(4) 393.3(4) 385.2(4) 378.0(4) 371.9(4) 364.9(4) 356.7(4) 350.8(4) 343.6(4) 337.4(4) 331.4(4) 0.45 0.53 388.5(4) 379.9(4) 370.2(4) 362.0(4) 354.2(4) 346.3(4) 338.1(4) 330.1(4) 323.1(4) 315.1(4) 308.0(3) 300.5(3) 294.3(3) 0.53 0.62 291.5(4) 282.7(4) 274.1(4) 265.9(4) 257.9(3) 250.3(3) 241.4(3) 234.4(3) 226.7(3) 220.1(3) 212.4(3) 206.3(3) 199.2(3) 0.62 0.71 154.4(3) 147.2(3) 139.9(3) 133.6(3) 126.7(2) 120.8(2) 114.8(2) 109.0(2) 103.2(2) 97.9(2) 92.8(2) 87.7(2) 82.8(2) 0.71 1.00 12.27(4) 11.01(4) 9.89(4) 8.88(4) 7.94(3) 6.97(3) 6.25(3) 5.51(3) 4.90(2) 4.32(2) 3.77(2) 3.28(2) 2.78(2) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>75 GeV, |ηj|<2.4,KR=KF= 0.5 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 11.1(1) 11.31(9) 11.12(9) 10.94(8) 10.97(9) 11.13(9) 10.99(8) 10.91(9) 10.93(8) 10.94(8) 10.98(7) 10.77(8) 10.75(8) 0.18 0.22 116.7(8) 115.9(7) 115.3(7) 113.7(7) 114.8(7) 111.8(7) 112.5(7) 109.1(7) 109.3(6) 108.1(6) 107.1(6) 106.3(6) 106.1(6) 0.22 0.27 222.4(1) 220.2(9) 217.2(9) 216.7(9) 215.3(9) 211.1(9) 208.0(8) 206.6(8) 204.6(8) 203.0(8) 199.0(8) 198.1(7) 195.2(7) 0.27 0.32 341.0(1) 334.0(1) 332.0(1) 327.0(1) 321.0(1) 317.0(1) 315.0(1) 310.0(1) 306.0(1) 304.4(1) 299.7(1) 292.1(9) 291.3(9) 0.32 0.38 440.0(1) 434.0(1) 428.0(1) 419.0(1) 413.0(1) 409.0(1) 400.0(1) 394.0(1) 392.0(1) 381.0(1) 378.0(1) 371.3(9) 365.2(9) 0.38 0.45 495.0(1) 487.0(1) 479.0(1) 471.0(1) 463.0(1) 456.0(1) 446.0(1) 438.0(1) 428.0(1) 423.1(1) 413.5(1) 407.1(9) 399.1(9) 0.45 0.53 478.0(1) 469.0(1) 455.0(1) 449.0(1) 435.0(1) 424.0(1) 417.0(1) 405.8(1) 397.6(9) 389.4(9) 383.5(9) 371.9(8) 363.9(8) 0.53 0.62 366.6(1) 356.7(9) 344.0(9) 332.4(8) 325.1(9) 314.0(8) 304.3(8) 296.0(8) 287.1(8) 278.6(7) 269.7(7) 261.3(7) 252.5(7) 0.62 0.71 197.2(7) 187.9(7) 181.0(7) 172.5(6) 164.2(6) 156.2(6) 147.5(6) 140.0(5) 132.1(5) 126.1(5) 119.6(5) 112.6(4) 107.2(4) 0.71 1.00 16.2(1) 14.7(1) 12.91(9) 11.87(8) 10.67(8) 9.47(7) 8.59(7) 7.60(6) 6.56(6) 5.76(5) 5.13(5) 4.45(4) 3.81(4) Table 9. Same as 7but for pj T>75 GeV. – 52 – JHEP05(2022)146 dσt¯ tj+X dρsdistribution with Nj≥1, pj T>100 GeV, |ηj|<2.4,KR=KF= 1 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 17.87(5) 17.62(5) 17.61(5) 17.33(5) 17.13(5) 17.14(5) 16.87(5) 16.62(4) 16.57(4) 16.43(4) 16.27(4) 16.13(4) 16.03(4) 0.18 0.22 128.7(4) 128.7(4) 126.6(4) 124.6(4) 123.8(4) 122.8(4) 121.2(4) 118.9(3) 117.8(3) 116.0(3) 115.5(3) 114.4(3) 112.5(3) 0.22 0.27 214.3(5) 211.6(5) 208.2(4) 205.2(5) 202.5(4) 199.7(4) 196.8(4) 194.6(4) 192.2(4) 189.7(4) 187.5(4) 184.7(4) 182.3(4) 0.27 0.32 293.0(6) 287.3(6) 284.0(5) 280.5(6) 276.6(5) 272.1(5) 268.5(5) 264.2(5) 259.8(5) 255.7(5) 252.5(5) 248.7(5) 244.0(5) 0.32 0.38 346.4(6) 340.2(6) 333.7(6) 329.8(6) 323.0(5) 317.9(5) 312.8(5) 308.8(5) 302.4(5) 297.7(5) 291.6(5) 287.7(5) 283.2(5) 0.38 0.45 356.1(6) 348.3(6) 342.1(5) 335.2(5) 328.7(5) 322.7(5) 315.8(5) 308.7(5) 303.2(5) 298.3(4) 292.0(4) 285.9(5) 281.1(4) 0.45 0.53 303.7(5) 296.3(5) 290.2(5) 281.8(5) 275.4(5) 267.7(4) 261.7(4) 255.4(4) 247.6(4) 241.9(4) 235.9(4) 229.9(4) 224.9(4) 0.53 0.62 194.3(4) 187.5(4) 180.2(4) 174.5(4) 168.0(3) 162.4(3) 156.0(3) 151.2(3) 144.9(3) 140.0(3) 134.4(3) 129.5(3) 124.5(3) 0.62 0.71 71.1(2) 66.8(2) 62.4(2) 58.2(2) 54.6(2) 50.7(2) 47.7(2) 44.4(2) 41.3(2) 38.1(2) 35.3(1) 32.8(1) 29.9(1) 0.71 1.00 1.73(2) 1.51(2) 1.27(1) 1.03(1) 0.87(1) 0.69(1) 0.560(8) 0.458(7) 0.356(6) 0.281(5) 0.215(4) 0.161(3) 0.116(2) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>100 GeV, |ηj|<2.4,KR=KF= 2 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 17.44(3) 17.38(3) 17.03(3) 16.90(3) 16.72(3) 16.53(3) 16.41(3) 16.23(3) 16.05(3) 15.87(3) 15.70(3) 15.56(3) 15.37(3) 0.18 0.22 120.0(2) 117.8(2) 116.6(3) 115.7(2) 114.2(2) 112.3(2) 110.6(2) 109.2(2) 108.6(2) 107.3(2) 105.4(2) 104.3(2) 103.0(2) 0.22 0.27 192.4(3) 189.6(3) 187.1(3) 184.5(3) 181.5(3) 179.8(3) 177.5(3) 174.7(3) 172.2(3) 170.2(3) 168.0(3) 165.4(2) 163.4(2) 0.27 0.32 260.5(4) 255.9(4) 252.3(4) 249.5(4) 244.4(3) 240.9(3) 237.4(3) 233.5(3) 229.9(3) 226.7(3) 222.7(3) 220.1(3) 216.9(3) 0.32 0.38 305.0(4) 299.5(4) 294.1(4) 288.5(4) 284.1(3) 278.7(3) 274.8(3) 270.0(3) 265.8(3) 260.7(3) 257.0(3) 252.0(3) 246.8(3) 0.38 0.45 310.2(4) 304.5(3) 298.0(3) 292.2(3) 286.3(3) 280.6(3) 275.3(3) 270.0(3) 263.8(3) 259.2(3) 253.3(3) 248.8(3) 244.2(3) 0.45 0.53 263.5(3) 256.9(3) 249.7(3) 244.6(3) 238.1(3) 232.4(3) 226.4(3) 220.5(3) 215.2(3) 209.4(3) 204.5(3) 199.2(2) 194.0(2) 0.53 0.62 167.2(3) 161.3(2) 155.9(2) 150.4(2) 145.0(2) 139.9(2) 134.3(2) 130.0(2) 124.7(2) 120.5(2) 115.8(2) 111.6(2) 107.3(2) 0.62 0.71 60.7(2) 57.2(2) 53.7(1) 50.3(1) 46.5(1) 43.6(1) 40.8(1) 37.9(1) 35.2(1) 32.5(1) 30.1(1) 27.75(9) 25.73(9) 0.71 1.00 1.42(1) 1.22(1) 0.99(1) 0.836(9) 0.689(7) 0.554(7) 0.443(6) 0.355(5) 0.277(4) 0.215(3) 0.153(3) 0.116(2) 0.083(2) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>100 GeV, |ηj|<2.4,KR=KF= 0.5 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 10.67(9) 10.79(9) 10.69(9) 10.36(8) 10.45(8) 10.65(8) 10.60(8) 10.43(8) 10.48(8) 10.48(7) 10.50(7) 10.37(7) 10.35(7) 0.18 0.22 105.5(7) 104.8(6) 104.3(6) 103.7(6) 103.7(6) 101.3(6) 101.8(6) 98.7(6) 98.9(6) 97.9(6) 97.7(5) 96.8(5) 96.2(5) 0.22 0.27 192.6(8) 190.7(8) 187.9(8) 188.3(7) 185.9(8) 182.4(7) 179.7(7) 180.1(7) 177.2(7) 176.2(7) 172.2(7) 171.6(6) 168.8(6) 0.27 0.32 283.0(1) 277.7(1) 273.6(1) 270.6(9) 264.8(9) 262.6(9) 259.6(9) 256.5(9) 253.5(9) 249.9(8) 248.7(8) 241.3(8) 241.1(8) 0.32 0.38 347.6(1) 341.9(1) 336.7(1) 329.0(9) 325.2(1) 322.2(9) 315.2(8) 310.1(8) 307.1(9) 299.6(9) 296.1(8) 292.4(7) 286.4(8) 0.38 0.45 368.3(1) 360.3(9) 356.8(9) 348.0(9) 342.8(9) 336.3(8) 329.1(8) 322.8(8) 316.7(8) 310.1(8) 303.8(8) 300.3(7) 294.0(7) 0.45 0.53 323.3(8) 315.7(8) 308.4(8) 301.9(8) 291.5(8) 286.0(8) 279.8(7) 271.4(7) 264.3(7) 259.0(7) 254.0(7) 245.9(6) 239.5(6) 0.53 0.62 210.4(7) 203.4(6) 196.5(6) 189.0(6) 183.1(6) 176.3(6) 170.6(5) 164.6(6) 158.2(5) 152.0(5) 147.6(5) 142.0(5) 136.1(4) 0.62 0.71 78.3(4) 73.9(4) 69.4(3) 65.6(3) 61.0(3) 56.8(3) 53.0(3) 48.4(3) 46.2(3) 42.7(3) 39.2(2) 36.3(2) 34.0(2) 0.71 1.00 2.17(3) 1.83(3) 1.49(2) 1.31(2) 1.09(2) 0.89(2) 0.75(1) 0.58(1) 0.484(9) 0.366(8) 0.293(6) 0.233(5) 0.182(4) Table 10. Same as 7but for pj T>100 GeV. – 53 – JHEP05(2022)146 dσt¯ tj+X dρsdistribution with Nj≥1, pj T>30 GeV, |ηj|<2.4,KR=KF= 1 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 26.70(8) 26.61(8) 26.24(7) 26.18(8) 25.77(8) 25.57(8) 25.25(7) 25.09(7) 24.94(7) 24.68(7) 24.50(7) 24.39(6) 24.03(6) 0.18 0.22 208.5(7) 206.6(7) 203.9(6) 201.7(7) 199.9(6) 197.8(7) 195.0(6) 192.9(6) 190.5(6) 189.5(6) 187.3(6) 185.5(6) 182.5(5) 0.22 0.27 381.3(9) 376.4(9) 371.6(9) 366.9(1) 362.8(9) 358.3(9) 353.8(9) 349.6(8) 344.2(8) 340.5(8) 335.0(8) 332.5(8) 327.8(8) 0.27 0.32 593.0(1) 582.0(1) 575.0(1) 569.0(1) 560.0(1) 551.0(1) 544.0(1) 539.0(1) 530.0(1) 522.0(1) 516.0(1) 512.0(1) 505.0(1) 0.32 0.38 807.0(1) 799.0(1) 786.0(1) 775.0(2) 762.0(1) 750.0(1) 740.0(1) 727.0(1) 716.0(1) 709.0(1) 698.0(1) 687.0(1) 676.0(1) 0.38 0.45 1002.0(2) 984.0(2) 968.0(1) 951.0(2) 936.0(2) 922.0(1) 907.0(1) 890.0(1) 876.0(1) 860.0(1) 849.0(1) 832.0(1) 820.0(1) 0.45 0.53 1110.0(2) 1083.0(2) 1064.0(2) 1044.0(2) 1025.0(2) 1003.0(2) 983.0(2) 969.0(1) 948.0(1) 929.0(1) 913.0(1) 896.0(1) 880.0(1) 0.53 0.62 1063.0(2) 1040.0(2) 1017.0(1) 993.0(2) 968.0(1) 947.0(1) 925.0(1) 901.0(1) 881.0(1) 862.0(1) 839.0(1) 819.0(1) 801.0(1) 0.62 0.71 853.0(1) 831.0(1) 804.0(1) 780.0(1) 754.0(1) 731.0(1) 707.0(1) 685.0(1) 664.0(1) 640.0(1) 617.0(1) 598.0(1) 577.1(1) 0.71 1.00 232.0(4) 218.9(4) 204.1(3) 192.0(4) 179.0(3) 168.4(3) 156.8(3) 146.7(3) 137.1(3) 127.8(3) 119.2(2) 111.3(2) 103.0(2) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>30 GeV, |ηj|<2.4,KR=KF= 2 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 26.63(5) 26.29(5) 25.96(5) 25.77(5) 25.63(5) 25.23(5) 25.08(5) 24.79(4) 24.35(5) 24.28(4) 24.06(4) 23.78(4) 23.53(4) 0.18 0.22 194.1(4) 191.4(4) 188.9(4) 187.6(4) 184.7(4) 182.6(4) 181.3(4) 177.8(4) 176.9(4) 174.0(4) 171.6(4) 169.6(4) 168.2(4) 0.22 0.27 345.6(6) 341.2(6) 336.0(6) 333.1(6) 329.4(6) 324.0(6) 319.3(6) 316.5(5) 311.9(5) 306.5(5) 304.0(5) 299.9(5) 296.6(5) 0.27 0.32 530.5(9) 521.4(9) 514.7(8) 507.1(9) 498.5(9) 493.2(8) 487.4(8) 479.2(8) 475.3(8) 467.5(7) 460.5(7) 456.3(7) 447.3(7) 0.32 0.38 715.1(1) 705.9(1) 695.4(1) 683.9(1) 676.7(9) 664.2(9) 651.3(9) 644.0(9) 633.1(9) 626.4(8) 615.9(8) 607.5(8) 599.0(8) 0.38 0.45 882.0(1) 867.0(1) 852.6(1) 837.0(1) 823.8(9) 813.3(9) 799.5(9) 783.5(9) 773.0(9) 758.0(8) 746.0(9) 734.0(8) 724.5(8) 0.45 0.53 973.0(1) 953.0(1) 937.0(1) 919.0(1) 903.0(1) 882.8(9) 866.8(9) 849.4(9) 831.9(9) 816.8(8) 802.1(9) 785.6(9) 769.5(8) 0.53 0.62 932.0(1) 912.0(1) 891.4(1) 869.1(1) 849.1(9) 830.8(9) 809.8(9) 790.0(9) 771.2(9) 754.5(8) 735.2(8) 718.8(8) 701.4(7) 0.62 0.71 750.6(1) 727.0(9) 704.0(9) 682.6(9) 658.4(9) 639.4(9) 619.2(8) 598.5(8) 579.0(8) 560.4(7) 541.8(7) 524.8(7) 506.4(7) 0.71 1.00 202.6(3) 190.5(2) 178.3(2) 167.4(2) 156.8(2) 147.1(2) 137.4(2) 128.5(2) 120.1(2) 111.9(2) 104.0(2) 96.9(2) 90.1(1) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>30 GeV, |ηj|<2.4,KR=KF= 0.5 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 15.0(1) 14.8(1) 14.8(1) 15.0(1) 15.0(1) 14.7(1) 14.7(1) 14.8(1) 14.8(1) 14.5(1) 14.6(1) 14.7(1) 14.5(1) 0.18 0.22 165.0(1) 163.0(1) 163.0(1) 164.0(1) 159.0(1) 159.0(1) 159.0(1) 155.0(1) 155.0(1) 153.9(9) 153.3(1) 151.4(9) 149.5(9) 0.22 0.27 338.0(2) 332.0(2) 331.0(2) 328.0(2) 325.0(2) 322.0(1) 318.0(1) 312.0(1) 310.0(1) 306.0(1) 305.0(1) 300.0(1) 298.0(1) 0.27 0.32 560.0(2) 555.0(2) 548.0(2) 535.0(2) 530.0(2) 522.0(2) 518.0(2) 514.0(2) 506.0(2) 503.0(2) 495.0(2) 486.0(2) 485.0(2) 0.32 0.38 798.0(3) 787.0(3) 775.0(3) 764.0(3) 754.0(2) 743.0(2) 734.0(2) 721.0(2) 712.0(2) 696.0(2) 695.0(2) 684.0(2) 672.0(2) 0.38 0.45 1017.0(3) 998.0(3) 981.0(3) 968.0(3) 952.0(3) 938.0(2) 922.0(2) 912.0(2) 893.0(2) 879.0(2) 860.0(2) 853.0(2) 836.0(2) 0.45 0.53 1151.0(3) 1129.0(3) 1105.0(3) 1086.0(3) 1061.0(3) 1046.0(2) 1025.0(3) 1005.0(2) 984.0(2) 972.0(2) 949.0(2) 935.0(2) 910.0(2) 0.53 0.62 1119.0(3) 1095.0(3) 1073.0(3) 1045.0(2) 1019.0(2) 996.0(2) 972.0(2) 952.0(2) 928.0(2) 907.0(2) 883.0(2) 859.0(2) 845.0(2) 0.62 0.71 912.0(2) 886.0(2) 858.0(2) 827.0(2) 803.0(2) 776.0(2) 751.0(2) 728.0(2) 704.0(2) 680.0(2) 660.0(2) 638.0(2) 616.0(2) 0.71 1.00 250.0(6) 234.8(6) 219.3(6) 205.9(6) 192.9(6) 180.6(5) 169.5(5) 157.8(4) 147.6(4) 138.2(4) 127.9(4) 118.8(4) 111.4(4) Table 11. NLO predictions for the ρsdistribution in the pole mass scheme for √S= 14 TeV using as input the static scale µ0=mtand the ABMP16 NLO PDF set. At least one jet with |ηj|<2.4and pj T>30 GeV is required. – 54 – JHEP05(2022)146 dσt¯ tj+X dρsdistribution with Nj≥1, pj T>50 GeV, |ηj|<2.4,KR=KF= 1 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 26.39(7) 26.18(7) 25.91(7) 25.82(7) 25.41(7) 25.23(7) 24.91(7) 24.72(7) 24.56(6) 24.31(7) 24.13(6) 23.99(6) 23.60(6) 0.18 0.22 199.9(6) 197.9(6) 194.2(6) 193.4(6) 191.3(6) 189.0(6) 186.9(5) 184.1(5) 181.7(5) 180.2(5) 178.8(5) 176.4(5) 174.4(5) 0.22 0.27 351.2(8) 346.0(8) 341.7(8) 336.8(8) 332.9(8) 330.0(7) 324.7(7) 320.1(7) 317.0(7) 312.8(7) 307.3(7) 306.1(7) 300.9(7) 0.27 0.32 521.0(1) 510.0(1) 505.0(1) 499.0(1) 490.0(1) 482.8(1) 477.0(1) 470.5(9) 463.8(9) 457.2(1) 450.9(9) 445.5(9) 440.8(9) 0.32 0.38 672.0(1) 663.0(1) 652.0(1) 643.0(1) 632.0(1) 622.0(1) 612.0(1) 602.0(1) 592.0(1) 584.9(1) 575.7(1) 566.7(9) 556.9(9) 0.38 0.45 778.0(1) 767.0(1) 751.0(1) 739.0(1) 723.0(1) 713.0(1) 701.0(1) 687.7(1) 675.9(1) 663.9(1) 651.6(9) 641.1(9) 630.1(9) 0.45 0.53 794.0(1) 776.0(1) 759.0(1) 744.0(1) 731.0(1) 713.0(1) 699.0(1) 685.8(9) 669.3(9) 655.8(9) 642.4(9) 629.5(9) 615.8(8) 0.53 0.62 681.8(1) 665.5(1) 647.7(9) 628.0(1) 612.1(1) 595.8(9) 580.5(9) 564.9(8) 549.5(8) 535.1(8) 519.4(8) 504.0(8) 491.2(7) 0.62 0.71 462.6(8) 445.6(8) 427.9(8) 411.9(8) 397.0(8) 382.3(8) 365.9(7) 353.3(7) 339.5(6) 325.5(6) 312.3(6) 300.6(6) 286.7(6) 0.71 1.00 77.2(2) 71.4(2) 66.5(2) 61.4(2) 56.5(2) 52.0(1) 48.1(1) 44.1(1) 40.4(1) 37.1(1) 34.1(1) 30.9(1) 28.35(9) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>50 GeV, |ηj|<2.4,KR=KF= 2 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 25.97(5) 25.68(5) 25.33(5) 25.13(5) 24.99(4) 24.60(4) 24.45(4) 24.15(4) 23.73(4) 23.65(4) 23.42(4) 23.16(4) 22.90(4) 0.18 0.22 184.1(4) 181.5(4) 179.2(4) 177.8(4) 175.2(4) 173.0(4) 171.7(4) 168.7(3) 167.3(4) 164.7(3) 162.4(3) 160.3(3) 158.8(3) 0.22 0.27 315.6(5) 311.7(5) 306.9(5) 303.8(5) 300.0(5) 295.5(5) 291.1(5) 288.0(5) 284.5(5) 279.2(4) 277.2(4) 273.0(4) 269.3(4) 0.27 0.32 462.4(7) 454.5(7) 448.8(7) 440.6(7) 434.2(7) 429.0(7) 423.3(6) 416.1(6) 412.1(6) 405.6(6) 398.6(6) 394.7(6) 387.9(6) 0.32 0.38 590.9(8) 582.0(7) 573.8(7) 564.1(7) 555.1(7) 545.1(7) 535.5(7) 529.1(7) 519.8(6) 513.3(6) 504.4(6) 496.3(6) 488.8(6) 0.38 0.45 681.6(8) 669.5(8) 656.2(7) 643.1(8) 631.8(7) 623.6(7) 611.8(7) 599.6(7) 590.8(6) 578.9(6) 570.1(6) 558.5(6) 549.8(6) 0.45 0.53 691.6(7) 675.0(7) 661.9(7) 649.2(7) 634.5(7) 620.4(7) 607.5(6) 596.5(6) 583.2(6) 570.2(6) 558.0(6) 546.4(6) 534.2(5) 0.53 0.62 589.0(7) 575.5(6) 560.5(6) 546.1(6) 531.1(6) 517.7(6) 504.1(6) 489.4(5) 475.2(6) 464.1(5) 449.9(5) 438.4(5) 426.5(5) 0.62 0.71 400.6(6) 385.2(6) 370.6(5) 357.4(5) 342.7(5) 331.1(5) 318.0(5) 305.7(4) 293.5(4) 282.2(4) 270.4(4) 259.7(4) 248.2(4) 0.71 1.00 66.5(1) 61.6(1) 57.0(1) 52.6(1) 48.7(1) 44.76(9) 41.32(9) 38.08(9) 34.83(8) 31.95(8) 29.28(7) 26.73(7) 24.33(7) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>50 GeV, |ηj|<2.4,KR=KF= 0.5 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 15.6(1) 15.5(1) 15.4(1) 15.6(1) 15.6(1) 15.3(1) 15.4(1) 15.3(1) 15.3(1) 15.1(1) 15.1(1) 15.3(1) 15.2(1) 0.18 0.22 163.0(1) 161.0(1) 162.0(1) 162.0(1) 157.0(1) 156.0(1) 156.6(1) 153.6(1) 153.0(1) 152.0(8) 151.5(8) 150.5(8) 147.3(8) 0.22 0.27 319.0(1) 314.0(1) 314.0(1) 309.0(1) 306.0(1) 303.0(1) 300.0(1) 294.0(1) 293.0(1) 290.0(1) 286.0(1) 283.0(1) 282.0(1) 0.27 0.32 503.0(2) 500.0(2) 492.0(2) 480.0(2) 478.0(2) 470.0(2) 464.0(2) 459.0(2) 450.0(2) 449.0(1) 441.0(1) 434.0(1) 432.0(1) 0.32 0.38 680.0(2) 666.0(2) 657.0(2) 646.0(2) 637.0(2) 627.0(2) 620.0(2) 609.0(2) 601.0(2) 591.0(2) 584.0(2) 574.0(2) 566.0(1) 0.38 0.45 808.0(2) 791.0(2) 777.0(2) 766.0(2) 751.0(2) 739.0(2) 728.0(2) 714.0(2) 703.0(2) 690.0(2) 678.0(2) 668.0(1) 657.0(1) 0.45 0.53 840.0(2) 823.0(2) 805.0(2) 792.0(2) 771.0(2) 757.0(2) 741.0(2) 725.0(2) 711.0(2) 699.0(2) 683.0(1) 670.0(1) 653.0(1) 0.53 0.62 731.0(2) 714.0(2) 696.0(2) 676.0(2) 662.0(2) 641.0(1) 627.0(1) 608.0(1) 592.0(1) 576.0(1) 560.0(1) 542.0(1) 531.0(1) 0.62 0.71 504.0(1) 487.0(1) 467.0(1) 448.0(1) 433.0(1) 416.0(1) 399.0(1) 384.0(1) 370.0(1) 353.0(1) 340.7(1) 326.9(1) 315.6(9) 0.71 1.00 85.6(3) 78.7(3) 73.2(3) 68.2(3) 62.5(3) 57.8(2) 53.3(2) 48.8(2) 45.2(2) 41.1(2) 37.6(2) 34.4(2) 31.3(1) Table 12. Same as figure 11 but for pj T>50 GeV. – 55 – JHEP05(2022)146 dσt¯ tj+X dρsdistribution with Nj≥1, pj T>75 GeV, |ηj|<2.4,KR=KF= 1 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 25.61(7) 25.42(7) 25.11(6) 25.06(7) 24.63(7) 24.45(7) 24.13(6) 24.00(6) 23.79(6) 23.61(6) 23.32(6) 23.22(6) 22.88(5) 0.18 0.22 185.5(6) 183.8(5) 180.1(5) 178.8(6) 176.8(5) 175.1(5) 173.5(5) 170.0(5) 168.5(5) 166.9(5) 165.8(5) 163.5(5) 161.3(4) 0.22 0.27 309.4(7) 305.6(7) 302.5(6) 297.7(7) 295.0(7) 290.6(6) 286.5(6) 282.8(6) 279.4(6) 275.1(6) 271.3(6) 268.2(6) 264.5(5) 0.27 0.32 437.3(9) 429.0(8) 423.8(8) 418.7(9) 410.7(8) 403.9(8) 397.5(8) 393.9(7) 387.6(7) 382.7(8) 377.5(7) 372.1(7) 367.3(7) 0.32 0.38 534.6(9) 525.0(9) 516.9(8) 509.5(9) 500.3(8) 492.8(8) 483.5(8) 474.9(7) 467.1(8) 461.6(8) 454.0(7) 446.0(7) 437.8(7) 0.38 0.45 574.9(9) 566.9(9) 555.5(8) 545.6(9) 534.3(8) 525.0(8) 515.2(8) 506.5(7) 495.8(7) 486.0(7) 476.5(7) 469.2(7) 460.2(6) 0.45 0.53 535.5(8) 523.1(8) 512.3(7) 499.8(8) 489.5(7) 476.7(7) 465.4(7) 455.3(7) 445.8(7) 435.1(6) 424.7(6) 415.7(6) 406.0(6) 0.53 0.62 400.0(7) 387.7(7) 375.4(6) 364.2(7) 352.8(6) 341.8(6) 332.5(6) 320.8(5) 310.7(5) 300.9(5) 292.3(5) 281.7(5) 273.8(5) 0.62 0.71 210.4(5) 201.3(5) 191.3(4) 181.7(5) 173.8(4) 165.4(4) 157.1(4) 149.2(4) 141.4(4) 133.7(4) 127.2(3) 120.5(3) 112.9(3) 0.71 1.00 16.76(7) 15.24(7) 13.69(6) 12.33(7) 10.93(6) 9.68(5) 8.67(5) 7.70(5) 6.78(4) 5.87(4) 5.16(4) 4.50(3) 3.85(3) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>75 GeV, |ηj|<2.4,KR=KF= 2 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 25.04(5) 24.76(5) 24.43(4) 24.25(4) 24.10(4) 23.74(4) 23.59(4) 23.31(4) 22.88(4) 22.81(4) 22.62(4) 22.35(4) 22.05(4) 0.18 0.22 170.5(4) 168.4(4) 166.0(4) 164.0(4) 161.6(3) 159.7(3) 158.3(3) 155.6(3) 154.3(3) 152.1(3) 150.0(3) 147.9(3) 146.5(3) 0.22 0.27 278.6(5) 275.2(5) 270.9(4) 267.7(4) 264.1(4) 260.4(4) 256.6(4) 253.5(4) 250.2(4) 245.6(4) 243.6(4) 240.0(4) 236.4(4) 0.27 0.32 387.5(6) 380.5(6) 375.3(5) 369.0(6) 363.3(5) 358.7(5) 353.5(5) 348.1(5) 343.6(5) 338.4(5) 333.1(5) 328.8(5) 323.2(5) 0.32 0.38 468.0(6) 461.3(6) 453.1(5) 445.7(6) 439.4(5) 430.4(5) 423.1(5) 417.2(5) 409.1(5) 403.5(5) 397.3(5) 389.9(5) 384.4(4) 0.38 0.45 503.3(6) 494.4(6) 484.1(5) 473.4(6) 465.7(5) 457.6(5) 449.1(5) 440.0(5) 431.7(5) 423.8(5) 415.8(5) 407.8(4) 399.8(4) 0.45 0.53 465.3(5) 453.9(5) 444.1(5) 434.2(5) 425.6(5) 413.4(5) 404.2(4) 395.4(4) 386.5(4) 376.4(4) 368.3(4) 359.7(4) 350.4(4) 0.53 0.62 344.1(4) 333.7(4) 323.8(4) 314.4(4) 304.5(4) 295.8(4) 286.9(4) 277.4(4) 269.0(3) 260.7(3) 252.2(3) 244.7(3) 236.6(3) 0.62 0.71 181.4(3) 173.5(3) 164.8(3) 156.9(3) 149.5(3) 142.4(3) 134.9(3) 128.5(2) 121.5(2) 115.0(2) 108.8(2) 103.5(2) 97.1(2) 0.71 1.00 14.35(5) 12.90(5) 11.59(4) 10.39(4) 9.21(4) 8.27(4) 7.34(3) 6.52(3) 5.71(3) 4.99(3) 4.39(2) 3.82(2) 3.30(2) dσt¯ tj+X dρsdistribution with Nj≥1, pj T>75 GeV, |ηj|<2.4,KR=KF= 0.5 xlxr165 GeV 166 GeV 167 GeV 168 GeV 169 GeV 170 GeV 171 GeV 172 GeV 173 GeV 174 GeV 175 GeV 176 GeV 177 GeV 0.00 0.18 15.6(1) 15.4(1) 15.3(1) 15.5(1) 15.5(1) 15.3(1) 15.3(1) 15.3(1) 15.3(1) 15.0(1) 15.0(1) 15.2(1) 15.0(1) 0.18 0.22 152.5(1) 150.8(1) 151.7(1) 150.7(9) 147.4(9) 146.5(9) 146.6(9) 144.4(9) 143.1(9) 142.3(8) 141.0(8) 140.3(8) 138.3(8) 0.22 0.27 284.0(1) 279.0(1) 278.0(1) 274.0(1) 271.0(1) 269.0(1) 266.0(1) 261.0(1) 260.0(1) 256.2(9) 254.4(1) 250.0(1) 250.0(1) 0.27 0.32 421.0(2) 421.0(2) 414.0(1) 405.0(1) 401.0(1) 395.0(1) 389.0(1) 385.0(1) 379.0(1) 377.0(1) 371.0(1) 365.0(1) 362.0(1) 0.32 0.38 541.0(2) 530.0(2) 521.0(2) 513.0(2) 508.0(1) 497.0(1) 492.0(1) 484.0(1) 476.0(1) 467.0(1) 462.0(1) 455.0(1) 450.0(1) 0.38 0.45 600.0(1) 588.0(1) 580.0(1) 568.0(1) 558.0(1) 548.0(1) 541.0(1) 529.0(1) 518.0(1) 510.0(1) 498.0(1) 490.0(1) 483.0(1) 0.45 0.53 571.0(1) 559.0(1) 545.0(1) 532.0(1) 520.0(1) 509.0(1) 497.0(1) 486.0(1) 475.0(1) 467.0(1) 454.5(1) 445.4(1) 434.7(1) 0.53 0.62 431.0(1) 420.0(1) 408.0(1) 396.0(1) 385.0(1) 371.7(1) 361.9(9) 349.9(9) 339.4(9) 328.1(8) 318.8(8) 306.4(8) 297.1(8) 0.62 0.71 232.2(8) 222.1(8) 209.8(8) 199.9(7) 190.7(7) 181.7(7) 172.2(6) 164.2(6) 157.1(6) 148.2(6) 140.2(5) 132.2(5) 125.5(5) 0.71 1.00 19.0(1) 17.0(1) 15.4(1) 13.9(1) 12.46(9) 11.11(9) 9.86(8) 8.65(7) 7.68(7) 6.71(6) 5.95(6) 5.21(5) 4.43(5) Table 13. Same as figure 11 but for pj T>75 GeV. – 56 –