Lepton-pair production in association with a \( b\overline{b} \) pair and the determination of the W boson mass

Journal of High Energy Physics, Jul 2018

Abstract We perform a study of lepton-pair production in association with bottom quarks at the LHC based on the predictions obtained at next-to-leading order in QCD, both at fixed order and matched with a QCD parton shower. We consider a comprehensive set of observables and estimate the associated theoretical uncertainties by studying the dependence on the perturbative QCD scales (renormalisation, factorisation and shower) and by comparing different parton-shower models (Pythia8 and Herwig++) and matching schemes (MC@NLO and POWHEG). Based on these results, we propose a simple procedure to include bottom-quark effects in neutral-current Drell-Yan production, going beyond the standard massless approximation. Focusing on the inclusive lepton-pair transverse-momentum distribution \( {p}_{\perp}^{\ell^{+}\ell -} \), we quantify the impact of such effects on the tuning of the simulation of charged-current Drell-Yan observables and the W-boson mass determination.

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Lepton-pair production in association with a \( b\overline{b} \) pair and the determination of the W boson mass

Received: March Lepton-pair production in association with a bb pair and the determination of the Emanuele Bagnaschi 0 2 3 4 Fabio Maltoni 2 3 4 Alessandro Vicini 2 3 4 Marco Zaro 1 2 3 4 Science Park 105, NL-1098 XG Amsterdam, The Netherlands 0 Deutsches Elektronen-Synchrotron (DESY) , Notkestra e 85, 22607 Hamburg , Germany 1 Sorbonne Universites, UPMC Univ. Paris 06 2 LPTHE , F-75005, Paris , France 3 Universite catholique de Louvain , Chemin du Cyclotron 2, 1348 Louvain-La-Neuve , Belgium 4 UMR 7589, LPTHE , F-75005, Paris , France 5 Sezione di Milano , Via Celoria 16, I-20133 Milano , Italy We perform a study of lepton-pair production in association with bottom quarks at the LHC based on the predictions obtained at next-to-leading order in QCD, both at xed order and matched with a QCD parton shower. We consider a comprehensive set of observables and estimate the associated theoretical uncertainties by studying the dependence on the perturbative QCD scales (renormalisation, factorisation and shower) and by comparing di erent parton-shower models (Pythia8 and Herwig++) and matching schemes (MC@NLO and POWHEG). Based on these results, we propose a simple proce- dNikhef - dure to include bottom-quark e ects in neutral-current Drell-Yan production, going beyond the standard massless approximation. Focusing on the inclusive lepton-pair transversemomentum distribution p`+` , we quantify the impact of such e ects on the tuning of the simulation of charged-current Drell-Yan observables and the W -boson mass determination. Setup of the simulations Lepton-pair production in association with bottom quarks in the 4FS 3.1 MG5 aMC and POWHEG-BOX implementations 3.2 Identi cation of the reference energy scale for lepton-pair production in association with a b-quark pair The `+` bb transverse-momentum distribution ? proximations The p`+` distribution, inclusive over b-quark contributions, in di erent ap4 Inclusive lepton-pair transverse-momentum distribution 4.1 Four- vs. ve- avour schemes 5 Interplay between NC-DY and CC-DY 4.1.1 4.1.2 Generalities Bottom-quark contributions to DY in 4FS and 5FS Merging 4FS and 5FS results: bottom-quark e ects on the p`+` distribution 19 ? Transferring the bottom-quark e ects to the simulation of charged-current Drell-Yan Template- t determination of mW 1 Introduction 2 3 3.3 3.4 4.2 5.1 5.2 6 Conclusions A Di erential observables in `+` bb production A.1 Jet multiplicities A.2 p`+` ? with extra tagged b jets A.3 Invariant mass of the two hardest b jets A.4 Invariant mass of the two hardest B hadrons with or without tagged b jets A.5 y A.6 y distance of the two hardest b jets distance of the two hardest B hadrons with or without tagged b jets A.7 Transverse momentum distributions of the two hardest b jets A.8 Pseudo-rapidity distributions of the two hardest b jets { 1 { Introduction The production of a pair of high-transverse-momentum leptons in hadron-hadron collisions is one of the historical testing grounds of perturbative Quantum Chromodynamics (QCD). At the lowest order (Born approximation), it proceeds through the parton level amplitude qq ! Z= ! `+` , which once folded with parton distributions, gives the ( rst order) prediction for the inclusive rate, the so called Drell-Yan (DY) process. As shown a long time ago, higher-order QCD corrections [ 1 ] are important and need to be included to improve both the precision and accuracy of the calculation. Predictions for more exclusive nal states can also be calculated in perturbative QCD, including for example QCD jets or heavy quarks (bottom o top quarks). The theoretical interest in this process is matched (or even surpassed) by the exper? imental one: dilepton pairs in high-energy collisions have been always considered golden nal states for Standard Model measurements as well as for new physics searches. In the long and impressive list of experimental results which feature an `+` nal state at hadron colliders, the measurement of the inclusive lepton-pair transverse-momentum distribution, conventionally dubbed p`+` , in neutral-current (NC) DY, has now reached an impressive level of accuracy at the LHC. Using 8 TeV measurements, ATLAS [2] and CMS [3] have attained a total experimental uncertainty below the 0.5% level in a large interval of transverse-momentum values, ranging between 2 and 50 GeV. These achievements represent a formidable challenge for the theoretical predictions which need to combine approximate results obtained with di erent techniques ( xed higher-order corrections vs resummation to all orders of logarithmically-enhanced terms) matched together, to perform a sensible test of the Standard Model (SM). ? As mentioned above, the DY processes start at Leading Order (LO) as a purely electroweak (EW) scattering, qq ! Z= ! `+` . The radiative corrections are exactly known up to O( s2) [4{8] in the strong-interaction coupling, while the O( s3) threshold corrections have been presented in refs. [9, 10] for the inclusive cross section and for the rapidity distribution of the dilepton pair, respectively. The corrections up to O( ) [11{14] in the EW coupling are available. The p`+` spectrum, at large transverse momenta, is known with next-to-next-to-leading (NNLO) QCD accuracy [15{19]. The approximate inclusion of initial-state logarithmically-enhanced corrections to all perturbative orders is necessary to perform a meaningful comparison with di erential distributions of the leptons and is known up to next-to-next-to-leading logarithmic (NNLL) QCD accuracy [20, 21] with respect to log(pV =mV ), where pV? is the lepton-pair transverse ? momentum and mV is the relevant gauge boson mass (V = W; Z); these corrections have been implemented in simulation codes such as ResBos [22] or DYqT/DYRes [23, 24]. The problem of merging xed-order and all-order results, avoiding double counting, has been separately discussed in the context of QCD [22, 25{30] and in the EW [31{33] computations. QCD and EW results have to be combined together to obtain a realistic description of the DY nal states: general-purpose Shower Monte Carlo programs, such as Py8 [34, 35], Hw++ [36] or Sherpa [37], include the possibility of multiple photon, gluon and quark emissions via a combined application of QCD and QED Parton Shower { 2 { initial-state quark cross section (pb) d u s c b total separated perturbative scales, mb and mZ , where QCD mb mZ , can potentially lead to large perturbative logarithmic corrections whose impact needs to be carefully assessed on a observable-by-observable basis. Starting from the pragmatic point of view that b quarks can be found as partons in the proton, it can be easily checked that at the LHC bb ! Z provides a small but non negligible fraction of the total cross section for inclusive lepton-pair production. As this contribution a ects both the normalisation and the shape of the kinematic distributions, a careful analysis is required that can also estimate bottom mass e ects. A rst estimate of the relative importance of contributions from di erent avours of quarks in DY processes can be obtained by computing the individual contributions of quarks to the total cross section for NC-DY in the so-called ve- avour scheme (5FS), i.e. in terms of ve massless active quarks. Results are shown in table 1. While this decomposition is not physical per se as it is ambiguous beyond NLO, it allows to appreciate the precision needed in the predictions of NC-DY production through heavy quark avours. Although in this case all the active avours in the proton are described as massless, in the case of heavy quarks the e ect of their mass mQ is introduced in an initial condition that controls the evolution equations of the respective parton densities; the latter start to be non-zero at an energy scale of O(mQ). These boundary conditions, combined with all { 3 { Flavour decomposition of the p`+` distribution computed with ve active massless the other constraints satis ed by the proton PDFs, result a heavy quark PDFs behaviour which is typically quite di erent from those of light quarks, leading to signi cant di erences in observables like the p`+` distribution. In gure 1 we appreciate the shape of the various contributions initiated by di erent quark avours, which display a harder spectrum in the case of heavy quarks. The di erence in shape can be noticed either from the rst inset, where the contributions of each avour are compared, or from the second one, where we show the shape of all avours except for the b or the b and c. We conclude that given the present experimental uncertainty, the bottom-quark contribution to the p`+` distribution deserves a dedicated study of the residual uncertainties due to the treatment of the bottom With the latter, we mean an improved description of the bottom-quark kinematics, including mass-dependent terms, at NLO-QCD. ? The p`+` distribution provides access to a large range of scales and therefore it o ers a stringent test of perturbative QCD in di erent regimes: in the low-momentum region it is sensitive to non-perturbative QCD contributions and possibly to the avour structure of the proton [43]. The precise knowledge of this part of the p`+` spectrum makes it possible to calibrate the non-perturbative models that describe the partonic transverse degrees of freedom inside the proton. Such models, implemented e.g. in QCD PS, are then used to ? { 4 { ? simulate other scattering processes, and their uncertainties propagate in the prediction ? of the new observables. A striking example is the determination of the W -boson mass mW [44{46], which relies on the p`+` input to obtain an accurate simulation of the W boson transverse-momentum spectrum and in turn of the leptonic nal state. Eventually, the extraction of mW displays a strong sensitivity to the modelling assumptions for the lowmomentum part of the p`? spectrum. In this context, it is important to remark that the heavy-quark contribution, in particular the one originated by the bottom which will be the main subject of this work, is di erent in CC- and NC-DY. This is due to the di erent initialstate avour structure, following from electric charge conservation and Cabibbo-KobayashiMaskawa (CKM) mixing. More speci cally, bottom-quark e ects which are present in improving the accuracy and precision of the heavy-quark contributions to the inclusive Zboson production, is relevant: i) to reduce the amount of information which has to be encoded in a model that describes the low-momentum part of the gauge-boson transversemomentum spectrum; ii) to capture some non-universal avour-dependent contributions, which distinguish massless and massive quarks. Understanding heavy-quark contributions to lepton-pair production bene ts also from the analysis of exclusive nal states where the leptons are associated to a pair of bottomantibottom quarks, which are explicitly tagged in terms of either b jets or B hadrons. The presence of additional energy scales, such as the masses and transverse momenta of the measured b quarks, imposes non-trivial constraints on the structure of the radiative corrections that have to be included in the simulations to obtain accurate predictions. Understanding these nal states is also propedeutic to that of other heavy systems, e.g., a Higgs boson or a tt pair, accompanied by a bb pair. The production of `+` bb, with the inclusion of NLO-QCD corrections has been discussed in refs. [47{51], and more recently in ref. [52], for nal states with at least one or with two tagged b-quark jets, in the so called four- avour scheme (4FS), namely using a parameterisation of the proton structure in terms of only four active quarks and considering bottom quarks in the nal state as massive. The matching of xed-order matrix elements with a Parton Shower has been implemented in ref. [53] in the MadGraph5 aMC@NLO framework [54] and in ref. [55] in the Sherpa framework. Given on the one hand the very high level of precision necessary to obtain sensible results in the description of p`+` ? the other hand, the link between exclusive and inclusive nal states characterised by the presence of heavy quarks, we deem necessary to scrutinise the theoretical uncertainties a ecting the prediction of the observables for `+` bb nal states. To this aim, we present a systematic comparison of two di erent schemes of matching between xed order results and eventually in the determination of mW , and, on { 5 { with a QCD PS,1 namely the MadGraph5 aMC@NLO (from now on MG5 aMC) and the POWHEG ones; we expose the impact of di erent treatments for the QCD PS phase space assignment and we present the phenomenological results obtained with two QCD PS models, namely Pythia8 and Herwig++ (dubbed Py8 and Hw++ in the following). To summarise we i) thoroughly compare the implementations of the production of a lepton pair in association with a bb pair in the 4FS between two available Monte Carlo event generators, with a systematic analysis of all the relevant QCD theoretical uncertainties; ii) consider the e ects of including bottom-quark-mass contributions on the inclusive transverse-momentum spectrum of the lepton pair; iii) estimate the impact that such contributions may have on the determination of the W -boson mass. The paper is structured as follows. In section 2 we describe the setup employed for the numerical simulations; in section 3 we study lepton-pair production in association with bottom quarks in the 4FS, we compare the implementations in the MG5 aMC and POWHEG frameworks and discuss several sources of theoretical uncertainties for inclusive observables. We defer to appendix A an extensive comparison of more exclusive observables. In section 4, in order to evaluate the e ects of the bottom-quark mass, we consistently combine the 4FS prediction for `+` bb with the usual 5FS inclusive lepton-pair calculation and study the transverse-momentum distribution p`+` . In section 5 we consider the impact of bottom-quark mass e ects on CC-DY observables and on the determination of the W ? boson mass. We draw our conclusions in section 6. 2 Setup of the simulations In this work we study the processes pp ! `+` + X; pp ! `+` + bb + X; pp ! ` + ` + X; (2.1) (2.2) (2.3) for one leptonic family, in a setup typical of the LHC, with p S = 13 TeV. Unless stated otherwise, the simulations have been run at NLO+PS accuracy with the codes MG5 aMC (all the processes have been generated within the same computational framework) and POWHEG-BOX. Both codes have been interfaced with the same QCDPS programs, namely Py8 (version 8.215, Monash tune) [35, 57] and Hw++ (version 2.7.1) [36, 58].2 We did not include any QED e ect via QED PS. The simulation of the 1For a similar study in the case of Higgs production in gluon fusion, cfr. ref. [56]. 2In the showering phase, the parton densities are always those speci c to the tune employed. In particular, for The Monash tune in Pythia8, they correspond to the NNPDF 2.3 LO set. For what concerns the value of s(mZ), for POWHEG-based simulations, the Monash tune value is also employed ( s(mZ) = 0:135). Instead, within MG5 aMC, in order not to spoil the cancelation between the rst emission from the S events and the H ones, the value s(mZ) = 0:118 is employed. We have veri ed that, when the same parton distributions and s(mZ) of the hard events are employed in the shower, or when MG5 aMC samples are showered with the default value of s from the Monash tune, e ects on the p`?+` distribution are di erences are smaller to those related to the choice of reference shower scale. { 6 { underlying event is not performed. For the proton parton-density parameterisation we use the NNPDF 3.0 NLO PDFs with s(mZ) = 0:118, with the same avour-number scheme as for the hard process [59]. The SM parameters are set to the following values [60, 61]: jVubj= jVcbj = jVtdj = jVtsj = 0: G It is understood that the quoted value of mb is employed only in the 4FS process, eq. (2.2). For the central value of the renormalisation and factorisation scales, we use for all samples the lepton-pair transverse mass divided by four: = 1 q 4 M 2(`+; ` ) + (p`+` )2 : ? For the 5FS NC-DY, this choice was advocated in ref. [62]. The only exception to what stated above is represented by the samples for charged-current Drell-Yan used in section 5.2, where the transverse mass of the (reconstructed) W boson is used: ? q ? CC DY = M 2(`+; ) + (p` )2 : ? In eqs. (2.5) and (2.6) M 2 and p`+` (p` ) are respectively the squared invariant mass and the transverse momentum of the lepton pair (lepton-neutrino pair). In the simulation of processes (2.1) and (2.2), a generation cut M (`+; ` ) > 30 GeV is applied in order to avoid the singularity related to the photon contribution. At the analysis level, for the processes (2.1) and (2.2), we apply a cut on the transverse momentum of each lepton, p?(` ) > 20 GeV, and on their pseudorapidity, (` ) < 2:5, together with an invariant-mass cut around the Z peak, jM (`+; ` ) mZ j < 15 GeV. In process (2.3) we impose a cut on the charged-lepton transverse momentum and on the missing transverse again a pseudorapidity cut j (`+)j < 2:5 for the charged lepton. energy (the transverse momentum of the neutrino), p?(`+) > 25 GeV; p?miss > 25 GeV, and jVtbj = 1; Lepton-pair production in association with bottom quarks in the 4FS In this section we study the process pp ! `+` bb + X in the 4FS. The bottom quarks, absent in the proton PDFs, are treated as massive nal-state hard partons. In section 3.1 we compare the formulation of two matching recipes to combine xed- and all-order results with NLO-QCD accuracy, with special attention to the details of the inclusion of multiple parton radiation and to the perturbative sources of uncertainty. We then discuss phenomenological results, obtained with the setup outlined in section 2: in section 3.2 we rst determine a typical scale which characterises the process and then in sections 3.3 { 7 { and 3.4 we compare respectively the results of the various matched schemes for the transverse momentum of the `+` bb system and the e ect of higher-order corrections and of the matching for the lepton-pair transverse momentum. The interested reader can nd more details on other di erential observables in the appendix A. In this section we compare two di erent matching schemes, namely those implemented in the MG5 aMC and POWHEG-BOX Monte Carlo event generators. The aim is to disentangle genuine bottom-quark e ects from those due to a di erent treatment of higherorder emissions in the two approaches. The simulation of scattering processes in hadron collisions requires not only the inclusion of xed-order corrections in order to obtain a reliable estimate of the overall normalisation of the cross sections, but also the inclusion of multiple parton emissions at all orders in order to achieve a realistic description of the shape of the distributions. The possibility of simultaneously preserving the NLO accuracy for all the observables that are regular when the radiative corrections are included, together with the description of multiple parton emissions, is achieved by matching xed- and all-order results. Di erent matching schemes have been proposed in the literature, and here we focus on MC@NLO [26] and POWHEG [27]; they share the same xed-order accuracy, but di er for the inclusion of subsets of higher-order terms. The latter are beyond the accuracy of the calculation with respect to the coupling constant expansion and are formally subleading in a logarithmic expansion in powers of log(pV =mV ), where pV represents a generic transverse-momentum ? ? variable that yields a singularity of the amplitude in the limit pV ? ! 0, and mV is the invariant mass of the system whose transverse momentum is described by pV ; although subleading, these terms can nevertheless have a sizeable numerical impact on the predic? tions, in particular for those observables that have only the lowest order accuracy. The matching of xed- and all-orders corrections should avoid double counting between the two contributions and respect the ordering of the emissions of QCD partons, in order to preserve the logarithmic accuracy of the results. In a Monte Carlo approach, the hardest QCD parton, with respect to the radiation ordering parameter t, plays a special role, for it receives the exact matrix element corrections of the xed-order calculation. The subsequent emissions are instead generated by the QCD-PS and the associated phase-space volume is part of the matching prescription. A generic scattering process, whose lowest order (LO) is characterised by the presence of k nal state particles, receives radiative corrections due to the emission of n additional partons. In the MC@NLO approach an event is generated according to the following steps: 1. The event weight is split in two contributions, called standard (S-events) and hard (H-events), which describe nal states with respectively k and k + 1 nal state particles; both standard and hard terms are matched with a QCD-PS that generates n additional parton emissions using: i) the standard Sudakov form factor computed in the collinear approximation; ii) an approximated phase-space measure; iii) an upper limit for the hardness of the emission set by a scale Qsh called shower scale. { 8 { HJEP07(218) 2. H-events account for the exact real matrix-element corrections describing the rst real emission, evaluated in the full phase space with exact integration measure. The double counting in the generation of the hardest parton between the PS and the exact matrix element is avoided with an appropriate counterterm. 3. S-events account for all the terms entering a NLO cross sections (Born, virtual, counterterms, etc.) except for the real-emission matrix elements and the corresponding counterterms. 4. The shower scale Qsh associated to each S-event is extracted from a probability distribution. The latter parametrically depends, event-by-event, on a reference scale, which we denote with the symbol sh and which is computed considering the S-event kinematics. For the corresponding H-event, the maximum of the allowed values by the same distribution is used. The details of this procedure and the functional form of the distribution are given in section 2.4.4 of ref. [54]. In the POWHEG approach an event with n additional partons is generated according to the following steps: 1. The weight B of each LO con guration is rescaled by a factor B~=B that accounts for virtual corrections and the integral over the rst real emission. This rescaling guarantees the full NLO accuracy for inclusive quantities. 2. The expression of the POWHEG Sudakov form factor depends on the splitting, in h2 the full real-emission matrix elements R, between the singular Rs and a remaining regular part Rf , controlled by a scale h according to: R = Rs + Rf ; Rs f (h; t)R; Rf = (1 f (h; t))R where the damping factor f (h; t) depends on the radiation variable t through the function f (h; t) = t+h2 : it goes to 1 in the collinear limit t ! 0 and vanishes for large t. The scale h de nes the region where the Sudakov suppression is active and the e ects of multiple parton emissions are systematically included. 3. Since Rf is non singular, it can be directly employed to generate part of the events, which are typically named \remnant" events. 4. The probability of the rst emission on top of the B~=B-rescaled LO con guration is evaluated using: i) the POWHEG Sudakov form factor; ii) the exact radiation phase space; iii) the singular part Rs of the exact matrix elements for the real emission. 5. Related to the previous point, the fact that the rst parton with emission variable t = t is by construction the hardest is obtained: i) computing the product of the Sudakov form factor for an emission with t with the corresponding real-emission matrix element; ii) limiting the QCD-PS phase space at the value t of the emission variable t. 6. The QCD-PS populates the available phase space, assigning to the ordering variable t a maximum value equal to the shower scale Qsh of the event; this value is by { 9 { the Qsh value, di erent than t, is allowed in the generation of the remnant events (based on the non-singular part Rf of the real-emission matrix element) without spoiling the accuracy of the calculation; once Qsh is assigned, the generation of n 1 additional emissions proceeds in the PS approximation for the branching probability and integration measure. These two approaches share the NLO-QCD accuracy in the prediction of the total inclusive cross section, and di er by the inclusion of terms of higher order in the perturbative expansion in powers of s. As already said, the latter are formally subleading with sizeable, depending on the phase space region under study. respect to the enhancement due to log(pV?=mV ) factors, but can nevertheless be numerically We note that in general when PS programs take in short-distance events, a comparison is performed between the shower scale Qsh provided in the event (in the event-record eld SCALUP) and the corresponding scale that would be associated by the code based on its own phase space evaluation. Since all the PS emissions must be ordered with respect to the hardness parameter t, the smallest value between the two is eventually used in the QCD-PS. We focus now our attention on the actual distribution of Qsh in the event samples produced in the two Monte Carlo frameworks, i.e. the distribution of the values of the SCALUP eld of the event records. In gure 2 we show the histograms obtained with MG5 aMC for di erent choices of sh, in the 4FS simulation.3 In gure 3 we show the distributions obtained in the POWHEG-BOX 4FS simulation, for the events describing the singular part and the regular reminder of the real matrix element (B~ and remnant events respectively), for di erent values of the damping factor h. In the POWHEG-BOX default setup, Qsh coincides with the transverse momentum of the rst emission. As said, it is possible to preserve the logarithmic accuracy of the 3The choice sh = p sh = HT =2 is the new default. In these newer versions, it is still possible to use sh = ps^, by setting the s^ has been the default in MG5 aMC up to version 2.5.2. From version 2.5.3, i scale=0 in the subroutine assign ref scale inside montecarlocounter.f. generated by the POWHEG-BOX . calculation with a di erent choice of Qsh for the remnant events.4 We recall that the generation probability of a remnant event depends also on the scale h introduced in the POWHEG Sudakov form factor, as it can be seen from the plot. Despite being related to the same entry in the Les-Houches event le, the quantities shown in the plots in gures 2 and 3 have di erent physical meaning: in the case of MG5 aMC, it can correspond either to the scale used by the shower in the rst real emission (on top of the S-events) or to the second (on top of the H-events). Instead in the case of POWHEG, it represents the scale employed by the shower for the second emission, which in the POWHEG method is set equal to the transverse momentum of the rst emission (for the B~ events). Both plots, on their own, show how di erent distributions of Qsh may appear in the event samples generated by the two frameworks. This fact and the di erent structure of the matching procedure itself, impact the nal results of the simulations (after showering), formally beyond the claimed accuracy but still in a phenomenologically relevant way. We also note that the nal numerical results, after showering, depend on the PS ordering variable, so that di erent QCD-PS models may yield di erent results even if using the same Qsh distribution. To summarise, the formulation of the matching and the interface with the QCD PS are closely entangled; their ambiguities and prescriptions represent an important source of theoretical uncertainty, beside the canonical ones, related e.g. to the choice of the renormalisation and factorisation scales. These additional uncertainties are relevant in the study of the shape of the kinematic distributions, in particular of those sensitive to the details of real radiation. 3.2 Identi cation of the reference energy scale for lepton-pair production in association with a b-quark pair In ref. [62], the production of a Higgs boson in association with a massive bb pair is considered and, following the discussion of ref. [63], a universal logarithmic factor L 4For example, we tested a di erent option in gure 16 where we also considered the distributions of gure 3 divided by a factor two. pzi z ) i : Event distribution, in `+` bb production, with respect to the variable M de ned in eq. (3.2). The red arrow corresponds to the peak of the distribution. log Q2(z)=mb2 , associated to each g ! bb splitting is identi ed. We adapt this approach to the case under study of the subprocess g=q(p1) g=q(p2) ! `+(q+)` (q )b(k1)b(k2) and obtain that the universal corrections have the form L = log M 2(`+; ` ) (1 m2 b zi)2 z i with zi = ; si = (q+ + q + ki)2 ; (3.1) where M 2(`+; ` ) is the squared lepton-pair invariant mass. The e ective scale that characterises the process is M M (`+; ` ) (3.2) In gure 4 we plot the distribution d =dM and observe the presence of a peak at M 25 GeV. We interpret this value as one of the typical energy scales that characterise the process and justify, following ref. [62], our choices described in section 2 for the renormalisation and factorisation scales in the 4FS. 3.3 The `+` bb transverse-momentum distribution In this section we consider the transverse-momentum distribution of the `+` bb system and present numerical results in di erent approximations in gure 5. This quantity is of technical interest, because it makes it possible to study the impact of QCD radiation on this system, with interesting features due to the presence of coloured particles in the nal state, whose emissions contribute to the recoil of `+` bb. In the left plot of gure 5 we show the distribution of the logarithm of the transversemomentum distribution of the `+` BB system5 computed with di erent combinations 5In the evaluation of the distribution we tag and analyse the b quarks at xed order, while for computations matched to parton shower we consider the two hardest B mesons that are present in the nal state. No other cut is applied besides those on the leptons, described in section 2. Transverse momentum distribution of the `+` BB nal state, computed in the 4FS. In the plot we represent the log(p?(`+` BB)) distribution. The left plot shows the distribution in di erent approximations; the lower inset shows the size of the corresponding PDF, factorisation and renormalisation scale uncertainty bands, with respect to the NLO prediction. The right plot shows the relative e ect of di erent approximations, with respect to the xed-order NLO results: di erent matching schemes (MG5 aMC vs POWHEG-BOX) with the corresponding matchingparameter uncertainty band, sh = ps^=2; ps^=4 or h = mZ=4; mZ (lower inset); di erent Parton Shower models (Py8 vs Hw++). of generators and of scales:6 in black we present the divergent distribution computed xed order NLO calculation (we stress that this quantity is only LO accurate in this calculation); in blue we show results obtained with MG5 aMC and Py8, using as reference shower scale, sh, the partonic variable show results obtained with MG5 aMC and Py8, using as sh the sum of the nal-state transverse masses HT =2 (dashed) or HT =4 (solid); in green we show results obtained with POWHEG and Py8, setting the value of the scale h in the damping factor equal to mZ ps^=2 (dashed) or (dashed) or mZ=4 (solid). In the upper inset of the right plot of gure 5, we compare the di erent combinations of tools and scales of the left plot with the xed-order NLO-QCD results. As a function of the transverse momentum of the system we observe: the Sudakov suppression at low momenta; the redistribution of the events due to the unitarity of the PS and of the matching procedure, yielding an increase of the distribution at intermediate momenta; a decrease of the distributions compared to the xed order results occurs at large momenta, for both MG5 aMC and the POWHEG-BOX and irrespective of the choices of the PS parameters. In the lower inset of the left plot we show the PDF and renormalisation/factorisation scale uncertainties, which are at the 20% level for transverse momenta smaller than 6The same colour codes, combinations of codes and approximations are valid also in gures 7, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 23, 27, 28, 29, 30. 120 GeV, but increase and reach the 45% level for transverse momenta close to 1 TeV, and are dominated by the scale uncertainties. In the lower inset of the right plot we compare again with the xed-order NLOQCD curve the result obtained with MG5 aMC combined with Py8 (blue) and with Hw++ (brown) using for sh the partonic variable ps^=2 (dashed) or ps^=4 (solid). We observe similar trends of the two QCD-PS models but a quantitative signi cant di erence range, in particular for the size of the bands corresponding to the ps^=2 scale choice. in the comparison of the bands obtained with a variation of the shower scale in the same distribution, inclusive over b-quark contributions, in di erent In this section we study the transverse-momentum distribution p`+` of the lepton pair, in presence of a bb pair in the nal state, inclusive over the b-quark contributions. The process pp ! `+` bb is studied in the 4FS in di erent perturbative approximations, namely at LO, at xed NLO-QCD, including QCD-PS e ects matched with the LO or with the NLO-QCD results. At variance with the 5FS case, where the p`+` distribution ? is divergent at xed-order O( s) when p`+` ? ! 0, this observable in the 4FS is regular in ? ! 0. ? for the singularity associated with the limit p`+` the same limit at xed order and a fortiori after matching with a QCD PS. The regular behaviour of p`+` in the 4FS is due to the bottom-quark mass, which acts as a regulator ? At NLO-QCD the p`+` distribution is sensitive to large logarithmic corrections due to QCD initial-state radiation, mostly from the gg-initiated subprocess.7 The origin of these large e ects can be understood by considering the two mechanisms that yield a transverse momentum of the lepton pair: i) the LO distribution of the =Z boson in a three-body nal state and ii) the recoil against QCD radiation of the `+` bb system. While the former is regular in the whole phase space, the latter is sensitive to the presence of collinear divergences due to initial-state radiation. In fact, the transverse-momentum distribution of the `+` bb system is divergent at xed order for vanishing transverse momentum and requires the resummation of logarithmically-enhanced terms to all orders to become regular. After the resummation, the transverse-momentum distribution of the `+` bb system is still sensitive to logarithmically-enhanced corrections, which contribute in turn to the second of the two mechanisms that yields the p`+` distribution, explaining why the prediction of ? ? the latter requires not only a xed-order calculation but also the matching with multiple parton emissions at all orders via QCD-PS. In gure 6 (left panel) we compare the 4FS distributions in di erent perturbative approximations: at xed-order LO and NLO and, after the matching of MG5 aMC with the Py8 QCD-PS, with LO+PS and NLO+PS accuracy. in section 2 and set sh = ps^=4 as the reference shower scale. In We use the inputs described gure 6 (right panel) we show the relative impact of the various approximations relative to the LO results. The NLO corrections (green) yield a large K-factor of O(70%), at almost the whole p`+` , with ? the exception of the low transverse-momentum region, where the corrections are smaller, 7A similar statement is also present in ref. [53] (cfr. gure 6). p`+` distribution in 4FS inclusive over the b-quarks contribution: comparison of predictions in di erent perturbative approximations: LO, NLO, LO matched with the Py8 QCDPS, NLO matched with a QCD-PS (left panel) and relative impact of higher-order corrections relative to the LO prediction (right panel). mations and colour codes as in gure 5. of O(50%). The action of a QCD-PS on top of the LO distributions strongly modi es the shape of the distribution, with a corrections which is negative and reaches 40% at After matching the NLO results with the QCD-PS, the relative impact of the latter with respect to the xed NLO results is similar to the di erence between LO+PS and LO for low p`+` , while for large p`+` a positive correction of O(+20%) remains. ? ? In gure 7 we study di erent sources of theoretical uncertainty, using the same colour code and comparing the same approximations as in gure 5. In the left plot, upper inset, ? very low p`+` O(+20%) at p`+` ? ? values, vanishes at p`+` ? 25 GeV, then increases and has a maximum of 35 GeV, decreases and eventually vanishes for larger values of p`+` . the results obtained with LO+PS and NLO+PS accuracy with Py8 and Hw++, for a given choice of the matching parameter. we compare the predictions obtained with MG5 aMC and POWHEG-BOX, both interfaced with Py8, using di erent variables and values for sh and h. We observe a global compatibility between the di erent options: if we consider the envelope of the di erent bands as an estimate of the matching and shower uncertainties, we conclude that they are at the O( 7%) level, with the exception of the rst two bins, as long as p`+` ? and slightly increase for larger values of p`+` . We observe a common trend of the correc< 100 GeV, tions due to multiple parton emissions, which are negative down to 30%, with respect to ? the xed NLO prediction, for p`+` ? From the upper inset of the right plot of gure 7 we observe that in the low-p`+` region the POWHEG-BOX corrections are slightly smaller in size than those of MG5 aMC. ? < 20 GeV and positive up to +15% for larger values. In the left plot, lower inset, we show the uncertainty bands associated to scale variations (the renormalisation and factorisation scales are varied independently in the interval [ =2; 2 ]) and to PDFs. As it can be seen, scale variations provide an uncertainty of O( 20%) which is quite independent on the value of p`+` , and represent by far the dom? inant source of uncertainty. PDF uncertainties are much smaller, below 3%. In the right plot, lower inset, we compare the PS Py8 and Hw++, both matched to MG5 aMC and with the same reference shower scale sh. We observe that accidentally the combination of MG5 aMC with Hw++ yields results which are in size and shape similar to those obtained with the POWHEG-BOX and Py8. sh = ps^=2; we show the relative di erence with respect to the NLO In gure 8 we compare the predictions of MG5 aMC at LO+PS and NLO+PS accuracy obtained with Py8 and Hw++ as QCD-PS (left plot), with either sh = ps^=4 or in the right panel. The di erences of order 15% at LO+PS (green and red bands) are reduced down to 7% at NLO+PS, because the rst real emission is described, in the latter case, by the exact matrix element and the PS di erences appear from the second emission. Inclusive lepton-pair transverse-momentum distribution In this section we discuss the prediction of the inclusive lepton-pair transverse-momentum distribution and propose a formulation that includes a re ned treatment of the bottomquark contributions, exploiting the advantages of both 4FS and 5FS formulations of the lepton-pair production process. Two procedures are commonly followed to calculate high-energy processes characterised by a hard scale Q, that involve the production of heavy quarks such as the bottom quark. In the so called \massive" or four- avour scheme (4FS) the heavy quarks do not contribute to the proton wave-function because the value of their mass, larger than the one of the proton, makes their creation in pairs possible only in high-energy interactions. In this scheme the active degrees of freedom are nf light quarks, while the heavy quarks are decoupled and do not contribute to the running of the strong coupling constant nor to the PDF evolution; in particular a bottom-quark PDF is absent. The validity of this approach is guaranteed when the hard scale Q of the process is comparable to the heavy-quark mass mb. The latter acts as a natural cut-o in the case of additional collinear emissions. In the case when a hierarchy between the heavy-quark mass and the hard scale of the process is present (mb Q) it is possible that large corrections enhanced by log2 Q2 in the cross sections, spoiling the convergence of the perturbative expansion, while powers of the ratio mb=Q are naturally suppressed. The initial-state logarithmic corrections can be resummed to all orders via the Altarelli-Parisi equations and reabsorbed in the de nition of a bottom-quark proton PDF, while in the nal-state case it is possible to introduce appropriate fragmentation functions. The bottom quark belongs then to the light quarks present in the proton (nf = 5) and contributes to the running of the strong coupling constant. This approach is called \massless" or ve- avour scheme (5FS). The advantages of the 5FS are related to the lower multiplicity of scattering particles: the simplicity of the nal-state structure makes it possible to include higher-order radiative corrections more easily than in the corresponding 4FS processes. In addition, the presence of a bottom PDF in the proton resums to all orders initial-state collinear logarithms due to gluon emissions. As of today, the nal-state higher multiplicity in the 4FS forbids the inclusion of corrections beyond NLO-QCD. On the other hand, the exact description of the massive-quark kinematics is already present at LO and can be analysed in detail upon inclusion of the NLO corrections and also after matching with a QCD PS. In addition, as argued in section 3.2 based on the results of ref. [62], possibly large logarithms log2 Q2 in fact suppressed by phase space e ects, and the e ective scale Q2 is parametrically lower than the vector boson mass. One therefore expects that in this region bottom mass e ects to be more relevant than the collinear logarithms and the 4FS could be preferred over the 5FS. Another motivation for employing the 4FS scheme is provided by the inclusive gaugeboson transverse-momentum distribution, which has a peak in the interval between three m2 are b m2 appear b to the bottom quark, the latter evaluated in di erent schemes and approximations. and eight GeV, namely at values comparable to the bottom-quark mass. The simulation of the lepton-pair transverse momentum distribution in shower Monte Carlo codes based on the 5FS description requires, in the case of the bottom-induced partonic contributions, that the emission of real radiation stops at a transverse momentum scale of O(mb), that the b parton be put on its mass shell and that the hadronisation of b quark into B hadron takes place. This is typically handled by an ad hoc procedure in the QCD PS, which features intrinsic ambiguities. In the 4FS instead, the lepton-pair transverse momentum distribution receives an exact matrix-element description including the O( s) corrections in the full range from zero GeV up to the kinematic limit. 4.1.2 Bottom-quark contributions to DY in 4FS and 5FS We are interested in combining the advantages of the 4FS and 5FS approaches, in order to improve the description of the bottom-quark e ects in the lepton-pair transversemomentum distribution.8 The merging of 4FS and 5FS results is in principle possible provided that double counting is avoided. To this aim, equivalent terms that contribute in the two schemes need to be identi ed, then subtracted from the 5FS description of the process and added back as evaluated in the 4FS. The rationale behind this combination is the possibility of exploiting the improved description o ered by the 4FS of the heavy-quark contribution to observables like the gauge-boson transverse momentum at low-/intermediate-momentum values. In the 5FS, at tree level, the DY process occurs through quark-antiquark annihilation, the partonic cross section starts at O(G2 ) and bottom-initiated subprocesses are already present. Since we are interested in the bottom contribution, we remark that this density, 8The formulation of ve- avour schemes retaining power-suppressed mass-e ects at some level of accuracy in inclusive or semi-inclusive observables has a long history, e.g., see section 2 of ref. [63]. For a recent proposal in the context of fully-exclusive predictions see ref. [64]. generated inside the proton by a radiative mechanism, is proportional to s and it contains, via Altarelli-Parisi evolution, the resummation to all orders in s of terms enhanced by a factor log( F =mb). In gure 9 we show, with NLO+PS accuracy, in black dashed the complete p`+` distribution in the 5FS and in red dashed the contribution given by the subprocesses initiated by at least one bottom PDF. The size of the latter is consistent with the overall larger value than the one of the all- avour p`+` distribution (10 GeV vs. 3 GeV). contribution of O(4%) to the total cross section, but the peak of the distribution is at a ? After the matching of exact NLO matrix elements with a QCD-PS that simulates ? parton radiation to all orders, we have to consider the possibility that the emitted gluons HJEP07(218) split into bb pairs, which appear as nal-state hard partons; such terms are of O( s2 G2 ) (when the initial state contains only light quarks) or higher. Since it is not possible to make a distinction between initial- and nal-state bottom contributions, we are lead to de ne the bottom contribution to DY in the 5FS as the one given by all the events that contain at least one B hadron in the nal state (generated in the hadronisation phase of the QCD-PS). We recall that in the 5FS the cross section is evaluated with ve active avours contributing to the strong coupling-constant running, inducing a bottom contribution also in the subprocesses initiated by light quarks and gluons; the latter are not tagged by the B hadron selection. In the 4FS, the bottom quark in the proton is by de nition absent; lepton-pair production in association with a bb pair starts at O( s2G2 ), with the strong coupling-constant running with four active avours. This LO cross section is exact in the description of the kinematics of the massive bb pair. In a NLO-QCD accurate calculations, also terms of O( s3G2 ) are exactly included. In this scheme, heavy-quarks contributions to the s running are decoupled and included in the renormalisation condition. After matching with a QCD PS, additional bb pairs might be created, although with suppressed rate, starting from O( s4G2 ). In gure 9 we show in green dotted the p`+` distribution in the 4FS inclusive over ? the b quarks, at NLO QCD, while in blue and in black solid we present the results with NLO+PS accuracy, for two di erent choices of the reference shower scale. The sizeable impact of the matching with a QCD-PS can be appreciated at glance. 4.2 Merging 4FS and 5FS results: bottom-quark e ects on the p`+` distri? bution 4FS results. { 19 { As discussed in section 4.1, the improvement over the plain 5FS description can be obtained by the subtraction of the bottom-related contributions and their replacement with the We de ne two physical distributions, namely the production of a lepton pair strictly without B hadrons (our B-vetoed 5FS calculation, that we label 5FS-Bveto) and the production of a lepton-pair accompanied by at least one B hadron (our 4FS results), which are complementary with respect to the additional particles beside the lepton pair.9 The 9A similar procedure has been proposed in ref. [65] in order to have an improved description of ttbb in (orange shaded areas). In the top row we show the POWHEG-BOX results, while in the bottom row we present the equivalent curves from MG5 aMC. In blue, red and green we display the curves obtained by using the reweighting function R, for several di erent setups. In the present study we use all our central choices for the input parameters, as described in section 2 to compute the templates for the CC-DY lepton transverse momentum and lepton-pair transverse mass in the plain 5FS using tune1, i.e. our default Py8 tune. We generate the distributions corresponding to di erent values of mW with the reweighting technique described in refs. [66, 67]. For illustration we show in gure 13, for the transverse mass of the W boson as well as for the transverse momentum of the charged lepton, the comparison of templates computed with di erent mW values in a range up to 20 MeV about the central PDG value: we observe a spread of the curves at the Jacobian peak in correspondence of the W resonance. The same gure also shows the e ect of the p` reweighting on these observables, according to our alternative predictions as shown in gure 10. These curves represent the pseudodata that will be employed in the t, which are obtained again in the plain 5FS with default Py8 tune, PDG values for the masses and the p`? -dependent reweighting by R(p`? ), described in section 5.1, to simulate the impact of the improved bottom-quark treatment. For both templates and pseudodata we consider the shape of the distributions: we de ne a range of values of the tted variable around its ? Jacobian peak and normalise the distribution to the corresponding integral. In this way we enhance the sensitivity of the template t procedure to the precise position of the peak. The level of agreement between templates and pseudodata can be assessed with the least squares method. The standard de nition of a 2 indicator, j2(O~ data) = X k2bins (Okdata Ok 2 k j;template)2 ; assumes that all the bins are uncorrelated and that each contributes according to its statistical error, represented by k2. For each template j we compute j2; as a function of j we should obtain a parabola whose minimum indicates the preferred value of the t parameter. We perform two independent ts on the lepton transverse momentum and on the W -boson transverse mass, which is de ned, starting from the transverse momentum of the charged lepton and of the neutrino,10 as (5.3) (5.4) (5.5) ? where `+ is the azimuthal angle between the two leptons. The t is performed in the following ranges, which correspond to the ones employed by ATLAS [46]: 32 GeV < p?(`+) < 45 GeV; 66 GeV < M?(`+; ) < 99 GeV : The granularity of the mW scan is of 1 MeV. In gure 14 we show the 2 parabolas and the shift induced by reweighting the p` description. The left column of the gure refers to distribution with our alternative p`+` ? the transverse mass, the right column to the lepton transverse momentum. Plots in the top row are obtained with the POWHEG-BOX, those in the bottom row with MG5 aMC. As far as the transverse mass is concerned, all induced shifts are compatible with zero. In fact this observable is known to be insensitive to the details of the p` modelling [68]. When the lepton transverse momentum is considered, the mass shifts are of the order ? mW mW 4 5 MeV when the 5FS is improved with the xed-order 4FS prediction, and of 1 2 MeV or mW 3 MeV for the predictions improved with the NLO+PS 4FS, in the POWHEG-BOX and MG5 aMC respectively. We take the results based on the xedorder NLO 4FS calculation as a technical benchmark, while we consider the results obtained matching xed- and all-orders calculations, discussed in detail in section 3.4, as more accurate in the description of these transverse-momentum distributions. In conclusion, our estimate of the mW mass shift due to b-quark e ects, in the measurement from the lepton transverse-momentum distribution, is in general smaller than 5 MeV. We conclude our section by investigating how the extracted W -mass shift depends on the details of the ? tting window, in particular on its boundaries. To do so, we compute the shift when the ? ? window boundaries [pmin; pmax] vary in the ranges 30 GeV < pmin < 35 GeV and 45 GeV < pmax < 50 GeV. The resulting values of the mass shift are shown in gure 15 and can ? be justi ed with a comparison with gure 13, where the reweighted distributions and the 10We identify the neutrino transverse momentum and the missing transverse energy, i.e. p?( ) p?miss. Result of the template t to distributions that include the improved bottom-quark e ects in di erent QCD approximations. templates are compared. We should consider, for a given template j, which bins contribute the most to the j2 value and this can be guessed at glance by checking when a reweighted distribution follows and when it deviates from the template. Above the Jacobian peak the reweighting procedure yields a shape qualitatively di erent than any of the templates, so that the inclusion of the bins where the deviation is more pronounced may a ect the position of the minimum of the 2 parabola. This problem is particularly evident when considering the xed-order results (green dots in gure 13), which correspond to the lower right plot of gure 15. Dependence on the t window of the template- t results, in the case of the lepton transverse-momentum distribution. The black marks correspond to the t range used in gure 14. The high luminosity of the LHC together with the stunning performances of the detectors (ATLAS, CMS, and also LHCb) have turned Drell-Yan processes into high-precision arenas where to test our understanding of the fundamental interactions on the one hand and to perform the most precise measurements of the parameters of the SM, on the other hand. At this level of precision one needs to control not only higher-order perturbative e ects, QCD as well as EW, but also less obvious ones, such as non-perturbative or parametric e ects. An interesting example, discussed in this work, is given by the contribution from bottom quarks to Drell-Yan processes which so far has been considered in the massless approximation. In fact, the associated production of a lepton pair together with a bb pair is a rather complicated process, featuring several (if not all the) aspects that make the description of nal states involving b quarks an interesting challenge for theorists as well as for experimentalists. In this paper we have considered `+` bb production in the 4FS at the LHC with the main goal of assessing the accuracy and precision currently achievable of `+` inclusive over the bottom quarks. To this aim we have employed state-of-the-art Monte Carlo tools accurate at NLO in QCD and matched to parton showers. We have shown that predictions from di erent NLO MC tools for quantities that are inclusive with respect to the bottom quark in the `+` bb nal state are in agreement within the expected uncertainties. We have employed a simple prescription which makes it possible to consistently include the contributions from massive bottom quarks into the inclusive DY production calculated in the 5FS, and studied their e ects together with the associated uncertainties. In so doing we have been able to estimate that, when the W -mass extraction is mostly sensitive to p?(`+) (as it is the case in the ATLAS measurement [46]), the residual uncertainties have a small but not negligible ( mW < 5 MeV) impact on the W -mass extraction. The stability of this prescription, with respect to the inclusion of higher-order QCD corrections, could be further explored with the help of codes which make it possible to match QCDPS with NNLO-QCD accurate predictions for Drell-Yan processes [28{30]. We have also performed an extensive study in the 4FS of the observables that are exclusive on the bottom quarks. This analysis, which is documented in appendix A, reveals di erences between formally equivalent methods that are larger than the (estimated) associated uncertainties, at least in some cases. A thorough comparison of many distributions has allowed us to identify the regions in phase space where the di erences arise. Assessing the origin of such discrepancies in the speci c case of `+` bb and providing a resolution will be an important task for the SM and BSM programme of the LHC (see e.g., the measurement of HZ-associated production at small transverse momentum and the search for dark matter in the missing-transverse energy +b-jet nal states are two examples directly related to `+` bb). A deeper understanding of the treatment of the bottom quark contributions can be crucial also for the precision prediction of very important nal states like ttbb. This, however, needs a dedicated e ort which goes beyond the scope of our work and it is left for future investigations. Finally, we would like to comment on the opportunity of performing a similar study for the charm-initiated contribution to NC-DY, which was brie y touched upon in section 1. First, because of the smaller mass of the charm, initial-state logarithms should be larger than for the bottom, even in the low-p`+` range considered in this paper. Second, powers of the charm mass, which typically enter the cross-section trough terms (mc=Q)k, will be more suppressed than the corresponding ones involving mb by a factor (mb=mc)k 3k, while the ratio between the relative contributions initiated by the charm and the bottom is 2:3. Both points suggest that a scheme where the charm quark is treated as massless should be more appropriate for these contributions. More detailed studies in this direction would be welcome. Acknowledgments We would like to thank Maarten Boonekamp, Stefano Camarda, Rikkert Frederix, Stefano Frixione, Luca Perrozzi, Paolo Torrielli, for many stimulating discussions. MZ is grateful to the Physics Department of the University of Milan for the hospitality in many di erent occasions. This work has received funding from the European Union's Horizon 2020 research and innovation programme as part of the Marie Sklodowska-Curie Innovative Training Network MCnetITN3 (grant agreement no. 722104). EB is supported by the Collaborative Research Center SFB676 of the DFG, \Particles, Strings and the early Universe". AV is supported by the Executive Research Agency (REA) of the European Commission under the Grant Agreement PITN-GA-2012-316704 (HiggsTools) and by the European Research Council under the Grant Agreement 740006NNNPDFERC-2016-ADG/ERC-2016ADG. MZ is supported by the Netherlands National Organisation for Scienti c Research (NWO), by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodovska-Curie grant agreement No 660171 and in part by the ILP LABEX (ANR-10-LABX-63), in turn supported by French state funds managed by the ANR within the \Investissements d'Avenir" programme under reference ANR-11-IDEX-0004-02. A Di erential observables in `+` bb production In this appendix we compare results obtained with di erent PS and/or matching schemes for various di erential observables in pp ! `+` bb production, possibly distinguishing di erent signatures depending on the number of tagged b-jets. We use the setup described in section 2. After parton shower and hadronisation, hadrons are clustered into jets using the anti-kT algorithm [69] as implemented in FastJet [70, 71], using a radius parameter R = 0:4. Jets are required to satisfy the following conditions p?(j) > 30 GeV ; j (j)j < 2:5 : (A.1) A jet is considered as a B-tagged jet if at least one B- avoured hadron is found among its constituents. For xed-order predictions we apply the same jet-clustering algorithm to QCD partons (gluons and quarks, including the b), and we consider a jet as B-tagged if at least one b quark appears among its constituents. In both cases we assume a 100% B-tagging e ciency and zero mis-tagging rate. The point of this comparison is to stress the fact that the di erences that emerge by employing di erent matching approaches and QCD PS models (as one can appreciate in gures 5, and 16{30), make it apparent that higher-order terms with respect to the s expansion, subleading in the counting of logarithmic enhancing factors, can nevertheless be numerically sizeable. We consider the width of the envelope of the di erent uncertainty bands presented in these gures as a conservative quantity useful to characterise our level of understanding of the observable under consideration and of the accuracy of our simulations. When we observe a similar shape in the correction factors expressing the impact of all the terms beyond NLO-QCD, we tend to consider the envelope a reliable conservative estimate of the residual uncertainties; when this is the case, in all the plots considered the envelope has a width typically of O( 20%) with respect to its mid point, a value that also represent the typical uncertainty from scale variations in most kinematic con gurations (scale and PDF uncertainties are shown for all di erential observables). When instead we observe di erent trends in the corrections, rather than quoting a very large uncertainty, we can only argue that the comparison is signalling the presence of a quantity whose description is very sensitive to the details of the radiation and deserves further analytical and numerical investigation. The colour code employed in all gures in this appendix is the same as in gure 5. For the sake of clarity, we report it again here: in black we present the xed-order NLO calculation; in blue we show results obtained with MG5 aMC and Py8, using as reference shower scale, sh, the partonic variable show results obtained with MG5 aMC and Py8, using as sh the sum of the transverse masses HT =2 (dashed) or HT =4 (solid); in green we show results obtained with POWHEG and Py8, setting the value of the scale h in the damping factor equal to mZ ps^=2 (dashed) or (dashed) or mZ=4 (solid). A.1 Jet multiplicities The rst observable we investigate is the number of reconstructed b jets, shown in gure 16. With respect to the normal layout of the gures, for this speci c observable we also show as a green-patterned band the uncertainty related to the variation of Qsh in the \remnant" events of the POWHEG-BOX samples, as described in section 2. Higher-order QCD corrections play a non trivial role in the jet reconstruction, yielding in turn sizeable e ects. The b-jet multiplicity is thus the rst quantity that has to be discussed, for a correct interpretation also of the other observables. The largest bin is the one with zero b-jets, because the production of b quarks is due to the collinear splitting of the incoming gluons, so that the transverse momentum of the jet that includes the b quark does not ful l the jet de nition. The number of events with 1 or 2 b jets depends on the transverse momentum distribution of the nal state b and b at NLO-QCD. Higher-order corrections beyond NLOQCD, simulated with a QCD-PS, yield a redistribution of the events. We observe that in MG5 aMC there is a moderate stability of the 0-jet and 1-jet bins (changes do not exceed the 5% level) and an increase of the 2-jets bin with respect to the xed NLO prediction. The precise description of the e ects in the rst two bins and their overall stability depend on the details of the QCD-PS and PS phase space adopted. The increase of the third bin is due to a migration of events from the 1-jet to the 2-jets bin. Even if the absolute number of events that migrate is not large, the percentage e ect is large, of O(+20%), because of the steeply-falling shape of the distribution. The hard recoil of the `+` bb system, compared to the xed-order prediction, in MG5 aMC at intermediate transverse momentum values (see gure 5) may explain the larger number of events with both b quarks passing the b-jet requirements. In the POWHEG-BOX case we observe an increase of the 0-jet bin and a corresponding reduction of the rates with 1 and 2 b jets (the observables with a genuine NLO accuracy), independently of the value of the h scale in the damping parameter, if the default prescription for Qsh (i.e. the transverse momentum of the rst, hardest emission) is used. Variations in the shower scale of the \remnant" events give instead e ects comparable with those from shower-scale variations in MG5 aMC , in the 2 b-jets bin. The latter is the most sensitive to changes in the treatment of the \remnant" events, characterised by a large transverse momentum of the rst parton. Although the variation of Qsh will not be shown for the other observables presented in this section, the reader should keep in mind that the h variation in the POWHEG-BOX may give only a partial estimate of the theoretical uncertainties, and that other sources of uncertainty exist. Events with 3 or 4 b jets are due to additional splittings via the QCD-PS and are a ected by large parametric uncertainties. A.2 with extra tagged b jets ? In gure 17 we show the results obtained for the p`+` distribution, where the lepton pair is detected in association with at least 1 b jet. The size of the higher-order corrections, ? ! 0, the choice in MG5 aMC of the variable used to select sh is very important, yielding positive (O(+20%)) or negative (O( 20%)) e ects with s^ or HT =2 respectively. At moderate or large p`+` (above 100 GeV), there is a good shape agreement among the > 50 GeV. In the limit di erent matched predictions, while di erences in normalisation re ect those the 1-jet bin in gure 16. In gure 18 we show the results obtained for the p`+` distribution, where the lepton pair is detected in association with at least 2 b jets. Corrections with respect to xed ? NLO span from +30% for MG5 aMC+Py8 down to -20% for POWHEG-BOX+Py8 but they are rather at in shape. This behaviour is associated to the positive impact of QCD-PS corrections in the value of the 2 b-jets multiplicity. In both gures 17 and 18 we observed a fair compatibility of the predictions computed with the di erent options of matching scheme and PS, once the di erences in normalisation are accounted for. Invariant mass of the two hardest b jets In gure 19 we consider a nal state with at least 2 b jets and study the invariant-mass distribution of the hardest b-jet pair. This observable is particularly relevant for Higgs searches in V H-associated production. Thee related observable in ttbb production is known to have large sensitivity to secondary g ! bb splittings generated by the PS [72, 73]. We have checked that such e ects are milder (at the 10% level) for `+` bb. However, the dependence on the details of the matching remain sizeable. At very low invariant masses we observe a large negative correction in all matching schemes, due to the de nition of b jet and to the action of the QCD-PS: at xed NLO the b jets contain, beside the b quarks, at most one additional parton and the jet mass is therefore rather close to the b-quark mass; the inclusion of additional partons via QCD-PS rapidly increases the total jet mass, with a consequent migration of events to the larger dijet-mass bins and a corresponding depletion of the rst ones. At larger invariant masses, for m(b1b2) > 50 GeV, we observe that the PS corrections obtained with MG5 aMC are positive, with the predictions matched to Py8 reaching the +40% level when m(b1b2) 500 GeV. The e ect of matching to Hw++ is milder and atter with respect to xed NLO, while the POWHEGBOX predictions lie below it. In gure 19 the di erences between the various matching options are a consequence of the jet de nition, because the largest fraction of the radiative e ects due to collinear emissions is integrated in the jet cone. Looking at the uncertainty bands due to shower-scale (in MG5 aMC) and h variations (in the POWHEG-BOX), we notice that the latter are visibly smaller than the former. A similar behaviour has been observed for the same observable in the context of the ttbb implementation in the POWHEG-BOX [73]. jets In gures 20, 21 and 22 we study a more exclusive observable with respect to the one of gure 19, i.e. we consider the production of a pair of B hadrons and plot the invariantmass distribution of the pair made by the two hardest B hadrons in the event, in events characterised by the presence of at least 0, 1 or 2 b jets respectively. We do not require that the two hardest B hadrons belong to any of the tagged jets, nor we ask that they satisfy any cut in order to be detected. In the case where no b jet is explicitly requested, shown in gure 20, we observe that the MG5 aMC+Py8 results are largely independent of the choice of the variable and of the interval used to extract Qsh, but that there is a strong sensitivity to the QCDPS model, with di erences between Py8 and Hw++ at the 20% level. Curiously enough, MG5 aMC+Hw++ is rather similar to POWHEG-BOX+Py8. All matched predictions are considerably softer than those at xed NLO, in which the B hadrons are replaced by the b quarks, because of the loss of energy due to the fragmentation of the latter into the former. In the case with at least 1 b jet, shown in gure 21, we observe in the MG5 aMC results that the choice of the variable used to extract Qsh, namely s^ vs HT =2, yields di erences at the 10 20% level at large invariant masses. From the MG5 aMC+Py8 histograms one can appreciate the fact that, once the matching scheme and the PS are xed, the pattern of the predictions closely follows those of the shower-scale distribution shown in gure 2, with the hardest prediction corresponding to the largest shower-scale. The di erences between Py8 and Hw++ are sizeable through the whole invariant mass spectrum, both in shape and in size of the corrections. We stress that these e ects are due to terms beyond NLOQCD in the perturbative expansion. As in the case without explicitly asking extra b jets, predictions with MG5 aMC+Hw++ are close to those with POWHEG-BOX+Py8. Finally, in the case with at least 2 b jets, shown in gure 22, we observe in the MG5 aMC results that there is a good agreement between the di erent options of matching xed- and all-orders results and between Py8 and Hw++. We also observe the large size of the radiative e ects in the rst two invariant mass bins, where the higher orders enhance the cross section up to a factor +120%, while at large invariant masses the corrections range from being negative (-20%) to being compatible with zero. This large correction is explained as due to the appearance via QCD-PS of events where both B hadrons belong to the same jet (because of secondary g ! bb splittings) and turn out to be the hardest pair, but, at the same time, have a small invariant mass. For what concerns the POWHEGBOX predictions, they fall below and are manifestly softer than xed NLO for values of the invariant mass starting at 100 GeV, with a depletion of rate that can reach 50% at m(B1; B2) 500 GeV. A.5 distance of the two hardest b jets We introduce the distance R(ij) p(yi yj )2 + ( i j )2 between particles i and j, whose rapidity and azimuthal angle are denoted with y and , and show, in gure 23, the R distribution of the hardest b-jet pair. distribution for the distance between the two hardest b jets. For this observable, matched predictions can display large di erences. More in detail, while the POWHEG-BOX+Py8 prediction is rather close to the xed-order one, with almost no visible shape distortion, the MG5 aMC ones show sizeable deviations, particularly for R(b1; b2) > . In this region, the MG5 aMC+Py8 prediction with the largest shower scale can lead to rates which are larger than the xed-order predictions by a factor 1.5-2. This is partially mitigated by the choice of smaller values for the shower scale HT or by matching with Hw++. In fact, in these two cases, predictions show a rather similar behaviour: up to R(b1; b2) = they lie quite close to the xed-order one; starting from respect to the xed-order one, up to a factor 1.4 at R(b1; b2) = the rate is enhanced with R(b1; b2) = 4:5. Finally, for very large R(b1; b2), these matched predictions seem suppressed with respect to the xed-order one, although the statistics for this kinematic region is quite poor. jets A.6 distance of the two hardest B hadrons with or without tagged b We show in gures 24, 25 and 26, the distributions with respect to the distance R(B1; B2) between the two hardest B hadrons, in presence of an increasing number of b jets (at least 0, 1 and 2). As it has been the case for the corresponding invariant-mass distributions ( gures 20{22) we do not require that the two hardest B hadrons belong to any of the tagged jets, nor any condition for tagging the B hadrons is required. When no b jet is explicitly required ( gure 24), the two B hadrons can reach quite large distances ( R 10) keeping sizeable rates. However, for such large distances, the two B hadrons are typically in an extreme forward-backward con guration, in kinematic regions where no or poor detector coverage exists. Nevertheless, it remain interesting to study how di erent predictions behave. Looking at the distributions, we observe important discrepancies among the various predictions: if we compare to the xed-order distribution, the prediction matched with the POWHEG-BOX shows the smallest deviations, which Figure 24. R distribution of the B hadrons pair in association with at least 0 b jets. Figure 25. R distribution of the B hadrons pair in association with at least 1 b jet. remain well below 10% over all the range that we display. Conversely, the MG5 aMCmatched predictions show quite large discrepancies: when Py8 is employed, the prediction is suppressed at small and large distances ( R < 3 and R > 6 7), while it is mildly enhanced (up to +10%) at intermediate distances. While at small and moderate distances the behaviour of the Py8-matched predictions is only marginally dependent on the choice of shower scale, and the departure from the xed-order prediction reaches at most 20% at very small distances, at large distances such a dependence is apparent, with larger shower scales leading to bigger suppressions, with the predictions suppressed by a factor two or more with respect to the xed-order one. If instead MG5 aMC+Hw++ is employed, the behaviour is even more complicated, but overall the deviations with respect to the xedorder predictions are smaller than with Py8: at very small distances the Hw++-matched Figure 26. R distribution of the B hadrons pair in association with at least 2 b jets. prediction lies below the xed-order one, with a suppression of 20%. For distances in the range 1 < R(B1; B2) < 4, the matched prediction lies 5% above the xed-order one, while R(B1; B2) < 9 it is again below, with a suppression between 10% and 15%. Finally, at very large distances R(B1; B2) > 9, the matched prediction returns 20% above the xed-order one. This enhancement of the MG5 aMC+Hw++ prediction has been also observed for bbH associated production [74] and for charged-Higgs production in association with a top quark [75]. Requiring at least one b jet partially mitigates these discrepancies: the POWHEGBOX prediction is very similar to the MG5 aMC+Hw++ one, and both are also similar to the one obtained with MG5 aMC+Py8 and with a shower scale HT , up to R(B1; B2) = 6 (for larger distances, the latter prediction predicts a suppressed rate with respect to the former ones). When these matched predictions are compared to the xed-order one, the behaviour is not much di erent from the case without extra jets: predictions are suppressed (up to 20%) for small distances ( R(B1; B2) < 1), for intermediate distances (1 < R(B1; B2) < 4) they behave similarly to the xed-order one, while at larger distances they are again suppressed ( 30% for the POWHEGBOX and MG5 aMC+Hw++ and up to Finally, the MG5 aMC+Py8 with Qsh p 50% for MG5 aMC+Py8 with Qsh HT ). s^ follows the other matched predictions up to R(B1; B2) = 3:5. It then keeps growing with respect to the xed-order one for about one unit of distances, where the enhancement with respect to the xed-order prediction reaches +10%, nally for large distances it predicts suppressed rates with respect to the xed order one, with the suppression reaching 40%. For this last prediction, a variation of Qsh by a factor two can have an e ect as large as 10% for R(B1; B2) > 4, while for the other predictions the shower-scale dependence is much smaller. Finally, when two b jets are required, gure 26, the R(B1B2) distributions closely follow the corresponding counterparts for the distance between the two b jets, shown in gure 23. Transverse momentum distributions of the two hardest b jets In gures 27, 28 we show the transverse momentum of the hardest and the second hardest b jets. For the transverse momentum of the hardest jet, the general behaviour of matched computations is to be softer than the xed-order prediction, and this e ect is more pronounced for predictions matched with Py8 than for the ones matched with Hw++. For small values of the transverse momentum, di erences among the matched simulations are moderate (at the level of 10%) and re ect the pattern observed for the one-jet multiplicity displayed in gure 16, while for larger values such di erences are mitigated. For the second-hardest b jet, no visible distortions of the matched spectra with respect to the xed-order one can be appreciated, and di erences in rate re ect those of the two-jet bin in gure 16. Pseudo-rapidity distributions of the two hardest b jets In gures 29, 30 we show the pseudo-rapidity distributions of the hardest and the second hardest b jets. As it has been the case for their transverse-momentum counterpart, differences in rate re ect the one-jet and two-jet bins of gure 16. Besides these di erences, it is worth to note that matched predictions have the general tendency to populate more the forward and backward regions with respect to the xed-order ones. 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Emanuele Bagnaschi, Fabio Maltoni, Alessandro Vicini, Marco Zaro. Lepton-pair production in association with a \( b\overline{b} \) pair and the determination of the W boson mass, Journal of High Energy Physics, 2018, 101, DOI: 10.1007/JHEP07(2018)101