Top-quark pair hadroproduction in association with a heavy boson at NLO+NNLL including EW corrections
Alessandro Broggio, Andrea Ferroglia, Rikkert Frederix, Davide Pagani,, Benjamin D. Pecjak, Ioannis Tsinikos
TL;DR
This paper provides precise theoretical predictions for top-quark pair production with a heavy boson at the LHC, including NLO electroweak and QCD corrections and soft gluon resummation, aiding experimental analyses.
Contribution
It offers the first comprehensive NLO+NNLL predictions for associated top-quark pair production with electroweak corrections at 13 TeV LHC.
Findings
Predicted total cross sections and differential distributions with high precision.
Quantified the impact of electroweak and QCD corrections on observables.
Analyzed the residual uncertainties from scale and PDF choices.
Abstract
This work studies the associated production of a top-quark pair with a , , or Higgs boson at the LHC. Predictions for the total cross sections as well as for several differential distributions of the massive particles in the final state are provided. These predictions, valid for the LHC operating at TeV, include without any approximation all the NLO electroweak and QCD contributions of with . In addition, the predictions presented here improve upon the NLO QCD results by adding the effects of soft gluon emission corrections resummed to next-to-next-to-leading logarithmic accuracy. The residual dependence of the predictions on scale and PDF choices is analyzed.
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aainstitutetext: Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italybbinstitutetext: INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italyccinstitutetext: Physics Department, New York City College of Technology, The City University of New York, 300 Jay Street, Brooklyn, NY 11201 USAddinstitutetext: The Graduate School and University Center, The City University of New York, 365 Fifth Avenue, New York, NY 10016 USAeeinstitutetext: Theoretical Particle Physics, Department of Astronomy and Theoretical Physics, Lund University, Sölvegatan 14A, SE-223 62 Lund, Swedenffinstitutetext: Technische Universität München, James-Franck-Str. 1, D-85748 Garching, Germanygginstitutetext: Institute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Department of Physics, University of Durham, Science Laboratories, South Rd, Durham DH1 3LE, United Kingdom111Preprint: TUM-HEP-1208/19, LU-TP 19-30, IPPP/19/57
Top-quark pair hadroproduction in association with a heavy boson at NLO+NNLL including EW corrections
Alessandro Broggio c,d
Andrea Ferroglia e
Rikkert Frederix f
Davide Pagani g
Benjamin D. Pecjak f
Ioannis Tsinikos
Abstract
This work studies the associated production of a top-quark pair with a , , or Higgs boson at the LHC. Predictions for the total cross sections as well as for several differential distributions of the massive particles in the final state are provided. These predictions, valid for the LHC operating at TeV, include without any approximation all the NLO electroweak and QCD contributions of with . In addition, the predictions presented here improve upon the NLO QCD results by adding the effects of soft gluon emission corrections resummed to next-to-next-to-leading logarithmic accuracy. The residual dependence of the predictions on scale and PDF choices is analyzed.
1 Introduction
The Run-II at the LHC, with a center of mass energy of TeV and a higher instant luminosity w.r.t. Run-I, made this collider a fully operational top-quark factory. Indeed, the heaviest of the Standard Model (SM) particles can be produced via different channels, many of which have been observed at the LHC. To date, not only top-quark pair Abe:1995hr ; D0:1995jca ; Aad:2010ey ; Khachatryan:2010ez and single top-production Aaltonen:2009jj ; Abazov:2009ii ; Chatrchyan:2011vp ; Aad:2012ux ; Chatrchyan:2014tua modes, but also the production of a top-quark pair in association with a heavy electroweak (EW) boson have been measured. The latter class involves Khachatryan:2015sha ; Aad:2015eua ; Aaboud:2016xve ; Sirunyan:2017uzs , Khachatryan:2015sha ; Aad:2015eua ; Aaboud:2016xve ; Sirunyan:2017uzs ; Sirunyan:2017nbr and Aaboud:2018urx ; Sirunyan:2018hoz production processes. These three processes are extremely important in the searches for beyond-the-SM (BSM) effects, both as components of the background and of the signal itself. For example, and production constitute the main backgrounds in the measurement of the leptonic signatures emerging from production Aaboud:2017jvq ; Sirunyan:2018shy ; Maltoni:2015ena , which in turn enables the direct measurement of the coupling of the top quark to the Higgs boson. Analogously, the production process can be employed for the measurement of the coupling of the top quark to the boson Sirunyan:2017uzs ; Bylund:2016phk . Finally, it is worth noting that very recently also single-top plus associated production was observed Sirunyan:2018zgs .
For a correct interpretation of current and future measurements and the possible identification of BSM effects, precise predictions for these processes, and, consequently, the study of their radiative corrections, are of paramount relevance. For top-quark pair and single-top production next-to-next-to-leading (NNLO) QCD corrections were computed in Czakon:2013goa ; Brucherseifer:2014ama ; Berger:2016oht ; Liu:2018gxa ; Catani:2019hip . For top pair production, also next-to-leading-order (NLO) electroweak (EW) corrections Czakon:2017wor ; Czakon:2017lgo and/or next-to-next-to-leading-logarithmic (NNLL) accuracy resummation of threshold and small-mass logarithms Czakon:2018nun ; Czakon:2019txp ) were accounted for. This level of accuracy is not yet achievable for processes with three massive particles (two of which are colored) in the final state, nor is it expected in the near future. Still, it is desirable to have the best possible current predictions, i.e. those which include all corrections of QCD and EW origin that can be calculated with current technology. In addition, it is necessary to thoroughly study the phenomenological impact of these predictions at the differential level.
In this paper we provide state-of-the-art SM predictions for top-quark pair hadroproduction in association with an EW heavy boson; we calculate the complete-NLO predictions for , and in proton–proton collisions at 13 TeV and we resum soft gluon emission effects at NNLL accuracy in QCD. All the EW and QCD contributions of with are evaluated without any approximation. In addition, in Mellin space, the resummation procedure accounts for terms proportional with at all orders () in , where with the Mellin parameter, and is the soft emission limit.
The calculation of the complete-NLO corrections to production is based on the work in Frederix:2017wme and has been carried out with the new public version of MadGraph5_aMC@NLO Frederix:2018nkq . This code was also used to obtain complete-NLO corrections to and production. The calculations of soft gluon effects to NNLL accuracy in QCD for , and are based on the work in Broggio:2015lya ; Broggio:2016zgg ; Broggio:2016lfj ; Broggio:2017kzi ; Broggio:2017oyu and on the in-house parton level Monte Carlo code that was developed for those papers. The resummation of soft emission effects was also studied in Kulesza:2017ukk ; Kulesza:2018tqz , where a resummation framework different from the one considered in Broggio:2015lya ; Broggio:2016zgg ; Broggio:2016lfj ; Broggio:2017kzi was employed. Very recently, in Ju:2019lwp also the resummation of Coulomb effects for production was studied.
The paper is organized as follows. In Section 2 we briefly summarize the salient features of the calculational framework used in order to evaluate the various corrections. Section 3 includes a description of the input parameters and PDF sets employed in the calculation, as well a discussion of the values chosen for the factorization and resummation scales. Predictions for the total cross section and differential distributions for the processes considered in this study are collected in Section 4. Finally, Section 5 contains our conclusions.
2 Calculational framework
In this section we describe the calculational framework on which the phenomenological predictions presented in Section 4 are based. In Sections 2.1 and 2.2 we briefly summarize the calculation of the complete-NLO corrections of QCD and EW origin Frederix:2017wme ; Frederix:2018nkq and the resummation of soft-gluon effects at NNLL accuracy Broggio:2015lya ; Broggio:2016zgg ; Broggio:2016lfj ; Broggio:2017kzi , respectively. In Section 2.3 we explain how the combination and matching of complete-NLO and resummation of soft-gluon effects is carried out. We will denote the class of processes considered in this work as , where can be or . In Section 2.4 we recall the most relevant phenomenological features of the different contributions entering the complete-NLO calculation, and comment on the implications for soft gluon resummation.
2.1 Complete-NLO
The fixed order expansion of a generic observable for the processes (where indicates that the process is inclusive over extra QCD and QED radiation) in powers of and can be expressed as
[TABLE]
with and positive integers. LO contributions consist of terms with and involve tree-level diagrams only. NLO corrections correspond to the terms with and are induced by the interference among all the possible one-loop and tree-level Born diagrams as well among all the possible tree-level diagrams involving one additional quark, gluon or photon in the final state.
In this work, “complete-NLO” is used to indicate the quantity , in which all terms with are included. On the other hand, a more user-friendly notation can be used to refer to any individual term in Eq. (1). We denote observables at LO as and further redefine the individual perturbative orders as
[TABLE]
Similarly, NLO corrections and their individual perturbative orders can be defined as
[TABLE]
In contrast to the notation used in previous works Frixione:2014qaa ; Frixione:2015zaa ; Pagani:2016caq ; Frederix:2016ost ; Czakon:2017wor ; Frederix:2017wme ; Frederix:2018nkq , here and in the rest of the text indicates a quantity that does not include any LO contribution, while indicates a quantity that does include LO contributions. In particular, all the are included for predictions beyond the LO, unless the subscript “QCD” is present; in this case only the is included. Consequently, with this convention an observable evaluated at complete-NLO accuracy can be written as
[TABLE]
Observe that the quantities and are in general defined in such a way that they do include the appropriate multiplicative factor of powers of and , as shown in Eqs. (2) and (3). We use the symbols or interchangeably their shortened aliases to indicate individual terms in the l.h.s. of Eqs. (2) and (3). It is important to remember that in the literature the term “LO” usually refers only to , which instead here is denoted by . Therefore, with this notation one has
[TABLE]
NLO EW corrections, which are of w.r.t. the observable, correspond to the terms, so we also denote them as . On occasion, we also refer to the set of and corrections with as “electroweak corrections” (not to be confused with corrections just defined). The prediction at complete-NLO accuracy, which is the sum of all the and () terms, is denoted as “” 222In Ref. Frederix:2017wme a slightly different notation has been used. Therein NLO and predictions refer to the corrections only, without including LO contributions. On the other hand, note that for the case of EW corrections also here .. Consistently with the notation introduced above, the sum of the corrections without the LO is indicated by (see Eq. (3)).
It is important to point out that for all the processes we do not include the (finite) contributions from the real-emission of heavy particles (, and bosons), also denoted in the literature as heavy-boson-radiation (HBR) contributions. Although they are formally part of the inclusive predictions at complete-NLO accuracy, these finite contributions in general lead to very different collider signatures and are typically small. For processes, the HBR contributions to were evaluated in Frixione:2015zaa .
The calculation of the complete-NLO predictions is carried out by employing the latest version of MadGraph5_aMC@NLO Frederix:2018nkq , which is now public. In MadGraph5_aMC@NLO, the FKS method Frixione:1995ms ; Frixione:1997np (automated in the module MadFKS Frederix:2009yq ; Frederix:2016rdc ) is used in order to deal with infrared singularities. One-loop amplitudes are evaluated by dynamically switching among different kinds of techniques for integral reduction, namely, the OPP method Ossola:2006us , the Laurent-series expansion Mastrolia:2012bu , and the tensor integral reduction Passarino:1978jh ; Davydychev:1991va ; Denner:2005nn . These techniques are automated in the module MadLoop Hirschi:2011pa , which is used for generating the amplitudes. We remind the reader that MadLoop employs CutTools Ossola:2007ax , Ninja Peraro:2014cba ; Hirschi:2016mdz and Collier Denner:2016kdg , and includes an in-house implementation of the OpenLoops optimization Cascioli:2011va .
2.2 Resummation
The resummation of the soft-gluon emission corrections to the production processes is carried out as described in detail in Broggio:2016zgg ; Broggio:2016lfj ; Broggio:2017kzi , with techniques based on Soft Collinear Effective Theory333For an introductory review of SCET, see Becher:2014oda (SCET) Bauer:2000yr ; Bauer:2001yt ; Beneke:2002ph and renormalization-group-improved perturbation theory. We summarize here the salient features of the resummation procedure. In production, the underlying partonic processes are of the form
[TABLE]
where indicates unobserved final-state light-quark and/or gluon radiation. The incoming partons which enter the production process depend on the boson under consideration. At lowest order in QCD, if then , where indicates the isospin partner of the quark . If instead, both the quark-annihilation channel and the gluon-fusion channel contribute to the process, so that .
One can then define the invariants444In Refs. Broggio:2015lya ; Broggio:2016lfj ; Broggio:2016zgg ; Broggio:2017kzi , as well as in a number of papers on top-quark pair production (see for example Ahrens:2009uz ; Ahrens:2010zv ; Broggio:2013uba ; Broggio:2014yca ), the invariant mass of the massive particles in the final states is indicated by , as it is done in this section. However, in Section 4 we discuss simultaneously results for and production. In that section, in order to avoid any possible source of confusion, we differentiate the three different processes considered by indicating the invariant mass of each one of them as , respectively.
[TABLE]
and starting from these quantities one can define the parameter
[TABLE]
At lowest order in QCD, , while beyond leading order . We define the “soft” or “partonic threshold” limit as the limit , since in this limit the final state radiation must be soft.
In the partonic threshold limit the production cross section factorizes as follows:
[TABLE]
In Eq. (9), indicates the square of the hadronic center of mass energy, the symbol is used to indicate the list of momenta , while
[TABLE]
The functions and are the hard function, the soft function and the parton luminosity function, respectively. These functions are channel dependent and therefore they appear in Eq. (9) with an subscript. The trace of the product of the hard and soft function is integrated over the phase space, whose integration measure is indicated by . The hard and soft functions are matrices in color space. Only partonic channels that are open at contribute to the cross section in the partonic threshold limit. In the quark-annihilation channel, which contributes to and , the hard and soft functions are two-by-two matrices in color space, while in the gluon-fusion channel, which contributes only to and , the hard and soft functions are three-by-three matrices. Details on the definition of the hard, soft and luminosity functions as well as on the final state phase space can be found in Refs. Broggio:2015lya ; Broggio:2016zgg ; Broggio:2016lfj ; Broggio:2017kzi .
It is important to observe that the soft functions are singular in the partonic threshold limit . They contain delta functions and plus distributions of the form
[TABLE]
The plus distributions are defined in such a way that they can be integrated up to ; if represents a smooth test function that is not singular in the limit, then one has
[TABLE]
At each fixed order in perturbation theory, the soft function involves terms proportional to , where indicates the order of QCD corrections and . For example, NLO QCD corrections include and distributions, NNLO QCD corrections include and distributions, etc. These terms arise from soft gluon emission corrections and provide numerically large contributions to the hadronic cross section and differential distributions. In a sense, the purpose of resummation is to account for some of the terms proportional to the plus distributions to all orders in perturbation theory. One convenient way of achieving this goal is to derive and solve the renormalization group equations satisfied by the hard and soft functions. The renormalization group equations are regulated by anomalous dimensions, which were computed to two loops in Refs. Ferroglia:2009ep ; Ferroglia:2009ii .
The hard functions and soft functions are free from large logarithmic corrections at appropriately chosen (and different) scales and . At those scales, the hard and soft functions are well behaved in fixed-order perturbation theory. In order to achieve NNLL accuracy, one needs to evaluate the hard and soft function up to NLO. The soft functions are process independent. The soft function for the quark-annihilation channel in is identical to the quark-annihilation channel soft function for or , up to a trivial replacement of the mass of the heavy boson or , respectively. Similarly, the soft functions for the gluon-fusion channel in and production are also identical. The NLO hard functions are instead process dependent. They receive contributions only from one-loop QCD corrections to the production channels that are already open at tree level in QCD: quark annihilation channel for production, quark-annihilation and gluon-fusion channels for and production. The hard functions needed for this work were evaluated by means of a customized version of the code Openloops Cascioli:2011va run in combination with the tensor reduction library Collier Denner:2016kdg .
The resummation of the soft emission corrections is carried out in Mellin space, where the integral form of the cross section becomes
[TABLE]
The Mellin parameter is indicated by and the threshold limit corresponds to the limit in Mellin space. The functions and are the Mellin transforms of the luminosity function and of the trace of the product of the hard and soft function, respectively. The plus distributions found in the soft function in momentum space are mapped into logarithms of the Mellin parameter in Mellin space, such that in Mellin space the QCD corrections contain terms of the form , with . Terms suppressed by inverse powers of in the partonic cross section in Mellin space are neglected in Eq. (13).
While the hard and soft functions included in are evaluated in fixed order perturbation theory at the scales and , their product is evolved to a common scale by solving the renormalization group equations satisfied by the functions. The scale is the scale which enters in the PDFs and, consequently, in the parton luminosity function . Ultimately, the resummed hard scattering kernels have the following structure
[TABLE]
with . The evolution factors include the full dependence on potentially large logarithms of the ratios and are, like the hard and soft functions, channel-dependent matrices in color space. The explicit expression for the evolution factors in terms of the anomalous dimensions regulating the renormalization group equations can be found in Eq. (3.7) in reference Broggio:2016zgg for the case. The evolution factors are identical also for the and cases, provided that one accounts for the fact that the explicit expressions of the anomalous dimensions are different for the quark-annihilation and gluon-fusion channels.
If all of the factors in the r.h.s. of Eq. (14) were known at all orders in perturbation theory, the l.h.s. of the equation would not depend on nor on . However, since the hard function, the soft function, and the anomalous dimensions entering in the evolution factor are all evaluated up to a certain order in perturbation theory, a residual numerical dependence on the choice of and remains in the predictions presented in Section 4. This residual dependence on the scale choices is used, as usual in QCD, to estimate the theoretical error induced by the truncation of the perturbative series in the calculation of the various elements in the resummation formula. As discussed above, one should choose the hard and soft scales and in such a way that the hard and soft functions are free from large logarithmic corrections and are therefore calculable, at their characteristic scales, in fixed-order perturbation theory. Reasonable choices for these scales are, e.g., or , where is the invariant mass of the final state and is the sum of the transverse mass of the top quark, antitop quark and heavy vector boson:
[TABLE]
The issue of scale choices is discussed in Section 3.2. However, at this stage, it is important to observe that, in order to eliminate large logarithms from the soft function in Mellin space, the soft scale must depend on the Mellin parameter . This fact gives rise to a branch cut in for large values of , which in turn is related to the Landau pole in . The integration path in the complex plane is chosen according to the Minimal Prescription Catani:1996yz . Notice that the ratio . An alternative to this approach is to perform the resummation directly in momentum space, fixing the soft scale at the hadronic level through a fitting procedure, see for example Becher:2007ty ; Ahrens:2008nc ; Ahrens:2010zv ; Broggio:2011bd ; Broggio:2013cia . When resummation is carried out up to NNLL accuracy, as it is the case in this work, one is accounting for terms proportional to with to all orders in in the partonic cross section in Mellin space. Finally, the parton luminosity functions in Mellin space, , that appear in Eq. (13) can be obtained using techniques described in Refs. Bonvini:2012sh ; Bonvini:2014joa .
NNLL corrections to differential distributions such as the top-quark transverse momentum distribution, the vector boson transverse momentum distribution, the top-pair invariant mass, the system invariant mass etc., can be obtained by evaluating Eq. (13) by means of the in-house Monte Carlo code developed for Broggio:2016zgg ; Broggio:2016lfj ; Broggio:2017kzi . The code evaluates the total cross section while simultaneously binning events w.r.t. variables which can be built out of the momenta, such as the ones listed above. However, it must be pointed out that, in its current implementation, the code calculates the Mellin transform of the luminosity function in Eq. (13) and loses the information about the values at which the PDFs are evaluated. Hence it cannot be employed to evaluate rapidity distributions to NNLL accuracy in the laboratory frame555Note that this is not a matter of principle, and indeed NNLL resummation for rapidity distributions was recently carried out in Pecjak:2018lif ..
Nevertheless, the NNLL resummation formula can also be employed to obtain approximate results, which are indicated by in this work. The cross section can be obtained by solving the renormalization group equation satisfied by the NLO soft function. The predictions discussed in Section 4 include, on top of the complete NLO, all of the terms of order () in the partonic QCD cross section in momentum space, as well as part of the terms proportional to . A detailed description of the terms of the latter class that are included in the calculations can be found in Section 3 in Broggio:2015lya . These calculations depend on a single scale , in contrast with resummed calculations, which have a residual dependence on the scales . In the context of this work, calculations allow us to obtain predictions also for rapidity distributions.
We conclude this section by returning to a point briefly mentioned in the discussion of the hard function. The resummation carried out in Refs. Broggio:2015lya ; Broggio:2016lfj ; Broggio:2016zgg ; Broggio:2017kzi and in this work deals with QCD corrections only, meaning that the resummation formulas are linear in the fine structure constant . While it is in principle possible to consider the resummation of soft-gluon emission corrections to contributions that are proportional to higher powers of , their implementation is not trivial. However, the contribution of these corrections is expected to be numerically smaller than the contribution of the soft emission to the QCD process. In addition, one can gain some rough sense of the size of neglected higher order mixed QCD-electroweak corrections by comparing the multiplicative and additive approaches to the matching of NLO and NNLL calculations, discussed in the next section. Results given in Section 4 indicate that the difference between the matched results in the additive approach and in multiplicative approach is, with few exceptions, a small effect.
2.3 Matching procedure
The main goal of this paper is to match the NLO QCD and electroweak corrections to production (i.e. the complete-NLO corrections) to the resummation of soft gluon emissions to NNLL accuracy in QCD. In order to achieve this goal, it is necessary to avoid the double counting of terms that are included in both the NLO QCD corrections and the NNLL resummation formula. The method that allows one to avoid such a double counting is well understood and goes under the name of matching procedure.
In order to understand the details of the matching procedure it is necessary to identify terms in the NLO QCD partonic cross section that are included in the resummation formula in Eq. (14). If one sets in that equation, the evolution factors become identity matrices in color space. In that situation, the trace of the hard function and Mellin-space soft function (both evaluated to NLO) includes terms proportional to and , as well as terms that do not depend on the Mellin parameter . The latter class of terms still depends on the Mandelstam invariants; nevertheless, those independent terms are referred to as “constant” terms. Terms proportional to inverse powers of the Mellin parameter, which are present in the full QCD partonic cross section at NLO in Mellin space, cannot be reconstructed starting from the NNLL resummation formula. The trace of the hard function and soft function at NLO in Mellin space, including the terms discussed above, can be inserted in Eq. (13) to obtain what is referred to as the approximate NLO QCD cross section, denoted here with the subscript . The cross section contains the contribution of all of the terms proportional to () and in the partonic cross section in momentum space. In analogy with the notation introduced in Section 2.1, we indicate the terms of included in the corrections to a given observable with . Consequently, we define the observable evaluated to as
[TABLE]
Once the QCD and EW complete-NLO corrections (whose sum will simply be referred to as NLO), the NNLL corrections, and the predictions for a given observable are available, it is straightforward to combine them into an NLO+NNLL prediction by using the matching formula
[TABLE]
The symbol indicates the numerical value of the resummed total cross section in Eq. (13) or, in the case of differential distributions, the value of that resummed cross section in a specific bin of the distribution. The terms included in square brackets in Eq. (17) are of and higher, and represent the NNLL corrections to be added to the NLO result. The quantity is defined in such a way as to include all of the corrections to the observable considered in this work. In discussing the results of this study, it is also useful to match the resummed formulas to the QCD cross section only, by excluding all the EW corrections. In that case, Eq. (17) must be modified by replacing in the first term in the r.h.s. of the equation. The predictions obtained in this way include only QCD effects and are indicated by the subscript:
[TABLE]
Calculations at accuracy correspond to the results presented in Refs. Broggio:2016lfj ; Broggio:2016zgg ; Broggio:2017kzi .
Analogously, it is possible to match predictions discussed at the end of Section 2.2 to the complete-NLO prediction. A given observable can be evaluated to by calculating the quantity
[TABLE]
In Eq. (19), includes terms of and terms of . contains terms of (), including the complete cross section. Consequently, the square bracket in Eq. (19) includes only the terms of that must be added to the complete-NLO calculation in order to evaluate the observable to nNLO. Finally, one can exclude the EW corrections from Eq. (19) by replacing in the first term on the r.h.s.: in this way one obtains approximate NNLO corrections to the QCD process, which are indicated with .
Eqs. (17) and (19) combine NLO to NNLL QCD or approximate NNLO QCD corrections in an additive approach, which is well defined in perturbation theory. However, it is possible to combine these contributions within a multiplicative approach, which is often employed in combining NLO QCD and NLO EW corrections, denoted in this work by and , respectively. While in the additive approach and are simply summed so that
[TABLE]
in the multiplicative approach these two corrections are combined via the prescription
[TABLE]
By comparing Eqs. (20) and (21), it is possible to see that differences between the two approaches only enters at the level of mixed QCD-EW NNLO corrections of relative to , i.e., in the case of cross sections at , which is beyond the accuracy of the calculations presented in this work. However, there are specific configurations where the multiplicative approach is well-motivated and expected to provide improved predictions. The typical case is when the contribution is dominated by soft-QCD physics, and the correction by large EW Sudakov logarithms. Indeed, these two classes of corrections factorize, and therefore the entire mixed QCD-EW NNLO corrections of relative to are expected to be well approximated by the difference between Eq. (21) and Eq. (20), namely
[TABLE]
The resummation procedure allows one to account for soft emission corrections at all orders in . In particular, the NNLL resummation discussed in this work accounts for terms in the partonic cross section in Mellin space that are proportional to with , where the soft configuration corresponds to the limit . Consequently, one can generalize the multiplicative approach to approximate not only the mixed QCD-EW NNLO corrections of relative to the observables, but also the corrections to the Mellin space partonic cross section proportional to with , for all orders in . A resummed observable can then be evaluated in the multiplicative approach at accuracy as follows:
[TABLE]
Similarly, it is also possible to combine predictions to the complete-NLO ones in the multiplicative approach by using the matching relation
[TABLE]
In the tail of the differential distributions for and productions, where Sudakov logarithms are large and QCD radiation is typically soft, predictions can be considered as an improvement w.r.t. those at accuracy. In the rest of the phase space this is not necessarily true. Therefore, the difference between the two approximations can be considered as an estimate of the impact of missing higher-order QCD-EW terms. The same argument holds for the comparison between and predictions.
The situation is completely different in the case of , where the contribution is dominated by hard radiation, as discussed in Section 2.4. In addition, the Sudakov logarithms present in are proportional to the contribution, which arises from a initial state, while the dominant contributions arise from quark radiation in initiated processes. Thus, in the case of production, the multiplicative approach cannot be motivated by sound theoretical arguments. This is particularly relevant in the tail of the distributions, where both the and corrections are large, the latter due to the presence of Sudakov logarithms. Therefore the multiplicative approach can lead to uncontrolled NNLO terms. Moreover, since in production the correction is numerically much larger than the contribution, even if the multiplicative approach as defined in Eqs. (23) and (24) were justified, it would probably not account for the dominant mixed QCD-EW NNLO contributions, which are expected to be those of relative to the cross section. For consistency, in Section 4 results in the additive and multiplicative approaches are shown and compared also for the process. However, one should bear in mind that only in the case of and production can the multiplicative approach be expected to improve the predictions.
2.4 Structure of the fixed-order corrections
This section describes the structures underlying the complete NLO corrections to production. We start by reviewing the most important features of and production, which are discussed in detail in Frederix:2017wme . Subsequently, we consider and production.
In () production at LO only () initial states contribute, where and are a generic up- and down-type quarks. The boson is radiated from the quark, while the pair is produced either via a gluon or a photon/ boson. The gluon mediated diagrams contribute to the cross section, while the diagrams involving a photon/ boson contribute to the cross section. The interference between these two classes of diagrams vanishes after summing over colors, so that the cross section also vanishes. On the contrary, all of the contributions are non-vanishing.
The contribution to the production process is in general large. It was calculated in Hirschi:2011pa ; Garzelli:2012bn ; Campbell:2012dh ; Maltoni:2014zpa and studied in detail in Maltoni:2015ena . Large QCD corrections are mainly induced by the opening of the channel, which depends on the gluon luminosity and therefore is enhanced in high-energy proton–proton collisions. Moreover, the radiation of quarks in is typically hard and in particular very large -factors are present in the tail of the distribution, which receives an additional enhancement on top of the one due to the luminosity (see left diagram in Figure 1 and Maltoni:2015ena for a detailed discussion). The impact of multiple soft-gluon emissions for this process is scale sensitive and non-negligible Li:2014ula ; Broggio:2016zgg ; Kulesza:2018tqz ; the predictions contained in Section 4 account for soft emission up to NNLL accuracy. However, it is important to observe that a large component of corrections, and therefore the associated scale uncertainties, originates from hard radiation in the channel. Therefore, the threshold resummation in the channels is not expected to drastically reduce the total scale uncertainty. A detailed discussion of the size of the various corrections can be found in Section 4.
For what concerns the EW contributions to production, the corrections were calculated for the first time in Frixione:2015zaa and further phenomenological studies were provided in deFlorian:2016spz . In a boosted regime, due to Sudakov logarithms, the corrections can be as large as the NLO QCD scale uncertainty. The contribution is sizable Frederix:2017wme since it contains real-emission channel that involves EW scattering Dror:2015nkp (see right diagram in Figure 1 and Frederix:2017wme for a detailed discussion). Similarly to what happens in the case of the corrections, this channel becomes even more relevant as the LHC center-of-mass energy grows, due to the presence of an initial-state gluon. Although scattering is present also in the corrections, in that case it is induced by a initial state. It is therefore suppressed w.r.t. scattering contributing to by the smaller luminosity of the photon and also by a factor . Similarly, all of the other terms are negligible since they are of .
In contrast to the case of production, tree-level Born diagrams for and production are induced by both and initial states. In particular, the gluon-fusion channel contributes only to the term and, due to the partonic luminosity, yields the largest part of the LO cross section. The initial states contribute also to via squared diagrams featuring pairs stemming from a photon or propagator. Similarly to the case, their interference with diagrams contributing to vanishes due to color. However, the contribution to the cross section is non-vanishing for these two processes. Indeed, some of the initial-state diagrams feature a -channel -boson that leads to non-vanishing interference contributions. Moreover, the initial-state processes contribute to via squared diagrams. As shown in Frederix:2018nkq , the and contributions to the cross section are numerically negligible.
All of the contributions are non-vanishing. The correction is in general large; it was calculated in Beenakker:2001rj ; Beenakker:2002nc ; Dawson:2002tg ; Dawson:2003zu for and in Hirschi:2011pa ; Lazopoulos:2008de ; Garzelli:2011is ; Kardos:2011na ; Garzelli:2012bn for . In addition, the correction was studied in detail in Maltoni:2015ena , where, as in the case of production, large -factors for the differential distribution were found. On the other hand, in presence of LO contributions involving two gluons in the initial state, the luminosity is not providing a significant enhancement of the cross section. Furthermore, in contrast to the case of production, the QCD emissions in the corrections are not typically hard. Also, the largest contribution from QCD emissions arises from the initial state, which is in general the dominant partonic channel both for the and the . This also applies to the corrections to all the and differential distributions considered in this work. For this reason, by resumming soft emission corrections to NNLL accuracy one observes, as expected, a sizable reduction of the residual scale uncertainty affecting the total cross section and differential distributions. Also this feature will be quantified in detail in Section 4.
For what concerns the EW corrections to and production, the corrections were calculated for the first time in Frixione:2015zaa and further phenomenological studies were carried out in deFlorian:2016spz . For the total cross section, the relative size of the corrections is smaller than in production. However, in the tail of the differential distributions the contribution can be non-negligible in comparison to the NLO QCD scale uncertainty.
For and production, the and corrections were calculated in Frederix:2018nkq ; a phenomenological study involving these contributions to the cross section is presented for the first time in this work. Compared to production, not only the but also the correction is small. At this order, () production involves () scattering in () real-emission channels. However, as discussed for the case of the enhancement, the luminosity is not providing a significant enhancement and therefore the relative size of correction in and production is smaller than in production.
The and processes share several features at the diagrammatic level and therefore also at the phenomenological level. This similarity, which is present also at the level of and corrections to these processes, was advocated as a possible proxy to be used to reduce the theoretical uncertainties in the measurements of the top-quark Yukawa coupling Plehn:2015cta . The main kinematic difference between these two processes concerns the rapidity of the bosons Maltoni:2015ena , because the bosons can be emitted both from initial-state quarks and final-state top-quarks, while the boson can be emitted only from the latter. This situation is markedly different from the case of production, where the bosons are emitted only from the initial state light quarks. Moreover, as discussed before, and corrections have a very different impact on predictions for production compared to and production. Similarly, the impact of soft gluon resummation is different for production and in or production.
3 Input parameters, scales and PDFs
The predictions presented in Section 4 depend on the numerical values of physical input parameters, on the PDFs employed in the calculations, and on the choice of the unphysical scales that enter in fixed-order and resummed calculations. The choices made in this work are listed and discussed in this section.
3.1 Masses and couplings
The masses of the heavy SM particles are set equal to
[TABLE]
whereas all the other masses are set equal to zero. The decay widths of all particles are also set to zero. In addition, we use the on-shell renormalization scheme for all masses. The strong coupling is renormalized in the -scheme with five active flavors, while the EW input parameters and the renormalization condition for are in the -scheme, with
[TABLE]
The CKM matrix is set equal to the unity matrix.
3.2 Scale choices and uncertainties
In the literature, two choices for the (functional form of the) central values for the factorization and renormalization scale entering these processes are commonly adopted. In particular, in Frederix:2018nkq it was argued that , with defined as in Eq. (15), is a reasonable choice for the factorization and renormalization scale. In Broggio:2016lfj ; Broggio:2016zgg ; Broggio:2017kzi , on the other hand, scales based on the top-antitop-heavy-boson invariant mass, , were used. The former work is based on fixed-order perturbation theory, while the latter also considers the resummation of soft emission corrections. An additional study of the different scale choices was carried out in Maltoni:2015ena .
Since the factorization/renormalization scale and the hard and soft scales are unphysical, it is acceptable and even recommendable to explore different scale choices. Numerical differences among values of the same observables evaluated for different scale choices can be used as an estimate of the uncertainty associated to the truncation of the perturbative series. In this work, we consider both and based scale choices. In particular, when relating the central value of the three scales involved in the calculations to the value of , we choose
[TABLE]
When we relate the scales to we set instead
[TABLE]
The uncertainty associated to missing higher-order corrections can be estimated by considering the dependence of the predictions for a given observable on the non-physical scales that enter the calculation. At fixed order, this is done by varying the renormalization and factorization scales in the range (). The uncertainty estimate is then given by the bin-by-bin envelope of the 9 predictions obtained in this way. For the resummed results, the hard, soft and factorization scales are varied in the range (). In particular, by introducing the notation (), one can rewrite Eq. (17) by making explicit the dependence of each element on :
[TABLE]
In contrast to the case of NLO calculations, in Eq. (29) no distinction is made between the renormalization and factorization scales. One can then define an upper and lower scale uncertainty for the variation of each scale in Eq. (29) as follows
[TABLE]
for . In Eqs. (30) the two scales that are not varied are kept fixed to their central values: if . The residual theoretical uncertainty affecting a given resummed observable is then obtained by combining in quadrature the uncertainties associated to each of the three scale variations in each of the histogram bins as done in Refs. Broggio:2016zgg ; Broggio:2016lfj ; Broggio:2017kzi . With reference to Eq. (30) one can then define the upper and lower scale uncertainty as
[TABLE]
As discussed in Section 4, results with the two scale choices in Eqs. (27) and (28) are compatible with each other at the level of total cross sections, although somewhat less so at the level of differential distributions. Since there is no conclusive argument in favor of either scale choice, we opt for taking the bin-by-bin average of the two results as the best prediction for the central value of each given observable. Moreover, we use the envelope of the uncertainty bands generated with the two scale choices as an estimate of the missing higher-order corrections. This combined-scale method is particularly relevant in the case of production, where for the two choices in Eqs. and individually one observes that the NNLL corrections lead to a reduction of scale uncertainty for the total cross section, but the difference between the central values obtained with the two scale choices increases when NNLL corrections are accounted for. Hence, we think that considering only a single central scale choice and its corresponding uncertainty band underestimates the uncertainties due to missing higher orders. On the other hand, we believe that our procedure of taking the envelope of the scale uncertainty bands of the two calculations considered results in a reliable and robust uncertainty estimate.
With the notation introduced above, one can rewrite Eq. (19) as
[TABLE]
The scale uncertainty associated to this quantity is obtained varying in the range . Finally, in calculations where the matching between NLO and NNLL corrections is carried out within the multiplicative approach, the scale uncertainty is obtained by applying the method described in Eqs. (29) and (32) to Eqs. (23) and (24).
3.3 PDF uncertainties
Results in Section 4 are obtained by using the LUXqed17_plus_PDF4LHC15_nnlo_100 PDF set Manohar:2016nzj ; Manohar:2017eqh , which in turn was obtained starting from the PDF4LHC PDF set Butterworth:2015oua ; Ball:2014uwa ; Harland-Lang:2014zoa ; Dulat:2015mca . The PDFs in Refs. Manohar:2016nzj ; Manohar:2017eqh include NLO QED effects in the DGLAP evolution deFlorian:2015ujt ; deFlorian:2016gvk and they provide the most precise determination of the photon PDF available to date. In complete-NLO calculations, the PDF uncertainties are evaluated by means of MadGraph5_aMC@NLO thanks to the procedure introduced in Frederix:2011ss . In this way, one can calculate an observable for each PDF replica in a given PDF set. The PDF uncertainties related to the NNLL resummation corrections, i.e., the terms between square brackets in Eq. (17), are instead evaluated with an approximation. This is necessary because the evaluation of the NNLL resummation formulas for all of the PDFs in a given set would require an excessive amount of computer time. The approximation relies on the assumption that the relative PDF uncertainty associated to the part of the corrections that does not depend on the luminosity, denoted here by , is the same relative PDF uncertainty affecting . In this approximation, corrections arising from quark-gluon channel diagrams are excluded from the calculation of the relative error induced by PDFs in resummed calculations because this channel is subleading in the threshold limit. Therefore, for each replica in the PDF set we assume that
[TABLE]
where the subscript “central” refers to the central PDF prediction. In conclusion, for each replica the value of is evaluated via MadGraph5_aMC@NLO, rescaled as prescribed by Eq. (33) and added back to , to provide an estimate of the NLO+NNLL calculation carried out with the replica in the PDF set. Once an NLO+NNLL prediction is available for each replica , PDF uncertainties are evaluated following the standard procedure for the PDF set considered.
The same procedure can be employed for the nNLO predictions via the substitution in Eq. (33). In the case of the combination of NLO predictions and NNLL corrections in the multiplicative approach we do not evaluate PDF uncertainties, but it is reasonable to think that they would be similar in size to those calculated in the additive approach.
4 Results
In this section we present predictions for each of the four processes considered in this work, namely , , , and production. We start by considering total cross sections and charge asymmetries, and then give results for differential distributions in Section 4.2.
4.1 Total cross sections and asymmetries
The total cross sections for the four processes, calculated within different perturbative approximations, can be found in the middle columns of Tables 1-4. Each table is subdivided in three sections:
- •
In the top section of each table the cross sections are evaluated with the -based scale choices listed in Eq. (27).
- •
In the middle section of the tables the cross sections are evaluated with the -based scale choices listed in Eq. (28).
- •
Finally, the lower section of each table shows the combination of the results for the two aforementioned scale choices. The results for the scale choices in Eqs. (27) and (28) are combined as explained in Section 3.2; the results for the total cross section listed in the lower portion of the tables represent one of the main results of this paper.
In each of the three parts of the tables, the predictions become more accurate as one moves from the highest line to the lowest line. Each line starts with a label that indicates the accuracy of the calculations found on that line. For convenience, we summarize the notation introduced in Section 2 and used to label the various rows:
- •
: rows labeled in this way are based on tree-level QCD calculations, i.e., they include only the contribution to the cross section.
- •
: lines labeled in this way include the calculation added to the NLO QCD corrections. For example, for the total cross section
[TABLE]
where, consistent with the notation introduced in Section 2, includes only terms of .
- •
NLO: rows labeled in this way correspond to complete-NLO results, which include NLO QCD corrections (), NLO EW corrections (), and further subleading contributions. For example, for the case of the total cross section, one has
[TABLE]
- •
: indicates the approximate-NNLO predictions evaluated by adding to the results the contribution of the corrections of relative to the LO QCD cross section that are obtained from the NNLL resummation formula for the QCD process.
- •
nNLO: same as , but including in addition the NLO EW corrections and further subleading contributions from the complete-NLO calculation.
- •
+NNLL: this label indicates the predictions improved by NNLL resummation.
- •
NLO+NNLL: lines labeled this way include the complete-NLO predictions improved by NNLL resummation. These must be considered our most accurate predictions.
In the cases where beyond-NLO predictions are combined with NLO results, i.e., nNLO and NLO+NNLL calculations, the matching is carried out with the additive approach discussed in Section 2.3. For total cross sections and charge asymmetries, which are the quantities considered in the Tables 1-4, results based on the multiplicative approach differ from the additive combination by less than 1%. For this reason, results obtained with the multiplicative approach are not shown in the tables. However, in the case of differential distributions discussed in Section 4.2, the differences between additive approach and multiplicative approach are sometimes larger. Therefore, in that case, results obtained with the multiplicative approach are presented in a separate ratio inset in the figures. The labels employed to identify calculations in the multiplicative approach are the following:
- •
: this label indicates a calculations that includes the same corrections found in NLO+NNLL calculations, but with and purely QCD corrections combined in the multiplicative approach, according to Eq. (23).
- •
: this label indicates calculations analogous to nNLO but with and QCD corrections combined with the multiplicative approach, according to Eq. (24).
4.1.1 and
Values for the total cross sections for the and processes are shown in Tables 1 and 2, respectively. The results for are quantitatively rather similar to the results for ; consequently, we comment almost exclusively on the latter.
The central value of the cross section for production is about fb or fb for the -based or -based scale choices, respectively. This difference is well captured by the uncertainty due to scale variation, which is roughly in both cases. Since the central values for the two scale choices differ by about , when combining the two scale choices the cross section gets a value of fb, with a slightly increased scale dependence of . As discussed in Section 2.4, the large NLO QCD corrections to this process are due to the opening of the -induced real-emission channel. This contribution is particularly sizable since the gluon luminosity is rather large at the LHC. Indeed, the predictions are more than 50% larger than the cross section. The corrections are only slightly larger for the -based scale choice than for the -based choice. This fact brings the two cross section results closer to each other (in relative terms) as compared to the cross sections, as expected from perturbation theory. At the relative uncertainty from scale variation is for the combined-scales result, which is significantly smaller than the corresponding uncertainty. The inclusion of the EW corrections further increases the cross section by about , mainly due to corrections. Even though this contribution is suppressed by a factor w.r.t. the corrections, there is a large enhancement due to the opening of the -channel-enhanced scattering contribution Dror:2015nkp ; Frederix:2017wme .
For both the and -based scale choices, approximate NNLO corrections increase the cross section. These corrections for the -based scale choice are slightly larger than for the -based scale choice. Therefore, the central values of the nNLO cross sections for the two scale choices are closer to each other than the values of the NLO cross sections. By including the approximate NNLO QCD corrections the cross-section scale dependence is reduced to for the combined-scale calculation. Hence, perturbation theory seems to converge well. On the other hand, if one considers NNLL resummed results, either matched to or to complete-NLO predictions, a slightly different picture emerges. For both scale choices, the corrections due to resummation are small and well-behaved. Moreover, they reduce significantly the scale dependence, to about and for the total cross section obtained with the and -based scales, respectively. However, the corrections move the cross-section central values further apart from each other than in the case of NLO calculations. The corrections for the -based scales are negative, while for the -based scales the corrections are positive. The net effect is that the central value for the combined-scale result is very much compatible with the corresponding NLO predictions, with a scale uncertainty equal to , which is larger than the NLO+NNLL scale uncertainties for the two separate scale choices. The reason behind this feature resides in the origin of the NLO QCD corrections. The term is dominated by hard radiation and especially by the initial-state contribution. Therefore, in this case, the reduction of the scale dependence due to the resummation of soft emission does not reflect the theoretical uncertainty associated to missing higher-order corrections. However, by combining the results obtained with the two different scale choices, one obtains a more reliable estimate of the uncertainty due to missing higher order corrections. Nevertheless, by following this approach an improvement in the scale uncertainty in the combined-scales results is observed; the scale uncertainty affecting the combined-scales NLO cross section is larger than the scale uncertainty for the NLO+NNLL cross section. Since scale uncertainties primarily affect QCD corrections, this argument still holds when EW effects are included, as can be seen by comparing the scale uncertainties of , , and calculations of the cross section.
The PDF uncertainties on the total cross section are significantly smaller than the corresponding scale uncertainties, and are at the level of for production and for production.
The third column in Tables. 1 and 2 shows predictions for the charge asymmetry . At the LHC, the charge asymmetry is defined as
[TABLE]
where , and () indicates the top-quark (antiquark) rapidity in the laboratory frame. Consequently, is positive when the top is emitted less centrally than the antitop. An analogous asymmetry was measured at the Tevatron for top-pair production666At colliders such as the Tevatron the relevant observable was the forward-backward asymmetry, defined as in Eq. (36) but with .. The Tevatron asymmetry received considerable attention due to a tension between the measured asymmetry and the SM predictions Aaltonen:2011kc , which were initially known only at the lowest non-vanishing order (NLO QCD) Kuhn:1998kw . The tension could be interpreted as a BSM effect. With improved measurements and especially with the calculation of NLO EW Hollik:2011ps and then NNLO QCD corrections Czakon:2014xsa ; Czakon:2017lgo , the tension between theory predictions and experimental data decreased considerably. It is therefore essential to have precise predictions for such observables.
At the LHC, the asymmetry for top-pair production is rather small (see e.g. Czakon:2017lgo ) due to the fact that the cross section is dominated by the gluon-fusion channel, which is charge symmetric. It is therefore interesting to consider top-pair production in association with a boson Maltoni:2014zpa ; Maltoni:2015ena . Since at lowest order the boson only couples to initial state quarks, the contribution of the channel to the asymmetry is suppressed and only enters at NNLO and beyond. As a result of this situation, the asymmetry in production is significantly larger than in the case of top-pair production. For production at , which is the lowest perturbative order for which the asymmetry is non-zero, one finds that is equal to about and for the and -based scale choices, respectively, with scale uncertainties of 777Both for scale and PDF uncertainties a full correlation is assumed for the numerator and the denominator of Eq. (36). In the case of combined-scale predictions we proceed similarly to the case of total cross sections, by looking at the envelope of the and -based scale choices directly for .. The inclusion of the EW corrections increases the asymmetry by a small amount, about percent.
In our framework, it is not possible to evaluate the charge asymmetry to or accuracy, since the resummation is carried out inclusively w.r.t. the rapidities. Hence, the nNLO calculations are the most accurate predictions for the charge asymmetry that we present in this paper. On the other hand, the NNLO QCD corrections to (and ) production involve processes, which are expected to be large due to the luminosity and are completely symmetric, so that they contribute only to the denominator of Eq. (36). These effects cannot be estimated via scale variations and may substantially alter the prediction for (and ). However, when the approximate NNLO corrections are included, the charge asymmetry increases by and percent for the and -based scale choices, bringing the central values for the two scale choices rather close to each other. More significantly, by including these terms, which constitute the first order correction to the asymmetry, the scale dependence is reduced to for the combined-scales prediction. In this case, the uncertainties coming from the PDFs can no longer be neglected, since they are similar in size the scale uncertainties ( for production). Similar remarks apply to the charge asymmetry calculation in production.
4.1.2
The total cross section for production is shown in the second column of Table 3. When NLO QCD corrections are included, the cross-section central values obtained with the two scale choices differ by fb, roughly half of the difference between the central values of the cross section calculated with the two scale choices at . For both scale choices, EW corrections increase the cross section by w.r.t. the result; this is a small correction when compared to the scale uncertainty. Indeed, although the scale uncertainty at is more than a factor two smaller than at , it remains of the order of .
When QCD corrections beyond NLO are included, the agreement between the predictions obtained with the two scale choices is very good: for and calculations the difference between the two values of the cross section is below the permille. The scale uncertainties, which are of the order of for calculations, are significantly reduced w.r.t. NLO calculations. Compared to these small uncertainties, EW corrections can no longer be neglected.
In contrast to the processes, the NLO+NNLL cross sections come with a larger scale uncertainty than the nNLO ones and the central values with and -based scale choices are further apart than at nNLO. By combining results for the two scale choices one obtains a total cross section of approximately fb with a scale uncertainty just below the level. The scale uncertainty in the resummed calculations is obtained by separately varying three different non-physical scales, while the scale uncertainty associated to approximate NNLO results is obtained by varying only one scale. For this and other reasons, as discussed in Refs. Broggio:2016zgg ; Broggio:2016lfj ; Broggio:2017kzi , we consider NLO+NNLL predictions to be more complete and reliable than the approximate NNLO predictions. Hence, the cross section in the last line of Table 3 should be considered the most accurate prediction for the total cross section presented in this paper.
The charge asymmetry for the top and the antitop quarks in production is given in the third column of Table 3. As expected, the asymmetry for production is smaller than for production, since the latter does not contain (up to NLO) the large and symmetric -induced contributions, which enter only in the denominator of Eq. (36). The difference between the central values of the asymmetry calculated with the and -based scale choices is small. By comparing NLO and predictions it is possible to see that the contribution of the EW corrections to the asymmetry is sizable. Going beyond NLO, the and predictions show very asymmetric uncertainty bands. For the -based scale choice, the central value of the asymmetry lies at the lower edge of the uncertainty band and the overall size of the band does not decrease significantly compared to NLO. For the -based scale choice, the band in the nNLO and calculations does decrease in size w.r.t. the NLO calculation, but its central value lies near the upper edge of the uncertainty band. The combination of the and -based calculations leads to a nNLO result that has a central value close to the NLO calculation, but with a significantly smaller scale dependence. However, overall, the charge asymmetry is rather small for production and can be challenging to measure.
The PDF uncertainties are small for production, of the order of for the total cross sections and slightly larger for the charge asymmetry.
4.1.3
Results for the production total cross section are listed in the second column of Table 4. Similarly to the processes considered so far, the predictions show good perturbative convergence for both the and -based scale choices. The difference between the values of the cross section obtained with the two scale choices is large at : at that order, the -based scale choice leads to a cross section that is about 9% larger than the one found with the -based scales. The difference between the two scale choices at NLO is reduced to less than 5%; nNLO calculations further reduce it to just over 1%. For all of the perturbative orders considered, the uncertainty bands obtained through scale variations are compatible with these differences. In the results that combine the two scale choices (lower part of Table 4) the uncertainty bands at are , and they reduce to at NLO and to at nNLO. Similar to production, the resummed calculation produces a larger difference in central values for the total cross section calculated with and -based scales than the nNLO one, and also has larger uncertainty bands. Again, this fact indicates that scale variation in nNLO calculations does not lead to a reliable estimate of the uncertainty associated to missing higher-order corrections. For this reason, the most accurate and reliable prediction for the total cross section in production is given by the combined-scales calculation at NLO+NNLL accuracy, yielding a total cross section of about fb, with an uncertainty from missing higher orders corrections of . This prediction includes the contributions of the EW corrections, which are rather small for the total cross sections. In fact, NLO calculations increase the total cross section by about 1% w.r.t. calculations; this difference falls well within the theory uncertainty band.
Similarly to the case of production, the charge asymmetry is small for production. This is expected since the cross section is dominated by the gluon-fusion channel.888However, observe that a small and negative charge asymmetry is present in production already at Maltoni:2015ena . The contribution of EW corrections to the asymmetry is not negligible. However, contrary to production, the approximate nNLO QCD corrections do play a prominent role: indeed they increase the asymmetry by about 20%. Furthermore, similarly to production, the asymmetry calculated to nNLO has smaller scale-uncertainty bands than the asymmetry calculated to NLO. The nNLO asymmetry in production is about 1% with a scale uncertainty of .
For the total cross section, the uncertainties due the PDFs are of the order of , independently from the level of accuracy in the theory predictions. They are therefore smaller than the residual scale uncertainty in all cases. However, as for production, they are slightly larger for the charge asymmetry. Given the fact that, for the charge asymmetry, the scale uncertainties are much smaller for production than for the process, the PDF uncertainty cannot be neglected for the former process.
4.2 Differential distributions
In this section, we present predictions for binned differential distributions for invariant mass, transverse momentum, and rapidity observables. Transverse momentum and invariant mass distributions are evaluated up to NLO+NNLL accuracy (NNLL-resummed results, matched to the complete-NLO predictions). Rapidity distributions are evaluated up to nNLO (approximate NNLO QCD results, matched to complete-NLO calculations), since the NNLL resummed results in our framework are integrated over rapidities. Only predictions that combine calculations carried out with -based and -based scale choices are shown in Figures 2-8.
In each of the figures there are four plots. Each of them corresponds to one of the four processes considered in this work: the top-left plot of each figure refers to production, the top-right plot shows the process, the bottom-left plot refers to production, and the bottom-right plot shows the process.
Each of these plots has the same layout, consisting of a main (top) panel and three ratio insets below it. The four components of each plot show the following information:
- •
The top panel shows the absolute predictions for the differential distribution. The central value of the distribution in each bin (calculated to NLO+NNLL accuracy for the invariant mass and transverse momentum distributions, and to nNLO for the rapidity distributions), is plotted in blue. The differential distribution calculated to complete-NLO (denoted simply as NLO) is plotted in red. In this panel the vertical axis indicates the cross section per bin. Consequently, the total cross section is simply the sum of the heights of the distribution in each bin (including the bins that fall outside the range shown). For the invariant mass and transverse momentum distributions the horizontal axis is logarithmic.
- •
In the ratio inset just below the main panel, the NLO+NNLL (or nNLO in the case of rapidity distributions) and NLO predictions are shown as a ratio w.r.t. the central value of the NLO calculations. Here, also the uncertainties from scale variations (dark shaded band) and PDFs (light shaded band) are shown, with the latter added linearly999By linearly adding scale and PDF uncertainties we adopt a conservative approach, assuming full correlation among the two classes of effects. to the former. The purpose of this inset is to show the impact of the soft emission corrections on the shape of the distribution.
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The middle ratio inset shows the difference between the additive and the multiplicative combination for the and the EW corrections, together with the corresponding scale uncertainties. PDF uncertainties are not shown in this panel. In particular, the inset shows the ratio between the distribution ( for rapidities) and the central value of the () calculation in each bin, as well as the ratio between the distribution ( for rapidities) and the central value of the () calculation in each bin. Multiplicative results are shown as a dark yellow band, while the additive-approach results are shown, as before, by a blue band.
- •
In the lower inset the effect of EW corrections is shown by plotting the ( for rapidity distributions) calculations and the +NNLL (or, in the case of rapidity distributions, to ) calculations, both divided by the central value of the +NNLL () calculation in each bin. Also in the lower inset, NLO+NNLL (nNLO) calculations are indicated by the blue band, while results at +NNLL () accuracy are shown by the brown band.
4.2.1 Invariant masses
The invariant mass distribution of the system is shown in Figure 2. The upper-left (upper-right) plot shows the invariant mass of the () system. From the first ratio-inset it can be seen that the resummation has a relatively small impact on the shape of the distribution w.r.t. the NLO calculation. Even though the uncertainties affecting the NLO+NNLL calculation are slightly smaller than the NLO uncertainties, they remain large, in particular for large values of the invariant mass. The reason for the rather large scale dependence at high invariant mass is that the predictions for the -based and -based scales differ significantly in this region of phase space. This observation applies to NLO calculations as well as to NLO+NNLL accuracy calculations. The tail of the invariant mass distribution is dominated by real radiation from quark emissions in the -initiated channel. Consequently, soft-gluon resummation cannot improve its description, since the channel is subleading in the threshold limit.
The difference between the additive and multiplicative methods of combining EW corrections and QCD resummed calculations is small, as it can be seen by examining the middle inset in the two plots at the top of Figure 2; indeed, the blue and dark-yellow bands overlap almost entirely. Moreover, from the lower inset in the same plots it can be seen that EW corrections have a significant impact on the distribution only for small invariant masses. Indeed, the expected EW Sudakov suppression at large invariant masses is not observed since the NLO QCD corrections are rather large and dominated by hard real-emission corrections. As shown in Frederix:2017wme , a jet-veto can suppress the large QCD corrections, which results in an enhancement of the relative impact of EW corrections.
The situation is somewhat different for the and invariant mass distributions, shown in the lower-left and lower-right plots of Figure 2, respectively. Since the NLO corrections for these processes are not dominated by the opening of new channels, the resummation of soft radiation reduces the scale dependence significantly. In addition, NLO+NNLL calculations lead to an increase of the cross-section central value in each bin, ranging from a few percents for small invariant masses, to about () for () production at 3 TeV. By looking at the first inset in the lower line of Figure 2 one sees that for production the entire uncertainty band at NLO+NNLL is contained in the NLO uncertainty band over the whole mass range shown in the figure. For production, the central value of the NLO+NNLL distributions remains well within the NLO uncertainty, as one would expect from a well-behaved perturbative expansion. However, the NLO+NNLL uncertainty band only has a partial overlap with the NLO uncertainty band in the far tail of the invariant mass distribution.
Given that resummation changes the central value of the NLO predictions, there is a slight dependence on how these effects are combined with the NLO corrections. As shown in the middle inset, the additive and multiplicative approaches lead to slightly different shapes for the invariant mass distribution. This difference is marginal, though, and remains well within the uncertainty band. Nevertheless, for large values of , this difference in shape amounts to a few percent. In this phase-space region, predictions in the multiplicative approach can be preferred, as discussed in Section 2.3.
EW corrections in the and invariant mass distributions are more relevant near the production threshold, as was already observed in Frixione:2015zaa . This effect is due to a Sommerfeld enhancement arising, e.g., from one-loop diagrams with Higgs propagators connecting two of the final-state particles Degrassi:2016wml ; Maltoni:2017ims , which contribute to the corrections. In contrast with the analysis in Frixione:2015zaa , here also the subleading EW contributions are included: the and contributions are positive and almost completely cancel the negative corrections at large invariant masses, resulting in a negligible difference between the NLO+NNLL and +NNLL predictions. Note that the multiplicative approach spoils this cancellation, since it rescales only the corrections according to Eq. (23).
The distributions differential w.r.t. the pair invariant mass are shown in Figure 3. For the and processes (upper-left and upper-right plots, respectively) the corrections from threshold resummation to NNLL matched to the NLO corrections reduce the uncertainty band over the whole invariant mass range considered. The effect of the resummation on the central value of each bin is negligible. In comparison to the invariant mass distributions (Figure 2), the scale-uncertainty band at large top-pair invariant masses is narrower. The reason is that the two scale choices (-based and -based) give predictions that lie much closer to each other, in comparison to the case of invariant mass distributions.
On the other hand, for and production, the corrections to the invariant mass distributions due to resummation are similar to the case of the invariant mass distribution, i.e., they enhance the cross section over the full invariant mass range shown in the figure, starting with small effects at threshold and reaching up to about at . The theory uncertainties at these large invariant masses are still larger than the difference in central values between NLO and NLO+NNLL calculations, resulting in a stable perturbative expansion. For all four processes, the difference between the additive and multiplicative combination is small, even though there is a trend: the multiplicative approach gives a slightly softer invariant mass spectrum in the tail of the distribution.
The lowest inset in each of the four plots in Figure 2 shows that the EW corrections, included in the NLO+NNLL predictions, distort the shape of the distributions calculated to +NNLL accuracy. For all four processes, the EW corrections result in a positive contribution to the cross section at small invariant masses, where they enhance the distribution in each bin by for GeV. At larger invariant masses the EW corrections are negligible in and production. For and production, the EW corrections remain positive up to TeV, where they turn negative, as expected from EW Sudakov suppression.
4.2.2 Transverse momenta
The distributions differential w.r.t. the top quark (Figure 4) and antitop quark (Figure 5) transverse momentum are very similar. For these two observables, the NNLL resummation reduces the scale uncertainty in comparison to NLO calculations. In addition, NNLL resummed results lead to slightly smaller cross sections at large transverse momenta ( GeV). For production and especially for production, NNLL resummation increases the cross section in comparison to NLO calculations for transverse momenta smaller than GeV. Hence, the corrections due to soft-gluon emission affect the shape of the distribution, even though the NLO+NNLL and NLO uncertainty bands always have a large overlap. Even more than in the case of invariant mass distributions, the differences between calculations carried out in the additive and multiplicative approaches are marginal. The EW corrections show their typical behavior: for small transverse momenta, they induce a constant upward shift in the central value of the distribution in each bin, of the order of for and production and somewhat larger, , for and production. These effects decrease rapidly as the transverse momentum increases, until the EW corrections start lowering the QCD cross section in each bin for large transverse momenta. The large positive correction at small transverse momenta in and production is mainly due to the large correction in these processes. In this region of phase space, the EW corrections are of a size similar to the scale uncertainties of the resummed calculations; this results in a NLO+NNLL prediction for the distribution whose central value in each bin lies just within the uncertainty band of the +NNLL calculation, and vice versa.
The distributions differential w.r.t. the transverse momentum of the heavy EW boson are shown in Figure 6. Qualitatively, the corrections beyond , either from resummation of soft emission, or from EW corrections in fixed order perturbation theory, are very similar to the corresponding corrections to the distributions for the transverse momenta of the top and the antitop quarks. For this reason, many of the remarks made in the discussion of those distributions apply to Figure 6 as well. There is, however, one important exception, i.e., the non-negligible difference ( at ) between the additive and multiplicative matching of the resummed results with the NLO corrections at very large transverse momenta. The reason is that in the tail of these distributions both the and corrections are large and therefore the difference between the two approaches, which is dominated by the product of these two corrections, is not negligible.
4.2.3 Rapidities
Figures 7 and 8 show the predictions for the distributions differential w.r.t. the top-quark and top-antiquark rapidities, respectively. In each individual process, antitop quarks are produced more centrally than top quarks. This fact is particularly evident for the and processes. Indeed, this property is responsible for the large charge asymmetry for these processes.
In addition, by comparing the nNLO predictions to the NLO predictions for the differential distribution, (as it is done in first inset in each of the plots in Figure 7), it can be seen that approximate NNLO corrections enhance the distributions at large forward and backward rapidities. On the contrary, the region of forward and backward rapidities in the differential distribution receives relatively large corrections only in and production, but not in production. For production, the different behavior of the nNLO corrections to the and differential distributions in the forward and backward rapidity regions is the cause of the relatively large nNLO corrections to the charge asymmetry in these processes.
For these distributions, the additive and multiplicative approaches to matching lead to almost identical results. The EW corrections do not have a large impact on the shape of the top and antitop rapidity differential distributions, apart from the case of production, where the EW corrections enhance the small rapidity region of the distributions. Finally we observe that, for the distributions shown in Figures 7 and 8, when one compares nNLO calculations to NLO calculations, distributions for and production show a larger reduction of the relative size of the uncertainty bands than distributions for production.
5 Conclusions
The purpose of this paper is to provide the most complete predictions to date for the total cross section and several differential distributions for the , , and production processes at the LHC operating at a center of mass energy of TeV. In order to achieve this goal, we combined complete-NLO corrections, accounting for both QCD and electroweak effects, with the resummation of soft emission corrections to NNLL accuracy in QCD. The complete-NLO calculations were carried out with the most recent version of MadGraph5_aMC@NLO, while the NNLL resummation formulas were evaluated with an in-house parton level Monte Carlo code.
After considering theoretical uncertainties related to the choice of PDFs and to the residual dependence of the calculation on unphysical scales, we find the following predictions for the total cross section of the three processes
[TABLE]
These predictions have NLO+NNLL accuracy. The number in parentheses next to the central value of the cross section indicates the statistical uncertainty due to the Monte Carlo integration. The first set of uncertainties is related to scale choices, and the second to PDFs.
Several differential distributions were analyzed in Section 4: the invariant masses of the and systems, the transverse-momenta of the top quark, antitop quark and vector boson, all calculated to NLO+NNLL accuracy, and the rapidities of the top quark and antitop quark, evaluated to nNLO. The behavior of the perturbative series and the relative size of the residual theoretical uncertainties indicate that the predictions for the observables considered here are stable and sufficiently accurate when compared to current and expected experimental errors.
Acknowledgments
The authors are grateful to Marco Zaro for useful conversations. The work of R.F., D.P. and I.T. is supported in part by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovalevskaja Award Project “Event Simulation for the Large Hadron Collider at High Precision”. D.P. has been also supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre SFB1258. The work of A.F. is supported in part by the National Science Foundation under Grant No. PHY-1417354 and PSC CUNY Research Award TRADA-61151-00 49. A.B. acknowledges the support by the ERC Starting Grant REINVENT-714788. A.B. would like to thank New York City College of Technology CUNY (grant PSC-CUNY 60185-00 48) for the kind hospitality in December 2018. R.F. is supported in part by the Swedish Research Council under contract number 2016-05996. A.F. would like to thank S. Alioli and his group at Università degli Studi di Milano-Bicocca and INFN for their kind hospitality in the final stage of this work.
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