# A unified NLO description of top-pair and associated Wt production

## Abstract

We present an NLO simulation of WWbb production with massive b-quarks at the LHC. Off-shell and non-resonant contributions associated with top-pair and single-top channels and with leptonic W-boson decays are consistently taken into account using the complex-mass scheme. Thanks to the finite b-quark mass, WWbb predictions can be extended to the whole b-quark phase space, thereby including Wt-channel single-top contributions that originate from collinear g \(\rightarrow \) bb splittings in the four-flavour scheme. This provides a consistent NLO description of tt and Wt production and decay, including quantum interference effects. The simulation is also applicable to exclusive 0- and 1-jet bins, which is of great importance for Higgs-boson studies in the H \(\rightarrow \) WW channel and for any other analysis with large top backgrounds and jet vetoes or jet bins.

## Keywords

Integrate Cross Section Subtraction Term Scale Choice Dipole Subtraction Good Perturbative Convergence## 1 Introduction

Top quarks are the heaviest known fundamental particles, and the precise theoretical understanding of their production and decay mechanism, within or beyond the Standard Model, has deep implications on countless aspects of the LHC physics programme. At the LHC, top quarks are mainly produced as \({\mathrm {t}}\bar{\mathrm {t}}\) pairs and via single-top production in the \(t\)-channel or in the associated Wt mode. At 8 TeV these latter single-top channels amount to 40 and \(10\,\%\) of the \({\mathrm {t}}\bar{\mathrm {t}}\) cross section, respectively. In spite of their smaller cross sections, they play an important role as direct probes of top-quark weak interactions and of their flavour structure. The separation of top-production into individual top-pair and single-top contributions poses non-trivial experimental and theoretical challenges, which are mainly due to the similarity among the final states associated with the various mechanisms of top-production and decay. In particular, the definition of \({\mathrm {t}}\bar{\mathrm {t}}\) and \({\mathrm {W}}{\mathrm {t}}\) production involves notorious and quite subtle theoretical issues [1].

In the five-flavour (5F) scheme, \({\mathrm {W}}{\mathrm {t}}\) production proceeds via b-quark induced partonic channels like \({\mathrm {g}}{\mathrm {b}}\rightarrow {\mathrm {W}}^-{\mathrm {W}}^{+}{\mathrm {b}}\), and the presence of a single b-jet represents a clearly distinctive feature with respect to \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) final states associated with \({\mathrm {t}}\bar{\mathrm {t}}\) production. However, beyond LO this separation ceases to exist, since \({\mathrm {g}}{\mathrm {g}}\rightarrow {\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) enters also the next-to-leading order (NLO) corrections to \({\mathrm {W}}{\mathrm {t}}\) production. The resulting \({\mathrm {t}}\bar{\mathrm {t}}\) contamination represents a huge NLO correction, which jeopardises the perturbative convergence of the \({\mathrm {W}}{\mathrm {t}}\) cross section in the 5F scheme. To circumvent this problem within the 5F scheme, various approaches have been proposed aimed at subtracting the contribution of a second top resonance in \({\mathrm {p}}{\mathrm {p}}\rightarrow {\mathrm {W}}{\mathrm {t}}+X\) [1]. However, these prescriptions either break gauge invariance or are not applicable to a realistic experimental setup. Moreover they neglect the quantum interference between top-pair and single-top contributions.

A theoretically more rigorous approach consists of adopting the four-flavour (4F) scheme, where initial-state b-quarks result from gluons via explicit \({\mathrm {g}}\rightarrow {\mathrm {b}}\bar{\mathrm {b}}\) splittings. In this framework, the process \({\mathrm {p}}{\mathrm {p}}\rightarrow {\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}+X\) provides a unified description of Wt and \({\mathrm {t}}\bar{\mathrm {t}}\) production [2], and the presence of the \({\mathrm {t}}\bar{\mathrm {t}}\)–Wt interference at LO stabilises the perturbative expansion. In the 4F scheme, treating finite-top-width effects in the complex-mass scheme [3] ensures a consistent off-shell continuation of top-quark propagators and allows one to include double-, single-, and non-resonant contributions to \({\mathrm {p}}{\mathrm {p}}\rightarrow {\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}+X\) with all relevant interferences. Moreover, the ill-defined separation of top-pair and Wt production can be replaced by a gauge-invariant separation of \({\mathrm {p}}{\mathrm {p}}\rightarrow {\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) into its narrow-top-width limit, which corresponds to on-shell top-pair production and decay, and a finite-width remainder that includes off-shell \({\mathrm {t}}\bar{\mathrm {t}}\) effects as well as single-top and non-resonant contributions plus related interferences.

The presence of four final-state particles and intermediate top-quark resonances render the simulation of \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) production quite challenging beyond LO. First NLO calculations with massless b-quarks have been presented in [4, 5, 6]. For \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) production with two hard b-jets, apart from a few noticeable exceptions [5], most observables turn out to be completely dominated by the on-shell \({\mathrm {t}}\bar{\mathrm {t}}\) contribution. In phase-space regions with unresolved b-quarks, the importance of off-shell and single-top contributions is expected to increase quite substantially. However, due to the presence of collinear singularities, such regions are not accessible in the massless b-quark approximation of [4, 5, 6]. To fill this gap, in this paper we present a complete NLO \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) calculation including off-shell W-boson decays and massive b-quarks in the 4F scheme. A similar calculation has been presented very recently in [7]. These simulations provide NLO accurate \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) predictions in the full phase space and allow one to investigate, for the first time, top-pair and single-top production in presence of jet vetoes or jet bins, such as in the case of the \({\mathrm {H}}\rightarrow {\mathrm {W}}^{+}{\mathrm {W}}^-\) analysis. An important advantage of NLO \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) predictions in the 4F scheme is that they provide a fully differential NLO description of both final-state b-jets and a correspondingly accurate modelling of jet vetoes, while in the 5F scheme a similar level of accuracy for spectator b-quarks in Wt production would require an NNLO calculation.

## 2 Technical tools and ingredients of the calculation

The entire calculation has been performed with highly flexible and automated NLO programs, and the high complexity resulting from the presence of multiple top- and W-resonances, as well as from the wide spectrum of involved scales, render \({\mathrm {p}}{\mathrm {p}}\rightarrow {\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) an excellent technical benchmark to test the performance of the employed tools. To evaluate tree, virtual, and real-emission amplitudes, we employed OpenLoops [8], a new one-loop generator that will become public in the next future. The OpenLoops program is based on a novel numerical recursion, which is formulated in terms of loop-momentum polynomials called ‘open loops’ and allows for a fast evaluation of scattering amplitudes with many external particles. It uses the Collier library [9] for the numerically stable evaluation of tensor integrals [10, 11] and scalar integrals [12]. Together with [13, 14], the present study is one of the very first applications of OpenLoops. Phase-space integration and infrared subtractions are performed with an in-house NLO Monte-Carlo framework [15], which is interfaced with OpenLoops and provides full automation along the entire chain of operations that are required for NLO calculations. This tool is applicable to any Standard-Model process at NLO QCD. Infrared singularities are handled with dipole subtraction [16, 17], and since collinear \({\mathrm {g}}\rightarrow {\mathrm {b}}\bar{\mathrm {b}}\) splittings are regularised by the finite b-quark mass, corresponding subtraction terms are not included. The phase-space integrator is based on the adaptive multi-channel technique [18] and implements dedicated channels for the dipole subtraction terms, which improve the convergence, especially for multi-resonance processes. Multiple scale variations in a single run are also supported. This tool has been validated in several NLO processes and, in combination with OpenLoops and Collier, it is also applicable to NNLO calculations [19]. The correctness of the results is supported by various checks: OpenLoops has been validated against an independent in-house generator for more than hundred partonic processes, including \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) production with massless b-quarks and various processes with massive heavy-quarks. For the process at hand we checked the cancellation of infrared and ultraviolet singularities. The correctness of phase-space integration and dipole subtraction was tested by means of a second calculation based on OpenLoops in combination with Sherpa [20, 21] and Amegic++ [22].

## 3 Input parameters, cuts and jet definition

^{1}that is obtained from a variable-flavour set with \(\alpha _{\mathrm {s}}^{(5)}(M_\mathrm {Z})=0.118\) via inverse 5F evolution down to \(\mu _\mathrm {F}=m_{{\mathrm {b}}}\) and subsequent upward evolution with four active flavours. Since the NNPDF2.3 release does not include LO parton distributions, for LO predictions we adopt the NNPDF21_lo_nf4_100 4F set, which corresponds to a reference strong-coupling value \(\alpha _{\mathrm {s}}^{(5)}(M_\mathrm {Z})=0.119\). While the 4F running of \(\alpha _{\mathrm {s}}\) misses heavy-quark-loop effects, corresponding \(\mathcal {O}(\alpha _{\mathrm {s}})\) contributions are consistently included in the virtual corrections via zero- momentum subtraction of the top- and bottom-quark loops in the renormalisation of \(\alpha _{\mathrm {s}}\).

## 4 Scale choice for top-pair and single-top production

^{2}The respective characteristic scales are the bottom- and the top-quark transverse energies, \(E_{\mathrm {T},{\mathrm {b}}}\ll E_{\mathrm {T},{\mathrm {t}}}\), and a QCD scale of type

^{3}

^{4}The tuning of \(R\) is performed in LO approximation on the fully inclusive level and yields \(R=7.96\). At NLO, the kinematic quantities that enter \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) are defined in terms of b- and \(\bar{\mathrm {b}}\)-jet momenta that are constructed with a modified jet algorithm where \({\mathrm {b}}\bar{\mathrm {b}}\) pairs are not clustered and light partons with \(|\eta |>4.5\) are excluded from the recombination procedure. The latter prescription guarantees the collinear safety of the reconstructed top mass, \((p_{{\mathrm {W}}}+p_{{\mathrm {b}}})^2\), with respect to collinear light-parton emission from the initial state. In the reconstruction of the top and anti-top masses \((p_{{\mathrm {W}}}+p_{{\mathrm {b}}})^2\) that enter (15), remaining hard jets are clustered with the \({\mathrm {t}}\)- or \(\bar{\mathrm {t}}\)- system if the resulting invariant mass turns out to be closer to \(m_{{\mathrm {t}}}\). Top-jet clusterings are applied only if they yield \(P_{\mathrm {t}}>0.5\). If that holds for \({\mathrm {t}}\)- and \(\bar{\mathrm {t}}\)- system, the clustering to maximise the \({\mathrm {t}}\bar{\mathrm {t}}\) probability, \(P_{{\mathrm {t}}}P_{\bar{\mathrm {t}}}\), is chosen.

## 5 Predictions for the LHC at 8 TeV

Predictions for the integrated cross section and in exclusive jet bins are listed in Table 1. To assess the influence of the scale choice, results based on \(\mu _0=\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) are compared to the case of the conventional scale \(\mu _0=m_{{\mathrm {t}}}\). For the total cross section we find positive corrections of about 40 %.^{5}

Scale uncertainties decrease from about 30 % at LO to 10 % at NLO, and the differences between the two scale choices are consistent within scale variations. The last three columns of Table display jet cross sections in bins with \(N_{j}=0,1\) and \(N_{j}\ge 2\) jets, where \(N_{j}\) refers to the total number of b-jets and light jets. The different bins receive quite different corrections, and the relative weight of the individual bins in percent changes from 3:30:67 at LO to 2:21:76 at NLO. This indicates that a significant fraction of the 0- and 1-jet bin cross sections migrates to the inclusive 2-jet bin. We attribute this feature to the rather high probability of light-jet emissions with \(p_\mathrm {T}\gtrsim 30\,\mathrm {GeV}\). While NLO scale uncertainties turn out to be fairly small in all jet bins, matching to the parton shower is certainly important for a more reliable description of such radiative processes. Comparing the two scale choices, also in jet bins we do not observe any dramatic difference: absolute LO and NLO results are well consistent within scale variations, and also K-factors and scale variations themselves turn out to be quite similar.

LO and NLO predictions for \({\mathrm {p}}{\mathrm {p}}\rightarrow {\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) at 8 TeV with scale variations and corrections, \(K=\sigma _{\mathrm {NLO}}/\sigma _{\mathrm {LO}}\), for different scale choices: total cross section with leptonic cuts and partial contributions with 0, 1 and \({\ge }2\) jets. Full \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) predictions (\(\sigma \)) are compared to finite-top-width contributions (\(\sigma ^{\mathrm {FtW}}\))

\(\mu _0\) | \(\sigma [\mathrm {fb}]\) | \(\sigma _{0}[\mathrm {fb}]\) | \(\sigma _{1}[\mathrm {fb}]\) | \(\sigma _{2^+}[\mathrm {fb}]\) | |
---|---|---|---|---|---|

\(\mathrm {LO}\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | \(1232^{+34\,\%}_{-24\,\%}\) | \( 37^{+38\,\%}_{-25\,\%}\) | \( 367^{+36\,\%}_{-24\,\%}\) | \( 828^{+33\,\%}_{-23\,\%}\) |

\(\mathrm {NLO}\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | \(1777^{+10\,\%}_{-12\,\%}\) | \( 41^{+3\,\%}_{-8\,\%}\) | \( 377^{+1\,\%}_{-6\,\%}\) | \( 1359^{+14\,\%}_{-14\,\%}\) |

\(K\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | 1.44 | 1.09 | 1.03 | 1.64 |

\(\mathrm {LO}\) | \(m_{{\mathrm {t}}}\) | \(1317^{+35\,\%}_{-24\,\%}\) | \( 35^{+37\,\%}_{-25\,\%}\) | \( 373^{+36\,\%}_{-24\,\%}\) | \( 909^{+35\,\%}_{-24\,\%}\) |

\(\mathrm {NLO}\) | \(m_{{\mathrm {t}}}\) | \(1817^{+8\,\%}_{-11\,\%}\) | \( 40^{+4\,\%}_{-8\,\%}\) | \( 372^{+1\,\%}_{-8\,\%}\) | \( 1405^{+13\,\%}_{-13\,\%}\) |

\(K\) | \(m_{{\mathrm {t}}}\) | 1.38 | 1.14 | 1.00 | 1.55 |

\(\mu _0\) | \({\sigma ^{\mathrm {FtW}}}[\mathrm {fb}]\) | \({\sigma ^{\mathrm {FtW}}_{0}}[\mathrm {fb}]\) | \({\sigma ^{\mathrm {FtW}}_{1}}[\mathrm {fb}]\) | \({\sigma ^{\mathrm {FtW}}_{2^+}}[\mathrm {fb}]\) | |
---|---|---|---|---|---|

\(\mathrm {LO}\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | \(91^{+41\,\%}_{-27\,\%}\) | \( 13^{+42\,\%}_{-27\,\%}\) | \( 71^{+40\,\%}_{-27\,\%}\) | \( 7^{+45\,\%}_{-29\,\%}\) |

\(\mathrm {NLO}\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | \(107^{+6\,\%}_{-11\,\%}\) | \( 13^{+1\,\%}_{-7\,\%}\) | \( 61^{+2\,\%}_{-16\,\%}\) | \( 33^{+51\,\%}_{-31\,\%}\) |

\(K\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | 1.18 | 0.99 | 0.86 | 4.70 |

\(\mathrm {LO}\) | \(m_{{\mathrm {t}}}\) | \(63^{+36\,\%}_{-25\,\%}\) | \( 8^{+36\,\%}_{-25\,\%}\) | \( 49^{+36\,\%}_{-24\,\%}\) | \( 6^{+46\,\%}_{-29\,\%}\) |

\(\mathrm {NLO}\) | \(m_{{\mathrm {t}}}\) | \(100^{+17\,\%}_{-16\,\%}\) | \( 13^{+14\,\%}_{-14\,\%}\) | \( 65^{+9\,\%}_{-12\%}\) | \( 23^{+42\,\%}_{-28\,\%}\) |

\(K\) | \(m_{{\mathrm {t}}}\) | 1.58 | 1.47 | 1.32 | 3.89 |

Full \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) predictions and finite-top-width contributions for bins with 0, 1 and \({\ge }2\) b-jets. Same conventions as in Table

\(\mu _0\) | \(\sigma [\mathrm {fb}]\) | \(\sigma _{0}[\mathrm {fb}]\) | \(\sigma _{1}[\mathrm {fb}]\) | \(\sigma _{2^+}[\mathrm {fb}]\) | |
---|---|---|---|---|---|

\(\mathrm {LO}\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | \(1232^{+34\,\%}_{-24\,\%}\) | \( 37^{+38\,\%}_{-25\,\%}\) | \( 367^{+36\,\%}_{-24\,\%}\) | \( 828^{+33\,\%}_{-23\,\%}\) |

\(\mathrm {NLO}\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | \(1777^{+10\,\%}_{-12\,\%}\) | \( 65^{+20\,\%}_{-17\,\%}\) | \( 571^{+14\,\%}_{-14\,\%}\) | \( 1140^{+7\,\%}_{-10\,\%}\) |

\(K\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | 1.44 | 1.73 | 1.56 | 1.38 |

\(\mathrm {LO}\) | \(m_{{\mathrm {t}}}\) | \(1317^{+35\,\%}_{-24\,\%}\) | \( 35^{+37\,\%}_{-25\,\%}\) | \( 373^{+36\,\%}_{-24\,\%}\) | \( 909^{+35\,\%}_{-24\,\%}\) |

\(\mathrm {NLO}\) | \(m_{{\mathrm {t}}}\) | \(1817^{+8\,\%}_{-11\,\%}\) | \( 63^{+20\,\%}_{-17\,\%}\) | \( 584^{+14\,\%}_{-14\,\%}\) | \( 1170^{+5\,\%}_{-9\,\%}\) |

\(K\) | \(m_{{\mathrm {t}}}\) | 1.38 | 1.80 | 1.56 | 1.29 |

\(\mu _0\) | \({\sigma ^{\mathrm {FtW}}}[\mathrm {fb}]\) | \({\sigma ^{\mathrm {FtW}}_{0}}[\mathrm {fb}]\) | \({\sigma ^{\mathrm {FtW}}_{1}}[\mathrm {fb}]\) | \({\sigma ^{\mathrm {FtW}}_{2^+}}[\mathrm {fb}]\) | |
---|---|---|---|---|---|

\(\mathrm {LO}\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | \(91^{+41\,\%}_{-27\,\%}\) | \( 13^{+42\,\%}_{-27\,\%}\) | \( 71^{+40\,\%}_{-27\,\%}\) | \( 7^{+45\,\%}_{-29\,\%}\) |

\(\mathrm {NLO}\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | \(107^{+6\,\%}_{-11\,\%}\) | \( 20^{+18\,\%}_{-17\,\%}\) | \( 82^{+4\,\%}_{-10\,\%}\) | \( 5^{+2\,\%}_{-10\,\%}\) |

\(K\) | \(\mu _{{\mathrm {W}}{\mathrm {W}}{\mathrm {b}}{\mathrm {b}}}\) | 1.18 | 1.49 | 1.16 | 0.77 |

\(\mathrm {LO}\) | \(m_{{\mathrm {t}}}\) | \(63^{+36\%}_{-25\%}\) | \( 8^{+36\%}_{-25\%}\) | \( 49^{+36\%}_{-24\%}\) | \( 6^{+46\%}_{-29\%}\) |

\(\mathrm {NLO}\) | \(m_{{\mathrm {t}}}\) | \(100^{+17\%}_{-16\%}\) | \( 16^{+22\%}_{-18\%}\) | \( 77^{+16\%}_{-15\%}\) | \( 6^{+12\%}_{-16\%}\) |

\(K\) | \(m_{{\mathrm {t}}}\) | 1.58 | 1.89 | 1.58 | 1.10 |

## 6 Summary and conclusions

We have presented a complete NLO simulation of \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) production at the LHC, including W-boson decays in the opposite-flavour di-lepton channel, finite W- and top-width effects, and massive b-quarks in 4F scheme. The finite b-quark mass acts as a regulator of collinear singularities and allows one to describe the full b-quark phase space, including single-top contributions that arise from initial-state \({\mathrm {g}}\rightarrow {\mathrm {b}}\bar{\mathrm {b}}\) splittings followed by \({\mathrm {g}}{\mathrm {b}}\rightarrow {\mathrm {W}}{\mathrm {t}}\) scattering. This yields a gauge-invariant description of top-pair, single-top, and non-resonant \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) production including all interferences at NLO QCD. We introduced a dynamical scale choice aimed at an improved perturbative stability of initial-state \({\mathrm {g}}\rightarrow {\mathrm {b}}\bar{\mathrm {b}}\) splittings in single-top contributions. Using this scale, the NLO \({\mathrm {W}}^+{\mathrm {W}}^-{\mathrm {b}}\bar{\mathrm {b}}\) cross section in bins with 0, 1 and 2 jets features NLO scale uncertainties at the 10–15 % level. The more conventional choice \(\mu _0=m_{{\mathrm {t}}}\) yields similarly small NLO uncertainties in jet bins. While providing further evidence of the good convergence of the perturbative expansion, this means that a sophisticated dynamical scale is unnecessary for the rather inclusive observables considered in this letter. However, such a dynamical scale might become important for more exclusive observables, like jet-\(p_\mathrm {T}\) distributions.

Finite-top-width corrections mainly originate from single-top and off-shell \({\mathrm {t}}\bar{\mathrm {t}}\) contributions. They represent 6 % of the integrated cross section and are strongly sensitive to the jet multiplicity. In the 2-jet bin they are as small as 2 %, while in the 1- and 0-jet bins they reach the 16 and 32 % level, respectively. Also NLO corrections vary quite strongly with the jet multiplicity. Moreover, finite-top-width contributions receive quite different corrections as compared to on-shell \({\mathrm {t}}\bar{\mathrm {t}}\) production.

The non-trivial interplay of NLO and finite-width effects is especially relevant for the 0- and 1-jet bins. It plays an important role for the accurate description of associated \({\mathrm {W}}{\mathrm {t}}\) production, as well as for top backgrounds to \({\mathrm {H}}\rightarrow {\mathrm {W}}^{+}{\mathrm {W}}^-\) and to other searches based on leptons, large missing energy and jet vetoes. All employed tools are fully automated and can easily be exploited to extend the present results to the like-flavour di-lepton channel or to simulate any other Standard-Model process at NLO QCD.

## Footnotes

- 1.
NNPDF23_nlo_FFN_NF4_as_0118.

- 2.
The charge-conjugate channels are implicitly understood.

- 3.
The \(\chi _{{\mathrm {b}}}\) and \(\chi _{{\mathrm {t}}}\) distributions are defined as dimensionless functions by introducing \(m_{{\mathrm {t}}}\)-terms in the numerator. This convention is, however, irrelevant, since the probabilities resulting from (16) and (17) are independent of the normalisation of \(\chi _{{\mathrm {b}}}\) and \(\chi _{{\mathrm {t}}}\).

- 4.
Here we assume that finite-top-width effects are dominated by non-\({\mathrm {t}}\bar{\mathrm {t}}\) contributions. Note also that the finite-top-width term on the left-hand side of (17) must be extracted through \(\Gamma _{{\mathrm {t}}}\rightarrow 0 \) extrapolation by keeping \(\Gamma _{{\mathrm {t}}}\) and \(R\) fixed in (15)–(16).

- 5.
We note that these results are not directly comparable to those of [5], which reports a significantly smaller \(K\)-factor. In particular, while we apply the same cuts on leptons, missing energy and jets, here we do not restrict ourselves to the case of two b-jets, we adopt a smaller jet-resolution parameter and a different QCD scale choice. Moreover we employ a 4F PDF set, which implies an enhancement of the gluon density due to the absence of \({\mathrm {g}}\rightarrow {\mathrm {b}}\bar{\mathrm {b}}\) splittings in the PDF evolution. The LO PDF sets used in [5] and in the present study feature also significantly different values of \(\alpha _{\mathrm {s}}\), which influences LO results and \(K\)-factors. Finally, in addition to uniform scale variations considered in [5], here also independent \(\mu _{\mathrm {R}}\) and \(\mu _{\mathrm {F}}\) variations are taken into account.

## Notes

### Acknowledgments

We thank A. Denner, S. Dittmaier and L. Hofer for providing us with the one-loop tensor-integral library Collier. We are grateful to S. Höche and F. Siegert for Sherpa technical support. Our research is funded by the SNSF and supported, in part, by the European Commission through the network PITN-GA-2010-264564 (*LHCPhenoNet*).

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