Weak Vector Boson Production with Many Jets at the LHC $\sqrt{s}= 13$ TeV

Signatures with an electroweak vector boson and many jets play a crucial role at the Large Hadron Collider, both in the measurement of Standard-Model parameters and in searches for new physics. Precise predictions for these multi-scale processes are therefore indispensable. We present next-to-leading order QCD predictions for $W^\pm/Z$+jets at $\sqrt{s}=13$ TeV, including up to five/four jets in the final state. All production channels are included and leptonic decays of the vector bosons are considered at the amplitude level. We assess theoretical uncertainties arising from renormalization- and factorization-scale dependence by considering fixed-order dynamical scales based on the $H_{\rm T}$ variable as well as on the MiNLO procedure. We also explore uncertainties associated to different choices of parton-distribution functions. We provide event samples that can be explored through publicly available $n$-tuple sets, generated with BlackHat in combination with SHERPA.

Precise calculations for vector-boson production with a single jet have reached a new era with the recent results at next-to-next-to-leading order (NNLO) QCD precision for V + 1 jet [7] as well as many other refinements, which for example have been used to precisely describe backgrounds for dark matter searches [8]. At next-to-leading-order (NLO) QCD, the state of the art is nowadays calculations for processes with four and five jets [9][10][11] in the final state. The matching to parton showers has been carried out for up to three jets in the final state [12], and multi-jet merging has been studied with up to two jets [13]. Nextto-leading order electroweak corrections have been computed for up to three jets [14], and combined with QCD merging [15].
In this article we extend the set of NLO QCD calculations by dedicated predictions for the 13 TeV LHC for W + n-jet and Z + m-jet production with n ≤ 5 and m ≤ 4. We employ the BlackHat library [16] for computing the required one-loop matrix elements. This library is based on on-shell and unitarity techniques (see for example [17]) which allows the extraction of loop amplitudes from simpler building blocks. It has been employed for many multi-light-jet studies at hadron colliders [9][10][11][18][19][20][21][22], and has recently been extended to applications for high-multiplicity processes including heavy jets [23]. The calculation is performed with the help of the Sherpa package [24], used for integration over phase space as well as calculation of real corrections employing the Catani-Seymour dipole subtraction [25] as implemented in the matrix-element generator Comix [26]. We have stored our results and made them publicly available as Root [27] n-tuple files. This file format [28] contains all information necessary to compute the NLO fixed-order predictions, to change the run-ning QCD coupling, renormalization and factorization scales, as well as parton-distribution functions (PDFs). We employ an extension [29] of the n-tuple file format which allows for extended reweighting procedures.
Due to its large center-of-mass energy, the LHC explores scales ranging from tens of GeV's to the multi-TeV regime. This large hierarchy of scales makes leading-order (LO) QCD predictions unreliable, as they tend to be very sensible to the unphysical renormalization and factorization scales, and also because they can miss important initial-state partonic configurations. Next-to-leading order QCD calculations are less affected by scale choices and give a first reliable estimate of the theoretical uncertainty * . We study in detail in this paper the theoretical uncertainties associated to our predictions that are related to scale sensitivity and PDF dependence. We explore in particular the scale dependence by using fixed-order dynamical scales based on the total partonic transverse energy as well as different variants of the MiNLO method [31]. In addition, we perform the conventional scale variation by constant factors around the central scales. A similar study was carried out recently for on-shell tt production in association with up to three light jets [32], and good agreement was found between NLO QCD results employing a similar set of dynamical scales. Our analysis is the first to compare results obtained with fixed-order scales and with the MiNLO method in processes with four or five light jets in the final state. This paper is organized as follows. In section II we summarize our calculational setup, showing all kinematical information employed both at the level of producing n-tuple files and for the distributions studied in the rest of the paper. We also show in this section our implementation of the dynamical scales employed. In section III we present our results for total and differential cross sections and study their scale dependence and uncertainties associated to PDFs. In section IV we show a series of observable ratios that can help to reduce the theoretical uncertainties in multi-jet environments. The paper ends with our conclusions in section V. * A notable exception to this are processes with large accessible phase space and missing partonic channels at the next-to-leading order. This has been pointed out for example in [30].

II. BASIC SETUP
We employ the Sherpa package [24] to manage the overall calculations. The required one-loop matrix elements are produced by the BlackHat library [16]. Born and realemission contributions are computed by the matrix-element generator Comix [26], which also provides the necessary Catani-Seymour subtraction terms [25]. More details of our computational setup can be found in Refs. [10] and [11]. We have included in all of our results all contributing subprocesses, confirming in particular that 8-quark finite contributions to the real part in W ± + 5-jet production are negligible [10].
We use the CT14 LO (CT14llo) and NLO (CT14nlo) PDFs [33] at the respective orders, including the corresponding definition of the strong coupling α s . We also employ the corresponding CT14nlo error set to explore PDF uncertainties, and compare to predictions generated with the PDF error sets of ABM [34], MMHT [35] and NNPDF 3.1 [36]. The lepton-pair invariant mass follows a relativistic Breit-Wigner distribution, with M W = 80.385 GeV and M Z = 91.1876 GeV, and the widths are given by Γ Z = 2.4952 GeV and Γ W = 2.085 GeV.
We employ a diagonal CKM matrix and use real values for the electroweak parameters.
All light quarks (u, d, s, c, b) are treated as massless. We do not include contributions from real or virtual top quarks, and we expect this to have a percent-level effect on crosssections [9,11,23,37]. The one-loop matrix elements for V + 4, 5-jets have been computed in the leading-color approximation [38] which we find to be precise at the level of 2% of the total cross section in lower jet-multiplicity calculations. Our results are quoted for a single lepton (pair) flavor. We treated both leptons as massless, an approximation that can be applied to the electron or muon families. Results presented are produced in fixed-order parton-level perturbation theory and we do not apply any non-perturbative corrections to account for effects associated to underlying event or hadronization.
In the following subsections we describe first the common setup to define the fiducial regions in which total and differential cross sections are computed. Second, we list the basic phase-space cuts applied at the level of producing the n-tuple sets and finally the dynamical scales that we employ to study the scale sensitivity of our results.

A. Kinematical Setup
In our study we consider the inclusive processes pp → V + n jets at the LHC with centerof-mass energy √ s = 13 TeV, with n ≤ 5 and n ≤ 4 for V = W ± and Z, respectively. We define jets using the anti-k T algorithm [39] with R = 0.4 and impose the kinematical cuts: We order the jets in p T and label them according to their hardness. For all charged leptons we require: For processes with W ± bosons we define its transverse mass by M W ). For these we impose the additional cuts: Finally for processes with a Z boson we impose the following constraint on the invariant mass of its decay products: (2.4)

B. Kinematical Setup for Public N -tuples
We have saved intermediate results in publicly available Root-format [27] n-tuple files [28] in order to facilitate new studies of infrared-safe observables with different cuts, different scale choices, jet algorithms or PDF sets. Our study uses an extension of the original file format which facilitates reweighting in the MiNLO procedure and enables extended reweighting procedures [29]. Due to the generation cuts applied to events stored in the n-tuple files, future studies should operate either with identical cuts or with tighter cuts.
We use the FastJet package [40] to define jets according to the algorithms anti-k T , k T and SISCone [39,41,42] with jet parameter R = 0.4, 0.5, 0.6 and 0.7. We set the recombination f -parameter for the SISCone algorithm to 0.75. We impose p jet T > 30 GeV on all jets and do not constrain their pseudorapidity.
For charged leptons we impose p l T > 20 GeV and |η l | < 3, while for neutrinos we require p ν T > 10 GeV. For processes with a Z boson, in which γ * contributions appear, we also constrain the associated invariant-lepton-pair mass by 60 GeV < M l + l − < 120 GeV.
We explore the renormalization and factorization scale dependence of the cross sections using a conventional variation of the central scale by factors (1/2, 1/ √ 2, 1, √ 2, 2), keeping factorization and renormalization scales equal. In addition, we explore the sensitivity of the calculation to the functional form of the scale using on one hand conventional fixed-order scales based on the H T variable and on the other hand the so-called MiNLO procedure [31].
In this subsection, we first define all the fixed-order scales employed, and second we present our variant of the MiNLO procedure, which we will label MiNLO . Finally we summarize the nomenclature used in this article to label all dynamical scales considered.

Fixed-order scales
We define the total partonic transverse energy variableĤ T according to: where the sum runs over all final-state partons and E V T ≡ M 2 V + (p V T ) 2 is the transverse energy of the vector boson (V either W or Z). The scale µ 0 =Ĥ T /2 has proven to be a sensitive choice as it tends to reduce the shape changes and global size of quantum corrections when going from leading to next-to-leading order (see for example [10,11,22]). In general, NLO corrections are less sensitive to the choices of scale, as long as the scale reflects the hardness of the Born process [19].
We introduce an additional scale, designed to match the invariant mass of the lepton pair in kinematic configurations with very small hadronic transverse energy and the transverse momentum of the hardest QCD jet in processes of di-jet type. This scale is denoted aŝ (2.6) In section II C 3 we give the standard notation used throughout this article to refer to the different scales employed.

MiNLO
The second type of dynamical scale considered in this study, schematically denoted as MiNLO , is based on the MiNLO reweighting procedure proposed in Ref. [31], which is inspired by the NLL branching formalism in [41]. It builds on an event-by-event identification of the most likely branching history leading to the full V + n−parton final state using a k T -type clustering algorithm. No-branching probabilities in the form of NLL Sudakov form factors are assigned to the intermediate "partons" in the branching tree, to reflect the fact that no radiation above a resolution scale, given by lowest nodal k T value in the clustering, should occur. The strong coupling associated to each node in the branching tree is evaluated at the respective transverse momentum, following [43]. This method can also be interpreted as a generalization of the CKKW procedure [44] to NLO QCD calculations, but without the possibility to further develop the intra-jet real radiation pattern through resolved emissions.
In particular, the MiNLO method accounts for the resummation of large logarithmic corrections associated to very disparate scales in high-energy collisions using the known universal factorization properties of the cross section in the collinear limit.
In order to reflect the nature of QCD interactions, only 1 → 2 branchings consistent with elementary interaction vertices are allowed in our MiNLO procedure. In addition, we require the branching history to be ordered, which implies that we terminate the clustering as soon as an inverted scale hierarchy is encountered. In this case, by default we set the scale of the remaining V + m-parton (with m < n) "core" interaction, µ core , toĤ T /2 (or when explicitly stated, toŜ T ). This biases the scale choice for events with many hard scales towardsĤ T /2 (Ŝ T ), an effect that will be further discussed in Sec. III. Note in particular that at very high energies there may be configurations where no clustering can be performed at all, for example a V + 2 jet event with p T,j1 ≈ p T,j2 m T,W or a V + 5 jet event where p T,j1 ≈ p T,j2 ≈ . . . ≈ p T,j5 and y j1 y j2 . . . y j5 . The large logarithms associated with such configurations cannot be resummed in QCD collinear factorization and are therefore not amenable to a treatment in the coherent branching formalism on which the MiNLO method is based. This is a considerable source of uncertainty because of the large available phase space at the LHC.
In general, the clustering procedure for a leading-order process of O(α N s ) will yield a branching history with M ≤ N ordered nodal scales q 1 . . . q M and a core interaction of connected to the i-th branching are multiplied by a factor ∆ a (q min , q i ), where the lowest branching scale q 1 = q min is identified as the resolution scale. Intermediate lines connecting nodes j < i are dressed by factors ∆ a (q min , q i )/∆ a (q min , q j ). Internal lines connected to the primary process are assigned form factors between their respective scales and µ core . The factorization scale µ F used in the evaluation of the PDFs is set to the lowest scale, µ F = q 1 .
In our MiNLO procedure, we use a physical definition of the Sudakov form factors, which is given by and a = g, q corresponds to massless gluons and quarks, respectively. Eq. (2.7) does not exceed unity and can therefore be interpreted as a no-branching probability between the scales Q 0 and Q, while maintaining the correct limiting behavior for Q 0 Q. The above definition is easily obtained from the known NLL expressions of [46] by employing the following symmetry of the LO DGLAP splitting functions b=q,g (2.9) Following standard practice, we include next-to-leading logarithms proportional to the twoloop cusp anomalous dimension [45].
The generalization of the MiNLO method to NLO requires some modifications [31]. Virtual corrections and integrated IR-subtraction terms are treated identically to the leading order case. Real-emission events have branching histories with M + 1 ≤ N + 1 ordered branchings, but they are treated as born-like M -parton events for the purpose of scale definition. This is achieved by discarding the softest branching, i.e. if the M + 1 step branching history is given by q 0 < q 1 < · · · < q K , we set the resolution scale to q 1 . Consequently, the softest emission at NLO (with scale q 0 ) is neither dressed with Sudakov factors nor does it enter the definitions of µ R and µ F . In order to retain NLO accuracy of the full calculation, the Born configuration receives correction terms that are proportional to the first-order expansion of the Sudakov factors, eq. (2.7). Note that in this case the scale of the strong coupling is set to µ R , and that the two-loop cusp term is neglected, as it contributes at O(α 2 s ). The value of the additional strong coupling at NLO (the N + 1-th power) appearing in both real and virtual corrections is set to the average of all other values of α s , i.e.  [32]. We perform them in the same manner, i.e. we set q 1 equal to µ F .

Nomenclature for dynamical scales explored
Throughout this paper, we set renormalization and factorization scales equal µ R = µ F = µ 0 . Our results labeled "LO" and "NLO" use the central scale µ 0 =Ĥ T /2 by default, wherê H T is defined in eq. (2.5). When necessary, to distinguish the usage of the fixed-order scales defined in section II C 1, we write "(N)LOĤ T /2" or "(N)LOŜ T ", whereŜ T is defined in eq. (2.6).
In our MiNLO procedure described in section II C 2 the default core scale is µ core = H T /2. When considering variations of this choice we explicitly write "Mi( We also compare to the original formulation of the MiNLO method [31] for processes with fewer than three jets in the final state. Compared to our implementation, this variant uses the Sudakov factors of [46] and unordered clustering histories are treated in a different manner. Following the previous naming convention, we label those results as "Mi(N)LOĤ T /2" or "Mi(N)LOŜ T " depending on the choice of core scale µ core employed.

A. Total cross sections
In tables I, II and III we present total cross sections for the production of a weak vector boson V in association with up to 5 jets for V = W + and W − , and with up to 4 jets for V = Z. Results with central scaleĤ T /2 and MiNLO are included (see section II C 3 for the nomenclature that we use for the different dynamical scales considered). We also show in table IV jet production ratios [22,47] for all the vector boson, that is the ratios of the total cross sections for the production for V + n jets to the production of V + (n − 1) jets.
We observe that LO cross sections for the scale choice µ 0 =Ĥ T /2 have a monotonic increase of scale dependence from about 20% for V + 2 jet up to 50% for V + 5 jets. We notice that the LO scale dependence for V + 1 jet, which appears at around 4% is not representative of the associated theoretical uncertainties, in particular due to kinematical constraints that are released at NLO. That is, at LO the p T of the vector boson matches the one from the unique jet. Real contributions at NLO release this constraint, producing a soft enhancement that tends to produce large corrections [19,30,48].
The central scale choice for the dynamical scale µ 0 =Ĥ T /2 falls near the plateau of the NLO scale dependence. This makes the uncertainty estimates based on lower/upper values of the cross sections seem slightly small. If we quote the absolute deviations with respect to this value, then we can estimate NLO scale sensitivity at the order of 10% (running from about 6% to 16%, depending on multiplicity).
We find similar total cross sections with MiLO to the corresponding LO results. Nevertheless, the absolute predictions are larger for MiLO , and the excess increases slightly with multiplicity. In the case of MiNLO results, the central predictions agree well with the NLO results. Moreover, their ratio is rather stable as a function of the jet multiplicity.
In figures 1-3 we display total cross sections and scale variations for all scales considered in this analysis. It can be seen that in particular the MiLO and MiNLO predictions exhibit a considerable variation, both in their central value and in the associated conventional scale uncertainty. This can be traced back to the procedure for the identification of ordered clustering hierarchies, and the associated value of the scale of the core interaction, µ core (cf. Sec. II C 2). Since we require the branching history to be ordered, we must terminate the      clustering as soon as an inverted scale hierarchy is encountered. This biases the scale choice for events with many hard scales towards µ core , and therefore the precise definition of µ core plays a significant role [49]. The choice ofŜ T increases µ core on average, thus permitting more clusterings in high-multiplicity final states, and therefore inducing more associated Sudakov form factors. On average this reduces both the central value of the prediction and the related scale uncertainty.
Jet-production ratios have been shown to be a good handle for precision tests of QCD.
In these ratios many uncertainties associated with scale sensitivity and PDF dependence cancel to a large extent. Also, in experimental analyses, systematic uncertainties associated with luminosity measurements are canceled. In table IV we observe a remarkable stability of these ratios at NLO for both scale choicesĤ T /2 and MiNLO , all of them falling around a value of 0.25. This universality is present for NLO results in V +jet even though the corresponding LO results deviate considerably. The universality of the jet production ratios can be very helpful for tests of the SM and can even be exploited to make extrapolations of total and differential cross sections to large jet-multiplicity processes [22]. We observe that the NLO results obtained with both dynamical scales are mostly consistent. For multiplicities with more than 2 jets, the differences between the two scale choices lie within the respective factor-two scale variations. In the case of W + 2-jet production, an approximately 15% discrepancy appears, which can be taken as an estimate for the total scale uncertainty associated with the prediction. Furthermore, we observe that with increas-  ing multiplicity, the bands associated with the uncertainty of the respective scale choice seem to behave differently betweenĤ T /2 and the MiNLO scheme. In general, the scale uncertainty obtained by factor-two variations grows with multiplicity, e.g. forĤ T /2, from around 4.5% for W − + 2j to about 15% for W − + 5j production. It is now interesting to observe that this increase in scale uncertainty is more pronounced for the MiNLO scale choice, where the uncertainty grows from around 6.5% (W − + 2j) to around 35% (W − + 5j).

C. Differential distributions
In this section we analyze several differential distributions and the impact that higher-  We study the distribution in the total jet hadronic transverse energy H T in Figure 6 as well as ratios thereof between the MiNLO andĤ T /2 scale choices in Figure 7. The format of Figure 6 is the same as in Figure 5. We show in side-by-side panels results for W − + n jets, with n = 2, 3, 4 and 5. In general we find that for n ≥ 3, the corrections to the H T spectrum change the shape of the distribution only mildly, which can be seen by looking at both the LO to NLO ratios as well as the MiNLO to NLO ratios. Notice that the fluctuations at NLO for small values of H T , as we increase multiplicity, are just due to the fact that near threshold the integration errors grow large. On the other hand, the n = 2 LO predictions have a large shape difference compared to the NLO results. These changes are similar to the corresponding observable for n = 1 for which it is well known that large corrections appear from configurations with many jets in the final state [30]. The widening of the NLO scale band indeed shows that real contributions are large in the tail of the distribution, making this observable sensitive to quantum corrections (an associated observation for the same observable but at a hadron collider running at 100 TeV was made in [50]). In principle a computation of NNLO QCD correction to V + 2 jets would be desirable in order to stabilize the predictions for this highly relevant process.    for BSM physics, but also because it is associated to having under control known missing energy signatures.
In Figure 9 we explore uncertainties associated to the PDFs and show the PDF uncertainty bands for nth-jet p T spectra in an inclusive sample of W − +jets forĤ T /2, analogous to the distributions shown in Figure 5. The additional bottom panel shows the NLO PDF uncertainty bands for the CT14 (gray) [33], ABM (magenta) [34], MMHT (green) [35] and NNPDF 3.1 (orange) [36] PDF sets, and we normalize to the central value obtained with CT14. The data for Figure 9 was generated by creating a fastNLO table [51] from the n-tuple data. We also show the central values for total cross sections obtained with the different choices of PDF sets, normalized to the results for CT14, in table V.
We observe that PDF uncertainties can reach values of up to 10% and that most of the errors sets overlap. Nevertheless, both central value and uncertainty bands of the ABM results lay outside the uncertainty bands of all other PDF sets. Also, the MMHT bands lay systematically higher than the others, a trend that is more pronounced in the largemultiplicity cases. All uncertainty bands increase at larger p T , as the effective mass sampled for the corresponding events grows and the PDFs are evaluated for larger values of the Bjorken x, with less data available to constrain the PDF fits. PDF uncertainties are thus of the same order as NLO scale uncertainties, in particular for the high-multiplicity processes, where we observe a considerable spread between the different PDF sets. At the level of normalized NLO QCD total cross sections this is shown in table V.

IV. CROSS SECTION RATIOS
In this section we study a series of observable ratios, which feature reduced uncertainties compared to basic observables such as cross sections or differential distributions. In particular, experimental uncertainties related to jet energy scale, lepton efficiency, acceptance and proton-proton luminosity should be greatly reduced. The observable ratios are also expected to suffer less from theoretical uncertainties from uncalculated higher-order corrections. In consequence, these ratios are helpful to better understand the structure of quantum corrections to processes with a vector boson in association with multiple jets. As shown in [22], they exhibit certain universal features that can be exploited in phenomenological studies at hadron colliders. In this section, we study the differential jet ratio for W − production as well as differential flavor ratios of both W − /W + and Z/W production.

A. Jet ratios
We first examine the dependence of the jet-production ratio on the W − -boson transverse momentum p W T and total hadronic transverse energy H T . We display the differential cross section ratios for the associated production with up to 5 jets in Figure 10 As a function of p W T , we observe that the W − + 2-jet / W − + 1-jet ratio (n = 2) shows large NLO corrections. This is mainly due to the large corrections appearing at NLO [30], which are stabilized at NNLO [7]. In the lowest p T region (up to the order of the W mass), the ratios lie around a value of 0.25, roughly independent of the number of jets. The NLO corrections are modest for all displayed multiplicities in this region, being in agreement with the total cross sections displayed in table II. With higher p W T , the ratios grow monotonically and for n ≥ 3, they stabilize for large values of p W T . For more jets, the increase is less pronounced and the ratio stabilizes for n = 5 around a value 0.5. For the low-multiplicity cases (n ≤ 3), and in particular for large values of p W T , the ratio for MiLO lies below that for LO. The ratios for higher multiplicities as well as all those of NLO and MiNLO agree very well.
Similar to the p W T ratios, we observe that the differential ratio in H T for n = 2 is not stable with the fixed-order quantum corrections. Even at NLO, the differential jet production ratio is larger than 1 for a large part of H T shown. We would expect this observable to stabilize as higher-multiplicity results are included either through NNLO or higher-order calculations, or through multi-jet merging at NLO. Around the threshold, all ratios lie at a value of the same order. With increasing H T they again increase monotonically and at NLO stabilize for large values of H T . Looking at the high H T region, the ratios show a characteristic behavior. Such events tend to be populated by multiple jets and the H T distributions in these jet bins tend to overlap [22], which results in the ratio tending to 1.0. As for the previous distributions, MiNLO and NLO ratios agree very well, with a slight discrepancy between LO and MiLO ratios for the lower multiplicity cases.

B. Flavor ratios
In this subsection we show flavor ratios of W − to W + production as well as Z to W + production. These ratios are of particular benefit as they are reliable observables for the extraction of valence-quark PDF information for large values of Bjorken x [22,52].
In Figures 11 and 12   We observe a similar behavior for the differential ratios as a function of the total hadronic energy H T . In particular, the ratios decrease monotonically with increasing H T , and take values of the same order at low H T for all multiplicity, also the results for both scale choices agree very well. However, we observe noticable NLO correction for the n = 1 case in the high H T tail The ratios of Z to W + production in Figure 12 are quite flat over the full range of variables shown, which is in contrast to the associated charge-asymmetry ratios in Figure 11. The values of Z/W + ratios as a function of both the variables shown lie around 0.2 for all multiplicities n shown. A small decrease of the Z/W + ratio can be observed in the low p T region in the ratio for n = 1. Also for all the Z/W + ratios, the results for both scale choices agree very well. Our results show that the quantum corrections in the Z/W + observable ratios are quite mild, which makes them excellent choices for searches for new physics.

V. CONCLUSIONS
In this paper we presented NLO QCD predictions for the production of an electroweak gauge boson with up to five jets for W ± , and with up to four jets for Z, in the final state for the LHC with √ s = 13 TeV, extending previous predictions [10,11] to the energy of LHC Run II. Since these processes constitute an important background to many new physics searches involving missing energy, as well as precise top-quark measurements, the theoretical predictions we provide are an important ingredient to fully exploit the discovery potential of the LHC in the upcoming years.
We observed that the scale dependence in NLO predictions for both total cross sections and differential distributions is strongly reduced compared to those at LO. Similar to the √ s = 7 TeV case, the scale dependence of total cross sections shrinks from up to 50% at LO to a sensitivity of about 15% at NLO. We also compared results obtained with several other functional forms of dynamical scales, most notably comparing results with fixed-order scales likeĤ T /2 with the MiNLO (original formulation in [31]) reweighting procedure. This is the first time this comparison is carried out in the context of processes with high jet multiplicity.
For total cross sections, as well as for differential distributions, NLO results obtained witĥ H T /2 and MiNLO are largely consistent in both shapes, normalization and scale sensitivity. In general the good agreement between both scale choices confirms that NLO QCD predictions in high multiplicity processes give the first numerically reliable predictions.
We furthermore computed several observable ratios and found that jet production ratios have an increased stability compared to the results at √ s = 7 TeV. We thereby have found a setup for which there is independence of the jet-production ratios on the number of jets.
The present study provides insight into the phenomenologically relevant process class of electroweak gauge boson (W ± , Z) production plus up to five jets at √ s = 13 TeV. We look forward to more comparisons of our results, as the ones recently shown in [5], with LHC data at the energy frontier.