HEJ 2.2: W boson pairs and Higgs boson plus jet production at high energies

We present version 2.2 of the High Energy Jets ( HEJ ) Monte Carlo event generator for hadronic scattering processes at high energies. The new version adds support for two further processes of central phenomenological interest, namely the production of a W boson pair with equal charge together with two or more jets and the production of a Higgs boson with at least one jet. Furthermore, a new prediction for charged lepton pair production with high jet multiplicities is provided in the high-energy limit. The accuracy of HEJ 2.2 can be increased further through an enhanced interface to standard predictions based on conventional perturbation theory. We describe all improvements and provide extensive usage examples. HEJ 2.2 can be obtained from https: // hej.hepforge.org.

High Energy Jets (HEJ) is both a framework and a flexible Monte Carlo generator for the all-order resummation of high-energy logarithms [1,2,3].In HEJ 2 [4], this highenergy resummation can additionally be matched to leading-order predictions obtained using conventional fixed-order generators [5].HEJ has been validated against data in experimental studies of pure multijet production [6,7,8,9], lepton pair production via a virtual W boson, photon, or Z boson in association with two or more jets [10,11,12,13], and Higgs boson production with jets [14].
In the following, we present HEJ 2.2.This new version implements high-energy resummation for the production of two leptonically decaying W bosons with the same charge in association with two or more jets.Moreover, the existing implementation for the production of a Higgs boson with jets has been extended to also cover Higgs boson production with a single jet.Resummation for the production of a charged lepton pair via a virtual photon or Z boson together with two or more jets is now also supported for higher multiplicities where no fixed-order prediction is available.Furthermore, new options have been added, for instance to facilitate differential next-to-leading-order matching and to separate events with soft jets.In section 2, we briefly summarise High Energy Jets and describe the various improvements in version 2.2.We give examples for the use of the new features in section 3 and conclude in section 4. Appendix A contains instructions for download and installation of HEJ 2.2.

HEJ in a nutshell
Before describing the changes in HEJ 2.2, let us briefly review the general formalism and program structure.As input, HEJ requires leading-order (LO) events, generated with e.g.Sherpa [15] or MadGraph5 aMC@NLO [16].For higher jet multiplicities exact fixed-order generation becomes increasingly time consuming.To address this problem, HEJ includes the fast HEJ fixed-order generator HEJFOG based on the high-energy approximation of the leading-order matrix elements.
Using the kinematics of each (approximate or exact) input event, we identify whether resummation is possible.For each event that permits resummation, HEJ generates a number of matching events in the resummation phase space, which include real and virtual corrections to all orders in the high-energy limit.Details are given in [17].Together with the unchanged non-resummable input events, the generated resummation events are then passed on to any number of output event files and/or analyses.This standard control flow is depicted in figure 1.It can be modified through HEJ options, such that e.g.nonresummable events are discarded.
The first type of event kinematics for which resummation is implemented are leadinglogarithmic (LL) configurations, which for pure multijet production have to fulfil the following constraints: 1.The flavour of the most backward outgoing parton has to match the flavour of the backward incoming parton.2. The flavour of the most forward outgoing parton has to match the flavour of the forward incoming parton.3.All other outgoing partons have to be gluons.
These criteria remain the same in processes involving virtual photons and/or Z bosons.
For virtual W bosons, the incoming and outgoing flavours in criteria 1 and 2 only have to match up to the change induced by W boson couplings.In the case of a final-state Higgs boson, configurations where the backward (forward) incoming parton is a quark or antiquark and the most backward (forward) outgoing particle is the Higgs boson are formally subleading.Nevertheless, we also implement resummation for such configurations.Depending on the process, resummation is also implemented for two further types of next-to-leading-logarithmic (NLL) configurations.These configurations differ from LL ones as follows.
• Unordered gluon: Either the most forward or most backward outgoing parton is a gluon, and the next outgoing parton in rapidity order is a quark or antiquark whose flavour matches the one of the respective incoming parton.
• Quark-antiquark: A pair of final-state gluons that are adjacent in rapidity is replaced by a quark-antiquark pair.
The current status of the implemented resummation is summarised in table 1.
The resummation events generated for the LL and supported NLL configurations are given a final matrix element weight of where M HEJ is the all-order scattering matrix element in the high-energy approximation, M HEJ,LO its leading-order truncation, calculated for the kinematics of the input event and |M LO | 2 is taken from the LO input.
To illustrate the structure of the HEJ matrix element, we first focus on LL configurations in pure multijet production.We denote these configurations as where f a is the flavour of the incoming parton in the backward direction with momentum p a .Correspondingly, we use f b and p b for the flavour and momentum of the forward incoming parton.order by rapidity, viz.
The most backward outgoing parton has flavour f a , the most forward one flavour f b , and all other outgoing partons are gluons.This implies that the outgoing parton with flavour f a has momentum p 1 and the outgoing parton with flavour f b has momentum p n .However, note that this identification does not necessarily hold at NLL.For example, if there is an unordered gluon in the backward direction then the outgoing parton with flavour f a has momentum p 2 .Using the introduced notation, we can write the general form of the squared HEJ matrix element as V(p a , p b , p 1 , p n , q i , q i+1 ) where q i = p a − i j=1 p j is the t-channel momentum after the emission of parton i. B fa,f b is derived from the modulus square of the Born-level matrix element for the process f a f b → f a f b , V accounts for the real emission of the n − 2 gluons in between f a and f b , and W incorporates the virtual and unresolved real corrections.
The Born-level function B fa,f b is given by where α s is the strong coupling constant and N C = 3 the number of colours.K fa and K f b are generalised colour factors depending on the respective parton flavour and, in the case of gluons, also the parton momentum.For quarks and antiquarks one finds 2N C ; the factor K g for gluons is derived in [2].S faf b →fa•••f b denotes the contraction of two currents: where j λ µ is the current j λ µ (p, q) = ūλ (p)γ µ u λ (q) (5) for helicity λ.HEJ employs the symbolic manipulation language FORM [18] to generate compact symbolic expressions for current contractions.
The real corrections are given by contractions of Lipatov vertices [5]: V µ (p a , p b , p 1 , p n , q i , q i+1 ) = − (q i + q i+1 ) µ with Finally, the virtual and unresolved real corrections W can be expressed in terms of the regularised Regge trajectory ω 0 : W(q j⊥ , y j , y j+1 ) = exp[ω 0 (q j⊥ )(y j+1 − y j )]. ( For a detailed discussion and an explicit expression for ω 0 see [17]. The generalisations to NLL configurations and additional non-partonic final state particles are derived in [10,13,5,19,20,21,14].In all cases one finds a factorisation into a Born-level function, resolved real emissions, and virtual and unresolved real corrections.In the absence of interference, one recovers the same structure as in equation (2).In particular, the functions V and W comprising the all-order corrections are universal, whereas the Born-level function B is process dependent.

High-energy resummation for W pair production
Based on the pure-QCD LL configurations f a f b → f a • • • f b , additional W bosons can be produced via emission off the partons f a and f b .In HEJ 2.2, we consider LL configurations with two leptonically decaying W bosons with equal charge.For definiteness, we discuss configurations where the rapidities of the final-state charged and neutral leptons do not necessarily respect any rapidity ordering.Note that the couplings to the W bosons induce flavour changes f a → f a ′ and f b → f b ′ .We use a diagonal CKM matrix and do not include third-generation quarks/anti-quarks, i.e. the number of active flavours is 4. The production of two positively charged W bosons and the decay of the two W bosons into the same lepton flavours is completely analogous.
We identify two contributions to the amplitude.Parton f a can either couple to the W boson that decays into an electron and its antineutrino or to the W boson decaying into a muon and its antineutrino.In the first case, the t-channel momenta are given by where p e is the momentum of the electron and p νe the momentum of its antineutrino.In the second case the t-channel momenta are with the muon momentum p µ and the corresponding antineutrino momentum p νµ .The resulting modulus square of the matrix element including interference is [21] M 1,e q 2 1,µ V (q i,e , q i+1,e ) • V (q i,µ , q i+1,µ ) W( √ q i,e⊥ q i,µ⊥ , y i , y i+1 ) .
Here, we have introduced contractions between generalised currents j c W,l accounting for the coupling between a parton with flavour f c and a W boson decaying into a charged lepton l and the corresponding antineutrino.The contractions are where the parton helicities λ a and λ b are determined by the flavour of the respective parton, namely λ c = − if f c is a quark and λ c = + if f c is an antiquark.The electron helicity λ e and the muon helicity λ µ correspond to the charge sign of the parent W boson, i.e. λ e = λ µ = − in the present case.We have introduced a generalised current j ρ,λaλ ℓ V for the coupling of a parton with helicity λ to a leptonically decaying vector boson1 V with lepton helicity λ ℓ .It is given by [20] p V = p ℓ + p νℓ is the vector boson momentum, g V its coupling to the fermion f a , M V its mass, and Γ V the width.

Higgs boson production with a single jet
In the gluon-fusion production of a Higgs boson together with one or more jets new LL configurations beyond those derived from pure multijet production (c.f.section 2.1) arise.In these configurations, one of the incoming partons is a gluon while the corresponding most forward or most backward outgoing particle is the Higgs boson, i.e.
Without loss of generality we consider the former configuration.The modulus square of the HEJ matrix element reads [14] W(q i⊥ , y i , y i+1 ), (16) with the universal real and virtual correction factors V and W defined in equations ( 6) and (8).The only differences to the pure QCD case in equation ( 2) are the replacement p 1 → p a in the third argument of V and the adjustment of the process-dependent Born function to [14] where ϵ λa (p a ) is the polarisation vector of the incoming gluon and V H the effective vertex coupling the Higgs boson to two gluons at one-loop, including finite quark-mass dependence.The structure of equation ( 16) then allows finite quark-mass dependence to be applied for arbitrarily high numbers of jets.

Spillover from small transverse momenta
As described in section 2.1, a number of all-order resummation events are generated for each resummable input event.Since the resummation events include real corrections, the resulting kinematics differ slightly from the kinematics of the input events.While jet rapidities are always preserved, this is generally neither true for transverse momenta nor for the rapidities of any other particles.This implies that cuts imposed on the leadingorder generation should be significantly looser than the final analysis cuts.Empirically, the difference in transverse momentum cuts should be about 10-20%, with a slight increase towards larger multiplicities.
Simply adjusting the cuts in the leading-order generation is correct, but inefficient: events with small transverse momenta dominate the leading-order prediction, but only give a small contribution to the final resummed results.It is therefore more efficient to split up the leading-order generation.One first generates a high-statistics sample in which all particles fulfil the transverse momentum cuts of the analysis.Then, one generates a second low-statistics sample where in each event there is at least one particle with small transverse momentum that does not pass the final cuts.Since the two samples are disjoint, one can separately apply HEJ resummation to each sample and add up the results.
However, implementing the requirement of at least one "soft" particle is often not straightforward with standard fixed-order generators.To facilitate resummation for the low transverse momentum sample, HEJ 2.2 introduces a new option for discarding events in which all jets are above the analysis threshold.An example is given in section 3.1.3.

Matching to Next-to-Leading Order
To improve the accuracy of the obtained total cross section to next-to-leading order (NLO), one can obviously multiply the HEJ prediction by a flat factor of σ NLO /σ HEJ , where σ HEJ is the leading-order accurate total cross section according to HEJ and σ NLO the total cross section at NLO. HEJ 2.2 enables us to achieve NLO accuracy also in differential distributions.Considering a distribution dσ/dO in some observable O, we can combine NLO and HEJ resummation through the reweighting Here, the subscript HEJ denotes the prediction before reweighting, NLO the NLO-accurate prediction, and HEJ,NLO the truncation of the HEJ prediction to NLO.In section 3.2.2,we show in an example how to truncate the HEJ prediction and obtain NLO-reweighted distributions.

Predictions without Fixed-Order Matching
The computational cost for generating the fixed-order input events rises steeply with the jet multiplicity.For this reason, HEJ includes the HEJFOG, a fast generator based on the leading-order truncation of the HEJ matrix element given in equation (2).The intended use is that one will generate exact low-multiplicity input events with a conventional generator and supplement them with approximate high-multiplicity events using the HEJFOG.In HEJ 2.2, the HEJFOG includes charged lepton pair production with jets as a new process.Furthermore, the generation efficiency for the production of a W boson with jets has been improved by aligning the rapidity of the W boson with its emitter, reducing the Monte Carlo uncertainty by a factor of up to 2.

Example Usage
In the following, we show how the new features in HEJ 2.2 can be used in practice.For concreteness, we will generate leading-order events with Sherpa 2.2 and analyse the output with Rivet 3 [22].However, we stress that any leading-order generator that can produce event files in the LHEF format [23] is supported.In addition to the direct Rivet interface, HEJ can write its output to event files in various formats and allows arbitrary custom analyses via plugins.Since these options are not new, we refer to the HEJ documentation on https://hej.hepforge.orgfor details.

Same-sign W pair production with jets
We first consider the process pp The various settings are explained in more detail in the Sherpa manual.Since HEJ treats all quarks as massless, we have to set the charm and bottom quark masses to zero for consistency.As explained in section 2.4, leading-order events containing particles with transverse momenta below the analysis cuts can still contribute to the resummed prediction.For this reason, we accept jets with transverse momenta as low as 15 GeV, despite having an analysis cut of 20 GeV.In section 3.1.3,we will discuss a computationally more efficient way to incorporate this contribution from leading-order events that do not pass the analysis transverse momentum cuts.
We can now generate input events for the case of two jets by running Sherpa in the directory containing Run.dat.With the Sherpa 2.2.15 installation inside the HEJ Docker image this yields a cross section of (1.88795 ± 0.109611) fb.Results may differ between versions and when re-using integration grids from previous Sherpa runs.
We should then also produce event files for higher jet multiplicities after adjusting the EVENT_OUTPUT, Process, and FastjetFinder entries in Run.dat.Increasing the number of jets to three leads to a cross section of ( Previous HEJ versions only accepted input from pipes if the total cross section was equal to the sum of event weights, which is not the case for Sherpa event files.This restriction is lifted in the new version 2.2, which removes a potential bottleneck in time and storage.

HEJ resummation
In addition to the leading-order event input, HEJ needs a configuration file.Adapting the template config.ymldistributed with the HEJ source code to the parameters listed in table 2  HEJ outputs a cross section of (1.094 ± 0.06614) fb.It also produces the Rivet analysis output file WW2j.yoda.We produce resummed predictions for the higher-multiplicity event files events WWnj.lhe in the same way, after changing the analyses entry in config WWjets.yml to analyses: − rivet: [MC_WWINC, MC_WWJETS] output: WWnj and adjusting the event file name when running HEJ.To guarantee statistically independent output, it is also recommended to change the seed entry for each run.If we nonetheless use the default seed with the previously generated event file events WW3j.lhe we obtain a cross section of (0.905 ± 0.07534) fb.
We then combine the results for the different jet multiplicities with As examples, we show the inclusive jet multiplicities and the distribution of the rapidity difference between the two W bosons obtained from resumming fixed-order predictions with two and three jets in figure 2. In contrast to the recommended usage, we have not modified the random number generator seeds between runs in order to facilitate reproducing this figure. 3We observe that the bulk of the events does not pass the analysis cuts.

Dedicated low transverse momentum runs
So far, we have generated the leading-order events with significantly looser transverse momentum cuts than wanted for the final analysis.As argued in section 2.4, it is more efficient to split the generation into a high-statistics run with the strict cuts used in the final analysis and a low-statistics run with loose cuts where in each leading-order event there is at least one particle that does not pass the final cuts.For the jet transverse momentum cuts, this separation is facilitated by a new option in HEJ 2.2 which ensures the presence of at least one "soft" jet in the input for the low-statistics run.Note that this option only refers to jets; any other particles should be generated according to the loose transverse momentum cuts in both samples.
In detail, one should go through the following steps for the present example of samesign W pair production with jets:  Inclusive jet multiplicity  The cross sections for the various runs are summarised in table 3. Here, "low p ⊥ " refers to the contribution from the phase-space region where at least one jet is softer than 20 GeV.Conversely, "high p ⊥ " implies that all jets are harder than 20 GeV.Since we do not impose an upper limit on the transverse momenta of the jets in the Sherpa runcard, the low-statistics run covers both of these regions.The "high p ⊥ " contribution is then removed when running HEJ with the require low pt jet option, which requires that at least one jet in the fixed-order input is below the analysis cut.While the contributions from the low transverse momentum run are negligible in this example, they can become relevant in high-statistics distributions that probe the phase space near the minimum jet transverse momentum.What is more, the impact tends to increase with the jet multiplicity.

Higgs boson production with one or more jets
We now consider the production of a Higgs boson together with one or more jets.We use the parameters listed in table 4. For the sake of simplicity, we first consider the limit of an infinitely heavy top quark.

Collider energy
Table 4: Parameters for the production of a Higgs boson together with at least one jet.
In close analogy with section 3.1, we first generate leading-order input events for Higgs boson production with a single jet.We use Sherpa with the following run card For the resummation, we use a similar HEJ configuration file as before.Anticipating further runs with higher multiplicities, we enable resummation for the supported NLL configurations.In the present case this concerns configurations involving an unordered gluon, which first contribute to Higgs boson plus dijet production, cf.section 2.1.Since there is no standard Rivet analysis for stable Higgs boson production, we use the generic MC JETS analysis.As described in section 3.1, we then add predictions for higher jet multiplicities.We show reference cross sections in table 5 and example distributions in figure 4. We stress that, as in most examples shown here, the Sherpa and HEJ cross sections are not directly This is because the Sherpa runcards use a lower transverse momentum cut than HEJ, for the reason explained in section 2.4.Therefore a significant fraction of the input events will not pass the cuts after resummation (see section 3.1.3for an example on how to deal with this in a more efficient way).

Run
H

Quark mass corrections
For accurate predictions in the high-energy region, we have to take into account the finite top quark mass.In the following, we assume a mass of 174 GeV.On the Sherpa side, we can add AMEGIC [26] and OpenLoops [27,28] to ME_SIGNAL_GENERATOR and insert the following lines into the (run) block: # finite top mass effects KFACTOR GGH OL_IGNORE_MODEL 1 OL_PARAMETERS preset 2 allowed_libs pph2,pphj2,pphjj2 psp_tolerance 1.0e−7 HEJ needs to be compiled with support for QCDLoop [29] to incorporate quark mass corrections in Higgs boson production.We can include them by adding to the configuration file.For higher jet multiplicities, we face the problem that it is no longer feasible to compute the leading-order input with exact dependence on the top quark mass m t .However, we can still retain this dependence and also include the dependence on the bottom-quark mass m b in the high-energy resummation.
As in equation ( 1), the weight w of a leading-order matched resummation event has the following dependence on the leading-order matrix element M LO and the all-order HEJ matrix element M HEJ (m b , m t ): For consistency, the values for the quark masses have to match those used in M LO .Therefore, if the leading-order input is only known for m b → 0, m t → ∞ the correct reweighting factor is Currently, there is no built-in HEJ option for choosing different quark mass values in M HEJ and M HEJ,LO .However, HEJ supports flexible custom analyses, which allow us to manually reweight by the correction factor |M HEJ,LO (m b , m t )| 2 /|M HEJ,LO (0, ∞)| 2 in equation (21).We can also use this opportunity to include bottom quark mass corrections in the case where only the exact leading-order dependence on the top quark mass is available.
Custom analyses are described in the HEJ user documentation on https://hej.hepforge.org,where also a template is provided.The reweighting can be implemented as shown here: higgs matching analysis.cc#include <string> #include <memory> #include "HEJ/Analysis.hh"#include "HEJ/Config.hh"#include "HEJ/event_types.hh"#include "HEJ/Event.hh"#include "HEJ/HiggsCouplingSettings.hh"#include "HEJ/MatrixElement.hh"#include "HEJ/RivetAnalysis.hh"#include "HEJ/YAMLreader.hh"#include "yaml−cpp/yaml.h"namespace LHEF { class HEPRUP; } // part 'tree_kin' here.// To account for the possibility of interference, 'tree_kin' returns // a 'std::vector' instead of a single value.Here, the 'std::vector' // has only a single element.const double me_exact = me_exact_.tree_kin(LO_event).front(); const double me_approx = me_approx_.tree_kin(LO_event) We can then compile the analysis into a shared object library with a compiler supporting C++17, for instance a recent version of g++: To use our custom analysis, we adjust the HEJ configuration file.We use YAML anchors (starting with &) and references (starting with * ) to ensure that the settings passed to the analysis are consistent.The following code listing is for the case of a leading-order prediction in the infinite top-quark mass limit.

Matching to next-to-leading order
Using HEJ 2.2, we can extend the fixed-order accuracy of distributions from LO to NLO, cf.section 2.5.In the following, we consider NLO matching for the production of a Higgs boson together with at least one jet.While we derive the matching in the limit of an infinitely heavy top quark, the resulting rescaling factors can also be used to improve HEJ predictions incorporating quark-mass corrections.We apply the matching bin-bybin in the resulting histograms.We can generate an NLO prediction using Sherpa and OpenLoops by adjusting the run card as follows.The resulting cross section is (18.4425 ± 3.83924) pb.
For the HEJ prediction, we add the option NLO truncation: enabled: true nlo order: 1 # number of jets to the configuration file config Hjets.ymlfrom section 3.2, change the name of the Rivet output file to Hj HEJ NLO 1j.yoda, and run HEJ on the previously generated leading-order input file assuming an infinitely heavy top quark.We obtain a cross section of (7.846 ± 1.091) pb.The result is exclusive in the number of jets; in contrast to the Sherpa prediction the contribution from events with two jets is not included, yet.To generate the missing piece, we remove the NLO process configuration from the above Sherpa run card, i.e. we change the ANALYSIS_OUTPUT to Hjj LO, remove OpenLoops from ME SIGNAL GENERATOR and delete the following lines.
NLO_QCD_Mode Fixed_Order NLO_QCD_Part BVIRS Integration_Error 0.02 Loop_Generator OpenLoops We then increase the number of jets to two and run Sherpa to generate a file Hjj LO.yoda, with a total cross section of (8.80745 ± 0.321808) pb.We add this contribution to the truncated HEJ result: To obtain the final NLO matched prediction, we should multiply each histogram in the original HEJ output by the ratio of the corresponding histograms in Hj NLO.yoda and Hj HEJ NLO.yoda.The following script gives an example of how this reweighting can be implemented.For the sake of brevity we have omitted the error handling code, which is of course essential in actual applications.Generating Hjets.yoda as described in section 3.2.1, the resulting NLO-matched rapidity distribution of the hardest jet is shown in figure 6.Note that the MC JETS Rivet analysis employs adaptive binning for a number of distributions, causing the division of the respective histograms to fail.This problem can of course be circumvented by using a custom analysis.

Charged lepton pair production with many jets
The production of two charged leptons with jets was first implemented in HEJ 2.1 [30] for at most moderate jet multiplicities, where exact fixed-order matching is feasible.To overcome this difficultly, in HEJ 2.2 approximate high-multiplicity events can be generated with the HEJ Fixed Order Generator.
In the following example, we consider the process pp → µ + µ − + ≥ 2 jets with the same parameters as in previous examples, see table 2. We assume that predictions including up to 4 jets have already been produced as described in sections 3.1 and 3.2.To generate input events with 5 jets, we adapt the configuration file configFO.ymldistributed together with HEJ: predictions for pp → µ + µ − + 5 jets.These could then be combined with the results for lower multiplicities as described in section 3.1.2.

Conclusions
The HEJ Monte Carlo event generator provides accurate high-energy descriptions for a steadily growing range of scattering processes.The new version 2.2 adds predictions for the QCD corrections to the production of two W bosons with the same charge together with two or more jets, which are pivotal for experimental measurements of weak boson fusion.A further major improvement concerns the gluon-fusion production of a Higgs boson with jets, where for the first time final states with only a single jet are included in the description.
While LO-matched predictions for charged lepton pair production with jets were already available in HEJ 2.1, the new release allows to supplement these with LL-accurate high-energy corrections for high multiplicities where exact LO generation may no longer be feasible.Furthermore, HEJ 2.2 allows NLO matching at the level of differential distributions and facilitates a computationally more efficient event input generation by separating off low-p ⊥ events with a small contribution to the final predictions.
We have given detailed examples for the new features.The program code as well as comprehensive documentation including options added in previous versions are available on https://hej.hepforge.org.
• Version 2 of QCDLoop [29] is required to include finite top mass corrections in Higgs boson + jets production.
• HepMC [41] versions 2 and 3 enable event output in the respective format.
• HighFive [42] has to be installed in order to read and write event files in the HDF5 [43]based format suggested in [44].
We strongly recommend to install these programs and libraries to standard locations: • The executable files should be inside one of the directories listed in the PATH environment variable.This concerns cmake, the C++ compiler, and the executables contained in autoconf and automake.
• The library header files ending with .h,.hh,or .hppshould be in a directory where they are found by the C++ compiler.For gcc or clang, custom locations can be specified using the CPLUS_INCLUDE_PATH environment variable.
• The compiled library files ending with .a,.so,or .dylibshould be in a directory where they are found by the linker.Custom locations can be set via the LIBRARY_PATH environment variable.For shared object libraries (.so or .dylib)custom locations should also be part of LD_LIBRARY_PATH on linux and DYLD_FALLBACK_LIBRARY_PATH or DYLD_LIBRARY_PATH on macOS.

Figure 2 :
Figure 2: Inclusive N -jet cross sections (left) and rapidity difference between the two W bosons (right) obtained with Sherpa and HEJ 2.2 for the production of two leptonically decaying W − bosons with at least two jets.

Figure 3 :
Figure 3: Inclusive N -jet cross sections (left) and rapidity difference between the two W bosons (right) combining dedicated high and low transverse momentum runs.

Figure 4 :
Figure 4: Hardest jet transverse momentum and rapidity for the production of a Higgs boson together with between one and three jets.

Figure 5 :
Figure 5: Hardest jet transverse momentum and rapidity for the production of a Higgs boson together with between one and three jets including finite quark mass effects.

Figure 6 :
Figure 6: NLO for the rapidity distribution of the hardest jet produced together with a Higgs boson.
pt, should be slightly below analysis cut peak pt: 20 # peak pt of jets, should be at analysis cut algorithm: antikt # jet clustering algorithm Rparticles: [p, p] ## PDF ID according to LHAPDF pdf: 13100

Table 1 :
The final state contains n partons with momenta p 1 , . . ., p n , which we Implemented processes and higher-order logarithmic corrections in HEJ.The "pure LL" column lists processes implemented in the HEJFOG.The NLL columns include both pure NLL and NLL matched to LO.

Table 2 :
jets with the parameters shown in table 2. H T = p e⊥ + p νe⊥ + p µ⊥ + p νµ⊥ + n i=1 p i⊥ is the sum of the scalar transverse momenta of all outgoing particles.Parameters for the production of multiple jets together with a same-sign W boson pair decaying to charged leptons and neutrinos.
1.47689 ± 0.10331) fb.To avoid the creation of large intermediate event files, we can use named pipes instead, i.e. we run Sherpa with we get Here, we choose to pass the resummed events directly to the Rivet analyses 2 MC WWINC and MC WWJETS.Using the Docker virtualisation software, we can run HEJ [25] with the following command.docker run −v $PWD:$PWD −w $PWD hejdock/hej HEJ config_WWjets.ymlevents_WW2j.lheAlternatively, after compiling and installing HEJ and its dependencies we can use HEJ config_WWjets.ymlevents_WW2j.lhe 1. Generate the high-statistics sample.(a)Change the minimum transverse momentum in the FastjetFinder entry in In the event treatment entry in config WWjets.yml,change all keep values to discard.Specifically, change (e) Run Sherpa and HEJ.After repeating these steps for higher jet multiplicities, the samples can again be combined with yodastack −o WWjets.yodaWW * j.yoda WW * j_lowpt.yoda

Table 3 :
Reference cross sections in femtobarn when using dedicated low transverse momentum runs.All runs use the default random number generator seeds and the code versions in the HEJ Docker image.Sherpa integration grids (Results.db)should be deleted between runs to reproduce the exact numbers.

Table 5 :
Reference cross sections in picobarn for Higgs boson plus jet production.
If the leading order prediction includes the exact dependence on the top-quark mass, but not the dependence on the bottom-quark mass, one should replace the LO Higgs coupling entry by

Table 6
Table Reference cross sections in picobarn for Higgs boson plus jets production with finite quark mass effects.All HEJ resummation results account for massive bottom and top quarks.The Sherpa prediction for H + 1 jet uses the physical top-quark mass, but assumes a massless bottom quark.The fixed-order predictions for higher multiplicities do not include finite quark mass effects.