## NLO Merging

Pythia offers two NLO merging approaches. Both of these methods have been presented in [Lon13]. The goal of NLO merging is to extend tree-level multi-jet merging methods to next-to-leading order accuracy in QCD, for every available jet multiplicity. If for example NLO calculations for Higgs + 0 jet, Higgs + 1 jet and Higgs + 2 jets were available, NLO merging allows to simultaneously describe 0-, 1- and 2-jet observables with NLO accuracy. Further jets can, depending on additional tree-level input, be described by additional tree-level matrix elements. In the example, it would be possible to achieve NLO accuracy for 0-, 1- and 2-jet observables, tree-level accuracy for 3-, 4- and 5-jet configurations, and use the parton shower approximation for events with more than five jets.

The two NLO merging methods implemented in Pythia are called NL3 (for Nils Lavesson + Leif Lönnblad) and UNLOPS (for unitarised NLO+PS merging). Both of these schemes require Les Houches Event File input that is generated by tree-level or NLO matrix element generators. Currently, Pythia requires NLO input generated within the POWHEG framework. The generation of sensible input will be discussed below. The two NLO merging methods are illustrated in the sample main programs `main87.cc` (introducing NL3 ) and `main88.cc` (introducing UNLOPS). Before describing these programs, we would like to outline the differences between the two approaches.

NL3 is a generalisation of CKKW-L tree-level merging. The main idea of NL3 is to start from CKKW-L-reweighted multi-jet merging, and replace the αsn+0- and αsn+1-terms by the NLO result of POWHEG. This "replacement" means that we subtract the αsn+0- and αsn+1-terms from the CKKW-L-reweighted tree-level samples, and add another sample -- the POWHEG input. All "higher orders" are unchanged w.r.t. CKKW-L. We have implemented the "inclusive" scheme of [Lon13] in Pythia. This means that the POWHEG input will contain contributions for hard, resolved real emission jets, which are already taken care of by higher-multiplicity samples in CKKW-L. Thus, explicit phase space subtractions are also included. The sample program `main87.cc`, together with the input file `main87.cmnd`, illustrates the procedure.

UNLOPS is a generalisation of the UMEPS multi-jet merging scheme. Since UMEPS is already slightly more complicated than CKKW-L, this makes UNLOPS complicated than NL3. The basic idea however remains the same: Start from a tree-level merging scheme (in this case UMEPS), remove all undesirable αsn+0- and αsn+1-terms from this result, and add back the "correct" description via POWHEG input samples. Again, since the "inclusive" scheme of [Lon13] was implemented in Pythia, it is necessary to handle explicit phase space subtractions. Similar to UMEPS, UNLOPS further ensures that the lowest-multiplicity cross section is given by the NLO result. This means that the UMEPS philosophy of "subtract what you add" needs to be extended to multi-leg NLO inputs.

UNLOPS is a theoretically more appealing definition of NLO merging than NL3, and should thus be considered the preferred choice. However, we believe it valuable to include both methods into Pythia, so that the variation of NLO merged results due to different NLO merging schemes can be studied in situ. Furthermore, NLO merging can be outlined more pedagogically when starting from NL3. The two NLO merging methods share parts of code with CKKW-L and UMEPS, and correspondingly share many input settings with these schemes. In particular,

The hard process (`Merging:Process`) needs to be defined exactly as in CKKW-L (see Defining the hard process in the CKKW-L documentation).

The merging scale value (`Merging:TMS`) has to be set.

The maximal number of additional partons in tree-level events (`Merging:nJetMax`) has to be set.

All settings listed under the sections "Matrix element merging and HepMC output for RIVET" and "Further variables" in the CKKW-L documentation can be accessed in NLO merging as well. Further, the `Merging:nRecluster` switch (see the UMEPS documentation) is important. Also, all `MergingHooks` routines that allow for user interference in CKKW-L merging are also usable for NLO merging -- with the exception of a user-defined merging scale. The NLO merging schemes currently implemented in Pythia do not allow for a merging scale definition that differs from the parton shower evolution variable. Since this merging scale definition is not completely obvious, the NLO merging schemes also share the `Merging:enforceCutOnLHE` switch with CKKW-L. In this way, it is possible to use LHE files that are regularised only with weak cuts as input, while the merging machinery imposes the stronger merging scale cut automatically. This means that no merging scale implementation is required from the user side, but also means that it is the user's responsibility to ensure that the cuts used for generating input LHE files are always looser than the cut given by the merging scale value `Merging:TMS`. This will lead to warnings of the form "`Les Houches Event fails merging scale cut. Cut by rejecting event`". These warning should rather be regarded as information. An example of inclusive matrix element generation cuts would be pTjet = 5 GeV, ΔRjetA jetB = 0.01 and QjetA jetB = 5 GeV, if NLO merging with a desired merging scale value of `Merging:TMS = 15` is attempted for Higgs + jets events at the LHC.

In the following, we will first describe the generation of NLO input samples, and list input settings for NLO merging in Pythia. Then, we will examine the sample main programs `main87.cc` and `main88.cc`, which implement NL3 and UNLOPS merging, respectively.

### Inputs for NLO merging

The NLO merging schemes in Pythia currently require Les Houches Event File input. To perform a merging with up to M additional partons described by tree-level matrix elements, and with up to N ≤ M-1 additional partons at NLO accuracy, the user needs to supply

LHE files for 0... M additional partons, taken from a tree-level matrix element generator, and

LHE files for 0... N additional partons, taken from a POWHEG NLO generator.

All input files need to be regularised, if they contain additional partons. Large files with fairly inclusive (i.e. loose) cuts are recommended. The input LHE files should further be generated with fixed renormalisation and factorisation scales. (In the POWHEG-BOX program, this means using the settings `runningscales 0, btlscalereal 1, btlscalect 1, ckkwscalup 0`. Some older processes in the POWHEG-BOX program need the input `runningscale 0` instead of `runningscales 0`.)

When attempting NLO merging, the following Pythia settings are relevant.

`mode` Merging:nJetMaxNLO (`default = 0`; `minimum = 0`)
The maximal number of additional jets for which NLO event samples are supplied by the user.

`parm` Merging:muFac (`default = -1.0`)
The fixed factorisation scale used in the hard process cross section, as needed to generate the leading-order weight, in case the factorisation scale cannot be inferred from Les Houches event input. (This is the case for files that have been generated with the POWHEG-BOX program, since this program prints the transverse momentum scale of the real emission into the LH events.). If the value is not set, the `SCALUP` variable of the current LH event will be used instead. If wimpy showers (see Timelike Showers and Spacelike Showers) are used together with multi-jet merging, then this scale further sets the parton shower starting scale (μQ) for the core hard process.

`parm` Merging:muRen (`default = -1.0`)
The fixed renormalisation scale used in the hard process cross section, as needed to generate the leading-order weight, in case the renormalisation scale cannot be inferred from Les Houches event input. (As mentioned above, this is the case for files generated with the POWHEG-BOX program.) If the value is not set, the `SCALUP` variable of the current LH event will be used instead.

`parm` Merging:muFacInME (`default = -1.0`)
The fixed factorisation scale used in the matrix element calculation. This information is needed if factorisation scale variations in NLO merged results are attempted. Depending on the matrix element generator, it might not be possible to infer the factorisation scale from Les Houches event input, and thus, setting an explicit value is required. (As mentioned above, this is the case for files generated with the POWHEG-BOX program.) If the value is not set, the `SCALUP` variable of the current LH event will be used instead.

`parm` Merging:muRenInME (`default = -1.0`)
The fixed renormalisation scale used in the matrix element calculation. This information is needed if renormalisation scale variations in NLO merged results are attempted, for the same reason as factorisation scales might be required. (As mentioned above, this is the case for files generated with the POWHEG-BOX program.) If the value is not set, the `SCALUP` variable of the current LH event will be used instead.

All further settings will be discussed while examining the sample main programs.

### NL3 merging with main87.cc

NL3-style NLO merging in Pythia is illustrated by the sample main program `main87.cc`. This program works together with an input file (e.g. `main87.cmnd`) for Pythia settings, and requires LHE input files that follow the naming convention name_tree_#nAdditionalJets.lhe (tree-level samples) and name_powheg_#nAdditionalJets.lhe (POWHEG NLO samples). `main87.cc` produces HepMC event output [Dob01], which can be used for analysis (e.g. using RIVET [Buc10]), or as input for detector simulations. For users not familiar with HepMC output, it is of course possible remove the HepMC code in the sample program, and use Pythia's histogramming routines instead. Histograms should then be filled as indicated for the histPTFirstSum histograms in `main84.cc`, i.e. using weightNLO*normhepmc.

If the user only wants to change the number of requested events (`Main:numberOfEvents`), the hard process (`Merging:Process`), the merging scale value (`Merging:TMS`) and the maximal number of additional tree-level or NLO-accuracte jets (`Merging:nJetMax` and `Merging:nJetMaxNLO`, respectively), and HepMC output is desired, then there is no need to change the `main87.cc` code. The input LHE files are also part of the (command line) input for `main87.exe`. The default settings in `main87.cmnd` are intended to work with the (very short) sample LHEF inputs (w_production_tree_0.lhe, w_production_tree_1.lhe, w_production_tree_2.lhe and w_production_powheg_0.lhe, w_production_powheg_1.lhe). For these input files, the `main87.exe` executable can be run with the command

`./main87.exe main87.cmnd w_production myhepmc.hepmc`

to produce a file myhepmc.hepmc of NLO merged HepMC event output. All mandatory Pythia input settngs have been outlined earlier. Please refrain from adding input switches than invoke any other merging scheme (e.g. e.g. `Merging:doKTMerging`) into the input file that you want to use in conjunction with `main87.cc`.

In the following, we will explain `main87.cc` in depth. Users who are willing to accept the default choices do not need to know all details, but are still encouraged to read further.

#### Program flow

`main87.cc` can be divided into four steps:

1. Estimate the cross section for tree-level and NLO samples after the merging scale cut.

2. Produce reweighted tree-level events, which do not contain αs0- and αs1-terms.

3. Add POWHEG NLO events.

4. Subtract phase space points with an extra (real-emission) jet above the merging scale from the POWHEG result, since such configurations have already been taken into account by processing other samples.

The first step is necessary to produce the correct weights for HepMC output events. The estimation of tree-level cross sections after the merging scale cut is generated by invoking the switch `Merging:doXSectionEstimate` together with `Merging:doNL3Tree`. In this configuration, the latter switch will only act to define the merging scale. After the tree-level cross sections have been estimated, `main87.cc` estimates the NLO cross sections after application of the merging scale cut, by inferring `Merging:doXSectionEstimate` together with `Merging:doNL3Loop`. Again, in this configuration, the latter switch only acts as the merging scale definition. When generating the estimates, all showering, multiparton interactions and hadronisation is turned off to not unnecessarily waste processor time. For all estimates, is further mandatory to set the value of `Merging:nRequested` to the jet multiplicity of the current event sample (e.g. to "2" for a sample containing W + 2 jet events). This is necessary in order to correctly apply the merging scale cut. POWHEG NLO input files for W + 1 jet e.g. contain W + 1 jet and W + 2 jet (i.e. real emission) kinematics. However, the merging scale cut aims at regularising the "underlying Born" configuration (i.e. the W + 1 states in our example). Setting `Merging:nRequested = 1` for the W + 1 jet POWHEG sample ensures that even for real-emission (W + 2 jet) kinematics, the merging scale cut is applied to W + 1 jet states.

After the cross section estimation step, `main87.cc` proceeds to perform the actual merging. Before explaining this part, we would like to make some comments about K-factors.

`main87.cc` is prepared to use fixed K-factors to rescale the weight of tree-level events. This rescaling does not affect the NLO accuracy of the method, and was investigated in [Lon13]. By default, `main87.cc` does not use K-factors. However, if the user wants to include K-factors, this can be done by using the following input settings.

`parm` Merging:kFactor0j (`default = 1.0`)
The k-Factor used to rescale the tree-level (i.e. CKKW-L or UMEPS) part of zero-jet tree-level events.

`parm` Merging:kFactor1j (`default = 1.0`)
The k-Factor used to rescale the tree-level (i.e. CKKW-L or UMEPS) part of one-jet tree-level events.

`parm` Merging:kFactor2j (`default = 1.0`)
The k-Factor used to rescale the tree-level (i.e. CKKW-L or UMEPS) part of two-jet tree-level events.

If the variables k0, k1, k2 in `main87.cc` are set to non-unity values, K-factors will be applied. The K-factor of highest jet multiplicity will then be used to also rescale tree-level samples with a number of additional jets beyond the number of the highest-multiplicity real-emission sample. If we, for example, attempt an NLO merging of W+0 jet and W+1 jet at NLO accuracy, and with W+≤4 jets at tree-level accuracy, then `Merging:kFactor2j` is used to rescale the W+2 jet, W+3 jets and W+4 jets tree-level samples. We recommend to not include a K-factor rescaling of the tree-level samples.

Let us turn to the production of NLO merged events. The first step in the procedure is to generate reweighted tree-level samples. This is implemented by using the following switch.

`flag` Merging:doNL3Tree (`default = off`)
This switch will allow the generation of the weight that should be applied to tree-level events in the NL3 merging scheme. Please note that, in order for this to work smoothly, the switch `Merging:doNL3Loop` and the switch `Merging:doNL3Subt` have to be turned off. As for the estimation of cross sections, it is mandatory to set the correct value of `Merging:nRequested`.

The weight of tree-level events can be accessed by calling the function double Info::mergingWeightNLO(). When printing (or histogramming) NLO merged events, this weight, multiplied with the estimated cross section of the current event sample, should be used as event weight (or weight of histogram bins). For `Merging:doNL3Tree = on`, the weight double Info::mergingWeightNLO() contains the CKKW-L weight, subtracted, if necessary, by αs0- and αs1-terms. This weight can become negative. As an example, imagine we attempt an NLO merging of W + 0 jet and W + 1 jet at NLO accuracy, and with W + 2 jets at tree-level accuracy. This weight will then be

Info::mergingWeightNLO() = CKKW-L-weight for zero jets - αs0-terms - αs1-terms for events in the zero-jet sample,

Info::mergingWeightNLO() = CKKW-L-weight for one jet - αs0-terms - αs1-terms for events in the one-jet sample, and

Info::mergingWeightNLO() = CKKW-L-weight for two jets for events in the two-jet sample.

After the tree-level events have been reweighted, `main87.cc` will move on to process the POWHEG NLO input. This is done by switching to the following flag.

`flag` Merging:doNL3Loop (`default = off`)
This switch will allow the processing of POWHEG NLO events in the NL3 merging scheme. Please note that, in order for this to work smoothly, the switch `Merging:doNL3Tree` and the switch `Merging:doNL3Subt` have to be turned off. As for the estimation of cross sections, it is mandatory to set the correct value of `Merging:nRequested`.

Also in this case, the NLO merging weight of the events can be accessed by calling the function double Info::mergingWeightNLO(). This weight should also be used when printing (or histogramming) events. For `Merging:doNL3Loop = on`, the weight double Info::mergingWeightNLO() is either one or zero (see Appendix E in [Lon13]). After the processing of POWHEG NLO events, `main87.cc` continues by generating explicit phase space subtractions. This is facilitated by the following switch.

`flag` Merging:doNL3Subt (`default = off`)
This switch will allow the processing of tree-level events, to produce explicit phase space subtractions in the NL3 merging scheme. Please note that, in order for this to work smoothly, the switch `Merging:doNL3Tree` and the switch `Merging:doNL3Loop` have to be turned off. As for the estimation of cross sections, it is mandatory to set the correct value of `Merging:nRequested`. Further, it is necessary to set the value of `Merging:nRecluster` to one.

These contributions are necessary because we have implemented the "inclusive scheme" of [Lon13] in Pythia. The benefit of this scheme is the user does not have to intrusively change the POWHEG-BOX program to implement very particular cuts. Let us explain this comment with an example (a more detailed explanation of the idea is given in Appendix A.2 of [Lon13]). When generating W + 0 jet events with the POWHEG-BOX program, the output LHE files will contain W + 1 jet real emission events. Some of these events will contain a jet above the merging scale. However, in NLO merging methods, such configurations have already been included by a separate W + 1 jet sample. Thus, to avoid counting such events twice, we have to remove the configurations from the POWHEG-BOX output. We choose to remove such events by explicit subtraction.

As always, the NLO merging weight of the events can be accessed by calling the function double Info::mergingWeightNLO(). This weight should also be used when printing (or histogramming) events. For `Merging:doNL3Subt = on`, the weight double Info::mergingWeightNLO() is either one or zero (see Appendix E in [Lon13]).

After these steps, all necessary events for NL3 merging have been produced. `main87.cc` finishes by returning the NL3-merged total cross section.

### UNLOPS merging with main88.cc

UNLOPS-style NLO merging in Pythia is illustrated by the sample main program `main88.cc`, which relies on an input file (e.g. `main88.cmnd`) for Pythia settings. As for all merging methods in Pythia, `main88.cc` requires LHE input files. To use `main88.cc` without any changes, these input files should follow the naming convention name_tree_#nAdditionalJets.lhe (for tree-level samples) and name_powheg_#nAdditionalJets.lhe (for POWHEG NLO samples). `main88.cc` produces HepMC event output, which can e.g. be analysed with RIVET, or used as input for detector simulations. For users not familiar with HepMC output, it is of course possible remove the HepMC code in the sample program, and use Pythia's histogramming routines instead. Histograms should then be filled as indicated for the histPTFirstSum histograms in `main84.cc`, i.e. using weightNLO*normhepmc.

As for NL3, it is not necessary to change `main88.cc` if the user is only interested in changing standard settings. Thus, if the user only wants to change the number of requested events (`Main:numberOfEvents`), the hard process (`Merging:Process`), the merging scale value (`Merging:TMS`) and the maximal number of additional tree-level or NLO-accuracte jets (`Merging:nJetMax` and `Merging:nJetMaxNLO`, respectively), and HepMC output is desired, then there is no need to change the `main88.cc` code. The input LHE files are also part of the (command line) input for `main88.exe`. The default settings in `main88.cmnd` are intended to work with the (very short) sample LHEF inputs (w_production_tree_0.lhe, w_production_tree_1.lhe, w_production_tree_2.lhe and w_production_powheg_0.lhe, w_production_powheg_1.lhe). For these input files, the `main88.exe` executable can be run with the command

`./main88.exe main88.cmnd w_production myhepmc.hepmc`

to produce a file myhepmc.hepmc of UNLOPS merged HepMC event output. Please refrain from adding input switches than invoke any other merging scheme (e.g. `Merging:doKTMerging`) into the input file that you want to use in conjunction with `main88.cc`.

In the following, we will explain `main88.cc` in depth. To not be overly repetitive, we will at times refer to the relevant parts in the discussion of `main87.cc`. Users who are willing to accept the default choices do not need to know all details, but are still encouraged to read further.

#### Program flow

`main88.cc` can be divided into five steps:

1. Estimate the cross section for tree-level and NLO samples after the merging scale cut.

2. Produce reweighted tree-level events, which do not contain αs0- and αs1-terms.

3. Add POWHEG NLO events.

4. Subtract integrated, reweighted tree-level events, to ensure that the inclusive NLO cross section remains intact upon inclusion of multi-jet tree-level events.

5. Subtract integrated POWHEG NLO events, to ensure that the inclusive NLO cross section remains intact upon inclusion of multi-jet tree-level events.

The estimation of cross sections after the application of the merging scale cut is nearly identical to the first step in `main87.cc`, and we refer to the first paragraph of the "Program flow" discussion for `main87.cc` for details. For `main88.cc`, the flags `Merging:doUNLOPSTree` or `Merging:doUNLOPSLoop` supply the merging scale definition used in the cross section estimation.

After the cross section estimation step, `main88.cc` proceeds to perform the actual NLO merging. The discussion of K-factors given in the NL3 section (i.e. of `Merging:kFactor0j`, `Merging:kFactor1j` and `Merging:kFactor2j`) also applies to `main88.cc`. Although UNLOPS is considerably stable than NL3 upon changing the K-factors, we do not recommend the use of K-factors.

The production of UNLOPS-merged events with `main88.cc` starts by generating reweighted tree-level events. The processing of tree-level events can be invoked by setting the following flag.

`flag` Merging:doUNLOPSTree (`default = off`)
This switch will allow the generation of the weight that should be applied to tree-level events in the UNLOPS merging scheme. Please note that, in order for this to work smoothly, the switches `Merging:doUNLOPSLoop`, `Merging:doUNLOPSSubt` and `Merging:doUNLOPSSubtNLO` have to be turned off. As for the estimation of cross sections, it is mandatory to set the correct value of `Merging:nRequested`.

The weight of tree-level events is returned by the function double Info::mergingWeightNLO(). When printing (or histogramming) NLO merged events, this weight, multiplied with the estimated cross section of the current event sample, should be used as event weight (or weight of histogram bins). For `Merging:doUNLOPSTree = on`, the weight double Info::mergingWeightNLO() contains the UMEPS weight, subtracted, if necessary, by αs0- and αs1-terms. This weight can become negative. As an example, assume that we attempt an UNLOPS merging of W + 0 jet and W + 1 jet at NLO accuracy, and with W + 2 jets at tree-level accuracy. This weight will then be

Info::mergingWeightNLO() = UMEPS-weight for one jet - αs0-terms - αs1-terms for events in the one-jet sample, and

Info::mergingWeightNLO() = UMEPS-weight for two jets for events in the two-jet sample.

After reweighted tree-level events have been generated, `main88.cc` processes the POWHEG NLO input files. This is facilitated by the following switch.

`flag` Merging:doUNLOPSLoop (`default = off`)
This switch will allow the processing of POWHEG NLO events in the UNLOPS merging scheme. Please note that, in order for this to work smoothly, the switches `Merging:doUNLOPSTree`, `Merging:doUNLOPSSubt` and `Merging:doUNLOPSSubtNLO` have to be turned off. As for the estimation of cross sections, it is mandatory to set the correct value of `Merging:nRequested`.

The NLO merging weight of the events can be accessed by calling the function double Info::mergingWeightNLO(). This weight should also be used when printing (or histogramming) events. For `Merging:doUNLOPSLoop = on`, the weight double Info::mergingWeightNLO() is either one or zero (see Appendix E in [Lon13]).

After processing the POWHEG NLO events, `main88.cc` continues by generating the reweighted subtraction terms of UMEPS. This part is implemented by setting the following flag.

`flag` Merging:doUNLOPSSubt (`default = off`)
This switch will allow the processing of tree-level events, to produce UMEPS subtraction terms for the UNLOPS merging scheme. Please note that, in order for this to work smoothly, the switches `Merging:doUNLOPSTree`, `Merging:doUNLOPSLoop` and `Merging:doUNLOPSSubtNLO` have to be turned off. As for the estimation of cross sections, it is mandatory to set the correct value of `Merging:nRequested`. Further, it is necessary to set the value of `Merging:nRecluster` to one.

By using this switch, `main88.cc` ensures that the inclusive cross section is preserved. At variance with UMEPS however, the event weight contains the UMEPS weight, subtracted, if necessary, by αs+0- and αs1-terms. Otherwise, αsn+0- and αsn+1-terms of the UMEPS procedure would be introduced, although our goal is to describe all αsn+0- and αsn+1-terms by n-jet POWHEG input. The weight of these integrated, subtractive tree-level events is, as always, returned by the function double Info::mergingWeightNLO(). When printing (or histogramming) NLO merged events, this weight, multiplied with the estimated cross section of the current event sample, and with -1, should be used as event weight (or weight of histogram bins). As for the case of tree-level events in UNLOPS, this weight can become negative. For the example given before, i.e. attempting an UNLOPS merging of W + 0 jet and W + 1 jet at NLO accuracy, and with W + 2 jets at tree-level accuracy, this weight will be

Info::mergingWeightNLO() = UMEPS-weight for the integrated one-jet sample - αs0-terms - αs1-terms for events in the integrated one-jet sample, and

Info::mergingWeightNLO() = UMEPS-weight for the integrated two-jet sample for events in the integrated two-jet sample.

This choice of weights already incorporates the fact that we have implemented the "inclusive scheme" of [Lon13], meaning that the "explicit phase space subtractions" of NL3 are (partially) included though these weights.

To ensure that the NLO inclusive cross section is unchanged, UNLOPS further requires the introduction of another sample. If POWHEG NLO events with one or more jets are included, it is necessary to subtract these samples in an integrated form. In `main88.cc`, this is done by setting the following flag.

`flag` Merging:doUNLOPSSubtNLO (`default = off`)
This switch will allow the processing of POWHEG NLO events, to produce NLO subtraction terms for the UNLOPS merging scheme. Please note that, in order for this to work smoothly, the switches `Merging:doUNLOPSTree`, `Merging:doUNLOPSLoop` and `Merging:doUNLOPSSubt` have to be turned off. As for the estimation of cross sections, it is mandatory to set the correct value of `Merging:nRequested`. Further, it is necessary to set the value of `Merging:nRecluster` to one.

This sample also provides some "explicit phase space subtractions" of NL3, which are necessary because we implemented the "inclusive scheme" of [Lon13]. Let us again look at the example of UNLOPS merging of W + 0 jet and W + 1 jet at NLO accuracy. The integrated W + 1 jet NLO events, which are produced by the setting `Merging:doUNLOPSSubtNLO = on`, contain a tree-level part. This part will exactly cancel the real-emission events with one jet above the merging scale in the W + 0 jet NLO events.

The NLO merging weight of these "integrated" events can be accessed by calling the function double Info::mergingWeightNLO(). This weight should also be used when printing (or histogramming) events. For `Merging:doUNLOPSSubtNLO = on`, the weight double Info::mergingWeightNLO() is either one or zero (see Appendix E in [Lon13]).

After these five steps (estimation of cross sections, tree-level processing, POWHEG processing, integrated tree-level processing, integrated POWHEG processing) we have produced a UNLOPS-merged HepMC event file. `main88.cc` finishes by returning the UNLOPS-merged total cross section.

### NLO merging and "exclusive" NLO inputs

Currently, both sample main programs for NLO merging (`main87.cc` and `main88.cc`) are intended for "inclusive" POWHEG input. Inclusive input means that all real emission phase space points are included in the POWHEG output files. In order to avoid double counting with higher-multiplicity matrix elements, it is then necessary remove phase space points with too many jets from the real-emission configurations. This can be done by introducing explicit phase space subtractions. Another way of removing the undesired configurations is by implementing a cut in the NLO generator. This is not a completely trivial task, since it is necessary to ensure numerical stability and the correct cancellation of (finite) dipole regularisation terms. One way of producing such exclusive NLO output is by setting the (tree-level) real-emission matrix element in the NLO generator to zero if the real-emission phase space point contains too many jets above the merging scale. This will however not be numerically stable for too low merging scale values.

We should be very clear that using exclusive NLO input is not recommended, since it requires hacking the NLO generator. Only for the expert user, we briefly summarise the necessary changes for using exclusive NLO input.

For the moment, assume that the NLO input has been produced in an "exclusive" way. This input can then be processed by some trivial changes in `main87.cc`: estimate the cross section for tree-level and NLO samples after the merging scale cut, still using inclusive NLO samples, remove the last part of `main87.cc`, i.e. the part that produces explicit phase space subtractions, and use the exclusive NLO files as input files for the processing of "POWHEG NLO files".

The changes to `main88.cc` (implementing UNLOPS) are slightly more complicated. This is the case because the weights of integrated tree-level samples change when using exclusive input, as can be seen in Appendix D in [Lon13]. The correct weights can be produced by Pythia by using the following flag.

`flag` Merging:doUNLOPSTilde (`default = off`)
This flag allows the UNLOPS machinery to produce the event weights if exclusive NLO input is used for the merging. This flag should be set to "on" directly after the cross section estimates have been produced.

Then, it is necessary to add code for processing another sample to `main88.cc`, since when using exclusive inputs, it is also necessary to enforce two integrations on tree-level events (the "↑"-contributions in Appendix D of [Lon13]). This can be achieved by adding the following code at the end of `main88.cc`.

      cout << endl << endl << endl;
      cout << "Shower subtractive events" << endl;

      // Switch on processing of counter-events.
      pythia.settings.flag("Merging:doUNLOPSTree",false);
      pythia.settings.flag("Merging:doUNLOPSLoop",false);
      pythia.settings.flag("Merging:doUNLOPSSubt",true);
      pythia.settings.flag("Merging:doUNLOPSSubtNLO",false);
      pythia.settings.mode("Merging:nRecluster",2);

      nMaxCT         = nMaxNLO+1;
      njetcounterCT  = nMaxCT;
      iPathSubt      = iPath + "_tree";

      while(njetcounterCT >= 2){

        // From njet, choose LHE file
        stringstream in;
        in   << "_" << njetcounterCT << ".lhe";
        string LHEfile = iPathSubt + in.str();

        cout << endl << endl << endl
             << "Start subtractive treatment for " << njetcounterCT << " jets\n"
             << "Recluster at least 2 times" << endl;

        pythia.readString("Beams:frameType = 4");
        pythia.settings.word("Beams:LHEF", LHEfile);
        pythia.settings.mode("Merging:nRequested", njetcounterCT);
        pythia.init();
        // Remember position in vector of cross section estimates.
        int iNow = sizeLO-1-njetcounterCT;

        // Start generation loop
        for( int iEvent=0; iEvent < nEvent; ++iEvent ){

          // Generate next event
          if( !pythia.next() ) {
            if( pythia.info.atEndOfFile() ) break;
            else continue;
          }

          // Get event weight(s).
          double weightNLO  = pythia.info.mergingWeightNLO();
          double evtweight  = pythia.info.weight();
          weightNLO        *= evtweight;
          // Do not print zero-weight events.
          if ( weightNLO == 0. ) continue;

          // Construct new empty HepMC event.
          HepMC::GenEvent* hepmcevt = new HepMC::GenEvent();
          // Get correct cross section from previous estimate.
          double normhepmc = -1*xsecLO[iNow] / nAcceptLO[iNow];
          // Set hepmc event weight.
          hepmcevt->weights().push_back(weightNLO*normhepmc);
          // Fill HepMC event.
          ToHepMC.fill_next_event( pythia, hepmcevt );
          // Add the weight of the current event to the cross section.
          sigmaTotal += weightNLO*normhepmc;
          errorTotal += pow2(weightNLO*normhepmc);
          // Report cross section to hepmc.
          HepMC::GenCrossSection xsec;
          xsec.set_cross_section( sigmaTotal*1e9, pythia.info.sigmaErr()*1e9 );
          hepmcevt->set_cross_section( xsec );
          // Write the HepMC event to file. Done with it.
          ascii_io << hepmcevt;
          delete hepmcevt;

        } // end loop over events to generate

        // print cross section, errors
        pythia.stat();

        // Restart with ME of a reduced the number of jets
        if( njetcounterCT > 2 )
          njetcounterCT--;
        else
          break;

      }

### Further variables

More advanced manipulations of the merging machinery are of course possible, and additional switches can be found at the end of the CKKW-L tree-level merging documentation. Here, we only document switches that only apply to NLO merging.

`flag` Merging:allowIncompleteHistoriesInReal (`default = off`)
If switched on, this will allow to keep states with incomplete parton shower histories (i.e. states that cannot be projected onto an allowed underlying Born process) in the real contributions of an NLO input sample. By default, such configurations will instead be included through higher-multiplicity tree-level matrix elements. However, NLO input samples can contain a significant number of such configurations if Diagram Subtraction (DS) techniques had been applied. In order not to change the DS scheme, it is important not to remove incomplete histories from the real-emission contribution. Note that furthermore, if this switch turned on, you will have to ensure yourself that no double-counting between states with incomplete histories will occur between NLO samples and higher-multiplicity tree-level samples. This might for example entail using the `MergingHooks` facilities, and the function `double MergingHooks::dampenIfFailCuts(const Event& event)"` in particular.

`mode` Merging:unlopsTMSdefinition (`default = -1`; `minimum = -1`)
The definition of the merging scale for UNLOPS merging. Any value larger or equal to zero means a user-defined merging scale function (to be defined by supplying a `MergingHooks` class) is used for UNLOPS.