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<h2>NLO Merging</h2>
Pythia offers two NLO merging approaches. Both of these methods have been
presented in [<a href="Bibliography.html" target="page">Lon13</a>]. 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.
<p/> The two NLO merging methods implemented in Pythia are called
NL<sup>3</sup> (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 <code>main87.cc</code>
(introducing NL<sup>3</sup> ) and <code>main88.cc</code> (introducing UNLOPS).
Before describing these programs, we would like to outline the differences
between the two approaches.
<p/> NL<sup>3</sup> is a generalisation of <a href="CKKWLMerging.html" target="page">CKKW-L
tree-level merging</a>. The main
idea of NL<sup>3</sup> is to start from CKKW-L-reweighted multi-jet
merging, and replace the
<i>α<sub>s</sub><sup>n+0</sup></i>- and
<i>α<sub>s</sub><sup>n+1</sup></i>-terms by the NLO result of POWHEG.
This "replacement" means that we subtract the
<i>α<sub>s</sub><sup>n+0</sup></i>- and
<i>α<sub>s</sub><sup>n+1</sup></i>-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 [<a href="Bibliography.html" target="page">Lon13</a>] 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
<code>main87.cc</code>, together with the input file <code>main87.cmnd</code>,
illustrates the procedure.
<p/> UNLOPS is a generalisation of the
<a href="UMEPSMerging.html" target="page">UMEPS multi-jet merging</a> scheme.
Since UMEPS is already slightly more complicated than
CKKW-L, this makes UNLOPS more complicated than NL<sup>3</sup>. The basic idea
however remains the same: Start from a tree-level merging scheme (in this case
UMEPS), remove all undesirable <i>α<sub>s</sub><sup>n+0</sup></i>- and
<i>α<sub>s</sub><sup>n+1</sup></i>-terms from this result, and add back
the "correct" description via POWHEG input samples. Again, since the
"inclusive" scheme of [<a href="Bibliography.html" target="page">Lon13</a>] 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.
<p/> UNLOPS is a theoretically more appealing definition of NLO merging than
NL<sup>3</sup>, 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 NL<sup>3</sup>. 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,
<p/>
The hard process
(<code>Merging:Process</code>) needs to be defined
exactly as in CKKW-L (see <strong>Defining the hard process</strong> in the
<a href="CKKWLMerging.html" target="page">CKKW-L documentation</a>).
<p/>
The merging scale value
(<code>Merging:TMS</code>) has to be set.
<p/>
The maximal number of additional partons in
tree-level events (<code>Merging:nJetMax</code>) has to be set.
<p/> All settings listed under the sections "<strong>Matrix element
merging and HepMC output for RIVET</strong>" and "<strong>Further
variables</strong>" in the <a href="CKKWLMerging.html" target="page">CKKW-L
documentation</a> can be accessed in NLO merging as well. Furthermore,
the <code>Merging:nRecluster</code> switch (see the
<a href="UMEPSMerging.html" target="page">UMEPS documentation</a>) is important.
Also, all <code>MergingHooks</code> 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
<code>Merging:enforceCutOnLHE</code> 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 <code>Merging:TMS</code>. This
will lead to warnings of the form "<code>Les Houches Event fails merging scale
cut. Cut by rejecting event</code>". These warning should rather be regarded as
information. An example of inclusive matrix element generation cuts would be
<i>pT<sub>jet</sub> = 5 GeV</i>, <i>ΔR<sub>jetA jetB</sub> =
0.01</i> and <i>Q<sub>jetA jetB</sub> = 5 GeV</i>, if NLO merging with a
desired merging scale value of <code>Merging:TMS = 15</code> is attempted for
Higgs + jets events at the LHC.
<p/> 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 <code>main87.cc</code> and
<code>main88.cc</code>, which implement NL<sup>3</sup> and UNLOPS merging,
respectively.
<br/><br/><hr/>
<h3>Inputs for NLO merging</h3>
The NLO merging schemes in Pythia currently require Les Houches Event File
input. To perform a merging with up to <i>M</i> additional partons described
by tree-level matrix elements, and with up to <i>N ≤ M-1</i> additional
partons at NLO accuracy, the user needs to supply
<p/>
LHE files for <i>0... M</i> additional
partons, taken from a tree-level matrix element generator, and
<p/>
LHE files for <i>0... N</i> additional
partons, taken from a POWHEG NLO generator.
<p/> 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 <code>runningscales 0, btlscalereal 1, btlscalect 1,
ckkwscalup 0</code>. Some older processes in the POWHEG-BOX program need the
input <code>runningscale 0</code> instead of <code>runningscales 0</code>.)
<p/> When attempting NLO merging, the following Pythia settings are relevant.
<p/><code>mode </code><strong> Merging:nJetMaxNLO </strong>
(<code>default = <strong>0</strong></code>; <code>minimum = 0</code>)<br/>
The maximal number of additional jets for which NLO event samples are
supplied by the user.
<p/><code>parm </code><strong> Merging:muFac </strong>
(<code>default = <strong>-1.0</strong></code>)<br/>
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 <code>SCALUP</code> variable of the current LH
event will be used instead.
If wimpy showers (see <a href="TimelikeShowers.html" target="page">Timelike Showers</a> and
<a href="SpacelikeShowers.html" target="page">Spacelike Showers</a>) are used together
with multi-jet merging, then this scale further sets the
parton shower starting scale (<i>μ<sub>Q</sub></i>) for the core hard
process.
<p/><code>parm </code><strong> Merging:muRen </strong>
(<code>default = <strong>-1.0</strong></code>)<br/>
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 <code>SCALUP</code> variable of the current LH
event will be used instead.
<p/><code>parm </code><strong> Merging:muFacInME </strong>
(<code>default = <strong>-1.0</strong></code>)<br/>
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 <code>SCALUP</code> variable of the current LH
event will be used instead.
<p/><code>parm </code><strong> Merging:muRenInME </strong>
(<code>default = <strong>-1.0</strong></code>)<br/>
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 <code>SCALUP</code> variable of the current LH
event will be used instead.
<p/> All further settings will be discussed while examining the sample main
programs.
<br/><br/><hr/>
<h3>NL<sup>3</sup> merging with main87.cc</h3>
NL<sup>3</sup>-style NLO merging in Pythia is illustrated by the sample
main program <code>main87.cc</code>. This program works together with an input
file (e.g. <code>main87.cmnd</code>) for Pythia settings, and requires LHE
input files that follow the naming convention
<i>name_tree_#nAdditionalJets.lhe</i> (tree-level samples) and
<i>name_powheg_#nAdditionalJets.lhe</i> (POWHEG NLO samples).
<code>main87.cc</code> produces HepMC event output [<a href="Bibliography.html" target="page">Dob01</a>], which can
be used for analysis (e.g. using RIVET [<a href="Bibliography.html" target="page">Buc10</a>]), 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 <i>histPTFirstSum</i> histograms in <code>main84.cc</code>, i.e.
using <i>weightNLO*normhepmc</i>.
<p/> If the user only wants to change the number of requested events
(<code>Main:numberOfEvents</code>), the hard process
(<code>Merging:Process</code>), the merging scale value
(<code>Merging:TMS</code>) and the maximal number of additional tree-level or
NLO-accuracte jets (<code>Merging:nJetMax</code> and
<code>Merging:nJetMaxNLO</code>, respectively), and HepMC output is desired,
then there is no need to change the <code>main87.cc</code> code. The input
LHE files are also part of the (command line) input for <code>main87.exe</code>.
The default settings in <code>main87.cmnd</code> are intended to work with the
(very short) sample LHEF inputs (<i>w_production_tree_0.lhe</i>,
<i>w_production_tree_1.lhe</i>, <i>w_production_tree_2.lhe</i> and
<i>w_production_powheg_0.lhe</i>, <i>w_production_powheg_1.lhe</i>). For
these input files, the <code>main87.exe</code> executable can be run with
the command
<p/>
<code>./main87.exe main87.cmnd w_production myhepmc.hepmc</code>
<p/> 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.
<code>Merging:doKTMerging</code>) into the input file that you want to use in
conjunction with <code>main87.cc</code>.
<p/> In the following, we will explain <code>main87.cc</code> 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.
<h4>Program flow</h4>
<code>main87.cc</code> can be divided into four steps:
<p/> 1. Estimate the cross section for
tree-level and NLO samples <i>after</i> the merging scale cut.
<p/> 2. Produce reweighted tree-level events,
which do not contain <i>α<sub>s</sub><sup>0</sup></i>- and
<i>α<sub>s</sub><sup>1</sup></i>-terms.
<p/> 3. Add POWHEG NLO events.
<p/> 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.
<p/> 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
<code>Merging:doXSectionEstimate</code> together with
<code>Merging:doNL3Tree</code>. In this configuration, the latter switch will
only act to define the merging scale. After the tree-level cross sections
have been estimated, <code>main87.cc</code> estimates the NLO cross sections
after application of the merging scale cut, by inferring
<code>Merging:doXSectionEstimate</code> together with
<code>Merging:doNL3Loop</code>. 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 <i>mandatory</i> to set the value of
<code>Merging:nRequested</code> 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 <code>Merging:nRequested = 1</code> 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.
<p/> After the cross section estimation step, <code>main87.cc</code> proceeds
to perform the actual merging. Before explaining this part, we would like to
make some comments about K-factors.
<p/> <code>main87.cc</code> 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
[<a href="Bibliography.html" target="page">Lon13</a>]. By default, <code>main87.cc</code> does not use K-factors.
However, if the user wants to include K-factors, this can be done by using
the following input settings.
<p/><code>parm </code><strong> Merging:kFactor0j </strong>
(<code>default = <strong>1.0</strong></code>)<br/>
The k-Factor used to rescale the tree-level (i.e. CKKW-L or UMEPS) part of
zero-jet tree-level events.
<p/><code>parm </code><strong> Merging:kFactor1j </strong>
(<code>default = <strong>1.0</strong></code>)<br/>
The k-Factor used to rescale the tree-level (i.e. CKKW-L or UMEPS) part of
one-jet tree-level events.
<p/><code>parm </code><strong> Merging:kFactor2j </strong>
(<code>default = <strong>1.0</strong></code>)<br/>
The k-Factor used to rescale the tree-level (i.e. CKKW-L or UMEPS) part of
two-jet tree-level events.
<p/> If the variables <i>k0, k1, k2</i> in <code>main87.cc</code> 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 <i>W+0 jet</i> and <i>W+1 jet</i> at NLO
accuracy, and with <i>W+≤4 jets</i> at tree-level accuracy, then
<code>Merging:kFactor2j</code> is used to rescale the <i>W+2 jet</i>,
<i>W+3 jets</i> and <i>W+4 jets</i> tree-level samples. We recommend to
not include a K-factor rescaling of the tree-level samples.
<p/> 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.
<p/><code>flag </code><strong> Merging:doNL3Tree </strong>
(<code>default = <strong>off</strong></code>)<br/>
This switch will allow the generation of the weight that should be applied to
tree-level events in the NL<sup>3</sup> merging scheme. Please note that, in
order for this to work smoothly, the switch <code>Merging:doNL3Loop</code> and
the switch <code>Merging:doNL3Subt</code> have to be turned off. As for the
estimation of cross sections, it is <i>mandatory</i> to set
the correct value of <code>Merging:nRequested</code>.
<p/>
The weight of tree-level events can be accessed by calling the function
<strong>double Info::mergingWeightNLO()</strong>. 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 <code>Merging:doNL3Tree = on</code>,
the weight <strong>double Info::mergingWeightNLO()</strong> contains the
CKKW-L weight, subtracted, if necessary, by
<i>α<sub>s</sub><sup>0</sup></i>- and
<i>α<sub>s</sub><sup>1</sup></i>-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
<p/> <i>Info::mergingWeightNLO() = CKKW-L-weight for zero jets
- <i>α<sub>s</sub><sup>0</sup></i>-terms
- <i>α<sub>s</sub><sup>1</sup></i>-terms
</i> for events in the zero-jet sample,
<p/> <i>Info::mergingWeightNLO() = CKKW-L-weight for one jet
- <i>α<sub>s</sub><sup>0</sup></i>-terms
- <i>α<sub>s</sub><sup>1</sup></i>-terms
</i> for events in the one-jet sample, and
<p/> <i>Info::mergingWeightNLO() = CKKW-L-weight for two jets </i>
for events in the two-jet sample.
<p/> After the tree-level events have been reweighted, <code>main87.cc</code>
will move on to process the POWHEG NLO input. This is done by switching to the
following flag.
<p/><code>flag </code><strong> Merging:doNL3Loop </strong>
(<code>default = <strong>off</strong></code>)<br/>
This switch will allow the processing of POWHEG NLO events in the
NL<sup>3</sup> merging scheme. Please note that, in order for this to work
smoothly, the switch <code>Merging:doNL3Tree</code> and the switch
<code>Merging:doNL3Subt</code> have to be turned off. As for the estimation
of cross sections, it is <i>mandatory</i> to set the correct value of
<code>Merging:nRequested</code>.
<p/>
Also in this case, the NLO merging weight of the events can be accessed by
calling the function <strong>double Info::mergingWeightNLO()</strong>. This
weight should also be used when printing (or histogramming) events.
For <code>Merging:doNL3Loop = on</code>,
the weight <strong>double Info::mergingWeightNLO()</strong> is either one or
zero (see Appendix E in [<a href="Bibliography.html" target="page">Lon13</a>]).
After the processing of POWHEG NLO events, <code>main87.cc</code> continues
by generating explicit phase space subtractions. This is facilitated by the
following switch.
<p/><code>flag </code><strong> Merging:doNL3Subt </strong>
(<code>default = <strong>off</strong></code>)<br/>
This switch will allow the processing of tree-level events, to produce explicit
phase space subtractions in the NL<sup>3</sup> merging scheme. Please note
that, in order for this to work smoothly, the switch
<code>Merging:doNL3Tree</code> and the switch
<code>Merging:doNL3Loop</code> have to be turned off. As for the estimation
of cross sections, it is <i>mandatory</i> to set the correct value of
<code>Merging:nRequested</code>. Furthermore, it is necessary to set the
value of <code>Merging:nRecluster</code> to one.
<p/> These contributions are necessary because we have implemented
the "inclusive
scheme" of [<a href="Bibliography.html" target="page">Lon13</a>] 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 [<a href="Bibliography.html" target="page">Lon13</a>]). 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.
<p/>
As always, the NLO merging weight of the events can be accessed by
calling the function <strong>double Info::mergingWeightNLO()</strong>. This
weight should also be used when printing (or histogramming) events.
For <code>Merging:doNL3Subt = on</code>,
the weight <strong>double Info::mergingWeightNLO()</strong> is either one or
zero (see Appendix E in [<a href="Bibliography.html" target="page">Lon13</a>]).
<p/> After these steps, all necessary events for NL<sup>3</sup> merging have
been produced. <code>main87.cc</code> finishes by returning the
NL<sup>3</sup>-merged total cross section.
<br/><br/><hr/>
<h3>UNLOPS merging with main88.cc</h3>
UNLOPS-style NLO merging in Pythia is illustrated by the sample
main program <code>main88.cc</code>, which relies on an input file (e.g.
<code>main88.cmnd</code>) for Pythia settings. As for all merging methods in
Pythia, <code>main88.cc</code> requires LHE input files. To use
<code>main88.cc</code> without any changes, these input files should follow
the naming convention <i>name_tree_#nAdditionalJets.lhe</i> (for tree-level
samples) and <i>name_powheg_#nAdditionalJets.lhe</i> (for POWHEG NLO
samples). <code>main88.cc</code> 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
<i>histPTFirstSum</i> histograms in <code>main84.cc</code>, i.e. using
<i>weightNLO*normhepmc</i>.
<p/> As for NL<sup>3</sup>, it is not necessary to change
<code>main88.cc</code> if the user is only interested in changing standard
settings. Thus, if the user only wants to change the number of requested
events (<code>Main:numberOfEvents</code>), the hard process
(<code>Merging:Process</code>), the merging scale value
(<code>Merging:TMS</code>) and the maximal number of additional tree-level or
NLO-accuracte jets (<code>Merging:nJetMax</code> and
<code>Merging:nJetMaxNLO</code>, respectively), and HepMC output is desired,
then there is no need to change the <code>main88.cc</code> code. The input
LHE files are also part of the (command line) input for <code>main88.exe</code>.
The default settings in <code>main88.cmnd</code> are intended to work with the
(very short) sample LHEF inputs (<i>w_production_tree_0.lhe</i>,
<i>w_production_tree_1.lhe</i>, <i>w_production_tree_2.lhe</i> and
<i>w_production_powheg_0.lhe</i>, <i>w_production_powheg_1.lhe</i>). For
these input files, the <code>main88.exe</code> executable can be run with
the command
<p/>
<code>./main88.exe main88.cmnd w_production myhepmc.hepmc</code>
<p/> 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. <code>Merging:doKTMerging</code>) into the input file that you
want to use in conjunction with <code>main88.cc</code>.
<p/> In the following, we will explain <code>main88.cc</code> in depth. To
not be overly repetitive, we will at times refer to the relevant parts in the
discussion of <code>main87.cc</code>. Users who are willing to accept the
default choices do not need to know all details, but are still encouraged to
read further.
<h4>Program flow</h4>
<code>main88.cc</code> can be divided into five steps:
<p/> 1. Estimate the cross section for
tree-level and NLO samples <i>after</i> the merging scale cut.
<p/> 2. Produce reweighted tree-level events,
which do not contain <i>α<sub>s</sub><sup>0</sup></i>- and
<i>α<sub>s</sub><sup>1</sup></i>-terms.
<p/> 3. Add POWHEG NLO events.
<p/> 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.
<p/> 5. Subtract integrated POWHEG NLO
events, to ensure that the inclusive NLO cross section remains
intact upon inclusion of multi-jet tree-level events.
<p/> The estimation of cross sections after the application of the merging
scale cut is nearly identical to the first step in <code>main87.cc</code>, and
we refer to the first paragraph of the "Program flow" discussion for
<code>main87.cc</code> for details. For <code>main88.cc</code>, the flags
<code>Merging:doUNLOPSTree</code> or <code>Merging:doUNLOPSLoop</code> supply
the merging scale definition used in the cross section estimation.
<p/> After the cross section estimation step, <code>main88.cc</code> proceeds
to perform the actual NLO merging. The discussion of K-factors given in the
NL<sup>3</sup> section (i.e. of <code>Merging:kFactor0j</code>,
<code>Merging:kFactor1j</code> and <code>Merging:kFactor2j</code>) also
applies to <code>main88.cc</code>. Although UNLOPS is considerably more stable
than NL<sup>3</sup> upon changing the K-factors, we do not recommend the
use of K-factors.
<p/> The production of UNLOPS-merged events with <code>main88.cc</code> starts
by generating reweighted tree-level events. The processing of tree-level
events can be invoked by setting the following flag.
<p/><code>flag </code><strong> Merging:doUNLOPSTree </strong>
(<code>default = <strong>off</strong></code>)<br/>
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
<code>Merging:doUNLOPSLoop</code>, <code>Merging:doUNLOPSSubt</code> and
<code>Merging:doUNLOPSSubtNLO</code> have to be turned off. As for the
estimation of cross sections, it is <i>mandatory</i> to set
the correct value of <code>Merging:nRequested</code>.
<p/>
The weight of tree-level events is returned by the function
<strong>double Info::mergingWeightNLO()</strong>. 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
<code>Merging:doUNLOPSTree = on</code>,
the weight <strong>double Info::mergingWeightNLO()</strong> contains the
UMEPS weight, subtracted, if necessary, by
<i>α<sub>s</sub><sup>0</sup></i>- and
<i>α<sub>s</sub><sup>1</sup></i>-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
<p/> <i>Info::mergingWeightNLO() = UMEPS-weight for one jet
- <i>α<sub>s</sub><sup>0</sup></i>-terms
- <i>α<sub>s</sub><sup>1</sup></i>-terms
</i> for events in the one-jet sample, and
<p/> <i>Info::mergingWeightNLO() = UMEPS-weight for two jets </i>
for events in the two-jet sample.
<p/> After reweighted tree-level events have been generated,
<code>main88.cc</code> processes the POWHEG NLO input files. This is
facilitated by the following switch.
<p/><code>flag </code><strong> Merging:doUNLOPSLoop </strong>
(<code>default = <strong>off</strong></code>)<br/>
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 <code>Merging:doUNLOPSTree</code>,
<code>Merging:doUNLOPSSubt</code> and <code>Merging:doUNLOPSSubtNLO</code>
have to be turned off. As for the estimation of cross sections, it is
<i>mandatory</i> to set the correct value of
<code>Merging:nRequested</code>.
<p/>
The NLO merging weight of the events can be accessed by
calling the function <strong>double Info::mergingWeightNLO()</strong>. This
weight should also be used when printing (or histogramming) events.
For <code>Merging:doUNLOPSLoop = on</code>,
the weight <strong>double Info::mergingWeightNLO()</strong> is either one or
zero (see Appendix E in [<a href="Bibliography.html" target="page">Lon13</a>]).
<p/>
After processing the POWHEG NLO events, <code>main88.cc</code> continues
by generating the reweighted subtraction terms of UMEPS. This part is
implemented by setting the following flag.
<p/><code>flag </code><strong> Merging:doUNLOPSSubt </strong>
(<code>default = <strong>off</strong></code>)<br/>
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
<code>Merging:doUNLOPSTree</code>, <code>Merging:doUNLOPSLoop</code> and
<code>Merging:doUNLOPSSubtNLO</code> have to be turned off.
As for the estimation of cross sections, it is <i>mandatory</i> to set the
correct value of <code>Merging:nRequested</code>. Furthermore, it is necessary
to set the value of <code>Merging:nRecluster</code> to one.
<p/>By using this switch, <code>main88.cc</code> ensures that the inclusive
cross section is preserved. At variance with UMEPS however, the event weight
contains the UMEPS weight, subtracted, if necessary, by
<i>α<sub>s</sub><sup>+0</sup></i>- and
<i>α<sub>s</sub><sup>1</sup></i>-terms. Otherwise,
<i>α<sub>s</sub><sup>n+0</sup></i>- and
<i>α<sub>s</sub><sup>n+1</sup></i>-terms of the UMEPS procedure would
be introduced, although our goal is to describe all
<i>α<sub>s</sub><sup>n+0</sup></i>- and
<i>α<sub>s</sub><sup>n+1</sup></i>-terms by n-jet POWHEG input.
The weight of these integrated, subtractive tree-level events is, as always,
returned by the function <strong>double Info::mergingWeightNLO()</strong>.
When printing (or histogramming) NLO merged events, this weight, multiplied
with the estimated cross section of the current event sample, and with <i>
-1</i>, 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
<p/> <i>Info::mergingWeightNLO() = UMEPS-weight for the integrated
one-jet sample - <i>α<sub>s</sub><sup>0</sup></i>-terms
- <i>α<sub>s</sub><sup>1</sup></i>-terms
</i> for events in the integrated one-jet sample, and
<p/> <i>Info::mergingWeightNLO() = UMEPS-weight for the integrated
two-jet sample</i>
for events in the integrated two-jet sample.
<p/>This choice of weights already
incorporates the fact that we have implemented the "inclusive scheme" of
[<a href="Bibliography.html" target="page">Lon13</a>], meaning that the "explicit phase space subtractions" of
NL<sup>3</sup> are (partially) included though these weights.
<p/> 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 <code>main88.cc</code>, this is done by setting the
following flag.
<p/><code>flag </code><strong> Merging:doUNLOPSSubtNLO </strong>
(<code>default = <strong>off</strong></code>)<br/>
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
<code>Merging:doUNLOPSTree</code>, <code>Merging:doUNLOPSLoop</code> and
<code>Merging:doUNLOPSSubt</code> have to be turned off.
As for the estimation of cross sections, it is <i>mandatory</i> to set the
correct value of <code>Merging:nRequested</code>. Furthermore, it is necessary
to set the value of <code>Merging:nRecluster</code> to one.
<p/> This sample also provides some "explicit phase space subtractions" of
NL<sup>3</sup>, which are necessary because we implemented the
"inclusive scheme" of [<a href="Bibliography.html" target="page">Lon13</a>]. 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
<code>Merging:doUNLOPSSubtNLO = on</code>, 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.
<p/>
The NLO merging weight of these "integrated" events can be accessed by
calling the function <strong>double Info::mergingWeightNLO()</strong>. This
weight should also be used when printing (or histogramming) events.
For <code>Merging:doUNLOPSSubtNLO = on</code>,
the weight <strong>double Info::mergingWeightNLO()</strong> is either one or
zero (see Appendix E in [<a href="Bibliography.html" target="page">Lon13</a>]).
<p/> 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. <code>main88.cc</code> finishes by returning the
UNLOPS-merged total cross section.
<br/><br/><hr/>
<h3>NLO merging and "exclusive" NLO inputs</h3>
Currently, both sample main programs for NLO merging (<code>main87.cc</code>
and <code>main88.cc</code>) 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.
<p/><i> 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.</i>
<p/> 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
<code>main87.cc</code>: estimate the cross section for tree-level and NLO
samples <i>after</i> the merging scale cut, still using inclusive NLO
samples, remove the last part of <code>main87.cc</code>, 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".
<p/> The changes to <code>main88.cc</code> (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 [<a href="Bibliography.html" target="page">Lon13</a>]. The correct weights can be produced by Pythia
by using the following flag.
<p/><code>flag </code><strong> Merging:doUNLOPSTilde </strong>
(<code>default = <strong>off</strong></code>)<br/>
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.
<p/> Then, it is necessary to add code for processing another sample to
<code>main88.cc</code>, since when using exclusive inputs, it is also
necessary to enforce two integrations on tree-level events (the
"↑"-contributions in Appendix D of [<a href="Bibliography.html" target="page">Lon13</a>]). This can be
achieved by adding the following code at the end of <code>main88.cc</code>.
<p/>
<pre>
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;
}
</pre>
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