FERMILAB-PUB-21-536-T, MCNET-21-30

Reducing negative weights in Monte Carlo event generation with Sherpa

K. Danziger1, S. Höche2, F. Siegert1*,

1 Institut für Kern- und Teilchenphysik, TU Dresden, Dresden, Germany

2 Fermi National Accelerator Laboratory, Batavia, USA

*<EMAIL_ADDRESS>
## Abstract

An increase in theoretical precision of Monte Carlo event generators is
typically accompanied by an increased need for computational resources. One
major obstacle are negative weighted events, which appear in Monte Carlo
simulations with higher perturbative accuracy. While they can be handled
somewhat easily in fixed-order calculations, they are a major concern for
particle level event simulations. In this article, the origin of negative
weights in the S-Mc@Nlo method is reviewed and mechanisms to reduce the
negative weight fraction in simulations with the Sherpa event generator are
presented, with a focus on $V$+jets and $t\bar{t}$+jets simulations.

###### Contents

  1. 1 Introduction
  2. 2 Impact of negative event weights
  3. 3 Sources of negative weights in S-Mc@Nlo and Meps@Nlo
  4. 4 Mechanisms to improve the negative weight fraction
    1. 4.1 Leading Colour Approximation
    2. 4.2 $\mathbb{H}$-Event Shower Interplay
    3. 4.3 Core local $K$-factor
  5. 5 Results
    1. 5.1 $Z$+jets
    2. 5.2 $t\bar{t}$+jets
  6. 6 Conclusions

##  1 Introduction

Computing challenges and limitations have a large impact on the feasibility
and ambition of the physics program at the Large Hadron Collider (LHC). A
significant part of the computing needs of the major LHC experiments arises
from the simulation of signal and background events and their interactions
with the detectors [1]. With more luminosity being delivered to the
experiments in future LHC data taking periods, the number of events that can
be simulated with the current resources and computing model will likely not
match the number of events recorded in experimental data [2].

The simulation of physics processes relies heavily on general-purpose Monte
Carlo event generators like Herwig7 [3], Pythia8 [4], or Sherpa [5]. They
provide a realistic simulation of the final state of proton-proton collisions
including effects that go beyond what is possible in fixed-order calculations,
like the resummation of soft or collinear QCD radiation in a parton shower
model, the fragmentation into primordial hadrons and their decays, and the
possibility of multiple partonic interactions within the same collision.

In the last decade, these generators have been extended to include a more
accurate calculation of the hard scattering process within the framework of
perturbative QCD. The matching of next-to-leading order (NLO) matrix elements
to parton showers (PS) [6, 7, 8, 9] and the subsequent merging of NLO+PS
simulations for several jet multiplicities into an inclusive sample [10, 11,
12, 13, 14] results in simulations, which achieve the highest precision for
many of the processes critical for the ongoing experimental analyses. At the
same time, these simulations come with practical disadvantages. Not only are
the calculations much more complex and thus require more computational
resources per event, but they also often contain events which are not
distributed naturally with unit weights (“unweighted”) but instead feature
negative weights that represent subtraction terms in the calculation. These
events reduce the statistical power of the event sample, necessitating the
simulation of larger samples using correspondingly larger computational
resources.

The problem has received much attention recently [15], and various solutions
have been proposed. Resampling methods [16, 17, 18] for example can be used to
redistribute the event weights a posteriori in a largely observable
independent manner. Generator or algorithm specific techniques such as the
Mc@Nlo-$\Delta$ method [19] can be used to eliminate a large fraction of
negative weighted events a priori. In this article we present various options
that are specific to S-Mc@Nlo and Meps@Nlo simulations with the Sherpa event
generator. After a brief introduction to the various sources of negative
weights we present the mitigation techniques and discuss the impact of their
implementation not only on the negative weight fraction but also on the
predictive power of the event sample. We focus on the two most relevant
background processes for LHC physics, $V$+jets and $t\bar{t}$+jets production.

##  2 Impact of negative event weights

Naively, each event in a simulation sample is associated to an event weight,
which corresponds to the differential cross section of the phase space point
the event represents. For experimental applications it is particularly
interesting to obtain event samples with equal weights, as then the events
represent a natural distribution of the simulated process like it would be
found in collision events. Thus, most event generators implement an
unweighting procedure to reject events in such a way, that the output sample
follows a natural distribution with unit weights.

As will be shown in Sec. 3, events can also be associated with negative
weights in some cases where a partial differential cross section becomes
negative. The unweighting procedure in such cases will reject positively and
negatively weighted events separately, such that each set is unweighted to
weights $w=\pm 1$, respectively.

Samples containing negatively weighted events have a reduced statistical
power. Considering a negative weight fraction $\varepsilon$ and, up to a sign,
equally weighted events with weights $w=\pm c$, where $c$ is a constant larger
than zero, the statistical accuracy of an event sample of size $N$ can be
evaluated as

$\frac{\sqrt{\sum_{i}w_{i}^{2}}}{\sum_{i}w_{i}}=\frac{\sqrt{c^{2}\cdot
N}}{c\cdot(1-2\varepsilon)\cdot
N}=\frac{1}{(1-2\varepsilon)}\cdot\frac{1}{\sqrt{N}}=\frac{1}{\sqrt{N_{\mathit{eff}}}}.$
(1)

This yields the effective number of events, given by
$N_{\mathit{eff}}=(1-2\varepsilon)^{2}\cdot N$, which incorporates the
diminished statistics in the presence of negatively weighted events. Hence,
one needs to generate a factor

$f(\varepsilon)=\frac{1}{(1-2\varepsilon)^{2}}$ (2)

more events to get the same statistical accuracy in comparison to only
positively weighted events. Figure 1 illustrates this factor $f$ as a function
of the negative weight fraction $\varepsilon$. As can be seen, this factor
increases dramatically with $\varepsilon$. For instance, if a quarter of the
events have a negative weight, a four times larger sample has to be produced,
and if the negative weight fraction rises to $\varepsilon=40\%$ one needs to
generate 25 times more events. While this raises the computational cost
already on the event generation side, the more pressing issue is the
subsequent detector simulation, making up the major part in terms of CPU
consumption. If the negative weight fraction is too large, using full detector
simulations becomes impractical. Thus, it is of utmost importance to eliminate
negative weights whenever possible.

Figure 1: Dependence of the factor $f$, quantifying the excess of events to be
generated for the same statistical accuracy as in the case of solely positive
weighted events, on the negative weight fraction $\varepsilon$.

##  3 Sources of negative weights in S-Mc@Nlo and Meps@Nlo

Negatively weighted events are commonplace when going beyond the leading order
in perturbation theory for non-trivial processes. To illustrate this, we give
a brief summary of their origin in the S-Mc@Nlo matching scheme implemented in
Sherpa.

Numerical NLO calculations are typically performed using infrared (IR)
subtraction techniques [20, 21, 22]. These methods isolate the IR divergences
in the real-emission corrections and virtual corrections, and allow to define
integrands that are finite in four dimensions for any IR safe observable, and
are thus integrable with Monte Carlo methods. In the dipole method [21, 22],
soft singularities of the real-emission matrix elements are mapped onto
collinear sectors and isolated into a sum of dipole terms $D^{(\mathcal{S})}$,
which can be used to render the real emission matrix elements $R$ finite. The
corresponding integrated subtraction terms $I^{(\mathcal{S})}$ in turn render
the virtual corrections finite, such that the differential dipole-subtracted
cross-section at NLO can schematically be written as

$\mathrm{d}\sigma^{\mathrm{NLO}}=\mathrm{d}\Phi_{B}\Big{[}B(\Phi_{B})+\tilde{V}(\Phi_{B})+I^{(\mathcal{S})}(\Phi_{B})\Big{]}+\mathrm{d}\Phi_{R}\Big{[}R(\Phi_{R})-D^{(\mathcal{S})}(\Phi_{R})\Big{]},$
(3)

Here $B$ and $\tilde{V}$ are the Born and UV renormalized virtual
contribution, while $\mathrm{d}\Phi_{B}$ and $\mathrm{d}\Phi_{R}$ denote the
phase-space element for the Born and real-emission configuration,
respectively. The dipole terms are determined by means of insertion operators
that correlate the color- and spin-dependent Born amplitudes according to the
structure of QCD in the soft-gluon limit and the collinear limit.
Schematically we have

$D^{(\mathcal{S})}(\Phi_{R})\to\sum_{ijk}\langle M_{B}(\Phi_{B})|\frac{{\bf
T}_{ij}{\bf T}_{k}}{{\bf
T}_{i}^{2}}V_{ijk}(\Phi_{R},\Phi_{B})|M_{B}(\Phi_{B})\rangle\;,$ (4)

where the sum runs over all parton indices, ${\bf T}_{i}$ is the color
insertion operator for a parton $i$ [23], and the spin-dependent splitting
kernels $V_{ijk}$ are given in [21]. We note that, depending on the precise
value of the color correlator ${\bf T}_{ij}{\bf T}_{k}$, individual dipole
terms can be either positive or negative valued.

Born-like events, corresponding to the first square bracket in Eq. (3), and
real-emission like events, corresponding to the second square bracket in Eq.
(3), are generated separately in practice. Negative event weights therefore
occur in the calculation if the dipole approximation overestimates the real-
emission matrix element.

This situation is complicated additionally by the matching to a parton shower
resummation. In the Mc@Nlo matching method, an additional subtraction term is
introduced, which removes the parton shower approximation from the real-
emission like events and adds it in integrated form to the Born-like events.
This modified subtraction technique removes the overlap between the parton-
shower approximation and the fixed-order NLO result. Denoting the
corresponding MC subtraction terms by $D^{(\mathcal{A})}$, Eq. (3) becomes

$\begin{split}\mathrm{d}\sigma^{\mathrm{NLO}}&=\mathrm{d}\Phi_{B}\Big{[}B(\Phi_{B})+\tilde{V}(\Phi_{B})+I^{(\mathcal{S})}(\Phi_{B})\Big{]}+\mathrm{d}\Phi_{R}\Big{[}D^{(\mathcal{A})}(\Phi_{R})-D^{(\mathcal{S})}(\Phi_{R})\Big{]}\\\
&+\mathrm{d}\Phi_{R}\Big{[}R(\Phi_{R})-D^{(\mathcal{A})}(\Phi_{R})\Big{]}.\end{split}$
(5)

Standard parton showers are based on the leading color and leading spin
approximation, and do therefore not capture the complete singularity structure
of Eq. (4). The corresponding parton-shower splitting kernels would not
suffice to define a locally IR finite differential cross section in Eq. (3).
In the S-Mc@Nlo prescription, $D^{(\mathcal{A})}$ is therefore defined with
the help of the dipole terms $D^{(\mathcal{S})}$, and set to zero outside the
resummation region [24]:

$D^{(\mathcal{A})}=D^{(\mathcal{S})}\Theta(\mu_{Q}^{2}-t).$ (6)

Applying the parton shower resummation using $D^{(\mathcal{A})}$ as splitting
kernels yields the matching formula

$\begin{split}\sigma_{\textsc{S-Mc@Nlo}}&=\int\mathrm{d}\Phi_{B}\bar{B}^{(\mathcal{A})}(\Phi_{B})\Bigg{[}\underbrace{\bar{\Delta}^{(\mathcal{A})}(t_{c},\mu_{Q}^{2})}_{\mathrm{unresolved}}+\underbrace{\int_{t_{c}}^{\mu_{Q}^{2}}\mathrm{d}\Phi_{1}\frac{D^{(\mathcal{A})}(\Phi_{B},\Phi_{1})}{B(\Phi_{B})}\
\bar{\Delta}^{(\mathcal{A})}(t,\mu_{Q}^{2})}_{\mathrm{resolved,\
singular}}\Bigg{]}\\\
&+\int\mathrm{d}\Phi_{R}\underbrace{\Big{[}R(\Phi_{R})-D^{(\mathcal{A})}(\Phi_{R})\Big{]},}_{\mathrm{resolved,\
non-singular,}\ \equiv\ H^{(\mathcal{A})}}\end{split}$ (7)

where $\bar{B}^{(\mathcal{A})}$ denotes the so-called next-to-leading order
weighted Born cross-section given by

$\bar{B}^{(\mathcal{A})}(\Phi_{B})=B(\Phi_{B})+\tilde{V}(\Phi_{B})+I^{(\mathcal{S})}(\Phi_{B})+\int\mathrm{d}\Phi_{1}\Big{[}D^{(\mathcal{A})}(\Phi_{B},\Phi_{1})-D^{(\mathcal{S})}(\Phi_{B},\Phi_{1})\Big{]}$
(8)

and $H^{(\mathcal{A})}=R-D^{(\mathcal{A})}$ describes the hard remainder term.
The first emission is described in terms of a modified Sudakov form factor

$\bar{\Delta}^{(\mathcal{A})}(t,t^{\prime})=\exp\Bigg{\\{}-\int_{t}^{t^{\prime}}\mathrm{d}\Phi_{1}\
\frac{D^{(\mathcal{A})}(\Phi_{B},\Phi_{1})}{B(\Phi_{B})}\Bigg{\\}}.$ (9)

The first line in Eq. (7) corresponds to so-called $\mathbb{S}$-events, where
the first emission is generated as in the standard parton shower, but
including the full colour structure. In contrast, the second line refers to
$\mathbb{H}$-events, where a hard emission is generated pursuant to the
subtracted real-emission matrix element and the parton shower generates
subsequent emissions only. It is apparent that the differential cross section
weight of $\mathbb{H}$-events can become negative due to the modified
subtraction procedure. Furthermore, also $\mathbb{S}$-events can induce
negative weights due to the exponentiation of negative valued subtraction
terms corresponding to sub-leading colour configurations in the approximated
real emission matrix elements of Eq. (4).

The Meps@Nlo merging procedure [11, 10] is an extension of both the S-Mc@Nlo
matching method and the Ckkw merging algorithm, which allows to combine
matched NLO calculations for varying jet multiplicity by an appropriate
overlap removal through Sudakov reweighting and explicit subtraction of NLO
fixed-order corrections already accounted for by the parton shower. As such,
when applied to NLO fixed-order inputs, the Meps@Nlo algorithm has the same
sources of negative weights as the S-Mc@Nlo matching method described above.
In addition, when applied to LO inputs, the Meps@Nlo algorithm generates a
Born-local $K$-factor that enables the smooth merging between the matrix
elements of varying jet multiplicity [11, 10] and is a further source of
negative weighted events, because it is evaluated in a Monte-Carlo fashion.

##  4 Mechanisms to improve the negative weight fraction

This section will discuss three mechanisms to reduce the negative weight
fraction within Sherpa S-Mc@Nlo and Meps@Nlo events. It is worth noting, that
all improvements relevant for S-Mc@Nlo simulations will also benefit Meps@Nlo
simulations, since the former are components of the latter.

To define effective approaches for the reduction of negative weights, it is
instructive to separate their two sources in S-Mc@Nlo events as described in
Sec. 3. This is straightforward, because these events can appear as
$\mathbb{S}$ and $\mathbb{H}$ events, which correspond to negative weights
from subleading colour configurations and from the subtraction procedure,
respectively. As can be seen in Table 1, $\mathbb{S}$-events take up a major
share of the negatively weighted events for typical $Z$+jets and
$t\bar{t}$+jets setups111The exact setup details will be defined in Sec. 5..

| Positive | Negative
---|---|---
| Fraction | Fraction
$\mathbb{S+H}$ | 82% | 18%
thereof $\mathbb{S}$ | 88% | 58%
thereof $\mathbb{H}$ | 12% | 42%
(a) $pp\to Z$+0,1,2 jets@LO+3 jets@NLO

| Positive | Negative
---|---|---
| Fraction | Fraction
$\mathbb{S+H}$ | 75% | 25%
thereof $\mathbb{S}$ | 91% | 72%
thereof $\mathbb{H}$ |  09% | 28%
(b) $pp\to t\bar{t}$+0,1 jets@LO+2,3 jets@NLO

Table 1: Percentage of positive and negative weighted events and their
composition of $\mathbb{S}$ and $\mathbb{H}$-events for proton-proton
collisions at a centre-of-mass energy of 13 TeV.

###  4.1 Leading Colour Approximation

One possibility to reduce the negative weight fraction is to establish and
validate a leading colour and leading spin approximation in the matching of
NLO calculations to the parton shower. This approximation is motivated by the
fact that corrections to the large $N_{C}$-limit in processes with two color
charged particles at the leading order are typically suppressed by
$1/N_{C}^{2}$, which would lead to corrections of the order of 10%. This
suppression is particularly relevant in the context of the standard candles
$pp\to W/Z$ at the LHC, and in the context of Higgs-boson production through
gluon fusion and the associated production $pp\to HW^{\pm}$ and $pp\to HZ$, as
well as the corresponding Standard Model backgrounds.

Using the S-Mc@Nlo algorithm, as described in Section 3, in a leading colour,
leading spin mode amounts to dropping the analytic weights introduced in
connection with the colour correct one-step parton shower [8], and restricting
the emitting color dipoles to the ones that are present in the large-$N_{c}$
limit. This requires a projection of the hard matrix element onto the possible
partial amplitudes, and selection of a configuration according to their
magnitude squared. The precise details are given in [25, 26].

We note that the elimination of spin and color correlations from the S-Mc@Nlo
procedure can be motivated theoretically by the fact that the parton shower is
activate in the region of phase space which is dominated by large logarithms.
The correct description of spin and color effects in this region would in
principle require to include them in the parton-shower simulation at all
orders. The quality of the matched simulation should therefore not be affected
significantly when a single-emission effect is altered. However, the actual
phase space covered by the parton shower does typically extend far into the
non-logarithmic region (e.g. $p_{T,V}\sim m_{V}$ in $pp\to V$), and therefore
the effects included through S-Mc@Nlo may play a non-negligible role. While
the structure of color singlet production at hadron colliders effectively
excludes any significant effect due to the factorization of the color matrix
(see for example [21]), more complex Born processes may require a different
treatment [27]. We therefore recommend the careful validation of the leading
colour, leading spin approximation on a process-by-process and observable-by-
observable basis. In this context it is important to have the complete
S-Mc@Nlo algorithm available.

###  4.2 $\mathbb{H}$-Event Shower Interplay

The $\mathbb{H}$-events in S-Mc@Nlo matched simulations are identified as hard
corrections to $\mathbb{S}$-events, and they do not exhibit any singularities
in the single soft or double collinear regions of the phase space. In a multi-
jet merged approach, it is therefore necessary to apply a Sudakov reweighting
to an $n$-jet $\mathbb{H}$-event between the scales of the zeroth and the
$n$-th jet, but a reweighting between the scale of the $n$-th jet and lowest
clustering scale is not strictly required. If applied, such a reweighting
corresponds to treating the $\mathbb{H}$-events like leading-order
contributions to a multi-jet merged prediction, rather than finite remainders
[10]. The additional Sudakov suppression reduces their cross section in the
soft region of phase space, and consequently reduces the negative weight
fraction. It neither influences the formal accuracy of the matched result nor
the negative weight fraction in the high-$p_{T}$ region. However, it is
preferred in order to achieve a continuous transition to higher-multiplicity
Born-level predictions. The implementation of the reweighting has also been
discussed in the context of Mc@Nlo matching at fixed multiplicity [19].

###  4.3 Core local $K$-factor

The negative weight fraction at next-to-leading order is increasing when going
to higher multiplicities in the final-state due to the higher complexity of
the processes. This fact can be exploited to reduce the negative weight
fraction by reducing the cases where high-multiplicity NLO matrix elements are
evaluated without being phenomenologically relevant.

In the context of multijet merging employing the Meps@Nlo method, a
differential factor $k_{n}^{(\mathcal{A})}$ is introduced for the purpose of a
smooth merging between the matrix elements of varying jet multiplicity at NLO
and LO [11, 10]. This K-factor is originally applied for high-multiplicity LO
matrix element events (e.g. $V$+4 jets) by clustering them with a reverse
shower algorithm until they correspond to a multiplicity for which an NLO
matrix element exists (e.g. $V$+2 jets, corresponding to
$k_{n=2}^{(\mathcal{A})}$).

One can instead determine $K^{(\mathcal{A})}$ from the lowest underlying core
process, e.g. $V$+0 jets ($k_{n=0}^{(\mathcal{A})}$). Given the smaller
negative weight fraction of this low-multiplicity process, this is expected to
further reduce $\varepsilon$ in higher transverse momentum regions where the
high-multiplicity LO matrix elements contribute most.

Electroweak virtual corrections can induce significant effects in the high
transverse momentum region of $V$+jets final states and similar processes.
These corrections originate in the emergence of electroweak Sudakov logarithms
[28], which have recently been implemented in Sherpa [29]. They can also be
modeled very efficiently in the EW${}_{\text{virt}}$ approximation [30].
Defining local $K$-factors based on the lowest multiplicity final state has
the drawback that such corrections cannot be applied. Consequently, using
$k_{n=0}^{(\mathcal{A})}$ may become undesirable if electroweak corrections
are relevant, and the applicability of the approximation needs to be assessed
depending on the process and observable under consideration.

##  5 Results

This section presents results obtained for the previously introduced
mechanisms with the aim of reducing the amount of negatively weighted events.
Two process setups are investigated, namely

  * $pp\to Z(\to e^{-}e^{+})$+0,1,2 jets@NLO+3 jets@LO

and

  * $pp\to t\bar{t}$+0,1 jets@NLO+2,3 jets@LO,

where the notation specifies, which jet multiplicity processes are simulated
at NLO accuracy and which ones are simulated at LO accuracy within the sample
merged according to the Meps@Nlo prescription [10]. Both processes are studied
at a centre-of-mass energy of $\sqrt{13}$ TeV using the NNPDF 3.0 NNLO PDF set
with $\alpha_{S}$ = 0.118. The merging cut $Q_{\mathrm{cut}}$ is set to 20 GeV
in the case of $Z$ production and to 30 GeV for $t\bar{t}$+jets. All results
are obtained using version 2.2.8 of Sherpa with virtual contributions provided
by OpenLoops version 1.3.1 [31]. In the following, all displayed uncertainties
correspond to the statistical Monte Carlo uncertainty.

Analyses of the physics performance and the negative weight fractions are
performed using the Rivet framework [32]. In the case of $Z$ production, an
invariant mass of $66~{}\mathrm{GeV}<m_{e^{-}e^{+}}<116~{}\mathrm{GeV}$ is
required in the analysis. Furthermore, jets are defined using the anti-$k_{t}$
clustering algorithm [33] with $R=0.4$ and $p_{T}>20$ GeV.

###  5.1 $Z$+jets

The study of vector boson production in association with jets plays an
important role in the ongoing physics program of the LHC, where they can be
measured with high precision and thus enable stringent tests of the Standard
Model. Furthermore, these processes contribute as background to some of the
most important searches for physics beyond the Standard Model, e.g. dark
matter or supersymmetric signatures. As a result, vector boson plus jet
processes constitute a vast amount of the Monte Carlo samples produced for the
ATLAS and CMS experiments at the LHC, making the reduction of negatively
weighted events for these processes a priority.

| Negative Weight Fraction
---|---
Default | 18.1%
Leading Colour Mode | 14.0%
\+ shower veto on $\mathbb{H}$-events |  09.6%
\+ local K-factor from core |  09.1%
Table 2: Evolution of the mean negative weight fraction for $pp\to Z$+0,1,2
jets@LO +3 jets@NLO.

Table 2 shows the evolution of the negative weight fraction $\varepsilon$ for
$Z$+jets using the approaches introduced in the previous section. As can be
seen, $\varepsilon$ is reduced by roughly a factor of two, from 18.1% down to
9.1%. A reduction of negative weights is only useful if the precision of the
physics prediction is not altered. Figures 2 and 3 show a comparison of the
discussed approaches for various Monte Carlo validation observables. There,
the lower two ratio plots display the negative fraction $\varepsilon$ and the
corresponding factor $f(\varepsilon)$, which indicates how many more events
are needed on average in order to achieve the same statistical accuracy as
compared to only positively weighted events in each bin.

Let us first review the physics performance of the various simplifications.
Various levels of agreement between the nominal and negative-weight-reduced
predictions can be observed. The leading-colour approximation performs very
well and reproduces the nominal prediction at the %-level, certainly much
better than the naive expectation of a 10% accuracy. The additional Sudakov
veto on $\mathbb{H}$-events causes a rate reduction of roughly $5\%$ at
intermediate values of $p_{\perp}^{Z}$ but otherwise the prediction is not
significantly modified. The core local $k$-factor does not show a strong
effect, implying a high level of universality between the $Z+n$-jet NLO
predictions. In summary, the observed deviations in all methods are small
compared to the systematic uncertainties that have to be taken into account in
such predictions, e.g. from perturbative scale variations or ambiguities in
the shower algorithm.

The leading colour mode reduces the number of negative weights in the low
$p_{T}$ region, while it leaves $\varepsilon$ unchanged for large transverse
momenta. The additional Sudakov suppression on $\mathbb{H}$-events further
decreases the negative fraction for low to intermediate $p_{T}$, in the region
where $\mathbb{H}$-events already contribute and resummation effects are still
relevant. Defining the local K-factor on the core process lowers $\varepsilon$
mainly for large transverse momenta, as expected in this region, where multi-
leg processes are dominant and taken into account through LO matrix elements.

It is essential for Monte Carlo event generators to describe the data as good
as possible. Measurements of the production of a $Z$ boson in association with
jets in proton-proton collisions at a centre-of-mass energy of 13 TeV using
3.16 fb-1 of data, collected by the ATLAS experiment at the LHC [34], are
compared to Sherpa in Figure 4. The Monte Carlo predictions are in good
agreement with the measurement, while again reducing the negative weight
fraction significantly by employing the above discussed methods.

(a) Transverse momentum of $Z$ $p_{\perp}^{Z}$ (b) Transverse momentum of
leading jet $p_{\perp}$(jet 1)

Figure 2: Monte Carlo validation observables comparing the different
mechanisms to reduce the number of negative weighted events, as discussed in
Section 4, for $pp\to Z$+0,1,2 jets@LO+3 jets@NLO. The lower two ratio plots
display the negative weight fraction $\varepsilon$ and the corresponding
factor $f(\varepsilon)$, which indicates how much more events need to be
generated on average in order to achieve the same statistical accuracy as
compared to only positive weighted events.

(a) Scalar sum of jet transverse momenta HT (b) Azimuthal separation between
jets $\Delta\phi$(jet 1, jet 2)

Figure 3: Monte Carlo validation observables comparing the different
mechanisms to reduce the number of negative weighted events, as discussed in
Section 4, for $pp\to Z$+0,1,2 jets@LO+3 jets@NLO. See Fig. 2 for details.

(a) Transverse momentum distribution of leading jet (b) Azimuthal separation
between jets $\Delta\phi$(leading, second jet) (c) $H_{T}$ distribution for
all jets (d) Jet multiplicity

Figure 4: Comparison of the different mechanisms as described in Section 4 to
experimental $Z$+jets data measured by ATLAS [34]. See Fig. 2 for details.

###  5.2 $t\bar{t}$+jets

The investigation of top quark pair production and decays in association with
jets is crucial in the context of LHC physics. This process is one of the
dominant backgrounds in almost all searches for new physics at the LHC, since
it can also produce many jets in the final state and a high amount of missing
transverse energy. For such large background samples it becomes particularly
relevant to reduce the negative weight fraction.

| Negative Weight Fraction
---|---
Default | 24.8%
Leading Colour Mode | 18.7%
\+ shower veto on $\mathbb{H}$-events | 14.5%
\+ local K-factor from core | 12.6%
Table 3: Evolution of the mean negative weight fraction for $pp\to
t\bar{t}$+0,1 jets@LO +2,3 jets@NLO.

As can be seen from Table 3, regular $t\bar{t}$+jets samples contain
approximately $25\%$ of events with negative weights. This fraction can be
approximately halved using the mechanisms as discussed in Section 4. In
Figures 5 to 7, the different approaches aiming for the reduction of
$\varepsilon$ are compared for various Monte Carlo validation observables.

Again, and despite now studying this non-trivial coloured core process, we
find an excellent performance of the leading-colour approximation, which
agrees with the nominal prediction in all observables. The same holds for the
shower veto on $\mathbb{H}$ events, which does not lead to a deviation from
the nominal prediction. A significantly stronger impact is observed in the
setup where the local $k$-factor is calculated from the inclusive $t\bar{t}$
process. Especially distributions of the transverse momenta of jets are
sensitive and yield a slope of up to 10% over the ranges studied here,
resulting in significantly softer distributions. It is important to keep in
mind, that especially these LO-accurate regions are plagued by large
perturbative uncertainties [35, 36], significantly larger than the deviations
found through the negative-weight reduction methods here.

A look at the differential impact on the negative weight fractions shows that
the leading colour approximation reduces $\varepsilon$ in the soft region, and
the shower veto on $\mathbb{H}$-events decreases the negatively weighted
events in the soft to intermediate region even further. In the high $p_{T}$
region, $\varepsilon$ can be lowered only by defining the local K-factor using
the lowest multiplicity process at NLO.

Figure 8 compares the Monte Carlo results to measurements of top quark pair
production in association with jets for $pp$ collisions at a centre-of-mass
energy of 13 TeV collected by the ATLAS detector, corresponding to an
integrated luminosity of 3.2 fb-1 [37]. In particular, the transverse momentum
of hadronically decaying top quarks as well as of the $t\bar{t}$-system are
investigated, where all tested methods are in fair agreement with the data.

(a) Transverse momentum distribution for jet 1 (b) Transverse momentum
distribution for $b$-jet 1

Figure 5: Monte Carlo validation observables comparing the different
mechanisms to reduce the number of negative weighted events, as discussed in
Section 4, for $pp\to t\bar{t}$+0,1 jets@LO+2,3 jets@NLO. See Fig. 2 for
details.

(a) $H_{\text{T}}$ distribution for all jets (b) Azimuthal separation between
$b$-jets $\Delta\phi(b_{1},b_{2})$

Figure 6: Monte Carlo validation observables comparing the different
mechanisms to reduce the number of negative weighted events, as discussed in
Section 4, for $pp\to t\bar{t}$+0,1 jets@LO+2,3 jets@NLO. See Fig. 2 for
details.

(a) Mass distribution for reconstructed top quark (b) Mass distribution for
$W$ bosons

Figure 7: Monte Carlo validation observables comparing the different
mechanisms to reduce the number of negative weighted events, as discussed in
Section 4, for $pp\to t\bar{t}$+0,1 jets@LO+2,3 jets@NLO. See Fig. 2 for
details.

(a) $p_{T}$ of hadronically decaying tops (b) $p_{T}$ of the $t\bar{t}$-system

Figure 8: Comparison of the different mechanisms as described in Section 4 to
experimental $t\bar{t}$+jets data measured by ATLAS [37]. See Fig. 2 for
details.

##  6 Conclusions

We have presented methods to improve the computational footprint of Sherpa
S-Mc@Nlo and Meps@Nlo simulations. The focus lies on a reduction of negatively
weighted events, since these drastically reduce the statistical power of event
samples of a given size and thus also impact upon the computational cost of
the experimental detector simulation.

The sources of negative weights in the S-Mc@Nlo matching algorithm have been
illustrated and their relative contributions quantified. Three mechanisms to
reduce negative weights have been proposed. Some methods correspond to
approximations which are valid within the claimed precision of the sample but
can change the physics predictions of the simulation.

These proposals have been validated in the two most relevant processes, the
production of a vector boson in association with jets and the production of a
top-quark pair. The physics performance has been studied in typical
observables to quantify the size of the changes induced. No critical
differences were identified. At the same time, these methods were able to
reduce the negative weight fraction by a factor of approximately two in the
inclusive phase space. With the resulting negative weight fractions of
$\varepsilon\approx 10\%$ the generation of large event samples becomes
feasible again, which is particularly important in light of future high-
luminosity LHC runs.

## Acknowledgments

We are grateful to our colleagues from the Sherpa collaboration for many
fruitful discussions. FS and KD were supported by the German Research
Foundation (DFG) under grant No. SI 2009/1-1. This research was supported by
the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No.
DE-AC02-07CH11359. We thank the Center for Information Services and High
Performance Computing (ZIH) at TU Dresden for generous allocations of
computing time.

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