On the maximal use of Monte Carlo samples: reweighting events at NLO accuracy
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Abstract
Accurate Monte Carlo simulations for highenergy events at CERN’s Large Hadron Collider, are very expensive, both from the computing and storage points of view. We describe a method that allows to consistently reuse partonlevel samples accurate up to NLO in QCD under different theoretical hypotheses. We implement it in MadGraph5_aMC@NLO and show its validation by applying it to several cases of practical interest for the search of new physics at the LHC.
Keywords
Transverse Momentum Monte Carlo Parton Shower Effective Field Theory Leading Order1 Introduction
The search of new physics is one of the main priorities of the LHC. The recent observation of an anomaly in the diphoton spectra [1, 2] gives hope that we might have a first evidence of Beyond Standard Model (BSM) physics very soon. In that case, we would only be at the beginning of a long program of investigations of what the underlying physics is. In any case, searches of new particles or modifications of the interactions among the SM particles will continue as well as progress associated to our ability to provide precise predictions to be compared with data.
In the recent years, efforts have focussed on providing accurate theoretical predictions for a large number of BSM models at Leading Order (LO), in the form of event generators. First, various programs such as FeynRules [3], LanHep [4] or Sarah [5] have automated the extraction of the Feynman rules from a given Lagrangian. Secondly several matrix element based generators like MadGraph5_aMC@NLO [6] (referred to as MG5_aMC later on), Sherpa [7] or Whizard [8] have extended the class of BSM model they support with extensions in various directions: high spins, high color representations and any kind of Lorentz structure [9, 10, 11]. More recently, automated NexttoLeading Order (NLO) prediction (in QCD) for BSM models are available thanks to the NLOCT [12] package of FeynRules which adds in the model the additional elements (R2 and UV counterterms) required by loop computations.
It is now possible to generate Monte Carlo sample for a large class of BSM theories at LO and for an increasing number at NLO accuracy. Even though technically possible, producing samples for many models and benchmark points down to full detector level at the high luminosity expected at the LHC would require an unmanageable number of computing and storage resources. However, the stages of a simulation (partonlevel generation, partonshower and hadronisation, detector simulation, and reconstruction) are independent and factorise. Therefore changes in local probabilities happening at very short distance, i.e. from BSM physics, decouple from the rest of the simulation stages. This is particularly interesting since the slowest part of the simulation is the full simulation of the detector.
A logical possibility therefore arises: one can generate large samples under a SM or basic BSM hypothesis and then continuously and locally deform the probability functions associated to the distributions of partonlevel events in the phase space by changing the “weight” of each event in a sample to account for an alternative theory or benchmark point. Under a nottoorestrictive set of hypotheses which are easy to list, such an eventbyevent reweighting can be shown to be exactly equivalent (at least in the infinite statistic limit) to a direct generation in the BSM. Note that such an eventbyevent reweighting is conceptually different from the very common yet very crude method where events are reweighted using a pivotal onedimensional distribution. Eventbyevent reweighting is a common practice in MC simulations, yet currently it has been only publicly available at LO [13, 14] or available at NLO for very specific cases (e.g. [15]) or in methods where NLO accuracy is far from ensured [16, 17]. It is the aim of this work to show that a consistent (and practical) reweighting of events can also be done at NLO accuracy.
The plan of this paper is as follows. Before introducing the NLO reweighting method, we will focus on the LO case in order to explain the intrinsic limitations of such types of methods (Sect. 2). In Sect. 3, we present three types of NLO reweighting, two of them correspond to methods already introduced in the literature [14, 18]. The third one is the NLO accurate reweighting method introduced here for the first time. In Sect. 4, we present some validation plots performed with MG5_aMC. We then present our conclusions in Sect. 5.
2 Reweighting at the leading order
 Helicity The helicity state of the external states of a partonlevel event is optional in the LHEF convention, yet some programs (e.g. [21]) use this information to decay the heavy state with an approximated spincorrelation matrix. In this case it is easy to modify Eq. 1 to correctly take into account the helicity information by using the following reweighting:where \(M^{h}_{new}^2\) and \(M^{h}_{orig}^2\) are the matrix elements associated to the event for a given helicity h – the one written in the LHEF – and for the corresponding theoretical hypothesis. This reweighting is allowed since the total crosssection is equal to the sum of the individual polarized crosssections.$$\begin{aligned} W_{new} = \frac{M^{h}_{new}^2}{M^{h}_{orig}^2}W_{orig}, \end{aligned}$$(4)

Colorflow A second piece of information presented in the LHEF is the color assignment in the large \(N_c\) limit. This information is used as the starting point for the dipole emission of the parton shower and therefore determines the result of the QCD evolution and hadronisation. Such information is untouched by the reweigthing limiting the validity of the method. For example, it is not possible to reweight events with a Higgs boson, with a process where the Higgs boson is replaced by a colored particle. One could think that, as for the helicity case, one could amend the reweighting formula to be able to handle modifications in the relative importance between various flows. While possible in principle, in practice such reweighting would require to store additional information (the relative probabilities of all color flows in the old model) in the LHEF, something that does not seem practical.

Internal resonances In presence of onshell propagators, the associated internal particle is written in the LHEF. This is used by the partonshower program to guarantee that the associated invariant mass is preserved during the reshuffling procedure intrinsic to the showering process. Consequently, modifying the mass/width of internal propagator should be done with caution since it can impact the partonshower behaviour. This information can not be corrected via a reweighting formula, as it links in a nontrivial way shortdistance with longdistance physics.
3 Next to leading order reweighting
In this section, we will present three reweighting methods for NLO samples. First we will present a LO type of reweighting that we dubbed “Naive LOlike” reweighting introduced in VBFNLO (i.e. REPOLO [17]) and MadSpin [22, 23]. As it will become clear later, this method is not NLO accurate and should be used only if the difference between the two theories factorizes from the QCD production. The second method that we propose is original and consists in a fully accurate and general NLO reweighting. Finally, we present the “loopimproved” reweighting method [18] to perform approximate NLO computation for loopinduced processes when the associated twoloop computations are not available.
3.1 Naive LOlike reweighting
3.2 NLO reweighting
3.3 Loop improved reweighting
4 Implementation and validation
The various methods of reweighting discussed in the previous section have been implemented in MG5_aMC and are publicly available starting from version 2.4.0. At the LO, the default reweighting mode is based on the helicity information present in the event (Eq. 4), while for NLO samples, the default reweighting mode is the NLO accurate one (Eq. 16). Fixedorder NLO generation can not be reweighted since no event generation is performed in this mode. A manual of the code is available online at the following address: https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/Reweight.
In this section, we will present four validation examples covering the various types of reweighting introduced in the previous section. Since the purpose of this section is mainly to validate our method, the details of the simulation used (cuts, type of scale, ...) are kept to a minimum. Unless otherwise stated, the settings used correspond to the default value of MG5_aMC (version 2.4.0). In particular the minimal transverse momentum on jet is of 20 GeV at LO and of 10 GeV at NLO.
4.1 ZW associated production in the effective field theory at the LO
In Fig. 1 we present the differential distributions for the transverse momenta of the Z boson at LO accuracy. Starting from a sample of Standard Model events (black solid curve), we have reweighted our sample to get the SM plus the interference term with the dimension six operator for two values of the associated coupling: \(c=50\, \text {TeV}^{2}\) (dashed blue) and \(c=500\, \text {TeV}^{2}\) (dashed green). This second value is clearly outside the validity region for the EFT approach as the differential distributions turns to be negative at low transverse momentum. Nevertheless, having such large effects is interesting for the validation of the reweighting method. The same differential distributions are generated with MG5_aMC (solid green and blue) and validates the reweighting method.
The ratios between the differential curves obtained with each method are presented in the second inset. This inset contains also the statistical uncertainty (yellow band) for the ratio of two independent SM samples. The compatibility of those two ratio plots with the expected statistical fluctuation validates our approach/code implementation. The first inset presents the ratio between the EFT and SM predictions. It shows that the method works correctly for quite small and quite large modifications of the differential distributions.
4.2 ZH associated production in the effective field theory at NLO
From the comparison of the two methods for the HD curve in the plot of the transverse momenta (top plot), we can observe that the statistical fluctuations are more pronounced for the curve obtained by reweighting. This is an example of enhancement of statistical uncertainty due to the reweighting as discussed around Eq. 3 since in the high \(p_T\) region, the HDder is suppressed compare to the other theories under consideration (HD and SM).
4.3 Effective field theory (\(t\bar{t}Z\)) at NLO
In Fig. 3, we present the transverse momentum of the Z boson in the associated production of this boson with a top/antitop quark pair. We present the result for both the full matrix element squared (labelled \(\sigma ^{(2)}\)) and for the SM contribution plus the interference with the dimension 6 operator only (labelled \(\sigma ^{(1)}\)).
4.4 Higgs plus one jet production at LO and NLO order
In this last example, we will present results for the associated production of a SM Higgs with one jet. In Fig. 4, we present the transverse momentum of the Higgs at both LO and NLO accuracy. For the LO case, we present three curves. The first one is the curved obtained within the heft model [36] featuring the dimension five operator obtained by integrating out the top quark (HEFT LO). The second line (SM LO/RWGT) is the one obtained by reweighting the previous curve by the full one loop matrix element squared which contain the complete topquark mass dependence. The last LO curve is the one obtained via direct integration of the oneloop amplitude squared by MG5_aMC [37] (SM LO). At NLO accuracy, we have the curve in the infinite top mass limit (HEFT NLO) using the Higgs characterization model [34]. This sample is then reweighted by the fullloop (Loop Improved) following the loopimproved method presented in the previous section. It is so far not possible to compute the NLO contribution directly in order to compare the accuracy of such method.
5 Conclusion
We have presented the implementation of several methods that can be used for reweighting LO and NLO samples and discuss the associated intrinsic limitations. We have released a new version of MG5_aMC that allows the users to employe the various reweighting methods presented in this paper in a fully automatic and userfriendly way. In particular we have introduced for the first time an NLO accurate reweighting method and compared it with the approximate methods available in the literature. Other reweighting methods like the Naive LOlike and the loopimproved are for the first time available in a public code.
The comparison between the various methods shows that the approximate method (the Naive LOlike reweighting) performs a satisfactory job. This indicates that the non locality of the MC counter terms is often more a theoretical problem than a contribution spoiling the NLO accuracy of the Naive LOlike reweighting. Therefore the Naive LOlike reweighting should be a good approximation in a quite large class of model/observable either when the virtual contribution is subdominant and/or when the effect of the BSM physics factorises. Consequently, we recommend phenomenologist to first test the Naive LOlike reweighting and in case of loss of accuracy move forward to the slower NLO method. On the other hand for mass production at the LHC, where the samples are often used for more than one study, we recommend to always use the NLO accurate method.
The framework introduced here is flexible enough to accommodate different types of reweighting approaches. In the near future we plan to capitalise on this to allow different type of functionalities. First we plan to implement a standard systematics uncertainty computation module as it is done in [16, 25, 38, 39, 40, 41, 42]. Compared to the existing module of MG5_aMC [25], this new module will allow to perform this determination independently of the event generation which will be extremely useful to evaluate the effect of a new PDF set/test a new scale scheme on existing samples. In a second stage, we plan to be able to compute the systematics uncertainty for the reweighted BSM sample at the time of the reweighting.
Footnotes
 1.
For the simplicity of the discussion, we will always consider that the sum of the weights is equal to the total crosssection of the sample.
 2.
The normalisation choice implies that the phasespace factor \( \Omega _{PS}\) is proportional to \(N^{1}\) where N is the number of phasespace points used to probe the phase space.
 3.
For intermediate particle a small variation of the mass – order of the width – is reasonable.
 4.
We also use the same (MC) counter terms as described in that paper.
 5.
Due to the presence of multiple couter terms, the kinematic configuration on which the matrixelement is evaluated is not unique: an implicit sum over such kinematical configurations is assumed here and in the rest of the paper.
 6.
One can notice that \(\mathcal {W}_{\beta ,V}^{\alpha } = \mathcal {W}_{\beta ,R}^{\alpha } = 0 \) for \(\beta = R,F\) due to the use of the EllisSexton scale [6].
 7.
There would also be quadratic contribution if we include the squared matrix element associated to the dimension six operator.
 8.
For non definite positive quantity the same idea holds by using \(\max _i (W^i_{orig})\).
Notes
Acknowledgements
I would like to thank all the authors of MG5_aMC for their discussions, help and support at many stages of this Project. I would like also to thank C. Degrande, R. Frederix and F. Maltoni to have read and comment on this manuscript, E. Vryonidou, F. DeMartin, I. Tsinikos, V. Hirschi for their help during the validation of this implementation. O.M. is supported by a Durham International Junior Research Fellowship. This work is supported in part by the IISN “MadGraph” convention 4.4511.10, by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P7/37, by the European Union as part of the FP7 Marie Curie Initial Training Network MCnetITN (PITNGA2012315877), and by the ERC Grant 291377 LHCtheory: Theoretical predictions and analyses of LHC physics: advancing the precision frontier.
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