Reweighting QCD matrixelement and partonshower calculations
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Abstract
We present the implementation and validation of the techniques used to efficiently evaluate parametric and perturbative theoretical uncertainties in matrixelement plus partonshower simulations within the Sherpa eventgenerator framework. By tracing the full \(\alpha _s\) and PDF dependences, including the partonshower component, as well as the fixedorder scale uncertainties, we compute variational event weights onthefly, thereby greatly reducing the computational costs to obtain theoreticaluncertainty estimates.
1 Introduction
The first operational run of the LHC collider during the years 2009–2013 was a tremendous success, clearly culminating in the announcement of the discovery of a Higgsboson candidate by the ATLAS and CMS collaborations in July 2012 [1, 2]. Through a large number of experimental analyses, focusing on a variety of final states and observables, the LHC experiments (re)established and underpinned to an unprecedented level of accuracy the validity of the Standard Model of particle physics (SM) [3].
When comparing theoretical predictions with actual collider data, MonteCarlo event generators prove to be an indispensable tool. In particular partonshower MonteCarlo programmes like Herwig [4, 5], Pythia [6] and Sherpa [7, 8] provide simulations at the level of exclusive particlelevel final states [9]. The cornerstones of these generators are their implementations of QCD partonshower algorithms and their modelling of the nonperturbative partontohadron fragmentation process. With the advent of sophisticated techniques to combine partonshower simulations with exact higherorder QCD calculations at leading [10, 11], nexttoleading [12, 13] and even nexttonexttoleading order [14, 15, 16, 17], MonteCarlo simulations have developed into highprecision tools, encapsulating the best of our current knowledge of perturbative QCD.

Parametric uncertainties reflecting the dependence of the prediction on input parameters such as couplings, particle masses or the partondensity functions (PDFs).

Perturbative uncertainties originating from the fact that perturbation theory is used in making predictions, to fixedorder in the matrix elements and resummed to allorders with a certain logarithmic accuracy in the showers, thereby, however, neglecting higherorder contributions. Similarly, the use of the large\(N_c\) approximation in the showers belongs in this category.

Algorithmic uncertainties corresponding to the actual choices made in the implementation of the shower algorithm, i.e. for the evolution variable, the inclusion of nonsingular terms in the splitting functions, or the employed matching/merging prescription. Per construction, for sensible choices, these systematics also correspond to higherorder perturbative corrections, but might be addressed separately.
This publication focuses on the efficient evaluation of parametric and (some) perturbative uncertainties in matrixelement plus partonshower simulations within the Sherpa eventgenerator framework. We present a comprehensive approach to fully trace the \(\alpha _s\) and PDF dependences in the matrixelement and partonshower components of particlelevel Sherpa simulations in leading [21] and nexttoleading [22] order merged calculations based on the Sherpa dipoleshower implementation [23]. Furthermore, we provide the means to quickly evaluate the renormalisation and factorisationscale dependence of the fixedorder matrixelement contributions. Our approach is based on eventwise reweighting and allows us to provide with a single generator run a set of variational event weights corresponding to the predefined parameter and scale variations, that would otherwise have to be determined through dedicated reevaluations. The alternative event weights can either be accessed through the output of a HepMC event record [24], or directly passed via the internal interface of Sherpa to the Rivet analysis framework [25].
The systematics of leadingorder partonshower simulations with Herwig 7 have recently been discussed in [18], a corresponding reweighting procedure has been presented in [26]. A similar reweighting implementation for the Pythia 8 parton shower has also appeared recently [27]. A discussion of uncertainty estimates for the Vincia shower model can be found in [28, 29, 30]. A comprehensive comparison of various generators is presented in [31]. The impact of PDFs in partonshower simulations has been discussed in [32, 33].
Our paper is organised as follows. In Sect. 2 we review the dependence structure of leadingorder (LO) and nexttoleadingorder QCD calculations on \(\alpha _s\), the PDFs and the renormalisation and factorisation scales, and introduce the reweighting approach. In Sect. 3 we extend this to partonshower simulations and in particular the algorithm employed in the Sherpa framework. In Sect. 4 we present the generalisation of the reweighting approach to multijetmerged calculations, based on leading and nexttoleadingorder matrix elements matched to the parton shower. Our conclusions are summarised in Sect. 6. In Appendix A we present CPU time measurements that assess the reduction in computational time when the reweighting is used. The technical details on enabling and accessing the variations considered in Sherpa runs are listed in Appendix B.
Note, while fixedorder reweighting is already available with Sherpa2.2, the general reweighting implementation described here, including parton showers and multijet merging, will be part of the next release, i.e. Sherpa2.3.
2 Reweighting fixedorder calculations
In order to reevaluate a QCD crosssection calculation for a new choice of input parameters, i.e. \(\alpha _\mathrm{s}\), PDFs or renormalisation and factorisation scales, it is necessary to understand and traceout its respective dependences. This is a rather easy task at leadingorder (LO) but is already more involved when considering nexttoleading order (NLO) calculations in a given subtraction scheme. However, these decompositions have been presented for Catani–Seymour dipole subtraction and the FKS subtraction formalism in [34, 35].
In this section, we briefly review the dependence structure and discuss the corresponding reweighting equations for LO and Catani–Seymour subtracted NLO calculations within the Sherpa framework. With this paragraph we also introduce the notation used in the later sections, which explore the reweighting of more intricate QCD calculations, involving QCD parton showers and merging different finalstate multiplicity processes.
2.1 The leadingorder case
2.2 The nexttoleadingorder case
The remaining pieces of Eq. (5) are the Born matrix element \(\text {B}\), the real emission contribution \(\text {R}\) and the differential dipole subtraction terms \(\text {D}_{S,j}\). The latter defines an underlying Born configuration \(\Phi _{B,j}\) through its dipoledependent phasespace map, employing the phasespace factorisation \(\Phi _R=\Phi _{B,j}\cdot \Phi _1^j\). While the transformation of \(\text {B}\) under the exchange of input parameters was detailed in Eq. (3), the transformation of \(\text {R}\) and the \(\text {D}_{S,j}\) contributions works identically, merely having to adjust the power of the strongcoupling factor.
2.3 Validation
The reweighting approach outlined above has been implemented in the Sherpa framework for the two matrixelement generators Amegic [38] and Comix [39, 40] in conjunction with the corresponding Catani–Seymour dipolesubtraction implementation [41]. The required decomposition of virtual amplitudes is generic and can be used for matrix elements from BlackHat [34, 42], OpenLoops [43], GoSam [44], Njet [45], the internal library of simple \(2\rightarrow 2\) processes, or, via the BLHA interface [46].
The treatment of partonic thresholds deserves a short discussion. While any flavour thresholds in the running of \(\alpha _s\) do not present any challenges to the reweighting algorithm as \(\alpha _s(\mu ^2)>0\) for all \(\mu ^2>0\) and any loop order, this is different for the PDFs, where crossing a parton threshold results in a vanishing PDF for that flavour. Hence, the cross section component of the given partonic channel may be zero if no other nonzero contribution exists. Such an event will be discarded and, thus, cannot be reweighted. If now the respective parton threshold of the target PDF is smaller than the target factorisation scale while the one of the nominal PDF is larger than the nominal factorisation scale we are in a region of phase space where the reweighting must fail to reproduce a dedicated calculation. This could be remedied by storing events as well which vanish solely due to crossing PDF thresholds. However, as only observables sensitive to onthreshold production of light quarks (typically bottom quarks) are susceptible to these effects, they are of little relevance to the vast majority of LHC observables.^{2}
Variations, which are used for studies in this publication, with two variants depending on the PDF choice. Note that each \(\alpha _\mathrm{s}(m_Z^2)\) value is used with its associated PDF set variant in the context of hadronic collisions
Nominal  Variations  Error band  

PDF sets  CT14  56 Hessian error sets with a 90% CL  Hessian 
NNPDF3.0  100 statistical replicas with a 68% CL  Statistical  
\(\alpha _\mathrm{s}(m_Z^2)\) value  0.118  0.115, 0.117, 0.119, 0.121  Envelope 
\(\mu _R\)/\(\mu _F\) factors  \(\left( 1, 1\right) \)  \(\left( \frac{1}{2}, \frac{1}{2}\right) \), \(\left( 1, \frac{1}{2}\right) \), \(\left( \frac{1}{2}, 1\right) \), \(\left( 2, 1\right) \), \(\left( 1, 2\right) \), \(\left( 2, 2\right) \)  Envelope 
Comparing the uncertainties for the W\(p_\perp \), we observe that the scale uncertainties are the largest, with relative deviations of \(\mathcal {O}\left( {10}\%\right) \). The relative deviations related to the PDF and the strong coupling do not exceed \(\sim \)3%. The scale uncertainty exhibits a minimum for \({100}{~{\text {TeV}}}< p_\perp ^{\mathrm{W}} < {200}{~{\text {TeV}}}\). The reason is that the variations of \(\mu _F\) alone cross the central value prediction in this range, such that only the \(\mu _R\) variation contributes to the overall scale uncertainty here.
Note that \(p_\perp ^{\mathrm{W}} = 0\) at \(\mathcal {O}\left( \alpha _\mathrm{s}^0\right) \), and therefore only realemission events contribute to the distribution. Hence, the observable is only described to leadingorder. We introduce it here as a reference for our later validations including the partonshower, which use this observable. For the current validation, we complement the discussion of the Wtransverse momentum with the one of the lepton it decays to, as the region below \(m_{\mathrm{W}}/2\) is already filled at \(\mathcal {O}\left( \alpha _\mathrm{s}^0\right) \), and therefore we have in part a true nexttoleading description for this observable. In fact, the scale uncertainties are much larger in that region, especially towards the \(m_{\mathrm{W}}/2\) threshold, and at the lepton \(p_\perp \) cut at 25 \({\text {TeV}}\). This gives a more realistic picture of the perturbative uncertainties than in the leadingorder region above the threshold.
The small panels on the right of Fig. 1 compare the uncertainty bands calculated using the reweighting approach to uncertainty bands where dedicated calculations have been done for each variation. We observe that all bands overlap perfectly for both observables. This is because the reweighting as presented above is exact and for all runs the same phasespace points could be used: The reweighted and the dedicated predictions for each variation are therefore equal, and so are the uncertainty bands.^{3}
3 Reweighting partonshower calculations
3.1 Partonshower dependence structure
Definition of the evolution and splitting variables for each dipole type. The fifth column lists the splitting process as seen from the Born process, c and \(c'\) refer to the flavour of the initial state before and after the splitting process, respectively. The variables \(y_{ij,k}\), \(\tilde{z}_i\), \(x_{ij,a}\), \(x_{jk,a}\), \(x_{j,ab}\), \(u_j\) and \(v_j\) are defined in [23, 36, 37]
Type  \(\displaystyle z\)  \(\displaystyle y\)  \(\displaystyle x\)  \(\displaystyle (ij,k)\rightarrow (i,j,k)\)  \(\displaystyle c,c'\) 

FF  \(\displaystyle \tilde{z}_i\)  \(\displaystyle y_{ij,k}\)  \(\displaystyle 1\)  \((ij,k)\rightarrow (i,j,k)\)  \(\displaystyle a,a\) 
FI  \(\displaystyle \tilde{z}_i\)  \(\displaystyle \frac{1x_{ij,a}}{x_{ij,a}}\)  \(\displaystyle x_{ij,a}\)  \((ij,a)\rightarrow (i,j,a)\)  \(\displaystyle a,a\) 
IF  \(\displaystyle x_{jk,a}\)  \(\displaystyle \frac{u_j}{x_{jk,a}}\)  \(\displaystyle x_{jk,a}\)  \((aj,k)\rightarrow (a,j,k)\)  \(\displaystyle aj,a\) 
II  \(\displaystyle x_{j,ab}\)  \(\displaystyle \frac{\tilde{v}_j}{x_{j,ab}}\)  \(\displaystyle x_{j,ab}\)  \((aj,b)\rightarrow (a,j,b)\)  \(\displaystyle aj,a\) 
3.2 Reweighting trial emissions
To numerically integrate Sudakov form factors typically the Sudakov Veto Algorithm is used [50, 51, 52, 53, 54, 55]. Therein the integrands \(\text {K}\) found in the Sudakov form factors are replaced with integrable overestimates \(\hat{\text {K}}\). This is balanced by only accepting a proposed emission with probability \(P_\text {acc}= \text {K}/ \hat{\text {K}}\). A multiplicative factor in \(\text {K}\) is therefore equivalent to a multiplicative factor in \(P_\text {acc}\) [52]. This observation is for example used to apply matrixelement corrections [54], where the splitting kernels are replaced with a realemissionlike kernel \(\text {R}/\text {B}\). This is done aposteriori, i.e. the event weight is multiplied by \((\text {R}/\text {B})/\text {K}\), the emission itself is unchanged. The same method is also used in the Vincia parton shower to calculate uncertainty variations for different scales, finite terms of the antenna functions, ordering parameters and subleading colour corrections [28]. Here we employ this technique to account for variations of the strongcoupling parameter and the PDFs in the shower evolution of LO and NLO QCD matrix elements.
3.3 Nexttoleadingorder matching
3.4 Validation
To validate the reweighting of scale and parameter dependences in CSShower and SMc@Nlo calculations within the Sherpa framework we perform closure tests between reweighting results and dedicated simulations.
Our implementation allows to constrain the maximum number of reweighted shower emissions per event. For a pure leadingorder partonshower run or \(\mathbb {H}\)like events in SMc@Nlo calculations this amounts to setting \(n_\text {PS}\in \{0,1,2,\ldots ,\infty \}\). When considering SMc@Nlo simulations in addition the parameter \(n_\textsc {NloPs} \in \{0,1\}\) can be used to disable the reweighting of the \(\mathcal {O}(\alpha _\mathrm{s})\) emission for \(\mathbb {S}\)events.
Of course the reweighting result will only coincide with a dedicated calculation if all emissions are reweighted, i.e. \(n_\textsc {NloPs} =1\) and \(n_\text {PS}=\infty \). However, by subsequently enabling the reweighting of more and more emissions the relevance of their dependences for the determination of the full uncertainty can be studied. A finite value of \(n_\text {PS}\) can also be useful in production, if the effect of reweighting higherorder emissions becomes negligible. The reduced amount of reweighting per event then allows for a faster event generation. An additional benefit would be that rare highmultiplicity shower histories do not spoil the statistical convergence of the reweighted result, even if their exact kinematics might be irrelevant for the studied observable.
3.5 The finalstate only case: Thrust in e\(^{+}\) e\(^{}\rightarrow \mathrm{q}\bar{\mathrm{q}}\) events
To validate LoPs reweighting, we consider two observables, which are complementary in their sensitivity to partonshower emissions. At first, we consider the eventshape variable thrust T [59] in hadronic events in \(e^+e^\)collisions at \({\sqrt{s}={91.2}{~{\text {TeV}}}}\). In this case QCD emissions are restricted to the finalstate. Accordingly, there appear no PDF factors in the shower reweighting, cf. Eq. (16), and thus no factorisation scale dependence. Moreover, as we consider the leadingorder matrix element for \(e^+e^\rightarrow q\bar{q}\) only, the renormalisation scale is also absent in the hardprocess component. Therefore, we can concentrate on the pure \(\alpha _\mathrm{s}\) uncertainty in the parton shower here. Leaving the perturbative order of its running invariant it is defined by its value at the input scale \(m_Z\).
However, for low values of T, the statistical fluctuations of the reweighting results with higher \(n_\text {PS}\) grow larger, corresponding to a widening of the distribution of reweighting factors. Low values of the thrust observable correspond to the emission of several hard partons, which is less probable in the partonshower approximation, and more appropriately modelled in multijetmerged calculations, cf. Sect. 4. In this phasespace region it is difficult for the reweighting to compensate the multitude of accepted soft emissions off these hard legs, that turn unstable for \(P_\text {acc}\rightarrow 1\), with rejected ones, cf. Eq. (17). This issue can be addressed by introducing a prefactor for the overestimator function \(\hat{\text {K}}\) in the reweighting runs, to ensure that \(P_\text {acc}\) does not approach 1, cf. [26, 27, 52]. This renders the Sudakov Veto Algorithm somewhat less efficient, but is shown to reduce statistical fluctuations in the reweighting.
3.6 The initialstate dominated case: \(p_\perp ^{\mathrm{W}}\) in pp \(\rightarrow \mathrm{W}[\mathrm{e}\nu ]\) events
The second observable considered to validate our CSShower and SMc@Nlo reweighting implementation is the Wboson transversemomentum distribution \(p_\perp ^{\mathrm{W}}\) in 13 \({\text {TeV}}\) protonproton collisions, that has already been used in Sect. 2.3 in the NLO case. The definitions for constructing the uncertainty bands used there are kept the same, and are stated in Table 1. We now use the CT14nlo PDF set, which uses a Hessian error representation at a 90% confidence level [60].^{5} Therefore the PDF error band will be larger than before, as it now corresponds to nearly two standard deviations instead of only one.
We now turn to the scale uncertainty band – which is entirely due to factorisation scale variations, because the LO matrix element is independent of \(\alpha _\mathrm{s}\) , and therefore the band underestimates the perturbative uncertainty. We also observe that the band is nearly flat. As we vary only the scales of the matrixelement calculation, the constant spread corresponds to the factorisationscale uncertainty of the Born configuration at \(p_\perp ^W=0\), merely propagating to higher \(p_\perp ^W\) bins through the parton shower, which is unaware of the scale variations. In the matrixelement reweighting, we can guarantee the same phasespace points as in the dedicated run, such that we see perfect agreement between dedicated and reweighted predictions. We therefore omit comparisons for different \(n_\text {PS}\) for the scaleuncertainty band.
Looking at the \(\alpha _\mathrm{s}\) uncertainty band, we can see that the envelope constricts at the position of the peak of the distributions. This reflects that the variation of \(\alpha _\mathrm{s}\) shifts the position of the peak, such that variations that are below the nominal distribution on the left side of the peak, are exceeding the nominal distribution on the right side, and vice versa. Comparing the reweighted prediction to the dedicated one, we find a flat band for \(n_\text {PS}=0\), corresponding to restricting the reweighting to the fixedorder matrix element. As the LO calculation is independent of \(\alpha _\mathrm{s}\), this only reflects the change of the PDFs, which are fitted to \(\alpha _\mathrm{s}(m_{\mathrm{Z}})\). The reproduction of the shape of the \(\alpha _\mathrm{s}\) uncertainty improves a lot when reweighting up to one emission (\(n_{\text {PS}}=1\)), and slightly more when adding another emission on top (\(n_{\text {PS}}=2\)).
For the PDF uncertainty, we see that the reweighting with \(n_\text {PS}=0\) underestimates it by at least 1–5% for small transverse momenta, and overestimate it around the Wmass. As for the \(\alpha _\mathrm{s}\) uncertainty, this improves for \(n_\text {PS}=1,2\).
The last depicted step, i.e. \(n_{\text {PS}}=3\), on the other hand, does not contribute further to the reproduction of the \(\alpha _\mathrm{s}\) and PDF uncertainties. No significant differences with respect to the \(n_\text {PS}=2\) case is observed. It can be concluded that it is sufficient to include up to two emissions to reproduce the uncertainty bands for this observable.
This is to be expected, as the gauge boson recoils against the shower emissions and is therefore mostly affected by the few hardest branchings. These mainly originate from the incoming hard virtual partons, so the generally softer finalstate emissions barely contribute. Although we do not reproduce this here, we confirmed this by entirely disabling finalstate emissions, which showed no effect on the results.
To assess the quality of the reweighting, we consider again different settings for the parameters \(n_\textsc {NloPs} \) and \(n_\text {PS}\). Assuming \(n_\textsc {NloPs} =n_\text {PS}=0\), only the scale variations of the hard process are considered and the partonshower contribution to the \(\mathcal {O}(\alpha _\mathrm{s})\) correction is not reweighted. Furthermore, we present results for \(n_\textsc {NloPs} =1\) and \(n_\text {PS}=0,1,2\). With these settings, the \(\mathcal {O}(\alpha _\mathrm{s})\) corrections get properly reweighted, but the number of subsequent shower emissions off the \(\mathbb {S}\) and \(\mathbb {H}\)like events treated correctly is varied. We observe a saturation for reproducing the dedicated calculations at \(n_\textsc {NloPs} + n_\text {PS} \ge 2\), with no further improvement when \(n_\text {PS}\) is increased from 1 to 2. This confirms the findings made when considering the LoPs setup in Fig. 3: the gaugeboson transversemomentum distribution is dominated by the few hardest emissions.
4 Reweighting multijetmerged calculations
In this section we address the reweighting of multijetmerged event generation runs. These approaches allow to combine LO or NLO QCD matrix elements of different multiplicity dressed with parton showers into inclusive samples. Accordingly, the production of jets associating a given core process can be modelled through exact matrix elements rather than relying on the logarithmic approximation of the parton shower only. In particular, when considering hard jet kinematics or angular correlations such techniques prove to be indispensable to properly describe experimental observations, see for instance [61, 62, 63, 64].
To first approximation the reweighting as described in the previous sections can be used without change, only that the perturbative order p is no longer a constant across the sample, but varies for each event, corresponding to the considered matrixelement parton multiplicity. However, there are also new algorithmspecific intricacies which complicate the dependence on the input parameters and need to be dealt with to allow for a consistent reweighting. The LO and NLO merging techniques employed within the Sherpa framework are presented in [21, 65, 66], respectively. They rely on the reconstruction of partonshower histories for multiparton amplitudes that set the partonshower initial conditions for their subsequent evolution. This is achieved by running a backwardclustering algorithm that identifies a corresponding core process and calculates hardparton splitting scales that serve as predetermined shower branchings. In the Sherpa approach the actual parton shower then starts off the reconstructed core process and implements the predetermined hard splittings based on a truncated shower. Furthermore, it is the purpose of the truncatedshower evolution to implement possible Sudakov vetoes for shower emissions above the phasespace separation or merging scale \(Q_\text {cut}\). It should be emphasised here, partonshower reweighting is vital when using modified input parameters in order to cancel the \(Q_\text {cut}\) dependence to the accuracy of the parton shower. In case only the hardprocess matrix element parameters get reweighted, the dependence on \(Q_\text {cut}\) is cancelled to leadinglogarithmic accuracy only, however, residual subleading contributions from the running coupling or the PDF evolution remain [65].
For the reweighting of the truncatedshower Sudakov veto probability the methods described in Sect. 3 can be applied. In what follows we will detail the specifics of the reweighting procedure for LO and NLO multijetmerging runs with Sherpa supplemented by an extensive validation of the implementation.
4.1 Preliminaries
The sequence \(\{t_i\}\) of reconstructed branching scales then may be either ordered or unordered, with an ordered history satisfying \(t_j<t_{j1}<\cdots<t_1<t_0=\mu _{F,\text {core}}^2\). The recombination probabilities in each clustering step are determined by the forwardsplitting probabilities and are therefore dependent on the parton shower and its parameters and choices. This is reflected, stepbystep, in the addition of one factor of \(\alpha _\mathrm{s}\) (when appropriate) at the reconstructed splitting scale, a ratio of PDFs at the reconstructed initial flavours and their momentum fractions, and a Sudakov form factor describing the evolution of each step.
In the Sherpa implementations the \(\alpha _\mathrm{s}\) and PDF factors are added explicitly onto the respective matrix elements and can therefore be reweighted directly. The Sudakov form factor, on the other hand, is implemented through a vetoed truncated parton shower [21, 22]. The truncated shower itself, accounting for the possibility of soft partonshower emissions between subsequent reconstructed hard emissions, i.e. with \(t_m<t<t_{m1}\) but \(Q<Q_\text {cut}\), can be reweighted with the methods described in Sect. 3. If, however, an emission with \(Q>Q_\text {cut}\) occurs the event is vetoed. Practically, this is accounted for through increasing \(n_\text {trial}\) of the next accepted event by \(n_\text {trial}\) of the vetoed event. Thus, \(n_\text {trial}\) becomes dependent on the partonshower parameters.
A special remark concerning unordered histories, i.e. histories whose sequence of \(\{t_i\}\) has at least one pair \(t_k\ge t_{k1}\), is in order. Such histories can be encountered in various configurations, e.g. when the last clustering step produces a splitting scale larger than the nominal starting scale of the core process^{7} or the flavour structure only allows further clusterings at scales \(t_{k1}\) lower than the last identified one \(t_k\).^{8} As such configurations cannot be generated by a strictly ordered parton shower, for each unordered step neither the accompanying PDF ratio nor Sudakov factor is therefore present in the calculation. More than one unordering in a cluster history of a given event is possible and in fact likely at high multiplicities. PDF ratios and Sudakov factors then of course only occur in the ordered subhistories in between the unorderings. For the sake of clarity and brevity we will omit this case from the discussion of the following subsections. Its implications to the algorithm, and therefore to the reweighting, are straightforward. If ordered histories are enforced, core configurations beyond the standard \(2\rightarrow 2\) processes occur. Independent of the presence and number of unorderings, the renormalisation and factorisation scales \(\mu _R\) and \(\mu _F\) are always set in the way, as will be detailed below.
4.2 The leadingorder case
Apart from the Sudakov form factors the softcollinear structure of the \(\text {B}^\text {merge}_j\) is now identical to the emission of j partons off a \(\text {B}_0\) configuration with the parton shower described in Sect. 3. In case of finalstate splittings the ratio of parton distribution functions is simply unity as neither the partonic \(x_{a/b,i}\) and \(x_{a/b,i1}\) nor the initialstate flavours \(a_i,\,b_i\) and \(a_{i1},\,b_{i1}\) differ. In principle, with every ratio of PDFs there is also a ratio of flux factors. However, all such factors cancel except for the outermost ones, corresponding to \(\Phi _j\) and, hence, are regarded as part of \(\text {B}^\prime _j\).
4.3 The nexttoleadingorder case
4.4 Validation
The reweighting for multijetmerged calculations as discussed in the previous sections has been implemented within Sherpa with the CSShower for leadingorder matrix elements (MePs@Lo), nexttoleading order matrix elements (MePs@Nlo) and nexttoleadingorder matrix elements with additional leadingorder ones on top (MeNloPs). For the validation, we again perform closure tests between reweighted and dedicated predictions for the transverse momentum of the Wboson in Figs. 6, 7 and 8. As before, the uncertainty bands are the ones defined in Table 1, with the CT14 PDF set. For all merged calculations, we employ a merging cut of \(Q_\text {cut}={20}{~{\text {TeV}}}\).
The same is true in the MePs@Nlo case depicted in Fig. 8. A direct comparison of the scale uncertainties to the MeNloPs case is not straightforward though, as we combine NLO matrix elements for the 0 and the 1jet multiplicity, where the virtual amplitudes are obtained from BlackHat [42]. Hence, the 2jet multiplicity is described at leading order through the 1jet \(\mathbb {H}\)events. As such, the setup is not a simple upgrade from our MeNloPs calculation.
In all multijetmerging validations, we find a similar behaviour with respect to the imprint of including emissions in the reweighting. For \(n_{\textsc {NloPs}} + n_\text {PS}= 2\), the dedicated calculations are well reproduced, and no further improvement is found for \(n_{\textsc {NloPs}} + n_\text {PS}= 3\). It is noteworthy, that for the MeNloPs case we find a worse reproduction for \(n_{\textsc {NloPs}} = 1\) and \(n_\text {PS}= 0\) compared to the NloPs and the MePs@Nlo cases. This originates in the fact that in the latter two cases, we enable the reweighting of emissions off \(\mathbb {S}\)events at all involved multiplicities, whereas in the MeNloPs case only the first of the three multiplicities is affected, because the other two are at LO and therefore do not have \(\mathbb {S}\)events. Thus, the overall importance of the \(\mathbb {S}\) emission reweighting gets restricted to the region below \(Q_\text {cut}\) of the 1jet configuration in the MeNloPs case.
5 Consistent variations
In general, the renormalisation and factorisation scales, \(\alpha _s\) and the PDFs should be varied consistently throughout any of the presented calculations. While at fixed order the situation is clear, the matched and merged approaches allow for some degree of freedom regarding partial variations while still retaining their respective accuracies.
In the simplest case, LOPS, \(\mu _R\) and \(\mu _F\) of the short distance cross section and the parton shower may be varied independently as these variations can be expressed as higherorder terms in a perturbative expansion in the coupling parameter \(\alpha _s\). This is not the case for \(\alpha _s\) itself and the PDFs as they are fixed through measured input values and parametrisations. Changes in these input values cannot be expressed as simple higherorder terms. Thus they need to be chosen consistently throughout.
Similarly, in NLOPS calculations, the renormalisation and factorisation scales may be varied in the matrix element (\(\overline{\text {B}}\) and \(\text {H}_A\)) or the parton shower (\(\text {PS}_{\textsc {NloPs}}\) and \(\text {PS}\)) separately without losing neither the fixedorder nor the resummation accuracy. As the pseudosubtraction through the \(\text {D}_A\) in any case employs different scales in \(\text {PS}_{\textsc {NloPs}}\) and the \(\overline{\text {B}}\) and \(\text {H}_A\) functions, it always leaves remainders of \(\mathcal {O}(\alpha _s^2)\). Hence, further scale variations in either one, the shortdistance cross sections or the \(\text {PS}_{\textsc {NloPs}}\), do not worsen the nominal accuracy of the method. Retaining the logarithmic accuracy of the parton shower on the other hand requires identical renormalisation and factorisation scales throughout all resummationrelevant components, i.e. \(\text {PS}_{\textsc {NloPs}}\) and \(\text {PS}\). Again, variations in \(\alpha _s\) or the PDFs need to be consistent throughout the calculation.
The multijetmerged calculations impose further constraints since they treat multijet matrix elements and partonshower emissions on the same footing. The notation of the scales already reflects this for \(\mu _R\) and \(\mu _F\). In their definitions only the core scales remain as free parameters and may be varied independently. Again, the \(\alpha _s\) and PDF parametrisations need to be the same throughout.
6 Conclusions
In this publication we have presented the implementation and validation of reweighting techniques allowing for the fast and efficient evaluation of perturbative systematic uncertainties in the Sherpa eventgenerator framework. We have lifted the available techniques for the determination of PDF, \(\alpha _s\) and scale uncertainties in leading and nexttoleading order QCD calculations to include the respective variations in partonshower simulations. In turn we provide the means to perform consistent uncertainty evaluations for multijetmerged simulations based on leading or nexttoleadingorder accurate matrix elements of varying multiplicity matched with parton showers. The foundation for our reweighting method is the knowledge of the very dependence structure of the perturbative calculations on the parameters to be varied. For the fixedorder components this amounts to the corresponding decomposition of the Catani–Seymour dipole subtraction terms. This needed to be supplemented by the reweighting of the parametric dependences of the parton shower treated through the Sudakov Veto Algorithm.
With our extensive validation we have been able to prove on the onehandside the correctness of the implementation and have, furthermore, been able to illustrate the importance of partonshower reweighting for reliable uncertainty estimates. With comparably little additional computational costs this allows for the onthefly determination of PDF, \(\alpha _S\) and scale uncertainties based on one single generator run, that, otherwise, would require explicit recomputations. The overall reduction in CPU time is by a factor of about 3–20, depending on the eventgeneration mode used, see Appendix A. The variational event weights provided are easily accessible through the HepMC event record and are furthermore consistently handed over to the Rivet analysis software by the corresponding Sherpa interface.
The methods presented in this publication are ideally suited for eventwise uncertainty estimates and can readily be used in arbitrary theoretical and experimental analyses. An extension to nexttonexttoleadingorder QCD calculations possibly dressed with parton showers, as presented in [16, 17, 71], is straightforward and planned for the near future.
The decomposition of the fixedorder part of QCD calculations employed here is also a necessary ingredient to produce crosssection grids as provided by the APPLgrid [72] and FastNLO [73, 74] tools. These store the perturbative coefficients for a certain observable calculation discretised in \(Q^2\) and x. Using interpolation methods, this allows for the a posteriori inclusion of PDFs, \(\alpha _\mathrm{s}\) and variations of the renormalisation and factorisation scales. In turn, such techniques are well suited for (combined) fits of PDFs and \(\alpha _\mathrm{s}\) that require a multitude of recomputations of the theoretical predictions. Over the last years, tools have been developed that automate the projection of arbitrary nexttoleadingorder QCD calculations onto such grids, namely the aMCfast [75] and the MCgrid [76, 77, 78] packages. The first one produces APPLgrid s with MadGraph5_aMC@NLO [79], the latter APPLgrid s or FastNLO grids from Sherpa events projected on the observables through Rivet. The APFELgrid tool [80] provides an improved convolution method for use with APPLgrid files that furthermore speedsup the reevaluations.
However, none of these approaches includes generic partonshower effects, i.e. the parametric dependence of the shower component on the PDFs and \(\alpha _\mathrm{s}\) is ignored. With the methods presented in this publication we are confident that we can surmount this limitation and in the future provide interpolation grids that properly reflect showerresummation effects and allow for the inclusion of the affected phasespace regions in PDF determinations.
Footnotes
 1.
The two parameters \(c_R^{\,\prime \,(i)}\) correspond to the single and double pole coefficients of the loop matrix element while the remaining sixteen coefficients are comprised of eight pairs of coefficients, \(\bar{c}_{F,a/b}^{\,(i)}\) and \(\tilde{c}_{F,a/b}^{\,(i)}\), corresponding to the \(\mu _F\)dependent and independent parts for all four flavour structures of each beam, respectively.
 2.
A typical example for the threshold problem in the reweighting would be the very low\(p_\perp \) part of bjet spectrum in Wb production in a calculation with five massless flavours. Any strong dependence on the bottom quark PDF threshold, however, also indicates the invalidity of a calculation with five massless flavour for this observable.
 3.
The reweighted and the dedicated calculations are implemented independently, such that their predictions can vary within the numerical uncertainties of the calculation. However, these lie several orders of magnitude below the physical uncertainties considered here.
 4.
Although the emission scales can not be reweighted themselves using the presented method, the input scales of the strong coupling and the PDFs can be changed, as indicated in the text. We focus on constant prefactors here, but the functional form can also be changed, although the overall functional form of \(k_{\alpha _\mathrm{s}}t\) should be restricted to the CMWlike rescaling [56].
 5.
The reason for switching from NNPDF to CT14 PDFs is the strict positivity of the latter. The CSShower rejects emissions when negative PDF values are involved, a behaviour which spoils the reweighting in regions where the original and the target PDF do not have the same sign. The deviations seem to be small in practical applications, but here we chose to establish closure in a clean context first.
 6.
An example here is the interpretation of a e\(^{+}\) e\(^{}\rightarrow \mathrm{gd}\bar{\mathrm{d}}\) configuration. Its matrix element does not contain terms/diagrams that allow the quarkantiquark pair to be clustered.
 7.
An example here is the interpretation of a \(\mathrm{g}q\rightarrow \mathrm{Z}_\mathrm{q}\) configuration. In regions of large transverse momenta of the final state parton its identified branching scale \(t_1\) is larger than the starting scale \(t_0\) of the core process \(q\bar{q}\rightarrow \mathrm{Z}\), usually defined as the Zvirtuality.
 8.
An example here is the interpretation of a e\(^{+}\)e\(^{}\rightarrow \mathrm{d}\bar{\mathrm{d}}\mathrm{u}\bar{\mathrm{u}}\) configuration. In a first step there are only two choices to cluster, resulting in an identified branching scale \(t_2\). There now is a finite region in phase space where the gluon can (only) be clustered with scale \(t_1<t_2\).
 9.
If NLO matrix elements at higher multiplicities are needed for an event generation, the time needed for the integrator optimisation and the process selection weight optimisation can be quite substantial, e.g. a couple of days. In the case of unweighted event generation, this even has to be redone for every single parameter variation, as the channel weights are used for the unweighting. When reweighting is used, this is not necessary and so even more CPU time is saved.
Notes
Acknowledgments
We acknowledge financial support from BMBF under Contract 05H15MGCAA, the Swiss National Foundation (SNF) under Contract PP00P2128552 and from the EU MCnetITN research network funded under Framework Programme 7 Contract PITNGA2012315877.
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