Abstract
We perform a detailed study of the sources of perturbative uncertainty in partonshower predictions within the Herwig 7 event generator. We benchmark two rather different partonshower algorithms, based on angularordered and dipoletype evolution, against each other. We deliberately choose leading order plus parton shower as the benchmark setting to identify a controllable set of uncertainties. This will enable us to reliably assess improvements by higherorder contributions in a followup work.
Introduction
General purpose Monte Carlo (MC) event generators [1,2,3,4,5,6] are central to both theoretical and experimental collider physics studies. Recent development of these simulations has seen improvements in various areas, both within perturbative calculations, through matching to fixed order [2, 7,8,9,10,11,12,13,14,15], combining higher jet multiplicities [16,17,18,19,20,21,22], as well as the allorder resummation with parton showers [23,24,25,26] and also within the nonperturbative, phenomenological models [27, 28]. While there are well established prescriptions on how to quantify the theoretical uncertainty of fixedorder calculations due to missing higherorder contributions [29,30,31,32,33,34,35]^{Footnote 1}, there is no such consensus for general resummed calculations [36,37,38,39,40,41], and partonshower algorithms in particular [42,43,44,45,46,47], since a number of ambiguities are present within the different schemes; however, there is progress in towards this goal. Given the perturbative improvements, and the expected precision from datataking at Run II of the Large Hadron Collider [48, 49], the task of assigning theoretical uncertainties to MC event generators is becoming increasingly crucial. This also applies to validating new approaches against existing data, as well as using predictions to design future observables and/or collider experiments. Phenomenological studies, for example, indicate that MC event generators can be used even in primarily data driven methods to perform powerful analyses once theoretical uncertainties are under control [50, 51]. It is therefore important to quantify the uncertainties associated with an event generator in a reliable way.
Uncertainties due to nonperturbative modelling have been addressed in [52, 53], as well as the impact of the parton shower on reconstructed observables [54]. Various ambiguities and sources of uncertainty have been addressed within the context of other multipurpose event generators as well; in particular recoil schemes [55, 56] and parton distribution functions (PDF) [57,58,59] have so far been considered, both for pure showers and in the context of matched or merged samples, see e.g. [60,61,62]. The eigentune method [63] has been applied by both ATLAS [64] and CMS [65] to determine systematic tune variations. All of these studies share a commonality in that they focus on a single source of uncertainty which is usually connected to the development/improvement studied. Contrary to this, the authors in [66,67,68,69] describe possible approaches to uncertainty handling for the Drell–Yan process. An even more systematic approach for how to handle the possible interplay between theoretical and experimental uncertainties can be found in [70]. Further in the direction of a systematic approach, CMS published a short guide on how to estimate MC uncertainties [71] and outlined some issues to address. Finally, new techniques of propagating uncertainties through the parton shower by means of an alternate event weight were proposed [72, 73].
In the present work, we address uncertainties of partonshower algorithms within the Herwig 7 event generator [1, 2]. Herwig 7 is a general purpose event generator that computes any observable at nexttoleading order (NLO) precision in perturbation theory automatically matched to a parton shower. It includes sophisticated modules for very different physics aspects ranging from interfaces for physics beyond the standard model and two independent partonshower algorithms [55, 74], to a detailed modelling of multiple particle interactions [75,76,77].
It is our aim to develop a consistent uncertainty evaluation for event generators, and Herwig 7 in particular. This work is a first step in this direction, concerning the partonshower part and will be extended by further detailed studies in the context of higherorder improvements and the interplay with nonperturbative, phenomenological models and parameter fitting. The present paper is therefore structured as follows: in Sect. 2 we classify all different types of uncertainties and their respective sources. We then argue as to why we start with a pure leading order (LO) plus partonshower (PS) study. The sources of uncertainty tested in this study are described in detail in Sect. 3. Our results are presented in Sect. 4 for \(e^+e^\) and fully inclusive pp production, while our findings including additional jet radiation are described in Sect. 5. The results establish a baseline of a set of controllable uncertainties, which can then be used to quantify the impact of higherorder corrections to be addressed in upcoming work. Finally, we present a summary and outlook in Sect. 6.
Context
Sources of uncertainty
For any general purpose event generator that is based on both perturbative input and phenomenological models, there are a number of different sources of uncertainty to be addressed:

Numerical Computational precision and statistical convergence. This is clearly a limitation which can be overcome by investing enough computing resources and will hence not be addressed further.

Parametric Quantities taken from measurements or fits beyond the event generator parameters. This includes masses, coupling constants, and PDFs, and the impact of these needs to be quantified separately and potentially on a processbyprocess basis watching out for maximum sensitivity.

Algorithmic The actual partonshower algorithm, matching and merging prescriptions, and phenomenological models considered. The last are not considered here, as we limit ourselves to the simulation available in Herwig 7.

Perturbative Truncation of expansion series in coupling or logarithmic order. The main purpose of this work is to elaborate on quantifying these uncertainties in the case of leading order plus partonshower simulation, which will be motivated in more detail below.

Phenomenological Goodness of fit uncertainties (e.g. the socalled eigentunes) regarding parameters in the nonperturbative models. We will argue that a remaining spread of predictions obtained by fitting parameters for each of the variations of controllable perturbative uncertainties is able to quantify the cross talk to nonperturbative models and a genuine model uncertainty.
In this study we will address perturbative uncertainties in the partonshower algorithms as a first piece of the chain of variations to be done. We will use two different shower algorithms to benchmark the uncertainty prescriptions against each other and to point out further interesting differences. The results considered here will serve as further input to identify improvements of NLO matching and merging, to be addressed in a separate paper.
Phenomenological uncertainties will be subject to future investigations. However, we will point out first hints towards their influence by considering variations of the shower infrared cutoff in a selected number of cases. The reasoning to this is twofold: On one hand, we want to stress the fact that parton level studies should typically be carried out with care, and their region of validity can by estimated by cutoff variations with large changes that indicate nonnegligible hadronisation corrections. On the other hand, this fact also indicates how cutoff variations, along with other variations, may actually point to the possibility of quantifying otherwise unknown, generic, model uncertainties and the interplay with nonperturbative corrections.
Why leading order?
We solely consider LO plus PS simulation in this work. The motivation to do so is as follows: With fixedorder improvements it is clearly very hard to disentangle sources of uncertainty stemming from pure parton showering, and those which have been potentially improved by higherorder corrections. In order to quantify genuine partonshower uncertainties in an improved setting one would typically need to look at jets beyond those that received fixedorder hard process input (e.g. the second jet from a leading order configuration in a NLO matched simulation). Not only is this computationally unnecessary for the sake of studying only partonshower uncertainties, it also introduces slightly different shower dynamics, the differences of which, with respect to leading order, would also need to be quantified carefully. Additionally, it is our aim to show where and how fixedorder input improves the simulation along with the expected reduction in uncertainty; stated otherwise: To use the NLO matched simulation in order to identify which of the nonfirstprinciple variations considered in this work are indeed reliable estimators of theoretical uncertainty in the perturbative part of event generator predictions.
Different algorithms or uncertainties?
To quantify to what extent commonly used recipes are a sensible measure of uncertainties in partonshower algorithms the first step is a clear distinction of what possible sources exist within a fixed algorithm, and what differences should actually be attributed to the consideration of distinct algorithms. Looking at different algorithms, we obtain a strong crosscheck on whether the uncertainties assigned to one algorithm are sensible, provided we consider algorithms that exhibit similar resummation properties. We will also show that changes to the algorithms that are naively expected to be subleading, can cause severe difference in the resummation properties. Similarly, kinematics parametrisations to convert onshell partons to offshell ones after multiple radiation are known to cause numerically significant differences [55, 56, 78].
Such details, as well as the choice of splitting kernels and evolution variable should not be considered a source of uncertainty within an algorithm but are details that fix a distinct algorithm; we therefore call them algorithmic uncertainties. An uncertainty band based on varying such details cannot serve as a systematic framework to quantify missing higherlogarithmic contributions. If differences between algorithms are not covered by variation of the scales involved, either the estimate of uncertainty or the resummation properties of the algorithms should be questioned.
The relevant scales for the study at hand are:

the hard scale \(\mu _\mathrm{H}\) (factorisation and renormalisation scale in the hard process);

the veto scale \(\mu _\mathrm{Q}\) (boundary on the hardness of emissions);

the shower scale \(\mu _\mathrm{S}\) (argument of \(\alpha _S\) and PDFs in the parton shower).
No a priori prescription can be obtained as to what these scale choices should optimally be; the first two are usually taken as ‘a typical scale of the hard process’, while the last one faces more constraints to guarantee resummation properties and the correct backward evolution [79] within the parton shower. Having made a central choice, we vary the scales by fixed factors to generate subleading terms with coefficients of order one as an initial guess on higherorder corrections and phasespace effects.
At least two parameters in our shower algorithms are typically obtained in the course of tuning to data, the strong coupling \(\alpha _s(M_Z)\), and the shower cutoff parameter.^{Footnote 2} Using the different tuned values (at least with the latter having, in general, a different meaning between the two showers), the predictions on parton level will differ, though fully simulated, hadronic events, will yield a comparable description of data. We argue that these differences should be evaluated carefully, but belong to a future study that will address the interplay with nonperturbative models in more detail.
Simulation setup
We consider both partonshower modules available in Herwig 7, the default angularordered shower [74] and the dipoletype shower based on [8, 55]; in addition to their default settings, which we have adjusted to make them as similar as possible by choosing the same \(p_\perp \) cutoff and \(\alpha _s\) running (the ‘baseline’ settings for this work), we consider a number of modifications mainly outlined in Sect. 3, all of which constitute different algorithms in the sense outlined above. The two showers are very different in their nature: The angularordered, QTilde, shower evolves on the basis of \(1\rightarrow 2\) splittings with massive DGLAP functions, using a generalised angular variable and employs a global recoil scheme once showering has terminated; its available phase space is intrinsically limited by the angularordering criterion, resulting in a ‘dead zone’, though it is able to generate emissions with transverse momenta larger than the hard process scale and so typically an additional veto on jet radiation is imposed (see Sect. 3 for more details). The dipolebased shower, Dipole, uses \(2\rightarrow 3\) splittings with Catani–Seymour kernels with an ordering in transverse momentum and so is able to perform recoils on an emissionbyemission basis; the splitting kernels naturally require the two possible emitting legs of each dipole to share their phase space and there is no a priori phasespace limitation, but the available phase space is controlled by the starting scale of the shower.
Using the baseline, we find very similar predictions despite the very different nature of these algorithms. As an example we show in Fig. 1 the predictions for the Higgs \(p_\perp \) spectrum at an LHC with \(\sqrt{s}=13\ \mathrm{TeV}\). The only difference between the algorithms we observe in the very low \(p_\perp \) region where the interplay with the treatment of the remnant and intrinsic transverse momentum smearing becomes important. For future reference, we have also included results running the showers at their default settings to highlight what level of interplay with tuned values and nonperturbative models can be expected. To be more precise, we use a twoloop running, \(\overline{\text {MS}}\), \(\alpha _s\) including CMW correction [80] with \(\alpha _s^{CMW}(M_Z)=0.126\) (which corresponds to \(\alpha _s^{\overline{\text {MS}}}(M_Z)=0.118\)), and the MMHT2014 NLO PDF set [81] with five active flavours, interfaced through LHAPDF 6 [82], as far as initialstate radiation is concerned.^{Footnote 3} Hard processes are simulated at leading order (see the previous discussion), using the Matchbox infrastructure powered by amplitudes generated by MadGraph5_aMC@NLO [11]. In \(e^+e^\) collissions, we consider dijet production; at hadron colliders, in addition, we consider stable Zboson Drell–Yan production, \((e^+e^j)\) production within the mass window \(66~\mathrm {GeV}< m_{ll} < 116~\mathrm {GeV}\) around the Z mass, as well as production of a stable, \(125\ \mathrm{GeV}\), Higgs accompanied by zero or one jet. In the presence of additional jets in the hard process we use FastJet [83, 84] to perform the generation cuts; analyses are performed throughout using the Rivet framework [85], with analysis modules based on existing experimental and generic Monte Carlo implementations. In \(e^+e^\) collisions, where we choose a centreofmass energy of \(\sqrt{s} = 100\) GeV as baseline, we reconstruct jets with the Durham algorithm [86], while the hadron collider setup reconstructs anti\(k_\perp \) jets with a radius of \(R=0.4\) within a rapidity range \(y<5\) and a transverse momentum threshold of \(p_{\perp } > 20\ \mathrm{GeV}\). Parton level without multiple interactions and hadronisation is employed, and partons up to and including bquarks are treated as massless objects. Both parton showers mentioned above use a \(p_\perp \) cutoff prescription with a value of \(\mu _{\text {IR}}=1\ \mathrm{GeV}\). Electroweak parameters are kept at their default values.
Consistency checks
The ability to compare different algorithms puts us into the unique position of performing a number of consistency checks for the uncertainty estimate that we advocate. In particular, perturbative error bands should cover algorithmic discrepancies, if these algorithms are expected to deliver the same accuracy. If that is not the case then the algorithm at hand is questionable. Furthermore, by construction the shower approximates emissions in the soft and collinear region. If we force the shower to produce hard emissions, larger uncertainties are to be expected by a controllable prescription. Another point is the possibility of double counting hard emissions. The shower should not cover phasespace regions that are already covered by the hard process input. This property is typically reflected in demanding that observables that receive input at fixed order are not significantly altered by subsequent showering. Clearly, the definition of ‘region’, which in this case is covered by the veto scale on hard emissions (see Sect. 3 for a more detailed discussion), is again only precise to the level of accuracy covered by the parton shower and varying this boundary should serve as a measure of missing logarithmic orders. We emphasise that a boundary chosen to be far away from the correct ordering behaviour may introduce severe double counting issues, ultimately impacting on a resummation of a tower of logarithms which is not typical to the process, i.e. not encountered in any higherorder corrections to an observable considered. Furthermore, the perturbative uncertainties for observables in phasespace regions that do not receive logarithmically enhanced contributions should be driven by the hard scale alone, while the other scales have negligible impact. Logarithmically sensitive observables, on the other hand, should be altered by the parton shower and the uncertainties should be driven by all possible scale variations together. The setting where this is least clear is pure jet production, which we will address amongst other ‘jetty’ processes in Sect. 5.
Scale choices, variations and profiles
Phasespace restrictions and profile choices
The quantity central to parton showers is the splitting kernel. Its exponentiation gives rise to the Sudakov form factor, which regulates the divergence of the splitting kernel for soft and/or collinear emissions. On top of this, there are two further crucial ingredients (besides formally subleading, though not necessarily small issues like kinematic parametrisations): The evolution variable chosen, and the phase space accessible at a fixed value of the evolution variable. Emissions are typically further subject to an upper bound on their hardness. This cannot be directly deduced from a priori principles but should be chosen in the order of magnitude of the typical hardness scale of the process being evolved to avoid the double counting issues mentioned before.
The central point we are concerned with in this section shall be summarised in a simplified model of finalstate radiation. Quite generally, we have to consider three different scales: a hard scale \(K_\perp \) defining the phase space available to emissions at a fixed transverse momentum; a veto scale \(Q_\perp \) defining the maximum transverse momentum available to emissions; and the kinematic limit of transverse momentum, \(R_\perp \). We consider the \(p_\perp \) spectrum of a single soft emission with splitting kernel (possibly after an appropriate transformation of the evolution variable into a transverse momentum)
where \(C_i\) is the colour factor associated with the emitting leg. The longitudinal momentum fraction, z, has limits that read
in the presence of a hard scale \(K_\perp ^2\). With emissions weighted by \(\kappa \), an arbitrary function of a veto scale \(Q_\perp ^2\), we find a \(p_\perp \) spectrum of the form
with the Sudakov form factor
\(R_\perp \) denotes the scale that makes all of phase space available to emissions, while we denote the infrared cutoff by \(\mu _{IR}^2\) (we have not shown the zero \(p_\perp \), nonradiating event contribution). Once a hard cutoff \(\kappa (Q_\perp ^2,p_\perp ^2)=\theta (Q_\perp ^2p_\perp ^2)\) is chosen, this setup is known to reproduce the right anomalous dimensions. It has to be applied to a full evolution in a hierarchy \(Q_\perp ^2\rightarrow q_\perp ^2\) where \(K_\perp ^2=Q_\perp ^2\) is chosen and the form of the z boundaries being crucial to produce the correct logarithmic pattern [25, 55]. Instead, if one desires to make all of the phase space available to partonshower emissions, \(K_\perp ^2=R_\perp ^2\) is chosen and no other than the kinematic constraint \(p_\perp ^2<R_\perp ^2\) is in place.^{Footnote 4}
We have here considered the freedom of ensuring suppression of such emissions by an arbitrary function \(\kappa \). We call this weighting function a profile scale choice. One of the subjects of the present study is to identify sensible profile scale choices; we stress that such a choice is of algorithmic nature and not an intrinsic source of uncertainty. We will consider the following choices, depicted in Fig. 2:

theta: \(\kappa (Q_\perp ^2,q_\perp ^2) = \theta (Q_\perp ^2  q_\perp ^2)\), which is expected to reproduce the correct tower of logarithms;

resummation: \(\kappa (Q_\perp ^2,q_\perp ^2)\) is one below \((12\rho )\ Q_\perp \), zero above \(Q_\perp \), and quadratically interpolating in between. This profile is expected to reproduce the correct towers of logarithms, and switches off the resummation smoothly towards the hard region (currently we use \(\rho =0.3\) ^{Footnote 5}):
$$\begin{aligned}&\kappa (Q_\perp ^2,q_\perp ^2)= \left\{ \begin{array}{ll} 1 &{} \quad q_\perp /Q_\perp \le 12\rho \\ 1  \frac{(12\rho q_\perp /Q_\perp )^2}{2\rho ^2} &{} \quad q_\perp /Q_\perp \in (12\rho ,1\rho ] \\ \frac{(1q_\perp /Q_\perp )^2}{2\rho ^2}&{} \quad q_\perp /Q_\perp \in (1\rho ,1] \\ 0 &{} \quad q_\perp /Q_\perp > 1 \end{array} \right. ;\nonumber \\ \end{aligned}$$(5) 
hfact: \(\kappa (Q_\perp ^2,q_\perp ^2) = \left( 1+q_\perp ^2/Q_\perp ^2\right) ^{1}\), which is also referred to as damping factor within the POWHEGBOX implementation [7]; and

power shower: imposing nothing but the phasespace restrictions inherent to the shower algorithm considered.
Different combinations of \(R_\perp ^2\) and \(K_\perp ^2\) can be achieved within the two showers. In particular, the dipole shower is able to populate the region up to \(K_\perp ^2=R_\perp ^2\) (‘power shower’), while, for \(2\rightarrow 1\) processes at hadron colliders the angularordered phase space, by construction, imposes \(K_\perp ^2=Q_\perp ^2\) to be the mass of the singlet which is produced.
The leading logarithmic contribution of the z integration at this simple qualitative level is given by
We shall illustrate the impact of the profile scale choice \(\kappa \) on the Sudakov form factor by considering a fixed \(\alpha _s\), and evaluate
To obtain the desired resummation properties, namely
a number of limitations on \(\kappa \) and the other scale choices need to be imposed. Clearly, the limiting cases for small and large transverse momenta need to be reproduced;
While this is the case for all of the profiles we considered in this study, it is not sufficient to produce the desired tower of logarithms. Imposing the former restriction we still require that:

\(K_\perp ^2\sim Q_\perp ^2\) is imposed by the z boundaries; and

\(\kappa (Q_\perp ^2,q_\perp ^2) \sim \text {const}\) whenever \(q_\perp ^2\) is not of the order of \(Q_\perp ^2\) for the term involving the derivative of \(\kappa \) to become subleading.
Specifically the first restriction is only guaranteed by either the angularordered phase space which naturally imposes this restriction, or the restricted phase space chosen for the dipole shower.^{Footnote 6} The second restriction also excludes choices of \(\kappa \) providing a ratio of logarithms to effectively replace \(K_\perp ^2\) by \(Q_\perp ^2\) in the first term in Eq. 7. To this extent, we conclude that only those profiles that are narrow smeared versions (in the sense of varying only in a region where \(Q_\perp ^2/q_\perp ^2\) is of order one) of a thetatype cutoff will provide the proper tower of logarithms. Choices such as the resummation profile are desirable to avoid discontinuities introduced by the thetatype cutoff which are beyond the accuracy considered, while keeping the resummation properties of the parton shower; the profile we consider here is only one such kind, and there is no restriction on the exact form considered. The name ‘profile’ is chosen since the treatment of the hard scale we consider here closely resembles prescriptions on scale variations within the analytic resummation context [87].
Identifying a ‘Resummation Scale’
The hard veto scale \(Q_\perp ^2\) is the scale that, when considering transverse momentum spectra as outlined in the previous section, is closest in role to a resummation scale in analytical resummation. However, it is not typically the same as a shower starting scale. Though in our case this statement is true for the dipole shower, no such notion exists for the angularordered shower where typically the masses of the emitting dipoles set the shower starting scale owing to the angularordered phase space. In the latter case, emissions exceeding the transverse momenta of jets present in the hard process are possible and an additional veto on transverse momenta generated by the shower is applied; the value of this veto is, in this case, the analogue of the starting scale of the dipole shower. The resummation scale of the typical \(q_\perp \)resummation can thus not directly be related to an analogue hard scale present in different shower algorithms, especially when they evolve in a variable different from the transverse momentum and hence built up the full spectrum from multiple, differently ordered emissions.
For both the showers considered here, transverse momenta of parton shower emissions are expected to be limited or suppressed by the scale \(Q_\perp ^2\), which on very general grounds should thus be of the order of a typical scale of the hard process, i.e. the factorisation scale. As with fixedorder calculations the residual dependence on \(Q_\perp ^2\) is expected to become smaller as more and more logarithmic orders are incorporated. This implies a pattern of scale compensation through successive logarithmic orders, which a parton shower can typically only guarantee at the level of at most nexttoleading logarithms (NLL).
Scale variations
Having chosen a reasonable profile scale and value of \(K_\perp ^2\sim Q_\perp ^2\), the leading behaviour of the Sudakov exponent takes the wellknown form
where the highest level of accuracy one can hope for with coherent evolution is NLL accuracy, neglecting subleading colour correlations, at least for some observables and typically limited phasespace regions [80],
along with a twoloop running of \(\alpha _s\).
In addition to variations of the scales in the hard process, \(\mu '_{R/F}=\xi _H \mu _{R/F}\), we vary both the hard veto scale, \(\mu _\mathrm{Q}=\xi _Q Q_{\perp }\), and the arguments of \(\alpha _s\) and the PDFs in the partonshower splitting kernels, \(\mu _\mathrm{S}=\xi _S q_\perp \). We constrain the number of possible variations to be \(\xi \in [1/2,1,2]\). This spans a cube of, \(\xi _H \otimes \xi _S \otimes \xi _Q\), 27 combinations. All these choices are connected to logarithmic scale choices. There is therefore no a priori way of reducing their number. We emphasise that in principle only the full 27point envelope constitutes a comprehensive uncertainty measure. We therefore always produce the full envelope along with envelopes for each of the individual variations. Using this it is possible to observe which scale drives the overall uncertainty in a particular region of phase space. While one expects the variation of \(Q_\perp ^2\) to cancel out to the level of NLL accuracy (if this is indeed resembled by the parton shower), the situation is less clear for the other variation and different proposals have been made as to what extent the contribution at the level of NLL contributions should be cancelled (see e.g. [88] for a discussion) or otherwise considered as a probe of where precisely higher accuracy of the shower is missing. We do not consider introducing any terms that cancel these variations to the NLL order, and postpone a detailed analysis of this issue to future work. We do, however, analyse these variations as we are convinced that they are another clean handle on controlling where we expect, specifically, soft emissions and contributions by the hadronisation model to dominate. A recent Les Houches study [89] has also shown that, when not taking into account the full variations of this kind, discrepancies between different shower algorithms, which are expected to be similar, are not covered within these variations.
Reallife constraints
Besides the unclear definition of a resummation scale in the context of different shower algorithms, another word of caution needs to be raised when considering the hard veto scales: There are cases in which there is no meaningful variation as the hard scale is a fixed quantity such as the mass of an independently evolving finalstate emitting system, e.g. showers in \(e^+e^\rightarrow \) hadrons. It is not clear how one would quantify the respective shower uncertainty in this case, besides looking at shape differences encountered at different centreofmass energies of the \(e^+e^\) collider to quantify the scaling of the predictions with respect to ratios of the hard scale to the infrared sensitive quantity considered. Already this observation clearly marks the fact that no claim of a full and wellunderstood uncertainty recipe can be made at this point, but only are we able to perform initial steps in this direction. Similarly for the power shower there is no meaningful variation of \(\mu _\mathrm{Q}\). It can also happen, as is the case for the angularordered shower, that the algorithm chosen naturally imposes an upper bound on the hardness of the emission. In the case of Drell–Yantype processes, the angularordered shower, for example, will only allow for a downvariation of \(Q_\perp ^2\) and is thus questionable as to whether this variation in these cases is the right measure.
As with the small scales probed in the evolution of the parton shower, \(\mu _{R,F}\) variations in the shower may actually encounter regions where typically some cutoff or freezinglike behaviour is imposed to both, the running of \(\alpha _s\) and the parton distributions functions, which may result in interesting dynamics when variation of such small scales is used to infer uncertainties – a variation of the freezing prescription may thus be desirable, as well.
Clean benchmarks
To begin exploring the uncertainties that arise from the considerations of Sect. 3 we start by studying ‘clean benchmarks’, i.e. hard processes with the least number of legs: \(e^+e^\) annihilation, and Drell–Yantype \(2\rightarrow 1\) processes producing either a Z or Higgs boson. For the case of \(e^+e^\) collisions, the notion of a hard veto scale does not directly exist owing to the fact that the phasespace boundary and relevant hard scale coincide. However, we can compare variations of the collision energy and quantify this impact at the level of normalised distributions to acquire a handle on variations of the logarithmic structure similar to hadron–hadron collisions^{Footnote 7}. On top of the three scales \(\mu _\mathrm{H,S,Q}\) described above, we vary the infrared cutoff of the shower by a factor of 1 / 2 and 2 for the \(e^+e^\) setting, in order to obtain a first indication of how much dynamics of the shower is expected to be absorbed into hadronisation effects; notice that varying the argument of \(\alpha _s\) may serve a similar purpose.
Finalstate showers
\(e^+ e^ \rightarrow qq\) provides the clean environment to study finalstate radiation. Note that in this case the power and theta profile coincide, which is also our choice in the following.
The Thrust distribution, Fig. 3, shows good agreement between showers; this is true both for the central prediction and its variations, and shows that they possess the same resummation accuracy. Differences that do emerge between the showers are related to cutoff effects and nonradiating events in the region towards \(T=1\); these offer no insight into the resummation properties. A further difference emerges from the deadzone of the QTilde shower, however, this is a region that can be supplemented by using matching or ME corrections. For this observable we note that the \(\sqrt{s}\) and \(\mu _\mathrm{S}\) variations are similar in magnitude.
In Fig. 4 we show results for the integrated twojet rate; the uncertainties are dominated by \(\sqrt{s}\) as well as cutoff variations at small \(y_\mathrm {cut}\). Again, the overall uncertainties are comparable between the showers; as expected, we obtain large uncertainties in the small \(y_\mathrm{cut}\) region, which is dominated by hadronisation effects.
Initialstate showers
As far as initialstate showering is concerned, we investigate a gluoninitiated process \(pp \rightarrow H\) (in the large\(m_t\) effective theory), and a quarkinitiated process \(pp \rightarrow Z\); these particles are set stable for simplicity. Inclusive observables, such as the rapidity of the resonance in this case, are quantities expected to be well described by the matrix element, and thus should be unmodified by the parton shower; this is reflected in Figs. 5 and 7 where both showers display good agreement, with uncertainties mainly driven by the hard process variation. The differences in magnitude should be attributed to different couplings for each process, with envelope shape differences attributed to the PDFs.
The jets in these samples are generated solely from the parton shower; therefore the \(p_\perp \) of the leading (hardest) jet directly probes the impact of the profile scales.^{Footnote 8}
Comparing Figs. 6, 7, 8, 9, 10, we find that the different profile choices exhibit significantly different behaviours, both amongst themselves as well as between different showers. The resummation and theta profiles, as intended, yield comparable results in terms of central predictions and uncertainties and across the different shower algorithms. This clearly shows that we can indeed expect the same resummation accuracy using these profiles. The variations towards high \(p_\perp \) for the theta profile expose the effect of the different phasespace limitations. In the QTilde shower the upward variation of the scales (\(\mu _\mathrm{Q}\)) is ultimately irrelevant, as there are no possible emissions at this scale; looking at the dipole shower one sees the effect of such variations. However, this is not the case for the resummation profile whose interpolating region is sensitive to such variations, and displays similar variations between showers.
For large transverse momenta, the uncertainties should reflect the case that partonshower emissions in these regions are unreliable. We observe this for both the theta and resummation profiles and to some extent for the hfact choice, though the variation is considerably smaller than indicated by the thetatype choices. The power shower, however, shows no increased uncertainty and in fact is dominated by variations of \(\mu _\mathrm{H}\), since by definition there is no variation of \(\mu _\mathrm{Q}\). Given the marked differences in the hardness of jets between the two showers, the power shower seems to offer no handle towards the assessment of shower uncertainties. We can also clearly observe the intrinsic limitation of the QTilde shower phase space, which in this case is not able to populate high\(p_\perp \) emissions which ultimately needs to be supplied by matching and/or matrix element corrections similarly to the ‘dead zone’ effect in \(e^+e^\) collisions.
We therefore conclude that within this basic setting the showers and profile scale choices do admit the expected behaviour, and the two showers using thetatype profiles exhibit similar central predictions and uncertainties.
Jetty processes
Having established shower uncertainties using simple benchmark processes, the next simplest examples are the processes studied in Sect. 4 with an additional hard emission off the hard process, e.g. H / Z plus one (inclusive) jet. In addition, pure dijet production is investigated because of the absence of a colour singlet setting a hard scale and the related ambiguities in possible hard scale choices. We do not investigate the shower cutoff as we shall now focus on properties which are not expected to be significantly altered by hadronisation effects.
As with the clean benchmarks presented in Sect. 4, we consider variations of the three relevant scales discussed in Sect. 3, changing them by factors 1 / 2 and 2, respectively, to span a cube of a total of 27 variations; we will also perform crossvalidations between both available showers. From arguments given in Sect. 3 we expect observables and/or regions in phase space where the uncertainty is mainly driven by \(\xi _\text {H}\), i.e. in the case of inclusive observables. As all uncertainties connected with scale choices stem from logarithmic arguments there is no a priori way to exclude any of the possible variations when determining shower uncertainties, unless one is able to identify scale compensation patterns between the different scales for which we see no evidence in the setting considered in this study.
For the rapidity distributions of the Higgs and Z boson, shown in Figs. 11 and 12, respectively, we find that the distributions are consistent with the prediction of the hard matrix element, as is expected from such inclusive quantities; this applies to all of the profile scales considered, with the power shower showing larger deviations in the forward region. Scale variations affect these observable mainly through variations present in the hard process.
Similarly to the rapidity distributions, we expect the \(p_\perp \)spectra of the leading jet to be predicted mainly by the hard matrix element, according to the consistency conditions discussed in Sect. 2.5. In Figs. 13 and 14 (H and Z production, respectively) we show the results for the QTilde shower, while Figs. 15 and 16 contain our findings for the Dipole shower. We again find that the uncertainties are dominated by the variation of the hard scale. For both showers the resummation profile is consistent with the hard matrix element prediction, except for jets close to the threshold where cut migration effects are being probed.^{Footnote 9} For the hfact profile with the QTilde shower we find a spectrum compatible with the one anticipated by the matrix element; for the dipole shower, a significantly harder spectrum is obtained. A similar, but even more dramatic picture emerges for the power shower setting. The spread of predictions for the QTilde shower is smaller than the spread for the dipole shower, owing to the intrinsic limitations of the phasespace volume available to angularordered emissions as already pointed out in the previous sections. The combinations QTilde plus power, and Dipole plus hfact or power contradict the criterion of controllable showering, which in this case is expected to not significantly alter the jet \(p_\perp \) spectrum. Combined with the empirical findings of Sect. 4, we will therefore not consider the power shower profile choice any further.
Turning to more exclusive observables^{Footnote 10}, we consider the angular separation between the boson and the leading jet \(\Delta R_{(H/Z)j}\), which probes both matrix element and shower dominated regions in a continuous observable: Matrix element emissions in this case can only populate the phasespace region \(\Delta R_{(H/Z)j} \ge \pi \). The region below is solely filled by the parton shower, typically operating at the boundary of validity of the underlying approximation as this phase space requires the shower to produce a hard, largeangle emission. Within the definition of controllable and consistent uncertainties, we therefore expect large uncertainties for \(\Delta R_{(H/Z)j} \le \pi \), while the shower should reproduce the matrix element dynamics above. Results for the QTilde shower are shown in Figs. 17 and 19 (H and Z production, respectively) and for the Dipole shower in Figs. 18 and 20. We place particular emphasis on the comparison of the resummation and hfact profiles. For all processes/showers we find that hfact predicts a small uncertainty band and produces slightly more hard jets; the latter can be attributed to the available phase space, while the former can be obtained by analysing Eq. 7, stressing the fact that the region in which the derivative of the profile is varying significantly extends over a larger region than for the other profiles, though with less overall variation implied. Contrary, and matching the expectations motivated by the logarithmic structure, the uncertainty for the resummation profile in the small \(\Delta R_{(H/Z)j}\) region is large and driven by all scale variations together. In addition, in the bottom ratio plot of Figs. 17, 18, 19 and 20 we show a subset of scale variations for the resummation profile choice, varying the hard and shower scales in a correlated and anticorrelated setting, at a fixed \(\mu _\mathrm{Q}\). This breakdown shows how different subsets of variations constitute the full uncertainty band. Besides the simple domination of the uncertainty by one variation, other regions of phase space show that the uncertainty is strongly underestimated by considering the variations separately. We therefore argue that only the full, combined, scale variation produces a reliable error band. As another probe of the interaction of shower emissions with the hardest jet, we consider \(k_\perp \)splitting scales, particularly the one in which an event with two jets would turn into an event with one jet as the jet \(p_\perp \) threshold passes through the scale obtained. These observables have also been proven to be accessible to analytic considerations [90], for which comparisons to full parton showers are highly desirable though are beyond the scope of this paper. In Fig. 21 we show our results for the QTilde shower for Higgs production.^{Footnote 11} Once again we compare the resummation profile choice with the hfact profile. It is noteworthy that the hfact profile introduces a strong change in the shape of the Sudakov peak, on top of the harder spectrum already observed for the first jet; besides the tail effects we are therefore concerned that profile scale choices along these lines may significantly impact the resummation properties of the parton shower, as may already be expected from the arguments presented in Sect. 3. We therefore conclude that, even with intrinsically restricted phase space, the hfact profile does not provide controllable uncertainties and will not be taken further into account in this study. We also use Fig. 21 to perform an comprehensive breakdown of the different variation directions in the ‘cube’ of possible variations, showing that no individual variation actually covers the full dynamics present. For LO plus PS simulations, we therefore argue that the full band is taken into consideration and improvements in the context of matching and merging will be subject to future work.
We have so far considered processes with a colourless, massive object that dominates the scale hierarchy at hand, and, even in the presence of an additional jet, makes the dynamics rather insensitive to additional radiation (as far as this radiation is confined to reasonable phasespace regions as identified above). A process where this is clearly not the case is pure jet production in hadron collisions, which also probes different colour structures that have not been encountered in the hard processes considered thus far. Owing to the backtoback configuration at lowest order, we expect considerable partonshower effects in comparison to the hard matrix element for a number of observables and expect to make a more detailed comparison to fixed order only once NLO improvement has been incorporated. Nevertheless, we can still test as to what extent the shower variations match up to expectations in signalling regions where the prediction should generally be considered unreliable. We also test, once more, if the two showers are comparable within their uncertainties. Following the previous arguments, we only consider the resummation profile, with a hard scale again given by the jet \(p_\perp \). Sample results comparing to the hard matrix element are shown in Figs. 22, 23 and 24, which show that the two showers preform in a very similar way both in their central predictions and variations; they also show that qualitatively we find a behaviour similar to the singlet plus jet benchmarks as if we had replaced the hard, colourless, object with a jet as hard probe. Quantitatively, however, we observe significant changes in the rates for the second jet, which need to be confronted with the impact of cut migration as well as the impact of higherorder corrections. We also note that choosing the hard veto scale in this setting has a significant impact on showered results.
With the transverse momentum of the third jet and the \(2\rightarrow 3\) resolution shown in Figs. 25 and 26 we consider purely shower driven quantities; both of these nicely reveal that the two showers, together with the resummation profile, are perfectly compatible with each other, exhibiting the same resummation accuracy.
Conclusions and outlook
We have performed a comprehensive and detailed study of the sources of uncertainty in parton showers, utilising the two partonshower algorithms available in Herwig 7. We have investigated different choices of profile scales to approach the boundary of hard emissions, as these are highly relevant to effects that appear in the context of NLO plus PS matching. We have systematically categorised the sources of uncertainty and outlined their interplay with other simulation components, putting this study into context of a bigger work programme to eventually establish uncertainties for event generators in total.
Focussing on the perturbative, partonshower part, of the simulation, we have deliberately chosen LO plus PS calculations to establish a baseline of controllable and consistent variations that will allow us to subsequently identify improvements and reduction in these uncertainties as higherorder corrections are included. We have found that profile scale choices are very constrained when applying consistency conditions on both central predictions (which should not alter input distributions of the hard process) and uncertainties (with large uncertainties to be expected in unreliable regions or regions dominated by hadronisation corrections), as well as stable results in the Sudakov region. Particularly the hfact and power shower configurations do not admit results compatible with these criteria. Utilising a resummation profile, which is very close to the theta cutoff for hard emissions as implemented in previous algorithms, we find that the angularordered and dipolebased shower algorithms are compatible with each other, both in central predictions and uncertainty claims, despite their very different nature.
Notes
 1.
While being based on scale compensation arguments, these methods are, however, not able to predict the impact of finite corrections.
 2.
One can argue that the tuning of \(\alpha _s(M_Z)\) is typically absorbing the CMW correction advertised in [80] which would have to be included otherwise to obtain a satisfactory description of data.
 3.
This setup has been chosen such as to later on enable a fair comparison to NLO improved simulation that necessitates these orders of running.
 4.
Typically, the splitting kernel for exact phasespace factorisation is then accompanied by a damping factor \(\sim 1  p_\perp ^2/R_\perp ^2\) towards the edge of phase space.
 5.
In principle \(\rho \) should be varied with a reasonable range, though we do not expect a big effect from this variation, given the similarities between \(\rho =0.3\) and \(\rho =0\) corresponding to the theta profile; see the following sections.
 6.
Lifting this restriction in the case of the dipole shower will induce logarithms of \(K_\perp /Q_\perp \) which can become parametrically as large as the leading \(Q_\perp /p_\perp \) ones, if these scales are not anymore of the same order.
 7.
We do not consider deep inelastic scattering, which is interesting in its own respect. Similarly, a (hypothetical) \(e^+e^\rightarrow gg\) collider setting should be explored to complement our studies of Z versus H production in pp collisions; we postpone these discussions to later work elaborating on the interplay with hadronizsation models, where these differences are expected to be more relevant; the reader is also referred to the Les Houches study [89] in this context.
 8.
Note that the profile scales, especially in the case of the QTilde shower, need to be applied to all emissions such as to make sure the hardest emission is corrected in the intented way.
 9.
Cut migration for jetty processes should actually be considered another source of uncertainty beyond the ones discussed here; however, we do not address these in detail but chose to use equal generation and analysis cuts to highlight these effects.
 10.
We remind the reader that ‘exclusive’ here means: potentially probing more and more shower emissions on top of the hard process.
 11.
The results for Z plus one jet and the Dipole shower yield similar observations and conclusions.
References
 1.
M. Bahr et al., Herwig++ Physics and Manual. Eur. Phys. J. C 58, 639–707 (2008). arXiv:0803.0883
 2.
J. Bellm et al., Herwig 7.0/Herwig++ 3.0 release note, Eur. Phys. J.C76 (2016)(4), 196. arXiv:1512.0117
 3.
T. Sjöstrand, S. Mrenna, P. Skands, Pythia 6.4 physics and manual. J. High Energy Phys. 2006(05), 026 (2006)
 4.
T. Sjöstrand, S. Ask, J.R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C.O. Rasmussen, P.Z. Skands, An Introduction to PYTHIA 8.2. Comput. Phys. Commun. 191, 159–177 (2015). arXiv:1410.3012
 5.
T. Sjostrand, S. Mrenna, P.Z. Skands, A. Brief, Introduction to PYTHIA 8.1. Comput. Phys. Commun. 178, 852–867 (2008). arXiv:0710.3820
 6.
T. Gleisberg, S. Hoeche, F. Krauss, M. Schonherr, S. Schumann, F. Siegert, and J. Winter, Event generation with SHERPA 1.1, JHEP02, 007 (2009). arXiv:0811.4622
 7.
S. Alioli, P. Nason, C. Oleari, E. Re, A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX. JHEP 06, 043 (2010). arXiv:1002.2581
 8.
S. Plätzer, S. Gieseke, Dipole Showers and Automated NLO Matching in Herwig++. Eur. Phys. J. C 72, 2187 (2012). arXiv:1109.6256
 9.
S. Hoeche, F. Krauss, M. Schonherr, F. Siegert, A critical appraisal of NLO+PS matching methods. JHEP 09, 049 (2012). arXiv:1111.1220
 10.
A.J. Larkoski, J.J. LopezVillarejo, P. Skands, Helicitydependent showers and matching with VINCIA. Phys. Rev. D 87(5), 054033 (2013). arXiv:1301.0933
 11.
J. Alwall et al., The automated computation of treelevel and nexttoleading order differential cross sections, and their matching to parton shower simulations. JHEP 07, 079 (2014). arXiv:1405.0301
 12.
K. Hamilton, P. Nason, E. Re, G. Zanderighi, NNLOPS simulation of Higgs boson production. JHEP 10, 222 (2013). arXiv:1309.0017
 13.
S. Höche, Y. Li, S. Prestel, DrellYan lepton pair production at NNLO QCD with parton showers. Phys. Rev. D 91(7), 074015 (2015). arXiv:1405.3607
 14.
M. Czakon, H.B. Hartanto, M. Kraus, M. Worek, Matching the NagySoper parton shower at nexttoleading order. JHEP 06, 033 (2015). arXiv:1502.0092
 15.
S. Jadach, W. Płaczek, S. Sapeta, A. Siódmok, M. Skrzypek, Matching NLO QCD with parton shower in Monte Carlo scheme–the KrkNLO method. JHEP 10, 052 (2015). arXiv:1503.0684
 16.
S. Hoeche, F. Krauss, M. Schonherr, F. Siegert, QCD matrix elements + parton showers: The NLO case. JHEP 04, 027 (2013). arXiv:1207.5030
 17.
R. Frederix, S. Frixione, Merging meets matching in MC@NLO. JHEP 12, 061 (2012). arXiv:1209.6215
 18.
S. Alioli, C.W. Bauer, C.J. Berggren, A. Hornig, F.J. Tackmann, C.K. Vermilion, J.R. Walsh, S. Zuberi, Combining higherorder resummation with multiple NLO calculations and Parton showers in Geneva. JHEP 09, 120 (2013). arXiv:1211.7049
 19.
L. Lonnblad, S. Prestel, Unitarising matrix element + Parton shower merging. JHEP 02, 094 (2013). arXiv:1211.4827
 20.
S. Plätzer, Controlling inclusive cross sections in parton shower + matrix element merging. JHEP 08, 114 (2013). arXiv:1211.5467
 21.
L. Lönnblad, S. Prestel, Merging Multileg NLO matrix elements with Parton showers. JHEP 03, 166 (2013). arXiv:1211.7278
 22.
R. Frederix, K. Hamilton, Extending the MINLO method. arXiv:1512.0266
 23.
S. Platzer, M. Sjodahl, Subleading \(N_c\) improved Parton Showers. JHEP 07, 042 (2012). arXiv:1201.0260
 24.
Z. Nagy, D.E. Soper, Effects of subleading color in a Parton Shower. JHEP 07, 119 (2015). arXiv:1501.0077
 25.
S. Höche, S. Prestel, The midpoint between dipole and Parton Showers. Eur. Phys. J. C 75(9), 461 (2015). arXiv:1506.0505
 26.
N. Fischer, P. Skands, Coherent Showers for the LHC (2016). arXiv:1604.0480
 27.
C. Bierlich, G. Gustafson, L. Lönnblad, A. Tarasov, Effects of overlapping strings in pp collisions. JHEP 03, 148 (2015). arXiv:1412.6259
 28.
J.R. Christiansen, P.Z. Skands, String Formation Beyond Leading Colour. JHEP 08, 003 (2015). arXiv:1505.0168
 29.
Particle Data Group Collaboration, K. A. Olive et al., Review of particle physics. Chin. Phys. C38, 090001 (2014)
 30.
P.M. Stevenson, Resolution of the renormalization scheme ambiguity in perturbative QCD. Phys. Lett. B 100, 61 (1981)
 31.
S. J. Brodsky, G. P. Lepage, P. B. Mackenzie, On the elimination of scale ambiguities in perturbative quantum chromodynamics, Phys. Rev. D28, 228–235 (Jul, 1983)
 32.
I.W. Stewart, F.J. Tackmann, Theory uncertainties for Higgs and other searches using jet bins. Phys. Rev. D 85, 034011 (2012). arXiv:1107.2117
 33.
M. Cacciari, N. Houdeau, Meaningful characterisation of perturbative theoretical uncertainties. JHEP 09, 039 (2011). arXiv:1105.5152
 34.
A. David, G. Passarino, How well can we guess theoretical uncertainties? Phys. Lett. B 726(1–3), 266–272 (2013)
 35.
E. Bagnaschi, M. Cacciari, A. Guffanti, L. Jenniches, An extensive survey of the estimation of uncertainties from missing higher orders in perturbative calculations. JHEP 02, 133 (2015). arXiv:1409.5036
 36.
G. Bozzi, S. Catani, D. de Florian, M. Grazzini, Transversemomentum resummation and the spectrum of the Higgs boson at the LHC. Nucl. Phys. B 737, 73–120 (2006). arXiv:hepph/0508068
 37.
C.F. Berger, C. Marcantonini, I.W. Stewart, F.J. Tackmann, W.J. Waalewijn, Higgs Production with a Central Jet Veto at NNLL+NNLO. JHEP 04, 092 (2011). arXiv:1012.4480
 38.
T. Becher, M. Neubert, Threshold resummation in momentum space from effective field theory. Phys. Rev. Lett. 97, 082001 (2006). arXiv:hepph/0605050
 39.
T. Becher, A. Broggio, A. Ferroglia, in Introduction to SoftCollinear Effective Theory, vol. 896. (Springer, 2015)
 40.
X. Liu, F. Petriello, Reducing theoretical uncertainties for exclusive Higgsboson plus onejet production at the LHC. Phys. Rev. D 87(9), 094027 (2013). arXiv:1303.4405
 41.
M. Bonvini, S. Marzani, C. Muselli, L. Rottoli, On the Higgs cross section at N \(^{3}\) LO+N\(^{3}\) LL and its uncertainty. JHEP 08, 105 (2016). arXiv:1603.0800
 42.
E. Bothmann, M. Schönherr, S. Schumann, Reweighting QCD matrixelement and partonshower calculations. arXiv:1606.0875
 43.
J. Bellm, S. Plätzer, P. Richardson, A. Siódmok, S. Webster, Reweighting Parton Showers. Phys. Rev. D 94(3), 034028 (2016). arXiv:1605.0825
 44.
J. Barnard, E. N. Dawe, M. J. Dolan, N. Rajcic, Parton Shower uncertainties in jet substructure analyses with deep neural networks. arXiv:1609.0060
 45.
S. Mrenna, P. Skands, Automated PartonShower variations in Pythia 8. Phys. Rev. D 94, 074005 (2016). arXiv:1605.0835
 46.
N. Fischer, S. Prestel, M. Ritzmann, P. Skands, Vincia for Hadron Colliders. arXiv:1605.0614
 47.
J. R. Andersen et al., Les Houches 2015: Physics at TeV Colliders Standard Model Working Group Report, in 9th Les Houches Workshop on Physics at TeV Colliders (PhysTeV 2015) Les Houches, France, June 1–19, 2015, 2016. arXiv:1605.0469
 48.
ATLAS Collaboration, G. Aad et al., Luminosity public results run2 (2016)
 49.
CMS Collaboration, V. Khachatryan et al., Public CMS luminosity information (2016)
 50.
C. Englert, T. Plehn, P. Schichtel, S. Schumann, Jets plus Missing energy with an autofocus. Phys. Rev. D 83, 095009 (2011). arXiv:1102.4615
 51.
C. Englert, T. Plehn, P. Schichtel, S. Schumann, Establishing Jet scaling patterns with a photon. JHEP 02, 030 (2012). arXiv:1108.5473
 52.
P. Richardson, D. Winn, Investigation of Monte Carlo Uncertainties on Higgs Boson searches using Jet Substructure. Eur. Phys. J. C 72, 2178 (2012). arXiv:1207.0380
 53.
M.H. Seymour, A. Siodmok, Constraining MPI models using \(\sigma _{eff}\) and recent Tevatron and LHC Underlying Event data. JHEP 10, 113 (2013). arXiv:1307.5015
 54.
M.H. Seymour, C. Tevlin, A Comparison of two different jet algorithms for the top mass reconstruction at the LHC. JHEP 11, 052 (2006). arXiv:hepph/0609100
 55.
S. Plätzer, S. Gieseke, Coherent Parton Showers with Local Recoils. JHEP 01, 024 (2011). arXiv:0909.5593
 56.
S. Hoeche, F. Krauss, M. Schonherr, Uncertainties in MEPS@NLO calculations of h+jets. Phys. Rev. D 90(1), 014012 (2014). arXiv:1401.7971
 57.
D. Bourilkov, Study of parton density function uncertainties with LHAPDF and PYTHIA at LHC, in LHC / LC Study Group Meeting Geneva, Switzerland, May 9, 2003, 2003. arXiv:hepph/0305126
 58.
S. Gieseke, Uncertainties of Sudakov formfactors. JHEP 01, 058 (2005). arXiv:hepph/0412342
 59.
A. Buckley, Sensitivities to PDFs in parton shower MC generator reweighting and tuning. arXiv:1601.0422
 60.
S. Hoeche, M. Schonherr, Uncertainties in nexttoleading order plus parton shower matched simulations of inclusive jet and dijet production. Phys. Rev. D 86, 094042 (2012). arXiv:1208.2815
 61.
B. Cooper, J. Katzy, M.L. Mangano, A. Messina, L. Mijovic, P. Skands, Importance of a consistent choice of alpha(s) in the matching of AlpGen and Pythia. Eur. Phys. J. C 72, 2078 (2012). arXiv:1109.5295
 62.
M. Rauch, S. Plätzer, Parton Shower Matching Systematics in VectorBosonFusion WW Production. arXiv:1605.0785
 63.
A. Buckley, H. Hoeth, H. Lacker, H. Schulz, J.E. von Seggern, Systematic event generator tuning for the LHC. Eur. Phys. J. C 65, 331–357 (2010). arXiv:0907.2973
 64.
ATLAS Run 1 Pythia8 tunes, Tech. Rep. ATLPHYSPUB2014021, CERN, Geneva (Nov, 2014)
 65.
C.M.S. Collaboration, V. Khachatryan et al., Event generator tunes obtained from underlying event and multiparton scattering measurements. Eur. Phys. J. C 76(3), 155 (2016). arXiv:1512.0081
 66.
N.E. Adam, V. Halyo, S.A. Yost, Evaluation of the theoretical uncertainties in the cross sections at the lhc. J. High Energy Phys. 2008(05), 062 (2008)
 67.
M.W. Krasny, F. Fayette, W. Placzek, A. Siodmok, Zboson as ’the standard candle’ for high precision Wboson physics at LHC. Eur. Phys. J. C 51, 607–617 (2007). arXiv:hepph/0702251
 68.
F. Fayette, M.W. Krasny, W. Placzek, A. Siodmok, Measurement of \(MW^+  MW^\) at LHC. Eur. Phys. J. C 63, 33–56 (2009). arXiv:0812.2571
 69.
M.W. Krasny, F. Dydak, F. Fayette, W. Placzek, A. Siodmok, \(\Delta M_{W} \le 10 MeV/c^{2}\) at the LHC: a forlorn hope? Eur. Phys. J. C 69, 379–397 (2010). arXiv:1004.2597
 70.
K. Cranmer, S. Kreiss, D. LopezVal, T. Plehn, Decoupling theoretical uncertainties from measurements of the Higgs Boson. Phys. Rev. D 91(5), 054032 (2015). arXiv:1401.0080
 71.
P. Bartalini, R. Chierici, A. de Roeck, Guidelines for the Estimation of Theoretical Uncertainties at the LHC, Tech. Rep. CMSNOTE2005013, CERN, Geneva (Sep, 2005)
 72.
P. Stephens, A. van Hameren, Propagation of uncertainty in a parton shower. arXiv:hepph/0703240
 73.
W.T. Giele, D.A. Kosower, P.Z. Skands, Higherorder corrections to Timelike Jets. Phys. Rev. D 84, 054003 (2011). arXiv:1102.2126
 74.
S. Gieseke, P. Stephens, B. Webber, New formalism for QCD parton showers. JHEP 12, 045 (2003). arXiv:hepph/0310083
 75.
M. Bahr, S. Gieseke, M.H. Seymour, Simulation of multiple partonic interactions in Herwig++. JHEP 07, 076 (2008). arXiv:0803.3633
 76.
M. Bahr, J.M. Butterworth, M.H. Seymour, The underlying event and the total cross section from Tevatron to the LHC. JHEP 01, 065 (2009). arXiv:0806.2949
 77.
S. Gieseke, C. Rohr, A. Siodmok, Colour reconnections in Herwig++. Eur. Phys. J. C 72, 2225 (2012). arXiv:1206.0041
 78.
M. Bengtsson, T. Sjostrand, A comparative study of coherent and noncoherent Parton Shower evolution. Nucl. Phys. B 289, 810 (1987)
 79.
T. Sjostrand, A model for initial state Parton Showers. Phys. Lett. B 157, 321 (1985)
 80.
S. Catani, B.R. Webber, G. Marchesini, QCD coherent branching and semiinclusive processes at large x. Nucl. Phys. B 349, 635–654 (1991)
 81.
L.A. HarlandLang, A.D. Martin, P. Motylinski, R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs. Eur. Phys. J. C 75, 204 (2015). arXiv:1412.3989
 82.
A. Buckley et al., LHAPDF6: parton density access in the LHC precision era. Eur. Phys. J. C 75, 132 (2015). arXiv:1412.7420
 83.
M. Cacciari, FastJet: a code for fast k(t) clustering, and more. arXiv:hepph/0607071
 84.
M. Cacciari, G.P. Salam, G. Soyez, FastJet user manual. Eur. Phys. J. C 72, 1896 (2012). arXiv:1111.6097
 85.
A. Buckley et al., Rivet user manual. Comput. Phys. Commun. 184, 2803–2819 (2013). arXiv:1003.0694
 86.
S. Catani, Y.L. Dokshitzer, M. Olsson, G. Turnock, B.R. Webber, New clustering algorithm for multi  jet crosssections in \(e^+\) \(e^\) annihilation. Phys. Lett. B 269, 432–438 (1991)
 87.
F. Tackmann, Private communication
 88.
M. Dasgupta, G.P. Salam, Resummation of the jet broadening in DIS. Eur. Phys. J. C 24, 213–236 (2002). arXiv:hepph/0110213
 89.
S. Hoeche, A. Jueid, G. Nail, S. Plätzer, et al., Towards parton shower variations, in Proceedings of the 2015 Les Houches workshop
 90.
E. Gerwick, P. Schichtel, Jet properties at highmultiplicity. arXiv:1412.1806
Acknowledgements
We are grateful to the other members of the Herwig collaboration for encouragement and helpful discussions; in particular we would like to thank Stefan Gieseke, Peter Richardson and Mike Seymour for a careful review of the manuscript. We also acknowledge fruitful exchange with Mrinal Dasgupta, Keith Hamilton and Frank Tackmann. The work of JB and PS has been supported by the European Union as part of the FP7 Marie Curie Initial Training Network MCnetITN (PITNGA2012315877). GN acknowledges a short term student visit funded by MCnetITN. SP acknowledges support by a FP7 Marie Curie Intra European Fellowship under Grant Agreement PIEFGA2013628739. We are also grateful to the Cloud Computing for Science and Economy project (CC1) at IFJ PAN (POIG 02.03.0300033/0904) in Cracow and the U.K. GridPP project whose resources were used to carry out some of the numerical calculations for this project. Thanks also to Mariusz Witek and Miłosz Zdybał for their help with CC1 and Oliver Smith for his help with grid computing.
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Bellm, J., Nail, G., Plätzer, S. et al. Partonshower uncertainties with Herwig 7: benchmarks at leading order. Eur. Phys. J. C 76, 665 (2016). https://doi.org/10.1140/epjc/s100520164506x
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Keywords
 Transverse Momentum
 Parton Shower
 Hard Process
 Hard Scale
 Parton Distribution Function