1 Introduction

Monte Carlo simulations are an essential tool in the analysis of modern collider experiments. These event generators contain a large number of both perturbative and non-perturbative parameters which are tuned to a wide range of experimental data. While significant effort has been devoted to the tuning of the parameters to produce a best fit there has been much less effort understanding the uncertainties in these results. Historically a best fit result, or at best a small number of tunes, are produced and used to predict observables making it difficult to assess the uncertainty on any prediction. The “Perugia” tunes [1, 2] have addressed this by producing a range of tunes by varying specific parameters in the Pythia [3] event generator to produce an uncertainty.

Here we make use of the Professor Monte Carlo tuning system [4] to give an assessment of the uncertainty by varying all the parameters simultaneously about the best-fit values by diagonalizing the error matrix. This then allows us to systematically estimate the uncertainty on any Monte Carlo prediction from the tuning of the event generator. We will illustrate this by considering the uncertainty on jet substructure searches for the Higgs boson at the LHC.

As the LHC takes increasing amounts of data the discovery of the Higgs boson is likely in the near future. Once we have discovered the Higgs boson, most likely in the diphoton channel, it will be vital to explore other channels and determine if the properties of the observed Higgs boson are consistent with the Standard Model. For many years it was believed that it would be difficult, if not impossible, to observe the dominant \(h^{0}\rightarrow b\,\bar{b}\) decay mode of a light Higgs boson. However, in recent years the use of jet substructure [520] offers the possibility of observing this mode. Jet substructure for \(h^{0}\rightarrow b\,\bar{b}\) as a Higgs boson search channel, was first studied in Ref. [5] building on previous work of a heavy Higgs boson decaying to W ± bosons [16], high-energy WW scattering [21] and SUSY decay chains [22], and subsequently reexamined in Refs. [8, 15]. Recent studies at the LHC [2325] have also shown this approach to be promising.

The study in Ref. [5] was carried out using the (FORTRAN) HERWIG 6.510 event generator [26, 27] together with the simulation of the underlying event using JIMMY 4.31 [28]. In order to allow the inclusion of new theoretical developments and improvements in non-perturbative modelling a new simulation based on the same physics philosophy Herwig++, currently version 2.6 [29, 30], is now preferred for the simulation of hadron–hadron collisions.

Herwig++ includes both an improved theoretical description of perturbative QCD radiation, in particular for radiation from heavy quarks, such as bottom, together with improved non-perturbative modeling, especially of multiple parton–parton scattering and the underlying event. In FORTRAN HERWIG a crude implementation of the dead-cone effect [31] meant that there was no radiation from heavy quarks for evolution scales below the quark mass, rather than a smooth suppression of soft collinear radiation. In Herwig++ an improved choice of evolution variable [32] allows evolution down to zero transverse momentum for radiation from heavy particles and reproduces the correct soft limit. There have also been significant developments of the multiple-parton scattering model of the underlying event [33, 34], including colour reconnections [35] and tuning to LHC data [36].

The background to jet substructure searches for the Higgs boson comes from QCD jets which mimic the decay of a boosted heavy particle. Although Herwig++ has performed well in some early studies of jet substructure [25, 37, 38], it is important that we understand the uncertainties in our modelling of the background jets which lie at the tail of the jet mass distribution.

In addition we improve the simulation of Higgs boson decay by implementing the next-to-leading-order (NLO) corrections to Higgs boson decay to heavy quarks in the POWHEG [39, 40] formalism.

In the next section we present our approach for the tuning of the parameters, which effect QCD radiation and hadronization, in Herwig++ together with the results of our new tune. We then recap the key features of the Butterworth, Davison, Rubin and Salam (BDRS) jet substructure technique of Ref. [5]. This is followed by our results using both the leading and next-to-leading-order matrix elements in Herwig++ with implementation of the next-to-leading-order Higgs boson decays and our estimate on the uncertainties.

2 Tuning Herwig++

Any jet substructure analysis is sensitive to changes in the simulation of initial- and final-state radiation, and hadronization. In particular the non-perturbative nature of the phenomenological hadronization model means there are a number of parameters which are tuned to experimental results. Herwig++ uses an improved angular-ordered parton shower algorithm [29, 32] to describe perturbative QCD radiation together with a cluster hadronization model [29, 41].

The Herwig++ cluster model is based on the concept of preconfinement [42]. At the end of the parton-shower evolution all gluons are non-perturbatively split into quark–antiquark pairs. All the partons can then be formed into colour-singlet clusters which are assumed to be hadron precursors and decay according to phase space into the observed hadrons. There is a small fraction of heavy clusters for which this is not a reasonable approximation which are therefore first fissioned into lighter clusters. The main advantage of this model, when coupled with the angular-ordered parton shower is that it has fewer parameters than the string model as implemented in the Pythia [3] event generator yet still gives a reasonable description of collider observables [43].

To tune Herwig++, and investigate the dependency of observables on the shower and hadronization parameters, the Professor Monte Carlo tuning system [4] was used. Professor uses the Rivet analysis framework [58] and a number of simulated event samples, with different Monte Carlo parameters, to parameterise the dependence of each observableFootnote 1 used in the tuning on the parameters of the Monte Carlo event generator. A heuristic chi-squared function

$$ \chi^{\prime\,2}(\boldsymbol{p}) = \sum_{\mathcal{O}}w_{\mathcal{O}} \sum_{b\in\mathcal{O}} \frac{ (f^b(\boldsymbol{p})-\mathcal{R}_b )^2}{\Delta^2_b}, $$
(1)

is constructed where p is the set of parameters being tuned, \(\mathcal{O}\) are the observables used each with weight \(w_{\mathcal{O}}\), b are the different bins in each observable distribution with associated experimental measurement \(\mathcal{R}_{b}\), error Δ b and Monte Carlo prediction f b(p). Weighting of those observables for which a good description of the experimental result is important is used in most cases. The parameterisation of the event generator response, f(p), is then used to minimize the χ ′ 2 and find the optimum parameter values.

There are ten main free parameters which affect the shower and hadronization in Herwig++. These are shown in Table 1 along with their default values and allowed ranges.

Table 1 The ten parameters to which the jet substructure is most sensitive with their default values, the allowed range of these values in Herwig++, the range scanned over and the new optimum value found from minimizing χ ′ 2

The gluon mass, GluonMass, is required to allow the non-perturbative decay of gluons into \(q\bar{q}\) pairs and controls the energy release in this process. PSplitLight, ClPowLight and ClMaxLight control the mass distributions of the clusters produced during the fission of heavy clusters. ClSmrLight controls the smearing of the direction of hadrons containing a (anti)quark from the perturbative evolution about the direction of the (anti)quark. AlphaMZ is strong coupling at the Z 0 boson mass and controls the amount of QCD radiation in the parton shower, while Qmin controls the infrared behaviour of the strong coupling. pTmin is the minimum allowed transverse momentum in the parton shower and controls the amount of radiation and the scale at which the perturbative evolution terminates. PwtDIquark and PwtSquark are the probabilities of selecting a diquark–antidiquark or \(s\bar{s}\) quark pair from the vacuum during cluster splitting, and affect the production of baryons and strange hadrons respectively.

Previous experience of tuning Herwig++ has found that Qmin, GluonMass, ClSmrLight and ClPowLight to be flat, and so it was chosen to fix these at their default values [29].

To determine the allowed variation of these parameters Professor was used to tune the variables in Table 1 to the observables and weights found in Appendix A in Tables 5, 6, 7 and 8. The dependence of χ ′2 on the various parameters, about the minimum χ ′2 value, is then diagonalized.

The variation of the parameters along the eigenvectors in parameter space obtained corresponding to a certain change, Δχ ′ 2, in χ ′ 2 can then be used to predict the uncertainty in the Monte Carlo predictions for specific observables.

In theory, if the χ ′ 2 measure for the parameterised generator response is actually distributed as a true χ 2, then a change in the goodness of fit of one will correspond to a one sigma deviation from the minima, i.e. the best tune. In practice, even the best tune does not fit the data ideally and nor is the χ ′ 2 measure actually distributed according to a true χ 2 distribution. This means that one cannot just use Professor to vary the parameters about the minima to a given deviation in the χ ′ 2 measure without using some subjective opinion on the quality of the results.

We simulated one thousand event samples with different randomly selected values of the parameters we were tuning. Six hundred of these were used to interpolate the generator response. All the event samples were used to select two hundred samples randomly two hundred times in order to assess the systematics of the interpolation and tuning procedure. A cubic interpolation of the generator response was used as this has been shown to give a good description of the Monte Carlo behaviour in the region of best generator response [4]. The parameters were varied between values shown in Table 1. The quality of the interpolation was checked by comparing the χ ′ 2/N df, where N df is the number of observable bins used in the tune, in the allowed parameter range on a parameter by parameter basis for the observables by comparing the interpolation response with actual generator response at the simulated parameter values. Bad regions were removed and the interpolation repeated leaving a volume in the 5-dimensional parameter space where the interpolation worked well.

Figure 1 shows the χ ′ 2/N df distributions for two hundred tunes based on two hundred randomly selected event samples points for the cubic interpolation. The spread of these values gives an idea of the systematics of the tuning process showing that we have obtained a good fit for our parameterisation of the generator response.

Fig. 1
figure 1

The χ ′ 2/N df distributions for the parameters that were varied from their default values whilst determining the error tune. The scatter of the results gives a representation of the systematics of tuning procedure

The line indicates the tune which is based on a cubic interpolation from six hundred event samples. It is this interpolation which was used to vary χ ′ 2 about the minimum to assess the uncertainty on the measured distributions. During the tune it was discovered that PSplitLight was relatively insensitive to the observables used in the tune. As such, PSplitLight was fixed at the default value of 1.20 during the tune and subsequent χ ′ 2 variation.

Professor was used to vary χ ′ 2 about the minimum value, as described above, determining the allowed range for the parameters. As five parameters were eventually varied, there are 10 new sample points—one for each of the parameters and one “+” and one “−” along each eigenvector direction in parameter space.

We follow the example set by the parton distribution function (PDF) fitting groups in determining how much to allow χ ′ 2 to vary. Our situation is different to the PDF fitters in that we are using leading-order calculations with leading-log accuracy in the parton shower, where they fit to next-to-leading order calculations which gives better overall agreement with the observables used. Generally, PDF groups fit to fully inclusive variables, where as we have fitted to more exclusive processes and by nature, these are more model dependent, in particular hadronization.

In Refs. [45, 46] these issues are explored in terms of PDFs and the allowed variation is related to a tolerance parameter T, where

$$ \Delta \chi^{\prime\,2}_{\mathrm{global}} \leq T^2 . $$
(2)

A tolerance parameter of T≈10 to 15 is generally chosen for the PDF groups, where they are fitting to around 1300 data points. As our fit is likley to have a higher χ 2 than their fit due to the aforementioned reasons, and that we fit to a greater number of parameters, we will have a higher tolerance parameter.

In our fit, we have 1665 degrees-of-freedom and we examined various changes in χ ′ 2, whilst considering the effects of the precision data from LEP. A variation of Δχ ′ 2/N df=5, equivalent to T≈90, seems, subjectively to keep the LEP data within reasonable limits while a variation of Δχ ′ 2/N df=10, i.e. T≈130 is too large. Anything less T≈40 had very little variation and was therefore deemed inappropriate. The values for both Δχ ′ 2/N df=5 and Δχ ′ 2/N df=10 are shown in are shown in Tables 2 and 3 respectively.

Table 2 The five directions corresponding to the error tune for a Δχ ′ 2/N df=5 and the values the parameters take in each direction
Table 3 The five directions corresponding to the error tune for a Δχ ′ 2/N df=10 and the values the parameters take in each direction

The Professor tune was then compared with the internal Herwig++ tuning procedure [29] as not all analyses that are in the internal Herwig++ tuning system are available in Rivet and subsequently accessible to Professor. Looking at Fig. 4 it is found that PSplitLight at a value of 0.90 is favoured and gives a significant reduction in the χ ′ 2/N df. It was therefore decided to use the values obtained from minimisation procedure, but using the value of 0.90 for PSplitLight to maintain a good overall description of the data. The new minima for the QCD parameters are summarized in the Table 1. Examples of the new tune and the uncertainty band are shown in Figs. 2 and 3 for the out-of-plane transverse momentum and thrust measured by DELPHI [44].

Fig. 2
figure 2

Results from the DELPHI [44] analysis of out-of-plane p T with-respect-to the thrust axis and 1-thrust showing the new tune and the envelopes corresponding to a change in Δχ ′ 2/N df=5

Fig. 3
figure 3

Results from the DELPHI [44] analysis of out of plane p T with-respect-to the thrust axis and 1-thrust showing the new tune and the envelopes corresponding to a change in Δχ ′ 2/N df=10

Fig. 4
figure 4

A scan of PSplitLight using the internal Herwig++ tuning system with the other parameters fixed at their new tuned value. From the total χ ′ 2/N df we see that a value of 0.90 for PSplitLight is favoured at the new tuned parameters driven by the multiplicities

These error tune values can now be used to predict the uncertainty from the tuning of the shower parameters on any observable. In the next section we will present an example of using these tunes to estimate the uncertainty on the predictions for searches for the Higgs boson using the BDRS jet substructure method.

3 Jet substructure boosted Higgs

The analysis of Ref. [5] uses a number of different channels for the production of the Higgs boson decaying to \(b\bar{b}\) in association with an electroweak gauge boson, i.e. the production of h 0 Z 0 and h 0 W ±. Reference [5] uses the fact that the Higgs boson predominantly decays to \(b\,\bar{b}\) in a jet substructure analysis to extract the signal of a boosted Higgs boson above the various backgrounds. Their study found that the Cambridge-Aachen algorithm [47, 48] with radius parameter R=1.2 gave the best results when combined with their jet substructure technique. For our study, we used the Cambridge-Aachen algorithm as implemented in the FastJet package [49]. Three different event selection criteria are used:

  1. (a)

    a lepton pair with \(80~\mathrm{GeV} < m_{l^{+}l^{-}} < 100~\mathrm{GeV}\) and \(p_{T} > p_{T}^{\min}\) to select events for Z 0 + ;

  2. (b)

    missing transverse momentum to select events with \(Z^{0}\to\nu\bar{\nu}\);

  3. (c)

    missing transverse momentum and a lepton with p T >30 GeV consistent with the presence of a W boson with \(p_{T} > p_{T}^{\min}\) to select events with Wℓν;

where \(p_{T}^{\min} = 200~\mathrm{GeV}\).

In addition the presence of a hard jet with \(p_{T_{j}} > p_{T}^{\min}\) with substructure is required. The substructure analysis of Ref. [5] proceeds with the hard jet j with some radius R j , a mass m j and in a mass-drop algorithm:

  1. 1.

    the two subjets which were merged to form the jet, ordered such that the mass of the first jet \(m_{j_{1}}\) is greater than that of the second jet \(m_{j_{2}}\), are obtained;

  2. 2.

    if \(m_{j_{1}} < \mu \,m_{j}\) and

    $$ y = \frac{\min(p_{Tj_1}^2,\,p_{Tj_2}^2)}{m_j^2}\Delta R^2_{j_1,j_2} > y_{\mathrm{cut}}, $$
    (3)

    where \(\Delta R^{2}_{j_{1},j_{2}}=(y_{j_{1}}-y_{j_{2}})^{2}+(\phi_{j_{1}}-\phi_{j_{2}})^{2}\), and \(p_{Tj_{1,2}}\), \(\eta_{j_{1,2}}\), \(\phi_{j_{1,2}}\) are the transverse momenta, rapidities and azimuthal angles of jets 1 and 2, respectively, then j is in the heavy particle region. If the jet is not in the heavy particle region the procedure is repeated using the first jet.

This algorithm requires that j 1,2 are b-tagged and takes μ=0.67 and y cut=0.09. A uniform b-tagging efficiency of 60 % was used with a uniform mistagging probability of 2 %. The heavy jet selected by this procedure is considered to be the Higgs boson candidate jet. Finally, there is a filtering procedure on the Higgs boson candidate jet, j. The jet, j, is resolved on a finer scale by setting a new radius \(R_{\mathrm{filt}}=\mathrm{min}(0.3,R_{b\bar{b}}/2)\), where from the previous mass-drop condition, \(R_{b\bar{b}} = \Delta R^{2}_{j_{1},j_{2}}\). The three hardest subjects of this filtering process are taken to be the Higgs boson decay products, where the two hardest are required to be b-tagged.

All three analyses require that:

  • after the reconstruction of the vector boson, there are no additional leptons with pseudorapidity |η|<2.5 and p T >30 GeV;

  • other than the Higgs boson candidate, there are no additional b-tagged jets with pseudorapidity |η|<2.5 and p T >50 GeV.

In addition, due to top contamination, criterion (c) requires that other than the Higgs boson candidate, there are no additional jets with |η|<3 and p T >30 GeV. For all events, the candidate Higgs boson jet should have \(p_{T} > p_{T}^{\mathrm{min}}\). The analyses were implemented using the Rivet system [58].

The plots shown in Fig. 5 use the leading-order matrix elements for the production and decay of Higgs boson but the W, Z and top [50] have matrix element corrections for their decays. The plots shown in Fig. 6 have leading-order \(t\bar{t}\) production, leading-order vector boson plus jet production (with the same matrix element corrections as the LO matrix elements) but the NLO vector boson pair production [51] and NLO vector and Higgs boson associated production [52]. In addition we have implemented the corrections to the decay \(h^{0} \rightarrow b \,\bar{b}\) in the POWHEG scheme, as described in Appendix B. The signal significances are outlined in Table 4.

Fig. 5
figure 5

Results for the reconstructed Higgs boson mass distribution using leading-order matrix elements. A SM Higgs boson was assumed with a mass of 115 GeV. In addition to the full result the contribution from top quark pair production (\(t\bar{t}\)), the production of W ± (W+Jet) and Z 0 (Z+Jet) bosons in association with a hard jet, vector boson pair production (VV) and the production of a vector boson in association with the Higgs boson (V+Higgs), are shown

Fig. 6
figure 6

Results for the reconstructed Higgs boson mass distribution using leading-order matrix elements for top quark pair production (\(t\bar{t}\)), and the production of W ± (W+Jet) and Z 0 (Z+Jet) bosons in association with a hard jet. The next-to-leading-order corrections are included for vector boson pair production (VV) and the production of a vector boson in association with the Higgs boson (V+Higgs) as well as in the decay of the Higgs boson, \(h^{0} \rightarrow b \,\bar{b}\). A SM Higgs boson was assumed with a mass of 115 GeV

Table 4 The significance of the different processes for the leading- and next-to-leading-order matrix elements. The significance is calculated using all masses in the range 112–120 GeV

The uncertainties due to the Monte Carlo simulation are shown as bands in Figs. 7 and 8. As there are correlations between the different processes the uncertainty is determined for the sum of all processes. Whilst it would be possible to show the envelope for the individual processes, this would not offer any information on the envelope for the sum of the processes which is the result of interest. In addition the uncertainty on the significance is shown in Table 4.

Fig. 7
figure 7

Results for the reconstructed Higgs boson mass distribution using leading-order matrix elements. A SM Higgs boson was assumed with a mass of 115 GeV. The envelope shows the uncertainty from the Monte Carlo simulation

Fig. 8
figure 8

Results for the reconstructed Higgs boson mass distribution using leading-order matrix elements for top quark pair production, and the production of W ± and Z 0 bosons in association with a hard jet. The next-to-leading-order corrections are included for vector boson pair production and the production of a vector boson in association with the Higgs boson as well as in the decay of the Higgs boson, \(h^{0} \rightarrow b \,\bar{b}\). A SM Higgs boson was assumed with a mass of 115 GeV. The envelope shows the uncertainty from the Monte Carlo simulation

4 Conclusions

Monte Carlo simulations are an essential tool in the analysis of modern collider experiments. While significant effort has been devoted to the tuning of the parameters to produce a best fit there has been much less effort understanding the uncertainties in these results. In this paper we have produced a set of tunes which can be used to assess this uncertainty using the Herwig++ Monte Carlo event generator.

We then used these tunes to assess the uncertainties on the mass-drop analysis of Ref. [5] using Herwig++ with both leading- and next-to-leading-order matrix elements including a POWHEG simulation of the decay \(h^{0}\rightarrow b \bar{b}\).

We find that while the jet substructure technique has significant potential as a Higgs boson discovery channel, we need to be confident of our tunes to investigate this with Monte Carlo simulations.

The error tunes and procedure here can now be used in other analyses where the uncertainty due to the Monte Carlo simulation is important.