Merging NLO multijet calculations with improved unitarization
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Abstract
We present an algorithm to combine multiple matrix elements at LO and NLO with a parton shower. We build on the unitarized merging paradigm. The inclusion of higher orders and multiplicities reduce the scale uncertainties for observables sensitive to hard emissions, while preserving the features of inclusive quantities. The combination allows further soft and collinear emissions to be predicted by the allorder partonshower approximation. We inspect the impact of terms that are formally but not parametrically negligible. We present results for a number of collider observables where multiple jets are observed, either on their own or in the presence of additional uncoloured particles. The algorithm is implemented in the event generator Herwig.
1 Introduction
Multi purpose Monte Carlo event generators [1, 2, 3, 4] are used in most LHC analyses to obtain predictions for a multitude of observables at the level of finalstate particles. The outstanding accuracy of the LHC experiments calls for predictions at the highest possible theoretical accuracy, where nexttoleading order (NLO) predictions in the perturbative expansion of quantum chromodynamics (QCD) in the strong coupling constant \(\alpha _S\) have become the de facto standard during the last decade.
NLO corrections are available for many standard model processes with a moderate number of additional parton emissions. Once a higher jet multiplicity is of interest, it will be increasingly important to simulate also processes with higher multiplicities at the NLO level. It is clear that the computational effort for corrections to processes with higher and higher multiplicities increases enormously with the number of emitted partons. With the help of a parton shower, it is, however, possible to consistently merge computations with different multiplicities into one inclusive sample that contains full final states at different jet multiplicity.
The first successful attempts at correcting the results of parton showers integrated a matrix element (ME) for the emission of one additional particle into the shower evolution, resulting in socalled matrixelement corrections [5, 6]. The next step from there has been the development of systematic merging prescriptions, either literally based on a jet definition as the socalled MLM^{1} method [7], or as an approach from the analytic formulation of emission probabilities [8], known as the CKKW^{2} method. An alternative formulation, CKKWL^{3} [9], was based on very similar ideas, although here the noemission probability was not computed based on assumptions on the partonshower formulation but rather literally taken from socalled trial parton showers, carried out in the very parton shower that was used for the merging of MEs itself. The latter approaches show that a residual merging scale dependence is beyond the level of logarithmic approximation of the parton shower. Following the first approaches for \(e^+e^\) annihilation final states, the CKKW algorithm has been generalized to hadronic collisions as well [10]. A systematic comparison of the different approaches has been carried out in [11].
As opposed to merging several treelevel MEs of different multiplicity, the development of the last 15 years has led to the fact that it has become standard to simulate collider processes at NLO. Two methods have been pioneered on which all of the following work is based. In the mc@nlo method [12] and implementation the partonshower contribution to a partonic final state is expanded in \(\alpha _S\) and subtracted from the corresponding contributions of the ME, such that a consistent matching of NLO MEs and parton showers becomes possible. The program package of the same name comes with a multitude of builtin processes that can be simulated with different partonshower programs. The other method powheg [13] rather changes the Sudakov form factor for the first emissions of a parton shower in a way that up to this perturbative order the partonlevel answer of the computation is consistent with an NLO calculation. The program package powhegbox [14] provides a large number of processes that can afterwards be showered with different partonshower programs. A number of processes have been implemented into herwig [15, 16, 17]. In [18] the soft region of the parton shower is singled out and resummed separately. NLO MEs have also been matched to antenna like showers [19]. First attempts at matching NLO calculations with an NLO parton shower have been made in [20].
While the two aforementioned packages address each process separately, the enormous technical progress of the last decade made possible to simulate practically any standard model process at colliders completely automatically. A multitude of programs are capable of providing matrix elements [21, 22, 23, 24, 25, 26] and cross section calculations/matching at NLO, e.g. [26]. The generated MEs from these programs provide information according to a common standard [27], which in turn will then be interfaced to the general purpose event generators to simulate full hadron level events at NLO [2, 28, 29]. In herwig 7 [2] the interface is merely used to compute the MEs, while the phase space sampling, subtraction terms, and matching algorithms are performed within herwig. A newer development is the possibility to provide theoretical uncertainties, based on scale variations within these programs and within the parton showers [30, 31, 32]. While a lot of experience with NLO matching has been accumulated, the first processes have been combined with NNLO MEs [33, 34, 35, 36].
At the same time, the merging method has evolved into a standard for the simulation of final states with a large number of jets. It has been found that for a consistent matching it is very important to understand the clustering and subsequent shower in great detail. In the end the phase space that is covered by the parton shower has to be matched to the ME phase space. In order to achieve this, socalled truncated showers have been introduced [37, 38].
With the advent of more and more partly automatically generated NLO matrix elements it has become possible to add virtual corrections to the merging as well [39, 40, 41]. At first, only the corrections to the process with lowest multiplicity have been added, such that the overall normalization of the inclusive cross section has been stabilized, while multiple jet emissions have still been described using treelevel matrix elements. This method can be seen as the unification of the previous matching and merging approaches. However, the approach has not been limited to the lowest multiplicity MEs but rather allowed the introduction of virtual corrections to higher multiplicity MEs as well, thereby reducing also the scale uncertainties for observables that previously have only been described at LO. In [31] the systematic uncertainties from perturbative and nonperturbative sources have been addressed.
While most of the work is concerned with NLO QCD corrections and merging of MEs with additional parton emission, the emission of weak bosons has been studied in [42]. A merged sample of V+jets was obtained by either using the electroweak process as a starting point and adding further hard emissions to it or, alternatively, starting from a multijet process and then producing the weak boson as a partonshower emission. Here, particularly the harder parts of the QCD phase space can be addressed consistently.
As outlined above, the merging of QCD MEs has matured significantly over the last years. While it is possible to improve on the independence on unphysical scales using the NLO merging, one shortcoming still are uncontrolled terms of higher orders within inclusive cross sections. This point has been addressed conceptually in [43], where, based on the partonshower formulation of higherorder contributions in [44], the problem of unitarity violations has been addressed. A formulation of the merging has been made that inherently preserves unitarity and thereby also preserves the inclusive cross section and its given accuracy. First implementations of this method are now known as the ulops and unlops approach [45, 46, 47].
In this paper we present a new implementation of the unitary merging algorithm that was outlined in [43], based on the dipole shower module [48, 49] of Herwig 7 [2]. We address all aspects of the merging algorithm from clustering to the assignment of subsequent partonshower phase space as well as a detailed discussion of perturbative scales in the merging algorithm. The presented algorithm is built upon a very detailed formal description of the partonshower contribution at any given order. This allows us to discuss not only the merging but also an indepth analysis of terms that are beyond the targeted approximation on the perturbative and logarithmic level. We test the sensitivity of our merging against variations of higherorder terms and choices of scales. This allows, in addition to a fullyrealistic simulation, also a somewhat reliable estimate of theoretical uncertainties. In this paper we study the canonical examples of \(e^+e^\) annihilation and single vector boson production, accompanied by a number of jets, at hadron colliders. Furthermore, we consider Higgs production, as here the higherorder corrections are known to be numerically large. Finally, we consider pure jet production, which, due to the complexity of colour structures and the ambiguous definition of a hard object, is the most difficult process from the perspective of merging.
The paper is organized as follows. In Sect. 2 we introduce a formalism and notation in order to describe the parton shower analytically and to formulate all aspects of the subsequent subtractions. In Sect. 3 we describe the unitary merging algorithm for LO MEs first, including details of the clustering and scale settings. Later, in Sect. 4, we extend the algorithm to the unitary merging with NLO MEs. After a discussion of the scales we are using in Sect. 5 we present some validation of our approach in Sect. 6. Finally, we present results obtained with our approach in Sect. 7.
2 Notations and definitions
In order to set the scene for describing the complex algorithms involved in merging higherorder cross sections in a detailed way, we start with a review of fixedorder cross sections and parton showers to fix our notation of basic quantities and definitions.
2.1 Basic notation
The higherorder contributions taken into account in Eq. (5) are typically a combination of individually divergent contributions. The ultra violet (UV) divergences are regularized and then removed by renormalization, introducing the dependence on the unphysical scale \(\mu _R\). Besides the UV divergences, which stem from large momentum components in loop diagrams, also the region of small components or collinear momentum configurations can produce divergences, the latter specifically for massless particles. These infrared and collinear (IRC) divergences cancel in IRC safe observables, when the higher multiplicity, real emission, contributions with the same coupling power are included in the calculation.
2.2 Subtraction
2.3 LO \(\oplus \) PS
starting from an initial weight \(X_0\). Here \(P^{\alpha _i}_{i,i1}(\phi ^{\alpha _i}_{i})\) are the splitting probabilities without the scale dependent functions \(\alpha _S(q^i)\) and the ratio of parton distributions functions (PDF) accounting for changes in the initial state, while \(\eta _i\) is the pair of momentum fractions after the ith emission.
2.4 Dividing and filling the phase space
In order to include corrections to the parton shower it is possible to match the partonshower results with higherorder MEs. Within these approaches, the double counting occurring beyond the leadingorder contributions is subtracted at the desired, typically nexttoleading, order in \(\alpha _S\). Contrary to this, merging approaches aim at combining parton showers and multijet final states into one inclusive sample which can describe observables across different jet bins. These algorithms typically build on dividing the phase space accessible to emissions into regions of jet production (hard, matrixelement region) and jet evolution (partonshower region).
There are multiple ways how the phase space can be partitioned into regions of soft (IRC divergent) and regions of hard, largeangle emission. Jet algorithms like the \(k_\perp \) algorithm [57, 58] might be a tool of choice to achieve such a separation, based on clustering partons according to an interparton separation measuring how soft and/or collinear an emission has been. Each clustering scale can then be subjected to a cut classifying it to either belong to the jet production or jet evolution regime. It proves useful to design such an iterative jet algorithm to correspond precisely to the inverse action of the partonshower emission, and taking the actual shower emission scales to separate phase space into regions of hard and soft emission. In our default implementation, we use the simple algorithm presented in Alg. 2.2 to define the ME region. In this region all scales \(p^T_{ij,k}\), defined by the transverse momentum in the shower approach^{4}, encountered in any possible clustering configuration^{5} need to be larger than a merging scale \(\rho \).
But in a standard shower setup competing channels can radiate within the same phase space region, e.g. emissions from the two legs of a dipole system. Such emissions can, as depicted in Fig. 1, typically not be uniquely assigned to one scale and one individual emitter: Phase space points after emission of one channel could, even though they were generated with a scale above \(\rho \), also be assigned to another splitting with a scale below \(\rho \). Even though the partonshower action was divided in Eq. (19) into two regions, above and below \(\rho \), the region populated by \(\mathcal {PS}_\rho \) is not suited to define the matrixelement region, since an emission above \(\rho \) from one dipole leg could exhibit a scale below \(\rho \) w.r.t. to a different dipole leg.
2.5 Scales

Renormalization scale shower\(\mu ^\mathcal {PS}_R\): Scale used to calculate the value of \(\alpha _S(\mu ^\mathcal {PS}_R)\) for shower emissions. It is usually related to the ordering variable and/or the respective transverse momentum of the shower emission. The scale can be varied by \(\xi _R^\mathcal {PS}\).

Renormalization scale ME\(\mu ^{ME}_R(\phi )\): Renormalization scale used to calculate the (N)LO cross section. In LO or matched simulations the scale is applied to the ME calculation only. The scale can be varied by \(\xi _R^{ME}\).

Factorization scale shower\(\mu ^\mathcal {PS}_F\): The scale used to calculate the value of parton distribution function factors \(f_{i}(\eta _{i},\mu ^\mathcal {PS}_F)/f_{i1}(\eta _{i1},\mu ^\mathcal {PS}_F)\) for shower emissions. As \(\mu ^\mathcal {PS}_R\), it is usually related to the ordering variable or the respective transverse momentum of the shower emission. The scale can be varied by \(\xi _F^\mathcal {PS}\).

Factorization scale ME\(\mu ^{ME}_F\): Renormalization scale used to calculate the (N)LO cross section. In LO or matched simulations the scale is applied to the cross section calculation only. The scale can be varied by \(\xi _F^{ME}\).

Shower Starting Scale\(Q_S\): The shower starting scale is used to restrict the phase space of shower emissions, both in their transverse momenta and, possibly also for their longitudinal momentum fraction. The scale can be varied by \(\xi _Q\).

Shower CutOff\(\mu \): The infrared cutoff on shower emissions; no emissions will be generated with transverse momenta below this scale. The scale can be varied by \(\xi _{IR}\), and variations here typically need to be accompanied by refitting the hadronization parameters.

Merging Scale\(\rho \): The merging scale is a technical parameter that divides the phase space into a ME and a shower region. The dependence of the merging scale is nonphysical and gives an indication on the quality of the merging algorithm.
3 Merging multiple LO cross sections
3.1 Conventional LO merging
We briefly establish LOmerging algorithms as the basis of the present work; however, the reader is referred to the original publications for specific details of a particular algorithm. LOmerging algorithms like the conventional CKKW [8], CKKWL [9] and MLM [7] can be summarized to include cross sections of higher multiplicities in regions of the phase space where the scales of the extra emissions are above a merging scale \(\rho \). The ME region is usually defined in terms of a jet clustering algorithm, which we discuss in detail in Sect. 3.3. To merge the cross sections of the different multiplicities the weights of the phase space configuration are multiplied by additional factors that reflect the probability that the parton shower would have generated the given configuration. These factors are built out of Sudakov factors accounting for the probability of no emission between intermediate scales. The intermediate scales are obtained by the clustering algorithm, which reflects the inverse of the partonshower evolution. The shower emission densities are usually used to select the most probable interpretation in terms of a shower history, though this procedure is not unique. The reweighted multiplicities now enter a modified/vetoed parton shower, not allowing emissions in the ME region except if it starts off a state of the highest ME multiplicity available to the algorithm.
We will detail the ingredients to this generic procedure in the following subsections. The definition of a ME region and its relation to the partonshower emission phase space was sketched in Sect. 2.4 with focus on regions which can be filled by more than one shower emission process. The reweighting procedure will be discussed in detail in the sections to follow.
3.2 Clustering probabilities
3.3 Clustering
We summarize the clustering algorithm employed in the merging strategy in algorithmic form in Algorithm 3.1. The probability to choose a given clustering has already been discussed in Sect. 3.2, and is based on the partonshower approximation. At the beginning of the algorithm we ensure that the phase space point \(\phi _n\) is contained inside the ME region for chosen merging scale \(\rho \). We now select all possible clusterings \(\alpha \) that provide scales and splitting variables that could have been used to perform a shower emission from the underlying kinematics \(\phi _{n1,\alpha }(\phi _n)\); this includes the restriction imposed by the shower starting scale Q assigned to the underlying configuration \(\phi _{n1,\alpha }(\phi _n)\). Q either is the initial shower scale if no further clustering is possible (i.e. we have reached the lowestorder input process) or the maximum scale \(p^{T,\alpha \beta ...}_{ij,k}\) assigned to all subsequent clusterings, e.g. for secondary clusterings \(p_T^{\alpha \beta }(\phi _{n2,\alpha \beta }(\phi _{n1,\alpha }(\phi _n)))\).
Configurations \(\phi _{nk,\alpha ...}(...(\phi _n))\) that are not ruled out by the constraints are inserted into a selector^{6} with the weight as it is described in Sect. 3.2. From these possible clusterings we choose one according to the weight of the shower approximation and continue this procedure iteratively^{7} from the state obtained through the clustering. If no possible clustering is found the algorithm terminates and the phase space point is returned to be the seed phase space point to initiate the modified showering process.
3.4 Starting scale
3.5 LO merging
3.6 Unitarized LO merging
Parton showers are built on factorization properties of treelevel amplitudes in singlyunresolved limits. The corresponding virtual contributions are derived by imposing a unitarity argument, and inclusive quantities are hence unchanged after partonshower evolution. Upon replacing the emission probability of the parton shower through a merging algorithm we expect a nonunitary action since the virtual contributions, present at leading logarithmic level to all orders in the Sudakov form factor, have not been changed such as to retain the correct resummation properties. The amount by which inclusive cross sections are changed through such a procedure have been addressed in [43], and they are essentially probing the quality of the partonshower approximation by integrating the difference between a parton shower approximated and a full real emission matrix element, weighted by the Sudakov form factor for the relevant singular limit considered. As such, no logarithmically enhanced terms^{8} are expected to contribute to inclusive cross sections. However, with NLO merging in reach for which a potentially much more dangerous miscancellation in inclusive quantities is expected, we first address how inclusive quantities can be constrained through appropriate subtractions within merged cross sections.
While the first approaches to unitarized merging suggested an exact restoration of inclusive quantities, we here relax this criterion such as to capture terms with a logarithmic enhancement, which a parton shower aims to resum, only. We want to make it clear that the restriction of the shower phase space is essential for the logarithmic structure of the partonshower evolution. As we understand the replacement of the shower emissions by the ME expressions we argue that by unitarization of possible shower configurations we preserve the initial shower structure. To this extent, for each clustering \(\phi _n\) to \({\tilde{\phi }}^\alpha _{n1}\), we determine the respective transverse momentum \(p_\perp ^\alpha \) and longitudinal fraction \(z^\alpha \) of the emission, which are required to be within the phase space available to shower emissions. If no such configuration is obtained for potential clustering we assume that a genuine correction of hard jet production has been found which will then set the initial conditions for subsequent showering, subject to acceptance of a hard trigger object like a Z boson or a jet within the generation cuts at the level of this particular hard process configuration.
Those logarithmically enhanced contributions which are subtracted from the lower multiplicity in order to constrain inclusive cross sections provide approximate NLO corrections in the same spirit as the LoopSim [44] method has established capturing the most enhanced corrections. Having identified and being able to control these contributions we are in the position to establish a unitarized merging of NLO multijet cross sections.
In Sect. 2.4 we have shown how the partonshower evolution can be factored into a region of jet production (MEregion) and jet evolution, subject to a merging scale \(\rho \) and including the fact that emissions with transverse momenta below or above \(\rho \) do not necessarily respect this ordering when the kinematic mapping is defined with respect to a different dipole configuration than the emitting one. It is therefore clear that shower emissions off a configuration which is contained in the MEregion (i.e. one with all evolution scales above \(\rho \)) cannot be subjected to a naive veto on transverse momenta. Only if the full kinematics of the emission are known can we test if one or more of the emitting configurations gave rise to an emission below \(\rho \) in case of which the emission is accepted as being contained in the shower region. We conjecture that this procedure of a modified vetoed shower is precisely equivalent to the truncated showering procedures employed elsewhere [15, 29].
3.7 Generation cuts
While generation cuts within a merging algorithm are not required to render cross section predictions with additional jets finite, they might still be desirable to enhance populating certain regions of phase space. Even more than with a fixed jet multiplicity cut migration does become an issue within this context, specifically as clustered phase space points which have been identified as seed processes, will or will not pass the generation cut criteria independently of the acceptance of the unclustered configuration. Just as with cut migration present in standard shower evolution off a hard seed process, we indeed require that cuts are solely applied to the seed process which has been identified after the clustering procedure.
This still requires that care needs to be taken in the region subject to the migration and results should only be considered away from these boundaries, just as is the case for showering jetty processes in a standard setup. Ideally, generation cuts should not be required but efficiency issues should be addressed through a biasing procedure complemented by a reweighting such as to ensure that no event which possibly could contribute to an observable of interest will be discarded. We will address this issue in more detail in future work.
4 NLO merging
In this section we explicitly construct the merging of multiple NLO cross sections. We will follow the proposal of unitarized merging algorithms, which specifically focus on potentially problematic terms arising in NLO multi jet merging. Based on the combination of multiple LO cross sections with the parton shower, which constitutes an improved, resummed prediction, the task actually boils down to matching such a calculation consistently to the first \({{{\mathcal {O}}}}(\alpha _S)\) correction in each jet multiplicity. These corrections can receive both virtual corrections to the multiplicity at hand and approximate contributions from the partonshower action on a lower multiplicity. The remaining dependence on these corrections in inclusive cross sections can be traced back to a mismatch of the leadingorder parton shower attempting to approximate a nexttoleading order correction, and it will explicitly be removed by the unitarization procedure.
4.1 Real emission contributions
 (A)
ME region If the phase space point \(\phi _{n+1}\) of the real emission is contained in the ME region, the LO merging has already populated the region with LO corrections to the parton shower. The \(\alpha _S\)expansion of the LOmerged contribution, see expressions proportional to \(\phi _{n+1}\) in Eq. (31), need to be subtracted in this region in order to solve the double counting of ME corrections in this region. The same double counting argument now requires that the second expression of Eq. (31), which is stemming from the unitarization expressions of the LO merging needs to be added to the expansion of \(\alpha _S\). This contribution is proportional to the real emission contribution, in the ME region and is clustered to the underlying Born phase space points \({\tilde{\phi }}^\alpha _{n}\) according to the weight \(w_{C,\alpha }/\sum _\beta w_{C,\beta }\). Note that this weight is implicitly generated by the clustering algorithm of the LO merging. The same weight needs to be included here, since it is not part of the \(\alpha _S\)expansion. In addition to the clustered real emission, the contributions from subtraction terms need to be constructed. Since in the CS dipole subtraction, the subtraction dipoles are evaluated according to the real emission phase space points, but observables are evaluated at one of the underlying Born phase space points, the subtraction terms can be calculated alongside the clustered real emission contribution. We generate this contribution as follows: \(\phi _{n+1}\) is clustered randomly to one of the underlying \({\tilde{\phi }}^\alpha _{n}\) (the point at which the dipole term \(D_{\alpha }\) is calculated) and in addition \(\phi _{n+1}\) is clustered with the same algorithm used in LO merging. Only if \({\tilde{\phi }}^\alpha _{n}\) coincides with the LO clustering, the real emission point is calculated including the subtraction dipole at this point. Otherwise only the dipole \(D_\alpha \) is retained. By multiplying the result with the number of dipoles we compensate for the random choice of the first clustering. With this strategy we generate the same clustering weights as used in the LO merging. Since the dipoles and the integrated counterparts must cancel, the at first randomly chosen kinematics \({\tilde{\phi }}^\alpha _{n}\) now needs to be subjected to the definition of the ME region of the process with n additional legs. If it is contained in this phase space volume, the point is accepted and the algorithm proceeds to generate a history for \({\tilde{\phi }}^\alpha _{n}\) and the virtual contributions.
 (B)
The differential PSregion This region of phase space is populated by the transparent veto algorithm \(\mathcal {\widetilde{PS}}\). The real emission contribution needs to be subtracted by a shower approximation above the shower cutoff. While in the fixedorder calculation the real emission is subtracted with the dipole expressions that are contributing to observables that are evaluated at the reduced phase space point \({\tilde{\phi }}^\alpha _{n}\), the expansion of the shower expansion contributes to the same real emission phase space point \(\phi _{n+1}\). Compared to the dipole expressions, the \({\mathcal {O}}(\alpha _S)\) expansion of the shower are restricted by ordering in the shower evolution scale. Only these expansions contribute, for which the clustering scale \(p^\alpha _T\) associated to the clustering \( {\tilde{\phi }}^\alpha _{n}\) is ordered with respect to the shower starting scale evaluated for \( {\tilde{\phi }}^\alpha _{n}\) if \(n=0\) or if any of the underlying scales \(p^{\alpha ,\beta }_T\) associated to any of the clusterings \( \tilde{{\tilde{\phi }}}^{\alpha ,\beta }_{n1}\) is larger than \(p^\alpha _T\).
 (C)
Clustered PSregion The expansion of the parton shower outside the ME region, which was calculated with the real emission in the previous region has a counterpart in the no emission probability in the shower algorithm. As for matching algorithms this no emission probability needs to be expanded and subtracted from the clustered phase space points \( {\tilde{\phi }}^\alpha _{n}\). These expansions are constructed like the subtraction expressions of the previous region but opposite in sign and are here calculated with the dipole expressions of the CS dipole subtraction. While the expansion is ordered in the evolution scale the dipole contributions do not require this ordering, as the integrated counterpart, which subtract the IRC singularities of the virtual corrections, have no restriction on the analytically integrated expressions.
4.2 Virtual contributions
We will now describe algorithms to calculate the \({\mathcal {O}}(\alpha _S)\) expansion of the history weights. The history weights consist of three contributions as discussed in Sect. 3.5. First we have the \(\alpha _S\) ratio, for which the expansion is simply given by the running coupling. Corrections which have been calculated in the \(\overline{\text {MS}}\) scheme require an additional compensating term for the CMW^{9} prescription used in the shower^{10}, cf. the discussion in Sect. 5.1.
Algorithm 4.2 is generating the contribution in Eq. (32) together with the corrections for using the CMW scheme in the partonshower evolution.
The second contribution to the history weight is the PDF ratio. This contribution can be obtained using the \({\mathcal {P}}\) operator of the subtraction formalism, Algorithm 4.3.
The third contribution in the expansion of the history weights is the expansion of the Sudakov form factors. As the computation of the Sudakov exponent is performed by sampling the exponent with MC methods, it is possible to use the same routines to evaluate the firstorder expanded form factor. In contrast to the Sudakov sampling the scale of \(\alpha _S\) is kept fix and the PDF ratio is evaluated at the scale of the last emission.^{11}

the interference of the loop diagrams and the treelevel contribution,

the UVcounterterms,

the integrated dipole counter terms together with the collinear counterterm from PDF renormalization as contained in \({\mathbf {I}},{\mathbf {P}}\) and \({\mathbf {K}}\) operators of the CS subtraction formalism [51, 54].
4.3 Unitarising the NLO corrections
Having the framework for unitarized LO merging at hand, along with the subtractions required to match NLO calculations for individual multiplicities, the unitarization of NLOmerged cross sections is now straightforward: In order to ensure that the dipole subtraction terms cancel, their integrated contribution and the differential counterparts to these need to be subject to the same reweighting procedure at the underlying Born phase space point encountered. The virtual contributions are treated by the algorithm the same as the respective Born contributions.
In the case of real emission contributions, we consider a clustered phase space point \(\phi _{n1}^\alpha \) just as a Bornlike contribution. We choose the index \(\alpha \) randomly and reweight with the number of possibilities, and therefore effectively integrate the dipole contribution over the full phase space to match their integrated counter parts. Secondary clusterings from the point \(\phi ^\alpha _{n1}\) to \(\phi ^{\alpha ,\beta }_{n2}\) above the merging scale will be considered as a virtual contribution, and their subtraction realizes the unitarization of NLO corrections as outlined in [43].
4.4 Ambiguities in \(\mathbf {\varvec{\alpha }_S}\) expansions
Combinations of partonshower calculations and fixedorder corrections like matching and merging are subject to a number of ambiguities. While there are both technical and algorithmic details that can turn out to be numerically significant, the by far most striking ambiguity is in terms which are beyond the control of the fixedorder input at hand. Specifically the weights applied within the NLOmerging algorithm can be adjusted already at one higher order in the strong coupling, which is beyond both the fixedorder and the partonshower accuracy.^{12} We address this ambiguity in detail and consider a number of different schemes of expanding the history weights to fixed order:
Scheme 0 is not NLO accurate as the expansion contains \({\mathcal {O}}(\alpha _S)\) expressions of the same order as the \({\mathcal {O}}(\alpha _S)\) corrections of the NLO contribution. We merely use this for illustrative proposes. Scheme 1 originates from Eq. (33). Since the history weight \(w^i_\alpha \) is in general larger than one, scheme 2 suppresses the contribution which originates from expanding the Sudakov form factors. We expect these terms, to first order, to be proportional to \(\alpha _S \log ^2(q_1/q_2)\) and as such the dominating part of the history weight expansion for large scale separations. Scheme 3 is similar to scheme 1 but suppresses the expansion by keeping the \(\alpha _S\) weight fixed. Scheme 4 was introduced as the opposite to scheme 2. Here we suppress the negative contribution of the \(\alpha _S\) ratio expansion, rather than the positive Sudakov expansion. Further schemes could be constructed by rearranging the expressions in a way that keeps the \(\alpha _S\) expansion fixed.
Expansion schemes considered for the treatment of terms beyond NLO
Scheme  Expression 

0  \(\prod _{X } w^i_X \) (no hist. exp., not NLO correct) 
1  \(\left[ 1  \sum _{X } \alpha _S w^i_{\partial _X}\right] \prod _{X } w^i_X \) 
2  \(\left[ 1  \alpha _S w^i_{\partial _\alpha } \alpha _S w^i_{\partial _f}  \alpha _S w^i_{\partial _\varDelta }/ w^i_\alpha \right] \prod _{X } w^i_X\) 
3  \(\left[ 1  \sum _{X } \alpha _S w^i_{\partial _X}/ w^i_\alpha \right] \prod _{X } w^i_X \) 
4  \(\left[ 1  \alpha _S w^i_{\partial _\alpha }/ w^i_\alpha  \alpha _S w^i_{\partial _f}  \alpha _S w^i_{\partial _\varDelta } \right] \prod _{X } w^i_X\) 
5 CMW scheme, scale variations and notation
In this section we discuss additional input to the algorithm and the freedom of choosing schemes beyond the accuracy of our merged cross section calculation. One of the choices is the running and input value of the strong coupling constant which we discuss in detail in Sect. 5.1. We then elaborate on scale variations which we have chosen to assess the theoretical uncertainty of the merged calculation, followed by a discussion of the functional choice of the merging scale cut. At the end of this section we introduce a short notation for merged simulations.
5.1 The CMW scheme
This modification is usually referred to as the CMW [59] or Monte Carlo scheme. Using the modified version of \(\alpha _S\) in the merging algorithm can help to describe data but changes the scheme used earlier to calculate the MEs. At LO the effect is beyond the claimed accuracy, but by merging NLO cross sections, the prior estimate of higherorder corrections needs to be subtracted to match the scheme used in the ME calculation. For every \(\alpha _S\) ratio in the shower history reweighting of the form \(\alpha '_S(q)/\alpha _S(\mu _R)\) the NLOmerged algorithm must subtract the \(\alpha _S\) expansion \({\mathrm {d}}\sigma _B K_g\alpha _S/2\pi \). Note that this additional linear term and scaling q with the factor k in Eq. (36) produce the same \({\mathcal {O}}(\alpha _S)\)expansion. Both schemes are implemented and can be studied using Herwig. A detailed study of uncertainties related to choices in this scheme will be published elsewhere.
5.2 Scale variations
All scales that have been used to evaluate the predictions can be varied to estimate theoretical uncertainties. Usually constant factors are used to alter the scales up and down. We include five different factors. At first we have the variation \(\xi _{R/F}^{ME}\) of the production process. We call the renormalization and factorization scales used here \(\mu _{R/F}^{ME}\), weighted with \(\xi _{R/F}^{ME}\). They apply to the production process and – if a full shower history is found by the clustering algorithm – the reweighting of the history that restores the weights for the assumed production process. Additional emissions are produced with the scale of the shower splitting. These scales are \(\mu _{R/F}^{\mathcal {PS}}\) and are altered by factors \(\xi _{R/F}^{\mathcal {PS}}\). The last scale we need is the shower starting scale \(Q_{S}\) varied by \(\xi _{Q}\). While \(\xi _{R/F}^{\mathcal {PS}}\) apply to any shower emission the scaling factor \(\xi _{Q}\) is only chosen for the initial emission.
One could also construct a scenario where the shower should make use of a different PDF set than the ME calculation, e.g. LO for the former and NLO PDFs for the latter. Then the part in the ratios that are rescaled with \(\xi _{F}^{\mathcal {PS}}\) belong to the shower related PDFs and the one with \(\xi _{F}^{ME}\) needs to be used in the ME related set. In NLOmerged samples this difference between the two sets needs to be corrected by subtracting the difference of the two sets in order to preserve NLO accuracy.
5.3 Functional form of the merging scale cut
5.4 Notation
It should be clear that the accuracy of the abovementioned example \(Z(0^*, 1^*, 2, 3)\) depends on the observable we consider. For the given example the inclusive cross section or differential cross section of the hardest jet’s transverse momentum would be described at NLO level, while the transverse momentum of the second jet and third jet would be described at LO only. The fourth and higher jets would certainly only be described at the LLA level of the parton shower. We may wonder about the accuracy of the inclusive cross section in this case. Albeit formally at the NLO level, the \(Z(0^*, 1^*, 2, 3)\) sample doubtlessly contains more perturbative information than the \(Z(0^*, 1)\), which is the smallest sample that contains an NLO description of the inclusive cross section. The former case contains almost all ingredients of the NNLO, except the twoloop virtual contributions, which are only represented by NLO plus leading logarithms from the parton shower.
6 Sanity checks
To validate the merging algorithm several sanity checks have been performed. For simple processes we check the Sudakov suppression from our implementation against an independent Mathematica implementation. Further we can check that the subtraction of the real emission contribution is performed in an IR safe way. To validate the weights of the shower history, we replace a LO ME by the corresponding dipole expression and expect similar results as we would with pure shower emissions. Since the merged cross section is corrected by the algorithm for hard emissions in the ME region, soft physics should be hardly affected by the algorithm. A quantity sensitive to soft physics in Herwig is the cluster mass spectrum, which will be discussed. The introduction of an auxiliary cross section can provide a reduction of events with negative weights. In the last part of this section we compare the various schemes introduced in Sect. 4.4.
6.1 Sudakov sampling
6.2 Subtraction plots
Performing NLO calculations with MC techniques requires the introduction of subtraction terms for the virtual and real contributions as described in Sect. 2.2. In the CS framework the real emission contributions are subtracted with auxiliary cross sections that cancel versus the integrated counterparts for the virtual contributions. In the NLOmerging algorithm the subtraction for the real emission contributions is more complicated; see Sect. 4.1. The expansion of the LO merging up to \({\mathcal {O}}(\alpha _S)\) produces similar subtraction terms. There are three different phase space regions, the MEregion (A), the differential PSregion (B) and the clustered PSregion (C) as described in Sect. 4.1 which can be checked for subtractive properties.
6.3 Simple check for \(\mathbf {\varvec{\alpha }_S}\) and PDF ratios and merging scale variation
In order to validate the implementation of \(\alpha _S\) and PDF ratios, we replace the MEs in LO \(Z^0\) production, merged with one additional emission, at the LHC, with the corresponding shower approximation. In Fig. 6^{14} we show the result with the same parameters as described in Sect. 7.1. The red distribution shows LO \(Z^0\) production with a socalled power shower, achieved by applying no restrictions on the emission phase space of the shower and starting the shower at the highest possible scale s. In purple the same reweights and clustering properties are used as one would use in merging the two processes with the difference that we replace the \((Z^0 + 1\,\text {jet})\) sample by the sum of the corresponding dipoles, which corresponds here to the shower approximation. The blue lines represent the LOmerged result Z(0, 1), see Sect. 5.4 for notation, with one additional multiplicity merged to the production process with merging scales: \(10\,\text {GeV}\) (solid), \(5\,\text {GeV}\) (dotted) and \(20\,\text {GeV}\) (dashed) and full MEs. The difference from the pure shower case or the replacement with the dipole expressions is due to the MEs. Similar checks have been performed for light and massive finalstate radiation where we find similar agreement between pure parton shower and merging with shower approximations for the first emission.
6.4 Cluster mass spectrum
When merging several multiplicities the description of hard emissions is improved with the information of higherorder MEs and approximations used by the parton shower are rectified in its hard emission region. Effects of soft physics models are expected to be hardly affected by the inclusion of these corrections. One of the observables sensitive to the soft physics is the cluster mass spectrum of Herwig. We expect the parton shower at the end of the merging to evolve the final state on the soft end into a cluster spectrum that is almost unchanged. The cluster mass spectrum is the essential input for the hadronization model and the description of e.g. hadron multiplicities would suffer and would require a retuning if this part of the simulation is strongly affected by the improvements made for hard emissions.
In Figs. 7 and 8 we compare the cluster mass spectrum of primary clusters in \(e^+e^\) collisions. Figure 7 shows the cluster spectra of LO+PS and LOmerged samples with up to two additional jets. While the massless case shows a minor spike near the bottom mass, the merging of multiple cross sections hardy alter the contributions. In the third ratio plot we split up the various contributions of the ee(0, 1, 2), namely the merging of three multiplicities leading in sum to the full result. Figure 8 compares the same spectrum for NLOmerged contributions and again only the massless showering varies in shape compared to the massive LO+PS contribution. In conclusion we do not expect major retuning of the details of the cluster model to be necessary as a result of the merging procedure.
In contrast to the cluster model of hadronization the tuning of underlying event could change due to the effects of the merging. As seen in [61] the cross talk between underlying event and hard process calculation is more relevant. With respect to Pythias interleaved showering, Herwigs MPI model is fundamentally different as the showering is performed in separate steps of the event generation. The effects of retuning the underlying event model needs to be addressed in future studies.
6.5 Reduction of negative weights
One potential technical problem of unitarized merging may be the appearance of negative weights. Since the higher multiplicities are clustered, weighted and subtracted, depending on the merging scale, strong compensation effects and therefore negative contributions are unavoidable. A method to reduce the negative contributions is to introduce an auxiliary cross section which cancels in the full result, as it is done in NLO calculations. Introducing such helper weights may also serve as a strong check for the consistency of the implementation. In [21] a cut on the dipole phase space for NLO calculations performed with CS dipoles was introduced to speed up the calculation. We can use these expressions to get the analytically integrated dipole phase space above the cuts imposed in [21] and subtract the same regions from the differential clustered contributions. Hereby one contribution is subtracted from the clustered higher multiplicity and the compensating piece is added to the lower multiplicity.
To achieve the same result, we change the algorithm for LO merging. The first backward clustering is made randomly and without phase space restrictions instead of clustering with the clustering algorithm described in Sect. 3.2. In addition we cluster the first step with the cluster algorithm of Sect. 3.2 and calculate the weight of the LO cross section only if the same underlying contribution was picked. For the chosen channel the dipole is calculated if the phase space point is above the phase space cut. The sum of dipole and, if picked, clustered and negative LO contributions is multiplied by the number of dipoles, which compensates for the random choice of the underlying configurations. The history weight is then calculated for the dipole up to the underlying contribution and for the LO contribution with one additional step. With these changes we integrate the dipole parts above the cut and weighted with the history weights to the underlying configuration.
The integrated counterparts, known in the analytic form can now be added to the points calculated for the \({\mathrm {d}}\sigma _{n1}\) Born configurations. As these parts are negative the \({\mathrm {d}}\sigma _{n1}\) is suppressed and parts of the no emission probability produced by clustering and subtracting of the higher multiplicity are compensated.
Cut as in [21]  Positive events  Negative events 

1  6981  3019 
0.1  7218  2782 
0.05  7313  2687 
We have also confirmed the expectation that the fraction of negative events is being reduced as the merging scale is increased. Note that although the proportion of events with negative weight is reduced the unweighting of final event samples may still not be improved much. The multiplication of the number of dipoles and therefore enhanced maximum weights may reduce performance. We therefore only use this for testing purposes.
6.6 Expansion schemes and scale variations
As described in Sect. 4.4 various schemes can be constructed to include the shower history expansion. In Figs. 9, 10 and 11 we choose a rather small merging scale of \(\rho =4\,\text {GeV}\) and show various choices of these schemes for jet production at LEP. All lines are variations of the merging of three LO cross sections, where the production process and the first emission process are corrected with NLO contributions. We set \(\alpha ^{\overline{\mathrm{MS}}}_S (M_Z)=0.118\). While the red line, corresponding to no expansion of the history weights, clearly overestimates the regions of high emission scales, the schemes with expansion tend to describe the data measured at LEP better. In the simulation the CMW modified strong coupling was used according to Sect. 5.1. The expansion of the shower \(\alpha _S\) compared to the \(\overline{\text {MS}}\) coupling suppresses the emission contribution, which leads to the observed behaviour. In addition the NLO correct schemes, which are all of the same accuracy, are performing rather differently in the comparison to data. While scheme 1 (all shower expansions weighted with the full reweights), scheme 3 (all expansions weighted with Sudakov suppression weights^{15}) or scheme 4 (\(\alpha _S\)ratio expansion only weighted with Sudakov suppression) tend to overestimate the data in the softer region, the choice of scheme 2 (Sudakov expansion weighted only with Sudakov suppression) is performing well over the full range of energies. The expansion of the Sudakov form factor is producing a squared logarithmic contribution, which is suppressed by scheme 2. This leads us to make scheme 2 our preferred choice albeit the other schemes are formally of the same accuracy. We propose to take all schemes into account as an evaluation of theoretical uncertainties.
6.7 Comparision to full unitarization
To compare the difference between the improved and the full unitarization we show in Figs. 14 and 15 the transverse momentum of the \(Z^0\) boson in the environment of a \(100\,\text {TeV}\) collider. We choose this energy and a—for this high energy—relatively small merging scale of \(15\,\text {GeV}\) to emphasize possible effects at the merging scale. In both plots we show and normalize the ratio plot to the nonunitarized merging procedure in blue. While at LO the nonunitarized merging has a smooth transition w.r.t. LO+PS (black line) at the merging scale, the unitarization procedures (green and red lines) produce ’kinks’ in the transition region. Already here we observe that the full unitarized distribution (red line), which does not consider the phase space boundaries of the shower, enhances the size of the ’kink’ as more clustered cross section is subtracted from the \(N1\) configurations (here \(pp\rightarrow Z\)). The underlying idea of unitarizing the cross sections in the first place is to be able to add NLO contributions without an ambiguity of \(\mathcal {O}({\alpha _S})\) corrections from the LO merging. The appearance of these kind of ’kinks’ is expected and one of the weak points of the unitarization procedure. We now consider adding NLO corrections as the full unitarized description requires one to add the real emissions with N partons outside the partonshower phase space to the clustered \(N1\) multiplicities. This might help the full unitarization when adding NLO corrections. Our unitarization procedure only adds those in the PSregion to the underlying process configurations.
7 Results
In this section we collect a number of results for simulations with the merging methods set up previously. We describe the simulation setup before we present the results. We begin with jet production from \(e^+e^\) annihilation at LEP before going towards W and Z boson production with additional jets at the LHC. Higgs boson production in association with additional jets has been chosen as a special case as here the higherorder corrections are known to be particularly sizeable. We finally present results for dijet production at the LHC where the Born process already has all legs colour charged. We focus on the effect of merging more and more processes to the Born setup, either with additional legs or with additional virtual corrections. We occasionally also vary the merging scale.
7.1 Simulation setup
We briefly summarize the parameters that we use in the following simulations which are important for our discussion from the point of view of perturbation theory. All parameters are used throughout unless explicitly stated.
We use the MMHT2014nlo68cl [65] NLO PDF set interfaced to Herwig via LHAPDF6 [66] for LO and NLO MEs in order to have a common basis for all samples. We use an implementation of \(\alpha _S\) with twoloop running and fixed \(\alpha _S(M_Z) = 0.118\). \(\alpha _S\) is modified with the CMW scheme, cf. Sect. 5.1. All LO MEs are obtained from MadGraph [26] via a dedicated interface. In addition, we use ColorFull [67] for colour correlations. The merging scale for LEP is always \(\rho = 4\,\text {GeV}\), while for LHC we use \(\rho = 10\,\text {GeV}\).
7.2 LEP
When comparing to data, we first consider data taken from hadron production processes in \(e^+e^\) annihilation at LEP. Here, we find the cleanest environment regarding the development of QCD cascades. When comparing results from our new simulations to the results at LEP but also relative to the LO only simulation we have to be aware of some caveats. As all other event generators, a large part of the simulation in herwig has been developed with LEP results as the first benchmark. Hence, a large part of the modelling, particularly of hadronic final states, has been adjusted with LEP data as the most important benchmark. Therefore, when we encounter a worsening of our description at the first sight we must not necessarily be surprised. We would expect an improvement of the description of many observables with our improved approach, particularly in regions where they should be dominated by perturbative physics. If this is not the case it might well be that the nonperturbative components of the program had previously been adjusted to compensate for shortcomings in the perturbative description of observables.
Figure 16 shows the differential threejet rate with the Durham jet algorithm as it was measured by the OPAL experiment at LEP. This observable measures the hardness of the second emission from a dijet system. We show the pure LO result ee(0) as well as a result with two extra emissions ee(0, 1, 2). Additional loop corrections are shown in the results \(ee(0^*, 1^*, 2)\) and \(ee(0^*, 1^*, 2^*, 3)\), where the latter is expected to describe even observables related to the fourth jet in the system at NLO accuracy. We vary the renormalization scale used to calculate the ME and the scale of the shower emissions only synchronized by factors of 2 up and down. Because the LO merging does not reduce the scale uncertainties the bands are overlapping at this level. Inclusion of NLO corrections to up to the second additional jet, so up to the \(2 \rightarrow 4\) process, successively improves the scaling behaviour of the simulation and hence reduces the differential uncertainty band from roughly 40% down to a 10% level. In this observable the two simulations with NLO merging give relatively similar results with slight improvements from \(ee(0^*, 1^*, 2^*, 3)\) concerning the scale variations.
Using an enhanced (“tuned”) \(\alpha _S\) and correcting the additional emissions in the \(\overline{\mathrm{MS}}\)scheme would lead to an over shooting in the data description and tuning would then require a reduced \(\alpha _S\) value. The various contributions are produced by using scheme 2 as described in Sect. 4.4.
7.3 \(\mathbf {Z^0}\) boson production at LHC
In this section we describe the simulation results for \(Z^0\) boson production, i.e. we consider final states with a lepton pair in the mass range \(m_{ll}\in [66,116]\,\text {GeV}\), which we call a \(Z^0\) boson in the following. We show three plots (Figs. 20, 21 and 22) that each exhibit a specific property of the unitarized merging. Note that in Fig. 6 we already show the transverse momentum of the \(Z^0\) boson in a merged simulation with two LO contributions, see Sect. 6.3, with a particular focus on the merging scale. All figures show distributions for Z(0), Z(0, 1, 2), \(Z(0^*,1^*,2)\) and \(Z(0^*,1^*,2,3)\) with scale variation bands as described above (cf. Sect. 5.2). For Z(0, 1, 2) we also show the variation of the merging scale, \(5\,\text {GeV}< \rho < 20\,\text {GeV}\), as a hashed blue band.
The rapidity of the \(Z^0\) boson, see Fig. 21, is an inclusive observable with respect to partonshower effects. Z(0, 1, 2) is flat compared to Z(0) and receives contributions from unordered histories above the hard scale of the shower, which leads to a slight enhancement of the cross section. The contributions with NLO corrections are enhanced due to the K factor of the NLO \(Z^0\) production process and are more stable with respect to scale variations.
In Fig. 22 we compare our simulation with the jet multiplicity as it was measured in [70]. Without the contribution of the \(pp\rightarrow Zjj\) cross sections the phase space for the second partonshower emission, which is not corrected for in either Z(0, 1) or in \(Z^0\) production in NLO matching, is not capable of describing higher jet multiplicities. The recoiling of the two jets is suppressed for shower emissions, leading to an undershooting in backtoback configurations with respect to the measured data.
7.4 \(\mathbf {W^\pm }\) boson production at LHC
Closely linked to \(Z^0\) boson production is the production of a single \(W^\pm \) boson, which we consider in the leptonic channel. As the neutrino of the W decay leaves the detector, the transverse momentum is less well measured. Closely linked to the transverse momentum but in this study more interesting are the splitting scales of the \(k_T\) algorithm. These are resolution scales \(d_i\) of the \(k_T\) algorithm at which the event switches from an i jet event to an \(i+1\) jet event in \(W^\pm \) boson production. Figure 23 shows our simulated result of the \(\sqrt{d_0}\) distribution, compared to data from ATLAS [71]. We show W(0), W(0, 1, 2) and \(W(0^*,1^*,2)\) with the full event generator setup including MPI and hadronization effects. In addition, we gradually switch off the nonperturbative effects for \(W(0^*,1^*,2)\). With neither MPI nor hadronization included, the pure QCD process tends to overshoot the data at low scales. Inclusion of MPI corrects for the medium range scales at approximately \(10\,\text {GeV}\) at LHC energies but undershoots the data for very low scales. Only the full simulation describes data from large scales down to very small splitting scales of a few GeV. Note that we do not show the scale variations in this case for the sake of clarity. The scale variation is very similar to the scale dependence in the case of \(Z^0\) production.
7.5 Higgs boson production in the LHC environment
Higgs production is delicate due to the enormous NLO corrections to the production process as well as for higher jet multiplicities. The production is simulated via an effective ggHvertex and has, due to the gluon initial state and the colour factor, a rather large emission probability. The Sudakov peak is around \(10\,\text {GeV}\). In spite of the need for resummation at the Sudakov peak we choose the merging scale to be of the same order as for the other processes at the LHC, \(\rho =15\,\text {GeV}\). In Fig. 25 the contributions for the transverse momentum of the Higgs boson are shown for LO and NLO merging with up to two additional legs and loop corrections for two different schemes.
For LO we show four distributions. The pure parton shower in black which is apart from statistical fluctuation identical to the merging with an additional jet multiplicity if the ME are replaced with the dipole content H(0, 1) in grey; see Sect. 6.3. The inclusion of the correct ME contributions for the first H(0, 1) or second emission H(0, 1, 2), red and green, respectively, slightly change the behaviour in the ME region, and due to unitarization also the region below the merging scale.
The inclusion of NLO corrections to the production process, \(H(0^*,1)\), enhances the contribution without introducing a kink at the merging scale \(\rho \). Further NLO corrections to the process with one additional jet \(H(0^*,1^*,2)\) are then scheme dependent, see Sect. 6.6. The preferred scheme 2 (in blue) for dijet production at LEP introduces a enhancement below the merging scale, which becomes visible only for low merging scales as it was chosen here. By using the alternative scheme 1, where the Sudakov expansion is treated as the expansion of the \(\alpha _S\) ratios, a smoother transition at the Sudakov peak is produced. Note, that the choice of scheme is above the claimed accuracy and needs to be treated as an uncertainty estimation. Further more we want to point out that the corrections to the production process, are allowed to emit into the full shower phase space for \(H(0^*,1)\), but are vetoed for the \(H(0^*,1^*,2)\) process. In the case of \(H(0^*,1^*,2)\) the \({\mathcal {O}}(\alpha _S)\) contributions to the process with an additional emission are unitarized to the H(0) phase space. The rather smooth transition at the merging scale is therefore due to a compensation of corrections of the production and the oneadditional emission process.
7.6 Dijet production at LHC
The last process we consider in this paper is the production of dijets at a hadron collider. In contrast to the production of a single vector boson the scale of the production process is more ambiguous in this case. Scale choices like the invariant dijet mass \(m_{jj}\), the scalar sum of the transverse momenta \(H_T\) and the transverse momentum of the hardest jet are reasonable for the production process. For the results shown here, we choose the transverse momentum of the hardest jet to be the shower starting scale as well as the renormalization and factorization scale. As there is only the dijet system in the first place at LO this coincides with the \(p_T\) of either one of the two partons. We require a \(p_T\) cut on a single inclusive jet of \(20\,\text {GeV}\) for the production process. Here we only show samples for J(0), J(0, 1) and \(J(0^*,1)\), while we leave a detailed study with higher multiplicities for future work.
The observable pictured in Fig. 27 reflects the R separation of the two leading jets and has two regions of interest. The first region is \(\varDelta R_{jj}\ge \pi \), which is already present in fixedorder dijet production at LO as \(\varDelta \phi _{jj}=\pi \) here. The region \(\varDelta R_{jj}<\pi \) can only be filled by additional emissions, either the fixedorder NLO or the parton shower. The merged samples hardly alter the region above \(\pi \), while the contribution below is modified by the inclusion of the cross section to the process with an additional emission. Note that the scale uncertainty band of the NLO corrected contribution shrinks in the ’inclusive’ region above \(\pi \) and shows larger variations below \(\pi \).
In Fig. 28 the previously described contributions are compared to data measured by the ATLAS collaboration [73]. While the pure LO+PS contribution J(0) tends to overshoot the data for the third jet, the merged samples provide a better description for this multiplicity. In all three cases the description of higher jet multiplicities tend to undershoot the measured data.
8 Conclusion and outlook
In this paper we have presented a new algorithm to combine NLO QCD calculations for the production of multiple jets together with a parton shower using a modified unitarized approach. We have implemented this algorithm based on the Matchbox NLO module and together with the dipole shower evolution as available in the Herwig event generator to obtain a flexible and fullyrealistic simulation of collider final states. The implementation will become available with the 7.1 release of Herwig.
Improving on just NLO matched simulations, the modified unitarization procedure allows us to remove contributions which can lead to spurious terms in inclusive cross sections at parametrically the same level as NLO QCD corrections. At the same time this allows us to preserve finite enhancements at higher jet multiplicity by only subtracting those contributions in the improved unitarization which the parton shower would have included in the Sudakov supression of the respective, lower multiplicity. A strict unitarization of these contributions would otherwise have lead to unphysical predictions. In order to arrive at this level of simulation it is crucial to identify which contributions can lead to logarithmic enhancements, and which momentum configurations are treated as contributing to an additional hard subprocess.
In comparison to other, existing schemes our approach is in the line of unitarized NLOmerging algorithms [43, 46] and is therefore fundamentally different to approaches like [37, 39, 40, 41]. For a further comparison to other unitarized approaches [43, 46] we summarize main differences: With the implementation in a dipole shower where momentum is preserved at each cluster and shower step the necessity of a careful evaluation at each step of the shower history required a new vetoing algorithm that allows emissions from lower multiplicities if the configuration cannot be identified as part of the ME region. We further loosen the unitarization requirement, by restricting the phase space of the unitarization. As a result, the identified clustered parts of the cross section are used to replace the shower accessible phase space only. In addition, the treatment of the real emission is differential below the merging scale [43].
We have performed a detailed comparison to available collider data. We have investigated the impact of formally subleading ambiguities in order to estimate the theoretical accuracy of the advocated procedure. We do find the expected improvements, namely an overall improved description of multiple jet emissions and a reduction of the associated scale uncertainties. We also find that remaining ambiguities from formally higherorder terms can be large and that our approach allows an estimate of the size of these effects. The method is expected to shed light on dominant contributions at even higher orders, with NNLO QCD corrections in reach for a combination with the modified unitarized merging algorithm.
Footnotes
 1.
MultiLeg Merging in [7].
 2.
Catani, Krauss, Kuhn and Webber [8].
 3.
Lönnblad [9].
 4.
Note that we are here restricted to \(p^T\)ordered shower algorithms like the Herwig dipole shower.
 5.
Possible clusterings are defined by the shower/subtraction algorithm. This includes e.g. for final states the combination of same flavour quarks (or two gluons) to a gluon or the clustering of a quark gluon system to a quark.
 6.
The entries of a selector can be accessed randomly according to the weight that has been assigned to the entry.
 7.
The procedure is performed on a stepbystep basis. We do not evaluate all possible chains. The change due to calculating the weights of full chains might be part of further investigation.
 8.
We here refer to logarithms in the partonshower’s resolution scale, i.e. jet resolution corresponding to a jet algorithm acting inverse to the parton shower. In general, claims of the logarithmic accuracy of a parton shower can only be made in a casebycase and observabledependent context.
 9.
Catani, Marchesini and Webber in [59].
 10.
In the CMW scheme parts of the NLO corrections for simple coloursinglet production and decays are already covered in the choice of \(\Lambda _{QCD}\).
 11.
Note that this is of the same level of accuracy for an NLO calculation.
 12.
For partonshower accuracy we require a leading log (double logarithmic) accuracy in terms of the shower evolution variable.
 13.
For numerical stability we choose \(\rho /3\) as a dynamical cut for the separation between the fully dipole subtracted and shower approximated subtracted region.
 14.
For most of the results made use of RIVET [60].
 15.
Note that at LEP no PDF reweighting is needed.
Notes
Acknowledgements
The authors want to thank the other members of the Herwig collaboration for continuous discussions and support. We especially thank Mike Seymour and Michael Rauch for a careful reading of the manuscript. Further we thank David Grellscheid for his support with technical aspects of the code. This work was supported by the European Union Marie Curie Research Training Network MCnetITN, under contract PITNGA2012315877. SP acknowledges support by a FP7 Marie Curie Intra European Fellowship under Grant Agreement PIEFGA2013628739 during part of this project, and the kind hospitality of the ErwinSchrödingerInstitute at Vienna while part of this work has been finalized.
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