The parton splittings in a parton shower are ordered according to an ordering variable, for example the transverse momentum of the daughter partons relative to the direction of the mother, the virtuality of the splitting, or the angle between the daughter partons. We analyze the choice of the ordering variable and conclude that one particular choice has the advantage of factoring softer splittings from harder splittings graph by graph in a physical gauge.
We present the determination of Transverse Momentum Dependent (TMD) parton distributions from Monte Carlo parton showers. We investigate the effective TMD distributions obtained from the PYTHIA8 and HERWIG6 parton showers and compare them to the TMD distributions determined within the Parton Branching method.
In this study we tried to emphasize the role parton shower plays in event generation, and in the physics of high energy event generation. We achieved this task by comparing the next-to-leading order and next-to-next-to-leading order results from EERAD3 with the next-to-leading order plus parton shower results from Vincia, and comparing both with real data.
When the initial state evolution of a parton shower is organized according to the standard "backward evolution'' prescription, ratios of parton distribution functions appear in the splitting probabilities. The shower thus organized evolves from a hard scale to a soft cutoff scale. At the end of the shower, one expects that only the parton distributions at the soft scale should affect the results. The other effects of the parton distributions should have cancelled. This means that the kernels for parton evolution should be related to the shower splitting functions. If the initial state partons …
In this paper we outline a new parton shower algorithm based on the Catani-Seymour dipole factorization. Our motivation is to have an algorithm which can naturally cooperate with the NLO calculations.
Presented here is a technique of propagating uncertainties through the parton shower by means of an alternate event weight. This technique provides a mechanism to systematically quantify the effect of variations of certain components of the parton shower leading to a novel approach to probing the physics implemented in a parton shower code and understanding its limitations. Further, this approach can be applied to a large class of parton shower algorithms and requires no changes to the underlying implementation.
A new calculation using off-shell matrix elements with TMD parton densities supplemented with a newly developed initial state TMD parton shower is described. The calculation is based on the KaTie package for an automated calculation of the partonic process in high-energy factorization, making use of TMD parton densities implemented in TMDlib. The partonic events are stored in an LHE file, similar to the conventional LHE files, but now containing the transverse momenta of the initial partons. The LHE files are read in by the CASCADE package for the full TMD parton shower, final state shower and…
We report on the possibility of reweighting parton-shower Monte Carlo predictions for scale variations in the parton-shower algorithm. The method is based on a generalization of the Sudakov veto algorithm. We demonstrate the feasibility of this approach using example physical distributions. Implementations are available for both the parton-shower modules in the Herwig 7 event generator.
One can use more than one scale variable to define the family of surfaces in the space of parton splitting parameters that define the evolution of a parton shower. In an earlier paper, we developed this idea for electron-positron annihilation. Here, we use multiple scale variables for a parton shower with initial state partons. Then we need a more sophisticated analysis because the evolution of parton distribution functions must be coordinated with the parton shower evolution. We make the needed connections more precise than in our earlier work, even for the case of just one scale variable. Th…
We present an algorithm for improving subsequent parton shower emissions by full SU(3) colour correlations in the framework of a dipole-type shower. As a proof of concept, we present results from the first implementation of such an algorithm for a final state shower.
We present an implementation of a parton shower algorithm for hadron colliders and electron-positron colliders based on the dipole factorisation formulae. The algorithm treats initial-state partons on equal footing with final-state partons. We implemented the algorithm for massless and massive partons.
Parton showers are widely used to generate fully exclusive final states needed to compare theoretical models to experimental observations. While, in general, parton showers give a good description of the experimental data, the precise functional form of the probability distribution underlying the event generation is generally not known. The reason is that realistic parton showers are required to conserve four-momentum at each vertex. In this paper we investigate in detail how four-momentum conservation is enforced in a standard parton shower and why this destroys the analytic control of the pr…
We derive a new method for initial-state collinear showering in Monte-Carlo event generators which is based on the use of unintegrated parton correlation functions. Combined with a previously derived method for final-state showering, the method solves the problem of treating both the hard scattering and the evolution kernels to be used in arbitrarily non-leading order. Although we only treat collinear showering, so that further extensions are needed for QCD, we have discovered several new results: (1) It is better to generate exact parton kinematics in the hard scattering rather than with the …
We present a systematic study of differences between NLL resummation and parton showers. We first construct a Markovian Monte-Carlo algorithm for resummation of additive observables in electron-positron annihilation. Approximations intrinsic to the pure NLL result are then removed, in order to obtain a traditional, momentum and probability conserving parton shower based on the coherent branching formalism. The impact of each approximation is studied, and an overall comparison is made between the parton shower and pure NLL resummation. Differences compared to modern parton-shower algorithms for…
We perform a detailed analysis of next-to-leading order plus parton-shower matching in vector-boson-fusion WW production including leptonic decays. The study is performed in the Herwig 7 framework interfaced to VBFNLO 3, using the angular-ordered and dipole-based parton-shower algorithms combined with the subtractive and multiplicative matching algorithms.
We report on re-calculation of the next-to-leading order DGLAP evolution kernels performed in a scheme suited for Monte Carlo simulations of parton cascades (parton showers).
A modified version of the CKKW matrix element merging algorithm is presented, suitable for use in an angular-ordered parton shower, using truncated showers and forced splittings. The algorithm is implemented in the Herwig++ Monte Carlo event generator for the benchmark process e+e- -> hadrons. Results are presented at parton and hadron levels, demonstrating a smooth merging between the matrix elements and parton shower and an improved description of LEP data.
In this talk, we discuss recent developments in combining parton showers and fixed-order calculations. We focus on the UNNLOPS method for matching next-to-next-to-leading order computations to the parton shower, and we present results from Sherpa for Drell-Yan lepton-pair and Higgs-boson production at the LHC.
Parton showers in Monte Carlo event generators reflect to a certain accuracy our understanding of QCD radiation at all orders. For observables sensitive to interjet energy flow in well defined regions of phase space, it has been known for some time that relevant all-order dynamics is substantially more complicated than that encoded via angular ordering in parton shower algorithms, even to leading logarithmic accuracy. Here we investigate the extent of the numerical mismatch between leading logarithmic analytical estimates (resummation) and parton showers in an effort to better understand the a…
This lecture discusses the physics implemented by Monte Carlo event generators for hadron colliders. It details the construction of parton showers and the matching of parton showers to fixed-order calculations at higher orders in perturbative QCD. It also discusses approaches to merge calculations for a varying number of jets, the interface to the underlying event and hadronization.
We perform a detailed study of the sources of perturbative uncertainty in parton shower predictions within the Herwig 7 event generator. We benchmark two rather different parton shower algorithms, based on angular-ordered and dipole-type evolution, against each other. We deliberately choose leading order plus parton shower as the benchmark setting to identify a controllable set of uncertainties. This will enable us to reliably assess improvements by higher-order contributions in a follow-up work.
Motivated by asking how to combine parton showers with nonleading QCD matrix elements, we discuss a subtractive technique based on gauge-invariant Wilson-line operators and how this can be used to treat the soft region.
We perform a detailed study of NLO parton shower matching uncertainties in Higgs boson pair production through gluon fusion at the LHC based on a generic and process independent implementation of NLO subtraction and parton shower matching schemes for loop-induced processes in the Sherpa event generator. We take into account the full top-quark mass dependence in the two-loop virtual corrections and compare the results to an effective theory approximation. In the full calculation, our findings suggest large parton shower matching uncertainties that are absent in the effective theory approximatio…
We consider idealized parton shower event generators that treat parton spin and color exactly, leaving aside the choice of practical approximations for spin and color. We investigate how the structure of such a parton shower generator is related to the structure of QCD. We argue that a parton shower with splitting functions proportional to $α_s$ can be viewed not just as a model, but as the lowest order approximation to a shower that is defined at any perturbative order. To support this argument, we present a formulation for a parton shower at order $α_s ^k$ for any $k$. Since some of the inpu…
Initial state evolution in parton shower event generators involves parton distribution functions. We examine the probability for the system to evolve from a higher scale to a lower scale without an initial state splitting. A simple argument suggests that this probability, when multiplied by the ratio of the parton distributions at the two scales, should be independent of the parton distribution functions. We call this the PDF property. We examine whether the PDF property actually holds using Pythia and Deductor. We also test a related property for the Deductor shower and discuss the physics be…
When parton distributions are falling steeply as the momentum fractions of the partons increases, there are effects that occur at each order in $α_s$ that combine to affect hard scattering cross sections and need to be summed. We show how to accomplish this in a leading approximation in the context of a parton shower Monte Carlo event generator.
We describe an algorithm for improving subsequent parton shower emissions by full SU(3) color correlations in the framework of a dipole-type shower. As a proof of concept, we present results from the first implementation of such an algorithm for a final state shower. The corrections are found to be small for event shapes and jet rates but can be more significant for tailored observables.
We have previously described a mathematical formulation for a parton shower based on the approximation of strongly ordered virtualities of successive parton splittings. Quantum interference, including interference among different color and spin states, is included. In this paper, we add the further approximations of taking only the leading color limit and averaging over spins, as is common in parton shower Monte Carlo event generators. Soft gluon interference effects remain with this approximation. We find that the leading color, spin averaged shower in our formalism is similar to that in othe…
A method is suggested to combine tree level QCD matrix for the production of multi jet final states and the parton shower in hadronic interactions. The method follows closely an algorithm developed recently for the case of $e^+e^-$ annihilations.
We propose a new approach for combining next-to-leading order (NLO) and parton shower (PS) calculations so as to obtain three core features: (a) applicability to general showers, as with the MCatNLO and POWHEG methods; (b) positive-weight events, as with the KrkNLO and POWHEG methods; and (c) all showering attributed to the parton shower code, as with the MCatNLO and KrkNLO methods. This is achieved by using multiplicative matching in phase space regions where the shower overestimates the matrix element and accumulative (additive) matching in regions where the shower underestimates the matrix …
We make a thorough comparison between different schemes of merging fixed-order tree-level matrix element generators with parton-shower models. We use the most basic benchmark of the O(alpha_S) correction to e+e- -> jets, where the simple kinematics allows us to study in detail the transition between the matrix-element and parton-shower regions. We find that the CKKW-based schemes give a reasonably smooth transition between these regions, although problems may occur if the parton shower used is not ordered in transverse momentum. However, the so-called Pseudo-Shower and MLM schemes turn out to …
We investigate angular correlations in multi-jet final states at high-energy colliders and discuss their sensitivity to initial-state showering effects, including QCD coherence and corrections to collinear ordering.
Programs that calculate observables in quantum chromodynamics at next-to-leading order typically generate events that consist of partons rather than hadrons -- and just a few partons at that. These programs would be much more useful if the few partons were turned into parton showers, which could be given to one of the Monte Carlo event generators to produce hadron showers. In a previous paper, we have seen how to generate parton showers related to the final state collinear singularities of the perturbative calculation for the example of e+ + e- --> 3 jets. This paper discusses the treatment of…
In this publication the implementation of a new parton shower model based on the Catani-Seymour dipole factorisation, as first suggested by Z. Nagy and D. E. Soper, is discussed. First results obtained with the new algorithm are compared with experimental data.
We introduce a method for the separation of soft and collinear logarithms in QCD parton evolution at $\mathcal{O}(α_s^2)$ and at leading color. Using an implementation of the technique in the Dire parton shower, we analyze the numerical impact of genuine triple-collinear corrections from quark pair emission in $e^+e^-\to$ hadrons.
We discuss the necessary steps for implementing an angularly ordered (AO) electroweak (EW) parton shower in Herwig 7 multi-purpose event generator. This includes calculating the helicity-dependent quasi-collinear EW branching functions that correspond to the full range of final-state EW parton shower, in addition to the initial-state EW gauge vector boson radiations. The results are successfully embedded in the AO Herwig 7 shower algorithm and have undergone a set of comprehensive and conclusive performance tests. Furthermore, we have used this EW parton shower algorithm, alongside the existin…
Substantial progress has been made recently in the formalisms which form the basis of event generators. I discuss some of the ideas involved, emphasizing the role of higher-order QCD corrections and their interplay with parton showers
We investigate the effects of combining next-to-leading order QCD results with parton-shower effects in $W^+W^-jj$ production via vector-boson fusion including leptonic decays. Using the Herwig 7 framework interfaced to VBFNLO 3, we compare the predictions obtained from the angular-ordered and dipole-based parton shower algorithms combined with subtractive, MC@NLO-type, and multiplicative, Powheg-type, matching. A consistent treatment of renormalisation and factorisation scale variations in the hard process and the parton shower allows to assign more reliable theory uncertainty predictions to …
We give a prescription for attaching parton showers to next-to-leading order (NLO) partonic jet cross sections in electron-positron annihilation. Our method effectively extends to NLO the scheme of Catani, Krauss, Kuhn, and Webber for matching between m hard jets and (m+1) hard jets. The matching between parton splitting as part of a shower and parton splitting as part of NLO matrix elements is based on the Catani-Seymour dipole subtraction method that is commonly used for removing the singularities from the NLO matrix elements.}
One can use more than one scale variable to specify the family of surfaces in the space of parton splitting parameters that define the evolution of a parton shower. Considering $e^+e^-$ annihilation, we use two variables, with shower evolution following a special path in this two dimensional space. In addition, we treat in a special way the part of the splitting function that has a soft emission singularity but no collinear singularity. This leads to certain advantages compared to the usual shower formulation with only one scale variable.
In a companion publication, we have explored how to examine the summation of large logarithms in a parton shower. Here, we apply this general program to the thrust distribution in electron-positron annihilation, using several shower algorithms. The method is to work with an appropriate integral transform of the distribution for the observable of interest. Then, we reformulate the parton shower calculation so as to obtain the transformed distribution as an exponential for which we can compute the terms in the perturbative expansion of the exponent.
It is useful to describe a leading order parton shower as the solution of a linear equation that specifies how the state of the partons evolves. This description involves an essential approximation of a strong ordering of virtualities as the shower progresses from a hard interaction to softer interactions. If this is to be the only approximation, then the partons should carry color and spin and quantum interference graphs should be included. We explain how the evolution equation for this kind of a shower can be formulated. We discuss briefly our efforts to implement this evolution equation num…
We present a study of Higgs hadroproduction through vector-boson fusion at the NLO in QCD matched with parton showers. We discuss the matching systematics affecting this process through a comparison of the aMC@NLO predictions with the POWHEG and the pure-NLO ones.
We present a process-independent technique to consistently combine next-to-leading order parton-level calculations of varying jet multiplicity and parton showers. Double counting is avoided by means of a modified truncated shower scheme. This method preserves both the fixed-order accuracy of the parton-level result and the logarithmic accuracy of the parton shower. We discuss the renormalisation and factorisation scale dependence of the approach and present results from an automated implementation in the Sherpa event generator using the test case of W-boson production at the Large Hadron Colli…
We propose an extension of matrix element plus parton shower merging at tree level to preserve inclusive cross sections obtained from the merged and showered sample. Implementing this constraint generates approximate next-to-leading order (NLO) contributions similar to the LoopSim approach. We then show how full NLO, or in principle even higher order, corrections can be added consistently, including constraints on inclusive cross sections to account for yet missing parton shower accuracy at higher logarithmic order. We also show how NLO accuracy below the merging scale can be obtained.
Several methods to improve the parton-shower description of hard processes by an injection of matrix-element-based information have been presented over the years. In this article we study (re)weighting schemes for the first/hardest emission. One objective is to provide a consistent matching of the POWHEG next-to-leading order generator to the Pythia shower algorithms. Another is to correct the default behaviour of these showers at large transverse momenta, based on a comparison with real-emission matrix elements.
We describe the \POWHEGBOX{} package, a general computer code framework for implementing NLO calculations in Shower Monte Carlo programs according to the \POWHEG{} method.
In this review a new method is presented for attaching parton shower algorithms to NLO partonic jet cross sections in electron-positron annihilation. Our method is based on the Catani-Seymour dipole subtraction method and also uses an adaptation of the matching scheme of Catani, Krauss, Kuhn, and Webber.
I describe MiNNLO$_{\rm PS}$, a novel method to match NNLO QCD computations with Parton Showers, and present a selection of results for color-singlet production at the LHC.
Programs that calculate observables in quantum chromodynamics at next-to-leading order typically suffer from the problem that, when considered as event generators, the events generated consist of partons rather than hadrons and just a few partons at that. Thus the events generated are completely unrealistic. These programs would be much more useful if the few partons were turned into parton showers, in the style of standard Monte Carlo event generators. Then the parton showers could be given to one of the Monte Carlo event generators to produce hadron showers. We show how to generate parton sh…
I give an overview of the recent progress on the matching of fixed-order calculations and parton showers. The focus is on the matching with NNLO QCD corrections as well as with NLO EW ones.
We present a parton-shower matched NNLO QCD calculation for hadronic final state production in Deep Inelastic Scattering. The computation is based on the UNLOPS method and is implemented in the publicly available event generation framework SHERPA. Results are compared to measurements performed by the H1 collaboration.
I discuss the recent implementation of medium-modified splitting functions within the HERWIG angular-ordered parton shower algorithm and present a few results on transverse momentum, energy and angular distributions.
We compute the production cross section of a top-antitop pair in association with a jet at hadron colliders at next-to-leading order accuracy matched with parton shower algorithms to make predictions at the hadron level. The parton shower allows for including the decay of the top quarks at the leading order accuracy. We use a framework based on three well established numerical codes, the POWHEG-BOX, used for the calculation of the cross section, HELAC, which generates the matrix elements for the Born-level, real emission and the virtual part, and finally a parton shower program, such as PYTHIA…
We introduce a backward evolution Monte Carlo algorithm implementation of the CCFM equation and present latest developments in phenomenology of hadron-hadron collisions for the Monte Carlo generator Cascade.
