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.
We propose a method to examine how a parton shower sums large logarithms. In this method, one works with an appropriate integral transform of the distribution for the observable of interest. Then, one reformulates the parton shower so as to obtain the transformed distribution as an exponential for which one can compute the terms in the perturbative expansion of the exponent. We apply this general program to the thrust distribution in electron-positron annihilation, using several shower algorithms. Of the approaches that we use, the most generally applicable is to compute some of the perturbati…
We explore jet physics in hadron collisions using the parton shower event generator Deductor. Of particular interest is the one jet inclusive cross section dsigma/dpT for jets of very high pT. Compared to the Born level, the cross section decreases substantially because of pT loss from the jet during showering. We compare to the same effect in Pythia and Dire. The cross section then increases substantially because of the summation of threshold logarithms included in Deductor. We also study the cross section to have a gap with no jets between two hard jets that are widely separated in rapidity.…
We study the logarithmic accuracy of angular-ordered parton showers by considering the singular limits of multiple emission matrix elements. This allows us to consider different choices for the evolution variable and propose a new choice which has both the correct logarithmic behaviour and improved performance away from the singular regions. In particular the description of $e^+e^-$ event shapes in the non-logarithmic region is significantly improved.
We demonstrate that all the known single- and double-logarithm summations for a parton distribution function can be unified in the Collins-Soper resummation technique by applying soft approximations appropriate in different kinematic regions to real gluon emissions. Neglecting the gluon longitudinal momentum, we obtain the $k_T$ (double-logarithm) resummation for two-scale QCD processes, and the Balitsky-Fadin-Kuraev-Lipatov (single-logarithm) equation for one-scale processes. Neglecting the transverse momentum, we obtain the threshold (double-logarithm) resummation for two-scale processes, an…
Parton showers are among the most widely used tools in collider physics. Despite their key importance, none so far has been able to demonstrate accuracy beyond a basic level known as leading logarithmic (LL) order, with ensuing limitations across a broad spectrum of physics applications. In this letter, we propose criteria for showers to be considered next-to-leading logarithmic (NLL) accurate. We then introduce new classes of shower, for final-state radiation, that satisfy the main elements of these criteria in the widely used large-$N_C$ limit. As a proof of concept, we demonstrate these sho…
We study the electron self--energy in a strong magnetic field when the parameter η\equiv (α/2π) \ln^2 (eB/m^2_0) \sim 1 and explore the transition between the perturbative regime η<<1 and the nonperturbative massless QED regime η>>1 where chiral symmetry is broken spontaneously and electrons acquire the dynamically induced mass.
Standard dipole parton showers are known to yield incorrect subleading-colour contributions to the leading (double) logarithmic terms for a variety of observables. In this work, concentrating on final-state showers, we present two simple, computationally efficient prescriptions to correct this problem, exploiting a Lund-diagram type classification of emission regions. We study the resulting effective multiple-emission matrix elements generated by the shower, and discuss their impact on subleading colour contributions to leading and next-to-leading logarithms (NLL) for a range of observables. I…
We show how perturbation theory may be reorganized to give splitting functions which include order by order convergent sums of all leading logarithms of $x$. This gives a leading twist evolution equation for parton distributions which sums all leading logarithms of $x$ and $Q^2$, allowing stable perturbative evolution down to arbitrarily small values of $x$. Perturbative evolution then generates the double scaling rise of $F_2$ observed at HERA, while in the formal limit $x\to 0$ at fixed $Q^2$ the Lipatov $x^{-λ}$ behaviour is eventually reproduced. We are thus able to explain why leading ord…
A factorization algorithm for a patron shower model based on the evolution of momentum distributions proposed in a previous work is studied. The scaling violation of initial state parton distributions is generated using parton showers to an accuracy of the next-to-leading logarithmic (NLL) order of quantum chromodynamics (QCD) using the information from only splitting functions and initial parton distributions at some fixed low energy. In the algorithm proposed in this paper, the total momentum of the initial state partons is conserved in any factorization scheme for mass singularities. As an …
We develop an event generator, NLL-QEDPS, based on the QED shower including the next-to-leading logarithm correction in the e^+e^- annihilation. The shower model is the Monte Carlo technique to solve the renormalization group equation so that they can calculate contributions of alpha^m log^n(S/m_e^2) for any m and n systematically. Here alpha is the QED coupling, m_e is the mass of electron and S is the square of the total energy in the e^+e^- system. While the previous QEDPS is limited to the leading logarithm approximation which includes only contributions of (alpha log(S/m_e^2))^n, the mode…
A new method to construct event-generators based on next-to-leading order QCD matrix-elements and leading-logarithmic parton showers is proposed. Matrix elements of loop diagrams as well as those of a tree level can be generated using an automatic system. A soft/collinear singularity is treated using a leading-log subtraction method. Higher order re-summation of the soft/collinear correction by the parton shower method is combined with the NLO matrix-element without any double-counting in this method.
Studying the Infrared Behaviour of Improved Logarithmic Accuracy Parton Showers with Herwig
Department of Physics and Astronomy, University of Manchester（曼彻斯特大学物理与天文系）；Institute of Physics, NAWI Graz, University of Graz（格拉茨大学NAWI格拉茨物理研究所）；Particle Physics, Faculty of Physics, University of Vienna（维也纳大学物理学院粒子物理）；Theoretical Physics Department, CERN（欧洲核子研究中心理论物理部）
In the empirically sensible limit in which QCD, t-quark Yukawa, and scalar-field-interaction coupling constants dominate all other Standard-Model coupling constants, we sum all leading-logarithm terms within the perturbative expansion for the effective potential that contribute to the extraction of the Higgs boson mass via radiative electroweak symmetry breaking. A Higgs boson mass of 216 GeV emerges from such terms, as well as a scalar-field-interaction coupling constant substantially larger than that anticipated from conventional spontaneous symmetry breaking. The sum of the effective potent…
As part of a programme to develop parton showers with controlled logarithmic accuracy, we consider the question of collinear spin correlations within the PanScales family of parton showers. We adapt the well-known Collins-Knowles spin-correlation algorithm to PanScales antenna and dipole showers, using an approach with similarities to that taken by Richardson and Webster. To study the impact of spin correlations, we develop Lund-declustering based observables that are sensitive to spin-correlation effects both within and between jets and extend the MicroJets collinear single-logarithmic resumm…
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…
We apply Laplace method for the tree diagrams to calculate the interference terms arising as a result of summation of particles permutations in the final state of scattering. These terms significantly influence the values of experimentally observed quantities. We show that the tree diagram with a single input line and an arbitrary number of output lines has a maximum value for the equal four-momenta of secondary particles. It means the tree diagrams of inelastic scattering describe the formation of parton and hadron showers. We also demonstrate the need to take into account the contributions f…
We introduce a simple procedure that resolves the long-standing question of how to account for single-logarithmic spin-correlation effects in parton showers not just in the collinear limit, but also in the soft wide-angle limit, at leading colour. We discuss its implementation in the context of the PanScales family of parton showers, where it complements our earlier treatment of the purely collinear spin correlations. Comparisons to fixed-order matrix elements help validate our approach up to third order in the strong coupling, and an appendix demonstrates the small size of residual subleading…
We formulate some first fundamental elements of an approach for assessing the logarithmic accuracy of parton-shower algorithms based on two broad criteria: their ability to reproduce the singularity structure of multi-parton matrix elements, and their ability to reproduce logarithmic resummation results. We illustrate our approach by considering properties of two transverse momentum ordered final-state showers, examining features up to second order in the strong coupling. In particular we identify regions where they fail to reproduce the known singular limits of matrix elements. The characteri…
Cross sections for physical processes that involve very different momentum scales in the same process will involve large logarithms of the ratio of the momentum scales when calculated in perturbation theory. One goal of calculations using parton showers is to sum these large logarithms. We ask whether this goal is achieved for the transverse momentum distribution of a Z-boson produced in hadron-hadron collisions when the shower is organized with higher virtuality parton splittings coming first, followed successively by lower virtuality parton splittings. We find that the virtuality ordered sho…
A new method to construct event-generators based on next-to-leading order QCD matrix-elements and leading-logarithmic parton showers is proposed. Matrix elements of loop diagram as well as those of a tree level can be generated using an automatic system. A soft/collinear singularity is treated using a leading-log subtraction method. Higher order re-summation of the soft/collinear correction by the parton shower method is combined with the NLO matrix-element without any double-counting in this method. An example of the event generator for Drell-Yan process is given for demonstrating a validity …
A method of exact all-order summation of leading infrared logarithms in two dimensional massless $Φ^4$-type non-renormalizable effective field theories (EFTs) is developed. The method is applied to the ${\rm O}(N)$-symmetric EFT, which is a two-dimensional sibling of the four dimensional ${\rm O}(N+1)/{\rm O}(N)$ sigma-model. For the first time the exact all-order summation of the $\left(E^{2} \ln(1/E)\right)^n$ contributions (chiral logarithms) for the $2 \to 2$ scattering amplitudes is performed in closed analytical form. The cases when the resulting amplitudes turn to be meromorphic functio…
We study the mismatch between a full calculation of non-global single-logarithms in the large-N_c limit and an approximation based on free azimuthal averaging, and the consequent angular-ordered pattern of soft gluon radiation in QCD. We compare the results obtained in either case to those obtained from the parton showers in the Monte Carlo event generators HERWIG and PYTHIA, with the aim of assessing the accuracy of the parton showers with regard to such observables where angular ordering is merely an approximation even at leading-logarithmic accuracy and which are commonly employed for the t…
The arbitrariness in how the logarithm is defined within the QCD series for the inclusive electroproduction cross-section is shown to affect the summation to all orders in $α_s$ of leading and successively-subleading logarithms within that perturbative series, even though such summations largely eliminate the residual dependence of the original series on the arbitrary renormalization scale $μ$. However, given that the original (unimproved) series is known to third-order in $α_s(μ)$, this logarithm ambiguity is shown not to enter the optimally improved summation-of-logarithms series until the t…
We formulate PanScales parton showers for hadron collisions so as to achieve next-to-leading logarithmic (NLL) accuracy across a broad set of observables. We do so specifically for colour singlet production. Relative to the existing PanScales final-state showers, the main new question is that of how to redistribute momentum imbalances from initial-state branching across the remainder of the event. We presents tests of the showers at fixed order, including the treatment of full colour for soft-collinear emissions and of spin correlations in both the soft and collinear domains. We also include c…
Starting from a factorization theorem in effective field theory, we derive a parton-shower equation for the resummation of non-global logarithms. We have implemented this shower and interfaced it with a tree-level event generator to obtain an automated framework to resum the leading logarithm of non-global observables in the large-$N_c$ limit. Using this setup, we compute gap fractions for dijet processes and isolation cone cross sections relevant for photon production. We compare our results with fixed-order computations and LHC measurements. We find that naive exponentiation is often not ade…
Three-particle correlations in quark and gluon jets are computed for the first time in perturbative QCD. We give results in the double logarithmic approximation and the modified leading logarithmic approximation. In both resummation schemes, we use the formalism of the generating functional and solve the evolution equations analytically from the steepest descent evaluation of the one-particle distribution. We thus provide a further test of the local parton hadron duality and make predictions for the LHC.
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.
A new method for combining QCD matrix elements and parton showers in Monte Carlo simulations of hadronic final states is outlined. The aim is to provide at least a leading-order description of all hard multi-jet configurations together with jet fragmentation to next-to-leading logarithmic accuracy, while avoiding the most serious problems of double counting.
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.
We study the simulation of initial-state radiation in angular-ordered parton showers in order to investigate how different interpretations of the ordering variable affect the logarithmic accuracy of such showers. This also enables us to implement a recoil scheme which is consistent between final-state and initial-state radiation. We present optimal values of the strong coupling and intrinsic transverse momentum to be used in each version of the parton shower, tuned using $Z^0$-boson production at the LHC at 7 TeV. With these tuned showers, we perform a phenomenological study of the Drell-Yan p…
We quantify uncertainties in the Monte-Carlo simulation of inclusive and dijet final states, which arise from using the MC@NLO technique for matching next-to-leading order parton level calculations and parton showers. We analyse a large variety of data from early measurements at the LHC. In regions of phase space where Sudakov logarithms dominate over high-energy effects, we observe that the main uncertainty can be ascribed to the free parameters of the parton shower. In complementary regions, the main uncertainty stems from the considerable freedom in the simulation of underlying events.
The interpretation of experimental measurements at the LHC requires accurate theoretical predictions for exclusive observables, and in particular the summation of soft and collinear radiation to all orders in perturbation theory. We report on recent progress towards the automated calculation of multi-parton LHC cross sections at next-to-leading order in QCD, including the summation of next-to-leading logarithmic corrections through the combination with parton showers.
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…
Invariance of the effective action under changes of the renormalization scale $μ$ leads to relations between those (presumably calculated) terms independent of $μ$ at a given order of perturbation theory and those higher order terms dependent on logarithms of $μ$. This relationship leads to differential equations for a sequence of functions, the solutions of which give closed form expressions for the sum of all leading logs, next to leading logs and subsequent subleading logarithmic contributions to the effective action. The renormalization group is thus shown to provide information about a mo…
We present a parameter-free scheme to combine fixed-order multi-jet results with parton-shower evolution. The scheme produces jet cross sections with leading-order accuracy in the complete phase space of multiple emissions, resumming large logarithms when appropriate, while not arbitrarily enforcing ordering on momentum configurations beyond the reach of the parton-shower evolution equation. This requires the development of a matrix-element correction scheme for complex phase-spaces including ordering conditions as well as a systematic scale-setting procedure for unordered phase-space points. …
We propose a method for combining QCD matrix elements and parton showers in Monte Carlo simulations of hadronic final states in $e^+e^-$ annihilation. The matrix element and parton shower domains are separated at some value $y_{ini}$ of the jet resolution, defined according to the $k_T$-clustering algorithm. The matrix elements are modified by Sudakov form factors and the parton showers are subjected to a veto procedure to cancel dependence on $y_{ini}$ to next-to-leading logarithmic accuracy. The method provides a leading-order description of hard multi-jet configurations together with jet fr…
Using renormalization-group methods, we derive differential equations for the all-orders summation of logarithmic corrections to the QCD series for R(s) = sigma(e^+ e^- --> hadrons)/sigma(e^+ e^- --> mu^+ mu^-), as obtained from the imaginary part of the purely-perturbative vector-current correlation function. We present explicit solutions for the summation of leading and up to three subsequent subleading orders of logarithms. The summations accessible from the four-loop vector-correlator not only lead to a substantial reduction in sensitivity to the renormalization scale, but necessarily impo…
We present the medium-modified energy collimation in the leading-logarithmic approximation (LLA) and next-to-leading-logarithmic approximation (NLLA) of QCD. As a consequence of more accurate kinematic considerations in the argument of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) fragmentation functions (FFs) we find a new NLLA correction ${\cal O}(α_s)$ which accounts for the scaling violation of DGLAP FFs at small $x$. The jet shape is derived from the energy collimation within the same approximations and we also compare our calculations for the energy collimation with the event ge…
The use of QCD calculations that include the resummation of soft-collinear logarithms via parton-shower algorithms is currently not possible in PDF fits due to the high computational cost of evaluating observables for each variation of the PDFs. Unfortunately the interpolation methods that are otherwise applied to overcome this issue are not readily generalised to all-order parton-shower contributions. Instead, we propose an approximation based on training a neural network to predict the effect of varying the input parameters of a parton shower on the cross section in a given observable bin, i…
We discuss the GENEVA Monte Carlo framework, which combines higher-order resummation (NNLL) of large Sudakov logarithms with multiple next-to-leading-order (NLO) matrix-element corrections and parton showering (using PYTHIA8) to give a complete description at the next higher perturbative accuracy in alpha_s at both small and large jet resolution scales. Results for e+e- -> jets compared to LEP data and for Drell-Yan production are presented.
Large logarithms that arise in cross sections due to the collinear and soft singularities of QCD are traditionally treated using parton showers or analytic resummation. Parton showers provide a fully-differential description of an event but are challenging to extend beyond leading logarithmic accuracy. On the other hand, resummation calculations can achieve higher logarithmic accuracy but often for only a single observable. Recently, there have been many resummation calculations that jointly resum multiple logarithms. Here we investigate the benefits and limitations of joint resummation in a c…
We present predictions for hadronic decays of the Higgs boson at next-to-next-to-leading order (NNLO) in QCD matched with parton shower based on the POWHEG framework. Those include decays into bottom quarks with full bottom-quark mass dependence, light quarks, and gluons in the heavy top quark effective theory. Our calculations describe exclusive decays of the Higgs boson with leading logarithmic accuracy in the Sudakov region and next-to-leading order (NLO) accuracy matched with parton shower in the three-jet region, with normalizations fixed to the partial width at NNLO. We estimated remaini…
We carry out extensive tests of the next-to-leading logarithmic (NLL) accuracy of the PanScales parton showers, as introduced recently for colour-singlet production in hadron collisions. The tests include comparisons to (semi-)analytic NLL calculations of a wide range of hadron-collider observables: the colour-singlet boson transverse momentum distribution; global and non-global hadronic energy flow variables related to jet vetoes and analogues of jettiness distributions; (sub)jet multiplicities; and observables sensitive to the DGLAP evolution of the incoming momentum fractions. In the tests,…
We introduce a new efficient algorithm for phase space generation. A parton shower is used to distribute events across all of multiplicity, flavor, and phase space, and these events can then be reweighted to any desired analytic distribution. To verify this method, we reproduce the e+e- -> n jets tree-level result of traditional matrix element tools. We also show how to improve tree-level matrix elements automatically with leading-logarithmic resummation. This algorithm is particularly useful in the context of a new framework for event generation called GenEvA. In a companion paper [arXiv:0801…
We present a method to combine next-to-leading order (NLO) matrix elements in QCD with leading logarithmic parton showers by applying a suitably modified version of the phase-space-slicing method. The method consists of subsuming the NLO corrections into a scale-dependent phase-space-slicing parameter, which is then automatically adjusted to cut out the leading order, virtual, soft and collinear contributions in the matrix element calculation. In this way a positive NLO weight is obtained, which can be redistributed by a parton shower algortihm. As an example, we display the method for single-…
Studies on jet substructure have evolved significantly in recent years. Jet substructure is essentially determined by QCD radiations and non-perturbative effects. Predictions of jet substructure are usually different among Monte Carlo event generators, and are governed by the parton shower algorithm implemented. For leading logarithmic parton shower, even though one of the core variables is the evolution variable, its choice is not unique. We examine evolution variable dependence of the jet substructure by developing a parton shower generator that interpolates between different evolution varia…
In the absence of a tree-level scalar-field mass, renormalization-group methods permit the explicit summation of leading-logarithm contributions to all orders of the perturbative series within the effective potential for $SU(2)\times U(1)$ electroweak symmetry. This improvement of the effective potential function is seen to reduce residual dependence on the renormalization mass scale. The all-orders summation of leading logarithm terms involving the dominant three couplings contributing to radiative corrections is suggestive of a potential characterized by a plausible Higgs boson mass of 216 G…
The dipole formalism provides a powerful framework from which parton showers can be constructed. In a recent paper, we proposed a dipole shower with improved colour accuracy and in this paper we show how it can be further improved. After an explicit check at $\mathcal{O}(α_{\mathrm{s}}^{2})$ we confirm that our original shower performs as it was designed to, i.e. inheriting its handling of angular-ordered radiation from a coherent branching algorithm. We also show how other dipole shower algorithms fail to achieve this. Nevertheless, there is an $\mathcal{O}(α_{\mathrm{s}}^{2})$ topology where…
In these proceedings we present three possible interpretations of the ordering variable implemented in the Herwig7 angular-ordered parton shower. Each interpretation determines a different recoil-scheme prescription and we show how it can impact the logarithmic accuracy of the algorithm. We also present comparisons with LEP data.
