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The Cascade3 Monte Carlo event generator based on Transverse Momentum Dependent (TMD) parton densities is described. Hard processes which are generated in collinear factorization with LO multileg or NLO parton level generators are extended by adding transverse momenta to the initial partons according to TMD densities and applying dedicated TMD parton showers and hadronization. Processes with off-shell kinematics within kt-factorization, either internally implemented or from external packages via LHE files, can be processed for parton showering and hadronization. The initial state parton shower is tied to the TMD parton distribution, with all parameters fixed by the TMD distribution.

The Parton Branching method offers a Monte Carlo solution to the DGLAP evolution equations by incorporating Sudakov form factors. In this approach, the Sudakov form factor can be divided into perturbative and non-perturbative components, with the non-perturbative part being analytically calculable under specific conditions. We first examine forward evolution and demonstrate that including soft and non-perturbative gluons (through the non-perturbative Sudakov form factor) is essential for the proper cancellation of divergent terms in parton density evolution. This non-perturbative component is also important for Transverse Momentum Dependent (TMD) parton distributions, and within the Parton Branching framework, it is constrained by fits to inclusive collinear parton densities. Additionally, we explore the impact of this non-perturbative Sudakov form factor on backward parton evolution and its effects on parton and hadron spectra originating from initial state showers. Our results show that soft and non-perturbative gluons significantly influence inclusive distributions, such as Drell–Yan transverse momentum spectra. However, we found that soft and non-perturbative gluons have a minimal impact on final state hadron spectra and jets.

A bstract Soft-gluon resolution scales characterize parton branching Monte Carlo implementations of the evolution equations for parton distribution functions in Quantum Chromodynamics (QCD). We examine scenarios with dynamical, i.e., branching-scale dependent, resolution scale, and discuss physical implications for both collinear and transverse-momentum dependent (TMD) distributions. We perform the first determination of parton distributions with dynamical resolution scale, at next-to-leading order (NLO) in perturbation theory, from fits to precision deep-inelastic scattering measurements from HERA. We present an application of TMD distributions with dynamical resolution scale to Drell-Yan lepton-pair transverse momentum spectra at the LHC and lower-energy experiments, and comment on the extraction of non-perturbative intrinsic- k T parameters from Drell-Yan data at small transverse momenta.

Understanding the intrinsic transverse momentum (intrinsic- k T ) of partons within colliding hadrons, typically modeled with a Gaussian distribution characterized by a specific width (the intrinsic- k T width), has been an extremely challenging issue. This difficulty arises because event generators like Pythia 8 require an intrinsic- k T width that unexpectedly varies with collision energy, reaching unphysical values at high energies. This paper investigates the underlying physics behind this energy dependence in Pythia 8, revealing that it arises from an interplay between two non-perturbative processes: the internal transverse motion of partons and non-perturbative soft gluon emissions. These contributions are most constrained in the production of Drell–Yan pairs with very low transverse momentum, where soft gluon effects become increasingly prominent with rising collision energy-contrary to initial expectations. Through a detailed analysis of the Sudakov form factor and its influence on intrinsic- k T width, we clarify the observed energy scaling behavior in Pythia 8, providing insight into a longstanding issue in parton shower modeling.

It has been observed in the literature that measurements of low-mass Drell–Yan (DY) transverse momentum spectra at low center-of-mass energies s are not well described by perturbative QCD calculations in collinear factorization in the region where transverse momenta are comparable with the DY mass. We examine this issue from the standpoint of the Parton Branching (PB) method, combining next-to-leading-order (NLO) calculations of the hard process with the evolution of transverse momentum dependent (TMD) parton distributions. We compare our predictions with experimental measurements at low DY mass, and find very good agreement. In addition we use the low mass DY measurements at low s to determine the width q s of the intrinsic Gauss distribution of the PB-TMDs at low evolution scales. We find values close to what has earlier been used in applications of PB-TMDs to high-energy processes at the Large Hadron Collider (LHC) and HERA. We find that at low DY mass and low s even in the region of p T / m DY ∼ 1 the contribution of multiple soft gluon emissions (included in the PB-TMDs) is essential to describe the measurements, while at larger masses ( m DY ∼ m Z ) and LHC energies the contribution from soft gluons in the region of p T / m DY ∼ 1 is small.

The azimuthal correlation, Δϕ12, of high transverse momentum jets in pp collisions at s=13 TeV is studied by applying PB-TMD distributions to NLO calculations via MCatNLO together with the PB-TMD parton shower. A very good description of the cross section as a function of Δϕ12 is observed. In the back-to-back region of Δϕ12→π, a very good agreement is observed with the PB-TMD Set 2 distributions while significant deviations are obtained with the PB-TMD Set 1 distributions. Set 1 uses the evolution scale while Set 2 uses transverse momentum as an argument in αs, and the above observation therefore confirms the importance of an appropriate soft-gluon coupling in angular ordered parton evolution. The total uncertainties of the predictions are dominated by the scale uncertainties of the matrix element, while the uncertainties coming from the PB-TMDs and the corresponding PB-TMD shower are very small. The Δϕ12 measurements are also compared with predictions using MCatNLO together Pythia8, illustrating the importance of details of the parton shower evolution.

We reconsider the associated Z boson and charm or beauty jet production at the LHC by paying special attention to the formation dynamics of heavy jets. Two different approaches are studied: first one, where heavy quarks are produced in the hard scattering subprocesses, implemented in the Monte-Carlo generator pegasus, and another method, where the hard scattering is calculated at NLO with MadGraph5_aMC@NLO and TMD parton shower is included (implemented in the Monte-Carlo generator Cascade3). We compare the predictions obtained in both schemes with latest experimental data for associated Z+b production cross sections and the relative production rate σ(Z+c)/σ(Z+b) collected by the ATLAS and CMS Collaborations at s=13 TeV. We introduce two kinematic observables (denoted as zb and pTrel) which can be used to discriminate between the heavy jet production mechanisms. Using these variables we trace the shape of the simulated b-jet events and recommend that these observables be taken into consideration in the forthcoming experimental analyses.

The internal motion of partons inside hadrons has been studied through its impact on very low transverse momentum spectra of Drell–Yan (DY) pairs created in hadron-hadron collisions. We study DY production at next-to-leading order using the Parton Branching (PB) method which describes the evolution of transverse momentum dependent parton distributions. The main focus is on studying the intrinsic transverse momentum distribution (intrinsic- k T ) as a function of the center-of-mass energy s . While collinear parton shower Monte Carlo event generators require intrinsic transverse momentum distributions strongly dependent on s , this is not the case for the PB method. We perform a detailed study of the impact of soft parton emissions. We show that by requiring a minimal transverse momentum, q 0 , of a radiated parton, a dependence of the width of the intrinsic- k T distribution as a function of s is observed. This dependence becomes stronger with increasing q 0 .

The Parton Branching (PB) method describes the evolution of transverse momentum dependent (TMD) parton distributions, covering all kinematic regions from small to large transverse momenta k T . The small k T -region is very sensitive both to the contribution of the intrinsic motion of partons (intrinsic k T ) and to the resummation of soft gluons taken into account by the PB TMD evolution equations. We study the role of soft-gluon emissions in TMD as well as integrated parton distributions. We perform a detailed investigation of the PB TMD methodology at next-to-leading order (NLO) in Drell–Yan (DY) production for low transverse momenta. We present the extraction of the nonperturbative “intrinsic- k T ” distribution from recent measurements of DY transverse momentum distributions at the LHC across a wide range in DY masses, including a detailed treatment of statistical, correlated and uncorrelated uncertainties. We comment on the (in)dependence of intrinsic transverse momentum on DY mass and center-of-mass energy, and on the comparison with other approaches.

Azimuthal correlations in Z+jet production at large transverse momenta are computed by matching Parton-Branching (PB) TMD parton distributions and showers with NLO calculations via MCatNLO. The predictions are compared with those for dijet production in the same kinematic range. The azimuthal correlations Δϕ between the Z boson and the leading jet are steeper compared to those in dijet production at transverse momenta O(100) GeV , while they become similar for very high transverse momenta O(1000) GeV . The different patterns of Z+jet and dijet azimuthal correlations can be used to search for potential factorization-breaking effects in the back-to-back region, which depend on the different color and spin structure of the final states and their interferences with the initial states. In order to investigate these effects experimentally, we propose to measure the ratio of the distributions in Δϕ for Z+jet- and multijet production at low and at high transverse momenta, and compare the results to predictions obtained assuming factorization. We examine the role of theoretical uncertainties by performing variations of the factorization scale, renormalization scale and matching scale. In particular, we present a comparative study of matching scale uncertainties in the cases of PB-TMD and collinear parton showers.