We revisit the basic steps necessary to obtain next-to-leading-logarithmic accurate small-x results for the DGLAP splitting functions, and their implementations within the HELL framework. We derive new analytical all-order results for the leading-logarithmic gg anomalous dimension, the qg and gg finite Green functions, and most importantly for the qg anomalous dimension, which allows us to arrive for the first time at a properly resummed qg splitting kernel. We use these results as cornerstones of a new implementation of small-x splitting-function resummation which is more solid and numerically better behaved with respect to those available thus far. All of these novelties are included in the upcoming 4.0 version of HELL.

We present a determination of the parton distribution functions (PDFs) of the proton from HERA data using a PDF parametrization inspired by a quantum statistical model of the proton dynamics. This parametrization is characterised by a very small number of parameters, yet it leads to a reasonably good description of the data, comparable with other parametrizations on the market. It may thus provide an alternative to standard parametrizations, useful for studying parametrization bias and to possibly simplify the fit procedure thanks to the small number of parameters. Interestingly, the model reproduces key physical features, such as a d̅ distribution larger than u̅ , that HERA data alone are not able to constrain when using more flexible parametrizations. Moreover, polarized distributions are described in the model by the same parameters of the unpolarized ones, giving us the possibility of extracting both types of distributions within the same fit.

We consider threshold resummation of rapidity distributions, for which various approaches exist in the literature. Recently, a work by Lustermans, Michel, Tackmann suggested that older approaches by Becher, Neubert, Xu (BNX) and Bonvini, Forte, Ridolfi (BFR) were wrong because they miss some leading power contributions at threshold. In this work, we prove and demonstrate that the BNX and BFR approaches are correct and able to resum threshold logarithms to leading power accuracy. We then show that the BNX and BFR approaches can provide rather good alternatives to more modern approaches to threshold resummation of rapidity distributions, provided the threshold logarithms are resummed according to the ψ -soft definition introduced in the context of Higgs production.

We consider the production of a heavy quark pair in proton-proton collisions. For bottom and charm quarks, the final state invariant mass is typically much smaller than the collider energy (e.g. at the LHC), so that high-energy logarithms may spoil the perturbativity of the theoretical prediction at fixed order. The resummation of these logarithms to all orders is thus needed to obtain reliable predictions. In this work, we extend previous results on high-energy (or small-x) resummation to differential distributions in rapidity, transverse momentum and invariant mass, and implement them in the public code HELL.

I review recent progress in the determination of PDFs with the inclusion of small-x resummation, and its impact in precision phenomenology, and discuss future prospects.

The Large Hadron electron Collider (LHeC) is designed to move the field of deep inelastic scattering (DIS) to the energy and intensity frontier of particle physics. Exploiting energy recovery technology, it collides a novel, intense electron beam with a proton or ion beam from the High Luminosity--Large Hadron Collider (HL-LHC). The accelerator and interaction region are designed for concurrent electron-proton and proton-proton operation. This report represents an update of the Conceptual Design Report (CDR) of the LHeC, published in 2012. It comprises new results on parton structure of the proton and heavier nuclei, QCD dynamics, electroweak and top-quark physics. It is shown how the LHeC will open a new chapter of nuclear particle physics in extending the accessible kinematic range in lepton-nucleus scattering by several orders of magnitude. Due to enhanced luminosity, large energy and the cleanliness of the hadronic final states, the LHeC has a strong Higgs physics programme and its own discovery potential for new physics. Building on the 2012 CDR, the report represents a detailed updated design of the energy recovery electron linac (ERL) including new lattice, magnet, superconducting radio frequency technology and further components. Challenges of energy recovery are described and the lower energy, high current, 3-turn ERL facility, PERLE at Orsay, is presented which uses the LHeC characteristics serving as a development facility for the design and operation of the LHeC. An updated detector design is presented corresponding to the acceptance, resolution and calibration goals which arise from the Higgs and parton density function physics programmes. The paper also presents novel results on the Future Circular Collider in electron-hadron mode, FCC-eh, which utilises the same ERL technology to further extend the reach of DIS to even higher centre-of-mass energies.

We consider the problem of quantifying the uncertainty on theoretical predictions based on perturbation theory due to missing higher orders. The most widely used approach, scale variation, is largely arbitrary and it has no probabilistic foundation, making it not suitable for robust data analysis. In 2011, Cacciari and Houdeau proposed a model based on a Bayesian approach to provide a probabilistic definition of the theory uncertainty from missing higher orders. In this work, we propose an improved version of the Cacciari–Houdeau model, that overcomes some limitations. In particular, it performs much better in case of perturbative expansions with large high-order contributions (as it often happens in QCD). In addition, we propose an alternative model based on the same idea of scale variation, which overcomes some of the shortcomings of the canonical approach, on top of providing a probabilistically-sound result. Moreover, we address the problem of the dependence of theoretical predictions on unphysical scales (such as the renormalization scale), and propose a solution to obtain a scale-independent result within the probabilistic framework. We validate these methods on expansions with known sums, and apply them to a number of physical observables in particle physics. We also investigate some variations, improvements and combinations of the models. We believe that these methods provide a powerful tool to reliably estimate theory uncertainty from missing higher orders that can be used in any physics analysis. The results of this work are easily accessible through a public code named THunc.

We introduce a new parametrization for the parton distribution functions (PDFs) designed to be flexible in the small-x region. We implement it in the xFitter open-source PDF fitting tool, and compare it to the default xFitter parametrization, widely used for many PDF studies, and notably for the HERAPDF determination. We find that we can describe the combined inclusive HERA I+II data using NNLO theory with a significantly higher quality than HERAPDF2.0: the $\chi^{2}$ is reduced by more than 60 units, having used only four more parameters. Our result highlights a significant parametrization bias in the default xFitter parametrization at small x, which would lead to even more dramatic effects when used for higher-energy colliders, where the small-x region is more relevant. We also find that the inclusion of small-x resummation, that was shown in previous studies to lead to similar improvements in the fit quality, further reduces the $\chi^{2}$ by approximately 30 extra units.

There has been a revived interest in small-x resummation in recent times. The main motivation was its success in describing small-x HERA data without the inclusion of non-perturbative corrections. In this contribution, I will review the recent developments in the field.

In response to the 2013 Update of the European Strategy for Particle Physics (EPPSU), the Future Circular Collider (FCC) study was launched as a world-wide international collaboration hosted by CERN. The FCC study covered an energy-frontier hadron collider (FCC-hh), a highest-luminosity high-energy lepton collider (FCC-ee), the corresponding 100km tunnel infrastructure, as well as the physics opportunities of these two colliders, and a high-energy LHC, based on FCC-hh technology. This document constitutes the third volume of the FCC Conceptual Design Report, devoted to the hadron collider FCC-hh. It summarizes the FCC-hh physics discovery opportunities, presents the FCC-hh accelerator design, performance reach, and staged operation plan, discusses the underlying technologies, the civil engineering and technical infrastructure, and also sketches a possible implementation. Combining ingredients from the Large Hadron Collider (LHC), the high-luminosity LHC upgrade and adding novel technologies and approaches, the FCC-hh design aims at significantly extending the energy frontier to 100TeV. Its unprecedented centre of-mass collision energy will make the FCC-hh a unique instrument to explore physics beyond the Standard Model, offering great direct sensitivity to new physics and discoveries.

This note summarizes the state-of-the-art predictions for the cross sections expected for Higgs boson production in the 27 TeV proton-proton collisions of a high-energy LHC, including a full theoretical uncertainty analysis. It also provides projections for the progress that may be expected on the timescale of the high-luminosity LHC and an assessment of the main limiting factors to further reduction of the remaining theoretical uncertainties. https://twiki.cern.ch/twiki/bin/view/LHCPhysics/LHCHXSWG

In response to the 2013 Update of the European Strategy for Particle Physics, the Future Circular Collider (FCC) study was launched, as an international collaboration hosted by CERN. This study covers a highest-luminosity high-energy lepton collider (FCC-ee) and an energy-frontier hadron collider (FCC-hh), which could, successively, be installed in the same 100 km tunnel. The scientific capabilities of the integrated FCC programme would serve the worldwide community throughout the 21st century. The FCC study also investigates an LHC energy upgrade, using FCC-hh technology. This document constitutes the second volume of the FCC Conceptual Design Report, devoted to the electron-positron collider FCC-ee. After summarizing the physics discovery opportunities, it presents the accelerator design, performance reach, a staged operation scenario, the underlying technologies, civil engineering, technical infrastructure, and an implementation plan. FCC-ee can be built with today’s technology. Most of the FCC-ee infrastructure could be reused for FCC-hh. Combining concepts from past and present lepton colliders and adding a few novel elements, the FCC-ee design promises outstandingly high luminosity. This will make the FCC-ee a unique precision instrument to study the heaviest known particles (Z, W and H bosons and the top quark), offering great direct and indirect sensitivity to new physics.

This report comprises the outcome of five working groups that have studied the physics potential of the high-luminosity phase of the LHC (HL-LHC) and the perspectives for a possible future high-energy LHC (HE-LHC).The working groups covered a broad range of topics: Standard Model measurements, studies of the properties ofthe Higgs boson, searches for phenomena beyond the Standard Model, flavor physics of heavy quarks and leptonsand studies of QCD matter at high density and temperature.The work is prepared as an input to the ongoing process of updating the European Strategy for Particle Physics,a process that will be concluded in May 2020.

The discovery of the Higgs boson in 2012, by the ATLAS and CMS experiments, was a success achieved with only a percent of the entire dataset foreseen for the LHC. It opened a landscape of possibilities in the study of Higgs boson properties, Electroweak Symmetry breaking and the Standard Model in general, as well as new avenues in probing new physics beyond the Standard Model. Six years after the discovery, with a conspicuously larger dataset collected during LHC Run 2 at a 13 TeV centre-of-mass energy, the theory and experimental particle physics communities have started a meticulous exploration of the potential for precision measurements of its properties. This includes studies of Higgs boson production and decays processes, the search for rare decays and production modes, high energy observables, and searches for an extended electroweak symmetry breaking sector. This report summarises the potential reach and opportunities in Higgs physics during the High Luminosity phase of the LHC, with an expected dataset of pp collisions at 14 TeV, corresponding to an integrated luminosity of 3 ab$^{-1}$. These studies are performed in light of the most recent analyses from LHC collaborations and the latest theoretical developments. The potential of an LHC upgrade, colliding protons at a centre-of-mass energy of 27 TeV and producing a dataset corresponding to an integrated luminosity of 15 ab$^{-1}$, is also discussed.

In response to the 2013 Update of the European Strategy for Particle Physics (EPPSU), the Future Circular Collider (FCC) study was launched as a world-wide international collaboration hosted by CERN. The FCC study covered an energy-frontier hadron collider (FCC-hh), a highest-luminosity high-energy lepton collider (FCC-ee), the corresponding 100 km tunnel infrastructure, as well as the physics opportunities of these two colliders, and a high-energy LHC, based on FCC-hh technology. This document constitutes the third volume of the FCC Conceptual Design Report, devoted to the hadron collider FCC-hh. It summarizes the FCC-hh physics discovery opportunities, presents the FCC-hh accelerator design, performance reach, and staged operation plan, discusses the underlying technologies, the civil engineering and technical infrastructure, and also sketches a possible implementation. Combining ingredients from the Large Hadron Collider (LHC), the high-luminosity LHC upgrade and adding novel technologies and approaches, the FCC-hh design aims at significantly extending the energy frontier to 100 TeV. Its unprecedented centre-of-mass collision energy will make the FCC-hh a unique instrument to explore physics beyond the Standard Model, offering great direct sensitivity to new physics and discoveries.

We present the first double-resummed prediction of the inclusive cross section for the main Higgs production channel in proton-proton collisions, namely, gluon fusion. Our calculation incorporates to all orders in perturbation theory two distinct towers of logarithmic corrections which are enhanced, respectively, at threshold, i.e., large x, and in the high-energy limit, i.e., small x. Large-x logarithms are resummed to next-to-next-to-next-to-leading logarithmic accuracy, while small-x ones to leading logarithmic accuracy. The double-resummed cross section is furthermore matched to the state-of-the-art fixed-order prediction at next-to-next-to-next-to-leading accuracy. We find that double resummation corrects the Higgs production rate by 2% at the currently explored center-of-mass energy of 13 TeV and its impact reaches 10% at future circular colliders at 100 TeV.

We consider the expansion of small-x resummed DGLAP splitting functions at next-to-leading logarithmic (NLL) accuracy to four-loop order, namely next-to-next-to-next-to-leading order (N3LO). From this, we extract the exact LL and NLL small-x contributions to the yet unknown N3LO splitting functions, both in the standard \( \overline{\mathrm{MS}} \) scheme and in the Q0\( \overline{\mathrm{MS}} \) scheme usually considered in small-x literature. We show that the impact of unknown subleading logarithmic contributions (NNLL and beyond) at N3LO is significant, thus motivating future work towards their computation. Our results will be also needed in future to match NLL resummation to N3LO evolution. In turn, we propose an improved implementation of the small-x resummation and therefore release a new version of the resummation code (HELL 3.0) which contains these changes.

We present a determination of the parton distribution functions of the proton in which NLO and NNLO fixed-order calculations are supplemented by NLLx small-x resummation. Deep-inelastic structure functions are computed consistently at NLO+NLL x or NNLO+NLL x , while for hadronic processes small-x resummation is included only in the PDF evolution, with kinematic cuts introduced to ensure the fitted data lie in a region where the fixed-order calculation of the hard cross-sections is reliable. In all other respects, the fits use the same methodology and are based on the same global dataset as the recent NNPDF3.1 analysis. We demonstrate that the inclusion of small-x resummation leads to a quantitative improvement in the perturbative description of the HERA inclusive and charm-production reduced cross-sections in the small x region. The impact of the resummation in our fits is greater at NNLO than at NLO, because fixed-order calculations have a perturbative instability at small x due to large logarithms that can be cured by resummation. We explore the phenomenological implications of PDF sets with small-x resummation for the longitudinal structure function F L at HERA, for parton luminosities and LHC benchmark cross-sections, for ultra-high-energy neutrino-nucleus cross-sections, and for future high-energy lepton-proton colliders such as the LHeC.

We discuss the recent results obtained in the context of PDF determination with the inclusion of logarithmic resummations and their phenomenological applications.