Resummation, Evolution, Factorization 2023 (REF2023)

In this talk I discuss applications of Soft Collinear Effective Theory (SCET) to TMD observables at leading and subleading power. In particular, I discuss matching between resummed and fixed order results to obtain cross sections accurate for both small and large transverse momentum simultaneously, and applications like the transverse lepton pT in W production. I also discuss the collinear and soft ingredients needed for bare factorization for SIDIS, Drell-Yan, and e+e- to dihadrons at subleading and subsubleading power.

We consider the factorization of transverse momentum-dependent azimuthal asymmetries in semi-inclusive DIS (SIDIS), Drell-Yan and e+e- to dihadron, which appear at the first subleading power in the expansion for small transverse momentum. These observables feature an intricate all-order resummation structure common to many applications of SCET at subleading power. In addition, and perhaps more importantly, they provide access to nonperturbative higher-point quark-gluon-quark (QGQ) correlation functions in the nucleon or in the fragmentation cascade. In this talk I discuss the role of soft contributions that give rise to a distinct term present only in SIDIS factorization theorems. This plays a crucial role for defining finite universal QGQ correlators.

In this talk I will review the recent progress and results concerning next-to-leading power factorization. I will mostly focus on semi-inclusive DIS process, showing the derivation of the structure functions at one-loop accuracy.
I will comment on some of the specific details that emerge at next-to-leading power, such as the cancellation of special rapidity divergences and the emergence of Qiu-Sterman-like contribution in the factorization formula.

In view of the increasing precision of theoretical calculations and experimental measurements, power corrections to transverse-momentum-dependent observables have become highly important. Power corrections for TMD observables are a rich and complex subject, and are difficult to grasp even in the most simple scenarios. Our goal is to understand TMD factorization at next-to-leading power for a relatively simple case: the semi-inclusive annihilation of an electron and a positron into two jets. By working with jets, everything is perturbatively calculable and there are substantial simplifications compared to the general next-to-leading power framework. We obtain a factorized expression for the cross section and identify the new jet functions that appear at next-to-leading power. As expected, we encounter special rapidity divergences and endpoint divergences. Importantly, our analysis with jets can be extended to semi-inclusive deep-inelastic scattering, with the future Electron-Ion Collider as key application.

In this talk, I will show a general expression for weighted cross sections in leptonic annihilation to hadrons based on time-ordered perturbation theory (TOPT). For any infrared-safe weight, the cancellation of infrared divergences is implemented locally at the integrand level, and in principle can be evaluated numerically in four dimensions. We go on to show that it is possible to eliminate unphysical singularities that appear in time-ordered perturbation theory for arbitrary amplitudes. This is done by reorganizing TOPT into an equivalent form that combines classes of time orderings into a “partially time-ordered perturbation theory”. This formalism is based on the poset structure inherently present in TOPT. I will apply this formalism to leptonic annihilation to show a formula for a generic weighted cross-section, only carrying singularities that correspond to unitarity cuts.

In the process of eliminating unphysical singularities, cuts that disconnect the graph into more than two pieces, we will find that long-time processes are encapsulated by real, non-local composite vertices. In the example of leptonic annihilation cross-sections, we will make this manifest through a local factorization between long-time and short-time processes. I will make some comments about potential connections between this formalism and the organization of power corrections in perturbation theory. I will also demonstrate how to carry out contour deformations in a generic Feynman parametrized amplitude, systematically, to all orders. We will see that our contour deformations automatically vanish on each pinch surface, admitting a physical singularity upon integrating around a pinch. We will find that, here, the difference of IR-safe weight functions vanishes in such a way that the integrand is locally finite.

We propose semi-inclusive diffractive deep inelastic scattering to investigate the gluon tomography in the nucleon and nuclei at small-x. The relevant diffractive quark and gluon parton distribution functions can be computed in terms of the color dipole S-matrices in the fundamental and adjoint representations, respectively.

I'm going to review recent results for forward jests at the LHC and EIC as obtained within small-x Improved Transverse Momentum Dependent factorization (ITMD). In addition to an elementary overview of various approaches to perturbative QCD at high energy, including High Energy Factorization, Color Glass Condensate, and ITMD, we describe the Monte Carlo implementation and discuss the existing and unpublished phenomenological results for forward dijets.

The talk is going to be based on
arXiv:2306.17513
A. van Hameren, H. Kakkad, P. Kotko, K. Kutak, S. Sapeta

One of the most intriguing phenomena of strong interaction is the so-called gluon saturation in nucleons and nuclei at very small value of the Bjorken-$x$. In the saturation regime, the density of partons, per unit of transverse area, in hadronic wavefunctions becomes very large leading to non-linear effects. These latter stop the strong growth of the gluon density at small-$x$ leading to a state characterized by a balancing of gluon emissions and recombinations.

With the aim of obtaining accurate theoretical predictions to test the physics of saturation, we calculate, at next-to-leading logarithmic accuracy, the cross-sections of diffractive single and di-hadron photo- or electroproduction with large $p_T$, on a nucleon or a nucleus. These new processes can be investigated at both the EIC, or, at the LHC in $p A$ and $A A$ scattering, using Ultra Peripheral Collisions (UPC).

We study diffractive (2+1)-jet production in photon nucleus collisions at high energy in the CGC framework. We focus in the regime in which two of the jets are hard, yet we show that large dipoles with size 1/Qs (with Qs the nucleus saturation momentum) control the cross section. These dominant contributions admit factorization in terms of parton TMDs of the Pomeron. The bulk of these TMDs is located at transverse momenta smaller or of the order of Qs and allows one to derive explicit results of the respective DPDFs.

We introduce SKMHS22, a novel set of diffractive parton distribution functions (PDFs) and their uncertainties, computed at both next-to-leading-order and next-to-next-to-leading-order accuracy within the framework of perturbative QCD in xFitter. Our analysis comprehensively addresses diffractive deep inelastic scattering (DIS) data from HERA, encompassing the latest H1/ZEUS measurements, while considering three distinct perspectives: the standard twist-2 contribution, the twist-4 correction involving longitudinal virtual photons, and the impact of subleading Reggeon exchange on the structure-function $F_2^D$. To handle heavy flavors, we employ the Thorne-Roberts scheme. Notably, the inclusion of twist-4 and subleading Reggeon contributions significantly enhances the description of the data, particularly in the high-$\beta$ region. We also investigate the stability of the SKMHS22 QCD fit with the addition of higher-order corrections, leading to subtle improvements in data agreement. These SKMHS22 PDFs exhibit strong agreement with a wide range of diffractive DIS data, outperforming prior fits, and are accessible via the LHAPDF interface. We conclude with forward-looking suggestions for further research in this domain.

QCD calculations for collider physics make use of perturbative solutions of renormalisation group equations (RGEs). RGE solutions can contribute significantly to systematic uncertainties of theoretical predictions for physical observables. We propose a method to express these systematic effects in terms of resummation scales, using techniques borrowed from soft-gluon resummation approaches. We discuss applications to collinear and Sudakov processes at hadron colliders, including deep-inelastic scattering (DIS) structure functions and the Drell-Yan (DY) vector-boson transverse momentum distribution.
The talk is based on work in progress in collaboration with V. Bertone
and F.Hautmann and on work published in Phys. Rev. D 105 (2022) 096003
"Perturbative hysteresis and emergent resummation scales".

The high level of experimental precision achieved at hadron colliders requires competitive theoretical predictions in order to extract fundamental SM parameters, such as the W boson mass.
Theoretical efforts should therefore be directed towards relevant kinematic distributions, such as the transverse momentum (pT) spectrum of the W and Z bosons and their ratio.
In this talk, essentially based on our paper (JHEP 07 (2023) 104), I will discuss the combined QCD-QED transverse-momentum resummation formalism for weak vector boson production at hadron colliders. In particular, logarithmically enhanced terms originating from soft and/or collinear QCD and QED radiation are consistently evaluated and resummed at NNLL+NNLO in QCD with the inclusion of mixed QCD-QED effects at LL and pure QED effects at NLL, matched with a fixed-order full-EW NLO contribution (i.e. at a loop). A naive abelianisation of the well-established QCD formalism, which has been used in the literature to obtain the QED contributions for on-shell Z boson production, is not sufficient for on-shell W boson production due to the electric charge of the final state. In particular, for this situation, we have performed an abelianisation of the heavy quark resummation formalism, and thus properly resum also the QED soft radiation of the final state. The inclusion of this last contribution is thus the main original feature of our study. Numerical results at hadron colliders (Tevatron, LHC) are presented for the on-shell Z and W boson transverse-momentum distributions and their ratio, potentially relevant in the context of W boson mass extraction, showing a non-trivial shape modification caused by final-state radiation and QED effects up to the percent level.

The determination of Parton Distribution Functions (PDFs) is an example of inverse problem: a model is sought knowing a finite set of experimental observations. Given the fact that the model is a continuous function, i.e. an element of an infinite dimensional space, its determination from a discrete set of data is notoriously a ill-posed problem. In the currently used methodologies for PDF determination, the model is parameterized in terms of a finite (albeit large) set of parameters, which are then fitted to the observed data. This procedure, known as parametric regression, reduces the problem to a finite dimensional and solvable one, but generally it has the drawback of introducing some bias.
A Bayesian approach provides a suitable alternative to address inverse problems, avoiding the need to introduce a finite-dimensional parameterization and recasting the problem in a probabilistic language. I will discuss a Bayesian methodology for the determination of PDFs, providing examples for the determination of PDFs from Deep Inelastic Scattering data and from lattice matrix elements.

We investigate the Higgs + hadron/jet inclusive production process at the LHC in the hybrid factorization scheme with full next-to-leading accuracy in the all-order resummation of energy logarithms. We present preliminary results for different key observables, with the aim of probing the impact of high-energy resummation in the Higgs production mechanism.

In recent articles, the discussion on the treatment of Power Corrections in the three-jet limit, along with its influence on precision $\alpha_s$ determinations from fits to event shape experimental data, has attracted a lot of attention. On the basis of a factorization formula for the $e^+e-$ thrust distribution, derived within soft-collinear effective theory, we discuss the treatment of Power Corrections in the tail region of the spectrum and the problems connected to it. In addition, we review past studies on the strong coupling determination from event shapes carried out by some of us, and provide updated results produced with a new numerical code which incorporates recent theoretical developments. Furthermore, we show the mild effect caused by the tau-dependent power correction recently advocated for in the literature.

In this talk we present results for soft gluon resummation applied to the production of four top quarks in the absolute mass threshold limit. The theoretical predictions at NLO, including a full set of strong and electroweak NLO corrections, yield a relatively large error due to scale variation. We compute the resummed cross section at next-to-leading logarithmic (NLL$^\prime$) accuracy, taking into account non-logarithmic $\mathcal{O}(\alpha_s)$ contributions that do not vanish at threshold. By considering the matching of the NLL$^\prime$ result to the available NLO result, we achieve NLO+NLL$^\prime$ precision for the total cross section. The threshold-resummed result shows an improved scale dependence when compared to the fixed-order calculation.

We study associated production of heavy quarkonia and D mesons in the improved color evaporation model using the high-energy factorization as it is formulated in the parton Reggeization approach. The last one is based on the modified Kimber-Martin-Ryskin-Watt model for unintegrated parton distribution functions and the effective field theory of Reggezied gluons and quarks, suggested by L.N. Lipatov. We predict cross section for associated J/ψ or Υ and D hadroproduction via the single and double parton scattering mechanisms using the set of model parameters which have been obtained early for the description of single and double J/ψ and Υ production at the LHC energies [1]. The numerical calculations are realized using the Monte-Carlo event generator KaTie. The calculation results are compared with the LHC data at the energies √s = 7 and 8 TeV.
[1]A. A. Chernyshev, V.A. Saleev, PHYSICAL REVIEW D 106, 114006 (2022)

I will present the Next-to-next-to-leading order QCD massive corrections,
induced by a loop of top quarks, to di-photon production in proton-proton
collisions. This process is very important as a test of perturbative QCD
and as a background process for the decay of a Higgs into two photons.
All the involved partonic subprocesses are taken into account; in
particular the quark-antiquark annihilation channel, which has required
the computation of new two-loop Master Integrals. These have been evaluated using the
differential equations method, in the semi-analitically approach with
generalised power series expansions.
I will then present a phenomenological study of the impact of our
corrections for different observables.

The measurement of the top-quark mass in top-quark pair production using the full Run 2 dataset collected with the ATLAS experiment at √s = 13 TeV will be presented. This measurement is carried out in the dilepton channel using a template method. The talk will have a special emphasis on the theoretical uncertainties affecting experimental measurements of the top-quark mass, as more and more often, these are becoming the limiting factors to further improve the precision of current measurements.

In this work we present the tools developed for the automatization of the calculation of the soft anomalous dimension matrices for the heavy-quark hadroproduction processes at one- and two-loop accuracy in QCD. The foundation of this procedure is the construction of orthogonal color basis for $2 \rightarrow n$ processes which will also be discussed. The application of these tools is demonstrated on the examples of $t\bar t j$ and $t\bar t t \bar t$ hadroproductions, making a step forward towards implementing a procedure for the resummation of soft-gluon emission logarithms for these processes.

TBA

In this talk, I will present an overview of heavy ion collisions. In particular, I will introduce the main phenomena in these collisions, and discuss them from the point of view of QCD, in order to highlight some of the possible approaches to tackle situations where collective phenomena play a crucial role.

Heavy (bottom and charm) quarks are ideal probes of the nonperturbative dynamics of confinement because they effectively serve as static color sources during the entire hadronization cascade. In this talk, I discuss the transverse momentum-dependent (TMD) fragmentation functions (FFs) for heavy quarks fragmenting into heavy hadrons as powerful tools for probing heavy-quark fragmentation in depth. I identify novel nonperturbative TMD matrix elements in heavy-quark effective theory (HQET) and present new TMD sum rules that arise from heavy-quark spin symmetry. These results lead to a rich phenomenology, especially for the heavy-quark Collins effect, at existing $B$ factories. In addition, I calculate all TMD parton distribution functions (PDFs) for the production of heavy quarks from polarized gluons within the nucleon and demonstrate the potential of the future EIC to resolve TMD heavy-quark fragmentation in semi-inclusive DIS.

Nearly ten years ago, Kang, Ma, Qiu, and Sterman derived an evolution equation for a heavy-quark pair fragmenting into a quarkonium. In this study we explore the consequence of this evolution for the color-evaporation model, focusing on $J/\psi$ transverse-momentum distributions in proton-proton collisions. In particular, I will show that it brings NLO calculations in agreement with data. Given the significant impact of the evolution on the transverse-momentum spectrum, our study suggests that this effect should be included in the fits of NRQCD long-distance-matrix elements.

We used a transverse-momentum-dependent formalism, and I will argue that quarkonium data could help constrain unintegrated PDFs.

Quarkonia are perfect laboratories to study the interplay between perturbative and nonperturbative QCD, due to the different momentum scales it provides. I am going to focus on the production of quarkonia through the fragmentation of a gluon, one of the possible mechanisms to quarkonia production. I am going to talk about the factorization of the TMD in terms of the short-distance coefficients and long-distance matrix elements in the framework of NRQCD, the theshold expansion in order to describe the heavy quarkonium states and I am going to show our recent result of the gluon TMD fragmentation function into heavy-quarkonium in the color-octet channel.

We calculate the production of polarized $J/\psi$ to probe the transverse momentum dependent (TMD) quark and gluon PDFs in the proton. Within the TMD framework we calculate the 18 leading order fragmentation functions for light quarks fragmenting to a $J/\psi$, considering all possible parton and hadron polarizations. We also calculate the production of $J/\psi$ from photon-gluon fusion within the collinear factorization framework. We use non-relativistic QCD (NRQCD) to write the cross sections in terms of perturbative coefficients and non-perturbative NRQCD matrix elements. We then make predictions for semi-inclusive deep inelastic scattering cross sections relevant for the future Electron Ion Collider, identifying the different kinematic regions where each production mechanism dominates.

We present an updated analysis of Belle data for the transverse $\Lambda$ polarization in $e^+e^−$ annihilation processes within a TMD factorization approach. Special attention to $SU(2)$ isospin symmetry and charm quark contribution will be paid, both concerning the description of the experimental data and the extraction of the polarizing fragmentation functions.
Predictions for SIDIS processes at typical energies of the future EIC will be presented.

In this talk we present the latest results about the extraction of unpolarized Transverse-Momentum-Dependent (TMD) distributions. We discuss the extraction of unpolarized quark TMD Parton Distribution Functions (TMD PDFs) in the proton, as well as TMD Fragmentation Functions (TMD FFs), from global fits of Drell-Yan and Semi-Inclusive Deep-Inelastic Scattering (SIDIS) data sets made by the MAP collaboration.

We present an extraction of unpolarized transverse momentum dependent parton distributions functions and Collins-Soper kernel from the fit of Drell-Yan and weak-vector boson production data at the N4LL order of perturbative accuracy. Based on this, Pion TMDPDFs as well as TMD fragmentation functions can be extracted, of which preliminary results are available.

In this talk, we present preliminary results about the path to a global Bayesian reweighting of TMD functions. Within an effective model as the Generalized Parton Model, and by adopting the Sivers, transversity and Collins functions as extracted from SIDIS and $e^+ e^-$ azimuthal asymmetries, we plan a simultaneous reweighting of these TMDs using data on Single-Spin Asymmetries measured at RHIC. The strategy for generalizing Bayesian reweighting to multiple fits is illustrated, together with compression techniques, that will allow us to improve the efficiency of the whole procedure. We expect this work to give some hints about TMD universality, the role of positivity bounds and to quantitatively estimate factorization breaking effects in inclusive processes.

The quark-gluon dynamics manifests itself in a set of non-perturbative functions describing all possible spin-spin and spin-orbit correlations defining azimuthal modulations in hard scattering processes. Production of correlated hadron pairs in current, as well as target fragmentation regions, plays an increasingly important role in the interpretation of pion electroproduction data in general, and hadronization process of quarks, in particular. More significant, than originally anticipated, fraction of pions and kaons coming from correlated di-hadrons, indicated by recent measurements at JLab, and supported by various realistic models describing the hadronization process, may have a significant impact on various aspects of data analysis, including the modeling, composition, and interpretation of semi-inclusive DIS data, as well as calculations of radiative corrections.
In this contribution, we will present ongoing studies, and some near term, as well as long term, future measurements related to studies of non-perturbative QCD effects at Jefferson Lab.

Pion TMDs are not as well defined as their proton counterparts, due to the small amount of available pion scattering data. The recent measurements of unpolarized Drell Yan cross section taken at COMPASS with different targets, could improve the current understanding of pion structures. In the near future also AMBER, a new fixed-target experiment at CERN SPS, will help understand the meson structure, thanks to its planned program of DY measurements involving high energy pions and kaosn. In this talk I will give an overview of the contribution that these experimental programs will give to the determination of the dynamics of partons and how it differs in proton and mesons.

We present a systematic formalism based on a factorization theorem in soft-collinear effective theory to describe non-global observables at hadron colliders, such as gap-between-jets cross sections. The cross sections are factorized into convolutions of hard functions, capturing the dependence on the partonic center-of-mass energy $\sqrt{\hat s}$, and low-energy matrix elements, which are sensitive to the low scale $Q_0\ll\sqrt{\hat s}$ characteristic of the veto imposed on energetic emissions into the gap between the jets. The scale evolution of both objects is governed by a renormalization-group equation, which we derive at one-loop order. By solving the evolution equation for the hard functions for arbitrary $2\to M$ jet processes in the leading logarithmic approximation, we accomplish for the first time the all-order resummation of the so-called ``super-leading logarithms'' discovered in 2006, thereby solving an old problem of quantum field theory. We study the numerical size of the corresponding effects for different partonic scattering processes and explain why they are sizable for $pp\to 2\,\text{jets}$ processes, but suppressed in $H/Z$ and $H/Z$\,+\,jet production. The super-leading logarithms are given by an alternating series, whose individual terms can be much larger than the resummed result, even in very high orders of the loop expansion. Resummation is therefore essential to control these effects. We find that the asymptotic fall-off of the resummed series is much weaker than for standard Sudakov form factors.

Using the analytical expression for traverse momentum dependent (TMD) gluon density in a proton , a self-consistent simultaneous description of low Q^2 data on proton structure function F_2(x,Q^2), reduced cross-section for the electron proton scattering at HERA and soft hadron production in pp collisions at the LHC is achieved in the framework of color dipole approach and modified quark-gluon string model.

Transverse Momentum Dependent (TMD) factorization leads to well established theorems for a restricted class of processes. In such cases, some of the effects produced by the soft radiation cancel out and the physics associated to them cannot be accessed. Extension of the factorization to non-standard processes inevitably leads to deal with the non trivial structure of the soft sector, revealing details that in the standard cases are not accessible. In particular, comparing the extraction of the TMD Fragmentation Function obtained from SIDIS with that extracted from single-inclusive $e^+e^-$ annihilation gives access to a new universal non-perturbative function describing the long-distance behavior of soft gluons and potentially revealing hidden properties of the QCD vacuum. In this talk, I present such comparison among the most recent phenomenological extractions and I will provide the first quantitative estimate of these hidden soft effects.

Track-based observables offer superior angular resolution and pile-up suppression. I developed a theoretical framework to account for the conversion to tracks using track functions, which are a kind of generalized fragmentation functions. I will discuss recent progress, including our determination of the evolution of track functions at order alpha_s^2, from which the evolution of multi-hadron fragmentation functions can be obtained. I will also discuss the use of track functions in new observables, focussing in particular on energy correlators.

In this talk I will give an overview of the latest developments on parton-showers within the PanScales framework. First, I will discuss the criteria that a parton shower needs to meet from a logarithmic resummation perspective. Then, I will introduce the next-to-leading logarithmic accurate PanScales showers for different collision systems: e+e-, DIS and colour singlet production in pp. Finally, I will comment on the challenges and first steps towards building parton showers an order higher in precision. This next-to-next-to-leading logarithmic accuracy will be fundamental in order to meet the expected precision of experimental analyses both at HL-LHC and FCC.

In this talk we present a procedure to deal with massive bubbles analoguos to the dispersive integral method but with the Mellin Barnes representation of the diagram. This allows us to simplify the needed previous loop order computations leading to the same ones as in renormalon calculus.
Furthermore, if we postpone the inverse Mellin integral until the very end we can easily get analytic results, not only for the diagrams but for the resummed expressions of the corresponding part of the matrix element that would otherwise involve a numerical integration. These results can always be expressed as fast-converging power series.
We applied the method to get the bHQET Hard and Jet functions, we reproduce other cases that were already known and it also could be used for massive bosons.

The Parton-Branching method (PB) allows us to determine Transverse Momentum Dependent (TMD) parton densities from very small to large transverse momentum ($k_T$). In the small $k_T$ range, both intrinsic parton motion and soft gluons contribute. Our study highlights the importance of soft gluons below a resolvable scale for integrated and TMD parton densities.

PB TMD parton densities, coupled with NLO calculations in MC@NLO style, are used to predict Drell-Yan transverse momentum spectra across a broad mass range. Using a free parameter, the intrinsic-$k_T$ width is extracted comparing world data to PB-NLO-HERAI+II-2018 set2 TMD based predictions. Unlike other approaches, the width of the intrinsic-$k_T$ distribution is found independent of the mass of the Drell-Yan pair and of the center-of-mass energy $\sqrt{s}$.

We generalize and update our former top quark mass calibration framework for Monte Carlo event generators based on the e+e- hadron-level 2-jettiness distribution in the resonance region for boosted top production. The updated framework includes the addition of the shape variables sum of jet masses, modified jet mass and the treatment of two more gap subtraction schemes to remove the leading renormalon. These generalizations entail implementing a more versatile shape-function fit procedure and accounting for a certain type of massive power corrections. The theoretical description employs boosted heavy-quark effective theory at NNLL matched to soft-collinear effective theory at NNLL and full QCD at NLO and includes the dominant top width effects. We update the top mass calibration results by applying the new framework to PYTHIA 8.205, HERWIG 7.2 and SHERPA 2.2.11.

In this talk, we will talk about the high-precision fixed order results for the Drell-Yan lepton pair production as well as W boson production with the fiducial cut of CDFII at N3LO accuracy in QCD. Such high precision is achieved by the qt slicing method, which sums up the large qt contribution obtained from event-generator NNLOJET and the small qt contribution from transverse-momentum-dependent factorization.

Geneva is a Monte Carlo event generator that matches fixed-order NNLO predictions with higher-order (NNLL') resummation in the jet resolution variable (zero-jettiness), and further combines them with parton showers. In this talk, I will first discuss recent improvements in the Geneva framework and demonstrate their impact in selected color-singlet processes. Furthermore, I will present results for the resummation of one-jettiness resolution variable, which is relevant for the production of a color-singlet state (V) together with a hard jet (j). These results constitute an indispensable ingredient and the first steps towards the complete construction of a V+j event generator such as Geneva.