We investigate the collinear matching of transverse momentum dependent (TMD) distributions at large values of $\textit{x}$, computing and resumming the leading large-$\textit{x}$ asymptotics for matching coefficients. The large-$\textit{x}$ resummation is done directly within TMD distributions, ensuring the process-independence of the result. The derived resummation formulas are valid for all TMD distributions (except the pretzelosity). Their application improves perturbative convergence, provides practical estimation for unknown higher-order contributions, and sets restrictions for the nonperturbative part of models. Using the known anomalous dimensions, resummation can reach N LL, often exceeding the accuracy of known coefficient functions.

Spain

Oscar Rio

United States

Alexei Prokudin

Ignazio Scimemi

Alexey Vladimirov

Transverse momentum distributions at large-$\textit{x}$

The transverse momentum dependent (TMD) factorization theorem accommodates various types of power corrections. Among them, the least studied are $\textit{q}$ $\textit{/Q}$ corrections, which become significant at large values of transverse momentum. These corrections partially originate from higher-twist TMD distributions, which exhibit singularity at small transverse distances. We propose a decomposition that reveals this singularity explicitly, and makes the $\textit{q}$ $\textit{/Q}$ correction manifest. As a concrete application, we consider the next-to-leading power correction for the angular distributions in Drell-Yan, and determine the leading $\textit{q}$ $\textit{/Q}$ corrections. These corrections are significant for the angular distributions $\textit{A}$ and $\textit{A}$ , in complete agreement with the data.

Arturo Arroyo-Castro

Leading $\textit{q}$

We establish robust relations between Transverse Momentum Dependent distributions (TMDs) and collinear distributions. We define weighted integrals of TMDs that we call Transverse Momentum Moments (TMMs) and prove that TMMs are equal to collinear distributions evaluated in some minimal subtraction scheme. The conversion to the modified minimal subtraction (<math><mover><mi>MS</mi><mo>¯</mo></mover></math>) scheme can be done by a calculable factor, which we derive up to three loops for some cases. We discuss in detail the zeroth, the first, and the second TMMs and provide phenomenological results for them based on the current extractions of TMDs. The results of this paper open new avenues for theoretical and phenomenological investigation of the three-dimensional and collinear hadron structures.

Oscar del Rio

Transverse momentum moments

We present an extraction of unpolarized transverse momentum dependent parton distributions functions (TMDPDFs) and Collins-Soper kernel from the fit of Drell-Yan and weak-vector boson production data. The TMDPDF are parameterized, as commonly done, using their (large transverse momentum) asymptotic matching to PDF. The analysis is done at the next-to-next-to-next-to-next-to leading logarithmic accuracy (N4LL) (performed only approximately because PDF evolution is known so far at next-to-next-to leading order (NNLO)). The non-perturbative model used for TMDPDF is flavor dependent to reduce the colllinear PDF bias. The estimation of uncertainties is done with the replica method and, for the first time, it includes the propagation of uncertainties due to the collinear distributions.

Germany

Valentin Moos

Pia Zurita

Extraction of unpolarized transverse momentum distributions from the fit of Drell-Yan data at N4LL

In view of the increasing precision of theoretical calculations and experimental measurements, power corrections to transverse-momentum-dependent observables are highly important. We study the next-to-leading power corrections for transverse momentum measurements in e + e  −  → 2 jets. We obtain a factorized expression for the cross section, which involve twist-2 and twist-3 operators, and identify the new jet functions that appear in it. We calculate these jet functions at order α  s  for a family of recoil-free schemes, and provide the corresponding anomalous dimensions at leading order. Additionally, we show that the (endpoint) divergences that typically arise in sub-leading-power factorization can be subtracted and cancel for our case. By working with jets, everything is perturbatively calculable and there are substantial simplifications compared to the general next-to-leading power framework. Importantly, our analysis with jets can be extended to semi-inclusive deep-inelastic scattering, with the future Electron-Ion Collider as key application.

Rafael Castillo

Netherlands

Max Jaarsma

Wouter Waalewijn

Transverse momentum measurements with jets at next-to-leading power

We compute the gluon transverse-momentum-dependent fragmentation function (TMDFF) at next-to-leading order (NLO) into heavy quarkonium in the color-octet   <math> <mmultiscripts> <msubsup> <mi>S</mi> <mn>1</mn> <mfenced> <mn>8</mn> </mfenced> </msubsup> <mprescripts></mprescripts> <none></none> <mn>3</mn> </mmultiscripts> </math>  $$ {}^3{S}_1^{\left[8\right]} $$  channel, based on the NRQCD factorization approach. The spurious rapidity divergences are explicitly shown to cancel in a well-defined TMDFF, which incorporates the needed soft factor. We also compute the integrated gluon FF at NLO in the same   <math> <mmultiscripts> <msubsup> <mi>S</mi> <mn>1</mn> <mfenced> <mn>8</mn> </mfenced> </msubsup> <mprescripts></mprescripts> <none></none> <mn>3</mn> </mmultiscripts> </math>  $$ {}^3{S}_1^{\left[8\right]} $$  channel, and show that the matching coefficient of the TMDFF onto the FF at large transverse momentum is the expected one. These results are relevant to perform precise and sensible phenomenological studies of transverse-momentum spectra of quarkonium production, for which the production mechanism through fragmentation plays a relevant role, like in the future Electron-Ion Collider.

Miguel Echevarria

Samuel Romera

Gluon TMD fragmentation function into quarkonium

Different exclusive processes have been proposed to access the generalized transverse momentum dependent distributions (GTMDs) with no proof of factorization, which allows to rigorously define the GTMDs. Using Soft Collinear Effective Theory we derive for the first time the factorization of the differential cross section for the exclusive double Drell-Yan process <math><mi>π</mi><mi>N</mi><mo>→</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>′</mo></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>→</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>)</mo></math>, for small transverse momenta of the photons in terms of a perturbatively calculable hard factor, GTMDs and light-cone wave functions (LCWFs). We find that the hard factor of the process can be obtained from single inclusive Drell-Yan production so that one can resum logarithms at high orders in QCD. We also discuss the evolution of the GTMDs and the LCWFs.

Miguel G. Echevarria

Patricia A. Gutierrez Garcia

GTMDs and the factorization of exclusive double Drell-Yan

Transverse momentum dependent (TMD) distributions match collinear parton density functions (PDF) in the limit of small transverse distances, which is accounted for by global extractions of TMD distributions. We study the influence of the collinear PDF value and uncertainties on the determination of unpolarized TMD distributions and the description of Drell-Yan (DY) and Z-boson production measurements at low transverse momenta. We take into account, for the first time in fits including TMD evolution, flavor-dependent non-perturbative TMD profiles. We carry out a Bayesian analysis to incorporate the propagation of PDF uncertainties into TMD extractions. We find that collinear PDF uncertainties and non-perturbative TMD flavor dependence are both essential to obtain reliable TMD determinations, and should be included in future global analyses.

Poland

Marcin Bury

Belgium

CERN

United Kingdom

Francesco Hautmann

Austria

Sergio Leal-Gomez

PDF bias and flavor dependence in TMD distributions

We discuss the measurement of gluon transverse momentum distribution (TMD) in dijet and heavy hadron pair (HHP) production in semi-inclusive deep inelastic scattering. The factorization of these processes in position space shows the appearance of a specific new soft factor matrix element on top of angular and complex valued anomalous dimensions. We show in detail how these features can be treated consistently and we discuss a scale prescription for the evolution kernel of the dijet soft function. As a result we obtain phenomenological predictions for unpolarized and angular modulated cross-sections for the electron-ion collider (EIC) using current available information on unpolarized TMD.

Italy

Yiannis Makris

Transverse momentum dependent distributions in dijet and heavy hadron pair production at EIC

We develop a method of transverse momentum dependent (TMD) operator expansion that yields the TMD factorization theorem on the operator level. The TMD operators are systematically ordered with respect to TMD-twist, which allows a certain separation of kinematic and genuine power corrections. The process dependence enters via the boundary conditions for the background fields. As a proof of principle, we derive the effective operator for hadronic tensor in TMD factorization up to the next-to-leading power (∼ q  T /Q) at the next-to-leading order for any process with two detected states.