The Collins-Soper kernel relates transverse momentum-dependent parton distribution functions (TMDPDFs) at different energy scales. For small parton transverse momentum $q_T\sim Λ_\text{QCD}$, this kernel is non-perturbative and can only be determined with controlled uncertainties through experiment or first-principles calculations. This work presents the first exploratory determination of the Collins-Soper kernel using the lattice formulation of Quantum Chromodynamics. In a quenched calculation, the $N_f=0$ kernel is determined at scales in the range 250 MeV $< q_T < 2$ GeV, and an analysis of the remaining systematic uncertainties is undertaken.
This work presents a lattice quantum chromodynamics (QCD) calculation of the nonperturbative Collins-Soper kernel, which describes the rapidity evolution of quark transverse-momentum-dependent parton distribution functions. The kernel is extracted at transverse momentum scales in the range 400 MeV $< q_T < 1.7$ GeV in a calculation with dynamical fermions and quark masses corresponding to a larger-than-physical pion mass, $m_π=538(1)$ MeV. It is found that different approaches to extract the Collins-Soper kernel from the same underlying lattice QCD matrix elements yield significantly different…
The Collins-Soper kernel, which governs the energy evolution of transverse-momentum dependent parton distribution functions (TMDPDFs), is required to accurately predict Drell-Yan like processes at small transverse momentum, and is a key ingredient for extracting TMDPDFs from experiment. Earlier we proposed a method to calculate this kernel from ratios of the so-called quasi-TMDPDFs determined with lattice QCD, which are defined as hadronic matrix elements of staple-shaped Euclidean Wilson line operators. Here we provide the one-loop renormalization of these operators in a regularization-indepe…
At small transverse momentum $q_T$, transverse-momentum dependent parton distribution functions (TMDPDFs) arise as genuinely nonperturbative objects that describe Drell-Yan like processes in hadron collisions as well as semi-inclusive deep-inelastic scattering. TMDPDFs naturally depend on the hadron momentum, and the associated evolution is determined by the Collins-Soper equation. For $q_T \sim Λ_\mathrm{QCD}$ the corresponding evolution kernel (or anomalous dimension) is nonperturbative and must be determined as an independent ingredient in order to relate TMDPDFs at different scales. We pro…
We present lattice results for the non-perturbative Collins-Soper (CS) kernel, which describes the energy-dependence of transverse momentum-dependent parton distributions (TMDs). The CS kernel is extracted from the ratios of first Mellin moments of quasi-TMDs evaluated at different nucleon momenta.The analysis is done with dynamical $N_f=2+1$ clover fermions for the CLS ensemble H101 ($a=0.0854\,\mathrm{fm}$, $m_π=m_K=422\,\mathrm{MeV}$). The computed CS kernel is in good agreement with experimental extractions and previous lattice studies.
In the framework of large-momentum effective theory at one-loop matching accuracy, we perform a lattice calculation of the Collins-Soper kernel which governs the rapidity evolution of transverse-momentum-dependent (TMD) distributions. We first obtain the quasi TMD wave functions at three different meson momenta on a lattice with valence clover quarks on a dynamical HISQ sea and lattice spacing $a=0.12$~fm from MILC, and renormalize the pertinent linear divergences using Wilson loops. Through one-loop matching to the light-cone wave functions, we determine the Collins-Soper kernel with transver…
The Collins-Soper kernel from a vacuum soft function
Institute of Physics, National Yang Ming Chiao Tung University（国立阳明交通大学物理研究所）；Centre for High Energy Physics, Chung-Yuan Christian University（中原大学高能物理中心）；Physics Division, National Centre for Theoretical Sciences（国家理论科学中心物理部）；Physics Division, Argonne National Laboratory（阿贡国家实验室物理部）
Unveiling the Collins-Soper kernel in inclusive DIS at threshold
INFN Sezione di Roma（罗马国家核物理研究所）；Christopher Newport University（克里斯托弗纽波特大学）；Jefferson Lab（杰斐逊实验室）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of Turin（都灵大学）；INFN Sezione di Torino（都灵国家核物理研究所）
Transverse-momentum-dependent parton distribution functions (TMDs) can be studied from first principles by a perturbative matching onto lattice-calculable quantities: so-called lattice TMDs, which are a class of equal-time correlators that includes quasi-TMDs and TMDs in the Lorentz-invariant approach. We introduce a general correlator that includes as special cases these two Lattice TMDs and continuum TMDs, like the Collins scheme. Then, to facilitate the derivation of a factorization relation between lattice and continuum TMDs, we construct a new scheme, the Large Rapidity (LR) scheme, inter…
Lattice QCD Determination of the Collins-Soper Kernel in the Continuum and Physical Mass Limits
Shanghai Jiao Tong University（上海交通大学）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交大李政道研究所）；South China Normal University（华南师范大学）；University of Maryland, College Park（马里兰大学帕克分校）；Chinese Academy of Sciences（中国科学院）；Shanghai Normal University（上海师范大学）；Universität Regensburg（雷根斯堡大学）
First simultaneous analysis of transverse momentum dependent and collinear parton distributions in the proton
Jefferson Lab（杰斐逊实验室）；Argonne National Laboratory（阿贡国家实验室）；Penn State University Berks（宾夕法尼亚州立大学伯克斯分校）；William & Mary（威廉与玛丽学院）；Lebanon Valley College（黎巴嫩谷学院）；Universidad Complutense de Madrid（马德里康普顿斯大学）
The transverse-momentum-dependent (TMD) soft function is a key ingredient in QCD factorization of Drell-Yan and other processes with relatively small transverse momentum. We present a lattice QCD study of this function at moderately large rapidity on a 2+1 flavor CLS dynamic ensemble with $a=0.098$ fm. We extract the rapidity-independent (or intrinsic) part of the soft function through a large-momentum-transfer pseudo-scalar meson form factor and its quasi-TMD wave function using leading-order factorization in large-momentum effective theory. We also investigate the rapidity-dependent part of …
Transverse-momentum-dependent parton distribution functions (TMDs) were investigated at the twists 3 and 4 for spin-1 hadrons in addition to the twist-2 ones. They were found by studying all the possible decomposition of a quark correlation function in a Lorentz-invariant way with the Hermiticity and parity invariance. The time-reversal invariance was not imposed for the TMDs due to an active role of gauge links; however, they were used for collinear parton distribution functions (PDFs) by integrating the TMDs over the transverse momentum. We found that 30 TMDs exist in the tensor-polarized sp…
Transverse-momentum-dependent parton distributions (TMDs) can be calculated from first principles by computing a related set of Euclidean lattice observables and connecting them via a factorization formula. This work focuses on the leading-power factorization formula connecting the lattice quasi-TMD and continuum Collins TMD for gluons. We calculate the one-loop gluon matching coefficient, which is known to be independent of spin and exhibits no mixing with quarks. We demonstrate that this coefficient satisfies Casimir scaling with respect to the quark matching coefficient at one-loop order. O…
Determination of charged pion unpolarised TMDPDFs from Drell-Yan measurements
Institute of Physics, Academia Sinica（中央研究院物理研究所）；Department of Physics, Chung-Yuan Christian University（中原大学物理系）；Centre for High Energy Physics, Chung-Yuan Christian University（中原大学高能物理中心）；Institute of Physics, National Yang Ming Chiao Tung University（国立阳明交通大学物理研究所）；Physics Division, National Centre for Theoretical Sciences（国家理论科学中心物理组）
We discuss the possibility of non-minimal gauge invariance of transverse-momentum-dependent parton densities (TMDs) that allows direct access to the spin degrees of freedom of fermion fields entering the operator definition of (quark) TMDs. This is achieved via enhanced Wilson lines that are supplied with the spin-dependent Pauli term $\sim F^{μν}[γ_μ, γ_ν]$, thus providing an appropriate tool for the "microscopic" investigation of the spin and color structure of TMDs. We show that this generalization leaves the leading-twist TMD properties unchanged but modifies those of twist three by contri…
We derive the leading order matching of the quark generated polarized transverse momentum dependent (TMD) distributions onto the collinear functions at small values of the transverse distance. Starting from the very definition of the TMD operator and performing the light-cone operator product expansion up to twist-3 order, we evaluate each distribution directly in position space. We primarily consider the cases of Sivers, Boer-Mulders and worm-gear functions. The effects of the TMD process dependence on the matching are explicitly shown. We also discuss the moments of TMD distributions which c…
Using soft collinear effective field theory, we derive the factorization theorem for the quasi-transverse-momentum-dependent (quasi-TMD) operator. We check the factorization theorem at one-loop level and compute the corresponding coefficient function and anomalous dimensions. The factorized expression is built from the physical TMD distribution, and a nonperturbative lattice related factor. We demonstrate that lattice related functions cancel in appropriately constructed ratios. These ratios could be used to explore various properties of TMD distributions, for instance, the nonperturbative evo…
The three-dimensional momenta of quarks inside a hadron are encoded in transverse momentum-dependent parton distribution functions (TMDs). This work presents an exploratory lattice QCD study of a TMD observable in the pion describing the Boer-Mulders effect, which is related to polarized quark transverse momentum in an unpolarized hadron. Particular emphasis is placed on the behavior as a function of a Collins-Soper evolution parameter quantifying the relative rapidity of the struck quark and the initial hadron, e.g., in a semi-inclusive deep inelastic scattering (SIDIS) process. The lattice c…
We present a first calculation of transverse momentum dependent nucleon observables in dynamical lattice QCD employing non-local operators with staple-shaped, "process-dependent" Wilson lines. The use of staple-shaped Wilson lines allows us to link lattice simulations to TMD effects determined from experiment, and in particular to access non-universal, naively time-reversal odd TMD observables. We present and discuss results for the generalized Sivers and Boer-Mulders transverse momentum shifts for the SIDIS and DY cases. The effect of staple-shaped Wilson lines on T-even observables is studie…
