We present the first lattice quantum chromodynamics (QCD) calculation of the pion valence-quark transverse-momentum-dependent parton distribution function (TMDPDF) within the framework of large-momentum effective theory (LaMET). Using correlators fixed in the Coulomb gauge (CG), we computed the quasi-TMD beam function for a pion with a mass of 300 MeV, a fine lattice spacing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.06</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, and multiple large momenta up to 3 GeV. The intrinsic soft functions in the CG approach are extracted from form factors with large momentum transfer, and as a byproduct, we also obtain the corresponding Collins-Soper (CS) kernel. Our determinations of both the soft function and the CS kernel agree with perturbation theory at small transverse separations (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>) between the quarks. At larger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math>, the CS kernel remains consistent with recent results obtained using both CG and gauge-invariant TMD correlators in the literature. By combining next-to-leading logarithmic factorization of the quasi-TMD beam function and the soft function, we obtain an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>-dependent pion valence-quark TMDPDF for transverse separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub><mo>≳</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. Interestingly, we find that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mo>⊥</mo></msub></math> dependence of the phenomenological parametrizations of TMDPDF for moderate values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> are in reasonable agreement with our QCD determinations. In addition, we present results for the transverse-momentum-dependent wave function for a heavier pion with 670 MeV mass.

United States

Dennis Bollweg

Xiang Gao

Jinchen He

Swagato Mukherjee

Yong Zhao

Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF

We outline a general framework for evaluating the nonperturbative soft functions in the quantum chromodynamics (QCD) instanton vacuum. In particular, from the soft function we derive the Collins-Soper (CS) kernel, which drives the rapidity evolution of the transverse-momentum-dependent parton distributions. The resulting CS kernel, when supplemented with the perturbative contribution, agrees well with recent lattice results and some phenomenological parametrizations. Moreover, our CS kernel depends logarithmically on the large quark transverse separation, providing a key constraint on its phenomenological parametrization. Finally, a lattice calculation can be directly compared to our generic results in Euclidean signature, thus providing a new approach for evalulating the soft function and extracting the CS kernel by analytical continuation.

Wei-Yang Liu

Ismail Zahed

Collins-Soper kernel in the QCD instanton vacuum

We consider evolution of observables which depend on a small but fixed value of longitudinal momentum fraction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>, to high rapidity, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>η</mi><mo>&gt;</mo><mi>ln</mi><mtext> </mtext><mn>1</mn><mo>/</mo><mi>x</mi></mrow></math>. We show that this evolution is not given by the Jalilian-Marian–Iancu–McLerran–Weigert–Leonidov–Kovner (JIMWLK) (or Balitsky-Kovchegov) equation. We derive the evolution Hamiltonian—<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>H</mi><mrow><mi>CSS</mi><mo>−</mo><mi>JIMWLK</mi></mrow></msub></math> which generates this evolution in the cases of dilute and dense projectile wave function. The two limits yield identical results for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>H</mi><mrow><mi>CSS</mi><mo>−</mo><mi>JIMWLK</mi></mrow></msub></math>. We show that the resulting evolution for the gluon Transverse-Momentum-Dependent is identical to the (double logarithmic) perturbative Collins-Soper-Sterman evolution equation in the longitudinal resolution parameter at a fixed and very large transverse resolution.

Haowu Duan

Alex Kovner

Israel

Michael Lublinsky

Collins-Soper-Sterman Hamiltonian: High energy evolution of rapidity dependent observables

New results are presented on a high-statistics measurement of Collins and Sivers asymmetries of charged hadrons produced in deep inelastic scattering of muons on a transversely polarized <math><mrow><mmultiscripts><mrow><mi>LiD</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>6</mn></mrow></mmultiscripts></mrow></math> target. The data were taken in 2022 with the COMPASS spectrometer using the 160 GeV muon beam at CERN, statistically balancing the existing data on transversely polarized proton targets. The first results from about two-thirds of the new data have total uncertainties smaller by up to a factor of three compared to the previous deuteron measurements. Using all the COMPASS proton and deuteron results, both the transversity and the Sivers distribution functions of the <math><mi>u</mi></math> and <math><mi>d</mi></math> quark, as well as the tensor charge in the measured <math><mi>x</mi></math> range are extracted. In particular, the accuracy of the <math><mi>d</mi></math> quark results is significantly improved.

G. D. Alexeev

Italy

M. G. Alexeev

C. Alice

A. Amoroso

V. Andrieux

V. Anosov

Armenia

S. Asatryan

Czechia

K. Augsten

Poland

W. Augustyniak

Portugal

C. D. R. Azevedo

B. Badelek

Germany

J. Barth

R. Beck

J. Beckers

France

Y. Bedfer

CERN

J. Bernhard

M. Bodlak

F. Bradamante

A. Bressan

Taiwan

W.C. Chang

C. Chatterjee

M. Chiosso

A. G. Chumakov

S.U. Chung

A. Cicuttin

P. M. M. Correia

M. L. Crespo

D. D’Ago

S. Dalla Torre

India

S. S. Dasgupta

S. Dasgupta

F. Delcarro

I. Denisenko

O. Yu. Denisov

S. V. Donskov

Japan

N. Doshita

Ch. Dreisbach

W. Dünnweber

R. R. Dusaev

D. Ecker

D. Eremeev

P. Faccioli

M. Faessler

M. Finger

H. Fischer

K. J. Flöthner

W. Florian

J. M. Friedrich

V. Frolov

L. G. Garcia Ordòñez

F. Gautheron

O. P. Gavrichtchouk

S. Gerassimov

J. Giarra

D. Giordano

A. Grasso

A. Gridin

M. Grosse Perdekamp

B. Grube

M. Grüner

A. Guskov

P. Haas

D. von Harrach

M. Hoffmann

A. Hoghmrtsyan

N. d’Hose

C.Y. Hsieh

S. Huber

S. Ishimoto

A. Ivanov

T. Iwata

V. Jary

R. Joosten

E. Kabuß

F. Kaspar

A. Kerbizi

B. Ketzer

A. Khatun

G. V. Khaustov

T. Klasek

F. Klein

J. H. Koivuniemi

V. N. Kolosov

K. Kondo Horikawa

I. Konorov

V. F. Konstantinov

A. Yu. Korzenev

A. M. Kotzinian

O. M. Kouznetsov

A. Koval

Z. Kral

F. Krinner

F. Kunne

K. Kurek

R. P. Kurjata

A. Kveton

K. Lavickova

S. Levorato

Y.S. Lian

High-Statistics Measurement of Collins and Sivers Asymmetries for Transversely Polarized Deuterons

This Letter presents a determination of the quark Collins-Soper kernel, which relates transverse-momentum-dependent parton distributions (TMDs) at different rapidity scales, using lattice quantum chromodynamics (QCD). This is the first such determination with systematic control of quark mass, operator mixing, and discretization effects. Next-to-next-to-leading logarithmic matching is used to match lattice-calculable distributions to the corresponding TMDs. The continuum-extrapolated lattice QCD results are consistent with several recent phenomenological parametrizations of the Collins-Soper kernel and are precise enough to disfavor other parametrizations.

Artur Avkhadiev

Phiala E. Shanahan

Michael L. Wagman

Determination of the Collins-Soper Kernel from Lattice QCD

We present a lattice QCD calculation of the rapidity anomalous dimension of quark transverse-momentum-dependent distributions, i.e., the Collins-Soper (CS) kernel, up to transverse separations of about 1 fm. This unitary lattice calculation is conducted, for the first time, employing the chiral-symmetry-preserving domain wall fermion discretization and physical values of light and strange quark masses. The CS kernel is extracted from the ratios of pion quasi-transverse-momentum-dependent wave functions (quasi-TMDWFs) at next-to-leading logarithmic perturbative accuracy. Also for the first time, we utilize the recently proposed Coulomb-gauge-fixed quasi-TMDWF correlator without a Wilson line. We observe significantly slower signal decay with increasing quark separations compared to the established gauge-invariant method with a staple-shaped Wilson line. This enables us to determine the CS kernel at large nonperturbative transverse separations and find its near-linear dependence on the latter. Our result is consistent with the recent lattice calculation using gauge-invariant quasi-TMDWFs, and agrees with various recent phenomenological parametrizations of experimental data.

Nonperturbative Collins-Soper kernel from chiral quarks with physical masses

The production of vector mesons in deep inelastic scattering is an interesting yet scarcely explored channel to study the transverse spin structure of the nucleon and the spin-dependence of fragmentation. The COMPASS collaboration has performed the first measurement of the Collins and Sivers asymmetries for inclusively produced <math><msup><mrow><mi>ρ</mi></mrow><mrow><mn>0</mn></mrow></msup></math> mesons. The analysis is based on the data set collected in deep inelastic scattering in 2010 using a <math><mn>160</mn><mspace width="0.2em"></mspace><mspace width="0.2em"></mspace><mrow><mi>GeV</mi></mrow><mo>/</mo><mi>c</mi></math> <math><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup></math> beam impinging on a transversely polarized <math><mi>N</mi><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub></math> target. The <math><msup><mrow><mi>ρ</mi></mrow><mrow><mn>0</mn></mrow></msup></math> mesons are selected from oppositely charged hadron pairs, and the asymmetries are extracted as a function of the Bjorken-x variable, the transverse momentum of the pair and the fraction of the energy z carried by the pair. Indications for positive Collins and Sivers asymmetries are observed.

G.D. Alexeev

M.G. Alexeev

C.D.R. Azevedo

V.E. Burtsev

A.G. Chumakov

P.M.M. Correia

M.L. Crespo

D. D'Ago

S.S. Dasgupta

F. Del Carro

O.Yu. Denisov

S.V. Donskov

R.R. Dusaev

A. Efremov

C. Elia

K.J. Flöthner

J.M. Friedrich

L.G. Garcia Ordòñez

O.P. Gavrichtchouk

M. Gorzellik

N. Horikawa

N. d'Hose

M. Jandek

G.V. Khaustov

J.H. Koivuniemi

V.N. Kolosov

V.F. Konstantinov

A.M. Korzenev

A.M. Kotzinian

O.M. Kouznetsov

R.P. Kurjata

Collins and Sivers transverse-spin asymmetries in inclusive muoproduction of Mesons

This work presents a determination of the quark Collins-Soper kernel, which relates transverse-momentum-dependent parton distributions at different rapidity scales, using lattice QCD. This is the first lattice QCD calculation of the kernel at quark masses corresponding to a close-to-physical value of the pion mass, with next-to-next-to-leading logarithmic matching to transverse-momentum-dependent parton distributions from the corresponding lattice-calculable distributions, and includes a complete analysis of systematic uncertainties arising from operator mixing. The kernel is extracted at transverse momentum scales <math><mrow><mn>240</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>≲</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>≲</mo><mn>1.6</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> with a precision sufficient to begin to discriminate between different phenomenological models in the nonperturbative region.

Collins-Soper kernel from lattice QCD at the physical pion mass

The Collins-Soper (CS) kernel is a nonperturbative function that characterizes the rapidity evolution of transverse-momentum-dependent parton distribution functions (TMDPDFs) and wave functions. In this paper, we calculate the CS kernel for pion and proton targets and for quasi-TMDPDFs of leading and next-to-leading power. The calculations are carried out on the CLS ensemble H101 with dynamical <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> clover-improved Wilson fermions. Our analyses demonstrate the consistency of different lattice extractions of the CS kernel for mesons and baryons, as well as for twist-two and twist-three operators, even though lattice artifacts could be significant. This consistency corroborates the universality of the lattice-determined CS kernel and suggests that a high-precision determination of it is in reach.

Hai-Tao Shu

Maximilian Schlemmer

Tobias Sizmann

Spain

Alexey Vladimirov

Lisa Walter

Michael Engelhardt

Andreas Schäfer

China

Yi-Bo Yang

Universality of the Collins-Soper kernel in lattice calculations

Two main frameworks for defining transverse momentum dependent (TMD) parton densities are the Collins-Soper-Sterman (CSS) formalism, and the Parton Branching (PB) approach. While PB-TMDs have an explicit dependence on a single scale which is used to evolve PB-TMDs in momentum space, TMDs defined in CSS formalism present a double-scale evolution in renormalization and rapidity scales, via a pair of coupled evolution equations. In this letter I leverage the Collins-Soper kernel determined from simulated Drell Yan transverse momentum spectra using PB-TMDs, and provide, for the first time, the transformation of TMD parton distributions from the PB framework to the CSS formalism. The evolved PB-TMDs in b-space are compared to the recently released, unpolarized TMD distribution ART23.

Armando Bermudez Martinez

Transformation of transverse momentum distributions from Parton Branching to Collins-Soper-Sterman framework