A striking prediction of QCD on the properties of the novel Transverse Momentum Dependent (TMD) distribution functions is that the time-reversal odd Sivers and Boer-Mulders functions extracted from semi-inclusive deep-inelastic scattering (SIDIS) will undergo a sign reversal in the Drell-Yan (DY) process. This prediction has been tested by experiments that have focused on the Sivers functions so far. We review the current status on the theoretical prediction and experimental extraction of the signs of the Boer-Mulders functions from SIDIS and DY. We note that the existing SIDIS and DY data are consistent with the predicted sign reversal of the Boer-Mulders functions for proton’s valence quark distribution. Prospects for future experiments at EIC capable of testing the sign reversal of the pion Boer-Mulders functions are also discussed.

United States

Taiwan

Jen-Chieh Peng

Ming-Xiong Liu

Guanghua Xu

Sign reversal of Boer-Mulders functions from semi-inclusive deep-inelastic scattering to the Drell-Yan process

The impact of the inclusion of transverse-momentum-dependent (TMD) shape functions on the transverse momentum spectrum in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production at the electron-ion collider (EIC) is investigated by considering the matching of the TMD factorization description at low transverse momentum with the collinear factorization description at high transverse momentum by means of the inverse-error weighting method. Despite large uncertainties from scale variations and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> long-distance matrix elements, predictions for the differential cross section and its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>ϕ</mi><mi>ψ</mi></msub><mo stretchy="false">)</mo></math> modulation are obtained. We find that physical constraints are satisfied in case a process-dependent term is included for color octet production, but not in all cases when it is excluded. These numerical results support the analytic calculations in the work by Boer et al. [<a href="http://dx.doi.org/10.1007/JHEP08(2023)105">J. High Energy Phys. 08 (2023) 105</a>.]. Future experimental data can thus test the validity of TMD factorization in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production and explore the presence of nontrivial process-dependent effects in the soft-gluon resummation. This will be crucial in the extraction of gluon unpolarized and linearly polarized TMD distributions of the proton. In addition, we suggest how the sign of the latter can be determined by investigating the presence of a node in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>ϕ</mi><mi>ψ</mi></msub><mo stretchy="false">)</mo></math> modulation as a function of transverse momentum.

Netherlands

France

Luca Maxia

Daniël Boer

Jelle Bor

Impact of the TMD shape function on matching the transverse momentum spectrum in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production at the EIC

We present the first lattice QCD calculation of the quark transverse spin-momentum correlation, i.e., the naive time-reversal-odd Boer-Mulders function, of the nucleon, using large-momentum effective theory (LaMET). The calculation is carried out on an ensemble with lattice spacing $\textit{a}$ = 0.098 fm and pion mass 338 MeV, at various proton momenta up to 2.11 GeV. We have implemented perturbative matching up to the next-to-next-to-leading order together with a renormalization-group resummation improvement. The result exhibits the expected decay behavior with increasing transverse separation $\textit{b}$ . We also compare the results in the nucleon and pion.

China

Lingquan Ma

Jun Hua

Germany

Andreas Schäfer

Hai-Tao Shu

Yushan Su

Peng Sun

Lisa Walter

Wei Wang

Xiaonu Xiong

Yi-Bo Yang

Jian-Hui Zhang

Qi-An Zhang

Quark transverse spin-momentum correlation of the nucleon from lattice QCD: the Boer-Mulders function

We present the first lattice QCD calculation of the quark transverse spin-momentum correlation, i.e., the T-odd Boer-Mulders function of the pion, using large-momentum effective theory. The calculation is done at three lattice spacings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0.098</mn><mo>,</mo><mn>0.085</mn><mo>,</mo><mn>0.064</mn><mo stretchy="false">}</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and pion masses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>350</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, with pion momenta up to 1.8 GeV. The matrix elements are renormalized in a state-of-the-art scheme and extrapolated to the continuum and infinite momentum limit. We have implemented the perturbative matching up to the next-to-next-to-leading order and carried out a renormalization-group resummation. Our results provide valuable input for phenomenological analyses of the Boer-Mulders single-spin asymmetry.

Sebastian Lahrtz

Hong Kong

Quark transverse spin-momentum correlation of the pion from lattice QCD: The Boer-Mulders function

A symmetry preserving treatment of a vector $$\otimes $$ vector contact interaction (SCI) is used as the basis for calculations of the two pion transverse momentum dependent parton distribution functions (TMDs); namely, that for unpolarised valence degrees-of-freedom and the analogous Boer–Mulders (BM) function. Amongst other things, the analysis enables the following themes to be addressed: the quark current mass dependence of pion TMDs; the impact of the gauge link model on the positivity constraint that bounds the BM function relative to the unpolarised TMD; the equivalence of direct diagrammatic and light-front wave function TMD calculations; and the size of the BM shift. Interpreted astutely, these SCI results enable one to draw insightful pictures of pion TMDs.

Dan-Dan Cheng

Zhu-Fang Cui

Minghui Ding

Craig Roberts

Sebastian Schmidt

Pion Boer–Mulders function using a contact interaction

Azimuthal modulations in lepton and heavy-quark pair production in ultraperipheral collisions (UPCs) of highly charged ions are investigated. The modulations in the azimuthal angles of the sum and difference of the transverse momenta of the pair of particles in the final state, as well as of the transverse impact parameter, arise from the collisions of unpolarized and polarized photons. A full description of the cross section in terms of Generalized Transverse Momentum Dependent parton distributions (GTMDs) for photons is given including a careful consideration of the Fourier transform to impact parameter space. In particular, this leads to a feed-in mechanism among harmonics of different orders, which in principle generates harmonics of all (even) orders. Wherever comparable, our analytical results for the azimuthal modulations agree with those presented in other papers on this topic. Compared to these other works, we separate effects that arise from the anisotropies of the GTMDs from those that do not and retain terms proportional to the mass of the produced particles, as they are relevant for muon, charm and bottom quark production. We show that the normalized differential cross section changes considerably with the produced particle mass, which should be discernible in UPCs at RHIC and LHC. For the numerical results we adopt several models for the photon GTMD correlator, and find that all of them are in fairly good agreement with each other and with UPC data from STAR. We also present results for various azimuthal modulations for RHIC kinematics, where we compare $\textit{e}$ $\textit{e}$ production with the production of heavier particles, and for LHC kinematics, focusing on $\textit{μ}$ $\textit{μ}$ production. These results exhibit interesting mass-dependent features in the asymmetries that may help study the anisotropies arising from the underlying photon GTMD description.

Italy

Cristian Pisano

Azimuthal asymmetries in lepton and heavy-quark pair production in UPCs

We extend a previous GTMD model to improve the description of the HERA data on diffractive dijet production, and include exclusive coherent diffractive $$J/\psi $$ <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> </mrow> </math> production data. We find that within our gluon GTMD model context and assumptions, there is considerable tension between the data for these two types of processes concerning the t dependence. Photo- and electroproduction data for protons and nuclei from EIC and UPC data from LHC and RHIC can help to establish whether a common GTMD description is possible, as one would expect, and to facilitate studies of such data we provide predictions for the various experiments. We point out explicitly in which sense this goes beyond the description in terms of GPDs.

Indonesia

Chalis Setyadi

Probing gluon GTMDs through exclusive coherent diffractive processes

It has been shown previously that the transverse momentum dependent (TMD) factorization of heavy quarkonium production requires a TMD shape function. Its perturbative tail can be extracted by matching the cross sections valid at low and high transverse momenta. In this article we compare the order-α s TMD expressions with the order- <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math> $$ {\alpha}_s^2 $$ collinear ones for J/ψ production in semi-inclusive deep inelastic scattering (SIDIS), employing nonrelativistic QCD in both cases. In contrast to previous studies, we find that the small transverse momentum limit of the collinear expressions contains discontinuities. We demonstrate how to properly deal with them and include their finite contributions to the TMD shape functions. Moreover, we show that soft gluon emission from the low transverse momentum Born diagrams provide the same leading order TMD shape functions as required for the matching. Their revised perturbative tails have a less divergent behavior as compared to the TMD fragmentation functions of light hadrons. Finally, we investigate the universality of TMD shape functions in heavy quarkonium production, identify the need for process dependent factorization and discuss the phenomenological implications.

Feng Yuan

Transverse momentum dependent shape function for J/ψ production in SIDIS

We compute the Sivers, Boer-Mulders, worm-gear (T and L) transverse momentum dependent distributions in terms of twist-two and twist-three collinear distributions in the small-b limit up to next-to-leading order (NLO) in perturbation theory.

Felix Rein

Simone Rodini

Spain

Alexey Vladimirov

Sivers, Boer-Mulders and worm-gear distributions at next-to-leading order

We obtain the leading twist T-odd quark transverse momentum-dependent parton distribution functions (TMDs) of the proton, namely, the Sivers function <math><msubsup><mi>f</mi><mrow><mn>1</mn><mi>T</mi></mrow><mrow><mo>⊥</mo><mi>q</mi></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msubsup><mi>p</mi><mo>⊥</mo><mn>2</mn></msubsup><mo>)</mo></math> and the Boer-Mulders function <math><msubsup><mi>h</mi><mn>1</mn><mrow><mo>⊥</mo><mi>q</mi></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msubsup><mi>p</mi><mo>⊥</mo><mn>2</mn></msubsup><mo>)</mo></math>, in a light-front quark-diquark model constructed with the wave functions predicted by the soft-wall AdS/QCD. The gluon rescattering is crucial to predict nonzero T-odd TMDs. We study the utility of a nonperturbative <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></math> gluon rescattering kernel going beyond the usual approximation of perturbative <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> gluons. The spin asymmetries in semi-inclusive deep inelastic scattering (SIDIS) associated with these T-odd TMDs are found to be consistent with HERMES and COMPASS data. We further predict the Sivers asymmetries of pion production in the SIDIS process in the kinematical region of the Electron-Ion Collider (EIC) and the EIC in China. We also evaluate the generalized Sivers and Boer-Mulders shifts and compare them with the available lattice QCD simulations.