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

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

<p>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.</p>

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

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.

India

Bheemsehan Gurjar

Dipankar Chakrabarti

Chandan Mondal

Sivers and Boer-Mulders TMDs of the proton in a light-front quark-diquark model

We compute the leading-twist T-odd quasidistributions of the proton in a spectator model with scalar and axial-vector diquarks: the quasi-Sivers function <math><mrow><msubsup><mrow><mover><mrow><mi>f</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn><mi>T</mi></mrow><mrow><mo>⊥</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>;</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>)</mo></mrow></math> and the quasi-Boer-Mulders function <math><mrow><msubsup><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow><mrow><mo>⊥</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>;</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>)</mo></mrow></math>. We obtain the quark-quark correlators in the four-dimensional Euclidian space by replacing <math><msup><mi>γ</mi><mo>+</mo></msup></math> and <math><msup><mi>σ</mi><mrow><mi>i</mi><mo>+</mo></mrow></msup></math> in the light-cone frame with <math><msub><mi>γ</mi><mi>z</mi></msub></math> and <math><msub><mi>σ</mi><mrow><mi>i</mi><mi>z</mi></mrow></msub></math>. We show by analytical calculation that the results of <math><mrow><msubsup><mrow><mover><mrow><mi>f</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn><mi>T</mi></mrow><mrow><mo>⊥</mo></mrow></msubsup></mrow></math> and <math><mrow><msubsup><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow><mrow><mo>⊥</mo></mrow></msubsup></mrow></math> derived from the correlators can reduce to the expressions of the corresponding standard T-odd distributions <math><mrow><msubsup><mrow><mi>f</mi></mrow><mrow><mn>1</mn><mi>T</mi></mrow><mrow><mo>⊥</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></mrow></math> and <math><mrow><msubsup><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>⊥</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msubsup><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></mrow></math> in the limit <math><msub><mi>P</mi><mi>z</mi></msub><mo>→</mo><mi>∞</mi></math>. The numerical results for these quasidistributions and their first transverse moments for the <math><mi>u</mi></math> and <math><mi>d</mi></math> quarks in different <math><mi>x</mi></math> and <math><msub><mi>P</mi><mi>z</mi></msub></math> regions are also presented. We find that <math><mrow><msubsup><mrow><mover><mrow><mi>f</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn><mi>T</mi></mrow><mrow><mo>⊥</mo><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>)</mo></mrow></math> and <math><mrow><msubsup><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow><mrow><mo>⊥</mo><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>)</mo></mrow></math> in the spectator model are fair approximations to the standard ones (within 20%–30%) in the region <math><mn>0.1</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>0.5</mn></math> when <math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>≥</mo><mn>2.5</mn><mi>–</mi><mn>3</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. This supports the idea of using T-odd quasidistributions to obtain standard distributions in the region <math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>&gt;</mo><mn>2.5</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> as fair approximation.

Chentao Tan

Zhun Lu

Quark quasi-Sivers function and quasi-Boer-Mulders function in a spectator diquark model

In this study, we investigate the Kotzinian-Mulders effect under semi-inclusive deep inelastic scattering (SIDIS) within the framework of transverse momentum dependent (TMD) factorization. The asymmetry is contributed by the convolution of the Kotzinian-Mulders function   $ g_{1T}$   and the unpolarized fragmentation function   $ g_{1T}$  . As a TMD distribution, the Kotzinian-Mulders function in the coordinate space in the perturbative region can be represented as the convolution of the C-coefficients and the corresponding collinear correlation function. The Wandzura-Wilczek approximation is used to obtain this correlation function. We perform a detailed phenomenological numerical analysis of the Kotzinian-Mulders effect in the SIDIS process within TMD factorization at the kinematics of the HERMES and COMPASS experiments. We observe that the obtained   $ g_{1T}$  -,   $ g_{1T}$  -, and   $ g_{1T}$  -dependent Kotzinian-Mulders effects are basically consistent with the HERMES and COMPASS measurements. We also make predictions at EIC and EicC kinematics.

Xuan Luo

Hao Sun

Kotzinian-Mulders effect in semi-inclusive DIS within TMD factorization  

We present a phenomenological analysis of the  $$q_{\text {T}}$$  <math> <msub> <mi>q</mi> <mtext>T</mtext> </msub> </math>  -weighted transverse spin dependent azimuthal asymmetry recently measured by the COMPASS Collaboration in polarized pion-proton Drell–Yan process. In the kinematical regimes explored by experiments, we consider the leading-twist contributions from the Boer–Mulders distribution functions  $$h_{1}^{\perp }(x,k_{\text {T}}^{2})$$  <math> <mrow> <msubsup> <mi>h</mi> <mrow> <mn>1</mn> </mrow> <mo>⊥</mo> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msubsup> <mi>k</mi> <mrow> <mtext>T</mtext> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </math>   of both the pion and the proton, the transversity distribution  $$h_{1}(x,k_{\text {T}}^{2})$$  <math> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msubsup> <mi>k</mi> <mrow> <mtext>T</mtext> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </math>   and the pretzelosity distribution  $$h_{1\text {T}}^{\perp }(x,k_{\text {T}}^{2})$$  <math> <mrow> <msubsup> <mi>h</mi> <mrow> <mn>1</mn> <mtext>T</mtext> </mrow> <mo>⊥</mo> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msubsup> <mi>k</mi> <mrow> <mtext>T</mtext> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </math>   of the proton, with the unpolarized transverse-momentum-dependent distribution function  $$f_{1}(x,k_{\text {T}}^{2})$$  <math> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msubsup> <mi>k</mi> <mrow> <mtext>T</mtext> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </math>   of the proton being also involved in the calculation. By comparing the data reported by the COMPASS Collaboration with theoretical calculated results, we find that the COMPASS measurements represent the first experimental evidence of the Boer–Mulders effect in polarized Drell–Yan process. We also test the universality between proton and pion Boer–Mulders functions.

Xiaonan Liu

Bo-Qiang Ma

Boer–Mulders function of the pion and pretzelosity distribution of the proton in the polarized pion-proton Drell–Yan process at COMPASS

In series of papers, the Boer-Mulders function for a given quark flavor has been extracted: (i) from data on semi-inclusive deep inelastic scattering, using the simplifying, but theoretically inconsistent, assumption that it is proportional to the Sivers function for each quark flavor and (ii) from data on Drell-Yan reactions. In earlier papers, using the semi-inclusive deep inelastic COMPASS deuteron data on the <math><mo>⟨</mo><mi>cos</mi><msub><mi>ϕ</mi><mi>h</mi></msub><mo>⟩</mo></math> and <math><mo>⟨</mo><mi>cos</mi><mn>2</mn><msub><mi>ϕ</mi><mi>h</mi></msub><mo>⟩</mo></math> asymmetries, we extracted the collinear <math><msub><mi>x</mi><mrow><mi>B</mi></mrow></msub></math> dependence of the Boer-Mulders function for the sum of the valence quarks <math><msub><mi>Q</mi><mi>V</mi></msub><mo>=</mo><msub><mi>u</mi><mi>V</mi></msub><mo>+</mo><msub><mi>d</mi><mi>V</mi></msub></math> using a small number of model dependent assumptions and found a significant disagreement with the analysis in (i). In the present paper, we provide a more complete analysis of the semi-inclusive deep inelastic scattering reaction, including a discussion of higher twist and interaction-dependent terms, and also a comparison with the Boer-Mulders function extracted from data on the Drell-Yan reaction. We confirm that the proportionality relation of the BM function to the Sivers function, for each quark flavor, fails badly, but find that it holds rather well if applied to the nonsinglet valence-quark combination, <math><msub><mi>Q</mi><mi>V</mi></msub></math>. We also find good agreement with the results of the Drell-Yan analysis. Furthermore, we obtain interesting information on the quark transverse momentum densities in the nucleon and on the hadron transverse momentum dependence in quark fragmentation.

Bulgaria

E. Christova

JINR

Poland

D. Kotlorz

United Kingdom

E. Leader

New study of the Boer-Mulders function: Implications for the quark and hadron transverse momenta