We point out a novel mechanism to generate <math altimg='si1.svg'><mi mathvariant='normal'>cos</mi><mo>&#8289;</mo><mi>&#981;</mi></math> two-particle azimuthal correlation (&#8216;directed flow&#8217;) in unpolarized proton-proton and proton-nucleus collisions in the forward rapidity region of the projectile proton. This is a direct consequence of the recently discovered strong spin-orbit coupling in gluons at small- <italic>x</italic>. The observable simultaneously serves as a unique probe into the double helicity parton distribution functions of the proton.

Croatia

Sanjin Benić

United States

Yoshitaka Hatta

Directed flow from parton spin-orbit coupling in <italic>pp</italic> and <italic>pA</italic> collisions

<p>Working in the hybrid framework of the high energy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>A</mi></math> collisions we identify a new contribution to transverse single spin asymmetry (SSA). The phase necessary for the SSA is provided by the Pomeron-odderon interference in the dense nuclear target. The complete formula for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>A</mi><mo stretchy="false">→</mo><mi>h</mi><mi>X</mi></math> polarized cross section also contains the transversity distribution for the polarized projectile as well as the real part of the twist-three fragmentation function. We numerically estimate the asymmetry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mi>N</mi></msub></math> and its nuclear dependence. Based on a model computation we find that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mi>N</mi></msub></math> can be a percent level in the forward and low-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>h</mi><mo>⊥</mo></mrow></msub></math> region. For large nuclei we find significant suppression, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mi>N</mi></msub><mo>∝</mo><msup><mi>A</mi><mrow><mo>−</mo><mn>7</mn><mo>/</mo><mn>6</mn></mrow></msup></math> parametrically. As a notable feature we find a node of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mi>N</mi></msub></math> as a function of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>h</mi><mo>⊥</mo></mrow></msub></math> around the values of the initial saturation scale that could be used to test this mechanism experimentally.</p>

Eric Andreas Vivoda

Single spin asymmetry in forward <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>A</mi></math> collisions from the Pomeron-odderon interference

<p>Exclusive production of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></math> axial vector quarkonia in photon-proton scattering at high energies requires a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi></math>-odd <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi></math>-channel exchange. In the limit of vanishing momentum transfer this occurs either via the exchange of a photon, the Primakoff process, where the spin of the proton does not change. For axial-vector meson production, as a consequence of the Landau-Yang theorem, the Primakoff cross section is finite as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></math>. Alternatively, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi></math>-odd spin dependent Odderon can be exchanged, which involves a spin flip of the proton. The resulting cross section is related to the square of the collinear trigluon correlator or the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>k</mi><mo>⊥</mo></msub></math>-moment of the gluon Sivers function. Using two models for the gluon Sivers function from the literature we compute the ratio of Sivers to Primakoff cross sections and the angular coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>λ</mi><mi>θ</mi></msub></math> governing the angular distribution of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo stretchy="false">→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><mi>γ</mi></math> decay as functions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>. We point out that these observables constrain the magnitude of the gluon Sivers function at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> which could be accessed in electron-proton scattering and ultraperipheral proton-proton and nucleus-proton collisions.</p>

Adrian Dumitru

Poland

Leszek Motyka

Tomasz Stebel

Gluon Sivers function from forward exclusive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>1</mn></mrow></msub></mrow></math> photoproduction on unpolarized protons

Exclusive <math><mi>C</mi><mo>=</mo><mo>+</mo><mn>1</mn></math> scalar, axial-vector, and tensor quarkonium production in high-energy electron-proton scattering requires a <math><mi>C</mi></math>-odd <math><mi>t</mi></math>-channel exchange of a photon or a three-gluon ladder. We derive the expressions for the corresponding amplitudes. The relative phase of the photon vs three-gluon exchange amplitudes is determined by the sign of the light-front matrix element of the eikonal color current operator <math><msup><mi>d</mi><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></msup><msup><mi>J</mi><mrow><mo>+</mo><mi>a</mi></mrow></msup><msup><mi>J</mi><mrow><mo>+</mo><mi>b</mi></mrow></msup><msup><mi>J</mi><mrow><mo>+</mo><mi>c</mi></mrow></msup></math> at moderate <math><mi>x</mi></math>, and is not affected by small-<math><mi>x</mi></math> QCD evolution. Model calculations predict constructive interference, which is particularly strong for momentum transfer <math><mrow><mo>|</mo><mi>t</mi><mo>|</mo><mo>∼</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> where the cross section for <math><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> production exceeds that for pure photon exchange by up to a factor of 4. Exclusive <math><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> electroproduction at the high luminosity Electron-Ion Collider occurs with well measurable rates and measurements of these processes should find evidence for the perturbative odderon exchange. We also compute the total electroproduction cross section as a function of energy and provide first estimates of the number of <math><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> events per month at the Electron-Ion Collider design luminosity.

Abhiram Kaushik

Photon-odderon interference in exclusive <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math> charmonium production at the Electron-Ion Collider

In a previous publication [Benić et al., Phys. Rev. D 104, 094027 (2021)], we have computed the perturbative QCD contribution to transverse single spin asymmetries (SSAs) in semi-inclusive deep inelastic scattering (SIDIS) involving the <math><msub><mi>g</mi><mi>T</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> distribution. In this paper, we first present a more efficient derivation of the asymmetries which is applicable to both transverse and longitudinal SSAs and correct some inconsistencies in our previous calculation. We then adapt the method to compute transverse SSAs in the Drell-Yan process proportional to <math><msub><mi>g</mi><mi>T</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> and its gluonic counterpart and discuss the crossing symmetry between the results for SIDIS and the Drell-Yan process. Finally, we present numerical results for various asymmetries measurable at the EIC, RHIC, COMPASS, and Fermilab (SpinQuest), including also part of the genuine twist-three corrections to <math><msub><mi>g</mi><mi>T</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> from a recent global analysis. We find that the asymmetries can reach percent-level magnitude, if the kinematics predominantly probes the large-<math><mi>x</mi></math> (valence) region of the polarized proton, but remain at subpercent levels otherwise.

China

Hsiang-nan Li

Perturbative QCD contribution to transverse single spin asymmetries in the Drell-Yan process and SIDIS

We compute exclusive <math><msub><mi>η</mi><mi>c</mi></msub></math> production in high-energy electron-nucleon and electron-nucleus collisions that is sensitive to the odderon. In perturbative QCD the odderon is a <math><mi>C</mi></math>-odd color singlet consisting of at least three <math><mi>t</mi></math>-channel gluons exchanged with the target. By using the Color Glass Condensate effective theory our result describes the odderon exchange at the high-collision energies that would be reached at a future electron-ion collider. The odderon distribution is evolved to small-<math><mi>x</mi></math> using the Balitsky-Kovchegov evolution equation with running coupling corrections. We find that while at low momentum transfers <math><mi>t</mi></math> the cross section off a proton is dominated by the Primakoff process, the odderon becomes relevant at larger momentum transfers of <math><mrow><mo>|</mo><mi>t</mi><mo>|</mo><mo>≥</mo><mn>1.5</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>. We point that the odderon could also be extracted at low-<math><mi>t</mi></math> using neutron targets since the Primakoff component is strongly suppressed. In the case of nuclear targets, the odderon cross section becomes enhanced thanks to the mass number of the nuclear target. The gluon saturation effect induces a shift in the diffractive pattern with respect to the Primakoff process that could be used as a signal for the odderon.

Davor Horvatić

Exclusive <math><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math> production from small-<math><mi>x</mi></math> evolved Odderon at an electron-ion collider

We compute the transverse single spin asymmetry in forward <math><msup><mi>p</mi><mo>↑</mo></msup><mi>p</mi><mo>→</mo><mi>h</mi><mi>X</mi></math> and <math><msup><mi>p</mi><mo>↑</mo></msup><mi>A</mi><mo>→</mo><mi>h</mi><mi>X</mi></math> collisions from the odderon mechanism originally suggested by Kovchegov and Sievert [Phys. Rev. D 86, 034028 (2012)]. Working in the hybrid approach of the color glass condensate effective theory, we first identify the relevant collinear parton distribution function (PDF) of the transversely polarized proton <math><msup><mi>p</mi><mo>↑</mo></msup></math> as the intrinsic twist-3 <math><msub><mi>g</mi><mi>T</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> distribution. We further argue that the complete polarized cross section also contains contributions from the kinematical and the dynamical twist-3 PDFs, in addition to the intrinsic twist-3 PDF. By restricting to the Wandzura-Wilczek approximation, where the dynamical twist-3 PDFs are dropped, we find that the odderon contribution to the polarized cross section for inclusive hadron production is exactly zero at the next-to-leading order in the strong coupling.

Odderon mechanism for transverse single spin asymmetry in the Wandzura-Wilczek approximation

We compute the isolated photon production in association with a charged hadron at midrapidity in <math><mi>p</mi><mi>p</mi></math> and <math><mi>p</mi><mi>A</mi></math> based on the Color Glass Condensate (CGC) framework of high energy Quantum Chromodynamics (QCD) where, for the first time, we incorporate the Sudakov effect of soft gluon emissions. Our results are based on the leading order <math><mi>q</mi><mi>g</mi><mo>→</mo><mi>q</mi><mi>γ</mi></math> channel in the CGC framework and confronted with the recent data from Relativistic Heavy-Ion Collider (RHIC) and Large Hadron Collider (LHC) concerning the angular distributions and out-of-plane transverse momentum distributions. We find that, while the CGC computation alone results in too narrow distributions, with the help of the Sudakov effect, we can get a satisfactory description of the data. With this as a benchmark, we provide predictions for the magnitude of the nuclear effect brought by the phenomena of gluon saturation in the CGC.

Germany

Oscar Garcia-Montero

Anton Perkov

Isolated photon-hadron production in high energy <math><mi>p</mi><mi>p</mi></math> and <math><mi>p</mi><mi>A</mi></math> collisions at RHIC and LHC

Motivated by a novel origin of transverse single spin asymmetry (SSA) in semi-inclusive deep inelastic scattering (SIDIS) uncovered by some of us, we quantitatively investigate its impact on the theoretical understanding of the mechanism responsible for SSA. This new contribution from the quark-initiated channel first appears in two-loop perturbation theory and involves the <math><msub><mi>g</mi><mi>T</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> distribution. We point out another entirely analogous piece from the gluon-initiated channel proportional to the gluon helicity distribution <math><mi>Δ</mi><mi>G</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>. Both contributions are solely expressed in terms of twist-two polarized parton distribution functions and twist-two fragmentation functions in the Wandzura-Wilczek approximation, such that they can be unambiguously evaluated without introducing free parameters. We make predictions for measurements of the asymmetries <math><msub><mi>A</mi><mrow><mi>U</mi><mi>T</mi></mrow></msub></math> at the future electron-ion collider (EIC), and find that <math><msub><mi>A</mi><mrow><mi>U</mi><mi>T</mi></mrow></msub></math> associated with the <math><mi>sin</mi><mo>(</mo><msub><mi>ϕ</mi><mi>h</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>S</mi></msub><mo>)</mo></math>, <math><mi>sin</mi><msub><mi>ϕ</mi><mi>S</mi></msub></math> and <math><mi>sin</mi><mo>(</mo><mn>2</mn><msub><mi>ϕ</mi><mi>h</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>S</mi></msub><mo>)</mo></math> harmonics can reach up to 1–2% even at the top EIC energy.

<math><msub><mi>g</mi><mi>T</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> contribution to single spin asymmetries in SIDIS

We find a novel mechanism for generating transverse single-spin asymmetry (SSA) in semi-inclusive deep inelastic scattering, distinct from the known ones which involve the Sivers and Collins functions, or their collinear twist-three counterparts. It is demonstrated that a phase needed for SSA can be produced purely within a parton-level cross section starting at two loops. We identify the complete set of two-loop diagrams for SSA, and discuss their gauge invariance and collinear factorization which features the <math><msub><mi>g</mi><mi>T</mi></msub></math> distribution function. In the <math><msub><mi>k</mi><mi>T</mi></msub></math> factorization framework, many more sources for SSA exist, and contributions from all possible two-parton transverse-momentum-dependent parton distribution functions are presented up to two loops and twist three.