<p>Future planned lepton colliders, both in the circular and linear configurations, can effectively work as virtual and quasireal photon-photon colliders and are expected to stimulate an intense physics program in the next few years. In this paper, we suggest to consider photon-photon scattering as a useful source of information on transverse-momentum-dependent fragmentation functions (TMD FFs), complementing semi-inclusive deeply inelastic scattering and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> annihilation processes, which provide most of the present phenomenological information on TMD FFs. As a first illustrative example, we study two-hadron azimuthal asymmetries around the jet thrust axis in the process <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">→</mo><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><mi>γ</mi><mo stretchy="false">→</mo><mi>q</mi><mover accent="true"><mrow><mi>q</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">→</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>X</mi></mrow></math>, in which in a circular lepton collider one tagged, deeply virtual photon scatters off an untagged quasireal photon, both originating from the initial lepton beams, producing inclusively an almost back-to-back light-hadron pair with large transverse momentum, in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>γ</mi><mo>*</mo></msup><mi>γ</mi></math> center-of-mass frame. Similar processes, in a more complicated environment due to the presence of initial hadronic states, can also be studied in ultraperipheral collisions at the LHC and the planned future hadron colliders.</p>

Italy

Simone Anedda

Francesco Murgia

Cristian Pisano

First insight into transverse-momentum-dependent fragmentation physics at photon-photon colliders

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 production with the production of heavier particles, and for LHC kinematics, focusing on production. These results exhibit interesting mass-dependent features in the asymmetries that may help study the anisotropies arising from the underlying photon GTMD description.

Netherlands

Daniël Boer

Luca Maxia

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

Within the framework of transverse momentum dependent factorization in combination with nonrelativistic quantum chromodynamics (QCD), we study charmonium and bottomonium production in hadronic collisions. We focus on quarkonium states with even charge conjugation, for which the color-singlet production mechanism is expected to be also dominant in the small transverse momentum region, <math><msubsup><mi>q</mi><mi>T</mi><mn>2</mn></msubsup><mo>≪</mo><mn>4</mn><msubsup><mi>M</mi><mrow><mi>c</mi><mo>,</mo><mi>b</mi></mrow><mn>2</mn></msubsup></math>. It is shown that the distributions of linearly polarized gluons inside unpolarized, longitudinally, and transversely polarized protons contribute to the cross sections for scalar and pseudoscalar quarkonia in a very distinctive, parity-dependent way, whereas their effects on higher angular momentum states are strongly suppressed. We derive analytical expressions for single and double spin asymmetries, which would allow for the direct extraction of the gluon transverse momentum dependent distributions, mirroring the phenomenological studies of the Drell-Yan processes aimed at the extraction of their quark counterparts. By adopting Gaussian models for the gluon transverse momentum dependent distributions, which fulfill without saturating everywhere their positivity bounds, we provide numerical predictions for the transverse single-spin asymmetries. These observables could be measured at LHCSpin, the fixed target experiment planned at the LHC.

Nanako Kato

Spin asymmetries for <math><mi>C</mi></math>-even quarkonium production as a probe of gluon distributions

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.

France

Jelle Bor

United States

Feng Yuan

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

We present a detailed phenomenological study of <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> polarization in semi-inclusive deep-inelastic scattering processes, focusing on the kinematics accessible at the future Electron-Ion Collider. We show theoretical estimates for the standard polarization parameters for different frames usually adopted in the literature, in the large <math><msub><mi>P</mi><mi>T</mi></msub></math> region, namely <math><msub><mi>P</mi><mi>T</mi></msub><mo>≫</mo><msub><mi>Λ</mi><mi>QCD</mi></msub></math>, where collinear factorization is expected to hold. We adopt both the color singlet model and the nonrelativistic QCD approach, paying special attention to the role of different sets of long distance matrix elements. Finally we present a preliminary analysis of some frame independent polarization invariants.

Umberto D’Alesio

India

Sangem Rajesh

<math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> polarization in large-<math><msub><mi>P</mi><mi>T</mi></msub></math> semi-inclusive deep-inelastic scattering at the EIC

We present the distribution of unpolarized quarks in a transversely polarized proton in three-dimensional momentum space. Our results are based on the extractions of the unpolarized and Sivers transverse momentum dependent parton distributions (TMDs) in a fully consistent TMD framework.

Alessandro Bacchetta

Filippo Delcarro

Marco Radici

The 3-dimensional distribution of quarks in momentum space

We study the polar and azimuthal decay angular distributions of J/ψ mesons produced in semi-inclusive, deep-inelastic electron-proton scattering. For the description of the quarkonium formation mechanism, we adopt the framework of nonrelativistic QCD, with the inclusion of the intermediate color-octet channels that are suppressed at most by a factor v 4 in the velocity parameter v relative to the leading color-singlet channel. We put forward factorized expressions for the helicity structure functions in terms of transverse momentum dependent gluon distributions and shape functions, which are valid when the J/ψ transverse momentum is small with respect to the hard scale of the process. By requiring that such expressions correctly match with the collinear factorization results at high transverse momentum, we determine the perturbative tails of the shape functions and find them to be independent of the J/ψ polarization. In particular, we focus on the cos 2ϕ azimuthal decay asymmetry, which originates from the distribution of linearly polarized gluons inside an unpolarized proton. We therefore suggest a novel experiment for the extraction of this so-far unknown parton density that could be performed, in principle, at the future Electron-Ion Collider.

J/ψ polarization in semi-inclusive DIS at low and high transverse momentum

Recently unpolarized and polarized <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>(</mo><mi>ϒ</mi><mo>)</mo></math> production at the Electron-Ion Collider (EIC) has been proposed as a new way to extract two poorly known color-octet NRQCD long-distance matrix elements: <math><mrow><mo>⟨</mo><mn>0</mn><mo>|</mo><msubsup><mrow><mi>O</mi></mrow><mrow><mn>8</mn></mrow><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></msubsup><mo>(</mo><mmultiscripts><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts><mo>)</mo><mo>|</mo><mn>0</mn><mo>⟩</mo></mrow></math> and <math><mrow><mo>⟨</mo><mn>0</mn><mo>|</mo><msubsup><mrow><mi>O</mi></mrow><mrow><mn>8</mn></mrow><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></msubsup><mo>(</mo><mmultiscripts><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts><mo>)</mo><mo>|</mo><mn>0</mn><mo>⟩</mo></mrow></math>. The proposed method is based on a comparison to open heavy-quark pair production ideally performed at the same kinematics. In this paper we analyze this proposal in more detail and provide predictions for the EIC based on the available determinations of the color-octet matrix elements. We also propose two additional methods that do not require comparison to open heavy-quark pair production.

Pieter Taels

Extracting color octet NRQCD matrix elements from <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production at the EIC

We consider the transverse single-spin asymmetry (SSA) for <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production in <math><msup><mi>p</mi><mo>↑</mo></msup><mi>p</mi><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><mi>X</mi></math> within a phenomenological transverse momentum dependent scheme in nonrelativistic QCD. Extending a previous study [U. D’Alesio et al., Eur. Phys. J. C 79, 1029 (2019)], we employ here the color-gauge invariant generalized parton model (CGI-GPM), in which spin and intrinsic transverse momentum effects are taken into account, together with leading-order initial- and final-state interactions (ISIs and FSIs). We find that, even when the heavy-quark pair is produced in a color-octet state, ISIs and FSIs lead to a nonvanishing SSA, allowing us, in principle, to test the process dependence of the gluon Sivers function (GSF). We show that of the two independent contributions, due to the so-called <math><mi>f</mi></math>- and <math><mi>d</mi></math>-type GSFs, appearing in the CGI-GPM, the <math><mi>d</mi></math>-type one turns out to be dynamically suppressed. Therefore, as already found adopting the Color-Singlet Model approach for the <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> formation [U. D’Alesio et al., Phys. Rev. D 96, 036011 (2017)], only the <math><mi>f</mi></math>-type GSF could play a role in phenomenology. A comparison with the corresponding results obtained in the generalized parton model, without the inclusion of ISIs and FSIs, is also carried out.

Process dependence of the gluon Sivers function in <math><msup><mi>p</mi><mo>↑</mo></msup><mi>p</mi><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><mi>X</mi></math> within a TMD scheme in NRQCD

We consider the transverse momentum spectrum and the cos 2ϕ azimuthal distribution of J/ψ mesons produced in semi-inclusive, deep-inelastic electron-proton scattering, where the electron and the proton are unpolarized. At low transverse momentum, we propose factorized expressions in terms of transverse momentum dependent gluon distributions and shape functions. We show that our formulae, at the order α  s  , correctly match with the collinear factorization results at high transverse momentum. The latter are computed at the order   <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math>  $$ {\alpha}_s^2 $$  in the framework of nonrelativistic QCD (NRQCD), with the inclusion of the intermediate   <math> <msup> <mrow></mrow> <mn>3</mn> </msup> <msubsup> <mi>S</mi> <mn>1</mn> <mfenced> <mn>1</mn> </mfenced> </msubsup> </math>  $$ {}^3{S}_1^{\left[1\right]} $$  color-singlet Fock state, as well as the subleading color-octet ones that are relatively suppressed by a factor v 4 in the NRQCD velocity parameter v. We show that the   <math> <msup> <mrow></mrow> <mn>1</mn> </msup> <msubsup> <mi>S</mi> <mn>0</mn> <mfenced> <mn>8</mn> </mfenced> </msubsup> </math>  $$ {}^1{S}_0^{\left[8\right]} $$  and   <math> <msup> <mrow></mrow> <mn>3</mn> </msup> <msubsup> <mi>P</mi> <mi>J</mi> <mfenced> <mn>8</mn> </mfenced> </msubsup> </math>  $$ {}^3{P}_J^{\left[8\right]} $$  (J = 0, 1, 2) contributions diverge in the small transverse momentum region and allow us to determine the perturbative tails of the shape functions, which carry the same quantum numbers. These turn out to be identical, except for the overall magnitude given by the appropriate NRQCD long distance matrix element.