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The 3d structure of hadrons and origin of the proton's spin The Fundamental Pieces of Visible Matter: Offering an Unprecedented Insight into the 3D Structure of Hadrons and the Proton Spin Puzzle. All the visible matter in the universe is composed of atoms, which consist of electrons orbiting a small but extremely dense nucleus. The nucleus contains protons and neutrons, which also have a substructure of particles called quarks. Quarks come in different types, called flavors: up, down, strange, charm, bottom, and top. The quarks are held together by the strong nuclear force, which is mediated by other particles called gluons. Quarks and gluons, collectively called partons, are some of the most elementary pieces of visible matter in the universe. A “zoo” of particles, generically called hadrons, is made up of quarks and gluons.

The worm-gear transverse momentum dependent (TMD) function g1T is one of the eight leading-twist TMDs and has the probabilistic interpretation of finding a longitudinally polarized quark inside a transversely polarized hadron. In this work, we present the first simultaneous extraction of g1T from all the available experimental measurements. The study involves the analysis of COMPASS, HERMES, and Jefferson Lab data on semi-inclusive deep-inelastic scattering using Monte Carlo techniques. We also compare g1T obtained from this experimental data with different theoretical approaches, such as the large-Nc approximation, the Wandzura-Wilczek-type approximation, and lattice QCD.

We present a Monte Carlo based analysis of the combined world data on polarized lepton-nucleon deep-inelastic scattering at small Bjorken $x$ within the polarized quark dipole formalism. We show for the first time that double-spin asymmetries at $x, 7 pages, 5 figures; v2: Discussion extended, typos corrected, data set expanded, figure added. To appear in Phys. Rev. D

In this paper we perform the first simultaneous QCD global analysis of data from semi-inclusive deep inelastic scattering, Drell-Yan, $e^+e^-$ annihilation into hadron pairs, and proton-proton collisions. Consequently, we are able to extract a universal set of non-perturbative functions that describes the observed asymmetries in these reactions. The outcome of our analysis indicates single transverse-spin asymmetries in high-energy collisions have a common origin. Furthermore, we achieve the first phenomenological agreement with lattice QCD on the up and down quark tensor charges., Comment: 8 pages, 6 figures, matches published version

We study transverse polarization of hyperons produced in unpolarized proton–proton collision, pp → Λ^↑X, based on the collinear twist-3 factorization. We focus on the twist-3 gluon fragmentation contribution to this process, which is potentially as important as the previously calculated contributions from the twist-3 distribution in the initial proton and the twist-3 quark-fragmentation for Λ^↑. After summarizing the complete set of the purely gluonic twist-3 fragmentation functions for Λ^↑, we provide a formalism of calculating its contribution to pp → Λ^↑X.

We discuss the production of transversely polarized hyperons in unpolarized proton-proton collisions in the framework of the collinear twist-3 factorization. We focus on the contribution from the twist-3 fragmentation functions for the final hyperon, that represent the effect of multi-parton correlations in the fragmentation process. This contribution consists of the quark fragmentation one and the gluon fragmentation one. We present the twist-3 cross section for the former in the Leading Order (LO) with respect to the QCD coupling constant. We also present a new formalism to calculate the cross section for the latter contribution. These contributions are as important as that from the twist-3 distribution in the initial proton combined with the transversity fragmentation function for the final hyperon.

Transverse momentum dependent (TMD) observables in high-energy particle collisions allow one to study the motion of hadron constituents (partons) in three dimensions. The intense theoretical, phenomenological, and experimental work over the last several decades has seen great progress in modeling and extracting TMD parton distribution functions (PDFs) and fragmentation functions (FFs) [1, 2]. The wealth of experimental data, both current and future [3–6], across various reactions and energy ranges, along with substantial progress in other approaches, such as Lattice QCD [7], has increased the demand to rigorously study the intrinsic transverse motion of hadrons. Such issues currently being addressed include factorization, evolution, and operator definitions of TMD distributions, as well as how they influence phenomenological and Lattice QCD studies. Therefore, in this special issue we attempt to further the effort of understanding TMD structure through a series of review articles that address some of the significant areas of ongoing research.

We compute the contribution from the twist-3 fragmentation function for light hadron production in collisions between transversely and longitudinally polarized protons, i.e., $p^\uparrow \vec{p} \to h\,X$, which can cause a double-spin asymmetry (DSA) $A_{LT}$. This is a na\"ive T-even twist-3 observable that we analyze in collinear factorization using both Feynman gauge and lightcone gauge as well as give a general proof of color gauge invariance. So far only twist-3 effects in the transversely polarized proton have been studied for $A_{LT}$ in $p^\uparrow \vec{p} \to h\,X$. However, there are indications that the na\"ive T-odd transverse single-spin asymmetry (SSA) $A_N$ in $p^\uparrow p \to h\,X$ is dominated not by such distribution effects but rather by a fragmentation mechanism. Therefore, one may expect similarly that the fragmentation contribution is important for $A_{LT}$. Given possible plans at RHIC to measure this observable, it is timely to provide a calculation of this term., Comment: 11 pages, 5 figures, Minor changes, version to appear in PLB

We review the theoretical framework underlying the description of transverse single-spin asymmetries (SSAs) for both collinear and transverse-momentum-dependent (TMD) processes. For the latter, we discuss modifications to the widely used CSS formalism that are needed to address certain issues that have arisen in TMD analyses. After these improvements, one can derive rigorous relations between collinear and TMD functions. Moreover, we show how both collinear and TMD SSA observables are driven by multi-parton correlations, which allows for a future global extraction to be performed including data from both types of measurements. QCD Evolution 2017 -QCDEV2017- 22-26 MAY 2017 Jefferson Lab (JLAB), Newport News Virginia, USA Speaker.

We determine the small-$x$ asymptotics of the gluon helicity distribution in a proton at leading order in perturbative QCD at large $N_c$. To achieve this, we begin by evaluating the dipole gluon helicity TMD at small $x$. We then construct and solve novel small-$x$ large-$N_c$ evolution equations for the operator related to the dipole gluon helicity TMD. Our main result is the small-$x$ asymptotics for the gluon helicity distribution: \begin{align}\label{dG_final} \Delta G \sim \left( \frac{1}{x} \right)^{\alpha_h^G} \ \ \ \mbox{with} \ \ \ \alpha_h^G = \frac{13}{4 \sqrt{3}} \, \sqrt{\frac{\as \, N_c}{2 \pi}} \approx 1.88 \, \sqrt{\frac{\as \, N_c}{2 \pi}}. \end{align} We note that the power $\alpha_h^G$ is approximately 20$\%$ lower than the corresponding power $\alpha_h^q$ for the small-$x$ asymptotics of the quark helicity distribution defined by \begin{align}\label{dq_final} \Delta q \sim \left( \frac{1}{x} \right)^{\alpha_h^q} \ \ \ \mbox{with} \ \ \ \alpha_h^q = \frac{4}{\sqrt{3}} \, \sqrt{\frac{\as \, N_c}{2 \pi}} \approx 2.31 \, \sqrt{\frac{\as \, N_c}{2 \pi}} \end{align} found in our earlier work.

We study the production of polarized $\Lambda$-hyperons in electron-positron annihilation. We are particularly interested in the transverse-spin dependence of the cross section for unpolarized incident electron-positron pairs. At high energies this process may be described in the collinear twist-3 framework, where the hadronization transition of partons into a transversely polarized $\Lambda$-hyperon can be written in terms of collinear twist-3 fragmentation matrix elements. We calculate the hard partonic cross sections and interference terms in perturbative QCD to next-to-leading order accuracy. We find that the QCD equation of motion plays a crucial role in our analysis. As a byproduct, assuming the validity of QCD factorization for twist-3 observables at next-to-leading order, we derive the evolution equation for the relevant twist-3 fragmentation matrix element., Comment: 43 pages, 25 figures

We determine the small-x asymptotics of the gluon helicity distribution in a proton at leading order in perturbative QCD at large Nc. To achieve this, we begin by evaluating the dipole gluon helicity TMD at small x. We then construct and solve novel small-x large-Nc evolution equations for the operator related to the dipole gluon helicity TMD. Our main result is the small-x asymptotics for the quark helicity distribution

We analyze the longitudinal-transverse double-spin asymmetry in lepton-nucleon collisions where a single hadron is detected in the final state, i.e., $\vec{\ell}\,N^\uparrow \rightarrow h\,X$. This is a subleading-twist observable in collinear factorization, and we look at twist-3 effects in both the transversely polarized nucleon and the unpolarized outgoing hadron. Results are anticipated for this asymmetry from both HERMES and Jefferson Lab Hall A, and it could be measured as well at COMPASS and a future Electron-Ion Collider. We also perform a numerical study of the distribution term, which, when compared to upcoming experimental results, could allow one to learn about the "worm-gear"-type function $\tilde{g}(x)$ as well as assess the role of quark-gluon-quark correlations in the initial-state nucleon and twist-3 effects in the fragmenting unpolarized hadron., Comment: 14 pages, 7 figures, minor changes to the text, version to appear in Phys. Lett. B

We present results for transverse single-spin asymmetries in proton-proton collisions at kinematics relevant for AFTER, a proposed fixed-target experiment at the Large Hadron Collider. These include predictions for pion, jet, and direct photon production from analytical formulas already available in the literature. We also discuss specific measurements that will benefit from the higher luminosity of AFTER, which could help resolve an almost 40-year puzzle of what causes transverse single-spin asymmetries in proton-proton collisions., 10 pages, 4 figures; more details/discussion added to the text, references added/updated, version to appear in Advances in High Energy Physics for the Special Issue "Physics at a Fixed-Target Experiment Using the LHC Beams"

We extend the improved Collins-Soper-Sterman (iCSS) $W+Y$ construction recently presented in~\cite{Collins:2016hqq} to the case of polarized observables, where we focus in particular on the Sivers effect in semi-inclusive deep-inelastic scattering. We further show how one recovers the expected leading-order collinear twist-3 result from a (weighted) $q_T$-integral of the differential cross section. We are also able to demonstrate the validity of the well-known relation between the (TMD) Sivers function and the (collinear twist-3) Qiu-Sterman function within the iCSS framework. This relation allows for their interpretation as functions yielding the average transverse momentum of unpolarized quarks in a transversely polarized spin-$\frac{1}{2}$ target. We further outline how this study can be generalized to other polarized quantities., 14 pages, Version to be published in PLB

We determine the small-$x$ asymptotics of the gluon helicity distribution in a proton at leading order in perturbative QCD at large $N_c$. To achieve this, we begin by evaluating the dipole gluon helicity TMD at small $x$. In the process we obtain an interesting new result: in contrast to the unpolarized dipole gluon TMD case, the operator governing the small-$x$ behavior of the dipole gluon helicity TMD is different from the operator corresponding to the polarized dipole scattering amplitude (used in our previous work to determine the small-$x$ asymptotics of the quark helicity distribution). We then construct and solve novel small-$x$ large-$N_c$ evolution equations for the operator related to the dipole gluon helicity TMD. Our main result is the small-$x$ asymptotics for the gluon helicity distribution: $\Delta G \sim \left( \tfrac{1}{x} \right)^{\alpha_h^G}$ with $\alpha_h^G = \tfrac{13}{4 \sqrt{3}} \, \sqrt{\tfrac{\alpha_s \, N_c}{2 \pi}} \approx 1.88 \, \sqrt{\tfrac{\alpha_s \, N_c}{2 \pi}}$. We note that the power $\alpha_h^G$ is approximately 20$\%$ lower than the corresponding power $\alpha_h^q$ for the small-$x$ asymptotics of the quark helicity distribution defined by $\Delta q \sim \left( \tfrac{1}{x} \right)^{\alpha_h^q}$ with $\alpha_h^q = \tfrac{4}{\sqrt{3}} \, \sqrt{\tfrac{\alpha_s \, N_c}{2 \pi}} \approx 2.31 \, \sqrt{\tfrac{\alpha_s \, N_c}{2 \pi}}$ found in our earlier work., Comment: 36 pages, 9 figures; v3: minus signs and factors of 2 corrected, main results remained the same

We present a new analysis of A N in p ↑ p → π X within the collinear twist-3 factorization formalism. We incorporate recently derived Lorentz invariance relations into our calculation and focus on input from the kinematical twist-3 functions, which are weighted integrals of transverse momentum dependent (TMD) functions. In particular, we use the latest extractions of the Sivers and Collins functions with TMD evolution to compute certain terms in A N . Consequently, we are able to constrain the remaining contributions from the lesser known dynamical twist-3 correlators.

Transverse single-spin asymmetries are among the most intriguing observables in hadronic physics. Though such asymmetries were already measured for the first time about four decades ago, their origin is still under debate. Here we consider transverse single-spin asymmetries in semi-inclusive lepton–nucleon scattering, in nucleon–nucleon scattering, and in inclusive lepton–nucleon scattering. It is argued that, according to recent work, the single-spin asymmetries for those three processes may be simultaneously described in perturbative QCD, where the re-scattering of the active partons plays a crucial role. A comparison of single-spin asymmetries in different reactions can also shed light on the universality of transverse momentum dependent parton correlation functions. In particular, we discuss what existing data may tell us about the predicted process dependence of the Sivers function.

We present a new collinear twist-3 analysis of the transverse SSA A N at RHIC. We use the TMD Sivers/Collins function to fix some of the relevant collinear twist-3 functions and perform a fit of the RHIC data with other parameterized twist-3 functions. This allows us to keep the consistency among descriptions in pp collision, SIDIS, and e + e − annihilation and thus could provide a unified description of the spin asymmetries in the low- and high-P T processes. By taking into account the twist-3 fragmentation contribution, we show for the first time this contribution could be the main source of A N in $${pp^{\uparrow} \rightarrow hX}$$ and its inclusion could provide a solution for the sign-mismatch problem.

In this Letter, we analytically solve the evolution equations for the small-$x$ asymptotic behavior of the (flavor singlet) quark helicity distribution in the large-$N_c$ limit. These evolution equations form a set of coupled integro-differential equations, which previously could only be solved numerically. This approximate numerical solution, however, revealed simplifying properties of the small-$x$ asymptotics, which we exploit here to obtain an analytic solution. We find that the small-$x$ power-law tail of the quark helicity distribution scales as $\Delta q^S (x, Q^2) \sim \left(\tfrac{1}{x} \right)^{\alpha_h}$ with $\alpha_h = \tfrac{4}{\sqrt{3}} \sqrt{\tfrac{\alpha_s N_c}{2\pi}}$, in excellent agreement with the numerical estimate $\alpha_h \approx 2.31\sqrt{\tfrac{\alpha_s N_c}{2\pi}}$ obtained previously. We then verify this solution by cross-checking the predicted scaling behavior of the auxiliary "neighbor dipole amplitude" against the numerics, again finding excellent agreement., Comment: 5 pages, 2 figures

We discuss two twist-3 spin asymmetries in $pp$ collisions. One is the transverse polarization of hyperons produced in unpolarized $pp$ collisions, and the other is the spin asymmetry for the hadron production in the collision between the longitudinally and transversely polarized protons. The former is a naively ``$T$-odd" observable, while the latter is a naively ``$T$-even" one, and this leads to inherently different form of the corresponding twist-3 cross sections. Since both asymmetries occur as a result of multi-parton correlations either in the initial protons or in the fragmentation process, they play a complementary role to test the theoretical framework and to give an insight on the hadron structure. Measurement of these asymmetries at RHIC will be particularly interesting.

It has been known for a long time that hyperons produced in hadronic collisions are polarized perpendicular to the production plane of the reaction. This effect cannot be described by using twist-2 collinear parton correlators only. Here we compute the contribution of twist-3 fragmentation functions to the production of transversely polarized hyperons in unpolarized proton-proton collisions in the framework of collinear factorization. By taking into account the relations among the relevant twist-3 fragmentation functions which follow from the QCD equation of motion and the Lorentz invariance property of the correlators, we present the leading-order cross section for this term., Comment: 9 pages, 6 figures

We extend our earlier results for the quark helicity evolution at small $x$ to derive the small-$x$ asymptotics of the flavor singlet and flavor non-singlet quark helicity TMDs and PDFs and of the $g_1$ structure function. In the flavor singlet case we re-derive the evolution equations obtained in our previous paper on the subject, performing additional cross-checks of our results. In the flavor non-singlet case we construct new small-$x$ evolution equations by employing the large-$N_c$ limit. All evolution equations resum double-logarithmic powers of $\alpha_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the single-logarithmic powers of $\alpha_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. We solve the linearized flavor non-singlet equation analytically, obtaining an intercept which agrees with the one calculated earlier by Bartels, Ermolaev and Ryskin using the infra-red evolution equations. Our numerical solution of the linearized large-$N_c$ evolution equations for the flavor singlet case is presented in the accompanying Letter and is further discussed here., Comment: 30 pages, 12 figures

We review what is currently known about the transverse spin structure of hadrons, in particular from observables that can be analyzed within a collinear framework. These effects have been around for 40 years and represent a critical test of perturbative QCD. We look at both proton-proton and lepton-nucleon collisions for various final states. While the main focus is on transverse single-spin asymmetries, we also discuss how longitudinal-transverse double-spin asymmetries offer a complimentary, yet equally important, source of information on the quark-gluon content of hadrons. We also summarize some recent progress in solidifying the theoretical formalism behind these observables and give an outlook on future directions of research., Typos corrected, references added/updated, invited review accepted in International Journal of Modern Physics A; 32 pages, 9 figures, 2 tables

We calculate the double spin asymmetry (DSA) ALT in proton-proton (pp) collisions for the production of photons, hadrons, and jets in the collinear twist-3 framework. These processes allows us for the first time to access a complete set of collinear twist-3 functions. We also give numerical estimates for the asymmetries of which photon production shows the most promise of a measurable effect. Furthermore, we comment on how these asymmetries would allow us to probe the gluon helicity down to as of yet unexplored x values, remark on the importance of these effects in determining the evolution of the Efremov-Teryaev-Qiu-Sterman (ETQS) function TF(x, x), and discuss their usefulness in resolving the true mechanism behind the large transverse single spin asymmetries (SSAs) measured in pp collisions.

Transverse single spin asymmetries in inclusive deep-inelastic lepton-nucleon scattering can be generated through multi-photon exchange between the leptonic and the hadronic part of the process. Here we consider two-photon exchange, and mainly focus on the transverse target spin asymmetry. In particular, we investigate the case where two photons couple to different quarks. Such a contribution involves a quark-photon-quark correlator in the nucleon, which has a (model-dependent) relation to the Efremov-Teryaev-Qiu-Sterman quark-gluon-quark correlator TF. Using different parameterizations for TF we compute the transverse target spin asymmetry for both a proton and a neutron target, and compare the results to recent experimental data. Potential implications on our general understanding of single spin asymmetries in hard scattering processes are discussed as well.

Recently, three twist-3 spin asymmetries for single-inclusive hadron production in deep-inelastic lepton-nucleon scattering have been computed using collinear factorization and the leading order approximation. Here we summarize the main findings of these studies.

We construct a numerical solution of the small-$x$ evolution equations recently derived in \cite{Kovchegov:2015pbl} for the (anti)quark helicity TMDs and PDFs as well as the $g_1$ structure function. We focus on the case of large $N_c$ where one finds a closed set of equations. Employing the extracted intercept, we are able to predict directly from theory the behavior of the helicity PDFs at small $x$, which should have important phenomenological consequences. We also give an estimate of how much of the proton's spin may be at small $x$ and what impact this has on the so-called "spin crisis.", Comment: 5 pages, 4 figures; minor changes, to appear in PRL

We investigate the behavior under Lorentz tranformations of perturbative coefficient functions in a collinear twist-3 formalism relevant for high-energy observables including transverse polarization of hadrons. We argue that those perturbative coefficient functions can, {\it a priori}, acquire quite different yet Lorentz-invariant forms in various frames. This somewhat surprising difference can be traced back to a general dependence of the perturbative coefficient functions on lightcone vectors which are introduced by the twist-3 factorization formulae and which are frame-dependent. One can remove this spurious frame dependence by invoking so-called Lorentz invariance relations (LIRs) between twist-3 parton correlation functions. Some of those relations for twist-3 distribution functions were discussed in the literature before. In this paper we derive the corresponding LIRs for twist-3 fragmentation functions. We explicitly demonstrate that these LIRs remove the lightcone vector dependence by considering transverse spin observables in the single-inclusive production of hadrons in lepton-nucleon collisions, $\ell N\to hX$. With the LIRs in hand, we also show that twist-3 observables in general can be written solely in terms of three-parton correlation functions., 17 pages, 6 figures, minor changes, final version, published in PRD