International Journal of Modern Physics: Conference Series

We outline key objectives and capabilities of an Electron-Ion Collider (EIC) — a high-energy and high-luminosity electron-proton/nucleus collider with polarized electron and proton beams. One of goals of a future EIC is to map the 3D (in configuration and momentum spaces) structure of sea quarks and gluons in the nucleon and nuclei. We briefly present and discuss key observables and measurements pertaining to the program of 3D imaging at an EIC.

The high-energy behavior of the SYM amplitudes in the Regge limit can be calculated order by order in perturbation theory using the high-energy operator expansion in Wilson lines. At large N_c, a typical four-point amplitude is determined by a single BFKL pomeron. The conformal structure of the four-point amplitude is fixed in terms of two functions: pomeron intercept and the coefficient function in front of the pomeron (the product of two residues). The pomeron intercept is universal while the coefficient function depends on the correlator in question. The intercept is known in the first two orders in coupling constant: BFKL intercept and NLO BFKL intercept calculated in Ref. [1]. As an example of using the Wilson-line OPE, we calculate the coefficient function in front of the pomeron for the correlator of four Z² currents in the first two orders in perturbation theory.

At high energy (small x) n-point coorelators of Wilson lines appear in calculation of physical observables. The energy dependence of these observables is determined by the solution of the evolution equations these correlators satisfy. The most common correlator is the two-point function, the imaginary part of the forward scattering amplitude of a quark anti-quark dipole scattering on a target. This appears in structure functions in DIS as well as single inclusive hadron production in proton-nucleus collisions. Higher point correlators of Wilson lines appear in less inclusive processes, such as two-hadron angular and rapidity correlations and satisfy the Balitski-JIMWLK evolution equation. Here we derive the evolution equation satisfied by the six point correlator of Wilson lines which appears in di-hadron angular correlations in proton-nucleus collisions at high energy.

I briefly review recent work, within the AdS/CFT correspondence, on the determination of the remainder function of the six point scattering amplitude in the multi-Regge limit, both at weak and at strong coupling.

We discuss some recent developments towards a quantitative understanding of the production and early-time evolution of bulk quark-gluon matter in ultrarelativistic heavy ion collisions.

Using the high-energy Operator Product Expansion of the T-product of two electromagnetic currents, we calculate the Photon Impact Factor for Deep Inelastic Scattering at small value of the Bjorken variable x_B at the next-to-leading order (NLO) accuracy in α_s. We provide for the first time an analytic expression in coordinate space and in Mellin space of the NLO impact factor for the forward unpolarized structure functions.

The high-energy factorization and the associated B-JIMWLK or BK evolution equations are presented, using the example of DIS structure functions. The necessity of taking gluon saturation into account is discussed, and also the various approximations underlying high-energy factorization. The appearance of large NLL corrections in such a framework with or without gluon saturation is recalled, and their physical origin is explained. Finally, old and new results are presented about the resummation of some of those large corrections, related to kinematical approximations.

Nonlinear evolution at small x was evaluated numerically with full dependence on impact parameter using the BK equation. several distinct behaviors were found and are presented for the leading logarithmic kernel in the BK evolution equation with both fixed and running coupling. The value of the saturation scale at various dipole sizes was found to agree with analytic expectations. Calculation of the F₂ structure function from the numerical solution of the evolution with running coupling were then compared to the HERA data and qualitative agreement found. The agreement is improved with inclusion of soft contributions and these are discussed.

We review and discuss the use of TMD, or "unintegrated", gluon distributions in the domain of small-x physics. The definitions employed, and the hazards of the naive applications of the TMD factorization and the associated gluon distributions are discussed.

I give an account of a new definition of transverse-momentum-dependent parton densities. The new definition solves a number of difficulties and inconsistencies in earlier definitions.

We discuss the application of transverse momentum dependent (TMD) factorization theorems to phenomenology. Our treatment relies on recent extensions of the Collins-Soper-Sterman (CSS) formalism. Emphasis is placed on the importance of using well-defined TMD parton distribution functions (PDFs) and fragmentation functions (FFs) in calculating the evolution of these objects. We explain how parametrizations of unpolarized TMDs can be obtained from currently existing fixed-scale Gaussian fits and previous implementations of the CSS formalism in the Drell-Yan process, and provide some examples. We also emphasize the importance of agreed-upon definitions for having an unambiguous prescription for calculating higher orders in the hard part, and provide examples of higher order calculations. We end with a discussion of strategies for extending the phenomenological applications of TMD factorization to situations beyond the unpolarized case.

We present a factorization theorem for the low transverse momentum (p_T) and rapidity (Y) distribution of the Higgs and electroweak gauge bosons using the Soft-Collinear Effective Theory. In the region M ≫ p_T ≫ Λ_QCD, where M denotes the mass of the electroweak object, the factorization formula is given in terms of perturbatively calculable functions and the standard PDFs. For p_T ~ Λ_QCD, the factorization theorem is given in terms of non-perturbative Impact-parameter Beam Functions (iBFs) and an Inverse Soft Function (iSF). The iBFs correspond to completely unintegrated PDFs and can be interesting probes of momentum distributions in the nucleon. The iBFs and the iSF are grouped together and written as a product of a gauge invariant and non-perturbative Transverse Momentum Function (TMF) with the standard PDFs. We present results of NLL resummation for the Higgs and Z-boson distributions and give a comparison with Tevatron data.

Gluonic pole matrix elements explain the appearance of single spin asymmetries (SSA) in high-energy scattering processes. They involve a combination of operators which are odd under time reversal (T-odd). Such matrix elements appear in principle both for parton distribution functions and parton fragmentation functions. We show that for parton fragmentation functions these gluonic pole matrix elements vanish as a consequence of the analytic structure of scattering amplitudes in Quantum Chromodynamics. We extend this analysis to the case of multi-partonic pole matrix elements. This result is important in the study of the universality of transverse momentum dependent (TMD) fragmentation functions.

Semi-inclusive deep inelastic scattering experiments allow us to probe the motion of quarks inside the proton in terms of so-called transverse momentum dependent parton distribution functions (TMD PDFs), but the information is convoluted with fragmentation functions (TMD FFs) and soft factors. It has long been known that weighting the measured event counts with powers of the hadron momentum before forming angular asymmetries de-convolutes TMD PDFs and TMD FFs in an elegant way, but this also entails an undesirable sensitivity to high momentum contributions. Using Bessel functions as weights, we find a natural generalization of weighted asymmetries that preserves the de-convolution property and features soft-factor cancellation, yet allows us to be less sensitive to high transverse momenta. The formalism also relates to TMD quantities studied in lattice QCD. We briefly show preliminary lattice results from an exploratory calculation of the Boer-Mulders shift using lattices generated by the MILC and LHP collaborations at a pion mass of 500 MeV.

We present a completely gauge-invariant operator definition of transverse-momentum-dependent parton densities (TMD), supplied with longitudinal lightlike gauge links as well as transverse gauge links at lightcone infinity. Within this framework, we consider the consistent treatment of specific divergences, emerging in the "unsubtracted" TMD beyond the tree approximation, and construct the soft factors to cancel unphysical singularities. We confront this approach with factorization schemes, which make use of covariant gauges with off-the-lightcone gauge links, and discuss their mutual connection.

In this talk, we introduce two sets of twist-3 quark-gluon correlation functions that are relevant to transverse single spin asymmetries, and present corresponding evolution equations at the leading order in strong coupling constant, α_s. The similarities and differences between the evolution of the leading power parton distributions and that of the twist-3 multiparton correlation functions are also discussed.

We report on a recent work concerning the scale dependence of twist-three correlation functions relevant for the single transverse spin asymmetry in the framework of collinear factorization. Evolution equations are presented for both the flavor–nonsinglet and flavor–singlet distributions. Our results do not agree with previous calculations of the evolution in the limit of vanishing gluon momentum. Possible sources for this discrepancy are identified.

We shortly review the status of perturbative corrections to hard exclusive processes, which were partially obtained by utilizing conformal symmetry. We outline the conformal partial wave expansion of generalized parton distributions (GPDs) and hard exclusive amplitudes, which can be used for a model dependent extraction of the two-dimensional momentum fraction shape of GPDs by means of evolution. We exemplify this method and point out it importance for the description of collider data in the small x_Bj region.

Dispersion relations have been applied to hard exclusive processes suggesting that generalized parton distributions can be fully determined from their values on a kinematical ridge. We show significant pitfalls associated that vitiate this approach. We suggest an improved treatment of this assumption. Secondly, in order for a partonic interpretation to hold throughout the accessible kinematic region, semi-disconnected diagrams should not contribute. We show, however, that this condition is not satisfied for non-forward kinematics at leading order. Lastly we present our recent flexible parameterization of GPDs and indicate its application to neutral pseudoscalar electroproduction.

We propose an extraction of the running coupling constant of QCD in the infrared region from experimental data on deep inelastic inclusive scattering at Bjorken x → 1. We first attempt a perturbative fit of the data that extends NLO PQCD evolution to large x values and final state invariant mass, W, in the resonance region. We include both target mass corrections and large x resummation effects. These effects are of order , and they improve the agreement with the Q² dependence of the data. Standard analyses require the presence of additional power corrections, or dynamical higher twists, to achieve a fully quantitative fit. Our analysis, however, is regulated by the value of the strong coupling in the infrared region that enters through large x resummation effects, and that can suppress, or absorb, higher twist effects. Large x data therefore indirectly provide a measurement of this quantity that can be compared to extractions from other observables.

Inclusion of transverse momentum effects into the description of the pion-photon transition form factor is discussed. At low k_⊥, a consistent description may be obtained within the light-front formalism. On the other hand, the large-k_⊥ limit is most conveniently treated within the Sudakov parameterization framework. The third way of introducing k_⊥ is based on the "local duality" approach motivated by QCD sum rules. It is shown that, while the local duality gives a correct (coinciding with the axial anomaly based prediction) result for the real photon point, a straightforward extension of the formulas of the modified factorization approach into the small virtualities domain produces a divergent result. It is pointed out that such lessons should be taken into account while constructing transverse-momentum-dependent schemes for inclusive processes.

We revisit the standard procedure to match non-perturbative models to perturbative QCD, using experimental data. The strong coupling constant plays a central role in the QCD evolution of parton densities. We will extend this procedure with a non-perturbative generalization of the QCD running coupling and use this new development to understand why perturbative treatments are working reasonably well in the context of hadronic models. Vice versa, this new procedure broadens the ways of analyzing the freezing of the running coupling constant.

A new analytical method of performing ERBL evolution is described. The main goal is to develop an approach that works for distribution amplitudes that do not vanish at the end points, for which the standard method of expansion in Gegenbauer polynomials is inefficient. Two cases of the initial DA are considered: a purely flat DA, given by the same constant for all x, and an antisymmetric DA given by opposite constants for x ≷ 1/2. For a purely flat DA, the evolution is governed by an overall dependence on the evolution parameter t times a factor that was calculated as an expansion in t. For an antisymmetric flat DA, an extra overall factor |1 - 2x|^2t appears due to a jump at x = 1/2. A good convergence was observed in the t ≲ 1/2 region. For larger t, one can use the standard method of the Gegenbauer expansion.