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The ultraviolet and rapidity divergences of transverse-momentum dependent parton distribution functions with lightlike and transverse gauge links is studied, also incorporating a soft eikonal factor. We find that in the light-cone gauge with q^--independent pole prescriptions extra divergences appear which amount, at one-loop, to a cusp-like anomalous-dimension. We show that such contributions are absent when the Mandelstam-Leibbrandt prescription is used. In the first case, the soft factor cancels the anomalous dimension defect, while in the second case its ultraviolet-divergent part reduces to unity.

We give an overview of the issue of anomalies in field theories with extra dimensions. We start by reviewing in a pedagogical way the computation of the standard perturbative gauge and gravitational anomalies on noncompact spaces, using Fujikawa's approach and functional integral methods, and discuss the available mechanisms for their cancellation. We then generalize these analyses to the case of orbifold field theories with compact internal dimensions, emphasizing the new aspects related to the presence of orbifold singularities and discrete Wilson lines, and the new cancellation mechanisms that are becoming available. We conclude with a very brief discussion on global and parity anomalies.

We consider the supergravity backgrounds that correspond to supersymmetric Wilson line operators in the context of AdS/CFT correspondence. We study the gravitino and dilatino conditions of the IIB supergravity under the appropriate ansatz, and obtain some necessary conditions for a supergravity background that preserves the same symmetry as the supersymmetric Wilson lines. The supergravity solutions are characterized by continuous version of Maya diagrams. This diagram is related to the eigenvalue distribution of the Gaussian matrix model. We also consider the similar backgrounds of the 11-dimensional supergravity.

The high-energy behavior of the of 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 first two orders in coupling constant : LO BFKL intercept and NLO BFKL 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 leading and next-to-leading order.

We discuss the possibility of quantum entanglement for pairs of charged particles produced in high-energy pp-collisions at the LHC. Using a framework of interacting Wilson lines, we calculate 2D and 1D two-particle angular correlation functions in terms of the differences of the pseudorapidities and azimuthal angles of the produced particles. The calculated near-side angular correlation shows a localized maximum around Δϕ≈0, though it is less pronounced compared to the peak observed by the CMS Collaboration. We argue that this soft correlation is universal and insensitive to the specific properties of the matter (quark–gluon plasma, QCD vacuum, etc.) used to describe hadronic states — though such properties can be included to further improve the results.

Calculation results for the HQET field anomalous dimension and the QCD cusp anomalous dimension, as well as their properties, are reviewed. The HQET field anomalous dimension ${\gamma }_h$ is known up to four loops. The cusp anomalous dimension $\mathrm{\Gamma }(\phi )$ is known up to three loops, and its small-angle and large-angle asymptotics up to four loops. Some (but not all) color structures at four loops are known with the full $\phi$-dependence. Some simple contributions are known at higher loops. For the $\phi \to \infty$ asymptotics of $\mathrm{\Gamma }(\phi )$ (the light-like cusp anomalous dimension) and the ${\phi }^2$-term of the small-$\phi$ expansion (the Bremsstrahlung function), the ${𝒩}=4$ SYM results are equal to the highest-weight parts of the QCD results. There is an interesting conjecture about the structure of $\mathrm{\Gamma }(\phi )$ which holds up to three loops; at four loops it holds for some color structures and breaks down for other ones. In the cases when it holds, it related highly nontrivial functions of $\phi$, and it cannot be accidental; however, the reasons of this conjecture and its failures are not understood. The cusp anomalous dimension at the Euclidean angle $\varphi \to \pi$ is related to the static quark–antiquark potential due to conformal symmetry; in QCD, this relation is broken by an anomalous term proportional to the $\beta$-function.

Some new results are also presented. Using the recent four-loop result for ${\gamma }_h$, here we obtain analytical expressions for some terms in the four-loop on-shell renormalization constant of the massive quark field $Z_Q^{\mathrm{\text{os}}}$ which were previously known only numerically. We also present two new contributions to ${\gamma }_h$, $\mathrm{\Gamma }(\phi )$ at five loops and to the quark–antiquark potential at four loops.

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.

The photon impact factor for the BFKL pomeron is calculated in the next-to-leading order (NLO) approximation using the operator expansion in Wilson lines. The result is represented as a NLO k_T-factorization formula for the structure functions of small-x deep inelastic scattering.

The anomalous dimensions of light-ray operators of twist two are obtained by analytical continuation of the anomalous dimensions of corresponding local operators. I demonstrate that the asymptotics of these anomalous dimensions at the "BFKL point" j → 1 can be obtained by comparing the light-cone operator expansion with the high-energy expansion in Wilson lines.

Wilson lines are key objects in many QCD calculations. They are parallel transporters of the gauge field that can be used to render non-local operator products gauge invariant, which is especially useful for calculations concerning validation of factorization schemes and in calculations for constructing or modelling parton density functions. We develop an algorithm to express Wilson lines that are defined on piecewise linear paths in function of their Wilson segments, reducing the number of diagrams needed to be calculated. We show how different linear path topologies can be related using their color structure. This framework allows one to easily switch results between different Wilson line structures, which is helpful when testing different structures against each other, e.g. when checking universality properties of non-perturbative objects.

The high-energy behavior of QCD amplitudes can be described in terms of the rapidity evolution of Wilson lines. I present the hierarchy of evolution equations for Wilson lines in the next-to-leading order.

We study how the rapidity evolution of gluon transverse momentum dependent distribution changes from nonlinear evolution at small x ≪ 1 to linear double-logarithmic evolution at moderate x ~ 1.

We present an algorithm to express Wilson lines that are defined on piecewise linear paths in function of their individual segments, reducing the number of diagrams needed to be calculated. The important step lies in the observation that different linear path topologies can be related to each other using their color structure. This framework allows one to easily switch results between different Wilson line topologies, which is helpful when testing different structures against each other.

At high energies, the relevant degrees of freedom are Wilson lines - infinite gauge links ordered along straight lines collinear to the velocities of colliding particles. The effective action for these Wilson lines is determined by the scattering of QCD shock waves. I develop the symmetric expansion of the effective action in powers of strength of one of the shock waves and calculate the leading term of the series. The corresponding first-order effective action, symmetric with respect to projectile and target, includes both up and down fan diagrams and pomeron loops.

The small-x deep inelastic scattering in the saturation region is governed by the non-linear evolution of Wilson-line operators. In the leading logarithmic approximation it is given by the BK equation for the evolution of color dipoles. I discuss recent calculation of the next-to-leading order evolution of color dipoles in QCD and SYM.

The high-energy behavior of the of 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 first two orders in coupling constant : LO BFKL intercept and NLO BFKL 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 leading and next-to-leading order.

We discuss the state-of-the-art of the theory of transverse-momentum dependent parton densities (TMDs), paying special attention to their renormalization properties, the structure of the gauge links in the operator definition, and the role of the soft factor in the factorization formula within the TMD approach to the semi-inclusive processes. We argue that the use of the lightcone axial gauge offers certain advantages for a consistent definition of TMDs as compared to the off-the-light-cone gauges, or covariant gauges with off-the-lightcone gauge links.