Narrow 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.