^{*}

^{3}.

The rapidity anomalous dimension (RAD), or Collins-Soper kernel, defines the scaling properties of transverse momentum dependent distributions and can be extracted from the experimental data. I derive a self-contained nonperturbative definition that represents RAD without reference to a particular process. This definition makes possible exploration of the properties of RAD by theoretical methods on one side, and the properties of QCD vacuum with collider measurements on another side. To demonstrate these possibilities, I compute the power correction to RAD, its large-

The nontrivial structure of the QCD vacuum raises a lot of fundamental and yet unsolved problems, such as mechanisms of quark confinement and hadronization. As a matter of fact, there is a little number of experimental observables that test properties of the QCD vacuum. In this Letter, I demonstrate that the evolution kernel of transverse momentum dependent (TMD) distributions is exclusively sensitive to the structure of QCD vacuum and thus is a valuable tool to study it.

The rapidity anomalous dimension (RAD), or Collins-Soper kernel, was introduced in Refs.

Despite the long history of RAD, very little is known about its NP nature from the theory side. Apart from a general identification that the NP part exists, I know only a few works that are dedicated to this problem at least partially

The cross section for the Drell-Yan pair production at small transverse momentum is described by the TMD factorization formula

A distinctive feature of TMD distributions is their dependence on two scales

Equation

Comparison of extracted values of RAD. The lines labeled as SV19, SV17, Pavia19, and Pavia17 correspond to Refs.

To derive the self-contained expression for RAD, I take a step backward in the derivation of Eq.

The TMD soft factor is defined as

Contours defining the TMD soft factor (in the Drell-Yan kinematics) and its derivatives. Axes

The expression

The regularized soft factor

Despite that the left-hand side of Eq.

The expression

RAD is very well studied in the perturbation theory, where it has been derived up to next-to-next-to-leading order (NNLO)

The perturbative calculation is made in the regime

The computation of

The LO contribution to term

In contrast to

Structure of the operator that describes the leading power correction to RAD. Blue lines are the gauge links, and dots are insertions of gluon strength tensors.

Using the parametrization

The function

These values are obtained with LO approximation at

One of the most promising applications of the expression

In SVM one assumes that the QCD dynamics is dominated by two-point correlators, whereas multipoint correlators give a negligible contribution. Additionally, one ignores the gauge links connecting fields, assuming their unimportance at large distances. In this way, all gluonic observables are written in terms of two functions

Considering various relations derived in SVM (in particular, the static interquark potential

The expression

To provide an elementary check and the demonstration of definition

The derived LO power correction

The possibility to investigate the QCD vacuum in high energy collisions sounds contradictory to the intuitive picture that the structure of accelerated particles is cleared from low-energy effects. Indeed, the partons do not interact with each other within a highly energetic hadron. Nonetheless, their temperate transverse motion is sensitive to the structure of the underlying vacuum. Therefore, measuring the low-

A. A. V. is thankful to V. Braun, O.Teryaev, A.Schäfer, and I.Scimemi for stimulating discussions. This work was supported by DFG (FOR 2926 “Next Generation pQCD for Hadron Structure: Preparing for the EIC,” Project No. 430824754).