^{3}.

We discuss conformal properties of TMD operators and present the result of the conformal rapidity evolution of TMD operators in the Sudakov region.

In recent years, the transverse-momentum dependent parton distributions (TMDs)

The TMDs are defined as matrix elements of quark or gluon operators with attached lightlike gauge links (Wilson lines) going to either

In our opinion, a good starting point is to obtain conformal leading-order evolution equations. It is well known that at the leading-order perturbative QCD (pQCD) is conformally invariant, so there is hope of get any evolution equation without explicit running coupling from conformal considerations. In our case, since TMD operators are defined with attached lightlike Wilson lines, formally they will transform covariantly under the subgroup of the full conformal group which preserves this lightlike direction. However, as we mentioned, the TMD operators contain rapidity divergencies which need to be regularized. At present, there is no rapidity cutoff which preserves conformal invariance, so the best one can do is to find the cutoff which is conformal at the leading order in perturbation theory. In higher orders, one should not expect conformal invariance since it is broken by the running of QCD coupling. However, if one considers corresponding correlation functions in

For definiteness, we will talk first about gluon operators with lightlike Wilson lines stretching to

The algebra of full conformal group

As we noted, infinite Wilson lines in the definition

In the next section, we demonstrate that the “small-

The rapidity evolution of the TMD operator

First, let us specify what we call a Sudakov region. A typical factorization formula for the differential cross section of particle production in hadron-hadron collision is

As we mentioned, TMD operators exhibit rapidity divergencies due to infinite lightlike gauge links. The “small-

The Sudakov region

In the next section, we demonstrate that the rapidity cutoff

In this section, we derive the evolution of gluon TMD operator

Typical diagrams for production (a) and virtual (b) contributions to the evolution kernel. The dashed lines denote gauge links.

Evolution equation

It is easy to see that the rhs of Eq.

A simple calculation of evolution of quark operator

As we mentioned above, the TMD factorization formula

As we mentioned in the Introduction, TMD evolution is analyzed by very different methods at small

To compare with conventional TMD analysis, let us write down the evolution of “generalized TMD”

The first result of our paper is finding the subgroup of

The second result is related to the fact that conformal invariance is violated by the rapidity cutoff (even in

Our main outlook is to try to connect to the small-

We thank V. M. Braun, A. Vladimirov, and A. Tarasov for discussions. The work of I. Balitsky is supported by Jefferson Science Associates, LLC under U.S. Contract No. DE-AC05-06OR23177 and by U.S. Grant No. DE-FG02-97ER41028.

Hereafter, we use the simplified notation

We assume that