^{3}.

Using soft collinear effective field theory, we derive the factorization theorem for the quasi-transverse-momentum-dependent (quasi-TMD) operator. We check the factorization theorem at one-loop level and compute the corresponding coefficient function and anomalous dimensions. The factorized expression is built from the physical TMD distribution, and a nonperturbative lattice related factor. We demonstrate that lattice related functions cancel in appropriately constructed ratios. These ratios could be used to explore various properties of TMD distributions, for instance, the nonperturbative evolution kernel. A discussion of such ratios and the related continuum properties of TMDs is presented.

Over the last years, continuous progress in theory and phenomenology of a transverse-momentum-dependent (TMD) factorization theorem made it a valuable tool for analysis and prediction of many observables (for a review see

There are eight leading twist TMD distributions, each of which depends on a transverse variable and longitudinal momentum fraction

Suggestions for lattice studies of TMD observables were made long ago

The expression derived here is only the leading term of the factorization theorem. The subsequent terms are formally suppressed by powers of hadron momentum. This fact should not be over interpreted because a closer analysis reveals the potential breakdown of this expansion. Namely, each next order term has a stronger small-

The paper is split into three sections. In Sec.

The considered lattice observables read

The space-time orientation of different quantities in

Illustration for the definition of the matrix element

The clear separation of collinear and soft-field modes within the hadron is a prerequisite for any TMD factorization theorem. It can be achieved by considering a fast moving hadron, for which anticollinear components of field momenta are suppressed in comparison to collinear ones. To quantify this condition we write the momentum of a hadron as

We also assume that the staple-shaped gauge links contour

In the notation

The analogy with the TMD hadron tensor

In SCET II, one distinguishes collinear, anticollinear and soft fields. In leading approximation, the fast-moving hadron is a composition of collinear fields (

Using these counting rules we write the leading power SCET operator that corresponds to the current

The coefficient

Expression

Illustration for Wilson lines structure for operators

Combining

Substituting the effective currents

The function

Unsubtracted TMD distributions have rapidity divergences that appear due to the presence of infinite lightlike Wilson lines separated in the transverse plane. Rapidity divergences are associated with the directions of Wilson lines. In the current case, there are two light-cone directions, and thus we introduce two regularization parameters

In the expression

Introducing the boost-invariant variables

Generally, the function

The final form of the factorized expression reads

There are two points in equation

The second point concerns the definition of the operator

Using these facts we rewrite

It is important to emphasize that the size of power corrections in

In this section we present the computation of elements of the factorization theorem at one loop. The calculation supports the correctness of the construction. The calculation presented here is done in the

To evaluate the matching coefficient between the QCD current

In the

Diagrams to be computed for evaluation of the hard matching coefficient. Solid (dashed) lines represent the quark field (Wilson line). In the case

In

Evaluating analogously the rest of the diagrams (note that the results for

The hard-matching coefficient is

We have also performed the same computation with a finite-length

The functions

All components of the current

The renormalization group requires

The factorized expression

There are two principal ways to bypass this problem. The first approach is to obtain the values of

In this section, we consider ratios of the form

For brevity of the formulas in this section we denote only those arguments of

The most elementary test of the factorization theorem

For example, considering

Similar measurements have been performed in

A great feature of lattice QCD is the possibility to measure objects unaccessible in an experiment directly. In particular, one can compare measurements of different Lorentz structures and check the TMD factorization theorem in a completely controlled environment. The Dirac structures of higher TMD twist must be suppressed due to dominance of the collinear components in the hadron. We have

Despite the apparent triviality of this statement the numerical evaluation of this ratio on the lattice is very important because it allows to estimate systematic uncertainties. In a sense, it directly measures the size of power corrections to the factorization theorem

The most exciting property of the ratios

A similar ratio is considered in detail in Ref.

To avoid these difficulties we suggest to consider the case of

The ratio

In Fig.

Functions

Comparison of the functions

To extract the rapidity anomalous dimension

Such simplified schemes should be applied with caution, because

There are two sources of small-

In this way, there is a certain hierarchy of

Convergence hierarchy for

Finally, let us stress again that the rapidity anomalous dimension is universal. Therefore, the ratios

In the present article, we have considered quasi-TMD operators, that can be investigated on the lattice. We pointed out the similarity of the lattice observable to the hadronic tensor of TMD processes, such as Drell-Yan or SIDIS. Using the method of soft-collinear effective field theory (SCET II) we derived the factorized expression for the lattice hadronic tensor in terms of physical TMD distributions, and the new instant-jet TMD distribution

Since the factorization formula contains an unknown nonperturbative function

The hard scale of the derived factorization theorem is the hadron momentum

Quite generally, many complications can be avoided if one considers the ratios of lattice observables. The information that could be extracted from ratios is limited, but still valuable. For instance, one can extract the nonperturbative rapidity anomalous dimension. The most simple observable in this case is the ratio of the first (and possibly the second) Mellin moments of quasi-TMDs at different hadron momenta. This ratio is almost exclusively dependent on

Authors are thankful to V. Braun, X. Ji, Y. Kovchegov, Y. Liu, Y.-S. Liu, M. Schlemmer, I. Stewart, and Y.-B. Yang for stimulating discussions. This work was supported by DFG (FOR 2926 “Next Generation pQCD for Hadron Structure: Preparing for the EIC”, Project No. 40824754).