Abstract
We develop a method of transverse momentum dependent (TMD) operator expansion that yields the TMD factorization theorem on the operator level. The TMD operators are systematically ordered with respect to TMD-twist, which allows a certain separation of kinematic and genuine power corrections. The process dependence enters via the boundary conditions for the background fields. As a proof of principle, we derive the effective operator for hadronic tensor in TMD factorization up to the next-to-leading power (∼ qT/Q) at the next-to-leading order for any process with two detected states.
Highlights
The transverse momentum dependent (TMD) factorization raises a series of problems, especially when extending the formalism beyond the leading power approximation
We develop a method of transverse momentum dependent (TMD) operator expansion that yields the TMD factorization theorem on the operator level
We have developed a method to derive the TMD factorization theorem
Summary
We provide the notation and the basic definitions that are used in this work. In order to apply the background-field method we need to write down the hadronic tensor in eq (2.1) as a functional integral and to identify the field modes relevant for the task. The distinctive feature of the TMD factorization from cited cases is that the background field has two independent components, collinear and anti-collinear. After implementing these definitions in the functional integral in eq (2.11), the hadronic tensor reads. ×eiSQ(+C)D[q,q,A]−iSQ(−C)D[q,q,A]Ψ∗p(−)[q, q, A]Oa[q, q, A]Ψ(p+)[q, q, A], and the fields in definition of Φ’s are just collinear or anti-collinear In these expressions the functions Φunsub (that are unsubtracted TMD distributions) are nonperturbative in the sense that they contain unknown information on the hadronic structure and low-energy QCD interactions. In the rest of the paper, the effective operator is the same for all cases
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