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The twist-3 collinear factorization framework has drawn much attention in recent decades as a successful approach in describing the data for single spin asymmetries (SSAs). Many SSAs data have been experimentally accumulated in a variety of energies since the first measurement was done in the late 1970s and it is expected that the future experiments like Electron-Ion-Collider will provide us with more data. In order to perform a consistent and precise description of the data taken in different kinematic regimes, the scale evolution of the collinear twist-3 functions and the perturbative higher-order hard part coefficients are mandatory. In this paper, we introduce the techniques for next-to-leading order (NLO) calculation of transverse-momentum-weighted SSAs, which can be served as a useful tool to derive the QCD evolution equation for twist-3 functions and to verify the QCD collinear factorization for twist-3 observables at NLO, as well as obtain the finite NLO hard part coefficients.

The large single transverse spin asymmetries (SSAs) have been a longstanding problem over 40 years since it was turned out that the conventional perturbative calculation based on the parton model picture failed to describe the large SSAs which were experimentally observed in pion and polarized hyperon productions [

The twist-3 collinear factorization framework is a natural extension of the conventional perturbative QCD framework and it could give a reasonable description of the large SSAs. Measurements of SSAs at Relativistic-Heavy-Ion-Collider (RHIC) [

Electron-Ion-Collider (EIC) is a next-generation hadron collider expected to provide more data in different kinematic regimes for SSAs. In order to extract the fundamental structure of the nucleon from the measurements at a future EIC, comprehensive and precise calculations for SSAs in transversely polarized lepton-proton collision are highly demanded. It is well known that nonperturbative functions in the perturbative QCD calculation, in general, receive logarithmic radiative corrections and the evolution equation with respect to this logarithmic scale is necessary for a systematic treatment of the cross sections in wide range of energies. Most famous example is the DGLAP evolution equation of the twist-2 parton distribution functions (PDFs). Correct description of the small scale violation of the structure function controlled by the DGLAP equation was an important success of the QCD phenomenology in the early days. The twist-3 function is expected to have similar logarithmic dependence and its evolution equation will play an important role in global fitting of the SSA data accumulated in different energies. Consistent description of the data will be a good evidence that the twist-3 framework, one of major fundamental developments in recent QCD phenomenology, is a feasible theory to solve the 40-year mystery in high energy physics. The evolution equations for the twist-3 functions have been derived in two different methods. The first method is a calculation of the higher-order corrections to the nonperturbative function itself [

The rest of the paper is organized as follows. In Section

In this paper, we take the process of SIDIS as an example to show the techniques of perturbative calculation for transverse-momentum-weighted differential cross section at twist-3. We start this section by specifying our notation and the kinematics of SIDIS and present the calculation for transverse-momentum-weighted SSA at leading order (LO).

We consider the scattering of an unpolarized lepton with momentum

The concept of the transverse-momentum-weighted technique is mostly the same with the twist-2 case. Notice that direct

We recall the cross section for SIDIS presented in [

We demonstrate how to derive the LO cross section for the transverse-momentum-weighted SSA based on the collinear twist-3 framework and show that the LO cross section is proportional to the first moment of the TMD Sivers function. The twist-3 calculation is well formulated in the diagrammatic method. We consider a set of the general diagrams shown in Figure

A series of LO diagrams in the diagrammatic method.

The NLO virtual correction diagrams in SIDIS.

In this section, we review the calculation for transverse-momentum-weighted SSA at NLO including both real and virtual corrections.

We first consider the NLO contribution from the virtual correction which is given by the

We now complete the NLO calculation by adding the real emission contribution represented by

Typical diagrams for soft-gluon pole (left), hard pole (middle), and another hard pole (right). The red barred propagator gives the pole term.

A lot of works on the transverse-momentum-weighted SSA have been done in recent years. We briefly summarize all related work in this section. The evolution equation of the Qiu-Sterman function (

The diagrams which give the gluon mixing contribution to the evolution equation of

The calculation technique for these diagrams was developed in [

The NLO transverse-momentum-weighted cross section has been completed for single inclusive hadron production in SIDIS and Drell-Yan process in proton-proton collisions by a series of work presented here. These results are useful not only for the derivation of the evolution equations, but also for the verification of twist-3 collinear factorization feasibility, as well as for the global analysis of the experimental data. Measurement of the weighted SSAs just began very recently [

The transverse-momentum-weighted technique has also been extended to study the transverse momentum broadening effect for semi-inclusive hadron production in lepton-nucleus scattering [

We reviewed the transverse-momentum-weighted technique as a useful tool to derive the scale evolution equation for the twist-3 collinear function which is expressed by the first moment of the TMD function. We first demonstrated the calculation of the LO cross section formula in a pedagogical way. Then we showed the basic techniques for the NLO calculation for both the virtual correction and real emission contributions. A lot of work have been done on the Qiu-Sterman function [

In the end, we would like to point out the importance of the

The authors declare that they have no conflicts of interest.

This research is supported by NSFC of China under Project no. 11435004 and research startup funding at SCNU.

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_{h⊥}-weighted single-transverse spin asymmetry in semi-inclusive deep inelastic scattering

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