^{1, 2}

^{3, 4}

^{1}

^{1}

^{2}

^{3}

^{4}

^{3}.

The inclusive large-

Transverse Single-Spin Asymmetries (TSSAs) have been abundantly observed in several inclusive proton-proton experiments for a long time; when reaching large enough energies and

In fact, large TSSAs cannot be generated by the hard elementary processes, because of helicity conservation (in the massless limit) typical of QED and QCD interactions; indeed, such asymmetries were expected to vanish at high energies. Their persisting must be related to nonperturbative properties of the nucleon structure, such as parton intrinsic and orbital motion. A true understanding of the origin of TSSAs would allow a deeper understanding of the nucleon structure.

Since the 1990s two different, despite being somewhat related, approaches have attempted to tackle the problem. One is based on the collinear QCD factorisation scheme and involves as basic quantities, which can generate single-spin dependences, higher-twist quark-gluon-quark correlations in the nucleon as well as higher-twist fragmentation correlators. The second approach is based on a physical, although unproven, generalisation of the parton model, with the inclusion, in the factorisation scheme, of transverse momentum dependent partonic distribution and fragmentation functions (TMDs), which also can generate single-spin dependences. The twist-3 correlations are related to moments of some TMDs. We refer to [

In this paper we consider TSSAs at the proposed AFTER@LHC experiment, in which high-energy protons extracted from the LHC beam would collide on a (polarised) fixed target of protons, with high luminosity. For a description of the physics potentiality of this experiment see [

We recall our formalism by considering the Transverse Single-Spin Asymmetry

In the GPM

Similarly, according to the Collins effect the number density of unpolarised hadrons

According to the GPM formalism [

For details and full explanation of the notations in the above equations we refer to [

The

The denominator of (

We present here our results for

All details concerning the motivations for such a choice, the values of the parameters, and their derivation can be found in [

We present our results on

Our theoretical estimates for

Our theoretical estimates for

Our theoretical estimates for

Notice that, for both our choices of the Sivers functions, the gluon Sivers distributions are taken to be vanishing, as suggested by data [

The analogous results for the single direct photon are shown in Figures

Our theoretical estimates for

Our theoretical estimates for

Our theoretical estimates for

Our theoretical estimates for

Our theoretical estimates for

Our theoretical estimates for

Drell-Yan (D-Y) processes are expected to play a crucial role in our understanding of the origin, at the partonic level, of TSSAs. For such processes, like for SIDIS processes and contrary to single hadron production, the TMD factorisation has been proven to hold, so that there is a general consensus that the Sivers effect should be visible via TSSAs in D-Y [

Predictions for Sivers

In [

Adopting for the unpolarised TMD and the Sivers function the same expressions as in (

Notice that we consider here the

Our results for the Sivers asymmetry

Our theoretical estimates for

Some final comments and further details might help in understanding the importance of the measurements of the TSSAs at AFTER@LHC:

Most predictions given show clear asymmetries, sufficiently large as to be easily measurable, given the expected performance of AFTER@LHC [

The values of

The values found here are in agreement, both in sign and qualitative magnitude, with the values found in [

The results for single photon production are interesting; they isolate the Sivers effect and our predictions show that they can reach values of about 5%, with a reduced uncertainty band. We find positive values of

Our results, obtained within the GPM, have a similar magnitude to those obtained in [

The values of

A measurement of

Both the results of [

In this paper we have also considered azimuthal asymmetries in polarised D-Y processes, related to the Sivers effect. As explained above, in this case, due to the presence of a large and a small scale, like in SIDIS, the TMD factorisation is valid, with the expectation of an opposite sign of the Sivers function in SIDIS and D-Y processes. Also this prediction can be checked at AFTER@LHC.

The authors declare that there is no conflict of interests regarding the publication of this paper.

M. Anselmino and S. Melis acknowledge support from the “Progetto di Ricerca di Ateneo/CSP” (code TO-Call3-2012-0103). U. D’Alesio is grateful to the Department of Theoretical Physics II of the Universidad Complutense of Madrid for the kind hospitality extended to him during the completion of this work.

^{+}e

^{−}data

^{+}e

^{−}data

^{+}e

^{−}annihilations

_{T}and transverse-momentum distributions on-the-light-cone