Advances in High Energy Physics

Volume 2015, Article ID 475040, 12 pages

http://dx.doi.org/10.1155/2015/475040

## Transverse Single-Spin Asymmetries in Proton-Proton Collisions at the AFTER@LHC Experiment in a TMD Factorisation Scheme

^{1}Dipartimento di Fisica, Università di Torino, Via P. Giuria 1, 10125 Torino, Italy^{2}INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy^{3}Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy^{4}INFN, Sezione di Cagliari, CP 170, 09042 Monserrato, Italy

Received 14 April 2015; Accepted 15 June 2015

Academic Editor: Jibo He

Copyright © 2015 M. Anselmino et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The inclusive large- production of a single pion, jet or direct photon, and Drell-Yan processes, are considered for proton-proton collisions in the kinematical range expected for the fixed-target experiment AFTER, proposed at LHC. For all these processes, predictions are given for the transverse single-spin asymmetry, , computed according to a Generalised Parton Model previously discussed in the literature and based on TMD factorisation. Comparisons with the results of a collinear twist-3 approach, recently presented, are made and discussed.

#### 1. Introduction and Formalism

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 values, their understanding from basic quark-gluon QCD interactions is a difficult and fascinating task, which has always been one of the major challenges for QCD.

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 [1–9], and references therein, for more detailed account of the two approaches and possible variations, with all relevant citations. Following [10], we denote by CT-3 the first approach while the second one is, as usual, denoted by GPM.

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 [11] and for the latest technical details and importance for TMD studies see, for example, [12]. Due to its features the AFTER@LHC is an ideal experiment to study and understand the origin of SSAs and, in general, the role of QCD interactions in high-energy hadronic collisions; AFTER@LHC would be a polarised fixed-target experiment with unprecedented high luminosity.

We recall our formalism by considering the Transverse Single-Spin Asymmetry , measured in inclusive reactions and defined aswhere , are opposite spin orientations perpendicular to the - scattering plane, in the c.m. frame. We define the direction as the -axis and the unpolarised proton is moving along the -direction. In such a process the only large scale is the transverse momentum of the final hadron.

In the GPM originates mainly from two spin and transverse momentum effects, one introduced by Sivers in the partonic distributions [13, 14] and one by Collins in the parton fragmentation process [15], being all the other effects strongly suppressed by azimuthal phase integrations [16]. According to the Sivers effect the number density of unpolarised quarks (or gluons) with intrinsic transverse momentum inside a transversely polarised proton , with three-momentum and spin polarisation vector , can be written aswhere is the proton light-cone momentum fraction carried by the quark, is the unpolarised TMD (), and is the Sivers function. and are unit vectors. Notice that the Sivers function is most often denoted as [17]; this notation is related to ours by [18]where is the proton mass.

Similarly, according to the Collins effect the number density of unpolarised hadrons with transverse momentum resulting in the fragmentation of a transversely polarised quark , with three-momentum and spin polarisation vector , can be written aswhere is the parton light-cone momentum fraction carried by the hadron, is the unpolarised TMD , and is the Collins function. and are unit vectors. Notice that the Collins function is most often denoted as [17]; this notation is related to ours by [18]where is the hadron mass.

According to the GPM formalism [1, 2, 16], can then be written asThe Collins and Sivers contributions were recently studied, respectively, in [1] and [2], and are given by

For details and full explanation of the notations in the above equations we refer to [16] (where is denoted as ). It suffices to notice here that is a kinematical factor, which at equals 1. The phase factor in (7) originates directly from the dependence of the Sivers distribution [, (2)]. The (suppressing) phase factor in (8) originates from the dependence of the unintegrated transversity distribution , the polarized elementary interaction, and the spin- correlation in the Collins function. The explicit expressions of , , and in terms of the integration variables can be found via (60)–(63) in [16] and (35)–(42) in [19].

The ’s are the three independent hard scattering helicity amplitudes describing the lowest order QCD interactions. The sum of their moduli squared is related to the elementary unpolarised cross section ; that is,The explicit expressions of the combinations of ’s which give the QCD dynamics in (7) and (8), can be found, for all possible elementary interactions, in [16] (see also [1] for a correction to one of the product of amplitudes). The QCD scale is chosen as .

The denominator of (1) or (6) is twice the unpolarised cross section and is given in our TMD factorisation by the same expression as in (7), where one simply replaces the factor with .

#### 2. for Single Pion, Jet, and Direct Photon Production

We present here our results for , (1), based on our GPM scheme, (6), (7), and (8). The TMDs which enter in these equations are those extracted from the analysis of Semi-Inclusive Deep Inelastic (SIDIS) and data [20–23], adopting simple factorised forms, which we recall here. For the unpolarised TMD partonic distributions and fragmentation functions we have, respectively,The Sivers function is parameterised aswherewith , andSimilarly, the quark transversity distribution, , and the Collins fragmentation function, , have been parametrized as follows:where is the usual collinear quark helicity distribution,with , and

All details concerning the motivations for such a choice, the values of the parameters, and their derivation can be found in [20–23]. We do not repeat them here, but in the caption of each figure we will give the corresponding references which allow fixing all necessary values.

We present our results on for the process at the expected AFTER@LHC energy ( GeV) in Figures 1–3. Following [1, 2], our results are given for two possible choices of the SIDIS TMDs and are shown as function of at two fixed values (Figure 1), as function of at two fixed rapidity values (Figure 2) and as function of rapidity at one fixed value (Figure 3). is the usual Feynman variable defined as where is the -component of the final hadron momentum. Notice that, in our chosen reference frame, a forward production, with respect to the polarised proton, means negative values of . The uncertainty bands reflect the uncertainty in the determinations of the TMDs and are computed according to the procedure explained in the appendix of [21]. More information can be found in the figure captions.