Sivers effect and the single spin asymmetry in processes
Abstract
The single spin asymmetry , for large single inclusive particle production in collisions, is considered within a generalised parton model and a transverse momentum dependent factorisation scheme. The focus is on the Sivers effect and the study of its potential contribution to , based on a careful analysis of the Sivers functions extracted from azimuthal asymmetries in semiinclusive deep inelastic scattering processes. It is found that such Sivers functions could explain most features of the data, including some recent STAR results which show the persistence of a non zero up to surprisingly large values.
pacs:
13.88.+e, 12.38.Bx, 13.85.NiI Introduction
Among the leadingtwist Transverse Momentum Dependent Partonic Distribution Functions (TMDPDFs, often shortly referred to as TMDs), the Sivers distribution Sivers:1989cc ; Sivers:1990fh ; Boer:1997nt is most interesting and widely investigated. It describes the number density of unpolarised quarks (or gluons) with intrinsic transverse momentum inside a transversely polarised proton , with threemomentum and spin polarisation vector ,
(1) 
where is the proton lightcone 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 Mulders:1995dh ; this notation is related to ours by Bacchetta:2004jz
(2) 
A knowledge of the Sivers distribution allows a modelling of the 3dimensional momentum structure of the nucleon Boer:2011fh and, possibly, an estimate of the parton orbital angular momentum Bacchetta:2011gx .
All the available information on the Sivers function has been obtained from SIDIS data, , and the study of the azimuthal distribution of the final hadron around the direction in the centre of mass (c.m.) frame. This analysis is based on the TMD factorisation scheme Ji:2004xq ; Ji:2004wu ; Bacchetta:2008xw ; Collins:2011book , according to which the SIDIS cross section is written as a convolution of TMDPDFs, Transverse Momentum Dependent Fragmentation Functions (TMDFFs) and known elementary interactions. Such a scheme holds in the kinematical region defined by
(3) 
where is the magnitude of the final hadron transverse momentum. The presence of the two scales, small and large , allows to identify the contribution from the unintegrated partonic distributions (), while remaining in the region of validity of the QCD parton model. The study of the QCD evolution of the Sivers and unpolarised TMDs – the socalled TMD evolution – has much progressed lately Ji:2004xq ; Ji:2004wu ; Collins:2011book ; Aybat:2011zv ; Aybat:2011ge ; Echevarria:2012pw ; Echevarria:2012qe , with the first phenomenological applications Aybat:2011ta ; Anselmino:2012aa ; Bacchetta:2013pqa ; Godbole:2013bca ; Sun:2013dya ; Boer:2013zca ; Sun:2013hua .
The extraction of the Sivers functions from SIDIS data can then be performed on a sound ground. This has been done for the first time in Refs. Anselmino:2005ea ; Anselmino:2008sga ; Vogelsang:2005cs ; Collins:2005ie ; Anselmino:2005an , exploiting HERMES Diefenthaler:2007rj and COMPASS Martin:2007au data, and resulting in a reasonable knowledge of the Sivers functions for and quarks, although in a limited range of values, .
Much literature has emphasised the special interest and the peculiar properties of the Sivers effect. Trying to understand its origin at the partonic level has related the possibility of a non zero Sivers function with final Brodsky:2002cx or initial Brodsky:2002rv state interactions, respectively in SIDIS and DrellYan (DY) processes. This, in turns, induces a process dependence of the effect itself. The most clearcut consequence is the prediction of an opposite sign of the Sivers functions when contributing to single spin asymmetries (SSAs) in SIDIS and DY processes Collins:2002kn ; as polarised DY experiments have never been performed so far, such a prediction has not been tested yet. Crucial information might be available in the future from experiments at RHIC, Fermilab or from the COMPASS hadronic run at CERN, with pions colliding on a polarised nucleon target. The TMD factorisation scheme, valid for SIDIS processes, holds for DY as well, where the small and large scale are respectively the total transverse momentum () and the invariant mass () of the leptonic pair.
In this paper we focus on another class of puzzling results which strongly challenge our understanding of high energy strong interactions, that is the SSAs, usually denoted by , measured in inclusive reactions and defined as:
(4) 
and where are opposite spin orientations perpendicular to the scattering plane, in the c.m. frame. differs from the SSAs of SIDIS and DY processes because in such a case there is only one large scale in the process – the transverse momentum of the final observed hadron – and there is no small scale related to the intrinsic motions, both in the distribution and fragmentation functions, which are integrated over. The TMD factorisation scheme used for SIDIS and DY processes has not been proven in this case.
Large values of have been measured since a long time in many different experiments. The first ones were at relatively low energy Klem:1976ui ; Krueger:1998hz ; Allgower:2002qi ; Antille:1980th ; Adams:1991rw ; Adams:1991cs ; Adams:1991rv ; Adams:1991ru , and the common expectation was that such asymmetries would vanish at higher energies; however, data from RHIC at Arsene:2008aa , 200 Adams:2003fx ; Adler:2005in ; Lee:2007zzh ; Abelev:2008af ; Adamczyk:2012xd or even 500 Igo:2012 ; Bland:2013pkt GeV, still show puzzling non zero values of .
Several approaches to understanding , within QCD and some sort of factorisation scheme, can be found in the literature. All of them, directly or indirectly, are related to the Sivers function or other TMDs.
A QCD collinear factorisation formalism at nexttoleadingpower (twist3) has been developed and used in the phenomenological studies of Efremov:1981sh ; Efremov:1984ip ; Qiu:1991pp ; Qiu:1998ia ; Kanazawa:2000hz ; Kanazawa:2000kp ; Kouvaris:2006zy ; Kanazawa:2010au ; Kanazawa:2011bg . In this approach the spin effect is not embedded in a spin dependent TMD, but the necessary phase for generating the nonvanishing SSAs arises from the quantum interference between an elementary scattering amplitude with one active collinear parton and an amplitude with two active collinear partons. The SSAs are therefore proportional to some nonprobabilistic threeparton correlation functions, which are convoluted with product of amplitudes, rather than cross sections. These amplitudes are process dependent, while the threeparton correlation functions are universal.
However, one can show that the twist3 threeparton correlation functions have a close connection with the moment of the TMDPDFs; in particular the quarkgluon correlator is related to the first moment of the SIDIS Sivers function Boer:2003cm . It has been recently pointed out Kang:2011hk that the quarkgluon correlation functions, as obtained from the Sivers functions extracted from SIDIS data Anselmino:2005ea ; Anselmino:2008sga , indeed lead to sizeable values of , which agree in magnitude with the measured ones, but with the wrong sign (the socalled sign mismatch problem). A recent analysis Gamberg:2013kla of the spin asymmetry for single inclusive jet production in collisions collected by the ADY experiment Nogach:2012sh does not show the same sign problem; however, the measured asymmetry is very small.
An alternative, more phenomenological approach, is based on the assumption of the validity of the TMD factorisation also for processes Sivers:1989cc ; Sivers:1990fh ; Anselmino:1994tv ; Anselmino:1998yz ; Anselmino:1999pw ; D'Alesio:2004up ; Anselmino:2005sh ; D'Alesio:2007jt ; it generalises the usual collinear factorisation scheme (Generalised Parton Model, GPM) and the single inclusive cross section is written as a convolution of TMDPDFs, TMDFFs and QCD partonic cross sections. In that it adopts the same scheme which holds for SIDIS and DY processes with one small and one large scale. In this model the spin effects are included in the TMDs, which are supposed to be processindependent.
More recently, a third approach has been proposed Gamberg:2010tj ; Ratcliffe:2009sz , which assumes the TMD factorisation as in the GPM, but takes into account and absorbs the initial and final state interactions, i.e. the process dependence of the Sivers function, in the elementary interactions. In such a scheme the cross section is a convolution of processindependent TMDs with processdependent hard parts; these modified hard parts are very similar in form to those in the twist3 collinear approach. It turns out that this modified GPM formalism leads to results and predictions opposite to those of the conventional GPM Gamberg:2010tj .
In this paper we explore the possibility of understanding the experimental results on in processes with the Sivers effect and within the generalised parton model of Refs. D'Alesio:2004up ; Anselmino:2005sh ; D'Alesio:2007jt . The first phenomenological applications of the Sivers effect Anselmino:1994tv ; Anselmino:1998yz ; D'Alesio:2004up in hadronic interactions considered the Sivers function as a free input, not constrained by SIDIS data. In Ref. Boglione:2007dm it was shown that the use of the Sivers functions, as extracted from SIDIS data, could in principle explain the SSAs observed at RHIC, both in size and in sign. We further pursue this study, with a careful analysis of the SIDIS extracted Sivers functions, with their uncertainties, and investigate whether such functions, assumed to be process independent, can explain the data on , including the most recent ones. A similar study has been recently completed Anselmino:2012rq regarding the Collins effect Collins:1992kk , with the conclusion that it cannot, alone, explain all the available data on .
Ii Sivers effect and in the Generalised Parton Model formalism
The Generalised Parton Model D'Alesio:2004up ; Anselmino:2005sh ; D'Alesio:2007jt ; Anselmino:2012rq , can be considered as a natural phenomenological extension of the usual collinear factorisation scheme, with the inclusion of spin and effects through the TMDs and the dependence of the elementary interactions on the parton intrinsic motions; it was actually first proposed, for unpolarised processes, in Ref. Feynman:1978dt . In this approach the single spin effect, , originates from the TMDs; in Ref. Anselmino:2004ky and its correction Anselmino:2012rq it was shown that the only non negligible contributions to are given by the Sivers TMDPDF and the Collins TMDFF,
(5) 
The Collins contribution was studied in Ref. Anselmino:2012rq , while this paper is devoted to the Sivers effect.
In our GPM scheme the contribution of the Sivers effect to the numerator of , for large processes, is given by
(6)  
where is the Sivers function for parton , Eqs. (1, 2), which couples to the unpolarised TMD for parton , , and the unpolarised fragmentation function of parton into the final observed hadron . is the transverse momentum of hadron with respect to the 3momentum of its parent fragmenting parton. is a kinematical factor, which at equals 1. For details and a full explanation of the notations we refer to Ref. Anselmino:2005sh (where is denoted as ).
The phase factor originates directly from the dependence of the Sivers distribution [, Eq. (1)], while the are the three independent hard scattering helicity amplitudes defined in Ref. Anselmino:2005sh , describing the lowest order QCD interactions. The sum of their moduli squared is proportional to the elementary unpolarised cross section , that is
(7) 
The explicit expressions of , which give the QCD dynamics in Eq. (6), can be found, for all possible elementary interactions, in Ref. Anselmino:2005sh . The QCD scale is chosen as .
The denominator of Eq. (4) or (5) is twice the unpolarised cross section and is given in our TMD factorisation by the same expression as in Eq. (6), where one simply replaces the factor with . In Ref. Boglione:2007dm it was shown that such an expression leads to results for the unpolarised cross section in agreement with data.
We can now use the information so far available on the Sivers functions as extracted from SIDIS data and give some realistic estimates for the Sivers contribution to for several singleinclusive large particle production in collisions. More specifically, we will consider the Sivers effect for inclusive pion, kaon, photon and jet production and will see how much it can contribute to the available experimental data on . The analogue of Eq. (6) for direct photon and inclusive jet production will be given below.
ii.1 The Sivers functions in SIDIS and processes
Let us start by considering the available information on the Sivers functions and the procedure followed to obtain them. The first extraction – from now on denoted as SIDIS1 fit – was presented in Ref. Anselmino:2005ea , where the MRST01 set for the unpolarised PDFs Martin:2002dr and the Kretzer set for the unpolarised FFs Kretzer:2000yf were adopted. An updated extraction of the Sivers functions – SIDIS2 fit – was presented in Ref. Anselmino:2008sga . In this case, the GRV98 set for the unpolarised PDFs Gluck:1998xa and the pion and kaon FFs by de Florian, Sassot and Stratmann (DSS) deFlorian:2007aj were considered. Notice that the use of different PDFs does not make any relevant difference; therefore, in the following, we will consider only the GRV98 set.
The main features of the parameterisations adopted in those studies are the following: the analysis of SIDIS data is performed at leading order, , within the proven TMD factorisation approach for SIDIS, where is the large scale in the process. A simple factorised form of the TMD functions was adopted, using a Gaussian shape for their dependent component. For the unpolarised parton distribution and fragmentation functions we have:
(8) 
where and have been fixed by analysing the Cahn effect in unpolarised SIDIS processes, see Ref. Anselmino:2005nn :
(9) 
The recently introduced TMD evolution was not taken into account, while we considered the DGLAP QCD evolution of the collinear factorised part.
The Sivers functions, , have been parameterised as follows:
(10) 
where
(11) 
with , and
(12) 
With these choices, the Sivers functions automatically fulfil their proper positivity bounds for any values. For the evolution of the Sivers function, as commented above, we consider the unpolarised DGLAP evolution of its collinear factor . Notice that in the SIDIS1 fit we actually exploited also a different (powerlike) functional form for , still controlled by a single parameter, leading to almost no differences in our results. In what follows we will only use the functional form given in Eq. (12).
In order to reduce the number of free parameters, some additional assumptions were adopted. Concerning the SIDIS1 fit, we considered only and quarks Sivers functions, with flavour dependent and parameters. This amounts to a total of 7 parameters:
(13) 
In the SIDIS2 fit, since we were aiming also at explaining some large kaon SIDIS azimuthal asymmetries, we tentatively included also the Sivers functions for antiquarks and strange quarks, , , and . To keep the number of parameters under control we then assumed flavour independent and parameters for the sea quarks (). Moreover, since the large behaviour of the Sivers function could not, and still cannot, be constrained by SIDIS data (see a more detailed comment below), we also assumed a single flavour independent parameter, equal for quarks and antiquarks. This amounts to a total of 11 free parameters:
(14) 
Notice that even with such a choice, our complete parameterisation of the Sivers functions, Eq. (10), allows for further differences among parton flavours, which are contained in the usual unpolarised PDFs.
Both fits gave good results. Nevertheless it is worth stressing the main differences in the two extractions, which indeed play an important role in the present study. In fact, a direct use of SIDIS1 results in the computation of SSAs in processes for RHIC kinematics, as presented in Ref. Boglione:2007dm , gave very encouraging results. Notice that at that time the Collins effect was believed to be suppressed Anselmino:2004ky . On the other hand, if we use the SIDIS2 fit to compute the same SSAs we would get too small values, the reason being the different values coming from the two fits.
More generally, as discussed in the context of the Collins SIDIS azimuthal asymmetries for the transversity distributions Anselmino:2012rq , a study of the statistical uncertainties of the best fit parameters clearly shows that SIDIS data are not presently able to constrain the large behaviour of the quark (, ) Sivers distributions, leaving a large uncertainty in the possible values of the parameter . This is due to the limited range of Bjorken values currently explored by HERMES and COMPASS experiments, . In this respect the large results expected from JLab 12 GeV experiments will be precious Gao:2010av ; Gao:2011zzb .
This uncertainty plays a crucial role when one tries to study the SSAs in hadronic collisions starting from the results obtained from SIDIS data, because the largest pion SSAs are measured at large Feynman values, , which implies .
To investigate the role of the Sivers effect in explaining the large value of in collisions we should therefore carefully explore the large behaviour of the Sivers functions. To this aim we follow the same strategy we have devised in the recent study of the contribution of the Collins effect to , the socalled “scan procedure” Anselmino:2012rq . Here we summarise schematically its main steps and motivations.

The parameters, which control the large behaviour of the TMDs and are largely undetermined by SIDIS data, play instead an important role in the computation of , which is sizeable mainly in the large region. We can notice this explicitly by comparing, as commented above, the different implications on of the SIDIS1 and the SIDIS2 fits. In our choice of the independent parameters it is then natural to allow for a flavour dependence of , limited, because of the relevance of the large region, to the valence quark contributions. More explicitly, we only use the PDFs for and valence quarks in the Sivers functions (10) and the contribution of sea quarks and gluons is neglected in the sum over partons in Eq. (6).

We start the scan procedure by performing a preliminary 7parameter [those of Eq. (13)] “reference fit” to SIDIS data. This reference best fit will have a total . We then let the two parameters and vary, choosing them in the range — by discrete steps of , and for each of the 81 pairs of fixed s we perform a new 5parameter fit to SIDIS data.

As a next step we select only those fits leading to a such that . Notice that, since the reference fit and the scan fits have a different number of free parameters, the selection criterion is applied to the total rather than to the per degree of freedom, . The chosen value of is the same as that used to generate the error band, following the procedure described in the Appendix A of Ref. Anselmino:2008sga . We find (for 217 data points) and . As expected from the arguments given above, all 81 fits lead to acceptable values for SIDIS data; this further confirms the observation that the SIDIS data are not sensitive to the large behaviour of the Sivers function.

We then compute, for each of the 81 selected sets, the contribution of the Sivers effect to , according to Eqs. (4)–(6). We do that for pion and kaon production in the kinematical regions of the STAR and BRAHMS experiments at RHIC. The corresponding results span the shaded areas (scan band) which are shown in the figures of our results. When compared with the experimental available data, the scan bands show the potentiality of the Sivers effect alone to account for the measured values of in processes, while preserving a fair description (quantified by of the SIDIS data on the Sivers azimuthal asymmetry.

We have considered in our scan procedure all available SIDIS data Airapetian:2009ae ; Alekseev:2008aa , with the exception of the recent ones by the COMPASS Collaboration off a transversely polarised proton target Adolph:2012sn . As shown in Refs. Aybat:2011ta ; Anselmino:2012aa the analysis of these data, reaching higher values, requires a careful use of the proper TMD evolution, which is ignored here, as a correct implementation of the TMD evolution in large processes is so far unknown. We have checked that the of our fits would be approximately (3040)% worse for the SIDIS data including the proton COMPASS results and no TMD evolution.

We study the contribution of the Sivers effect to the SSA at RHIC energies only, although it might contribute also to the (larger) SSAs measured at lower energies Klem:1976ui ; Krueger:1998hz ; Allgower:2002qi ; Antille:1980th ; Adams:1991rw ; Adams:1991cs ; Adams:1991rv ; Adams:1991ru . The reason is that we consider only the processes for which our GPM and TMD factorisation can reasonably well reproduce the unpolarised cross section Boglione:2007dm .
ii.1.1 Results from the scan procedure
Some of our results for RHIC experiments are shown in Figs. 14. We have computed adopting, as explained above, a single set of collinear parton distributions Gluck:1998xa and two different sets for the pion and kaon collinear FFs Kretzer:2000yf ; deFlorian:2007aj ; the results shown correspond to the Kretzer set. Other results not shown are very similar and would not add any significant information.
Let us start by considering the case of inclusive pion production. This will also help a direct comparison with the corresponding study on the potential role of the Collins contribution to the same observable Anselmino:2012rq .
In Fig. 1 the scan band for , as a function of at fixed scattering angles, is shown for charged pions and BRAHMS kinematics, while in Fig. 2 the same result is given, at fixed pseudorapidity values, for neutral pions and STAR kinematics. We also give the scan band, as a function of at several fixed values, for STAR kinematics in Fig. 3. All these results are given at GeV. We then consider the latest and interesting preliminary data obtained by STAR at large and GeV Igo:2012 , and show our scan band in Fig. 4 for different values of .
From these results we can conclude that the Sivers effect alone might in principle be able to explain the BRAHMS charged pion results on in the full kinematical range so far explored, as well as almost the full amount of STAR data on . This is to be contrasted with the analogous study of the Collins effect Anselmino:2012rq , with the conclusion that such effect alone cannot explain the observed values of in the mediumlarge region.
This can be understood as follows. In the case of SSAs for neutral pion production the Collins effect suffers from two possible cancellations: the opposite sign between the and quark transversity distributions and the opposite sign between the favoured and disfavoured Collins FFs (necessary to build the Collins FF for ); instead, for the Sivers effect only a cancellation between and flavours in the distribution sector may play a role, as it couples to the unpolarised TMDFF.
A further remark concerns the values of the parameters and the area spanned by the bands: the upper borderlines of the scan bands for neutral and positively charged pions correspond to the set of Sivers functions with (up quark unsuppressed) and (down quark strongly suppressed), while the lower borderlines correspond to the case where the values of are interchanged. Notice that larger values of would not change this picture. For negative pions the situation is just reversed since to get the largest values, in size, of (lower border) the down quark should dominate (that is and ).
The results obtained with a different choice of the fragmentation functions (the DSS set) are qualitatively very similar in the large regions. They are instead smaller in size at smaller , due to the large gluon contribution in the leading order (LO) DSS fragmentation functions. They are not shown here.
The case of SSAs for kaon production would require a further study of the corresponding unpolarised fragmentation functions, which represents an open issue by itself and falls outside the purposes of this paper. However, for completeness and a qualitative estimate, we consider for production as measured by the BRAHMS Collaboration Lee:2007zzh , with the kaon set of fragmentation function as given in Ref. Kretzer:2000yf . Our results for the scan band, compared with the data, are shown in Fig. 5.
The results in Figs. 15 show that the Sivers effect alone, as computed in our GPM scheme, Eqs. (4)–(6), and based on the Sivers functions extracted from SIDIS data and assumed to be universal, can be large enough to explain alone the pion SSAs observed at RHIC. One should not forget that, indeed, the phenomenology of the Sivers effect was originally generated in the attempt to explain the large values of observed by the E704 Collaboration Sivers:1989cc ; Sivers:1990fh ; Anselmino:1994tv . However, the amount of uncertainty in the scan bands, due to lack of precise SIDIS data at large , is still much too large to draw any definite conclusions.
A full understanding of the SSAs in inclusive processes should also take into account the contribution of the Collins effect, which might be small, but not entirely negligible. Rather than addressing the issue of a best fit of SIDIS + data with Collins and Sivers effects, which is premature at this stage, we now adopt a more pragmatic attitude. We wonder whether, among the 81 sets of parameters which build up the possible results on contained in the scan bands, we can find some which give a good description of all the data.
ii.1.2 Results with a selected set of parameters and its statistical uncertainty bands
Among the full set of curves produced by the scan procedure, we have isolated the set leading to the best description of (actually one could find more than a single set); we have then evaluated, as in Appendix A of Ref. Anselmino:2008sga , the corresponding statistical error band. Our results are presented in Figs. 6, 7, 8 and 9, respectively for BRAHMS data vs. at fixed angles, for STAR results vs. at fixed pseudorapidities, for STAR results vs. at different values, and for BRAHMS data vs. at a fixed angle, all of them at GeV. The corresponding values of the parameters are given in Table 1. From these results one can see that it is possible to find a set of Sivers functions for and quarks which, while describing well the SIDIS data, can also describe fairly well, alone, the SSAs for pion production, as measured both by BRAHMS and STAR Collaborations at 200 GeV.
The preliminary STAR data at 500 GeV Igo:2012 deserve a dedicated comment. Quite surprisingly, they show values of of the order of few percents, with a flat behaviour as a function of at fixed , up to GeV. Such a trend is well reproduced by our set of chosen best parameters; however, the computed magnitude of is smaller than data, as shown in Fig. 10, left plots. As the asymmetry is so small, we have also computed the Collins contribution to , following Ref. Anselmino:2012rq . It turns out that, for some sets of the parameters, the Collins contribution has a similar trend and magnitude as the Sivers one, as shown in Fig. 10, right plots. Then, an appropriate sum of the two contributions, according to Eq. (5), might well explain also this new puzzling data.
Another cautious comment about the STAR data on at 500 GeV concerns the large value of their QCD scale, . As we noticed for the COMPASS proton data, at such values the TMD evolution might play an important role. Our results should then be taken as an indication in favour of a combined Collins + Sivers effect, rather than a proof. Qualitatively, one expects from TMD evolution an increase of the average value of the Sivers distribution, which would help increasing the corresponding value of .
GeV 
ii.2 SSAs for and processes
In these processes no fragmentation mechanism is required, so that, within the GPM and the TMD factorisation approach, one can access directly the spin and properties of the partonic distributions. After integration over the intrinsic azimuthal phases only the Sivers effect survives, which is then best studied in these processes, as discussed, e.g., in Refs. D'Alesio:2004up ; D'Alesio:2010am . Notice that, for the same reasons, the SSAs for inclusive jet or photon production can be used to test the process dependence of the Sivers functions in a modified generalised parton model with inclusion of initial and final state interactions Gamberg:2010tj ; D'Alesio:2011mc or within the twist3 approach Gamberg:2013kla .
The numerator of for the inclusive jet production can be obtained from Eq. (6) simply replacing the TMD fragmentation function, , with a factor (and identifying now the final hadron momentum, , with the jet momentum ). More explicitly the numerator of for inclusive jet production reads
(15)  
Notice that the elementary hard scattering interactions are exactly the same as those for the inclusive hadron production and the jet, at LO, is identified with the final parton .
Concerning the direct photon production the basic partonic processes are the Compton process . In this case one can formally use the above equation replacing the partonic unpolarised cross section, Eq. (7), with the corresponding one for the process (see also Ref. D'Alesio:2004up ). and the annihilation process
No SSA data are so far available for direct photon production, while very recently some preliminary data for inclusive jet production have been released by the ADY Collaboration at GeV Bland:2013pkt . The values measured for are very tiny, but very precise and might indicate a non zero asymmetry.
In the left plot of Fig. 11 we show our estimate, based on the chosen best set parameters of Table 1, for in processes at a fixed pseudorapidity value and = 500 GeV, and compare it with the ADY data Bland:2013pkt . In the right plot we give our corresponding estimates for in processes at a fixed pseudorapidity value and = 200 GeV.
Iii Conclusions
The origin of the azimuthal asymmetries in SIDIS processes is considered to be well understood and related to TMDPDFs and TMDFFs, via the QCD TMD factorisation scheme. Indeed, the measurement of such asymmetries has been used to extract information on the TMDs. A reasonable knowledge of the Sivers TMDPDF is by now available and confirmed by independent groups Anselmino:2005ea ; Anselmino:2008sga ; Vogelsang:2005cs ; Collins:2005ie ; Anselmino:2005an . However, due to the kinematical range of the data, this knowledge is limited to the small region, . The same TMD factorisation is expected to hold also in DrellYan and processes.
The situation is not so clear concerning the oldest and largest single spin asymmetries measured in several hadronic processes, in particular in . Due to the presence of a single large scale – the of the final hadron – one cannot extend to these processes the proof of the QCD TMD factorisation theorem, which requires the presence of two separate scales, a large and a small one. As explained in the Introduction, a twist3 collinear and factorised approach has been proposed Efremov:1981sh ; Efremov:1984ip ; Qiu:1991pp ; Qiu:1998ia ; Kanazawa:2000hz ; Kanazawa:2000kp ; Kouvaris:2006zy ; Kanazawa:2010au ; Kanazawa:2011bg , which introduces new threeparton correlation functions, related to the moment of the TMDPDFs. However, it seems to predict values of opposite to the observed ones Kang:2011hk . Thus, the true origin of the large values of remains obscure.
Among the first attempts to explain Sivers:1989cc ; Sivers:1990fh ; Anselmino:1994tv ; Anselmino:1998yz ; Anselmino:1999pw ; D'Alesio:2004up ; Anselmino:2005sh ; D'Alesio:2007jt , and the unpolarised cross section Feynman:1978dt , one should consider the simple extension of the collinear QCD factorisation to the TMD case, the socalled Generalised Parton Model (GPM) in which one assumes TMD factorisation and the universality of the TMFPDFs and TMDFFs. Although such a factorisation has not been proven, it is worth exploring its phenomenological consequences. In this paper we have studied, within the GPM, the contribution of the Sivers effect to the single spin asymmetry as measured by RHIC Collaboration experiments. The Sivers functions are the same as those which explain azimuthal asymmetries in SIDIS processes. A similar analysis was performed in Ref. Anselmino:2012rq , concerning the Collins effect.
Our results, limited to the contribution of the valence quarks, to the SIDIS cases where the TMD evolution is not expected to be relevant and to the large processes for which we can well reproduce the unpolarised cross sections, are rather encouraging. For most pion data the Sivers effect alone could explain the observed values of in magnitude and, in particular, in sign. This is in contrast to other approaches, also related to the Sivers effect, which seem to have severe problems Kang:2011hk ; Gamberg:2010tj in explaining the sign of the observed .
We have performed our analysis by varying the parameters of the Sivers functions which are not well fixed by the SIDIS data, due to their limited kinematical range, obtaining the socalled scan bands. In particular, we have let the power which fixes the large behaviour of the and quark Sivers functions, , vary between 0 and 4 in steps of 0.5. Thus, we have 81 different sets of Sivers functions; each of them still fits well the SIDIS data. The bands, which appear in Figs. 15, are the envelope of the 81 different curves, and , obtained in our GPM approach.
Then, among the explored sets of parameters, we have chosen a particular one, which gives one of the best descriptions of the data. We have used such a set to compute estimates for SSAs in and processes. Such measurements will further allow one to discriminate between our approach and others. Predictions for in different processes and kinematical regions can be easily obtained, if necessary.
While encouraged by the results of our analysis we should avoid making definite conclusions at this stage. This work shows that the GPM TMD factorisation scheme could explain at the same time the main features of the SSAs measured in SIDIS and hadronic processes. While such a scheme is well justified for SIDIS it can, so far, only be considered as a phenomenological model for hadronic processes, which needs further confirmation or disproval from data and further theoretical work. Our choice of the sets of parameters given in Table 1 is not meant to be interpreted as the final best set of Sivers functions. A full analysis, including TMD evolution, of all data involving the Sivers effect – i.e. all the SSAs in several different processes – would require much more attention and work.
Acknowledgements.
We would like to thank L. Bland for information on the ADY data and S. Heppelmann for information on the STAR data on at 500 GeV. We acknowledge support of the European Community under the FP7 “Capacities  Research Infrastructures” program (HadronPhysics3, Grant Agreement 283286). Some of us (M.A., M.B., U.D., F.M.) acknowledge partial support from MIUR under Cofinanziamento PRIN 2008. U.D. 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.References
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