Transverse single-spin asymmetries in within a TMD approach:
Role of quasireal photon exchange
We present an updated study of transverse single-spin asymmetries for the inclusive large- processes and , within a transverse momentum-dependent approach, including the contribution of quasireal (Weizsäcker-Williams) photons. In the spirit of a unified transverse momentum-dependent scheme, predictions are obtained adopting the Sivers and transversity distributions and the Collins fragmentation functions as extracted from fits to the azimuthal asymmetries measured in semi-inclusive deep inelastic scattering and annihilation processes. The description of the available data is extremely good, showing a clear general improvement with respect to the previous leading-order analysis. Predictions for unpolarized cross sections and single-spin asymmetries for ongoing and future experiments are also given.
The role played by transverse single-spin asymmetries (SSAs) in our understanding of the nucleon structure is nowadays well consolidated and, at the same time, still source of challenging issues. Indeed, SSAs observed in processes where two energy scales (a large and a small one) are detected are unambiguously studied within an approach based on factorization theorems in terms of transverse momentum dependent distributions (TMDs). On the other hand, the description of the large data sets for the SSA measured in inclusive pion production in collisions, where only one energy scale is present, is still under debate (see for instance Refs. D’Alesio and Murgia (2008); Aschenauer et al. (2016) for general overviews, and Refs. Adams et al. (1991a, b, 1992a, 1992b, 2004); Adler et al. (2005); Lee and Videbaek (2007); Abelev et al. (2008); Adamczyk et al. (2012); Igo et al. (2012); Bland et al. (2013) for the experimental results).
In Refs. Anselmino et al. (2010); Anselmino et al. (2014a) this issue was investigated in a somehow theoretically more simple single-inclusive process, , still characterized by a single large energy scale, but very close to the semi-inclusive deep inelastic scattering (SIDIS) process, for which TMD factorization has been proved Collins (2002); Collins and Metz (2004); Ji et al. (2005, 2004); Bacchetta et al. (2008); Collins (2013); Echevarria et al. (2012, 2014).
This process indeed can be considered a sort of bridge between the and processes: it is single inclusive with a single large energy scale (as the process), and at the same time, at leading order, is controlled by the colour-blind electromagnetic interaction (as the SIDIS process). This should reduce the role played by initial/final state interactions leading to potential factorization breaking effects. On the other hand, adopting the relevant TMDs (Sivers and Collins functions), as extracted from SIDIS data, in the inclusive hadron production in lepton-proton collisions represents an attempt towards (and a test for) a unified TMD scheme. It is worth mentioning that the same process was also considered in Refs. Koike (2003); Gamberg et al. (2014) in the framework of collinear factorization with twist-three correlation functions, while inclusive jet production was studied in Ref. Kang et al. (2011).
In Refs. Anselmino et al. (2010); Anselmino et al. (2014a), to which we refer the reader for all details of the approach, SSAs were computed assuming a TMD factorization scheme at leading order (LO), that is considering only the elementary partonic channel . In particular, in Ref. Anselmino et al. (2014a) the theoretical estimates were compared with a selection111Only data for inclusive events in the backward target hemisphere at large and tagged events (deep inelastic scattering category) were considered. of the experimental results by the HERMES Collaboration Airapetian et al. (2014), showing a good agreement in sign and size. In spite of this, it was also pointed out that some of the discrepancies still present between theory and experiment could be ascribed to effects neglected in a LO treatment.
Here we want to extend this LO study including the contribution from quasireal photon exchange, in the Weizsäcker-Williams approximation, potentially relevant in the kinematical configuration dominated by small . This will allow us, still within a TMD scheme, to improve the description of the fully-inclusive data and consider, for the first time, the HERMES anti-tagged data set, dominated by events in which the final lepton (not observed) has a very small scattering angle. Notice that this data category was not included in the previous analysis because a simple LO approach (namely via ) is expected to be not adequate.
In this respect we will benefit from the study performed in Ref. Hinderer et al. (2015), even if with a different perspective and approach. In this work the authors, within a collinear-factorization scheme, computed the next-to-leading order (NLO) corrections to the unpolarized cross sections for the same process and discussed the role of quasireal photon exchange. In most kinematical configurations they found that this contribution represents only a small part of the NLO corrections. They then concluded that only a full NLO treatment could be considered complete.
On the other hand, within a TMD scheme, as well as in the twist-three approach, NLO corrections are still not available for such a process and it is then worth seeing to what extent the quasireal photon exchange could play a role in the computation of spin asymmetries. On top of that, and relevant from our point of view, by including transverse momentum effects the estimates of unpolarized cross sections are enhanced w.r.t. those computed in a collinear framework. Experimental data, still not available, would definitely help in this respect. Notice that for the process the estimates of unpolarized cross sections in a TMD approach at leading order show a reasonable agreement with available data from the Relativistic Heavy Ion Collider (RHIC); see Ref. Boglione et al. (2008).
The main aim of this study will be then to provide the complete calculation within a TMD formalism of the quasireal photon exchange in and processes and to compute the unpolarized cross sections and the SSAs for various experimental setups.
The paper is organized as follows: in Section II we recall the general formalism, deriving and discussing all new theoretical results. In particular, in Section II.2 we present, for the first time, the full TMD expressions for the quasireal photon contribution to unpolarized and transversely polarized cross sections for inclusive hadron and inclusive jet production. In Section III we show our phenomenological results, starting with the unpolarized cross sections for HERMES, Jefferson Lab (JLab), COMPASS and Electron-Ion Collider (EIC) experiments, and then focusing on transverse SSAs, with special emphasis on the comparison with HERMES data. Predictions for other experimental setups are also given and discussed. Conclusions and final comments are gathered in Section IV.
We consider the transverse single-spin asymmetry, , for the process in the proton-lepton center-of-mass (c.m.) frame,
and and are respectively the three-momentum of the final hadron and its vector transverse component. The polarized proton (or nucleon) is in a pure transverse spin state and is assumed to move along the positive axis, while the lepton is taken unpolarized. We define as transverse polarization for the proton the direction, with and respectively for protons polarized along or opposite to . The axis is defined in such a way that a hadron with is produced to the left of the incoming proton (see also Fig. 1 of Ref. Anselmino et al. (2010)).
Notice that for a generic transverse polarization, , along an azimuthal direction in the chosen reference frame, in which the direction is given by , one has:
where is the proton momentum. Following the usual definition adopted in SIDIS experiments, one simply obtains:
In order to include effects from quasireal photon exchange, adopting the Weizsäcker-Williams (WW) approximation, within a TMD approach, we write the SSA under consideration as follows:
Proper definition of all functions and variables appearing in the above equations can be found in Ref. Anselmino et al. (2010) and its Appendices and in Ref. Anselmino et al. (2006). For a better understanding we recall here their physical meaning.
and are respectively the transverse momentum of the parton in the proton and of the final hadron with respect to the direction of the fragmenting parent parton, with momentum . Notice that and are different vectors.
The extra factors are the unpolarized elementary interaction () and the unpolarized fragmentation function ; in the chosen reference frame, where , the correlation factor gives the modulation .
The second and third terms (this last one numerically negligible) on the r.h.s. of Eq. (8) represent the contribution to of the Collins effect, given respectively as a convolution of the unintegrated transversity distribution, , and the pretzelosity distribution, , with the Collins function Collins (1993); Bacchetta et al. (2004),
The product is related to the spin transfer elementary interaction (, while the factors and arise from phases in the -dependent transversity and pretzelosity distributions, the Collins function and the elementary polarized interaction.
The first (and dominant) term on the r.h.s. of Eq. (9) is the convolution of the unpolarized TMD parton distribution and fragmentation functions with the unpolarized partonic interactions, while the second one, numerically negligible, represents the Boer-Mulders mechanism Boer and Mulders (1998); Boer (1999), with the corresponding function defined as
In the following Sections we discuss in detail the Weizsäcker-Williams approximation and its role in the (un)polarized process under consideration.
ii.1 Weizsäcker-Williams approximation
As shown in Ref. Hinderer et al. (2015), in a NLO treatment of the inclusive process , the collinear lepton singularities could be regularized, and opportunely redefined, by introducing a QED parton distribution for the lepton, in strong analogy with the ordinary nucleon’s parton distributions. The only difference is that in such a case the partons are the lepton itself and the photon. Without entering into many details, we can say that at order there will be a contribution from the photon acting as a parton of the lepton and entering the hard scattering process. This can be represented as a Weizsäcker-Williams contribution von Weizsacker (1934); Williams (1934), where the lepton acts as a source of real photons (see also Refs. Brodsky et al. (1971); Terazawa (1973); Kniehl (1991)). We then assume the following factorization formula for the WW contribution to the process :
where is the number density of photons inside the lepton, carrying a lepton-momentum fraction () and is the cross section for the process initiated by a real photon.
For the WW distribution we follow Ref. Hinderer et al. (2015), adopting
where is the electromagnetic coupling constant, the factorization scale and the lepton mass. We have also tried an alternative form for the WW distribution, like the one proposed in Refs. Brodsky et al. (1971); Terazawa (1973); Kniehl (1991) and adopted, in the context of SSA studies, in Refs. Godbole et al. (2012, 2013). In both cases we have considered two choices of the factorization scale, namely or . Since these choices do not lead to any significant differences we will present our estimates only for the form in Eq. (14) with .
ii.2 Quasireal photon contribution to SSAs for inclusive particle production
In order to compute the WW contribution to , based on the factorized expression (13), we start with the general treatment for the cross section, in a TMD scheme, of the large- inclusive polarized process Anselmino et al. (2006), adapted here to the process :
which can be written schematically as
Notice that in Eq. (II.2) we have consistently adopted a collinear WW distribution, as properly defined for the case of a scattered lepton, and a photon, almost collinear with the initial lepton and that now can be a quark (antiquark) or a gluon (this is at variance w.r.t. the LO calculation where only quark TMDs are involved).
For the notation and the meaning of the quantities entering Eq. (II.2) we refer the reader to Refs. Anselmino et al. (2010, 2006). Here we only note that the Mandelstam variables for the process are defined using and that the ’s and the ’s are respectively the helicity density matrices of partons (photons) inside a polarized hadron (an unpolarized lepton) and the helicity amplitudes for the elementary processes and . We further recall that the ’s are defined in the proton-lepton c.m. frame, where the processes are not planar. They can be expressed in terms of the corresponding canonical helicity amplitudes in the - c.m. frame by performing proper boost and rotations as described in Ref. Anselmino et al. (2005a); Anselmino et al. (2006) (see also Appendix A).
By summing over the helicities, using the proper definition of the helicity density matrices for spin-1/2 and spin-1 partons, and exploiting the parity properties of the helicity amplitudes, we obtain the following expressions for the kernels :
where can be either a quark or an antiquark and
where again can be either a quark or an antiquark and
These can be obtained from the processes by interchanging in the two above equations with (that is and ) and with .
In the above equations stand for the quark, gluon and photon polarization vector components and for the gluon and photon linear polarization ones, while are the azimuthal phases of the helicity amplitudes (see Appendix A for details).
We are now ready to compute the WW contributions to . By choosing in the adopted reference frame, we have
and once again in Eqs. (26)-(29) can be either a quark or an antiquark, while for the channel one can use the last two relations replacing with . In Eqs. (26) and (27) we have redefined , consistently, and in agreement, with the notation adopted in the LO expressions222Notice that the explicit calculation of the azimuthal phases given in Ref. Anselmino et al. (2010) leads to the same results obtained following the boost-rotation procedure described in Refs. Anselmino et al. (2005a); Anselmino et al. (2006)..
In Eqs. (26) and (28) we recognize the Sivers and Collins effects. Once again, as for the LO piece, the terms involving the pretzelosity in Eq. (26) and the Boer-Mulders function in Eq. (27) are numerically negligible (even saturating their positivity bounds). On the other hand, at variance with the leading-order analysis, we have also a potential contribution from the gluon Sivers function (see Eq. (30)). Notice that all contributions from linearly polarized gluons () appearing in Eq. (LABEL:ggaqqb) disappear since they are coupled to linearly polarized photon () distributions that are identically zero for an unpolarized initial lepton.
ii.2.1 SSAs in single-inclusive jet production at large transverse momentum
Inclusive jet production in lepton-proton collisions, although more difficult to measure, could be an invaluable tool to access the Sivers effect, as the lack of any fragmentation process forbids other contributions. In Ref. Anselmino et al. (2010) this case was discussed and some results for a high-energy electron-nucleon collider were presented. In the same spirit here we extend this analysis including the quasireal photon contribution. The expressions can be directly obtained from the case of inclusive hadron production by replacing the fragmentation functions with proper Dirac delta functions. We report here the main results for the WW contribution, referring to Ref. Anselmino et al. (2010) for the LO piece. For the master formula we have
while for the contributions to
with Eq. (25) still valid also for jet production. For the sums and differences of the kernels we can use the same expressions as given in Eqs.(26)-(31) replacing with 1 and with 0. In this case, obviously, there is no fragmentation process and only the Sivers effect contributes to . Notice that in the present treatment the jet coincides with a single final parton.
Iii Phenomenological results, comparison with data and predictions
In this Section we present our theoretical estimates of the unpolarized cross sections and the SSAs for inclusive pion production in lepton-proton collisions, focusing on the role of the WW contribution and its relevance w.r.t. the LO approximation. In particular, we will discuss in some detail HERMES kinematics, for which transverse SSA data are available. We will then give predictions for experiments at JLab with the upgrade at 12 GeV, for COMPASS at CERN, and for a future Electron-Ion Collider. In this last case we will also show some estimates for inclusive jet production.
Before presenting our results, it is worth giving some comments on the adopted kinematical configuration w.r.t. usual experimental setups.
According to the HERMES analysis Airapetian et al. (2014), for instance, the lepton is assumed to move along the positive axis, so that we should consider the processes , rather than . In this reference frame the () direction is still along the () axis and, keeping the usual definition of , where is the longitudinal momentum of the final hadron, only the sign of is reversed.
The azimuthal dependent cross section measured by HERMES is defined as Airapetian et al. (2014):
coincides with our of Eq. (3), as and (respectively, the proton and the lepton 3-momenta) are opposite vectors in the lepton-proton c.m. frame and one has:
where is the SSA that we compute here, and is the quantity measured by HERMES Airapetian et al. (2014).
In the following, to keep uniform the presentation of our results, we will show our predictions adopting the HERMES setup also for JLab and COMPASS experiments. For EIC we prefer to keep the other configuration, with the proton moving along the positive axis, since it allows to emphasize the strong analogies with the SSAs observed in processes.
Finally, we notice that at relatively low , around 1-2 GeV, due to the inclusion of transverse momentum effects one or more of the partonic Mandelstam variables might become smaller than a typical hadronic scale. This configuration would correspond to a situation where the propagator of the exchanged particle in the partonic scattering becomes soft. In order to avoid such a potential problem, following Ref. D’Alesio and Murgia (2004), we have introduced an infrared regulator mass ( GeV). We have checked that shifting the partonic Mandelstam invariants by this quantity squared or cutting them out below it gives similar results. Estimates will be shown adopting the shifting procedure.
iii.1 Unpolarized cross sections
For the computation of the unpolarized cross sections within the adopted TMD approach we will use the following factorized expressions for the unpolarized TMDs:
with GeV and GeV as extracted in Ref. Anselmino et al. (2005b). For the collinear parton distributions, , we adopt the GRV98 set Gluck et al. (1998), while for the collinear fragmentation functions (FFs), , we use the Kretzer set Kretzer (2000) and the one by de Florian, Sassot and Stratmann (DSS) de Florian et al. (2007). The reasons for this choice are the following: these sets were adopted in the extraction of the Sivers and Collins functions we use here for the calculation of the SSAs (next Section); they are characterized by a different role of the gluon fragmentation function, that could play a role in the WW contribution.
In Figs. 1 and 2 we present our estimates for the unpolarized cross sections for (left panels) and (right panels) production at GeV, respectively at fixed as a function of , and at fixed GeV as a function of . The thin curves refer to the LO calculation, while the thick ones to the total (LO+WW) contribution. In particular, the blue dashed lines are obtained adopting the Kretzer set for the fragmentation functions, while the red solid lines with the DSS set.