First multidimensional, high precision measurements of semi-inclusive π^+ beam single spin asymmetries from the proton over a wide range of kinematics
S. Diehl, A. Kim, G. Angelini, K. Joo, S. Adhikari, M. Amaryan, M. Arratia, H. Atac, H. Avakian, C. Ayerbe Gayoso, N.A. Baltzell, L. Barion, S. Bastami, M. Battaglieri, I. Bedlinskiy, F. Benmokhtar, A. Bianconi, A.S. Biselli, M. Bondi, F. Bossu, S. Boiarinov, K.-T. Brinkmann, W.J. Briscoe, W. Brooks, D. Bulumulla, V.D. Burkert, D.S. Carman, J.C. Carvajal, A. Celentano, P. Chatagnon, T. Chetry, G. Ciullo, L. Clark, B.A. Clary, P.L. Cole, M. Contalbrigo, G. Costantini, V. Crede, A. D'Angelo, N. Dashyan, R. De Vita, M. Defurne, A. Deur, C. Dilks, C. Djalali, M. Dugger, R. Dupre, M. Ehrhart, A. El Alaoui, L. El Fassi, L. Elouadrhiri, S. Fegan, A. Filippi, T. Forest, G. Gavalian, G.P. Gilfoyle, F.X. Girod, D.I. Glazier, A.A. Golubenko, R.W. Gothe, Y. Gotra, K.A. Griffioen, M. Guidal, K. Hafidi, H. Hakobyan, M. Hattawy, T.B. Hayward, D. Heddle, K. Hicks, A. Hobart, M. Holtrop, C.E. Hyde, D.G. Ireland, E.L. Isupov, H.S. Jo, R. Johnston, S. Joosten, D. Keller, M. Khachatryan, A. Khanal, W. Kim, A. Kripko, V. Kubarovsky, S.E. Kuhn, L. Lanza, M. Leali, S. Lee, P. Lenisa, K. Livingston, Z. Lu, I.J.D. MacGregor, D. Marchand, N. Markov, L. Marsicano, V. Mascagna, B. McKinnon, Z.E. Meziani, R.G. Milner, T. Mineeva, M. Mirazita, et al. (54 additional authors not shown)
FFirst multidimensional, high precision measurements of semi-inclusive π + beam singlespin asymmetries from the proton over a wide range of kinematics S. Diehl,
34, 6
A. Kim, G. Angelini, K. Joo, S. Adhikari, M. Amaryan, M. Arratia, H. Atac, H. Avakian, C. Ayerbe Gayoso, N.A. Baltzell, L. Barion, S. Bastami, M. Battaglieri,
44, 17
I. Bedlinskiy, F. Benmokhtar, A. Bianconi,
47, 20
A.S. Biselli, M. Bondi, F. Boss`u, S. Boiarinov, K.-T. Brinkmann, W.J. Briscoe, W. Brooks, D. Bulumulla, V.D. Burkert, D.S. Carman, J.C. Carvajal, A. Celentano, P. Chatagnon, T. Chetry,
27, 32
G. Ciullo,
15, 10
L. Clark, B.A. Clary, P.L. Cole, M. Contalbrigo, G. Costantini,
47, 20
V. Crede, A. D’Angelo,
18, 37
N. Dashyan, R. De Vita, M. Defurne, A. Deur, C. Dilks, C. Djalali, M. Dugger, R. Dupre, M. Ehrhart,
1, 21
A. El Alaoui, L. El Fassi, L. Elouadrhiri, S. Fegan, A. Filippi, T. Forest, G. Gavalian, G.P. Gilfoyle, F.X. Girod, D.I. Glazier, A.A. Golubenko, R.W. Gothe, Y. Gotra, K.A. Griffioen, M. Guidal, K. Hafidi, H. Hakobyan,
45, 52
M. Hattawy, T.B. Hayward, D. Heddle,
4, 44
K. Hicks, A. Hobart, M. Holtrop, C.E. Hyde, D.G. Ireland, E.L. Isupov, H.S. Jo, R. Johnston, S. Joosten, D. Keller, M. Khachatryan, A. Khanal, W. Kim, A. Kripko, V. Kubarovsky, S.E. Kuhn, L. Lanza, M. Leali,
47, 20
S. Lee, P. Lenisa,
15, 10
K. Livingston, Z. Lu, I.J.D. MacGregor, D. Marchand, N. Markov,
44, 6
L. Marsicano, V. Mascagna,
46, 20
B. McKinnon, Z.E. Meziani,
1, 43
R.G. Milner, T. Mineeva, M. Mirazita, V. Mokeev, P. Moran, A. Movsisyan, C. Munoz Camacho, P. Nadel-Turonski, P. Naidoo, K. Neupane, S. Niccolai, G. Niculescu, T.R. O’Connell, M. Osipenko, M. Paolone,
30, 43
L.L. Pappalardo,
15, 10
R. Paremuzyan,
44, 29
E. Pasyuk, W. Phelps, O. Pogorelko, Y. Prok, A. Prokudin, B.A. Raue,
11, 44
M. Ripani, J. Ritman, A. Rizzo,
18, 37
C.D. Roberts, P. Rossi,
44, 16
J. Rowley, F. Sabati´e, C. Salgado, A. Schmidt, E.P. Segarra, Y.G. Sharabian, U. Shrestha, O. Soto,
16, 45
N. Sparveris, S. Stepanyan, P. Stoler, I.I. Strakovsky, S. Strauch, K. Tezgin, A. Thornton, N. Tyler, R. Tyson, M. Ungaro, L. Venturelli,
47, 20
H. Voskanyan, A. Vossen, E. Voutier, D.P. Watts, K. Wei, X. Wei, S.-S. Xu, B. Yale, N. Zachariou, and J. Zhang (The CLAS Collaboration) Argonne National Laboratory, Argonne, Illinois 60439 Arizona State University, Tempe, AZ 85281 IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France Christopher Newport University, Newport News, Virginia 23606 University of California, Riverside, CA 92521 University of Connecticut, Storrs, Connecticut 06269 Duke University, Durham, North Carolina 27708-0305 Duquesne University, 600 Forbes Avenue, Pittsburgh, PA 15282 Fairfield University, Fairfield CT 06824 Universita’ di Ferrara , 44121 Ferrara, Italy Florida International University, Miami, Florida 33199 Florida State University, Tallahassee, Florida 32306 The George Washington University, Washington, DC 20052 Idaho State University, Pocatello, ID 83209 INFN, Sezione di Ferrara, 44100 Ferrara, Italy INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy INFN, Sezione di Genova, 16146 Genova, Italy INFN, Sezione di Roma Tor Vergata, 00133 Rome, Italy INFN, Sezione di Torino, 10125 Torino, Italy INFN, Sezione di Pavia, 27100 Pavia, Italy Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Institute fur Kernphysik (Juelich), Juelich, Germany James Madison University, Harrisonburg, Virginia 22807 Kyungpook National University, Daegu 41566, Republic of Korea Lamar University, 4400 MLK Blvd, PO Box 10046, Beaumont, Texas 77710 Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 Mississippi State University, Mississippi State, MS 39762-5167 National Research Centre Kurchatov Institute - ITEP, Moscow, 117259, Russia University of New Hampshire, Durham, New Hampshire 03824-3568 New Mexico State University, PO Box 30001, Las Cruces, NM 88003, USA Norfolk State University, Norfolk, Virginia 23504 Ohio University, Athens, Ohio 45701 a r X i v : . [ h e p - e x ] J a n Old Dominion University, Norfolk, Virginia 23529 II. Physikalisches Institut der Universit¨at Gießen, 35392 Gießen, Germany Rensselaer Polytechnic Institute, Troy, New York 12180-3590 University of Richmond, Richmond, Virginia 23173 Universita’ di Roma Tor Vergata, 00133 Rome Italy School of Physics, Southeast University, Nanjing 211189, Jiangsu, China School of Physics and Institute for Nonperturbative Physics, Nanjing University, Nanjing 210093, Jiangsu, China School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu, China Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119234 Moscow, Russia University of South Carolina, Columbia, South Carolina 29208 Temple University, Philadelphia, PA 19122 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V Valpara´ıso, Chile Universit`a degli Studi dell’Insubria, 22100 Como, Italy Universit`a degli Studi di Brescia, 25123 Brescia, Italy University of Glasgow, Glasgow G12 8QQ, United Kingdom University of York, York YO10 5DD, United Kingdom University of Virginia, Charlottesville, Virginia 22901 College of William and Mary, Williamsburg, Virginia 23187-8795 Yerevan Physics Institute, 375036 Yerevan, Armenia
High precision measurements of the polarized electron beam-spin asymmetry in semi-inclusivedeep inelastic scattering (SIDIS) from the proton have been performed using a 10.6 GeV incidentelectron beam and the CLAS12 spectrometer at Jefferson Lab. We report here the first multidi-mensional study of single π + SIDIS data over a large kinematic range in z , x B , P T and virtualities Q ranging from 1 GeV up to 7 GeV . In particular, the structure function ratio F sin φLU /F UU hasbeen determined, where F sin φLU is a twist-3 quantity that can reveal novel properties of quark-gluoncorrelations within the nucleon. The impact of the data on the evolving understanding of the un-derlying reaction mechanisms and their kinematic variation is explored using theoretical models forthe different contributing transverse momentum dependent parton distribution functions. PACS numbers: 75.25.-j, 13.60.-r, 13.88.+e, 24.85.+p
Many decades of experiments in deep inelastic scat-tering (DIS) of lepton beams off nucleons have mappedout the momentum distributions in the nucleon in termsof one-dimensional (1-D) parton distribution functions(PDFs) [1–3]. While these measurements provided sig-nificant insight into the structure of the nucleon, manyimportant and interesting aspects of the nucleon struc-ture cannot be revealed in this 1-D picture since PDFsare essentially averaged over all degrees of freedom ex-cept the longitudinal momentum. Therefore, they can-not address questions such as: Do quarks undergo orbitalmotion? Is there a connection between the motion ofquarks, their spin and the spin of the proton? How is thetotal spin of the proton built up from the spin and theorbital angular momentum of partons? Today, the pos-sibility of three-dimensional (3-D) imaging exists, whichallows such questions to be addressed [4–9]. Remark-able theoretical advances over the past decade have ledto a rigorous framework where information on the con-fined motion of the partons inside a fast moving nucleonis matched to transverse momentum dependent partondistribution functions (TMDs) [6–8, 10]. In particular,TMDs can encode information about the orbital motionof quarks in the parent nucleon and correlations betweenthe motion of partons and their spin.Semi-inclusive DIS (SIDIS), where a specified hadron is detected in the final state, is a powerful tool to studythe transverse structure of the nucleon. Spin asymme-tries in polarized SIDIS are directly related to TMDsand fragmentation functions (FFs), and are the subjectof intense theoretical and experimental studies [11–16].Recently, sizable non-vanishing single spin asymmetries(SSAs) have been observed in SIDIS with longitudinallypolarized electron beams and unpolarized targets (in thefollowing referred to as beam SSAs) [17–21]. Since beamSSAs are subleading twist-3 objects, they are expected tobe suppressed by O ( M/Q ), where M is the target massand Q is the photon virtuality. However, with the ener-gies available at existing fixed-target facilities, contribu-tions of the order O ( M/Q ) could be sizable, making suchtwist-3 contributions accessible and thereby providing ac-cess to the information they contain about quark-gluoncorrelations.In this Letter, for the first time, we present beam SSAsand extract the sin φ moment A sin φLU in π + SIDIS of longi-tudinally polarized electrons off unpolarized protons witha wide range of fully differential multidimensional kine-matics in the Q range from 1.7 to 7.0 GeV , x B from0.13 - 0.52, z from 0.18 - 0.7 and P T up to 0.85 GeV (c =1). Here we define the fraction of the proton’s momentumcarried by the struck quark as x B , the energy fraction ofthe incoming lepton carried by the virtual photon as y ,the energy fraction of the virtual photon carried by theoutgoing hadron as z , the transverse momentum of thefinal state hadron as P T and the virtuality of the collisionas Q . The diagram in Fig. 1(a) shows the SIDIS scatter-ing process, including the involved TMDs and FFs, andFig. 1(b) shows the definition of the reaction kinematics. FIG. 1: (a) Schematic diagram of the single pion semi-inclusive deep inelastic scattering process with the involvedparton distributions and fragmentation functions. (b) Defini-tion of the reaction kinematics of single pion SIDIS.
In the one-photon exchange approximation beam SSAsare defined as follows:
SSA ( z, P T , φ, x B , Q ) = dσ + − dσ − dσ + + dσ − = A sin φLU sin φ A cos φUU cos φ + A cos 2 φUU cos 2 φ , (1)where dσ ± is the differential cross section for each beamhelicity state ( ± ). For the positive / negative helicity thespin is parallel / anti-parallel to the beam direction. Thesubscripts of the moments A ij represent the longitudi-nally polarized (L) or unpolarized (U) state of the beamand the target, respectively. φ is the azimuthal anglebetween the electron scattering plane and the hadronicreaction plane (see Fig. 1 (b)).The sin φ moment A sin φLU is proportional to the polar-ized structure function F sin φLU : A sin φLU = (cid:112) (cid:15) (1 − (cid:15) ) F sin φLU F UU,T + (cid:15)F UU,L , (2)where the structure functions in F UU = F UU,T + (cid:15)F UU,L correspond to the longitudinal and transverse polariza-tions of the virtual photon, and (cid:15) is the ratio of theirfluxes.This study focuses on the A sin φLU asymmetry and thuson the structure function F sin φLU , which is related to thequark-gluon-quark correlations in the proton (twist-3).Assuming factorization, it can be expressed as a convo- lution, denoted by C , of TMDs and FFs [8, 22]: F sin φLU = 2 MQ C (cid:34) − ˆ h · k T M h (cid:32) x B eH ⊥ + M h M f ˜ G ⊥ z (cid:33) + ˆ h · P T M (cid:32) x B g ⊥ D + M h M h ⊥ ˜ Ez (cid:33) (cid:35) . (3)Here e is a twist-3 PDF, H ⊥ is the Collins FF, f is anunpolarized distribution function, ˜ G ⊥ is a twist-3 FF, g ⊥ is a twist-3 T-odd distribution function, D is an un-polarized FF, h ⊥ is the Boer-Mulders function and ˜ E is a twist-3 FF. k T is the transverse quark momentum, M h is the pion mass and ˆ h is a unit vector in the direc-tion of the pion. Every term in the structure function F sin φLU is a so-called genuine twist-3 term, e.g. relatedto quark-gluon correlators (or current quark mass terms)[8]. Hence, the often used Wandzura-Wilczeck approxi-mation, which neglects all interaction-dependent parts inthe twist-3 terms in a structure function, is not valid inthis case as it would demand the entire asymmetry to bezero [23], which is not the case.Since the magnitude of the observed asymmetry ofseveral percent cannot be explained by perturbativeQCD, several mechanisms have been proposed to gen-erate such an asymmetry. One mechanism involves the eH ⊥ term [24, 25], which indicates that the asymme-try results from the coupling of the distribution function e ( x ) with the Collins FF H ⊥ . Other mechanisms relateto the convolution of the Boer-Mulders function h ⊥ withthe FF ˜ E and to the coupling between the unpolarizeddistribution function f and the twist-3 FF ˜ G ⊥ . Apartfrom the ones mentioned above, a mechanism involvingthe poorly known twist-3 TMD distribution function g ⊥ can also give rise to the beam SSA. g ⊥ appears in the de-composition of the quark correlator if the dependence onthe light-cone vector is included. In order to model thetwist-3 T-odd chiral-even TMD g ⊥ it is necessary to takeinto account final state interactions, which can be takeninto account by one-gluon exchange. Therefore, studyingbeam SSAs provides a unique opportunity to unravel therole of genuine twist-3 quark-gluon correlation effects.SIDIS π + electroproduction was measured at JeffersonLab with CLAS12 (CEBAF Large Acceptance Spectrom-eter for experiments at 12 GeV) [26]. Beam single spinasymmetries were extracted over a wide range in Q , x B , z , P T and φ . The incident electron beam was longitu-dinally polarized and had an energy of 10.6 GeV. Thetarget was unpolarized liquid hydrogen. The CLAS12forward detector consists of six identical sectors within atoroidal magnetic field. The momentum and the chargeof the particles were determined by 3 regions of driftchambers from the curvature of the particle trajecto-ries in the magnetic field. The electron identificationwas based on a lead-scintillator electromagnetic samplingcalorimeter in combination with a Cherenkov counter.Positive pions were identified by time-of-flight measure-ments. For the selection of deeply inelastic scatteredelectrons, cuts on Q > , y < .
75 and on theinvariant mass of the hadronic final state
W > e (cid:48) π + X missing mass be larger than 1.5 GeV to minimize the con-tribution from exclusive channels.Figure 2 shows the new CLAS12 data as a function of x B , z and P T integrated over all other kinematic variablesin comparison to the available world data for F sin φLU /F UU from previous experiments. Details on the multidimen-sional analysis for CLAS12 follow. Even though F sin φLU FIG. 2: CLAS12 data (filled squares), compared with theavailable world data from HERMES [19] (open squares),COMPASS [20] (open triangles) and CLAS [21] (filled trian-gles) for F sin φLU /F UU as a function of x B , z and P T integratedover all other kinematic variables. It has to be noted thatthe different experiments apply slightly different cuts on thekinematic variables and that in the case of COMPASS all pos-itive hadrons are considered. The A sin φLU values stated in theReferences were transformed to F sin φLU /F UU following Eq. (2).The grey histogram shows the systematic uncertainty of thepresent data. has been studied at HERMES [18, 19], COMPASS [20]and CLAS [21] during the last two decades, there is stillno consistent understanding of the contribution of eachpart to the total structure function. One of the mainreasons for this can be seen in the low statistics and theresulting large uncertainties or limited kinematic cover-age of many previous experiments. The high statisticson an extended kinematic range, which is available withthe new CLAS12 data, enables a fully differential multi-dimensional analysis for the first time and therefore pro-vides an excellent basis for the extraction of TMDs andFFs.For the multidimensional binning, first the electronvariables are sorted in 9 bins in Q and x B (see Fig. 3).For each of these Q - x B bins a binning is applied to z and P T as shown for the example of Q - x B bin 1 in Fig.3. The beam SSA and its statistical uncertainty were de-termined experimentally from the number of counts withpositive and negative helicity ( N ± i ) in a specific bin i as: SSA = 1 P e N + i − N − i N + i + N − i , σ SSA = 1 P e (cid:115) − ( P e SSA ) N + i + N − i , (4) FIG. 3: Left: Correlation between Q and x B . The bin bor-ders are shown as black lines and the bin numbering is given.Right: Correlation between z and P T for Q - x B - bin 1. Theblack lines indicate the bin borders. where P e is the average magnitude of the beam polar-ization. P e was measured with a Møller polarimeter up-stream of CLAS12 and was 86.3% ± φ moment, A sin φLU , the beam SSAwas measured as a function of the azimuthal angle φ .Then the data was fit with a sin φ function. Figure 4shows the beam SSA as a function of φ for two typicalmultidimensional bins. As expected the φ -dependence FIG. 4: Beam SSA as a function of φ for two typical bins (left: Q = 1.98 GeV , x B = 0.20, P T = 0.25 GeV, z = 0.65; right: Q = 6.5 GeV , x B = 0.53, P T = 0.29 GeV, z = 0.44). Thevertical bars show the statistical uncertainty of each point,while the horizontal bars correspond to the bin width. Thered line shows the fit with a sin φ function. can be well described by a sin φ function. The obtained A sin φLU moment is then related to F sin φLU /F UU via Eq. (2).Several sources of systematic uncertainty were investi-gated, including beam polarization, radiative effects, par-ticle identification and contamination from baryon reso-nances and exclusive ρ meson production. A detailedMonte Carlo simulation was performed to study accep-tance and bin-migration effects, which were both foundto be negligible compared to the other contributions. Theinfluence of additional azimuthal modulations cos φ andcos 2 φ on the extracted sin φ amplitude was also evalu-ated, and found to be negligible. The total systematicuncertainty in each bin is defined as the square-root ofthe quadratic sum of the uncertainties from all sources.It is typically on the order of 6.4% and dominated by theuncertainty from radiative effects (3.0%) and from thebeam polarization (3.0%).The structure function ratio F sin φLU /F UU has been ex-tracted for each of the obtained 344 bins. The resultfor each bin, as well as the mean value of the kinematicvariables in each bin, are provided in the supplementalmaterial and in the CLAS physics database [27, 28]. Fig-ure 5 (6) shows the z ( P T ) dependence for selected P T ( z )bins in different bins of Q and x B , which represent thecharacteristics of the different kinematic regions. The re-sults are compared to theoretical predictions, which havebeen calculated based on the models presented in Refs.[29] and [30] (models 1 and 2) and Ref. [31] (model 3).The first two models describe the proton as an activequark plus (inert) spectator scalar and axial-vector di-quarks. Both models include the eH ⊥ and g ⊥ D terms ofthe structure function, while the other terms are assumedto be small. Model 1 uses a complicated propagator forthe axial-vector diquark and the ratio of axial-vector-diquark/scalar-diquark strengths fitted [32] to ZEUS [33]and GRSV01 [34] distribution functions (DFs). Diquarkmasses and various cutoffs are also parameters includedin the fit to the DFs. The fit produces | I = 1 , I z = 1 (cid:105) and | I = 1 , I z = 0 (cid:105) axial-vector diquarks with very dif-ferent masses. Notably, this outcome conflicts with di-rect calculations of diquark masses, in which these cor-relations are typically degenerate [35]. Model 2 uses asimple propagator for axial-vector diquarks and the ratioof axial-vector-diquarks and scalar-diquarks is fixed bySU(4) spin-flavour symmetry. The model also uses massdegenerate axial-vector diquarks. The models differ mostsignificantly in the form of the propagator for the axial-vector diquark (complex versus simple) and the massesof these correlations (different versus degenerate). TheFFs used in both models are described in Ref. [36].Model 3 includes only the eH ⊥ term with the Collinsfunction taken from the parameterization of Ref. [15] and e ( x ) based on the chiral quark soliton model [25, 37, 38].This is the only model predicting the experimentally notmeasurable δ ( x )-contribution in e ( x ) expected in QCDand related to the pion-nucleon sigma term [24].For the experimental data it can be observed that the z dependence changes from a more flat behaviour at small P T , Q , and x B values to a steep increase at large P T , Q , and x B values. Also for the P T dependence a smallmagnitude with a nearly flat behaviour can be observedat small Q , x B , and z values, while for increasing z values a peaking structure with varying mean value andwidth can be observed at small Q and x B , while anincreasing trend becomes dominant at large Q and x B values.The best reproduction of the general trend is providedby model 2. For this model, the comparison of the dif-ferent kinematic regions shows, that while eH ⊥ is thedominant term at small Q and x B values, g ⊥ D be-comes more and more important for the description ofthe behaviour at large Q and x B . Model 1 mixes a FIG. 5: z dependence of F sin φLU /F UU for increasing P T bins(left to right) and for different Q - x B bins (bin 1: Q = 1.71GeV , x B = 0.13, bin 2: Q = 2.02 GeV , x B = 0.19, bin 7: Q = 4.89 GeV , x B = 0.39, bin 9: Q = 6.55 GeV , x B =0.52) . The systematic uncertainty is given by the histogramjust above the horizontal axis. The predictions of the differenttheoretical models are shown by the bold lines (blue: model1, red: model 2, magenta: model 3). For models 1 and 2 thecontribution from eH ⊥ (dashed line) and g ⊥ D (dotted line)are shown in the same color as the final result. complicated axial-vector diquark propagator with a sim-ple proton wave function and uses vastly different massesfor diquarks within the same isospin multiplet. Here itis seen to be challenged by high precision experiments.The simplicity and, hence, internal consistency of model2 is more natural beginning for phenomenological explo-rations. Even though model 3 uses a different approach,it provides results that are very similar to the eH ⊥ termof model 2, which provides an additional support for thismodel and, possibly, points to an important role for axial-vector diquarks in the proton’s wave function.Importantly, these theory-experiment comparisonshighlight the discriminating power of a fully multidimen-sional analysis with high statistics over a wide kinematicrange. Such data provides both the means of verifyingthe reliability of different models and their underlyingassumptions, and the ability to place increasingly tightconstraints on the TMDs involved.Since a fully multidimensional analysis is herein madeavailable for the first time, some issues with model 2 arealso exposed, especially in the finer details. This in-dicates that either the parametrisation of the involvedTMDs and FFs has to be improved or that additionalterms from Eq. (3) besides the two that have been used FIG. 6: P T dependence of F sin φLU /F UU for increasing z bins(left to right) and for different Q - x B bins (bin 1: Q = 1.71GeV , x B = 0.13, bin 2: Q = 2.02 GeV , x B = 0.19, bin 7: Q = 4.89 GeV , x B = 0.39, bin 9: Q = 6.55 GeV , x B =0.52) . The systematic uncertainty is given by the histogramjust above the horizontal axis. The predictions of the differenttheoretical models are shown by the bold lines (blue: model1, red: model 2, magenta: model 3). For models 1 and 2 thecontribution from eH ⊥ (dashed line) and g ⊥ D (dotted line)are shown in the same color as the final result. provide sizable contributions in some kinematic regions.Therefore, including the multidimensional data pointspresented in this work will help to further constrain theTMDs and FFs in global fits.In addition to the z and P T dependence, the x B de-pendence can also provide valuable insights into the kine-matic dependence of the involved TMDs and FFs. Theresult for the x B dependence are shown in Fig. 7. To ob-tain these dependences, the same multidimensional bin-ning is used. Owing to the correlation between x B and Q , the x B dependence is integrated/averaged over Q .Therefore, only discrete points are shown for the theorycalculations. Also as a function of x B a strong kinematicdependence of the behaviour can be observed, with amore flat behaviour at small z and P T and an increasingtrend for larger P T and z values. As for the z and P T dependence, the best agreement is provided by model 2.The x B dependence clearly shows that model 3, whichuses only the eH ⊥ term, provides a sufficient descriptionat small z and P T , but cannot reproduce the trend at thelargest P T and z values.The structure function ratio F sin φLU /F UU correspond-ing to the polarized electron beam single spin asymme-try in semi-inclusive deep inelastic scattering has been FIG. 7: x B dependence of F sin φLU /F UU for selected P T and z bins. The result is integrated over Q . The systematic un-certainty is given by the histogram just above the horizontalaxis. The predictions of the different theoretical models areshown as open symbols (blue triangles: model 1, red squares:model 2, magenta circles: model 3). measured over a wide range of kinematics in a fully mul-tidimensional study for the first time. The comparisonwith calculations allows a clear differentiation betweencompeting reaction models, e.g. highlighting the impor-tance of the poorly known T-odd chiral-even TMD g ⊥ atlarge P T and z , while providing new empirical informa-tion in support of an important role for axial-vector di-quark correlations in the proton’s wave function. There-fore, including this multidimensional measurement intoglobal fits, in combination with future measurements ofunpolarized cross sections, as well as polarized target spinasymmetries, will provide new, strong constraints on theparticipating TMDs and FFs. Such progress will set usfirmly on the path to a deeper understanding of nucleonstructure in the 3-D space most natural to picturing com-posite objects in relativistic quantum field theory.We acknowledge the outstanding efforts of the staffof the Accelerator and the Physics Divisions at Jeffer-son Lab in making this experiment possible. We owemuch gratitude to P. Schweitzer for many fruitful discus-sions concerning the interpretation of our results. 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