Azimuthal asymmetries in SIDIS di-hadron muoproduction off longitudinally polarized protons at COMPASS
AAzimuthal asymmetries in SIDIS di-hadron muoproduction off longitudinallypolarized protons at COMPASS
Stefan Sirtl ∗ Physics Department, Albert-Ludwigs-University, Freiburg 79104, Germany † (On Behalf of the COMPASS Collaboration) (Dated: October 8, 2018)In this review a first comprehensive study of azimuthal asymmetries in semi-inclusive deep inelasticmuoproduction of hadron pairs off longitudinally polarized protons at COMPASS is presented. Thestudy is based on data taken in 2007 and 2011, obtained by impinging a high-energetic µ + beam of160 GeV/ c , respectively 200 GeV/ c , momentum on a solid ammonia target. The discussion is focusedon both leading and subleading longitudinal target-spin-dependent asymmetries arising in the di-hadron SIDIS cross section, addressing the role of spin-orbit couplings and quark-gluon correlationsin the framework of collinear or transverse momentum dependent factorization. I. INTRODUCTION
Azimuthal cross section asymmetries in semi-inclusivedeep inelastic scattering (SIDIS) of polarized leptons offpolarized nucleons are key observables to investigate thespin dependent substructure of the nucleon. Assumingfactorization, azimuthal asymmetries can be theoreti-cally connected to combinations of parton distributionfunctions (PDFs) and fragmentation functions (FFs),encoding information about the partonic substructureof the nucleon and the fragmentation mechanism,respectively. In this review we present a first study ofazimuthal target-spin-dependent asymmetries, arising inthe di-hadron SIDIS cross section, measured on longitu-dinally polarized protons at COMPASS. This includesthe measurement of a set of nine azimuthal asymmetries,appearing in a transverse-momentum-dependent (TMD)approach at leading-twist, which can be consequentlyrelated to spin-orbit couplings. In particular, theseasymmetries are sensitive to the TMD PDFs g L and h L , describing the helicity distribution, respectivelythe distribution of transversely polarized quarks in alongitudinally polarized proton.Moreover, also azimuthal modulations at subleading-twist are considered within this work, which survivethe integration over quark transverse momenta. Mea-suring respective collinear cross section asymmetries atsubleading-twist can provide new understanding of so farunresolved quark-gluon correlation mechanisms. In par-ticular, such results can help to access the yet unknowncollinear PDFs h L and e L . Our results are characterizedby an unprecedented precision, covering a wide kinematicrange. For more comprehensive information on this anal-ysis the reader may be referred to Ref. [1]. A similarstudy has been already presented by the CLAS collabo-ration, focused on collinear asymmetries[2]. ∗ The author acknowledges financial support by the German Bun-desministerium f¨ur Bildung und Forschung (BMBF) † [email protected] Figure 1. Sketch of the considered two di-hadron SIDIS pro-cess, including the relevant azimuthal angles. The nucleon isassumed to be longitudinally polarized either along or againstthe direction of the incoming lepton.
II. THEORETICAL FRAMEWORK
This work considers the di-hadron SIDIS process µ ( l ) + p ( P ) → µ ( l (cid:48) ) + h ( P ) + h ( P ) + X, (1)where a beam muon µ probes a target proton p withmass M via the exchange of a virtual photon. Thecorresponding four-momenta are given in parenthesisin the above formula. The struck quark subsequentlyfragments into two unpolarized final state hadrons h and h and any, not necessarily detected, rest X in thefinal state. In particular, this work considers SIDISof longitudinally polarized muons off longitudinallypolarized protons, producing hadron pairs of oppositecharge. In order to keep orientations well defined, h is defined to be the positively and h the negativelycharged hadron.The di-hadron cross-section is modulated in two az-imuthal angles, φ h and φ R [3–5]. As sketched in Fig. 1 a r X i v : . [ h e p - e x ] F e b they are both enclosed by the scattering plane, spreadby the incoming lepton and the virtual photon direction,and a hadronic plane, spread by the virtual photon di-rection and either P h = P + P or R = 12 ( P − P ) . (2)The azimuthal angles can be consequently calculated via φ h = ( q × l ) · P h | ( q × l ) · P h | arccos (cid:18) ( q × l ) · ( q × P h ) | q × l | · | q × P h | (cid:19) (3) φ R = ( q × l ) · R ⊥ | ( q × l ) · R ⊥ | arccos (cid:18) ( q × l ) · ( q × R ⊥ ) | q × l | · | q × R ⊥ | (cid:19) . (4)where the bold variables indicate corresponding mo-menta. Here, R ⊥ describes the transverse component of R with respect to the virtual photon. It is calculatedfrom R ⊥ = z P ⊥ − z P ⊥ z + z (5)in order to ensure the invariance of φ R against boosts inthe direction of the virtual photon, where z / = E / /ν is the energy fraction of a hadron with respect to thevirtual photon energy. This definition of R ⊥ coincideswith the general one up to corrections of order 1 /Q [4, 6], where Q = − ( l − l (cid:48) ) .Asymmetries are defined as ratios of structure func-tions A m ( φ h ,φ R ) XY = F m ( φ h ,φ R ) XY F UU,T + εF UU,L , (6) where the subscripts indicate the polarization of thebeam (X) and the target (Y), here either unpolarized(U) or longitudinally polarized (L). The third subscriptrefers to longitudinally (L) or transversely (T) polarizedvirtual photons. The ratio of the corresponding photonfluxes is given by ε = 1 − y − γ y − y + y + γ y , (7)where y = νE is the fractional energy of the virtualphoton and γ = MxQ . The superscript m ( φ h , φ R ) inEq.(6) indicates the respective azimuthal modulation.In a TMD approach, i.e. when taking into accounttransverse momenta of quarks p T , a set of seven az-imuthal single spin asymmetries (SSAs) and two doublespin asymmetries (DSAs) can be measured at leadingtwist. They are sensitive to p T -dependent convolutionsof TMD PDFs, in particular the helicity distribution g L ( x, p T ) or the still unknown Boer-Mulders function h L ( x, p T ), coming with FFs. Detailed formulas can befound in Ref. [1].Considering the di-hadron cross-section in a collinearapproach, two longitudinal target spin asymmetries ariseat subleading twist: A sin( φ R ) UL = − MQ | R | M h (cid:80) q e q (cid:104) xh qL ( x ) H ∠ q,sp ( z, M h ) + M h Mz g q ( x ) ˜ G ∠ q,sp ( z, M h ) (cid:105)(cid:80) q e q f q ( x ) D q,ss + pp ( z, M h ) (8) A cos( φ R ) LL = MQ | R | M h (cid:80) q e q (cid:104) xe qL ( x ) H ∠ q,sp ( z, M h ) − M h Mz g q ( x ) ˜ D ∠ q,sp ( z, M h ) (cid:105)(cid:80) q e q f q ( x ) D q,ss + pp ( z, M h ) . (9)Their interpretation in the framework of the partonmodel involve flavor sums of simple products of PDFs andFFs, in particular interference FFs ( ∠ ), whereas the su-perscripts s and p indicate the contributing partial wavecharacteristics. The electric charge of a particular fla-vor q is denoted by e q . Assuming Wandzura-Wilzcekapproximation, the genuine twist-3 terms marked with atilde can be neglected, leaving a pure sensitivity to therespective leading products. Measuring these asymme-tries, and including recent results for the interference FF H ∠ from BELLE [7], hence provides a clean way to ac- cess the still unknown twist-3 PDF h L ( x ), respectively e L ( x ). The first can be interpreted as the distribution oftransversely polarized quarks in a nucleon with longitu-dinal spin orientation, which is among the missing puzzlepieces to complete the one-dimensional picture of the pro-ton at subleading twist. Assuming the gauge-link to bethe only source of t-odd behavour, the PDF e L ( x ) andwith it the respective asymmetry should vanish. Measur-ing these asymmetries can hence provide further insightinto Q -suppressed spin dependent mechanisms and serveto corroborate common theoretical assumptions. III. DATA ANALYSIS
This work comprises the analysis of combined data,obtained by scattering naturally polarized µ + with anominal momentum of 160 GeV/ c during a dedicateddata taking in 2007, respectively of 200 GeV/ c in 2011,off a longitudinally polarized solid state NH target. Apriori the Q -evolution and the kinematic dependencesof the considered asymmetries are unknown. Still, fromgeneral considerations, these kind of effects are expectedto be small or negligible within experimental accuracy.Hence, we find it reasonable to merge both data sets,although different beam energies were used.The standard COMPASS DIS cuts were applied. Inparticular was the four-momentum transfer limited to Q > c ) , the fractional energy transfer of themuon set to 0 . < y < . W > c . To matchCOMPASS kinematics, the Bjorken variable was limitedto 0 . < x < .
7. Per selected event, all possible com-binations of hadron pairs were included in the analysis.The fractional energy for each hadron was required tobe z / > . x F , / > . E miss = ( P + q − P h ) − q M = M X − M M , (10)was required to fulfill E miss > M and M X stand for the mass of the proton, respectively themass of the undetected recoiling system. Finally, a cut R T > .
07 was applied, to ensure the well-definition ofthe corresponding hadronic plane, hence the angle φ R .A further remark should be given concerning thepolarization of the target. Since it is practically po-larized along beam direction, there enters a transversespin contribution when considering the frame where thez-axis points along the direction of the virtual photon. Inthis analysis, this contribution of transverse polarizationcomponents along the photon axis is neglected due to itsstrong suppression in COMPASS kinematics.All azimuthal asymmetries are extracted in bins of x , z = z + z and the invariant mass M inv , including acorrection per kinematic bin regarding the beam polar-ization, the target polarization, the dilution of the target,as well as for respective depolarization factors. IV. RESULTS
Our results for the asymmetries arising at leadingtwist are shown in Fig. 3 and Fig. 4, where the statisticalerrors are represented by the error bars and the system-atic uncertainties are indicated by color bands on the æ A Æ - - UL ) R f - h f sin( A UL ) R f -2 h f sin(2 A UL ) h f sin(2 A UL ) R f + h f sin( A UL ) R f sin(2 A UL ) R f - h f sin(3 A UL ) R f -2 h f sin(4 A LL ) R f - h f cos( A LL ) R f - h f cos(2 A UL ) R f sin( A LL ) R f cos( A COMPASSpreliminary
Figure 2. Measured integrated azimuthal asymmetries arisingin the di-hadron cross-section up to subleading twist, consid-ering scattering off longitudinally polarized protons. Shownare the mean values when integrating over the entire kine-matic range. The upper nine values correspond to asymme-tries arising in a TMD approach at leading twist while the lasttwo refer to the asymmetries at subleading twist in a collinearapproach. bottom of each plot. No eminent kinematic dependenceis observed on any of the considered variables. Theasymmetries are found to be quite narrowly distributedaround zero over the entire kinematic ranges.Fig. 5 shows our results for the two asymmetries atsubleading twist. The single spin asymmetry A sin( φ R ) UL isfound to be clearly positive within experimental preci-sion, averaging A sin( φ R ) UL = 0 . ± . ± . . (11)This measurement confirms non-zero results from CLAS,measured in the high x -region. As already motivated inSec. II the presented results can serve to access the stillunknown PDF h L ( x ).The double spin asymmetry A cos( φ R ) LL was found to av-erage A cos( φ R ) LL = − . ± . ± . . (12)The fact, that this asymmetry is found to be smallwithin the experimental precision could consequentlycorroborate the Wandzura-Wilzcek assumption of negli-gible quark-gluon correlations on the fragmentation side, x - - U L ) R f - h f s i n ( A - z ] c [GeV/ inv M U L ) R f - h f s i n ( A - U L ) R f s i n ( A - U L ) R f + h f s i n ( A - U L ) h f s i n ( A - U L ) R f - h f s i n ( A - U L ) R f - h f s i n ( A - COMPASSpreliminary
Figure 3. Results for azimuthal SSAs arising in the TMD di-hadron cross-section at leading twist, considering scattering offlongitudinally polarized protons. x - - LL ) R f - h f c o s ( A - z ] c [GeV/ inv M LL ) R f - h f c o s ( A - COMPASSpreliminary
Figure 4. Results for azimuthal DSAs arising in the TMD di-hadron cross-section at leading twist, considering scattering offlongitudinally polarized protons. x - - LL ) R f c o s ( A - - z ] c [GeV/ inv M U L ) R f s i n ( A - COMPASSpreliminary
Figure 5. Results for azimuthal asymmetries arising in the collinear di-hadron cross-section at subleading twist, consideringscattering off longitudinally polarized protons. here encoded in the pure twist-3 interference FF ˜ D ∠ .The whole set of mean asymmetries measured withinthis work is shown in Fig. 2.Summarizing the presented analysis, the obtained results provide an abundance of new information on spinrelated mechanisms inside hadrons and fragmentationprocesses. They can serve as valuable input for globalanalyses of PDFs and FFs as well as for the validationof theoretical model approaches. [1] S. Sirtl, Azimuthal asymmetries in semi-inclusive deep-inelastic hadron muoproduction on longitudinally polarizedprotons , Ph.D. thesis, University of Freiburg (2016).[2] S. A. Pereira, PoS (DIS2014) , 231 (2014).[3] A. Bacchetta and M. Radici, Phys. Rev. D , 094002(2003), hep-ph/0212300.[4] S. Gliske, A. Bacchetta, and M. Radici, Phys. Rev. D , 114027 (2014), 1408.5721.[5] A. Bacchetta and M. Radici, Phys.Rev. D69 , 074026(2004), arXiv:hep-ph/0311173 [hep-ph].[6] A. Kotzinian et al. , Proceedings of Transversity 2014(2014), 10.1051/epjconf/20158502026, 1407.6572.[7] The BELLE Collaboration, Phys. Rev. Lett.107