The Electron Ion Collider (EIC)

7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 1

Abhay Deshpande

Lecture 2: What do we need polarized beams? Nucleon Spin Structure: What’s the problem? 2. Quantum Chromodynamics: The Fundamental Description of the Heart of Visible Matter

Sidebar 2.6: Nucleon Spin: So Simple and Yet So Complex The simple fact that the proton carries spin 1/2, quark spins to account for most of the proton’s spin measured in units of Planck’s famous constant, is but were quite surprised when experiments at CERN exploited daily in thousands of magnetic resonance and other laboratories showed that the spins of all imaging images worldwide. Because the proton is a quarks and antiquarks combine to account for no more composite system, its spin is generated from its quark than about 30% of the total. This result has led nuclear 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT and gluon constituents. Physicists’ evolving appreciation scientists2. Quantum to address Chromodynamics: the more daunting The Fundamental challenges2 Description of the Heart of Visible Matter of how the spin might be generated, and of how much involved in measuring the other possible contributions to we have yet to understand about it, is an illustrative the spin illustrated in Figure 1. The gluons also have an Thecase study nucleon of how seemingly simple spin properties of visible intrinsic spin (1 unit) and might be “polarized” (i.e., might matter emerge from complex QCD interactions. have a preferential orientation of their spins along or puzzle…. opposite the proton spin). And just as the earth orbits Quarks are also spin 1/2 particles, and at any given around the sun while simultaneously spinning about its moment an individual quark may spin in the same or own axis, the quarks and gluons in a proton could have Stheince opposite direction1988.. of its proton host, respectively, orbital motion that would also contribute to the overall making a positive or negative contribution to the total proton spin. proton spin. Physicists originally expected the sum of

Figure 2.15: Te increased coverage (colored bands) over existing experiments (point symbols) that EIC polarized electron-proton collisions will provide in parton momentum fraction and resolving power.

Figure 1: A schematic view of the proton and its potential spin contributions. Figure 2.14: Te projected reduction in the uncertainties of the net gluon and net quark spin contributions to the proton’s spin for EIC electron So how much of the spin comes from the spin of gluons? xmin indicated on the horizontal axis. For each dataset, The first important constraints have come from recent the uncertaintiesenergies of 5 (red) grow and 20 significantly (yellow) GeV. at low x as the 1 1 min measurements at RHIC, which provides the world’s contributions have not been directly measured there. first and= only polarized⌃ proton+ colliderG +capabilityLQ at + LG Tomographic Images of the Proton high2 energy.2 There, the STAR and PHENIX experiments In a significantDeep inelastic breakthrough, scattering (DIS)the RHIC experiments results tocarried date out at EIC collision rates will provide for the first time 3D have used the polarized quarks in one proton as a (light blue band in Figure 2) indicate that the gluon spins ? ? images of gluons in the proton’s internal landscape. Of scattering probe to reveal that the gluons in the other do indeed have a non-negligible orientation preference for xparticular above 5%. interest But they are tell exclusive us very measurements, little about whether where proton are indeed polarized. Just as critical has been the one detects an outgoing meson in coincidence with the development of the theoretical framework to integrate the much more abundant lower-momentum gluons may scattered electron with sufcient resolution to confirm the RHIC measurements into a global QCD analysis to reinforce or counterbalance this preference. This leaves that the proton has been left intact by the scattering constrain both quark and gluon spin preferences. Results a large uncertainty in the total gluon spin contribution, process. For example, the detection of exclusive J/ of those analyses are shown in Figure 2. indicated at the left edge of the plot, which can be reducedmeson by productionanalysis of wouldanticipated provide RHIC unprecedented polarized maps The protons at RHIC move at nearly the speed of light, proton(Figure data 2.17)(the showingdarker blue how band). the gluons However, are distributed this still and each quark and gluon inside carries a fraction x wouldin spaceleave thewithin overall a plane uncertainty perpendicular at a level to the that parent proton’s motion. These particular maps encode vital of the proton’s overall momentum. The widths of the remains larger than the entire proton spin itself. Only Figure 2.16: A simulation based on projected EIC data of the transverse colored bands in Figure 2 illustrate the uncertainties in an EICinformation, (yellow band) inaccessible can uniquely without settle the howEIC, onmuch the of amount motion preferences of an up sea quark within a proton moving out of the summed spin of all gluons that carry more than a the overallof proton spin spin is contributedassociated withby the the spins gluons’ of quarks,orbital motion. the page, with its spin pointing upward. Te color code indicates the probability of fnding the up quarks. fraction xmin of the proton’s momentum, with the value of antiquarks, and gluons combined. Proton Spin at the EIC and Lattice QCD with such detailed EIC measurements of proton spin The ability of LQCD calculations to reproduce many structure as the images and distributions in Figures 2.16 features of the hadron spectrum is a testimony to striking 36 and 2.17. Such comparisons will not only bring deep recent advances in our treatment of quark and gluon insight into the origin of the spin but will also shed light interactions from first principles. An important recent on the role the abundant gluons play in generating the breakthrough in LQCD methodology now provides the proton’s mass and confining quarks and gluons inside promise of precision future comparisons of theory the proton.

34 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 3

A brief history of nucleon spin: “Crisis” à “Puzzle” 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 4

Some equations… Assume only γ* exchange

• Lepton Nucleon Cross Section Nucleon spin

Lepton spin

• Lepton tensor Lµν affects the kinematics (QED) • Ηadronic tensor Wµν has information about the hadron structure 5332 D. ADAMS et al. 56

target. Both inclusive and semi-inclusive data were obtained, and polarized H and D targets will be used in the future. In this paper, we present SMC results on the spin- p p dependent structure functions g1 and g2 of the proton, obtained from data taken in 1993 with a polarized butanol target. First results from these measurements were published in Refs. ⌦9, 10. We use here the same data sample, but present a more refined analysis; in particular, the influence of the radiative corrections on the statistical error on the asymmetry is now properly taken into account, resulting in an observable increase of this error at small x, and we allow for a Q2 p evolution of the g1 structure function as predicted by pertur- FIG. 1. Lepton and nucleon kinematic variables in polarized bative QCD. SMC has also published results on the deuteron lepton scattering on a fixed polarized nucleon target. d structure function g1 ⌦11–13 and on a measurement of semi-inclusive cross section asymmetries ⌦14. For a test of The spin-dependent part of the cross section can be writ- d the Bjorken sum rule, we refer to our measurement of g1 . ten in terms of two structure functions g1 and g2 which The paper is organized as follows. In Sec. II we review describe the interaction of lepton and hadron currents. When the theoretical background. The experimental setup and the the lepton spin and the nucleon spin form an angle , it can data-taking procedure are described in Sec. III. In Sec. IV we be expressed as ⌦16 discuss the analysis of cross section asymmetries, and in Sec. ⇤ ⌅ V we give the evaluation of the spin-dependent structure ⌅ cos ⌅⇥ sin cos ⌅' , ⇥2.4⇧ function g p and its first moment. The results for g p are dis- 1 2 where is the azimuthal angle between the scattering plane cussed in Sec. VI. In Sec. VII we combine proton and deu- n and the spin plane ⇥Fig. 1⇧. teron results to determine the structure function g1 of the The cross sections ⌅ and ⌅ refer to the two configu- 5332 D. ADAMS et al. 56 ⇥ ' 5332 D. ADAMS et al. neutron and to test the Bjorken56 sum rule. In Sec. VIII we rations where the nucleon spin is ⇥anti⇧parallel or orthogonal 5332 D. ADAMS et al.interpret our results in terms of the spin structure of the pro- to the lepton56 spin; ⌅ is the difference between the cross ton. Finally, we present our conclusions in Sec. IX. ⇥ target. Bothtarget. inclusive Both inclusive and semi-inclusive and semi-inclusive data were data obtained, were obtained, sections for antiparallel and parallel spin orientations and and polarizedand polarized H and D H targets and D targets will be will used be in used the in future. the future. ⌅'⇤⇥hl⌅T /cos the difference between the cross sec- target. Both inclusive and semi-inclusive data were obtained, II. THEORETICAL OVERVIEW tions at angles and ⌅⌃. The corresponding differential In thisIn paper, this paper, we present we present SMC SMC results results on the on spin- the spin- and polarized H and Dp targetsp will be used in the future. cross sections are given by dependent structure functionsp g1 pand g2 of the proton, ob- A. Cross sections for polarized lepton-nucleon scattering dependent structure functions g1 and g2 of the proton, obtained fromIn this data taken paper, in 1993 we with present a polarized SMC butanol results tar- on the spin- The polarized deep-inelastic lepton-nucleon inclusive d2⌅ 16⌃⌥2y y 2y 2 2y tained from data taken in 1993 with a polarized butanol tar- ⇥ get. First results from these measurements werep publishedp in scattering cross section in the one-photon-exchange approxi- 2 ⇤ 4 1⇥ ⇥ g1⇥ g2 get. First resultsdependent from these structure measurements functions wereg published1 and g in2 of the proton, ob- dxdQ Q ⇧⌅ 2 4 ⇤ 2 Refs. ⌦9, 10. We use here the same data sample, but present mation can be written as the sum of a spin-independent term ⇥2.5⇧ Refs. ⌦9,a 10 moretained. We refined use from here analysis; data the same taken in particular, data in sample, 1993 the with but influence present a polarized of the butanol tar- ¯⌅ and a spin-dependent term ⌅ and involves the lepton a moreradiative refinedget. Firstanalysis; corrections results in on particular, the from statistical these the error influencemeasurements on the of asymmetry the were published in helicity hl⇤1: and radiative corrections on the statistical error on the asymmetry is nowRefs. properly⌦9, 10 taken. We into use account, here the resulting same in data an observ- sample, but present ⌅⇤¯⌅⇥ 1 h ⇤⌅. ⇥2.1⇧ 3 2 2 2 is now properly taken into account, resulting in an observ- 2 2 l d ⌅T 8⌥ y y y ablea increase more refinedof this error analysis; at small x in, and particular, we allow for the a Q influence of the 2 ⇤⇥cos 4 A1⇥y⇥ g1⌅g2 . able increase of this errorp at small x, and we allow for a Q2 dxdQ d Q 4 ⌅ 2 ⇤ evolution of the g1 structure function as predicted by pertur- FIG. 1. Lepton and nucleonFor longitudinally kinematic variables polarized in leptons polarized the spin Sl is along the radiativep corrections on the statistical error on the asymmetry ⇥2.6⇧ evolutionbative of the QCD.g1 structure SMC has function also published as predicted results by on pertur-the deuteron FIG.lepton 1. scatteringLepton and on nucleona fixed polarizedlepton kinematic momentum nucleon variables target.k. The in polarized spin-independent cross section for is now properlyd taken into account, resulting in an observ- bative QCD.structure SMC function has alsog published⌦11–13 resultsand on on a the measurement deuteron oflepton scattering on a fixed polarizedparity-conserving nucleon target. interactions can be expressed in terms of For a high beam energy E, is small since either x is small 1 2 two unpolarized structure functions F and F . These func- 2 able increased of this error at small x, and we allow for a Q 1 2 or Q high. The structure function g1 is therefore best mea- structuresemi-inclusive function g1 cross⌦11–13 section and asymmetries on a measurement⌦14. For a of test of The spin-dependent part of the cross section can be writ- 2 p d tions depend on the four-momentum transfer squared Q and sured in the ⇥anti⇧parallel configuration where it dominates Theten spin-dependent in terms of two part structure of the cross functions sectiong canand be2 g writ-which semi-inclusivetheevolution Bjorken cross sum section of rule, the asymmetries weg1 referstructure to our⌦14 measurement function. For a test as of of predictedg1 . by pertur- FIG.the 1. scaling Lepton variable and1 x⇤ nucleonQ /22M↵, kinematic where ↵ is the variables energy of inthe polarized spin-dependent cross section; g is best obtained from a d describe the interaction of lepton and hadron currents. When 2 the BjorkenThebative sum paper rule, QCD. is we organized refer SMC to has as our follows. also measurement published In Sec. of IIg we results1 . review onten the in terms deuteron of two structureleptonthe scattering functions exchanged ong virtual1 aand fixed photong2 polarizedwhich and M is nucleon the nucleon target. mass. measurement in the orthogonal configuration, combined with the lepton spin and the nucleonThe double-differential spin form an angle cross, section it can can be written as a The paperthe theoretical is organized background. as follows. Thed In experimental Sec. II we setup review and thedescribe the interaction of lepton and hadron currents. When a measurement of g1 . In all formulas used in this article, we structure function g1 ⌦11–13 and on a measurement of 2 the theoreticaldata-taking background. procedure The are described experimental in Sec. setup III. In and Sec. the IV wethe leptonbe expressed spin and as the⌦16 nucleon function spin form of x and anQ angle⌦15: , it can consider only the single-virtual-photon exchange. The inter- discusssemi-inclusive the analysis of cross cross section section asymmetries, asymmetries and in⌦ Sec.14. For a test of The spin-dependent part of the cross section canference be writ- effects between virtual Z0 and photon exchange in data-taking procedure are described in Sec. III. In Sec. IV we be7/18/16 expressed as ⌦16 EIC Lecture2 2 at NNPSS2 2016 at MIT 2 6 d ⌅⇤cos ⌅ ⌅sin dcos¯⌅ 4⌅⌃⌥, ⇥2.42m⇧ l V we give the evaluation of the spin-dependent structure ten in⇥ terms of two' structure2 functions2 g1 and gdeep-inelastic2 which muon scattering have been measured ⌦17 and discuss thethe analysis Bjorken of cross sum section rule, asymmetries, we refer to and our in Sec. measurement of g1 . 2 ⇤ 4 xy 1⇥ 2 F1⇥x,Q ⇧ function g p and its first moment. The results for g p are dis- ⌅⇤cos ⌅ ⌅sin cosdxdQ ⌅ Q, x ⇧ ⇥⌅2.4⇧ Q ⇤ found to be small and compatible with the standard model V we give theThe evaluation1 paper is of organized the spin-dependent as follows. structure2 In Sec. IIwhere we review is the azimuthaldescribe⇥ angle the between interaction the' scattering of lepton plane and hadron currents.expectations. When They can be neglected in the kinematic range cussedp in Sec. VI. In Sec. VII we combine protonp and deu- 2y 2 function g theand theoretical its first moment. background. The results The for g experimentalare dis- setupand andthe spin the plane the⇥Fig. lepton 1⇧. spin and the nucleon spin form2 an angle of, current it can experiments. 1 2 n Polarized lepton-nucleon⌅ cross1⇥y⇥ sectionF2⇥x,Q ⇧ , … ⇥2.2⇧ teron results5332 to determine the structure function gD.1 ADAMSof theetwhere al. is the azimuthal angle between56 the scattering⌅ plane4 ⇤ cussed in Sec.data-taking VI. In Sec. procedure VII we combine are described proton and in deu- Sec. III. In Sec.The IV cross we sectionsbe expressed⌅⇥ and ⌅' asrefer⌦16 to the two configu- neutron and to test the Bjorken sum rule. Inn Sec. VIII weand the spin plane ⇥Fig. 1⇧. B. Cross section asymmetries teron results to determinetarget. Both the inclusive structure and semi-inclusive function datag wereof obtained, the rations where the nucleon spin is ⇥anti⇧parallel or orthogonal discuss the analysis of cross section1 asymmetries, and in Sec. ⌅ where⌅ ml is the lepton mass, y⇤↵/E in the laboratory sys- interpret our resultsand polarized in terms H and D of targets the will spin be structureused in the future. of the pro- Theto cross the lepton sections spin; ⇥⌅andis the' differencerefer to the between two configu- the cross The spin-dependent cross section terms, Eqs. ⇥2.5⇧ and neutron and to test the Bjorken sum rule. In Sec. VIII we ⇥ tem, and⌅⇤cos ⌅⇥⌅sin cos ⌅' , ⇥2.4⇧ ton.V Finally, we give weIn present this the paper, evaluation our we conclusions present SMC of results in the Sec. on spin-dependent IX. the spin- rations where structure the nucleon spin is ⇥anti⇧parallel or orthogonal ⇥2.6⇧, make only a small contribution to the total deep- interpret our results independent terms structure of the functions spin structureg p and g p of of the the proton, pro- ob- sections for antiparallel and parallel spin orientations and p 1 2 to thep lepton spin; ⌅ is the difference between the cross inelastic scattering cross section and furthermore their con- ton. Finally,function we presenttainedg1 our fromand conclusions data its taken first in 1993 moment. in with Sec. a polarized IX. The butanol results tar- for g⌅2 'are⇤⇥ dis-hl⌅T /cos⇥ the difference between the2Mx crossA sec-Q2 get.II. First THEORETICAL results from these measurements OVERVIEW were published in sections for antiparallelwhere and parallel is the spin azimuthal orientations⇤ angle⇤ and between. the scattering⇥2.3⇧ tribution plane is, in general, reduced by incomplete beam and tar- cussed in Sec. VI. In Sec. VII we combine protontions and at deu- angles and ⌅⌃. The corresponding differential2 Refs. ⌦9, 10. We use here the same data sample, but present ⌅ ⇤⇥h ⌅ /cos andthe difference the spin between plane ⇥ theFig. crossA 1Q⇧. sec-↵ get polarizations. Therefore they can best be determined A. Cross sectionsa more refined for polarized analysis; in lepton-nucleon particular, the influence scattering of the ' crossn sectionsl T are given by teronII. THEORETICAL resultsradiative corrections to determine OVERVIEW on the statistical the error structure on the asymmetry function g1 of the tions at angles and ⌅⌃The. The cross corresponding sections differential⌅⇥ and ⌅' refer to the two configu- is now properly taken into account, resulting in an observ- 2 2 2 2 2 Theneutron polarized and deep-inelasticto test the Bjorken lepton-nucleon sum rule. inclusive2 In Sec. VIIId we⌅ 16⌃⌥ y y y For highy energy is small A. Cross sections forable polarized increase of this lepton-nucleon error at small x, and scattering we allow for a Q cross sections are⇥ givenrations by where the nucleon spin is ⇥antiγ ⇧parallel or orthogonal scattering cross section inp the one-photon-exchange approxi- 2 ⇤ 4 1⇥ ⇥ g1⇥ g2 interpretevolution our results of the g1 structure in terms function of as predicted the spin by pertur- structureFIG. 1. of Lepton thedxdQ and pro- nucleon kinematicQ variables⇧⌅ in polarized2 4 ⇤ 2 Themation polarized can be deep-inelasticbative written QCD. as SMC the has lepton-nucleon sum also published of a spin-independent results on inclusive the deuteron termlepton scatteringd2 on⌅ a fixed16 polarized⌃⌥2 nucleonyto the target. leptony 2y 2 spin; 2⌅y ⇥ is the difference between the cross ton. Finally,structure we function presentgd ⌦11–13 our and conclusions on a measurement of in Sec. IX. ⇥ ⇥2.5⇧ ¯⌅ and a spin-dependent term1 ⌅ and involves the lepton 2 ⇤ 4 sections1⇥ ⇥ for antiparallelg1⇥ g2 and parallel spin orientations and scattering cross sectionsemi-inclusive in the one-photon-exchange cross section asymmetries ⌦14. approxi- For a test of The spin-dependentdxdQ part ofQ the cross⇧⌅ section can2 be writ-4 ⇤ 2 d ten in terms of two structure functions g and g which mation canhelicity be writtenhl⇤the1: as Bjorken the sum rule, of we a referspin-independent to our measurement of termg1 . and ⌅'1 ⇤⇥2hl⌅T /cos the difference between the cross sec- The paper is organized as follows. In Sec. II we review describe the interaction of lepton and hadron currents. When ⇥2.5⇧ ¯⌅ and a spin-dependent termII. THEORETICAL⌅ and involves the OVERVIEW lepton the lepton spin and the nucleon spin form an angle , it can the theoretical background.⇤¯⇥ 1 The experimental setup and the 3 tions2 at angles and2 2 ⌅⌃. The corresponding differential data-taking procedure⌅ ⌅ are2 describedhl⇤⌅. in Sec. III. In Sec. IV we ⇥2.1be⇧ expressedd as ⌦16⌅ T 8⌥ y y y helicity hl⇤1: and cross sections are given by A. Crossdiscuss sections the analysis of for cross polarized section asymmetries, lepton-nucleon and in Sec. scattering2 ⇤⇥cos 4 A1⇥y⇥ g1⌅g2 . dxdQ⌅⇤cosd ⌅⇥⌅sin cos Q⌅' , ⇥2.4⇧ 4 ⌅ 2 ⇤ V we give the evaluation of the spin-dependent structure For longitudinally polarized1 leptons the spin Sl is along the function⌅⇤¯⌅g p⇥and itsh first⇤⌅ moment.. The results for g p ⇥are2.1 dis-⇧ 3 2 2 2 1 2 l 2 whered is the⌅T azimuthal angle between8⌥ they scattering2 plane y y2 ⇥2.6⇧ 2 2 2 lepton momentumThecussed polarized ink. Sec. The VI. spin-independent In deep-inelastic Sec. VII we combine cross proton lepton-nucleon and section deu- for inclusive⇤⇥cos d 1⌅⇥⇥y⇥ 16⌃⌥ yg ⌅g . y y y n and the spin2 plane ⇥Fig. 1⇧. 4 A 1 2 parity-conservingteron results interactions to determine can the be structure expressed function g in1 of terms the ofdxdQ d Q ⇤ 4 ⌅ 2 1⇥⇤ ⇥ g ⇥ g scattering cross section in the one-photon-exchangeThe crossFor sections approxi- a high⌅ beam⇥ and ⌅ energy' refer to theE, two is configu- small2 since4 either x is small 1 2 For longitudinally polarizedneutron and leptons to test the theBjorken spin sumS rule.l is In along Sec. VIII the we dxdQ Q ⇧⌅ 2 4 ⇤ 2 two unpolarized structure functions F1 and F2 . These func-rations whereor Q the2 nucleonhigh. spin The is structure⇥anti⇧parallel functionor orthogonalg is therefore best mea- lepton momentummationk can.interpret The be spin-independent our written results in terms as of the the cross spin sum structure section of of a the spin-independent for pro- to the lepton spin; term⌅ is the difference between the cross 1 ⇥2.6⇧ 2 ⇥ ⇥2.5⇧ tions depend onton. the Finally, four-momentum we present our conclusions transfer in Sec. squared IX. Q andsectionssured for antiparallel in the and⇥anti parallel⇧parallel spin orientations configuration and where it dominates parity-conserving¯⌅ and interactions a spin-dependent can2 be expressed term in⌅ termsand of involvesFor a the high lepton beam energy E, is small since either x is small the scaling variable x⇤Q /2M↵, where ↵ is the energy of⌅'⇤⇥thehl⌅ spin-dependentT /cos the difference cross between section; the cross sec-g is best obtained from a two unpolarized structure functionsII. THEORETICALF and OVERVIEWF . These func- 2 ⌅ 2 thehelicity exchangedh virtual⇤1: photon and1 M is2 the nucleon mass.tionsor atQ angleshigh. and The ⌃ structure. The correspondingand function differentialg1 is therefore best mea- l 2 cross sectionsmeasurement are given by in the orthogonal configuration, combined with tions depend on the four-momentumA. Cross sections for polarized transfer lepton-nucleon squared scatteringQ and sured in the ⇥anti⇧parallel configuration where it dominates The double-differential2 cross section can be written as a a measurement of g . In all formulas used in this article, we The polarized2 deep-inelastic lepton-nucleon1 inclusive d2⌅ 16⌃⌥2y y 21y 2 2y the scaling variable x⇤Q /2M↵, where ↵ is the energy of the spin-dependent⇥ cross section;3 g is best obtained from2 a 2 2 function of x and Q ⌦15: ⌅⇤¯⌅⇥ h ⇤⌅. consider2 ⇤ ⇥4 2.1 only⇧1⇥ the⇥ single-virtual-photong1⇥ g2 2 exchange. The inter- scattering cross section in the one-photon-exchange2 l approxi- dxdQ Q ⇧⌅ 2 4 ⇤d 2⌅T 8⌥ y y y the exchanged virtualmation photon can be written and asM theis sum the of a nucleonspin-independent mass. term measurement in the orthogonal configuration,0 combined with ference effects between virtual2 Z⇥2.5⇤⇥⇧andcos photon exchange4 in 1⇥y⇥ g1⌅g2 . The double-differential2¯¯⌅ and cross a spin-dependent2 section term can2⌅m beand2 involveswritten the as lepton a dxdQ d Q A 4 ⌅ 2 ⇤ d ⌅ 4⌃⌥ l a measurementdeep-inelastic of g muon1 . In scattering all formulas have used been in measured this article,⌦17 we and For longitudinally2helicity hl⇤1: polarized2 leptons the2 spinandS is along the function of x and Q ⌦215⇤: 4 xy 1⇥ 2 F1⇥x,Q ⇧ considerl only the single-virtual-photon exchange. The inter- dxdQ Q x ⇧ ⌅ 1 Q ⇤ found to be small and compatible with the standard model ⇤¯⇥ 3 2 2 2 lepton momentum k.⌅ The⌅ 2 h spin-independentl⇤⌅. ⇥2.1⇧ crossd ⌅T section8 for⌥ y y y 0 ⇥2.6⇧ ferenceexpectations.⇤⇥ effectscos between They 1⇥ cany virtual⇥ be neglectedZg ⌅gand. photon in the kinematic exchange range in 2 2 2 2 2 dxdQ2d Q4 A 4 ⌅ 2 1 2 ⇤ dparity-conserving¯⌅ 4⌃For⌥ longitudinally interactions polarized2my leptons the can spin S beis along expressed the in terms of 2 l 22 l deep-inelasticof current muon experiments. scatteringFor a high have beambeen measured energy ⌦E17,andis small since either x is small 2 ⇤ lepton4 momentum⌅xy1⇥1yk⇥.⇥ The spin-independent2 FF12⇥⇥xx,,QQ cross⇧⇧ , section for ⇥2.2⇧ 2 ⇥2.6⇧ dxdQtwo unpolarizedQparity-conservingx ⇧ ⌅ ⌅ structure interactionsQ4 ⇤ can⇤ functions be expressed inF terms1 and of Ffound2 . These to be func- small and compatible with the standard model For a high beam energy E, is smallor sinceQ eitherhigh.x is small The structure function g1 is therefore best mea- two unpolarized structure functions F and F . These func- 2 2 1 2 orexpectations.Q high. The structure They function canB.g Cross1 beis therefore neglected section best asymmetriesmea- in the kinematic range tions depend on the four-momentum transfer2 squared Q and sured in the ⇥anti⇧parallel configuration where it dominates tions depend on the2y four-momentum2 transfer squared Q and sured in the ⇥anti⇧parallel configuration where it dominates where ml is thethe scaling lepton variable mass,x⇤yQ⇤2/2M↵/↵2E, where2in the↵ is laboratory the energy of sys-of current experiments. the scaling⌅ 1⇥ variabley⇥ xF⇤⇥Qx,Q/2⇧M, ↵, where⇥2.2⇧↵ isthe spin-dependentthe energyThe cross spin-dependent of section; g2theis best spin-dependent cross obtained section from a terms, cross Eqs. section;⇥2.5⇧ and g is best obtained from a tem, and the⌅ exchanged virtual4 ⇤ photon2 and M is the nucleon mass. measurement in the orthogonal configuration, combined with 2 The double-differential cross section can be written as a ⇥2.6⇧, make only a small contribution to the total deep- the exchanged virtual photon and M is thea measurement nucleon of g mass.1 . In all formulasmeasurement used in this article, we in the orthogonal configuration, combined with function of x and Q2 ⌦15: B. Cross section asymmetries 2 considerinelastic only the single-virtual-photon scattering cross exchange. section The inter- and furthermore their con- The double-differential2Mx A crossQ section can beference written effects between as virtual a Z0 anda measurement photon exchange in of g . In all formulas used in this article, we where ml is the lepton mass,2 y⇤↵/E2 in the laboratory2 sys- tribution is, in general, reduced by incomplete1 beam and tar- d¯⌅⇤ 4⌃2⌥ ⇤ 2. ml ⇥2.3deep-inelastic⇧ muon scattering have been measured ⌦17 and function of x and⇤Q 2 ⌦15xy2 1:⇥ F ⇥x,Q2⇧ The spin-dependent cross section terms, Eqs. ⇥2.5⇧ and tem, and dxdQ2 AQ4x ⇧ ⌅↵ Q2 ⇤ 1 found toget be small polarizations. and compatible with Thereforeconsider the standard they only model can the best single-virtual-photon be determined exchange. The inter- expectations.⇥2.6⇧, make They can only be neglected a small in the contribution kinematic range to the total deep- 0 2y 2 ference effects between virtual Z and photon exchange in 2 2 2 2 2 ofinelastic current experiments. scattering cross section and furthermore their con- d 2¯⌅Mx ⌅A4Q⌃1⇥⌥y⇥ F2⇥x,Q ⇧ , 2m ⇥2.2⇧ ⌅ 4 ⇤ 2 l tribution2 is, in general,deep-inelastic reduced by incomplete muon beam scattering and tar- have been measured ⌦17 and ⇤ ⇤⇤ . xy 1⇥ ⇥2.3⇧F ⇥x,Q ⇧ B. Cross section asymmetries 22 ↵ 4 2 1 wheredxdQmAl Qis the leptonQ mass,x y⇧⇤↵/E in⌅ the laboratoryQ sys-⇤ get polarizations. Thereforefound they to be can small best be and determined compatible with the standard model tem, and The spin-dependent cross section terms, Eqs. ⇥2.5⇧ and ⇥2.6⇧, make only a small contributionexpectations. to the total deep- They can be neglected in the kinematic range 2 2 inelastic scattering cross section and furthermore their con- 2Mx AQ2 y ⇤ ⇤ . ⇥2.3⇧ 2tribution is, in general, reduced by incompleteof current beam and experiments. tar- ⌅ 1⇥2 y⇥↵ F ⇥x,Q ⇧ , ⇥2.2⇧ ⌅ AQ 4 ⇤ 2 get polarizations. Therefore they can best be determined B. Cross section asymmetries where ml is the lepton mass, y⇤↵/E in the laboratory system, and The spin-dependent cross section terms, Eqs. ⇥2.5⇧ and ⇥2.6⇧, make only a small contribution to the total deep- inelastic scattering cross section and furthermore their con- 2Mx AQ2 ⇤ ⇤ . ⇥2.3⇧ tribution is, in general, reduced by incomplete beam and tar- AQ2 ↵ get polarizations. Therefore they can best be determined 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5333 from measurements of cross section asymmetries in which respectively, where F1 is usually expressed in terms of F2 the spin-independent contribution cancels. The relevant and R: asymmetries are 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZE2 D... 5333 ↵⌃ ↵⌃ 1⇤ ⇤ ' F ⇥ F . ⇥2.16⌥ A⇤⇥ , A'⇥ , ⇥2.7⌥ 1 2 2¯⌃ from2¯⌃ measurements of cross section asymmetries in which respectively,2x⇥ where1⇤RF⌥1 is usually expressed in terms of F2 the spin-independent contribution cancels. The relevant and R: which are related to the virtual photon-protonasymmetries are asymmetries These relations are used in the present analysis2 for the evalu- ↵⌃ ↵⌃ 1⇤ A1 and A2 by ⇤ 56 ' SPIN STRUCTURE2 OFF THE⇥ PROTON FROMF . POLARIZED... ⇥2.16⌥ 5333 A⇤⇥ , A'⇥ ation, of g ⇥2.7in⌥ bins of x and Q , starting1 from2 the asymme- 2¯⌃ 2¯⌃ 1 2x⇥1⇤R⌥ A ⇥D⇥A ⇤⌅A ⌥, A ⇥d⇥A A ⌥, ⇥2.8⌥ fromtries measurements measured of cross in section the asymmetries parallel in spin which configurationrespectively, where andF1 usingis usually expressed in terms of F2 ⇤ 1 2 'which are2 related1 to the virtual photon-protonthe spin-independent asymmetries contribution cancels.2 The relevant and2 R: asymmetriesparametrizations are ofTheseF2( relationsx,Q ) are and usedR in(x the,Q present). analysis for the evalu- A1 and A2 by 2 The virtual photon-protonation of g in bins asymmetry of x and Q , startingA is from evaluated the asymme-2 where ↵⌃ ↵⌃ 1 2 1⇤ ⇤ tries measured' in the parallel spin configurationF ⇥ and using F . ⇥2.16⌥ A⇤⇥ , A'⇥ , ⇥2.7⌥ 1 2 A⇤⇥D⇥A1⇤⌅A2⌥, A'⇥d⇥Afrom2 A1 the⌥, measured⇥2.82¯⌃⌥ transverse2¯⌃ and longitudinal2 asymmetries2 2x⇥1⇤R⌥ 2 parametrizations of F2(x,Q ) and R(x,Q ). ⌃ ⌃ g g A⇤ and A' : 1/2 3/2 where1 2 which are related to the virtual photon-protonThe virtual asymmetries photon-proton asymmetry A2 is evaluated A ⇥ ⇥ These relations are used in the present analysis for the evalu- 1 , A and A by from the measured transverse and longitudinal asymmetries ⌃ ⇤⌃ F 1 2 ation of g in bins of x and Q2, starting from the asymme- 1/2 3/2 1 2 A and A : 1 ⌃1/2⌃3/2 g1 g2 ⇤ ' tries measured in the parallel spin configuration and using A1⇥ ⇥ , A⇤⇥D⇥A1⇤⌅A2⌥, A'⇥d⇥A21 A1⌥, A'⇥2.8⌥ A⇤ 2 2 ⌃ ⇤⌃ F parametrizations of F2(x,Q ) and R(x,Q ). TL 1/2 3/2 1 A2⇥ ⇤ . ⇥2.17⌥ 2⌃ g1⇤g2 The virtual photon-proton asymmetry A2 is evaluated where 1⇤⌅ ⇧ d 1 D ⌅A' A⇤ A ⇥ ⇥ . ⇥2.9⌥ from the measured transverse and longitudinal asymmetries 2 TL A2⇥ ⇤ . ⇥2.17⌥ ⇤ 2⌃ g1⇤g2 2 ⌃1/2 ⌃3/2 F1 1⇤⌅ A⇧ ⇤ dand A D: ⌅ A ⇥ ⇥ . ⌃⇥1/22.9⌥⌃3/2 g1 g2 ' 2 ⇤ A ⇥ ⇥ , ⌃1/2 ⌃3/2 F1 1 ⌃ ⇤⌃ F From Eqs. ⇥1/22.3⌥3/2and ⇥2.91 ⌥, A has an explicit 1/AQ2 depen- In Eqs. ⇥2.8⌥ and ⇥2.9⌥, D is the depolarization factor of the 2 12 A' A⇤ In Eqs. ⇥2.8⌥ and ⇥2.9⌥, D is the depolarization factor of theTL From Eqs. ⇥2.3⌥ and ⇥2.9⌥, A2 has an explicitA 1/⇥AQ depen-⇤ . ⇥2.17⌥ dence and is2 therefore⌃ g1⇤ expectedg2 to be small at high energies.2 1⇤⌅ ⇧ d D ⌅ virtual photon defined below and virtuald, ⌅, photon and definedare the below kine- and d, ⌅, and areA the2⇥ kine- ⇥dence and. is therefore⇥2.9 expected⌥ to be small at high energies. ⌃1/2⇤⌃3/2 F1 matic factors: The structure functionThe structureg2 is function obtainedg2 is from obtained the from measured the measured matic factors: 2 In Eqs.asymmetries⇥2.8⌥ and ⇥2.9⌥, D usingis theasymmetries depolarization Eqs. ⇥2.9 using⌥ factorand Eqs. of⇥ the2.17⇥2.9⌥⌥From.and ⇥ Eqs.2.17⇥⌥2.3. ⌥ and ⇥2.9⌥, A2 has an explicit 1/AQ depen- 2 2 dence and is therefore expected to be small at high energies. 2 2 A1y y virtual/4 photon defined below and d, ⌅, and are the kine- A1y y /4 d⇥ maticD, factors: ⇥2.10⌥ The structure function g2 is obtained from the measured d⇥ D, 1⇥2.10y/2⌥ C. Spin-dependentasymmetries structure function using Eqs.g1 ⇥2.9⌥ and ⇥2.17⌥. 2 2 1y/2 C.A1 Spin-dependenty y /4 structure function g1 2 2 d⇥ TheD significance, ⇥ of2.10 the⌥ spin-dependent structure function ⇥1y y /4⌥ 1y/2 C. Spin-dependent structure function g ⇥ g1 can be understood from the virtual photon asymmetry A1 . 1 2 2 ⌅ 2 , The significance⇥2.11⌥ of the spin-dependent structure function ⇥1y/2⌥⇥1⇤ y/2⌥ As shown in Eq. ⇥2.9⌥, A g /TheF or significance⌃ ⌃ of⇧ theg . spin-dependent In or- structure function ⇥1y y /4⌥ ⇥1y2y 2/4⌥ 1 1 1 1/2 3/2 1 ⌅⇥ , ⇥2.11⌥ g1 can be⌅⇥ understoodder to from conserve, the virtual angular⇥2.11⌥ momentum, photong1 can be asymmetry understood a virtual photon fromA the1 . with virtual photon asymmetry A1 . 2 ⇥1y/2⌥⇥1⇤2y/2⌥ ⇥1y/2⌥⇥1⇤ y/2⌥ ⇥1y/2⌥ helicity ⇤1 or 1 can onlyAs be shown absorbed in Eq. by⇥2.9 a quark⌥, A1 withg1 /F a1 or ⌃1/2⌃3/2⇧g1 . In or- ⇥ . As shown⇥ in2.12 Eq.⌥ ⇥2.9⌥, A1g1 /F1 or ⌃1/2 ⌃3/2⇧g1 . In or- 1⇤2y/2 spin projection of 1 or ⇤ 1 ,der respectively, to conserve if angular the quarks momentum, have a virtual photon with der to conserve⇥1 angulary/2⌥ momentum,2 2 ahelicity virtual⇤1 photon or 1 can with only be absorbed by a quark with a ⇥ . ⇥2.12⌥ ⇤ no2 orbital angular momentum. Hence, g1 contains 1 ⇤ informa-1 ⇥1 y/2⌥ helicity ⇤1 or11y/2 can only be absorbedspin by projection a quark of 2 withor 2 , a respectively, if the quarks have The cross sections ⌃1/2 and ⌃3/2 refer to the absorption of a tion on the quark spin orientations with respect to the proton ⇥ 2 . ⇥2.12⌥ 1 1 no orbital angular momentum. Hence, g1 contains informa- 1⇤ ytransversely/2 polarized virtual photonThe by crossspin a polarized sections projection⌃ proton1/2 and of⌃3/2spinrefer2 direction.or to the⇤ absorption2 , respectively, of a tion if on the the quark quarks spin orientations have with respect to the proton for total photon-proton angular momentumtransversely component polarized along virtual photon by a polarized proton spin direction. no orbital angular momentum.In the simplest Hence, quark-partong1 contains model, the informa-quark densities the virtual photon axis of 1/2 and 3/2,for respectively; total photon-proton⌃TL is angular an momentumdepend only component on the along momentumIn fraction the simplestx carried quark-parton by the model, the quark densities The cross sections ⌃ and ⌃ refer to the absorption of a TL 1/2 3/2 interference cross section due to thethe helicity virtualtion spin-flip photon on the axis ampli- quark of 1/2 and spin 3/2, respectively; orientations⌃ is with an respectdepend only to on the the proton momentum fraction x carried by the interference cross section due toquark, the helicity and g spin-flip1 is given ampli- by transversely polarized virtual photontude in by forward a polarized Compton scattering proton 18.spin The depolarization direction. quark, and g1 is given by tude in forward Compton scattering 18. The depolarization n f n for total photon-proton angular momentumfactor D depends component on y and along on the ratiofactorR⇥D ⌃dependsInL /⌃ theT of on simplest longi-y and on7/18/16 the quark-parton ratio R⇥⌃ /⌃ of longi- model,1 theEIC Lecture quark 2 at NNPSS densities 2016 at MIT1 f 7 L T g ⇥x⌥⇥ e2↵q ⇥x⌥, ⇥2.182⌥ tudinal and transverse photoabsorptionTL tudinal cross and sections: transverse photoabsorption cross sections:1 ⌦ i i g1⇥x⌥⇥ ⌦ ei ↵qi⇥x⌥, ⇥2.18⌥ the virtual photon axis of 1/2 and 3/2, respectively; ⌃ is an depend only on the momentum fraction2 i⇥1 x carried by the2 i⇥1 2 interference cross section due to the helicity spin-flipy⇥2 ampli-y⌥⇥1⇤2y/2quark,⌥ and yg⇥12isy⌥⇥ given1⇤ y/2 by⌥ D⇥ Relation to spin. structure function g1 D⇥ 2 2 ⇤ 2 .2 2 ⇤ 2 2 ⇤ where tude in forward Compton scattering 18y2.⇥1 The⇤2⌥ depolarization⇥12m /Q2⌥⇤2⇥1yy⇥21y 2/4⌥⇥⌥⇥11⇤R2m⌥ l /Q ⌥ where2⇥1 y y /4⌥⇥1 R⌥ l n f ⇥2.13⌥ factor D depends on y and on the ratio R⇥⌃L /⌃T of longi- ⇥2.13⌥ 1 2 ↵q ⇥x⌥⇥q⇤⇥x⌥q⇥x⌥⇤¯q ⇤⇥x⌥¯q⇥x⌥, ⇥2.19⌥ tudinal and transverse photoabsorption cross sections: From Eqs. ⇥2.8⌥ and ⇥2.9⌥,g we1⇥ canx⌥⇥ express⌦ thee virtual⇤i ↵qi⇥x⌥, ¯ ⇤ i ¯i ⇥2.18i ⌥ i i ↵q2i⇥ix⇥⌥⇥1 qi ⇥x⌥qi ⇥x⌥⇤q i ⇥x⌥qi ⇥x⌥, ⇥2.19⌥ From Eqs. ⇥2.8⌥ and ⇥2.9⌥, wephoton-proton can express asymmetry the virtualA1 in terms of g1 and A2 and find photon-proton asymmetry A1 in termsthe of followingg1 and A relation2 and for find the longitudinal asymmetry: q⇤ (¯q ⇤) and q(¯q ) are the distribution functions of 2 Quarki andi anti-quarki withi spin orientation along and y⇥2y⌥⇥1⇤the followingy/2⌥ relation for the longitudinal asymmetry: ⇤ ⇤ quarks ⇥antiquarks⌥ with spin parallel and antiparallel to the qi (¯q i ) and qi (¯q i ) are the distribution functions of ⇥ A⇤ g1 against the proton spin. D 2 2 2 2 2 2 . 2 nucleon spin, respectively, ei is the electric charge of the ⇤ ⇤ ⇤ where ⇥⇥1⇤ ⌥ quarks⇤⇥⌅⇥⌥antiquarksA2 . ⌥ ⇥with2.14⌥ spin parallel and antiparallel to the y ⇥1 ⌥⇥1 2ml /Q ⌥ 2⇥1 y yA /4⌥⇥1 Rg⌥ D F1 quarks of flavor i, and n is the number of quark flavors ⇤ 2 1 f ⇥⇥1⇤ ⌥ ⇤⇥⌅⌥A . ⇥2.14⌥ nucleon spin, respectively, einvolved.i is the electric charge of the D ⇥2.13F ⌥ 2 • In QCD quarks interact with each other through gluons, which 1 The virtual-photon asymmetriesquarks are bounded of flavor by positivityi, and n f is theIn QCD, number quarks of interact quark flavors by gluon exchange, which gives ⇤ ⇤2 2 relations ⇥A1⇥⇤1 and↵q⇥A⇥2⇥x⇤⌥A⇥Rgivesinvolved.q19.⇥ When xrise⌥ thetoq terma⇥ weakx propor-⌥⇤¯q Q ⇥dependencexrise⌥ to¯q a weak⇥xQ⌥, ofdependence structure⇥2.19⌥ of thefunctions structure functions. The The virtual-photon asymmetries are bounded by positivityi i i i i From Eqs. ⇥2.8⌥ and ⇥2.9⌥, we can express the virtual tional to A2 is neglected in Eqs. ⇥2.8In⌥ QCD,and ⇥2.14 quarks⌥, the interact longi- bytreatment gluon exchange, of g1 in perturbative which gives QCD follows closely that of tudinal asymmetry is related to A and g by 2 relations ⇥A1⇥⇤1 and ⇥A2⇥⇤AR 19. When the term propor- rise1 to a1 weak Q dependenceunpolarized of the structure parton functions. distributions The and structure functions 20. photon-proton asymmetry A1 in terms of g1 and A2 and find 2 2 tional to A is neglected in Eqs. ⇥2.8⌥ and ⇥2.14⌥, the longi- • At any given Q the spin Atstructure a given scale functionQ , g1 is is related related to the to polarized quark and 2 ⇤ ⇤ A g treatment1 A of g1 in perturbative QCD follows closely that of the following relation for the longitudinal asymmetry: ¯ ⇤ 1 ¯ ⇤ gluon distributions by coefficient functions Cq and Cg tudinal asymmetry is related to A1 and gq1 iby (q iA1) and, qpolarizediunpolarized(q i2 ), arequark parton the & distributions⇥ 2.15gluon distribution⌥ distributions and structure functions by functions coefficients of20. Cq and Cg D F1 1⇤ D 2 through 20 quarks ⇥antiquarksAt⌥ with a given spin scale parallelQ , g1 is related and antiparallel to the polarized to quark the and A g A⇤ g1 1 A⇤ ⇤ 2 1 gluon distributions by coefficient functions Cq and Cg ⇥⇥1⇤ ⌥ ⇤⇥⌅⌥A . A1 , ⇥2.14 ⌥ 2 nucleon, spin,⇥2.15⌥ respectively, ei is the electric charge of the 2 D F1 1⇤ D through 20 D F1 quarks of flavor i, and n f is the number of quark flavors involved. The virtual-photon asymmetries are bounded by positivity In QCD, quarks interact by gluon exchange, which gives 2 relations ⇥A1⇥⇤1 and ⇥A2⇥⇤AR 19. When the term propor- rise to a weak Q dependence of the structure functions. The tional to A2 is neglected in Eqs. ⇥2.8⌥ and ⇥2.14⌥, the longi- treatment of g1 in perturbative QCD follows closely that of tudinal asymmetry is related to A1 and g1 by unpolarized parton distributions and structure functions 20. 2 At a given scale Q , g1 is related to the polarized quark and A g 1 A ⇤ 1 ⇤ gluon distributions by coefficient functions Cq and Cg A1 , 2 , ⇥2.15⌥ D F1 1⇤ D through 20 5334 D. ADAMS et al. 56

n f e2 1 x x x 1 k dy S x 0,S 0,NS ⌅ C , s ⌅C , s ⌅⇧ 1⇤ , g1⇤x,t⌥ Cq , s⇤t⌥ ✏⇤y,t⌥ q ⌃ y ⌅ q ⌃ y ⌅ ⌃ y ⌅ 2 k⌅1 n ⇤x y ⌥ ⌃ y ⌅ f x x 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 8 0 ⇧2n C , ⇤t⌥ g⇤y,t⌥ Cg , s ⌅0. ⇤2.26⌥ f g⌃ y s ⌅ ⌃ y ⌅ Composition & Q2 or t dependence of Structure Functions 5334 D. ADAMS xet al. Note that g561 decouples from g in this scheme. ⇧CNS , ⇤t⌥ qNS⇤y,t⌥ . ⇤2.20⌥ Beyond leading order, the coefficient functions and the q ⌃ y s ⌅ n splitting functions are not uniquely defined; they depend on f e2 1 x x x 1 k dy S x In this equation:0,S 0,NS the renormalization scheme. The complete set of coefficient g ⇤x,t⌥⌅ C , ⇤t⌥ ✏⇤y,t⌥ Cq , s ⌅Cq , s ⌅⇧ 1⇤ , 1 q Ins this equation, t⌅ln(Q2/⌦2), 2 is⌃2y the strong⌅ coupling⌃ y ⌅ con-⌃ y ⌅ 2 k⌅1 n f ⇤x y ⌥ ⌃ y ⌅ t = ln(Q /sΛ ) functions has been computed in the modified minimal sub- stant, and ⌦ is the scale parameter of QCD. The superscripts traction (MS) renormalization scheme up to order 2 ↵23. = strong interactionx constant s x ‘‘S’’ and ‘‘NS,’’α respectively,S indicate flavor-singlet0 and 2 ⇧2n C , ⇤t⌥ g⇤y,t⌥ Cg , s ⌅0. The O(⇤2.26s )⌥ corrections to the polarized splitting functions f g⌃ y s ⌅ nonsinglet parton S distributions & NS stand and for coefficient flavor⌃ y functions;singlet⌅ & Pqq and Pqg have been computed in Ref. ↵23 and those to g(x,t) is the polarized gluon distribution, and ✏ and P and P in ↵24,25. This formalism allows a complete x flavorNote thatnon-singletg1 decouples from g in this scheme.gq gg NS NS NS ⇧Cq , s⇤t⌥ q ⇤y,t⌥ .q are the⇤2.20 singlet⌥ andBeyond nonsinglet leading combinations order, the coefficient of the po- functionsnext-to-leading and the order ⇤NLO⌥ QCD analysis of the scaling ⌃ y ⌅ larized quark and antiquarksplitting distributions: functions are not uniquely defined; theyviolations depend ofon spin-dependent structure functions. 2 the renormalization scheme. The complete set ofIn coefficient QCD, the ratio g1 /F1 is Q dependent because the 2 2 In this equation, t⌅ln(Q /⌦ ), s is the strong coupling con- functionsn f has been computed in the modifiedsplitting minimal sub-functions, with the exception of Pqq , are different stant, and ⌦ is the scale parameter of QCD. The superscripts 2 traction⌅ (MS) renormalization scheme up to orderfor polarized s ↵23. and unpolarized parton distributions. Both Pgq ✏⇤x,t⌥ 2 qi⇤x,t⌥, ⇤2.21⌥ ‘‘S’’ and ‘‘NS,’’ respectively, indicate flavor-singlet and The O( i⌅s1) corrections to the polarized splittingand P functionsgg are different in the two cases because of a soft gluon nonsinglet parton distributions and coefficient functions; Pqq and Pqg have been computed in Ref. ↵23singularityand those to at x⌅0, which is only present in the unpolarized g(x,t) is the polarized gluon distribution, and ✏ and P and P in ↵24,25. This formalism allows a complete n f gq n f gg n f case. However, in kinematic regions dominated by valence qNS are the singlet and nonsinglet combinations of the po- next-to-leading1 order1 ⇤NLO⌥ QCD analysis of the scaling 2 NS 2 2 2 quarks, the Q dependence of g1 /F1 is expected to be small larized quark and antiquark distributions: q ⇤x,t⌥⌅ eiviolations⇤ ofek spin-dependent ek structureqi⇤x, functions.t⌥. ⌥ i⌅1 ⌃ n f k⌅1 ⌅ ⇧ n f k⌅1 ↵26. 2 In QCD, the ratio g1 /F1 is Q ⇤2.22dependent⌥ because the n splitting functions, with the exception of P , are different f qq D. Small-x behavior of g1 for polarized and unpolarized parton distributions. Both P ✏⇤x,t⌥⌅ qi⇤x,t⌥, ⇤2.21⌥ gq i⌅1 The t dependence of theand polarizedP are different quark in and the gluon two cases distribu- because of aThe soft gluonmost important theoretical predictions for polarized 2 gg 2 To study the Q evolution experimentally:tions follows you the need Gribov-Lipatov-Altarelli-Parisisingularity a wide atQx ⌅range0, which in measurements is only⇤GLAP present⌥ in thedeep-inelastic unpolarized scattering are the sum rules for the nucleon structure functions g . The evaluation of the first moment of n f n f equationsn f ↵21,22. Ascase. for However,the unpolarized in kinematic distributions, regions dominated the by valence 1 1 1 2 NS 2 2 polarized2 singlet andquarks, gluon distributions the Q dependence are coupled of g1 /F by1 is expectedg1 to, be small q ⇤x,t⌥⌅ ei ⇤ ek ek qi⇤x,t⌥. ⌥ i⌅1 ⌃ n f k⌅1 ⌅ ⇧ n f k⌅1 ↵26. ⇤2.22⌥ 1 1 2 2 d s⇤t⌥ dy D.x Small-x behavior of g ⇥1⇤Q ⌥⌅ g1⇤x,Q ⌥dx, ⇤2.27⌥ S 1 ⇤0 ✏⇤x,t⌥⌅ Pqq , s⇤t⌥ ✏⇤y,t⌥ The t dependence of the polarized quark and gluondt distribu- 2 The⇤x mosty ⌥ important⌃ y theoretical⌅ predictions for polarized tions follows the Gribov-Lipatov-Altarelli-Parisi ⇤GLAP⌥ deep-inelastic scattering are the sum rules forrequires the nucleon knowledge of g1 over the entire x region. Since the x equations ↵21,22. As for the unpolarized distributions, the structure functions g1 . The evaluation of the firstexperimentally moment of accessible x range is limited, extrapolations ⇧2n f Pqg , s⇤t⌥ g⇤y,t⌥ , ⇤2.23⌥ polarized singlet and gluon distributions are coupled by g1 , ⌃ y ⌅ to x⌅0 and x⌅1 are unavoidable. The latter is not critical because it is constrained by the bound ⇥A1⇥⌅1 and gives 1 1 2 2 only a small contribution to the integral. However, the small d s⇤t⌥ dy x 1 ⇥1⇤Q ⌥⌅ g1⇤x,Q ⌥dx, ⇤2.27⌥ S d s⇤t⌥ dy x ⇤0 x behavior of g (x) is theoretically not well established and ✏⇤x,t⌥⌅ Pqq , s⇤t⌥ ✏⇤y,t⌥ ⌅ 1 dt 2 ⇤x y ⌥ ⌃ y ⌅ g⇤x,t⌥ Pgq , s⇤t⌥ ✏⇤y,t⌥ dt 2 ⇤x y ⌥ ⌃ y ⌅ evaluation of ⇥ depends critically on the assumption made 1 requires knowledge of g1 over the entire x region. Since the x for this extrapolation. experimentallyx accessible x range is limited, extrapolations 2 ⇧2n f Pqg , s⇤t⌥ g⇤y,t⌥ , ⇤2.23⌥ From the Regge model it is expected that for Q ⌃ y ⌅ ⇧Pto x⌅,0 and⇤t⌥ x⌅g1⇤ arey,t⌥ unavoidable., ⇤ The2.24⌥ latter is not critical p n p n ⇤ gg⌃ y s ⌅ 2M, i.e., x 0, g ⇧g and g ⇤g behave like x ↵27, 1 1 1 1 because it is constrained by the bound ⇥A1⇥⌅1 and gives only a small contribution to the integral. However,where the smallis the intercept of the lowest contributing Regge d ⇤t⌥ 1 dy x s x behavior of g1(x) is theoretically not well establishedtrajectories. and These trajectories are those of the pseudovector g⇤x,t⌥⌅ Pgq , whereass⇤t⌥ ✏⇤ they,t nonsinglet⌥ distribution evolves independently of p n dt 2 ⇤x y ⌥ ⌃ y ⌅ evaluation of ⇥ depends critically on the assumptionmesons madef for the isosinglet combination, g ⇧g and of a the singlet and gluon distributions: 1 1 1 1 1 for this extrapolation. for the isotriplet combination, g p⇤gn , respectively. Their x From the Regge model it is expected that for Q2 1 1 ⇧P , ⇤t⌥ g⇤y,t⌥ , ⇤2.24⌥ 1 p n p n intercepts⇤ are negative and assumed to be equal and in the gg y s d s⇤t⌥2M,dy i.e., x 0,x g ⇧g and g ⇤g behave like x ↵27, ⌃ ⌅ qNS⇤x,t⌥⌅ PNS , 1 ⇤t⌥1 qNS1⇤y,t1⌥. range ⇤0.5⇥ ⇥0. Such behavior has been assumed in most qq s dt 2where⇤x yis the⌃ intercepty ⌅ of the lowest contributinganalyses. Regge trajectories. These trajectories are those of the pseudovector whereas the nonsinglet distribution evolves independently of ⇤2.25⌥ A flavor-singlet contribution to g (x) that varies as mesons f for the isosinglet combination, g p⇧gn and of a 1 the singlet and gluon distributions: 1 1 ↵21 ln(1/x)⇤1 ↵28 was obtained from a model where an for the isotriplet combination, g p⇤gn , respectively. Their Here P are the QCD splitting functions for polarized1 parton1 exchange of two nonperturbative gluons is assumed. Even ij intercepts are negative and assumed to be equal and in the d ⇤t⌥ 1 dy x 2 ⇤1 NS s NS distributions.NS ⇤ ⇥ ⇥ very divergent dependences like g1(x)⌃(x ln x) were q ⇤x,t⌥⌅ Pqq , s⇤t⌥ q ⇤y,t⌥. range 0.5 0. Such behavior has been assumed in most dt 2 ⇤x y ⌃ y Expressions⌅ ⇤2.20⌥analyses., ⇤2.23⌥, ⇤2.24⌥, and ⇤2.25⌥ are valid in considered ↵29. Such dependences are not necessarily con- all orders of⇤2.25 perturbative⌥ QCD. The quark and gluon distri- sistent with the QCD evolution equations. A flavor-singlet contribution to g1(x) that varies as butions, coefficient functions,↵2 ln(1/x) and⇤1 splitting↵28 was functions obtained depend from a modelExpectations where an based on QCD calculations for the behavior on the mass factorization scale and on the renormalization 2 Here Pij are the QCD splitting functions for polarized parton exchange of two nonperturbative gluons is assumed.at small Evenx of g1(x,Q ) are twofold. 2 ⇤1 distributions. scale; we adopt here thevery simplest divergent choice, dependences setting both like scalesg1(x)⌃(x ln Resummationx) were of standard Altarelli-Parisi corrections Expressions ⇤2.20⌥, ⇤2.23⌥, ⇤2.24⌥, and equal⇤2.25⌥ toareQ valid2. At in leadingconsidered order, the↵29 coefficient. Such dependences functions are are not necessarilygives ↵30–32 con- all orders of perturbative QCD. The quark and gluon distri- sistent with the QCD evolution equations. butions, coefficient functions, and splitting functions depend Expectations based on QCD calculations for the behavior 2 on the mass factorization scale and on the renormalization at small x of g1(x,Q ) are twofold. scale; we adopt here the simplest choice, setting both scales Resummation of standard Altarelli-Parisi corrections equal to Q2. At leading order, the coefficient functions are gives ↵30–32 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 9

Spin Crisis àPuzzle (1960-2000) Limitations of the Quark Parton Model (QPM) & lessons learnt Early (fixed target) spin experiments and their limitations 7/18/16 a"brief"historyEIC Lecture 2 at NNPSS 2016 "at MIT 10 55"the"particle"zoo"55" “If I had known this earlier, @#"of""hadrons">"#"of"chemical"elements" I would have done Biology” --W. Pauli "baryons""("J"="n/2")n=1,2,..."

"mesons:""("J"="n"")n=0,1,..."

all"baryons"and"mesons"grouped"in"multiplets"" """"""""""""""""""""""""""""""""""""""""""""""""""""""""""""definded"by"SU(3)"symmetry"" e.g.,"baryons:"

Jp"="1/2" Jp"="3/2" (same"spin" and"parity)"

isolines"of"same"charge"and"strangeness" 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 11

Understanding the proton structure:

Friedman, Kendall, Taylor: 1960’s SLAC Experiment 1990 Nobel Prize: "for their pioneering investigations concerning deep inelastic scattering of electrons on protons and bound neutrons, which have been of essential importance for the development of the quark model in particle physics".

Obvious next Question: • predict: quarks carry the proton spin Could we understand other properties of proton, e.g. SPIN, in the quark-parton model? Proton Spin = ½, each quark is a spin ½ particle… = +

proton u d

• expect higher Fock states :

• model expectation: ~ 70% of proton spin due to quark and anti-quark spins

Measure in deeply-inelastic processes! ! 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 12

Structure Functions & PDFs

• The F1 and F2 are unpolarized structure functions or momentum distributions

• The g1 and g2 are polarized structure functions or spin distributions • In QPM

• F2(x) = 2xF1 (Calan Gross relation)

• g2 = 0 (Twist 3 quark gluon correlations)

Remember this

7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 13

Nucleon spin & Quark Probabilities

• Define

• With q+ and q- probabilities of quark & anti-quark with spin parallel and anti-parallel to the nucleon spin

• Total quark contribution then can be written as:

• The nucleon spin composition

7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 14 Nucleon’s Spin: Naïve Quark Parton Model • Protons and Neutrons are spin 1/2 particles • Quarks that constitute them are also spin 1/2 particles • And there are three of them in the Proton: u u d Neutron: u d d

S proton = Sum of all quark spins!

? 1/2 = 1/2 + 1/2 + 1/2

1/2 = 1/2 - 1/2 + 1/2 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 15

How was the Quark Spin measured?

• Deep Inelastic polarized electron or muon scattering

µ Spin 1/2 quarks Spin 1 γ*

µ 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 16

Measurements of spin structure functions: What issues we need to worry about?

1) Design of experiments, operational issues 2) Calculations of spin structure functions 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 17

Experimental Needs in DIS

Polarized target, polarized beam • Polarized targets: hydrogen (p), deuteron (pn), helium (3He: 2p+n) • Polarized beams: electron,muon used in DIS experiments

Determine the kinematics: measure with high accuracy: • Energy of incoming lepton • Energy, direction of scattered lepton: energy, direction • Good identification of scattered lepton

Control of false asymmetries: • Need excellent understanding and control of false asymmetries (time variation of the detector efficiency etc.) 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 18

Experimental issues detector beam target

Possible sources of false asymmetries: • beam flux • target size • detector size • detector efficiency 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 19 An Ideal Situation

If all other things are equal, they cancel in the ratio and…. 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 20

A Typical Setup

• Experiment setup (EMC, SMC, COMPASS@CERN)

• Target polarization direction reversed every 6-8 hrs • Typically experiments try to limit false asymmetries to be about 10 times smaller than the physics asymmetry of interest 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 21 Asymmetry Measurement

• f = dilution factor proportional to the polarizable nucleons of

interest in the target “material” used, for example for NH3, f=3/17

• D is the depolarization factor, kinematics, polarization transfer from polarized lepton to photon, D ~ y2

7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 22

First Moments of SPIN SFs

• With

a a a3=ga 8 0 ΔΣ Neutron decay (3F-D)/3 Hyperon Decay 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 23

p First moment of g1 (x) : Ellis-Jaffe SR

gA a3 = = F + D =1.2601 0.0025 gV ± a =3F D = F/D =0.575 0.016 8 ) ± Assuming SU(3) & s = 0 , Ellis & Jaffe: p f Δ 1 =0.170 0.004 ± Measurements were done at SLAC (E80, E130) Experiments: Low 8-20 GeV electron beam on fixed target -2 Did not reach low enough x à xmin ~ 10 Found consistency of data and E-J sum rule above 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 24

European Muon Collaboration at CERN

• 160 GeV muon beam (lower intensity), but significantly higher energy -3 • Significantly LOWER X reach à xmin ~ 10

• Polarized target

• Repeated experiment for A1 and measured g1 of the proton! 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 25

Proton Spin Crisis (1989)!

ΔΣ = (0.12) +/- (0.17) (EMC, 1989) Ashman et al., EMC Collaboration, NPB 328 (1989) 1 ΔΣ = 0.58 expected from E-J sum rule…. 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 26 How significant is this?

“It could the discovery of the century. Depending, of course on how far below it goes…” 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 27 Insights from the semi-inclusive measure- q and q¯ appear with di↵erent weights in ments are complementary to those from the Eq. (2.11). A large body of semi-inclusive inclusive measurements. Speciﬁcally, they data sensitive to nucleon helicity structure makeFixed it possible totarget delineate the experiments: quark and has been collected by the experiments at anti-quark spin contributions by ﬂavor, since CERN [42, 43, 44] and DESY [45].

10 3 Current polarized DIS data: CERN DESY JLab SLAC

Current polarized BNL-RHIC pp data: PHENIX π0 STAR 1-jet

2 ) 10 2 0.95 ≤ y 0.95 ≤ ≤ y ≤ (GeV 2 Q 10 √s= 140 GeV, 0.01 √s= 45 GeV, 0.01 EIC EIC

-4 -3 -2 -1 10 10 10 10 1 x

Figure 2.5: Regions in x, Q2 covered by previous spin experiments and anticipated to be accessible at an EIC. The values for the existing ﬁxed-target DIS experiments are shown as 2 2 data points. The RHIC data are shown at a scale Q = pT ,wherepT is the observed jet (pion) transverse momentum, and an x value that is representative for the measurement at that scale. The x-ranges probed at di↵erent scales are wide and have considerable overlap. The shaded regions show the x, Q2 reach of an EIC for center-of-mass energy ps = 45 GeV and ps = 140 GeV, respectively.

A further milestone in the study of the in the proton is a major focus and strength of nucleon was the advent of RHIC, the world’s RHIC. Here the main tools are spin asymme- ﬁrst polarized proton+proton collider. In the tries in the production of inclusive pions [46, context of the exploration of nucleon spin 47, 48, 49, 50] and jets [51, 52, 53, 54, 55] structure, the RHIC spin program is a log- at large transverse momentum perpendicular ical continuation. Very much in the spirit to the beam axis, which sets the hard scale of the unpolarized hadron colliders in the Q in these reactions. Their reach in x and 1980’s, RHIC entered the scene to provide Q2 is also indicated in Fig. 2.5. Unlike DIS, complementary information on the nucleon the processes used at RHIC do not probe that is not readily available in ﬁxed-target the partons locally in x, but rather sample lepton scattering. The measurement of the over a region in x. RHIC also provides com- spin-dependent gluon distribution g(x, Q2) plementary information on u, u,¯ d, d¯

23 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 28

Extrapolations! The most simplistic but intuitive theoretical predictions for the polarized deep inelastic scattering are the sum rules for the nucleon structure function g1.

Due to experimental limitations, accessibility of x range is limited, and extrapolations to x= 0 and x = 1 are unavoidable.

Extrapolations to x = 1, are somewhat less problematic: A 1 Small contribution to the integral | 1|  Future precisions studies at JLab 12GeV of great interest

Low x behavior of g1(x) is theoretically not well established hence of significant debate and excitement in the community

7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 29 A collection of low x behaviors: gp 20 1 QCD-fit 1 15 Extrapolation HERA • Low x behavior all over the x(ln(x))2 SMC 10 place

5 Regge Extrapolation 0 • No theoretical guidance for -5 2 1.75 which one is correct 1.5 -10 1.25 1 -15 0.75 0.5 0.25 • Only logical path is though -20 0 -0.25 QCD-Extrapolation -0.5 measurements. -25 -2 -1 10 10 • -30 Polarized HERA not easy -5 -4 -3 -2 -1 10 10 10 10 10 X • Now being considered in view Deshpande, Hughes, Lichtenstadt, HERA low x WS (1999) of the Electron Ion Collider Simulated data for polarized e-p scattering shown in the figure.

A. De Roeck, A. Deshpande, V. Hughes, J. Lichtenstadt, G. Radel, EPJ, C6 (1999), 121 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 30 Aftermath of the EMC Spin “Crisis” Naïve quark model yields: u =4/3 and d = 1/3= ⌃ =1 ) Relativistic effects included quark model: ⌃ =0.6 After much discussions, arguments an idea that became emergent, although not without controversy: “gluon anomaly” • True quark spin is screened by large gluon spin: Altarelli, Ross 2 Carlitz, Collins 2 ↵S(Q ) 2 Mueller et al. ⌃(Q )=⌃0 N g(Q f 2⇡ • But there were strong alternative scenarios proposed that blamed the remaining spin of the proton on: Jaffe, Manohar • Gluon spin (same as above) Ji et al • Orbital motion of quarks and gluons (OAM) It became clear that precision measurements of nucleon spin constitution was needed! 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 31

Experiments

HERMES at DESY

E155 etc. SLAC

Hall A at Jlab

SMC,COMPASS at CERN 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 32

singlet axial-vector current

p 0.08 1 EMC SMC Spin structure¯ Functionsxg Jµ5 =¯uγµγ5u + dγµγ5d +¯sγµγ5s (16) E143 0.06 E155 which is close to one and which goes to one in the limit HERMES 2 • Measurements over the Q .Thesymbolβ denotes the QCD beta func- 0.04 CLAS →∞ COMPASS 2 2 tion β(αs)= 11 f (α /2π)+... and γ is given last twenty odd years − − 3 s ! 2 " 0.02 by γ(αs)=f(αs/π) + ... where f (=3) is the number of active ﬂavors (Kodaira, 1980). The singlet ax- • SMC measure lowest x (0) 0 ial charge, gA inv, is independent of the renormaliza- | (0) 2

• d SLAC/Jlab1 measure tion scale µ and corresponds to gA (Q ) evaluated in the SMC 2 (3) limit Q . The ﬂavor non-singlet axial-charges gA xg E143 →∞ highest0.03 E155 x and g(8) are renormalization group invariants. We are A HERMES free to choose the QCD coupling α (µ) at either a hard s 0.02 CLAS or a soft scale µ. The perturbative QCD expansion of • Data COMPASSconsistent amongst E(αs) remains close to one – even for large values of 0.01 α .Ifwetakeα 0.6 as typical of the infra-red then all experiments over a s s ∼ E(αs) 1 0.13 0.03 + ... =0.84 + ... where 0.13 and 2 ≃ − − 2 − large0 range of Q 0.03 are the (αs)and (αs) corrections respectively. − O O (0) In the naive parton model gA is interpreted as the

fraction of the proton’s spin which is carried by the in- JLAB Hall A n trinsic spin of its quark and anti-quark constituents. The 1 • 0.02 E142 (0) Butxg there should be experimental value of gA is obtained through measuring E154 HERMES g1 and combining the first moment integral in Eq.(12) 0.01 (3) (8) differences if gluon with knowledge of gA and gA from other processes plus theoretical calculation of the perturbative QCD Wilson polarization0 is sizable, the coefficients. −0.01 The isovector axial-charge is measured independently the spin structure functions in neutron β-decays (g(3) =1.270 0.003 (Beringer et al., −0.02 A 2 2012)) and the octet axial charge± is commonly taken should evolve with Q to be the value extracted from hyperon β-decays as- −0.03 (8) −2 −1 suming a 2-parameter SU(3) fit (g =0.58 0.03 10 10 1 A x (Close and Roberts, 1993)). However, it should be± noted (8) the uncertainty quoted for gA has been a matter of FIG. 4 World data on xg1 as a function of x for the proton some debate (Jaffe and Manohar, 1990; Ratcliffe, 2004). (top), the deuteron (middle) and the neutron (bottom) at the SU(3) symmetry may be badly broken and some have Q2 of the measurement. Only data points for Q2 > 1GeV2 (8) and W>2.5GeVareshown.Errorbarsarestatisticalerrors suggested that the error on gA should be as large as 25% (Jaffe and Manohar, 1990). A recent re-evaluation only. of the nucleon’s axial-charges in the Cloudy Bag model taking into account the effect of the one-gluon-exchange hyperfine interaction and the pion cloud plus kaon loops (8) 1999), HERMES (Airapetian et al.,2007a),JLab led to the value gA =0.46 0.05 (Bass and Thomas, (Dharmawardane et al.,2006;Zhenget al.,2004),and ± (8) 2010). The model reduction of gA from the SU(3) value COMPASS (Alekseev et al., 2010d; Alexakhin et al., (3) comes primarily from the pion cloud with gA taking its 2007). There is a general consistency among all data physical value. sets. The kinematic reach of the different experiments is Deep inelastic measurements of g1 have been per- visible in Fig. 5. COMPASS have the smallest-x data, formed in experiments at CERN, DESY, JLab and down to x 0.004. ∼ SLAC. An overview of the world data on the nu- There are several striking features in the data. COM- cleon’s g1 spin structure function is shown in Fig. 4. PASS measurements of the deuteron spin structure func- d d This data is published in EMC (Ashman et al.,1989), tion g1 show the remarkable feature that g1 is consis- SMC (Adeva et al.,1998b),E142(Anthonyet al.,1996), tent with zero in the small-x region between 0.004 and E143 (Abe et al.,1998),E154(Abeet al.,1997), 0.02 (Alexakhin et al., 2007). In contrast, the isovec- p n E155 (Anthony et al.,2000),E155(Anthonyet al., tor part of g is observed to rise at small x as g − 1 1 ∼

13 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 33

How to extract the polarized gluon distribution? 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT

Similar to extraction of PDFs at HERA (RECALL)

NLO pQCD analyses: fits with linear DGLAP* equations

Gluon dominates

34 *Dokshitzer, Gribov, Lipatov, Altarelli, Parisi 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 35 F vs. g structure function measurements 2 1 Aidala et al.1209.2803v2 (This deep inelastic quantity misses any contribution to

i 4 EMC (0) x=0.006 SMC gA inv from a possible delta function at x =0).When

)+c | (8) (0) 2 E143 combined with gA =0.58 0.03, the value of gA pDIS 3.5 E155 ± | x=0.015 in Eq.(17) corresponds to a negative strange-quark po- (x,Q p

1 HERMES g x=0.025 CL A S larization 3 COMPASS x=0.035 1 (0) (8) LSS 05 ∆sQ2 = (gA pDIS,Q2 gA ) →∞ 3 | →∞ − x=0.049 2.5 = 0.08 0.01(stat.) 0.02(syst.)(18) x=0.077 − ± ± – that is, polarized in the opposite direction to the spin F 2 x=0.120 2 g1 of the proton. With this ∆s, the following values for the x=0.170 up and down quark polarizations are obtained 1.5 x=0.240 ∆uQ2 =0.84 0.01(stat.) 0.02(syst.) →∞ ± ± 1 ∆dQ2 = 0.43 0.01(stat.) 0.02(syst.)(19) x=0.340 →∞ − ± ±

The non-zero value of ∆s 2 in Eq.(18) is known as x=0.480 Q 0.5 the violation of the Ellis-Jaff→∞e sum-rule (Ellis and Jaffe, x=0.740 1974). 0 (0) 3 5 2 The extracted value of g pDIS required to be un- 10 10 10 1 1 10 10 10 102 A | Q2 (GeV2) derstood by theory, and the corresponding polarized 2 2 2 2 (8) Q (GeV ) Q (GeV ) strangeness, depend on the value of gA . If we in- 2 2 (8) FIG. 5 World data for g1(x, Q )fortheprotonwithQ > 2 stead use the value gA =0.46 0.05 the correspond- 1GeV and W>2.5GeV.Forclarityaconstantci = ± Large amount of polarized data since 1998… but not in NEW kinematic region! ing experimental value of g(0) would increase to 0.28(11.6 i)hasbeenaddedtotheg1 values within a par- A pDIS − 2 (0) | Large uncertainty in gluon polarization (+/-1.5)ticular x bin results starting with fromi =0for lackx of=0 .wide006. Error Q bars arm gA pDIS =0.36 0.03 0.05 with are statistical errors only. (Also shown is the QCD fit of | ± ± Leader et al. (2006).) ∆s 0.03 0.03. (20) ∼− ± We shall discuss the value of ∆s in more detail in Sections

0.22 0.07 V and VI in connection with more direct measurements x− ± (Alekseev et al., 2010d) and is much bigger d from semi-inclusive deep inelastic scattering plus global than the isoscalar g1 . This compares to the situation in fits to spin data, models and recent lattice calculations the unpolarized structure function F2 where the small-x with disconnected diagrams (quark sea contributions) in- region is dominated by isoscalar gluonic exchanges. cluded. The Bjorken sum-rule (Bjorken, 1966, 1970) for the isovector part of g follows from current algebra and is A. Spin sum-rules 1 a fundamental prediction of QCD. The first moment of the isovector part of g is determined by the nucleon’s To test deep inelastic sum-rules it is necessary to have 1 isovector axial-charge all data points at the same value of Q2. Next-to-leading order (NLO) QCD-motivated fits taking into account the 1 p n 1 (3) ℓ scaling violations associated with perturbative QCD are dxg − = g 1+ cNSℓ α (Q) . (21) 1 6 A s used to evolve all the data points to the same Q2. First 0 ℓ 1 ! " #≥ $ moment sum-rules are then evaluated by extrapolating these fits to x =0andtox = 1, or using a Regge- up to a 1% correction from charge symmetry violation motivated extrapolation of the data. Next-to-leading or- suggested by a recent lattice calculation (Cloet et al., der (NLO) QCD-motivated fits discussed in Section V.C 2012). It has been confirmed in polarized deep inelas- (3) are used to extract from these scaling violations the par- tic scattering at the level of 5%. The value of gA ex- ton distributions and in particular the gluon polarization. tracted from the most recent COMPASS data is 1.28 ± Polarized deep inelastic scattering experiments are in- 0.07(stat.) 0.010(syst.) (Alekseev et al.,2010d)and ± terpreted in terms of a small value for the flavor-singlet compares well with the Particle Data Group value 1.270 ± axial-charge. For example, COMPASS found using the 0.003 deduced from neutron beta-decays (Beringer et al., SU(3) value for g(8) (Alexakhin et al., 2007) and no lead- 2012). A 1 p n The evolution of the Bjorken integral dxg − ing twist subtraction constant xmin 1 as a function of xmin as well as the isosinglet integral (0) 1 p+n % gA pDIS,Q2 =0.33 0.03(stat.) 0.05(syst.). (17) x dxg1 are shown in Fig. 6. The Bjorken sum-rule | →∞ ± ± min % 14 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 36 Insights from the semi-inclusive measure- q and q¯ appear with di↵erent weights in ments are complementary to those from the Eq. (2.11). A large body of semi-inclusive inclusive measurements. Specifically, they data sensitive to nucleon helicity structure makeFixed it possible totarget delineate the experiments: quark and has been collected by the experiments at anti-quark spin contributions by flavor, since CERN [42, 43, 44] and DESY [45].

10 3 Current polarized DIS data: CERN DESY JLab SLAC

Current polarized BNL-RHIC pp data: PHENIX π0 STAR 1-jet

2 ) 10 2 0.95 ≤ y 0.95 ≤ ≤ y ≤ (GeV 2 Q 10 √s= 140 GeV, 0.01 √s= 45 GeV, 0.01 EIC EIC

-4 -3 -2 -1 10 10 10 10 1 x

23 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 37 Global analysis of Spin SF SMC PRD 58 112002 (1998)

• World’s all available g1 data • Coefficient and splitting functions in QCD at NLO • Evolution equations: DGLAP

• Quark distributions fairly well determined, with small uncertainty • ΔΣ = 0.23 +/- 0.04 • Polarized Gluon distribution has largest uncertainties • ΔG = 1 +/- 1.5 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 38 Evolution: Our Understanding of Nucleon Spin

? ANGULAR 1980s 1990/2000s MOMENTUM? GLUONS?

ΔΣ = 0.12 +/- 0.17

Spin Crisis for the prevailing models …. But their limitations were quickly appreciated and discussed.... 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 39

Limitations of fixed target experiments: • Kinematics of fixed target experiments: Does not allow exploration of low-x region • Extraction of gluon polarization needed large Q2 arm, and fixed target experiments did not allow that either….

• In 1990’s ideas to achieve high energy polarized proton beams evolved… Siberian Snake Magnets • High energy polarized proton beam polarimetry was developed as a future need…

• Ideally we needed a polarized e-p collider to overcome this, but non was under consideration! (although polarized HERA was proposed but failed) 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 40

NEED FOR A POLARIZED COLLIDER WAS UNDERSTOOD, TO MEASURE THE EXPECTED LARGE GLUON POLARIZATION

HOWEVER NO PLANS EXISTED FOR SUCH A COLLIDEER.

RHIC: RELATIVISTIC HEAVY ION COLLIDER WAS BEING PLANNED FOR THE INVESTIGATIONS OF QUARK GLUON PLASMA (AT BNL)

THE JAPANESE (RIKEN INSTITUTE) JUMPED IN WITH T.D. LEE AND DIR. SAMIOS, TO INSTALL SIBERIAN SNAKE MAGNETS AND SPIN ROTATOR MAGNETS TO ENABLE POLARIZED PROTON COLLISIONS IN RHIC. 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 41

Motivation for RHIC Spin:

• If gluons really carry the bulk of nucleon’s spin, why not use polarized proton (known by then to be predominantly made of gluons!)? • Technical know-how (Siberian Snakes, Spin Rotators, polarimetry ideas) to do this at high energy evolved around the time (mid/late-1990s)

• Why ΔΣ (quark + anti-quark’s spin) small? Are quark and anti- quark spins anti-aligned? Polarized p+p at high energy, through W+/- production could address this

• A severe need for investigations of the surprising transverse spin effects was naturally possible and needed with the proposed polarized p+p collider… 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 42 RHIC as a Polarized Proton Collider Absolute Polarimeter (H↑ jet) pC Polarimeters

PHOBOS BRAHMS

Siberian Snakes Siberian Snakes PHENIX STAR

Spin Rotators Spin flipper (longitudinal polarization) Spin Rotators (longitudinal polarization)

LINAC BOOSTER 5.9% Helical Partial Siberian Snake

- Pol. H Source AGS Internal Polarimeter 200 MeV Polarimeter pC Polarimeter 10-25% Helical Partial Siberian Snake

Without Siberian snakes: νsp = Gγ = 1.79 E/m → ~1000 depolarizing resonances ¡ With Siberian snakes (local 180 spin rotators): νsp = ½ → no first order resonances Two partial Siberian snakes (11¡ and 27¡ spin rotators) in AGS 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 43

Measuring ALL

dσ ++ − dσ +− 1 N++ − RN+− L++ ALL = = ; R = dσ ++ + dσ +− | P1P2 | N++ − RN+− L+−

(N) Yield (R) Relative Luminosity (P) Polarization

Exquisite control over false asymmetries due to ultra fast rotations of the target and probe spin.

ü Bunch spin configuration alternates every 106 ns ü Data for all bunch spin configurations are collected at the same time ⇒ Possibility for false asymmetries are greatly reduced 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 44 ThePolarized probes pp scatteringand techniques (RHIC) :at! RHIC NLO:

Jäger, Schäfer, Stratmann, WV!

Jäger,Stratmann, WV; Signer et al.

Gordon, WV; Contogouris et al.; Gordon, Coriano; Frixione. WV

Stratmann, Bojak

Weber; Gehrmann; Kamal; Smith, van Neerven, Ravindran; Nadolsky, Yuan; de Florian, WV and Figure 3-2 also include the preliminary point, illustrates the importance of the 510 GeV PHENIX and STAR results for inclusive π0 and RHIC measurements in extending the kinematic jet ALL measurements at √s = 510 GeV from Run- reach. Both new data sets also exhibit high statis- 2013 and Run-2012 (red points), respectively, as tical precision and small systematic uncertainties, a function of xT compared to three recent QCD which is crucial for making a significant impact global analyses (red curves). Taking xT as a in upcoming global QCD analyses of helicity rough proxy for the smallest momentum fraction PDFs. x at which Δg can be sampled by a given data

Figure 3-3: 90% C.L. areas in the plane spanned by the truncated moments of Δg computed for 0.05 < x < 1 and 0.001 < x < 0.05 at Q2=10 GeV2. Results for DSSV, DSSV* (including preliminary and final RHIC data up to Run-2006, respectively) and the new DSSV analysis based on RHIC data up to Run-2009 are shown, with the and Figure 3-2 also include the preliminary point, illustrates the importance of the 510 GeV symbols corresponding to the respective central values of 0 RHIC measurements in extending the kinematic PHENIX and STAR results for inclusive π and each central fit. jet ALL measurements at √s = 510 GeV from Run- reach. Both new data sets also exhibit high statis- 2013 and Run-2012 (red points), respectively, as tical precision and small systematic uncertainties, a function of xT compared to three recent QCD which is crucial for making a significant impact global analyses (red curves). Taking xT as a in upcoming global QCD analyses of helicity rough proxy for the smallest momentum fraction PDFs. x at which Δg can be sampled by a given data and Figure 3-2 also include the preliminary point, illustrates the importance of the 510 GeV PHENIX7/18/16 and STAR results for inclusive π0 and EIC LectureRHIC 2 measurementsat NNPSS 2016 at MITin extending 45 the kinematic 0.003 jet ALL measurements at √s = 510 GeV from Run- reach. Both new data sets also exhibit high statis- 2013 and Run-2012 (red points), respectively, as tical precision0 PHENIX and small MPC systematic 2013 uncertainties, 0.002 A a function of x compared to three recent QCD whichLL is crucialAnticipated for making X ~10a significant-3 impact g @ RHICT Current status: Figure 3-3: 90% C.L. areas in the planemin spanned by the globalΔ analyses (red curves). Taking xT as a 0.001truncatedin upcoming moments of globalΔg computed QCD for 0.05 analyses < x < 1 and of helicity rough proxy for the smallest momentum fraction 0.001PDFs. < x < 0.05 at Q2=10 GeV2. Results for DSSV, 0.05 < x < 0.2 DSSV* (including preliminary and final RHIC data up to 0 x at which Δg can be sampled by a given data Run-2006, respectively) and the new DSSV analysis based on RHIC data up to Run-2009 are shown, with the -0.001symbols corresponding to the respective central values of each central fit.DSSV 2014 with 90% C.L. band

-0.002 proj. incl. 0 data: 510 GeV, 3.1 < || < 3.9 -0.003 345 pT [GeV] Figure 3-3: 90% C.L. areas in the plane spanned by the STAR 2015

truncated moments of Δg computed for 0.05 < x < 1 and 0.003 2 2 à 0.001 < x < 0.05 at Q =10 GeV . Results for DSSV, 0 0.002 DSSV* (including preliminary and final RHIC data up to ALL Figure 3-4: The anticipated statistical precision for Figure 3-5: The projected statistical precision for ALL vs. pT for 0 inclusiveRun-200 π A6LL, measurements respectively) atand √s=510 the GeV new in DSSVthe inclusive analysis jets in 200 GeV p+p collisions based on the com- 0.001 pseudobasedrapidity on RHIC range data 3.1

Unmeasuredx with 90% C.L. band -0.002 proj. incl. 0 data: 510 GeV, 3.1 < || < 3.9 -0.003 Figure 3-4 illustrates the anticipated statistical electromagnetic calorimeter (MPC). The esti- 345 0 pT [GeV] precision for future inclusive π ALL measure- mates are based on the recorded luminosity in Measured x à ments at √s=510 GeV in the pseudorapidity Run-2013. Also shown is a theoretical expecta- range 3.1 < η < 3.9 with the PHENIX forward tion based on the latest DSSV analysis including

0.003 Figure 3-4: The anticipated statistical precision for Figure 3-5: The projected statistical precision for ALL vs. pT for 15 0 inclusive π ALL measurements at √s=510 GeV in the inclusive jets in 200 GeV p+p collisions based on the com- 0 bined data from the 2009 and 2015 RHIC runs, compared to 0.002 pseudorapidity range 3.1 < η < 3.9 with the PHE- ALLNIX MPC in 2013 compared to the theoretical ex- the uncertainties from the DSSV-2014 fit, which included the pectation based on the latest DSSV-2014 PDFs in- 2009 inclusive jet results among the inputs. 0.001 cluding 90% C.L. uncertainties.

0 Figure 3-4 illustrates the anticipated statistical electromagnetic calorimeter (MPC). The esti- 0 precision for future inclusive π ALL measure- mates are based on the recorded luminosity in -0.001 ments at √s=510 GeV in the pseudorapidity Run-2013. Also shown is a theoretical expecta- DSSV 2014 tion based on the latest DSSV analysis including rangewith 90% 3.1 C.L. band < η < 3.9 with the PHENIX forward -0.002 proj. incl. 0 data: 15 510 GeV, 3.1 < || < 3.9 -0.003 345 pT [GeV]

Figure 3-4: The anticipated statistical precision for Figure 3-5: The projected statistical precision for ALL vs. pT for 0 inclusive π ALL measurements at √s=510 GeV in the inclusive jets in 200 GeV p+p collisions based on the com- pseudorapidity range 3.1 < η < 3.9 with the PHE- bined data from the 2009 and 2015 RHIC runs, compared to NIX MPC in 2013 compared to the theoretical ex- the uncertainties from the DSSV-2014 fit, which included the pectation based on the latest DSSV-2014 PDFs in- 2009 inclusive jet results among the inputs. cluding 90% C.L. uncertainties.

Figure 3-4 illustrates the anticipated statistical electromagnetic calorimeter (MPC). The esti- 0 precision for future inclusive π ALL measure- mates are based on the recorded luminosity in ments at √s=510 GeV in the pseudorapidity Run-2013. Also shown is a theoretical expecta- range 3.1 < η < 3.9 with the PHENIX forward tion based on the latest DSSV analysis including 15 2 2 the current uncertainties on Δg. STAR performed tion of xmin at Q = 10 GeV and uncertainty esti- a similar measurement utilizing its forward elec- mates at 90% C.L. based on DSSV framework tromagnetic calorimeter (FMS) at 2.5 < η < 3.8 with and without including the various sets of [10]. The significant impact of these new meas- pseudo-data for PHENIX and STAR. The solid urements is obvious from comparing the antici- line and the outer band reflect our current pated statistical precision with the current range knowledge of Δg based on published RHIC and of theoretical uncertainties, which are mainly other world data similar to what was already driven by our ignorance of Δg at momentum shown in Figure 3-3. The inner uncertainty band fractions in the range from a few times 10-3 to illustrates the reduction of uncertainties due to about 0.01. Data on ALL at forward rapidities will the combined set of the projected RHIC meas- be an ideal addition to the suite of measurements urements discussed above. As can be seen, the performed so far. preliminary and upcoming RHIC data are ex- The projected statistical precision for ALL vs. pected to reduce the present uncertainties on the pT for inclusive jets in 200 GeV p+p collisions truncated integral by about a factor of 2 at xmin = based on the combined data from the 2009 and 10-3 where RHIC data can provide an experi- 2015 RHIC runs is shown in Figure 3-5 com- mental constraint. For illustration, Figure 3-6 -6 pared to theoretical expectation based on the lat- extends down to an xmin of 10 well outside the x- est DSSV-2014 analysis. region covered by RHIC data. It is interesting to To estimate and demonstrate the potential im- notice that the optimum fit of DSSV prefers an pact of all these preliminary and forthcoming integrated gluon helicity distribution of about RHIC data for ALL at √s = (200) 510 GeV up to 0.36, i.e., the gluons carry more than 70% of the Run-2015 (see Table 3-1) on our knowledge of proton’s spin in this scenario at a scale Q2 = 10 7/18/16 Δg, the DSSV group has performed aEIC global Lecture 2 at NNPSSGeV 20162. About at MIT half46 of this value already stems analysis with pseudo-data which reflect the antic- from the x-range covered by published RHIC ipated experimental precision. Figure 3-6 shows data, as was demonstrated in Figure 3-3. ! the running integral !"∆!(!, !!) as a func- RHIC’s limit…!!"# no/limited handle on low-x

) 1 2 2 2 Q = 10 GeV DSSV 2014 with 90% C.L. band

g(x,Q 0.8 projection with RHIC data 2015: STAR: Figure 3-6: The running integral for Δg 1-jet 200 & 510 GeV 2 2 min 1

dx as a function of xmin at Q = 10 GeV as x PHENIX: 0.6 incl. 0 510 GeV at obtained in the DSSV global analysis mid & fwd rapidity framework. The inner and outer uncertainty bands at 90% C.L. are estimated with and without including the combined 0.4 set of projected pseudo-data for preliminary and future RHIC measurements up to Run-2015 (see Table 3-1), respective-

0.2 RHICly. Spin

White Paper (2015) For NSAC LRP 0 -6 -5 -4 -3 -2 -1 10 10 10 10 10 10 x 1 min

Figure 3-7 provides a complementary view of happens outside of that interval, rather than com- the already achieved and still to be expected im- bining all contributions from xmin to 1 simultane- provements on our knowledge of Δg from the ously as in Figure 3-6. Results are shown for the RHIC spin program. Here we look into the max- original DSSV’08 analysis based on PHENIX imum allowed variation by the data in a small bin and STAR data only up to Run-2006, the recent of momentum fraction x and regardless of what update including the published Run-2009 data, 16 The latest: W-boson production at RHIC

7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 47 Anti-Quark Polarization measurement via u W production and decay unpol.The latest: W-boson production at RHIC • Large parity violating effect anticipated

! large Parity Violation effect u ! The latest: W-boson productioncomplementary at RHIC to SIDIS,• Measurementbut cleaner complimentary to SIDIS, but devoid of fragmentation unpol. ! new NLO for polarized case:function de makesFlorian, itWV cleaner! tailored for inclusion in global analysis u • NLO analyses about now available ! large Parity Violation effect unpol. ! complementary to SIDIS, but cleaner

! new NLO for polarized case: de Florian, WV tailored for inclusion in global analysis ! large Parity Violation effect

! complementary to SIDIS, but cleaner

! new NLO for polarized case: de Florian, WV tailored for inclusion in global analysis 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 48

Some insight in to what goes on….

1 2 1 2 1 2 1 2 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 49 W production @ RHIC

W AL 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 50

What about the anti-quark polarization?

• DIS probe (γ*) doesn’t distinguish q from qbar • Has to take measure semi-inclusive (π, K production) • Uncertainties in fragmentation functions • High energy p-p collisions enable probing q,qbar through W+/- production –> Plan at RHIC 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 51

In parallel: The Transverse Spin Puzzle Had been observed but ignored for almost 3 decades… 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 52

Transverse spin introduction

!" L! NL NR AN = R NL + NR

mq AN ↵S Kane, Pumplin, Repko 1978 ⇠ pT “Single-spinPRL 41 1689 asymmetry” (1978)

• Since people starved to measure effects at high pT to interpret them in pQCD frameworks, this was “neglected” as it was expected to be small….. However…. • Pion production in single transverse spin collisions showed us something different….

• expect AN ~ in simple parton model Kane, Pumplin, Repko ‘78 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 53 Pion asymmetries: at most CM energies! max xF = PL/PL =2PL/ps

ZGS/ANL AGS/BNL FNAL RHIC √s=4.9 GeV √s=6.6 GeV √s=19.4 GeV √s=62.4 GeV

Suspect soft QCD effects at low scales, but they seem to remain relevant to perturbative regimes as well Sivers effect: due to transverse motion of quarks in the nucleon: initial state effect Phys Rev D41 (1990) 83; Phys Rev D43 (1991) 261

xˆ zˆ yˆ x is longitudinal momentum Top view fraction.

Quark transverse momentum in Front view transversely polarized proton.

INITIAL STATE EFFECT: Orbital angular momentum? EIC Lecture 2 at NNPSS 2016 at MIT 7/18/16 54 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 55 What does “Sivers effect” probe? Top view, Breit frame Quarks orbital motion adds/ subtracts longitudinal momentum xˆ for negative/positive xˆ . Red shift zˆ yˆ PRD66 (2002) 114005 r Hard probe Parton Distribution S1 (Parton, γ*) Functions rapidly fall in longitudinal momentum Blue shift fraction x. Final State Interaction between outgoing quark and target spectator.

hep-ph/ Quark Orbital angular Sivers function 0703176 momentum Generalized Parton Distribution Functions PRD59 (1999) 014013 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 56 Collins (Heppelmann) effect: Asymmetry in the fragmentation hadrons

Nucl Phys B396 (1993) 161, Example: ↑ Nucl Phys B420 (1994) 565 p + p → h1 + h2 + X r S1 r pq' r ⊥ Sq r h 2 r h 1 r ⊥ k h Polarization of struck quark which fragments to hadrons. r r h1 + h 2 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 57 π π,K,p π 200 & 62.4 200 GeV GeV 62.4 GeV

K • Scales on plots K different

200 GeV • Kaon 62.4 GeV asymmetries not predicted • Unfortunately p no anti-proton p measurement 200 GeV 62.4 GeV 57 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 58

Although not expected, at any observable level, 400+ times the expected values of asymmetries have been routinely seen experimentally: both in ep and pp systems.

• Transverse motion/momentum of partons (indirect evidence for orbital angular momentum of the quarks & gluons?)

• Asymmetry in fragmentation process (final state) or • Both …. May be responsible. 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 59 Summary

• RHIC Spin program made great strides in furthering our knowledge of nucleon, but we need something more to address some of the still open issues:

• • ΔG at low x? Spin structure functions and its behavior at low x? • If orbital angular motion plays a role, what is the orbital contribution from Gluons? Precision measurements at a future facility are essential…

THE ELECTRON ION COLLIDER.... Can we do better? 7/18/16 EIC Lecture 2 at NNPSS 2016 at MIT 60 Study of internal structure of a watermelon:

A-A (RHIC) 1) Violent collision of melons

2) Cutting the watermelon with a knife Violent DIS e-A (EIC) 3) MRI of a watermelon Non-Violent e-A (EIC)