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Nucleon Spin Structure from Polarized Deep Inelastic

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Nucleon Spin Structure from Polarized Deep Inelastic EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH 4 Decemb er 1995 CERN{PPE/95{178 NUCLEON SPIN STRUCTURE FROM POLARIZED DEEP INELASTIC MUON-NUCLEON SCATTERING AT CERN Vernon W. Hughes (On b ehalf of the Spin Muon Collab ora tio n) Physics Department, Yale University, 260 Whitney Avenue, 465 J.W. Gibbs, New Haven, CT 06520-8121 , USA E-mail: [email protected] Abstract Polarized deep inelastic inclusivemuon-nucleon scattering has b een measured at + CERN by the Spin Muon Collab oration (SMC) with incident energies of 100 and 190 GeV and with p olarized proton and deuteron targets in an x interval of 0.003 2 2 to 0.7, with Q ranging from 1 to 60 GeV . The spin dep endent structure functions g have b een determined for the proton, deuteron and neutron. The fundamental 1 Bjorken p olarization sum rule has b een con rmed, and the Ellis-Ja e sum rules have b een found to b e invalid, with the consequences that the fraction of the nucleon spin contributed by the quark spins is small and strange quarks have a net negative p olarization. 1 INTRODUCTION The spin structure of the nucleon can b e studied from p olarized deep inelastic inclu- sivemuon-nucleon scattering [1{5]. Spin structure functions have not yet b een calculated from basic QCD, but are useful for testing nucleon mo dels. With their rst moments a basic test of QCD is made and imp ortant information ab out the comp osition of nucleon spin is obtained. The intrinsic nucleon spin angular momentum of 1/2 must equal the sum of the spin contributions from the quark spins , from the gluon spins G, and from the orbital angular momentum L of quarks and gluons, i.e. 1 +G+L =1=2 (1) z 2 The quark contribution has b een obtained. The kinematics and variables for p olarized inclusivemuon-proton scattering in the 2 2 single photon exchange approximation are given in Fig. 1. G (; Q ) and G (; Q ) are 1 2 spin dep endent structure functions. They are related in the scaling limit to the two spin dep endent structure functions g (x) and g (x). 1 2 2 2 M G (; Q )= g (x) 1 1 2 2 M G (; Q )= g (x) 2 2 Spin dep endent structure functions are determined by measuring di erences or asymme- tries in the di erential scattering cross sections for the antiparallel and parallel spin cases. These determine the virtual photon-proton absorption asymmetry A . 1 The counting rate asymmetry is related to the intrinsic muon-proton scattering cross-section asymmetry as shown in Fig. 2. A is a virtual photon-proton asymmetry 2 arising from the interference of longitudinal and transverse photon p olarizatio ns, and the kinematic factor is small for longitudinal muon and proton p olarizations. The kine- matic dep olarization factor D relates the muon b eam p olarizatio n to the virtual photon p olarizatio n. A should scale, and p ositivity limits are indicated for A and A . The usual 1 1 2 + expressions for F and g in the quark-parton mo del are given, in which q (x)[q (x)] is 1 1 i i the probability that a quark of avour i has its spin parallel [antiparallel] to the proton spin. Spin p olarization sum rules are given in Fig. 3. The Bjorken p olarization sum rule is a rigorous consequence of QCD in the scaling limit, indep endent of the mo del of the nucleon [6]. Perturbative QCD corrections to the Bjorken sum rule are given [7]. This sum rule assumes only that a general quark-parton picture of the nucleon applies and that the weak interactions of quarks and leptons are the same. The Ellis-Ja e sum rules for the rst moments of the proton and neutron structure functions separately are mo del-dep endent and assume that strange quarks in the nucleon sea are unp olarized and that SU(3) symmetry applies [8]. Perturbative QCD corrections F are given [9]. Formulae are given in Fig. 4 for the rst moments of the proton and neutron spin structure functions based on the op erator pro duct expansion which is basically a p er- 2 turbative QCD approach excluding non-p erturbativelowQ e ects. Assuming isospin invariance and SU(3) symmetry and using measured values for g =g from neutron b eta F A V decay and of F/D from hyp eron b eta decay, the fraction of the nucleon spin contributed by the spins of all quarks, , can b e determined. Also the fractional angular momenta due to u,d, and s quarks separately can b e evaluated. 1 k, sµ ′ Polarized k Scattered muon muon Virtual q photon P, S Px Hadron final state Polarized Wµν nucleon Wµν (F1, F2, G1, G2) Kinematic variables (Lab oratory frame) m muon rest mass M proton rest mass s electron spin-four-vector S =(O; S ) proton-spin four-vector k =(E; k) four-momentum of incidentmuon 0 0 0 k =(E ;k ) four-momentum of scattered muon P =(M; O) four-momentum target proton 0 q = k k =(; q ) four-momentum transfer 2 2 2 0 2 Q = q =4EE sin (=2) {(invariant mass) of virtual photon 0 energy of the virtual photon in the lab oratory = P q=M = E E scattering angle in the lab oratory 2 x = Q =2M Bjorken scaling variable y = =E fractional energy loss of scattered electron P four-momentum of hadronic nal state x 0 1 " 2 2 2 B C a cos d F F 2 1 2 2 0 2 B C = + 2 tan 2 tan (E + E cos ) @ A 0 d dE 2 M 2 2 4 4E sin 2 # +(A) 0 2 2 MG 8EE tan sin G 1 2 2 2 (P ) 2 2 d d ("#) ("") 0 2 MG (E + E cos ) Q G 0 0 1 2 d dE d dE A = 2 2 F F d d l 2 + ("#)+ ("") 2 0 0 M 2 tan d dE d dE 2 Virtual photon-proton absorption P γ ⇒ z–axis collision P Y J =+1=2 J =1 z z " " Y(J = +1); Y + P ! z 3=2 # " Y (J = 1); Y + P ! z 1=2 1=2 3=2 A = 1 + 1=2 3=2 Figure 1: Polarized muon-proton scattering: kinematics and asymmetries. 2 N("#) N ("") = Counting rate asymmetry N ("#)+("") = P P fA p P = Muon b eam p olarization ' 0:8 P = Proton target p olarization ' 0:8 p f =Fraction of p olarizable protons in target (dilution factor) ' 0:1 Hence ' 0:06 A A = D (A + A ) 1 2 A ' DA 1 2 1=2 3=2 TL A = ; A = 1 2 + + 1=2 3=2 1=2 3=2 D and are kinematic quantities 0 2 D = (E E ")=E (1 + "R)=y(2 y )=[y + 2(1 y )(1 + R)] q q 0 2 2 = " Q =(E E ") = [2(1 y )=y (2 y )]( Q =E ) where " # 1 2 Q 2 2 " = 1 + 2(1 + =Q ) tan 2 and R = = L T Scaling relation Positivity relations p 2 A (; Q ) ! A (x) jA j 1; jA j R 1 1 1 2 2 (; Q ) !1 ; x f ixed A F 1 2 g = = A F 1 1 1 2x(1 + R) X 1 2 + F (x)= e [q (x)+q (x)] 1 i i i 2 i X 1 2 + g (x)= e [q (x)q (x)] 1 i i i 2 i Figure 2: Measurement and prop erties of spin-dep endent quantities. 3 Z 1 g 1 A p p n n =0:210(1) (Bjorken) = (g g )dx = 1 1 1 1 6 g 0 V 3 With O ( ) p erturbative QCD corrections S 2 3 ! ! ! 2 3 2 2 2 (Q ) (Q ) (Q) 1 g S S S A p n 4 5 1+C = +C +C 1 2 3 1 1 6 g V 2 2 = 0:187(3) at Q = 10 GeV With C = 1; C =3:58 ; C = 20:21 . 1 2 3 --------------------------------------------------------- # " g 5 3F=D 1 1 A p =0:185(3) (Ellis-Ja e) 1+ = 1 12 g 3 F=D +1 V # " 1 g 5 3F=D 1 A n = =0:024(3) (Ellis-Ja e) 1+ 1 12 g 3 F=D +1 V 3 With O ( ) p erturbative QCD corrections for the non-singlet co ecient C and with NS S 2 O ( ) for the singlet co ecient C . SI S 1 1 (3F D ) p(n) = C (F + D )+ (3F D ) + C NS SI 1 12 36 9 p 2 n 2 =0:170(5) = 0:017(4) Q = 10 GeV 1 1 --------------------------------------------------------- g =g =1:2573(28) (Ratio of axial to vector weak coupling constant in neutron A V b eta decay.) 2 (M )=0:117(5); (10 GeV )= 0:24(2) (Strong coupling constant.) [25] S z S F=D =0:575(16) (Co ecients in hyp eron semileptonic decay.) [14] Three quark avours Figure 3: Spin p olarization sum rules. 4 Z 1 1 1 1 p(n) +() g dx = C (F + D )+ (3F D ) + C NS SI 1 12 36 9 0 3 With O ( ) p erturbative QCD corrections for the non-singlet co ecient C and with NS S 2 O ( ) for the singlet co ecient C , using three quark avours: SI S 2 3 p(n) g 1 1 A S S S 3:58 20:21 +() + = 1 S 1 g 12 36 V 2 1 3 S S + 1 0:33 0:55 + O ( ) S 9 g A =3F D =1:2573(28) Isospin invariance a = 3 g V a = 3F D =0:579(32) SU (3) symmetry g In the Quark Parton Mo del Z n o 1 + + q = [q (x) q (x)+q (x) q (x)] dx 0 a = (u +d+s)= 0 a = (u d) 3 a = (u +d2s) g For the Ellis-Ja e sum rule, s = 0 and a = a =(u+d).
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