陽子スピンの分解 八田佳孝 ( 京大基研 )

Transcription

1 陽子スピンの分解 八田佳孝 ( 京大基研 )

2 Outline QCD spin physics Proton spin decomposition: Problems and resolution Orbital angular momentum Twist analysis Transverse polarization Method to compute G on a lattice (PRD) (PLB) (JHEP) with Shinsuke Yoshida ( 吉田信介 ) (JHEP) with Kazuhiro Tanaka ( 田中和廣 )and S. Yoshida (PRD) with Xiangdong Ji ( 季向東 )and Yong Zhao ( 趙勇 )

3 The proton spin problem The proton has spin ½. The proton is not an elementary particle. Quarks helicity Gluons helicity Oribital angular Momentum (OAM) Quark model prediction: ¼ 0:6 with relativistic effects

4 `Spin crisis In 1987, EMC (European Muon Collaboration) announced a very small value of the quark helicity contribution!? Recent results from NLO QCD global analysis

5 World s facilities for high energy spin physics BNL-RHIC : PHENIX, STAR, BRAHMS pp Single (double) spin asymmetry CERN-SPS : EMC, COMPASS ¹ p DIS DESY-HERA : HERMES ep DIS, DVCS J-Lab : CLAS, Hall-A ep DVCS, SIDIS J-PARC : pp Drell-Yan, SSA

6 How to measure Longitudinal spin asymmetry in polarized DIS A LL ¹" p " ¹ " p # ¹ " p " + ¹ " p # = µ 1 + ¾ L ¾ T g1 F 1

7 Determination of G PHENIX, STAR, COMPASS, JLab Longitudinal spin asymmetry in pp A ¼ LL = d¾++ d¾ + d¾ ++ + d¾ +» X a;b f a - f b - ¾ a;b - D ¼

8 Latest global QCD analysis DeFlorian, Sassot, Stratmann, Vogelsang,

9 QCD angular momentum tensor QCD Lagrangian Lorentz invariant Noether current QCD angular momentum tensor quark spin gluon spin canonical energy momentum tensor Quark OAM Gluon OAM

10 Jaffe-Manohar decomposition (1990) 1 2 = G + Lq can + L g can Based on the canonical energy momentum tensor Operators NOT gauge invariant. Partonic interpretation in the light-cone gauge A + = 0

11 Ji decomposition (1997) Improved (Belinfante) energy momentum tensor et ¹º = T ¹º can ½ G ½¹º One can add a total derivative. = ¹ Ãi (¹Ã! D º) à F ¹½ F º ½ g ¹º L quark part gluon part Further decomposition in the quark part ¹Ãi (¹Ã! D º) à = ¹ Ãi ¹Ã! D º à 1 4 ½ ( ¹ à 5 ¾ Ã) J q = L q

12 Generalized parton distributions (GPD) Non-forward proton matrix element J q = 1 2 Z dxx(h q (x) + E q (x)) J g = 1 4 Z dx(h g (x) + E g (x)) Measurable in Deeply Virtual Compton Scattering

13 Jaffe-Manohar Two spin communities divided measured by PHENIX, STAR, COMPASS, HERMES 1 2 = G + Lq can + L g can Ji common and well-known not measured yet not even well-defined? Define rigorously. Must be related to GPDs! accessible from GPD at COMPASS, HERMES, JLab, J-PARC also calculated in lattice QCD

14 Complete decomposition Chen, Lu, Sun, Wang, Goldman (2008) Wakamatsu (2010) Y.H. (2011) My choice A ¹ phys = 1 D + F +¹ D ¹ pure = D ¹ ia ¹ phys Gauge invariant completion of Jaffe-Manohar 1 2 = G + Lq can + L g can

15 OAM from the Wigner distribution Wigner distribution in QCD Belitsky, Ji, Yuan (2003) W(x; q) = Z d 4 z (2¼) 4 eiqz à ¹ ³x z ¹ à 2 ³ x + z 2 position momentum Need a Wilson line! Define ~L = Z dq ~x ~q hw(x; q)i Lorce, Pasquini (2011) Which OAM is this??

16 Canonical OAM from the light-cone Wilson line Z YH (2011) dq ~x ~q hw light cone (x; q)i = h ¹ Ã ¹ ~x i Ã! D pure Ãi Naturally defined in infinite momentum frame. Parton interpretation possible. Kinetic OAM from the straight Wilson line Z Ji, Xiong, Yuan (2012) dq ~x ~q hw straight (x; q)i x x + z 2 = h ¹ Ã ¹ ~x i Ã! D Ãi x z 2

17 Twist analysis 1 2 = G + Lq can + L g can Understand these relations at the density level YH, Yoshida (2012) = X f Z dx q f (x) G = Z dx G(x)?? c.f. q(x) = 1 4¼S + Z dze ixp z hpsj ¹ Ã(z) + 5 Ã(0)jPSi

18 Density of OAM Ji s OAM canonical OAM `potential OAM A ¹ phys = 1 D + F +¹ For a 3-body operator, it is natural to define the double density. Z d d¹e i 2 (x 1+x 2 )+i¹(x 1 x 2 ) hp 0 S 0 j ¹ Ã( =2)D i (¹)Ã( =2)jP Si» ² ij j S + D (x 1 ; x 2 ) ``D-type ``F-type

19 The D-type and F-type correlators are related. Eguchi, Koike, Tanaka (2006) doubly-unintegrate The gluon has zero energy density interpretation Canonical OAM density No unique density for Ji s OAM.

20 Relation to twist-3 GPD twist-2 twist-3 From the equation of motion,

21 Quark canonical OAM density Wandzura-Wilczek part First moment: genuine twist-three The bridge between JM and Ji

22 Gluon canonical OAM density Relation between F- and D-type three-gluon correlators Related to a twist-3 gluon GPD First moment:

23 Transverse spin decomposition It s important to use the Pauli-Lubanski vector W ¹ = 1 Z 2 ²¹º½¾ P º d 3 xm + ½¾ instead of the angular momentum tensor M +¹º itself. Ji (1996) Ji, Xiong, Yuan (2012) YH, Tanaka, Yoshida (2012) Leader (2012) Frame dependent!

24 Longitudinal Complete transverse spin decomposition? 1 2 = G + Lq can + L g can Transverse same! 1 2 = 1 + G + Lq+g can 2 cannot be separated in a frame-independent way

25 New development: Measuring PDF on a lattice? Consider the usual quark distribution function. q f (x; Q 2 ) = 1 4¼ R dy e ixp + y hpj¹q f (0)W[0; y ] + q f (y )jpi Q 2 Nonlocal operator along the light-cone Real-time problem Local operator after taking the moment q f (j; Q 2 ) = = Z 1 0 dxx j 1 q f (x; Q 2 ) 1 2(P + ) j hp j¹q f(0) + (id + ) j 1 q f (0)jP i

26 G = = Z Z Computing dx G(x) Z i dy dx 2xP + 2¼ e ixp G on a lattice Ji, Zhang, Zhao (2013) YH, Ji, Zhao (2013) + y hp SjF +¹ (y )W [y ; 0] F ~ ¹ + (0)jP Si Nonlocal even though it is a x-moment!?? e G(P z ) = 1 2P 0 hpsj²ij F i0 A j jpsi Local, but gauge variant. Calculable on a lattice. Calculate G(P e z ) in a universality class of gauges and do the matching. ~G(P z ; ¹) = Z gg (P z =¹) G(¹) + Z gq (P z =¹) (¹) x-dependence may also be calculable Ji (2013)

27 Summary Complete gauge invariant decomposition of nucleon spin now available in QCD, even at the density level. 1 2 = G + Lq can + L g can Relation between the two decomposition schemes (JM vs Ji) fully revealed. The connection to twist-3 GPDs clarified.