Nucleon spin decomposition at twist-three Yoshitaka Hatta (Yukawa inst., Kyoto U.)
Outline Ji decomposition Complete decomposition Canonical orbital angular momentum Twist analysis Transversely polarized case Refs: 1101.5989 (PRD) 1111.3547 (PLB) 1207.5332 (JHEP) with Shinsuke Yoshida 1211.2918 (JHEP) with Shinsuke Yoshida and Kazuhiro Tanaka
Ji decomposition J q = 1 2 dxx(h q (x) + E q (x)) J g = 1 4 dx(h g (x) + E g (x)) Based on the Belinfante-improved energy momentum tensor Operators local and gauge invariant Related to twist-two GPDs, measurable in DVCS Further decomposition in the quark part J q = 1 2 + L q
Frequently Asked Questions G G Where is? Everybody says is important to understand the nucleon spin structure. Can we interpret the integrand as the `angular momentum density of partons with momentum fraction? Commutation relation ~L L ~ = i L ~ must be abandoned?* Would be nice to have it at least at tree level. x [D ¹ ; D º ] = igf ¹º 6= 0 Is this valid for the longitudinal polarization, or transverse, or both? In any frame? Go to twist-3
Twist-2 and twist-3 GPDs Early hint at twist-three Penttinen, Polyakov, Shuvaev, Strikman (2000) In the parton model, dxxg 3 (x) = L q related to OAM
Complete decomposition Chen, Lu, Sun, Wang, Goldman Wakamatsu Define the `physical and `pure gauge parts of the gauge field Y.H. A ¹ = A ¹ phys + A¹ pure My choice A ¹ phys (x) = Ã dy µ(y x )P exp ig x! A + (z )dz 0 F +¹ (y ) y
Remarks Gauge invariant completion of Jaffe-Manohar Gluon helicity part coincides with G Bashinsky, Jaffe (1998); Y.H. (2011) 1 2 = 1 2 + G + Lq can + L g can A lot of confusion about what A phys is. My is the infinite momentum limit of Chen et al. Each component measurable on the lattice!? A phys Ji, hang, hao, 1304.6708
Orbital angular momentum kinetic OAM ¹Ã ¹ (x º id x id º )à = canonical OAM ¹ à ¹ (x º id pure x id º pure)ã + ¹ à ¹ (x A º phys x º A phys)ã potential OAM F +x» F 0x + F zx = E x + B y = E x + (v B) x ~x ~ A phys» ~x dy ~ F Torque acting on a quark Burkardt
Potential angular momentum Nonforward generalization of the Qiu-Sterman matrix element single spin asymmetry OAM L pot = dx 1 dx 2 (x 1 ; x 2 ) x 1 x 2
OAM from the Wigner distribution Wigner distribution in QCD Lorce, Pasquini (2011) ~L = = dq ~x ~q hw(x; q)i dxd 2 k T Phase-space distribution of quarks k 2 T M 2 F 1;4(x; k 2 T ) related to a generalized TMD by Meissner, Metz, Schlegel
Canonical OAM from the Wigner distribution YH (2011) dq ~x ~q hw light cone (x; q)i = h à ¹ ¹ ~x i Ã! D pure Ãi cf. Boer, Mulders, Pijlman (2003) Kinetic OAM also follows Ji, Xiong, Yuan (see also, Lorce) dq ~x ~q hw straight (x; q)i = h à ¹ ¹ ~x i Ã! D Ãi x x z 2 x + z 2
Twist analysis YH, Yoshida (2012) see also, Ji, Xiong, Yuan (2012) Understand these relations at the density level etc.
Twist three matrix elements F-type D-type
Relation between F- and D-type correlators W xz D i (z)w zy = W xy D i (y) + i z y dt W xt gf +i (t)w ty Eguchi, Koike, Tanaka (2006) kinetic OAM canonical OAM potential OAM doubly-unintegrated The gluon has zero energy density interpretation
Canonical OAM and twist-3 GPD From the equation of motion, integrate dxxg 3 (x) = L q
Quark canonical OAM density Wandzura-Wilczek part First moment: genuine twist-three
Gluon canonical OAM Relation between F- and D-types Twist-3 gluon GPD
Relation between gluon canonical OAM and twist-3 GPD WW part first moment: genuine twist-three
Transverse spin decomposition It s important to use the Pauli-Lubanski vector in order to obtain a frame-independent sum rule. W i = ² ij µ P W ¹ = 1 2 e¹º½¾ P º M ++ j P + d 3 xm + ½¾ M + j hw i q;gi = J q;g S i = 1 2 (A q;g + B q;g )S i Higher twist? Equally important! Ji, Xiong, Yuan
Frame-dependence endures noncovariant! Bakker, Leader, Trueman YH, Tanaka, Yoshida Leader
Gluon spin in the transversely polarized nucleon d 2¼ ei x hp SjF + (0)F + ( )jp Si unpol longitudinal YH, Tanaka, Yoshida = 1 2 xg(x)g T i 2 x G(x)² S + ixg 3T (x)² + S? transverse hp SjF + A phys jp Si = 1 2 ² S + dx G(x) + ² + S j dxg 3T (x) = ² + ¹ S ¹ G g(x) = G 3T (x) = G g 1 (x) = g T (x) =
Longitudinal Complete transverse spin decomposition? 1 2 = 1 2 + G + Lq can + L g can Transverse same! 1 2 = 1 2 + G + Lq+g can cannot be separated in a frame-independent way
Frequently Asked Questions : Answers Where is G? We do have G in the complete decomposition Can we interpret the integrand as the `angular momentum density? Ambiguous for kinetic OAM. Unambiguous for canonical OAM. Commutation relation must be abandoned? Canonical OAMs do satisfy commutation relations. Is this valid for the longitudinal polarization, or transverse? Only for the longitudinal pol. Contaminated by a `higher twist term in the transverse case. Complete, gauge invariant decomposition in pqcd language now available