Nucleon spin and tomography. Yoshitaka Hatta (Yukawa institute, Kyoto U.)

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1 Nucleon spin and tomography Yoshitaka Hatta (Yukawa institute, Kyoto U.)

2 Outline Nucleon spin decomposition Jaffe-Manohar vs. Ji Complete gauge invariant decomposition Orbital angular momentum Relation to twist-3 GPDs 5D tomography of the nucleon Wigner distribution Husimi distribution

3 The proton spin problem The proton has spin ½. The proton is not an elementary particle. 1 2 = G + Lq z + L g z Quarks helicity Gluons helicity Orbital angular Momentum (OAM) Quark model prediction: ¼ 0:7 with relativistic effects

4 `Spin crisis In 1987, EMC (European Muon Collaboration) announced a very small value of the quark helicity contribution!? Latest results from NLO global analysis Z 1 0:05 dx G(x) ¼ 0:2 DeFlorian, Sassot, Stratmann, Vogelsang (2014)

5 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

6 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

7 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

8 Generalized parton distributions (GPD) Non-forward proton matrix element Twist-two GPDs J q = 1 2 Z dxx(h q (x) + E q (x)) J g = 1 4 Z dx(h g (x) + E g (x))

9 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 GPD! accessible from GPD at JLab, COMPASS, HERMES, J-PARC also calculated in lattice QCD

10 Complete decomposition Chen, Lu, Sun, Wang, Goldman (2008) Wakamatsu (2010) Y.H. (2011) where (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

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

12 Canonical OAM from the light-cone Wilson line YH (2011) Z dq ~ b ~ k hw light cone (b; k)i = h à ¹ ¹ ~ Ã! b i D pure Ãi Kinetic OAM from the straight Wilson line Z Ji, Xiong, Yuan (2012) dk ~ b ~ k hw straight (b; k)i x b + z 2 x + = h ¹ à ¹ ~ b i Ã! D Ãi b z 2 Difference between the two OAMs Z L pot = L L can = ~ b dx F ~ ` Potential OAM Torque acting on a quark Burkardt (2012)

13 Twist analysis YH, Yoshida (2012) see, also, Ji, Xiong, Yuan (2012) 1 2 = G + Lq can + L g can Understand this relation at the density level = X Z Z dx q f (x) G = f?? dx G(x) c.f. q(x) = 1 4¼S + Z dz e ixp + z hpsj ¹ Ã(z ) + 5 Ã(0)jPSi

14 `Density of OAM Ji s OAM canonical OAM `potential OAM h ¹ Ãb DÃi = h ¹ Ãb D pure Ãi + igh ¹ Ãb A phys Ãi 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

15 The D-type and F-type correlators are related. Eguchi, Koike, Tanaka (2006) h ¹ Ãb DÃi = h ¹ Ãb D pure Ãi + igh ¹ Ãb A phys Ãi doubly-unintegrate The gluon has zero energy partonic interpretation! Canonical OAM density x 2 x 1 = 0

16 Relation between L q can(x) and twist-3 GPD From the equation of motion, twist-3 twist-2 integrate Z dxxg 3 (x) = L q Penttinen, Polyakov, Shuvaev, Strikman (2000)

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

18 Gluon canonical OAM density L g can(x) twist-three gluon GPD WW part first moment: genuine twist-three

19 Complete transverse spin decomposition? Longitudinal YH, Tanaka, Yoshida (2012) 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

20 Summary so far 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. Progress towards calculating spin components on a lattice. Ji, Zhang, Zhao (2013,2014); YH, Ji, Zhao (2013)

21 Husimi distribution for nucleon tomography Hagiwara and YH, arxiv:

22 Wigner distribution Phase space distribution in quantum mechanics f W (q; p) = = Z 1 1 Z 1 1 dxe ipx=~ hãjq x=2ihq + x=2jãi dxe ipx=~ hq + x=2j^½jq x=2i density matrix ^½ = jãihãj Eugene Wigner ( ) Z dq 2¼~ f W (q; p) = jhãjpij 2 ; Z dp 2¼~ f W (q; p) = jhãjqij 2

23 Wigner distribution for the harmonic oscillator H = p2 2m + m!2 x 2 2 µ 4H f W (q; p) = 2( 1) n e 2H=~! L n ~! Laguerre polynomial n=0 n=1 n=4 n=2 Probability distribution? No way!

24 5D imaging of the nucleon: Wigner distribution in QCD Wigner distribution of quarks in the nucleon Ji (2003) Belitsky, Ji, Yuan (2003) W (x; ~ b? ; ~ k? ) = Z dz d 2 z? 16¼ 3 d 2? (2¼) 2 ei(xp + z ~ k? ~z? ) hp + 2 j ¹ Ã(b z 2 ) LÃ(b + z 2 )jp 2 i `Mother distribution ~ b? ~ k? integral TMD integral GPD ¹ = (0; 0; ~? ) momentum recoil (relativistic effect)

25 Model calculation Lorce, Pasquini, (2011) light-cone quark models (no gluons included)

26 Husimi distribution f H (q; p) = 1 ¼~ Z dq 0 dp 0 e m!(q0 q) 2 =~ (p 0 p) 2 =m!~ f W (q 0 ; p 0 ) Gaussian smearing of the Wigner distribution q = p ~=2m! p = p ~m!=2 q p = ~=2 Positive definite! f H (q; p) = h j^½j i = jhãj ij 2 0 aj i = j i coherent state Kodi Husimi ( ) 伏見康治

27 Husimi distribution for the harmonic oscillator n=4 f H (q; p) = 1 n! e H ~! µ n H ~! µ 4H f W (q; p) = 2( 1) n e 2H=~! L n ~! Localized around the classical orbit H = p2 2m + m!2 x 2 2 ¼ ~!(n )

28 Husimi distribution in QCD Hagiwara and YH, arxiv: Define H (x; ~ b? ; ~ k? ) 1 Z d 2 b 0 ¼?d 2 k?e `2 (~ b? ~ b 0? )2 `2( ~ k? ~ k 0? )2 W (x; ~ b 0?; ~ k?) 0 The parameter ` is arbitrary, but it is natural to take `. R hadron

29 Positivity? In the A + = 0 gauge Z H» d 2? e i~? ~ b? ` 2 2? 4 hp + =2jà y + ±(K+ (1 x)p + )e `2( ~ K? + ~ k? ) 2 à + jp =2i `good component Positive definite if it were not for the momentum recoil However, the Gaussian factor suppresses

30 Perturbative calculation Wigner distribution for an on-shell quark W(x; ~ b? ; ~ k? ) A + = 0 gauge = Z dz d 2 z? 16¼ 3 d 2? (2¼) 2 ei(xp + z ~ k? ~z? ) hp + 2 j ¹ Ã(b z 2 ) + LÃ(b + z 2 )jp 2 i Zeroth order W[x; ~ b? ; ~ k? ] = ±(x 1)± (2) ( ~ b? )± (2) ( ~ k? ) Nonsense =) H(x; ~ b? ; ~ k? ) = ±(1 x) e b2? =`2 `2k 2? ¼ 2 Physical

31 First order in s cf. Mukherjee, Nair, Ojha, W + [x; ~ b? ; ~ k? ] = Z sc F d 2? 2¼ 2 (2¼) 2 e i ~? ~ b? ³ 2 k? 2 2? 4 (1 x) P qq (x) + m 2 (1 x) 3 (q+ 2 + m 2 (1 x) 2 )(q 2 + m 2 (1 x) 2 ) ~q = ~ k? ~? 2 (1 x) splitting function P qq (x) = 1 + x2 1 x Divergent when ~ b? = 0 Behaves like crazy

32 Husimi distribution: Numerical result Before (Wigner) After (Husimi) x = 0:5; m 2 = 0:1 GeV 2 ; ` = 1 GeV 1

33 Before (Wigner) After (Husimi)

34 Speculations : relation to Color Glass? At small-x, the gluons can be treated as a classical coherent state McLerran, Venugopalan (1993) Husimi distribution is the coherent state expectation value. Any relation between the two?

35 A tantalizing hint b-moment of the QCD Husimi distribution does not reduce to the TMD. Z d 2 b? H (x; ~ b? ; ~ k? ) = Z dz d 2 z? 16¼ 3 e i(xp + z ~ k? ~z? ) e z2? 4`2 hp j Ã( z=2) LÃ(z=2)jP ¹ i identify `! 1 Q s (x) saturation scale e z2? =4`2! e Q2 s z2? =4 dipole S-matrix What is computed from classical gluon fields could be interpreted as the Husimi distribution?

36 Summary of the second part Wigner distribution is often unphysical and badly-behaved. Husimi distribution much better behaved, can be interpreted as a probability distribution. Proof of positivity spoiled by the relativistic kinematical effect, yet a model calculation shows no sign of negative regions.