Orbital Angular Momentum Feng Yuan Lawrence Berkeley Lab

Orbital Angular Momentum Feng Yuan Lawrence Berkeley Lab 9/15/2020 1

Proton Spin Sum Gluon spin~ 0 -70% RHIC, EIC, … Quark spin ~30% DIS, and pp coll. q ? Lq G LG 2 ?

http: //www. int. washington. edu/PROGRAMS/12 -49 w/ 9/15/2020 3

Relevance of OAM in hadron physics n GPD E and Nucleon Pauli Form Factor ¨ Magnetic n Higher-twist functions ¨ g. T n structure function, g 2 Single spin asymmetries ¨ T-odd n moments TMDs … 9/15/2020 4

Access to the OAM n Generalized Parton Distributions ¨ Exp. n And/or Lattice Transverse Momentum Dependent Distributions ¨ Complementary n Wigner Distributions ¨ Model calculations (so far) 9/15/2020 5

DVCS with transversely polarized target from HERMES & Jlab JLab Hall A Jq input parameter in the GPD ansatz, need More sophisticted model for GPDs HERMES Jlab Hall A, 0709. 0450; HERMES, 0802. 2499

TMDs

Wigner Distributions Define the net momentum projection Quark oribital angular momentum Lorce, Pasquini, ar. Xiv: 1106. 0139 Lorce, Pasquini, Xiong, Yuan, ar. Xiv: 1111. 482 Hatta, ar. Xiv: 1111. 3547 9/15/2020 8

Outline Angular momentum tensor and parton interpretation n Transverse polarized nucleon n ¨ Ji n sum rule via GPDs Longitudinal polarized nucleon ¨ Helicity, OAM via twist-three GPDs ¨ Wigner distributions Ji, Xiong, Yuan, ar. Xive: 1202. 2843; to be published 9/15/2020 9

Angular momentum density n From energy-momentum tensor n Off-forward matrix ¨ Take p=p’ in the end 10 Jaffe-Manohar Ji

Angular momentum density Partonic interpretation works in the infinite momentum frame (light-front) n In this frame, the leading component is P+, S+ n Next-to-leading component, ST n 9/15/2020 11

Leading component M++T Because of anti-symmetric α, β, the leading term is α=+, β=T, which related to the transverse spin of the nucleon n Transverse spin of nucleon has leadingtwist interpretation in parton language n However, individual spin is obscure n 9/15/2020 12

Next-to-leading: M+TT Because of two transverse indices, it inevitably involves twist-three operators n However, it does lead to the individual spin contribution, e. g. , from the quark n ¨ Jaffe-Manohar spin decomposition 9/15/2020 13

Spin sum is frame-independent Ji, 96 n A and B are form factors Works for both longitudinal and transverse spin of nucleon n Partonic interpretation is, … n 9/15/2020 14

Transversely polarized nucleon n Ji sum rule n Burkardt picture: rest frame (2005) ¨ Angular momentum obtained by impact parameter dependent PDF ¨ Wave packet determines the center of the proton 9/15/2020 15

Angular Momentum density (T) n Define the momentum density (T++) n AM depending momentum fraction x, Which gives the angular momentum density for quark with longitudinal momentum x 9/15/2020 16

In more detail n Calculate ρ+(x, ξ, ST) Integrate out ξ, second term drops out, we obtain the momentum density n Integral with weight ξT’, the first term drops out, Angular Momentum density n 9/15/2020 17

Where is the quark transverse spin? n Higher-twist in natural (M+-T) ¨ g. T n structure function Has nothing to do with the quark transversity 9/15/2020 18

Longitudinal (helicity) spin sum Quark spin explicitly n OAM, twist-three nature n ¨ Direct n extraction from twist-three GPDs Of course, we still have 9/15/2020 19

OAM Related to the Wigner Distribution Lorce-Pasquini, 2011 n Lorce-Pasquini-Xiong-Yuan, 2011 n Hatta 2011 n 9/15/2020 20

Wigner distribution for the quark n The quark operator n Wigner distributions Ji: PRL 91, 062001(2003) After integrating over r, one gets TMD After integrating over k, one gets Fourier transform of GPDs 21

Importance of the gauge links Gauge invariance n Depends on processes n Comes from the QCD factorization n n And partonic interpretation as well 9/15/2020 22

Fixed point gauge link Becomes unit in ξ. A=0 gauge n Moment gives the quark OAM n n OPE 9/15/2020 23

Quark OAM n Any smooth gauge link results the same OAM for the partons 9/15/2020 24

Light-cone gauge link n it comes from the physical processes ¨ DIS: future pointing ¨ Drell-Yan: to -∞ Cautious: have light-cone singularities, and need to regulate n Moments related to twist-three PDFs, and GPDs n 9/15/2020 25

Light-cone decomposition Bashinsky-Jaffe n Quark OAM only contains the partial derivative 9/15/2020 26

Gauge Invariant Extension n GIE is not unique n Canonical OAM can be calculated 9/15/2020 27

OAM from Wigner distribution Can be measured from hard processes n Moments relate to the canonical OAM n In the end of day, depends on twist-3 GPDs n ¨ Might be studied in many processes 9/15/2020 28

Road map to OAM n Model dependent determinations ¨ GPDs fit to the DVCS and other processes ¨ TMDs study in the SIDIS ¨ Wigner distributions, and many others, … n Toward a model-independence? ¨ With great efforts, we may converge to a model-independent constraint ¨ + the helicity Spin sum rule 9/15/2020 29

Conclusions Partonic interpretation of the proton spin can be understood through quantum phase space Wigner distribution n Transverse polarized nucleon induces the angular momentum density (x) n Longitudinal polarized nucleon depends on twist-2 and 3 GPDs n Canonical OAM can be measured from experiments n 9/15/2020 30

My wish In 5 (10) years? Lg=Jg-ΔG n Lq 9/15/2020 31

Comments on Chen et al. decomposition Start with Coulomb gauge results n Extend to other gauge by using the gauge invariant extension n Not associated with the physical observable, can not be evaluated in lattice QCD n Not gauge invariant n 9/15/2020 32