Gluon orbital angular momentum in the nucleon Contents

Gluon orbital angular momentum in the nucleon 八田佳孝 京都大学　基礎物理学研究所

Contents • Nucleon spin decomposition • PDF for orbital angular momentum • Method to measure gluon OAM

The proton spin problem The proton has spin ½. The proton is not an elementary particle. Quarks’ helicity Orbital angular Gluons’ helicity Momentum (OAM) In the quark model, with relativistic effects, EMC result Current value

Gluon polarization Result from the NLO global analysis after the RHIC 200 Ge. V pp data Note that the uncertainty from the small-x region is huge! RHIC 510 Ge. V, Electron-Ion Collider De. Florian, Sassot, Stratmann, Vogelsang (2014)

Lattice calculation of Compute the naïve gluon helicity in some gauge. Boost it to large momentum and do the matching Ji, Zhang, Zhao (2013) YH, Ji, Zhao (2013) QCD collaboration, ar. Xiv: 1609. 05937

Jaffe-Manohar decomposition Based on the canonical energy momentum tensor Operators NOT gauge invariant. cf. Partonic interpretation in the light-cone gauge

Ji decomposition Based on the improved (Belinfante) energy momentum tensor One can add a total derivative. Further decomposition in the quark part (but not in the gluon part) Components measurable in DVCS and lattice QCD

Gauge invariant completion of Jaffe-Manohar Chen, Lu, Sun, Wang, Goldman (2008) 　　　　　　　 Wakamatsu ( YH (2011) where (my choice)

Can we measure ? A big challenge for the whole community. No observable proposed so far…although OAM is the future of spin physics! Hint 1: We need to introduce the x-distribution cf. Hint 2: for OAMs , is related to the Wigner distribution The gluon Wigner distribution is measurable at small-x. YH, Xiao, Yuan (2016) Recent breakthroughs small-x YH, Nakagawa, Yuan, Zhao ar. Xiv: 1612. 02445 moderate-x Ji, Yuan, Zhao ar. Xiv: 1612. 02438

OAM at small-x: Does it matter?

OAM from the Wigner distribution in QCD Belitsky, Ji, Yuan (2003) Need a Wilson line ! Define Lorce, Pasquini (2011); YH (2011); Lorce, Pasquini, Xiong, Yuan (2011) Which OAM is this? ?

Canonical OAM from the light-cone Wilson line YH (2011) Ji’s OAM from the straight Wilson line Ji, Xiong, Yuan (2012) `Potential’ OAM Torque acting on a quark Burkardt (2012)

vs. on a lattice Engelhardt, talk at “ 3 D parton distributions” (2016) Ji staple length `angle’ of the staple Jaffe-Manohar corresponds to a lightlike Wilson line

OAM parton distribution Understand this relation at the density level Define the x-distribution . YH, Yoshida (2012), Ji, Xiong, Yuan (2012) Natural, because Jaffe-Manohar decomposition has a partonic interpretation. ? ?

Deconstructing OAM Ji’s OAM canonical OAM `potential OAM’ For a 3 -body operator, it is natural to define the double density. ``D-type” ``F-type”

The D-type and F-type correlators are related. Eguchi, Koike, Tanaka (2006) doubly-unintegrate The gluon has zero energy partonic interpretation! Canonical OAM density It coincides with defined via the Wigner distribution

Twist structure of OAM distributions YH, Yoshida (2012) Wandzura-Wilczek part Genuine twist-three part

Gluon Wigner distribution There are two ways to make it gauge invariant Bomhof, Mulders, Pijlman (2006) Dominguez, Marquet, Xiao, Yuan (2011) Weizsacker-Williams distribution Dipole distribution

WW vs. Dipole Wigner OAM naturally defined via the WW Wigner YH (2011) On the other hand, the dipole Wigner has a better chance to be measured YH, Xiao, Yuan (2016) One can prove that We shall propose an observable sensitive to at small-x

Dipole gluon Wigner distribution at small-x YH, Xiao, Yuan (2016) Approximate Use the identity ``Dipole S-matrix”

Probing Wigner in Diffractive dijet production YH, Xiao, Yuan (2016) see also, Altinoluk, Armesto, Beuf, Rezaeian (2015) Jet 1 Proton recoil momentum Jet 2 Dijet relative momentum Fourier transform of (for small)

Where is spin dependence? “Pomeron” “odderon” cannot contain the structure forbidden by PT symmetry Reason: eikonal approximation All the information about spin is lost (in the longitudinally polarized case) cf. spin-dependent odderon Zhou (2013)

OAM as a next-to-eikonal effect Parameterization

Next-to-eikonal cross section Green’s function cf. Altinoluk, Armesto, Beuf, Martinez, Salgado (2014)

Longitudinal single spin asymmetry 2 - 2 LO NLO Interference between eikonal and next-to-eikonal contributions. OAM function interferes with odderon, but Pomeron usually dominates…

Look at the kinematic regions Pomeron drops out because

Comments Experimental signature: What is ? We don’t know… Structure dependence in one-loop calculation. follows from parity Two jets are indistinguishable.

Expected x-dependence Odderon intercept is unity. Bartels, Lipatov, Vacca (1999)

Summary Gauge invariant canonical OAMs in QCD now available, even their x-distribution well defined. First-ever experimental signature of parton OAM—Holy Grail in spin physics Odderon—Holy Grail in small-x physics Kill two birds with one stone!