Introduction to
nucleon spin sum rule
Yoshitaka Hatta
Yukawa institute, Kyoto U.
Contents
• Introduction
• Jaffe-Manohar and Ji decomposition
• Complete gauge invariant decomposition
• Orbital angular momentum and Wigner distribution
• Gluon OAM at small-x
The proton spin problem
The proton has spin ½.
Quarks’ helicity
Gluons’ helicity
Orbital angular Momentum (OAM)
with relativistic effects, In the quark model,
The proton is not an elementary particle.
Longitudinal double spin asymmetry in polarized DIS
from polarized DIS
Polarized quark distribution function
p X
`Spin crisis’
In 1987, EMC (European Muon Collaboration) announced a very small value of the quark helicity contribution
Recent value from NLO QCD global analysis
!?
Gluon polarization
1st moment of the polarized gluon distribution
Operator definition
Determination of
Longitudinal double spin asymmetry in pp collisions at RHIC
central
Evidence of nonzero
HUGE uncertainty from the small-x region RHIC 510GeV,
Electron-Ion Collider
DeFlorian, Sassot, Stratmann, Vogelsang (2014)
Result from the NLO global analysis after the RHIC 200 GeV pp data
Maybe after including the small-x contribution. OAM not needed?
nonlocal in the light-cone direction. Local only in the light-cone gauge Real-time problem, impossible to simulate on a lattice.
Instead try a naïve definition
…gauge dependent, frame dependent.
Boost it to large momentum and do the matching Ji, Zhang, Zhao (2013) YH, Ji, Zhao (2013)
from lattice QCD
First lattice study by xQCD collaboration
Yang, et al. (2016)
QCD angular momentum tensor
QCD Lagrangian Lorentz invariant Noether current
quark helicity gluon helicity QCD angular momentum tensor
canonical energy momentum tensor
Quark OAM Gluon OAM
Based on the canonical angular momentum tensor Operators NOT gauge invariant.
Partonic interpretation in the light-cone gauge
Jaffe-Manohar decomposition (1
990)
Ji decomposition (1997)
Improved (Belinfante) energy momentum tensor
One can add a total derivative.
Further decomposition in the quark part possible (but not in the gluon part)
Relation to Generalized parton distribution (GP D)
non-forward matrix element
GPD definition
Ji’s sum rule
measurable in
Deeply Virtual Compton Scattering (DVCS)
Lattice results on Ji sum rule
Alexandrou et al. (2013)
Deka, Doi et al. (2015)
`Disconnected insertions’ included
Two spin communities divided
accessible from GPD measured at JLab, COMPASS, HERMES, J-PARC also calculable in lattice QCD
not measured yet
not even well-defined?
common and well-known
Define rigorously.
Must be related to GPD!
Jaffe-Manohar
Ji
measured by PHENIX, STAR, COMPASS, HERMES
Nonexistent in Ji’s scheme…
Complete gauge invarian t decomposition
where
(my choice)
Chen, Lu, Sun, Wang, Goldman (2008) YH (2011) Add a different surface term to get
Gauge invariant completion of Jaffe-Manohar.
Transverse spin decomposition
Boost and rotation do not commute.
Boost invariance becomes a nontrivial issue.
Longitudinal
Transverse
cannot be separated in a frame-independent way
same!
Note: Ji sum rule violated for transverse polarization.
YH, Tanaka, Yoshida (2012)
The QCD Wigner distribution
Form factor
charge
TMD GPD
Phase space distribution of partons in QCD—the `mother distribution’
Lorce, Pasquini (2011);
YH (2011);
Nice, but which OAM is this??
Define
Go to the momentum space and look for the component
Then
OAM from the Wigner distribution
YH (2011)
Canonical OAM from the light-cone Wilson line
Ji, Xiong, Yuan (2012)
Kinetic (Ji’s) OAM from the straight Wilson line
Torque acting on a quark Burkardt (2012)
`Potential’ OAM
Jaffe-Manohar vs. Ji First latt ice result
Engelhardt, 1701.01536
Staple length
Can we measure ?
A big challenge for the spin community.
No observable proposed so far…although OAM is the future of spin physics!
Hint1: We need to introduce the x-distribution for OAMs
Hint2: is related to the Wigner distribution.
The gluon Wigner distribution is measurable at low-x. YH, Xiao, Yuan (2016) cf. ,
Hagler, Schafer (1998)
Harindranath, Kundu (1999) YH, Yoshida (2012)
`Parton distribution’ for OAM
??
Define the x-distribution .
Caveat: It’s not an ordinary (twist-two) parton distribution function.
Similar to the structure function.
Deconstructing OAM
Ji’s OAM canonical OAM `potential OAM’
For a 3-body operator, it is natural to define the double density.
The gluon has zero energy,
partonic interpretation!
Canonical OAM density
YH, Yoshida (2012)
doubly-unintegrate
It coincides with defined via the Wigner distribution
Ji’s OAM canonical OAM `potential OAM’
Relation to twist-three GPD
twist-2
integrate
Penttinen, Polyakov, Shuvaev, Strikman (2000)
YH. Yoshida (2012)
Quark canonical OAM density
First moment:
Wandzura-Wilczek part
genuine twist-three
The bridge between JM and Ji
Gluon canonical OAM density
WW part
genuine twist-three
first moment:
`DGLAP’ equation for polarize
d PDFs
QCD at small-x
A myriad of small-x gluons
in a high energy hadron/nucleus!
!?
The gluon number eventually saturates, forming the universal
QCD matter at high energy called the Color Glass Condensate.
Spin at small-x?
unpolarized splitting function polarized splitting function
No enhancement,
spin effects are always suppressed by Consider q qg splitting
HOWEVER, they can be enhanced by double logarithms
The large uncertainty in in the small-x region is a current major problem.
Bartels, Ermolaev, Ryskin (1996), Kovchegov, Pitonyak, Sievert (2015~)
Gluon Wigner distribution
Weizsacker-Williams distribution
Dipole distribution
There are two ways to make it gauge invariant
Bomhof, Mulders, Pijlman (2006)
Dominguez, Marquet, Xiao, Yuan (2011)
One can prove that
Dipole gluon Wigner distribution at s mall-x
Approximate
``Dipole S-matrix”
cannot depend on spin—forbidden by PT symmetry
Lesson: All information about spin and OAM is lost in the eikonal approximation.
Measurable in diffractive dijet production in ep scattering.
YH, Xiao, Yuan (2016)
Hagiwara, YH, Pasechnik, Tasevsky, Teryaev (2017)
OAM as a next-to-eikonal ef fect
Go to next-to-eikonal
YH, Nakagawa, Xiao, Yuan, Zhao (2017)
Can have spin-dependent matrix elements, involves half-infinite Wilson lines
Polarized TMD gluon distribution
Exactly the same matrix element appears.
Linear relation between and
YH, Nakagawa, Xiao, Yuan, Zhao (2017)
small?
More, Mukherjee, Nair, 1709.00943
Summary
Complete gauge invariant decomposition of the nucleon spin now available in QCD, even at the density level.
OAM—Holy grail in spin physics. A lot of progress in theory recently, includ ing the first wave of ideas to access the OAM in experiments.
Progress also in the small-x evolution of helicity distributions. Possi ble large contribution from the small-x region?
small-x YH, Nakagawa, Yuan, Xiao, Zhao arXiv:1612.02445 moderate-x Ji, Yuan, Zhao arXiv:1612.02438
quark OAM Bhattacharya, Metz, Zhou arXiv:1702.04387
Kovchegov, Pitonyak, Sievert (2015~)
