Xiangdong JiXiangdong Ji

University of Maryland/SJTUUniversity of Maryland/SJTU

Observing the protonObserving the protonOff the light-coneOff the light-cone

JLab Physics Colloquium, April 23, 2013

Collaborators and papersCollaborators and papers

Feng Yuan, LBLFeng Yuan, LBL Jianhui Zhang, Shanghai Jiao Tong Jianhui Zhang, Shanghai Jiao Tong

UniversityUniversity Xiaonu Xiong, Peking UniversityXiaonu Xiong, Peking University Yong Zhao, U. MarylandYong Zhao, U. Maryland

11. Phys. Rev. Lett. 109, 152005 (2012). Phys. Rev. Lett. 109, 152005 (2012)

2. Phys.Lett. B717 (2012) 214-218 2. Phys.Lett. B717 (2012) 214-218

3. Parton picture for longitudinal polarization, Phys. Rev. D 3. Parton picture for longitudinal polarization, Phys. Rev. D

4. Physics of gluon helicity4. Physics of gluon helicity

5. parton distribution from Euclidean lattice5. parton distribution from Euclidean lattice

OutlineOutline

Frame dependence in relativistic quantum Frame dependence in relativistic quantum theoriestheories

Two special frames for understanding the Two special frames for understanding the proton structure: proton structure: the lab frame the lab frame and and infinite momentum frameinfinite momentum frame (IMF) (IMF)

The proton spin structure in IMFThe proton spin structure in IMF Lattice QCD calculation of parton physics Lattice QCD calculation of parton physics

in IMFin IMF ConclusionsConclusions

Important missionImportant mission

An important mission of Jlab 12 GeV An important mission of Jlab 12 GeV upgrade and EIC is to study the internal upgrade and EIC is to study the internal structure of proton and neutron at a new structure of proton and neutron at a new level. level.

A complex system of the quarks and A complex system of the quarks and gluonsgluons Strong interactions, relativistic Test the fundamental theory: QCD

Constraints of Relativity Constraints of Relativity

The proton wave function is a frame The proton wave function is a frame dependent conceptdependent concept

Boost operators, KBoost operators, Kii, are interaction-, are interaction-dependentdependent

|P|P˃ = U(˃ = U(ΛΛ(p)) |p=0>(p)) |p=0>

U is not just kinematical, it is dynamical!U is not just kinematical, it is dynamical!

Wave functions in Relativistic QMWave functions in Relativistic QM

Wave function is not a Lorentz invariant Wave function is not a Lorentz invariant concept as it is defined by observations of concept as it is defined by observations of different space points in a fixed time different space points in a fixed time (simultaneously) at a particular frame.(simultaneously) at a particular frame.

Simultaneity for two events in one frame Simultaneity for two events in one frame does not mean simultaneity in a different does not mean simultaneity in a different frame.frame.

In general, WF has not been very a In general, WF has not been very a popular concept in field theory. However, popular concept in field theory. However, it is an important and intuitive concept. it is an important and intuitive concept.

Two special framesTwo special frames

Lab frameLab frame: where the proton is at rest, or : where the proton is at rest, or low momentum, relatively speaking low momentum, relatively speaking

Probe frameProbe frame: in which the electron or virtual : in which the electron or virtual photon has the smallest momentum. photon has the smallest momentum.

In high-energy limit, the probe frame In high-energy limit, the probe frame reaches the infinite momentum frame (IMF), reaches the infinite momentum frame (IMF), i.e., the proton is probed as it travels at the i.e., the proton is probed as it travels at the speed of light!speed of light!

Electron scatteringElectron scattering

four momentum transfer qfour momentum transfer qµµ = (v, q) is a space = (v, q) is a space like vector vlike vector v22-q-q2 2 < 0 and fixed.< 0 and fixed.

smallest momentum happens when v=0, smallest momentum happens when v=0, QQ22=q=q22

Pq = PPq = P33Q = QQ = Q22/2x, thus P/2x, thus P3 3 = Q/2x. = Q/2x.

In the scaling limit, PIn the scaling limit, P33 -> infinity. -> infinity.

Reconcile physics in two framesReconcile physics in two frames

Physics shall be frame-independent! Physics shall be frame-independent! In the rest frame, one probes the light-cone,

time-dependent correlations (light-front) In the IMF, one probes the static correlations. The two different correlations are related by

Lorentz boost. If one uses the light-front quantization in the If one uses the light-front quantization in the

rest frame (an effective theory), one gets rest frame (an effective theory), one gets the same physics as IMF! the same physics as IMF!

Proton WF in the IMF is what high-energy Proton WF in the IMF is what high-energy scattering study. scattering study.

IMF and parton physicsIMF and parton physics

In the IMF, the interactions between particles In the IMF, the interactions between particles are Lorentz-dilated, and thus the systems are Lorentz-dilated, and thus the systems appear as if interaction free: appear as if interaction free: the proton is the proton is made of free partons. made of free partons.

This is only true to a certain degree: This is only true to a certain degree: leading leading twisttwist. The so-called higher-twist contributions . The so-called higher-twist contributions are sensitive to parton off-shellness, are sensitive to parton off-shellness, transverse momentum and parton transverse momentum and parton correlations. correlations.

Partons provide a useful language to Partons provide a useful language to understand the proton structure.understand the proton structure.

Spin of the proton in QCDSpin of the proton in QCD

The spin of the proton is generated from The spin of the proton is generated from the angular momentum of the internal the angular momentum of the internal quarks and gluonsquarks and gluons

1/ 2 ( ) ( )q gJ J J

From general consideration, From general consideration, one can write down, one can write down,

The quark orbital The quark orbital contribution can be probed contribution can be probed through GPDsthrough GPDs

These are frame-These are frame-independent results. independent results.

The spin structure in IMFThe spin structure in IMF

How is the spin of the proton generated How is the spin of the proton generated from the AM of the underlying partons? from the AM of the underlying partons?

Longitudinal polarization Longitudinal polarization Transverse polarizationTransverse polarization

Longitudinal polarizationLongitudinal polarization

Quark orbital AM: Quark orbital AM: intrinsically twist-threeintrinsically twist-three

Orbital AM is nominally a leading twist, Orbital AM is nominally a leading twist, however, it is actually a high-twist however, it is actually a high-twist contribution because it involves the parton contribution because it involves the parton transverse momentum. transverse momentum.

The total contribution can be probed through The total contribution can be probed through the GPD sum rule (L=J-the GPD sum rule (L=J-ΔΣΔΣ. However, the . However, the individual parton contributions do not yet individual parton contributions do not yet have a simple parton picture. have a simple parton picture.

OAM parton distribution can be probed OAM parton distribution can be probed through through twist-three GPDstwist-three GPDs. (Ji, xiong, & yuan). (Ji, xiong, & yuan)

Gluon polarization Gluon polarization

In IMF, however, the gluon partons have In IMF, however, the gluon partons have well-defined helicity ± 1 and densities gwell-defined helicity ± 1 and densities g±±

(x)(x)

Gluon helicity distribution is Gluon helicity distribution is

g(x) = g+(x) – g-(x) and

G = ʃdx g(x)

1/2+1 or -1

Total gluon helicityTotal gluon helicity

The total gluon helicity The total gluon helicity ΔGΔG is gauge is gauge invariant, and has a complicated invariant, and has a complicated expression on the rest frame expression on the rest frame

In light-cone gauge AIn light-cone gauge A++=0, it reduces to a =0, it reduces to a simple expression , which is the spin simple expression , which is the spin of the gluon, but has no gauge-symmetric of the gluon, but has no gauge-symmetric notionnotion

(Laudau, Jackson). (Laudau, Jackson).

AALLLL from RHIC 2009 from RHIC 2009

1717

Perspective on Delta GPerspective on Delta G

1818

Total gluon helicity in the IMFTotal gluon helicity in the IMF

A gauge potential can be decomposed into A gauge potential can be decomposed into longitudinal and transverse parts (R.P. longitudinal and transverse parts (R.P. Treat,1972),Treat,1972),

The transverse part is gauge covariant, The transverse part is gauge covariant,

One can define the gauge-symmetric gluon One can define the gauge-symmetric gluon spin as ExAspin as ExA┴ ┴ (X.Chen et al, 2009) (X.Chen et al, 2009)

It be shown that ΔG is the matrix element of about operator in the IMF. (Ji, Zhang, Zhao)

Reproducing the light-cone gauge Reproducing the light-cone gauge result by boostresult by boost

In fact, transforming to the rest frameIn fact, transforming to the rest frame

Which in the light-cone gauge, reduces to Which in the light-cone gauge, reduces to the standard expression ExA. the standard expression ExA.

Transverse polarizationTransverse polarization

Transverse polarization in term of the Transverse polarization in term of the polarization vector S is of subleading, thus polarization vector S is of subleading, thus does not seem to have a simple parton picturedoes not seem to have a simple parton picture

However, this is incorrect.However, this is incorrect. There are many misconceptions and pitfalls There are many misconceptions and pitfalls

about transverse pol.about transverse pol. The best language is not the transverse AM, The best language is not the transverse AM,

rather, it shall be transverse Pauli-Lubanski rather, it shall be transverse Pauli-Lubanski spin. [Transverse AM does not commute with spin. [Transverse AM does not commute with the hamiltonian]the hamiltonian]

Transverse polarizationTransverse polarization

There is a leading twist-contribution, the There is a leading twist-contribution, the subleading contributions either are cancelled or subleading contributions either are cancelled or related to the leading one by symmetry. related to the leading one by symmetry.

The leading contribution has a partonic The leading contribution has a partonic interpretation. interpretation.

First pointed out by Burkardt (2005), but the First pointed out by Burkardt (2005), but the connection to the spin sum rule was not rigourous. connection to the spin sum rule was not rigourous.

works for transverse spin!works for transverse spin!

A plane-wave derivation of transverse-A plane-wave derivation of transverse-spin parton spin parton sum rulesum rule

Consider parton momentum density Consider parton momentum density

It has a “distribution” term depending on It has a “distribution” term depending on coordinate coordinate ξξ, ,

Calculating its contribution to the transverse Calculating its contribution to the transverse spin densityspin density

Parton distributionsParton distributions

The Feynman momentum is easier to understand in the infinite momentum frame: fraction of the longitudinal momentum carried by quarks x = kz/Pz, 0<x<1

Lattice QCD calculationLattice QCD calculation

Current approach is based on the rest Current approach is based on the rest frame formalism. frame formalism.

To compute light-cone correlation difficult To compute light-cone correlation difficult because the lattice cannot handle real because the lattice cannot handle real time correlationstime correlations

One can calculate the moments (2-3) of One can calculate the moments (2-3) of parton distributions: local operators.parton distributions: local operators.

Lattice QCD calculationsLattice QCD calculations

Parton distributions are calculated using Parton distributions are calculated using the rest-frame formalism, the the rest-frame formalism, the corresponding operators are light-cone corresponding operators are light-cone correlationscorrelations

Since lattice cannot handle real-time Since lattice cannot handle real-time dependence, only moments are calculated dependence, only moments are calculated on lattice. on lattice.

Usually only a few moments (2-3) are Usually only a few moments (2-3) are calculated. calculated.

A new proposalA new proposal

Using the IMF formalism. Using the IMF formalism. Start with static correlation in the z-Start with static correlation in the z-

direction. direction.

X. Ji, to be publishedX. Ji, to be published

CommentsComments

This can be proved using the local matrix This can be proved using the local matrix elements formulationelements formulation

Approaching the light-cone through boostApproaching the light-cone through boost

One-loop calculation demonstrates how One-loop calculation demonstrates how this work. this work.

One-loop calculationOne-loop calculation

The extension of the approachThe extension of the approach

GPDsGPDs

TMDsTMDs

The extension of the approachThe extension of the approach

Wigner distributionWigner distribution

Light-cone amplitudesLight-cone amplitudes

Light-cone wave functions, higher-twists….Light-cone wave functions, higher-twists….

ConclusionsConclusions

In high-energy scattering, one probes In high-energy scattering, one probes parton physics, which can be understood parton physics, which can be understood both in the rest frame and IMF formulationboth in the rest frame and IMF formulation

Understanding the spin structure of the Understanding the spin structure of the proton in term of partons needs more proton in term of partons needs more theoretical and experimental effortstheoretical and experimental efforts

Parton physics is best calculated in lattice Parton physics is best calculated in lattice QCD using IMF formalism. QCD using IMF formalism.