Xiangdong JiXiangdong Ji

University of Maryland/SJTUUniversity of Maryland/SJTU

Physics of gluon Physics of gluon polarizationpolarization

Jlab, May 9, 2013

Gluon polarizationGluon polarization

Ever since the EMC “spin crisis”, the gluon Ever since the EMC “spin crisis”, the gluon polarization has been one of the most polarization has been one of the most important pursuits of hadron physics important pursuits of hadron physics communitycommunity HERMES COMPASS RHIC SPIN “HERA-N” “EIC…”

Physics argumentsPhysics arguments

ΔΔG G is an obvious contribution to the spin of is an obvious contribution to the spin of the proton. the proton.

Can contribute to the quark helicity Can contribute to the quark helicity through axial anomaly (Altarelli & Ross, through axial anomaly (Altarelli & Ross, Carlitz, Collins, & Mueller,…)Carlitz, Collins, & Mueller,…)

Its contribution to the spin grows like 1/Its contribution to the spin grows like 1/ααSS

However, there are a number of However, there are a number of theoretical puzzles about it! theoretical puzzles about it!

Axial AnomalyAxial Anomaly

It was argued that Δq probed by DIS is not It was argued that Δq probed by DIS is not entirely due to the quark contribution. entirely due to the quark contribution. There is a gluon anomaly contribution. There is a gluon anomaly contribution. This contribution is proportional to (This contribution is proportional to (ααS S

/2/2ππ))ΔΔGG For the anomaly to have a large For the anomaly to have a large

contribution contribution

ΔΔG must be on the order few unit of hbar. G must be on the order few unit of hbar. Thus, Thus, ΔΔG could be large even at non-G could be large even at non-

perturbative scale. perturbative scale.

Physics: Feynman parton picture Physics: Feynman parton picture

A fast moving proton is a beam of free quarks A fast moving proton is a beam of free quarks and gluons.and gluons.

The gluon partons have well-defined helicity ± The gluon partons have well-defined helicity ± 1 and densities g1 and densities g±±(x) in wavelength(x) in wavelength

Gluon helicity distribution is Gluon helicity distribution is

g(x) = g+(x) – g-(x) and G = ʃdx g(x) is the fraction of the proton helicity

carried in the gluon.

1/2 +1 or -1

QCD expressionQCD expression

The total gluon helicity The total gluon helicity ΔGΔG is gauge is gauge invariant quantity, and has a complicated invariant quantity, and has a complicated expression in QCD factorization (Manohar, expression in QCD factorization (Manohar, 1991) 1991)

It does not look anything like gluon spin or It does not look anything like gluon spin or helicity! Not in any textbook! helicity! Not in any textbook!

Light-cone gauge Light-cone gauge

In light-cone gauge AIn light-cone gauge A++=0, the above =0, the above expression reduces to a simple form expression reduces to a simple form

which is the spin of the photon (gluon) which is the spin of the photon (gluon)

(J. D. Jackson, CED), (J. D. Jackson, CED),

but is not gauge-symmetric: There is no but is not gauge-symmetric: There is no gauge symmetry notion of the gluon spin! gauge symmetry notion of the gluon spin!

(J. D. Jackson, L. Landau & Lifshitz). (J. D. Jackson, L. Landau & Lifshitz).

Don’t know how to calculate Don’t know how to calculate

ΔΔG involves explicit light-cone correlation G involves explicit light-cone correlation or real time. No one knows how to or real time. No one knows how to calculate this in lattice QCD (Models: RL calculate this in lattice QCD (Models: RL Jaffe, Chen & Ji) Jaffe, Chen & Ji)

One can consider A+=0 gauge, but no one One can consider A+=0 gauge, but no one knows how to fix this gauge in lattice QCDknows how to fix this gauge in lattice QCD

Thus there is no way to confront theory Thus there is no way to confront theory with experiment: with experiment: G = ʃdx g(x)

Is there a large contribution from small Is there a large contribution from small x?x?

AALLLL from RHIC 2009 from RHIC 2009

99

Electric field of a charge Electric field of a charge

A moving chargeA moving charge

Gauge potentialGauge potential

ObservationsObservations

Although the transverse part of the vector Although the transverse part of the vector potential is gauge invariant, the separately potential is gauge invariant, the separately EE┴┴ does not transform properly, under does not transform properly, under Loretez transformation, and is not a physical Loretez transformation, and is not a physical observable observable (X. Chen et al, x. Ji, PRL) (X. Chen et al, x. Ji, PRL) E ┴ generated from E ║ from Lorentz boost. A lorentz-transformed E has different

decomposition E = E┴ + E║ in different frames.

There is no charge that separately responds to E┴

and E║

Large momentum limitLarge momentum limit

As the charge velocity approaches the As the charge velocity approaches the speed of light, Espeed of light, E┴ ┴ >>E>>E║║, B ~ E, B ~ E┴┴, thus, thus E┴ become physically meaningul

The E┴ & B fields appear to be that of the free radiation

Weizsacker-William equivalent photon Weizsacker-William equivalent photon approximation (J. D. Jackson)approximation (J. D. Jackson)

Thus gauge-invariant AThus gauge-invariant A┴┴ appears to be now appears to be now physical which generates the Ephysical which generates the E ┴ ┴ & B. & B.

Gauge invariant photon helicityGauge invariant photon helicity

X. Chen et al (PRL, 09’) proposed that a X. Chen et al (PRL, 09’) proposed that a gauge invariant photon angular momentum gauge invariant photon angular momentum can be defined ascan be defined as

ExAExA┴┴

This is not an observable when the system move at finite momentum because this is only a part of the contribution which cannot be measured separately.

However, it becomes an observable in the However, it becomes an observable in the IMF when Weizsacker-William’s picture is IMF when Weizsacker-William’s picture is true!true!

TheoremTheorem

Thus, one would expect that the total gluon helicity Thus, one would expect that the total gluon helicity ΔΔG G must be the matrix element of ExAmust be the matrix element of ExA┴ ┴ in a large momentum in a large momentum nucleon. nucleon.

X. Ji, J. Zhang, and Y. Zhao (arXiv:1304.6708)X. Ji, J. Zhang, and Y. Zhao (arXiv:1304.6708)

is just the IMF limit of the matrix element is just the IMF limit of the matrix element

of ExAof ExA┴┴

QCD caseQCD case

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,

In the IMF, the gauge-invariant gluon spin In the IMF, the gauge-invariant gluon spin becomesbecomes

One-loop exampleOne-loop example

The result is frame-dependent, with log The result is frame-dependent, with log dependences on the external momentumdependences on the external momentum

Anomalous dimension coincides with X. Chen et al. Anomalous dimension coincides with X. Chen et al.

Taking large P limitTaking large P limit

If one takes P-> ∞ first before the loop If one takes P-> ∞ first before the loop integral, one finds integral, one finds

This is exactly photon (gluon) helicity This is exactly photon (gluon) helicity calculated in QCD factorization! Has the calculated in QCD factorization! Has the correct anomalous dimension. correct anomalous dimension.

Subtlety of the limiting proceduresSubtlety of the limiting procedures

There are two possible limits,There are two possible limits, Taking p->∞ before UV regularization (physical

case, light-cone) Taking UV regularization before p-> ∞

(practical calculation, time-independent) Two limits get the same IR physicsTwo limits get the same IR physics One can get one limit from the other by a One can get one limit from the other by a

perturbative matching condition, Z. perturbative matching condition, Z.

Lattice QCDLattice QCD

ExAExA┴┴ is perfectly fit for lattice QCD is perfectly fit for lattice QCD calculation of calculation of ΔΔG!G!

To get large momentum nucleon, one has To get large momentum nucleon, one has to have a fine lattice in the z-direction: to have a fine lattice in the z-direction:

P ~ 1/aP ~ 1/a To separate excited states of the moving To separate excited states of the moving

nucleon, one also needs fine lattice nucleon, one also needs fine lattice spacing in the time direction. spacing in the time direction.

323222X64X642 2

What a lattice calculation of What a lattice calculation of ΔΔG G implies?implies?

Settles if axial anomaly plays an important Settles if axial anomaly plays an important role in the quark helicity measurement, by role in the quark helicity measurement, by determining how large is determining how large is ΔΔGG

Since the experimental data says, Since the experimental data says,

how much how much ΔΔG sits at very small x?G sits at very small x? How much the gluon helicity contributes to How much the gluon helicity contributes to

the proton helicity at small scale. the proton helicity at small scale.

x-dependence? x-dependence?

X-dependence of a parton distribution has X-dependence of a parton distribution has been very difficult to calculate in the past. been very difficult to calculate in the past. The only approach is through the local The only approach is through the local moments. moments.

However, it is very difficult to calculate However, it is very difficult to calculate higher moments numerically. higher moments numerically.

It will be nice to find a way to directly It will be nice to find a way to directly calculate the x-dependence on latticecalculate the x-dependence on lattice

arXiv1305.1539arXiv1305.1539

observationobservation

Usual parton distributionUsual parton distribution

Consider instead Consider instead

RelationshipRelationship

The matching condition is perturbativeThe matching condition is perturbative

The correction is power-suppressed. For The correction is power-suppressed. For practical calculation, a momentum of 5 practical calculation, a momentum of 5 GeV might be good enough. GeV might be good enough.

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 Light-cone wave functions Higher-twists….Higher-twists….

ConclusionsConclusions

We find the gluon helicity measured in high-We find the gluon helicity measured in high-energy scattering is just EXAenergy scattering is just EXA┴ ┴ in the large in the large momentum limit, momentum limit, This gives the gauge invariant and physically

manifest notion of the total gluon helicity This gives practical way to calculate ΔG

We find a practical way how to calculate light-We find a practical way how to calculate light-cone distributions:cone distributions:

PDFs, TMDs, GPDs, HTs, LCWFs, LCDAs, etc…PDFs, TMDs, GPDs, HTs, LCWFs, LCDAs, etc…

ten years from now there will be a lot of ten years from now there will be a lot of lattice result. lattice result.