Xiangdong JiXiangdong Ji

University of MarylandUniversity of Maryland

Shanghai Jiao Tong UniversityShanghai Jiao Tong University

Parton Physics on a Bjorken-frame lattice

July 1, 2013

Knowledge of parton Knowledge of parton distributions is data-drivendistributions is data-driven─── ─── Paul Reimer from the prevous talk of Paul Reimer from the prevous talk of

this workshopthis workshop

OutlineOutline

Review of Bjorken frame and parton Review of Bjorken frame and parton physicsphysics

Why parton physics is hard to calculate? Why parton physics is hard to calculate? A new proposal, resource requirement, A new proposal, resource requirement,

and applicabilityand applicability Gluon polarization: its physics and Gluon polarization: its physics and

calculation calculation OutlooksOutlooks

High-energy scattering High-energy scattering and Bjorken frameand Bjorken frame

In high-energy scattering, the nucleon has a In high-energy scattering, the nucleon has a large momentum relative to the probes. large momentum relative to the probes.

In the Bjorken frameIn the Bjorken frame, the probes (electron or , the probes (electron or virtual photon) may have the smallest virtual photon) may have the smallest momentum, but the proton has a large momentum, but the proton has a large momentum (momentum (infinite momentum frameinfinite momentum frame, IMF) , IMF) relative to the observer and travels at near relative to the observer and travels at near the speed of light. the speed of light.

This frame has been used frequently in the This frame has been used frequently in the old literature, but giving away to rest-frame old literature, but giving away to rest-frame light-front quantization in recent years. light-front quantization in recent years.

Electron scattering in Bjorken Electron scattering in Bjorken frameframe

4-momentum transfer q4-momentum transfer qµµ = (v, q) is a space like = (v, q) is a space like vector vvector 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 /2x, thus PP3 3 = Q/2x= Q/2x. .

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

Bjorken frame and Parton physicsBjorken frame and Parton physics

The interactions between particles are Lorentz-dilated, The interactions between particles are Lorentz-dilated, and thus the system appears as if interaction-free: and thus the system appears as if interaction-free: the the proton is probed as free partons. proton is probed as free partons. In QCD, parton physics emerges when working in light-cone

gauge. A+=0. In field theory, parton physics is cut-off dependent. This is only true to a certain degree: leading twist. The

so-called higher-twist contributions are sensitive to parton off-shellness, transverse momentum and correlations.

Quark and gluon parton Quark and gluon parton distributionsdistributions

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

Parton PhysicsParton Physics

Light-cone wave function, Light-cone wave function, ψψnn(x(xii, k, k⊥⊥ii))

Distributions amplitudes, Distributions amplitudes, ψψnn(x(xii)) Parton distributions, f(x)Parton distributions, f(x) Transverse momentum dependent (TMD) Transverse momentum dependent (TMD)

parton distributions, f(x, kparton distributions, f(x, k⊥⊥))

Generalized parton distributions, F(x,Generalized parton distributions, F(x,𝜉,𝜉,rr⊥⊥))

Wigner distributions, W(x, kWigner distributions, W(x, k⊥⊥, r, r⊥⊥)) Fragmentation functions…Fragmentation functions…

Frame-independent formulation Frame-independent formulation of parton physicsof parton physics

Over the years, the parton physics have been Over the years, the parton physics have been formulated in a boost-invariant way. In formulated in a boost-invariant way. In particular it can be described particular it can be described as the physics in as the physics in the rest frame. the rest frame.

In this frame, the probe appears a In this frame, the probe appears a light-front light-front (light-like) correlation(light-like) correlation. .

Thus light-cone quantization is the essential Thus light-cone quantization is the essential tool tool (S. Brodsky)(S. Brodsky)! !

Light-front quantization Light-front quantization

Unique role of lattice QCD (1974)Unique role of lattice QCD (1974)

Lattice is the only non-perturbative Lattice is the only non-perturbative approach to solve QCD approach to solve QCD Light-front quantization: many years of efforts

but hard for 3+1 physics AdS/CFT: no exact correspondence can be

established, a model. An intrinsically Euclidean approachAn intrinsically Euclidean approach

“time” is Eucliean 𝜏=i t, no real time A4 = iA0 is real (as oppose to A0 is real) No direct implementation of physical time.

Ken Wilson (1936-2013)Ken Wilson (1936-2013)

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

Parton physics? Light-like correlationsParton physics? Light-like correlations

For parton distributions & distribution For parton distributions & distribution amplitudes: moments are ME of local operators, amplitudes: moments are ME of local operators, 2-3 moments. Very difficult beyond that…2-3 moments. Very difficult beyond that…

For parton physics that cannot be reduced to For parton physics that cannot be reduced to local operators, local operators, there is no way to calculatethere is no way to calculate! !

𝜉3

𝜉0 𝜉+𝜉-

A Euclidean distributionA Euclidean distribution

Consider space correlation in a large Consider space correlation in a large momentum P in the z-direction.momentum P in the z-direction.

Quark fields separated along the z-direction The gauge-link along the z-direction The matrix element depends on the momentum P. This distribution can be calculated using standard

lattice method.

𝜉3𝜉0

Z0

Taking the limit P-> ∞ firstTaking the limit P-> ∞ first

After renormalizing all the UV divergences, After renormalizing all the UV divergences, one has the standard quark distribution!one has the standard quark distribution! One can prove this using the standard OPE One can also see this by writing |P˃ = U(Λ(p)) |p=0> and applying the boost operator on the gauge link

The Altarelli-Parisi evolution was derived this way!

𝜉3𝜉0 𝜉+𝜉-

Finite but large PFinite but large P

The distribution at a finite but large P is The distribution at a finite but large P is the most interesting because it is the most interesting because it is potentially calculable in lattice QCD. potentially calculable in lattice QCD.

Since it differs from the standard PDF by Since it differs from the standard PDF by simply an infinite P limit, it shall have the simply an infinite P limit, it shall have the same infrared (collinear) physics. same infrared (collinear) physics.

It shall be related to the standard PDF by a It shall be related to the standard PDF by a matching condition in the sense that the matching condition in the sense that the latter is an effective theory of the former. latter is an effective theory of the former.

Relationship: factorization Relationship: factorization theoremtheorem

The matching condition is perturbativeThe matching condition is perturbative

The correction is power-suppressed. The correction is power-suppressed.

Pictorial factorizationPictorial factorization

q(x, μ)

ZZ(P, μ)

q(x, P, μ)

One-loop exampleOne-loop example

PPzz dependence is mostly isolated in the dependence is mostly isolated in the large logs of the loop integral. large logs of the loop integral.

Practical considerationsPractical considerations

For a fixed x, large PFor a fixed x, large Pzz means large k means large kzz, thus, , thus, as Pas Pzz gets larger, the valence quark gets larger, the valence quark distribution in the z-direction get Lorentz distribution in the z-direction get Lorentz contracted, z~1/kcontracted, z~1/kzz. .

Thus one needs increasing resolution in Thus one needs increasing resolution in the z-direction for a large-momentum the z-direction for a large-momentum nucleon. Roughly speaking: anucleon. Roughly speaking: aLL/a/aTT ~ ~ γγ

z

x,y

One needs special kinds of lattices γ=2

z

x,y

γ=4

Small x partonsSmall x partons

The smallest x partons that one access for The smallest x partons that one access for a nucleon momentum P is roughly, a nucleon momentum P is roughly,

xxminmin = = ΛΛQCDQCD/P~ 1/3/P~ 1/3γγ

small x physics needs large small x physics needs large γγ as well. as well. Consider x ~ 0.01, one needs a Consider x ~ 0.01, one needs a γγ factor factor

about 10~30. This means 100 lattice about 10~30. This means 100 lattice points along the z-direction. points along the z-direction.

A large momentum nucleon costs A large momentum nucleon costs considerable resources!considerable resources!

Ideal lattice configurationsIdeal lattice configurations

Time direction also needs longer evolution Time direction also needs longer evolution because the energy difference between because the energy difference between excited states and the ground state goes excited states and the ground state goes like 1/like 1/ γ γ

Thus ideal configurations for parton Thus ideal configurations for parton physics calculations will be physics calculations will be

242422x(24x(24γγ))2 2 or 36 or 3622x(36x(36γγ))22

There are not yet available!There are not yet available!

Sea quarksSea quarks

The parton picture is clearest in the axial The parton picture is clearest in the axial gauge Agauge AZ Z =0. =0.

In this gauge, see quarks correspond to In this gauge, see quarks correspond to backward moving quarks (Pbackward moving quarks (Pzz>0, k>0, kzz<0) or <0) or forward moving antiquark, but otherwise forward moving antiquark, but otherwise having arbitrary transverse momentum having arbitrary transverse momentum (with cut-off (with cut-off μμ) and energy (off-shellness). ) and energy (off-shellness).

In the limit of PIn the limit of Pzz->∞, the contribution does ->∞, the contribution does not vanish. not vanish.

Flavor structure? (Hueywen Lin’s talk)Flavor structure? (Hueywen Lin’s talk)

1/P1/P22 correction correction

Two types (to be published)Two types (to be published) The nucleon mass corrections in the traces The nucleon mass corrections in the traces

of the twsit-2 matrix elements can easily of the twsit-2 matrix elements can easily be calculated. be calculated.

Corrections in twist-four contributions can Corrections in twist-four contributions can also be directly calculated on lattice. The also be directly calculated on lattice. The contribution is suspected to be smaller contribution is suspected to be smaller than the mass correction. than the mass correction.

Higher-order corrections can similarly be Higher-order corrections can similarly be handled. handled.

Other applications Other applications

This approach is applicable for all parton This approach is applicable for all parton physicsphysics

Recipe: Recipe: Replace the light-cone correlation by that in

the z-direction. Replace the gauge link in the light-cone

direction by that in the z-direction. Derive factorizations of the resulting

distributions in terms of light-cone parton physics.

GPDs and TMDsGPDs and TMDs

GPDsGPDs

TMDsTMDs

Wigner distributions and LC Wigner distributions and LC amplitudesamplitudes

Wigner distributionWigner distribution

Light-cone amplitudesLight-cone amplitudes

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

Gluon helicity distributionGluon helicity distribution

∆∆g(x)g(x)

An important part of the nucleon spin An important part of the nucleon spin structurestructure

Much attention has been paid to this Much attention has been paid to this quantity experimentallyquantity experimentally DIS semi-inclusive RHIC spin …

In principle, it can be calculated from the In principle, it can be calculated from the approach discussed previously. However, approach discussed previously. However, it is still difficult to get ∆G, the integral. it is still difficult to get ∆G, the integral.

AALLLL from RHIC 2009 from RHIC 2009

3636

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).

Two long-standing problemsTwo long-standing problems

∆∆G does not have a gauge-invariant notion G does not have a gauge-invariant notion of the gluon spin.of the gluon spin.

There is no direct way to calculate ∆G, There is no direct way to calculate ∆G, unlike ∆∑, and orbital angular momentum. unlike ∆∑, and orbital angular momentum.

Electric field of a charge Electric field of a charge

A moving chargeA moving charge

Gauge potentialGauge potential

Suggestion by X. Chen et alSuggestion by X. Chen et al

Although the transverse part of the vector Although the transverse part of the vector potential is gauge invariant, the potential is gauge invariant, the separately Eseparately E┴┴ does not transform properly, does not transform properly, under Loretez transformation, and is not a under Loretez transformation, and is not a physical observable physical observable (X. Chen et al, x. Ji, (X. Chen et al, x. Ji, PRL) PRL)

Gauge invariant photon helicityGauge invariant photon helicity

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

ExAExA┴┴

This is not an observable when the system This is not an observable when the system move at finite momentum because (X. Ji)move at finite momentum because (X. Ji)

AA ┴ ┴ generated from A generated from A║║ from Lorentz boost. from Lorentz boost. A lorentz-transformed A has different

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

There is no charge that separately responds to AThere is no charge that separately responds to A┴ ┴ and Aand A║║

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.

TheoremTheorem

The total gluon helicity The total gluon helicity ΔΔG shall beG shall be the matrix element of the matrix element of ExAExA┴ ┴ in a large momentum nucleon. in a large momentum nucleon.

We proved in the following paper We proved in the following paper 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.

Matching conditionMatching condition

Taking UV regularization before p-> ∞ Taking UV regularization before p-> ∞ (practical calculation, time-independent)(practical calculation, time-independent)

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.

AA┴┴ can be obtained from Coulomb gauge can be obtained from Coulomb gauge fixing on lattice. fixing on lattice.

ConclusionsConclusions

Parton physics can be explored in lattice Parton physics can be explored in lattice QCD calculations using the Bjorken frame. QCD calculations using the Bjorken frame. This opens the door for precision This opens the door for precision comparisons of high-energy scattering comparisons of high-energy scattering data and fundamental QCD calculations. data and fundamental QCD calculations.

It will be a while before that “It will be a while before that “data driving” data driving” era is overera is over. However, we know how to get . However, we know how to get there. there.

”” He (Wilson) was decades ahead of his time with respect to computing and networks.”.”

─── ─── Paul Ginsparg, CornellPaul Ginsparg, Cornell