QCD Map of the ProtonQCD Map of the Proton

Xiangdong JiXiangdong Ji

University of MarylandUniversity of Maryland

OutlineOutline

An Alternative Formulation of Quantum An Alternative Formulation of Quantum MechanicsMechanics

Wigner parton distributions (WPD)Wigner parton distributions (WPD)– mother of all distributions!mother of all distributions!

Transverse-momentum dependent Transverse-momentum dependent parton distributions and pQCD parton distributions and pQCD factorizationfactorization

GPD & quantum phase-space GPD & quantum phase-space tomographytomography

SummarySummary

Alternative Formulations of Alternative Formulations of Quantum MechanicsQuantum Mechanics

Quantum mechanical wave functions are not Quantum mechanical wave functions are not directly measurable in experimentdirectly measurable in experiment. But is it . But is it possible to formula quantum mechanics in possible to formula quantum mechanics in terms of observables?terms of observables?– Heisenberg’s matrix mechanics (1925)Heisenberg’s matrix mechanics (1925)

– Wigner’s phase-space distributions (1932)Wigner’s phase-space distributions (1932)

– Feynman path integrals (1948)Feynman path integrals (1948)

– ……

QM with phase-space distributionQM with phase-space distribution

Phase-space formulation is based on the Phase-space formulation is based on the statistical nature of quantum mechanics. statistical nature of quantum mechanics. – The state of a classical particle is specified by its The state of a classical particle is specified by its

coordinate and momentum (x,p): coordinate and momentum (x,p): phase-spacephase-space

A state of classical identical particle system A state of classical identical particle system can be described by a phase-space distribution can be described by a phase-space distribution f(x,p). Time evolution of f(x,p) obeys f(x,p). Time evolution of f(x,p) obeys Boltzmann equation. Boltzmann equation.

Many identical copies of a quantum system Many identical copies of a quantum system can be described by a similar phase-space can be described by a similar phase-space distributiondistribution..

Wigner functionWigner function

Define as Define as

When integrated over p, one gets the When integrated over p, one gets the coordinate space density coordinate space density ρρ(x)=|(x)=|ψψ(x)|(x)|22

– Measurable in elastic scatteringMeasurable in elastic scattering

When integrated over x, one gets the When integrated over x, one gets the coordinate space density n(p)=|coordinate space density n(p)=|ψψ(p)|(p)|22

– Measurable in knock-out scatteringMeasurable in knock-out scattering

Uncertainty principleUncertainty principle Not positive definite in Not positive definite in general. But it is in the classical limitgeneral. But it is in the classical limit!!

Wigner DistributionWigner Distribution

Wigner distributions are physical observables Wigner distributions are physical observables – Real (hermitian)Real (hermitian)– Super-observable!Super-observable!

Many applicationsMany applications– heavy-ion collisions, heavy-ion collisions, – quantum molecular dynamics, quantum molecular dynamics, – signal analysis, signal analysis, – quantum information, quantum information, – optics, optics, – image processing…image processing…

),(),(),( pxWpxdxdpOpxO

Simple Harmonic OscillatorSimple Harmonic Oscillator

N=0 N=5

•Phase-space distribution gives a vivid “classical” picture.•Non-positive definiteness is the key for quantum interference

Phase-space tomographyPhase-space tomography

Phase-space distribution (a map) can be constructed from slices with fixed momentum. – For small p, the oscillator is at the turning point of the

oscillator potential.

– For large p, the oscillator is at the middle of the potential

– For every p, we have a topographic picture of the system which give a much detailed map of the system.

This information cannot be obtained from the densities in space or momentum alone!

Measuring Wigner function Measuring Wigner function of Quantum Lightof Quantum Light

Measuring Wigner function Measuring Wigner function of the Vibrational State in a Moleculeof the Vibrational State in a Molecule

Quantum State Tomography of Quantum State Tomography of Dissociateng moleculesDissociateng molecules

Skovsen et al. Skovsen et al. (Denmark) PRL91, 090604(Denmark) PRL91, 090604

Wigner distributions for quarks in Wigner distributions for quarks in protonproton

Wigner operator (Wigner operator (X. Ji,PRL91:062001,2003X. Ji,PRL91:062001,2003))

Wigner distribution: “Wigner distribution: “densitydensity” for quarks ” for quarks having having position position rr and 4-momentum k and 4-momentum k (off-(off-shell)shell)

No known experiment can measure this!7-dimensional distributions

a la Saches

Custom-made for high-energy processes Custom-made for high-energy processes (I)(I)

In high-energy processes, one cannot measure In high-energy processes, one cannot measure kk = (k = (k00–k–kz)z) and therefore, one must integrate and therefore, one must integrate this out. this out.

The reduced Wigner distribution is a function The reduced Wigner distribution is a function of 6 variables [of 6 variables [r,k=(r,k=(kk++ kk)].)].

Mother of all SP distributionsMother of all SP distributions!!

Integrating over z, resultingIntegrating over z, resulting a phase-space distribution a phase-space distribution q(x, rq(x, r kk) ) through which parton saturation at small x is through which parton saturation at small x is

easy to see.easy to see.2 2 4

( , , )

2cNdxq x r k

d r d k

Custom-made for high-energy processes Custom-made for high-energy processes (II)(II)

Integrating over Integrating over rr, , resulting resulting transverse-momentum transverse-momentum

dependent (TMD) parton distributionsdependent (TMD) parton distributions! !

q(x, kq(x, k))

Measurable in semi-inclusive DIS & Drell-Yan &..Measurable in semi-inclusive DIS & Drell-Yan &..

A major subject of this meeting…A major subject of this meeting…

Integrating over Integrating over kk, resulting a, resulting a reduced Wigner reduced Wigner

distributiondistribution

The above are not related by Fourier transformation!The above are not related by Fourier transformation!

q(x,r)

k

Wigner parton distributions & Wigner parton distributions & offspringsoffsprings

Mother Dis. W(r,p)

q(x, rq(x, r, , kk))

TMDPD q (x, kTMDPD q (x, k))

Reduced Reduced wigner diswigner dis

q(x,r)q(x,r)

PDF q(x) Density ρ(r)

TMD Parton DistributionTMD Parton Distribution

Appear in the processes in which hadron Appear in the processes in which hadron transverse-momentum is measured, often transverse-momentum is measured, often together with TMD fragmentation functions. together with TMD fragmentation functions.

The leading-twist ones are classified by Boer, The leading-twist ones are classified by Boer, Mulders, and Tangerman (1996,1998)Mulders, and Tangerman (1996,1998)– There are 8 of them There are 8 of them

q(x, kq(x, k┴┴), ), qqTT(x, k(x, k┴┴),),

ΔΔqqLL(x, (x, kk┴┴), ), ΔΔqqTT(x, (x, kk┴┴), ),

δδq(x, q(x, kk┴┴)),, δδLLq(x, q(x, kk┴┴), ), δδTTq(x, q(x, kk┴┴), ), δδT’T’q(x, q(x, kk┴┴))

Factorization for SIDIS with PFactorization for SIDIS with P┴┴

For traditional high-energy process with one For traditional high-energy process with one hard scale, inclusive DIS, Drell-Yan, jet hard scale, inclusive DIS, Drell-Yan, jet production,…production,…soft divergences typically cancel,soft divergences typically cancel, except at the edges of phase-spaceexcept at the edges of phase-space. .

At present, we have At present, we have two scales, Q and Ptwo scales, Q and P┴┴

((could be softcould be soft). Therefore, besides the collinear ). Therefore, besides the collinear divergences which can be factorized into TMD divergences which can be factorized into TMD parton distributions (not entirely as shown by parton distributions (not entirely as shown by the energy-dependence), there are also soft the energy-dependence), there are also soft divergences which can be taken into account divergences which can be taken into account by by the soft factorthe soft factor. .

X. Ji, F. Yuan, and J. P. Ma, X. Ji, F. Yuan, and J. P. Ma, PRD71:034005,2005 PRD71:034005,2005

Example IExample I

Vertex correctionsVertex corrections

Four possible regions of gluon momentum k: 1) k is collinear to p (parton dis) 2) k is collinear to p′ (fragmentation) 3) k is soft (wilson line) 4) k is hard (pQCD correction)

p

p′q

k

A general reduced diagramA general reduced diagram

Factorization theoremFactorization theorem

For semi-inclusive DIS with small pFor semi-inclusive DIS with small pTT

~

• Hadron transverse-momentum is generated from multiple sources.• The soft factor is universal matrix elements of Wilson lines and spin-independent.• One-loop corrections to the hard-factor has been calculated

Spin-Dependent processesSpin-Dependent processes

Ji, Ma, Yuan, PLB597, 299 (2004); Ji, Ma, Yuan, PLB597, 299 (2004); PRPRD70:074021(2004)

Reduced Wigner Distributions and Reduced Wigner Distributions and GPDsGPDs

The 4D reduced Wigner distribution f(The 4D reduced Wigner distribution f(rr,x) is ,x) is related torelated to Generalized parton distributions Generalized parton distributions (GPD)(GPD) H and E through simple FTH and E through simple FT,,

t= – q2

~ qz

H,E depend only on 3 variables. There is a rotational symmetry in the transverse plane..

What is a GPD? What is a GPD?

A proton matrix element which is a hybrid of A proton matrix element which is a hybrid of elastic form factor and Feynman distributionelastic form factor and Feynman distribution

Distributions depending on Distributions depending on

xx: : fraction of the longitudinal momentum fraction of the longitudinal momentum carried carried

by partonby parton

t=qt=q22: : t-channel momentum transfer squaredt-channel momentum transfer squared

ξξ: : skewness parameter (a new variable coming skewness parameter (a new variable coming from selection of a light-cone direction)from selection of a light-cone direction)

Review: Review: M. Diehl, Phys. Rep. 388, 41 (2003)M. Diehl, Phys. Rep. 388, 41 (2003)

X. Ji, Ann. Rev. Nucl. Part. Sci. 54, 413 (2004)X. Ji, Ann. Rev. Nucl. Part. Sci. 54, 413 (2004)

Charge and Current Distributions Charge and Current Distributions in Phase-spacein Phase-space

Quark charge distributions at fixed xQuark charge distributions at fixed x

Quark current at fixed x in a spinning nucleonQuark current at fixed x in a spinning nucleon

A GPD or W-Parton Distribution A GPD or W-Parton Distribution ModelModel

A parametrization which satisfies the A parametrization which satisfies the following following Boundary Conditions: Boundary Conditions: ((A. Belitsky, X. Ji, A. Belitsky, X. Ji,

and F. Yuan, PRD 69,074014,2004and F. Yuan, PRD 69,074014,2004))– Reproduce measured Feynman distributionReproduce measured Feynman distribution

– Reproduce measured form factorsReproduce measured form factors

– Polynomiality condition Polynomiality condition

– PositivityPositivity

RefinementRefinement– Lattice QCDLattice QCD

– Experimental dataExperimental data

Imaging quarks at fixed Feynman-xImaging quarks at fixed Feynman-x

For every choice of x, one can use the Wigner For every choice of x, one can use the Wigner distributions to picture the nucleon in 3-space; distributions to picture the nucleon in 3-space; quantum phase-space tomography!

z

bx

by

CommentsComments

If one puts the pictures at all x together, one If one puts the pictures at all x together, one gets a spherically round nucleon! (Wigner-gets a spherically round nucleon! (Wigner-Eckart theorem)Eckart theorem)

If one integrates over the distribution along the If one integrates over the distribution along the z direction, one gets the 2D impact parameter z direction, one gets the 2D impact parameter space pictures of Burkardt and Soper.space pictures of Burkardt and Soper.

Impact parameter space distributionImpact parameter space distribution

Obtained by integrating over z, (Soper, Obtained by integrating over z, (Soper, Burkardt)Burkardt)

x and b are in different directions and x and b are in different directions and therefore, there is no quantum mechanical therefore, there is no quantum mechanical constraint.constraint.– It is a true densityIt is a true density

– Momentum density in the z-directionMomentum density in the z-direction

– Coordinate density in the transverse plane.Coordinate density in the transverse plane.

( , ) ( , , )f x b dz f x b r z WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

QCD-Map: how to obtain it?QCD-Map: how to obtain it?

DataData

ParametrizationsParametrizations

Lattice QCDLattice QCD

Mass distributionMass distribution

Gravity plays an important role in cosmos and Gravity plays an important role in cosmos and at Plank scale. In the atomic world, the gravity at Plank scale. In the atomic world, the gravity is too weak to be significant (old view).is too weak to be significant (old view).

The phase-space quark distribution allows to The phase-space quark distribution allows to determine the determine the mass distributionmass distribution in the proton in the proton by integrating over x-weighted density, by integrating over x-weighted density,

– Where A, B and C are gravitational form factorsWhere A, B and C are gravitational form factors

Spin of the ProtonSpin of the Proton

Was thought to be carried by the spin of the Was thought to be carried by the spin of the three valence quarksthree valence quarks

Polarized deep-inelastic scattering found that Polarized deep-inelastic scattering found that only 20-30% are in the spin of the quarks.only 20-30% are in the spin of the quarks.

Integrate over the x-weighted phase-space Integrate over the x-weighted phase-space current, one gets the current, one gets the momentum currentmomentum current

Spin sum ruleSpin sum rule

One can calculate the total quark (orbital + One can calculate the total quark (orbital + spin) contribution to the spin of the protonspin) contribution to the spin of the proton

Amount of proton angular momentum carried Amount of proton angular momentum carried by quarks isby quarks is

2 2

2 2

1 10, 0,

2 21

[ ( , ,0, ) ( , ,0, )2

q q qJ A B

dxx H x E x

SummarySummary

One of the central goals for 12 GeV upgrade is One of the central goals for 12 GeV upgrade is to obtain a QCD map of the proton: DNA to obtain a QCD map of the proton: DNA sequencing in biologysequencing in biology

TMD parton distributions: semi-inclusive TMD parton distributions: semi-inclusive processesprocesses

Quantum phase-space tomographyQuantum phase-space tomography– Mass and spin of the protonMass and spin of the proton