Wigner Functions; Wigner Functions; PPTT-Dependent Factorization in SIDIS-Dependent Factorization in SIDIS

Xiangdong JiXiangdong Ji

University of MarylandUniversity of Maryland

— COMPASS Workshop, Paris, March 1-3, 2004 —

OutlineOutline

1.1. Quantum phase-space (Wigner) Quantum phase-space (Wigner) distributions.distributions.

2.2. GPD and phase-space picture of the GPD and phase-space picture of the nucleon.nucleon.

3.3. Transverse-momentum-dependent Transverse-momentum-dependent (TMD) parton distributions.(TMD) parton distributions.

4.4. Factorization theorem in semi-inclusive Factorization theorem in semi-inclusive DIS.DIS.

5.5. Single spin asymmetry: Single spin asymmetry: transversity and transversity and the connection between Collins, Sivers the connection between Collins, Sivers and Efremov-Teryaev-Sterman-Qiu.and Efremov-Teryaev-Sterman-Qiu.

MotivationMotivation

Elastic form-factorsElastic form-factors provide static provide static coordinate-coordinate-spacespace charge and current distributions (in the charge and current distributions (in the sense of Sachs, for example), but no sense of Sachs, for example), but no information on the dynamical motion.information on the dynamical motion.

Feynman parton densitiesFeynman parton densities give give momentum-momentum-space space distributions of constituents, but no distributions of constituents, but no information of the spatial location of the information of the spatial location of the partons.partons.

But sometimes, we need to know the But sometimes, we need to know the position position and momentum and momentum of the constituents. of the constituents. – For example, one need to know For example, one need to know rr and and p p to to

calculate calculate L=r×p !L=r×p !

Phase-space Distribution?Phase-space Distribution?

TheThe state state of a classical particle is specified of a classical particle is specified completelycompletely by its coordinate and momentum by its coordinate and momentum (x,p): (x,p): phase-spacephase-space– A state of classical identical particle system can be A state of classical identical particle system can be

described by a phase-space distribution described by a phase-space distribution f(x,p).f(x,p).

In quantum mechanics, because of the In quantum mechanics, because of the uncertainty principle, the phase-space uncertainty principle, the phase-space information is a “luxury”, but…information is a “luxury”, but…

Wigner introduced the first phase-space Wigner introduced the first phase-space distribution in quantum mechanics (1932) distribution in quantum mechanics (1932) – Heavy-ion collisions, quantum molecular Heavy-ion collisions, quantum molecular

dynamics, signal analysis, quantum info, dynamics, signal analysis, quantum info, optics, image processing…optics, image processing…

Wigner functionWigner function

Define as Define as

– When integrated over x (p), one gets the When integrated over x (p), one gets the momentum (probability) density. momentum (probability) density.

– Not positive definite in general, but is in Not positive definite in general, but is in classical limit.classical limit.

– Any dynamical variable can be calculated as Any dynamical variable can be calculated as ),(),(),( pxWpxdxdpOpxO

Short of measuring the wave function, the Wigner functioncontains the most complete (one-body) info about a quantum system.

Simple Harmonic OscillatorSimple Harmonic Oscillator

Husimi distribution: positive definite!Husimi distribution: positive definite!

N=0 N=5

Measuring Wigner function Measuring Wigner function of a quantum Light!of a quantum Light!

Quarks in the ProtonQuarks in the Proton

Wigner operatorWigner operator

Wigner distributionWigner 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 distribtuion

a la Saches

Ji (2003)

Custom-made for high-energy Custom-made for high-energy processes processes

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

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

1.1. After integrating over After integrating over r, r, one gets one gets transverse-momentum dependent (TDM) transverse-momentum dependent (TDM) parton distributions.parton distributions.

2.2. Alternatively, after integrating over Alternatively, after integrating over kk, , one gets a one gets a spatial distribution of spatial distribution of quarks with fixed Feynman quarks with fixed Feynman momentummomentum k k++=(k=(k00+k+kzz)=xM.)=xM.

f(r,x)

Proton images at a fixed xProton images at a fixed x

For every choice of x, one can use the Wigner For every choice of x, one can use the Wigner distribution to picture the quarks; distribution to picture the quarks; This is This is analogous to viewing the proton through the analogous to viewing the proton through the x x (momentum(momentum) filters!) filters!

The distribution is related to The distribution is related to Generalized Generalized parton distributionsparton distributions (GPD) (GPD) through through

t= – q2

~ qz

A GPD or Wigner Function ModelA GPD or Wigner Function Model

A parametrization which satisfies the A parametrization which satisfies the following following Boundary Conditions: Boundary Conditions: ((A. Belitsky, A. Belitsky, X. Ji, and F. Yuan, hep-ph/0307383, to appear X. Ji, and F. Yuan, hep-ph/0307383, to appear in PRDin PRD))– 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

x

y

z

Up-Quark Charge Density at x=0.4Up-Quark Charge Density at x=0.4

Up-Quark Charge Denstiy at x=0.01Up-Quark Charge Denstiy at x=0.01

Up-Quark Density At x=0.7Up-Quark Density At x=0.7

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 M. Burkardt (2000) and Soper.space pictures of M. Burkardt (2000) and Soper.

TMD Parton DistributionTMD Parton Distribution

Appear in the process in which hadron Appear in the process 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┴┴))

UV Scale-dependenceUV Scale-dependence

The ultraviolet-scale dependence is very The ultraviolet-scale dependence is very simple. It obeys an evolution equation simple. It obeys an evolution equation depending on the anomalous dimension of the depending on the anomalous dimension of the quark field in the v·A=0 gauge. quark field in the v·A=0 gauge.

However, we know the integrated parton However, we know the integrated parton distributions have a complicated scale-distributions have a complicated scale-dependence (dependence (DGLAP-evolutionDGLAP-evolution) )

Additional UV divergences are generated Additional UV divergences are generated through integration over transverse-through integration over transverse-momentum, which implies that momentum, which implies that

∫∫µ µ dd22kk┴┴ q(x, k q(x, k┴┴) ) q(x,µ) q(x,µ)

Consistency of UV RegularizationConsistency of UV Regularization

Feynman parton distributions are available in Feynman parton distributions are available in the scheme: the scheme: dimensional regularization, dimensional regularization, minimal subtractionminimal subtraction..

This cannot be implemented for TMD parton This cannot be implemented for TMD parton distributions because d=4 before the distributions because d=4 before the transverse-momentum is integrated.transverse-momentum is integrated.

On the other hand, it is difficult to implement a On the other hand, it is difficult to implement a momentum cut-off scheme for gauge theories…momentum cut-off scheme for gauge theories…

( I love to have one for many other reasons!)( I love to have one for many other reasons!)

Therefore, it is highly nontrivial thatTherefore, it is highly nontrivial that

∫ ∫ dd22kk┴┴ q(x, k q(x, k┴┴) ) Fey. Dis. known from fits? Fey. Dis. known from fits?

Gauge Invariance?Gauge Invariance?

Can be made gauge-independent by inserting a Can be made gauge-independent by inserting a gauge link going out to infinity in some direction gauge link going out to infinity in some direction vv (in non-singular gauges). (in non-singular gauges).

In singular gauges, the issue is more complicated In singular gauges, the issue is more complicated – Ji & Yuan (2003) conjectured a link at infinity to Ji & Yuan (2003) conjectured a link at infinity to

reproduce the SSA in a model by Brodsky et. al.reproduce the SSA in a model by Brodsky et. al.– Belitsky, Ji & Yuan (2003) derived the gauge linkBelitsky, Ji & Yuan (2003) derived the gauge link– Boer, Mulders, and Pijlman (2003): implications for real Boer, Mulders, and Pijlman (2003): implications for real

processesprocesses

If the link is not along the light-cone (used by If the link is not along the light-cone (used by Collins and Soper, and others…). The integration Collins and Soper, and others…). The integration over kover k┴ ┴ does not recover the usual parton does not recover the usual parton distribution.distribution.

Evolution In Gluon RapidityEvolution In Gluon Rapidity

The transverse momentum of the quarks can The transverse momentum of the quarks can be generated by soft gluon radiation. As the be generated by soft gluon radiation. As the energy of the nucleon becomes large, more energy of the nucleon becomes large, more gluon radiation (larger gluon rapidity) gluon radiation (larger gluon rapidity) contributes to generate a fixed transverse-contributes to generate a fixed transverse-

momentummomentum. . The evolution equation in energy or gluon The evolution equation in energy or gluon

rapidity has been derived by Collins and Soper rapidity has been derived by Collins and Soper (1981), but is non-perturbative if k(1981), but is non-perturbative if k┴,┴, is small. is small.

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 (to be published)X. Ji, F. Yuan, and J. P. Ma (to be published)

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

Example IIExample II

Gluon RadiationGluon Radiation

The dominating topology is the quark carrying most of the energy and momentum 1) k is collinear to p (parton dis) 2) k is collinear to p′ (fragmentation) 3) k is soft (Wilson line)

p

p′q

k

The best-way to handle all these is the soft-collineareffective field theory… (Bauer, Fleming, Steward,…)

A general leading region in non-singular A general leading region in non-singular gaugesgauges

J

J

s

H H

P P

PhPh

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

Sudakov double logs and soft Sudakov double logs and soft radiationradiation

Soft-radiation generates the so-called Sudakov Soft-radiation generates the so-called Sudakov double logarithms lndouble logarithms ln22QQ22/p/p22

TT and makes the and makes the hadrons with small-phadrons with small-pT T exponentially exponentially suppressed. suppressed.

Soft-radiation tends to wash out the Soft-radiation tends to wash out the (transverse) spin effects at very high-energy, (transverse) spin effects at very high-energy, de-coupling the correlation between spin and de-coupling the correlation between spin and transverse-momentum.transverse-momentum.

Soft-radiation is calculable at large pSoft-radiation is calculable at large pTT

What is a Single Spin Asymmetry What is a Single Spin Asymmetry (SSA)?(SSA)?

Consider scattering of a transversely-polarized Consider scattering of a transversely-polarized spin-1/2 hadron (spin-1/2 hadron (SS, , pp) with another hadron (or ) with another hadron (or photon), observing a particle of momentum photon), observing a particle of momentum kk

S

kp

The cross section can have a term depending on the azimuthal angle of k

)(~ kpSd

which produce an asymmetry AN when S flips: SSA

p’z

x

y

Why Does SSA Exist? Why Does SSA Exist?

Single Spin Asymmetry is proportional to Single Spin Asymmetry is proportional to

Im (FIm (FNN * F * FFF))

where where FFNN is the normal helicity amplitude is the normal helicity amplitude

and and FFFF is a spin flip amplitude is a spin flip amplitude

– Helicity flipHelicity flip: one must have a reaction mechanism : one must have a reaction mechanism for the hadron to change its helicity (in a cut for the hadron to change its helicity (in a cut diagram).diagram).

– Final State InteractionsFinal State Interactions (FSI) (FSI): to general a phase : to general a phase difference between two amplitudes. difference between two amplitudes.

The phase difference is needed because the The phase difference is needed because the structure structure

SS ··((pp ×× k k) formally violate time-reversal invariance.) formally violate time-reversal invariance.

Parton Orbital Angular Momentum Parton Orbital Angular Momentum and Gluon Spinand Gluon Spin

The hadron helicity flip can be generated by The hadron helicity flip can be generated by other mechanism in QCDother mechanism in QCD– Quark orbital angular momentum (OAM): Quark orbital angular momentum (OAM):

the quarks have transverse momentum in the quarks have transverse momentum in hadrons. Therefore, the hadron helicity flip hadrons. Therefore, the hadron helicity flip can occur without requiring the quark can occur without requiring the quark helicity flip. helicity flip.

1/2 −1/2

1/2 1/2−1

Beyond the naïve parton model in which quarks are collinear

Novel Way to Generate PhaseNovel Way to Generate Phase

Some propagators in the tree diagrams go on-shell

)(11 22

2222mki

mkP

imk

No loop is needed to generate the phase!

Coulombgluon

Efremov & Teryaev: 1982 & 1984Qiu & Sterman: 1991 & 1999

Single Target-Spin Asymmetry in Single Target-Spin Asymmetry in SIDISSIDIS

Observed in HERMES exp. Observed in HERMES exp.

At low-PAt low-Ptt, this can be generated from Siver’s distribution , this can be generated from Siver’s distribution function and Collins fragmentation function (twist-2).function and Collins fragmentation function (twist-2).

At large-PAt large-Ptt, this can be generated from Efremov-, this can be generated from Efremov-Taryaev-Qiu-Sterman (ETQS) effect (twist-3).Taryaev-Qiu-Sterman (ETQS) effect (twist-3).

Boer, Mulders, and Pijlman (2003) observed that the Boer, Mulders, and Pijlman (2003) observed that the moments of Siver’s function is related to the twist-3 moments of Siver’s function is related to the twist-3 matrix elements of ETQS. matrix elements of ETQS.

k’

P

k

X

Low PLow P┴┴ Factorization Factorization

If PIf P┴┴ is on the order of the intrinsic transverse- is on the order of the intrinsic transverse-momentum of the quarks in the nucleon. Then momentum of the quarks in the nucleon. Then the factorization theorem involved un-the factorization theorem involved un-integrated transversity distribution, integrated transversity distribution,

1) One can measure the un-integrated transversity 2) To get integrated one, one can integrate out PP┴┴

with p┴ ┴ weighted. (soft factor disappears…)

When PWhen P┴ ┴ is large…is large…

Soft factor produces most of the transverse-Soft factor produces most of the transverse-momentum, and it can be lumped to hard contribution.momentum, and it can be lumped to hard contribution.

The transverse-momentum in the parton distribution The transverse-momentum in the parton distribution can be integrated over, yielding the transversity can be integrated over, yielding the transversity distribution. distribution. Or when the momentum is large, it can be Or when the momentum is large, it can be factorized in terms of the transversity distribution. factorized in terms of the transversity distribution.

The transverse-momentum in the Collins The transverse-momentum in the Collins fragmentation function can also be integrated out, fragmentation function can also be integrated out, yielding the ETSQ twist-three fragmentation matrix yielding the ETSQ twist-three fragmentation matrix elements. elements. Or when the momentum is large, it can be Or when the momentum is large, it can be factorized in terms of the transversity distribution. factorized in terms of the transversity distribution.

Likewise for Siver’s effect. Likewise for Siver’s effect.

The SSA is then a twist-three observable. The SSA is then a twist-three observable.

Physics of a Sivers FunctionPhysics of a Sivers Function

Hadron helicity flipHadron helicity flip– This can be accomplished through non-perturbative This can be accomplished through non-perturbative

mechanics (chiral symmetric breaking) in hadron mechanics (chiral symmetric breaking) in hadron structure.structure.

– The quarks can be in both s and p waves in The quarks can be in both s and p waves in relativistic quark models (MIT bag).relativistic quark models (MIT bag).

FSI (phase)FSI (phase)– The hadron structure has no FSI phase, therefore The hadron structure has no FSI phase, therefore

Sivers function vanish by time-reversal (Collins, Sivers function vanish by time-reversal (Collins, 1993)1993)

– FSI can arise from the scattering of jet with FSI can arise from the scattering of jet with background gluon field in the nucleon (collins, 2002)background gluon field in the nucleon (collins, 2002)

– The resulting gauge link is part of the parton dis. The resulting gauge link is part of the parton dis.

ConclusionConclusion

GPDs are quantum phase-space distributions, and GPDs are quantum phase-space distributions, and can be used to visualize 3D quark distributions at can be used to visualize 3D quark distributions at fixed Feynman momentumfixed Feynman momentum

There is now a factorization theorem for semi-There is now a factorization theorem for semi-inclusive hadron production at low pt, which inclusive hadron production at low pt, which involves soft gluon effects, allowing study pQCD involves soft gluon effects, allowing study pQCD corrections systematically.corrections systematically.

According to the theorem, what one learns from According to the theorem, what one learns from SSA at low pt is unintegrated transversity SSA at low pt is unintegrated transversity distribution.distribution.

At large pt, SSA is a twist-three effects, the At large pt, SSA is a twist-three effects, the factorization theorem reduces to the result of factorization theorem reduces to the result of Efremov-Teryaev-Qiu-Sterman. Efremov-Teryaev-Qiu-Sterman.