Transverse Partonic Structure of the Proton

1
Transverse Partonic Structure of the Proton

• Feng Yuan
• Lawrence Berkeley National Laboratory
• RBRC, Brookhaven National Laboratory

2
Transverse spin physics
RHIC
RHIC
Bunce,Lambda
E704,pp to Pi
JLab
COMPASS
HERMES

BELLE
BHS
Factorization
unify
(non)Universality
Efremove-Teryav
Sivers
Jaffe-Ji
Collins
Mulders et al.
CSS
Ralston-Soper
Collins-Soper
Collins
Qiu-Sterman
BJY
3
Transverse spin physics

• Goal
• Quark transversity distributions
• Orbital motion of quarks and gluons?
• Transverse partonic structure of proton
• Various transverse momentum dependent physics
  (additional information on nucleon structure)
• Sivers function (PDF)
• Collins function (FF)

4
Transverse Momentum Dependent Parton Distributions

• Nucleon Structure, connection to GPDs, quantum
  phase space distribution
• Single spin asymmetry phenomena
• Nontrivial QCD dynamics, and fundamental test of
  the factorization, and the universality of PDFs,
  FFs,

5
Connection between TMD and GPD

• Wigner distributions
• After integrating over r, one gets TMD
• After integrating over k, one gets Fourier
  transform of GPDs

Ji PRL91,062001(2003)
6
Quantum Phase-Space Distributions of Quarks
Probability to find a quark u in a nucleon P with
a certain polarization in a position r and
momentum k
Wpu(k,r) Mother Wigner distributions
d3r
d2kT dr
GPDs Hpu(x,x,t), GDAs Fudp(x,x)
FT
TMD PDFs fpu(x,kT), TMD FFs Dup(kT,z)
IPDs Wpu(x,rT),
GPD
x0
x0,t0
d2r
dx
d2kT
Measure momentum transfer to target Direct info
about spatial distributions
Measure momentum transfer to quark Direct info
about momentum distributions
PDFs fpu(x), FFs Dup(z)
Form Factors F1pu(t),F2pu(t )..
Some PDFs same in exclusive and semi-inclusive
analysis
7
TMD Distribution the definition
Gauge Invariance requires the Gauge Link
Brodsky,Hwang,Schmidt 02 Collins
02 Belitsky,Ji,Yuan 02 Boer,Mulders,Pijlman, 03
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Polarized TMD Quark Distributions
Nucleon
Unpol.
Long.
Trans.
Quark
Unpol.
Long.
Trans.
Boer, Mulders, Tangerman (9698)
9
Three classes in the view of a quark model

• S-wave
• Unpolarized, helicity, transversity
• S-P interference
• g_1T,h_1L
• f_1T\perp, h_1\perp
• P-P or S-D interference
• h_1T\perp
• Miller 07, Burkardt 07, Avakian et al 08.

10
Lz?0 Amplitude and Sivers Function

• All distributions can be calculated using the
  wave function. Sivers function
• Similar expressions for others

Lz1
Lz0
Ji, Ma, Yuan, Nucl. Phys. B (2003)
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Where can we learn TMDs

• Semi-inclusive hadron production in deep
  inelastic scattering (SIDIS)
• Drell-Yan lepton pair production in pp scattering
• Relevant ee- annihilation processes
• Many others

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Inclusive and Semi-inclusive DIS
Inclusive DIS Partonic Distribution depending on
the longitudinal momentum fraction
Q
Semi-inclusive DIS Probe additional information
for partons transverse distribution in nucleon
Q
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Azimuthal Dependence in SIDIS

• Transverse Momentum Dependent (TMD) Parton
  Distributions and Fragmentations

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SIDIS Cross Section
At leading power of 1/Q
The structure functions depend on Q2, xB, z, PhT
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Two major contributions

• Sivers effect in the distribution
• Collins effect in the fragmentation
• Other contributions

ST (PXkT)
P
(zkpT)
(k,sT)
pTXsT
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Universality of the Collins Fragmentation
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Collins effects in ee-

• Reliable place to extract the information on the
  Collins fragmentation function

Belle Col., PRL 06
18
Collins asymmetry in pp collisions
Collins Fragmentation function
Quark transversity distribution
FY, arXiv0709.3272 hep-ph
19
Simple model a la Collins 93
Phase information in the vertex or the quark
propagator Collins-93
Universality of the Collins Function!!
20
One-gluon exchange (gauge link)?
Metz 02, Collins-Metz 02 Gamberg-Mukherjee-Mulder
s, 08 Universality of the Collins function!!
21
Similar arguments for pp collisions
Conjecture the Collins function will be the same
as ee- and SIDIS
22
Key observations

• Final state interactions DO NOT provide a phase
  for a nonzero SSA
• Eikonal propagators DO NOT contribute to a pole
• Ward identity is applicable to warrant the
  universality arguments

23
Sivers effect is different

• It is the final state interaction providing the
  phase to a nonzero SSA
• Ward identity is not easy to apply
• Non-universality in general
• Only in special case, we have
• Special Universality

24
DIS and Drell-Yan

• Initial state vs. final state interactions
• Universality fundamental QCD prediction

?
?
DIS
Drell-Yan
HERMES
25
Experiment SIDIS vs Drell Yan
HERMES Sivers Results
RHIC II Drell Yan Projections
0
Markus Diefenthaler DIS Workshop Munich, April
2007
0
0.1 0.2 0.3 x
http//spin.riken.bnl.gov/rsc/
26
Study QCD dynamics at different PT Region

• Integrate out PT (w/o weight)
• -- normal factorization, similar to inclusive
  DIS
• Large PT (gtgt?QCD)
• -- hard gluon radiation, can be calculated from
  perturbative QCD
• Low PT (?QCD)
• -- nonperturbative information TMD
  factorization formula

27
A unified picture for SSA

• In DIS and Drell-Yan processes, SSA depends on Q
  and transverse-momentum P?
• At large P?, SSA is dominated by twist-3
  correlation effects
• At moderate P?, SSA is dominated by the
  transverse-momentum-dependent parton
  distribution/fragmentation functions
• The two mechanisms at intermediate P? generate
  the same physics!
• Ji-Qiu-Vogelsang-Yuan,Phys.Rev.Lett
  .97082002,2006

28
A difficulty at next-leading-power (1/Q)

• Mismatch at low and high transverse momentum
  SIDIS at 1/Q
• Bacchetta-Boer-Diehl-Mulders, 0803.0227
• The factorization needs to be carefully examined
  at this order
• Earlier works indicates possible problems
• Afanasev-Carlson, PRD, 2006
• Gamberg-Hwang-Metz-Schlegel, PLB, 2006

29
Final PT Distribution

• PT dependence
• Which is valid for all Pt range
• SSA is suppressed by 1/Pt at large Pt

Sivers function at low Pt
Qiu-Sterman Twist-three
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Extend to all other TMDs large Pt power counting

• kt-even distributions have the same dependence on
  kt
• kt-odd distributions are suppressed at large kt
• Power Counting Rule
• kt-even 1/kt2
• kt-odd 1/kt4

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SIDIS cross sections at large Pt
1/Pt2
1/Pt4
1/Pt3
1/Pt5
32
Transition from Perturbative region to
Nonperturbative region?

• Compare different region of PT

Nonperturbative TMD
Perturbative region
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pp scattering experiments

• RHIC and RHIC II
• Collins effects, quark transversity
• Drell-Yan, quark Sivers
• Heavy flavor, gluon Sivers
• JPARC (pp collision at low energy)
• Drell-Yan, quark Sivers effects
• GSI-FAIR (ppbar collision)
• Drell-Yan, quark transversity
• Quark Sivers effects

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Summary

• We are in the early stages of a very exciting era
  of transverse spin physics studies, where the
  future JLAB, RHIC, and EIC experiments will
  certainly play very important roles
• We will learn more about QCD dynamics and nucleon
  structure from these studies, especially for the
  quark orbital motion
• More work needs to be done

35

• Quark distribution is deformed when nucleon is
  transversely polarized, because of the orbital
  motion
• This deformation will lead to a single transverse
  spin asymmetry (SSA), because of final state
  interactions
• Burkardt, 2003

y
x
y
z
36
Semi-Inclusive DIS

• Transverse Momentum Dependent (TMD) Parton
  Distributions and Fragmentations
• Novel Single Spin Asymmetries

U unpolarized beam T transversely polarized
target
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Whats Single spin asymmetry?
Transverse plane
Final state particle is Azimuthal symmetric
Single Transverse Spin Asymmetry (SSA)
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SSAs in Modern era RHIC, JLab, HERMES,
STAR
Central rapidity!!
BRAHMS
Large SSA continues at DIS ep and collider pp
experiments!!
39
Naïve parton model fails

• If the underlying scattering mechanism is hard,
  the naïve parton model generates a very small
  SSA (G. Kane et al, 1978),
• It is in general suppressed by aSmq/Q
• We have to go beyond this naïve picture

40
Two mechanisms in QCD

• Spin-dependent transverse momentum dependent
  (TMD) function
• Sivers 90
• Brodsky,Hwang,Schmidt, 02 (FSI)
• Gauge Property Collins 02Belitsky-Ji-Yuan,NPB03
• Boer-Mulders-Pijlman
  ,03
• Factorization Ji-Ma-Yuan,PRD04Collins,Metz,04
• Twist-3 quark-gluon correlations (coll.)
• Efremov-Teryaev, 82, 84
• Qiu-Sterman, 91,98

Sivers function ST (PXkT)
.
P
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What can we learn from SSA

• Quark Orbital Angular Momentum
• e.g, Sivers function the wave function
  amplitude with nonzero orbital angular momentum!
• Vanishes if quarks only in s-state!

Ji-Ma-Yuan, NPB03 Brodsky-Yuan, PRD06
42
Take Drell-Yan as an example(with non-zero
transverse momentum q?)

• We need a loop to generate a phase

-
-


Twist-three Correlations Efremov-Teryaev, 82,
84 Qiu-Sterman, 91,98
Kane et al., hard parton model
43
Further factorization (q?ltltQ)

• The collinear gluons dominate

q?ltltQ
Transverse Momentum Dependent distributions Sivers
, 90, Collins, 93,02 Brodsky-Hwang-Schmidt,02 Ji-Q
iu-Vogelsang-Yuan,06
Twist-three Correlations Efremov-Teryaev, 82,
84 Qiu-Sterman, 91,98
44
Quantum Phase Space Distribution

• Wigner operator
• Wigner distribution density for quarks having
  position r and 4-momentum k? (off-shell)

a la Saches
7-dimensional distribtuion
No known experiment can measure this!
45
Custom-made for high-energy processes

• In high-energy processes, one cannot measure k?
  (k0kz) and therefore, one must integrate this
  out.
• The reduced Wigner distribution is a function of
  six variables r,k(k k?).
• After integrating over r, one gets
  transverse-momentum dependent parton
  distributions
• Alternatively, after integrating over k?, one
  gets a spatial distribution of quarks with fixed
  Feynman momentum k(k0kz)xM.

46
Collins from HERMES

• Large, positive p asymmetries
• no surprise from u-quark dominance
• Large, negative p- asymmetries
• first a surprise, now understood by large,
  negative disfavored Collins function

47
Collins from COMPASS
PRL 94, 202002 (2005) and Nucl.Phys.B76531-70,200
7

• Smaller asymmetries than in proton case

48
First extraction of Collins functions and
transversity distributions from fitting HERMES
COMPASS BELLE data By Anselmino et al., PRD 75
(07)
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Comparison with some models
1 Soffer et al. PRD 65 (02) 2 Korotkov et al.
EPJC 18 (01) 3 Schweitzer et al., PRD 64 (01)
4 Wakamatsu, PLB 509 (01) 5 Pasquini et al.,
PRD 72 (05) 6 Anselmino et al., PRD 75 (07)
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Non-universality Dijet-correlation at RHIC

• Proposed by Boer-Vogelsang
• Pheno. studies Vogelsang-Yuan 05
• Bomhof-Mulders-Vogelsang-Yuan 07
• Initial state and/or final state interactions?
• Bacchetta-Bomhof-Mulders-Pijlman hep-ph/0406099,
  hep-ph/0505268, hep-ph/0601171, hep-ph/0609206
• Qiu-Vogelsang-Yuan, arXiv0704.1153 0706.1196
• Collins-Qiu, arXiv0705.2141 
• Voglesang-Yuan, arXiv0708.4398
• Collins, arXiv0708.4410
• Bomhof-Mulders, arXiv0709.1390
• Factorization? Universality?

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The asymmetry could be related to that in DIS,
only at the leading order (one-gluon exchange),

• qTDIS--- Sivers function from DIS
• qt--- imbalance of the dijet
• Hsivers depends on subprocess

Qiu,Vogelsang,Yuan, 07
52
This simple picture does not hold for two-gluon
exchanges
Vogelang-Yuan, 0708.4398 Qiu,Collins,
0705.4121 Collins, 0708.4410
Integrated over transverse momentum
Similar calculations can be Shown for QCD