Transversity and Transverse-Momentum-Dependent Partonic Functions

1
Transversity andTransverse-Momentum-Dependent
Partonic Functions

• Alessandro Bacchetta

2
Outline

• Theoretical framework
• Transversity
• Sivers function
• Boer-Mulders function

3
Theoretical framework
4
Semi-inclusive Deep Inelastic Scattering
proton
lepton
pion
5
Feynman diagrams Factorization
lepton
lepton
proton
pion
SIDIS
6
Feynman diagrams Factorization
lepton
lepton
proton
pion
SIDIS
7
Feynman diagrams Factorization
lepton
lepton
proton
pion
SIDIS
proton
pion
proton
pion
p-p to pions
8
Factorization proofs

• Factorization proofs for
• involving TMD functions
• LEADING TWIST (1/Q0), NOT for pp to pions
• Collins, Soper, NPB 193 (81)
• Ji, Ma, Yuan, PRD 71 (04)
• Ji, Ma, Yuan, PLB 597 (04)
• Collins, Metz, PRL 93 (04)

See talk by A. Metz
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Nonperturbative elements
2
10
Origin of the gauge link
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Different processes
SIDIS
See talk by C. Bomhof
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The correlation functionintegrated over kT
x- axis
0
a
SIDIS
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The correlation functionintegrated over kT
x- axis
0
a
Drell-Yan
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The correlation functionUNintegrated over kT
a
xT
0
x- axis
SIDIS
Belitsky, Ji, Yuan, NPB656 (03)
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The correlation functionUNintegrated over kT
a
0
Drell-Yan
Ji, Yuan, PLB 543 (02) Belitsky, Ji, Yuan,
NPB656 (03)
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Practical consequences

• Some TMD functions will not be sensitive to the
  differences in the structure of the gauge link
  (i.e. they will be connected to standard partonic
  cross-sections)
• Some TMD functions will be sensitive to it and
  will be multiplied by prefactors to be computed
  for each partonic subprocess (i.e. they will be
  connected to gluonic-pole cross-sections)

See talk by C. Bomhof
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Decomposition of the correlation function
integrated over kT
Leading twist only
Unpolarized distr. func. Helicity distr.
func. Transversity distr. func.
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Decomposition of the correlation function
UNintegrated over kT
Mulders, Tangerman, NPB 461 (96) Goeke, Metz,
Schlegel, PLB 618 (05)
Sivers, PRD 43 (91) Boer, Mulders, PRD 57 (98)
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Parton distribution functions with transverse spin
Photon moves into the screen/ proton moves out
of the screen
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Transversity
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Definition of transversity distribution function
Probability to find a quark with momentum xP and
transverse spin sq in a proton with transverse
spin S
NOTE transverse momentum is integrated over!
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Transversity birth and growth
Ralston, Soper, NPB 152 (79)
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Helicity and transversity
?
?
?
?
?
?
?
?
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Transversity vs helicity

• Different evolution

g1
h1_at_ 0.079 GeV2
g1
h1_at_ 0.75 GeV2
S. Scopetta, V. Vento,PLB 424 (1997)
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Transversity vs helicity

• Different evolution
• Different integrals (axial and tensor charge of
  the nucleon). E.g. from lattice QCD

S. Aoki et al., PRD 56 (1997) M. Göckeler et al.
QCDSF/UKQCD, PLB (05)
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Transversity vs helicity

• Different evolution
• Different integrals (axial and tensor charge of
  the nucleon). E.g. from lattice QCD
• Different sum-rules Bakker, Leader, Trueman,
  PRD 70 (04)

S. Aoki et al., PRD 56 (1997) M. Göckeler et al.
QCDSF/UKQCD, PLB (05)
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Comparison of tensor charges
c quark
spectator
non-relativistic
lattice
bag
c quark soliton 2
c quark soliton 1
Barone, Drago, Ratcliffe, PR 359 (2002)
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First data Collins asymmetry
Airapetian et al. HERMES, PRL 94 (05)
Alexakhin et al. COMPASS, PRL 94 (05)
hadron plane
See talk by G. Schnell, A. Martin
lepton plane
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First attempt to combine BELLE and HERMES data
Efremov, Goeke, Schweitzer, PRD 73 (06)
1. Get Collins function from BELLE
Talk by R. Seidl
Talks by P. Schweitzer and M. Boglione
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First attempt to combine BELLE and HERMES data
2. Check size of transversity at HERMES
See talk by P. Schweitzer
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Where to observe it
Talks by
Schnell,Martin, Avakian, Elschenbroich
Martin, Radici, Giordano
Heppelman, Dalpiaz
Koch, Fischer
Heppelman, Aidala
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The Sivers function
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Introducing the Sivers function
Probability to find a quark with momentum xP and
kT in a proton with transverse spin S
see e.g. Bacchetta, DAlesio, Diehl, Miller, PRD
70 (04)
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Sivers function birth and growth
D. Sivers, PRD41 (90)
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Two ingredients

• Final-state interactions (included in the gauge
  link)
• Ji, Yuan, PLB 543 (02) Belitsky, Ji, Yuan,
  NPB656 (03)
• Transverse-spin dependent distribution of quarks
  in transverse space
• Burkardt, PRD 66 (02) Diehl, Hägler, EPJ C44 (05)

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Final-state interactions
Side view
Front view
quarks
proton
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Final-state interactions
Side view
Front view
photon
NOTE QCD tells us that the FSI has to be
attractive, since quark and remnants form a color
antisymmetric state
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Final-state interactions
Side view
Front view
photon
See talk by M. Burkardt
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Change of sign in Drell-Yan
Side view
Front view
quarks
antiquark
proton
photon
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Distortions in transverse space
Side view
Front view
quarks
proton
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Distortions in transverse space
Side view
Front view
quarks
?
proton
The presence of spin can distort the distribution
of quarks in transverse space (orbital angular
momentum of quarks is required)
A distortion in the distribution of quarks in
transverse space can give rise to a nonzero
Sivers function
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Relation to GPDs

• The transverse-space deformation of unpolarized
  quarks in a transversely polarized nucleon is
  described by the generalized parton distribution
  function E

Work of Burkardt
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Lattice-QCD studies
up quarks
Sivers function for up quarks expected to be
NEGATIVE
See talk by P. Hägler
Preliminary results by QCDSF Collab. (see e.g.
hep-ph/05110032)
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Model calculations
See talk by L. Gamberg
Brodsky, Hwang, Schmidt PLB530 (02) Gamberg,
Goldstein, Oganessyan, PRD 67 (03)
spectator
proton
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Fits to HERMES and COMPASS data
Airapetian et al. HERMES, PRL 94 (05) Anselmino
et al., PRD72 (05)
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Comparison with COMPASS data
deuteron
Alexakhin et al. COMPASS, PRL94 (05), Anselmino
et al., PRD72 (05)
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Different fits
Anselmino et al., hep-ph/0511017 20 Anselmino
et al., PRD72 (05) 21 Vogelsang, Yuan, PRD72
(05) 23 Collins et al., hep-ph/0510342
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New preliminary kaon data
See talk by U. Elschenbroich
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Future experimental data
See talk by U. DAlesio

• HERMES_at_DESY p0 and weighted asymmetries (this
  year?)
• COMPASS_at_CERN proton data (this year?)
• CLAS_at_Jlab
• STAR,PHENIX_at_RHIC jet-production in pp collisions
• EIC (DIS with polarized protons)
• FAIR_at_GSI (Drell-Yan with polarized protons)

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The Boer-Mulders function
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Definition of the Boer-Mulders function
Probability to find a quark with momentum xP and
kT and spin sq
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Boer-Mulders function birth and growth
Boer, Mulders, PRD57 (98)
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Relation to transverse space
Side view
Front view
quarks
proton
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Relation to transverse space
Side view
Front view
quarks
?
?
?
?
?
?
?
proton
Quark spin can be unevenly distributed in
transverse space (orbital angular momentum of
quarks is required)
A distortion in the distribution of quark spin in
transverse space can give rise to a Boer-Mulders
function Burkardt, hep-ph/0510408
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Lattice-QCD studies
up quarks
Boer-Mulders function for up quarks expected to
be NEGATIVE
Preliminary results by QCDSF Collab. (see e.g.
hep-ph/05110032)
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Model calculations
See talk by L. Gamberg, G. Goldstein
Bacchetta, Schäfer, Yang, PLB 578 (04)
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Where to look for it

• Unpolarized semi-inclusive DIS (HERMES, COMPASS,
  H1, ZEUS, EIC, LHEC)
• Unpolarized Drell-Yan (GSI)
• Jet-production in unpolarized pp collisions
  (RHIC, LHC)

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Conclusions

• Many progresses from the theoretical,
  experimental, phenomenological sides
• ...Pay attention to the rest of the workshop!

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Semi-inclusive Deep Inelastic Scattering
proton
?
lepton
pion