Nonuniversality of Transverse Momentum Dependent Parton Distribution Functions

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Non-universality of Transverse Momentum
Dependent Parton Distribution Functions
TRENTO - 2008
GPD workshop ECT, Trento June 2008

• Piet Mulders

mulders_at_few.vu.nl
See also Mulders, WHEPP X talk, arXiv0806.1134
hep-ph
2
Outline
OUTLINE

• Introduction partons in high energy scattering
  processes
• (Non-)collinearity collinear and non-collinear
  parton correlators
• Distribution functions (collinear, TMD)
• Observables azimuthal asymmetries
• Time reversal odd phenomena/single spin
  asymmetries
• Gauge links
• Resumming multi-gluon interactions Initial/final
  states
• Color flow dependence
• Applications
• Universality an example gq?gq
• Conclusions

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QCD Standard Model
INTRODUCTION

• QCD framework (including electroweak theory)
  provides the machinery to calculate cross
  sections, e.g. gq ? q, qq ? g, g ? qq, qq ?
  qq, qg ? qg, etc.
• E.g.
• qg ? qg
• Calculations work for plane waves

_
_
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Confinement in QCD
INTRODUCTION

• Confinement limits us to hadrons as quark
  sources or targets (with PX P-p)
• These involve nucleon states
• At high energies interference terms between
  different hadrons disappear as 1/P1.P2
• Thus, the theoretical description/calculation
  involves instead of ui(p)uj(p) for hard
  processes, a forward matrix element of the form

_
quark momentum
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Leading partonic structure of hadrons
INTRODUCTION
Need PH.Kh s (large) to get separation of soft
and hard parts Allows ? ds ? d(p.P)
hard process
p
fragmentation correlator
distribution correlator
D(z, kT)
F(x, pT)
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Partonic correlators
INTRODUCTION

• The cross section can be expressed in hard
  squared QCD-amplitudes and distribution and
  fragmentation functions entering in forward
  matrix elements of nonlocal combinations of quark
  and gluon field operators (f ? y or G).

Distribution functions
Fragmentation functions
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Probing intrinsic transverse momenta
NON-COLLINEARITY

• In a hard process one probes quarks and gluons
• Momenta fixed by kinematics (external momenta)
• DIS
• SIDIS
• Also possible for transverse momenta
• SIDIS
• 2-particle inclusive hadron-hadron scattering

We need more than one hadron and knowledge of
hard process(es)!
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TMD correlators quarks
NON-COLLINEARITY
TMD
collinear

• Gauge link essential for color gauge invariance
• Arises from all leading matrix elements
  containing y A...A y
• Basic (simplest) gauge links for TMD correlators

F-
F
T
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TMD correlators gluons
NON-COLLINEARITY

• The most general TMD gluon correlator contains
  two links, possibly with different paths.
• Note that standard field displacement involves C
  C
• Basic (simplest) gauge links for gluon TMD
  correlators

Fg,
Fg-,-
Fg-,
Fg,-
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Process dependence of gauge link
GAUGE LINKS

• In order to determine gauge link for F(x1,p1T)
  one needs to consider all collinear gluon
  insertions
• For a given hard subprocess, only the insertions
  attached to (other) external partons contribute
• They give links to lightcone infinity for parton
  fields in F(x1,p1T) and connecting gauge links
  U0,x in correlator

Two color-flow possibilities
Link structure for fields in correlator 1
C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596
(2004) 277 hep-ph/0406099 EPJ C 47 (2006) 147
hep-ph/0601171
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Process dependence of gauge link
GAUGE LINKS

• E.g. qq-scattering as hard subprocess
• The correlator F(x1,p1T) enters for each
  contributing term in squared amplitude with
  specific link

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example qq ? qq in pp
GAUGE LINKS
Bacchetta, Bomhof, Pijlman, M, PRD 72 (2005)
034030 hep-ph/0505268
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TMD master formula
APPLICATIONS
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Result for integrated cross section
APPLICATIONS
Integrate into collinear cross section
?
(partonic cross section)
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COLLINEAR DISTRIBUTION AND FRAGMENTATION FUNCTIONS
even
odd
c
flip
U
U
L
L
T
T
even
odd
c
U
L
T
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Result for weighted cross section
APPLICATIONS
Construct weighted cross section (azimuthal
asymmetry)
Why?

• New info on hadrons (cf models/lattice)
• Allows T-odd structure (exp. signal SSA)

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Result for weighted cross section
APPLICATIONS
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Result for weighted cross section
APPLICATIONS
?
?
universal matrix elements
T-even
T-odd
(operator structure)
FG(x,x) is gluonic pole matrix element
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TMD DISTRIBUTION AND FRAGMENTATION FUNCTIONS
even
odd
c
U
T even
L
T odd
T
even
odd
c
U
?
L
T
Gamberg, Mukherjee, Mulders, arXiv0803.2632
hep-ph
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Result for weighted cross section
APPLICATIONS
?
?
universal matrix elements
Examples are CGU 1, CG U- -1, CGU?
U 3, CG Tr(U?)U Nc
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Result for weighted cross section
APPLICATIONS
?
?
(gluonic pole cross section)
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example qq ? qq in pp
APPLICATIONS
Bacchetta, Bomhof, Pijlman, M, PRD 72 (2005)
034030 hep-ph/0505268
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Gluonic pole cross sections
APPLICATIONS

• for pp
• etc.
• for SIDIS
• for DY

Bomhof, Mulders, JHEP 0702 (2007) 029
hep-ph/0609206
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example qq ? qq in pp
APPLICATIONS
weighted
Transverse momentum dependent
Bacchetta, Bomhof, Pijlman, M, PRD 72 (2005)
034030 hep-ph/0505268
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example qg ? qg in pp
APPLICATIONS
weighted
Transverse momentum dependent
D1
Only one factor, but more DY-like than SIDIS
D2
D3
Note also etc.
D4
Bacchetta, Bomhof, DAlesio, M, Murgia PRL 99
(2007) 212002
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example qg ? qg
UNIVERSALITY
weighted
Transverse momentum dependent
D1
D2
D3
D4
D5
e.g. relevant in Bomhof, M, Vogelsang, Yuan, PRD
75 (2007) 074019
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example qg ? qg
UNIVERSALITY
weighted
Transverse momentum dependent
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example qg ? qg
UNIVERSALITY
weighted
Transverse momentum dependent
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example qg ? qg
UNIVERSALITY
weighted
Transverse momentum dependent
It is possible to group the TMD functions in a
smart way into two particular TMD
functions (nontrivial for eight diagrams/four
color-flow possibilities)
We get process dependent and (instead of
diagram-dependent) But still no factorization!
Bomhof, Mulders, NPB 795 (2008) 409 arXiv
0709.1390
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Universality for TMD correlators?
UNIVERSALITY

• We can work with basic TMD functions F(x,pT)
  junk
• The junk constitutes process-dependent residual
  TMDs
• Thus
• The junk gives zero after integrating (dF(x) 0)
  and after weighting (dF?(x) 0), i.e. cancelling
  kT contributions moreover it most likely also
  disappears for large pT

Bomhof, Mulders, NPB 795 (2008) 409 arXiv
0709.1390
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(Limited) universality for TMD functions
UNIVERSALITY
QUARKS
GLUONS
Bomhof, Mulders, NPB 795 (2008) 409 arXiv
0709.1390
32
TMD DISTRIBUTION FUNCTIONS
even
odd
c
odd
even
U
L
T
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Conclusions
CONCLUSIONS

• Transverse momentum dependence, experimentally
  important for single spin asymmetries,
  theoretically challenging (consistency, gauges,
  universality, factorization)
• For leading integrated and weighted functions
  factorization is possible, but it requires
  besides the normal partonic cross sections use
  of gluonic pole cross sections
• For TMDs there is no simple universality, but the
  non-universality can be made explicit through the
  process dependence of gauge links.
• This is a step towards factorization of TMD
  correlators

Related work Bacchetta et al., JHEP 02 (2007)
093 Bacchetta, Boer, Diehl, Mulders,
arXiv0803.0227 Qiu, Vogelsang, Yuan, PLB 650
(2007) 373 arXiv0704.1153 Collins, Qiu, PRD 75
(2007) 114014 arXiv0705.2141 Qiu, Vogelsang,
Yan, PRD 76 (2007) 074029 arXiv0706.1196 Meissn
er, Metz, Goeke, PRD 76 (2007) 034002
hep-ph/0703176 Collins, Rogers, Stasto, PRD
arXiv0708.2833
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CONCLUSIONS 24/24
END