Appearance of single spin asymmetries in hard scattering processes

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Appearance of single spin asymmetries in hard
scattering processes

• Piet Mulders

mulders_at_few.vu.nl
Trento June 13, 2007
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Content

• Introduction the partonic structure of hadrons
• Transverse momentum dependence (TMD)
• Gauge links and dependence on hard processes
• Electroweak processes (SIDIS, Drell-Yan and
  annihilation)
• Hadron-hadron scattering processes
• Gluonic pole cross sections
• Factorization breaking for TMDs made explicit
• Conclusions

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IntroductionThe partonic structure of hadrons

• For (semi-)inclusive measurements, we want to
  investigate the factorization of cross sections
  in hard scattering processes into 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)

lightcone
lightfront x 0
TMD
FF
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Quarks

• Integration over x- x.P allows twist
  expansion
• Gauge link essential for color gauge invariance
• Arises from all leading matrix elements
  containing y A...A y

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Generic hard processes

• E.g. qq-scattering as hard subprocess
• Matrix elements involving parton 1 and additional
  gluon(s) A A.n appear at same (leading) order
  in twist expansion
• insertions of gluons collinear with parton 1 are
  possible at many places
• this leads for correlator involving parton 1 to
  gauge links to lightcone infinity

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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SIDIS
SIDIS ? FU F
DY ? FU- F-
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A 2 ? 2 hard processes qq ? qq

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

U? UU-
FTr(U?)U(x,pT)
FU?U(x,pT)
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Gluons

• Using 3x3 matrix representation for U, one finds
  in gluon correlator appearance of two links,
  possibly with different paths.
• Note that standard field displacement involves C
  C

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Sensitivity to intrinsic transverse momenta

• In a hard process one probes partons (quarks and
  gluons)
• Momenta fixed by kinematics (external momenta)
• DIS x xB Q2/2P.q
• SIDIS z zh P.Kh/P.q
• Also possible for transverse momenta
• SIDIS qT q xBP Kh/zh ? -Kh?/zh
• kT pT
• 2-particle inclusive hadron-hadron scattering
• qT K1/z1 K2/z2- x1P1- x2P2 ? K1?/z1
  K2?/z2
• p1T p2T k1T k2T
• Sensitivity for transverse momenta requires ? 3
  momenta
• SIDIS g H ? h X
• DY H1 H2 ? g X
• ee- g ? h1 h2 X
• hadronproduction H1 H2 ? h1 h2 X
• ? h X (?)

p ? x P pT k ? z-1 K kT
Knowledge of hard process(es)!
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TMD correlation functions (unpolarized hadrons)
quark correlator
F(x, pT)

• T-odd
• Transversely
• polarized quarks

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Integrating F(x,pT) ? F(x)
?
collinear correlator
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Integrating F(x,pT) ? F?a(x)
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Gluonic poles

• Thus FU(x) F(x)
• F?Ua(x) F?a(x) CGU pFGa(x,x)
• Universal gluonic pole m.e. (T-odd for
  distributions)
• pFG(x) contains the T-odd functions h1?(1)(x)
  Boer-Mulders and (for transversely polarized
  hadrons) the function f1T?(1)(x) Sivers
• F?(x) contains the T-even functions h1L?(1)(x)
  and g1T?(1)(x)
• For SIDIS/DY links CGU 1
• In other hard processes one encounters different
  factors
• CGU? U 3, CGTr(U?)U Nc

Efremov and Teryaev 1982 Qiu and Sterman
1991 Boer, Mulders, Pijlman, NPB 667 (2003) 201
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A 2 ? 2 hard processes qq ? qq

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

U? UU-
FTr(U?)U(x,pT)
FU?U(x,pT)
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examples qq?qq in pp
Bacchetta, Bomhof, Pijlman, Mulders, PRD 72
(2005) 034030 hep-ph/0505268
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examples qq?qq in pp
Bacchetta, Bomhof, DAlesio,Bomhof, Mulders,
Murgia, hep-ph/0703153
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Gluonic pole cross sections

• In order to absorb the factors CGU, one can
  define specific hard cross sections for gluonic
  poles (which will appear with the functions in
  transverse moments)
• for pp
• etc.
• for SIDIS
• for DY

Bomhof, Mulders, JHEP 0702 (2007) 029
hep-ph/0609206
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examples qg?qg in pp
D1
Only one factor, but more DY-like than SIDIS
D2
D3
D4
Note also etc.
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examples qg?qg
e.g. relevant in Bomhof, Mulders, Vogelsang,
Yuan, PRD 75 (2007) 074019
D1
D2
D3
D4
D5
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examples qg?qg
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examples qg?qg
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Pictures for qg?qg part in pp
(9 diagrams)
gluonic pole cross section
normal partonic cross section
Both are gauge-invariant combinations (of
course but non-trivial)
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examples qg?qg
It is also possible to group the TMD functions in
a smart way into two! (nontrivial for nine
diagrams/four color-flow possibilities)
But still no factorization!
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Residual TMDs

• We find that we can work with basic TMD functions
  F(x,pT) junk
• The junk constitutes process-dependent residual
  TMDs
• The residuals satisfies Fint ?(x) 0 and pFint
  G(x,x) 0, i.e. cancelling kT contributions
  moreover they most likely disappear for large kT

no definite T-behavior
definite T-behavior
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Conclusions

• Appearance of single spin asymmetries in hard
  processes is calculable
• For integrated and weighted functions
  factorization is possible
• For TMDs one cannot factorize cross sections,
  introducing besides the normal partonic cross
  sections some gluonic pole cross sections
• Opportunities the breaking of factorization can
  be made explicit and be attributed to specific
  matrix elements

Related Qiu, Vogelsang, Yuan, hep-ph/0704.1153 Co
llins, Qiu, hep-ph/0705.2141 Qiu, Vogelsang, Yan,
hep-ph/0706.1196 Meissner, Metz, Goeke,
hep-ph/0703176