Universality of single spin asymmetries in hard processes

1
Universality of single spin asymmetries in hard
processes
DIS2006, Tsukuba April 20-24, 2006

• Cedran Bomhof and Piet Mulders

mulders_at_few.vu.nl
2
Content

• Universality of Single Spin Asymmetries (SSA) in
  hard processes
• Introduction
• SSA and time reversal invariance
• Transverse momentum dependence (TMD)
• Through TMD distribution and fragmentation
  functions to transverse moments and gluonic poles
• Electroweak processes (SIDIS, Drell-Yan and
  annihilation)
• Hadron-hadron scattering processes
• Gluonic pole cross sections
• Conclusions

3
Introduction partonic structure of hadrons

• For (semi-)inclusive measurements, cross
  sections in hard scattering processes factorize
  into a hard squared amplitude 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
TMD
FF
4
Introduction partonic structure of hadrons

• Quark distribution functions (DF) and
  fragmentation functions (FF)
• unpolarized
• q(x) f1q(x) and D(z) D1(z)
• Polarization/polarimetry
• Dq(x) g1q(x) and dq(x) h1q(x)
• Azimuthal asymmetries
• g1T(x,pT) and h1L(x,pT)
• Single spin asymmetries
• h1?(x,pT) and f1T(x,pT) H1?(z,kT) and D1T(z,kT)
• Form factors
• Generalized parton distributions

FORWARD matrix elements x section one hadron in
inclusive or semi-inclusive scattering
NONLOCAL lightcone
NONLOCAL lightfront
OFF-FORWARD Amplitude Exclusive
LOCAL
NONLOCAL lightcone
5
SSA and time reversal invariance

• QCD is invariant under time reversal (T)
• Single spin asymmetries (SSA) are T-odd
  observables, but they are not forbidden!
• For distribution functions a simple distinction
  between T-even and T-odd DFs can be made
• Plane wave states (DF) are T-invariant
• Operator combinations can be classified according
  to their T-behavior (T-even or T-odd)
• Single spin asymmetries involve an odd number of
  (i.e. at least one) T-odd function(s)
• The hard process at tree-level is T-even higher
  order as is required to get T-odd contributions

6
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 kT pT
• q xBP Kh/zh ? -Kh?/zh
• 2-particle inclusive hadron-hadron scattering
• qT p1T p2T k1T k2T
• K1/z1 K2/z2- x1P1- x2P2 ?
  K1?/z1 K2?/z2
• 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 ? h X
• ? h1 h2 X

p ? x P pT k ? z-1 K kT
7
TMD correlation functions (unpolarized hadrons)
quark correlator
F(x, pT)

• T-odd
• Transversely
• polarized quarks

8
Color gauge invariance

• Nonlocal combinations of colored fields must be
  joined by a gauge link
• Gauge link structure is calculated from collinear
  A.n gluons exchanged between soft and hard part
• Link structure for TMD functions
• depends on the hard process!

DIS ? FU
SIDIS ? FU F
DY ? FU- F-
9
Integrating F(x,pT) ? F(x)
?
collinear correlator
10
Integrating F(x,pT) ? F?a(x)
11
Gluonic poles

• Thus
• F?a(x) F?a(x) CG pFGa(x,x)
• CG 1
• with universal functions in gluonic pole m.e.
  (T-odd for distributions)
• There is only one function h1?(1)(x)
  Boer-Mulders and (for transversely polarized
  hadrons) only one function f1T?(1)(x) Sivers
  contained in pFG
• These functions appear with a process-dependent
  sign
• Situation for FF is more complicated because
  there are no T constraints

What about other hard processes (in particular pp
scattering)?
Efremov and Teryaev 1982 Qiu and Sterman
1991 Boer, Mulders, Pijlman, NPB 667 (2003) 201
12
Other hard processes
C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596
(2004) 277

• qq-scattering as hard subprocess
• insertions of gluons collinear with parton 1 are
  possible at many places
• this leads for external parton fields to a
  gauge link to lightcone infinity

13
Other hard processes
C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596
(2004) 277

• qq-scattering as hard subprocess
• insertions of gluons collinear with parton 1 are
  possible at many places
• this leads for external parton fields to a
  gauge link to lightcone infinity
• 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)
14
Gluonic pole cross sections

• Thus
• F?Ua(x) F?a(x) CGU pFGa(x,x)
• CGU 1
• CGU? U 3, CGTr(U?)U Nc
• with the same uniquely defined functions in
  gluonic pole m.e. (T-odd for distributions)

15
examples qq?qq
Bacchetta, Bomhof, Pijlman, Mulders, PRD 72
(2005) 034030 hep-ph/0505268
16
Gluonic pole cross sections

• In order to absorb the factors CGU, one can
  define specific hard cross sections for gluonic
  poles (to be used with functions in transverse
  moments)
• for pp
• etc.
• for SIDIS
• for DY
• Similarly for gluon processes

Bomhof, Mulders, Pijlman, EPJ hep-ph/0601171
17
examples qq?qq
18
Conclusions

• Single spin asymmetries in hard processes can
  exist
• They are T-odd observables, which can be
  described in terms of T-odd distribution and
  fragmentation functions
• For distribution functions the T-odd functions
  appear in gluonic pole matrix elements
• Gluonic pole matrix elements are part of the
  transverse moments appearing in azimuthal
  asymmetries
• Their strength is related to path of color gauge
  link in TMD DFs which may differ per term
  contributing to the hard process
• The gluonic pole contributions can be written as
  a folding of universal (soft) DF/FF and gluonic
  pole cross sections

Belitsky, Ji, Yuan, NPB 656 (2003) 165 Boer,
Mulders, Pijlman, NPB 667 (2003) 201 Bacchetta,
Bomhof, Pijlman, Mulders, PRD 72 (2005)
034030 Bomhof, Mulders, Pijlman, EPJ
hep-ph/0601171 Eguchi, Koike, Tanaka,
hep-ph/0604003 Ji, Qiu, Vogelsang, Yuan,
hep-ph/0604023