1. Universality of single spin asymmetries in hard processes
DIS2006, Tsukuba
April 20-24, 2006
Universality of single spin asymmetries in hard processes
Cedran Bomhof and Piet Mulders

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
TMD
lightfront 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 DF’s 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
e+e-: g*  h1 + h2 + X
hadronproduction: H1 + H2  h + X
 h1 + h2 + X
p  x P + pT
k  z-1 K + kT
K2
K1
f2 - f1 df
pp-scattering

7. TMD correlation functions (unpolarized hadrons)
quark correlator
In collinear cross section
In azimuthal asymmetries
F(x, pT)
T-odd
Transversely
polarized quarks
Transverse moment

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  F[U]
SIDIS  F[U+] = F[+]
DY  F[U-] = F[-]

9. Integrating F[±](x,pT)  F[±](x)
collinear correlator

10. Integrating F[±](x,pT)  Fa[±](x)
transverse moment
FG(p,p-p1)
T-even
T-odd

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 qq-scattering as hard subprocess
C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277
Other hard processes
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
Link structure for fields in correlator 1

13. Other hard processes F[Tr(U□)U+](x,pT) F[U□U+](x,pT)
C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277
Other hard processes
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□ = U+U-†
F[Tr(U□)U+](x,pT)
F[U□U+](x,pT)

14. Gluonic pole cross sections
Thus
F[U]a(x) = Fa(x) + CG[U] pFGa(x,x)
CG[U±] = ±1
CG[U□ U+] = 3, CG[Tr(U□)U+] = Nc
with the same uniquely defined functions in gluonic pole m.e. (T-odd for distributions)

15. examples: qqqq CG [D1] D1 = CG [D2] = CG [D4] CG [D3] D2 D3 D4
Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) ; hep-ph/
examples: qqqq
CG [D1] D1
= CG [D2]
= CG [D4]
CG [D3] D2 D3 D4

16. Gluonic pole cross sections
In order to absorb the factors CG[U], 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
(gluonic pole cross section) y
Bomhof, Mulders, Pijlman, EPJ; hep-ph/

17. examples: qqqq D1
For Nc:
CG [D1]  -1
(color flow as DY)

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)
Bomhof, Mulders, Pijlman, EPJ; hep-ph/
Eguchi, Koike, Tanaka, hep-ph/
Ji, Qiu, Vogelsang, Yuan, hep-ph/