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 mulders@few.vu.nl

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

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

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

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

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

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

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[-]

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

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

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

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

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)

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)

examples: qqqq CG [D1] D1 = CG [D2] = CG [D4] CG [D3] D2 D3 D4 Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268 examples: qqqq CG [D1] D1 = CG [D2] = CG [D4] CG [D3] D2 D3 D4

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/0601171

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

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