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Trento July 2-6, 2006. Single spin asymmetries in pp scattering. _. Piet Mulders. mulders@few.vu.nl. Content. Single Spin Asymmetries (SSA) in pp scattering Introduction: what are we after? SSA and time reversal invariance Transverse momentum dependence (TMD)

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## Single spin asymmetries in pp scattering

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**Trento**July 2-6, 2006 Single spin asymmetries in pp scattering _ Piet Mulders mulders@few.vu.nl**Content**Single Spin Asymmetries (SSA) in pp scattering • Introduction: what are we after? • SSA and time reversal invariance • Transverse momentum dependence (TMD) Through TMD distribution and fragmentation functions totransverse momentsandgluonic poles • Electroweak processes (SIDIS, Drell-Yan and annihilation) • Hadron-hadron scattering processes • Gluonic pole cross sections • What can pp add? • Conclusions _ _**Introduction: what are we after?The 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**pictures?**appendix The 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 (i.e. at least one) of T-odd function(s) • The hard process at tree-level is T-even; higher order as is required to get T-odd contributions • Leading T-odd distribution functions are TMD functions**f2 - f1**K1 df K2 pp-scattering Intrinsic transverse momenta • In a hard process one probes partons (quarks and gluons) • Momenta fixed by kinematics (external momenta) DISx = 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**In collinear cross section**In azimuthal asymmetries Transverse moment pictures? TMD correlation functions (unpolarized hadrons) quark correlator F(x, pT) • T-odd • Transversely • polarized quarks**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**transverse moment**FG(p,p-p1) T-even T-odd Integrating F[±](x,pT) Fa[±](x)**_**What about other hard processes (e.g. pp and pp scattering)? Gluonic poles • Thus F[±]a(x) = Fa(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 onefunction f1T(1)(x) [Sivers] contained in pFG • These functions appear with a process-dependent sign • Situation for FF is (maybe) more complicated because there are no T-constraints Efremov and Teryaev 1982; Qiu and Sterman 1991 Boer, Mulders, Pijlman, NPB 667 (2003) 201 Metz and Collins 2005**C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277**Link structure for fields in correlator 1 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**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) = Fa(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 matrix elements (T-odd for distributions)**Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030;**hep-ph/0505268 D1 CG [D1] = CG [D2] D2 D3 CG [D3] = CG [D4] D4 examples: qqqq**(gluonic pole cross section)**y Gluonic pole cross sections • In order to absorb the factors CG[U], 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: • Similarly for gluon processes Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171**D1**For Nc: CG [D1] -1 (color flow as DY) examples: qqqq**end**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**D**P P’ Local – forward and off-forward Local operators (coordinate space densities): Form factors Static properties: Examples: (axial) charge mass spin magnetic moment angular momentum**Nonlocal - forward**Nonlocal forward operators (correlators): Specifically useful: ‘squares’ Selectivity at high energies: q = p Momentum space densities of f-ons: Sum rules form factors**Nonlocal – off-forward**Nonlocal off-forward operators (correlators AND densities): Selectivity q = p Sum rules form factors GPD’s b Forward limit correlators**back**Caveat • We study forward matrix elements, including transverse momentum dependence (TMD), i.e. f(p||,pT) with enhanced nonlocal sensitivity! • This is not a measurement of orbital angular momentum (OAM). Direct measurement of OAM requires off-forward matrix elements, i.e. GPD’s. • One may at best make statements like: linear pT dependence nonzero OAM no linear pT dependence no OAM**unpolarized**hadrons back Interpretation unpolarized quark distribution need pT T-odd helicity or chirality distribution need pT T-odd need pT transverse spin distr. or transversity need pT need pT

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