Transverse Spin Effects and k T -dependent Functions

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1 Transverse Spin Effects and k T -dependent Functions Daniël Boer Free University, Amsterdam Outline Left-right single spin asymmetries Azimuthal spin asymmetries; Sivers and Collins effects Transversity k T -odd transverse spin functions Theoretical issues Experimental opportunities and difficulties e p e π X (A LU ) Λ polarization 10 years perspective Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

2 Left-right asymmetries In scattering processes the distribution of produced particles can be asymmetric This may depend on the polarization of one or more particles in the collision Large single spin asymmetries in p + p π + X E704 Collab. ( 91); AGS ( 99); STAR ( 02) A left-right asymmetry Pion distribution is asymmetric depending on transverse spin direction and on pion charge It is not clear how to explain such SSA on the quark-gluon level In the theoretical descriptions based on k T -dependent functions these left-right asymmetries originate from azimuthal spin asymmetries Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

3 Sivers effect First proposal of a k T & S T dependent distribution function by Sivers (PRD 41 ( 90) 83) f 1T = P q k T s T q k T s T x x x,k T x,k T absent + + Sivers + Inspired by data on p + p π 0 + X (Antille, PLB 94 ( 80) 523) and the advent of E704 Intended as a test of perturbative QCD for large p T hadron production Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

4 Azimuthal spin asymmetries Sivers effect in semi-inclusive DIS e + p e + π + X P h h ^ k P φ h k q ^z lepton scattering plane dσ(e p e πx) dφ e πd P π 2 {1 + S T sin(φ e π φ e S) A T } where the spin of the proton is orthogonal to direction of γ A T f 1T D 1 Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

5 Collins effect (H 1 ) in semi-inclusive DIS Azimuthal spin asymmetries Collins, NPB 396 ( 93) 161 P h h ^ k P φ h k q ^z lepton scattering plane dσ(e p e πx) dφ e πd P π 2 {1 + S T sin(φ e π+φ e S) A T }, A T h 1 H 1 A T may be nonzero offering a probe of transversity (h 1 or δq) Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

6 Transversity h 1 or δq: how much of the transverse spin of the proton is transferred to its quarks P h P h x x + + q k k + h 1( x) + P p Φ p P Collins 93: can be probed if there is another helicity flip in the fragmentation process π H 1 = k T - s T π k T s T Helicity flip quantities (h 1, H 1 ), also called chiral-odd quantities, always appear in pairs Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

7 Importance of transversity The integral over x is a fundamental charge, like the electric and the axial charge δq dx δq(x) = tensor charge P, S ψ q σ µν γ 5 ψ q (0) P, S δq [P µ S ν P ν S µ ] Not measurable in elastic scattering or in inclusive deep inelastic scattering However, it can be probed by using other hadrons: e.g. in e p e π X or p p Completely new information on the proton spin structure Transverse polarization parton distributions are needed for the next step: polarized hadron colliders polarized quark colliders Also, h 1 0 is related to chiral symmetry breaking Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

8 k T -dependent transverse spin functions f 1T and H 1 proposed because of SSA and h 1, but are of interest in their own right Furthermore, there are two other k T -odd functions D 1T = Λ k T - Λ k T S T h 1 = P q kt P q kt s T Mulders & Tangerman, NPB 461 ( 96) 197; D.B. & Mulders, PRD 57 ( 98) 5780 These functions have unexpected properties, not yet fully studied Recent developments spurred on by a model calculation of Brodsky, Hwang & Schmidt (PLB 530 ( 02) 99) Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

9 Properties of f 1T Collins ( 93): f 1T = 0 Brodsky, Hwang & Schmidt ( 02); Belitsky, Ji & Yuan ( 02); Collins ( 02): (f 1T ) DIS = (f 1T ) DY 0 Burkardt (hep-ph/ ): a=q,g dx f (1)a 1T (x) = 0, f (1)a 1T (x) d 2 k T kt 2 a 2M 2f1T (x, kt 2 ) A cousin of the Schäfer-Teryaev sum rule (PRD 61 ( 00) ) h dz zh (1) 1 (z) = 0, H (1) 1 (z) z 2 These sum rules are of limited use for model calculations d 2 k T k 2 T 2M 2H 1 (z, z 2 k 2 T ) Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

10 Link structure Gauge invariant definition of Sivers function in DIS contains a future pointing Wilson line, whereas in Drell-Yan (DY) it is past pointing f 1T P, S T ψ(0) U(0, ξ) n ψ(ξ) P, S T Belitsky, Ji & Yuan, NPB 656 ( 03) 165 ξ T ξ T ξ ξ As a consequence (Collins, PLB 536 ( 02) 43): (f 1T ) DIS = (f 1T ) DY Open problem: what about more complicated processes? Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

11 Further properties of f 1T There is a direct (gauge invariant) relation between the Sivers and Qiu-Sterman effects f (1)[+] 1T (x) = g 2M S 2 T T (V ) F (x, x) [+] indicates a future pointing Wilson line D.B., Mulders, Pijlman, NPB 667 ( 03) 201 T (V ) F (x, x) leads to power suppressed single spin asymmetries (twist-3) Qiu, Sterman, PRL 67 (1991) 2264; PRD 59 (1999) A relation to the GPD E has been put forward f (1) 1T (x) ɛij T S T i Burkardt, hep-ph/ ; Burkardt & Hwang, hep-ph/ d 2 b I(b ) b j E(x, b ) Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

12 Theoretical problems The process dependence or universality of fragmentation functions On the basis of symmetry restrictions alone one finds (H 1 ) SIDIS A + B (H 1 ) e + e = A B D.B., Mulders, Pijlman, NPB 667 ( 03) 201 Metz (PLB 549 ( 02) 139): model calculation shows that B = 0 Evolution, energy scale (Q 2 ) dependence Very involved, not leading twist operators in OPE sense Sudakov suppression of azimuthal asymmetries D.B., NPB 603 ( 01) 195; D.B. & Vogelsang, PRD 04 More generally, factorization theorems Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

13 Experimental opportunities Azimuthal sin φ single spin asymmetries studied in e p scattering A UL in e + p e + π + X HERMES ( 99); CLAS ( 02) A LU in e + p e + π + X CLAS ( 02) A UT in e + p e + π + X HERMES ( 03); COMPASS ( 04) HERMES isolated Sivers and Collins A UT asymmetries separately Data to be expected: Sivers and Collins asymmetries in SIDIS H1 from e + e (π + ) jet1 (π ) jet2 X Using off-resonance data of B-factories (BELLE) Grosse Perdekamp et al., NPA 711 ( 02) 69c CLAS, COMPASS, HERMES Single and double transverse spin asymmetries from RHIC p p l l X, p p 2 jets X,... Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

14 Actual extraction One problem is that the k T -dependent functions appear in convolution integrals For example, Sivers effect in SIDIS: dσ(e p e π X) d 2 S T q T Q T F [w f D] sin(φ e π φ e S) F [ qt p T M f 1T D 1 ] d 2 p T d 2 k T δ 2 (p T + q T k T ) w f(x, p 2 T )D(z, z 2 k 2 T ) One solution would be to measure jet SIDIS : e p e jet X dσ(e p e jet X) d 2 q T S T sin(φ e π φ e S) Q T M f 1T (x,q 2 T ), Q 2 T = P jet 2 One can probe the k T -dependence of the Sivers function in this way! For asymmetries involving chiral-odd quantities this cannot be done A more general solution is to consider weighted asymmetries Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

15 Jet SIDIS e p e jet X Cross sections integrated, but weighted with function of observed transverse momentum W UT dz d 2 P jet W dσ[e p e jet X] dx dy dz dφ e jet d P jet 2 Weighted asymmetries become expressions in terms of transverse moments cos φ e P jet jet M sin UT φe S a,ā e 2 a x f (1)a 1T (x) D.B. & Mulders, PRD 57 ( 98) 5780 No suppression by 1/Q and no Sudakov suppression either Certain weighted asymmetries are free from Sudakov suppression Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

16 HERMES data 0.2 π + HERMES PRELIMINARY not corrected for smearing and acceptance effects P h /M P weighted 8% scale uncertainty 0.5 π + HERMES PRELIMINARY not corrected for smearing and acceptance effects P h /M π weighted 8% scale uncertainty sin(φ-φs) A UT 0.1 sin(φ+φs) A UT π - Sivers angle π - Collins angle sin(φ-φs) A UT sin(φ+φs) A UT π 0 Maximum possible effect of exclusive vector mesons 0.5 π 0 Maximum possible effect of exclusive vector mesons sin(φ-φs) A UT sin(φ+φs) A UT x B z x B z Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

17 Actual extraction Even if one deconvolutes asymmetries by weighting, in case of chiral-odd quantities, one is always dealing with products of functions h 1 H (1) 1, h 1 H 1, h 1 h1, h 1 h (1) 1,... Almost no experiment aiming to extract h 1 will be self-sufficient Would only apply to (h 1 ) 2 observables: p p l l X p p jet (high p T ) X Jaffe & Saito, PLB 382 ( 96) 165; Vogelsang ( 00) A global transversity analysis is needed The issue of process dependence of k T -dependent functions is very important Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

18 A LU is problematic A LU in e p e π X receives gluon radiation contributions (O(α 2 s)) Hagiwara, Hikasa, Kai, PRD 27 ( 83) 84; Ahmed & Gehrmann, PLB 465 ( 99) 297 A LU λ e M Q e H 1, λ e M Q h 1 E Levelt & Mulders, PLB 338 ( 94) 357; Yuan, hep-ph/ Since e and E are twist-3, factorization is much less certain dσ dq 2 dq 2 d σ d 2 q T LT " " NLT? " "? NNLT X X Collins & Soper, NPB 193 (1981) 381 Qiu & Sterman, NPB 353 (1991) 137 A UT A & A LU UL Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

19 Λ polarization Large asymmetries seen in p p Λ X ( 75) and in ν µ p µ Λ X (NOMAD 00) p y^ x^ p Λ P Λ > 0 z^ Explanation in terms of D 1T (Anselmino, D.B., D Alesio & Murgia, PRD 63 ( 01) ) D 1T = Λ k T - Λ k T S T Unfortunately data is not yet in the appropriate kinematic region Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

20 Λ polarization Since D 1T is k T -odd, it can probe the first derivative of the partonic cross section p A Λ X: at very high energies or for very large A, the asymmetry is sensitive to gluon saturation and the associated scale Q s P Λ D.B. & Dumitru, PLB 556 ( 03) Q s = 2 GeV Q s = 3 GeV l t (GeV) A similar result may be expected for e A Λ X Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

21 10 years perspective On the theoretical side I expect clarification of: Process dependence/universality Factorization Energy scale dependence of functions and asymmetries On the experimental side I expect: Reasonably accurate determination of Sivers and Collins function: Sivers from SIDIS and p p; Collins from e + e Some progress towards extracting h 1 Some progress on D 1T Furthermore, I expect: Trustworthy lattice determination of first moments of h 1 Better understanding of what is the main cause of p p πx asymmetries Some progress on p p Λ X Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,

22 Conclusions k T -dependent functions can be useful tools: transversity, saturation,... may explain left-right single spin asymmetries can lead to unsuppressed azimuthal spin asymmetries Measurements of azimuthal spin asymmetries hint at nonzero Sivers and Collins effects k T -dependence of Sivers function can be directly accessed in jet SIDIS Process dependence, scale dependence and factorization requires further study Twist-3 asymmetries like A LU may not allow a factorized description The road to a percent level transversity extraction is still quite long Second Electron Ion Collider Workshop (EIC2004), Jefferson Lab, March 16,