Lepton scattering and the structure of nucleons and

‘Lepton scattering and the structure of nucleons and nuclei’ September 16 -24, 2004 Polarized structure functions Piet Mulders pjg. mulders@few. vu. nl

Content • Spin structure & transversity • Transverse momenta & azimuthal asymmetries • T-odd phenomena & single spin asymmetries

DIS • Known leptonic part • Completeness allows reduction in hadronic tensor to commutator [Jm(x), Jn(0)] • Known structure of current in terms of quarks • OPE • ….

Deep inelastic scattering (DIS)

Lepton tensor • Lepton tensor can also be expanded using the spacelike and timelike vectors • Tensor encompasses many ‘polarization options’

Polarized DIS

Semi-inclusive deep inelastic scattering • Known lepton part with much flexibility (unused in DIS) • Involves two hadrons and hence a much more complex hadronic tensor

SIDIS

(calculation of) cross section in DIS Full calculation + PARTON MODEL + + +…

Lightcone dominance in DIS

Leading order DIS • In limit of large Q 2 the result of ‘handbag diagram’ survives • … + contributions from A+ gluons ensuring color gauge invariance A+ Ellis, Furmanski, Petronzio Efremov, Radyushkin A+ gluons gauge link

Parametrization of lightcone correlator • M/P+ parts appear as M/Q terms in s • T-odd part vanishes for distributions but is important for fragmentation leading part Jaffe & Ji NP B 375 (1992) 527 PRL 71 (1993) 2547

Basis of partons § ‘Good part’ of Dirac space is 2 -dimensional § Interpretation of DF’s unpolarized quark distribution helicity or chirality distribution transverse spin distr. or transversity

Matrix representation for M = [F(x)g+]T Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712 Quark production matrix, directly related to the helicity formalism Anselmino et al. § Off-diagonal elements (RL or LR) are chiral-odd functions § Chiral-odd soft parts must appear with partner in e. g. SIDIS, DY

Results for DIS • Structure functions in (sub)leading order in 1/Q • Two of three (Polarized) quark densities for each flavor: Not accessible in DIS

(calculation of) cross section in SIDIS “Full” calculation + PARTON MODEL + + +…

Lightfront dominance in SIDIS Three external momenta P Ph q transverse directions relevant q. T = q + x. B P – Ph/zh or q. T = -Ph /zh

Leading order SIDIS • In limit of large Q 2 only result of ‘handbag diagram’ survives • Isolating parts encoding soft physics ? ?

Lightfront correlators Collins & Soper NP B 194 (1982) 445 no T-constraint T|Ph, X>out = |Ph, X>in Jaffe & Ji, PRL 71 (1993) 2547; PRD 57 (1998) 3057

Distribution including the gauge link (in SIDIS) A+ One needs also AT G+a = +A a T ATa(x)= ATa(∞) + dh G+a Belitsky, Ji, Yuan, hep-ph/0208038 Boer, M, Pijlman, hep-ph/0303034 From <y(0)AT( )y(x)> m. e.

Parametrization of F(x, p. T) • • • Link dependence allows also T-odd distribution functions since T U[0, ] T = U[0, - ] Functions h 1 and f 1 T (Sivers) nonzero! These functions (of course) exist as fragmentation functions (no T-symmetry) H 1 (Collins) and D 1 T

Interpretation unpolarized quark distribution need p. T T-odd helicity or chirality distribution need p. T T-o dd need p T transverse spin distr. or transversity need p. T

Matrix representation for M = [F[±](x, p. T)g+]T T-odd: g 1 T – i f 1 T and h 1 L + i h 1 § p. T-dependent functions (imaginary parts) Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

T-odd single spin asymmetry § Wmn(q; P, S; Ph, Sh) = -Wnm(-q; P, S; Ph, Sh) * § Wmn (q; P, S; Ph, Sh) = Wnm(q; P, S; Ph, Sh) _ __ __ § Wmn(q; P, S; Ph, Sh) = Wmn(q; P, -S; Ph, -Sh) ___ _ _ * * § Wmn (q; P, S; Ph, Sh) = Wmn(q; P, S; Ph, Sh) _ _ symmetry structure hermiticity parity time reversal • with time reversal constraint only even-spin asymmetries • the time reversal constraint cannot be applied in DY or in 1 -particle inclusive DIS or e+e • In those cases single spin asymmetries can be used to select T-odd quantities

Leptoproduction of pions H 1 is T-odd and chiral-odd

COLLINS ASYMMETRY RESULTS OF COMPASS Acoll depends on ph. T, zh, x. Bj with more statistics, the full analysis is foreseen from 2002 data: Acoll vs x. Bj ! n g Si

COLLINS ASYMMETRY RESULTS OF COMPASS from 2002 data: AColl vs zh ! n g Si all the tests made are consistent with the fact that systematic effects, if present, are smaller than statistical errors

Distribution including the gauge link (in SIDIS or DY) A+ SIDIS A+ DY SIDIS F[-] DY F[+]

Difference between F[+] and F[-] upon integration Back to the lightcone (theoretically clean) integrated quark distributions twist 2 transverse moments measured in azimuthal asymmetries twist 2 & 3 ±

Difference between F[+] and F[-] upon integration In momentum space: gluonic pole m. e. (T-odd) Conclusion: T-odd parts are gluon-driven (QCD interactions)

Time reversal constraints for distribution functions T-odd (imaginary) p. FG Time reversal: F[+](x, p. T) F[-](x, p. T) F [+] F F [-] T-even (real) Conclusion: T-odd effects in SIDIS and DY have opposite signs

Time reversal constraints for fragmentation functions T-odd (imaginary) p. DG Time reversal: D[+]out(z, p. T) D[-]in(z, p. T) D [+] D D [-] T-even (real)

Time reversal constraints for fragmentation functions T-odd (imaginary) p. DG out Time reversal: D[+]out(z, p. T) D[-]in(z, p. T) D [+]out D [-]out T-even (real) Conclusion: T-odd effects in SIDIS and e+e- are not related

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 gauge link to lightcone infinity e. g.

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 gauge link to lightcone infinity • The correlator F(x, p. T) enters for each contributing term in squared amplitude with specific link • The link may enhance the effect of the (T-odd) gluonic pole contribution involving also specific color factors • Finding the right observables, however is crucial

Conclusions • Hard processes quark and gluon structure of hadrons (quark distributions, their chirality and transverse polarization) • Many new observables accessible when going beyond collinearity, often in combination with (transverse) polarization (among others the simplest access to transverse quark polarization) • Going beyond collinearity gives access to gluon dynamics in hadrons, which can be done in a controlled way via weighted asymmetries (twist limited, t 3), use of chirality, and the specific time-reversal behavior of single spin asymmetries.