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The present and (near) future of Di hadron F ragmentation F unctions ( DiFF )

The present and (near) future of Di hadron F ragmentation F unctions ( DiFF ). Transverse momentum, spin, and position distributions of partons in hadrons. ECT* (Trento), 11-15 June 2007. Marco Radici. Pavia. In collaboration with: A. Bacchetta (DESY)

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The present and (near) future of Di hadron F ragmentation F unctions ( DiFF )

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  1. The present and (near) future of Dihadron Fragmentation Functions(DiFF) Transverse momentum, spin, and position distributions of partons in hadrons ECT* (Trento), 11-15 June 2007 Marco Radici Pavia In collaboration with: A. Bacchetta (DESY) F. A. Ceccopieri (Univ. Parma)

  2. Outline - History: why DiFF ?

  3. Need DiFF in e+e-  h1h2 X Konishi, Ukawa, Veneziano P.L. B78 (78) 243 A B i h2 B A h1 h2 k rT h1 de Florian & Vanni, P.L. B578 (04) 139 a b l 1/rT collinear singularities cancelled by D(i h1h2) time-like analogue of Fracture Functions in SIDIS Trentadue & Veneziano, P.L. B323 (94) 201 DiFF evolution equations B A see also de Majumder & Wang, P.R. D70 (04) 014007; D72 (05) 034007

  4. Exp. info mostly on nontrivial spectrum in pair invariant mass Mh e+e- +- √s=91.2 GeV z>0.1 Abreu et al. (DELPHI), P.L. B298 (93) 236 background subtracted pp +- X √s=200 GeV Adams et al. (STAR), P.R.L. 92 (04) 092301 ep ! e’ (+-) X √s=7.5 GeV PYTHIA output at HERMES kin. Q2 >1 GeV2 W2 >4 GeV2 no elastic and diffractive events Mh(GeV)

  5. Leading-twist analysis: full dependence of DiFF T Ph=P1+P2 R=(P1-P2)/2 + Bianconi et al., P.R. D62 (00) 034008 Subleading-twist Bacchetta & Radici, P.R. D69 (04) 074026 \newcommand{\open}{< \kern -0.3 em \scriptscriptstyle{)}} Radici et al., P.R. D65 (02) 074031 Bacchetta & Radici P.R. D74 (06) 114007 Ph

  6. Outline - History: why DiFF ? • SIDIS: extract transversity • with leading twist SSA via • DiFF • (HERMES, COMPASS)

  7. SIDIS e p"! e’ (h1h2) X RT scattering plane LM 1 2 partial wave (LM)

  8. Partial wave expansion Bacchetta & Radici, P.R. D67 (03) 094002  c.m. frame SSA Collins et al., N.P. B420 (94) 565 naïve T-odd from interference |()L=0ih()L=1|

  9. Outline - History: why DiFF ? • SIDIS: extract transversity • with leading twist SSA via • DiFF • (HERMES, COMPASS) • e+e- : determine unknown • DiFF • (BELLE)

  10. DiFF from e+e-! (+-)jet 1 (+-)jet 2 X [ (Ph,R) (Ph,k) ]= R - l Leading-twist d very complicated; take Boer, Jakob, Radici, P.R. D67 (03) 094003

  11. “Artru-Collins” azimuthal asymmetry Artru & Collins, Z.P. C69 (96) 277 same as in SIDIS Boer, Jakob, Radici, P.R. D67 (03) 094003

  12. Outline - History: why DiFF ? • SIDIS: extract transversity • with leading twist SSA via • DiFF • (HERMES, COMPASS) • e+e- : determine unknown • DiFF • (BELLE)

  13. History of SSA: sign change at Mh=m? Jaffe, Jin, Tang, P.R.L. 80 (98) 1166 • no calculation of qI (z) → SSA  f(0,1) • “figureof merit” • s-pinterference from - elastic scattering • phase shifts only; sign change from Re[] Radici,Jakob, Bianconi, P.R. D65 (02) 074031 interference ~ Im[sp*] spectator model including full (z,Mh2~m2) dependence • uncertainty band from • different fp / fs strength ratio • PDF input for h1(x) p s Bacchetta et al., P.R. D70 (04) 117504 Let’s apply Trento conventions

  14. SSA : history of conventions (following Trento conventions) Bacchetta and Radici, P.R. D67 (03) 094002 (before Trento conventions) Radici, Jakob, Bianconi P.R. D65 (02) 074031 ( Hermes analysis: no dcos ) Van der Nat Transversity 2005 (Como – Italy) hep-ex/0512019

  15. History of SSA continues: upgrading the spectator model ep"! e’ (+-) X PYTHIA output on HERMES kinematics: 0.1<y<0.85 Q2 > 1 GeV2 0.023<x<0.4 W2>4 GeV2 excludes elastic and single/double diffractive events → semi-inclusive DIS Mh 2. q →  X2 → +- X2 (14.81%) 395 114 events 3. q →  X3 → +- X3 (0.31%) 239 043 events 4. q →  X’4 → +- (0 X’4) (8.65%) Total 2 667 889 (π+π-) pairs 5. q → η X’5→π+π- (X X5) (2.05%) 6. q → K0 X6→π+π- X6 (3.41%) 1. All – (2+..+6) = background (70.77%) 1 888 065 events

  16. The spectator model : 1. background ≡ q →π+π- X1 no resonance → real s-wave channel 2. q →ρ X2→π+π- X2 X1 = X2 = X3 = X4 = X p-wave channel = coherent sum |2.+3.+4.| 3. q →ω X3→π+π- X3 4. q →ω X’4→π+π- (π0 X’4) Warning: ω→ [(ππ)L=1π]J=1 X4 max number of (π+π-) pairs in s-p interference ~ Im [ p-wave channel ] parameters s-wave p-wave narrow-width approx. M3~m (m–m – Mh) Bacchetta and Radici, P.R. D74 (06) 114007

  17. Fit PYTHIA distributions in z and Mh fix parameters 2.+3.+4. p-wave 1. background s-wave Total Total + 5.+6. • PYTHIA: statistical error  1% • no - interference at Mh~0.65 • 2 / dof ~ 20 but model stat. error also  1% (~1/√#bin) Mh z

  18. Model parameters fixed  prediction of SSA @ z x Mh Bacchetta and Radici, P.R. D74 (06) 114007 6.6% scale uncertainty

  19. Theoretical uncertainty on SSA mostly from transversity input Strat Kor Soffer et al., Korotkov et al., P.R. D65 (02) 114024 E.P.J. C18 (01) 639 Schw Wak Schweitzer et al. Wakamatsu P.R. D64 (01) 034013 P.L. B509 (01) 59 Mh u from GRV98-LO @ Q2 = 2.5 GeV2 d x

  20. Flavor symmetry of model DiFF

  21. Future improvements upon the spectator model ρ → π+π- data ω → (π+π-) π0 “bin” the model and fit the exp. data Mh model prediction bins 0.15 0.15 0.22 1.23 Strat Soffer et al., P.R. D65 (02) 114024 Wak Wakamatsu P.L. B509 (01) 59 •  3 beyond narrow-width approx. •  cure dip at Mh ~ 0.65 GeV

  22. Prediction of SSA @ : deuteron target Q2>1 GeV2 s=604 GeV2 but no Trento conventions! Martin – SPIN Praha 2006 hep-ex/0702002 isospin symmetry in d={p,n} x Mh z 0.004<x<0.4 0.03<x<0.4 Mh Mh

  23. Prediction of SSA @ : proton target 0.1<y<0.9 Q2>1 GeV2 s=301 GeV2 0.004<x<0.4 ….. Mh 0.03<x<0.4 Mh

  24. Outline - History: why DiFF ? • SIDIS: extract transversity • with leading twist SSA via • DiFF • (HERMES, COMPASS) • e+e- : determine unknown • DiFF • (BELLE) evolution

  25. DiFF D(z1,z2,Q2) evolution equations h1 h2 i B A de Florian & Vanni, P.L. B578 (04) 139 k rT l B A which evolution equations for extended DiFF (extDiFF) D(z1,z2,Mh2,Q2)? • recover DiFF evolution with Jet Calculus technique • deduce extDiFF evolution “ “ “ “ “  LL

  26. single-hadron fragmentation in Jet Calculus Konishi, Ukawa, Veneziano N.P. B157 (79) 45 i j i j no interf. • Eij i j strong ordering i j+k qj2, qk2 << qi2 <<1 LL branching ladder diagram Eij j i h Eij time-like perturbative parton fragmentation function Q2 Q02 scale calculable unknown [at LL, resums all collinear sn logn(Q2/Q02) ]

  27. 2-h fragmentation in Jet Calculus Eb1a1 Eij  Eia1,a2  Eb2a2 hard process Q2 S2 Q02 multi-ladder diagram Eij h1 i j h1 A : h2 + B : h2

  28. introducing the scale RT  Mh h1 u • take Q02<RT2 < Q2 B: rT RT 1-u h2 Q2 S2Q02 fixing RT2 scale kj2S2 no longer arbitrary h1 j i 2. take RT2 Q02 A: RT scale kj2arbitrary  set to Q02 h2

  29. evolution equations for extDiFF Ceccopieri, Radici, Bacchetta, P.L. B650 (07) 81 same for polarized extDiFF with Pij RT2 breaks degeneracy of terms A-B  homogeneous evolution conjecture: at LL, factorization with same kernel as 1h-fragmentation

  30. effect of evolution on extDiFF 2. partial-wave expansion “commutes” with evolution h1 i j h2 1. evolution affects “perturbative sector” z, but not the “soft” 

  31. effect of evolution on extDiFF D1ss+pp Q02 = 2.5 GeV2 HERMES Q2 = 100 GeV2 BELLE D1ss+pp z PRELIMINARY ! Mh

  32. Outline - History: why DiFF ? • SIDIS: extract transversity • with leading twist SSA via • DiFF • (HERMES, COMPASS) • e+e- : determine unknown • DiFF • (BELLE) evolution • pp collision: determine all • the unknowns with two • measurements • self extraction of transversity

  33. A + B(" )! (C1C2)C + X PC=PC1+PC2k jet axis PCT=0 but PC¢PA/PC? large PC? hard scale analysis at leading twist o(1/|PC?|) N.B. Sb around PB defined as SB Sc around PC defined as RC Sc = Sb ab(" )! c(" ) d pa = xa PA pb = xb PB pc = PC / zc

  34. A B(" )! (C1 C2)C X Bacchetta, Radici, P.R. D90 (04) 094032

  35. A B ! (C1 C2)C (D1 D2)D X Bacchetta, Radici, P.R. D90 (04) 094032 self-consistent extraction of

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