Transversity and TransverseMomentumDependent Partonic Functions Alessandro Bacchetta Outline

Transversity and Transverse-Momentum-Dependent Partonic Functions Alessandro Bacchetta

Outline • • Theoretical framework Transversity Sivers function Boer-Mulders function Alessandro Bacchetta - Transverse-momentumdependent functions 2

Theoretical framework

Semi-inclusive Deep Inelastic Scattering proton lepton pion Alessandro Bacchetta - Transverse-momentumdependent functions 4

Feynman diagrams & Factorization lepton proton pion SIDIS Partonic scattering amplitude Fragmentation amplitude Distribution amplitude Alessandro Bacchetta - Transverse-momentumdependent functions 5

Feynman diagrams & Factorization lepton proton pion proton antilepton SIDIS electron Drell-Yan pion Partonic scattering amplitude Fragmentation amplitude positron pion e –e + to pions Alessandro Bacchetta - Transverse-momentumdependent functions Distribution amplitude 6

Feynman diagrams & Factorization lepton proton pion proton antilepton SIDIS Drell-Yan electron pion proton pion positron pion proton pion e–e+ to pions p-p to pions Alessandro Bacchetta - Transverse-momentumdependent functions 7

Factorization proofs for involving TMD functions LEADING TWIST (1/Q 0), NOT for pp to pions – – Collins, Soper, NPB 193 (81) Ji, Ma, Yuan, PRD 71 (04) Ji, Ma, Yuan, PLB 597 (04) Collins, Metz, PRL 93 (04) Alessandro Bacchetta - Transverse-momentumdependent functions See talk by A. Metz 8

Nonperturbative elements 2 = Alessandro Bacchetta - Transverse-momentumdependent functions 9

Origin of the gauge link . . . Alessandro Bacchetta - Transverse-momentumdependent functions 10

Different processes SIDIS Drell-Yan pp to hadrons See talk by C. Bomhof Bacchetta, Bomhof, Mulders, Pijlman, hep-ph/0505268 Alessandro Bacchetta - Transverse-momentumdependent functions 11

The correlation function integrated over k. T x- axis 0 a SIDIS Alessandro Bacchetta - Transverse-momentumdependent functions 12

The correlation function integrated over k. T 0 a x- axis Drell-Yan Alessandro Bacchetta - Transverse-momentumdependent functions 13

The correlation function UNintegrated over k. T x. T 0 a x- axis SIDIS Belitsky, Ji, Yuan, NPB 656 (03) Alessandro Bacchetta - Transverse-momentumdependent functions 14

The correlation function UNintegrated over k. T 0 a x. T x- axis Drell-Yan Ji, Yuan, PLB 543 (02); Belitsky, Ji, Yuan, NPB 656 (03) Alessandro Bacchetta - Transverse-momentumdependent functions 15

Practical consequences • Some TMD functions will not be sensitive to the differences in the structure of the gauge link (i. e. they will be connected to standard partonic cross-sections) • Some TMD functions will be sensitive to it and will be multiplied by prefactors to be computed for each partonic subprocess (i. e. they will be connected to gluonic-pole cross-sections) See talk by C. Bomhof Alessandro Bacchetta - Transverse-momentumdependent functions 16

Decomposition of the correlation function integrated over k. T Leading twist only Unpolarized distr. func. Helicity distr. func. Transversity distr. func. Alessandro Bacchetta - Transverse-momentumdependent functions 17

Decomposition of the correlation function UNintegrated over k. T Mulders, Tangerman, NPB 461 (96) Goeke, Metz, Schlegel, PLB 618 (05) rs e v i S s r e d l u M r e Bo Sivers, PRD 43 (91) Boer, Mulders, PRD 57 (98) Alessandro Bacchetta - Transverse-momentumdependent functions 18

Parton distribution functions with transverse spin Photon moves into the screen/ proton moves out of the screen Transversity - Sivers - Boer-Mulders - Alessandro Bacchetta - Transverse-momentumdependent functions 19

Transversity

Definition of transversity distribution function Probability to find a quark with momentum x. P+ and transverse spin sq in a proton with transverse spin S Notation of Anselmino et al. NOTE: transverse momentum is integrated over! Alessandro Bacchetta - Transverse-momentumdependent functions 21

Transversity: birth and growth Ralston, Soper, NPB 152 (79) Alessandro Bacchetta - Transverse-momentumdependent functions 22

Helicity and transversity Alessandro Bacchetta - Transverse-momentumdependent functions 23

Transversity vs helicity • Different evolution g 1 h 1@ 0. 75 Ge. V 2 h 1@ 0. 079 Ge. V 2 g 1 S. Scopetta, V. Vento, PLB 424 (1997) Alessandro Bacchetta - Transverse-momentumdependent functions 24

Transversity vs helicity • Different evolution • Different integrals (axial and tensor charge of the nucleon). E. g. from lattice QCD: S. Aoki et al. , PRD 56 (1997) M. Göckeler et al. [QCDSF/UKQCD], PLB (05) Alessandro Bacchetta - Transverse-momentumdependent functions 25

Transversity vs helicity • Different evolution • Different integrals (axial and tensor charge of the nucleon). E. g. from lattice QCD: S. Aoki et al. , PRD 56 (1997) M. Göckeler et al. [QCDSF/UKQCD], PLB (05) • Different sum-rules Bakker, Leader, Trueman, PRD 70 (04) Alessandro Bacchetta - Transverse-momentumdependent functions 26

Comparison of tensor charges c quark spectator lattice non-relativistic bag c quark soliton 2 c quark soliton 1 Barone, Drago, Ratcliffe, PR 359 (2002) Alessandro Bacchetta - Transverse-momentumdependent functions 27

First data: Collins asymmetry Airapetian et al. [HERMES], PRL 94 (05) Alexakhin et al. [COMPASS], PRL 94 (05) ha dr on pla ne lep ton pla ne Alessandro Bacchetta - Transverse-momentumdependent functions See talk by G. Schnell, A. Martin 28

First attempt to combine BELLE and HERMES data Efremov, Goeke, Schweitzer, PRD 73 (06) 1. Get Collins function from BELLE Talk by R. Seidl Talks by P. Schweitzer and M. Boglione Alessandro Bacchetta - Transverse-momentumdependent functions 29

First attempt to combine BELLE and HERMES data 2. Check size of transversity at HERMES See talk by P. Schweitzer Alessandro Bacchetta - Transverse-momentumdependent functions 30

Where to observe it Talks by Schnell, Martin, Avakian, Elschenbroich Martin, Radici, Giordano Heppelman, Dalpiaz Koch, Fischer Heppelman, Aidala Alessandro Bacchetta - Transverse-momentumdependent functions 31

The Sivers function

Introducing the Sivers function Probability to find a quark with momentum x. P+ and k. T in a proton with transverse spin S T-odd see e. g. Bacchetta, D’Alesio, Diehl, Miller, PRD 70 (04) Alessandro Bacchetta - Transverse-momentumdependent functions 33

Sivers function: birth and growth D. Sivers, PRD 41 (90) Alessandro Bacchetta - Transverse-momentumdependent functions 34

Two ingredients • Final-state interactions (included in the gauge link) Ji, Yuan, PLB 543 (02); Belitsky, Ji, Yuan, NPB 656 (03) • Transverse-spin dependent distribution of quarks in transverse space Burkardt, PRD 66 (02); Diehl, Hägler, EPJ C 44 (05) Alessandro Bacchetta - Transverse-momentumdependent functions 35

Final-state interactions Side view Front view up quarks proton down Alessandro Bacchetta - Transverse-momentumdependent functions 36

Final-state interactions Side view Front view up photon down NOTE: QCD tells us that the FSI has to be attractive, since quark and remnants form a color antisymmetric state Alessandro Bacchetta - Transverse-momentumdependent functions 37

Final-state interactions Side view Front view up photon down Chromodynamic lensing Burkardt, PRD 66 (02) Alessandro Bacchetta - Transverse-momentumdependent functions See talk by M. Burkardt 38

Change of sign in Drell-Yan Side view Front view up quarks antiquark proton photon down Clear-cut prediction of QCD Collins, PLB 536 (02) Alessandro Bacchetta - Transverse-momentumdependent functions 39

Distortions in transverse space Side view Front view up quarks proton down Alessandro Bacchetta - Transverse-momentumdependent functions 40

Distortions in transverse space Side view Front view up quarks proton down The presence of spin can distort the distribution of quarks in transverse space (orbital angular momentum of quarks is required) A distortion in the distribution of quarks in transverse space can give rise to a nonzero Sivers function Alessandro Bacchetta - Transverse-momentumdependent functions 41

Relation to GPDs • The transverse-space deformation of unpolarized quarks in a transversely polarized nucleon is described by the generalized parton distribution function E Work of Burkardt Alessandro Bacchetta - Transverse-momentumdependent functions 42

Lattice-QCD studies up quarks Sivers function for up quarks expected to be NEGATIVE See talk by P. Hägler Preliminary results by QCDSF Collab. (see e. g. hep-ph/05110032) Alessandro Bacchetta - Transverse-momentumdependent functions 43

Model calculations Brodsky, Hwang, Schmidt PLB 530 (02) Gamberg, Goldstein, Oganessyan, PRD 67 (03) See talk by L. Gamberg down spectator up proton Bacchetta, Schäfer, Yang, PLB 578 (04) Alessandro Bacchetta - Transverse-momentumdependent functions 44

Fits to HERMES and COMPASS data Airapetian et al. [HERMES], PRL 94 (05) Anselmino et al. , PRD 72 (05) Alessandro Bacchetta - Transverse-momentumdependent functions 45

de ut er on Comparison with COMPASS data Alexakhin et al. [COMPASS], PRL 94 (05), Anselmino et al. , PRD 72 (05) Alessandro Bacchetta - Transverse-momentumdependent functions 46

Different fits Anselmino et al. , hep-ph/0511017 [20] Anselmino et al. , PRD 72 (05) [21] Vogelsang, Yuan, PRD 72 (05) [23] Collins et al. , hep-ph/0510342 Alessandro Bacchetta - Transverse-momentumdependent functions 47

New preliminary kaon data See talk by U. Elschenbroich Alessandro Bacchetta - Transverse-momentumdependent functions 48

Future experimental data See talk by U. D’Alesio • HERMES@DESY: p 0 and weighted asymmetries (this year? ) • COMPASS@CERN: proton data (this year? ) • CLAS@Jlab • STAR, PHENIX@RHIC: jet-production in pp collisions • EIC (DIS with polarized protons) • FAIR@GSI (Drell-Yan with polarized protons) Alessandro Bacchetta - Transverse-momentumdependent functions 49

The Boer-Mulders function

Definition of the Boer-Mulders function Probability to find a quark with momentum x. P+ and k. T and spin sq T-odd Alessandro Bacchetta - Transverse-momentumdependent functions 51

Boer-Mulders function: birth and growth Boer, Mulders, PRD 57 (98) Alessandro Bacchetta - Transverse-momentumdependent functions 52

Relation to transverse space Side view Front view up quarks proton down Alessandro Bacchetta - Transverse-momentumdependent functions 53

Relation to transverse space Side view Front view up quarks proton down Quark spin can be unevenly distributed in transverse space (orbital angular momentum of quarks is required) A distortion in the distribution of quark spin in transverse space can give rise to a Boer-Mulders function Burkardt, hep-ph/0510408 Alessandro Bacchetta - Transverse-momentumdependent functions 54

Lattice-QCD studies up quarks Boer-Mulders function for up quarks expected to be NEGATIVE Preliminary results by QCDSF Collab. (see e. g. hep-ph/05110032) Alessandro Bacchetta - Transverse-momentumdependent functions 55

Model calculations See talk by L. Gamberg, G. Goldstein down up Bacchetta, Schäfer, Yang, PLB 578 (04) Alessandro Bacchetta - Transverse-momentumdependent functions 56

Where to look for it • Unpolarized semi-inclusive DIS (HERMES, COMPASS, H 1, ZEUS, EIC, LHEC) • Unpolarized Drell-Yan (GSI) • Jet-production in unpolarized pp collisions (RHIC, LHC) Alessandro Bacchetta - Transverse-momentumdependent functions 57

Conclusions Many progresses from theoretical, experimental, phenomenological sides…. . . Pay attention to the rest of the workshop! Alessandro Bacchetta - Transverse-momentumdependent functions 58

Semi-inclusive Deep Inelastic Scattering up proton lepton pion Alessandro Bacchetta - Transverse-momentumdependent functions down 59