TMD phenomenology: from HERMES and COMPASS to JLAB and EIC. Alexei Prokudin. Alexei Prokudin 1

Transcription

1 TMD phenomenology: from HERMES and COMPASS to JLAB and EIC Alexei Prokudin Alexei Prokudin

2 Semi Inclusive Deep Inelastic Scattering l W P q l xp Cross section xp + q X h dσ Y l + P l + h + X Kinematical variables: Q = q, x Bj = Q y = P q, Q (s mp )x, Bj z = P h P q, W = Q x Bj x Bj d 5 σ lp l hx dx Bj dydz h d P T L µν W µν + m p Leptonic tensor L µν = (l µ l ν + l νl ν l l g µν ) + iλ e ɛ µνρσ l ρ l σ Hadronic tensor W µν d 3 P X P δ (4) (q+p P X P h ) PS J µ () h, X h, X J ν () PS X X Alexei Prokudin

3 TMD Factorization in Semi Inclusive Deep Inelastic Scattering l P q l k + q H k h D S f dσ L µν W µν = q h H k P e q TMD factorization Ji, Ma, Yuan 5 is valid for soft P T Λ QCD Q Λ QCD and describes SIDIS cross section as convolution of Transverse Momentum Dependent (TMD) distribution and fragmentation functions. d k d p d l f q/p (x, k,...) S(l,...)HH (Q )D h q (z, p,...)δ () (zk + p + l P Th ) Intrinsic transverse momenta k and p are of non perturbative origin. Polarised case is explored in TMD factorization formalism Kotzinian 995; Mulders, Tangerman 995; Goeke, Metz, Schlegel 5, Bacchetta et al 7 Alexei Prokudin 3

4 Collinear Factorization in Semi Inclusive Deep Inelastic Scattering l P q l k h Cross section h dσ L µν W µν = q k P Collinear factorization Collins, Soper, Sterman 98, Meng, Olness, Soper 996 is valid for hard P T Q Λ QCD and describes SIDIS cross section as convolution of usual integrated distribution and fragmentation functions. Gluon radiation generates P T of the produced hadron. f q/p (x,...) σ D h q (z,...) In polarized case multy-parton correlations play role in collinear factorization formalism Efremov, Teryaev 98; Qiu, Sterman 99; Ji, Qui, Vogelsang, Yuan 6. Alexei Prokudin 4

5 Quark-quark Correlator and TMDs k k P P Φ ij (P, k, S) = d 4 ξ (π) 4 eik ξ P, S ψ j ()W(, ξ n )ψ i (ξ) P, S Mulders, Tangerman 995; Goeke, Metz, Schlegel 5, Bacchetta et al 7 Gauge link W(, ξ n ) ensures gauge invariance of the correlator. TMD distribution functions can be found via k integration Φ(P, k, S) = dk Φ(P, k, S) Dirac decomposition is done by Φ [Γ] (P, k, S) = Tr(Φ(P, k, S)Γ) Alexei Prokudin 5

6 Quark-quark Correlator and TMDs k k P P Φ ij (P, k, S) = d 4 ξ (π) 4 eik ξ P, S ψ j ()W(, ξ n )ψ i (ξ) P, S Mulders, Tangerman 995; Goeke, Metz, Schlegel 5, Bacchetta et al 7 Parametrization of the correlator is done by using hermiticity and parity invariance ( p = (p, p) ): Φ(P, k, S) = γ Φ (P, k, S)γ ; Φ(P, k, S) = γ Φ( P, k, S)γ Among TMDs there are T-odd and T-even functions: φ T even(t odd) (P, k, S) = +( )iγ γ 3 φ T even(t odd) ( P, k, S)iγ γ 3 Final state interactions allow existance of T-odd functions. T-odd functions are connected to multyparton correlators in collinear factorization. Alexei Prokudin 6

7 Quark-quark Correlator and TMDs k k P P Φ ij (P, k, S) = Mulders, Tangerman 995; Goeke, Metz, Schlegel 5, Bacchetta et al 7 Twist- decomposition contains 8 functions: d 4 ξ (π) 4 eik ξ P, S ψ j ()W(, ξ n )ψ i (ξ) P, S Φ [γ+] (P, k, S) = f (x, k ) ɛij T k is Tj M f T (x, k ) Φ [γ+ γ 5 ] (P, k, S) = S L g L (x, k ) k S T M g T (x, k ) Φ [iσi+ γ 5 ] (P, k, S) = ST i h k i +S L M h L ki kj /k g ij T M Amsterdam notation is used for the TMDs S Tj ht ɛij T k j M h Alexei Prokudin 7

8 Parton distribution functions Collinear Parton Distribution Functions: f a/a (x a ), q(x a ), }{{} T q(x a ) }{{} g Transversity h Transverse Momentum Dependent (TMD) PDFs Kotzinian 995; Mulders, Tangerman 995; Bacchetta et al 7; Anselmino et al 6 Hadron A unpolarized: ˆf a/a (x a, k a ), ˆf sy /A(x a, k a ) }{{} Boer-Mulders h Transversely polarised hadron A : ˆf a/a (x a, k a ), ˆf q s x (x /A a, k a ), ˆf sy /A (x a, k a ), }{{} Sivers f T }{{} Transversity h, h T ˆf sz /A (x a, k a ) }{{} gt 3 Longitudinally polarized hadron A : ˆf sz /+(x a, k a ), ˆf sx /+(x a, k a ) } {{ } q } {{ } hl Alexei Prokudin 8

9 Parton distribution functions Collinear Parton Distribution Functions: f a/a (x a ), q(x a ), }{{} T q(x a ) }{{} g Transversity h Transverse Momentum Dependent (TMD) PDFs Kotzinian 995; Mulders, Tangerman 995; Bacchetta et al 7; Anselmino et al 6 Hadron A unpolarized: ˆf a/a (x a, k a ), ˆf sy /A(x a, k a ) }{{} Boer-Mulders h Transversely polarised hadron A : ˆf a/a (x a, k a ), ˆf q s x (x /A a, k a ), ˆf sy /A (x a, k a ), }{{} Sivers f T }{{} Transversity h, h T ˆf sz /A (x a, k a ) }{{} gt 3 Longitudinally polarized hadron A : ˆf sz /+(x a, k a ), ˆf sx /+(x a, k a ) } {{ } q } {{ } hl Alexei Prokudin 9

10 Parton distribution functions Collinear Parton Distribution Functions: f a/a (x a ), q(x a ), }{{} T q(x a ) }{{} g Transversity h Transverse Momentum Dependent (TMD) PDFs Kotzinian 995; Mulders, Tangerman 995; Bacchetta et al 7; Anselmino et al 6 Hadron A unpolarized: ˆf a/a (x a, k a ), ˆf sy /A(x a, k a ) }{{} Boer-Mulders h Transversely polarised hadron A : ˆf a/a (x a, k a ), ˆf q s x (x /A a, k a ), ˆf sy /A (x a, k a ), }{{} Sivers f T }{{} Transversity h, h T ˆf sz /A (x a, k a ) }{{} gt 3 Longitudinally polarized hadron A : ˆf sz /+(x a, k a ), ˆf sx /+(x a, k a ) } {{ } q } {{ } hl Alexei Prokudin

11 Parton distribution functions Collinear Parton Distribution Functions: f a/a (x a ), q(x a ), }{{} T q(x a ) }{{} g Transversity h Transverse Momentum Dependent (TMD) PDFs Kotzinian 995; Mulders, Tangerman 995; Bacchetta et al 7; Anselmino et al 6 Hadron A unpolarized: ˆf a/a (x a, k a ), ˆf sy /A(x a, k a ) }{{} Boer-Mulders h Transversely polarised hadron A : ˆf a/a (x a, k a ), ˆf q s x (x /A a, k a ), ˆf sy /A (x a, k a ), }{{} Sivers f T }{{} Transversity h, h T ˆf sz /A (x a, k a ) }{{} gt 3 Longitudinally polarized hadron A : ˆf sz /+(x a, k a ), ˆf sx /+(x a, k a ) } {{ } q } {{ } hl Alexei Prokudin

12 Polarised Semi Inclusive Deep Inelastic Scattering Asymmetry in γ p cm frame of lp l hx TMD functions can be studied in asymmetries A UT = dσ dσ (dσ + dσ ) Unpolarised electron beam, Transversely polarised proton. Azimuthal dependence on Φ h and Φ S singles out different combinations. l, E l q θ PT Ph φh P y z x Contributions at leading twist dσ dσ N f q/p d ˆσ D h/q sin(φ h φ S ) + }{{} Sivers effect + T q ˆσ N D h/q sin(φ h + φ S ) + }{{} Collins effect +... Kotzinian 995; Mulders, Tangerman 995; Bacchetta et al 7 Alexei Prokudin

13 The fundamental distributions of partons inside a nucleon Unpolarised Distribution f (x) or q(x) Helicity Distribution g (x) or q(x) Transversity Distribution h (x) or T q(x) Distribution of unpolarised partons in an unpolarised nucleon. Well known Distribution of longitudinally polarised partons in a longitudinally polarised nucleon. Known Distribution of transversely polarised quarks in a transversely polarised nucleon. Little known! HERMES and COMPASS experimental measurements Alexei Prokudin 3

14 The fundamental distributions of partons inside a nucleon Unpolarised Distribution Helicity Distribution Transversity Distribution f (x) or q(x) σ r,nc (x,q ) x i x =.5, i= x =.8, i= x =.3, i=9 x =., i=8 x =.3, i=7 x =.5, i=6 H and ZEUS x =.8, i=5 x =.3, i=4 x =., i=3 x =.3, i= x =.5, i= x =.8, i= x =.3, i=9 x =., i=8 x =.3, i=7 x =.5, i=6 Distribution of - unpolarised partons in -3 an unpolarised nucleon. Well known HERA I NC e + p Fixed Target HERAPDF. x =.8, i=5 x =.3, i=4 g (x) or q(x) x =.8, i=3 x =.5, i= x =.4, i= Distribution of x =.65, i= longitudinally polarised partons Q in a / GeV longitudinally polarised nucleon. Known h (x) or T q(x) Distribution of transversely polarised quarks in a transversely polarised nucleon. Little known! HERMES and COMPASS experimental measurements Alexei Prokudin 4

15 The fundamental distributions of partons inside a nucleon Unpolarised Distribution f (x) or q(x) Helicity Distribution g (x) or q(x) Transversity Distribution h (x) or T q(x) Distribution of unpolarised partons in an unpolarised nucleon. Well known Distribution of longitudinally polarised partons in a longitudinally polarised nucleon. Known Distribution of transversely polarised quarks in a transversely polarised nucleon. Little known! HERMES and COMPASS experimental measurements Alexei Prokudin 5

16 The fundamental distributions of partons inside a nucleon Unpolarised Helicity!"#$%&'()*&+,#-.&/(** Distribution Distribution Transversity Distribution f (x) or q(x) g (x) or q(x) h (x) or T q(x) Distribution of unpolarised partons in an unpolarised nucleon. Well known ))&% Distribution of longitudinally polarised partons in a longitudinally polarised nucleon. Known +,-%./% #%79%35#9794$#4%:74;%4;$%<4% *)% Jianwei Qiu% Distribution of transversely polarised quarks in a transversely polarised nucleon. Little known! HERMES and COMPASS experimental measurements Alexei Prokudin 6

17 The fundamental distributions of partons inside a nucleon Unpolarised Distribution f (x) or q(x) Helicity Distribution g (x) or q(x) Transversity Distribution h (x) or T q(x) Distribution of unpolarised partons in an unpolarised nucleon. Well known Distribution of longitudinally polarised partons in a longitudinally polarised nucleon. Known Distribution of transversely polarised quarks in a transversely polarised nucleon. Little known! HERMES and COMPASS experimental measurements Alexei Prokudin 7

18 TRANSVERSITY Helicity distribution: g positive helicity and - negative helicity. Transversity distribution: = h = g = h in non-relativistic limit. In helicity basis ( ) = ( + ± i ) Transversity is a chiral-odd function. h = + + Alexei Prokudin 8

19 TRANSVERSITY Transversity cannot be studied in DIS as QED and QCD interactions conserve helicity up to corrections O(m q /E). + + Transversity can be measured if coupled with another chiral-odd function. This can be done in Semi Inclusive DIS (SIDIS), quark fragments into unpolarised hadron. It couples to so called Collins Fragmentation function that describes how a polarised quark fragments into unpolarised hadron Alexei Prokudin 9

20 TMDs and Structure functions Unpolarised hadron F UU = f D F cos φ h UU = Q (f D + h H +...) F cos φ h UU = h }{{} H }{{} Boer Mulders Collins FF F AB sin(cos)φ X beam polarization A = U, L, hadron polarization B = U, L, T and contribution to cross section σ sin(cos)φ X F AB sin(cos)φ X Boer-Mulders distribution (Boer and Mulders 998) describes distribution of transversely polarised quarks in unpolarised hadron. Measuring F UU we access non perturabtive dynamics of parton motion. Intermediate P T both TMD and Collinear factorizations are valid. Using a simple gaussian approximation we can obtain widths k =.5 GeV p =. GeV (Anselmino et al 7) of f and D Where are some hints that k u k d and on x and z dependence of k and p. Alexei Prokudin

21 From low to high P T Anselmino et al 7. - (GeV/c) ZEUS 96 <cos φ h > -. dσ/dp T /σ DIS cut P T (GeV/c) (GeV/c) P T TMD σ and collinear QCD σ cross sections are included: dσ = α s dσ + α s dσ + α s dσ αs dσ + Kαs dσ f (x, k ) = f (x) π k e k / k, D (z, p ) = D (z) π p e k / p K.5, The data are from ZEUS and E665. Fermilab E665 Collaboration, M.R. Adams et al., Phys. Rev. D48 (993) 557. M. Derrick et al, ZEUS Collaboration, Z. Phys. C7 (996). Alexei Prokudin

22 From low to high P T Anselmino et al 7. - (GeV/c) ZEUS 96 <cos φ h > -. dσ/dp T /σ DIS cut P T (GeV/c) (GeV/c) P T TMD σ and collinear QCD σ cross sections are included: dσ = α s dσ + α s dσ + α s dσ α s dσ + Kα s dσ k =.5 GeV p =. GeV in accordance with lattice results Bernhard Musch et al, 8 Alexei Prokudin

23 From low to high P T Anselmino et al 7. - (GeV/c) ZEUS 96 <cos φ h > -. dσ/dp T /σ DIS cut P T (GeV/c) (GeV/c) P T TMD σ and collinear QCD σ cross sections are included: dσ = α s dσ + α s dσ + α s dσ α s dσ + Kα s dσ k =.5 GeV p =. GeV in accordance with lattice results Bernhard Musch et al, 8 Alexei Prokudin 3

24 TMDs: Transversely polarized hadron F sin(φ h φ S ) UT = f T D Sivers effect F sin(φ h+φ S ) UT = h H Collins effect F sin(3φ h φ S ) UT = h T H F cos(φ h φ S ) LT = g T D Sivers function ft describes distribution of unpolarised quarks in transversely polarized hadron. Transversity h describes distribution of transversely polarized quarks in transversely polarized hadron. Survives k integration. ht describes distribution of transversely polarized quarks in transversely polarized hadron, vanishes under k integration. ht = g h in some models. Some preliminary data are available from HERMES and COMPASS. gt describes distribution of longitudinally polarized quarks in a transversely polarized hadron. No experimental data are available! Alexei Prokudin 4

25 How to measure transversity? SIDIS and e + e annihilation SIDIS ln l H X e + e H H X l l N D h q H q q + e N D h q H q δq P y x N D h q e y x z H z Collins effect gives rise to azimuthal Single Spin Asymmetry = T q(x, Q ) = N D h/q (z, Q ) Collins effect gives rise to azimuthal asymmetry, q and q Collins functions are present in the process: N D h/q (z, Q ) N D h/ q (z, Q ) D. Boer, R.Jacob and P. J. Mulders Nucl. Phys. B54 (997) 345 J. C. Collins, Nucl. Phys. B396 (993) 6 Alexei Prokudin 5

26 Description of the data HERMES A sin(φ h+φ S ) UT COMPASS A sin(φ h+φ S +π) UT ) S + φ h sin (φ A UT ) S + φ h sin (φ A UT ) S + φ h sin (φ A UT. π π + π HERMES preliminary x z (GeV) P T + π) + π) S + φ h sin (φ A UT S + φ h sin (φ A UT. + π.. π. COMPASS x z P T (GeV) ep eπx, p lab = 7.57 GeV. µd µπx, p lab = 6 GeV M.Anselmino et al, Nucl.Phys.Proc.Suppl.9:98-7,9 HERMES, M. Diefenthaler, (7),arXiv:76.4 COMPASS, M. Alekseev et al., (8), Phys.Lett.B673:7-35,9 Alexei Prokudin 6

27 Description of the data HERMES A sin(φ h+φ S ) UT COMPASS A sin(φ h+φ S +π) UT ) S + φ h sin (φ A UT ) S + φ h sin (φ A UT ) S + φ h sin (φ A UT. π π + π HERMES preliminary x z (GeV) P T + π) + π) S + φ h sin (φ A UT S + φ h sin (φ A UT. + π.. π. COMPASS x z P T (GeV) ep eπx, p lab = 7.57 GeV. µd µπx, p lab = 6 GeV HERMES - PROTON TARGET, COMPASS - DEUTERON PROTON + NEUTRON T u D T u + T d Alexei Prokudin 7

28 h h Description of the data Predictions for COMPASS operating on PROTON target COMPASS A sin(φ h+φ S +π) UT COMPASS A sin(φ h+φ S +π) UT + π) + φ S sin (φ A UT + π) + φ S sin (φ A UT. + π.. π. Anselmino et al x z P T (GeV) + π) + π) S + φ h sin (φ A UT S + φ h sin (φ A UT. + h.. h. COMPASS preliminary x z P T (GeV) Comparison with preliminary COMPASS data arxiv:88.86 Alexei Prokudin 8

29 Transversity vs. helicity.5 Solid red line transversity distribution x T d(x) x T u(x) x T q(x) this analysis at Q =.4 GeV. Solid blue line Soffer bound T q(x) < q(x) + q(x) GRV98LO + GRSV98LO 3 Dashed line helicity distribution q(x) GRSV98LO Alexei Prokudin 9

30 Transversity x T d(x) x T u(x) x T d(x, k ) x T u(x, k ) x =. x = x k (GeV) This is the extraction of transversity from existing experimental data. Anselmino et al 9 T u(x) > and T d(x) < T q(x) < q(x). JLab will provide wider region of x for tensor charge extraction. Alexei Prokudin 3

31 Transversity, comparison with models New extraction is close to most models. q(x) x T x Barone, Calarco, Drago PLB (97) Soffer et al. PRD 65 () Korotkov et al. EPJC 8 () 3 Schweitzer et al. PRD 64 () 4 Wakamatsu, PLB B653 (7) 5 Pasquini et al., PRD 7 (5) 6 Cloet, Bentz and Thomas PLB 659 (8) 7 Anselmino et al 9. Alexei Prokudin 3

32 Tensor charges δq = dx ( T q T q) = dx T q T u = , T d = at Q =.8 GeV our result δ T u δ T d Quark-diquark model: Cloet, Bentz and Thomas PLB 659, 4 (8), Q =.4 GeV CQSM: M. Wakamatsu, PLB 653 (7) 398. Q =.3 GeV 3 Lattice QCD: M. Gockeler et al., Phys.Lett.B67:3-3,5, Q = 4 GeV 4 QCD sum rules: Han-xin He, Xiang-Dong Ji, PRD 5:96-963,995, Q GeV 5 Constituent quark model: B. Pasquini, M. Pincetti, and S. Boffi, PRD7(5)949 and PRD76(7)34, Q.8 GeV Alexei Prokudin 3

33 Sivers effect The azimuthal asymmetry A sin(φ h φ S ) UT 9) arises due to Sivers function (Sivers f q/p (x, k ) = f q/p (x, k ) + N f q/p (x, k ) S T (ˆP ˆk ) Spin sum rule: = Σ + G+ < Lq, q z > + < L G z > EMC result on Σ = q, q q.3 triggered so called Spin crisis only 3% of the spin of the proton is carried by quarks. Leader, Anselmino A Crisis In The Parton Model: Spin? Z.Phys.C4:39,988 Where, Oh Where Is The Proton s S T (ˆP ˆk ) correlation between the spin (S T ) and angular momentum (L q ˆP ˆk ) implies non zero contribution < L q, q z > Data are available from HERMES and COMPASS. u and d Sivers functions are non zero thus L u,d. Alexei Prokudin 33

34 Sivers function: process dependence Sivers function can be measured in both SIDIS and DY process: A B l + l X B(P B) l + b(pb) γ (q) A(P A) a(pa) l A sin(φγ φ S ) UT SIDIS lp l hx A sin(φ H φ S ) UT N f DY q/a (x, k ) f q/b (x, p ) N f SIDIS q/p (x, k ) D h/q (z, p ) Alexei Prokudin 34

35 Sivers function: process dependence Operator definitions of distributions in SIDIS and DY: f SIDIS q/p (x, k, S) = dy d y (π) 3 e ixp+ y ik y P, S ψ()w SIDIS (, y n )ψ(y) P, S x SIDIS f DY q/p (x, k, S) = dy d y (π) 3 e ixp+ y ik y P, S ψ()w DY (, y n )ψ(y) P, S From parity and time reversal invariance one can show (Collins ) and f SIDIS q/p (x, k, S) = f DY q/p (x, k, S) N f SIDIS q/p (x, k ) = N f DY q/p (x, k ) y DY Alexei Prokudin 35

36 Relation between SIDIS and DY Sivers function is process dependent. Collins N f q/p DY = N f q/p SIDIS Let s consider a simple model of Final State Interactions as in Brodsky, Hwang, Schmidt, proton = quark + + antiquark + + SIDIS - attractive DY - repulsive Experimental test of this relation is fundamental for our understanding of the origin of the correlation between parton angular momentum and the spin of the proton and the gauge link formalism itself. Experimental DY data are not available, experiments are planned. + Alexei Prokudin 36

37 HERMES and COMPASS DATA HERMES ep eπx, p lab = 7.57 GeV. COMPASS µd µπx, p lab = 6 GeV. - φ S ) sin (φ h A UT.5 π HERMES π -. π M. Anselmino et al x z P T (GeV) - φ S ) sin (φ h A UT - φ S ) sin (φ h A UT. π+ COMPASS π M. Anselmino et al x z P T (GeV) lp lπ + X N u D u/π + > lp lπ X 4 N u D u/π + N d D d/π ld lπ + X ( N u + N d) D u/π + Alexei Prokudin 37

38 HERMES and COMPASS DATA HERMES ep eπx, p lab = 7.57 GeV. HERMES ep eπx, p lab = 7.57 GeV. - φ S ) sin (φ h A UT.5 π HERMES π sin(φ-φ A S ) UT.5..5 proton π + vs. HERMES preliminary π M. Anselmino et al x z P T (GeV) x Arnold et al 8 lp lπ + X N u D u/π + > lp lπ X 4 N u D u/π + N d D d/π ld lπ + X ( N u + N d) D u/π + Alexei Prokudin 38

39 f Sivers functions N f q () (x) d k k 4m p N f q/p (x, k ) = f ()q T (x). x (x) d u s s () N u d ) s x f(x, k d u N s d u.6.4. x =. x = x =. -.. x =. -.. x =. -.. x =. Sivers functions for u, d and sea quarks are extracted from HERMES and COMPASS data. N f u >, N f d <, first hints on nonzero sea quark Sivers functions x k (GeV) Alexei Prokudin 39

40 Constraint on Gluon Sivers Function Burkardt sum rule dx d k k f a/p (x, k ) a a M. Burkardt Phys.Rev.D69:95,4 k a = k a is related to the first moment of the Sivers function. k u = (MeV) kd = (MeV) The sum rule is almost saturated by u and d quarks at Q =.4 GeV : k u + kd = (MeV) kū + k d + k s + k s = (MeV). thus leaving little room for the gluon Sivers function k g 48 (MeV) Agrees with Anselmino et al 6; Brodsky, Gardner 6 But SIDIS measurements cover restricted region in x:. x.4 Still to be investigated... JLab will widen the region of x. Alexei Prokudin 4

41 Three dimentional picture of the proton The proton moves along Z direction (into the screen) and S T is along Y. S T y P z x This is the three dimentional view of the proton as seen by the virtual photon. Red color more quarks. Blue color less quarks. Distributions of quarks are not symmetrical and shifted due to final state interactions. x =. Alexei Prokudin 4

42 Three dimentional picture of the proton The proton moves along Z direction (into the screen) and S T is along Y. Sivers functions for u, d and sea quarks are extracted from HERMES and COMPASS data. Red color more quarks. Blue Color less quarks. Sivers functions is a left right asymmetry of quark distribution. x =. More information on sea quarks. EIC and JLab will contribute. Alexei Prokudin 4

43 Boer-Mulders effect Boer-Mulders function f q /p(x, k ) = Sivers function [ ] f q/p (x, k ) h q (x, k ) s (ˆP k ) M f q/p (x, k ) = f q/p (x, k ) f q T (x, k ) S T (ˆP k ) M Both functions measure correlation of (ˆP k ) and S T or s. Burkhadt (Burkhardt 5) conjecture: h q (x, k ) f q T (x, k ), h u,d (x, k ) <. Expected values: h u /f u T.8, h d /f d T Data are available from HERMES and COMPASS. The best fit is (Barone, AP, Melis ): h u /f u T =. ±., h d /f d T =. ±. A good accordance with expectations. Alexei Prokudin 43

44 COMPASS SIDIS UNPOLARISED ld lπ X s = (GeV). < y <.9 < x <. < z <.85 < Q < e+ (GeV ) < e+ (GeV ) < x F <. < P T <.5 (GeV) HERMES UNPOLARISED + lp lπ X s = (GeV).3 < y <.85.3 < x <. < z < < Q < e+ (GeV ) < W < e+ (GeV ) < E < e+ (GeV) h. < x < F.5 < P <.4 (GeV) T Boer-Mulders data COMPASS A cosφ h UU HERMES A cosφ h UU > < cos φ..5 5 < W < E h < e+ (GeV) < cos φ > A.5..5 A x x Barone, Melis, AP arxiv:9.594 Barone, Melis, AP arxiv:9.594 F cos φ S UU = h H + Q f D Twist- contribution (the dashed line) is comparable to higher twist (the dotted line) contribution at low Q. EIC at high Q allows to measure h H without higher twists. Alexei Prokudin 44

45 COMPASS SIDIS UNPOLARISED ld lπ X s = (GeV). < y <.9 < x <. < z <.85 < Q < e+ (GeV ) < e+ (GeV ) < x F <. < P T <.5 (GeV) Boer-Mulders data COMPASS A cosφ h UU > < cos φ..5 5 < W < E h < e+ (GeV) Boer-Mulders functions A.5 3 x Barone, Melis, AP arxiv:9.594 F cos φ S UU = h H + Q f D Twist- contribution (the dashed line) is comparable to higher twist (the dotted line) contribution at low Q. EIC at high Q allows to measure h H without higher twists. Alexei Prokudin 45

46 COMPASS SIDIS UNPOLARISED ld lπ X s = (GeV). < y <.9 < x <. < z <.85 < Q < e+ (GeV ) < e+ (GeV ) < x F <. < P T <.5 (GeV) Boer-Mulders data COMPASS A cosφ h UU > < cos φ..5 5 < W < E h < e+ (GeV) Boer-Mulders functions A.5 3 x Barone, Melis, AP arxiv:9.594 F cos φ S UU = h H + Q f D Twist- contribution (the dashed line) is comparable to higher twist (the dotted line) contribution at low Q. EIC at high Q allows to measure h H without higher twists. Alexei Prokudin 46

47 COMPASS SIDIS UNPOLARISED ld lπ X s = (GeV). < y <.9 < x <. < z <.85 < Q < e+ (GeV ) < e+ (GeV ) < x F <. < P T <.5 (GeV) Boer-Mulders data COMPASS A cosφ h UU > < cos φ..5 5 < W < E h < e+ (GeV) Boer-Mulders functions A.5 3 x Barone, Melis, AP arxiv:9.594 F cos φ S UU = h H + Q f D Twist- contribution (the dashed line) is comparable to higher twist (the dotted line) contribution at low Q. EIC at high Q allows to measure h H without higher twists. Alexei Prokudin 47

48 Some snapshots of EIC and JLab HERMES s = 7.5 GeV Q vs x Q x Alexei Prokudin 48

49 Some snapshots of EIC and JLab COMPASS s = 7.3 GeV Q vs x Q x Alexei Prokudin 49

50 Some snapshots of EIC and JLab JLab will mesure TMD in the region of high x JLAB s = 4.63 GeV Q vs x Q x Alexei Prokudin 5

51 Some snapshots of EIC and JLab EIC will widen kinematical coverage in x and Q JLAB+HERMES+COMPASS Q vs x Q x Alexei Prokudin 5

52 Some snapshots of EIC EIC will widen kinematical coverage in x and Q s = GeV Q vs x Q x Alexei Prokudin 5

53 Some snapshots of EIC s = GeV s = 65 GeV Q vs x Q 7 6 Q vs x Q x 3 x Biggest asymmetries are at x.. Wide range of Q at some fixed x is plausible. Increasing the energy we go to the low-x region, but loose Q range at moderate x. The energy of the collider should be chosen with great care to accommodate appropriate x Q range. Alexei Prokudin 53

54 Some snapshots of EIC s = GeV s = 65 GeV Q vs x Q 7 6 Q vs x Q x 3 x Biggest asymmetries are at x.. Wide range of Q at some fixed x is plausible. Increasing the energy we go to the low-x region, but loose Q range at moderate x. The energy of the collider should be chosen with great care to accommodate appropriate x Q range. Alexei Prokudin 54

55 Some snapshots of EIC s = GeV s = 65 GeV Q vs x Q 7 6 Q vs x Q x 3 x Biggest asymmetries are at x.. Wide range of Q at some fixed x is plausible. Increasing the energy we go to the low-x region, but loose Q range at moderate x. The energy of the collider should be chosen with great care to accommodate appropriate x Q range. Alexei Prokudin 55

56 Some snapshots of EIC s = GeV s = 65 GeV Q vs x Q 7 6 Q vs x Q x 3 x Biggest asymmetries are at x.. Wide range of Q at some fixed x is plausible. Increasing the energy we go to the low-x region, but loose Q range at moderate x. The energy of the collider should be chosen with great care to accommodate appropriate x Q range. Alexei Prokudin 56

57 Some snapshots of EIC and JLab s = GeV s = 65 GeV vs z P T z 45 vs z P T z P T P T High P T range allows to test intermidiate region where both TMD and collinear factorizations are valid. In these estimates only non perturbative TMD cross sections were used. Actual distributions will be shifted towards higher P T. Alexei Prokudin 57

58 Some snapshots of EIC and JLab s = GeV s = 65 GeV vs z P T z 45 vs z P T z P T P T High P T range allows to test intermidiate region where both TMD and collinear factorizations are valid. In these estimates only non perturbative TMD cross sections were used. Actual distributions will be shifted towards higher P T. Alexei Prokudin 58

59 Some snapshots of EIC and JLab s = GeV s = 65 GeV vs z P T z 45 vs z P T z P T P T High P T range allows to test intermidiate region where both TMD and collinear factorizations are valid. In these estimates only non perturbative TMD cross sections were used. Actual distributions will be shifted towards higher P T. Alexei Prokudin 59

60 Some snapshots of EIC and JLab s = GeV s = 65 GeV vs z P T z 45 vs z P T z P T P T High P T range allows to test intermidiate region where both TMD and collinear factorizations are valid. In these estimates only non perturbative TMD cross sections were used. Actual distributions will be shifted towards higher P T. Alexei Prokudin 6

61 Timeline of experimental measurements SIDIS experiments lh lhx : HERMES, DESY, Germany (terminated) s = 7.5 GeV COMPASS, CERN, Switzerland s = 7.3 GeV (ongoing) JLAB, USA (ongoing p lab = 6 GeV /planned p lab = ) EIC at medium - high energy s > 5 GeV. (planned) Transversity, Sivers, Collins etc DY experiments H ( ) H l + l X COMPASS, CERN, Switzerland πd >, PAX, GSI, Germany P P > 6, RHIC, BNL, USA collider P P > 3 Distribution function study, transversity at P P is the main goal. Hadron experiments H ( ) H hx RHIC, BNL, USA collider P P is a rich source of new data s = GeV. e + e h h X BELLE ongoing data analysis, BABAR planned data analysis. Alexei Prokudin 6

62 Timeline of experimental measurements SIDIS experiments lh lhx : HERMES, DESY, Germany (terminated) s = 7.5 GeV COMPASS, CERN, Switzerland s = 7.3 GeV (ongoing) JLAB, USA (ongoing p lab = 6 GeV /planned p lab = ) EIC at medium - high energy s > 5 GeV. (planned) Transversity, Sivers, Collins etc DY experiments H ( ) H l + l X COMPASS, CERN, Switzerland πd >, THANK YOU! PAX, GSI, Germany P P > 6, RHIC, BNL, USA collider P P > 3 Distribution function study, transversity at P P is the main goal. Hadron experiments H ( ) H hx RHIC, BNL, USA collider P P is a rich source of new data s = GeV. e + e h h X BELLE ongoing data analysis, BABAR planned data analysis. Alexei Prokudin 6

63 SIDIS kinematics We choose a coordinate system for SIDIS in which proton is moving along z-direction. Introduce light cone vectors: p = Λ(,,,, ) Λ( +,, ) n = (,,,, )/Λ (,, )/ Λ p n =, p = n = Then we can define decomposition A = αp µ + βn µ + A µ = A n pµ + A p n µ + A µ thus P µ = p µ + M nµ, q µ = x B p µ + Q x B k µ = xp µ + k +k x n µ + k µ, δ((k + q) ) x x B n µ, here M /Q is neglected. x z plane can be fixed as lepton scattering plane. Alexei Prokudin 63

64 TMDs: Longitudinally polarized hadron F LL = g L D F sin φ h UL = h L H hl describes distribution of transversely polarized quarks in longitudinally polarized hadron. g L can be accessed at low x. Study of k dependence of g L is possible. Is its width equal to that of f? HERMES, JLab and COMPASS measure F sin φ h UL and F LL A π + π - π sinφ A UL..5 HERMES CLAS.4 JLAB GeV, JLAB GeV, P T (GeV) π + π x π.5.5 A UL sinφ x data: HERMES, Airapetian et al, JLab, Avakian et al 9, theory: Anselmino et al 7, Boffi et al 9, Gamberg et al 8 Alexei Prokudin 64