TMDs and the Drell-Yan process

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1 TMDs and the Drell-Yan process Marc Schlegel Theory Center Jefferson Lab Jefferson Lab upgrade at 12 GeV, INT

2 Kinematics (less intuitive than DIS): The Drell Yan process d¾ d 4 l d 4 l 0 = d¾ d 4 q d 4 l / ±+ (l 2 )± + ((q l) 2 ) 4F X jmj 2 ± (4) (P a + P b q P X ) Disentangling the -functions Dilepton rest frame X Gottfried-Jackson frame and Collins-Soper frame: Lepton angles: d 4 l! d = dá d cos µ Diff. CS including angular dependences: d 6 ¾ d 4 q d = 2 s d 6 ¾ dx a dx b d 2 ~q T d Separation of the leptonic part (generated by one photon): d¾ d 4 qd / L ¹ºW ¹º with: L ¹º = 4 ³l ¹ lº 0 + l ºl¹ 0 Q2 2 g ¹º Limited number of structure function

3 Structure functions Hadronic Tensor W ¹º (P a ; S a ; P b ; S b ; q) = Z d 4 x (2¼) 4 eiq x ha; bjj ¹ (0)J º (x)ja; bi Constraints by current conservation, hermiticity and parity (not for Z-bosons); q ¹ W ¹º = W ¹º q º = 0 W ¹º = [W º¹ ] W ¹º (P a ; S a ; P b ; S b ; q) = W ¹º ( ¹P a ; ¹S a ; ¹P b ; ¹S b ; ¹q) Decomposition into = 48 structure functions [Arnold, Metz, M.S., PRD 79, ] e.g. unpolarized Drell-Yan d¾ UU dx a dx b d 2 q T d = 2 s 2Q 2 F ³ (1 + cos 2 )F 1 UU + (1 cos 2 )F 2 UU + sin 2 cos ÁF cos Á UU Huge amount of observables (even more for W- or Z-bosons) not all independent for identical hadrons or anti-hadrons Classification of structure functions helpful for data analysis Parton model: 24 leading twist structure functions F (x a ; x b ; q 2 T ; Q 2 ) + sin2 cos 2ÁF cos 2Á UU

4 Collinear Parton Model Drell-Yan in the parton model at leading order: q T -integrated DY Cross-Section: Z d¾ dx a dx b d d 2 d¾ q T dx a dx b d 2 q T d = 2 X 6sQ 2 q;¹q e 2 q h (1 + cos 2 µ)(f q 1 (x a)f ¹q 1 (x b) a bg q 1 (x a)g ¹q 1 (x b)) i +j~s at jj~s bt j sin 2 µ cos(2á Á a Á b )h q 1 (x a)h ¹q 1 (x b) Double transversely polarized proton-antiproton DY: ideal process to measure transversity possibly at the future PAX facility at GSI. Unintegrated DY-cross section at high q T ~ Q ( perturbative tail ): F i Z» x a x b p¹p or p¼ Z d» a» a pp ³ X d» b» b ± (» a x a )(» b x b ) q2 T s ) q;¹q e 2 q Contributions to all 4 structure functions (unpolarized): F 1 UU ; F 2 UU ; F cos Á UU ; F cos 2Á UU ³ f q 1 (» a)f ¹q 1 (» b) ^F q¹q i + f q 1 (» a)f g 1 (» b) ^F qg i +f g 1 (» a)f ¹q 1 (» b) ^F g¹q i

5 dn d DY data for the Lam Tung relation ³ d¾ d 4 qd.³ d¾ d 4 q Lam-Tung relation: = 3 4¼ h 1 + cos 2 µ + ¹ sin 2µ cos Á + º 2 sin2 µ cos 2Á 1 2º = 0 () FUU 2 cos 2Á = 2FUU LT-relation even valid after resummation (Extension of coll. parton model to lower q T ) has been measured in - N + - X [NA10 Coll. ('86/'88) & E615 Coll. ('89)] i Large angular distribution violates Lam-Tung relation, far from pqcd result. However, recent FermiLab data on proton-deuterium DY agree with LT-relation [FNAL-E866/NuSea Coll.]

6 k T -dependent Parton Distributions (TMDs) Possible candidate for violation of LT-relation: intrinsic transv. parton momenta q T Q Semi-inclusive processes at small transverse final state momenta: ij (x; ~ k T ; S) = R dz d 2 z T 2(2¼) 3 e ik z hp; Sj à ¹ j (0)W SIDIS=DY [0; z] à i (z) jp; Si z+ =0 Project out polarizations: 8 leading twist TMDs unpolarized: longitudinally: (f 1 ; f? 1T )(x; ~ k 2 T ) (g 1L ; g 1T ) transversely [chirally odd]: (h 1 ; h? 1T ; h? 1L ; h? 1 ) Appear in SIDIS and DY!

7 T-odd TMDs Neglect gauge link operator: Sivers function Time-reversal forbids Boer-Mulders function If T-odd TMDs 0: Gauge Link not neglegible, physical effect: Initial / Final state interactions Wilson line process-dependent: W[z 1 ; z 2 ] = Pe ig R z 2 z 1 ds A(s) f? 1T DIS Time reversal = f? 1T DY h? 1 switches sign: DIS = h? 1 DY

8 TMDs in Drell Yan Tree level formalism: [Ralston, Soper; Pire, Ralston; Tangerman, Mulders; Boer; Arnold, Metz, M.S.] Z W ¹º» d 2 k at d 2 k bt ± (2) ( ~ k at + ~ k bt ~q T )Tr[ (x a ; ~ k at ) ¹ ¹ (x b ; ~ k bt ) º ] + [ $ ¹ ] Only constraint: ~ k 2 at» ~ k 2 bt» q 2 T Q 2 q 2 Contraction with leptonic tensor Generates angular dependencies: (1 + cos 2 sin 2 cos(2 sin 2 sin( Structure functions in the parton model at leading order: F i / f A f ¹ B X Z d 2 k at d 2 k bt ± (2) (~q T ~ k at ~ k bt ) w( ~ k at ; ~ k bt )f A (x a ; ~ kat 2 ) f ¹ B (x b ; ~ kbt 2 ) Drell Yan factorization theorems: [Collins, Soper, Sterman; Ji, Ma, Yuan; Collins, Metz] d¾ d 4 qd / H f A ¹ f B S q Soft factor due to soft gluon radiation Wilson lines slightly modified

9 Prominent Observables in Drell Yan 24 leading twist structure functions in the parton model, but only 8 TMDs Higher twist contributions usually less important than for SIDIS. Each TMD appear at least twice: Drell-Yan self-consistent, cross-checks possible Some interesting observables: (among others...) º(q T Q) = 2F cos 2Á UU F 1 UU + O(1=Q) = 2(qT k Ch at )(q T k bt ) q 2 T k2 at k2 bt M a M b qt 2 C[f q 1 f ¹q 1 ] i h?;q 1 h?;¹q 1 Boer-Mulders effect might explain anomalously large cos(2 ) dependence Double transverse polarization: (GSI, RHIC) F cos 2Á Á a Á b T T / cos(2á Á a Á b )h 1 ¹ h 1 Single transverse polarization: Collins-like F sin(2á Á a) T U / sin(2á Á a )h 1 ¹ h? 1 Sivers asymmetry: Investigate sign change F 1 T U / Ch qt k at q T M a f?;q 1T f ¹q 1 i

10 Semi inclusive DIS Experimentally easier: SIDIS (measured at HERMES, COMPASS, JLAB) TMDs accessible: j ~P h? j Q Structure functions: [Bacchetta, Diehl, Goeke, Metz, Mulders,M.S., 2006] d¾ / F UU + (2 y) cos ÁF cos Á cos(2á) dxdydzd 2 UU + (1 y) cos(2á)fuu + ::: ~P h? dá s Parton Model: 8 leading twist spin observables, 8 TMDs complete set of observables Fragmentation need input from other processes. Factorization theorems for structure functions: F (x; z; ~P 2 h? ) / H f D S Relatively low Q higher twist effects more important e.g. longitudinal single-spin HERMES and JLAB

11 Prominent Observables in SIDIS 1.) Sivers asymmetry: ¾ SIDIS UT / sin(á h Á S )f? 1T D 1 2.) Collins asymmetry: ¾ SIDIS UT / sin(á h + Á S )h 1 H? 1 3.) Boer-Mulders: ¾ SIDIS UU / cos(2á)h? 1 H? 1 4.) pretzelosity : ¾ SIDIS UT / sin(3á h Á S )h? 1T H? 1 Collins-function important input from e + e - - data BELLE data

12 Phenomenology for the Sivers effect Deconvolution of k T integrals Gaussian ansatz: Most recent fits to HERMES and COMPASS data: [Anselmino et al., EPJA 39, 89] f?;q 1T (x) = 2N qx q (1 ( q + q) ( q+ q) x) q f q 1 (x) q q q q f(x; ~ h kt 2 ) = f(x)exp ~ i kt 2 =hkt 2 i [Arnold, Efremov, Goeke, MS, Schweitzer, arxiv: ] f?;q 1T (x) = Aq f q 1 (x) u- and d-sivers roughly equal, but opposite sign non-zero sea-quark Sivers functions. both ansätze: small statistical errors, / d.o.f. ~ 1 Fits not constraint for x >

13 Predictions for Sivers effect in DY Extractions of Sivers functions from SIDIS allows predictions for Drell-Yan Test of sign change [Anselmino et al., PRD 79, ], [Collins et al., PRD73, ], [Vogelsang, Yuan, PRD 72, ] Integration over CS - angles: d¾ h T U d 4 q = 8¼ 2 3sQ 2 j~ qt k i at S at j sin(á a Á )C f?;q 1T jq T jm f ¹q 1 a Weighted transverse SSA in the c.m.-frame: A sin(á a Á ) N = R q bin T q bin T Q R 2¼ dq 0 T q T dá 0 sin(á a Á )(d¾ " d¾ # ) R 1 q bin R T 2¼ 2 dq 0 T q T dá 0 (d¾ " + d¾ # ) Predictions for future DY-experiments: q bin T = 1 GeV; p s = 200 GeV COMPASS: (fixed proton target) ¼p "! ¹ + ¹ X = 1 GeV; p s = 17:4 GeV q bin T pos. x F : smaller x B < 0.4 in SIDIS neg. x F : larger x B > 0.4 in SIDIS - and + beam flavour separation

14 RHIC: (collider) p " p! l + l X q bin T = 1 GeV; p s = 200 GeV Expands the SIDIS x-region beyond x B > 0.4 Maximum at x ~ 0.2 valence quark effect GSI: (collider) p " ¹p! ¹ + ¹ X q bin T = 1 GeV; p s = 14:14 GeV neg. x F : valence (SIDIS) region pos. x F : large x B -region Asymmetry large > 10% in valence region proton antiproton: larger number of DY - events

15 GSI: (fixed proton target) ¹pp "! ¹ + ¹ X q bin T = 0:4 GeV; p s = 5:47 GeV Experiment covers the SIDIS x B range Consistency check Asymmetry sizeable up to 10% J-PARC: (collider) p " p! ¹ + ¹ X q bin T = 1 GeV; p s = 9:78 GeV Experiment covers the x B range > 0.4 Complementary to SIDIS data Asymmetry smaller

16 JINR: (fixed proton target) pp "! ¹ + ¹ X q bin T = 1 GeV; p s = 20 GeV large, neg. x F : large x B -region -0.4 < x F < 0.1: valence region x F > 0.1: sea-contribution IHEP: (fixed proton target) pp "! ¹ + ¹ X ¼p "! ¹ + ¹ X qt bin qt bin = 0:4 GeV; p s = 10:4 GeV = 0:4 GeV; p s = 8 GeV mainly valence region proton pion Many possibilities to explore TMDs in future Drell-Yan experiments

17 Summary: TMDs from SIDIS and DY at low final state transverse momenta T-odd TMDs through Initial/Final state interactions Single-spin asymmetries Drell-Yan self-consistent, clean process, complementary to SIDIS TMDs provide a possibly explanation for Lam-Tung violation Sign switch of T-odd TMDs can be verified experimentally Many possibilities to measure Drell-Yan at existing and future facilities

18 Inclusive Deep-Inelastic Scattering Most knowledge about partonic substructure of hadrons from a basic process DIS l + h! l + X Decomposition into structure functions: d¾ dx B dy! F 1(x B ; Q 2 ); F 2 (x B ; Q 2 ); g 1 (x B ; Q 2 ); g 2 (x B ; Q 2 ) Parton Model in the Bjorken-limit (leading order): Collinear PDFs f 1 (x; ¹); g 1 (x; ¹); h 1 (x; ¹) no transversity in DIS... x B = Q2 2P q y = P q P l X F 1 = 2x B F 2 = 1 2 e 2 qf q 1 X q g 1 = 1 2 e 2 q gq 1 q X g 1 + g 2 = 1 2 e 2 qg q T q Second-basic process with parton distributions h + h! l + ¹ l + X Drell-Yan process

19 Lam-Tung relation (similar to Callan-Gross): F 2 UU = 2F cos 2Á UU Valid in the parton model at leading order O( s ) Only slightly broken at NLO Extension to lower q T through resummation: (most events in this regime) Gluon radiation generates logarithms of the form k s ln m qt 2 at lower q T large logarithms s not a good expansion parameter resum large logarithms b-space [Collins, Soper, Sterman, NPB250,199] Z d¾ d 4 q» F UU 1» d 2 b e i~q T ~ b T (C f A )(1=b)(C f B )(1=b)e S Z Q 2 Sudakov form factor: S(q T ; Q) = qt 2 Q 2 q T -space e.g. [Ellis, Veseli, NPB511,649], [Kulesza, Stirling, JHEP12, 056] FUU 1» d h i (C f A )(q T )(C f B )(q T )e S F (q T ) dq 2 T d¹ 2 ¹ [A( s(¹ 2 )) ln( Q 2 ) + B( 2 ¹ 2 s (¹ 2 ))] Convolution functions C, coefficients A, B perturbatively calculable Non-pert. Sudakov form factor F(q T ) extension to very low q T fit to data Set up for d /d 4 q resummation more difficult for angular dependence F cos2 investigated at least up to LL [Berger, Qiu, Rodriguez-Pedraza, PLB656, 74; PRD76, ] Lam-Tung relation should be unaffected by resummation [Boer, Vogelsang, PRD74, ]

20 unpolarized f 1 Sivers helicity g 1 transversity h 1 Boer Mulders pretzelosity