Relations between GPDs and TMDs (?)
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1 Relations between GPDs and TMDs (?) Marc Schlegel Tuebingen University Ferrara International School Niccolo Cabeo, May 28, 2010
2 Generalizations of collinear Parton Distributions Collinear PDFs f 1 (x; ¹); g 1 (x; ¹); h 1 (x; ¹) Theory well-understood, delivers a one-dimensional picture of nucleon structure Measurable in DIS (but not sensitive to transversity...)? GPDs (off-diagonal): measurable in exclusive processes parameterize amplitudes provides 3-dim. spatial picture Theory well-understood, calc. on lattice Experiments difficult TMDs (small transv. deviations): measurable in semi-inclusive processes parameterize cross sections provides 3-dim. momentum picture Theory more complicated Experiments easier
3 Generalized Parton Distributions (for quarks) Deep-Virtual Compton Scattering (DVCS) + Bethe-Heitler-term: P = 1 2 (p + p0 ) = p 0 p + = 2»P + t = 2 T DVCS» 1 1 Skewtness parameter ξ Bjorken-variable x B GPD(x;»; t) dx x» + i²! <T» P 1 1 GPD(x;»; t) dx x» ; =T» GPD(»;»; t) Hard Exclusive Meson Production:
4 DVCS Interference term I: measured in azimuthal asymmetries I / cos Á[:::] + sin Á[:::] + sin(á Á s )[:::] + ::: Beam charge asym. GPD - matrix element spin asymmetries F ij (x;»; ~ T ) = R dz 2¼ eixp + z hp 0 j ¹q j ( z 2 ) z 2 ; z 2 Eight GPDs most prominent H and E h P + Tr[F + ] = ¹u(p 0 ) + H(x;»; t) + i¾+¹ ¹ unpolarized: 2M Long. pol. <T» P 1 1 Tr[F + 5 ]! ~H; ~E GPD(x;»; t) dx x» ; =T» GPD(»;»; t) Transv. pol. qi ( z 2 )jpi E(x;»; t) i u(p) Tr[F i¾ i+ 5 ]! H T ; E T ; ~H T ; ~E T
5 Properties of GPDs x > ξ (DGLAP): Evolution like a collinear parton distribution x < ξ (ERBL): Evolution like a light-cone wave function Polynomiality: Consequence of Lorentz-invariance + time reversal dx x N 1 H(x;»; t) = h (N) 0 (t) +» 2 h (N) 1 (t) + ::: +» N h (N) (t) dx x N 1 E(x;»; t) = e (N) 0 (t) +» 2 e (N) 1 (t) + ::: +» N e (N) (t) N = 1: F 1 (t) = E no correspondence in the collinear limit. Ji-Angular momentum sum rule J q = 1 2 dxh(x;»; t) F 2 (t) = 1 1 dxe(x;»; t) dx x[h q (x; 0; 0) + E q (x; 0; 0)] N N N even For n odd: ξ n-1 highest power H(x; 0; 0) = f 1 (x)
6 Impact Parameter Space Impact Parameter Space: (ξ=0) Impact parameter b T and transv. momentum transfer Δ T FT F ij (x; ~ b T ) = R d 2 T (2¼) 2 e i~ T ~ b T F ij (x; 0; ~ T ) Impact parameter space diagonal matrix element z 1=2 = z 2 n + b T F ij (x; ~ b T ) = R dz 2(2¼) eixp + z hp + ;~0 T j ¹ Ã j (z 1 ) [z 1 ; z 2 ] Ã i (z 2 )jp + ;~0 T i (ξ=0): probability density of partons in transverse plane.
7 k T Semi-inclusive processes at small transverse final state momenta: -dependent Parton Distributions (TMDs) q T ; P h? Q 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 Eight TMDs: 1 2 Tr[ + ] = f 1 (x; ~ kt 2 ) "ij T ki T Sj T M f 1T? (x;~ kt 2 )
8 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 Time reversal = f 1T DIS? h DY? 1 switches sign: DIS = h? 1 DY
9 TMD observables Semi-inclusive DIS at small transverse momentum: Structure functions: d¾ dxdydzd ~ P 2 h? dádá s / F UU + (2 y) cos ÁF cos Á UU cos(2á) + (1 y) cos(2á)fuu + ::: Leading order parton model structure functions as k T - convolutions: F (x; z; P 2 h? ) / f D d 2 k T d 2 p T ± (2) ( ~ k T ~p T + ~q T ) w( ~ k T ; ~p T )f(x; ~ k 2 T ) D(z; ~p2 T ) Sivers effect: HERMES, COMPASS, JLAB F sin(á Á s) UT / sin(á h Á S )f? 1T D 1 Boer-Mulders effect: F cos(2á) UU / cos(2á)h? 1 H? 1 Also available in Drell-Yan (RHIC, COMPASS, GSI,...)
10 Relations found in models Example: Diquark model Sivers function: GPD E: f 1T? (x; k2 T ) = e qe s g 2 (1 x)m(xm + m q ) 4(2¼) 2 kt 2 (k2 T + ~m2 ) ln kt 2 + ~m2 kt 2 E(x; 0; 2 ) = g2 (1 x) 2 (2¼) 3 d 2 k T M(xM + m q ) [k 2 T + ~m2 ][(k T + (1 x) T ) 2 + ~m 2 ] hkt r i T = d 2 k T k r T ³ ²ij T ki T Sj T M f? 1T (x; k2 T ) = e qe s g 2 4(2¼) 4 ²ij T Sj T (1 x)(xm + m q) = d 2 l ³ T (2¼) 2 e qe s 4 l r T l 2 T ² ij T li T Sj T (1 x)m = M² ij d 2 k T? T Sj T f?(1) 1T k i T k 2 T + ~m2 (x) ³ ³ kd 2 2 ln T l+ ~m 2 T 4¼ (2¼) 2 kt r kt 2 E(x; 0; l2 T (1 x) ) = 2 k 2 T l r T l 2 T [(l T + k T ) 2 + ~m 2 ] d 2 b T ³ eq e s 4¼ (1 x)b r T b 2 T ²ij T bi T Sj T M E0 (x; b 2 T ) Lensing function I: : represents FSI
11 Model relations also for Boer-Mulders function: 2M h?;(1) 1 = d 2 b T ~b T ~I M ET + 2 ~ H T 0 Relations between arbitrary moments: f?;(n) 1T (x) / E (n) (x) ; 0 n 1 TMD: GPD: f (n) (x)» R d 2 k T ( ~ k 2 T )n f(x; ~ k 2 T ) E (n) (x)» R d 2 T (~ 2 T )n 1 E(x; 0; ~ 2 T (1 x) 2 ) Relation between GPDs and T-even TMDs: h?;(n) 1T (x)» H ~ (n) T (x) No FSI / Lensing function needed! valid also in a light-cone constituent quark model Relations for gluon-gpds and gluon-tmds. Relations are likely to be broken for higher order diagrams.
12 Illustrative picture of the Sivers Effect GPD E Situation in SIDIS: Unpolarized nucleon quarks are equally distributed in transverse plane b T Final State interactions are assumed to be attractive here Lensing effect, but no observable net effect!
13 Sivers asymmetry Transversly polarized nucleon spatial distortion of the quark distribution in the transverse plane Impact parameter GPD E spin polarization Spatial distortion + FSI lead to observable net effect non-zero Left-Right (Sivers) asymmetry attractive FSI: correct prediction of the sign of SSA
14 Matrix elements Sivers E Non-trivial relations for T-odd parton distributions: hk i T i T (x) = Average transverse momentum of unpolarized partons in a transversely polarized nucleon: d 2 k T k i T 1 2 h Tr[ + ( ~ S T )] Tr[ + ]( ~ S T ) i / f?;(1) 1T (x) hk i T i(x) = Manipulation of Gauge Links + Impact parameter representation d 2 b T dz 2(2¼) eixp + z hp + ;~0 T ; S T j ¹ Ã(z 1 ) + [z 1 ; z 2 ] I i (z 2 ) Ã(z 2 ) jp + ;~0 T ; S T i z 1=2 = z 2 n + b T Impact parameter representation for GPD E I i (z ) = R dy [z ; y ]gf +i (y )[y ; z ] coll. soft gluon pole matrix element
15 (Semi-) classical approximation: ^I i ' I i (x; ~ b T )1 + ::: factorization of final state interactions and spatial distortion: hk i T i = M² ij T Sj T f?;(1) 1T (x) ' d 2 b T I i (x; ~ b T ) ~ b T ~ S 2 T E(x; ~ b 2 T ) I i (x; ~ b 2 T ) : Lensing Function = net transverse momentum Distortion effect given by flavor dipole moment: d q;i = R dx R d 2 b T b i T ~ bt ~ S T M with flavor an. magn. moment E0 (x; ~ b 2 T ) = ²ij T Sj T 2M u=p ' 1:7 d=p ' 2:0 R dxe q (x; 0; 0) = ²ij T Sj T 2M q f?;(1) 1T (x) / R d 2 b T I(x; ~ b T ) ~ b T S ~ T M E0 (x; ~ b 2 T ) Predicts opposite signs of u- and d- Sivers functions.
16 Classification of Mother Function (GTMDs): Wigner functions Fourier-Transf. GTMDs / k T -dep. GPDs (G)TMDs: (Generalized) Transv. Momentum Depend. R d 2 k T R d 2 k T = 0 Impact Parameter GPDs Fourier-Transf. GPDs? TMDs R dx = 0 R d 2 k T Form factors PDFs
17 Generalized Quark Quark correlator: ~W ij (x; k T ;»; T ) = dz d 2 z T (2¼) 3 e ik z hp 0 j ¹ Ã j ( z 2 )W( z 2 ; z 2 j n)ã i( z 2 ) jpi z+ =0 GTMDs parameterization: Tr ~W 0; + = 1 P + ¹u(p 0 ; 0) h P + M F 1 + i¾i+ kt i M F 1? + i¾i+ i T M F 1 + P + M 4 GTMDs: Amount of structures are doubled, GTMD are complex-valued. i i¾ ij kt i j T M F? 2 1 u(p; ) TMD-limit: = 0 f? 1T (x; ~ k 2 T ) = =[F? 1 ](x; 0; ~ k 2 T ; 0; 0) GPD-limit: R d 2 k T ;» = 0 (E + H)(x; 0; ~ 2 T ) = 2 d 2 k T <[F 1 ] + ~ k T ~ T ~ 2 T <[F? 1 ] No model-independent exact relation! Approximate relations? How approximate?
18 Non-perturbative Lensing function Relativistic Eikonal models: Treat FSI non-perturbatively. Work in the spectator picture (Lens. funct. identifyable) more realistic for a pion, works also for a diquark Only diagrams that reflect the naive picture. I i (x; ~q T ) = Separation between FSI and GPD E via Cutkosky-cut. d 2 p ³ T (2¼) (2p 2 T q T ) i =Mbc ab (j~p T j) (2¼) 2 ± ac ± (2) (~p T ~q T )+<Mda cd (jp T q T j) Scattering amplitude of two highly energetic particles at small momentum transfer: Summation of gluons possible if: 1) Particles are eikonalized (not even an appr. for quark...) 2) Generalized Ladder Approximation 3) IR-finite gluon propagator Dyson-Schwinger approach
19 Testing the relation The remaining piece of the puzzle: GPD E Scalar diquark with a multi-pole diquark vertex factor 2( 1) k 2 m 2 q [k 2 2 ] Need to fit the model parameters: GPD-limits: Form Factors F 1, F 2 t-dependence + valence u(x) (GRV) x-dependence Difficult to fit all limits for scalar diquark only Form Factor fit preferred. Other GPD parameterizations: [Diehl et al., EPJ C39 (2005) 1-39] E u v (x; 0; t) = (N u ux (1 x) )e tgu (x) [Guidal et al., PRD72, ] q(1 Ev q (x; 0; t) = x) q q v (x) R 1 0 dx (1 x) q qv (x) x 0 (1 x)t Extraction by Torino-Group N f (1) (x) = f? 1T (x)
20 Pion Boer-Mulders function Pion relation one chiral-odd GPD + one chiral-odd TMD (Boer-Mulders function) No input from data for chiral-odd GPD Spectator model Fit of parameters to pion Form Factor + PDF Can (in principle) be tested at future COMPASS Drell-Yan measurements... More things to do: Combine eikonal methods and LC wave function formalism better theoretical insights Eikonal methods possibly useful to study the soft factor (made of 2 eikonal lines)
21 Summary: GPDs from exclusive processes 3-dim. spatial picture, encode OAM TMDs from semi-inclusive processes 3-dim. momentum picture GTMD analysis no rigorous, exact relation between GPDs and TMDs Relation Sivers E via separation of FSI + spatial distortion of parton dist. Relation is not rigorous, model-dependent. Holds for lowest order spectator models. Relativistic Eikonal model: Non-perturbative, field-theoretical model of FSI Lensing Function. Relation reproduces the right order of magnitude of the Sivers effect.
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