Lattice QCD investigations of quark transverse momentum in hadrons. Michael Engelhardt New Mexico State University

Transcription

1 Lattice QCD investigations of quark transverse momentum in hadrons Michael Engelhardt New Mexico State University In collaboration with: B. Musch, P. Hägler, J. Negele, A. Schäfer J. R. Green, S. Meinel, A. Pochinsky, S. Syritsyn T. Bhattacharya, R. Gupta, B. Yoon S. Liuti, A. Rajan

2 Fundamental TMD correlator Φ [Γ] unsubtr.(b,p,s,...) P,S q(0) Γ U[0,...,b] q(b) P,S 2 Φ [Γ] (x,k T,P,S,...) d 2 b T d(b P) (2π) 2 (2π)P + exp (ix(b P) ib T k T ) Φ [Γ] unsubtr.(b,p,s,...) S(b 2,...) b + =0 Soft factor S required to subtract divergences of Wilson line U S is typically a combination of vacuum expectation values of Wilson line structures Here, will consider only ratios in which soft factors cancel

3 Relation to physical processes Context: All this is largely academic if we can t connect it to a physical measurement. Not least, this should inform choice of gauge link U[0,...,b]... Factorization theorem which allows one to separate cross section into hard amplitude, fragmentation function, TMD? For example, SIDIS: l + N(P) l + h(p h ) + X Note final state effects in SIDIS

4 Relation to physical processes In general, no factorization framework with well-defined TMDs exists (e.g., processes with multiple hadrons in both initial and final state)! SIDIS and DY: Factorization framework has been given, which in particular includes: Specific form of the gauge link U[0,b] Accounts for final state interactions Further regularization required! Staple-shaped gauge link U[0, ηv, ηv + b, b]

5 Gauge link structure motivated by SIDIS Beyond tree level: Rapidity divergences suggest taking staple direction slightly off the light cone. Approach of Aybat, Collins, Qiu, Rogers makes v space-like. Parametrize in terms of Collins-Soper parameter ˆζ P v P v Light-like staple for ˆζ. Perturbative evolution equations for large ˆζ. Modified universality, f T-odd, SIDIS = ft-odd, DY

6 Fundamental TMD correlator Φ [Γ] unsubtr.(b,p,s,...) P,S q(0) Γ U[0,ηv,ηv + b,b] q(b) P,S 2 Φ [Γ] (x,k T,P,S,...) d 2 b T (2π) 2 d(b P) (2π)P + exp (ix(b P) ib T k T ) Φ [Γ] unsubtr.(b,p,s,...) S(b 2,...) b + =0 Soft factor S required to subtract divergences of Wilson line U S is typically a combination of vacuum expectation values of Wilson line structures Here, will consider only ratios in which soft factors cancel

7 Decomposition of Φ into TMDs All leading twist structures: Φ [γ+] = f ǫ ij k i S j m H f T odd Φ [γ+ γ 5] = Λg + k T S T m H g T Φ [iσi+ γ 5] = S i h + (2k ik j k 2 T δ ij)s j 2m 2 H h T + Λk i m H h L + ǫ ij k j m H h odd

8 TMD Classification All leading twist structures: q H U L T U f h Boer-Mulders (T-odd) L g h L T f T g T h h T Sivers (T-odd)

9 Decomposition of Φ into amplitudes Φ [Γ] unsubtr.(b,p,s, ˆζ,µ) P,S q(0) Γ U[0,ηv,ηv + b,b] q(b) P,S 2 Decompose in terms of invariant amplitudes; at leading twist, 2P + Φ [γ+ ] unsubtr. = A 2B + im H ǫ ij b i S j A 2B 2P + Φ [γ+ γ 5 ] unsubtr. = ΛA 6B + i[(b P)Λ m H (b T S T )] 2P + Φ [iσi+ γ 5 ] unsubtr. = im H ǫ ij b j A 4B S i A 9B A 7B im H Λb i A 0B + m H [(b P)Λ m H (b T S T )]b i A B (Decompositions analogous to work by Metz et al. in momentum space)

10 Fourier-transformed TMDs f(x,b 2 T,...) d 2 k T exp(ib T k T )f(x,k 2 T,...) In limit b T 0, recover k T -moments: f (n) (x, 0,...) f (n) (x,b 2 T,...) n! 2 m 2 b 2 H T n f(x,b 2 T,...) d 2 k T k 2 T 2m 2 H n f(x,k 2 T,...) f (n) (x) ill-defined for large k T, so will not attempt to extrapolate to b T = 0, but give results at finite b T. In this study, only consider first x-moments (accessible at b P = 0), rather than scanning range of b P: f [] (k 2 T,...) dxf(x,k 2 T,...) Bessel-weighted asymmetries (Boer, Gamberg, Musch, Prokudin, JHEP 0 (20) 02)

11 Relation between Fourier-transformed TMDs and invariant amplitudes A i Invariant amplitudes directly give selected x-integrated TMDs in Fourier (b T ) space (showing just the ones relevant for Sivers, Boer-Mulders shifts), up to soft factors: f [](0) (b 2 T, ˆζ,...,ηv P) = 2 A 2B ( b 2 T, 0, ˆζ,ηv P)/ S(b 2,...) f []() T (b 2 T, ˆζ,...,ηv P) = 2A 2B ( b 2 T, 0, ˆζ,ηv P)/ S(b 2,...) h []() (b 2 T, ˆζ,...,ηv P) = 2 A 4B ( b 2 T, 0, ˆζ,ηv P)/ S(b 2,...)

12 Generalized shifts Form ratios in which soft factors, (Γ-independent) multiplicative renormalization factors cancel Boer-Mulders shift: k y UT m h []() H f [](0) = dx d2 k T k y Φ [γ+ +s j iσ j+ γ 5] (x,k T,P,...) dx d 2 k T Φ [γ+ +s j iσ j+ γ 5] (x,k T,P,...) s T =(,0) Average transverse momentum of quarks polarized in the orthogonal transverse ( T ) direction in an unpolarized ( U ) hadron; normalized to the number of valence quarks. Dipole moment in b 2 T = 0 limit, shift. Issue: k T -moments in this ratio singular; generalize to ratio of Fourier-transformed TMDs at nonzero b 2 T, k y UT (b 2 T,...) m H h []() (b 2 T,...) f [](0) (b 2 T,...) (remember singular b T 0 limit corresponds to taking k T -moment). Generalized shift.

13 Generalized shifts from amplitudes Now, can also express this in terms of invariant amplitudes: k y UT (b 2 T,...) m H h []() (b 2 T,...) f [](0) (b 2 T,...) = m H Analogously, Sivers shift (in a polarized hadron): A 4B ( b 2 T, 0, ˆζ,ηv P) A 2B ( b 2 T, 0, ˆζ,ηv P) k y TU (b 2 A T,...) = m 2B ( b 2 T, 0, ˆζ,ηv P) H A 2B ( b 2 T, 0, ˆζ,ηv P) Worm-gear (g T ) shift: k x TL (b 2 A T,...) = m 7B ( b 2 T, 0, ˆζ,ηv P) N A 2B ( b 2 T, 0, ˆζ,ηv P) Generalized tensor charge (no k-weighting) : h [](0) f [](0) A = 9B ( b 2 T, 0, ˆζ,ηv P) (m 2 N b2 /2) A B ( b 2 T, 0, ˆζ,ηv P) A 2B ( b 2 T, 0, ˆζ,ηv P)

14 τ Lattice setup Evaluate directly Φ [Γ] unsubtr.(b,p,s, ˆζ,µ) 2 P,S q(0) Γ U[0,ηv,ηv + b,b] q(b) P,S Euclidean time: Place entire operator at one time slice, i.e., b, ηv purely spatial Since generic b, v space-like, no obstacle to boosting system to such a frame! Parametrization of correlator in terms of A i invariants permits direct translation of results back to original frame; form desired A i ratios. Use variety of P, b, ηv; here b P, b v (lowest x-moment, kinematical choices/constraints) Extrapolate η, ˆζ numerically.

15 Initial Numerical Investigation Use three MILC 2+-flavor gauge ensembles with a 0.2 fm: m π = 369 MeV ; ; 284 samples m π = 369 MeV ; ; 5264 samples m π = 58 MeV ; ; 3888 samples Sink momenta P: (0, 0, 0), (, 0, 0), ( 2, 0, 0), (,, 0) Variety of b, ηv; note b P, b v (lowest x-moment, kinematical choices/constraints) Largest ˆζ = 0.78

16 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T Ζ 0.39, DY SiversShift, udquarks b T 0.2 fm, m Π 58 MeV SIDIS Ηvlattice units

17 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T Ζ 0.39, DY SiversShift, udquarks b T 0.24 fm, m Π 58 MeV SIDIS Ηvlattice units

18 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T Ζ 0.39, DY SiversShift, udquarks b T 0.36 fm, m Π 58 MeV SIDIS Ηvlattice units

19 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T Ζ 0.39, DY SiversShift, udquarks b T 0.47 fm, m Π 58 MeV SIDIS Ηvlattice units

20 Results: Sivers shift Dependence of SIDIS limit on b T GeV f 0 mn f T Sivers ShiftSIDIS, udquarks Ζ 0.39, m Π 58 MeV b T fm

21 Dependence on staple extent; flavor separated Results: Sivers shift GeV f 0 m N f T Sivers Shift DY preliminary downquarks, Ζ 0.38, l 0.36 fm SIDIS GeV f 0 m N f T Sivers Shift DY preliminary upquarks, Ζ 0.38, l 0.36 fm SIDIS staple extentηvlattice units staple extentηvlattice units

22 Results: Sivers shift Dependence on staple extent; sequence of panels at different ˆζ GeV f 0 mn f T Sivers ShiftSIDIS, udquarks Ζ 0, b T 0.36 fm, m Π 58 MeV Ηvlattice units

23 Results: Sivers shift Dependence on staple extent; sequence of panels at different ˆζ GeV f 0 mn f T Ζ 0.39, DY SiversShift, udquarks b T 0.36 fm, m Π 58 MeV SIDIS Ηvlattice units

24 Results: Sivers shift Dependence on staple extent; sequence of panels at different ˆζ GeV f 0 mn f T Ζ 0.78, DY SiversShift, udquarks b T 0.36 fm, m Π 58 MeV SIDIS Ηvlattice units

25 Results: Sivers shift Dependence of SIDIS limit on ˆζ GeV f 0 m N f T total contrib. A 2 b T 0.36 fm m Π 58 MeV Sivers ShiftSIDIS, udquarks Ζ

26 Dependence of SIDIS limit on ˆζ, all three ensembles Results: Sivers shift GeV f 0 mn f T m Π 58MeV 20 3 m Π 369MeV 20 3 m Π 369MeV 28 3 b T 0.36 fm Sivers ShiftSIDIS, udquarks Ζ

27 Results: Boer-Mulders shift Dependence on staple extent GeV f 0 mn h DY BoerMulders Shift, udquarks Ζ 0.39, b T 0.36 fm, m Π 58 MeV SIDIS Ηvlattice units

28 Results: Boer-Mulders shift Dependence of SIDIS limit on b T GeV f 0 mn h BoerMulders ShiftSIDIS, udquarks Ζ 0.39, m Π 58 MeV b T fm

29 Dependence on staple extent; flavor separated Results: Boer-Mulders shift 0.4 BoerMulders Shift total 0.4 BoerMulders Shift total GeV 0.2 contrib. from a 4 GeV 0.2 contrib. from a 4 f 0 f 0 m N h downquarks, Ζ 0.39, b 0.36 fm DY connected only SIDIS m N h upquarks, Ζ 0.39, b 0.36 fm DY connected only SIDIS staple extentηvlattice units staple extentηvlattice units

30 Results: Boer-Mulders shift Dependence of SIDIS limit on ˆζ GeV 0. total contrib. A 4 f 0 m N h b T 0.36 fm m Π 58 MeV BoerMulders ShiftSIDIS, udquarks Ζ

31 Dependence of SIDIS limit on ˆζ, all three ensembles Results: Boer-Mulders shift GeV f m Π 58MeV 20 3 m Π 369MeV 20 3 m Π 369MeV 28 3 mn h b T 0.36 fm 0.2 BoerMulders ShiftSIDIS, udquarks Ζ

32 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, udquarks f 0 0 h DY Ζ 0.39, b T 0.2 fm, m Π 58 MeV SIDIS Ηv lattice units

33 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, udquarks f 0 0 h DY Ζ 0.39, b T 0.24 fm, m Π 58 MeV SIDIS Ηv lattice units

34 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, udquarks f 0 0 h DY Ζ 0.39, b T 0.36 fm, m Π 58 MeV SIDIS Ηv lattice units

35 Results: Transversity Dependence of SIDIS/DY limit on b T.5 h, udquarks f 0 h Ζ 0.39, m Π 58 MeV b T fm

36 Results: Transversity Dependence of SIDIS/DY limit on ˆζ f 0 h total contrib. A 9 m b T 0.36 fm m Π 58 MeV h, udquarks Ζ

37 Results: Transversity Dependence of SIDIS/DY limit on ˆζ, all three ensembles f 0 h m Π 58MeV 20 3 m Π 369MeV 20 3 m Π 369MeV 28 3 b T 0.36 fm h, udquarks Ζ

38 Results: g T worm gear shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn g T g T Shift, udquarks Ζ 0.39, b T 0.2 fm, m Π 58 MeV DY SIDIS Ηv lattice units

39 Results: g T worm gear shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn g T g T Shift, udquarks Ζ 0.39, b T 0.36 fm, m Π 58 MeV DY SIDIS Ηv lattice units

40 Dependence of SIDIS/DY limit on b T Results: g T worm gear shift GeV f 0 mn g T g T Shift, udquarks Ζ 0.39, m Π 58 MeV b T fm

41 Results: g T worm gear shift Dependence of SIDIS/DY limit on ˆζ 0.30 GeV f 0 mn g T 0.25 total contrib. A b T 0.36 fm m Π 58 MeV 5 g T Shift, udquarks Ζ

42 Results: g T worm gear shift Dependence of SIDIS/DY limit on ˆζ, all three ensembles GeV f m Π 58MeV 20 3 m Π 369MeV 20 3 m Π 369MeV 28 3 mn g T b T 0.36 fm 5 g T Shift, udquarks Ζ

43 Challenges The limit ˆζ : Approaching the light cone Discretization effects, soft factor cancellation on the lattice in TMD ratios Progress toward the physical pion mass

44 Approaching the light cone (with a pion)

45 Results: Boer-Mulders shift (pion) Dependence on staple extent; sequence of panels at different b T

46 Results: Boer-Mulders shift (pion) Dependence on staple extent; sequence of panels at different b T

47 Results: Boer-Mulders shift (pion) Dependence on staple extent; sequence of panels at different b T

48 Results: Boer-Mulders shift (pion) Dependence on staple extent; sequence of panels at different b T

49 Dependence of SIDIS limit on b T Results: Boer-Mulders shift (pion)

50 Results: Boer-Mulders shift (pion) Dependence of SIDIS limit on ˆζ; open symbols: contribution A 4 only

51 Results: Boer-Mulders shift (pion) Dependence of SIDIS limit on ˆζ; open symbols: contribution A 4 only

52 Dependence of SIDIS limit on ˆζ; fit function a + b/ˆζ GeV f 0 mπ h Results: Boer-Mulders shift (pion) uquarks BoerMuldersShiftSIDIS Ζ m Π 58MeV 0.36 fm

53 Dependence of SIDIS limit on ˆζ; fit function a + b/ˆζ 0 Results: Boer-Mulders shift (pion) GeV f 0 mπ h uquarks BoerMuldersShiftSIDIS m Π 58MeV b T 0.34 fm Ζ

54 Discretization effects: Comparison of RBC/UKQCD DWF ensemble (m π = 297MeV, a = 84fm) with clover ensemble (m π = 37MeV, a = 0.4fm) produced by K. Orginos and JLab collaborators

55 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.4, b T 0.2 fm, m Π 297 MeV SIDIS GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.32, b T 0. fm, m Π 37 MeV SIDIS Ηvlattice units Ηvlattice units

56 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.4, b T 0.25 fm, m Π 297 MeV SIDIS GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.32, b T 0.23 fm, m Π 37 MeV SIDIS Ηvlattice units Ηvlattice units

57 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.4, b T 0.34 fm, m Π 297 MeV SIDIS GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.32, b T 0.34 fm, m Π 37 MeV SIDIS Ηvlattice units Ηvlattice units

58 Dependence of SIDIS limit on b T Results: Sivers shift

59 Dependence of SIDIS limit on ˆζ Results: Sivers shift

60 Dependence of SIDIS limit on b T Results: Boer-Mulders shift

61 Dependence of SIDIS limit on ˆζ Results: Boer-Mulders shift

62 Dependence of SIDIS limit on b T Results: Generalized Transversity

63 Dependence of SIDIS limit on ˆζ Results: Generalized Transversity

64 Dependence of SIDIS limit on b T Results: g T worm gear shift

65 Dependence of SIDIS limit on ˆζ Results: g T worm gear shift

66 Dependence on b T Results: Generalized Transversity, straight link

67 Results: g T worm gear shift, straight link Dependence on b T Evidence of operator mixing? Lattice perturbation theory M. Constantinou et al.

68 Dependence on the pion mass

69 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.27, b T 0.4 fm, m Π 70 MeV SIDIS Ηvlattice units GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.4, b T 0.2 fm, m Π 297 MeV SIDIS Ηvlattice units

70 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.27, b T 0.2 fm, m Π 70 MeV SIDIS Ηvlattice units GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.4, b T 0.7 fm, m Π 297 MeV SIDIS Ηvlattice units

71 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.27, b T 0.29 fm, m Π 70 MeV SIDIS Ηvlattice units GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.4, b T 0.25 fm, m Π 297 MeV SIDIS Ηvlattice units

72 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.27, b T 0.4 fm, m Π 70 MeV SIDIS Ηvlattice units GeV f 0 mn f T DY SiversShift, udquarks Ζ 0.4, b T 0.42 fm, m Π 297 MeV SIDIS Ηvlattice units

73 Results: Sivers shift Dependence of SIDIS limit on b T GeV 0 5 Sivers ShiftSIDIS, udquarks GeV 0 5 Sivers ShiftSIDIS, udquarks f f mn f T Ζ 0.27, m Π 70 MeV mn f T Ζ 0.29, m Π 297 MeV b T fm b T fm

74 Results: Sivers shift Dependence of SIDIS limit on ˆζ GeV 0. total contrib. A 2 GeV 0. total contrib. A 2 f 0 m N f T b T 0.29 fm m Π 70 MeV f 0 m N f T b T 0.25 fm m Π 297 MeV 0.5 Sivers ShiftSIDIS, udquarks 0.5 Sivers ShiftSIDIS, udquarks Ζ Ζ

75 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, udquarks.4 h, udquarks.2.2 f f h DY Ζ 0.27, b T 0.4 fm, m Π 70 MeV SIDIS h DY Ζ 0.4, b T 0.2 fm, m Π 297 MeV SIDIS Ηvlattice units Ηvlattice units

76 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, udquarks.4 h, udquarks.2.2 f f h DY Ζ 0.27, b T 0.2 fm, m Π 70 MeV SIDIS h DY Ζ 0.4, b T 0.7 fm, m Π 297 MeV SIDIS Ηvlattice units Ηvlattice units

77 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, udquarks.4 h, udquarks.2.2 f f h DY Ζ 0.27, b T 0.29 fm, m Π 70 MeV SIDIS h DY Ζ 0.4, b T 0.25 fm, m Π 297 MeV SIDIS Ηvlattice units Ηvlattice units

78 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, udquarks.4 h, udquarks.2.2 f f h DY Ζ 0.27, b T 0.4 fm, m Π 70 MeV SIDIS h DY Ζ 0.4, b T 0.42 fm, m Π 297 MeV SIDIS Ηvlattice units Ηvlattice units

79 Results: Transversity Dependence of SIDIS/DY limit on b T.5 h SIDIS, udquarks.5 h SIDIS, udquarks f 0.0 f 0.0 h Ζ 0.27, m Π 70 MeV h Ζ 0.29, m Π 297 MeV b T fm b T fm

80 Results: Transversity Dependence of SIDIS/DY limit on ˆζ.5 total.5 total f 0 h contrib. A 9 m b T 0.29 fm m Π 70 MeV f 0 h contrib. A 9 m b T 0.25 fm m Π 297 MeV h SIDIS, udquarks Ζ h SIDIS, udquarks Ζ

81 Results: Sivers shift summary Dependence of SIDIS limit on ˆζ Experimental value from global fit to HERMES, COMPASS and JLab data, M. Echevarria, A. Idilbi, Z.-B. Kang and I. Vitev, Phys. Rev. D 89 (204) 07403

82 2 = 2 2 = 2 q q Proton spin decompositions q + L q + J g (Ji) q q + L q + g + L g (Jaffe-Manohar) q... and many more (in fact, we will see a continuous interpolation between the two... ) There isn t one unique way of separating quark and gluon orbital angular momentum the different decompositions have different, legitimate meanings.

83 Interpreting terms in the energy-momentum tensor: Quark orbital angular momentum L q iψ ( r D) z ψ Can be obtained from L q = J q S q, where S q and J q can be related to GPDs (Ji sum rule) this has been used in Lattice QCD. Hitherto not accessed in Lattice QCD. L q iψ ( r ) z ψ in light cone gauge

84 Quark Orbital Angular Momentum L U 3 = dx d 2 k T d 2 r T (r T k T ) 3 W U (x,k T,r T ) Wigner distribution = dx d 2 k T k 2 T m 2 F 4(x,k 2 T,k T T, 2 T) T = 0 Generalized transverse momentum-dependent parton distribution (GTMD) = ǫ ij z T,i p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s T,j z + =z =0, T =0, z T 0 Y. Hatta, X. Ji, M. Burkardt: Staple-shaped U[ z/2, z/2] Jaffe-Manohar OAM Straight U[ z/2, z/2] Ji OAM Connection to GTMDs A. Metz, M. Schlegel, C. Lorcé, B. Pasquini...

85 Direct evaluation of quark orbital angular momentum L U 3 = dx d 2 k T d 2 r T (r T k T ) 3 W U (x,k T,r T ) Wigner distribution L U 3 n = ǫ ij z T,i p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s T,j z + =z =0, T =0, z T 0 p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s z + =z =0, T =0, z T 0 n : Number of valence quarks p = P + T /2, p = P T /2, P,S in 3-direction, P This is the same type of operator as used in TMD studies generalization to off-forward matrix element adds transverse position information

86 L U 3 n = ǫ ij z T,i Direct evaluation of quark orbital angular momentum p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s T,j z + =z =0, T =0, z T 0 p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s z + =z =0, T =0, z T 0 Role of the gauge link U: Y. Hatta, M. Burkardt: Straight U[ z/2, z/2] Ji OAM Staple-shaped U[ z/2, z/2] Jaffe-Manohar OAM Difference is torque accumulated due to final state interaction z ηv + z 2 2 η z 2 ηv z 2 v

87 L U 3 n = ǫ ij z T,i Direct evaluation of quark orbital angular momentum p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s T,j z + =z =0, T =0, z T 0 p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s z + =z =0, T =0, z T 0 Role of the gauge link U: Direction of staple taken off light cone (rapidity divergences) Characterized by Collins-Soper parameter ˆζ P v P v Are interested in ˆζ ; synonymous with P in the frame of the lattice calculation (v = e 3 ) z ηv + z 2 2 η z 2 ηv z 2 v

88 L U 3 n = ǫ ij z T,i Direct evaluation of quark orbital angular momentum p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s T,j z + =z =0, T =0, z T 0 p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s z + =z =0, T =0, z T 0 Parameters to consider:, ˆζ, z, η z ηv + z 2 2 η v z 2 ηv z 2

89 L U 3 n = ǫ ij z T,i Direct evaluation of quark orbital angular momentum p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s T,j z + =z =0, T =0, z T 0 p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s z + =z =0, T =0, z T 0 Dataset contains only one value of T = 4π/aL GeV Substantial underestimate of f/ T by using f T,j T,j =0 = 2 T,j (f( T,j ) f( T,j ))

90 Direct evaluation of quark orbital angular momentum f z T,i z T,i =0 = 2da (f(dae i) f( dae i ))

91 L U 3 n = ǫ ij z T,i Direct evaluation of quark orbital angular momentum p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s T,j z + =z =0, T =0, z T 0 p,s ψ( z/2)γ + U[ z/2,z/2]ψ(z/2) p,s z + =z =0, T =0, z T 0 Remaining parameters to consider: ˆζ, η z ηv + z 2 2 η v z 2 ηv z 2

92 Ji quark orbital angular momentum: η = udquarks m Π 58 MeV L 3 Η0 n Η Ζ Signature of underestimate of f/ T

93 From Ji to Jaffe-Manohar quark orbital angular momentum

94 From Ji to Jaffe-Manohar quark orbital angular momentum

95 From Ji to Jaffe-Manohar quark orbital angular momentum

96 Burkardt s torque extrapolation in ˆζ Τ3L 3 Η0 n Η udquarks m Π 58 MeV Ζ τ 3 = (L (η= ) 3 /n (η= ) ) (L (η=0) 3 /n (η=0) ) Integrated torque accumulated by struck quark leaving proton

97 Flavor separation from Ji to Jaffe-Manohar quark orbital angular momentum

98 Conclusions and Outlook Continued exploration of TMDs using bilocal quark operators with staple-shaped gauge link structures. Soft factors, multiplicative renormalizations are canceled by constructing appropriate ratios of Fourier-transformed TMDs / GTMDs. Exploration of challenges posed by ˆζ limit, discretization effects, physical pion mass limit. Generalization to mixed transverse momentum / transverse position observables (Wigner functions / GT- MDs) gives direct access to quark orbital angular momentum and related observables such as quark spin-orbit coupling. A first comparison with experiment (Sivers shift) is encouraging. Current efforts concentrate on approaching the physical pion mass, improving the treatment of momentum transfer in GTMDs, and exploring further new TMD/GTMD observables (longitudinal polarization, twist-3 GTMDs).