PARTON OAM: EXPERIMENTAL LEADS

Transcription

1 PARTON OAM: EXPERIMENTAL LEADS 7 TH GHP WORKSHOP FEBRUARY 1-3, 2017 WASHINGTON, DC Simonetta Liuti University of Virginia

2 2/1/17 2 and A. Rajan et al., soon to be posted

3 2/1/17 3 Outline 1. Definitions 2. Lorentz Invariant Relations!OAM is given by a twist three distribution 3. Equations of Motion Relations 4. A probe of QCD at the amplitude level 5. Process dependence of OAM distributions/universality 6. Conclusions

4 2/1/ DEFINITIONS

5 2/1/17 Simonetta Liuti 5 The spin crisis in a cartoon Jaffe Manohar 1 2 = 1 2 ΔΣ + L q + ΔG + L g Ji 1 2 = 1 2 ΔΣ + L q + J g ΔG L g 33% Σ 33% Σ J g L q L q L q L q J g L g + ΔG

6 2/1/17 Simonetta Liuti 6 Angular Momentum Budget Gluon J Gluon L Quark L Gluon Spin Quark Spin 20 0 Ji Jaffe Manohar

7 2/1/17 Simonetta Liuti 7 The Starting Point Jaffe and Manohar s field theoretical description of the quark and gluon orbital angular momentum through its relation to the QCD Energy Momentum Tensor. T µν M µνλ = x ν T µλ x λ T µν Angular Momentum density Energy density (mass) Momentum density T 00 T 01 T 02 T 03 T 10 T 11 T 12 T 13 T 20 T 21 T 22 T 23 T 30 T 31 T 32 T 33 Shear stress Pressure T µν = 1 4 iqψ ( γ µ D ν + γ ν D µ )ψ + Tr F µα F ν α 1 2 gµν F 2

8 Jaffe Manohar: M +12 = ψ t σ 12 ψ + ψ t ΔΣ x ( i ) [ ] 3 ψ + Tr(ε + ij F + j A j ) + 2iTrF + j ( x )A j L q ΔG L g * Ji: M +12 = ψ t σ 12 ψ + ψ t x i D ( ) 3 ψ + x E B ( ) 3 ΔΣ L q J g J q *Chen, Goldman et al., are consistent with this definition (see K.F.Liu et al.) 2/1/17 Simonetta Liuti 8

9 2/1/17 9 The first step towards an observable effect

10 Partonic OAM: Wigner Distributions L U q = Z Z dx d 2 k T Z d 2 b (b k T ) z W U (x, k T, b) 2/1/17 10 Hatta Lorce, Pasquini, Xiong, Yuan k T b L P 2/1/17 Simonetta Liuti 10

11 2/1/17 11 Wigner Distribution U = 1 2 Z d 2 Z T (2 ) 2 ei T b dz d 2 z T e ikz hp, 0 q(0) + U(0,z)q(z) P, i z + =0 11 GTMD 2/1/17 p-p =Δ " Δ T Fourier conjugate: b = transverse position of the quark inside the proton " k T Fourier conjugate: z T = transverse distance traveled by the struck quark between the initial and final scattering

12 2/1/17 12 Which GTMD? The quark-quark correlator for a spin ½ hadron has been parametrized up to twist four in terms of GTMDs, TMDs and GPDs, in a complete way in:

13 F 14 helicity non-flip UL correlation: unpolarized quark density in a longitudinally polarized proton

14 2/1/17 14 G 11 (L S)! 1 2 " W + 5 = 1 0 2M U(p0, 0 i ij T ) ki T apple i(k1 2 k 2 1 ) M 2 G 11 + G Z d 2 b (b k T ) z h q(0) + 5q(z)i # j T M 2 G 11 + i i+ 5 kt i P + G 12 + i i+ 5 i T P + G 13 + i + 5 G 14 U(p, ) apple 1 + i 2 G 13 + i (k 1 2 k 2 1 ) M 2M 2 G 11 + k 1 + i k 2 M G 12, 0 helicity non-flip UL correlation: longitudinally polarized quark density in an unpolarized proton

15 2/1/17 Simonetta Liuti 15 Integral relations Z 1 Z L q = dx 0 d 2 k T k 2 T M 2 F 14 = Z 1 0 dx F (1) 14 q S q = Z 1 0 dx Z d 2 k T k 2 T M 2 G 11 = Z 1 0 dx G (1) 11 Lorce, Pasquini, Xiong, Yuan Hatta, Yoshida Ji, Xiong, Yuan

16 2/1/ LIR

17 2/1/17 17 Lorentz Invariance Relations (LIR) (see D. Pitonyak and A. Rajan s talks) LIR in the off-forward sector: relations between twist-3 GPDs (!PDFs) and k T moments of GTMDs (!TMDs) Based on the most general Lorentz invariant decomposition of the fully unintegrated quark-quark correlator LIRs are a consequence of there being a smaller number of independent unintegrated terms in the decomposition than the number of GTMDs

18 2/1/17 Simonetta Liuti 18 Φ U = d 4 z 2π e i(k z) P', Λ' ψ (0)γ + U (0, n)ψ (z) P, Λ ( ) 4! parametrized in terms of invariant functions A 1, A 2, (Meissner, Metz and Schlegel (2009), Mulders, Tangerman, Pijlman, Bacchetta.)!Φ U = dk Φ U! parametrized in terms of invariant functions F 11, F 12,, F 21, F 22 Specifically, one finds the following relations (A. Rajan s talk) Z k T T F 12 + F 13 =2P + kt T dk tw 3 tw 2 2 T F 14 =2P + Z k T 2 T T dk (A 8 + xa 9 ) F 27 + F 28 =2P + Z dk kt 2 T 2 T T A 5 + A 6 xp 2 k P M 2 (A 8 + xa 9 ) A 5 + A M 2 (kt T ) 2 2 T k 2 T A 9

19 2/2/17 Simonetta Liuti 19 Generalized LIR for straight gauge link (no FSI) 0,0 z -,z T,0,z T Obtained by studying in detail the k T structure of GTMDs and twist 3 GPDs for a straight gauge link (Ji s definition) 0,0,0 d dx Z L q (x) d 2 k T OAM is given by a twist 3 GPD k 2 T M 2 F 14 = e E 2T + H + E k T moment of a GTMD d dx Z d 2 k T k 2 T M 2 G 11 = twist 3 GPD 2 H 2T 0 + E2T 0 + H A. Rajan, A. Courtoy, M. Engelhardt, S.L., PRD (2016), arxiv:

20 2/2/17 20 Integrating in dx one finds the OAM distribution function L q (x) L q F (1) 14 = Z 1 x dy (Ẽ2T + H + E) ) L q = Z 1 0 dxf (1) 14 = Z 1 0 dxxg 2 ~ F 14 and E 2T give us similar information on the distribution in x of OAM! new result In addition: we confirm and corroborate the global/integrated OAM result deducible from Ji et al Different notation! G 2 E! 2T + H + E Polyakov et al. Meissner, Metz and Schlegel, JHEP(2009)

21 2/2/17 21 Generalized LIR for a staple link 0,0 z -,z T,z T,0 d dx Z d 2 k T k 2 T M 2 F 14 = Ẽ2T + H + E + A LIR violating term

22 2/1/ EQUATIONS OF MOTION

23 2/1/17 Simonetta Liuti 23 Equations of Motion (EoM) relation Now insert the EoM in the correlator for a longitudinally polarized proton Z dz d 2 z T (2 ) 3 e ixp + z ik T z T hp 0, 0 ( z/2)( U i! 6D+i6D U) (z/2) p, i z+ =0 =0 We find 2T = Z 1 x dy y (H + E)+ " H x Z 1 x # dy y H 2 + apple 1 x M F 14 Z 1 x dy y 2 M F 14 1 dx... 0 relates to L = = J - S + 0

24 Consistent with OPE based relation Z 1 0 dx xg 2 = 1 2 Z 1 0 Polyakov et al.(2000), Hatta(2012) Z 1 dx x(h + E) dx H J q = L q q A generalized Wandzura Wilczeck relation obtained using OPE for twist 2 and twist 3 operators for the off-forward matrix elements

25 2/1/17 Simonetta Liuti 25 Validation of Ji s Sum Rule: J q = L q + 1 independently measured quantities 2 u-d GPD model O. Gonzalez Hernandez et al., Phys. Rev. C88; arxiv: J u d Experimental info -0.6 d 2 data Lattice: F 14 q L u d Σ/2 GPD model Lattice: GPD moments DVCS data through three A. Rajan et al, PRD (2016) arxiv: HLzênL ê»lz Hh=0L ên Hh=0L» Ê JLAB, Mazous et al. PRL (2007) DVCS + VGG u-d quarks m p = 518 MeV z`=0.39 arxiv: M. Engelhardt h»v» Hlattice unitsl Ê

26 2/1/ A PROBE OF QCD AT THE AMPLITUDE LEVEL

27 2/3/17 27 large effect from lattice (M. Engelhardt, arxiv: ) 0.0 HLzênL ê»lz Hh=0L ên Hh=0L» Ê u-d quarks m p = 518 MeV z`=0.39 F 14 Ê PRD, arxiv: h»v» Hlattice unitsl f 1T perp insight into non-perturbative aspects of QCD associated with dynamical chiral symmetry breaking

28 2/2/17 28 PT transformation Forward case: Sivers function (J. Collins, 2002) PT: hp, S (0) + (z) P,Si = hp, S (0) + (z) P, Si M + f 1T perp = M + -M - = 0 M - z -,z T,z T -,z T z -,z T PT: 0,0,0 -,0 0,0 P, S (0) + U(v, z) (z) P,Si = hp, S (0) + U( v, z) (z) P, Si M +v -M - -v = 0 f 1T perp,sidis = M +v -M -v = -f 1T perp,dy =M + -v -M - -v

29 2/2/17 29 Off forward case: F 14 PT: P, S (0) + U(v, z) (z) P,Si = hp, S (0) + U( v, z) (z) P, Si L + v,δ L - -v,-δ L + v,δ -L - -v,-δ = 0 (k T xδ T ) F 14 SIDIS = L + v, Δ -L - v, Δ = (k T xδ T ) F 14 DY =L + -v, Δ -L v, Δ

30 2/2/17 Simonetta Liuti 30 Genuine/intrinsic twist three term in Equation of Motion relation i 0 = 1 4 Z dz d 2 z T (2 ) 3 hp 0, 0 ( z/2) e ixp + z ik T z T apple (! /@ ig/a)u z/2 + U( /@ + ig/a) z/2 (z/2) p, i z+ =0 xẽ2t = H xẽ2t = H Z Z d 2 kt 2 k T M 2 F staple 14 + M staple F 14 d 2 kt 2 k T M 2 F straight 14 + M straight F 14 A = d dx Mstaple M straight LIR violating term

31 2/1/17 31 Generalized Qiu Sterman term Z d 2 k T kt 2 M 2 F 14 JM Z d 2 k T k 2 T M 2 F Ji 14 = T F (x, x, )

32 2/1/17 32 F 14 (1) with B. Kriesten and A. Rajan, using diquark model Ji Ratio L Ji /L JM =0.72 JM Lattice Ratio L Ji /L JM =0.62±0.16(stat) (extrapolated at ζ= )

33 2/1/17 Simonetta Liuti 33 Low x! L q (x) F 14! G Abha Rajan et al., arxiv: Q 2 = 0.1 GeV 2 genuine u quark tw WW Sum Rule Integrand LC model xg WW total x -1 xg 2 xg2 tw 3 xg2 tw x

34 2/1/ PROCESS DEPENDENCE

35 35 -,z T z -,z T,0 z -,z T,z T -, ,0 0,0,0 HLzênL ê»lz Hh=0L ên Hh=0L» Ê u-d quarks m p = 518 MeV z`=0.39 Ê h»v» Hlattice unitsl

36 2/2/17 36 q k+q q =q+δ k l k-l-δ SIDIS-like P P-k P-k+l P =P-Δ -k+q q k l k-l-δ DrellYan-like P P-k P-k+l P =P-Δ

37 37 Two additional processes: DVCS and TCS twist three contributions spacelike q k+q DVCS -k+q q timelike TCS Extracting twist 3 GPDs from these processes will allow us to zoom into aspects of the sign change

38 Dustin Keller & U.Va. Polarized Target Group

39 Deeply Virtual Exclusive Photoproduction d 5 dx Bj dq 2 d t d d S = (s M 2 ) 2p 1+ 2 T 2, T (k, p, k 0,q 0,p 0 )=T DV CS (k, p, k 0,q 0,p 0 )+T BH (k, p, k 0,q 0,p 0 ),

40 BASIC MODULE (based on helicity amplitudes) 1 1 Q 2 1 X 0, 0 T h 0 DV CS, 0 T h 0 DV CS, 0 = (F F 11 + F F 1 1 )+ (F F 00 ) +2 p (1 + )Re( F + 01 F 01 + F F )+2 Re (F + + F 1 1 hp ) +(2h) 1 2 (F F 11 F F 1 1 ) 2 p io polarized lepton (1 )Re(F F 01 + F F 0 1 )

41 Helicity amplitudes Virtual Photon helicities Λ (1) (2 ) F γ * Λ γ * ΛΛ' ( Λ (1) = f γ * Λ γ ' ΛΛ' )* (2 Λ ) f γ * Λ γ ' ΛΛ' Λ γ ' Initial and final proton helicities

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43 Phase dependence f e i Λ γ * Λ γ ' (Λ Λ') φ The phase is determined entirely by the virtual photon helicity which can be different for the amplitude and its conjugate

44 Twist 3 Bad component (exchanged gluon flips the quark chirali q, Λ γ q, Λ γ g Λ γλ γ λλ k, λ k, λ q, Λ γ =0 g 0Λ γ q, Λ λλ γ k, λ l, Λ g k,λ P, Λ P, Λ A Λ λ,λλ A Λ λ,λλ P, Λ P, Λ

45 The unpolarized cross section: example Twist 2 Twist 4 Twist 3 Photon helicity flip: transverse gluons

46 We connect the tw 3 amps DVCS formalism with the TMD, GPD, GTMD comprehensive parametrization in Meissner Metz and Schlegel, JHEP08 (2009) Example A +,++ = 1 2 Ẽ2T E 2T + Ẽ0 2T + E 0 2T A +,++ = 1 2 Ẽ2T + E 2T + Ẽ0 2T + E 0 2T # # # Orbital angular momentum Spin Orbit interaction

47 DVCS: bilinears of tw 2 and tw 3 CFFs F = P h H (Ẽ2T i Ē 2T +...),... Extraction from experiment using Wandzura Wilczek approximation A.Courtoy, G.Goldstein, O.Gonzalez Hernandez, S.L. and A.Rajan, PLB 731(2014)

48 2/3/17 Our proposal: GTMDs from Double DVCS hadron production (off-forward SIDIS) Simonetta Liuti 48 ep! e 0 + µ + µ p 0 P h t P h -Δ Δ q k k-δ q =q+δ-δ p p-δ t,t,p h2 <Q 2, Q 2 Φ P P t

49 2/1/17 49 Helicity amplitude formalism for DDVCS hadron production " To measure F 14 one has to be in a frame where the reaction cannot be viewed as a two-body quark-proton scattering " In the CoM the amplitudes are imaginary!ul term goes to 0 unless one defines two hadronic planes y photon-proton plane φ Δ p Δ e iϕ Δ k T eiϕ k Δ 1 iδ 2 ( )( k 1 ik 2 ) ik T Δ T z θ e k q h 1 φ k k h 2 two hadrons plane x q

50 2/1/17 Simonetta Liuti 50 Conclusions and Outlook The connection we established through the new relations between (G)TMDs and Twist 3 GPDs, not only allows us to evaluate the angular momentum sum rule, it also opens many interesting avenues: It allows us to study in detail the role of quark-gluon correlations, in a framework where the role of k T and off-shellness, k 2, is manifest. OAM was obtained so far by subtraction (also in lattice). We can now both calculate OAM on the lattice (GTMD) and validate this through measurements (twist 3 GPD) It provides an ideal setting to test renormalization issues, evolution etc QCD studies at the amplitude level shed light on chiral symmetry breaking TWIST THREE GPDs ARE CRUCIAL TO STUDY QCD AT THE AMPLITUDE LEVEL

51 2/3/17 51 Back up

52 2/1/17 Simonetta Liuti 52 Helicity and Transverse Spin Structures of H+E ( A ++,++ + A +,+ + A +, + + A, ) + ( A ++, + + A +, A,+ A +,++ ) ( A X + ++,++ AX + +,+ AX + +, + AX, ) + ( A X + ++,++ AX +,+ AX +, + AX, ) H iδ 2 E = Helicity flips Transv. spin conserved non flip E measures J, not L, but a change of one unit of L (because of the helicity flip) S z = 1/2! 1/2 ) L z =1 at fixed J k, λ k = k Δ, λ P, Λ P = P-Δ, Λ Brodsky and Drell 80s, Belitsky, Ji and Yuan, 90 s