Transverse Momentum Parton Distributions and Gauge Links. 31 May 2010 The College of William and Mary. Leonard Gamberg Penn State University

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1 Transverse Momentum Parton Distributions and Gauge Links 31 May 2010 The College of William and Mary Leonard Gamberg Penn State University

2 Outline Transverse spin Effects in TSSAs Gauge links-color Gauge Inv.- T-odd TMDs T-odd PDFs via FSIs & Transverse distortion QCD calc FSIs Gauge Links-Color Gauge Inv. T-odd TMDs Pheno -Transverse Structure TMDs and TSSAs-b and k asymm An improved dynamical approach for FSIs & model building f 1T (x, k 2 ) E(x, b 2 ) spin pol

3 Transverse SPIN Observables SSA (TSSA) P P πx Single Spin Asymmetry Rotational invariance σ (x F, p ) = σ (x F, p ) Left-Right Asymmetry P S T Parity Conserving interactions: SSAs Transverse Scattering plane σ is T (P P π ) L P R A N = σ (x F,p ) σ (x F, p ) σ (x F,p )+σ (x F, p ) σ P S T π P P π π P π

4 Reaction Mechanism Co-linear factorized QCD-parton dynamics σ pp πx f a f b ˆσ D q π Requires helicity flip-hard part ˆσ ˆσ ˆσ TSSA requires relative phase btwn different helicity amps ± ˆ â N = ˆσ ˆσ ˆσ +ˆσ Im ( M + M ) M M 2 f D M M / =( + ± i ) f

5 Factorization Theorem in QCD Helicity limit...triviality m + x QCD interactions conserve helicity m q 0 and Born amplitudes real A N m qα s E Kane, Repko, PRL:1978 Twist three and trival?! Not the full Twist 3 approach ETQS approach Phases in soft poles of propagator in hard subprocess Efremov & Teryaev :PLB 1982 Qiu-Sterman:PLB 1991, 1999, Koike et al. PLB , Ji,Qiu,Vogelsang,Yuan:PR 2006,

6 P S T Large Transverse SSA s at s = 62.4 & 200 GeV at RHIC L P R STAR PRL101, (2008) BRAHMS Preliminary GeV 62.4 GeV 0.1 A N (!) 0! + -! E704 E <p (!)<0.8 GeV/c T

7 lp l πx Hermes PRL 2009

8 Collins Asymmetry Compass-proton data 2007 comparison w/ HERMES-Collins D. Hasch INT-12 GeV

9 TSSAs thru T-odd non-pertb. spin-orbit correlations... Sensitivity to p T k T << Q 2 Sivers PRD: 1990 TSSA is associated w/ correlation transverse spin and momenta in initial state hadron P k x S T P k S T σ pp πx D f f ˆσ Born f (x, k ) = is T (P k ) f 1T (x, k ) Collins NPB: 1993 TSSA is associated with transverse spin of fragmenting quark and transverse momentum of final state hadron k P π k x s T k P π k s T σ ep eπx D f ˆσ Born D (x, p H1 ) = is T (P p ) (x, p )

10 Reaction Mechanism-FSI phases in TSSAs at unsupressed Brodsky, Hwang, Schmidt PLB: 2002 SIDIS w/ transverse polarized nucleon target e p eπx Collins PLB Gauge link Sivers function doesn t vanish Ji, Yuan PLB: Sivers fnct. FSI emerge from Color Gauge-links LG, Goldstein, Oganessyan 2002, 2003 PRD Boer-Mulders Fnct, and Sivers -spectator model LG, M. Schlegel, PLB 2010 Boer-Mulders Fnct, and Sivers beyond summing the FSIs through the gauge link

11 Factorization & Sensitivity to P T k TMDs John Collins Nuclear Physics B396 (1993) FACTORIZATION WITH INTRINSIC TRANSVERSE MOMENTUM AND POLARIZATION Ralston Spoper NPB 1979, Collins NPB 1993 Fig. 2. Parton model for semi-inclusive deeply inelastic scattering. lation [17,28]. A similar theorem should apply here. An obvious ansatz is E EBd3I,d3 = Efd~f~fd2ka± fd2kbl fa/a(~ k~1) Collins Soper NPB 1981, & Sterman NPB 1985 d6~ xe Ekh d 31 d~kb~ ~ khl) + Y(xB~,Q, z, q 1/Q). scattering function. The function fa/a defined earlier gives the intrinsic transverse-momentum dependence of partons in the initial-state hadron. Similarly, DB/a gives the distribution of hadrons in a parton, with kbl being the transverse momentum of the parton relative to the hadron.

12 Factorization parton model when PT of the hadron small W µν (q, P, S, P h ) a e 2 d 2 p T dp dp + (2π) 4 d 2 k T dk dk + (2π) 4 δ(p + x B P + )δ(k P h z )δ2 (p T + q T k T ) Tr [Φ(p, P, S)γ µ (k, P h )γ ν ] W µν (q, P, S, P h )= d 2 p T (2π) 4 d 2 k T (2π) 4 δ2 (p T P h z [( k T )Tr ) ( dp Φ γ µ ) ] dk + γ ν Integrate out small longitudinal momenta components Φ(x, p T,S) dp Φ(p, P, S), p+ =x B P + (z,k T ) dk + (k, P h ) P h Small transverse momentum k = P z h Integration support for integrals is where transverse momentum is small- cov parton model e.g. Landshoff Polkinghorne NPB28, 1971 (γ, ɛ) q (k, µ) P X (k, µ ) (p, λ) (p, λ ) (P, Λ) P X Φ (P, Λ )

13 What about FSIs and TSSAs? Extend Parton Model result-gauge Links What are the leading order gluons that implement color gauge invariance? How is the correlator modified? ν q µ k P h H ρ,ν = γ ν H ρν;a (k) H µ p p 1 p 1 p P Φ aρ A (p,p 1)

14 T-Odd Effects From Color Gauge Inv. Via Gauge links Gauge link determined re-summing gluon interactions btwn soft and hard Efremov,Radyushkin Theor. Math. Phys Belitsky, Ji, Yuan NPB 2003, Boer, Bomhof, Mulders Pijlman, et al NPB, PLB, PRD Φ [U[C]] (x, p T ) = ν q µ H ρν;a k P h (k) H µ dξ d 2 ξ T 2(2π) 3 eip ξ P ψ(0)u [C] [0,ξ] ψ(ξ, ξ T ) P ξ + =0 P p p 1 p 1 Φ aρ A (p,p 1) p P R Summing gauge link with color LG, M. Schlegel PLB 2010 The path [C] is fixed by hard subprocess within hadronic process. d 4 pd 4 kδ 4 (p + q k)tr [ ] Φ [U C [ ;ξ] (p)h µ (p, k) (k)h ν (p, k) ξ T U [+] ξ

15 Wilson Line = Gauge links... Path ordered Eikonal U[z s 1,z 2 ] = W[z 1; z 2 ] = [z 1 ; z 2 ]=Pe ig R z 2 z 1 ds A(s) z T z 2 = (0, z 2, z T 2 ) (0, z 2, z T 2 ) W TMD z z 1 = (0, z 2, z T 2 ) (0, +, z T 2 ) Process Dependence, Collins plb 02, Brodsky et al. NPB 02, Boer Mulders Pijlman Bomhoff 03, SIDIS ξ T ξ P&T ξ T ξ DY Φ [+] (x, p T ) = iγ 1 γ 3 Φ [ ] (x, p T )iγ 1 γ 3

16 Ji, Ma, Yuan: PLB, PRD 2004, 2005 Extend factorization of CS-NPB: 81 Also see Bacchetta Boer Diehl Mulders JHEP 08 F UU,T (x, z, Ph,Q 2 2 )=C [ ] f 1 D 1 = d 2 p T d 2 k T d 2 l T δ (2)( p T k T + l T P h /z ) x a e 2 a f a 1 (x, p 2 T,µ 2 ) D a 1(z, k 2 T,µ 2 ) U(l 2 T,µ 2 )H(Q 2,µ 2 ) TMD PDF TMD FF Soft factor Hard part Collins, Soper, NPB 193 (81) Ji, Ma, Yuan, PRD 71 (05)

17 Leading Twist TMDs from Correlator is Matrix in Dirac space { [ { { [ ( ) ] Φ [γ+] (x, p T ) f 1 (x, p 2 T) + ɛij T p Ti S Tj { M f 1T (x, p 2 T) Φ [γ+ γ 5 ] (x, p T ) λ g 1L (x, p 2 T) + p T S T M g 1T (x, p 2 T) { [( ( Φ [iσi+ γ 5 ] (x, p T ) S i T h 1T (x, p 2 T) + pi T M } λ h 1L(x, p 2 T) + p T S T M h 1T (x, p 2 T) ) ( + ɛij T p j T M h 1 (x, p 2 T) { Avakian Mulders-tableau

18 TSSAs in SIDIS P h d 6 σ =ˆσ hard C[wfD] Structure functions that are extracted q k F AB = C[w f D] p Φ P C[wfD] = a xe 2 a d 2 p T d 2 k T δ (2) ( p T k T P h z ) f a (x, p 2 T )D a (z, k 2 T )

19 Transverse Spin Observables and TMD Correlators in SIDIS Φ(x, p T )= 1 nf 1 (x, p T ) /P + ih 1 2 (x, p T ) [ /p T, /P ] 2M (z, k T )= 1 nzd 1 (z, k T ) /P h + izh 1 4 (z, k T ) [k T, /P h ] 2M h f 1T (x, p T ) ɛij T p T is T j M /P o zd 1T (z, k T ) ɛij T k T is T j /P h + M h { } SIDIS cross section dσ{λ,λ} ln lπx f 1 dˆσ lq lq D 1 + k Q f 1 dˆσ lq lq D 1 cos φ [ ] k 2 + Q 2f 1 dˆσ lq lq D 1 + h 1 dˆσ lq lq H1 cos 2φ + S T h 1 dˆσ lq lq H 1 sin(φ + φ S ) Collins + S T f 1T dˆσ lq lq D 1 sin(φ φ S ) Sivers Boer-Mulders o transversity + S L h 1L dˆσ lq lq H 1 sin(2φ) Kotzinian MuldersPLB

20 Spec. model workbench for ISI/FSI TSSAs &TMDs f1t,h 1,D1T,H1 & gluonic pole MEs / calculation Quark-Quark Correlator in Full QCD Φ [U[C]] (x, p T ) = Z dξ d 2 ξt 2(2π) 3 eip ξ P ψ(0)u [C] [0,ξ] ψ(ξ, ξ T ) P ξ + =0 Use Spectator Framework Develop a QFT to explore and estimates these effects with gauge links BHS FSI/ISI Sivers fnct, -PLB 2002, NPB 2002 Ji, Yuan PLB Sivers Function Metz PLB Collins Function L.G. Goldstein, 2002 ICHEP- Boer Mulders Function L.G. Goldstein, Oganessyan TSSA & AAS PRD 2003-SIDIS Boer Brodsky Hwang PRD 2003-Drell Yan Boer Mulders Bacchetta Jang Schafer PLB, Flavor-Sivers, Boer Mulders Lu Ma Schmidt PLB, PRD, 2004/2005 Pion Boer Mulders L.G. Goldstein DY and higher twist, PLB 2007 LG, Goldstein, Schlegel PRD 2008-Flavor dep. Boer Mulders cos 2φ SIDIS Conti, Bacchetta, Radici, Ellis, Hwang, Kotzinian 2008 hep-ph...! Spectator Model Field Theoretic used study Universality of T-odd Fragmentation ij Metz PLB 2002, Collins Metz PRL 2004 Bacchetta, Metz, Yang, PBL 2003, Amrath, Bacchetta, Metz 2005, Bacchetta, L.G. Goldstein, Mukherjee, PLB 2008 Collins Qui, Collins PRD 2007,2008 Yuan 2-loop Collins function PRL 2008 L.G., Mulders, Mukherjee Gluonic Poles PRD 2008 Mulders & Rogers Fact. breaking PRD 2010

21 Studies FSIs in 1-gluon exchange approx. LG, G. Goldstein, M. Schlegel PRD , Bacchetta Conti Radici PRD 78, 2008 P p + l Υ l P p l Γ l P p Build the T-odd TMD PDF with Final State Interactions-- one gluon exchange approx of Gauge link W i (P, k, S) Z ¼ ie q e dq d 4 l g ax ððp þ lþ 2 Þ ð2þ 4 p ffiffiffi ffiffiffi 3 ½g v ð2p 2p lþþð1þþðv ðp p þ lþ þ v ðp p 2lÞ ÞŠ ½½l v þ i0š½l 2 þ i0š½ðl þ pþ Þ 2 m 2 q þ i0š ðp6 þ6 l þ m q Þ 5 P R g uðp; SÞ ; M ɛ ij T ki T S j T f 1T (x, kt 2 M )= 8(2π) 3 (1 x)p + ( " ðp p;þd ax ðp p lþ W γ + W i ST W γ + W ) ST Many model calculations studying dynamics of FSIs Brodsky, Hwang et al, Bacchetta & Radici, et al, Pasquini et al, Courtoy et al,...

22 Flavor Dependence: Results & Phenomenology Flavor-dependent PDFs from diquark models: u = 3 2 s + 1 2a, d = a, moments: h (1/2) 1 (x) = d 2 p p T T M h 1 (x, p 2 T ) L.G. Goldstein, Schlegel PRD 2008 Comparison to f (u,d) normalization... xf(x) 1 (Glück, Reya, Vogt) parameters of the model, xf 1 (u) xf 1 (d) xf 1 (u,grv) xf 1 (d,grv) xf(x) xf(x) x κ = 1.0 (1/2,u) xf 1T (1/2,d) xf 1T (1/2,u) xh 1 (1/2,d) xh xf(x) x x κ = (1/2,u) xf 1T (1/2,d) xf 1T (1/2,u) xh 1 (1/2,d) xh xf(x) κ = 1.0 xf 1T (1,u) xf 1T (1,d) (1,u) xh 1 (1,d) xh x x -0.2 PRD Boer Mulders up and down are negative is spectator model and f (u) 1T x h (u) 1

23 Sivers Anselmino et al. PRD 05, EPJA 08 Gamberg, Goldstein,Schlegel PRD 77, 2008 x f (x) x f (x) (1) u N (1) d N Q = 2.4 GeV x f d (x, k ) x f u(x, k ) N N 0.6 x = xf(x) κ = µ 2 =0.26 GeV 2 x xf 1T (1,u) xf 1T (1,d) (1,u) xh 1 (1,d) xh x k (GeV) FIG. 5 (color online). The first moment of the Boer-Mulders and Sivers functions versus x for ¼ 1:0. Fig. 7. The Sivers distribution functions for u and d flavours, at the scale Q 2 =2.4 (GeV/c) 2, as determined by our present fit (solid lines), are compared with those of our previous fit [2] of SIDIS data (dashed lines), where π 0 and kaon productions were not considered and only valence quark contributions were taken into account. This plot clearly shows that the Sivers functions previously found are consistent, within the statistical uncertainty bands, with the Sivers functions presently obtained.

24 Boer Mulders effect Proton Target HERMES L Pappalardo-jlab exclusive wksp

25 Beyond One Loop Approximation So far: Most phenomenological approaches to T-odd TMDs! Final state interactions modeled by a one-gluon exchange e.g. Diquark-model, MIT-Bag model etc. Sivers-effect ~5%,!!!!"#"!# "$! "!"!" # %!!"# #!"$ strength of FSI BHS (02) Ji-Yuan (02) LG, G. Goldstein (02,03.. LG, GRG, Schlegel (08) Bacchetta,Conti,Radici et al. (08, 10) Lu Schmidt (05, 06)

26 Explore non-pertb. FSIs-Links & Gauge Link remnant W~ Non-perturbative calculation of FSIs M M L.G. & Marc Schlegel Phys.Lett.B685:95-103, 2010 & Mod.Phys.Lett.A24: ,2009 ɛ ij T ki T S j T f 1T (x, kt 2 M )= 8(2π) 3 (1 x)p + ( W γ + W ST W γ + W ) ST

27 Fruitful to exploit 2+1 Dimension Transverse Structure and TSSAs and TMDs Intuitive picture of Sivers asymmetry: Spatial distortion in transverse plane due to polarization+ FSI leads to observable effect Non-zero Left Right (Sivers) momentum asymmetry M. Burkardt [Nucl.Phys. A735, 185], [PRD66, ] Gockler et al. PRL07 x-moments of IP-GPDs b spin pol ST S ( ˆP k ) f 1T (x, k 2 ) ( ) FIG. 4: Lowest moment (n = 1) of the densities of unpolarized S quarks in a transversely polarized nucleon (left) and transversely ( ˆP polarized b) quarks ine(x, an unpolarized b 2 nucleon ) (right) for up (upper plots) and down (lower plots) quarks. The quark spins (inner arrows) and nucleon spins (outer arrows) are oriented in the transverse plane as indicated. κ up T,latt 3.0 and κdown T,latt 1.9. Similarly large positive

28 Used to predict Esign of TSSA-Sivers Burkardt 02,04 NPA PRD d q y 2M 1 dx d 2 b E q x,b 2M 1 dxe q x,0,0 F 2,q 0 = 2M, κ 2M κ p =1.79, κ n = 1.91 κ u/p = 1.67, κ d/p = 2.03 w/ attractive interactions r pion and kaon Anselmino production semi-inclusivet deep al. inelastic PRD scattering 05, 95 EPJA 08 x = 0.1 f (u) = neg & f (d) 1T 1T x f (x) x f (x) (1) u N (1) d N Q = 2.4 GeV x f d (x, k ) x f u(x, k ) N N 0.6 x = x k (GeV) Fig. 7. The Sivers distribution functions for u and d flavours, at the scale Q 2 =2.4 (GeV/c) 2, as determined by our present fit (solid lines), are compared with those of our previous fit [2] of SIDIS data (dashed lines), where π 0 and kaon productions were not considered and only valence quark contributions were taken into account. This plot clearly shows that the Sivers functions previously found are consistent, within the statistical uncertainty bands, with the Sivers functions presently obtained Sivers = pos FIG. 4: Lowest moment (n = 1) of the densities of unpolarized quarks in a transversely polarized nucleon (left) and transversely polarized quarks in an unpolarized nucleon (right) for up (upper plots) and down (lower plots) quarks. The quark spins (inner arrows) and nucleon spins (outer arrows) are oriented in the transverse plane as indicated. 3.0 and κdown T,latt 1.9. Similarly large positive GAMBERG, Gamberg, GOLDSTEIN, valuesgoldstein, AND have been obtained Schlegel SCHLEGEL in a PRD recent 77, model 2008 PHYS calculation 0.2 [25]. Large N c considerations predict κ up T κdown T [26]. Let us now discuss our results for ρ n (1,u) B. Single-spin as (b xf κ = 1.0 1T, s, S ) in (1,d) Since we have calc Eq. (1). For the numerical evaluation we xf Fourier 1T transform the p-pole parametrization to impact parameter parton distribution h 0.1? 1L (1,u) xh (b ) space. The parametrizations of the impact 1 parameter dependent GFFs then depend only xh on the p-pole Collins function to giv this result together wi (1,d) 1 0 masses m p and the forward values F 0. Before showing ment of the single-spin a our final 0.2 results, 0.4 we would0.6like to note 0.8 that the 1 moments of the transverse spin density can be writtenaccount as the flavor depen polarized target. In pa sum/difference of the corresponding moments for quarks -0.1 a similar procedure for and antiquarks, ρ n = ρ n q + ( 1) n ρ n q, because vector and for treating the leading tensor operators transform identically under charge conjugation. Although we expect contributions from anti- A decomposition int -0.2 quarks to be small in general, only the n-even moments section of semi-inclusi must be strictly positive. In Fig. 4, we show the lowest ized target reads (see e x moment n = 1 of spin densities for up and down quarks d in the nucleon. Due to the large anomalous magnetic UL FIG. 5 (color online). The first moment of the Boer-Mulders moments κ u,d, we find strong distortions for unpolarizeddxdydzd h dp 2 and Sivers functions versus x for ¼ 1:0. h? quarks in transversely polarized nucleons (left part of the xf(x) κ up T,latt

29 Spin-Orbit kinematics Analysis of correlators for TMDs and IP-GPDs similar forms Burkhardt-02 PRD &... Diehl Hagler-05 EPJC, Meissner, Metz, Goeke 07 PRD Φ q (x, k T ; S) F q (x, b T ; S) = = f q 1 (x, k 2 T ) ɛij T ki T Sj T M H q (x, b 2 T ) + ɛij T bi T Sj T M f q 1T (x, k 2 T ), ( E q (x, b 2 T ) ), k T b T Not conjugates (!) and... f1t (x, kt 2 ) (E(x, ) b 2 T ) Naive T-odd Naive T-even FSIs needed... Burkardt PRD 02 & NPA 04 How do we test this further?

30 Summing Gauge Link, Impact parameter & spectator remnant bt 1 x remnantspectator

31 Fourier transform of GPD F (x, 0, T ξ =0 Burkardt PRD 00, 02, P + P T =0 F(x, b)= b T Localizing partons: impact parameter states with definite light-cone momentum p + and transverse position (impact parameter): Soper PRD1977 dz + (2π) 2 eixp z P + ; 0 T q (z 1 ) W(z 1,z 2 )q (z 2 ) P + ; 0 T z 1/2 = ± z 2 n + b T 2 p +, b = d 2 p e ib p p +, p F(x, b) = d 2 T (2π) 2 ei T b F (x, 0, T ) = H(x, b)+ ɛij T bi T Sj T M ( E(x, ) b) F.T. b T Prob. of finding unpol. quark w/ long momentum x at position bt in trans. polarized ST nucleon: spin independent and spin flip part H E

32 Observable to test this possible connection btnw TMD and Impact par. picture? Gluonic Pole ME!!! " " "!"" #! #!! "!! " "! " $%&$ # '!& % " "( # $%&$ # '(!#& % # " " k i T (x) = d 2 b T dz 2(2π) eixp + z P + ; 0 T ; S T ψ(z 1 )γ + [z 1 ; z 2 ]I i (z 2 )ψ(z 2 ) P + ; 0 T ; S T z 1/2 = z 2 n + b T Impact parameter rep for GPD E I i (z ) = dy [z ; y ]gf +i (y )[y ; z ] Soft gluonic pole op Phases in soft poles of propagator in hard subprocess Efremov & Teryaev :PLB 1982 b... spin pol

33 hk q;i T x i UT Z Z! Note on Distortion and FSIs M. Burkardt [Nucl.Phys. A735, 185], [PRD66, ] #!! "!! " "! " # $%&$ # '!%& " "( # $%&$ # '(!#%& " " Manipulate gauge link and trnsfm to Z b space 1) hk q;i T x i UT 1 2 Z d 2 ~b T Z dz 2 eixp z hp ; ~0 T ; Sj z 1 W z 1 ; z 2 I q;i z 2 z 2 jp ; ~0 T ; Si; 2) Z h j F q x; ~b T ; S 1 Z dz z hp ; ~0 2 2 eixp T ; Sj z 1 W z 1 ; z 2 z 2 jp ; ~0 T ; Si; Γ γ + j i Comparing expressions difference Z is additional factor, I q,i and integration over b

34 Conjecture: factorization FSI and spatial distortion: k i T (x) =Mɛij T Si T f (1) 1T d 2 b T I i (x, b 2 T ) b T S T M b 2 T E(x, b 2 T ) I i (x, b 2 T ) Lensing Function bt 1 x remnantspectator

35 Boer Mulders as well... Av. transv. momentum of transv. pol. partons in an unpol. hadron:!!! " "#!"" #! #!! "!! " "! " $ #!$!! % "$!$" % $ #!$!! % "$ #!#$" #&%! &!&%"& "!"" $ $ #! ' " ( '" % (&!") ( ' " " ' '( # " "' " % & '( " #!") ( '! " "! " # Diehl & Hagler EJPC (05), Burkardt PRD (04)

36 ( )!! W!!!! We calc W again... ecify the form factor! that we attach to the nucleon-quark-diquark vertex w ( " # "'&# )!(""! ## # (" *"! #* $ + " #,""-.#) # " $% %& "#- "! /# ) " (0 # ""! /! ## $ $%)(""! ## # $ + # " $ $%)(##! + # ' $ $%) W (P, k) =! ɛ ij T ki T S j T f 1T (x, kt 2 M )= 8(2π) 3 (1 x)p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γ + W ST W γ + W ) ST

37 Conjecture born out factorization FSI and spatial distortion in eikonal + spectator approximation k i T (x) =Mɛij T Si T f (1) 1T d 2 b T I i (x, b 2 T ) b T S T M b 2 T E(x, b 2 T ) I i (x, b 2 T ) Lensing Function bt 1 x remnantspectator

38 Eikonal Color calculation and path ordered gauge link # +, '()!$% &#'" #!(!" $!!!! " %! % Abarabanel Itzykson PRL 69 Gamberg Milton PRD 1999 Fried et al. 2000,- &!#$")!!#%$ + "#$!$! &! -"" "&!#-! "!./ %#'# "0&/%$ 1 ## +, & G#=$1%>6%H=@)7>"7*()%>')%IJA=)(H%"7H%>')%$6(6#%4">#=$)@%>F%K37$>=67"(%KG "&!#-! "!./ %#'# "0&/%$ 1 ##!! +, # % $ &. &/ & #!.2 3 $ "2$4 $ "2 $ & #-! ".2 3 $ "2$ %#' $ "0&2%$ & #! # "!.2 1$ 4 $ +, "2$ & &! " #

39 FLOW CHART for calculation of Boer Mulders L.G. & Marc Schlegel 2m 2 π h (1) 1 (x) Phys.Lett.B685:95-103,2010 & Mod.Phys.Lett.A24: ,2009. d 2 b T b T I(x, b T ) I i (x, q T ) = 1 d 2 p T N c (2π) 2 (2p T q T ) ( i I[ M eik ] ) αδ ( p T ) δβ ( ( ) βγ ) (2π) 2 δ αβ δ (2) ( p T q T ) + R[ M eik ] ( p T q T ). b 2 T γα H π 1 (x, b 2 T ), Non-pertb FSIs in here ( ) M eik αδ (x, q (1 x)p+ T + k T ) = d 2 z T e i z T ( q T + k T ) (20) δβ m s d N2 c d 1 Nc 2 1 u α ( e iα u e ) ( ) iχ( z T )t α e it u (2π) δ αβ N2 c 1 αδ δβ. COLOR Integral f αβ (χ) d N2 c 1 α d Nc 2 1 u ( e iα u e ) iχ( z T )t α (2π) N2 c 1 αδ ( e it u ) δβ δ αβ

40 Lensing Function & untangling the COLOR FACTOR I i (x, b i (1 x) bt χ ] χ T ) = 2N c b T 4 C[, 4 [ ] [ ( ) [ ( ) 1 f αβ (χ) = n=1 (iχ) n (n!) 2 N c 2 1 a 1 =1... N c 2 1 a n =1 P n (t a1...t an t apn (1)...t apn(n) ) αβ.

41 Eikonal Phase and FSIs χ DS ( z T ) = 2 0 dk T k T α s (k 2 T )J 0( z T k T )Z(k 2 T, Λ2 QCD )/k2 T. α s (µ 2 ) = α s (0) ln [ ]. (35) e + a 1 (µ 2 /Λ 2 ) a 2 + b1 (µ 2 /Λ 2 ) b 2 The values for the fit parameters are Λ = 0.71 GeV, a 1 = 1.106, a 2 = 2.324, b 1 = and b 2 = These calculations 5 4 3![z T ] 2 #=0.2 GeV #=0.5 GeV #=0.7 GeV #=0.0 GeV Z(p 2,µ 2 ) = p 2 D 1 (p 2,µ 2 ) Fisher & Alkofer prd 03, Annal of phys. 09 ( αs (p 2 ) 1+2δ ) = α s (µ 2 ) c ( p 2 Λ 2 ) κ + d ( p 2 Λ 2 ) 2κ 1 + c ( p 2 Λ 2 ) κ + d ( p 2 Λ 2 ) 2κ 2, (36) with the parameters c = 1.269, d = 2.105, and δ = These fits for the running coupling and the gluon propaga- use running coupling extended to non-perturbative regime gluon non-perturbative gluon propagator z T

42 FIG. 4 (color online). Pion mass dependence of B Tn0 ðt ¼ 0Þ=m. The shaded bands represent fits as explained in the text.!!!""#$!!"# $ # " # $ % $!! " # " # $ # " %!! "$ %! " ##& " $ & '() %! "$ %! " ##& "!"#$%#&'()#*+%,#!"#$%&&'(%)&*)+',-)./*0)'*)'/%$1&'02'!*30)45'674&%8!!!""%$!!"# $ # " # $ * $!! # " # $ # " %!! "$ %! " ##& " +!$ & '() ' & " & # ' & $ % # # ),- ' ' & & # ),- ' $ "! "$ %! " & ##& "!!!""'$!!"# $ # " # -./01,') L.G. & Marc Schlegel Phys.Lett.B685:95-103,2010 & Mod.Phys.Lett.A24: , (2008) P H Y S I C A L R E V I E W L E T T E R S Spin Structure of the Pion mmel, 1,2 M. Diehl, 1 M. Göckeler, 2 Ph. Hägler, 3, * R. Horsley, 4 Y. Nakamura, 5 D. Pleiter, 5 P. E. L. Rakow, 6 A. Schäfer, 2 G. Schierholz, 1,5 H. Stüben, 7 and J. M. Zanotti 4 (QCDSF/UKQCD Collaborations) 1 Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany 2 Institut für Theoretische Physik, Universität Regensburg, Regensburg, Germany 3 Institut für Theoretische Physik T39, Physik-Department der TU München, Garching, Germany 4 School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom PRL 5 John 101, von Neumann-Institut (2008) für Computing P H Y NIC/DESY, S I C A L R E V Zeuthen, I E W Germany L E T T E R S 6 Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom 7 Konrad-Zuse-Zentrum für Informationstechnik Berlin, Berlin, Germany (Received 26 August 2007; published Spin 17Structure September 2008) of the Pion PRL 08 week ending 19 SEPTEMBER 2008 week ending 19 SEPTEMBER 2008 We present the first calculation D. Brömmel, 1,2 of the M. Diehl, 1 transverse spin M. Göckeler, 2 structure of Ph. Hägler, 3, * the pion in lattice R. Horsley, 4 QCD. Our Y. Nakamura, 5 D. Pleiter, 5 P. E. L. Rakow, 6 simulations are based on two flavors of nonperturbatively A. Schäfer, 2 improved Wilson G. Schierholz, 1,5 fermions, with H. Stüben, 7 pion masses as and J. M. Zanotti 4 low as 400 MeV in volumes up to ð2:1 fmþ 3 and lattice spacings below 0.1 fm. We find a characteristic asymmetry in the spatial distribution of transversely polarized quarks. This asymmetry is very similar in magnitude to the analogous asymmetry we previously (QCDSF/UKQCD obtained for quarks Collaborations) in the nucleon. Our results support the hypothesis that all Boer-Mulders functions are alike. 1 Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany DOI: /PhysRevLett Institut für Theoretische Physik, Universität PACSRegensburg, numbers: Gc, Aq Regensburg, Germany 3 Institut für Theoretische Physik T39, Physik-Department der TU München, Garching, Germany 4 School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom 5 John von Neumann-Institut für Computing NIC/DESY, Zeuthen, Germany tion. Since their discovery FIG. 5 (color 6 in the late 1940s, Department online). of Mathematical The lowest Sciences, moment University of the of Liverpool, densities Liverpool of L69 3BX, United Kingdom played a central role in nuclear unpolarized (left) and 7 and particle n ðb? ;s? Þ¼ Z 1 dx x n 1 ðx; b? ;s? Þ Konrad-Zuse-Zentrum für Informationstechnik 1 Berlin, Berlin, Germany s pseudo-goldstone bosons of spontaneously transversely polarized (right) up quarks in a þ (Received 26 August 2007; published 17 September 2008) ral symmetry, they together are at We with the core present corresponding of the lowtor of quantum chromodynamics (QCD). Since 2 ðb2? Þ si? ij b j? the first calculation profile the plots. ¼ transverse The 1 Aquark spin n0 structure spin is B of the pion 0 Tn0 m ðb2 in? ; Þ (1) lattice QCD. Our as spin zero, oriented its longitudinal in simulations the transverse spin are structure basedplane on in twoas flavors indicated of nonperturbatively by the arrow. improvedthe Wilson fermions, with pion masses as d error bands in lowthe as 400 profile MeV in plots volumes show upwhere to the ð2:1b uncertainties fmþ 0 Tn0 ark and gluon degrees of freedom is trivial. Pion 3 ¼ blattice 2?B Tn0 in spacings. The B ;u b?-dependent T10 ðt below ¼ 0.1 fm. vector We find and a characteristic ten- ments of asymmetry in the spatial distribution d 0Þ=m quark and gluon the p-pole helicitymasses operatorsat m phys of transversely polarized quarks. This asymmetry is very similar in to parity invariance: magnitude hðp to the analogous asymmetry we previously obtained for quarks in the nucleon. Our results e- tion. The dashed-dotted 0 Þj 3 jðpþi ¼ 0, B Tn0, for quarks 3 ¼ q 3 support the hypothesis linesthat show all Boer-Mulders the uncertainty functions from are alike. 5 q. An instructive quaning the extrapolation spin structure DOI: of (light hadrons /PhysRevLett Z 1 a ChPT shaded is the probaty ðx; b? Þ of quarks in impact parameter space Z 1 band). dx x n 1 H ðx; ¼ 0;b 2? Þ¼A n0 PACS ðb2? Þ; 1 numbers: Gc, Aq (2) is the longitudinal momentum fraction carried dx x n 1 E ðx; ¼ 0;b 2 Þ¼B ðb 2 Þ: FSI + distortion FSIs sor are generalized negative form factors (GFFs) of the pion A grow with Color! n0 from a linear extrapola- and, respectively, are moments of the GPDs: I(x=0.2,b T ) U(1) SU(2) SU(3) Perturbative!= b T xm 2 " h# Figure 3: Left: The lensing function 0

43 Prediction for Boer-Mulders Function of PION L.G. & Marc Schlegel Phys.Lett.B685:95-103,2010 & Mod.Phys.Lett.A24: , xm 2 " h# U(1) SU(2) SU(3) x Relations produce a BM funct. approx equiv. to Sivers from HERMES Expected sign i.e. FSI are negative Answer will come from pion BM from COMPASS πn Drell Yan

44 Results for u & d-quark Sivers f 1T (1) u(x) Diehl SU(3) Guidal SU(3) -up quark Guidal SU(3)-down quark Diquark x Relations produce a Sivers effect Nc=1 to 3 Torino extraction ~ 0.05 SU(3)! agrees with Chromodynamic LENSING Sivers effect increases with color Color tracing gives result of N c counting of Pobylitsa however there are subleading contributions that are non-trivial in performing color tracing x! f (x) x! f (x) (1) u N (1) d N !0.02!0.04!0.06! Q = 2.4 GeV!2 10! x

45 Sivers function increases with color SU(2) SU(3) U(1) xf 1T x

46 Reality Check Parm. of GTMD correlator hermiticity parity time-reversal from Andreas Metz INT talk (x, ξ, k T, T ) W q = 1 2 Z Z dz 2π d 2 z T eik z p ; λ ψ z γ + z W (2π) 2 GT MD ψ p; λ z+=0 2 2 Projection onto GPDs and TMDs F q = 1 2 Z = Z dz 2π eik z p ; λ ψ d 2 kt W q z z γ + W GP D ψ p; λ z+=zt 2 2 =0 Φ q = 1 2 Z dz 2π d 2 z T eik z p; λ ψ z z γ + W (2π) 2 T MD ψ p; λ z+=0 2 2 = W q =0

47 GTMD-Wigner Function Correlator Parameterization of GTMD-correlator Miessner Metz & Schlegel JHEP 2008 & 2009 Example: W q [γ+] = 1 2M ū(p, λ )» F 1,1 + iσi+ k i T P + F 1,2+ iσi+ i T P + F 1,3 + iσij k i T j T F M 2 1,4 u(p, λ) GTMDs are complex functions: F 1,n = F e 1,n + if o 1,n Implications for potential nontrivial relations Relations of second type E(x, 0, 2 T ) = Z d 2 kt» F e 1,1 +2 kt T 2 T F e 1,2 + F e 1,3 «f 1T (x, k 2 T ) = F o 1,2 (x, 0, k 2 T, 0, 0) No model-independent nontrivial relation between E and f 1T possible Relation in spectator model due to simplicity of the model No information on numerical violation of relation Likewise for nontrivial relation involving h 1

48 Conclusions Going beyond one loop in spectator framework transverse distortion of T-odd TMDs from FSIs as path ordered Gauge link, factorize into Lensing function times transverse distortion Approximate dynamical relation good for phenomenological approach for model builders Pheno-Transverse Structure TMDs and TSSAs b and k asymm. An improved dynamical approach for FSI & model building QCD calc FSIs Gauge Links-Color Gauge Inv. T-odd TMDs

49 Calculation of M!! Calculate the amplitude M in a relativistice eikonal model: [1970's: Fried, Quiros, Levy, Sucher, Zuber, etc...]!! (! Exact 4-point " function for quark-diquark scattering: # ( # #(!&' "! (! & " ' "" %!$!" " # " $ " )) $ "( $ $ # " %& '( "! ( ) " # & (! & " '!! '!$!* " # * $ " )) " "(! $ $%& " %& '( #! ( )! # &&& " )(" # $ # *( )! *+ neglect neglect! " #! Linkage operator:! 1 &, # # 2-3 $ " 4 )) &!3 $ " (!$!+ $ # 3 " " 4 )),!3 " "! # Eikonal approximation: 1 && %!$ # +

50 TMDs & Impact GPDs Project from GTMDs W [Γ] λ,λ (P, x, k T,,n)= dk W [Γ] λ,λ (P, k,,n) d 2 k T Integ. small component, GTMD--Meissner Metz Schlegel, 07 =0 F [Γ] λ,λ (x, ξ, t) = d 2 k T W [Γ] λ,λ (P, x, k T,,n) ξ =0 W [Γ] λ,λ (P, x, k T, 0,n) =Φ( x, k T ) TMD FT : T b T F λ,λ (x, b)= Impact-GPD d 2 T (2π) 2 ei T b F λ,λ (x, 0, T )

51 Unifying Transverse Structure of Nucleon GTMDs GTMD--Meissner Metz Schlegel 07, 08 W [Γ] λ,λ (P, x, k T,,n)= dk W [Γ] λ,λ (P, k,,n) Integ. small component!!! FT : b W [Γ] λ,λ (P, k, ; n) W [Γ] λ,λ (P, k, b; n) Wigner functions--belitsky Ji Yuan, 04 Reduce to TMDs, GPDs, Impact GPDs Relations among them?