The transverse spin and momentum structure of hadrons. 03/26/10 talk #3 Parton Model Gauge Links T-odd TMDs. Leonard Gamberg Penn State University

Transcription

1 The transverse spin and momentum structure of hadrons 03/26/10 talk #3 Parton Model Gauge Links T-odd TMDs Leonard Gamberg Penn State University

2 T-Odd Effects From Color Gauge Inv. via Wilson Line 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 ) = 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 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

3 The Transverse Spin and Momentum Structure of Hadrons Details TMDs from parton model framework extended to incorporate transverse momentum SIDIS Transverse spin kinematics- diagramatic factorization Levelt & Levelt, Mulders &Tangerman, Kotzinian, Boer & Mulders et al. 1994,1995,1996,

4 and more... Color Gauge Invariance & Links and FSI mechanism T-odd TMDS and TSSAs

5 Message: the PT of the hadron is 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 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 )

6 Correlator is Matrix in Dirac space Φ ji (p; P, S) = d 4 ξ eip ξ (2π) 4 PS ψ i (0)ψ j (ξ) PS Φ ji (x, p T )= Φ ji (x, p T )= dp 2 Φ ji(p, P, S) p+ =xp + dξ d 2 ξ eip ξ 2(2π) 3 PS ψ i (0)ψ j (ξ) PS x+ =0 Decompose into basis of Dirac matricies 1! 5!!!!! 5 i"!#! 5 Hermiticity: Φ(p, P, S ) = γ 0 Φ (p, P, S ) γ 0, parity: Φ(p, P, S ) = γ 0 Φ( p, P, S ) γ 0 and so forth for the other vectors.

7 Example for Unpolarized Target Φ(p, P) = M A A 2 /P + A 3 /p + A 4 M σ µνp µ p ν are real scalar functions ( 2 ) with dim Use Hierarchy in hard scale P + M : 1 : M P + Keeping only leading terms in P +

8 Convenient to Introduce Light-Like 4-vectors Light cone 4 vectors defining the light like directions n µ + = (1, 0, 0) n µ = (0, 1, 0) e.g., q µ =( xp +, P h z, 0) = xp + n µ + P h z nµ + where n + n + =1, n ± n ± =0 and /P = P µ γ µ P + γ = P + /n +

9 Leading Order Result Φ(p, P) P + (A 2 + xa 3 ) /n + + P + i [ ] /n+, /p T A4, 2M Φ(x, p T ) dp Φ(p, P) = 1 2 { f 1 /n + + ih 1 [ ] /pt, /n } +. 2M { [ ]} the parton distribution functions where, f 1 (x, p 2 T ) = 2P+ dp (A 2 + xa 3 ), h 1 (x, p2 T ) = 2P+ dp ( A 4 ). clear what variables the functions depend on, it is better to introduce the symb Original work Levelt Mulders PRD49,1994&B338,1994, Mulders Tangerman PLB461,1995&NPB461,1996 T-odds Boer and Mulders PRD57,1998 For more details see Barone,Drago, Ratcliffe Phys. Rep (1) Bacchetta s Ph.D thesis at Most upto date listing of TMDs, Bacchetta, Diehl, Goeke, Metz, Mulders, Schlegel JHEP 0702:093,2007

10 Leading Twist TMDs from Correlator { 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 ( { [ ( } Avakian tableau 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) {

11 Integrated pdfs f(x) = d 2 p T f(x, p 2 T) Transversity h 1 (x) = d 2 p T ( h 1T (x, p 2 T) + p2 T 2M 2 h 1T (x, p 2 T) )

12 What about dynamics ie FSIs--- necessary for TSSAs

13 Gauge Invariance must be implemented in the correlators Φ ji (p; P, S) = d 4 ξ (2π) 4 eip ξ PS ψ i (0)ψ j (ξ) PS Φ ji (x, p T )= Φ ji (x, p T )= dp 2 Φ ji(p, P, S) p + =xp + dξ d 2 ξ 2(2π) 3 PS ψ i (0)ψ j (ξ) PS x+ =0 not gauge invariant

14 Fix: by putting gauge link between quark fields Φ [U[C]] (x, p T ) = dξ d 2 ξ T 2(2π) 3 eip ξ P ψ(0)u [C] [0,ξ] ψ(ξ, ξ T ) P ξ + =0... Φ [U[C]] (z, k T ) = dξ + d 2 ξ T 4z(2π) 3 eik ξ 0 U [C] [0,ξ] ψ(0) X; P h X; P h ψ(ξ +, ξ T ) 0 ξ =0... Φ Efremov,Radyushkin Theor. Math. Phys Belitsky, Ji, Yuan NPB 2003, Boer, Bomhof, Mulders Pijlman, et al NPB, PLB, PRD

15 Must extend Parton Model result 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)

16 Jet SIDIS p k P (z,k T ) (/k + m)δ(z 1)δ (2) (k 2 T ) W µν (q, P, S, P jet ) = Tr (Φ(x, p T ) γ µ (/k + m)γ ν ) Note, leave unintegrated on pt see Boer Pijlman and Mulders NPB

17 Leading order contribution p p 1 P k p 1 p 1 Assumptions: 1) parton lines connecting to soft blob are approx. on shell 2) parton lines approx. colinear with parent hadron 3) these assumption are exploited by using light-like vectors such that Light cone 4 vectors n µ + = (1, 0, 0) n µ = (0, 1, 0) P µ = P + n µ + + O(M 2 /P + )n µ P µ h = P h nµ + O(M 2 /P h )nµ + where n + n + =1, n ± n ± =0

18 Leading order contribution cont. k p 1 p p 1 p 1 P Jet SIDIS W µν (q, P, S, P h ) dp d 4 p 1 Tr ( γ α ) /k /p 1 + m (k p 1 ) 2 m 2 + iɛ γν Φ α A(p, p p 1 )γ µ (/k + m) quark gluon correlation function Φ α A(p, p p 1 ) = d 4 ξ (2π) 4 d 4 η (2π) 4 eiξ (p p 1) e iη p 1 P, S ψ i (0)gA α (η)ψ j (ξ) P, S

19 Look for the leading contribution from hadronic tensor k p 1 p p 1 p 1 P W µν (q, P, S, P h ) dp dp + 1 d2 p 1 d 4 ξ dη d 2 η eiξ (p p 1) (2π) 4 (2π) 3 e iη p 1 P, S ψ i (0)γ µ /k /p 1 + m (/k + m)γ α (k p 1 ) 2 m 2 + iɛ γν ga α (η)ψ j (ξ) P, S η+ =ξ +,p + =xp +

20 perform the following steps which I will be glad to go over with you 1) p δ(ξ + ) 2) p 1 δ(η + ) 3) also now k + = q + + p + = xp + + p + =0

21 Look for dominant contribution /k /p 1 + m (k p 1 ) 2 m 2 + iɛ γ+ k γ p + 1 /p 1T + m k+ =0 2p + 1 k + iɛ γ + k 2p + 1 k + iɛ = 1 2 γ + p iɛ

22 Projecting A + Gluons (/k + m)γ α γ + p ga α iɛγν (k γ + )γ γ + p ga + iɛγν = k γ+ ga + p iɛγν γ ± = γ0 ± γ 3 2 Note: ( 1 2 γ γ + ) 2 = 1 2 γ γ + (γ + ) 2 =(γ ) 2 =0

23 So here we are... W µν (q, P, S, P h ) dp + 1 d2 p 1 dξ d 2 ξ T (2π) 3 P, S ψ i (0)γ µ k γ+ ga + p ψ j (ξ) P, S iɛγν dη d 2 η (2π) 3 eiξ (p p 1) e iη p 1 η+ =ξ + =0,p + =xp +

24 Finally we use these integrals d 2 p 1 e ip 1T (η T ξ T ) = (2π) 2 δ 2 (η T ξ T ) dp + 1 eip+ 1 (η ξ ) ga + p + 1 iɛ =2πigA+ (η)θ(η ξ )

25 Corresponds to the O(g) term in expansion of long. part of gauge link W µν (q, P, S, P h ) dξ d 2 ξ T (2π) 3 dη d 2 η (2π) 3 eiξ P, S ψ i (0)γ µ γ + γ ν ( ig) ξ A + [ ] (η)ψ j (ξ) P, S η + =ξ + =0,η T =ξ T,p + =xp + U [,ξ] [ = P exp ig ξ ] dη A + (η, 0, ξ T ) 1 ig ξ dη A + (η, 0, ξ T )

26 Not the full story: the transverse gluon field at infinity does not vanish and there are leading contributions to the gauge link that must be included Ji and Yuan PLB 543, 2002, Boer Mulders Pijlman NPB dp dξ ξt eip ξ P, S ψ(0) γ (2π) 4 µ γ + γ ν ( ig) dη T A T (, η +, η T ) ψ(ξ) P, S, T η + =0; η T =ξ T (4.27) see also A. Bacchetta Trento Lecutre Notes

27 Taking into account multiple exchange and summing all terms yields the gauge invariant correlator Φ ji (p; P, S) = dξ d 2 ξ T (2π) 3 e ip ξ PS ψ j (0)U [0, ;0] U [,ξ,ξ T ] ψ i(ξ) PS ξ+ =0 ξ T (ξ, 0, ξ ) ξ gauge link in SIDIS

28 IDIS Hadronic Tensor Drell-Yan Hadronic Tensor (ξ, 0, ξ ) ξt (ξ, 0, ξ ) ξt T-Odd Effects From Color Gauge Inv. via Wilson Line T-Odd Effects From Color Gauge Inv. via Wilson Line Gauge link for TMDs [ ] pastpointing Generalized Universality Φ[+] futurepointing Fund. ξ ξ fφ1t (x, k ) = f T 1TDY (x, kt ) SIDIS Prediction of QCD Leading twist Gauge Invariant Distribution Functions Leading twist Gauge Invariant Distribution Functions link for TMDs Gauge "! 2 Boer, Mulders: 2000, & Pijlman (BPM) NPB" 2003, Belitsky Ji Yuan NPB 20 dξ dnpb ξtpijlman Boer, Mulders: NPB 2000, & (BPM) NPB 2003, Belitsky Ji Yuan NPB 2003 ip ξ PLB " Process Collins 02, Brodsky, Hwang, Schmidt NPB 02 Φij (x, pt Dependence )= e!p ψ (0)U ψ (ξ) P " j [0,ξ] i " 3 8π + TMDs µν dσ = L W ξ =0! µν µν dσ = Lµν W dξ d2 ξ " ξt "!"#"! "... (0)U [0,ξ] ψi (ξ) P ""... ξ Φ ξ + =0 Φ "! 2 " dξ d ξt ip ξ!"#"! " = e!p ψ (0)U ψ (ξ) P " j i [0,ξ] "+ 8π 3 ξ =0 ξt SIDIS Hadronic Tensor U #$%&&'()* SIDIS Hadronic [+] Tensor ξ ξt T (ξ, 0, ξ ) ξt ξ ξ!+,-+.)/$-*0 (ξ, 0, ξ ) P&T ξ U[+] #$%&&'()* Φ 3 8π Φ ψ j (0)U[0,ξ] ψi eip ξ!p U[+]... Φ ξt... Φ ξ Drell-Yan Hadronic Tens U [ ] Drell-Yan Hadronic Tensor ξt ξt, 0, ξ ) (ξ ξt (ξ, 0, ξ ) ξ ξ Process Dependence Collins PLB 02, Brodsky, Hwang, Schmidt N U[!] U[+] U[ ] f (x, k ) = f (x, kt ) T 1T 1T SIDIS DY Process Dependence Collins PLB 02, Brodsky, Hwang, Schmidt NPB 02 ξ Φ[+] futurepointing Φ[+] futurepointing ξt ξ Process Dependence ξt T Φij (x, pt ) = Φ[ ] pastpointing Φ[ ] pastpointing ξ 1+0%2%$)&+-,.%$0 Collins PLB 02, Brodsky, Hwang, Schmidt NPB 02 3 U[ ] = iγ 1γ"#$%#&'()*+,-./'(012+$34'(05"(678(9:;<( Φ [ ](x, pt )iγ 1γ 3 Φ[+] (x, pt )!!+,-+.)/$-*0 ξ ξt ξ

29 Mechanism FSI produce phases in TSSAs at Leading Twist 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

30 Summary These lecture notes serve as an overview/introduction to T-odd effects in TSSAs T-odd TMD PDFs contain the essential ingredients to describe TSSAs at moderate PT Such effects are the object of present/future experimental studies, Hall A & CLAS (Hall B) JLAB 12 GeV upgrade will be an critical to these studies An EIC presents future opportunity to deepen our understanding of T-odd effects by increased luminosity and range in Q 2