The transverse spin and momentum structure of hadrons 03/26/10 talk #2 Parton Model, SIDIS & TMDs Leonard Gamberg Penn State University
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, 1998-...
and more... Color Gauge Invariance & Links and FSI mechanism T-odd TMDS and TSSAs
Factorization Sensitivity to P T k TMDs John Collins Nuclear Physics B396 (1993) 161 182 3.4. FACTORIZATION WITH INTRINSIC TRANSVERSE MOMENTUM AND POLARIZATION Fig. 2. Parton model for semi-inclusive deeply inelastic scattering. We now explain factorization for the semi-inclusive deep inelastic cross section when the incoming hadron A is transversely polarized but the lepton remains unpolarized. (It is left as an exercise to treat the most general case.) The factorization theorems, eq. (12) and eq. (14), continue to apply when we include polarization for the incoming hadron, but with the insertion of helicity density matrices for in and out quarks; this is a simple generalization of the results in refs. [10,231.... Ralston Spoper NPB 1979, Collins NPB 1993 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.
How to motivate the PDFs and FFs from SIDIS at moderate PT γ Consider the limit where Q 2 is large and assumed to scatter incoherently off constituents. Currents treated as in free field theory. Interactions btwn. struck quark and target remnant are neglected at least most of the time---color gauge links will modify this statement l X, P h S h J ν (0) PS l P h P P X
SIDIS and the parton model l P γ l P h Hand Bag diagram q (γ, ɛ) (k, µ) P h P X (k, µ ) (p, λ) (p, λ ) 1) Parton model assumption: virtual photon strikes quark inside nucleon assumed to scatter incoherently off constituents P X 0) Consider the limit where Q 2 is large 2) A DIS reaction where hadron in current region is detected in final: state-rapidity sep. between the target remnant and the current hadron 3) In case of SIDIS the tagged final state hadron comes from fragmentation of struck quark Hand Bag diagram 4)The scattering process can be factorized into two soft hadronic parts connected by a hard scattering piece (P, Λ) P X Φ γ (P, Λ )
Kinematics and Geometry of SIDIS Consider the limit where Q 2 is large l(l)+n(p ) l(l )+h(p h )+X, q 2 =(l l ) 2, s =(P + l) 2 Lepton and hadron planes in semi-inclusive leptoproduction. q 2 = q 2 0 q 2 3 =2q + q Light Cone Vectors q 2 Q 2 where Q 2 0 V + = V 0 + V 3 2, V = V 0 V 3 2, V T q µ =( Q 2, Q 2, 0) V µ =(V +,V, V T )
Light-cone kinematics Photon-Hadron Frame Interested in limit where Q 2 P.q, Ph.q and P.Ph become large while xb and zh remain finite x B = Q2 2 P q, y = P q P l, Photon Hadron Frame and given, x B,z h and q µ = ( Q 2, ) Q, 0, 2 z h = P P h P q P µ = (P +, M 2 ) 2P +, 0, q µ = ( x B P +, Q 2 ) 2x B P +, 0 P h =( m2 h + P 2 h 2P h,p h, P h ) note from z h = P P h P q P h q ( q µ = x B P +, P ) h, 0 = Q2 z h 2x B P + = P z h
dσ = 1 4 P Diagramatic Factorization Cross Section for SIDIS Sum over spin of out going electron Differential in phase space of final detected electron and hadron Sum over all unobserved final states X { s l X d 3 P X (2π) 3 2E X (2π) 4 δ 4 (P + l P X P h l ) M 2 } d 3 l (2π) 3 2E d 3 P h (2π) 3 2E h dσ = 1 4 P M 2 = e4 q 4 L µνw µν W µν = 1 (2π) 4 X Leptonic & Hadronic Tensor d 3 P X (2π) 3 2E X (2π) 4 δ 4 (P + l P X P h l ) PS J µ (0) X, P h S h X, P h S h J ν (0) PS e 4 q 4 L µνw µν (2π) 4 d 3 l (2π) 3 2E d 3 P h (2π) 3 2E h l l P h 2E h dσ d 3 P h de dω = α2 em 2MQ 4 E E L µνw µν P P X
Cross Section for SIDIS in terms of invariants x B = Q2 2 P q, y = P q P l, z h = P P h P q In terms of invariants 2E h dσ d 3 P h dxdy = πα2 emy Q 4 L µν W µν When P h P h E h = d 3 P h 2E h = dz hd 2 P h 2z h Compact expression dσ dxdydz h d 2 P h = πα2 em 2Q 4 y z h L µν W µν
Comment on DIS from SIDIS W µν (q, P, S, P h )= 1 (2π) 4 X d 3 P X (2π) 3 2E X (2π) 4 δ 4 (P + l P X P h l ) PS J µ (0) X, P h S h X, P h S h J ν (0) PS Summing over all possible hadrons and integrating over the final state hadron recover inclusive DIS result h d 3 P h (2π) 3 2E h W µν (q, P, S, P h )=W µν (q, P, S) l l l P h l P P P X P X
P Parton Model Factorization of SIDIS W µν = 1 (2π) 4 a d 4 p (2π) 4 e 2 X d 3 P X (2π) 3 2E X X d 3 P X (2π) 3 2E X d 4 k (2π) 4 (2π)4 δ 4 (P p P X )(2π) 4 δ 4 (p + q k)(2π) 4 δ 4 (k P h P X ) [ χ(k; P h,s h )γ µ φ(p; P, S)] [ χ(k; P h,s h )γ ν φ(p; P, S)] Hadron Matrix element of quark fields P h Levelt & Mulders PLB 94 see also Mulders, Pramana 72,2009 0 ψ(0) P h X P X P X φ(p; P, S) = X ψ(0) PS k p χ(k; P h,s h )= 0 ψ(0) P h S h ; X X ψ(0) P P X P X
Non-perturbative quark-quark Correlators from Hadron matrix elements of quark fields Φ ji (p; P, S) = d 3 P X (2π) 3 (2π) 4 δ 4 (P p P X )φ i (p; P, S) 2E φ j (p; P, S) X X exponentiate delta function = d 4 ξe ip ξ PS ψ j (0)ψ i (ξ) PS and ij (k; P h S h ) = X = X d 3 P X (2π) 3 2E X d 3 P X (2π) 3 2E X (2π) 4 δ 4 (P h + P X k)χ i (k; P h S h ) χ(k; P h S h ) d 4 ξe ik ξ 0 ψ i (ξ) P h S h,x P h S h,x ψ j (0) 0 0 ψ(0) P h X W µν (q, P, S, P h ) = a e 2 d 4 p (2π) 4 d 4 k (2π) 4 δ4 (p + q k)tr [Φγ µ γ ν ] k P p X ψ(0) P Φ
cut quark hadron scattering amplitudes 0 ψ(0) P h X k P p X ψ(0) P Φ
Light-cone kinematics quark momenta QUARK Momenta p µ =(p +, p2 + p 2 2p +, p ) k µ =( k2 + k 2 2k,k, k ) -initial quark -fragmenting quark Parton assumptions p + P +, k P h LARGE Momenta Q therefore p 1/P +, k + 1/P h SMALL Momenta 1/Q
Exploiting delta function in Long. Momenta W µν (q, P, S, P h )= a e 2 d 4 p (2π) 4 d 4 k (2π) 4 δ4 (p + q k)tr [Φγ µ γ ν ] δ 4 (p + q k) =δ(p + + q + k + )δ(p + q k )δ 2 (p T + q T k T ) Parton model assumptions p + P +, k P h LARGE Momenta Q therefore p 1/P +, k + 1/P h SMALL Momenta 1/Q = δ 4 (p + q k) δ(p + x B P + )δ(k P h z h )δ 2 (p T + q T k T )
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 h )δ 2 (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 h k T )Tr [Φ(x, p T )γ µ (z,k T )γ ν ] Φ(x, p T )= dp 2 Φ(p, P, S) p + =x B P +, (z,k T )= dk 2 (k, P h) k = P z h Small transverse momentum P 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 )
Comment on Photon-Hadron & Hadron-Hadron Frame q P P h q P P h Connection neglecting subleading twist q T z h P h
Correlator is Matrix in Dirac space 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 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.
Example for Unpolarized Target Φ(p, P) = M A 1 1 + 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 +
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 =( x B P +, P h z h, 0) = x B P + n µ + + P h z h n µ where n + n + =1, n ± n ± =0 and /P = P µ γ µ P + γ = P + /n +
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. 359 2002 (1) Bacchetta s Ph.D thesis at http://www.jlab.org/~bacchett/index.html Most upto date listing of TMDs, Bacchetta, Diehl, Goeke, Metz, Mulders, Schlegel JHEP 0702:093,2007
Leading Twist TMDs from Correlator Φ [γ+] (x, p T ) f 1 (x, p 2 T) + ɛijp T 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 ( } { [ H. Avakian tableau ( λ h 1L(x, p 2 T) + p T S T M h 1T (x, p 2 T) ) Integrated pdfs f(x) = d 2 p T f(x, p 2 T) + ɛij T p j T M h 1 (x, p 2 T) { h 1 (x) = d 2 p T ( h 1T (x, p 2 T) + p2 T 2M 2 h 1T (x, p 2 T) ) Transversity
What about dynamics ie FSIs--- necessary for TSSAs
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 2002- Gauge link Sivers function doesn t vanish Ji, Yuan PLB: 2002 -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
T-Odd Effects From Color Gauge Inv. via Wilson Line ν q µ H ρν;a k P h (k) Gauge link determined re-summing gluon interactions btwn soft and hard Efremov,Radyushkin Theor. Math. Phys. 1981 Belitsky, Ji, Yuan NPB 2003, Boer, Bomhof, Mulders Pijlman, et al. 2003-2008- NPB, PLB, PRD H µ P p p 1 p 1 Φ aρ A (p,p 1) p Summing gauge link with color LG, M. Schlegel PLB 2010 Φ ij (x, p T )= dz d 2 z T P R (8π 3 e ip z P q j (0)U [+] ) [0,ξ] q i(z) P z + =0 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) Next time ξ T U [+] ξ