QCD Factorization and Transverse Single-Spin Asymmetry in ep collisions

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1 QCD Factorization and Transverse Single-Spin Asymmetry in ep collisions Jianwei Qiu Brookhaven National Laboratory Theory seminar at Jefferson Lab, November 7, 2011 Jefferson Lab, Newport News, VA Based on work with many people

2 Outline of my talk q Transverse single-spin asymmetry in ep collisions Ideal observable to go beyond the leading power collinear factorization q Role of fundamental symmetries q QCD TMD factorization approach q QCD collinear factorization approach q Connection between these two approaches q Predictive power of QCD factorization approach q Summary

3 Electron-proton collisions q Cross sections: = ² Every parton can participate the hard collision! ² Cross section depends on matrix elements of all possible fields q Approximation single large momentum transfer: Q >> 1/fm σ(q) =σ LP (Q)+ Q s Q σnlp (Q)+ Q2 s Q 2 σnnlp (Q)+... σ LP (Q) q Leading power QCD factorization - approximation: σ(q) σ LP (Q) ˆσ(Q) p, s φ (k) φ(k) p, s +... q How good the approximation is? Universal parton distributions Hadron s partonic structure!

4 q From HERA: Inclusive DIS cross section

5 Inclusive single jet hadronic cross section q To Tevatron: With one set universal PDFs QCD is successful in last 30 years we now believe it

6 Fact and questions q FACT: ² LP QCD collinear factorization/calculations have been very successful in interpreting HEP scattering data if Q > 2 GeV ² QCD should be correct for the asymptotic regime: r < 1/10 fm! q QUESTIONS: < 1/10 fm ² How much have we learned about hadron s partonic structure? Collinear PDFs, Helicity PDFs, But, Not enough for the structure, ² How to test/explore QCD beyond the leading power formalism? Parton s transverse motion, and multiparton correlation beyond 1/10 fm?

7 Go beyond the LP collinear factorization q Recall: = ² LP collinear term dominates the single scale cross section q Need additional parameter the LP term is not sensitive to: ² Nuclear A-dependence: result of multiple scattering and multiparton correlations R A (Q) σ A (Q)/σ N (Q), qt 2 A qt 2 A qt 2 N,... ² Transverse-spin: power of fundamental symmetries cancels the LP collinear term A N (Q, s T ) σ(q, s T ) σ(q, s T ), A N (Q, q T,s T ) σ(q, q T,s T ) σ(q, q T, s T ),...

8 Transverse SSA in collinear parton model q SSA corresponds to a naively T-odd triple product: A N =[σ(p, s T ) σ(p, s T )]/[σ(p, s T )+σ(p, s T )] A N i s p (p ) i µναβ p µ s ν α p β Novanish A N requires a phase, enough vectors to fix a scattering plan, and a spin flip at the partonic scattering q Leading power in QCD: 2 Kane, Pumplin, Repko, PRL, 1978 σ AB (p T, s) = α s m q +... p T Need parton s transverse motion to generate the asymmetry!

9 Power of fundamental symmetries q Factorized cross sections asymmetries: A σ h(p) (Q, s) σ h(p) (Q, s) p, s O(ψ q,a µ ) p, s p, s O(ψ q,a µ ) p, s q Parity and Time-reversal invariance: q IF: or Operators lead to the + sign Operators lead to the - sign q Example: spin-averaged cross sections spin asymmetries

10 A N = 0 for inclusive DIS q DIS cross section: σ(q, s T ) L µν W µν (Q, s T ) q Leptionic tensor is symmetric: q Hadronic tensor: L µν = L νµ q Polarized cross section: q P and T invariance: W µν (Q, s T ) P, s T j µ(0)j ν (y) P, s T σ(q, s T ) L µν [W µν (Q, s T ) W µν (Q, s T )] P, s T j µ(0)j ν (y) P, s T = P, s T j ν(0)j µ (y) P, s T A N (Q, s T ) DIS =0

11 Advantage of SIDIS q Dominated by events with two different scales: (l, s e )+A(P A,s) (l )+h(p h )+X ² A large momentum transfer: Localized probe, suppress contribution of complicate matrix elements ² A small momentum scale: Sensitive to parton s motion inside a hadron TMD distributions ² Change from a two-scale problem to an one-scale problem ² Separation of various TMDs and spin states Q = (l l ) 2 p ht 1/fm q Power of varying p ht 1/fm p ht Q 1/fm Transition from TMD factorization to Collinear factorization q Two natural scattering planes:

12 TMD factorization SIDIS L µν W µν (Q, p BT,s T ) Collins book σ 0 φ(x, µ) D(z,µ)δ 2 (p BT ) q TMD parton distribution: F f/p (x, k 1T,S,µ,ζ F )=Tr color Tr Dirac γ + 2 q TMD fragmentation function: k 1 2π Gauge links D h/f (z,k 2T,µ,ζ D )= Tr color N c Tr Dirac 4 q TMDs are more fundamental if we can measure them: γ + z k 2 2π Carry more information on hadron s partonic structure Phase for SSA

13 q Gauge link QCD phase: Color flow gauge links + + Summation of leading power gluon field contribution produces the gauge link: Gauge invariant PDFs: Collinear PDFs: Localized operator with size ~ 1/xp ~ 1/Q q Universality of PDFs: localized color flow Gauge link should be process independent!

14 q Quark TMD distributions: TMD parton distributions ˆk µ = xp µ + k2 T 2xp + nµ + k µ T dk 2 dk + δ(x k + /P + ) Total 8 TMD quark distributions q Gluon TMD distributions, Production of quarkonium, two-photon,

15 Most notable TMDs q Sivers function transverse polarized hadron: Sivers function q Boer-Mulder function transverse polarized quark: Boer-Mulder function Affect angular distribution of Drell-Yan lepton pair production

16 Most notable TMDs II q Collins function FF of a transversely polarized parton: q Fragmentation function to a polarized hadron: Collins function Unpolarized parton fragments into a polarized hadron -

17 TMDs and spin asymmetries q Sivers effect Sivers function: Hadron spin influences parton s transverse motion q Collin s effect Collin s function: Transversity Parton s transverse spin affects its hadronization q TMD factorization is relevant for two-scale problems in QCD: Q 1 Q 2 Λ QCD Separation of different effects?

18 SIDIS is ideal for studying TMDs q SIDIS has the natural kinematics for TMD factorization: (s e )+p(s p ) + h(s h )+X Natural event structure: high Q and low p T jet (or hadron) q Separation of various TMD contribution by angular projection: Lepton plane vs. hadron plane l l 1 N N AUT ( ϕh, ϕs ) = P N + N Collins Sivers = A sin( φ + φ ) + A sin( φ φ ) UT Pretzelosity + A sin(3 φ φ ) UT h h S S UT h S Collins A sin( φ + φ ) h H A A UT h S Sivers UT sin( φ φ ) h S UT 1 1 1T sin(3 φ φ ) h H Pretzelosity UT h S UT 1T 1 UT f D 1

19 Our knowledge of TMDs q Sivers function from low energy SIDIS: EIC can do much better job in extracting TMDs q NO TMD factorization for hadron production in p+p collisions! Collins and Qiu, 2007, Vogelsang and Yuan, 2007, Mulders and Rogers, 2010,

20 Critical test of TMD factorization q TMD distributions with non-local gauge links: SIDIS: DY: q Parity + Time-reversal invariance: The sign change is a critical test of TMD factorization approach

21 Another critical test of TMD factorization q Predictive power of QCD factorization: ² Infrared safety of short-distance hard parts ² Universality of the long-distance matrix elements ² QCD evolution or scale dependence of the matrix elements q QCD evolution: If there is a factorization/invariance, there is an evolution equation q Collinear factorization DGLAP evolution: σ phy (Q, Λ QCD ) f ˆσ f (Q, µ) φ f (µ, Λ QCD ) d dµ σ phy(q, Λ QCD )=0 Scaling violation of nonperturbative functions Evolution kernels are perturbative a test of QCD

22 Evolution equations for TMDs q Collins-Soper equation: b-space quark TMD with γ + Boer, 2001, 2009, Idilbi, et al, 2004 Aybat, Rogers, 2010 Kang, Xiao, Yuan, 2011 Aybat, Collins, Qiu, Rogers, 2011 q RG equations: q Evolution equations for Sivers function: CS: RGs:

23 Scale dependence of Sivers function q Kernel is not perturbative for all b: Aybat, Collins, Qiu, Rogers, 2011 CSS prescription: (not unique) q Q 2 -dependence of Sivers function: Evolved Sivers function q Small-b perturbative contribution match to twist-3: Kang, Xiao, Yuan, 2011

24 Gaussian ansatz for input distributions q Up quark Sivers function: Aybat, Collins, Qiu, Rogers, 2011 Very significant growth in the width of transverse momentum

25 q TMD factorization to collinear factorization: A N (Q 2,p T ) p T Q TMD Q s Collinear Factorization q QCD collinear factorization: From low p T to high p T p T Q p T Ji,Qiu,Vogelsang,Yuan, Koike, Vogelsang, Yuan Two factorization are consistent in the overlap region where Λ QCD p T Q Efremov, Teryaev, 82; Qiu, Sterman, 91, etc. 2 p, s k σ(q, s) t 1/Q = σ LP (Q, s)+ Q s Q σnlp (Q, s)+... σ(s T ) T (3) (x, x) ˆσ T D(z)+δq(x) ˆσ D D (3) (z,z)+... T (3) (x, x) D (3) (z,z) Qiu, Sterman, 1991, Kang, Yuan, Zhou, 2010

26 Twist-3 correlation functions q Twist-2 parton distributions: Kang, Qiu, PRD, 2009 ² Unpolarized PDFs: ² Polarized PDFs: q Two-sets Twist-3 correlation functions: Role of color magnetic force!

27 Evolution equations and kernels q Evolution equation is a consequence of factorization: Factorization: DGLAP for f 2 : Evolution for f 3 : q Evolution kernel is process independent: ² Calculate directly from the variation of process independent twist-3 distributions ² Extract from the scale dependence of the NLO hard part of any physical process Kang, Qiu, 2009 Yuan, Zhou, 2009 Vogelsang, Yuan, 2009 ² Renormalization of the twist-3 operators Braun et al, 2009

28 Variation of twist-3 correlation functions q Closed set of evolution equations (spin-dependent): Kang, Qiu, 2009 Plus two more equations for: and

29 Scale dependence ² Follow DGLAP at large x ² Large deviation at low x (stronger correlation)

30 A sign mismatch q Sivers function and twist-3 correlation: Kang, Qiu, Vogelsang, Yuan, UVCT q direct and indirect twist-3 correlation functions: Calculate T q,f (x,x) by using the measured Sivers functions indirect direct direct indirect

31 Possible interpretations q A node in k T -distribution: Kang, Qiu, Vogelsang, Yuan, 2011 ² Like the DSSV s G(x) ² HERMES vs COMPASS ² Physics behind the sign change? EIC can measure TMDs for a wide range of k T COMPASS HERMES q Large twist-3 fragmentation contribution in RHIC data: If Sivers-type initial-state effect is much smaller than fragmentation effect and two effects have an opposite sign Can be tested by A N of single jet or direct photon at RHIC q A node in x-dependence of Sivers or twist-3 distributions Physics behind the node if there is any Boer,

32 Propose new observables for ep collisions q Process: e()+h(p) jet(p j )(or π,...)+x Kang, Metz, Qiu, Zhou, 2011 Lepton-hadron scattering without measuring the scattered lepton Single hard scale: p jt in lepton-hadron frame q Complement to SIDIS: e()+h(p) e ( )+jet(p j )(or π,...)+x Two scales: Q, p jt in virtual-photon-hadron frame q Key difference in theory treatment: Collinear factorization for TMD factorization for e()+h(p) jet(p j )(or π,...)+x e()+h(p) e ( )+jet(p j )(or π,...)+x Test the consistency between TMD and Twist-3 to SSA in the same experimental setting Jlab, Compass, Future EIC,

33 Analytical formulae q Factorization is valid: Same as hadron-hadron collision to jet + X Kang, Metz, Qiu, Zhou, 2011 q Leading order results: a = l, γ, q, q, g b = q, q, g λ l, λ p : S T : Lepton, hadron helicity, respectively Hadron s transverse spin vector

34 Numerical results q Asymmetries: q Double spin asymmetries very small: s = 50 GeV s = 100 GeV Wandzura-Wilczek approximation: A LT 0.001

35 Good probe of Sivers function q Independent check of the sign mismatch : Red line: T F (x, µ) extracted from fitting SSA in hadronic collisions Blue line: Sivers function Excellent test for the mechanism of SSA possibly at Jlab, surely at future EIC

36 More on future directions q RHIC spin, JLab at 12 GeV, possibly at Compass, q Future EIC: a dedicated QCD machine for the visible matter Yellow book on EIC physics from INT workshop is available: arxiv: submit/ [nucl-th] q A white paper on EIC physics: a writing group appointed by BNL and Jlab is working hard q Physics opportunities at EIC: ² Inclusive DIS Spin, F L, ² SIDIS TMDs, spin-orbital correlations, ² One jet or particle inclusive multiparton quantum correlation, ² GPDs parton spatial distributions ²

37 Summary q QCD factorization/calculation have been very successful in interpreting HEP scattering data q What about the hadron structure? Not much! Thank you! < 1/10 fm q RHIC spin, Jlab12, a future EIC with a polarized hadron beams opens up many new ways to test QCD and to study hadron structure: TMDs, GPDs, q The challenge for theorists: to indentify new and calculable observables that carry rich information on hadron s partonic structure to make measureable predictions

38 Backup slices

39 EIC Kinematics q DIS kinematics: k q EIC (erhic ELIC) basic parameters: Q 2 = q 2 = x B ys x B = Q2 2p q y = p q p k S =(p + k) 2 ² ² ² ² ² ² ² E e = 10 GeV (5-30 GeV available) E p = 250 GeV ( GeV available) S = 100 GeV ( GeV available) localized probe: Q 2 1 GeV x min 10 4 Luminosity ~ 100 x HERA Polarization, heavy ion beam,

40 Interpretation of twist-3 correlation functions q Measurement of direct QCD quantum interference: T (3) (x, x, S ) Qiu, Sterman, 1991, Interference between a single active parton state and an active two-parton composite state q Expectation value of QCD operators: P, s ψ(0)γ + ψ(y ) P, s P, s ψ(0)γ + γ 5 ψ(y ) P, s P, s ψ(0)γ + P, s ψ(0)γ + αβ s T α ig αβ s T α dy ψ(y 2 F + ) P, s β (y 2 ) dy 2 F + β (y 2 ) ψ(y ) P, s How to interpret the expectation value of the operators in RED?

41 A simple example q The operator in Red a classical Abelian case: q Change of transverse momentum: q In the c.m. frame: q The total change: Net quark transverse momentum imbalance caused by color Lorentz force inside a transversely polarized proton

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