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
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
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!
q From HERA: Inclusive DIS cross section
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
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?
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 ),...
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!
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
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
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:
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
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!
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,
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
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 -
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?
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
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,
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
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
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:
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
Gaussian ansatz for input distributions q Up quark Sivers function: Aybat, Collins, Qiu, Rogers, 2011 Very significant growth in the width of transverse momentum
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
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!
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
Variation of twist-3 correlation functions q Closed set of evolution equations (spin-dependent): Kang, Qiu, 2009 Plus two more equations for: and
Scale dependence ² Follow DGLAP at large x ² Large deviation at low x (stronger correlation)
A sign mismatch q Sivers function and twist-3 correlation: Kang, Qiu, Vogelsang, Yuan, 2011 + 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
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,
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,
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
Numerical results q Asymmetries: q Double spin asymmetries very small: s = 50 GeV s = 100 GeV Wandzura-Wilczek approximation: A LT 0.001
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
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/0295324 [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
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
Backup slices
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 (50-325 GeV available) S = 100 GeV (30-200 GeV available) localized probe: Q 2 1 GeV x min 10 4 Luminosity ~ 100 x HERA Polarization, heavy ion beam,
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?
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