QCD Factorization and PDFs from Lattice QCD Calculation

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1 QCD Evolution 2014 Workshop at Santa Fe, NM (May 12 16, 2014) QCD Factorization and PDFs from Lattice QCD Calculation Yan-Qing Ma / Jianwei Qiu Brookhaven National Laboratory ² Observation + Motivation ² Our proposal ² Case study ² Summary and outlook Based on work done with Tomomi Ishikawa, Yan-Qing Ma, Shinsuke Yoshida, arxiv: ,

2 Global QCD analyses a successful story q World data with Q > 2 GeV + Factorization: DIS: H-H: F 2 (x B,Q 2 )=Σ f C f (x B /x, µ 2 /Q 2 ) f(x, µ 2 ) dσ dydp 2 T =Σ ff f(x) dˆσ ff dydp 2 T f (x ) + DGLAP Evolution: f(x, µ 2 ) ln µ 2 =Σ f P ff (x/x ) f (x,µ 2 )

3 Global QCD analyses a successful story q World data with Q > 2 GeV + Factorization: DIS: H-H: F 2 (x B,Q 2 )=Σ f C f (x B /x, µ 2 /Q 2 ) f(x, µ 2 ) dσ dydp 2 T =Σ ff f(x) dˆσ ff dydp 2 T f (x ) + DGLAP Evolution: f(x, µ 2 ) ln µ 2 =Σ f P ff (x/x ) f (x,µ 2 )

4 Uncertainties of PDFs singlet sector non-singlet sector

5 PDFs at large x q Testing ground for hadron structure at x è 1: ² d/u 1/2 SU(6) Spin-flavor symmetry ² d/u 0 Scalar diquark dominance ² ² d/u 1/5 d/u 4µ2 n/µ 2 p 1 4 µ 2 n/µ 2 p 0.42 pqcd power counting Local quark-hadron duality

6 PDFs at large x q Testing ground for hadron structure at x è 1: ² d/u 1/2 SU(6) Spin-flavor symmetry ² u/u 2/3 d/d 1/3 ² d/u 0 Scalar diquark dominance ² u/u 1 d/d 1/3 ² d/u 1/5 pqcd power counting ² u/u 1 d/d 1 ² d/u 4µ2 n/µ 2 p 1 4 µ 2 n/µ 2 p 0.42 Local quark-hadron duality ² u/u 1 d/d 1 Can lattice QCD help?

7 PDFs from lattice q Moments of PDFs matrix elements of local operators x n (µ 2 ) q 1 0 dx x n q(x, µ 2 ) q Works, but, hard and limited moments: x 2 q x 3 q Dolgov et al., hep-lat/ Gockeler et al., hep-ph/

8 PDFs from lattice q How to get x-dependent PDFs with a limited moments? ² Assume a smooth functional form with some parameters ² Fix the parameters with the lattice calculated moments xq(x) =ax b (1 x) c (1 + x + γx) W. Dermold et al., Eur.Phys.J.direct C3 (2001) 1-15 Cannot distinguish valence quark contribution from sea quarks

9 Ji s new idea q Quasi quark distribution (spin-averaged): f q ( x, µ 2,P z )= q Cut-vertex notation: dξz 4π e i xp zξ z P ψ(ξ z )γ z exp ig ξz γ n z δ x k n z 2P n z P n z 0 dη z A z (η z ) Ji, arxiv: ψ(0) P d 4 k (2π) 4 q Features: Quark fields separated along the z-direction not boost invariant! Perturbatively UV power divergent: quark gluon This distribution can be calculated using standard lattice method quasi-pdfs è normal PDFs when P z è 1 q(x, µ 2 dy x,p z )= y Z y, µ Λ q(y, µ 2 2 Matching: )+O P z x µ/p z (µ/p z ) 2 P 2 z, M 2 P 2 z

10 Quasi-PDFs have no parton interpretation q Normal PDFs conserve parton momentum: M = q 1 dx xf q (x)+ 0 q Quasi-PDFs do not conserve parton momentum: 1 dx xf q (x) + 0 = dx xf q (x)+ 1 dx xf g (x) q 2 1 = 2(P + ) 2 P T ++ (0) P = constant M = q = q = 0 dx x f q ( x)+ dx x f q ( x) dx x f q ( x) + dx x f g ( x) 1 0 dx xf g (x) 1 2(P z ) 2 P [T zz (0) g zz (...)] P = constant Note: Quasi-PDFs are not boost invariant 0 T µν Energy-momentum tensor dx x f g ( x)

11 The first try Lin et Ji, al., arxiv:1305. arxiv:

12 Observation q Lattice QCD calculates single hadron matrix elements: 0 O(ψ, ψ, A) 0 = 1 DADψDψ e is(ψ,ψ,a) O(ψ, ψ, A) Z P P P P P P With an Euclidean time P z O(ψ, ψ, A) P z q Collinear divergence (CO) from the region when k T è 0: F.T. of P z O(ψ, ψ) P z Same CO divergence regardless Minkowski or Euclidean time q PDFs should cover all leading power CO divergences of F.T. of P z O(ψ, ψ, A) P z Single hadron matrix elements with a large momentum scale

13 Our proposal - a slightly modified idea Ma and Qiu, arxiv: q Not try to go the light-cone Less ambiguous: Lattice cross Finite P z + QCD factorization ² Identify single-hadron cross sections, calculable in Lattice QCD, and factorizable into a convolution with normal PDFs σ i ( x, Q 2, { ν}) =Σ f Cf i ( x/x, µ 2 / Q 2, { ν}) f(x, µ 2 )+O (1) 1 Q α e.g. σ i ( x, µ 2,P z ) f i ( x = k z /P z, µ 2,P z ) Ji s quasi-pdfs σ i ( x, µ 2,P z ) P z O(ψ, A) P z Need a large scale if O(ψ, A) is made of conserved currents KEY: all order factorization into the Normal PDFs with µ 2 ( xp z ) 2 Λ 2 QCD ² Calculate the Lattice cross sections in Lattice QCD the LHS of (1) Measure high energy scattering cross sections on Lattice ² Global analysis of Lattice cross sections to extract the normal PDFs Note: x, µ, { ν} are finite parameters: rapidity, hard scale,

14 Lattice cross section q Definition: σ F.T. of P z O(ψ, ψ, A) P z ² Calculable in lattice QCD with an Euclidean time, E ² Its continuum limit is UV and IR safe perturbatively ² All CO divergences can be factorized into the normal PDFs with perturbatively calculable hard coefficient functions Collision energy P z s rapidity x y Hard momentum transfer 1/a µ Q q Factorization:

15 Case study factorization of quasi-pdfs q The Quasi-quark distribution, as an example: q( x, µ 2,P z )= dyz 4π ei xp zy z P ψ(y z )γ z exp ² Feynman diagram representation: Σ cut k k γ z 2P z δ ig yz 0 x k z P z P z s µ/ x Ma and Qiu, arxiv: dyza z (yz) ψ(0) P P ² Like PDFs, it is IR finite P P 2 0 ² Unlike PDFs, it is linear UV divergent (quadratic UV divergent for gluon) Potential trouble! - show power UV div. decouple from Log UV of PDFs ² CO divergence is the same as that of normal PDFs Show to all orders in perturbation theory

16 All order QCD factorization of CO divergence q Generalized ladder decomposition in a physical gauge n A = A + =0 Ma and Qiu, arxiv: Mueller, PRD 1974 q C 0, K 0 : 2PI kernels ² Only process dependence: ² 2PI are finite in a physical gauge Ellis, Georgi, Machacek, Politzer, Ross, 1978, 1979

17 All order QCD factorization of CO divergence q 2PI kernels Diagrams: q Ordering in virtuality: P 2 k 2 µ 2 + power suppressed Cut-vertex for normal quark distribution Logarithmic UV and CO divergence q Renormalized kernel - parton PDF: K d 4 k i δ x i k+ γ n p + Tr 2p n K 0 γ p 2 + UVCT

18 All order QCD factorization of CO divergence q Projection operator for CO divergence: P K Pick up the logarithmic CO divergence of K q Factorization of CO divergence: Normal Quark distribution CO divergence free coefficient All CO divergence of quasi-quark distribution

19 One-loop example: quark è quark q Expand the factorization formula: Ma and Qiu, arxiv: q Feynman diagrams: Same diagrams for both f q/q and f q/q But, in different gauge q Gauge choice: n z A = 0 for fq/q n A = 0 for f q/q Gluon propagator: d αβ (l) = g αβ + lα n β z + n α z l β l z n2 z l α l β l 2 z with n 2 z = 1

20 One-loop quasi-quark distribution in a quark q Real + virtual contribution: Ma and Qiu, arxiv: where q Cancelation of CO divergence: Only the first term is CO divergent for 0 < y < 1, which is the same as the divergence of the normal quark distribution necessary! q UV renormalization: Different treatment for the upper limit of l 2 z integration - scheme Here, a UV cutoff is used other scheme is discussed in the paper

21 One-loop coefficient functions q MS scheme for f q/q (x, µ 2 ): Ma and Qiu, arxiv: where q Generalized + description: t = x/x For a testing function h(t) q Explicit verification of the factorization at one-loop: Coefficient functions for all partonic channels are IR safe and finite! C (1) i/j (t, µ2,µ,p z ) with i, j = q, q, g

22 From Lattice x-sections to PDFs q BNL RBRC efforts: Case study Factorization of Quasi-PDFs Ma, et al. On-going Lattice perturbation theory q To do list: ² Identify more good lattice cross sections ² Extend to GPDs (effectively collinear factorization), and ² TMDs (operators with two momentum or position scales)

23 Summary and outlook q lattice cross sections = hadronic matrix elements that are calculable in Lattice QCD and factorizable in QCD factorization, without requiring P z è (a parameter) e.g. σ i ( x, µ 2, { ν}) f i ( x = k z /P z, µ 2,P z ) Ji s quasi-pdfs q Extract PDFs by global analysis of data on Lattice cross sections. Same should work for other distributions q Conservation of difficulties complementarity: High energy scattering experiments less sensitive to large x parton distribution/correlation Lattice factorizable cross sections more suited for large x partons q Lattice QCD can calculate PDFs, but, more works are needed! Thank you!

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25 Partonic luminosities q - qbar g - g