QCD Collinear Factorization for Single Transverse Spin Asymmetries

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1 INT workshop on 3D parton structure of nucleon encoded in GPD s and TMD s September 14 18, 2009 QCD Collinear Factorization for Single Transverse Spin Asymmetries Iowa State University Based on work with Collins, Ji, Kang, Kouvaris, Sterman, Vogelsang, Yuan, 1

2 Outline Factorization vs non-factorization Necessary condition for pqcd factorization Collinear factorization for ONE hard scale processes Collinear factorization for TWO scale processes Collinear factorization + Sudakov resummation =\= TMD factorization TMD vs Collinear factorization Scale evolution in the collinear factorization Summary and outlook 2

3 Factorization vs non-factorization Cross section with identified hadron(s): PQCD does not work for the dynamics at hadronic scale: 1/fm PQCD alone cannot predict such a cross section Cross section with ONE identified hadron: Factorization to all powers OPE Function of Cross section with TWO or more identified hadron: OPE does not help for factorization Inclusive Drell-Yan cross section is not fully factorizable! 3

4 The necessary condition for factorization Experiments do not see partons directly: Perturbative pinch singularity: Dominated by k 2 ~0 Similar approximation for the p-integration Cross section is factorized into 3 parts: They are separated by long-lived parton states comparing to 1/Q Long-lived parton state necessary condition for Factorization 4

5 Collinear gluons: Other leading contributions Collinear longitudinally polarized gluons do not change the collinear collision kinematics Soft interaction: If the interaction between two jet functions can resolve the details of the jet functions, the jet functions could be altered before hard collision break of the factorization 5

6 Pinch => On-shell parton: Collinear Factorization Collinear approximation: if DIS: q k~x p p Hadron is approximated by a beam of partons of momentum fraction x i Parton s transverse motion is integrated into parton distributions: Parton distributions are process independent, and QCD collinear factorization has been very successful 6

7 Why Drell-Yan factorization makes sense? Pinch singularities Long-lived partonic states lowest order kinematics determines the process Approximation: Drell-Yan formula 7

8 QCD dynamics is rich and complicate Leading pinch surface: Analysis of leading (pinch or singular) integration regions gives the following: one-pair physical parton from each hadron Factorization: Long-distance distributions are process independent 8

9 Eikonalization of collinear gluons 9

10 Factorization of PDFs 10

11 Trouble from soft gluons Pinch from spectator interaction: 11

12 Soft gluons take care of themselves Most technical part of the factorization Sum over all final states to remove all poles in one-half plane no more pinch poles Deform the q ± integration out of the trapped soft region Eikonal approximation, unitarity, causality, and gauge invariance 12

13 Factorization high P T single particle Nayak, Qiu, Sterman, Eikonalization of gluons collinear to the final-state hadron Factorization of the fragmentation function Factorization of gluons from the initial-state hadrons same as the factorization of Drell-Yan Normalization of short-distance hard parts is fixed by the definition of the universal PDFs and FFs 13

14 Factorization beyond leading power I Cross section has the same dimension: Dimension of the power suppression is matched by the dimension of high twist matrix elements multi-parton correlation functions In collinear factorization, hadron mass does not enter the power expansion of partonic scattering: One active parton subprocess contributes to all power 14

15 Factorization beyond leading power II Multiparton correlation functions more active partons: Gluon field operator is not a good operator Need contribution from two parton process building blocks: Kinematics fix only one active momentum fraction: Let either or can be very soft Key difficulty in factorization beyond leading power: Impossible to separate the soft gluon from the zero momentum active parton in multiparton correlation functions Only collinearly factorizable power correction: Power correction with only one non-leading distribution! Qiu and Sterman, 1991 Factorize the soft interaction from all leading PDFs 15

16 Collinear factorization for TWO scale process Collinear factorization formalism should work for processes with TWO or more large observed scales But, perturbative expansion in powers of may not converge fast enough due to large logarithms Improvement resummation of large logs: Example: Sudakov resummation, Note: Resummation is simply a reorganization of the perturbative series in collinear factorization It is still the collinear factorization, even though it improves its predictive power to a wider kinematic regime Collinear factorization + Sudakov resummation =\= TMD factorization Note: There is NO Boer-Bulders function in collinear factorization 16

17 TMD vs collinear factorization - I TMD factorization and collinear factorization cover different regions of kinematics One complements the other, but, cannot replace the other! Predictive power of both formalisms relies on the validity of the factorization Very limited processes with valid TMD factorization Drell-Yan transverse momentum distribution: o quark Sivers function Semi-inclusive DIS for light hadrons: o mixture of quark Sivers and Collins function Collins, Qiu, 2007 Vogelsang, Yuan, 2007 e+e- to two light hadrons: o Collins function TMD formalism is more sensitive to non-perturbative dynamics 17

18 TMD vs collinear factorization - II Overlap region between TMD and collinear factorization: A perturbative region where collinear factorization works and the TMD distributions are generated by a perturbative mean A consistency between two formalisms in this region is crucial: o a consistent phenomenological matching of two regions o role of additional functions in TMD approaches o explore the rich dynamics in the transition region JLab12,, and EIC can help in a great deal here! Recall Wednesday discussion: Drell-Yan angular distribution at low q T Four helicity structure functions reduces to Two singular asymptotic functions Role of power corrections to the resummation formalism 18

19 Collinear factorization for SSA QCD Collinear factorization approach is more relevant Expansion Too large to compete! Three-parton correlation SSA difference of two cross sections with spin flip is power suppressed compared to the cross section Sensitive to twist-3 multi-parton correlation functions Integrated information on parton s transverse motion 19

20 SSA in QCD Collinear Factorization I All scales >> QCD : Factorization at twist-3 initial-state: Twist-3 quark-gluon correlation: Normal twist-2 distributions 20

21 SSA in QCD Collinear Factorization II Qiu, Sterman, 1998 Factorization formalism for SSA of single hadron: Only one twist-3 distribution in each term! 1 st term: Collinear version of Sivers effect 2 nd term: Collinear version of transversity + BM function 3 rd term: Collinear version of Collins effect 21

22 Scale dependence of SSA Almost all existing calculations of SSA are at LO: Strong dependence on renormalization and factorization scales Artifact of the lowest order calculation Improve QCD predictions: Complete set of twist-3 correlation functions relevant to SSA LO evolution for the universal twist-3 correlation functions NLO partonic hard parts for various observables NLO evolution for the correlation functions, Current status: Two sets of twist-3 correlation functions LO evolution kernel for and NLO hard part for SSA of p T weighted Drell-Yan Kang, Qiu, 2009 Vogelsang, Yuan,

23 Two sets of twist-3 correlation functions Twist-2 distributions: Unpolarized PDFs: Polarized PDFs: Two-sets Twist-3 correlation functions: Kang, Qiu, PRD,

24 Evolution equations and evolution kernels Evolution equation is a consequence of factorization: Factorization: DGLAP for f 2 : Evolution for f 3 : Evolution kernel is process independent: Calculate directly from the variation of process independent twist-3 distributions Extract from the scale dependence of the NLO hard part of any physical process Both approaches should give the same kernel Kang, Qiu, 2009 Yuan, Zhou, 2009 Vogelsang, Yuan,

25 Evolution equations I Feynman diagram representation of twist-3 distributions: Kang, Qiu, 2009 Different twist-3 distributions diagrams with different cut vertices Collinear factorization of twist-3 distributions: Cut vertex and projection operator in LC gauge: 25

26 Evolution equations II Closed set of evolution equations (spin-dependent): Plus two more equations for: and 26

27 Evolution equations III Distributions relevant to SSA: Important symmetry property: These two correlation functions do not give the gluonic pole contribution directly 27

28 Evolution kernels Feynman diagrams: Kang, Qiu, PRD, 2009 LO for flavor non-singlet channel: 28

29 Quark: Leading order evolution equations I Kang, Qiu, PRD, 2009 Antiquark: All kernels are infrared safe Diagonal contribution is the same as that of DGLAP Quark and antiquark evolve differently caused by tri-gluon 29

30 Leading order evolution equations II Gluons: Kang, Qiu, PRD, 2009 Similar expression for Kernels are also infrared safe diagonal contribution is the same as that of DGLAP Two tri-gluon distributions evolve slightly different has no connection to TMD distribution Evolution can generate as long as 30

31 Leading order evolution equations III Evolution equations for diagonal correlation functions are not closed! Model for the off-diagonal correlation functions: For the symmetric correlation functions: 31

32 Scale dependence of twist-3 correlations Follow DGLAP at large x Large deviation at low x (stronger correlation) 32 Kang, Qiu, PRD, 2009

33 D-meson production at EIC Dominated by the tri-gluon subprocess: Kang, Qiu, PRD, 2008 Active parton momentum fraction cannot be too large Intrinsic charm contribution is not important Sufficient production rate Single transverse-spin asymmetry: SSA is directly proportional to tri-gluon correlation functions 33

34 Features of the SSA in D-production at EIC Dependence on tri-gluon correlation functions: Separate and by the difference between and Model for tri-gluon correlation functions: Kinematic constraints: Note: The has a maximum SSA should have a minimum if the derivative term dominates 34

35 Minimum in the SSA of D-production at EIC SSA for D 0 production ( only): Kang, Qiu, PRD, 2008 COMPASS EIC Derivative term dominates, and small dependence Asymmetry is twice if, or zero if Opposite for the meson Asymmetry has a minimum ~ z h ~

36 Maximum in the SSA of D-production at EIC SSA for D 0 production ( only): Kang, Qiu, PRD, 2008 The SSA is a twist-3 effect, it should fall off as 1/P T when P T >> m c For the region, P T ~ m c, 36

37 Rapidity: SSA of D-meson production at RHIC Solid: Dashed: Dotted: 37 No intrinsic Charm included Kang, Qiu, Vogelsang, Yuan, 2008

38 SSA of D-meson production at RHIC P T dependence: Solid: Dashed: Dotted: 38 No intrinsic Charm included Kang, Qiu, Vogelsang, Yuan, 2008

39 Summary and outlook Collinear factorization and TMD factorization cover different regions of kinematics Two are complementary to each other One cannot replace the other! TMD factorization seems to involve more TWO-parton correlations, and TMD distributions resum a subset of power corrections of collinear formalism Exploring the transition region between the TMD and Collinear factorization should be very interesting JLab12,, and EIC can definitely help! Thank you! 39