Transverse Momentum Dependent Factorization

1. Transverse Momentum Dependent Factorization Feng Yuan Lawrence Berkeley National Laboratory RBRC, Brookhaven National Laboratory 2014/8/21 1

2. Outline Introduction Great progress has been made recently Transverse momentum dependent factorization Twist-three collinear factorization framework Next-to-leading order corrections Outlook 2 2014/8/21

3. Where can we learn TMDs Semi-inclusive hadron production in deep inelastic scattering (SIDIS) Drell-Yan lepton pair production in pp scattering Relevant e+e- annihilation processes Many others… 2014/8/21 EIC Meeting 2008 3

4. Inclusive and Semi-inclusive DIS Inclusive DIS: Partonic Distribution depending on the longitudinal momentum fraction Q Semi-inclusive DIS: Probe additional information for partons’ transverse distribution in nucleon Q 4

5. Why Worry about Factorization? Safely extract nonperturbative information Theoretically under control No breakdown by un-cancelled divergence NLO correction calculable Estimate the high order corrections 5

6. What to Worry for Factorization? Correct definition of TMD parton distributions Gauge Invariance? Soft divergence gets cancelled Hard Part can be calculated perturbatively The cross section can be separated into Parton Distribution, Fragmentation Function, Hard and Soft factors 6

7. Previous Works on Factorization(basis of the present work) Factorization for back-back jet production in e+e- annihilation (in axial gauge) -- Collins-Soper, NPB, 1981 Factorization for inclusive processes -- Collins, Soper, Sterman, NPB, 1985 -- Bodwin, PRD, 1985 -- Collins, Soper, Sterman, in Perturbative QCD, Mueller ed., 1989 7

8. New Context for SIDIS The Gauge Invariant definition for the TMDs -- Brodsky, Hwang, Schmidt, 2002 -- Collins, 2002, 2003 -- Ji, Belitsky, Yuan, 2002 Final state interaction induced by the gauge link has novel effects 8

9. TMD Factorization After soft and collinear factorizations Collins-Soper, 81 Ji-Ma-Yuan, 04 Collins-Metz 04 Scherednikov-Stefanis, 07 • Leading order in pt/Q • Additional soft factor

10. Few words on TMDs Light-cone gauge: Additional Gauge link at -=±∞ Definition 2014/8/21 10

11. TMDs are process dependent (Fragmentation is different) • Gauge link direction changes from DIS to Drell-Yan process • More complicated structure for dijet-correlation in pp collisions, standard factorization breaks • Light-cone singularity beyond Born diagram • Transverse momentum resummation 2014/8/21 11

12. TMD: Naïve Factorization • SIDIS Cross section • Naïve factorization (unpolarized structure function) Hadron tensor TMD distr. TMD frag. Mulders, Tangelman, Boer (96 & 98)

13. QCD Factorization • Factorization for the structure function • q: TMD parton distribution • q hat: TMD fragmentation function • S: Soft factors • H: hard scattering. Impact parameter space

14. One-loop Factorization Purpose: • Verify the factorization • Deduce the correct definition of TMD parton dis. • Estimate of one-loop correction to H Procedure: • Take an on-shell quark as target • Calculate dis. and frag. to one-loop order • Define and calculate the soft factor • Full QCD calculation at one-loop order • Extract the relevant hard part

15. TMD: the definition In Feynman Gauge, the gauge link v is not n to avoid l.c. singularity !!

16. One-Loop Virtual Contribution Double logs : Energy of the hadron

17. One-Loop Real Contribution energy dep.

18. Energy Dependence • The TMD distributions depend on the energy of the hadron! (or Q in DIS) • Introduce the impact parameter representation • One can write down an evolution equation in ζ • K and G obey an RG equation, Collins and Soper (1981) μ independent!

19. TMD Fragmentation functions • Can be defined in a similar way as the parton distribution • Have similar properties as TMD dis.

20. One-loop Factorization (virtual gluon) • Vertex corrections (single quark target) q p′ k p Four possible regions for the gluon momentum k: 1) k is collinear to p (parton distribution) 2) k is collinear to p′ (fragmentation) 3) k is soft (Wilson line) 4) k is hard (pQCD correction)

21. One-Loop Factorization (real gluon) • Gluon Radiation (single quark target) q p′ k p Three possible reagions for the gluon momentum k: 1) k is collinear to p (parton distribution) 2) k is collinear to p′ (fragmentation) 3) k is soft (Wilson line)

22. At one-loop order, we verified the factorization The hard part at one-loop order,

23. All Orders in Perturbation Theory Consider an arbitrary Feynman diagram Find the singular contributions from different regions of the momentum integrations (reduceddiagrams) Power counting to determine the leading regions Factorize the soft and collinear gluons contributions Factorization theorem.

24. Subtract the soft factor in the Dis. • TMD distribution contains the soft contribution • Subtract the soft contribution Zero bin subtraction: Monahar, Stewart, 06; Lee, Sterman, 06; Idilbi, Mehen, 07;

25. Reduced (Cut) Diagrams Leading contribution to a cross section from a diagram. Can be pictured as real spacetime process (Coleman and Norton)

26. Leading Regions The most important reduced diagrams are determined from power counting. No soft fermion lines No soft gluon lines attached to the hard part Soft gluon line attached to the jets must be longitudinally polarized In each jet, one quark plus arbitrary number of collinear long.-pol. gluon lines attached to the hard part. The number of 3-piont vertices must be larger or equal to the number of soft and long.-pol. gluon lines.

27. Leading Region

28. Collinear And Soft Gluons The collinear gluons are longitudinally polarized Use the Ward identity to factorize it off the hard part. The result is that all collinear gluons from the initial nucleon only see the direction and charge of the current jet. The effect can be reproduced by a Wilson line along the jet (or v) direction. The soft part can be factorized from the jet using Grammer-Yennie approximation The result of the soft factorization is a soft factor in the cross section, in which the target current jets appear as the eikonal lines.

29. Factorization After soft and collinear factorizations, the reduced diagram become which corresponds to the factorization formula stated earlier.

30. Large Logarithms Resummation Factorization form inbspace, Large logs: Differential equation respect to The Solution No Large logs.

31. Further factorization for the TMD distribution at large1/b (Collins&Soper 81) Integrated dis. CSS resummation at large 1/b (CSS’85) C functions Our one-loop results for the TMD dis. and frag. can reproduce the C functions,

32. Polarized TMD Quark Distributions Nucleon Unpol. Long. Trans. Quark Unpol. Long. Trans. Boer, Mulders, Tangerman (96&98) 32

33. Summary and Outlook Based on recent developments, we have shown that the TMD and collinear factorization approaches are consistent in the intermediate transverse momentum region Next-to-leading order QCD corrections to the Drell-Yan SSA have been calculated for the first time Extension to other processes and observables shall follow accordingly 33