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Three partons in kT factorization

Three partons in kT factorization. Hsiang- nan Li Academia Sinica May 16, 2012 Ref: Chen and Li, 1104.5398; 1112.5059. Outlines . Introduction Gauge invariance 3-parton contributions B -> pi form factors Summary. Introduction .

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Three partons in kT factorization

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  1. Three partons in kT factorization Hsiang-nan Li Academia Sinica May 16, 2012 Ref: Chen and Li, 1104.5398; 1112.5059

  2. Outlines • Introduction • Gauge invariance • 3-parton contributions • B -> pi form factors • Summary

  3. Introduction • kT factorization has been pushed to subleading level • NLO for pion transiton, EM form factors, B->pi form factors • Next-to-leading power in 1/Q needs to be examined too • Have examined 2-parton twist-3 • Consider 3-parton contributions, which should not be separated from 2-parton twist-3

  4. Power expansion in kT • kT is kept in propagator denominators • Can this be extended to higher power consistently? • Will there be double counting? • Is there gauge invariance at higher power?

  5. Form factors • Pion EM form factor is symmetric under flip of initial and final states • 3 partons on both sides, power of 1/Q^2 • B->pi form factor is not symmetric • 3 partons on one side only, power of 1/mB • 3-parton contribution vanishes as mB->0 • Need to confirm gauge invariance first • 3-parton contributions negligible, few percents

  6. Gauge invariance

  7. Gauge dependence • Two sources of gauge dependence: • Transverse momenta of 2-parton state • 3-parton state • The two sources cancel as combined into

  8. kinematics • LO diagrams for pion EM form factor • kinematics

  9. Fermion and color flows • Fierz transformation • Color identity focus on this one 2-parton 3- parton j i j i l k l k

  10. 2-to-2 gauge dependence • Spin projectors for initial and final state in LO diagrams • Gluon propagator in covariant gauge gauge parameter

  11. Amplitude from Fig. 1(a) • Gauge dependent piece • Extract term proportional to k1 and k2, ie., partial derivative of quark fields Ward identity valence quark valence anti-quark

  12. Amplitude from Fig.1(b) valence quark valence anti-quark

  13. 3-to-2 gauge dependence • Diagrams • A, B,…, and H represent attachments of additional valence gluon from initial state • Attachments to initial valence lines should be included for U(1) gauge invariance, which lead to 2-parton twist-3 DAs

  14. Attachment A as an example • Color factorization • Initial-state spin projector a b

  15. Extraction of gauge dependence • Amplitude from Attachment A • Extract term proportional to k2

  16. Other 3-to-2

  17. Gauge invariance • Sum over all attachments • A and B added into with color factor • Second term of G and H added into • Sum is independent of l1, which can be integrated out, • Equation of motion for

  18. 2-to-3 and 3-to-3 • 2-to-3 gauge dependence • 3-to-3 • Use equation of motion again

  19. 3-parton contributions

  20. Three-parton contributions • Consider the matrix element • Insert does not change power behavior • Employ . Just need to consider 3-parton state • gives 3-parton twist-4 • does not contribute

  21. Parton momenta and structures • Initial quark, anti-quark, gluon carry • Structures for initial- and final-states

  22. Dominant diagram • With 4-gluon vertex

  23. Factorization formula • For the dominant diagram obey equation of motion with 2-parton DAs

  24. Other diagrams

  25. More diagrams

  26. Numerical results

  27. B -> pi form factors

  28. Gauge dependence from 2 partons • LO diagrams for B->pi form factor • kinematics

  29. Amplitude from Fig.1(a) • Spin projectors for initial and final states • Gauge dependence • Extract term proportional to k2

  30. Amplitude from Fig.1(b) • Gauge dependent piece • Extract term proportional to k2 • Gauge dependence from Figs.1(a) and 1(b) cancel

  31. Gauge dependence from 3 partons • 2-to-3 diagrams with one additional valence gluon from the pion side • Spin projector for the pion replaced by • Color factorization for Attachment A

  32. Amplitudes from all attachments • Other attachments vanish • They cancel each other. No need of equation of motion

  33. 2-to-3 contribution • B -> pi form factors • Hard kernels proportional to mB

  34. 3-parton B wave function • 3-parton matrix elements • Sum rules by Grozin, Neubert • Nishikawa, Tanaka

  35. 3-to-2 contribution • Adopt 3-parton B meson wave function • 3-to-2 hard kernel, also proportional to mb

  36. Wave functions

  37. Numerical results • Cancellation between 2-to-3 and 3-to-2 contributions same order of magnitude as from Gegenbauer terms in 2-parton pion DAs

  38. Figures • Contributions from GN parameters larger than NT parameters LO

  39. Summary on various contributions • B meson spin projector for 2 partons • 1st , leading power; 2nd, 30%, 3rd, few percents • 3-parton contributions are also few percents • 3-parton contributions are of the same order of magnitude as higher Gegenbauer terms of 2-parton DAs integration of

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