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M ethod of R egions and I ts A pplications

M ethod of R egions and I ts A pplications. Graduate University of the CAS Deshan Yang. Outline. Introduction Examples of Method of Regions Connections to Effective Field Theory Applications Summary. Victor Frankenstein’s Idea of Science. Modern Physics

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M ethod of R egions and I ts A pplications

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  1. Method of Regionsand Its Applications Graduate University of the CAS Deshan Yang 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  2. Outline Introduction Examples of Method of Regions Connections to Effective Field Theory Applications Summary 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  3. Victor Frankenstein’s Idea of Science • Modern Physics • Understand the nature of the Universe • qualitatively and quantitatively. • What can we do? • Anatomy--approaching to the truth gradually • Cut the body into pieces and study each part • Stitch them together and hope for the best • Scientist: Frankenstein • To create the Frankenstein’s monster or an angel? 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  4. Beauty charmless decay • Many scales • Many couplings • Many hadrons • Difficulties： • Strong interactions • Way-out： • Factorization 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  5. Factorization 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  6. Questions to be answered • How to separate the contributions from the different scales? • How to establish the RGEs to resum the large logarithms? • How to estimate or compensate the loss due to the power corrections? Method of regions can help! 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  7. Integration by regions For a Feynman integral containing small parameters (multiple-scale problem) in dimensional regularization • Divide the space of the loop momentainto variousregions and , in eachregion, expand the integrand into a Taylor series with respect to the parameters that are considered small there; • Integrate the integrand, expanded in the appropriate way in every region, over the whole integration domain of the loop momenta; • Add up all the expanded integrals in all regions, we reproduce the Taylor series of the original Feynman integral with respect to the small parameters exactly. • Finally, a multiple-scale problem is divided into single (less) scale problems. 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  8. Example 1: Two-masses dependent integral 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  9. Cut-off regularization IR div. UV div. 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  10. Dimensional regularization • The expansion is valid up to any order of a; • The integral in each region is the function of only one scale and simpler than the original integral; • The factious divergence in each region is cancelled after adding up the contributions from large scale region and small scale region. IR div. UV div. 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  11. Example 2: Threshold Expansion Beneke & Smirnov, NPB1998 • Small parameter: • Hard region: • Potential region: • Soft/Ultra-soft region: or Tadpole diagrams: 0 in DR 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  12. Adding up 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  13. Remarks on method of regions 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  14. Effective Field Theory 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  15. Application 1: Effective weak Hamiltonian 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  16. Effective operators 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  17. First step factorization in B decays 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  18. Example of matching : Tree-level 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  19. One-loop level matching equation 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  20. One-loop matching equation 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  21. Hard part 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  22. Putting together 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  23. Renormalization 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  24. Application 2: Heavy-to-light Form-factors 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  25. Factorization formula There’sanother factorization formula in which the transverse momenta of the patrons are invoked to avoid the endpoint singularity. Kurimoto, Li, Sanda 2002 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  26. Factorization formula in SCET 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  27. Matching procedure 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  28. More on matching 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  29. “Hard” contribution 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  30. Wilson coefficients 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  31. Wilson coefficients 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  32. RGEs 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  33. Jet functions 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  34. Application 3: B two-body charmless decay 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  35. Matching onto SCETII 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  36. Factorization formula 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  37. Hard-spectator interaction 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  38. NNLO vertex corrections Complete NNLO: G.Bell, 2009; Beneke,Li,Huber 2009 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  39. Application 4: Exclusive single quarkonium production 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  40. NRQCD factorization For single quarkonium production • : NRQCD operator with definite velocity power counting • multi-scale problem: Q>>m • stability of the perturbation: large log(Q/m) may need the resummation. 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  41. Refactorization • At the leading power of velocity, • The hard kernel is the same as the similar process in which the quarkonium is replaced by a flavor singlet light meson. • Since , the LCDA of bounded heavy quark and anti-quark can be calculated perturbatively. • Ma and Si, PRD 2006; Bell and Feldmann, JHEP 2007; 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  42. Example: • NRQCD factorization up to leading power of velocity: • The short-distance contribution is parameterized as • The equivalent computation is to calculate the on-shell heavy • quark anti-quark pair with equal momentum and the same • quantum number as the quarkonium. At the tree level, 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  43. One-loop level  Sang, Chen, arXiv:0910.4071; Li, He, Chao arXiv:0910.4155 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  44. Leading regions • Hard Region: • Collinear region: • Anti-collinear region: • Potential region: • Soft region: • Ultra-soft region: NRQCD regions Non-perturbative 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  45. Form factor • NRQCD: • Collinear factorization: • Hard-kernel: • at tree level • Light-cone distribution amplitude Ma and Si, PRD 2006; Bell and Feldmann, JHEP 2007; 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  46. RGE for LCDA Brodsky-Lepage kernel: Resum the leading logarithms where 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  47. NLO results (preliminary) • Hard Part • Collinear Part • Total Results Braaten, PRD 1981; Ma and Si, PRD 2006; Bell and Feldmann, JHEP 2007; Sang, Chen, arXiv:0910.4071; Li, He, Chao arXiv:0910.4155 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  48. Summary • Method of regions: Not mathematically proved, but no counter-examples so far. • Intimately connected to the calculation of the matching coefficients in EFT. • Advantages: Multiple scale problems simplified to single scale problems; • Disadvantages: How to find the relevant regions? (No general procedure!) 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

  49. 谢谢！ 2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC

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