1. 1 Sudakov and heavy-to-light form factors in SCET Zheng-Tao Wei Nankai University 2009.9.9, KITPC, Beijing

2. 2 Wei, PLB586 (2004) 282, Wei, hep-ph/0403069, Lu, et al., PRD (2007)  Introduction to SCET  Sudakov form factor  Heavy-to-light transition form factors  Summary

3. 3 I. Soft-Collinear Effective Theory  The soft-collinear effective theory is a low energy effective theory for collinear and soft particles. (Bauer, Stewart, et al.; Beneke, Neubert….) (1) It simplifies the proof of factorization theorem at the Lagrangian and operator level. (2) The summation of large-logs can be performed in a new way. 2009.9.9, KITPC, Beijing

4. 4 Diagrammatic analysis and effective Lagrangian eikonal approximation 2009.9.9, KITPC, Beijing

5. 5 Transforming the diagrammatic analysis into an effective Lagrangian LEET 2009.9.9, KITPC, Beijing

6. 6 Power counting Momentum Field Degrees of freedom Reproduce the full IR physics 2009.9.9, KITPC, Beijing

7. 7 The effective Lagrangian: The effective interaction is non-local in position space. Two different formulae: hybrid momentum-position space and position space representation. 2009.9.9, KITPC, Beijing

8. 8 Gauge invariance The ultrasoft field acts as backgroud field compared to collinear field. The collinear and ultrasoft gauge transformation are constrained in corresponding regions, 2009.9.9, KITPC, Beijing

9. 9 Wilson lines Gauge invariant operators: (basic building blocks) No interaction with usoft gluons 2009.9.9, KITPC, Beijing

10. 10 Matching: mismatch? New mode, such as soft-collinear mode proposed by Neubert et al.? Endpoint singularity? 2009.9.9, KITPC, Beijing

11. 11 SCET(I) SCET(II) Two step matching: 1.Integrate out the high momentum fluctuations of order Q, 2. Integrate out the intermediate scale (hard-collinear field) 2009.9.9, KITPC, Beijing

12. 12 12  Form factor The matrix elements of current operator between initial and final states are represented by different form factors. Form factors are important dynamical quantity for describing the inner properties of a fundamental or composite particle. 2009.9.9, KITPC, Beijing III. Sudakov form factor

13. 13 The form factor (only the first term F 1 (Q 2 )) in the asymptotic limit q 2 →∞ is called Sudakov form factor. (in 1956) The interaction of a fermion with EM current is represented by At q 2 =0, the g-factor is given by and the anomalous magnetic moment is 2009.9.9, KITPC, Beijing

14. 14  The large double-logarithm spoils the convergence of pertubative expansion.  The summation to all orders is an exponential function.  The form factor is strongly suppressed when Q is large.  In phenomenology, it relates to most high energy process in certain momentum regions, DIS, Drell-Yan, pion form factor, etc. The naïve power counting is strongly modified (at tree level F=1). 2009.9.9, KITPC, Beijing

15. 15 Methods of momentum regions (by Beneke, Smirnov, etc) The basic idea is to expand the Feynman diagram integrand in the momentum regions which give contributions in dimensional regularization. Each region is involved by one scale. 2009.9.9, KITPC, Beijing

16. 16 Regularization method Introduce a cutoff scale δ in both k + and k -. DOF Bauer (2003)

17. 172009.9.9, KITPC, Beijing

18. 18 Factorization Step 1: the separation of hard from collinear contributions. Step 2: the separation of soft from collinear functions. 2009.9.9, KITPC, Beijing

19. 19 Evolution The anomalous dimension depends on the renormalization scale. The exponentiation is due to the RGE. The suppression is caused by the positive anomalous dimension. Two-step running: 2009.9.9, KITPC, Beijing

20. 20 Exponentiation and scaling Exponential of logs can be considered as a generalized scaling. 2009.9.9, KITPC, Beijing

21. 21 Comparisons with other works 1.The leading logarithmic approximation method sums the leading contributions from ladder graphs to all orders. The ladder graphs constitutes a cascade chain: qq->qq->…->qq. There are orderings for Sudakov parameters. 2.Korchemsky et al. used the RGE for a soft function whose evolution is determined by cusp dimension. The cusp dimension contains a geometrical meaning. 2009.9.9, KITPC, Beijing

22. 22 3. CSS use a diagrammatic analysis to prove the factorization. The RGEs are derived from gauge- dependence of the jet and hard function. The choice of gauge is analogous to the renormalization scheme. 2009.9.9, KITPC, Beijing

23. 23 IV. Heavy-to-light transition form factor The importance of heavy-to-light form factors:  CKM parameter V ub  QCD, perturbative, non-perturbative  basic parameters for exclusive decays in QCDF or SCET  new physics… At large recoil region q 2 <<m b 2, the light meson moves close to the light cone. Light cone dominance 2009.9.9, KITPC, Beijing

24. 24 Hard gluon exchange: soft spectator quark → collinear quark Perturbative QCD is applicable. Hard scattering 2009.9.9, KITPC, Beijing

25. 25 Endpoint singularity endpoint singularity Factorization of pertubative contributions from the non-perturbative part is invalid. There are soft contributions coming from the endpoint region. 2009.9.9, KITPC, Beijing

26. 26 Hard mechanism -- PQCD approach The transverse momentum are retained, so no endpoint singularity. Sudakov double logarithm corrections are included. 2009.9.9, KITPC, Beijing  Momentum of one parton in the light meson is small (x->0). Soft interactions between spectator quark in B and soft quark in light meson.  Methods: light cone sum rules, light cone quark model… (lattice QCD is not applicable.) Soft mechanism

27. 27 Spin symmetry for soft form factor In the large energy limit (in leading order of 1/m b ), The total 10 form factors are reduced to 3 independent factors. 3→1 impossible! J. Charles, et al., PRD60 (1999) 014001. 2009.9.9, KITPC, Beijing

28. 28 Definition 2009.9.9, KITPC, Beijing

29. 29 QCDF and SCET In the heavy quark limit, to all orders of α s and leading order in 1/m b, Sudakov corrections Soft form factors, with singularity and spin symmetry Perturbative, no singularity  The factorization proof is rigorous.  The hard contribution ~ (Λ/m b ) 3/2, soft form factor ~ (Λ/m b ) 2/3 (?)  About the soft form factors, study continues, such as zero-bin method… 2009.9.9, KITPC, Beijing

30. 30 Zero-bin method by Stewart and Manohar  A collinear quark have non-zero energy. The zero-bin contributions should be subtracted out.  After subtracting the zero-bin contributions, the remained is finite and can be factorizable. For example, 2009.9.9, KITPC, Beijing

31. 31 Soft overlap mechanism The soft part form factor is represented by the convolution of initial and final hadron wave functions. 2009.9.9, KITPC, Beijing

32. 32 Dirac’s three forms of Hamiltonian dynamics ( S. Brodsky et al., Phys.Rep.301(1998) 299 ) 2009.9.9, KITPC, Beijing

33. 33 Advantage of LC framework  LC Fock space expansion provides a convenient description of a hadron in terms of the fundamental quark and gluon degrees of freedom.  The LC wave functions is Lorentz invariant. ψ(x i, k ┴i ) is independent of the bound state momentum.  The vacuum state is simple, and trivial if no zero-modes.  Only dynamical degrees of freedom are remained. for quark: two-component ξ, for gluon: only transverse components A ┴. Disadvantage  In perturbation theory, LCQCD provides the equivalent results as the covariant form but in a complicated way.  It’s difficult to solve the LC wave function from the first principle. 2009.9.9, KITPC, Beijing

34. 34 LC Hamiltonian KineticVertex Instantaneous interaction LCQCD is the full theory compared to SCET. Physical gauge is used A + =0. 2009.9.9, KITPC, Beijing

35. 35 LC time-ordered perturbation theory Diagrams are LC time x + -ordered. (old-fashioned) Particles are on-shell. The three-momentum rather than four- is conserved in each vertex. For each internal particle, there are dynamic and instantaneous lines. 2009.9.9, KITPC, Beijing

36. 36 Perturbative contributions: Only instantaneous interaction in the quark propagator. The exchanged gluons are transverse polarized. Instantaneous, no singularity break spin symmetry have singularity, conserve spin symmetry 2009.9.9, KITPC, Beijing

37. 37 Basic assumptions of LC quark model  Valence quark contribution dominates.  The quark mass is constitute mass which absorbs some dynamic effects.  LC wave functions are Gaussian. 2009.9.9, KITPC, Beijing

38. 38 LC wave functions In principle, wave functions can be solved if we know the Hamiltonian (T+V). Choose Gaussian-type Power law: The scaling of soft form factor depends on the light meson wave function at the endpoint. 2009.9.9, KITPC, Beijing

39. 39 Melosh rotation 2009.9.9, KITPC, Beijing

40. 40 Numerical results The values of the three form factors are very close, but they are quite different in formulations. 2009.9.9, KITPC, Beijing

41. 41 Comparisons with other approaches 2009.9.9, KITPC, Beijing

42. 42Summary  SCET provides a model-independent analysis of processes with energetic hadrons: proof of factorization theorem, Sudakov resummation, power corrections.  SCET analysis of Sudakov form factor emphasizes the scale point of view.  LC quark model is an appropriate non-perturbative method to study the soft part heavy-to-light form factors at large recoil.  How to treat the endpoint singularity is still a challenge. 2009.9.9, KITPC, Beijing

43. Thanks 43