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Sudakov and heavy-to-light form factors in SCET

Sudakov and heavy-to-light form factors in SCET. Zheng-Tao Wei Nankai University. Introduction to SCET Sudakov form factor Heavy-to-light transition form factors Summary. Wei, PLB586 (2004) 282, Wei, hep-ph/0403069, Lu, et al., PRD (2007). I. Soft-Collinear Effective Theory.

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Sudakov and heavy-to-light form factors in SCET

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

  2. Introduction to SCET • Sudakov form factor • Heavy-to-light transition form factors • Summary Wei, PLB586 (2004) 282, Wei, hep-ph/0403069, Lu, et al., PRD (2007) 2009.9.9, KITPC, Beijing

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

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

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

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

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

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

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

  12. III. Sudakov form factor • 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 2009.9.9, KITPC, Beijing 12

  13. The interaction of a fermion with EM current is represented by At q2=0 , the g-factor is given by and the anomalous magnetic moment is The form factor (only the first term F1(Q2)) in the asymptotic limit q2→∞ is called Sudakov form factor. (in 1956) 2009.9.9, KITPC, Beijing

  14. The naïve power counting is strongly modified (at tree level F=1). • 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. 2009.9.9, KITPC, Beijing

  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. Bauer (2003) Regularization method Introduce a cutoff scale δ in both k+ and k-. DOF 2009.9.9, KITPC, Beijing

  17. 2009.9.9, KITPC, Beijing

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

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

  21. Comparisons with other works • 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. • 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. 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. IV. Heavy-to-light transition form factor • The importance of heavy-to-light form factors: • CKM parameter Vub • QCD, perturbative, non-perturbative • basic parameters for exclusive decays in QCDF or SCET • new physics… Light cone dominance At large recoil region q2<<mb2, the light meson moves close to the light cone. 2009.9.9, KITPC, Beijing

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

  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. Hard mechanism -- PQCD approach • The transverse momentum are retained, so no endpoint singularity. • Sudakov double logarithm corrections are included. Soft mechanism • 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.) 2009.9.9, KITPC, Beijing

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

  28. Definition 2009.9.9, KITPC, Beijing

  29. QCDF and SCET In the heavy quark limit, to all orders of αs and leading order in 1/mb, Sudakov corrections Soft form factors, with singularity and spin symmetry Perturbative, no singularity • The factorization proof is rigorous. • The hard contribution ~ (Λ/mb)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. 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. 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. Dirac’s three forms of Hamiltonian dynamics( S. Brodsky et al., Phys.Rep.301(1998) 299 ) 2009.9.9, KITPC, Beijing

  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(xi, 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. Kinetic Vertex LC Hamiltonian Instantaneous interaction • LCQCD is the full theory compared to SCET. • Physical gauge is used A+=0. 2009.9.9, KITPC, Beijing

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

  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. 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. Melosh rotation 2009.9.9, KITPC, Beijing

  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. Comparisons with other approaches 2009.9.9, KITPC, Beijing

  42. Summary • 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 2009.9.9, KITPC, Beijing

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