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Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo U)

Quark-gluon correlation inside the B-meson from QCD sum rules based on heavy quark effective theory. Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo U). Motivation. Exclusive decay of B meson provides important information for understanding CP violation.

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Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo U)

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  1. Quark-gluon correlation inside the B-meson from QCD sum rulesbased on heavy quark effective theory Tetsuo Nishikawa (Ryotokuji U) Kazuhiro Tanaka (Juntendo U)

  2. Motivation • Exclusive decay of B meson provides important information for understanding CP violation. • In the description of exclusive B-decay based on QCD factorization, a very important role is played by the light cone distribution amplitude (LCDA) of B-meson. • However, surprisingly, little attention to B-meson’s LCDA was received in past. Our poor knowledge about it limits to extract important physics from experimental data. • This work is a part of an attempt to precisely determine B-meson’s LCDA based on QCD.

  3. Heavy quark field Exclusive decay of B-meson Beneke, Buchalla, Neubert, Sachrajda (’99) Bauer, Pirjol, Stewart (’01) • QCD factorization of exclusive B-decay: • B-meson’s LCDA in HQET

  4. OPE of B meson’s LCDA Kawamura and Tanaka, PLB 673(2009)201 dim.3 dim.4 dim.5

  5. λE and λH: quark-gluon correlation inside the B-meson “Chromo-electric” “Chromo-magnetic” λE、λH〜 strength of the color-electric (-magnetic) field inside the B-meson play an important role for the determination of exclusive B-decay amplitude But, almost unknown (only one estimate by HQET sum rule) (F(μ): B meson’s decay constant)

  6. “3” LO “３＋４＋5” “３＋4” L-N Behavior of B-meson’s LCDA Kawamura and Tanaka, PLB 673(2009)201 • NLO perturbative corrections: • very large for τ→０ and 10-30% level for moderate τ • Nonperturbative corrections (dim. 5 and dim. 4 operators) • are important (20-30% level) • Effects from are significant in dim. 5 contributions.

  7. Extrapolation to long distance region Kawamura and Tanaka, PLB 673(2009)201 • In the long distance region, OPE diverges. • For large distances, we must rely on a model (Lee-Neubert’s ansatz is employed here). • smoothly connect the OPE and the model descriptions at certain distance LCDA for entire distances OPE L-N ansatz OPE up to dim. 5 ops. Model (Lee-Neubert ansatz)

  8. Inverse moment of LCDA Kawamura and Tanaka, PLB 673(2009)201 • LCDA enters the B-decay amplitude through its inverse moment. • Stable behavior for • Switching off λE and λH, stable behavior is not seen. The above results demonstrate the impact of Reliable and precise determinations of is necessary.

  9. Only one HQET sum rule estimate by Grozin and Neubert (1997) is known. The sum rule analysis for λE and λH is not complete, unless the calculation at NLO accuracy (dim.6 and O(αs) correction to dim.5) is carried out. Updating the estimate of λE and λH is needed. Estimate of λE and λH dominant

  10. In a heavy(Q)-light(q) system, Q is nearly on-shell: This is equivalent to write HQET (Heavy Quark Effective Theory) Light quark cloud Q Heavy quark

  11. Pair creation of QQ cannot occur. The new field hv is constrained to satisfy (neglect Q degree of freedom) QCD Lagrangian can be simplified to HQET (Heavy Quark Effective Theory) extract the physics of heavy-light mesons

  12. Basic object of the QCD sum rule • Current correlation function • j(x): “interpolating field” ex. meson: Interaction between quarks and with vacuum fluctuation

  13. Correlation function at = Operator Product Expansion (OPE)

  14. Bound state pole continuum Imaginary part of the correlation function • :spectral function • Using analyticity, we can relate and the spectral function as (Dispersion relation)

  15. QCD (Borel) Sum rule • Applying “Borel transform” on the dispersion relation, we obtain a sum rule: • Physical quantities extracted from the sum rule have mild M-dependence. ∵truncation of OPE, incompleteness of the spectral ansatz choice of a reasonable range of M Borel mass (arbitrary parameter) approximate ansatz

  16. HQET sum rule for λE,H Non-diagonal correlation function Representation of Π with hadronic states B-meson pole at (not mB !) 2-independent Lorentz structures

  17. Dispersion relation for two Lorentz structure Borel transform HQET sum rule for λE,H Spectral ansatz OPE of LHS HQET sum rules for

  18. HQET sum rule for λE,H Sum rules for Decay constant is independently determined from an HQET sum rule. Neubert, 1992 Bagan, Ball, Braun and Dosch, 1992 up to dim.6 operators, up to O(αs) Wilson coefficients

  19. OPE light quark heavy quark = Grozin&Neubert + + This work +・・・

  20. Renormalization of the interpolating field Counter term = UV-pole

  21. Renormalization of the interpolating field + + + Counter term= + + + + UV-pole

  22. O(αs) correction to dim5 term • UV-divergence is subtracted by counter terms. • Remaining IR-divergence is absorbed into the vacuum condensate.

  23. Results for λH2(μ=1GeV) (preliminary) ωth:continuum threshold : Grozin&Neubert : +dim6 : +dim6 +O(αs) correction

  24. Results for λH2 - λE2(μ=1GeV) (preliminary) ωth:continuum threshold : Grozin&Neubert : +dim6 : +dim6 +O(αs) correction

  25. Choice of the reasonable M-range • Criterion for M: Both of • Higher order power corrections in OPE • Continuum contribution should not be large (less than 30-50%). • Reasonable range of M • In this range,

  26. Summary λE and λH (quark-gluon correlation inside the B-meson) play important role in B-meson’s LCDA. HQET sum rule for λE and λH up to dim.6 operator in OPE radiative correction to the mixed condensate Small contribution of dim.6 term OPE seems to converge at this order. Radiative correction significantly lowers λE and λH. Renormalization group improvement etc. Matching the OPE of LCDA Estimation of the inverse moment of LCDA ( )

  27. On the results • Contribution of dim.6 is less than 1% OPE seems to converge at this order. • O(αs)-correction to dim.5 is significantly large and tends to suppress λH and λE. • After inclusion of O(αs)-correction, stability of the splitting becomes worse.

  28. Implication to B-meson wave function

  29. O(αs) correction to dim5 term + + + + + + + + + + + (counter term)

  30. Formulation of B-meson’s HQET sum rule • Correlation function • C.F. evaluated by OPE is related to B-meson’s physical quantities through the dispersion relation

  31. Correlation function Representation of Π with hadronic states B-meson pole at Formulation of HQET sum rule for B-meson

  32. Matrix elements • Two-body operator • Three body operator

  33. B-meson pole • 2-independent Lorentz structures • Write dispersion relations for

  34. Borel transform HQET sum rule for λE,H Spectral ansatz OPE of LHS HQET sum rules

  35. Results for λH2(μ=1GeV) (preliminary) ωth:continuum threshold : Grozin&Neubert : +dim6 : +dim6 +O(αs) correction

  36. Results for λH2 - λE2(μ=1GeV) (preliminary) ωth:continuum threshold : Grozin&Neubert : +dim6 : +dim6 +O(αs) correction

  37. In a heavy(Q)-light(q) system, Pair creation of QQ cannot occur. The new field hv is constrained to satisfy QCD Lagrangian can be simplified to HQET (Heavy Quark Effective Theory) Heavy quark Light quark cloud Q

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