Theoretical tools for non-leptonic B decays

1. Theoretical tools fornon-leptonic B decays Ru Min Wang Yonsei University

2. Outline • Effective Hamiltonian • Theoretical tools for hadronic matrix elements • Naïve Factorization (NF) • Generalized Factorization (GF) • QCD Factorization (QCD) • Perturbative QCD (PQCD) • Soft-Collinear Effective Theory (SCET) • Summary

3. Effective Hamiltonian Three fundamental scales: weak interaction scale b-quark mass scaleQCD scale Two tools of quantum field theory: • Operator Product Expansion To separate the full problem into two distinct parts • Long-distance physics at scale lower than µ is contained in operator matrix elements. • Short-distance physics at scale higher than µ is described by the Wilson coefficients. • Renormalization Group Improved Perturbation Theory To transfer the coefficients from scale mW down to the appropriate low energy scale.

4. Low-energy effective weak Hamiltonian at the scale µ: Separation of the hard and harder O(αs) contributions in effective field theory. Full theory Effective theory μindep. Low-energy < μ high-energy > μ 4-fermion operator (Qi(μ)) Wilson coefficient (Ci (μ)) Sum asln(mW/) to all orders The factorization scale  is arbitrary, and its dependence cancels between Ci() and Qi(). Fermi constantCKM matrixWilson CoefficientFour-Fermion Operator

5. 1 10-1 10-2 10-3 CKM matrix

6. Wilson coefficients can be calculated by Perturbation Theory and Renormalization Group Improved Perturbation Theory. Wilson Coefficients

7. Current-current operators: QCD penguins operators: Electroweak penguins operators: Magnetic penguins operators: α,β are the SU(3) color indices, andtaαβare the Gell-Mann matrices. The cross indicates magnetic penguins originate from the mass-term on the external line in the usual QCD or QED penguin diagrams

8. Is determined Is calculated efficiently inthe Renormalization Group Improved Perturbation Theory Decay amplitude: Hadronic matrix element Naïve Factorization (NF) Generalized Factorization (GF) QCD Factorization (QCDF) Perturbative QCD (PQCD) Soft-Collinear Effective Theory (SCET) Light-Cone Sum Rules (LCSR) …… Different approaches have been developed

9. Naïve Factorization M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29, 637 (1985), M. Bauer, B. Stech and M.Wirbel, Z. Phys. C 34, 103 (1987). Decay Amplitude:

10. u d c u Two color traces, Tr(I)Tr(I)=Nc2 Color-allowed One color trace, Tr(I)=Nc Color-suppressed b c b d Class I:Color-allowedA(B0→π+D–) ∝ C1(µ)+ C2(µ)/Nc= a1 Class II: Color-suppressedA(B0→π0D0) ∝ C2(µ)+ C1(µ)/Nc = a2 Class III: Color-allowed + Color-suppressed A (B+→π+D0)∝[C2(µ)+ C1(µ)] (1+/Nc ) = a1 + a2 - Three classes of decays Color flows(Q1(C)) NC is the number of colors with NC=3 in QCD a1,a2 are universal parameters ---is not successful

11. a1(mb)≈1.1 >> 0.1 a2(mb)≈0.1 ~ 0.1 The failure of NF implies the importance of non-factorizable corrections to color-suppressed modes NF Non-Factorizable Color-allowed Color-suppressed

12. c q unknown Summary of NF • NF is very simple and provides qualitative estimation of branching ratios of various two body nonleptonic B decays • Breaks the scale independence of decay amplitudes • Decay constant and form factor are notscale-dependent • Wilson coefficientsCi(µ)arescale-dependent • Physical decay amplitudes are dependent on renormalization scale and renormalization scheme • Non-factorizable amplitudes are not always negligible • Expected to work well for color-allowed modes • Fails for color-suppressed modes as expected • Evaluation of strong phasesis ambiguous Strong phase arises from charm penguin loop

13. NF Generalized Factorization How to calculate g()? A. J. Buras and L. Silvestrini, Nucl. Phys. B 548, 293 (1999), H. Y. Cheng, H. n. Li and K. C. Yang, Phys. Rev. D 60, 094005 (1999). Scale independence: Before applying factorization, extract the scale  dependence from the matrix element. dependences cancel Neff is treated as a phenomenological parameter which models the non-factorizable contributions to the hadronic matrix elements, which have not been includes into Cieff.

14. One proposal by Cheng, Li, Yang in Phys.Rev.D60:094005,1999. • Gauge dependent assume external quarks are on shell • Infrared cutoff absorb Infrared divergence into meson wave functions Gauge invariant Scale invariant Infrared finite Infrared cutoff and gauge dependences The extraction ofg() involves the one-loop corrections to the four quark vertex, which are infrared divergent. Cieffare μ and renormalization scheme independent, but are both gauge and infrared regulator dependent. Nceff The scale dependence is just replaced by the cutoff and gauge dependences.

15. Summary of GF • NCeff=2may explain many nonleptonic B decays • Decays of Class I and III are consistent with data • GF can not work well in Decays of Class II • GF can not offer convincing means to analyze the physics of non-factorizable contributions to non- leptonic decays • Strong phases still come from charm penguin loop • CPV can not be estimated correctly

16. Naïve Factorization QCD Factorization Beneke, Buchalla, Neuber and Sachrajda BBNS approach: PRL 83,1914(1999); NPB 591,313(2000). QCDF = NF + as and ΛQCD/mb non-factorizable corrections Spectator scattering Form Factor term Up to power Corrections ofΛQCD/mb

17. Other corrections which aren’t included in They are suppressed by ΛQCD/mb TI,II are the hard scattering kernels includes: : tree diagram :non-factorizable gluon exchange includes: : hard spectator scattering

18. End-point divergences Also in annihilation amplitudes At twist-3 for spectator amplitudes Parameterization(Phase parameters are arbitrary) Summary of QCDF • At the zeroth order of , QCDF reduces to naive factorization. At the higher order, the corrections can be computed systematically. • The renormalization scheme and scale dependence of is restored. In the heavy quark limit, the “non-factorizable” contributions is calculable perturbatively. It does not need to introduce NCeff. • Non-factorizable contribution to NF and strong phase from the penguin loop diagrams can be computed. QCDF is a breakthrough!

19. Has Naïve Factorization been so successful that what we need to do is only small corrections ? One proposal could be realized in an alternative way, the Perturbative QCD approach. The leading term is further factorized, and naïve factorization prediction could be modified greatly.

20. Φ:Meson distribution amplitude H: Hard scattering kernel • e-S:Sudakov factor • Describe the meson Distribution in kT • kT accumulates after infinitely many gluon exchanges Perturbative QCD H-n. Li and H. L. Yu, Phys. Rev. Lett. 74, 4388; Phys. Lett. B 353, 301. Y. Y. Keum, H-n. Li, and A. I. Sanda, Phys. Lett. B 504, 6; Phys. Rev. D 63, 054008 . C. D. Lu, K. Ukai, and M. Z. Yang, Phys. Rev. D 63, 074009 .

21. singularity at endpoint M. Beneke (Flavour Physics and CP violation, Taipeh, May 5-9, 2008): There is no end-point singularity because of the Sudakov factor. The gluon propagator: x,y are integral variables from 0 to 1 The arbitrary cutoffs in QCDF are not necessary.

22. Factorizable Non-factorizable Factorizable annihilationNon-factorizable annihilation The Feynman diagrams for calculating H

23. mb→∞ PQCDF Results as→0 QCDFF Results in PQCD in QCDF Power and Order Counting Rule Foctorizable:Annihilation:Non-Factorizable m0 is the chiral symmetry breaking scale (~1.8GeV)

24. Summary of PQCD • Incorporating Sudakov suppression makes form factors perturbatively calculable. • Hard spectator scattering and annihilation topology are calculable, no end-point divergences. • The prediction of large direct CP asymmetries are strongly model dependent. • The assumption of Sudakov suppression in hadronic B decays is questionable: PQCD calculation are very sensitive to details of kT dependenceof the wave function

25. Soft Collinear Effective Theory C.W. Bauer, S. Fleming, M. Luke, Phys. Rev. D 63, 014006 (2001). C.W. Bauer, S. Fleming, D. Pirjol, I.W. Stewart, Phys. Rev. D 63, 114020 . • New approach to the analysis of infrared divergences in QCD. • SCET has a transparent power counting in λ = ΛQCD/Q (or λ = (ΛQCD/Q)1/2) so that power corrections can be investigated in a systematic way. • Also introduces transverse momentum kT. • Many proofs of factorization are already complete:

26. Summary • The calculation of hadronic matrix elements is an great challenge for non-perturbative. • NFwas employed for a long time (since 80s). • Different approaches have been developed: GF, QCDF, PQCD, SCET......

27. Thanks!