Theoretical tools for non-leptonic B decays Ru Min Wang Yonsei University

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

Effective Hamiltonian Three fundamental scales: weak interaction scale b-quark mass scale QCD 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.

Fermi constant CKM matrix Wilson Coefficient Four-Fermion Operator 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 constant CKM matrix Wilson Coefficient Four-Fermion Operator

CKM matrix 1 10-1 10-2 10-3

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

α,β are the SU(3) color indices, and taαβ are the Gell-Mann matrices. Current-current operators: Magnetic penguins operators: Electroweak penguins operators: QCD penguins operators: α,β are the SU(3) color indices, and taαβ are 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

Decay amplitude: Naïve Factorization (NF) Is determined Is calculated efficiently in the Renormalization Group Improved Perturbation Theory 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

Naïve Factorization Decay Amplitude: 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:

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

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

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 not scale-dependent Wilson coefficients Ci(µ) are scale-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 phases is ambiguous c q unknown Strong phase arises from charm penguin loop

Generalized Factorization 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. NF  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. How to calculate g()?

Gauge invariant Scale invariant Infrared cutoff and gauge dependences The extraction of g() involves the one-loop corrections to the four quark vertex, which are infrared divergent. Cieff are 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. 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

Summary of GF NCeff=2 may 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

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

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

At twist-3 for spectator amplitudes Also in annihilation amplitudes 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! End-point divergences At twist-3 for spectator amplitudes Also in annihilation amplitudes Parameterization (Phase parameters are arbitrary)

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.

Perturbative QCD Φ: Meson distribution amplitude 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 . 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

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

The Feynman diagrams for calculating H Factorizable Non-factorizable Factorizable annihilation Non-factorizable annihilation

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

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 dependence of the wave function

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:

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

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