Theories of exclusive B meson decays

1. Theories of exclusive B meson decays Hsiang-nan Li Academia Sinica (Taiwan) Presented at Beijing Aug. 13-17, 2005

2. Titles of Lectures • I: Naïve factorization and beyond • II: QCDF and PQCD • III: SCET • IV: Selected topics in B Physics • Will not cover SU(3), QCD sum rules, determination of CKM, specific modes, new physics,…

3. Lecture I Naïve Factorization and beyond

4. Outlines • Introduction • Weak Hamiltonian • Naïve factorization • Diagnose FA • A plausible proposal • Summary

5. Introduction • Missions of B factories: Constrain standard-model parameters Explore heavy quark dynamics Search for new physics Must handle QCD eventually for precision measurement • Several theories have been developed recently, which go beyond the naïve factorization. • Semileptonic B decays (B meson transition form factors) are inputs to the above theories. • Predictions are then made for nonleptonic B decays. • Will discuss their ideas, differences, applications

6. sin (21)=0.685± 0.032 @LP 2005 Determine 1 using the golden mode B! J/ K. Penguin pollution ~5%. When reaching this precision, need a QCD theory.

7. Isospin relation A(D0-) p 2A(D00)/ a2 60o A(D+-)/ a1 a1, a2: the BSW parameters |a2|=0.35-0.6, much larger than expectation. Arg(a2/a1)» 60o is generated by decay dynamics. Their understanding requires a theory.

8. Mixing-induced CP Penguin-dominated Tree-dominated 4 S0 due to new physics? Need a theory for tree Pollution.

9. Complexity

10. Weak Hamiltonian IR finite difference High-energy» mW Effective theory Low-energy< mW Full theory indep. Sum ln(mW/) to all orders Weak Hamiltonian Heff = 4-fermion operator O() Wilson coefficient C() ­ The factorization scale  is arbitrary, and its dependence cancels between C() and O()

11. Penguins At O(s) or O(), there are also penguin diagrams (1-5) ! V-A  (1+5)! V+A b s b s  q q q q 2 Color flows: 2TaijTakl = -ijkl/Nc + ilkj QCD penguin: g EW penguin: greplaced by , Z

12. Example: Heff for b! s I, j label different color flows

14. Naïve factorization The decay amplitude for B! D h D|Heff|Bi/ C() hD|O()|Bi nonperturbative perturbative Must deal with the hadronic matrix element. The factorizationassumption (FA) was the first try. Decay constant and form factor are physical. No  dependence. To make physical prediction, must assume C to be constant, and It is better to be universal.

15. Color flows O1(C) u d c u b c b d One color trace, Tr(I)=Nc1 Color-suppressed two color traces, Tr(I)Tr(I)=Nc2 Color-allowed RHS is down by 1/N_c compared to LHS

16. Decay amplitudes Class 1: Color-allowed Class 2: Color-suppressed a1, a2: universal parameters

17. Success of FA a1 and a2 seem to be universal! Success due to “color transparency” D B Lorentz contraction Small color dipole Decoupling in space-time From the BD system FA is expected to work well for color-allowed modes with a light meson emitted from the weak vertex.

18. Failure of FA a2(D)a2(J/ K) is not a surprise B D Large correction in color-suppressed modes due to heavy D, large color dipole

19. The failure of FA implies the importance of nonfactorizable correction to color-suppressed modes, for which a2(mb)» 0.1<a2(J/ K), a2(D) In terms of Feynman diagrams, nonfactorizable correction is not universal. a1(mb)» 1.1 (J/) (K)

20. Nonfactorizable corrections Generalized naïve factorization Exp shows that Wilson coefficients are not really universal Due to nonfactorizable correction? Fine tune the mode-dependent parameters to fit data Equivalently, effective number of colors in Not very helpful in understanding decay dynamics How to calculate nonfactorizable correction?

21. Diagnose FA • FA should make some sense (color transparency). • The assumption of constant a1, a2 is not successful. • FA fails for color-suppressed modes as expected (small a2» nonfactorizable correction). • Stop data fitting. How to go beyond FA?

22. Scale dependence Decay constant and form factor are physical. No  dependence. • Problem of FA • Before applying factorization, extract the  dependence from the matrix element • The question is how to calculate g() C()h O()i¼ C()h O i|FA • dependence in C() remains  dependences cancel |FA independent

23. IR cutoff and gauge dependences • Look at the derivation of Weak Hamiltonian again • Considering off-shell external quarks, the constant a is gauge dependent, which is also hidden into the matrix element. • When extracting g(), one also extracts the dependences on cutoff and on gauge. The scale dependence is just replaced by the cutoff and gauge dependences. Dead end? Evaluated between external quark states -p2 is the off-shell IR cutoff hidden into matrix element ln(MW2/-p2)=ln(MW2/2)+ln(2/-p2) Absorbed into C(\mu)

24. Strong phase and CP asymmetry CP asymmetries in charmless decays can be measured at B factories Tree Penguin Interference of T and P Data Extraction Theory

25. Bander-Silverman-Soni mechanism In FA, strong phase comes from the BSS mechanism q It gives a small phase. Only source? Important source? Im/ Moreover, what is the gluon invariant mass q2? Can not compute thye strong phase unambiguoysly.

26. A plausible proposal • Recalculate O(s) corrections with on-shell quarks (Cheng, Li, Yang, May, 99) They are gauge invariant. But ln(2/-p2)! 1/IR How to deal with this IR pole?

27. IR divergence q • IR divergence is physical! • It’s a long-distance phenomenon, related to confinement, the hadronic bound state. • All physical hadronic high-energy processes involve both soft and hard dynamics. weak decay occurs g q Soft dynamics t=0 t=1

28. Factorization theorem • The idea is to absorb IR divergence into meson distribution amplitudes  • Factorization theorem • f: factorization scale. Its role is like . mW>>mb, mb>f> • H is the IR finite hard kernel. • The matrix element • A scale-independent, gauge-invariant, IR finite theory is possible! Scale dependence cancel All allowed decay topologies

29. Factorization vs. factorization • Factorization in “naïve factorization” means breaking a decay amplitude into decay constant and form factor. • Factorization in “factorization theorem” means separation of soft and hard dynamics in decay modes. • After 2000, factorization approach to exclusive B decays changed from 1st sense to 2nd.

30. Summary • FA is a simple model for nonleptonic B decays based on color transparency. • Its application is limited to branching ratios of color-allowed modes. • It can not describe color-suppressed modes, neglects nonfactorizable contributions, and has incomplete sources of strong phases. • Theoretically, it is not even a correct tool due to scale or gauge dependence. • A proposal for constructing a theory with the necessary merits has been made.