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

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,…

Lecture I Naïve Factorization and beyond

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

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

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

Isospin relation A(D +  - )/ a 1 p 2A(D 0  0 )/ a 2 A(D 0  - ) a 1, a 2 : the BSW parameters |a 2 |= , much larger than expectation. Arg(a 2 /a 1 )» 60 o is generated by decay dynamics. Their understanding requires a theory. 60 o

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

Complexity

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

Penguins    (1-  5 ) ! V-A   (1+  5 )! V+A At O(  s ) or O(  ), there are also penguin diagrams  QCD penguin: g EW penguin: g replaced by , Z 2 Color flows: 2T a ij T a kl = -  ij  kl /N c +  il  kj bs b q q s q q

Example: H eff for b! s I, j label different color flows

Naïve factorization h D  |H eff |Bi/ C(  ) hD  |O(  )|Bi perturbative nonperturbative Must deal with the hadronic matrix element. The factorization assumption (FA) was the first try. The decay amplitude for B! D  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.

Color flows bc d u O 1 (C) two color traces, Tr(I)Tr(I)=N c 2 Color-allowed b c d u One color trace, Tr(I)=N c 1 Color-suppressed RHS is down by 1/N_c compared to LHS

Decay amplitudes a 1, a 2 : universal parameters Class 1: Color-allowed Class 2: Color-suppressed

B D a 1 and a 2 seem to be universal! Success due to “color transparency” 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. Success of FA

Large correction in color-suppressed modes due to heavy D, large color dipole B D a 2 (D  )  a 2 (J/  K) is not a surprise Failure of FA

The failure of FA implies the importance of nonfactorizable correction to color-suppressed modes, for which a 2 (m b )» 0.1< a2(J/  K), a2(D  ) In terms of Feynman diagrams, nonfactorizable correction is not universal. a 1 (m b )» 1.1 (J/  ) (K)

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? Nonfactorizable corrections

Diagnose FA FA should make some sense (color transparency). The assumption of constant a 1, a 2 is not successful. FA fails for color-suppressed modes as expected (small a 2 » nonfactorizable correction). Stop data fitting. How to go beyond FA?

Scale dependence Problem of FA Before applying factorization, extract the  dependence from the matrix element The question is how to calculate g(  ) C(  )h O(  )i ¼ C(  )h O i| FA Decay constant and form factor are physical. No  dependence.  dependences cancel  dependence in C(  ) remains | FA  independent

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 ln(M W 2 /-p 2 )=ln(M W 2 /  2 )+ln(  2 /-p 2 ) Absorbed into C(\mu) -p 2 is the off-shell IR cutoff hidden into matrix element

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

In FA, strong phase comes from the BSS mechanism It gives a small phase. Only source? Important source? Bander-Silverman-Soni mechanism Im/ Moreover, what is the gluon invariant mass q 2 ? Can not compute thye strong phase unambiguoysly. q

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

IR divergence 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. q q g t=1t=0 weak decay occurs Soft dynamics

Factorization theorem The idea is to absorb IR divergence into meson distribution amplitudes  Factorization theorem  f : factorization scale. Its role is like . m W >  >m b, m b >  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

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.

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.