Non-factorizable Contributions in B  D (*) M Decays

Non-factorizable Contributions in B  D(*)M Decays C.-D. Lü (IHEP, Beijing) Collaborate with Keum, Kurimoto, H.n.Li, A.I.Sanda • Factorization Approach • Formalism of Perturbative QCD (PQCD) • Numerical Results • Summary Moriond -- CD Lu

Operator Product Expansion • VcbVud* • Multiple scales MW >> mb p2 << MW2 • VcbVud* u b d c Moriond -- CD Lu

Four Quark Operators u b d c L=1–5 Moriond -- CD Lu

Two kinds of diagrams contribute to B0→ π+D– decay: π+π+ u ubb B0D– B0 D– O2 O1 color enhancedcolor suppressed C2 ~ 1>C1/3 ~ – 0.2/3 d d Moriond -- CD Lu

Naïve Factorization Approach Decay matrix element can be separated into two parts: • Short distance Wilson coefficients and Hadronic parameters: form factor and decay constant < π+D–|Heff|B> = a1<π|V-A|0><D|V-A|B> = (C2+C1 /3)fπFB→D First class of decays: ∝ a1 ~ 1 non-factorizable contributions small Moriond -- CD Lu

Two kinds of diagrams contribute to B0→π0decay: u B0π0B0π0 color enhancedcolor suppressed C1 ~ – 0.2~ C2(1/3 ) ≡ C2/Nc ~ 1/3 d d +s8 Moriond -- CD Lu

Two kinds of diagrams contribute to B0→π0decay: u B0π0B0π0 color enhancedcolor suppressed C1 ~ – 0.2~ C2(1/3 +s8) ≡ C2/Nc ~ 1/3 d d Moriond -- CD Lu

Class II Decays Factorization approach: = a2<D| |0><π| |B> = (C1+C2 /Nc)fDF0B→π Class II decay： ∝ a2= C1+C2 /Nc ，small， Exp. B→π0D0decay with large branching ratio non-factorizable contribution may be large，  Factorization approach may not be good Moriond -- CD Lu

Class III Decays B+ B+ u u B+ B+ u u Moriond -- CD Lu

Class III Decays For charged B±decays，both a1 and a2 contribute = a2<D| |0><π| |B> +a1<π| |0><D| |B> = a2fDF0B→π +a1fπF0B→D Class III decays： ∝ a1+r a2 (no relative phase) Since a1 >>a2，they are similar to Class I decays, non-factorizable contribution not important Moriond -- CD Lu

Factorization Approach • class I decays ： M (B0 →π +D–) ∝ C1 + C2 /Nceff = a1 • class II decays ： M (B0 →π 0D0) ∝ C2 + C1 /Nceff = a2 • class III decays ： M (B+ →π +D0) ∝ (C2 + C1) (1+/Nceff ) = a1 + r a2 Moriond -- CD Lu

BD(*)branching ratios(x10–4) a1=1.08 a2=0.21 M. Neubert, B.Stech, hep-ph/9705292 Recent exp. implies phase between a1 & a2 Moriond -- CD Lu

Shortcoming of Factorization Approach • The non-factorizable contributions are not predictable • Form factors are not calculable, depend on experiments or other models • There are difficulties in calculation of the annihilation type diagram (Form factors unknown) • The strong phase can not be calculated well—which is essential for CP violation prediction Moriond -- CD Lu

Improve Factorization Approach: • QCD factorization (BBNS) • Perturbative QCD approach (PQCD) • Soft-collinear effective theory Moriond -- CD Lu

Picture of PQCD Approach 4-quark operator Six quark interaction inside the dotted line Moriond -- CD Lu

PQCDapproach • A ~ ∫d4k1 d4k2 d4k3 Tr [ C(t)B(k1) (k2) (k3)H(k1,k2,k3,t) ] exp{-S(t)} • (k3) are the wave functions for mesons • C(t)is Wilson coefficient of 4-quark operator • exp{-S(t)} is Sudakov factor，to relate the short- and long-distance interaction • H(k1,k2,k3,t) is perturbative calculation of six quark interaction Moriond -- CD Lu

Perturbative Calculation of H(t) in PQCD Approach Form factor—factorizable Non-factorizable Moriond -- CD Lu

Perturbative Calculation of H(t) in PQCD Approach Non-factorizable annihilation diagram Factorizable annihilation diagram Moriond -- CD Lu

For B0 D00 decay, these two diagrams do not cancel arg (a2/a1) ~ – 41° Moriond -- CD Lu

Comparison • a1=1.08, a2=0.21 relative phase not known for Factorization Approach • |a2/a1| ~ 0.47 arg (a2/a1) ~ – 41°in PQCD • The number for a1, a2 are extracted from exp. in Factorization approach • While in PQCD it is predictive Moriond -- CD Lu

Branching ratios of BD(*)(10–3) Y.Y. Keum et al, hep-ph/0305335 Moriond -- CD Lu

Branching ratios of BD(*)(‘)(10–4) CD Lu, PRD 68, 097502 (2003) Moriond -- CD Lu

Pure annihilation type decay BDsK K Ds– b d s b c s B+ B0 u c Ds+ u d s K + • B+ Ds+ K0B0 Ds– K+ • ∝VubVcd*∝ 4 ∝VcbVud*∝ 2 Moriond -- CD Lu

Branching Ratios • B0→DS–K+ ∝ VcbVud* ∝ 2 • Lü, Ukai, EPJ C28, 305(2003):Br ≈ (4±1)x10–5 • Belle: Br = 3.2 x 10–5 • BaBar: Br = 4.6 x 10–5 Moriond -- CD Lu

Summary • PQCD is useful in calculation of B decays to charmed mesons as well as for light mesons • PQCD can get a right number for a2/a1 and the relative phase • Higher order terms may be important. • since mc/mb is not a very small number Moriond -- CD Lu

Thank you! Moriond -- CD Lu

Contributions of different αs in H(t) calculation Fraction αs/ Moriond -- CD Lu

Non-factorizable Contributions in B  D (*) M Decays