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Factorization Approach for Hadronic B Decays

Factorization Approach for Hadronic B Decays. Hai-Yang Cheng Factorization ( and history) General features of QCDF Phenomenology CPV, strong phases & FSIs. November 19, 2004, Mini-workshop on Flavor Physics. Two complementary approaches for nonleptonic weak decays of heavy mesons:

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Factorization Approach for Hadronic B Decays

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  1. Factorization Approach for Hadronic B Decays • Hai-Yang Cheng • Factorization ( and history) • General features of QCDF • Phenomenology • CPV, strong phases & FSIs November 19, 2004, Mini-workshop on Flavor Physics

  2. Two complementary approaches for nonleptonic weak decays of heavy mesons: • Model-independent diagrammatical approach • Effective Hamiltonian & factorization (QCDF, pQCD,…)

  3. Diagrammatic Approach All two-body hadronic decays of heavy mesons can be expressed in terms of six distinct quark diagrams [Chau, HYC(86)] Chiang,Gronau,Rosner,… (tree) (color-suppressed) (exchange) (or Pa) (penguin) (annihilation) All quark graphs are topological and meant to have all strong interactions included and hence they are not Feynman graphs. And SU(3) flavor symmetry is assumed.

  4. Effective Hamiltonian • Effective Hamiltonian for nonleptonic weak decays was first put forward by Gaillard, Lee (74), and developed further by Shifman, Vainshtein, Zakharov (75,77); Gilman, Wise (79). • At scale , integrate out fermions & bosons heavier than  •  Heff=c()O() O(): 4-quark operator renormalized at scale  • operators with dim > 6 are suppressed by (mh/MW)d-6 • Why effective theory ? When computing radiative corrections to 4-quark operators, the result will depend on infrared cutoff and choice of gluon’s propagator, etc. The merit of effective theory allows factorization: WCs c() do not depend on the external states, while gauge & infrared dep. are lumped into hadronic m.e. • Radiative correction to O1=(du)V-A(ub)V-A will induce O2=(db)V-A(uu)V-A - - - -

  5. Penguin Diagram Penguin diagram [dubbed by John Ellis (77)] was first discussed by SVZ (75) motivated by solving I=1/2 puzzle in kaon decay • It is a local 4-quark operator since gluon propagator 1/k2 is cancelled by • (k k-gk2) arising from quark loop as required by gauge invariance • Responsible for direct CPV in K & B decays as dynamical phase can be generated when k2>4m2 (time-like) Bander,Silverman,Soni (79) • Fierz transformation of (V-A)(V+A)  -2(S-P)(S+P)  chiral enhancement of scalar penguin matrix elements  dominant contributions in many S=1 rare B decays

  6. QCD penguins Gilman, Wise (79) • EW penguins induce four more EW penguin operators • Effective Hamiltonian Buras et al (92)

  7. WC c()’s at NLO depend on the treatment of 5 in n dimensions: • NDR (naïve dim. regularization) {5, }=0 • HVBM (‘t Hooft, Veltman; Breitenlohner, Maison) =mb LO NDR HV c1 1.144 1.082 1.105 c2 -0.308 -0.185 -0.228 c3 0.014 0.014 0.013 c4 -0.030 -0.035 -0.029 c5 0.009 0.009 0.009 c6 -0.038 -0.041 -0.033 c7/ 0.045 -0.002 0.005 c8/ 0.048 0.054 0.060 c9/ -1.280 -1.292 -1.283 c10/ 0.328 0.263 0.266 • Results of WCs ci(i=1,…,10) were first obtained by Buras et al (92) • For details about WCs, see Buras et al. RMP, 68, 1125 (96) • In s 0 limit, c1=1, ci=0 for i1 • c3  c5  –c4/3  –c6/3 • c9 is the biggest among EW penguin WCs

  8. Naïve Factorization For a given effective Hamiltonian, how to evaluate the nonleptonic decay B M1M2 ? In mb limit, M2 produced in point-like interactions carries away energies O(mb) and will decouple from soft gluon effect M2 B M1 M2 is disconnected from (BM1) system  factorization amplitude  creation of M2 BM1 transition  decay constant  form factor Naïve factorization = vacuum insertion approximation

  9. - - - - Consider B--0 and H=c1O1+c2O2=c1(du)(ub)+c2(db)(uu) - d u u b B- 0 u from O1 color allowed 0 u u d b - B- u from O2 color suppressed Neglect nonfactorizable contributions from O1,2 ~

  10. Two serious problems with naïve factorization: • Empirically, it fails to describe color-suppressed modes for c1(mc)=1.26 and c2(mc)=-0.51, while Rexpt=0.55 • Theoretically, scheme and scale dependence of ci() doesn’t get compensation from Of as V and A are renor. scale & scheme independent  unphysical amplitude from naïve factorization

  11. How to overcome aforementioned difficulties ? • Bauer, Stech, Wirbel (87) proposed to treat ai’s as effective parameters and extract them from experiment. (Of course, they should be renor. scale & scheme indep.) • If ai’s are universal (i.e. channel indep)  generalized factorization • Test of factorization means a test of universality of a1,2 • Problems: • Penguin ai’s are difficult to determine • Cannot predict CPV • How to predict ai from a given effective Hamiltonian ?

  12. For problem with color-suppressed modes, consider nonfactorizable contributions • To accommodate DK data   -0.35 • In late 70’s & early 80’s, it was found empirically by several groups that discrepancy is greatly improved if Fierz-transformed 1/Nc terms are dropped so that a1  c1, a2  c2. Note that c2+c1/Nc=-0.09vs. c2=-0.51 • [Fukugita et al (77); Tadic & Trampetic (82); Bauer & Stech (85)] • This is understandable as 1/Nc+  0 ! • Buras, Gerard, Ruckl  large-Nc (or 1/Nc) approach (86) •  for charm decays has been estimated by Shifman & Blok (87) using QCD sum rules Nowadays, it is known that one needs sizable nonfactorizable effects & FSIs to describe hadronic D decays

  13. If large-Nc approach is applied to B decays  a1eff=c1(mb)  1.10, a2eff=c2(mb)  -0.25  destructive interference in B-D0- just like D+K0+ A(B-D0-)= a1O1+a2O2, while A(B0D-+) = a1O1 supported by sum-rule calculations (Blok, Shifman; Khodjamirian, Ruckl; Halperin) Bigsurprise from CLEO (93): constructive interference as B-D0- > B0D-+ Generalized factorization(I)[HYC (94), Kamal (96)] with 1/Nceff=1/Nc+  determined from experiment For BD decays, Nceff 2 rather than  ,  is positive ! _

  14. For problem with scheme and scale dependence, consider vertex and penguin corrections to four-quark matrix elements penguin corrections Apply factorization to Otree rather than to O()

  15. Z, Compute corrections to 4-quark matrix elements in the same 5 scheme as ci() : NDR or ‘t Hooft-Veltman Then, in general Ali, Greub (98) Chen,HYC,Tseng,Yang (99) V: anomalous dim., rV: scheme-dep constant, Pi: penguin Gauge & infrared problems with effective WCs [Buras, Silvestrini (99)] are resolved using on-shell external quarks [HYC,Li,Yang (99)]

  16. It is more convenient to define ai=ci+ci1/Nc for odd (even) i CF=(Nc2-1)/(2Nc) • Scale independence of ai or cieff • Scheme independence can be proved analytically for a1,2 and • checked numerically for other ai’s (Vertex & penguin corrections have not been considered in pQCD approach) A major progress before 1999!

  17. Generalized Factorization (II) Generalized factorization (II): Some of nonfactorizable effects are already included in cieff • Difficulties: • Gluon’s momentum k2 is unknown, often taken to be mB2/2. It is OK for BRs, but not for CPV as strong phase is not well determined • a6 & a8 are associated with matrix elements in the form mP2/[mb()mq()], which is not scale independent ! • a2,3,5,7,10 (especially a2, a10) are sensitive to Nceff. For example, Nceff 2 3 5  a2(=mb) 0.219 0.024 -0.131 -0.365 Expt’l data of charmless B decay  a2  0.20  Nceff 2

  18. QCD Factorization PRL, 83, 1914 (99) Beneke, Buchalla, Neubert, Sachrajda (BBNS) TI: TII: hard spectator interactions At O(s0) and mb, TI=1, TII=0, naïve factorization is recovered At O(s), TI involves vertex and penguin corrections, TII arises from hard spectator interactions M(x): light-cone distribution amplitude (LCDA) and x the momentum fraction of quark in meson M

  19. twist-2 & twist-3 LCDAs: Twist-3 DAs p &  are suppressed by /mb with =m2/(mu+md) Cn: Gegenbauer poly. with 01 du (u)=1, 01 du p,(u)=1

  20. In mb limit, only leading-twist DAs contribute The parameters ai are given by ai are renor. scale & scheme indep except for a6 & a8 strong phase from vertex corrections

  21. Penguin contributions Pi have similar expressions as before except that G(m) is replaced by Gluon’s virtual momentum in penguin graph is thus fixed, k2 xmb2 • Hard spectator interactions (non-factorizable) : not 1/mb2 power suppressed: i). B() is of order mb/ at =/mb   d/ B()=mB/B ii). fM  , fB  3/2/mb1/2, FBM  (/mb)3/2  H  O(mb0) [ While in pQCD, H  O(/mb) ]

  22. Power corrections 1/mb power corrections: twist-3 DAs, annihilation, FSIs,… We encounter penguin matrix elements from O5,6 such as formally 1/mb suppressed from twist-3 DA, numerically very important due to chiral enhancement: m2/(mu+md)  2.6 GeV at =2 GeV Consider penguin-dominated mode B K A(BK)  a4+2a6/mb where 2/mb 1 & a6/a4 1.7 Phenomenologically, chirally enhanced power corrections should be taken into account  need to include twist-3 DAs p &  systematically OK for vertex & penguin corrections

  23. Not OK for hard spectator interactions: • The twist-3 term is divergent as p(y) doesn’t vanish at y=1: Logarithmic divergence arises when the spectator quark in M1 becomes soft • Not a surprise ! Just as in HQET, power corrections are a priori nonperturbative in nature. Hence, their estimates are model dependent & can be studied only in a phenomenological way • BBNS model the endpoint divergenceby • with h being a typical hadron scale  500 MeV. • Relevant scale for hard spectator interactions • h=(h)1/2 (hard-collinear scale),s=s(h) • as the hard gluon is not hard enough • k2=(-pB+xp1)2xmB2 QCDmb 1 GeV2

  24. ai for B K at different scales black: vertex & penguin, blue: hard spectatorgreen: total

  25. Annihilation topology Weak annihilation contributions are power suppressed • ann/tree  fBf/(mB2 F0B)/mB • Endpoint divergence exists even at twist-2 level. In general, ann. amplitude contains XA and XA2 with XA10 dy/y • Endpoint divergence always occurs in power corrections • While QCDF results in HQ limit (i.e. leading twist) are model independent, model dependence is unavoidable in power corrections

  26. Classify into (i) (V-A)(V-A), (ii) (V-A)(V+A), (iii) (S-P)(S+P) • (V-A)(V-A) annihilation is subject to helicity suppression, in analog to the suppression of e relative to  • Helicity suppression is not applicable to (V-A)(V+A) & penguin- induced (S-P)(S+P) annihilation  dominant contributions • Since k2  xymB2 with x,y O(1), imaginary part can be induced from the quark loop bubble when k2> mq2/4 Gerard & Hou (91)

  27. Comparison between QCDF & generalized factorization • QCDF is a natural extension of generalized factorization with the following improvements: • Hard spectator interaction, which is of the same 1/mb order as vertex & penguin corrections, is included  crucial for a2 & a10 • Include distribution of momentum fraction  1. a new strong phase from vertex corrections 2. fixed gluon virtual momentum in penguin diagram • For a6 & a8, V=6 without log(mb/) dependence !So unlike other ai’s, a6 & a8 must be scale & scheme dependent  Contrary to pQCD claim, chiral enhancement is scale indep.

  28. Form factors • B D form factor due to hard gluon exchange is suppressed by wave function mismatch  dominated by soft process • For B , k2  h2  mb. Let FB=Fsoft+Fhard • It was naively argued by BBNS that Fhard=s(h)(/mB)3/2 & Fsoft=(/mB)3/2 • so that B to  form factor is dominated by soft process • In soft-collinear effective theory due to Bauer,Fleming,Pirjol,Stewart(01), • B light M form factor at large recoil obeys a factorization theorem • Writing FB(0)=+J, Bauer et al. determined  & J by fitting to B data • and found   J  (/mb)3/2 • In pQCD based on kT factorization theorem, <<J Beneke,Feldmann (01)

  29. In short, for B M form factor QCDF: Fsoft>> Fhard, SCET: Fsoft Fhard, pQCD: Fsoft<< Fhard However, BBNS (hep-ph/0411171) argued that Fsoft>>Fhard even in SCET We compute form factors & their q2 dependence using covariant light-front model [HYC, Chua, Hwang, PR, D69, 074025 (04)] BSW=Bauer,Stech,Wirbel MS=Melikhov,Stech LCSR=light-cone sum rule B+ +0  F0B(0)  0.25 B0   A0B(0)  0.29 CLF BSW MS LCSR FB(0) 0.25 0.33 0.29 0.31 FBK(0) 0.35 0.38 0.36 0.35 A0B(0) 0.28 0.28 0.29 0.37 A0BK*(0) 0.31 0.32 0.45 0.47 Light meson in B M transition at large recoil (i.e. small q2) can be highly relativistic  importance of relativistic effects

  30. Phenomenology: B PP BRs in units of 10-6 • For FB(0)=0.25, predicted BRs for K modes are (15-30)% smaller than expt. • A longstanding puzzle for the enormously large rate of K’. Same puzzle occurs for f0(980)K. Note that ’ & f0(980) are SU(3) singlet A LD rescattering (e.g. B DD+-) is needed to interfere destructively with +-. This will give rise to observed BR of 00 (annihilation doesn’t help)

  31. Phenomenology: B VP • For penguin-dominated modes, VP < PP due to destructive interference between a4 & a6 terms (K) or absence of a6 terms (K*) • Br(00)=1.40.7<2.9 by BaBar & 5.1 1.8 by Belle • Final-state rescattering will enhance 00 from 0.6 to 1.30.3. The pQCD prediction  0.2 is too small • QCDF predictions for penguin dominated modes K*, K are consistently too small  power corrections from penguin-induced annihilation and/or FSIs such as LD charming penguins

  32. Phenomenology: B VV average QCDF pQCD (a) (b) QCDF results from HYC & Yang, PL, B511, 40 (a): BSW, (b): LCSR • Tree-dominated modes tend to have large BRs • BRs can differ by a factor of 2 in different form factor models • The predicted K* & K* by QCDF are too small

  33. Direct CP violation in B decays Direct CPV: • Direct CPV (5.7) in B0 K+-was established by BaBar and Belle First confirmed DCPV observed in B decays ! 2nd evidence at Belle !! • Combined BaBar & Belle data  3.6 DCPV in B0 -+

  34. Direct CP violation in QCDF • For DCPV in B +-, 5.2  effect claimed by Belle(03), not yet confirmed by BaBar  QCDF predictions for DCPV disagree with experiment !

  35. ACP  sin sin : weak phase : strong phase • “Simple” CP violation from perturbative strong phases: penguin (BSS) vertex corrections (BBNS) annihilation (pQCD) • “Compound” CP violation from LD rescattering:[Atwood,Soni] strong weak

  36. Beneke & Neubert: Penguin-dominated VP modes & DCPV can be accommodated by having a large penguin-induced annihilation topology with A=1, A=-55 (PP), A=-20 (PV), A=-70 (VP) Sign of A is chosen so that sign of A(K+-) agrees with data • Difficulties: • The origin of strong phase is unknown & its sign is not predicted • The predicted ACP(K+)=0.10 is in wrong sign: expt= -0.510.19 • Annihilation doesn’t help explain tree-dominated modes 00 & 00  necessity of another power correction: FSI

  37. FSI as rescattering of intermediate two-body states [HYC, Chua, Soni; hep-ph/0409317] • Strong phases O(s,1/mb) • FSI is assumed to be dominated by rescattering of two-body intermediate states with one particle exchange in t-channel. Its absorptive part is computed via optical theorem: • Strong coupling is fixed on shell. For intermediate heavy mesons, • apply HQET+ChPT (for soft Goldstone boson) • Cutoff must be introduced as exchanged particle is off-shell • and final states are hard • Alternative: Regge trajectory [Nardulli,Pham][Falk et al.] [Du et al.] …

  38. Form factor is introduced to render perturbative calculation meaningful •  = mexc + rQCD (r: of order unity) •  or r is determined form a 2 fit to the measured rates •  r is process dependent • n=1 (monopole behavior), consistent with QCD sum rules Once cutoff is fixed  CPV can be predicted Dispersive part is obtained from the absorptive amplitude via dispersion relation subject to large uncertainties and will be ignored in the present work

  39. Final-state rescattering effects on decay rates • Penguin-dominated B K, K’, K*, K, K, K* receive significant LD charm intermediate states (i.e. charming penguin) contributions. Such FSIs contribute to penguin-induced annihilation topologies • Tree-dominated B 00 is enhanced by LD charming penguins to (1.30.3)10-6 to be compared with (1.91.2)10-6: (1.4 0.7)<2.9 10-6 from BaBar & (5.11.8)10-6 from Belle • Charming penguin contributions to B 00 are CKM suppressed. • B0D00 and its strong phase relative to B0D-+ are well accounted for by FSI  non-negligible annihilation E/T = 0.14 exp(i96) B0D-sK+ can proceed only via annihilation is well predicted • FSI can be neglected for tree-dominated color-allowed modes

  40. Final-state rescattering effects on DCPV • Strong phases are governed by final-state rescattering. • Signs of DCPV are in general flipped by FSIs.

  41. References for QCDF QCDF by BBNS: NP, B591, 313 (00): B  D NP, B606, 245 (01): B  K,  NP, B651, 225 (03): B  P’ NP, B675, 333 (03): B, Bs PP, VP & DCPV QCDF by Du et al.: PR, D64, 014036 (01): B  PP (a detailed derivation of ai) PR, D65, 074001 (02): B  PP PR, D65, 094025 (02): B  VP PR, D68, 054003 (03): Bs  PP,VP K.C. Yang, HYC: PR, D63, 074011 (01) : B  J/K PR, D64, 074004 (01) : B K PL, B511, 40 (01) : BVV

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