1. 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. 2 Two complementary approaches for nonleptonic weak decays of heavy mesons: 1.Model-independent diagrammatical approach 2.Effective Hamiltonian & factorization (QCDF, pQCD,…)

3. 3 All two-body hadronic decays of heavy mesons can be expressed in terms of six distinct quark diagrams [Chau, HYC(86)] 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. Diagrammatic Approach (penguin) (or P a ) (tree) (color-suppressed) (exchange) (annihilation) Chiang,Gronau, Rosner,…

4. 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   H eff =c(  )O(  ) O(  ): 4-quark operator renormalized at scale  operators with dim > 6 are suppressed by (m h /M W ) 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 O 1 =(du) V-A (ub) V-A will induce O 2 =(db) V-A (uu) V-A - - - -

5. 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/k 2 is cancelled by (k  k -g  k 2 ) arising from quark loop as required by gauge invariance Responsible for direct CPV in K & B decays as dynamical phase can be generated when k 2 >4m 2 (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. 6 QCD penguins EW penguins induce four more EW penguin operators Effective Hamiltonian Buras et al (92) Gilman, Wise (79)

7. 7 WC c(  )’s at NLO depend on the treatment of  5 in n dimensions: i)NDR (naïve dim. regularization) {  5,   }=0 ii)HVBM ( ‘ t Hooft, Veltman; Breitenlohner, Maison)  m b LO NDR HV c 1 1.144 1.082 1.105 c 2 -0.308 -0.185 -0.228 c 3 0.014 0.014 0.013 c 4 -0.030 -0.035 -0.029 c 5 0.009 0.009 0.009 c 6 -0.038 -0.041 -0.033 c 7 /  0.045 -0.002 0.005 c 8 /  0.048 0.054 0.060 c 9 /  -1.280 -1.292 -1.283 c 10 /  0.328 0.263 0.266 Results of WCs c i (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, c 1 =1, c i =0 for i  1 c 3  c 5  –c 4 /3  –c 6 /3 c 9 is the biggest among EW penguin WCs

8. 8 Naïve Factorization B M1M1 M2M2 In m b  limit, M 2 produced in point-like interactions carries away energies O(m b ) and will decouple from soft gluon effect M 2 is disconnected from (BM 1 ) system  factorization amplitude  creation of M 2  B  M 1 transition  decay constant  form factor Naïve factorization = vacuum insertion approximation For a given effective Hamiltonian, how to evaluate the nonleptonic decay B  M 1 M 2 ?

9. 9 Consider B -  -  0 and H=c 1 O 1 +c 2 O 2 =c 1 (du)(ub)+c 2 (db)(uu) B-B- -- 00 b u u u d B-B- 00 b u u d u -- Neglect nonfactorizable contributions from O 1,2  - - - - ~ from O 1 from O 2 color allowed color suppressed

10. 10 Two serious problems with naïve factorization: Empirically, it fails to describe color-suppressed modes for c 1 (m c )=1.26 and c 2 (m c )=-0.51, while R expt =0.55 Theoretically, scheme and scale dependence of c i (  ) doesn’t get compensation from  O  f as V  and A  are renor. scale & scheme independent  unphysical amplitude from naïve factorization

11. 11 How to overcome aforementioned difficulties ? Bauer, Stech, Wirbel (87) proposed to treat a i ’s as effective parameters and extract them from experiment. (Of course, they should be renor. scale & scheme indep.) If a i ’s are universal (i.e. channel indep)  generalized factorization Test of factorization means a test of universality of a 1,2 Problems: Penguin a i ’s are difficult to determine Cannot predict CPV How to predict a i from a given effective Hamiltonian ?

12. 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/N c terms are dropped so that a 1  c 1, a 2  c 2. Note that c 2 +c 1 /N c =-0.09 vs. c2=-0.51 [Fukugita et al (77); Tadic & Trampetic (82); Bauer & Stech (85)] This is understandable as 1/N c +   0 !  Buras, Gerard, Ruckl  large-N c (or 1/N c ) 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. 13 If large-N c approach is applied to B decays  a 1 eff =c 1 (m b )  1.10, a 2 eff =c 2 (m b )  -0.25  destructive interference in B -  D 0  - just like D +  K 0  + A(B -  D 0  - )= a 1  O 1  +a 2  O 2 , while A(B 0  D -  + ) = a 1  O 1  supported by sum-rule calculations (Blok, Shifman; Khodjamirian, Ruckl; Halperin) Big surprise Big surprise from CLEO (93): constructive interference as B -  D 0  - > B 0  D -  + Generalized factorization (I) [HYC (94), Kamal (96)] with 1/N c eff =1/N c +  determined from experiment For B  D  decays, N c eff  2 rather than  ,  is positive ! _

14. 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. 15 Compute corrections to 4-quark matrix elements in the same  5 scheme as c i (  ) : NDR or ‘t Hooft-Veltman Then, in general Ali, Greub (98) Chen,HYC,Tseng,Yang (99)  V : anomalous dim., r V : scheme-dep constant, P i : penguin Z,  Gauge & infrared problems with effective WCs [Buras, Silvestrini (99)] are resolved using on-shell external quarks [HYC,Li,Yang (99)]

16. 16 Scale independence of a i or c i eff Scheme independence can be proved analytically for a 1,2 and checked numerically for other a i ’s C F =(N c 2 -1)/(2N c ) A major progress before 1999! It is more convenient to define a i =c i +c i  1 /N c for odd (even) i (Vertex & penguin corrections have not been considered in pQCD approach)

17. 17 Generalized Factorization (II) Generalized factorization (II): Some of nonfactorizable effects are already included in c i eff Difficulties:  Gluon’s momentum k 2 is unknown, often taken to be m B 2 /2. It is OK for BRs, but not for CPV as strong phase is not well determined  a 6 & a 8 are associated with matrix elements in the form m P 2 /[m b (  )m q (  )], which is not scale independent !  a 2,3,5,7,10 (especially a 2, a 10 ) are sensitive to N c eff. For example, N c eff 2 3 5  a 2 (  =m b ) 0.219 0.024 -0.131 -0.365 Expt’l data of charmless B decay  a 2  0.20  N c eff  2

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

19. 19 twist-2 & twist-3 LCDAs: Twist-3 DAs  p &   are suppressed by   /m b with   =m  2 /(m u +m d ) with  0 1 du  (u)=1,  0 1 du  p,  (u)=1 C n : Gegenbauer poly.

20. 20 In m b  limit, only leading-twist DAs contribute The parameters a i are given by strong phase from vertex corrections a i are renor. scale & scheme indep except for a 6 & a 8

21. 21 Hard spectator interactions (non-factorizable) : not 1/m b 2 power suppressed: i).  B (  ) is of order m b /  at  =  /m b   d  /   B (  )=m B / B ii). f M  , f B   3/2 /m b 1/2, F BM  (  /m b ) 3/2  H  O(m b 0 ) [ While in pQCD, H  O(  /m b ) ] Penguin contributions P i have similar expressions as before except that G(m) is replaced by Gluon’s virtual momentum in penguin graph is thus fixed, k 2  xm b 2

22. 22 Power corrections 1/m b power corrections: twist-3 DAs, annihilation, FSIs,… We encounter penguin matrix elements from O 5,6 such as formally 1/m b suppressed from twist-3 DA, numerically very important due to chiral enhancement:    m  2 /(m u +m d )  2.6 GeV at  =2 GeV Consider penguin-dominated mode B  K A(B  K)  a 4 +2a 6   /m b where 2   /m b  1 & a 6 /a 4  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. 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 M 1 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 divergence by 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 k 2 =(-  p B +xp 1 ) 2   xm B 2   QCD m b  1 GeV 2

24. 24 m b /2 m b 2m b a1a1 1.073+ i0.048 -0.086 0.986+ i0.048 1.054+ i0.026 -0.061 0.993+ i0.026 1.037+ i0.015 -0.045 0.992+ i0.015 a2a2 -0.039- i0.113 0.231 0.192-i0.113 0.005-i0.084 0.192 0.197-i0.084 0.045-i0.066 0.167 0.212-i0.066 a4ua4u -0.031+i0.023 0.004 -0.027+i0.023 -0.029+i0.017 0.003 -0.026+i0.017 -0.027+i0.014 0.002 -0.025+i0.014 a5a5 -0.011+i0.005 0.016 0.004+i0.005 -0.007+i0.003 0.010 0.003+i0.003 -0.004+i0.001 0.008 0.004+i0.001 a6ua6u -0.052+i0.017 -0.052+i0.018 -0.052+i0.019 a 10 /  0.062+i0.168 -0.221 -0.161+i0.004 0.018+i0.121 -0.182 -0.164+i0.121 -0.028+i0.093 -0.157 -0.185+i0.093 black: vertex & penguin, blue: hard spectator green: total a i for B  K  at different scales

25. 25 Annihilation topology Weak annihilation contributions are power suppressed ann/tree  f B f  /(m B 2 F 0 B     /m B Endpoint divergence exists even at twist-2 level. In general, ann. amplitude contains X A and X A 2 with X A   1 0 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. 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 k 2  xym B 2 with x,y  O(1), imaginary part can be induced from the quark loop bubble when k 2 > m q 2 /4 Gerard & Hou (91)

27. 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/m b order as vertex & penguin corrections, is included  crucial for a 2 & a 10  Include distribution of momentum fraction  1. a new strong phase from vertex corrections 2. fixed gluon virtual momentum in penguin diagram  For a 6 & a 8, V=6 without log(m b /  dependence ! So unlike other a i ’s, a 6 & a 8 must be scale & scheme dependent  Contrary to pQCD claim, chiral enhancement is scale indep.

28. 28 Form factors  B  D form factor due to hard gluon exchange is suppressed by wave function mismatch  dominated by soft process  For B  , k 2   h 2  m b . Let F B  =F soft +F hard It was naively argued by BBNS that F hard =  s (  h )(  /m B ) 3/2 & F soft =(  /m B ) 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 F B  (0)=  +  J, Bauer et al. determined  &  J by fitting to B  data and found    J  (  /m b ) 3/2 In pQCD based on k T factorization theorem,  <<  J Beneke,Feldmann (01)

29. 29 In short, for B  M form factor QCDF: F soft >> F hard, SCET: F soft  F hard, pQCD: F soft << F hard However, BBNS (hep-ph/0411171) argued that F soft >>F hard even in SCET We compute form factors & their q 2 dependence using covariant light- front model [HYC, Chua, Hwang, PR, D69, 074025 (04)] CLF BSW MS LCSR F B  (0) 0.25 0.33 0.29 0.31 F BK (0) 0.35 0.38 0.36 0.35 A 0 B  (0) 0.28 0.28 0.29 0.37 A 0 BK* (0) 0.31 0.32 0.45 0.47 BSW=Bauer,Stech,Wirbel MS=Melikhov,Stech LCSR=light-cone sum rule B +   +  0  F 0 B  (0)  0.25 B 0       A 0 B  (0)  0.29 Light meson in B  M transition at large recoil (i.e. small q 2 ) can be highly relativistic  importance of relativistic effects

30. 30 Phenomenology: B  PP For F B  (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 f 0 (980)K. Note that  ’ & f 0 (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) BRs in units of 10 -6

31. 31 Phenomenology: B  VP For penguin-dominated modes, VP < PP due to destructive interference between a 4 & a 6 terms (K  ) or absence of a 6 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. 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. 33 Direct CP violation in B decays Direct CPV (5.7  ) in B 0  K +  - was established by BaBar and Belle Direct CPV: First confirmed DCPV observed in B decays ! 2nd evidence at Belle !! Combined BaBar & Belle data  3.6  DCPV in B 0   -  +

34. 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. 35 “Simple” CP violation from perturbative strong phases: penguin (BSS) vertex corrections (BBNS) annihilation (pQCD) “Compound” CP violation from LD rescattering: [Atwood,Soni] weak strong A CP  sin  sin   : weak phase  : strong phase

36. 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 A CP (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. 37 FSI as rescattering of intermediate two-body states [HYC, Chua, Soni; hep-ph/0409317]  Strong phases  O(  s,1/m b )  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. 38 Dispersive part is obtained from the absorptive amplitude via dispersion relation  = m exc + 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 subject to large uncertainties and will be ignored in the present work Form factor is introduced to render perturbative calculation meaningful

39. 39 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. B 0  D 0  0 and its strong phase relative to B 0  D -  + are well accounted for by FSI  non-negligible annihilation E/T = 0.14 exp(i96  )  B 0  D - s K + can proceed only via annihilation is well predicted FSI can be neglected for tree-dominated color-allowed modes Final-state rescattering effects on decay rates

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

41. 41 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, B s  PP, VP & DCPV QCDF by Du et al.: PR, D64, 014036 (01): B  PP (a detailed derivation of a i ) PR, D65, 074001 (02): B  PP PR, D65, 094025 (02): B  VP PR, D68, 054003 (03): B s  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 References for QCDF