1. 1 QCD Factorization with Final-State Interactions Chun-Khiang Chua Academia Sinica, Taipei 3rd ICFP, Cung-Li, Taiwan

2. 2 Factorization in B decays We basically have three scales in a non-leptonic B decay: m W >> m B >>  QCD Integrating out d.o.f. above m B : H=c i (  ) Q i (  ) Naïve factorization: A  B  M 1  0  M 2  a i (c j ) FF BM1 f M2 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 Bjorken

3. 3 Na ï ve factorization in B Decays For color allowed processes the naïve factorization approx. works well. However,  Corrections (non factorization contributions) are incalculable. Neglected.  Dependence of scale  in amp. from a i (  ) cannot be cancelled. BR(Theory)≈3  10 -3 BR(Expt.)=(2.76±0.25)  10 -3

4. 4 B f One needs at least two different B  f paths with distinct weak & strong phases  strong phase  weak phase e i(  +  ) BaBar Belle Average B 0 →K -  + -0.13  0.03-0.10  0.25-0.11  0.02 B0→+-B0→+- -0.47  0.16-0.53  0.30-0.47  0.14 B0→+-B0→+- 0.09  0.16 0.56  0.13 0.37  0.10 first confirmed DCPV (5.7  ) in B decays (2004) _ _ _ We do have 2 different paths Direct CP violations strong phase ?

5. 5 penguin corrections Ali, Greub (98) Chen,Cheng,Tseng,Yang (99) Generalize factorization For problem with scheme and scale dependence, consider vertex and penguin corrections to four-quark matrix elements Strong phase from the BSS cut: k 2 ~m 2 B /4  m 2 B /2 gives large uncertainty Corrections (non-fac. Contributions) are still incalculable. Parameterized.

6. 6 QCD Factorization Beneke, Buchalla, Neubert, Sachrajda (99) T I : T II : hard spectator interactions  M (x): light-cone distribution amplitude (LCDA) and x the momentum fraction of quark in meson M 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 (New)

7. 7 Comparison between QCDF & generalized fac. QCDF is a natural extension of generalized factorization with the following improvements: Corrections to naïve factorization are calculable [1+O(  s )] Hard spectator interaction, which is of the same 1/m b order as vertex & penguin corrections, is included (new)  crucial for a 2 & a 10 Include distribution of meson momentum fraction   1. a new strong phase from vertex corrections  2. fixed gluon virtual momentum in penguin diagram (imp.for  CP ) Except a6 and a8 all effective wilson coefficients are gauge and scheme independent.  a 6 and a 8 come with   /m B =m 2  /(m u +m d ) mB. Power correction. QCDF is model independent in the large mB limit and reduces to naïve fac. in the O(  s 0 ) limit.

8. 8 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 important (  enhancement) :   (2GeV)  m  2 /(m u +m d )  2.6 GeV, 2    m b For example, in the penguin-dominated mode B  K A(B  K)  a 4 +(2   /m b ) a 6 where 2   /m b  1 & a 6 /a 4  1.7 Phenomenologically, power corrections should be taken into account  need to include twist-3 DAs  p &   systematically OK for vertex & penguin corrections: (    m b ) a 6,8 : scale independent.

9. 9 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

10. 10 Endpoint divergence in hard spectator and annihilation 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. For annihilation contributions endpoint divergence starts at twist-2 term. Both endpoint divergences occur as 1/m B power corrections (model dependent). FSI could be important. Several hints…

11. 11 pQCD (Keum, Li, Sanda): A sizable strong phase from penguin- induced annihilation by introducing parton’s transverse momentum QCD factorization (Beneke, Buchalla, Neubert, Sachrajda): Because of endpoint divergences,  QCD /m b power corrections due to annihilation and twist-3 spectator interactions can only be modelled QCDF (S4 scenario): large annihilation with phase chosen so that a correct sign of A(K -  + ) is produced (  A =1,  A = -55  for PP,  A =-20  for PV and  A =-70  for VP) 1. Large strong phases in charmless modes are needed input

12. 12 Some decay modes do not receive factorizable contributions e.g. B  K  c0 with sizable BR though  c0 |c   (1-  5 )c|0  =0. Color-suppressed modes: B 0  D 0 h 0 (  0, ,  0, ,  ’),  0  0,  0  0 have the measured rates larger than theoretical expectations. Penguin-dominated modes such as B  K* , K , K , K*  predicted by QCDF are consistently lower than experiment by a factor of 2  3  importance of power corrections (inverse powers of mb) e.g. FSI, annihilation, EW penguin, New Physics, … 2. Rate enhancements in color-suppressed, fac.-forbidden or penguin-dominated modes

13. 13 FSI as rescattering of intermediate two-body states [Cheng, CKC, Soni]  FSIs via resonances are assumed to be suppressed in B decays due to the lack of resonances at energies close to B mass.  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 Form factor or cutoff must be introduced as exchanged particle is off-shell and final states are necessarily hard Alternative: Elastic Rescattering [CKC, Hou Yang] Regge trajectory [Nardulli,Pham][Falk et al.] [Du et al.] …

14. 14 Dispersive part is obtained from the absorptive amplitude via dispersion relation  = m exc + r  QCD (r: of order unity)  or r is determined by a fit to the measured rates  r is process dependent  n=1 (monopole behavior), consistent with QCD sum rules Once cutoff is fixed  direct 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  LD amp. vanishes in HQ limit

15. 15 BR SD (10 -6 ) BR with FSI (10 -6 ) BR Expt (10 -6 ) DCPV SD DCPV with FSI DCPV Expt BB 16.6 22.9 +4.9 -3.1 24.1  1.3 0.01 0.026 +0.00 -0.002 -0.02  0.03 B0B0 13.7 19.7 +4.6 -2.9 18.2  0.8 0.03-0.15 +0.03 -0.01 -0.11  0.02 B0B0 9.3 12.1 +2.4 -1.5 12.1  0.8 0.17-0.09 +0.06 -0.04 0.04  0.04 B0B0 6.0 9.0 +2.3 -1.5 11.5  1.0 -0.040.022 +0.008 -0.012 -0.09  0.14 For simplicity only LD uncertainties are shown here FSI yields correct sign and magnitude for A(  + K - ) ! K  anomaly: A(  0 K - )  A(  + K - ), while experimentally they differ by 3.4  SD effects?   Fleischer et al, Nagashima Hou Soddu, H n Li et al.] Final state interaction is important. _ _ _ _

16. 16 BR SD (10 -6 ) BR with FSI (10 -6 ) BR Expt (10 -6 ) DCPV SD DCPV with FSI DCPV Expt B 0  +   8.3 8.7 +0.4 -0.2 10.1  2.0 -0.01 -0.43  0.11 -0.47 +0.13 -0.14 B 0    + 18.0 18.4 +0.3 -0.2 13.9  2.1 -0.02 -0.25  0.06-0.15  0.09 B 0  0  0 0.44 1.1 +0.4 -0.3 1.8  0.6 -0.005 0.53  0.01 -0.49 +0.70 -0.83 B     0 12.3 13.3 +0.7 -0.5 12.0  2.0 -0.04 0.37  0.10 0.01  0.11 B      6.9 7.6 +0.6 -0.4 9.1  1.3 0.06 -0.58  0.15 -0.07 +0.12 -0.13 Sign and magnitude for A(  +  - ) are nicely predicted ! DCPVs are sensitive to FSIs, but BRs are not (r D =1.6) For  0  0, 1.4  0.7 BaBar Br(10 -6 )= 3.1  1.1 Belle 1.6 +2.2 -1.6 CLEO Discrepancy between BaBar and Belle should be clarified. ﹣ _ _ B   _

17. 17 Mixing induced CP violation Oscillation, e i  m t (V tb * V td ) 2 =|(V tb * V td ) 2 | e -i 2  Bigi, Sanda 81 Quantum Interference

18. 18  sin2  eff CKM phase is dominated. Look for small effects. Measuring the deviation of sin2  eff in charmonium and penguin modes (  w  0) is important in the search of NP [new physics (phase)] Deviation  NP How robust is the argument? Originally, FSI was totally ignored.

19. 19 In general, S f  sin2  eff  sin(2  +  W ). For b  sqq modes, Since a u is larger than a c, it is possible that S will be subject to significant “tree pollution”. However, a u here is color-suppressed. Penguin contributions to  K S and  0 K S are suppressed due to cancellation between two penguin terms (a 4 & a 6 )  relative importance of tree contribution  may have large deviation of S from sin2  Time-dependent CP asymmetries:

20. 20 FSI effects on sin2  eff (Cheng, CKC, Soni 05) FSI can bring in additional weak phase -- B→K * , K  contain tree V ub V us *=|V ub V us |e -i 

21. 21 FSI effects in rates FSI enhance rates though rescattering of charmful intermediate states [rates are used to fixed cutoffs (  =m + r  QCD, r~1)].

22. 22 FSI effects on direct CP violation Large CP violation in the  K mode.

23. 23 FSI effect on  S Theoretically and experimentally cleanest modes:  ’K s  K s Tree pollutions are diluted for non pure penguin modes.  K S,  0 K S sin2  =0.685  0.032 Input CKM sin2  =0.724

24. 24 FSI effects in mixing induced CP violation of penguin modes are small The reason for the smallness of the deviations:  The dominated FSI contributions are of charming penguin like. Do not bring in any additional weak phase.  The source amplitudes (K * ,K  ) are small (Br~10 -6 ) compare with Ds*D (Br~10 -2,-3 )  The source with the additional weak phase are even smaller (tree small, penguin dominate) If we somehow enhance K * ,K  contributions ⇒ large direct CP violation (A  Ks ). Not supported by data

25. 25 Conclusion QCDF improve naïve and generalized factorizations. It is model independent in the large m B limit. FSI should play some (sub-leading) role in B decays. (finite m B )  Rates are enhanced: PP modes K ,  ’K…; PV modes  0  0  K,  K,  0 K…  Large direct CP violation in K -         K   The deviation of sin2  eff from sin2  = 0.685  0.032 are at most O(0.1) in penguin-dominated B 0  K S,  K S,  0 K S,  ’K S,  0 K S, f 0 K S (w/wo FSI) sin2  eff on penguin modes are still good places to look for new phase. We should also try to look for them in other places.

26. 26 Back up slides

27. 27 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.

28. 28 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

29. 29 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 responsible for enhancement of color-suppressed graphs (see a 2 below)

30. 30 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