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

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 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  BR(Expt.)=(2.76±0.25)  10 -3

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 -     0.02 B0→+-B0→+    0.14 B0→+-B0→+    0.10 first confirmed DCPV (5.7  ) in B decays (2004) _ _ _ We do have 2 different paths Direct CP violations strong phase ?

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 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 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 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 m b /2 m b 2m b a1a i i i i i i0.015 a2a i i i i i i0.066 a4ua4u i i i i i i0.014 a5a i i i i i i0.001 a6ua6u i i i0.019 a 10 /  i i i i i i0.093 black: vertex & penguin, blue: hard spectator green: total a i for B  K  at different scales

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 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 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 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 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 BR SD (10 -6 ) BR with FSI (10 -6 ) BR Expt (10 -6 ) DCPV SD DCPV with FSI DCPV Expt BB   0.03 B0B0   0.02 B0B0   0.04 B0B0   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 BR SD (10 -6 ) BR with FSI (10 -6 ) BR Expt (10 -6 ) DCPV SD DCPV with FSI DCPV Expt B 0  +     B 0       0.09 B 0  0    B        0.11 B        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 CLEO Discrepancy between BaBar and Belle should be clarified. ﹣ _ _ B   _

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

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  Input CKM sin2  =0.724

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 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  =  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 Back up slides

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 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 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 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