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

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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  10 -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 -  + -0.13  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 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 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 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 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 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 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  0.032 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  = 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 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

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