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QCD Factorization with Final-State Interactions

QCD Factorization with Final-State Interactions. Chun-Khiang Chua Academia Sinica, Taipei 3rd ICFP, Cung-Li, Taiwan. M 2. B. M 1. Factorization in B decays. We basically have three scales in a non-leptonic B decay: m W >> m B >> L QCD Integrating out d.o.f. above m B :

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QCD Factorization with Final-State Interactions

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

  2. M2 B M1 Factorization in B decays • We basically have three scales in a non-leptonic B decay: mW >> mB >> LQCD • Integrating out d.o.f. above mB: H=ci(m) Qi(m) • Naïve factorization: A  BM1 0 M2 ai(cj)FFBM1 fM2 In mb limit, M2 produced in point-like interactions carries away energies O(mb) and will decouple from soft gluon effect Bjorken

  3. Naïve 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 m in amp. from ai(m) cannot be cancelled. BR(Theory)≈3 10-3 BR(Expt.)=(2.76±0.25)10-3

  4. Direct CP violations : strong phase : weak phase B f ei(+) One needs at least two different B  f paths with distinct weak & strong phases _ first confirmed DCPV (5.7) in B decays (2004) _ _ We do have 2 different paths strong phase ?

  5. Generalize factorization Ali, Greub (98) Chen,Cheng,Tseng,Yang (99) penguin corrections • For problem with scheme and scale dependence, consider vertex and penguin corrections to four-quark matrix elements • Strong phase from the BSS cut: k2~m2B/4  m2B/2 gives large uncertainty • Corrections (non-fac. Contributions) are still incalculable. Parameterized.

  6. QCD Factorization Beneke, Buchalla, Neubert, Sachrajda (99) TI: TII: hard spectator interactions • M(x): light-cone distribution amplitude (LCDA) and x the momentum fraction of quark in meson M • 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(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(as)] • Hard spectator interaction, which is of the same 1/mb order as vertex & penguin corrections, is included (new)  crucial for a2 & a10 • Include distribution of meson momentum fraction  • 1. a new strong phase from vertex corrections • 2. fixed gluon virtual momentum in penguin diagram (imp.for dCP) • Except a6 and a8 all effective wilson coefficients are gauge and scheme independent. • a6 and a8 come with mc/mB=m2p/(mu+md) mB. Power correction. • QCDF is model independent in the large mB limit and reduces to naïve fac. in the O(as0) limit.

  8. 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 important (c enhancement) : (2GeV)m2/(mu+md)  2.6 GeV , 2 mc  mb For example, in the penguin-dominated mode B K A(BK)  a4+(2/mb) a6 where 2/mb 1 & a6/a4 1.7 • Phenomenologically, power corrections should be taken into account •  need to include twist-3 DAs p &  systematically OK for vertex & penguin corrections: (mc/mb) a6,8: scale independent.

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

  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 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. • For annihilation contributions endpoint divergence starts at twist-2 term. • Both endpoint divergences occur as 1/mB power corrections (model dependent). • FSI could be important. Several hints…

  11. 1. Large strong phases in charmless modes are needed input • 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/mb 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)

  12. 2. Rate enhancements in color-suppressed, fac.-forbidden or penguin-dominated modes • 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: B0  D0 h0 (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, …

  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. Form factor is introduced to render perturbative calculation meaningful  LD amp. vanishes in HQ limit •  = mexc + 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 Dispersive part is obtained from the absorptive amplitude via dispersion relation subject to large uncertainties and will be ignored in the present work

  15. _ _ _ _ • For simplicity only LD uncertainties are shown here • FSI yields correct sign and magnitude for A(+K-) ! • K anomaly: A(0K-)  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. B   B  ﹣ _ _ _ • Sign and magnitude for A(+-) are nicely predicted ! • DCPVs are sensitive to FSIs, but BRs are not (rD=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.

  17. Mixing induced CP violation Bigi, Sanda 81 Quantum Interference Oscillation, eiDm t (Vtb*Vtd)2 =|(Vtb*Vtd)2| e-i 2b

  18. D sin2beff • CKM phase is dominated. Look for small effects. • Measuring the deviation of sin2beff in charmonium and penguin modes (dw0) is important in the search of NP [new physics (phase)] • Deviation  NP • How robust is the argument? • Originally, FSI was totally ignored.

  19. Time-dependent CP asymmetries: In general, Sf sin2eff sin(2+W). For bsqq modes, Since au is larger than ac, it is possible that S will be subject to significant “tree pollution”. However, au here is color-suppressed. • Penguin contributions to KS and 0KS are suppressed due to cancellation between two penguin terms (a4 & a6) • relative importance of tree contribution • may have large deviation of S from sin2

  20. FSI effects on sin2beff(Cheng, CKC, Soni 05) • FSI can bring in additional weak phase -- B→K*p, Kr contain tree Vub Vus*=|Vub Vus|e-ig

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

  22. FSI effects on direct CP violation • Large CP violation in the rK mode.

  23. FSI effect on DS sin2b=0.6850.032 Input CKM sin2b=0.724 • Theoretically and experimentally cleanest modes: h’KsfKs • Tree pollutions are diluted for non pure penguin modes. wKS, r0KS

  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*p,Kr) 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*p,Kr contributions ⇒ large direct CP violation (AfKs). Not supported by data

  25. Conclusion • QCDF improve naïve and generalized factorizations. It is model independent in the large mB limit. • FSI should play some (sub-leading) role in B decays. (finite mB) • Rates are enhanced: PP modes Kp, ’K…; PV modes r0p0K, K, 0K… • Large direct CP violation in K-p+, r+p-, r0K0... • The deviation of sin2beff from sin2 = 0.6850.032 are at most O(0.1) in penguin-dominated B0KS, KS, 0KS, ’KS, 0KS, f0KS (w/wo FSI) • sin2beff 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 /mb with =m2/(mu+md) Cn: Gegenbauer poly. with 01 du (u)=1, 01 du p,(u)=1

  28. 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

  29. 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) ] responsible for enhancement of color-suppressed graphs (see a2 below)

  30. 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

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