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C.D. LuSino-German1 Hadronic B Decays in Perturbative QCD Approach Formalism of perturbative QCD (PQCD) based on k T factorization Direct CP asymmetry Polarization in B VV decays Summary Thank colleagues: Keum, Li, Sanda, Ukai, Yang, … Cai-Dian Lü (IHEP, Beijing)

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C.D. LuSino-German2 Naïve Factorization Approach + u B 0 – d Decay matrix element can be separated into two parts: Short distance Wilson coefficients and Hadronic parameters: form factor and decay constant Idea borrowed from semi-leptonic decay (BSW)

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C.D. LuSino-German3 QCD factorization approach Based on naïve factorization ， expand the matrix element in 1/m b and α s = [1+∑r n α s n +O(Λ QCD /m b )] Keep only leading term in Λ QCD /m b expansion and sub-leading order in α s expansion

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C.D. LuSino-German4 QCDF OCD-improved factorization = naïve factorization + QCD correction Factorizable emission Leading Vertex Non-spectator Exchange & correction Annihilation Sub-leading

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C.D. LuSino-German5 Two concerns: The emission diagram is certainly leading…. But why must it be written in the BSW form ? Has naïve factorization been so successful that what we need to do is only small sub-leading correction ? QCDF amplitude: Both answers are “No”

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C.D. LuSino-German6 Picture of PQCD Approach Six quark interaction inside the dotted line 4-quark operator b

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C.D. LuSino-German7 PQCD approach A ~ ∫d 4 k 1 d 4 k 2 d 4 k 3 Tr [ C(t) B (k 1 ) (k 2 ) (k 3 ) H(k 1,k 2,k 3,t) ] exp{-S(t)} (k 3 ) are the light-cone wave functions for mesons: non-perturbative, but universal C(t) is Wilson coefficient of 4-quark operator exp{-S(t)} is Sudakov factor ， to relate the short- and long-distance interaction H(k 1,k 2,k 3,t) is perturbative calculation of six quark interaction channel dependent

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C.D. LuSino-German8 Perturbative Calculation of H(t) in PQCD Approach Form factor — factoriz able Non- factori zable

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C.D. LuSino-German9 Perturbative Calculation of H(t) in PQCD Approach Non- factorizable annihilation diagram Factorizable annihilation diagram D (*)

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C.D. LuSino-German10 Do not need form factor inputs All diagrams using the same wave functions (same order in s expansion) All channels use same wave functions Number of parameters reduced

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C.D. LuSino-German11 Feynman Diagram Calculation Wave function k 2 =m B (y,0,k 2 T ), k 1 =m B (0,x,k 1 T ) k 2 ·k 1 = k 2 + k 1 – - k 2 T ·k 1 T ≈ m B 2 xy

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C.D. LuSino-German12 Endpoint Singularity x,y are integral variables from 0 1, singular at endpoint In fact, transverse momentum at endpoint is not negligible then no singularity The gluon propagator

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C.D. LuSino-German13 After including the quark transverse momentum there is no endpoint singularity large double logarithm are produced after radiative corrections, they should be resummed to generate the Sudakov form factor

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C.D. LuSino-German14 Sudakov factor The soft and collinear divergence produce double logarithm ln 2 Pb ， Summing over these logs result a Sudakov factor. It suppresses the endpoint region

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C.D. LuSino-German15 There is also singularity at non-factorizable diagrams But they can cancel each other between the two diagrams ， that is why QCD factorization can calculate these two without introducing k T

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C.D. LuSino-German16 D(K) meson with asymmetric wave function emitted, they are not canceled between the two diagrams that is why QCDF can not do this kind of decays It is also true for annihilation type diagrams DD u u Endpoint Singularity

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C.D. LuSino-German17 Endpoint singularity in collinear factorization The sub-leading calculation shows an end-point singularity Need to introduce arbitrary cutoffs

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C.D. LuSino-German18 Power Counting--QCDF Form factor diagrams are leading All others are s suppressed Annihilation-type are even power suppressed (small)

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C.D. LuSino-German19 Power Counting--PQCD All diagrams are at the same order of s Some non-factorizable diagram contributions are suppressed due to cancellations power suppressed

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C.D. LuSino-German20 QCDF vs PQCD Form factor input No need form factor Wave function input Wave function input Parameterize Annihilation /exchange diagram Annihilation /exchange diagram calculable B D 0 pi 0 not calculable Most modes calculable

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C.D. LuSino-German21 Two operators contribute to decay: B 0 0 B 0 0 color enhanced color suppressed C 1 ~ – 0.2 ~ 1/3 C 2 ~ 1/3 dd u u

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C.D. LuSino-German22 arg (a 2 /a 1 ) ~ – 41° For B 0 D 0 0, non-factorizable diagrams do not cancel PQCDExp. B 0 D – + 2.8±0.43.0±0.4 B + D 0 + 5.5±0.45.3±0.5 B 0 D 0 ± ±0.05

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C.D. LuSino-German23 Branching Ratios Some of the branching ratios agree well with experiments for most of the methods Since there are always some parameters can be fitted : Form factors for factorization and QCD factorization Wave functions for PQCD, but CP ….

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C.D. LuSino-German24 Direct CP Violation Require two kinds of decay amplitudes with: Different weak phases (SM) Different strong phases – need hadronic calculation, usually non-perturbative

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C.D. LuSino-German25 B→ , K Have Two Kinds of Diagrams with different weak phase W b u Tree ∝ V ub V ud * (s) B d(s) (K ) W b t Penguin ∝ V tb V td * (s) B O 3,O 4,O 5,O 6 O 1,O 2 (K )

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C.D. LuSino-German26 Strong phase is important for direct CP But usually comes from non-perturbative dynamics, for example D K K K For B decay, perturbative dynamic may be more important

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C.D. LuSino-German27 Main strong phase in FA When the Wilson coefficients calculated to next-to-leading order, the vertex corrections can give strong phase

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C.D. LuSino-German28 Strong phase in QCD factorization It is small, since it is at α s order Therefore the CP asymmetry is small The strong phase of Both QCD factorization and generalized factorization come from perturbative QCD charm quark loop diagram

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C.D. LuSino-German29 Cut quark diagram ~ Sum over final-state hadrons ~ On-shell Off-shellhadrons Inclusive Decay

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C.D. LuSino-German30 Annihilation-Type diagram Very important for strong phases Can not be universal for all decays, since not only one type ----sensitive to many parameters

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C.D. LuSino-German31 Annihilation-Type diagram W annihilation W exchange Time-like penguin Space-like penguin

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C.D. LuSino-German32 Naïve Factorization fail ? Momentum transfer:

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C.D. LuSino-German33 pseudo-scalar B requires spins in opposite directions, namely, helicity conservation momentum B fermion flow spin (this configuration is not allowed) p1p1 p2p2 Annihilation suppression ~ 1/m B ~ 10% Like B e e For (V-A)(V-A), left-handed current

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C.D. LuSino-German34 PQCD Approach Two diagrams cancel each other for (V-A)(V-A) current — dynamical suppression (K)

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C.D. LuSino-German35 W Exchange Process V cb * V ud ~ 2

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C.D. LuSino-German36 W exchange process Results: Reported by Ukai in BCP4 (2001) before Exps:

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C.D. LuSino-German37 V tb * V td, small br, 10 –8 d s u K+K+ B K + K – decay K–K– d s Time-like penguin Also (V-A)(V-A) contribution Comparing B(B pi pi): 10 –6, 1%

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C.D. LuSino-German38 Chiral Enhancement Two penguin operators: O 4 ~(V-A)(V-A) O 6 ~(V-A)(V+A) bs q t R,L=1 5 Fiertz trans. O(1) (S+P)(S-P) 2(m K 2 /m s ) x 1/m B

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C.D. LuSino-German39 No suppression for O 6 Space-like penguin Become (s-p)(s+p) operator after Fiertz transformation Chirally enhanced No suppression, contribution “big” (20-30%) d u d + (K + ) ––

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C.D. LuSino-German40 CP Violation in B (K) (real prediction before exp.) CP(%)FABBNSPQCDExp + K – +9±3+5±9–17±5–11.5±1.8 + K ± 0.11 ±1– 1.0 ±0.5– 2 ±4 0 K + +8 ± 27 ±9– 13 ±4 +4 ± 4 + – –5±3–6±12+30±10+37±10 (2001)

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C.D. LuSino-German41 Annihilation in QCDF Power (1/m B ) suppressed and s suppressed Should not be large But has to be large from exp.

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C.D. LuSino-German42 Operator O 6 is very important Important for I = 1/2 rule in history B , K -- direct CP K* K K* K* branching ratio too small in QCDF polarization problem

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C.D. LuSino-German43 How about mixing induced CP? Dominant by the B-B bar mixing Most of the approaches give similar results Even with final state interactions: B + –, K, ’K, KKK …

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C.D. LuSino-German44 For Example: (From Yossi Nir)

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C.D. LuSino-German45 Polarization of B VV decays

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C.D. LuSino-German46 Helicity flip suppression of the transverse polarization amplitude Naïve counting rule H = M N M T

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C.D. LuSino-German47 Counting Rules for B VV Polarization The measured longitudinal fractions R L for B are close to 1. R L ~ 0.5 in K * dramatically differs from the counting rules. Are the K* polarizations understandable?

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C.D. LuSino-German48 Theoretical attempts to solve these puzzles Currents that breaks the naïve cutting rule: a) new physics b) the magnetic penguin c) the annihilation diagrams …… Nonperturbative corrections: a) the charming penguin b) the final state interactions ……

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C.D. LuSino-German49 There are still problems for some of the explanations The perpendicular polarization is given by: Final state interaction can not explain R N = R T and some others are difficult to explain the relative phase Naïve Babar and Belle Avg.

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C.D. LuSino-German50 Fierz Transformation The annihilation diagram The (S+P)(S-P) current can break the counting rule, The annihilation diagram contributes equally to the three polarization amplitudes

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C.D. LuSino-German51 No suppression for O 6 Space-like penguin Become (s-p)(s+p) operator after Fiertz transformation Chirally enhanced No suppression, contribution “big” (20-30%) d s d K*

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C.D. LuSino-German52 Annihilation can enhance transverse contribution: R L = 59% (exp:50%) and also right ratio of R =, R and right strong phase =, d s d Large transverse component in B K * decays K*K* H-n Li, Phys. Lett. B622, 68, 2005

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C.D. LuSino-German53 Alex Kagan ’ s study in QCDF

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C.D. LuSino-German54 Polarization for B ( ) ( ) Phys.Rev.D73:014024,2006

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C.D. LuSino-German55 Polarization of B K * ( ) Decay modesR L (exp)RLRL R=R= RR 66%76%13%11% 96%78%11% 78%12%10% 72%19%9% Phys.Rev.D73:014011,2006

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C.D. LuSino-German56 Transverse polarization is around 35% d s s Time-like penguin in B decays ( 10 – 8 ) Eur. Phys. J. C41, , 2005

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C.D. LuSino-German57 V tb * V td, small br, 10 – 7 d s u K*K* B K * K * decay K*K* d s Time-like penguin Also (V-A)(V-A) contribution

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C.D. LuSino-German58 Polarization of B K * K * Decay modesRLRL R=R= RR 67%18%15% 75%13%12% 99%0.5% Tree dominant Phys.Rev.D72:054015,2005

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C.D. LuSino-German59 and is the first measured channel in Bs decays, which is useful to determine the Bs wave function Experiment :

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C.D. LuSino-German60 Summary The direct CP asymmetry measured by B factories provides a test for various method of non-leptonic B decays PQCD can give the right sign for CP asymmetry the strong phase from PQCD should be the dominant one. The polarization in B VV decays can also be explained by PQCD Important role of Annihilation type diagram

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C.D. LuSino-German61 Thank you! Vielen Dank ！ 谢谢！

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C.D. LuSino-German62 Contributions of different α s in H(t) calculation Fraction αs/αs/

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C.D. LuSino-German63 Error Origin in PQCD The wave functions The decay constants CKM matrix elements High order corrections CP is sensitive to

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C.D. LuSino-German64 Branching ratio in NLO(10 -6 ) Li, Mishima, Sanda hep-ph/

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C.D. LuSino-German65 NLO direct CP asymmetry

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C.D. LuSino-German66 B K puzzle Their data differ by 3.6 A puzzle? K + - and K + 0 differ by sub-leading amplitudes P ew and C. Their CP are expected to be similar.

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