Polarizations of B V V in QCD factorization

Transcript

1 Polarizations of B V V in QCD factorization Based on collaborations with M.Beneke and J.Rohrer April

2 Motivation B V V decays share the roles of B PP, PV decays in the determination of CKM matrix elements, especially the angles(phases): sin2β and cos2β from B J/ΨK ; sin2α from B ρρ; new physics search direct search for the unexpected large branching ratios of rare decays in the SM; more new physics sensitive observables in B V V : polarizations, phases (phase diffences) of helicity amplitudes April

3 Plan of talk 1. Helicity amplitudes of B V V 2. Current experimental status 3. QCD factorization formula for B V V 4. Phenomenologies of B V V Tree-dominated decays (B ρρ) Penguin-dominated decays (B φk and ρk ) 5. Summary Based on the works collaborated with M.Beneke and J. Rohrer Branching fractions, polarization and asymmetries of B V V decays, Nucl. Phys. B774, , 2007; Enhanced electroweak penguin amplitude in B V V decays, Phys. Rev. Lett. 96, , April

4 Helicity amplitudes of B V V General decay amplitude of B(p B ) V 1 (p 1, η )V 2 (p 2, ǫ ) ( A(B V 1 V 2 ) = i η µ ǫ ν p Bµ p Bν p ρ S 1 g µν S 2 m 2 + is 3 ε 1 pσ 2 µνρσ B p 1 p 2 With definite helicity, A 0 = A(B V 1 (p 1, η0 )V 2(p 2, ǫ 0 ) = im2 B S 1 S 2 2m 1 m 2 2 A ± = A(B V 1 (p 1, η± )V 2(p 2, ǫ ± ) = i(s 1 S 3 ). Or we can define transversity amplitudes A 0, A are CP-even, and A is CP-odd. A, = ( A + ± A ) / 2. 6 flavor-tagged helicity amplitudes, total 10 independent observables; ( ) ), April

5 Definitions of observables flavor-tagged definitions 2 A0,, fl,, B = A A + A 2, 2 f B Ā 0,, L,, = Ā 2 Ā Ā 2, flavor-averaged quantities and asymmetries φb, = arg A, A 0, φ B, = arg Ā, Ā 0, f h = 1 2 ( f B h + f B h φ h φ B h φ h (mod 2π) ), A h CP = f B h f B h f B h + f B h φ B h + φ h (mod 2π), π 2 φ h < π 2 h = L,, In absence of CP violation, A h CP = 0 and δφ h = 0. April

6 Definitions of observables (Belle) Time dependent observables: with Γ( B 0 (B 0 )(t) V 1 V 2 ) = e Γ Bt λ σ (Λ λσ ± Σ λσ cos( m B t) ρ λσ sin( m B t))g λ g σ, Λ λλ = 1 2 ( A λ 2 + Ā λ 2 ), Σ λλ = 2 1 ( A λ 2 Ā λ 2 ), Λ i = Im(A A i Ā Ā i ), Λ 0 = Re(A A 0 + Ā Ā 0 ), Σ i = Im(A A i + Ā Ā i ), Σ 0 = Re(A A 0 Ā Ā 0 ), ρ i = Re ( q p (A Āi + A iā ) ), ρ 0 = Im ( q p (A Ā0 + A 0Ā ) ) ρ = Im ( q ) p A Ā, ),, ρ ii = Im ( q p A iāi where i = 0,. These 18 observables can have connections with the observables used by BaBar collaborations if one uses relevant normalization and neglects the small CP asymmetries. April

7 dγ(b V 1 V 2 (P 1 P 2 )(Q 1 Q 2 )) dcos ϑ 1 dcos ϑ 2 dϕ A 0 2 cos 2 ϑ 1 cos 2 ϑ sin2 ϑ 1 sin 2 ϑ 2 ( A+ 2 + A 2 ) cos ϑ 1 sin ϑ 1 cos ϑ 2 sin ϑ 2 [ Re ( e iϕ A 0 A +) + Re ( e +iϕ A 0 A sin2 ϑ 1 sin 2 ϑ 2 Re ( e 2iϕ A + A ), )] P 1 ϕ V 1 ϑ 1 Q 1 ϑ 2 V 2 Q 2 P 2 April

8 Picture of B non-leptonic two-body decays B Weak scale: Hard scale: Hard-collinear scale: Soft (or collinear) scale: m W m b M 2 M 1 m b Λ QCD Λ QCD Basic idea: to separate the contribution from different scales; Separation of m W and m b scale by the weak Hamiltonian H eff = C i (µ)q i (µ) i Separation of m b and lower scales M 1 M 2 Q i (µ) B =? April

9 Tree operators: Q u 1 = (ū αb α ) V A ( q β u β ) V A Q c 1 = ( c αb α ) V A ( q β c β ) V A Q u 2 = (ū αb β ) V A ( q β u α ) V A Q c 2 = ( c αb β ) V A ( q β c α ) V A QCD penguin operators: Q 3 = ( q α b α ) V A ( q q β q β ) V A Q 4 = ( q β b α ) V A ( q αq q β ) V A Q 5 = ( q α b α ) V A ( q q β q β ) V +A Q 6 = ( q β b α ) V A ( q αq q β ) V +A EW penguin operators: Q 7 = 3 2 ( q αb α ) V A e q ( q q β q β ) V +A Q 8 = 3 2 ( q βb α ) V A e q ( q αq q β ) V +A Q 9 = 3 2 ( q αb α ) V A e q ( q q β q β ) V A Q 10 = 2 3 ( q βb α ) V A e q ( q αq q β ) V A Dipole operators: Q 7γ = e 8π 2m b q α σ µν (1 + γ 5 )b α F µν Q 8G = g 8π 2m b q α σ µν t a αβ (1 + γ 5)b β G a µν April

10 Polarization in B V V Polarization: approximately, V (ǫ 0 ) v ū + v ū, V (ǫ + ) v ū, V (ǫ ) v ū u R (p) u (p), u L (p) u (p), v L (p) v (p), v R (p) v (p) each helicity flip costs the suppression m/2e. Ā 0 : Ā : Ā + 1 : Λ m B : Λ2 m 2 B f L = 1 O(m 2 V /m2 B ), f f O(m 2 V /m2 B ) April

11 Tree-dominated processes B ρρ, ωρ; 1 f L = O(Λ 2 /m 2 b ) about few percent; Decay Modes P.F. Belle BaBar HFAG B + ρ 0 ρ + f L 0.95 ± 0.11 ± ± B 0 ρ 0 ρ 0 f L 0.70 ± 0.14 ± ± 0.15 B 0 ρ + ρ f L ± ± B 0 K 0 K 0 f L ± B + ωρ + f L 0.82 ± 0.11 ± ± 0.11 Obtain sin2α from the time-dependent measurements of B ρρ; April

12 Penguin-dominated processes Case 1: B φk 1 f L = O(1) about a half (polarization puzzles); Decay Modes P.F. Belle BaBar HFAG B + φk + f L 0.52 ± 0.08 ± ± 0.12 ± ± 0.07 f 0.19 ± 0.08 ± ± 0.08 φ 2.10 ± 0.28 ± ± 0.31 φ 2.31 ± 0.30 ± ± 0.31 B 0 φk 0 f L 0.45 ± 0.05 ± ± ± ± f ± ± ± ± φ ± ± 0.14 ± φ 2.51 ± 0.25 ± ± 0.15 ± ± 0.14 April

13 Penguin-dominated processes Case 2: B ρk * 1 f L = O(Λ 2 /m 2 b ) for B+ ρ 0 K + * 1 f L = O(1) for B + ρ + K 0 and B 0 ρ 0 K 0 * Even more puzzling than B φk! Decay Modes P.F. Belle BaBar HFAG B + ρ 0 K + f L B + ρ + K 0 f L 0.43 ± ± ± 0.10 ± ± 0.08 B 0 K 0 ρ 0 f L 0.57 ± 0.09 ± ± 0.12 April

14 Polarization puzzles B φk : enhanced penguin amplitude with negative-helicity * new physics effects (scalar and tensor current-current coupling) Kagan, 2004;Das and Yang, 2004 * large charming penguin C.W.Bauer et al, 2003 * final state interactions H.Y.Cheng, 2004 * smaller A 0, larger A 1 and V H.N.Li et al, 2004 * large annihilation effects B ρk : * Expected to follow the same pattern of B φk ; * 1 f L = O(Λ 2 /m 2 b ) for B+ ρ 0 K + Kagan, 2004; Beneke, Rohrer and Yang, 2006 * 1 f L = O(1) for B + ρ + K 0 and B 0 ρ 0 K 0 * Large electroweak penguin in negative-helicity amplitude or new physics? April

15 QCD factorization formula for B V V Kagan, 2004; M.Beneke, J.Rohrer and DY, 2005 V 1,h V 2,h Q i B = F B V 1,h T I,h i f h V 2 Φ h V 2 + (V 1 V 2 ) + T II,h i f B Φ B f V1 Φ V1 f h V 2 Φ h V 2 + O(1/m b ). where h = 0,, and in terms of α-convention in [Beneke& Neubert, 2003] A h ( B V 1 V 2 ) A h α h i (V 1V 2 ) i A 0 A B V 1 0 ( ) Λ 3/2 mb A m 2 m B ( (1 + m 1 m B )A B V (1 m 1 A m 2 m B ( (1 + m 1 m B )A B V 1 1 (1 m 1 m B )V B V 1 m )V B V ) ( ) 1 Λmb 5/2 B ) ( Λmb ) 7/2 Positive helicity amplitude is highly suppressed and cannot be calculated in same way. f = f, φ = φ. April

16 Penguin weak annihilations X A ln m B Λ A (1 + ρ A e iφ A ), X 2 A, X L m B Λ L (1 + ρ L e iφ L ); P h = A h V 1 V 2 [ α h 4 + β h 3], P P 0 A α c A 0 ρk α c 4 (π K) + β3 c(π K) = 0.09 {0.02[ 0.01,0.05]}, α c0 4 (ρ K ) + β3 c0(ρ K ) = 0.03 {0.00[ 0.00,0.00]}, α c 4 (ρ K ) + β3 c (ρ K ) = 0.05 {0.03[ 0.04,0.10]}. ρk 4 + β3 c α c, [ 0.04,0.10] 0.12, with A ρk A 0 ρk 1 4. Negative-helicity penguin amplitude can be (but need not) be enhanced by the penguin annihilation! April

17 Enhanced EW penguin in B V V H eff = G F 2 p=u,c λ (D) p a= C a 7γ Qa 7γ +..., Q 7γ = e m b 8π 2 Dσ µν (1 ± γ 5 )F µν b λ (D) p = VpD V pb. introduces the additional EW penguin amplitude for the decays to the neutral vector meson (ρ 0,ω, φ), The resulted amplitudes for different helicities P0 EW : P EW 1 : m b/λ (Here we neglect the contribution from Q + 7γ.) April

18 The representative coefficient in QCDF α p 3,EW (V 1V 2 ) = 2α em 3π C 7γ,eff R m B m b m 2 V 2 with R = 1 in the heavy quark limit. Numerically, we have α p 3,EW (K ρ) 0.02, meanwhile the uncorrected EW penguin and leading QCD penguin Effectively, α p 3,EW (K ρ) = C 7 + C 9 + C 8 + C 10 N c , α c 3,EW (K V 2 ) ˆα c 4 (ρk ) = C 4 + C 3 N c = 2α em 3π R m 2 B m 2 V 2 with T1 K (0) 0.28, we get α c 3,EW (K ρ) = Γ(B K γ) G 2 F V ts V tb 2 α em 8π 3 4π m 5 B T 1 K (0)2 1/2 April

19 Tree-dominated decays BrAv(B ρ ρ 0 ) = V ub (0) 0.30 A B ρ 2 ( ) 10 6 BrAv / 10 6 A CP / percent Theory Experiment Theory Experiment B ρ ρ ± ± 13 B 0 ρ + ρ ± 13 B 0 ρ 0 ρ ± n/a B ωρ ± 18 B 0 ωρ < 1.5 no prediction n/a B 0 ωω < n/a April

20 f L / percent A 0 CP / percent Theory Experiment Theory B ρ ρ B 0 ρ + ρ ± B 0 ρ 0 ρ ± B ωρ ± B 0 ωρ n/a B 0 ωω n/a April

21 Strategies in penguin-dominated decays QCDF loses predictive power in penguin annihilations with transverse polarization; Use information from experiments as much as we can; Strategy 1: fit only the penguin annihilation from B φk measurements; Strategy 2: fit the whole penguin amplitude from B φk ; Trust the predictions for other topological amplitudes using QCDF; Constrained X A : A = 0.5 ± 0.2 exp. ϕ A = ( 43 ± 19 exp. ), ˆα c 4 = αc 4 + β 3 from data: with α c 3 Ā = A K φλ (s) c P K φ, P K φ = ( ± 0.008(exp) (th)) from QCDF ˆα c 4 +i(0.021 ± 0.015(exp) (th)), = ( 0.08 ± 0.02) + i(0.03 ± 0.02). April

22 Observable Theory Experiment BrAv /10 6 φk φ K A CP /% φk φ K f L /% φk φ K A 0 CP /% φk φ K (f f )/% φk φ K (A CP A CP )/% φk 0 +0 φ K φ / φk default constrained X A ˆα c 4 from data φ K φ / φk 0 +0 φ K (φ φ )/ φk φ K ( φ φ )/ φk 0 +0 φ K ± ± ± ± ± ± 4.0 n/a ± n/a n/a ± ± n/a April

23 BrAv /10 6 Theory Experiment default B K φ B 0 K 0 φ B K ω B 0 K 0 ω B K 0 ρ B K ρ B 0 K ρ B 0 K 0 ρ B s φφ ˆα c 4 from data ± ± < < ± < n/a ± April

24 f L / percent Theory Experiment default B K φ B 0 K 0 φ B K ω B 0 K 0 ω B K 0 ρ B K ρ B 0 K ρ B 0 K 0 ρ ˆα c 4 from data ± ± n/a 32 n/a ± n/a ± 12 April

25 More on B ρk system A h (ρ K 0 2Ah ) = P h (ρ 0 K ) = [P h + Ph EW ] + e iγ [T h + C h ] A h (ρ + K ) = P h + e iγ T h 2A h (ρ 0 K 0 ) = [P h Ph EW ] + e iγ [ C h ] and define x h = X h /P h (h = 0, 1). Γ (ρ K 0 ) : 2 Γ (ρ 0 K ) : 2 Γ (ρ 0 K 0 ) 1 : 1 + p EW 2 : 1 p EW 2 B K ρ 0 B0 K 0 ρ 0 incl. excl. exp. incl. excl. exp. BrAv / < ± 1.6 f L / % ± 12 A CP / % ± 19 QCDF predicts f L (ρ 0 K ) > f L (ρ K 0 ) > f L (ρ 0 K 0 ). It is against current measurements. April

26 Conclusions and perspective QCD factorization loses predictive power for penguin-dominated B V V decays; Penguin weak annihilation could be an answer to polarization puzzle of B φk ; Enhanced electroweak penguin with negative-helicity could explain the polarization puzzles in B + ρ + K 0 and B + ρ 0 K +, but not for polarization of B 0 ρ 0 K 0 ; Polarization puzzles of B ρk are challenging for new physics model building; New measurements on polarizations in B AV and TV will shed more light on research of chirality structure of interaction, but QCD effects are still crucial; April

27 THANKS! April