Polarizations of B V V in QCD factorization

Polarizations of B V V in QCD factorization Based on collaborations with M.Beneke and J.Rohrer April 2008 1

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 2008 2

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, 64-101, 2007; Enhanced electroweak penguin amplitude in B V V decays, Phys. Rev. Lett. 96, 141801, 2006. April 2008 3

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 2008 4

Definitions of observables flavor-tagged definitions 2 A0,, fl,, B = A 0 2 2 + A + A 2, 2 f B Ā 0,, L,, = Ā 2 Ā 2 0 + + 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 2008 5

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 2008 6

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 2 + 1 4 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 + 1 2 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 2008 7

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 2008 8

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 2008 9

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 2008 10

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 ± 0.02 0.905 ± 0.042 +0.023 0.027 0.912 +0.044 0.045 B 0 ρ 0 ρ 0 f L 0.70 ± 0.14 ± 0.05 0.70 ± 0.15 B 0 ρ + ρ f L 0.941 +0.034 0.040 ± 0.030 0.992 ± 0.024+0.026 0.013 0.978 +0.025 0.022 B 0 K 0 K 0 f L 0.80 +0.10 0.12 ± 0.06 0.80+0.12 0.13 B + ωρ + f L 0.82 ± 0.11 ± 0.02 0.82 ± 0.11 Obtain sin2α from the time-dependent measurements of B ρρ; April 2008 11

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 ± 0.03 0.46 ± 0.12 ± 0.03 0.50 ± 0.07 f 0.19 ± 0.08 ± 0.02 0.19 ± 0.08 φ 2.10 ± 0.28 ± 0.04 2.10 ± 0.31 φ 2.31 ± 0.30 ± 0.07 2.31 ± 0.31 B 0 φk 0 f L 0.45 ± 0.05 ± 0.02 0.506 ± 0.040 ± 0.015 0.491 ± 0.032 f 0.31 +0.06 0.05 ± 0.02 0.227 ± 0.038 ± 0.013 0.252 ± 0.031 φ 2.40 +0.28 0.24 ± 0.07 2.31 ± 0.14 ± 0.08 2.37+0.14 0.13 φ 2.51 ± 0.25 ± 0.06 2.24 ± 0.15 ± 0.09 2.36 ± 0.14 April 2008 12

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 0.96 +0.04 0.15 B + ρ + K 0 f L 0.43 ± 0.11 +0.05 ± 0.04 0.96+0.06 0.16 0.02 0.52 ± 0.10 ± 0.04 0.48 ± 0.08 B 0 K 0 ρ 0 f L 0.57 ± 0.09 ± 0.08 0.57 ± 0.12 April 2008 13

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 2008 14

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 1 + (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 2008 15

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 4 0.05 + [ 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 2008 16

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 2008 17

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 +... 0.01, α c 3,EW (K V 2 ) ˆα c 4 (ρk ) = C 4 + C 3 N c +... 0.055. = 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) = 0.023. 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 2008 18

Tree-dominated decays BrAv(B ρ ρ 0 ) = V ub 3.53 10 3 2 0 (0) 0.30 A B ρ 2 ( 18.8 0.4 +0.4 3.9 +3.2 ) 10 6 BrAv / 10 6 A CP / percent Theory Experiment Theory Experiment B ρ ρ 0 18.8 +0.4 0.4 +3.2 3.9 18.2 ± 3.0 0 +0 0 8 ± 13 B 0 ρ + ρ 23.6 +1.7 1.9 +3.9 3.6 23.1 +3.2 3.3 1 +0 0 +4 8 +11 ± 13 B 0 ρ 0 ρ 0 0.9 +0.6 0.3 +1.9 0.9 1.07 ± 0.38 +28 +5 7 +53 29 n/a B ωρ 12.8 +1.1 1.3 +2.0 8 +4 ± 18 B 0 ωρ 0 0.2 +0.1 0.1 +0.3 0.1 < 1.5 no prediction n/a B 0 ωω 0.9 +0.5 0.3 +1.5 0.9 < 4.0 29 +9 n/a 2.4 10.6 +2.6 2.3 8 +3 2 +5 6 +25 44 April 2008 19

f L / percent A 0 CP / percent Theory Experiment Theory B ρ ρ 0 95.9 +0.2 0.3 +3.4 5.9 91.2 +4.4 4.5 B 0 ρ + ρ 91.3 +0.4 0.3 +5.6 6.4 96.8 ± 2.3 2 +0 B 0 ρ 0 ρ 0 90 +3 4 +8 56 87 ± 14 8 +2 0 0 +4 2 1 +59 28 B ωρ 93.7 +1.1 1.0 +4.7 8.1 82 ± 11 2 +1 B 0 ωρ 0 49 +11 11 +47 23 n/a +35 +25 B 0 ωω 93 +2 4 +5 22 n/a +6 +1 0 +7 6 15 +47 84 1 +14 24 April 2008 20

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 = ( 0.084 ± 0.008(exp) +0.008 0.009 (th)) from QCDF ˆα c 4 +i(0.021 ± 0.015(exp) +0.003 0.002 (th)), = ( 0.08 ± 0.02) + i(0.03 ± 0.02). April 2008 21

Observable Theory Experiment BrAv /10 6 φk 10.1 0.5 +0.5+12.2 φ K 0 9.3 0.5 +0.5+11.4 A CP /% φk 0 +0 0 +2 φ K 0 1 +0 0 +1 f L /% φk 45 +0 0 +58 φ K 0 44 +0 0 +59 A 0 CP /% φk 1 +0 0 +2 φ K 0 0 +0 0 +1 (f f )/% φk 0 +0 0 +2 φ K 0 0 +0 0 +2 (A CP A CP )/% φk 0 +0 φ K 0 0 +0 φ / φk 41 +0 0 +84 default constrained X A ˆα c 4 from data φ K 0 42 +0 0 +87 φ / φk 0 +0 φ K 0 0 +0 (φ φ )/ φk 0 +0 0 +1 φ K 0 0 +0 0 +1 ( φ φ )/ φk 0 +0 φ K 0 0 +0 7.1 10.1 0.5 +0.5 4.8 +7.2 10.4 0.5 +0.5+5.2 6.5 9.3 0.5 +0.5 4.5 +6.7 9.6 0.5 +0.5+4.7 1 0 +0 0 0 +0 0 +3 0 1 +0 0 1 +0 0 +2 36 45 +0 0 +35 31 44 +0 0 +23 36 44 +0 0 +35 31 43 +0 0 +23 1 1 +0 0 +1 1 1 +0 0 +2 2 1 0 +0 0 +1 0 0 +0 0 +1 3.9 9.7 ± 1.5 3.6 9.50 ± 0.90 2 5 ± 11 1 0.0 ± 7.0 23 50.0 ± 7.0 23 49.0 ± 4.0 n/a 2 1.0 ± 8.0 2 0 +0 0 +2 2 0 +0 0 +2 2 12 +17 17 2 0 +0 0 +2 2 0 +0 0 +2 2 3.0 7.2 +8.9 0 0 +0 0 0 +0 0 n/a 0 0 +0 0 0 +0 0 32 +36 36 53 41 +0 0 +35 30 40 +0 0 +21 54 42 +0 0 +35 30 42 +0 0 +21 21 42 +10 9 1 0 +0 0 0 +0 0 n/a 0 0 +0 0 0 +0 1 2 ± 10 21 60 ± 16 1 0 +0 0 +1 1 0 +0 0 +1 1 12 +24 24 1 0 +0 0 +1 1 0 +0 0 +1 1 6 +14 13 0 0 +0 0 0 +0 0 n/a 0 0 +0 0 0 +0 0 0 +15 15 April 2008 22

BrAv /10 6 Theory Experiment default B K φ 10.1 +0.5 B 0 K 0 φ 9.3 0.5 +0.5+11.4 B K ω 2.4 0.7 +0.8+2.9 B 0 K 0 ω 2.0 0.1 +0.1+3.1 B K 0 ρ 5.9 0.3 +0.3+6.9 B K ρ 0 4.5 1.3 +1.5+3.0 B 0 K ρ + 5.5 +1.7 B 0 K 0 ρ 0 2.4 0.1 +0.2+3.5 B s φφ 21.8 +1.1 1.1 +30.4 ˆα c 4 from data 0.5 7.1 +12.2 10.4 0.5 +0.5+5.2 6.5 9.6 0.5 +0.5+4.7 1.3 2.3 0.7 +0.8+1.4 1.4 1.9 0.1 +0.1+1.5 3.7 5.8 0.3 +0.3+3.1 3.9 9.7 ± 1.5 3.6 9.50 ± 0.90 0.7 < 3.4 0.7 < 4.2 1.9 9.2 ± 1.5 1.0 < 6.1 1.4 4.5 1.3 +1.5+1.8 1.5 2.9 +5.7 5.4 1.5 +1.7 1.5 +2.6 n/a 2.0 2.3 0.1 +0.2+1.1 0.8 5.6 ± 1.6 17.0 19.5 1.0 +1.0 8.0 +13.1 14.0 7.0 +8.0 April 2008 23

f L / percent Theory Experiment default B K φ 45 +0 0 +58 B 0 K 0 φ 44 +0 0 +59 B K ω 53 +8 B 0 K 0 ω 40 +4 3 +77 B K 0 ρ 56 +0 0 +48 B K ρ 0 84 +2 3 +16 B 0 K ρ + 61 +5 7 +38 B 0 K 0 ρ 0 22 +3 3 +53 ˆα c 4 from data 36 44 +0 0 +23 36 43 +0 0 +23 11 +57 39 56 +8 11 +22 43 43 +4 3 +38 30 57 +0 0 +21 23 50.0 ± 7.0 23 49.0 ± 4.0 19 n/a 32 n/a 18 48.0 ± 8.0 25 85 +2 3 +9 11 96 +6 16 28 62 +5 6 +17 15 n/a 14 22 +3 3 +21 13 57 ± 12 April 2008 24

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 /10 6 4.5 5.4 < 6.1 2.4 1.4 5.6 ± 1.6 f L / % 84 70 96 +6 16 22 37 57 ± 12 A CP / % 16 14 20 +32 29 15 24 9 ± 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 2008 25

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 2008 26

THANKS! April 2008 27