Rescattering and Chiral Dynamics in B Decay Susan Gardner SLAC - PowerPoint PPT Presentation

Rescattering and Chiral Dynamics in B → ρπ Decay Susan Gardner SLAC & University of Kentucky ✘ Why B → ρπ : Isospin Analysis for α ☞ Isospin violation and ways to mimic it. Why σ ? ✘ Evaluating B → π + π − π 0 Decay ☞ Scalar and vector form factors consistent with chiral symmetry ✘ Mitigating unwanted contributions in the ρπ phase space [in collaboration with Ulf-G. Meißner (Univ. Graz) and Jusak Tandean (Univ. of Kentucky)] S.G. and U. Meißner, hep-ph/0112281, PRD, in press; J. Tandean and S.G., hep-ph/0204147. S . Gardner , B → ρπ, May , 2002, 1

Testing the Standard Model 2 sin γ sin 2 β sin γ 1 ∆ m s & ∆ m d K + →π + νν sin 2 β CK M ∆ m d f i t t e r 0.8 1 ∆ m d 0.6 |ε , / ε K | , K 0 →π 0 νν η |ε K | 0.4 0 η K 0 →π 0 νν | V ub /V cb | |ε K | 0.2 | V ub /V cb | sin 2 α sin 2 β WA sin 2 α 0 -1 -1 -0.5 0 0.5 1 K + →π + νν ρ |ε K | -2 -2 -1 0 1 2 ρ [ CKMfitter: H¨ ocker, Lacker, Laplace, Diberder, hep-ph/0104062; March, 2002 update ] S . Gardner , B → ρπ, May , 2002, 2

B → ρπ : Isospin Analysis for α [ Lipkin, Nir, Quinn, Snyder, 1991; Snyder and Quinn, 1993; Quinn and Silva, 2000. ] Isospin: ρπ state = ⇒ I f = 0 , 1 , 2 ; B = ⇒ I i = 1 / 2 ; | ∆ I | = 1 / 2 , 3 / 2 , 5 / 2 . Parametrize the amplitudes by A | ∆ I | ,I f : 1 3[ A 3 / 2 , 2 + A 5 / 2 , 2 ] + 1 2[ A 3 / 2 , 1 + A 1 / 2 , 1 ] + 1 a + − ≡ A ( B 0 → ρ + π − ) = √ √ 6 A 1 / 2 , 0 2 1 3[ A 3 / 2 , 2 + A 5 / 2 , 2 ] − 1 2[ A 3 / 2 , 1 + A 1 / 2 , 1 ] + 1 a − + ≡ A ( B 0 → ρ − π + ) = √ √ 6 A 1 / 2 , 0 2 a 00 ≡ A ( B 0 → ρ 0 π 0 ) = − 1 3[ A 3 / 2 , 2 + A 5 / 2 , 2 ] + 1 √ √ 6 A 1 / 2 , 0 , Two key assumptions: ☞ A ( B 0 → π + π − π 0 ) = f + a + − + f − a − + + f 0 a 00 , f i describes ρ i → ππ . ⇒ P 00 = 1 2 ( P + − + P − + ) . ☞ The penguin is | ∆ I | = 1 / 2 = S . Gardner , B → ρπ, May , 2002, 3

B → ρπ : Isospin Analysis for α CKM unitarity: = ⇒ 2 weak phases: V ∗ V ∗ ub V ud tb V td ub V ud | = e iγ tb V td | = e − iβ ; | V ∗ | V ∗ and α = π − β − γ , so that e iβ a + − = T + − e − iα + P + − e iβ a − + = T − + e − iα + P − + e iβ a 00 = T 00 e − iα + P 00 With | ∆Γ | ≪ Γ , we have q/p = exp( − 2 iβ ) and q ¯ a ij /p ∝ exp( − iβ ) , so that the isospin analysis determines sin(2 α ) . With the penguin assumption, there are 10 parameters in all, to be determined in an analysis of the Dalitz plot. S . Gardner , B → ρπ, May , 2002, 4

B → ρπ : Isospin Violation? Isospin is merely an approximate symmetry. In the SM, ✘ Isospin violation can generate a | ∆ I | = 5 / 2 amplitude. √ ☞ via O ( α ) × A 1 / 2 , 0 or O ( ǫ (2) ) × A 3 / 2 , 2 ; ǫ (2) = m )) . 3( m d − m u ) / (4( m s − ˆ ✘ Isospin violation can distinguish the form factors f i . ✘ Penguin contributions of | ∆ I | = 3 / 2 character can occur. These uncertainties can be mitigated in an empirically driven way. S . Gardner , B → ρπ, May , 2002, 5

B → ρπ : Isospin Violation? Isospin is merely an approximate symmetry. In the SM, ✘ Isospin violation can generate a | ∆ I | = 5 / 2 amplitude. √ ☞ via O ( α ) × A 1 / 2 , 0 or O ( ǫ (2) ) × A 3 / 2 , 2 ; ǫ (2) = m )) . 3( m d − m u ) / (4( m s − ˆ ✘ Isospin violation can distinguish the form factors f i . ✘ Penguin contributions of | ∆ I | = 3 / 2 character can occur. These uncertainties can be mitigated in an empirically driven way. ✘ The | ∆ I | = 5 / 2 and | ∆ I | = 3 / 2 amplitudes have the same weak phase? ☞ True to the extent that A 3 / 2 , 2 > A 1 / 2 , 0 . No “ | ∆ I | = 1 / 2 rule”? ✘ f ± vs. f 0 ? Compare hadronic τ and e + e − data. ☞ Note ρ 0 − ω mixing is reflective of B → ωπ 0 → π + π − π 0 . ✘ ? Not “enhanced” in any way. electroweak penguins or O ( α, ǫ (2) ) corrections to the ☞ Sources: hadronic m.e.’s [ Deshpande & He, 1995; S.G., 1999; S.G. & Valencia, 1999 ] S . Gardner , B → ρπ, May , 2002, 5

B → ρπ cf. B → ππ In B → ρπ decay the combination A 3 / 2 , 2 + A 5 / 2 , 2 appears throughout. However, in B → ππ decay b + − ≡ A ( B 0 → π + π − ) = − 1 3 A 1 / 2 , 0 + 1 √ √ 6[ A 3 / 2 , 2 − A 5 / 2 , 2 ] � b 00 ≡ A ( B 0 → π 0 π 0 ) = − 1 2 √ 3 A 1 / 2 , 0 − 3[ A 3 / 2 , 2 − A 5 / 2 , 2 ] √ 2 A 3 / 2 , 2 + 1 3 b +0 ≡ A ( B + → π + π 0 ) = √ 2 A 5 / 2 , 2 √ Isospin analysis in B → ππ relies on ( b + − − b 00 ) / 2 − b +0 = 0 [Gronau & London, 1990] If A 1 / 2 , 0 /A 3 / 2 , 2 is small, neglecting the A 5 / 2 , 2 amplitude can incur significant errors in sin(2 α ) and can break bounds on the hadronic uncertainty. [S.G., 1999] If A 5 / 2 , 2 and A 3 / 2 , 2 share the same weak phase, the B → ρπ analysis is insensitive to this effect. S . Gardner , B → ρπ, May , 2002, 6

B → ρπ : Mimicking Isospin Violation What other contributions enter the ρπ phase space? Note recent studies of D − → π − π + π − decay: D − → π − σ (500) → π − π + π − accounts for ≈ half of the total decay rate. [E791 Collaboration, Phys. Rev. Lett. 86, 765 (2001).] The B → σπ channel can contribute to the ρπ phase space in B → ππ + π − and modifies Br( ¯ B 0 → ρ ∓ π ± ) / Br( B − → ρ 0 π − ) to yield better agreement with experiment. [Deandrea & Polosa, Phys. Rev. Lett. 86, 216 (2001).] Using CLEO [PRL 85, 2881 (2000)] and BaBar data [hep-ex/0107058] yields B 0 → ρ ∓ π ± ) R = Br( ¯ Br( B − → ρ 0 π − ) = 2 . 7 ± 1 . 2 . This ratio of ratios is ≈ 6 if one works at tree level in factorization. [Bauer, Stech, and Wirbel, 1987.] How does this contribution impact the B → ρπ → π + π − π 0 analysis? The σπ contributes preferentially to the ρ 0 π 0 final state and can break the assumed relation between the penguin contributions. S . Gardner , B → ρπ, May , 2002, 7

B → ρπ : Extension to include σπ The B → σπ channel has definite properties under CP. The B → ρπ analysis can be extended to include this channel, if need be. 00 = A ( B 0 → σπ 0 ) , we have Defining a σ σ e − iα + P 00 e iβ a σ 00 = T 00 σ . T 00 σ and P 00 ⇒ four additional hadronic parameters. More observables = σ are present as well. Including the scalar channel, A 3 π ≡ A ( B 0 → π + π − π 0 ) = f + a + − + f − a − + + f 0 a 00 + f σ a σ 00 , where f σ describes σ → π + π − . j contained in | A 3 π | 2 and | ¯ A 3 π | 2 are distinguishable The products f i f ∗ through the Dalitz plot of this decay — the coefficients of these functions are distinct observables. Sufficient observables exist to determine all the parameters. S . Gardner , B → ρπ, May , 2002, 8

Evaluating B → π + π − π 0 Decay The effective, | ∆ B | = 1 Hamiltonian for b → dq ¯ q decay is given by � 10 � H eff = G F � λ u ( C 1 O u 1 + C 2 O u 2 ) + λ c ( C 1 O c 1 + C 2 O c √ 2 ) − λ t C i O i , 2 i =3 where λ q ≡ V qb V ∗ qd . C 1 ∼ O (1) and C 1 > C 2 ≫ C 3 ... 10 . For a light, narrow resonance, the product Ansatz A σ ( B → π + π − π ) ≡ � ( σ → π + π − ) π |H eff | B � = � σπ |H eff | B � Γ σππ , should work well. Γ σππ describes σ → ππ . Use factorization to compute � σπ |H eff | B � , � ρπ |H eff | B � . Focus on Γ σππ , Γ ρππ . Compute the effective B → ρπ branching ratio via M = A σ ( B → π + π − π ) + A ρ ( B → π + π − π ) + .... S . Gardner , B → ρπ, May , 2002, 9

Coupled-channel pion and kaon scalar form factors Earlier analyses (E791) use the σ → π + π − vertex function � x/ 4 − M 2 � 1 � M σ π √ x Γ σππ ( x ) = g σπ + π − ; Γ σ ( x ) = Γ σ . x − M 2 σ + i Γ σ ( x ) M σ � M 2 σ / 4 − M 2 π [Blatt & Weisskopf, 1952] FSI in the J = 0 I = 0 channel are very strong. S � π ≃ 0 . 6 fm 2 and � r 2 ρ ≃ 0 . 4 fm 2 ; fixed-order ChPT ☞ cf. � r 2 V � π ≃ 6 /M 2 fails more rapidly in scalar-isoscalar channel ☞ The σ need not be a “pre-existing” resonance; can be generated thr. FSI. [Oller & Oset, 1999] Need to invoke a resummation approach, consistent with unitarity (including both ππ and K ¯ K channels) and chiral symmetry.... [Meißner & Oller, 2001] S . Gardner , B → ρπ, May , 2002, 10

Coupled-channel pion and kaon scalar form factors We have Γ σππ ( x ) − → χ Γ n ∗ 1 ( x ) where √ √ 2 B 0 Γ n 2 B 0 Γ n � 0 | ¯ nn | ππ � = 1 ( s ) , � 0 | ¯ nn | KK � = 2 ( s ) , √ √ 2 B 0 Γ s 2 B 0 Γ s � 0 | ¯ ss | ππ � = 1 ( s ) , � 0 | ¯ ss | KK � = 2 ( s ) , B 0 is the vacuum quark condensate, B 0 = −� 0 | ¯ qq | 0 � /F 2 π . Start with the scattering T -matrix [Babelon, Basdevant, Caillerie, Mennessier, 1976; Oller & Oset, 1997] T ( s ) = [ I + K ( s ) · g ( s )] − 1 · R ( s ) K ( s ) from O ( p 2 ) CHPT Lagrangian, e.g., K ( s ) 11 = ( s − M 2 π / 2) /F 2 π . g ( s ) is the scalar loop integral. Fix R ( s ) by matching Γ( s ) to the next-to-leading order (one loop) CHPT ππ and K ¯ K scalar form factors. [Gasser & Leutwyler, 1985; Meißner & Oller, 2001] Γ( s ) valid from threshold up to energies of about 1.2 GeV. Resultant form factor consistent with dispersion relation approach. [Donoghue, Gasser, Leutwyler, 1990] S . Gardner , B → ρπ, May , 2002, 11