QCD factorisation and flavour symmetries illustrated in B d , s → KK decays S´ ebastien Descotes-Genon Laboratoire de Physique Th´ eorique CNRS & Universit´ e Paris-Sud 11, 91405 Orsay, France 9 October 2006 S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 1 / 21
Two-body nonleptonic B decays B d very much SM-like (Babar, Belle) B d still room for New Physics ? (CDF, D0, LHCb) Predict SM correlations between B d and B s decays and see whether these correlations are upset by New Physics S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 2 / 21
Two-body nonleptonic B decays B d very much SM-like (Babar, Belle) B d still room for New Physics ? (CDF, D0, LHCb) Predict SM correlations between B d and B s decays and see whether these correlations are upset by New Physics A ( B → H ) = G F � √ i λ i C i ( µ ) � H |O i | B � ( µ ) 2 π To compute � H |O i | B � (Lattice) b c Light-cone sum rules B D QCD factorisation Flavour symmetries S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 2 / 21
QCD factorisation For some classes of decays, in the heavy-quark limit m b → ∞ , M M 2 2 T I T II B B F M M 1 1 Long distance Short distance Nonperturbative Perturbative � H |O i | B � = ⊗ Universal Process dependent form factors F hard-scattering distrib amplitude φ kernel T S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 3 / 21
QCD factorisation For some classes of decays, in the heavy-quark limit m b → ∞ , M M 2 2 T I T II B B F M M 1 1 Long distance Short distance Nonperturbative Perturbative � H |O i | B � = ⊗ Universal Process dependent form factors F hard-scattering distrib amplitude φ kernel T Good : 1 / m b , α s expansions with control of short-distance physics Bad : Some numerically significant long-distance 1 / m b corrections left out: weak annihilation, spectator-quark interaction S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 3 / 21
Flavour symmetries Isospin symmetry ( u ↔ d ) : B d → π + π − , B d → π 0 π 0 , B − → π 0 π − U -spin symmetry ( d ↔ s ) : B d → π + π − ↔ B s → K + K − Good : Global symmetries of QCD, including long- and short-distances Bad : Only approximate, with potentially large corrections, e.g. SU (3) symmetry O (30%) S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 4 / 21
Flavour symmetries Isospin symmetry ( u ↔ d ) : B d → π + π − , B d → π 0 π 0 , B − → π 0 π − U -spin symmetry ( d ↔ s ) : B d → π + π − ↔ B s → K + K − Good : Global symmetries of QCD, including long- and short-distances Bad : Only approximate, with potentially large corrections, e.g. SU (3) symmetry O (30%) Idea : Combine the two where they are accurate to extract stringent SM correlations between B d and B s decays Illustration : B d → K 0 ¯ K 0 and B s → KK S. Descotes-Genon, J. Matias and J. Virto Phys.Rev.Lett.97:061801,2006 S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 4 / 21
B q → K 0 ¯ K 0 : interesting penguin decays Conventional tree and penguin decomposition B q → K 0 ¯ uq T q 0 + V cb V ∗ A ≡ A (¯ ¯ K 0 ) V ub V ∗ cq P q 0 = A ≡ A ( B q → K 0 ¯ ub V uq T q 0 + V ∗ K 0 ) V ∗ cb V cq P q 0 = Only penguin diagrams no contribution from O 1 and O 2 Difference between tree and penguin from the u , c quark in loop S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 5 / 21
B q → K 0 ¯ K 0 : interesting penguin decays Conventional tree and penguin decomposition B q → K 0 ¯ uq T q 0 + V cb V ∗ A ≡ A (¯ ¯ K 0 ) V ub V ∗ cq P q 0 = A ≡ A ( B q → K 0 ¯ ub V uq T q 0 + V ∗ K 0 ) V ∗ cb V cq P q 0 = Only penguin diagrams no contribution from O 1 and O 2 Difference between tree and penguin from the u , c quark in loop ⇒ T q 0 − P q 0 dominated by short-distance physics = computed fairly accurately within QCDF T d 0 − P d 0 (1 . 09 ± 0 . 43) · 10 − 7 + i ( − 3 . 02 ± 0 . 97) · 10 − 7 GeV = T s 0 − P s 0 (1 . 03 ± 0 . 41) · 10 − 7 + i ( − 2 . 85 ± 0 . 93) · 10 − 7 GeV = S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 5 / 21
T − P : A sum rule for B d → K 0 ¯ K 0 observables A ( t ) = Γ( B d ( t ) → K ¯ K ) − Γ(¯ B d ( t ) → K ¯ A d 0 dir cos(∆ M · t )+ A d 0 mix sin(∆ M · t ) K ) K ) = Γ( B d ( t ) → K ¯ K )+Γ(¯ B d ( t ) → K ¯ cosh(∆Γ d t / 2) − A d 0 ∆ sinh(∆Γ d t / 2) dir = | A | 2 −| ¯ � A | 2 mix = − 2 e − i φ d A ∗ ¯ A d 0 A | 2 , A d 0 ∆ + iA d 0 A | A | 2 + | ¯ | A | 2 + | ¯ with CP asymmetries A | 2 ∆ | 2 + | A d 0 dir | 2 + | A d 0 mix | 2 = 1 | A d 0 S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 6 / 21
T − P : A sum rule for B d → K 0 ¯ K 0 observables A ( t ) = Γ( B d ( t ) → K ¯ K ) − Γ(¯ B d ( t ) → K ¯ A d 0 dir cos(∆ M · t )+ A d 0 mix sin(∆ M · t ) K ) K ) = Γ( B d ( t ) → K ¯ K )+Γ(¯ B d ( t ) → K ¯ cosh(∆Γ d t / 2) − A d 0 ∆ sinh(∆Γ d t / 2) dir = | A | 2 −| ¯ � A | 2 mix = − 2 e − i φ d A ∗ ¯ A d 0 A | 2 , A d 0 ∆ + iA d 0 A | A | 2 + | ¯ | A | 2 + | ¯ with CP asymmetries A | 2 ∆ | 2 + | A d 0 dir | 2 + | A d 0 mix | 2 = 1 | A d 0 T d 0 − P d 0 related to B d → K 0 ¯ K 0 observables (also true for B s ) | T d 0 − P d 0 | 2 = BR d 0 × 32 π M 2 √ Bd M 2 Bd − 4 M 2 τ d K ×{ x 1 + [ x 2 sin φ d − x 3 cos φ d ] A d 0 mix − [ x 2 cos φ d + x 3 sin φ d ] A d 0 ∆ } where x 1 , x 2 , x 3 depend on CKM factors only, φ d B -¯ B mixing angle S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 6 / 21
T − P : A sum rule for B d → K 0 ¯ K 0 observables A ( t ) = Γ( B d ( t ) → K ¯ K ) − Γ(¯ B d ( t ) → K ¯ A d 0 dir cos(∆ M · t )+ A d 0 mix sin(∆ M · t ) K ) K ) = Γ( B d ( t ) → K ¯ K )+Γ(¯ B d ( t ) → K ¯ cosh(∆Γ d t / 2) − A d 0 ∆ sinh(∆Γ d t / 2) dir = | A | 2 −| ¯ � A | 2 mix = − 2 e − i φ d A ∗ ¯ A d 0 A | 2 , A d 0 ∆ + iA d 0 A | A | 2 + | ¯ | A | 2 + | ¯ with CP asymmetries A | 2 ∆ | 2 + | A d 0 dir | 2 + | A d 0 mix | 2 = 1 | A d 0 T d 0 − P d 0 related to B d → K 0 ¯ K 0 observables (also true for B s ) | T d 0 − P d 0 | 2 = BR d 0 × 32 π M 2 √ Bd M 2 Bd − 4 M 2 τ d K ×{ x 1 + [ x 2 sin φ d − x 3 cos φ d ] A d 0 mix − [ x 2 cos φ d + x 3 sin φ d ] A d 0 ∆ } where x 1 , x 2 , x 3 depend on CKM factors only, φ d B -¯ B mixing angle SM consistency test between BR d 0 , | A d 0 dir | and A d 0 mix (id. for B s ) mix | ) from the two others ( BR d 0 and A d 0 SM value of one (say | A d 0 dir ) S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 6 / 21
T − P : Hadronic parameters for B d → K 0 ¯ K 0 To extract the hadronic parameters of this decay Unknowns | T | , | P / T | and arg ( P / T ) Observables Br = (0 . 96 ± 0 . 26) · 10 − 6 , A dir (broad range), A mix S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 7 / 21
T − P : Hadronic parameters for B d → K 0 ¯ K 0 To extract the hadronic parameters of this decay Unknowns | T | , | P / T | and arg ( P / T ) Observables Br = (0 . 96 ± 0 . 26) · 10 − 6 , A dir (broad range), T − P But A mix very hard to measure precisely, hence trade it for T − P S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 7 / 21
T − P : Hadronic parameters for B d → K 0 ¯ K 0 To extract the hadronic parameters of this decay Unknowns | T | , | P / T | and arg ( P / T ) Observables Br = (0 . 96 ± 0 . 26) · 10 − 6 , A dir (broad range), T − P But A mix very hard to measure precisely, hence trade it for T − P In the plane P = ( x P + iy P ) · 10 − 6 y_P 2 GeV 1.5 1 Br + ( T − P ) = ⇒ a circle 0.5 A dir + ( T − P ) = ⇒ a strip x_P -2 -1 1 2 -0.5 From left to right -1 A dir = − 0 . 17 , − 0 . 03 , 0 . 10 -1.5 -2 (QCDF : A dir ≃ 0 . 20) Intersection : hadronic parameters up to a two-fold ambiguity S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 7 / 21
B d → K 0 ¯ K 0 and B s → K 0 ¯ K 0 : U -spin Final state K 0 ¯ K 0 invariant = ⇒ Most long-distance effects (rescattering) identical S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 8 / 21
B d → K 0 ¯ K 0 and B s → K 0 ¯ K 0 : U -spin Final state K 0 ¯ K 0 invariant = ⇒ Most long-distance effects (rescattering) identical U -spin breaking only in a few places : ¯ ¯ B s → K B d → K Difference in form factors f = M 2 (0) / [ M 2 B s F B d F (0)] 0 0 S´ ebastien Descotes-Genon (LPT-Orsay) QCDF and flavour sym.: B d , s → KK 9/10/06 8 / 21
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