Determination of B 0 s and B 0 d mixing parameters using lattice QCD Elvira G´ amiz In collaboration with: Christine T.H. Davies, G. Peter Lepage, Junko Shigemitsu, Howard Trottier and Matthew Wingate HPQCD Collaboration Madison, 11 May 2009
1. New Physics effects on B 0 − ¯ B 0 mixing W q u, c, t q b b B 0 B 0 ¯ B 0 B 0 ¯ u, c, t u, c, t W W q q q q u, c, t q q b b W • B 0 mixing parameters determined by the off diagonal elements of the mixing matrix � � | B s/d ( t ) � | B s/d ( t ) � i d M s/d − i 2 Γ s/d = | ¯ | ¯ dt B s/d ( t ) � B s/d ( t ) � ∆ M s/d ∝ | M s/d ∆Γ s/d ∝ | Γ s/d 12 | 12 |
1. New Physics effects on B 0 − ¯ B 0 mixing W q u, c, t q b b B 0 B 0 ¯ B 0 B 0 ¯ u, c, t u, c, t W W q q q q u, c, t q q b b W • B 0 mixing parameters determined by the off diagonal elements of the mixing matrix � � | B s/d ( t ) � | B s/d ( t ) � i d M s/d − i 2 Γ s/d = | ¯ | ¯ dt B s/d ( t ) � B s/d ( t ) � ∆ M s/d ∝ | M s/d ∆Γ s/d ∝ | Γ s/d 12 | 12 | New physics can significantly affect M s/d ∝ ∆ M s/d 12 * Γ 12 dominated by CKM-favoured b → c ¯ cs tree-level decays.
# Hints of discrepancies between SM expectations and some flavour observables ( see, for example, E. Lunghi , talk at BEACH08 ) * sin (2 β ) E. Lunghi and A. Soni , PLB 666 (2008) 162 * B 0 s mixing phase Nierste and Lenz , JHEP 0706 , UTfit coll. , arXiv:0803.0659 d − ¯ * CP violating effects B 0 B 0 d , Buras and Guadagnoli , PRD78(2008)033005
# Hints of discrepancies between SM expectations and some flavour observables ( see, for example, E. Lunghi , talk at BEACH08 ) * sin (2 β ) E. Lunghi and A. Soni , PLB 666 (2008) 162 * B 0 s mixing phase Nierste and Lenz , JHEP 0706 , UTfit coll. , arXiv:0803.0659 d − ¯ * CP violating effects B 0 B 0 d , Buras and Guadagnoli , PRD78(2008)033005 # These analyses depend on several theoretical inputs: V cb , V ub , ˆ B K and the SU(3) breaking mixing parameter ξ : � � � � � V td ∆ M d M B s � � � = ξ � V ts ∆ M s M B d * Comparison of ∆ M and ∆Γ with experiment also provides bounds for NP effects
# Hints of discrepancies between SM expectations and some flavour observables ( see, for example, E. Lunghi , talk at BEACH08 ) * sin (2 β ) E. Lunghi and A. Soni , PLB 666 (2008) 162 * B 0 s mixing phase Nierste and Lenz , JHEP 0706 , UTfit coll. , arXiv:0803.0659 d − ¯ * CP violating effects B 0 B 0 d , Buras and Guadagnoli , PRD78(2008)033005 # These analyses depend on several theoretical inputs: V cb , V ub , ˆ B K and the SU(3) breaking mixing parameter ξ : � � � � � V td ∆ M d M B s � � � = ξ � V ts ∆ M s M B d * Comparison of ∆ M and ∆Γ with experiment also provides bounds for NP effects Improvement in B 0 − ¯ B 0 mixing parameters which enter on those analyses is crucial.
2. Mixing parameters in the Standard Model W # In the Standard B 0 B 0 ¯ Model W H ∆ B =2 eff ∆ M q | theor. = G 2 F M 2 B q ˆ | V ∗ tq V tb | 2 η B 2 S 0 ( x t ) M B s f 2 W B B q 6 π 2
2. Mixing parameters in the Standard Model W # In the Standard B 0 B 0 ¯ Model W H ∆ B =2 eff ∆ M q | theor. = G 2 F M 2 B q ˆ | V ∗ tq V tb | 2 η B 2 S 0 ( x t ) M B s f 2 W B B q 6 π 2 * Non-perturbative input B q = � ¯ O L ≡ [ b i q i ] V − A [ b j q j ] V − A 8 3 f 2 B q B B q ( µ ) M 2 q | O L | B 0 B 0 q � ( µ ) with In terms of decay constants and bag parameters � ξ = f B s B B s � B B d f B d * Many uncertainties in the theoretical (lattice) determination cancel totally or partially in the ratio = ⇒ very accurate calculation
3. Some details of the lattice formulations and simulations Unquenched : Fully incorporate vacuum polarization effects MILC N sea = 2 + 1 f ⇒ errors O ( a 2 α s ) , u,d,s Asqtad action: improved staggered quarks = O ( a 4 ) * good chiral properties * accessible dynamical simulations NRQCD : Non-relativistic QCD improved through O (1 /M 2 ) , O ( a 2 ) b and leading relativistic O (1 /M 3 ) * Simpler and faster algorithms to calculate b propagator Improved gluon action * For further reduction of discretization errors
Parameters of the simulation # Lattice spacing: Two different values a ≃ 0 . 12 fm, 0 . 09 fm . Extracted from Υ 2S-1S splitting. # Bottom mass: Fixed to its physical value from Υ mass.
Parameters of the simulation # Lattice spacing: Two different values a ≃ 0 . 12 fm, 0 . 09 fm . Extracted from Υ 2S-1S splitting. # Bottom mass: Fixed to its physical value from Υ mass. # Light masses: We work with full QCD points ( m valence = m sea ). * Strange mass: Very close to its physical value (from Kaon masses). * up, down masses: six different values ( m min. ≃ 230MeV ) π → chiral regime
Parameters of the simulation # Lattice spacing: Two different values a ≃ 0 . 12 fm, 0 . 09 fm . Extracted from Υ 2S-1S splitting. # Bottom mass: Fixed to its physical value from Υ mass. # Light masses: We work with full QCD points ( m valence = m sea ). * Strange mass: Very close to its physical value (from Kaon masses). * up, down masses: six different values ( m min. ≃ 230MeV ) π → chiral regime # Renormalization and matching to the continuum: One-loop. < O L > MS ∝ (1 + ρ LL α s ) < O L > latt. + ρ LL α s < O S > latt � ¯ � � ¯ � with O S = b (1 − γ 5 ) q b (1 − γ 5 ) q .
Q = Q X , Q 1 j # Need 3-point (for any ˆ X ) and 2-point correlators ˆ Q ¯ B 0 B 0 � x 2 , t 2 � x 1 , t 1 t = 0 � � � (0)Φ † C (4 f ) ( t 1 , t 2 ) = ˆ � 0 | Φ ¯ B q ( � x 1 , t 1 ) Q B q ( � x 2 , − t 2 ) | 0 � ¯ x 1 ,� � x 2 � x, t )Φ † C ( B ) ( t ) = B q ( � � 0 | Φ ¯ B q ( � 0 , 0) | 0 � ¯ � x x, t ) = ¯ x, t ) is an interpolating operator for the B 0 • Φ ¯ B q ( � b ( � x, t ) γ 5 q ( � q meson. We carried out simultaneous fits of the 3-point and 2-point correlators using bayesian statistics to the forms → extract < O X > and f B s ( d ) .
4. Results � M B q B B q Results for f B q 1.5 Coarse Lattice Coarse Lattice 1.4 Fine Lattice 1.6 Fine Lattice Physical point Physical point (1/2) (1/2) (3/2) f Bd (M Bd B Bd ) 1.3 (3/2) f Bs (M Bs B Bs ) 1.5 1.2 1.4 1.1 1.3 r 1 r 1 1 1.2 0.9 1.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 r 1 m q r 1 m f � � ˆ ˆ f B s B B s = 266(6)(17)MeV f B d B B d = 216(9)(12)MeV Chiral+continuum extrapolations : NLO Staggered CHPT. * accounts for NLO quark mass dependence. � � α 2 s a 2 Λ 2 * accounts for light quark discretization effects through O QCD → remove the dominant light discretization errors
� M Bs Results for ξ M Bd 1.5 Coarse Lattice 1.4 Fine Lattice Continuum Physical point 1/2 ξ 1.3 (M Bs / M Bd ) 1.2 1.1 1 0.9 0 0.02 0.03 0.04 0.07 0.08 0.01 0.05 0.06 r 1 m q √ � � � � f Bs B Bs � V td √ B Bd ξ = = 1 . 258(25)(21) = ⇒ � = 0 . 214(1)(5) V ts f Bd * Previous value used in UT fits and another analyses ( HPQCD/JLQCD ): ξ = 1 . 20 ± 0 . 06
� M B q Results for f B q 1.5 Coarse Lattice Coarse Lattice 1.4 Fine Lattice 1.6 Fine Lattice Physical point (1/2) Physical point (1/2) (3/2) f Bd (M Bd B Bd ) 1.3 (3/2) f Bs (M Bs B Bs ) 1.5 1.2 1.4 1.1 1.3 r 1 r 1 1 1.2 0.9 1.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 r 1 m q r 1 m f f B s = 231(15) MeV f B d = 190(13) MeV f Bs f Bd = 1 . 226(26) * HPQCD previous numbers are f B s = 260(29) MeV , f B d = 216(22) MeV and f B s /f B d = 1 . 20(3)(1) . ** chiral extrapolation based only on coarse lattice, no continuum extrapolation.
Bag parameters: Calculation of B r ( B → µ + µ − ) # Very interesting place to look for the effect of new operators in the effective Hamiltonian. Hurth et al, NPB 808 (2009); Buras, arXiv:0904.4917 . * Scalar operators can enhance branching ratios to current experimental bounds.
Bag parameters: Calculation of B r ( B → µ + µ − ) # Very interesting place to look for the effect of new operators in the effective Hamiltonian. Hurth et al, NPB 808 (2009); Buras, arXiv:0904.4917 . * Scalar operators can enhance branching ratios to current experimental bounds. # The most precise way of extracting this branching ratio is from � � 2 B r ( B q → µ + µ − ) Y 2 ( x t ) = τ ( B q ) 6 π η Y α 1 m 2 µ ˆ 4 πM W sin 2 θ W ∆ M q η B S ( x t ) B q * Using our value of ˆ B r ( B s → µ + µ − ) = (3 . 2 ± 0 . 3) × 10 − 9 B s → CDF bound B r ( B s → µ + µ − ) ≤ 6 × 10 − 8 to be compared with
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