Irreducible theory errors in α and γ extraction Jure Zupan Carnegie Mellon University Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 1
Motivation Assume infinite statistics, what is the ultimate error on γ and α ? Will discuss only the (theoretically) most precise methods Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 2
Outline γ from B → DK α from B → ππ, ρρ, ρπ conclusions Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 3
B ± → DK ± Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 4
0 � D K A A B D graphically... � f B i ( Æ � � ) B A r e A B B D 0 � D K Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 5 1
Different methods methods can be grouped by the choice of final state f CP- eigenstate (e.g. K S π 0 ) Gronau, London, Wyler (1991) flavor state (e.g. K + π − ) Atwood, Dunietz, Soni (1997) singly Cabibbo suppressed (e.g. K ∗ + K − ) Grossman, Ligeti, Soffer (2002) many-body final state (e.g. K S π + π − ) Giri, Grossman, Soffer, JZ (2003) other extensions: many body B final states (e.g. B + → DK + π 0 ) Aleksan, Petersen, Soffer (2002) use D 0 ∗ in addition to D 0 use self tagging D 0 ∗∗ Sinha (2004) neutral B decays (time dependent and time-integrated) many refs. Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 6
Theory errors CP conserving D − ¯ D mixing does not change the methods CP violation in D sector the only uncertainty (!) in SM λ 6 ∼ 10 − 4 suppressed only relevant if beyond SM CP viol. in D is it present? compare (time integrated) D 0 and ¯ D 0 decays to f , ¯ f Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 7
Including CP viol. in D enters in two ways direct CP viol. A ( D 0 → f ) � = A ( ¯ D 0 → ¯ f ) through D − ¯ D mixing, q/p � = 1 can it be included in the analysis? first focus on 2-body final states with f � = ¯ f most general parametrization of direct CP viol. A ( D 0 → f ) = A f + B f , D 0 → ¯ A ( ¯ f ) = A f − B f A ( D 0 → ¯ D 0 → f ) = A ¯ A ( ¯ f ) = A ¯ f + B ¯ f , f − B ¯ f Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 8
Enough info? k different channels ( f, ¯ f ) 8 k observables: Γ( D 0 → f ) , Γ( D 0 → ¯ D 0 → f ) , Γ( ¯ D 0 → ¯ f ) , Γ( ¯ f ) Γ( B ± → f D K ± ) , Γ( B ± → ¯ f D K ± ) 7 k + 6 unknowns: 7 k channel specific: 4 magnitudes and 3 relative phases for each channel A f , A ¯ f , B f , B ¯ f 6 common real parameters: � � ∗ q p − p γ, A B , r B , δ B , ( x + iy ) q k ≥ 6 general analysis possible Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 9
Multibody decay B ± → ( K S π + π − ) D K ± 2 k bins in the Dalitz plot placed π + ↔ π − symmetrically 2 � GeV 2 � m K S �Π � 3 too many unknowns 2.5 � 2 i A f A ∗ T i , T ¯ i , f = c i + s i ¯ 1.5 � � � � i A f B ∗ f B ∗ i A f B ∗ f B ∗ 1 f , f , f , i A ¯ i A ¯ ¯ ¯ ¯ ¯ f 0.5 2 3 m K S �Π � � GeV 0.5 1 1.5 2 2.5 8 k observ. ⇔ 12 k + 6 unknowns model independent method possible if B f = 0 , even for q/p � = 1 B f can be fit to BW forms Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 10
Theory errors in α extraction B ( t ) → ππ B ( t ) → ρρ B ( t ) → ρπ Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 11
Isospin breaking the most useful methods for α extraction use isospin relations isospin breaking the limiting factor for precision measurements typical effect of isospin breaking ∼ ( m u − m d ) / Λ QCD ∼ α 0 ∼ 1% Questions: Are the isospin breaking effects that we can calculate of this order? Does any of the methods fare better? Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 12
Manifestations of isospin breaking sources of isospin breaking d and u charges different m u � = m d Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 13
Manifestations of isospin breaking sources of isospin breaking d and u charges different m u � = m d extends the basis of operators to EWP Q 7 ,..., 10 mass eigenstates do not coincide with isospin eigenstates: π − η − η ′ and ρ − ω mixing reduced matrix elements between states in the same isospin multiplet may differ e.g. � π + π − | Q 1 | B 0 � � = 1 2 � π + π 3 | Q 1 | B 0 � √ may induce ∆ I = 5 / 2 operators not present in H W Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 13
Electroweak penguins neglecting Q 7 , 8 Neubert, Rosner; Gronau, Pirjol, Yan; Buras, Fleischer (1999) V ∗ eff , EWP = − 3 C 9 + C 10 tb V td H ∆ I =3 / 2 H ∆ I =3 / 2 eff , c − c V ∗ 2 C 1 + C 2 ub V ud δα = (1 . 5 ± 0 . 3 ± 0 . 3) ◦ ⇒ conservatively ∼ 2( | c 7 | + | c 8 | ) / ( | c 9 | ) < 0 . 2 the same shift in ππ , ρρ and ρπ systems Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 14
π 0 − η − η ′ mixing π 0 w.f. has η, η ′ admixtures | π 0 � = | π 3 � + ǫ | η � + ǫ ′ | η ′ � where ǫ = 0 . 017 ± 0 . 003 , ǫ ′ = 0 . 004 ± 0 . 001 Kroll (2004) GL triangle relations in B → ππ no longer hold √ √ A + − + 2 A 00 − 2 A +0 � = 0 √ √ ¯ 2 ¯ 2 ¯ A + − + A 00 − A +0 � = 0 previous analysis Gardner (1999) estimated using generalized factorization obtained ∆ α ∼ 0 . 1 ◦ − 5 ◦ (including EWP) SU(3) decomposition for A 0 η ( ′ ) , A + η ( ′ ) M. Gronau, J.Z. (2005) + exp. information | ∆ α π − η − η ′ | < 1 . 4 ◦ Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 15
Snyder-Quinn Snyder, Quinn (1993), Lipkin et al. (1991), Gronau (1991) B 0 → 3 π ) ) model the Dalitz plot (similarly for A ( ¯ A + � �� � A ( B 0 → π + π − π 0 ) = A ( B 0 → ρ + π − ) D ρρ ( s + ) cos θ + + + A ( B 0 → ρ − π + ) D ρρ ( s − ) cos θ − + A ( B 0 → ρ 0 π 0 ) D ρρ ( s 0 ) cos θ 0 � �� � � �� � A − A 0 A i ) → e iβ A i ( e − iβ ¯ rotate A i ( ¯ A i ) tree and penguin defined according to CKM A ± , 0 = e − iα T ± , 0 + P ± , 0 , A ± , 0 = e + iα T ± , 0 + P ± , 0 ¯ an isospin relation only between penguins P 0 + 1 2 ( P + + P − ) = 0 (EWP and isospin breaking neglected) Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 16
Effect of isospin breaking isospin breaking affects only the relation between penguins! largest shift δα = (1 . 5 ± 0 . 3 ± 0 . 3) ◦ due to EWP because they are related to tree P − + P + + 2 P 0 = P EW other isospin breaking effects are P/T ∼ 0 . 2 suppressed using similar approach of SU(3) relations as in ππ to estimate shift due to π 0 − η − η ′ mixing | ∆ α π − η − η ′ | ≤ 0 . 1 ◦ Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 17
Conclusions the methods based on B ± → f D K ± for measuring γ contain no theory error, even CP violation in D sector can be accomodated isospin breaking effect on α extraction from B → ρπ is P/T ∼ 0 . 2 suppressed compared to B → ππ, ρρ Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 18
Backup slides Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 19
Effect of CP conserving D − ¯ D mixing in case of no D − ¯ D mixing � � | A f | 2 = f | 2 = dτ | A even dτ | A even ( t ) | , | A ¯ ( t ) | ¯ f f � A f A ∗ dτA even ( t ) A even ( t ) ∗ f = ¯ ¯ f f CP even D − ¯ D mixing the same with replacement A f = A f − 1 � q p + p � A f → ˜ 4( y + ix ) A ¯ f q f − 1 � q p + p � f → ˜ A ¯ A ¯ f = A ¯ 4( y + ix ) A f q Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 20
Using data for π − η − η ′ mixing M. Gronau, J.Z. (2005) use SU(3) decomposition for A 0 η ( ′ ) , A + η ( ′ ) + neglect annihilation-like contributions ⇒ A 33 = 1 ⇒ A +3 = 1 SU (2) SU (2) √ √ A + − = t + p ⇐ = = 2( c − p ) ⇐ = = 2( t + c ) ⇑ SU (3) ⇓ A 3 η = 1 A 3 η ′ = 1 √ √ 6(2 p + s ) 3( p + 2 s ) A + η = 1 A + η ′ = 1 √ √ 3( t + c + 2 p + s ) 6( t + c + 2 p + 4 s ) Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 21
Using data II triangle relation is modified only slightly √ √ A + − + 2 A 00 − 2 A +0 (1 − e 0 ) = 0 � � 3 ǫ ′ = 0 . 016 ± 0 . 003 2 1 where e 0 = 3 ǫ + A +0 is a sum of pure ∆ I = 3 / 2 amplitude A +3 with weak phase γ and isospin-breaking terms √ √ 2 ǫ ′ A 0 η ′ . A +0 = A +3 (1 + e 0 ) + 2 ǫA 0 η + while e iγ A +3 = e − iγ ¯ A +3 no longer e iγ A +0 = e − iγ ¯ A +0 also | A +0 | � = | ¯ A +0 | ⇐ exp. check Irreducible theory errors in α and γ ... J. Zupan Hawaii, 04/20/05 – p. 22
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