Isospin breaking effects in B ππ, ρρ, ρπ

Isospin breaking effects in B ππ, ρρ, ρπ Jure Zupan Carnegie Mellon University based on M. Gronau, J.Z., hep-ph/0502139 J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 1

Outline Effect of isospin breaking in α extraction from B(t) ππ B(t) ρρ B(t) ρπ Conclusions J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 2

Motivation 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? J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 3

Manifestations of isospin breaking sources of isospin breaking d and u charges different m u m d J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 4

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 J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 4

B(t) ππ J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 5

B ππ Gronau, London (1990) completely general isospin decomposition A + = π + π H B 0 = A 1/2 + 1 2 A 3/2 1 2 A 5/2 A 00 = π 0 π 0 H B 0 = 1 2 A 1/2 + A 3/2 A 5/2 A +0 = π + π 0 H B + = 3 2 A 3/2 + A 5/2 neglecting A 5/2 αa 1/2 ( i.e. 1% correction) A + + 2A 00 = 2A +0 Ā + + 2Ā00 = 2Ā+0 neglecting EWP A +0 only tree contribs. e iγ A +0 = e iγ Ā +0 A +0 = Ā+0 J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 6

Gronau-London triangle sin 2α from Γ(B 0 (t) π + π ) [1 + C ππ cos mt S ππ sin mt] sin(2α eff ) = S ππ / 1 C 2 ππ 2α = 2α eff 2θ 1 2 _ A + _ A 00 1 2 A + 2θ A 00 _ A = +0 A +0 J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 7

Electroweak penguins separate triangle relations still hold neglecting Q 7,8 H I=3/2 eff,ewp = 3 2 C 9 + C 10 Vtb V td C 1 + C 2 Vub V ud Neubert, Rosner; Gronau, Pirjol, Yan; Buras, Fleischer (1999) H I=3/2 eff,c c e iγ A +0 = e i(γ+2δ) Ā +0, but still A +0 = Ā+0 α = α eff θ δ with δ = (1.5 ± 0.3 ± 0.3) conservatively 2( c 7 + c 8 )/( c 9 )< 0.2 the same relation e iγ T = e i(γ+2δ) T holds for I = 3/2 (tree) amplitudes in ρρ and ρπ system J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 8

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 no longer hold A + + 2A 00 2A +0 0 Ā + + 2Ā00 2Ā+0 0 previous analysis Gardner (1999) estimated using generalized factorization obtained α 0.1 5 (including EWP) can we say more using wealth of data from Belle and BaBar? J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 9

Using data M. Gronau, J.Z. (2005) use SU(3) decomposition for A 0η ( ), A +η ( ) + neglect annihilation-like contributions A + = t+p SU(2) == A 33 = 1 2 (c p) SU(2) == A +3 = 1 2 (t+c) SU(3) A 3η = 1 6 (2p + s) A 3η = 1 3 (p + 2s) A +η = 1 3 (t + c + 2p + s) A +η = 1 6 (t + c + 2p + 4s) J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 10

Using data II triangle relation is modified only slightly where e 0 = A + + 2A 00 2A +0 (1 e 0 ) = 0 2 3 ǫ + 1 3 ǫ = 0.016 ± 0.003 A +0 is a sum of pure I = 3/2 amplitude A +3 with weak phase γ and isospin-breaking terms A +0 = A +3 (1 + e 0 ) + 2ǫA 0η + 2ǫ A 0η. while e iγ A +3 = e iγ Ā +3 no longer e iγ A +0 = e iγ Ā +0 also A +0 Ā+0 exp. check J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 11

Using data III varying the phases of A 0η ( ), Ā0η ( ) gives bound ) α π η η 2 τ + τ0 (ǫ B0η B +0 + ǫ B0η B +0 at 90% CL using WA values α π η η < 1.05ǫ + 1.28ǫ = 1.6 the bound can be improved using the SU(3) relations A +η ( ) = 2 3 A +0 + 2A 0η ( ) leading to α π η η < 1.4 J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 12

Intermediate summary isospin breaking due to π η ( ) mixing is of expected order bound α π η η < 1.4 does not encompass all isospin breaking still assumed SU(2) for A +,A +3,A 33 reduced matrix elements A 5/2 set to zero J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 13

B(t) ρρ J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 14

Generalities Isospin analysis in the same spirit as for B ππ 3 separate isospin relations (for each polarization) almost completely longitudinally polarized since Γ ρ 0 I = 1 contributions possible Falk, Ligeti, Nir, Quinn (2003) O(Γ 2 ρ/m 2 ρ) effect possible to constrain experimentally J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 15

Isospin breaking shift due to EWP exactly the same as in ππ ρ ω mixing couplings g I g(ρ I π + π ) and g c g(ρ + π + π 3 ) may not be equal g c = g(ρ + π + π 0 ) + O(ǫ 2 ) PDG: g c /g I 1 = (0.5 ± 1.0)% as in ππ not much can be said about breaking in the reduced matrix elements A 5/2 0 J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 16

Effect of ρ ω mixing only relevant for B + ρ + ρ 0 and B 0 ρ 0 ρ 0 the effect of ρ ω can be determined by fits to m π+ π distributions focusing on B + ρ + ρ 0 A +0 (s 12,s 34 ) = g c g I [A(B + ρ + ρ I )D ρρ (s 34 ) + A(B + ρ + ω I ) D ] ρω (s 34 ) the scalar part of ρ propagator near the pole D ρρ (s) = 1 s m 2 ρ + im ρ Γ ρ the mixed ρω contribution has a double pole D ρω (s) = Π ρω (s)d ρρ (s)d ωω (s) J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 17

large effect... Π ρω (s) from pion form factor Gardner, O Connell (1998) large effect expected because A(B + ρ + ω)/a(b + ρ + ρ 0 ) = 0.69 ± 0.14 an estimate: use SU(3) relation & P/T = 0.2 & same strong phase for P,T events in a. u. 1.75 1.5 1.25 0.75 0.5 0.25 B decays B + decays similar effect seen in D 0 K S π + π 1 650 700 750 800 850 900 s 1 2 MeV integrated effect of ω resonance is < 2% J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 18

B(t) ρπ J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 19

B π + π π 0 Dalitz plot model the Dalitz plot (similarly for A( B 0 3π)) A(B 0 π + π π 0 ) = A + {}}{ 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 other resonances need to be included in the fit ρ ω mixing treated in the same way as in ρρ possible to determine A +, Ā+, A, Ā, A 0, Ā0 up to overall phase 11 independent measurables J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 20

Snyder-Quinn Snyder, Quinn (1993), Lipkin et al. (1991), Gronau (1991) rescale A i (Āi) e iβ A i (e iβ Ā i ) tree and penguin defined according to CKM A ±,0 = e iα T ±,0 + P ±,0, Ā ±,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) 10 unknowns: e.g. α, t ±, t 0, arg t ±, p ±, arg p ± enough info to determine them J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 21

Effect of isospin breaking isospin breaking affects only the relation between penguins! largest contribution from EWP because they are related to tree P + P + + 2P 0 = P EW where P EW can be obtained from and A + + A + 2A 0 = Te iα + P EW J. Charles, PhD thesis P EW T = 3 2 ( c9 + c 10 c 1 + c 2 ) Vtb V td V ub V ud = +0.013sin(β + α) sin β J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 22

Other isospin breaking 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 = ǫp ρη + ǫ P ρη T 0.024ǫ + 0.069ǫ 0.1 the suppression P/T 0.2 applies also for shift due to isospin breaking in reduced matrix elements A 5/2 0 J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 23

Conclusions shift due to π η η mixing in α extracted from ππ is < 1.4 and is of expected size close to ω mass the effect of ρ ω mixing in B ρρ can be large can be excluded using fits the uncertainty due to uncalculated isospin breaking effects in B ρπ is suppressed and most probably below 0.5 J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 24

Backup slides J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 25

SU(3) decomp. for ππ, interm. result A +0 = A +3 + ǫa +η + ǫ A +η = 1 2 (t + c)(1 + e 0 ) + 1 3 ǫ(2p + s) + 2 3 ǫ (p + 2s) A 00 = A 33 + 2ǫA 3η + 2ǫ A 3η = 1 2 (c p) + 1 3 ǫ(2p + s) + 2 3 ǫ (p + 2s) J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 26

g(ρ + π + π 0 ) π η η mixing induces g(ρ + π + π 0 ) =g(ρ + π + π 3 ) + ǫg(ρ + π + η) + ǫ g(ρ + π + η ) at (84%CL) g(ρ + π + η) g(ρ + π + π 3 ) = [( 1 m2 η m 2 ρ ) ] 1/2 Br(ρ + π + η) Br(ρ + π + < 0.055 π 3 ) J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 27

First bound B ππ ( α α 0 ) π η η 1 2 Arg(e2iγ Ā 0 A +0) [ ǫ ( Ā0η sin ψ ) η A 0η sin ψ η = 1 2 A+0 + ǫ ( Ā0η sin ψ η A 0η sin ψ η extreme at ψ η ( ) = ψ η ( ) = π/2 ( α α 0 ) π η η ǫ ( ) A0η + Ā0η 2 A+0 ( ) + ǫ A0η + Ā0η 2 A+0 experimental input τ + τ 0 = 1.081 ± 0.015, B 0η < 2.5 10 6 (90% CL) B +0 = (5.5 ± 0.6) 10 6 B 0η < 3.7 10 6 (90% CL) J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 28

Improved bound ππ using A +η ( ) = 2 3 A +0 + 2A 0η ( ) and ingoring info from CP asymmetries gives a bound ( ) ( α α 0 ) π η η 2 τ + B 0η ǫ (1 r η ) + ǫ B 0η (1 r η ) τ 0 B +0 B +0 with r η = 3 16 r η = 3 8 [ ] 2 τ0 (B τ +η 2 B τ+ + 3 +0) 2 B τ 0η 0 B +0 B 0η [ τ0 τ + (B +η 1 3 B +0) 2 B +0 B 0η ] 2 τ+ B τ 0 0η J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 29

π η ( ) in B ρπ A + + A + 2A 0 = Te iα + P EW + ǫp ρη + ǫ P ρη SU(3) decomposition T = t P + t V + c P + c V P ρη = 1 6 ( p P p V s V ) P ρη = 1 2 3 (p P + p V + 4s V ) using T t V, p V t V = 0.2 and from global SU(3) fits arg t P arg t V c P, c V smaller p P /t P p V /t V 0.2, p V p P, s V smaller P ρη 1 6 s V 0.3 6 p V P ρη 2 3 s V 0.6 3 p V α π η η = ǫp ρη+ǫ P ρη T 0.024ǫ + 0.069ǫ 0.1 J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 30

SCET B ππ in SCET A = G Fm 2 { 1 B f M1 2 +f M1 ζ BM 2 0 1 0 dudzt 1J (u,z)ζ BM 2 J (z)φ M 1 (u) } T 1ζ (u)φ M 1 (u) + { } 1 2 + λ (f) c A M 1,M 2 c c. isospin breaking in T ij (u),t iζ (u) is 1/m B suppressed remaining isospin violation is encoded in ζ BM, ζj BM (z), f M φ M (u), and A M 1,M 2 c c, with M 1,2 isospin eigenstates J. Zupan Isospin breaking effects in B ππ,ρρ,ρπ CKM2005 p. 31