Introduction to 2 from charmless...

NH1/M

O.N� 1

Introduction to φ2 from charmless decays

Blazenka Melic— Rudjer Boskovic Institute, Zagreb —

〈[email protected]〉

B. Melic CKM2006, Nagoya 12 - 16 Dec 2006

NH1/M

O.N� Contents 2


NH1/M

O.N� Contents 2

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3


NH1/M

O.N� Contents 2

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 CONSTRAINTS ON α FROM B → ππ . . . . . . . . . . . . . . . . . . . . . . . 4


NH1/M

O.N� Contents 2

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3


3 CONSTRAINTS ON α FROM B → πρ . . . . . . . . . . . . . . . . . . . . . . . 9


NH1/M

O.N� Contents 2

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3



4 CONSTRAINTS ON α FROM B → ρρ . . . . . . . . . . . . . . . . . . . . . . . 11

◦ ISOSPIN BREAKING: ELECTROWEAK PENGUINS ◦ ISOSPIN BREAKING:∆I = 5/2 corrections ◦ ISOSPIN BREAKING: MIXING CORRECTIONS


NH1/M

O.N� Contents 2

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3





5 CONSTRAINTS ON α FROM OTHER CHARMLESS DECAYS . . . . . . . . . . 17


NH1/M

O.N� Contents 2

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3





5 CONSTRAINTS ON α FROM OTHER CHARMLESS DECAYS . . . . . . . . . . 17

6 UTfit and CKMfitter CONSTRAINTS ON α . . . . . . . . . . . . . . . . . . . . 19


NH1/M

O.N� INTRODUCTION 3

R. Fleischer, hep-ph/0608010

φ2 ≡ α ≡ arg[−VtdV∗tb/VudV∗

ub|

SM prediction (indirect measurements of α):-combining measurements of |Vus|, |Vud|, |Vub|, |Vcb|, CPV from K0 − K

0, Bd,s − Bd,s mixing and

sin 2β:

α = (98.2± 7.7)◦ (UTfit) α = (100.0+4.5−7.3)

◦ (CKMfitter)


NH1/M

O.N� CONSTRAINTS ON α FROM B → ππ 4

M.Gronau, D.London, Phys.Rev.Lett. 65, 3381 (1990)

in general there are two amplitudes with different WEAK phases:

(a) tree diagram (b) penguin dia-grams

CP-ASYMMETRY for tagged B0 (B0):

aCP(t) =Γ [B(t) → π+π−] − Γ [B0(t) → π+π−]

Γ [B(t) → π+π−] + Γ [B0(t) → π+π−]= S+− sin(∆mt) − C+− cos(∆mt)

= a±mixed sin(∆mt) − a±dir cos(∆mt)


NH1/M


S+− ≡ a±mixed =2Imλ

1 + |λ|2C+− ≡ a±dir =

1 − |λ|2

1 + |λ|2

λ is related to the B0-B0

mixing through p/q and to the ratio of the amplitudes:

λ =q

p

A

A

NO PENGUIN AMPLITUDES: S+− = sin 2α C+− = 0EXPERIMENT: S+− = −0.59± 0.09 C+− = −0.39± 0.07this can be parametrized as

sin(2αeff) = S+−

/√1 − C2

+−

αeff = α − θ = α − arg(A+−/A+−)

α from aCP(t) → requires the knowledge P/T which includes several nonperturbativeamplitudes:

A(B → π+π−) = −λu(T + Pu) − λcPc − λtPt= e−iγTππ + eiφPππ√2A(B → π0π0) = λu(−C + Pu) + λcPc + λtPt = e−iγCππ − eiφPππ= e−iγ(T−0 − Tππ) − eiφPππ√

2A(B− → π−π0) = −λu(T + C)= e−iγT−0


NH1/M


HOW TO DETERMINE α ? → BY USING FLAVOUR (ISOSPIN) SYMMETRIES !see talk by J. Zupan at this workshop


NH1/M


HOW TO DETERMINE α ? → BY USING FLAVOUR (ISOSPIN) SYMMETRIES !see talk by J. Zupan at this workshop

ISOSPIN FOR ππ SYSTEM:

• u-d symmetry - broken by the quark masses - neglected

• EW penguins are neglected

• ππ system - I = 0, 2 due to the Bose statistics

weak Hamiltonian contains tree-level operators O1,2 (∆I = 1/2, 3/2), gluonic penguins O3,..,6

(∆I = 1/2) and electroweak penguins O7,..,10 (∆I = 1/2, 3/2 - neglected):

Hweak =GF√

2

∑p=u,c

VpbV∗pd

{C1(µ)Op

1 + C2(µ)Op2 +

∑i=3,..,10

Ci(µ)Oi

}

1√2A(B → π+π−) = AI=2 − AI=0

A(B → π0π0) = 2AI=2 + AI=0

A(B− → π−π0) = 3AI=2

if there are no EW penguins, AI=2 receives contributions only from tree operators


NH1/M


ISOSPIN RELATION BETWEEN A AMPLITUDES (and their CP-conjugate amplitudes):

1√2A(B → π+π−) + A(B → π0π0) = A(B− → π−π0)

A+−

A−0~

A+−~

A00

A00~

12 1

2

+0A , A = e2iγA

mixed CP asymmetries:

S+− ∼ Imλ+− = Im[e2iα

[1−A0/A21−A0/A2

]]== Im

[e2iα

[1−|A0/A2|e±iθ

1−|A0/A2|e±iθ

]]S00 ∼ Imλ00 = Im

[e2iα

[1+1/2|A0/A2|e±iθ

1+1/2|A0/A2|e±iθ

]]. we have eightfold discrete ambiguity in the determination of α


NH1/M


CONCLUSIONS:

. the theoretical uncertainty due to the penguin diagrams (gluonic) are eliminated - angleα can be determined by tagged measurement of all ππ modes

. in the absence of the full information (in particular present measurement of a00dir does

not provide significant bound on α), one can set several bounds on θ angle(αeff = α − θ) by predicting amplitudes T , P, C by using models and some assumptions:

M.Gronau, D. London, N. Sinha, R. Sinha, Phys.Lett.B 514, 315 (2001)D. Pirjol, Phys.Rev.D 60, 054020 (1999); R. Fleischer, Phys.Lett.B 459, 306 (1999)G. Buchalla, A.S. Safir, Phys. Rev. Lett. 93, 021801 (2004)

Y. Grossman, A. Hocker, Z. Ligeti, D. Pirjol, Phys.Rev.D 72, 094033 (2005)

see talk by A. Bevan at this workshop


NH1/M

O.N� CONSTRAINTS ON α FROM B → πρ 9

A.E.Snayder, H.R.Quinn, Phys.Rev.D 48, 2139(1993); H.R.Quinn, J.P.Silva, Phys.Rev.D 62, 054002 (2000)

ISOSPIN FOR ρπ SYSTEM:

• much more complicated since ρ±π∓ final states are not CP eigenstates

• ρπ system has I = 0, 1, 2 - four independent isospin amplitudes

• pentagon isospin analysis or a Dalitz plot analysis of π+π−π0 final decay state

A(B+ → ρ+π0) =12

√32A2,3/2 −

12

√12A1,3/2 +

√12A1,1/2

[−

√16A2,5/2

]

A(B+ → ρ0π+) =12

√32A2,3/2 +

12

√12A1,3/2 −

√12A1,1/2

[−

√16A2,5/2

]

A(B0 → ρ+π−) =12

√13A2,3/2 +

12A1,3/2 +

12A1,1/2 +

√16A0,1/2

[+

12

√13A2,5/2

]

A(B0 → ρ−π+) =12

√13A2,3/2 −

12A1,3/2 −

12A1,1/2 +

√16A0,1/2

[+

12

√13A2,5/2

]

A(B0 → ρ0π0) =

√13A2,3/2 −

√16A0,1/2

[+

√13A2,5/2

].


NH1/M

O.N� CONSTRAINTS ON α FROM B → πρ 10

• tree level contributions: A2,3/2 and A1,3/2

• other pieces: tree level + penguins (gluonic); EW penguins - neglected• ∆I = 5/2 from electromagnetic rescattering effects: A2,5/2 - neglected

instead of triangular isospin relations as in B → ππ here we have PENTAGON in the complexplane (and similarly for CP-conjugate amplitudes):√

2(A(B+ → ρ+π0) + A(B+ → ρ0π+)

)= A(B0 → ρ+π−) + A(B0 → ρ−π+) + 2A(B0 → ρ0π0)

• time dependent Dalitz analysis of B0 → ρπ → π+π−π0 removes penguin contributions andextract α (no discrete ambiguities for 2α)• there are two different ways how to do time dependent Dalitz analysis:- Quasi-2-body approach - uses quasi-two body representations of B0 → (π±π0)π∓ decayscorresponding to distinct bounds in the 3-pion Dalitz plot in the vicinity of the ρ resonances- full time-dependent Dalitz analysis of B0 → π±π∓π0, accounting for the interference betweenintersecting ρ resonance bands and other resonances

BaBar obtained (hep-ex/0408099): α = (113+27−17 ± 6)◦ see talk by G.Cavoto at this workshop

Belle obtained (hep-ex/0609003): α = (83+12−23)

◦see talk by C-C. Wang at this workshop

using BaBar analysis a model dependent constraint can be set (M.Gronau, J.Zupan, Phys.Rev.D

70, 074031 (2004)):α = (93± 4exp ± 6th)

◦


NH1/M

O.N� CONSTRAINTS ON α FROM B → ρρ 11

G.Kramer, W.Palmer, Phys.Rev.D 45, 193 (1992)

ISOSPIN FOR ρρ SYSTEM:

• ρπ system has I = 0, 2 states only → one can do similar isospin analysis as for ππ

system

• S=0 particle decays into two S=1 particles → three helicity states H = 0 (longitudinal L),±1 (transversal T)

• H = 0 (L state) is CP-even state and dominates in the decay

π0

π+

ρ+

0π

θ 2

ρ_

_

π

θ 1

φ

d2Γ

Γ d cos θ1 d cos θ2=

94

[fL cos2 θ1 cos2 θ2 +

14(1 − fL) sin2 θ1 sin2 θ2

]fL = fraction of the H = 0 state in the decay


NH1/M

O.N� CONSTRAINTS ON α FROM B → ρρ 12

• by measuring BR’s and aCP ’s of B → ρ0ρ0 and B → ρ±ρ∓ one can determine α

• there are some corrections:- due to the final width of the ρ meson (I = 1 contributions), but there are expected to be oforder of 4%- isospin violation

see talk by A. Somov at this workshop


NH1/M

O.N� ISOSPIN BREAKING: ELECTROWEAK PENGUINS 13

if we include EW penguins (∆I = 1/2, 3/2):

A(B → π+π−) = e−iγTππ + eiφPππ ,√2A(B → π0π0) = e−iγ(T−0 − Tππ) − eiφPππ ,√

2A(B− → π−π0) = e−iγT−0 + eiβPEW.

SU(2) triangular relations still hold true:

1√2A(B → π+π−) + A(B → π0π0) = A(B− → π−π0)

1√2A(B → π+π−) + A(B → π0π0) = A(B+ → π+π0)

but now A(B+ → π+π0) 6= A(B− → π−π0)

• there are model-dependent estimates of PEW/T−0 giving SU(2) breaking corrections to α

determination:Deshpande, He, hep-ph/9408404; Gronau et al, hep-ph/9504327; Fleischer, hep-ph/9509204; Charles, hep-ph/9806468;

Neubert, Rosner,hep-ph/9808493; Buras, Fleischer,Eur.Phys.J C 11, 93 (1999); Gronau, Pirjol, Yan, hep-ph/9810482;

Beneke et al, hep-ph/0104110; etc....


NH1/M

O.N� ISOSPIN BREAKING: ELECTROWEAK PENGUINS 14

−A(B−∼ ππ )

A(B−∼ −ππ+ )

ππ )A(B )π π

A(B )+π π−

T+C

θ

∆θ

0

0 0 00 0

0

0

0∼ −A(B

0ππ+ )A(B+

)( uc dc CEW

PPEW

( + )−

)+(EW

P PEWCcdcu( )− ∼ ∼

Figure 2: Isospin analysis of B → ππ decays with the inclusion of electroweak penguins.The amplitudes A(B → ππ) are defined as exp(2iγ)A(B → ππ), with similar definitionsfor PEW and P C

EW .

observing the time-dependence of B0(t)→ π+π−. The amplitudes of these six processesform two triangles, as shown in Fig. 2, in which the CP-conjugate amplitudes havebeen rotated by a common phase A(B → ππ) ≡ exp(2iγ)A(B → ππ) (and similarlyfor PEW and P C

EW ). The CKM phase α is measured from the time-dependent rate ofB0(t)→ π+π−, which involves a term

∣

∣

∣

∣

∣

A(B0 → π+π−)

A(B0 → π+π−)

∣

∣

∣

∣

∣

sin(2α + θ) sin(∆mt) , (4)

where ∆m is the neutral B mass difference. The angle θ is measured as shown in Fig. 2.The effect of the EWP amplitudes on determining θ and correspondingly fixing α is

rather clearly represented by the small vectors at the right bottom corner of the Fig. 2.These terms, given by (cu−cd)(PEW +P C

EW) and its CP-conjugate, have unknown phases

relative to the T + C term which dominates A(B+ → π+π0) and its charge-conjugate.This leads to a very small uncertainty in the relative orientation of the two triangles. [Inthe limit of neglecting EWP amplitudes, one would have A(B− → π−π0) = A(B+ →

π+π0)]. The uncertainty in measuring θ, and consequently in determining α, is given by

∆α ≈1

2∆θ ≤

∣

∣

∣

∣

∣

(cu − cd)(PEW + P C

EW)

T + C

∣

∣

∣

∣

∣

. (5)

We therefore conclude that the effects of EWP amplitudes on the measurement of α areat most of order λ2 and are negligible.

Since a different conclusion has been claimed in [11, 17], let us clarify the apparentdisagreement. The authors of [11, 17] have only shown that the error in determining α

10

• the correction due to the EW penguins is given by the ratio PEW/T−0, i.e. by the angle ∆θ ;bounds can be set:

cos ∆θ > 1 − 2∣∣∣∣PEW

T−0

∣∣∣∣2

or |∆θ| 6 2∣∣∣∣PEW

T−0

∣∣∣∣Gronau et al., Phys.Rev.D 52, 6374 (1995)

. effect of EW penguins on α is < few degrees


NH1/M

O.N� ISOSPIN BREAKING: ∆I = 5/2 corrections 15

S. Gardner, U. Meissner, Phys.Rev.D 65, 094004 (2002); J.Charles et al., hep-ph/0406184

it comesi) from the insertion of the d − u mass difference ∆I = 1 operatorii) by electromagnetic corrections ( O(α) corrections )

without any assumptions on the operators one can write:

1√2A(B → π+π−) =

(AI=2 − A

5/2I=2

)− AI=0

A(B → π0π0) = 2(AI=2 − A

5/2I=2

)+ AI=0

A(B− → π−π0) = 3AI=2 +√

6A5/2I=2

the isospin triangular relation is broken:

1√2A(B → π+π−) + A(B → π0π0) − A(B− → π−π0) ∝ A

5/2I=2

. effect of ∆I = 5/2 corrections to α is < then the current experimental precision


NH1/M

O.N� ISOSPIN BREAKING: MIXING CORRECTIONS 16

π0 − η − η′

MIXING IN B → ππ; ρ0 − ω MIXING IN B → ρπ AND B → ρρ

S. Gardner, Phys.Rev.D 59, 077502 (1999); M. Gronau, J. Zupan, hep-ph/0502139


NH1/M


π0 − η − η′



• π0 − η − η′

: if the d − u mass difference is introduced, π0 becomes contaminated by theisospin singlet components:

|π0〉 = |π3〉+ ε|η〉+ ε′|η′〉

with ε = 0.017± 0.003, ε′= 0.004± 0.001

. effect of π0 − ηη′

mixing in B → ππ decays to α determination is < 1.4◦


NH1/M


π0 − η − η′



• π0 − η − η′

: if the d − u mass difference is introduced, π0 becomes contaminated by theisospin singlet components:

|π0〉 = |π3〉+ ε|η〉+ ε′|η′〉

with ε = 0.017± 0.003, ε′= 0.004± 0.001

. effect of π0 − ηη′

mixing in B → ππ decays to α determination is < 1.4◦

• ρ0 − ω : ρ0 and ω are mixtures of an isovector ρI and an isoscalar ωI:

|ρ0〉 = |ρI〉− ε1|ωI〉 , |ω〉 = |ωI〉+ ε2|ρI〉

with ε1, 2 < 0.01.. effect of ρ0 − ω mixing in B → ρπ and B → ρρ decays to α determination is expected to besmall


NH1/M

O.N� CONSTRAINTS ON α FROM OTHER CHARMLESS DECAYS 17

. ISOSPIN BREAKING EFFECTS IN B → ργ, B → ωγ

A.Ali, A. Parkhomenko, hep-ph/0610149

∆ ≡ 12

Γ(B+ → ρ+γ)

Γ(B0 → ρ0γ)− 1 = −0.36± 0.27 (BaBar, hep − ex/0607099)

• calculated at the leading order:∆LO = −2εA|λu| cos α + εA|λu|2

where εA parametrizes interference between penguin and annihilation contributions whichcan be calculated in some model• since: λu =

VubV∗ud

VtbV∗td

=∣∣∣−VubV∗

udVtbV∗

td

∣∣∣ eiα one can in principle get an independent

determination of α


NH1/M


. ISOSPIN BREAKING EFFECTS IN B → ργ, B → ωγ

A.Ali, A. Parkhomenko, hep-ph/0610149

∆ ≡ 12

Γ(B+ → ρ+γ)

Γ(B0 → ρ0γ)− 1 = −0.36± 0.27 (BaBar, hep − ex/0607099)

• calculated at the leading order:∆LO = −2εA|λu| cos α + εA|λu|2

where εA parametrizes interference between penguin and annihilation contributions whichcan be calculated in some model• since: λu =

VubV∗ud

VtbV∗td

=∣∣∣−VubV∗

udVtbV∗

td

∣∣∣ eiα one can in principle get an independent

determination of α

. B0 → a±1 (1260)π∓ DECAYS AND SU(3)f SYMMETRYR. Aleksan et al, Nucl.Phys.B 361, 141 (1991); M. Gronau, J. Zupan, hep-ph/0512148

• recently B0 → a±1 π∓ was observed• the situation is similar to B → ρπ, so one can apply isospin analysis and use SU(3) relationscorrections from penguin amplitudes are bounded by relating B0 → a±1 π∓ with corresponding∆S = 1 decays, B → a1K and B → a1K1A (K1A is admixture of the K1(1270) and K1(1400)

resonances )


NH1/M


. B0 → ρρ AND B+ → K∗0ρ+ DECAYS WITH SU(3)f SYMMETRYM. Beneke, M. Gronau, J. Rohrer, M. Spranger, hep-ph/0604005

• using of SU(3)f relations to extract penguin amplitudes and QCD factorization as atheoretical model they obtain:

α =[91.2+9.1

−6.6(exp)+1.2−3.9(th)

]◦


NH1/M




α =[91.2+9.1

−6.6(exp)+1.2−3.9(th)

]◦

. B0 → K0K0

A. Datta, M. Imbeault, D. London, J. Matias, hep-ph/0611280

• they set a relation for extracting α from B0 → K0K0

decay

2 sin2 α =κB

|V∗ubVud|2|T − P|2

• with present data no bound on α can be set


NH1/M




α =[91.2+9.1

−6.6(exp)+1.2−3.9(th)

]◦

. B0 → K0K0

A. Datta, M. Imbeault, D. London, J. Matias, hep-ph/0611280

• they set a relation for extracting α from B0 → K0K0

decay

2 sin2 α =κB

|V∗ubVud|2|T − P|2

• with present data no bound on α can be set

. AND FROM OTHER MODES ... see talk by F. Palombo at this workshop


NH1/M

O.N� UTfit and CKMfitter CONSTRAINTS ON α 19see talk by V. Lubicz at this workshop see talk by S. T’Jampens at this workshop

]o[α0 50 100 150

Pro

bab

ility

den

sity

0

0.002

0.004

0.006

0.008

]o[α0 50 100 150

Pro

bab

ility

den

sity

0

0.002

0.004

0.006

0.008

]o[α0 50 100 150

Pro

bab

ility

den

sity

0

0.002

0.004

0.006

0.008

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120 140 160 180

B →

ππB

→ ρπ (Babar)

B →

ρρCombined

CKM fit

α (deg)

1 –

CL

WACK Mf i t t e r

Beauty06

indirect measurements of α:α = (98.2± 7.7)◦ (UTfit) α = (100.0+4.5

−7.3)◦ (CKMfitter)

direct measurements of α:α = (92± 7)◦ (UTfit) α = (92.6+10.7

−9.3 )◦ (CKMfitter)

theoretical model dependent precisions:precision from B → ππ = 13◦

precision from B → ρπ = 15.5◦

precision from B → ρρ = 9◦


Introduction to 2 from charmless...

Documents

Transcript of Introduction to 2 from charmless...