PDF Effect of charm mixing and CPV in the

Effect of charm mixing and CPV in the extraction of 𝛾 with 𝐵 ± → 𝐷 ∗ 𝐾 ∗ ±

Matteo Rama

Laboratori Nazionali di Frascati

ICHEP 2014

July 2-9 2014, Valencia

Measurement of g with B  DK

color allowed

𝐴₁ ∝ 𝑉_𝑐𝑏𝑉_𝑢𝑠^∗ ∼ 𝐴𝜆³

color suppressed

𝐴₂ ∝ 𝑉_𝑢𝑏𝑉_𝑐𝑠^∗ ∼ 𝐴𝜆³ 𝜌 + 𝑖𝜂

• g is measured in the interference of the two amplitudes

• unknowns: 𝜸, 𝑟

_𝑏

, 𝛿

_𝑏

+𝛿

_𝐷

• theoretically clean

• most sensitive method to constrain g at present

• similar principle applies to several other processes: 𝐵

⁰

→ 𝐷

^(∗)+

𝜋

⁻

, 𝐵

⁰

→ 𝐷

⁰

𝐾

^{∗ 0}

, 𝐵

_𝑠

→ 𝐷

_𝑠

𝐾, …

𝜙 = −𝜸 + 𝛿 𝛿_𝑏 + 𝛿_𝐷= strong phase

from B and D decay

𝐴_𝑡𝑜𝑡 ² = 𝐴₁ + 𝐴₂ ² = 𝐴₁ ² + 𝐴₂ ² + 2 𝐴₁ 𝐴₂ cos 𝜙

𝑟_𝑏 ≡ 𝐴 𝐵⁺→ 𝐷⁰𝐾⁺

𝐴 𝐵⁺→ 𝐷⁰𝐾⁺ ∼ 0.1

B hadronic parameters extracted with g

→ 𝑓

→ 𝑓

measured at charm factories

Main methods

• GLW

– D⁰ to two-body CP eigenstates K⁺K^-, p⁺p^- (even), K_sp⁰, K_sw (odd)

• ADS

– D⁰ to doubly Cabibbo suppressed decays K⁺p^-, K⁺p^-p⁰, …

• GGSZ (Dalitz)

– D⁰ to n-body decays K_sp⁺p^-, K_sK⁺K^-, p⁺p^-p⁰, etc.

• Dalitz plot fitted to determine how the strong phase of D⁰ decay amplitude varies over the Dalitz plane

• in alternative, model independent analysis

[M. Gronau, D. London, D. Wyler, PLB253,483 (1991); PLB 265, 172 (1991)]

[D. Atwood, I. Dunietz, A. Soni, PRL 78, 3357 (1997)]

[D. Atwood et al., PRL78, 3257 (1997); A. Giri et al., PRD68, 054018 (2003)]

at present the most sensitive method

• Different B decays D

⁰

K

^±

, D*

⁰

K

^±

, D

⁰

K*

^±

and flavour-tagged D

⁰

K*

⁰

. They depend on mode-dependent hadronic factors (r

_b

, d

_b

)

• Strategy: combine as many channels as possible to improve the overall sensitivity

Different methods depending on D final state:

Theoretical and experimental precision

Irreducible theoretical precision: ≲ O(10

^-7

) rad

[Brod,Zupan, JHEP 01 (2014) 051, 1308.5663]

Current world average experimental precision: 7-8

^o

[CKMfitter, UTfit]

Expected precision at upgraded LHCb and Belle2:

𝐷 − 𝐷 mixing, possible CPV in D decays, CPV in K

⁰

system have usually been neglected so far. They can all be taken into account, therefore no impact on the theoretical cleanness of the method.

What is the bias on g when these effects are ignored?

(*) Belle2 projections from arxiv:1002.5012 (**) LHCb projections from LHCb-PUB-2013-015 (***) combination

assumes no correlations

s(g) [deg]

B  DK rates

𝐴_𝐷 = 𝐵⁻ → 𝐷⁰𝐾⁻ decay amplitude 𝐴_𝐷 = 𝐵⁻ → 𝐷⁰𝐾⁻ decay amplitude 𝐴_𝑓 = 𝐷⁰ → 𝑓 decay amplitude

𝐴 _𝑓 = 𝐷⁰ → 𝑓 decay amplitude

The ‘classic’ formulation ignores 𝐷 − 𝐷 mixing (and CPV in D decays)

Γ(𝐵⁺ → 𝑓 _DK⁺) is obtained with 𝐴_𝐷 → 𝐴 _𝐷 = 𝐴_𝐷, 𝐴_𝐷 → 𝐴 _𝐷, 𝐴_𝑓 → 𝐴 _𝑓 = 𝐴_𝑓, 𝐴 _𝑓 → 𝐴_𝑓 = 𝐴 _𝑓

∝ 𝑟_𝑓𝑒^{−𝑖𝛿𝑓} + 𝑟_𝐵𝑒^{𝑖𝛿𝐵−𝛾 2} with

B  DK rates

Γ(𝐵⁺ → 𝑓 _DK⁺) is obtained with 𝐴_𝐷 → 𝐴 _𝐷, 𝐴_𝐷 → 𝐴 _𝐷, 𝐴_𝑓 → 𝐴 _𝑓 , 𝐴 _𝑓 → 𝐴_𝑓 , 𝑞/𝑝 → 𝑝/𝑞

𝐴_𝐷 = 𝐵⁻ → 𝐷⁰𝐾⁻ decay amplitude 𝐴_𝐷 = 𝐵⁻ → 𝐷⁰𝐾⁻ decay amplitude 𝐴_𝑓 = 𝐷⁰ → 𝑓 decay amplitude

𝐴 _𝑓 = 𝐷⁰ → 𝑓 decay amplitude

∝ 1 − 𝑂(𝑥² + 𝑦²)

∝ 𝑂(𝑥² + 𝑦²)

∝ 𝑥

∝ 𝑦 (HFAG)

𝜖 𝑡 = reco. efficiency as a function of the D proper time 𝑔_±(𝑡) include the effect of 𝐷 − 𝐷 mixing

𝑥, 𝑦 = 𝐷 − 𝐷 mixing parameters 𝑥 = 0.39 ± 0.17 10⁻²

𝑦 = 0.67 ± 0.08 10⁻² 𝑥² + 𝑦² 𝑟_𝐵 ~ 0.1

[see for example

Meca,Silva PRL81,1377(1998),

Amorim,Santos,Silva PRD59,056001(1999), MR PRD89,014021(2014)]

We want to quantify the impact of these additional terms on the extraction of g Including the neglected terms:

Charm mixing in GLW method

exact relation, no quadratic or higher-order terms in 𝑥, 𝑦 are neglected

Measured observables: ^{𝑓 = 𝑓𝐶𝑃± 𝐾+𝐾−, 𝜋+𝜋−, … , 𝐾−𝜋+}

With 𝐷 − 𝐷 mixing the rate of the 𝑓_𝐶𝑃± modes are multiplied by a factor 1/(1 ± 𝑦):

and are unaffected because the factors cancel out in the ratio Charm mixing terms appear in but are 𝑂(𝑟_𝐵 𝑥² + 𝑦²)

•

•

classic GLW observables

Therefore:

The GLW method is almost unaffected by 𝐷 − 𝐷 mixing

and analogously for 𝐵⁻ → 𝑓_{𝐶𝑃± 𝐷}𝜋⁻

Charm mixing in ADS method

Ignoring charm mixing corresponds to measuring 𝑥_𝐵±^′ , 𝑦_𝐵±^′ instead of 𝑥_𝐵±, 𝑦_𝐵±

The 𝑅^∓ ADS observables are defined as

Including 𝐷 − 𝐷 mixing the ratios can be rewritten as

𝑥_𝐵±^′ = 𝑥_𝐵± − 𝑦/2 𝑦_𝐵±^′ = 𝑦_𝐵± − 𝑥/2 𝑥_𝐵± = 𝑟_𝐵cos(𝛿_𝐵 ± 𝛾) 𝑦_𝐵± = 𝑟_𝐵sin(𝛿_𝐵 ± 𝛾)

However, the bias of g is accidentally suppressed. See slide 12.

(BD⁰K)

Δ𝑅^∓ ≈ 1/3 stat error of 𝑅^∓ measured at LHCb [LHCb, PLB 712 (2012) 203, 1203.3662]

already non-negligible

[MR, 1307.4384 PRD89,014021(2014]

x,y = D mixing parameters

Charm mixing in model dependent GGSZ method

Neglecting the 𝐷 − 𝐷 mixing the rate is

Γ 𝐵⁻ → 𝑓 _𝐷𝐾⁻ ∝ 𝑓₋ ² + 𝑟_𝐵² 𝑓₊ ² + 2𝑥_𝐵−𝑅𝑒 𝑓₋𝑓₊^∗ + 2𝑦_𝐵−𝐼𝑚 𝑓₋𝑓₊^∗

With mixing the rate can be rewritten as

Γ 𝐵⁻ → 𝑓 _𝐷𝐾⁻ ∝ 𝑓₋ ² + 𝑟_𝐵^′2 𝑓₊ ² + 2𝑥′_𝐵−𝑅𝑒 𝑓₋𝑓₊^∗ + 2𝑦′_𝐵−𝐼𝑚 𝑓₋𝑓₊^∗

𝑓_∓ = 𝑓 𝑚2_±, 𝑚2_∓ = decay amplitudes of 𝐷⁰/𝐷⁰ over the Dalitz plot

Ignoring charm mixing corresponds to measuring 𝑥_𝐵±^′ , 𝑦_𝐵±^′ instead of 𝑥_𝐵±, 𝑦_𝐵± .

Note: if 𝑓_∓ are measured from flavor-tagged, time-integrated 𝐷 → 𝑓 decays without taking charm mixing into account the estimate of the bias on 𝑥_𝐵±, 𝑦_𝐵± requires a simulation, though in general the magnitude is expected to be reduced.

𝑥_𝐵±^′ = 𝑥_𝐵± − 𝑦/2

𝑦_𝐵±^′ = 𝑦_𝐵± − 𝑥/2 [MR, 1307.4384

PRD89,014021(2014]

Charm mixing in model independent GGSZ method

measured 𝑩^∓ → 𝑫𝑲^∓ yield flav-tagged DK_Spp yield

extracted from fit to the B^ yields measured by CLEO/BESIII

for each bin J in the Dalitz plot

no 𝐷 − 𝐷 mixing

(and no CPV in 𝐷 decay)

(1) With mixing the rate can be rewritten in terms of 𝑥_𝐵±^′ , 𝑦_𝐵±^′

(2) ^𝑥𝐵±

′ = 𝑥_𝐵± − 𝑦/2 𝑦_𝐵±^′ = 𝑦_𝐵± − 𝑥/2

If eq. (1) is used and mixing is ignored in the extraction of 𝐾_±𝑗 , the neglected terms are O 𝑥, 𝑦 × 𝑂(𝑟_𝐵) and hence further suppressed. In this case Δ𝛾 ≲ 0.2^o

[Bondar,Poluektov,Vorobiev PRD82,034033(2010), 1004.2350]

Measurement of input parameters 𝐶_𝑗, 𝑆_𝑗, 𝐾_±𝐽 :

• 𝐶_𝑗, 𝑆_𝑗 at Ψ(3770) unaffected by 𝐷 − 𝐷 mixing

• 𝐾_±𝑗 from flav-tagged 𝐷 → 𝐾_𝑆𝜋𝜋 includes mixing effects unless explicitly taken into account

Otherwise, 𝑥_𝐵±^′ , 𝑦_𝐵±^′ are measured instead of 𝑥_𝐵±, 𝑦_𝐵± as in ADS and GGSZ model-dep

D flight distance significance >2 to remove charmless Bhhh background

Time acceptance

Discussions so far assumed uniform selection efficiency as a function of D proper time With non uniform 𝜖(𝑡) the terms linear in 𝑥, 𝑦 are multiplied by a correction factor 𝛼 > 1 Academic example:

select 𝑡 > 𝑡_𝑐 to suppress background

corr. factor for terms

∝ 𝑥 or ∝ 𝑦

corr. factor for terms

∝ 𝑂(𝑥²+ 𝑦²)

𝛼 = 1 + 𝑡_𝑐/𝜏_𝐷 Real example:

analysis of 𝐵 → 𝐷𝑕, 𝐷 → 𝑕𝑕 at LHCb (GLW and ADS)

[PLB726 (2013) 151, 1305.2050]

𝛼 = 1.20 ± 0.04 Used in 𝛾 combination with 𝐵 → 𝐷𝑕

[LHCb, PLB712 (2012) 203, 1203.3662]

𝜖(𝑡) derived from acceptance and resolution function

Note: if 𝛼₁, 𝛼₂ are the correction factors in the determination of 𝑁_𝑗^∓and 𝐾_±𝑗 in GGSZ model-independent method (eq. (1) slide 10), terms ∝ 𝛼₁ − 𝛼₂ × 𝑂(𝑥, 𝑦) appear

[MR, 1307.4384 PRD89,014021(2014]

however, significant reduction if the d_B range measured in BDK is considered

Bias in the extraction of g from 𝑥 𝐵± ′ , 𝑦 𝐵± ′

(𝛿₀ = atan 𝑦/𝑥) (𝛼 ≥ 1 is the time acceptance correction factor)

max bias

≈ ±3^o !

in all cases |Dg|≲1^o using the measured ranges of d_B

r_B=0.1, g=70^o, a=1

BDK d_B=(115±9)^o

similarly for BD*K and BDK*

Let’s suppose that g is extracted from 𝑥_𝐵±^′ , 𝑦_𝐵±^′ while believing they are 𝑥_𝐵±, 𝑦_𝐵±

Potential bias is quite large 𝑥_𝐵±^′ = 𝑥_𝐵± − 𝑦/2

𝑦_𝐵±^′ = 𝑦_𝐵± − 𝑥/2

[MR, 1307.4384 PRD89,014021(2014]

Direct CPV in D decays

Δ𝐴^𝑑𝑖𝑟_𝐶𝑃 = 𝐴_𝐶𝑃 𝐾⁺𝐾⁻) − 𝐴_𝐶𝑃 𝜋⁺𝜋⁻ = (−0.253 ± 0.104 %

[HFAG June 2014]

Impact of direct CPV in Dh⁺h^- on GLW analysis:

𝑅_𝐶𝑃+ unaffected

𝐴_𝐶𝑃+ 𝑕𝑕 ≅ 2𝑟_𝐵 sin 𝛿_𝐵 sin 𝛾

1 + 𝑟_𝐵² + 2𝑟_𝐵 cos 𝛿_𝐵cos 𝛾 + 𝐴_𝐶𝑃^𝑑𝑖𝑟(𝑕𝑕)

[Bhattacharya,Gronau, London, Rosner PRD87, 074002 (20013); Martone, Zupan PRD87, 034005 (2013); Wang, PRL110, 061802 (2013)]

•

•

(case where only 𝐴_𝐶𝑃+ is used;

Dg scales approximately as 𝐴_𝐶𝑃+/𝑟_𝐵)

Effect of direct CPV in DK_Spp on mod-independent Dalitz method: [Bondar,Dolgov,Poluetkov, Vorobiev EPJ C(2013) 73]

Systematic error from current exp limits on direct CPV in DK_Spp: |Dg|_max < 3^o

Generalization of the method to allow possible CPV in Dn-body decays (e.g. DK_Spp) Loss of statistical precision due to additional free parameters limited to <10%

Effect of direct CPV in Dh⁺h^- on GLW method:

If CPV is allowed in CP-tag (used to measure C_i) additional data-driven input is necessary

•

•

•

•

If 𝐴_𝐶𝑃^𝑑𝑖𝑟 correction ignored: ^{Δ𝛾𝜋𝜋~ 0 ± 1 𝑜}

Δ𝛾_𝐾𝐾~ −1 ± 1 ^𝑜

CPV in neutral kaon system

[Grossman,Savastio, JHEP03(2014) 008]

GLW method, CP-odd decays (e.g. 𝐷⁰ → 𝐾_𝑆𝜋⁰) 𝐴_𝐶𝑃− ~ 2𝑟_𝐵(sin 𝛾 sin 𝛿_𝐵 + 𝑅𝑒(𝜖)/𝑟_𝐵)

GGSZ mod-independent method, e.g. DK_Sp⁺p^-

for 𝐵 → D⁰𝐾: 𝑅𝑒(𝜖) 𝑟_𝐵~1.6 10⁻² In 𝑅_𝐶𝑃− the effect cancels out

Otherwise, if 𝐾_𝑖 and 𝐾 _−𝑖 are determined separately in 𝐷^∗± → 𝐷𝜋^± data samples the neglected terms are 𝑂(𝑟_𝐵 𝜖 )  Δ𝛾 ~𝑂 0.1 ^𝑜

Possible to apply the mod-independent method with CPV allowed discussed in sl. 13

no bias of g at the price of < 10% loss of statistical power

•

If 𝐾 and 𝐾 are assumed to be equal (no CPV) the bias can be 𝑂 1 ^𝑜

𝐾_𝑖 = 𝐾 _−𝑖 if CP conserved (see also sl. 10)

What about 𝐵 ± → 𝐷 ∗ 0 𝜋 ± ?

Effects of charm mixing scale as 𝑥² + 𝑦² 𝑟_𝐵,𝜋 ~ 𝑂(1)

𝐵𝐹 𝐵⁻ → 𝐷⁰𝜋⁻ ~ 13 × 𝐵𝐹(𝐵⁻ → 𝐷⁰𝐾⁻) 𝑟_𝐵,𝜋 ~ 𝜆²𝑟_𝐵 ~ 0.005 (not measured yet) Fundamental role as control sample. Can it be also used to measure g?

Effects of direct CPV in 𝐷 decay scale as 𝐴_𝐶𝑃 𝑟_𝐵,𝜋 (could be 𝑂(1) in SCS decays) Effects of CPV in 𝐾⁰ system scale as |𝜖| 𝑟_𝐵,𝜋 ~ 𝑂(1)

•

•

•

In principle all these effects can be corrected for, but extra care is needed compared to 𝐷𝐾

[LHCb, PLB726 (2013) 151, 1305.2050]

LHCb has combined 𝐵 → 𝐷⁰𝜋 GLW and ADS (pp, KK, Kp, K3p) with 𝐵 → 𝐷⁰𝐾 taking 𝐷 − 𝐷 mixing and 𝐴_𝐶𝑃 𝑕𝑕 into account.

First use of BDp to constrain g at LHCb:

Possible hidden sources of exp bias might be amplified by factors O(10) in Dp

•

Assumptions usually considered robust in DK might become questionable (e.g.: can CPV in DCS and CA decays be safely neglected in the ADS method?)

•

Summary

•

With the start of the LHC Run2 in 2015 and the planned Belle2 physics run in 2016 we are entering a phase where a number of tiny effects ignored so far in the

extraction of g from BD⁽*⁾K⁽*⁾ will become important

– 𝐷 − 𝐷 mixing

– possible CP violation in D decays

– CP violation in the neutral kaon system

•

These effects can be corrected for, hence no practical impact on the cleanness of the method

•

When they are neglected the bias of g is usually O(1^o) (with a few notable exceptions)

•

Since the shift of g scales as 1/𝑟_𝐵, in BD⁽*⁾p the effect is enhanced by a factor O(10): the corrections cannot be ignored even in the current datasets.

In general, if this decay is used to constrain g extreme care is needed.

backup

Dg from 𝑥 𝐵± ′ , 𝑦 𝐵± ′

𝑥_𝐵^′ = 𝑥_𝐵 − 𝑦/2

𝑦_𝐵^′ = 𝑦_𝐵 − 𝑥/2 ^(𝛿0 = atan 𝑦/𝑥)

geometric representation of Dg assuming r_B=0.1 and g=70^o. Left: d_B=115^o; right: d_B=(115-90)^o

d_B=115^o d_B=25^o

When d_B=(115-90)^o |Dg| is visibly larger compared to the case d_B=115^o

[MR, 1307.4384 PRD89,014021(2014]

When is the effect of charm mixing quadratic in x,y?

[Grossman/Soffer/Zupan PRD72,031501 (2005)]

If one measures directly Γ_𝑓 , Γ _𝑓 and 𝛿 _𝑓 from data and assumes 𝜖_𝑓 = 0, the shift Δ𝛾 in eq.

(1) is quadratic in 𝑥/𝑟_𝑓 and 𝑦/𝑟_𝑓

The rate of 𝐵⁻ → 𝐷⁰𝐾⁻ in presence of charm mixing can be written as:

𝜖_𝑓 = 𝑂((𝑥²+𝑦²) 𝑟 ) _𝑓² where terms linear in x,y are absorbed in Γ_𝑓, Γ _𝑓 and 𝛿 _𝑓

(1)

Two practical limitations (ADS method):

What is measured in charm mixing combinations (e.g. by HFAG) is 𝑟_𝑓², not Γ_𝑓 Γ _𝑓. Γ_𝑓 Γ _𝑓 could be measured for this purpose but it would be experiment-dependent:

attention to possible terms linear in x,y and weighted by 𝛼₁ − 𝛼₂ (sl. 11)

•

𝛿 _𝑓 differs from 𝛿_𝑓 by terms 𝑂( 𝑥² + 𝑦² 𝑟_𝐵) and should be obtained directly from the fit to the 𝐵^± rates with consequent reduction of sensitivity to 𝛾

•

GLW results

D0 K* D*0 K D0 K CP- CP+ CP- CP+ CP- CP+

ADS results (B  DK)

DK DK DK* D*[Dg]K D*[Dp0]K DK K3p Kpp0 Kp Kp Kp Kp DK DK* D*[Dg]K D*[Dp0]K DK K3p Kp Kp Kp Kp

LHCb dominates the BDK, DKp mode.

Final states with neutrals difficult in hadronic environment

ADS results (B  D p )

Dp Dp D*[Dg]p D*[Dp0]p Dp K3p Kpp0 Kp Kp Kp Dp Dp D*[Dg]p D*[Dp0]p Dp K3p Kpp0 Kp Kp Kp

LHCb dominates the BDp, DKp mode.

Final states with neutrals difficult in hadronic environment

𝐴₋ 𝑚₋², 𝑚₊² = |𝐴(𝐵⁻ → 𝐷⁰𝐾⁻) 𝐴_𝐷 𝑚₋², 𝑚₊² + 𝑟_𝑏𝑒^{𝑖(𝛿𝑏−𝛾)}𝐴_𝐷(𝑚₊², 𝑚₋²)

The GGSZ method

• The interference varies as function of the position in the D

⁰

Dalitz plot

• A

_D

(m

_-²

,m

₊²

) is measured with a Dalitz plot analysis of high statistics samples of flavour-tagged D

⁰

and D

⁰

• The B

⁺

and B

^-

yields are measured as a function of the position in the D

⁰

Dalitz plot (ML fit)

• Unknowns: g, rb and d

_b

2 2 0 2

( _S )

m_ m K p^

0 0

D  KSp p^{ } m_2

m2

0 0

D KSp p^{ }

Vcb



K_S⁰pp

Vub _ ⁰_p^_p^

KS

+

m_2

m_2

𝐴_𝐷 𝑚₊, 𝑚₋ ²

𝐴_𝐷 𝑚₋, 𝑚₊ ²

𝐴₊ 𝑚₋², 𝑚₊² = |𝐴(𝐵⁺ → 𝐷⁰𝐾⁺) 𝐴_𝐷 𝑚₊², 𝑚₋² + 𝑟_𝑏𝑒^{𝑖(𝛿𝑏+𝛾)}𝐴_𝐷(𝑚₋², 𝑚₊²)

Model independent analysis

•

divide the DK_Spp Dalitz plot in 2k bins (symmetric w.r.t. the 𝑚₊² vs 𝑚₋² axis)

•

express the 𝐵^± → 𝐷𝐾^± yields in each bin in terms of 𝑥_±, 𝑦_± and 2k parameters 𝑐_𝑖, 𝑠_𝑖

•

𝑐_𝑖, 𝑠_𝑖 are measured by CLEO exploiting the quantum coherence in 𝜓 3770 → 𝐷⁰𝐷⁰

•

extract 𝑥_±, 𝑦_± from ML fit to 𝐵^± → 𝐷𝐾^± yields in all bins

Model-independent measurement of g.

Proposed by A. Giri et al.

[Phys Rev. D68 054018 (2003)].

Pioneered by Belle.

𝑁

_𝑖^±

= 𝑕

_𝐵

𝐾

_±𝑖

+ 𝑟

_𝑏²

𝐾

_∓𝑖

+ 2 𝐾

_𝑖

𝐾

_−𝑖

𝑥

_±

𝑐

_𝑖

± 𝑦

_±

𝑠

_𝑖

measured by CLEO [PRD82, 112006 (2010)]

from flav.-tagged DKspp

𝑩^±→ 𝑫𝑲^± yields

extracted from fit to the B^± yields

GLW method

• D

⁰

to K

⁺

K

^-

, p

⁺

p

^-

(CP+) and Ksp

⁰

, Ksw , Ksf (CP-)

• measure B

⁺

and B

^-

yields to determine the GLW observables:

 











  ^{ } _^{ }_ ^{ }

 2 ( )

) (

) (

0

0 0

K D B

K D B

K D

R_CP B ^{CP CP}

1  2 r

_b

cos g cos d

_b

 r

_b²

 

















  _

 

 















 ( ) ( )

) (

) (

0 0

0 0

K D B

K D B

K D B

K D A B

CP CP

CP CP

CP

 2 r

_b

sin g sin d

_b

R

_CP

• 4 observables, 3 independent unknowns: g , d

_b

, r

_b

ADS method

• D

⁰

to K

⁺

p

^-

, K

⁺

p

^-

p

⁰

, K

⁺

p

⁺

p

⁺

p

^-

, … (doubly-Cabibbo-supp.)





 D

^(*)0

K

^(*)

B





 D

^(*)0

K

^(*)

B

f D

⁰



f D

⁰



suppressed

suppressed favored

favored +

+

same final state

large interference ~O(1)

• Measures B

⁺

and B

^-

yields to determine the ADS observables:

) ] [ (

) ] [ (

) ] [ (

) ] [ (











































 

K f D B

K f D B

K f D B

K f D

A_ADS B  2r_br_D sin(d_b d_D)sing /R_ADS )

] [

( ) ] [

(

) ] [

( ) ] [

(











































 

K f D B

K f D B

K f D B

K f D

R_ADS B  r_b² r_D² 2r_br_D cos(

d

_b 

d

_D)cos

g

) (

) (

0 0

f D A

f D r_D A



 



 

 ( )

arg

0 f

D d A

(r_D(K⁺p^-)=0.06)