Constraint on the CKM angle from the experimental measurements of

PHYSICAL REVIEW D 66, 074011 共2002兲

Constraint on the CKM angle ␣ from the experimental measurements of CP violation in B 0d \ ␲ ¿ ␲ À decay Cai-Dian Lu¨* CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China and Institute of High Energy Physics, P.O. Box 918(4), Beijing 100039, China†

Zhenjun Xiao‡ Department of Physics, Nanjing Normal University, Nanjing, Jiangsu 210097, China 共Received 28 June 2002; published 17 October 2002兲 In this paper, we study and try to find the constraint on the Cabibbo-Kobayashi-Maskawa 共CKM兲 angle ␣ from the experimental measurements of C P violation in B 0d → ␲ ⫹ ␲ ⫺ decay, as reported very recently by the BaBar and Belle Collaborations. After considering the uncertainties of the data and the ratio r of the penguin to the tree amplitudes, we found that a strong constraint on both the CKM angle ␣ and the strong phase ␦ can be obtained from the measured C P asymmetries S ␲␲ and A ␲␲ : 共a兲 The ranges of 87°⭐ ␣ ⭐131° and 36° ⭐ ␦ ⭐144° are allowed by 1 ␴ of the averaged data for r⫽0.31; 共b兲 for the Belle result alone, the limits on ␣ and ␦ are 104°⭐ ␣ ⭐139° and 42°⭐ ␦ ⭐138° for 0.32⭐r⭐0.41; and 共c兲 an angle ␣ larger than 90° is preferred. DOI: 10.1103/PhysRevD.66.074011

PACS number共s兲: 13.25.Hw, 12.15.Hh, 12.15.Ji, 12.38.Bx ⫹0.25 A ␲␲ ⫽⫹0.94⫺0.31 共 stat兲 ⫾0.09共 syst兲 .

I. INTRODUCTION

To study the C P violation mechanism is one of the main goals of the B factory experiments. In the standard model 共SM兲, C P violation is induced by the nonzero phase angle appearing in the Cabibbo-Kobayashi-Maskawa 共CKM兲 mixing matrix. Recent measurements of sin 2␤ in the neutral B 0 by the BaBar 关1,2兴 and Belle meson decay B 0d →J/ ␺ K S,L 关3,4兴 Collaborations established a third type of C P violation 共interference between decay and mixing兲 of the B d meson system. The two new measurements of sin 2␤ as reported this year by the BaBar 关2兴 and Belle 关5兴 Collaborations are sin共 2 ␤ 兲 ⫽0.75⫾0.09共 stat兲 ⫾0.04共 syst兲 ,

共1兲

sin共 2 ␤ 兲 ⫽0.82⫾0.12共 stat兲 ⫾0.05共 syst兲 ,

共2兲

with an average sin共 2 ␤ 兲 ⫽0.78⫾0.08,

共3兲

which is very consistent with last year’s world average, sin(2␤)⫽0.79⫾0.10, and leads to the bounds on the angle ␤ :

␤ ⫽ 共 26°⫾4° 兲 ⵪ 共 64°⫾4° 兲 .

共4兲

Despite the well measured CKM angle ␤ , we have very poor knowledge of the other two angles ␣ and ␥ . Very recently, the Belle Collaboration reported their first measurements of the C P violation of B 0d → ␲ ⫹ ␲ ⫺ decay 关5兴: ⫹0.38 ⫹0.16 S ␲␲ ⫽⫺1.21⫺0.27 共 stat兲 ⫺0.13 共 syst兲 ,

*Email address: [email protected] †

Mailing address. ‡ Email address: [email protected] 0556-2821/2002/66共7兲/074011共7兲/$20.00

共5兲

The probability for S ␲␲ ⫽0 and A ␲␲ ⫽0 is 99.9% 关5兴. Based ¯ decays, on a data sample of about 88 million ⌼(4S)→BB the BaBar Collaboration updated their measurement of C P violating asymmetries of B→ ␲ ⫹ ␲ ⫺ decay1 关6兴: S ␲␲ ⫽0.02⫾0.34共 stat兲 ⫾0.05共 syst兲 , C ␲␲ ⫽⫺0.30⫾0.25共 stat兲 ⫾0.04共 syst兲 .

共6兲

The uncertainties of BaBar’s new results are smaller than those of their previous results 关7,8兴. It is easy to see that the experimental measurements of the BaBar and Belle Collaborations are not fully consistent with each other: BaBar’s results are still consistent with zero, while Belle’s results strongly indicate nonzero S ␲␲ and A ␲␲ . Further improvement of the data will enable us to draw definite conclusions about the values of both S ␲␲ and A ␲␲ . Inspired by the recent measurements, attempts have been made to obtain information on strong phases and CKM phases from the recent experimental measurements 关9,10兴. In Ref. 关9兴, Gronau and Rosner examined the time-dependent measurements of B→ ␲ ⫹ ␲ ⫺ decay to gain information on strong and weak phases and found the following: 共a兲 if sin ␦ is small a discrete ambiguity between ␦ ⯝0 and ␦ ⯝ ␲ could be resolved by comparing the measured branching ratio Br(B→ ␲ ⫹ ␲ ⫺ ) with that predicted in the absence of the penguin amplitude; 共b兲 if A ␲␲ is nonzero, the discrete ambiguity between ␦ and ␲ ⫺ ␦ becomes harder to resolve, but its effect on the CKM parameters becomes less important; and 共c兲 the sign of the quantity D ␲␲ ⫽2Re(␭ ␲␲ )/(1⫹ 兩 ␭ ␲␲ 兩 2 ) is 1

For the parameters A ␲␲ ,C ␲␲ , there is a sign difference between the conventions of the Belle and BaBar Collaborations A ␲␲ ⫽ ⫺C ␲␲ . We here use the Belle convention 关5兴. 66 074011-1

©2002 The American Physical Society

¨ AND Z. XIAO C.-D. LU


always negative for the allowed range of CKM parameters and therefore a positive value of D ␲␲ would signify new physics beyond the SM. In Ref. 关10兴, Fleischer and Matias investigated the allowed regions in observable space of B → ␲ K, B d → ␲ ⫹ ␲ ⫺ , and B s →K ⫹ K ⫺ decays. They considered the correlations between these three kinds of decay modes implied by the SU共3兲 flavor symmetry and the U spin symmetry and found new constraint on the CKM angle ␥ by using the B-factory measurements of C P violation of B d → ␲ ⫹␲ ⫺. It is well known that C P asymmetry measurements in B → ␲␲ decays play an important role in extracting the CKM angle ␣ . In this paper, we focus on the B→ ␲ ⫹ ␲ ⫺ decay and try to extract a constraint on the angle ␣ from the measured S ␲␲ and A ␲␲ and the ratio r of the penguin to the tree amplitude fixed by theoretical arguments. Taking into account both the Belle and BaBar Collaborations’ newest measurements 关5,6兴, the weighted averages of S ␲␲ and A ␲␲ are expt ⫽⫺0.57⫾0.25, S ␲␲

expt A ␲␲ ⫽0.57⫾0.19.

FIG. 1. Unitarity triangle in ¯␳ -¯␩ plane.

V CKM

⫽

共7兲

We will treat the above averages as the measured asymmetries of B d → ␲ ⫹ ␲ ⫺ decay in the following analysis. We also investigate what happens if only Belle’s measurements are taken into account. For the case of S ␲␲ ⬇0 and A ␲␲ ⬇0 as indicated by BaBar’s results alone, one can see the discussion in Ref. 关9兴. This paper is organized as follows. In Sec. II we present a general description of the C P asymmetries of B→ ␲ ⫹ ␲ ⫺ decay. In Sec. III we consider the new BaBar and Belle measurements of S ␲␲ and A ␲␲ to draw constraints on the CKM angle ␣ and the strong phase ␦ . The conclusions are included in the final section.

冉

1⫺

␭2 2

␭

⫺␭

1⫺

A␭ 3 共 1⫺¯␳ ⫺i ¯␩ 兲

冉冊冉 s⬘

b⬘

⫽

V ub

V cd

V cs

V cb

V td

V ts

V tb

冊冉冊 d s

.

共8兲

⫺A␭ 2

1

冊

,

共9兲

共10兲

This unitary triangle is just a geometrical presentation of this equation in the complex plane. We show it in the ¯␳ -¯␩ plane in Fig. 1. The three unitarity angles are defined as

冉冉冉

In the SM with SU(2)⫻U(1) as the gauge group, the quark mass eigenstates are not the same as the weak eigenstates. The mixing between the 共down type兲 quark mass eigenstates was described by the CKM matrix 关11兴. The mixing is expressed in terms of a 3⫻3 unitary matrix V CKM operating on the down type quark mass eigenstates (d, s, and b):

V us

A␭ 2

* ⫹V cd V cb * ⫹V td V tb * ⫽0. V ud V ub

II. CP ASYMMETRIES OF B\ ␲ ¿ ␲ À DECAY

V ud

␭2 2

where A, ␭, ¯␳ , and ¯␩ are the Wolfenstein parameters. The unitarity of the CKM matrix implies six ‘‘unitarity triangles.’’ One of them applied to the first and third columns of the CKM matrix yields

␣ ⫽arg ⫺

d⬘

A␭ 3 共 ¯␳ ⫺i ¯␩ 兲

␤ ⫽arg ⫺

␥ ⫽arg ⫺

V* tb V td

* V ud V ub * V cd V cb * V td V tb * V ud V ub * V cd V cb

冊冊冊

,

共11兲

,

共12兲

.

共13兲

The above definitions are independent of the parametrization of the CKM matrix elements. Thus they are universal. In the Wolfenstein parametrization, in terms of (¯␳ , ¯␩ ), sin(2␾i) ( ␾ i ⫽ ␣ , ␤ , ␥ ) can be written as

b

As a 3⫻3 unitary matrix, the CKM mixing matrix V CKM is fixed by four parameters, one of which is an irreducible complex phase. Using the generalized Wolfenstein parametrization 关12兴, V CKM takes the form 074011-2

sin共 2 ␣ 兲 ⫽

2 ¯␩ 共 ¯␩ 2 ⫹¯␳ 2 ⫺¯␳ 兲 , ␩ 2 兲关共 1⫺¯␳ 兲 2 ⫹ ¯␩ 2 兴共 ¯␳ 2 ⫹ ¯

共14兲

sin共 2 ␤ 兲 ⫽

2 ¯␩ 共 1⫺¯␳ 兲 , ¯ 兲 2 ⫹ ¯␩ 2 共 1⫺%

共15兲

CONSTRAINT ON THE CKM ANGLE ␣ FROM . . .

sin共 2 ␥ 兲 ⫽


2¯␳ ¯␩ . ¯␳ 2 ⫹ ¯␩ 2

共16兲

The SM predicts C P-violating asymmetries in the timedependent rates for initial B 0 and ¯B 0 decays to a common C P eigenstate f C P . In the case of f C P ⫽ ␲ ⫹ ␲ ⫺ , the timedependent rate is given by f ␲␲ 共 ⌬t 兲 ⫽

e ⫺ 兩 ⌬t 兩 / ␶ B 0 4 ␶ B0

兵 1⫹q• 关 S ␲␲ sin共 ⌬m d ⌬t 兲

III. CONSTRAINT ON ␣ AND ␦

⫹A ␲␲ cos共 ⌬m d ⌬t 兲兴其 ,

共17兲

where ␶ B 0 is the B 0d lifetime, ⌬m d is between the two B 0d mass eigenstates, 0

the mass difference ⌬t⫽t C P ⫺t tag is the ¯ 0 ) and the accomtime difference between the tagged B (B panying ¯B 0 (B 0 ) with opposite b flavor decaying to ␲ ⫹ ␲ ⫺ at the time t C P , and q⫽⫹1(⫺1) when the tagged B meson is ¯ 0 ). The C P-violating asymmetries S ␲␲ and A ␲␲ are a B 0 (B defined as S ␲␲ ⫽

2 Im共 ␭ ␲␲ 兲 1⫹ 兩 ␭ ␲␲ 兩 2

A ␲␲ ⫽

,

兩 ␭ ␲␲ 兩 2 ⫺1

1⫹ 兩 ␭ ␲␲ 兩 2

* V td V tb

冋

* T ␲␲ e i ␦ 1 ⫺V tb V td * P ␲␲ e i ␦ 2 V ub V ud

* V ud T ␲␲ e i ␦ 1 ⫺V tb * V td P P ␲␲ e i ␦ 2 V tb V * td V ub

⫽e 2i ␣

冋

with r⫽

1⫹re i( ␦ ⫺ ␣ ) 1⫹re i( ␦ ⫹ ␣ )

冏冏冏

册

共18兲

,

where the parameter ␭ ␲␲ is ␭ ␲␲ ⫽

guin contributions, we have A ␲␲ ⫽0, S ␲␲ ⫽sin(2␣ef f ), where ␣ e f f depends on the magnitudes and strong phases of the tree and penguin amplitudes. In this case, the C P asymmetries cannot give the size of the angle ␣ directly. A method has been proposed to extract the CKM angle ␣ using B ⫹ → ␲ ⫹ ␲ 0 and B 0 → ␲ 0 ␲ 0 decays together with B 0d → ␲ ⫹ ␲ ⫺ decay by the isospin relation 关13兴. However, it will take quite some time for the experiments to measure the three channels together.

册共19兲

,

冏

* P ␲␲ V tb V td , T ␲␲ V ub V ud *

␦ ⫽ ␦ 2⫺ ␦ 1 ,

共20兲

In this section, we will show that the only measured C P asymmetries of B 0d → ␲ ⫹ ␲ ⫺ decay can at least provide some constraint on the angle ␣ . From Eqs. 共21兲, 共22兲, one can see that the asymmetries S ␲␲ and A ␲␲ generally depend on three ‘‘free’’ parameters: The CKM angle ␣ with ␣ ⫽ 关 0,␲ 兴 , the strong phase ␦ with ␦ ⫽ 关 ⫺ ␲ , ␲ 兴 , and the ratio r as defined in Eq. 共20兲. We cannot solve these two equations with three unknown variables. However, by the following study, we can at least give some constraint on the angle ␣ and strong phase ␦ . Since the penguin contributions are loop order corrections ( ␣ s suppressed兲 compared with the tree contribution, we can assume 0⬍r ⬍0.5, in a reasonable range. Now we are ready to extract ␣ through the general parametrization of S ␲␲ and A ␲␲ in terms of ( ␣ , ␦ ,r) as given in Eqs. 共21兲, 共22兲. As discussed previously 关9兴, there may exist some discrete ambiguities between ␦ and ␲ ⫺ ␦ for the mapping of S ␲␲ and A ␲␲ onto the ␦ -␣ plane. expt at 1 ␴ level and First, because of the positiveness of A ␲␲ the fact that sin ␣⬎0 for ␣ ⫽(0,␲ ), the ranges ⫺ ␲ ⬍ ␦ ⬍0 and ␦ ⫽0, ⫾ ␲ are excluded, and therefore only the range 0°⬍ ␦ ⬍180° needs to be considered here. For the special case of ␦ ⫽90°, the discrete ambiguity between ␦ and ␲ ⫺ ␦ disappears and the expressions for S ␲␲ and A ␲␲ can be rewritten as S ␲␲ ⫽

where T ␲␲ and P ␲␲ describe the tree and penguin contributions to the B 0d → ␲ ⫹ ␲ ⫺ decay, and ␦ is the difference between the corresponding strong phases of the tree and penguin amplitudes. By explicit calculations, we find that S ␲␲ ⫽

A ␲␲ ⫽

sin 2 ␣ ⫹2r cos ␦ sin ␣ 1⫹r 2 ⫹2r cos ␦ cos ␣ 2r sin ␦ sin ␣ 1⫹r 2 ⫹2r cos ␦ cos ␣

,

共21兲

.

共22兲

If we neglect the penguin diagram contribution 共which is expected to be smaller than the tree diagram contribution兲, we have A ␲␲ ⫽0, S ␲␲ ⫽sin(2␣). That means we can measure sin(2␣) directly from the B 0d → ␲ ⫹ ␲ ⫺ decay. This is the reason why B 0d → ␲ ⫹ ␲ ⫺ decay was previously assumed to be the best channel for measuring the CKM angle ␣ . With pen-

A ␲␲ ⫽

sin 2 ␣ 1⫹r 2

共23兲

,

2r sin ␣ 1⫹r 2

.

共24兲

The range 0°⭐ ␣ ⭐90° is excluded by the negativeness of expt , and the ranges sin ␣⬍0.19(1⫹r 2 )/r and sin ␣ S ␲␲ ⬎0.38(1⫹r 2 )/r are excluded by the measured A ␲␲ ⫽0.57 ⫾0.19 at the 1 ␴ level. In Fig. 2a, we show the ␣ dependence of S ␲␲ for given ␦ ⫽90° and for r⫽0.1 共dotted curve兲, 0.2 共small-dashed curve兲, 0.3 共solid curve兲, 0.4 共dashed curve兲, and 0.5 共dashdotted curve兲. The band between the two horizontal dotted expt ⫽ lines shows the allowed range from the measured S ␲␲ ⫺0.57⫾0.25 at 1 ␴ level. Figure 2b shows the ␣ dependence of S ␲␲ for fixed r⫽0.3 and for ␦ ⫽30° 共dotted curve兲, 60° 共small-dashed curve兲, 90° 共solid curve兲, 120° 共dashed curve兲, and 150° 共dash-dotted curve兲. The differences be-

074011-3



FIG. 2. Plots of S ␲␲ vs the angle ␣ . 共a兲 The dotted, smalldashed, solid, dashed, and dash-dotted curves correspond to r ⫽0.1, 0.2, 0.3, 0.4, and 0.5, respectively. 共b兲 The five curves from left to right are for ␦ ⫽150°,120°, 90°, 60°, and 30°. The band between the two horizontal dotted lines shows the experimental 1 ␴ expt allowed range ⫺0.82⭐S ␲␲ ⭐⫺0.32.

FIG. 3. Plots of A ␲␲ vs the angle ␣ . In 共a兲 the dotted, smalldashed, solid, dashed, and dash-dotted curves correspond to r ⫽0.1, 0.2, 0.3, 0.4, and 0.5, respectively. In 共b兲 the same symbols are for 30°, 60°, 90°, 130°, and 150°, respectively. The band between the two horizontal dotted lines shows the experimental 1 ␴ ex pt allowed range 0.38⭐A ␲␲ ⭐0.76.

tween the curves for ␦ ⫽30° and ␦ ⫽150° ( ␦ ⫽60° and ␦ ⫽120°) show the effects of the discrete ambiguity between ␦ and ␲ ⫺ ␦ . The constraint on the CKM angle ␣ from the measured S ␲␲ alone can be read off directly from Fig. 2. For r⫽0.3, for example, the allowed ranges for the CKM angle ␣ are

curve兲. The band between the two horizontal dotted lines expt shows the region allowed by the measured A ␲␲ ⫽0.57 ⫾0.19 at 1 ␴ level. Figure 3b shows the ␣ dependence of A ␲␲ for given r⫽0.3 and for ␦ ⫽30° 共dotted curve兲, 60° 共small-dashed curve兲, 90° 共solid curve兲, 120° 共dashed curve兲, and 150° 共dash-dotted curve兲. It is easy to see that most of the allowed ranges of ␣ as given in Eqs. 共25兲–共27兲 can be excluded by the inclusion of the measured A ␲␲ . The second solutions as given in Eqs. 共25兲–共27兲 are removed by taking the measured A ␲␲ into account. For the case of r ⫽0.3 and ␦ ⭐30° or ␦ ⭓150°, the whole range of ␣ is excluded by the measured S ␲␲ and A ␲␲ , as illustrated in Fig. 3b. From the above analysis, we can see that a strong constraint on the CKM angle ␣ can be obtained by using the experimental measurements of S ␲␲ and A ␲␲ as well as the ¯ pair production and ratio r. With the rapid increase of BB decay events collected at B factory experiments, the difference between the central values of S ␲␲ and A ␲␲ and the experimental uncertainties will become smaller within two years. The third input parameter r can be fixed through avail-

109°⭐ ␣ ⭐128°⵪153°⭐ ␣ ⭐171°

共25兲

101°⭐ ␣ ⭐121°⵪149°⭐ ␣ ⭐169°

共26兲

for ␦ ⫽60°,

for ␦ ⫽90°, and 92°⭐ ␣ ⭐112°⵪146°⭐ ␣ ⭐168°

共27兲

for ␦ ⫽120°. In general, the current experimental measurements of S ␲␲ prefer ␣ ⬎90°. In Fig. 3a, we show the ␣ dependence of A ␲␲ for given ␦ ⫽90° and r⫽0.1 共dotted curve兲, 0.2 共small-dashed curve兲, 0.3 共solid curve兲, 0.4 共dashed curve兲, and 0.5 共dash-dotted

074011-4



able data or reliable theoretical considerations. From Eqs. 共3兲 and 共15兲, the measured sin(2␤) leads to an equation between ¯␳ and ¯␩ : ¯␩ ⫽ 共 1⫺¯␳ 兲 ␰ ⫽ 共 1⫺¯␳ 兲

1⫾ 冑1⫺sin2 共 2 ␤ 兲 . sin共 2 ␤ 兲

共28兲

The solution with the ⫹ sign in the numerator of ␰ is totally inconsistent with the global fit results and can be dropped. Numerically, ⫹0.09 ␰ ⫽ 共 1⫺ 冑1⫺sin2 共 2 ␤ 兲兲 /sin共 2 ␤ 兲 ⫽0.48⫺0.07

共29兲

for the measured sin(2␤)⫽0.78⫾0.08. There exists quite a lot of information about the CKM matrix elements as reported by the Particle Data Group 关15兴 and in other recent papers 关14,16 –19兴. The parameter ␭⫽ 兩 V us 兩 is known from K l3 decay with good precision, ␭⫽0.2196⫾0.0023.

共30兲

In terms of (¯␳ , ¯␩ ), the parameter r as defined in Eq. 共20兲 can be rewritten as r⫽z

⫽

冑共 1⫺¯␳ 兲 2 ⫹ ¯␩ 2

冑¯␳ 2 ⫹ ¯␩ 2 共 1⫺␭ 2 /2兲 z

共 1⫺¯␳ 兲冑1⫹ ␰ 2

1⫺ 共 ␭ 2 /2兲

冑¯␳ 2 ⫹ 共 1⫺¯␳ 兲 2 ␰ 2

, 共31兲

where z⫽ 兩 P ␲␲ /T ␲␲ 兩 measures the relative sizes of tree and penguin contributions to the studied decay. From general considerations, z may be around 20%. By employing the QCD factorization approach 关20兴 and/or the perturbative QCD approach 关21兴, one can fix r to rather good occuracy. By using the QCD factorization approach, the estimated value of 兩 P ␲␲ /T ␲␲ 兩 is found 关20兴 to be z⫽0.285⫾0.077,

共32兲

where the contribution from weak annihilation has been taken into account and the dominant error comes from the uncertainties of m s and the renormalization scale ␮ 关20兴. From the numbers given in Eqs. 共29兲, 共30兲, 共32兲 and ¯␳ ⫽0.20⫾0.16, we have numerically ⫹0.01 r⫽0.31⫾0.09共 ⌬z 兲 ⫾0.01共 ⌬ ␰ 兲 ⫺0.03 共 ⌬¯␳ 兲

⫾0.0002共 ⌬␭ 兲 ⫽0.31⫾0.10.

FIG. 4. Contour plots of the asymmetries S ␲␲ and A ␲␲ versus the strong phase ␦ and CKM angle ␣ for r⫽0.21 关the small circles in 共a兲兴 and 0.31 关the large circles in 共a兲兴, and 0.41 共b兲. The dotted circles show the effects of discrete ambiguity. The regions inside ex pt each circle are still allowed by both ⫺0.82⭐S ␲␲ ⭐⫺0.32 and ex pt ⭐0.76, which is the experimental 1 ␴ allowed range. 0.38⭐A ␲␲

关circles in 共b兲兴, respectively. The regions inside each circle expt expt are still allowed by both S ␲␲ ⫽⫺0.57⫾0.25 and A ␲␲ ⫽0.57⫾0.19 共experimental 1 ␴ allowed ranges兲. The discrete ambiguities between ␦ and ␲ ⫺ ␦ are shown by the solid and dotted circles in Fig. 4. For ␦ ⫽90°, these discrete ambiguities disappear. If we take the theoretically fixed value of r⫽0.31⫾0.10 as a reliable estimation of r, the constraints on the CKM angle ␣ and the strong phase ␦ can be read off directly from Fig. 4. Numerically, the allowed regions for the CKM angle ␣ and the strong phase ␦ are

Here the estimated result r⭐0.41 is in good agreement with our general argument of r⬍0.5. Thus our analysis in this paper is meaningful. The common range of ␣ allowed by both the measured S ␲␲ and A ␲␲ is what we are trying to find. Figure 4 shows the contour plots of the C P asymmetries S ␲␲ and A ␲␲ versus the strong phase ␦ and CKM angle ␣ for r⫽0.21 关the small circles in 共a兲兴, 0.31 关the larger circles in 共a兲兴, and 0.41

68°⭐ ␦ ⭐112°

共34兲

87°⭐ ␣ ⭐131°, 36°⭐ ␦ ⭐144°

共35兲

97°⭐ ␣ ⭐113°,

共33兲 for r⫽0.21,

for r⫽0.31, and finally

074011-5

80°⭐ ␣ ⭐138°⵪143°⭐ ␣ ⭐155°, 24°⭐ ␦ ⭐151°

共36兲



FIG. 6. Contour plot of the asymmetries S ␲␲ and A ␲␲ versus the strong phase ␦ and CKM angle ␣ for r⫽0.36 共small solid circle兲, 0.41 共middle-sized solid circle兲, and 0.51 共large solid circle兲. The dotted circles show the effects of discrete ambiguity. The regions ex pt inside each circle are still allowed by the Belle limits S ␲␲ ⭐ ex pt ⫺0.67 and A ␲␲ ⭓0.54.

S ␲␲ and A ␲␲ only, and take the direct sum of the statistical and systematic errors as the total 1 ␴ error, then the experimental limits on both S ␲␲ and A ␲␲ take the form expt ⭐⫺0.67, S ␲␲

FIG. 5. Contour plots of the asymmetries S ␲␲ and A ␲␲ versus the CKM angle ␣ and the ratio r for ␦ ⫽60° and 120° 共a兲 and 90° 共b兲, respectively. The dotted semiclosed curve in 共a兲 shows the effects of discrete ambiguity. The regions inside the semiclosed curves are still allowed by the data.

for r⫽0.41. There is a twofold ambiguity in the determination of the angle ␣ for r⬇0.4. In fact, the CKM angle ␣ in the second region in Eq. 共36兲 is too big to be consistent with the standard model unitarity relation ␣ ⫹ ␤ ⫹ ␥ ⫽180°. One can see from Fig. 4 that if we take the weighted average of the BaBar and first Belle measurements of the asymmetries S ␲␲ and A ␲␲ as reliable measured values of S ␲␲ and A ␲␲ , we can obtain a strong constraint on both the strong phase ␦ and the CKM angle ␣ . Even considering the uncertainties of the input parameters, most of the parameter space is excluded. In order to show more details of the r dependence of the constraint on ␣ , we draw Fig. 5. The semiclosed regions shown in Fig. 5a 共for ␦ ⫽60° and 120°) and Fig. 5b 共for ␦ ⫽90°) are still allowed by the measured S ␲␲ and A ␲␲ as given in Eq. 共7兲. As shown in Fig. 5, the region of r⭐0.2 is excluded by the data. The effects of discrete ambiguity are also shown in Fig. 5. The solid semiclosed region in Fig. 5a corresponds to ␦ ⫽60°, while the dotted semiclosed region refers to ␲ ⫺ ␦ ⫽120°. For ␦ ⫽90°, this discrete ambiguity disappears. As discussed in the previous section, there are some discrepancies between the BaBar and Belle measurements of S ␲␲ and A ␲␲ 共or C ␲␲ ). If we use Belle’s measurement of

expt A ␲␲ ⭓0.54.

共37兲

The corresponding contour plots of the asymmetries S ␲␲ and A ␲␲ versus the strong phase ␦ and CKM angle ␣ are illustrated in Fig. 6 for r⫽0.36 共the small solid circle兲, 0.41 共the middle-sized solid circle兲, and 0.51 共the large solid circle兲, respectively. For r⭐0.32, the whole ‘‘␦ -␣ ’’ plane is excluded. The dotted circles correspond to the discrete ambiguity between ␦ and ␲ ⫽ ␦ . Numerically, we find that the allowed ranges for the CKM angle ␣ and the strong phase ␦ are 54°⭐ ␦ ⭐126°

共38兲

104°⭐ ␣ ⭐139°, 42°⭐ ␦ ⭐138°

共39兲

108°⭐ ␣ ⭐130°, for r⫽0.36,

for r⫽0.41, and finally 95°⭐ ␣ ⭐152°, 28°⭐ ␦ ⭐152°

共40兲

for r⫽0.51, although we do not expect such a large value of the ratio r. If we use the Belle measurement of S ␲␲ and A ␲␲ only, and take the square root of the statistical and systematic errors as the total 1 ␴ error, then the experimental limits on both S ␲␲ and A ␲␲ will be expt ⭐⫺0.80, S ␲␲

expt A ␲␲ ⭓0.62.

共41兲

The whole ‘‘␦ -␣ ’’ plane will be excluded even for r⫽0.51. In other word, the Belle result has to be changed in the future, otherwise, new physics may be required to explain the data.

074011-6



Since the discrete ambiguity between ␦ and ␲ ⫺ ␦ vanishes when ␦ ⫽ ␲ /2, the contour plots shown in Figs. 4 and 6 are symmetric with respect to the axis of ␦ ⫽90° in the ␦ -␣ plane. This discrete ambiguity can alter the constraints on ␦ by about 7°, but has little effect on the possible limits on the CKM angle ␣ derived from the measured S ␲␲ and A ␲␲ if we fix the value of r and treat ␦ as a free parameter varying in the range 0°⬍ ␦ ⬍180°, as can be seen in Figs. 4 and 6. It is worth mentioning that the constraint on the angle ␣ from one recent global fit is 82°⭐ ␣ ⭐126° as given in Ref. 关14兴. The constraint from the measured S ␲␲ and A ␲␲ is comparable to or stronger than the global fit result. IV. CONCLUSION

In this paper, we studied the B 0d → ␲ ⫹ ␲ ⫺ decay and tried to find constraints on the CKM angle ␣ and the strong phase ␦ from the measured asymmetries S ␲␲ and A ␲␲ as reported by the BaBar and Belle Collaborations. If we take the weighted average of the BaBar and Belle measurements of S ␲␲ and A ␲␲ as the measured results, strong constraints on both the CKM angle ␣ and the strong phase ␦ can be obtained. The range of ␦ ⭐0 is excluded by the positiveness of measured A ␲␲ . The range 0°⭐ ␣ ⭐90° is excluded by the negativeness of the measured A ␲␲ for ␦ ⫽90°. Within the parameter space of S ␲␲ ⫽⫺0.57⫾0.25, A ␲␲ ⫽0.57⫾0.19, and r⫽0.31⫾0.10, most of the ‘‘␦ -␣ ’’ plane is excluded, as shown in Figs. 3– 6. For fixed r ⫽0.31, for example, the ranges 87°⭐ ␣ ⭐131° and 36°⭐ ␦

关1兴 BaBar Collaboration, B. Aubert et al., Phys. Rev. Lett. 87, 091801 共2001兲; BaBar Collaboration, J. Dorfan, ‘‘BaBar Results on C P Violation,’’ talk given at LP 2001, Roma, 2001, Report No. BaBar-talk-01/77. 关2兴 BaBar Collaboration, B. Aubert et al., ‘‘Improved Measurement of C P Violating Asymmetry Amplitude sin 2␤,’’ hep-ex/0203007. 关3兴 Belle Collaboration, K. Abe et al., Phys. Rev. Lett. 87, 091802 共2001兲; Belle Collaboration, S. L. Olsen, ‘‘Measurement of sin 2␾1,’’ talk given at LP 2001, Roma, 2001, Belle Report No. LP-01. 关4兴 Belle Collaboration, K. Abe et al., Phys. Rev. D 66, 032007 共2002兲. 关5兴 Belle Collaboration, K. Abe et al., Phys. Rev. Lett. 89, 071801 共2002兲. 关6兴 BaBar Collaboration, B. Aubert et al., ‘‘Measurements of Branching Fractions and C P Violating Asymmetries in B 0 → ␲ ⫹ ␲ ⫺ ,K ⫹ ␲ ⫺ ,K ⫹ K ⫺ Decays,’’ Report No. SLAC-PUB9317, hep-ex/0207055. 关7兴 BaBar Collaboration, B. Aubert et al., Phys. Rev. D 65, 051502共R兲共2002兲. 关8兴 BaBar Collaboration, B. Aubert et al., ‘‘Measurements of Branching Fractions and C P Violating Asymmetries in B 0

⭐144° are allowed by 1 ␴ of the averaged S ␲␲ and A ␲␲ values. In general the data prefer ␣ ⬎90°. The discrete ambiguity between ␦ and ␲ ⫺ ␦ will disappear for ␦ ⫽90° and has little effects on the possible limits on the CKM angle ␣ if we fix the value of r and treat ␦ as a free parameter varying in the range 0°⬍ ␦ ⬍180°, as shown in Figs. 4 and 6. If we consider only the Belle measurements, a very narrow range in the ‘‘␦ -␣ ’’ plane is allowed, as illustrated in Fig. 6. The limits on ␣ and ␦ are 104°⭐ ␣ ⭐139° and 42° ⭐ ␦ ⭐138° for 0.32⭐r⭐0.41. Considering the previous sin 2␤ measurement ( ␤ ⫽26°⫾4°), we can conclude that the other CKM angle ␥ should be smaller than 90°. We know that the current data for S ␲␲ and A ␲␲ are still preliminary experimental measurements with large uncertainties. The apparent large difference between the BaBar and Belle measurements and the corresponding experimental uncertainties will become smaller along with the rapid increase of the observed B decay events. Therefore we will soon be able to extract the angle ␣ with good accuracy. ACKNOWLEDGMENTS

This work was partly supported by National Science Foundation of China under Grants No. 90103013 and No. 10135060. Z.J.X. acknowledges support by the National Natural Science Foundation of China under Grant No. 10075013, and by the Research Foundation of Nanjing Normal University under Grant No. 214080A916.

关9兴关10兴关11兴关12兴

关13兴关14兴关15兴关16兴关17兴关18兴关19兴关20兴关21兴

074011-7

→␲⫹␲⫺,K⫹␲⫺,K⫹K⫺ Decays,’’ hep-ex/0205082. M. Gronau and J. Rosner, Phys. Rev. D 65, 093012 共2002兲. R. Fleischer and J. Matias, Phys. Rev. D 66, 054009 共2002兲. M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 共1973兲. A.J. Buras, in ‘‘Flavour Physics: C P Violation and Rare Decays,’’ lectures given at International School of Subnuclear Physics, Erice, Italy, 2000, hep-ph/0101336. M. Gronau and D. London, Phys. Rev. Lett. 65, 3381 共1990兲. A. Ho¨cker, H. Lacker, S. Laplace, and F. le Diberder, Eur. Phys. J. C 21, 225 共2001兲. Particle Data Group, D.E. Groom et al., Eur. Phys. J. C 15, 1 共2000兲. X.G. He, Y.K. Hsino, J.Q. Shi, Y.L. Wu, and Y.F. Zhou, Phys. Rev. D 64, 034002 共2001兲. A.F. Falk, ‘‘Flavor Physics and the CKM Matrix: An Overview,’’ hep-ph/0201094. Z.J. Xiao and M.P. Zhang, Phys. Rev. D 65, 114017 共2002兲. D. Atwood and A. Soni, Phys. Lett. B 508, 17 共2001兲. M. Beneke, G. Buchalla, M. Neubert, and C.T. Sachrajda, Nucl. Phys. B601, 245 共2001兲. C.D. Lu¨, K. Ukai, and M.Z. Yang, Phys. Rev. D 63, 074009 共2001兲.