Annihilation contributions and CP asymmetries in B� ! ��K0, K�K0 and B0 ! K0 �K0

Xi-Qing Hao,1 Xiao-Gang He,1,2 and Xue-Qian Li11Department of Physics, Nankai University, Tianjin

2Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei(Received 10 January 2007; published 20 February 2007)

Recently the branching ratios for B� ! K� �K0 and B0 ! K0 �K0 have been measured. Data indicate thatthe annihilation amplitudes in these decays are not zero. A nonzero annihilation amplitude plays animportant role in CP violation for B� ! ��K0, K� �K0. Using the measured branching ratios for thesedecays, we show that there is an absolute bound of 5% for the size of CP asymmetry in B� ! ��K0 froma relation between the amplitudes of these decays. The size of CP asymmetry in B� ! K� �K0 can,however, be as large as 90%. Future experimental data will test these predictions.

DOI: 10.1103/PhysRevD.75.037501 PACS numbers: 13.20.He, 11.30.Er, 11.30.Hv, 14.40.Nd

Rare B decays provide much information about thestandard model (SM) of the strong and electroweak inter-action. So far the SM is perfectly consistent with data on Bdecays. With more data becoming available from BABARand Belle experiments, B physics has entered an era of aprecision test. Not only stringent constraints on new phys-ics beyond the SM have been derived, but also informationabout some of the more subtle aspects on the low energystrong interaction responsible for B hadronization andrelated hadronic decay amplitudes has been extracted, inparticular, forB to two SU�3� octet pesudoscalarsPP [1,2].In this work we study information about decay amplitudesand CP asymmetries using the recently measured branch-ing ratios for [3] B� ! ��K0, K� �K0 and B0 ! K0 �K0.

There are several reasons why B� ! ��K0, K� �K0 andB0 ! K0 �K0 decays are interesting to study [4]. The treecontributions are usually thought to be small for thesedecays since they come from the so-called annihilationdiagrams and are neglected in many of the previous stud-ies. If the annihilation contributions are neglected,~B�B� ! ��K0�= ~B�B� ! K� �K0� (here the branching ra-tios are averaged over B and �B decays) is approximatelyequal to jVts=Vtdj2 which can be tested. A method usingB! �K decays to determine the CP violating phase � inthe Cabbibo-Kobayashi-Maskawa (CKM) matrix as dis-cussed in Ref. [5] crucially depends on the assumption thatthe annihilation contribution to B� ! ��K0 is zero.Therefore it is important to find out whether the annihila-tion contributions are zero or not. Using recent experimen-tal data, we find that the annihilation contributions to rare Bdecays may not be negligible after all. We also find thatusing a relation in the amplitudes for B� ! ��K0 andB� ! K� �K0, with nonzero annihilation contributionsthere is a relation for CP asymmetries in B� ! ��K0

and B� ! K� �K0. From this relation, we obtain an upperbound of 5% for the size of CP asymmetry in B� !��K0. The size of CP asymmetry in B� ! K� �K0 can,however, be as large as 90%. In the following we providedetails.

In the SM the leading order decay amplitude for thecharmless B decays can be decomposed into two terms

proportional to V�ubVuq and V�tbVuq, respectively. For B� !��K0 and B� ! K� �K0, we have

A�B� ! ��K0� � V�ubVusTs � V�tbVtsPs;

A�B� ! K� �K0� � V�ubVudTd � V�tbVtdPd:

(1)

In the SU�3� limit, Ts � Td and Ps � Pd. In terms of thediagram amplitudes discussed in Ref. [6], one can parame-trize them as Ts � A and Ps � P� PCEW=3, where P, PCEW,and A stand for penguin, electroweak penguin, and anni-hilation amplitudes, respectively. When SU�3� breakingeffects are included, there are modifications. It is expectedthat a large fraction of the SU�3� breaking effects may beabsorbed into the relevant meson decay constants. This canbe understood from the PQCD calculation for B! PPdecay amplitudes [7]. In this approach quarks are viewedas partons forming the initial and final mesons and areconvoluted with the relevant hadron light-cone wave func-tions which are normalized to their decay constants. To theleading order one would have fKTd � f�Ts and fKPd �f�Ps. We can then further simplify the above decay am-plitudes to

A�B� ! ��K0� � Ps�V�ubVusa� V�tbVts�;

A�B� ! K� �K0� �fKf�Ps�V�ubVuda� V

�tbVtd�;

(2)

where a � Ts=Ps is a complex number. We parametrize itas a � rei�r for our later discussions.

Since only small annihilation amplitude A contributes toterms proportional to VubV�uq in the above two processes,these terms are usually neglected. If applicable, there areseveral interesting experimental consequences with one ofthem being that CP asymmetries in B� ! ��K0 andB� ! K� �K0 are zero, and also that R � ~B�B� !�� �K0�= ~B�B� ! K� �K0� would give a good determinationfor �f2

�=f2K�jVts=Vtdj

2.Recent experiments at the BABAR and Belle have

measured branching ratios of these decays with [3]~B�B� ! ��K0� � �23:1� 1:0� � 10�6 and ~B�B� !K�K0� � �1:36�0:29

�0:27� � 10�6. Using these numbers and

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fK=f� � 1:198� 0:003�0:016�0:005 [8,9], we obtain

jVtdj2

jVtsj2� 0:041� 0:009; (3)

in accordance with the best global fit value for the CKMparameters which gives [8] jVtdj2=jVtsj2 � 0:036.

If the SU�3� correction factor fK=f� in Eq. (2) is notincluded, one would obtain a value jVtdj2=jVtsj2 around0.059 which is substantially away from the best global fitvalue. This supports the expectation that a large fraction ofthe SU�3� breaking effects are taken care of by the lightmeson decay constants. We note that the central value ofjVtdj2=jVtsj2 determined using f2

�=f2KR and using global fit

deviate from each other. Of course it is too early to saythere is a definitive difference due to large errors.

The available experimental data also allow us to extractmore detailed information about the decay amplitudes. Theamplitudes Td;s need not be zero. In has been pointed outbefore that the data allow nonzero values for Td;s [2]. Thediscussion above also does not rule out this possibility. Wewould like to point out that there is another indication thatthe amplitude A is not zero. This is from a comparison ofthe observed branching ratios for B� ! K� �K0 and B0 !K0 �K0. One can write the amplitudes for these decays as

A�B� ! K� �K0� � V�ubVudA� V�tbVtd�P�

13P

CEW�;

A�B0 ! K0 �K0� � V�tbVtd�P�13P

CEW � PA�:

(4)

Note that one needs to introduce a new amplitude PA whichis called the penguin-annihilation amplitude, for B0 !K0 �K0. Since the Wilson coefficients involved in PA aresmaller than those in A, PA is expected to be smaller than Ain size. If annihilation amplitudes A and PA are negligiblysmall, one would obtain S � ��B0=�B��� ~B�B

� !K� �K0�= ~B�B0 ! K0 �K0�� � 1. However, the measuredbranching ratios [3] ~B�B0 ! K0 �K0� � �0:95�0:20

�0:19� � 10�6

and ~B�B� ! K� �K0� � �1:36�0:29�0:27� � 10�6 gives S �

1:32� 0:39. The central value of S is substantially awayfrom 1. This indicates that the annihilation amplitude A orPA (or both of them) may not be zero. To have somedetailed idea about what the allowed ranges are for r and�r, we consider the case with PA neglected in more detail.We have

S � 1� r2jV�ubVudV�tbVtd

j2 � 2r cos�rRe�V�ubVudV�tbVtd

�: (5)

Using the above we can obtain the allowed ranges for r and�r. In Fig. 1 we show r as a function of �r for several

values of S. In obtaining Fig. 1, we have treated the CKMmatrix elements as known and used the most recent valuesgiven by the Particle Data Group with [8] � � 0:2262, A �0:815 �� � 0:235, and �� � 0:349 (the corresponding sinesof mixing angles and phase are: s12 � 0:2262, s13 �0:0039, s23 � 0:0417, and �13 � 0:9781).

There are theoretical calculations [10,11] trying to esti-mate the decay amplitudes for B� ! ��K0 and B� !K� �K0. One of the popular methods used to estimate suchcontributions is the QCD factorization (QCDF) [11]. In thismethod, the annihilation amplitudes have end point diver-gences which are usually indicated by a quantity XA �R

10 dy=y. In Ref. [11] this is phenomenologically regulated

by a cut off �b parametrized as XA � �1� �Aei�A��

ln�mB=�b� and �A < 1. The resulting annihilation ampli-tude A is in general complex. The typical value from such acalculation gives r of order 10% with a cut off �b 0:5 GeV. If a smaller cut off �b is used, r can be larger.Owing to the uncertainties in treating the divergences, onecannot give a precise value for a � re�r . Therefore in ourlater discussions, we will not use a theoretical value for abut use an allowed range from data. The important point isthat one should not set r to zero for precision studies. Wenow study some implications with nonzero annihilationcontributions. Before carrying out a numerical analysis ofthe allowed ranges for r and �r, we study some implica-tions forCP violation with nonzero values of r and �r. Oneconsequence is that CP violating asymmetries ACP�B� !��K0; K� �K0� defined in the following will not be zero,

ACP�B� ! ��K0� �B�B� ! �� �K0� � B�B� ! ��K0�

B�B� ! ��K0� � B�B� ! �� �K0�� �

2r sin�rIm �V�ubVusVtbV

�ts�

jV�tbVtsj2 � r2jV�ubVusj

2 � 2r cos�Re �V�ubVusVtbV�ts�;

ACP�B� ! K� �K0� �

B�B� ! K�K0� � B�B� ! K� �K0�

B�B� ! K�K0� � B�B� ! K� �K0�� �

2r sin�rIm �V�ubVudVtbV�td�

jV�tbVtdj2 � r2jV�ubVudj

2 � 2r cos�Re �V�ubVudVtbV�td�:

(6)

0 1 2 3 4 5 6δ r

0

0.5

1

1.5

2

2.5

rS=1.336

S=1.73

S=1.01S=0.999

S=0.969

FIG. 1. Allowed ranges for r and � for different values of S ���B0=�B��� ~B�B

� ! K� �K0�= ~B�B0 ! K0 �K0��. There is no solu-tion if S is smaller than 0.968.

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It has been pointed out before [12] that, due to theproperty Im�V�ubVusVtbV

�ts� � �Im�V

�ubVudVtbV

�td� of the

CKM matrix, there are relations between rate differences��PP� � �� �B! �P �P� � ��B! PP� for some B! PPdecays. For ACP�B

� ! ��K0� and ACP�B� ! K� �K0�,

we have

ACP�B� ! ��K0� � �f2�

f2K

1

RACP�B� ! K� �K0�: (7)

Since the maximal size for ACP�B� ! K� �K0� can atmost be 1, one immediately obtains an upper bound of 5%for the size of CP asymmetry in B� ! ��K0. The presentdata 0:009� 0:025 on ACP�B

� ! ��K0� is consistentwith the upper bound. Measurement of CP asymmetry inB� ! ��K0 therefore can test the relation in Eq. (7).

Using experimental data for ACP�B� ! ��K0�, one wouldobtain, ACP�B� ! K� �K0� �0:22� 0:62. This is to becompared with the data 0:12�0:17

�0:18 for ACP�B� ! K� �K0�.Naively it seems that there is a potential problem since thecentral experimental data has opposite sign as that pre-dicted. However, the error bar is still too large to draw aconclusion. Improved data on B� ! K� �K0 can providemore crucial information about hadronic parameters in Bdecays.

To find out how large CP asymmetry in B� ! K� �K0

can be and how it is correlated to information from B� !��K0, more information is needed. Again we use theexperimental data on the branching ratios and knownvalues for the CKM matrix elements to constrain theallowed regions. Theoretically, we have

R �~B�B� ! ��K0�

~B�B� ! K� �K0��f2�

f2K

jV�tbVtsj2 � r2jV�ubVusj

2 � 2r cos�rRe �V�ubVusVtbV�ts�

jV�tbVtdj2 � r2jV�ubVudj

2 � 2r cos�rRe �V�ubVudVtbV�td�: (8)

Comparing the above expression with data, we obtainthe allowed ranges of the parameter space for r and cos�rin Fig. 2. We find that not the entire allowed one- range ofR � 13:41 20:55 has solutions for r and �r. With in-creased R the allowed range becomes smaller and even-tually shrinks to a point at R � 19:44. There are nosolutions beyond that point. If future experimental datadetermine a R larger than 19.44, it is a strong indicationthat there is new physics beyond the SM so that the relationin Eq. (2) is badly broken. Within the one- allowed rangefor R, there is a solution for r � 0 where R � 18:65. Withmore precise determination of R, one can have betterinformation about the size of r. At present we see thatthere is a large allowed range for r and �r. Although large ris unlikely from theoretical considerations, this possibilityis not ruled out yet at the present. We note that the allowedrange is more restrictive than that shown in Fig. 1 if both Rand S are restricted to be within their one- ranges. In thefollowing discussions we will use the ranges for r and �r

constrained from R instead that from S. The reason is twofolds with one of them being that the constraint from R ismore stringent, and another being that in obtaining allowedranges for r and �r in Fig. 1, PA has been neglected. Anonzero PA can change the ranges.

With the allowed ranges for r and �r fixed, one canobtain the allowed range for CP asymmetry in B� !K� �K0. We present the predicted CP asymmetry for B� !K� �K0 in Fig. 3. CP asymmetry for B� ! ��K0 can beeasily obtained from Fig. 3 using Eq. (7). We see that thesize of CP asymmetry ACP�B� ! K� �K0� becomes largeras R decreases. At the lower end of the one- allowed R,the size of ACP�B� ! K� �K0� can be as large as 0.92.Consequently the size of ACP�B� ! ��K0� can almostreach its upper bound of 0.05. Measurement on CP asym-metries in these decay modes can provide valuable infor-mation about the hadronic parameters in B decays.

To summarize, in this work we have shown that therecently measured branching ratios for B� ! K� �K0 and

0 1 2 3 4 5 6δr

0

0.5

1

1.5

2

r

R=16.98

R=13.41

R=18.44R=19.00

R=19.40

FIG. 2. Allowed ranges for r and � for different values of R �~B�B� ! ��K0�= ~B�B� ! K� �K0�.

0 1 2 3 4 5 6δ r

-1

-0.5

0

0.5

1

Acp

(K+

K–0)

R=16.98R=13.41

R=18.44

R=19.00R=19.40

FIG. 3. ACP�B� ! K� �K0� as a function of � for several al-lowed values R.

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B0 ! K0 �K0 indicate that the annihilation amplitudes inthese decays may be not zero. This observation has im-portant impacts on CP violation in B� ! ��K0. Using themeasured branching ratios for these decays, we find thatthere is an absolute bound of 5% for the size of CPasymmetry in B� ! ��K0 from a relation in the ampli-

tudes of these decays. The size of CP asymmetry in B� !K� �K0 can, however, be as large as 90%. Future experi-mental data will test these predictions.

This work was supported in part by NNSF and NSC.X. G. H. thanks NCTS for partial support.

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