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A Determination of the CKM-angle $alpha$ using Mixing-induced CP Violation in the Decays $B

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MPI-PhT/95-72TUM-T31-96/95TTP95-30hep-ph/9507460July 1995A Determination of the CKM-angle αusing Mixing-induced CP Violation in the Decays B d →π+π?and B d →K 0ˉK 0?Andrzej J.Buras Technische Universit¨a t M¨u nchen,Physik Department D–85748Garching,Germany Max-Planck-Institut f¨u r Physik –Werner-Heisenberg-Institut –F¨o hringer Ring 6,D–80805M¨u nchen,Germany Robert Fleischer Institut f¨u r Theoretische Teilchenphysik Universit¨a t Karlsruhe D–76128Karlsruhe,Germany Abstract

We present a method of determining the CKM-angle αby performing simultaneous measurements of the mixing-induced CP asymmetries of the decays B d →π+π?and B d →K 0ˉK 0.The accuracy of our approach is limited

by SU (3)-breaking e?ects originating from ˉb →ˉds ˉs QCD-penguin https://www.wendangku.net/doc/0316280864.html,ing plausible power-counting arguments we show that these uncertainties are expected to be of the same order as those arising through electroweak penguins in the standard Gronau-London-method in which αis extracted by means of isospin relations among B →ππdecay amplitudes.Therefore our approach,which does not involve the experimentally di?cult mode B d →π0π0and is essentially una?ected by electroweak penguins,may be an interesting alternative to determine α.

CP-violating asymmetries arising in nonleptonic B-decays(see e.g.refs.[1]-[8])will play a central role in the determination of the anglesα,βandγin the unitarity triangle [9,10]at future experimental B-physics projects.Unfortunately,these asymmetries are in general not related to the CKM-angles in a clean way,but su?er from uncertainties originating from so-called penguins.These contributions preclude in particlular a clean determination of the CKM-angleαby measuring the mixing-induced CP-violating asymmetry A mix-ind

(B d→π+π?).In a pioneering paper[11],Gronau and London

have presented a method to eliminate the uncertainty in this determination ofαthat is related to QCD-penguins.It uses isospin relations among B d→π+π?,B d→π0π0and B±→π±π0decay amplitudes and requires besides a time-dependent study of B d→π+π?yielding A mix-ind

(B d→π+π?)a measurement of the corresponding branching

ratios.

However,there are not only QCD-but also electroweak penguin operators. Although one would expect na¨?vely that electroweak penguins should only play a minor role in nonleptonic B-decays,there are certain transitions that are a?ected signi?cantly by these operators which become important in the presence of a heavy top-quark.This interesting feature has?rst been pointed out in refs.[12]-[14]and has been con?rmed later by the authors of refs.[15]-[17].As has been stressed?rst by Deshpande and He [18],the in?uence of electroweak penguins on the extraction ofαby using the stan-dard Gronau-London-method[11]could also be sizable.A more elaborate analysis[19] shows,however,that this impact is expected to be rather small,at most a few per cent.

In a recent paper[20]we have presented strategies for the experimental determina-tion of electroweak penguin contributions to nonleptonic B-decays.These strategies allow in particular to control the electroweak penguin uncertainty a?ecting the extrac-tion of the CKM-angleαin the Gronau-London-method[11].Although this method of determiningαis very clean from the theoretical point of view,it requires the mea-surement of the decay B d→π0π0which is rather di?cult.The very recent analysis by Kramer and Palmer[21]indicates a branching ratio BR(B d→π0π0)<～O(10?6).There-fore,it is important to search for other methods that allow a clean determination ofα. Such methods are also needed for overconstraining the shape of the unitarity triangle.

Motivated by this experimental situation,Dunietz[22]has suggested an alternative way of extractingαthat is based on the SU(3)?avour symmetry of strong interactions [23]-[27]and uses time-dependent measurements of the modes B d→π+π?and B s→K+K?.However in view of the large B0s–ˉB0s–mixing,the time-dependent analysis of the transition B s→K+K?with the expected branching ratio at the O(10?5)level may be di?cult as well.

In this letter we would like to propose a di?erent method of extractingα.In or-

der to eliminate the penguin contributions,we use time-dependent measurements of

the modes B d→π+π?and B d→K0ˉK0yielding the corresponding mixing-induced CP-violating asymmetries and employ the SU(3)?avour symmetry of strong inter-actions[23]-[27]to derive relations among the corresponding decay amplitudes.The transition B d→K0ˉK0is–in contrast to B s→K+K?–a pure penguin-induced mode with a branching ratio O(10?6)[21,28].Yet because of smaller B0d–ˉB0d–mixing, time-dependent studies of this channel may probably be easier for experimentalists than those of the decay B s→K+K?.As we will see in a moment,our approach is essentially una?ected by electroweak penguins.

In the previous literature it has been claimed by several authors that the Standard Model predicts vanishing CP-violating asymmetries for decays such as B d→K S K S or B d→K0ˉK0(the CP asymmetries of both channels are equal)because of the cancellation of weak decay-and mixing-phases(see e.g.refs.[3,7,8]).This result is however only correct,if theˉb→ˉd QCD-penguin amplitudes are dominated by internal top-quark exchanges.As has been pointed out in refs.[28,29],QCD-penguins with internal up-and charm-quarks may generally also play a signi?cant role and in the case of B d→K0ˉK0could lead to rather large CP asymmetries of O(10?50)% [28].Unfortunately,these asymmetries su?er from large hadronic uncertainties and

are therefore not related to CKM-angles in a clean way.Nevertheless,A mix-ind

(B d→K0ˉK0)may be combined with additional inputs to determineαin a clean way as we will demonstrate in this letter.

In our discussion it is convenient to use the description of B→P P decays given by Gronau,Hern′a ndez,London and Rosner in refs.[19]and[30]-[35].Using the same notation as these authors,the B0d→π+π?and B0d→K0ˉK0decay amplitudes take the form

A(B0

d→π+π?)=? (T+E)+(P+P A)+c u P C EW

A(B0d→K0ˉK0)= (P+P A+P3)+c s P C EW ,(1) where T and E describeˉb→ˉu uˉd colour-allowed tree-level and exchange amplitudes, respectively,P denotesˉb→ˉd QCD-penguins,P A is related to QCD-penguin annihi-lation diagrams and P C EW to colour-suppressedˉb→ˉd electroweak penguins.The term P3describes SU(3)-breaking e?ects that are introduced through the creation of a sˉs pair in theˉb→ˉd QCD-penguin diagrams[35].If we follow the plausible arguments of Gronau et al.outlined in[19,35],we expect the following hierarchy of the di?erent topologies present in(1):

1:|T|

O(ˉλ):|P|

O(ˉλ2):|E|,|P3| O(ˉλ3):|P A|, P C EW .(2)

Note that the parameterˉλ=O(0.2)appearing in these relations is not related to the usual Wolfenstein parameterλ.It has been introduced by Gronau et al.just to keep track of the expected orders of magnitudes.We have named this quantityˉλin order not to confuse it with Wolfenstein’sλ.

Consequently,if we neglect the terms of O(ˉλ3),we obtain

A(B0d→π+π?)=?[(T+E)+P]

A(B0

d→K0ˉK0)=P+P3.(3)

Within this approximation,terms of O(ˉλ4),i.e.SU(3)-breaking corrections to the P A and P C EW amplitudes,which have not been written explicitly in(1),have also to be neglected.

Rotating theˉB0d→π+π?andˉB0d→K0ˉK0amplitudes by the phase factor e?2iβ, we?nd

e?2iβA(ˉB0d→π+π?)=? e2iα(T+E)+e?2iβˉP

e?2iβA(ˉB0d→K0ˉK0)=e?2iβ ˉP+ˉP3 ,(4) where we have used the relation

e?2iβ(ˉT+ˉE)=e?2i(β+γ)(T+E)=e2iα(T+E).(5) Using(3)and(4)it is an easy exercise to eliminate P andˉP and to derive the following relations:

A(B0

d→K0ˉK0)+(T+E)?P3+A(B0d→π+π?)=0(6) e?2iβA(ˉB0d→K0ˉK0)+e2iα(T+E)?e?2iβˉP3+e?2iβA(ˉB0d→π+π?)=0,(7) which have been represented graphically in the complex plane in Fig.1.If theˉb→ˉd QCD-penguins were dominated by internal top-quark exchanges,we would have e?2iβˉP3=P3.However,as has been shown in refs.[28,29],QCD-penguins with internal up-and charm-quarks are expected to lead to sizable corrections to this relation.

The anglesψandφappearing in Fig.1can be determined directly by measuring the mixing-induced CP asymmetries of the decays B d→K0ˉK0and B d→π+π?, respectively,which are given by[20]

A mix-ind

CP (B d→K0ˉK0)=?2|A(

ˉB0

d→K0ˉK0)||A(B0d→K0ˉK0)|

|A(ˉB0d→π+π?)|2+|A(B0d→π+π?)|2sinφ(9)

and enter the formulae for the corresponding time-dependent CP asymmetries in the following way:

a CP(t)≡

Γ(B0d(t)→f)?Γ(ˉB0d(t)→f)

Here,A dir CP (B d →f )describes direct CP violation and is given by

A dir CP (

B d →f )=|A (B 0d →f )|2?|A (ˉB 0d

→f )|22|P 3|+|ˉP 3|

where we have introduced the quantities a,b and c through

a≡ˉAˉB cosφ?AB cosψ(18)

b≡ˉAˉB sinφ?AB sinψ(19)

c≡

a2+b2?c2

a2+b2?c2(21) and?xes tanσup to a two-fold ambiguity corresponding to“+”and“?”,respectively. Consequently,σcan be determined up to a four-fold ambiguity.Note that there would be no ambiguity in tanσin the special cases c=0,which corresponds to the limit of no direct CP violation in the decays B d→π+π?and B d→K0ˉK0,and a2+b2?c2=0. The anglesψandφdetermined by using(8)and(9),respectively,su?er also from two-fold ambiguities which are a characteristic feature of the determination of angles by using CP-violating asymmetries or amplitude relations.Taking into account additional information from other processes,it should be possible to exclude certain solutions and to resolve these ambiguities.In particular the future knowledge of the shape of the unitarity triangle obtained from loop induced transitions(see e.g.[36])should be useful in this respect.

Usingσdetermined by means of eq.(21),both the angleαand the quantity|T+E| can be extracted in the limit of vanishing SU(3)-breaking,i.e.P3=ˉP3=0,as can be seen from Fig.1.One could easily generalize the equations above by including the e?ect of P3andˉP3.This would modifyσand consequentlyαby corrections of O(ˉλ2). Due to the lack of knowledge of the exact values of P3andˉP3this generalization would not improve the accuracy of our method at present.

Consequently,combining all these considerations(see also eq.(13)),we expect the uncertainty in the determination ofαin our approach to be of O(ˉλ2).It should be stressed–as has already been done in refs.[19,35]–that this estimate should not be taken too literally sinceˉλ=O(0.2)is not a small number.Therefore,in practice the accuracy of our approach may well be of O(ˉλ2±1).In order to control it in a quantitative way,we have to deal with the SU(3)-breaking contributions P3andˉP3which is beyond the scope of this letter.In this respect the O(ˉλ2)electroweak penguin uncertainty a?ecting the determination ofαin the Gronau-London-method[11]is in better shape as we have shown in ref.[20].Performing measurements of the branching ratios of certain B→πK channels,which are expected to be of O(10?5),these electroweak penguin e?ects can be determined in principle.

In summary we have presented a determination of the CKM-angleαby using mixing-induced CP violation in the decays B d→π+π?and B d→K0ˉK0.Interestingly enough,the accuracy of our method,which is limited by SU(3)-breaking e?ects related to the creation of sˉs pairs inˉb→ˉd QCD-penguin processes,is expected to be of the same order inˉλ,i.e.O(ˉλ2),as the one arising from electroweak penguins in the original B→ππapproach of Gronau and London.As we stated above,the electroweak penguin uncertainties in the latter method can be brought under control as demonstrated in ref.[20],whereas this is not the case of the O(ˉλ2)SU(3)-breaking e?ects present in the method described here.Despite of this our method may be an interesting alternative to determine the CKM-angleαin a rather clean way.An advantage of our approach is the fact that it does not involve a measurement of the decay B d→π0π0which is considered to be di?cult.However,we need instead a time-dependent analysis of the pure penguin-induced mode B d→K0ˉK0.Experimentalists will?nd out which method can be performed easier in practice.It is needless to say that a comparison of αdeterminations by means of these two methods would give another test of the CKM picture of CP violation.

A.J.

B.would like to thank Iris Abt for illuminating discussions.

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Figure Caption

Fig.1:A di?erent strategy for determining the CKM-angleα.

Figure1:

This figure "fig1-1.png" is available in "png" format from: https://www.wendangku.net/doc/0316280864.html,/ps/hep-ph/9507460v1