New Strategies to Extract Weak Phases from Neutral B Decays

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CERN-TH/2003-241hep-ph/0310081New Strategies to Extract Weak Phases from Neutral B Decays Robert Fleischer Theory Division,CERN,CH-1211Geneva 23,Switzerland Abstract We discuss new,theoretically clean strategies to determine the angle γof the uni-tarity triangle from B d →DK S(L),B s →Dη(′),Dφ,...decays,and point out that the B s →DK S(L)and B d →Dπ0,Dρ0,...modes allow very interesting determi-nations of the B 0q –

1Introduction

The time-dependent CP asymmetries of neutral B q-meson decays(q∈{d,s})into CP

eigenstates,which satisfy(CP)|f =±|f ,provide valuable information

[1]:

Γ(B0q(t)→f)?Γ(

Γ(B0q(t)→f)+Γ(

cosh(?Γq t/2)?A?Γsinh(?Γq t/2)

.(1) Here the CP-violating observables

A dir CP≡1?|ξ(q)f|2

1+|ξ(q)f|2

(2)

originate from“direct”and“mixing-induced”CP violation,respectively,and are gov-erned by

ξ(q)f=?e?iφq A(A(B0q→f) ,(3) where

φq SM=2arg(V?tq V tb)= +2β(q=d)

?2λ2η(q=s)

(4) is the CP-violating weak B0q–

B0q(t)→f)

∝[cosh(?Γq t/2)?A?Γsinh(?Γq t/2)]e?Γq t.(5) 2B d→DK S(L),B s→Dη(′),Dφ,...and B s→DK S(L),B d→Dπ0,Dρ0,...

Let us consider in this section B0q→D0f r transitions,where r∈{s,d}distinguishes between b→Ds and b→Dd processes[2,3].If we require(CP)|f r =ηf r CP|f r ,B0q and B0q mixing and decay processes,which involve the weak phaseφq+γ:

?For r=s,i.e.B d→DK S(L),B s→Dη(′),Dφ,...,these e?ects are governed by a hadronic parameter x f

s

e iδ

f s∝R b≈0.4,and are hence favourably large.

?For r=d,i.e.B s→DK S(L),B d→Dπ0,Dρ0...,these e?ects are tiny because of x f

d

e iδ

f d∝?λ2R b≈?0.02.

2.1B d→DK S(L),B s→Dη(′),Dφ,...

Let us?rst focus on r=s.If we make use of the CP eigenstates D±of the neutral D-meson system satisfying(CP)|D± =±|D± ,we obtain additional interference e?ects between B0q→D0f s and B0q→

B0q(t)→D±f s)

?Γq=0

= Γ(B0q→D±f s)+Γ(

Γ(B q→D+f s) + Γ(B q→D?f s)

.(7) Interestingly,already this quantity o?ers valuable information onγ,since bounds on this angle are implied by

|cosγ|≥|Γf s+?|.(8)

Moreover,if we?x the sign of cosδf

s

with the help of the factorization approach,we obtain

sgn(cosγ)=?sgn(Γf s+?),(9) i.e.we may decide whetherγis smaller or larger than90?.If we employ,in addition,the

mixing-induced observables S f s±≡A mix

CP (B q→D±f s),we may determineγ.To this end,

it is convenient to introduce the quantities

S f

s ±≡

S f s+±S f s?

Γf s+?

+[ηf s S f s ??sinφq],(11)

whereηf

s

≡(?1)Lηf s CP,with L denoting the Df s angular momentum[2].If we use this simple–but exact–relation,we obtain the twofold solutionγ=γ1∨γ2,with γ1∈[0?,180?]andγ2=γ1+180?.Since cosγ1and cosγ2have opposite signs,(9)allows us to?xγunambiguously.Another advantage of(11)is that S f

s

+andΓf s+?are both proportional to x f

s

≈0.4,so that the?rst term in square brackets is of O(1),whereas the

second one is of O(x2f

s ),hence playing a minor r?o le.In order to extractγ,we may also

employ D decays into CP non-eigenstates f NE,where we have to deal with complications originating from D0,

2.2B s→DK S(L),B d→Dπ0,Dρ0,...

The r=d case also has interesting features.It corresponds to B s→DK S(L),B d→Dπ0,Dρ0...decays,which can be described through the same formulae as their r=s

counterparts.Since the relevant interference e?ects are governed by x f

d ≈?0.02,these

channels are not as attractive for the extraction ofγas the r=s modes.On the other hand,the relation

ηf

d S f

d

?=sinφq+O(x2f

d

)=sinφq+O(4×10?4)(12)

o?ers very interesting determinations of sinφq[2].Following this avenue,there are no penguin uncertainties,and the theoretical accuracy is one order of magnitude better than in the“conventional”B d→J/ψK S,B s→J/ψφstrategies.In particular,φSM s=?2λ2ηcould,in principle,be determined with a theoretical uncertainty of only O(1%), in contrast to the extraction from the B s→J/ψφangular distribution,which su?ers from generic penguin uncertainties at the10%level.

3B s→D(?)±

s

K?,...and B d→D(?)±π?,...

Let us now consider the colour-allowed counterparts of the B q→Df q modes discussed above,which we may write generically as B

q

→D q

B0q meson may decay into D q

B0q mixing and decay processes,which involve the weak phase φq+γ:

?In the case of q=s,i.e.D s∈{D+s,D?+s,...}and u s∈{K+,K?+,...},these e?ects are favourably large as they are governed by x s e iδs∝R b≈0.4.

?In the case of q=d,i.e.D d∈{D+,D?+,...}and u d∈{π+,ρ+,...},the interference e?ects are described by x d e iδd∝?λ2R b≈?0.02,and hence are tiny.

We shall only consider B q→D q u q states is a pseudoscalar meson;otherwise a complicated angular analysis has to be performed.

It is well known that such decays allow determinations of the weak phasesφq+γ, where the“conventional”approach works as follows[6,7]:if we measure the observables C(B q→D q D q u q)≡

u q)≡S q and S(B q→S q associated with the sin(?M q t)

terms of the time-dependent rate asymmetries must be measured,where it is convenient to introduce

.(13)

S q ±≡

2

If we assume that x q is known,we may consider

s+≡(?1)L 1+x2q

2x q S q ?=?sinδq cos(φq+γ),(15) yielding

1+s2+?s2?(1+s2+?s2?)2?4s2+

sin2(φq+γ)=

u q angular momentum, has to be properly taken into account.

Let us now discuss the new strategies to explore the B q→D q

u q) = Γ(B q→D q

u q)sinh(?Γq t/2)]

and their CP conjugates provide A?Γ(B s→D s D s u s)≡

A?Γs ?

A?Γs + =+

u q) + Γ(B q→

which can straightforwardly be converted into bounds on φq +γ.If x q is known,stronger constraints are implied by

|sin(φq +γ)|≥|s +|,|cos(φq +γ)|≥|s ?|.(20)

Once s +and s ?are known,we may of course determine φq +γthrough the “conventional”approach,using (16).However,the bounds following from (20)provide essentially the same information and are much simpler to implement.Moreover,as discussed in detail in

[5]for several examples,the bounds following from the B s and B d modes may be highly complementary,thereby providing particularly narrow,theoretically clean ranges for γ.

Let us now further exploit the complementarity between the B 0s →D (?)+s K ?and

B 0d →D (?)+π?modes.If we look at their decay topologies,we observe that these channels are related to each other through an interchange of all down and strange quarks.Consequently,the U -spin ?avour symmetry of strong interactions implies a s =a d and δs =δd ,where a s =x s /R b and a d =?x d /(λ2R b )are the ratios of hadronic matrix elements entering x s and x d ,respectively.There are various possibilities to implement these relations.A particularly simple picture emerges if we assume that a s =a d and δs =δd ,which yields

tan γ=?

sin φd ?S sin φs cos φd ?S .(21)Here we have introduced

S =?R S d +

λ2

1f π 2 Γ(

Γ(B s →D (?)+s K ?) + Γ(B s →D (?)?s K +) .(24)

Alternatively,we may only assume that δs =δd or that a s =a d .Apart from features related to multiple discrete ambiguities,the most important advantage with respect to the “conventional”approach is that the experimental resolution of the x 2q terms is not required.In particular,x d does not have to be ?xed,and x s may only enter through a 1+x 2s correction,which can straightforwardly be determined through untagged B s rate measurements.In the most re?ned implementation of this strategy,the measurement of x d /x s would only be interesting for the inclusion of U -spin-breaking e?ects in a d /a s .Moreover,we may obtain interesting insights into hadron dynamics and U -spin-breaking e?ects.

4Conclusions

We have discussed new strategies to explore CP violation through neutral B q decays. In the?rst part,we have shown that B d→DK S(L),B s→Dη(′),Dφ,...modes pro-vide theoretically clean,e?cient and unambiguous extractions of tanγif we combine an “untagged”rate asymmetry with mixing-induced observables.On the other hand,their B s→D±K S(L),B d→D±π0,D±ρ0,...counterparts are not as attractive for the determi-nation ofγ,but allow extremely clean extractions of the mixing phasesφs andφd,which may be particularly interesting for theφs case.In the second part,we have discussed

interesting new aspects of B s→D(?)±

s K?,...and B d→D(?)±π?,...decays.The observ-

ables of these modes provide clean bounds onφq+γ,where the resulting ranges forγmay be highly complementary in the B s and B d cases,thereby yielding stringent constraints onγ.Moreover,it is of great advantage to combine the B d→D(?)±π?modes with

their U-spin counterparts B s→D(?)±

s K?,allowing us to overcome the main problems of

the“conventional”strategies to deal with these modes.We strongly encourage detailed feasibility studies of these new strategies.

References

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