CP VIOLATION AND CKM PHASES FROM TIME-DEPENDENCES OF UNTAGGED B s DECAYS a ROBERT FLEISCHER

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1 hep-ph/ September 1996 CP VIOATION AND CKM PASES FROM TIME-DEPENDENCES OF UNTAGGED B s DECAYS a ROBERT FEISCER Institut ür Theoretische Teilchenphysik, Universität Karlsruhe, D Karlsruhe, Germany The B s system is analyzed in light o a possible width dierence Γ s between its mass eigenstates. I Γ s is sizable, untagged B s-meson decays may allow a probe o CP violation and moreover the extraction both o the Wolenstein parameter η and o the notoriously diicult to measure angle γ o the unitarity triangle. To accomplish this ambitious task, time-dependent angular distributions or untagged B s decays into admixtures o CP eigenstates and channels that are caused by b cu s quark-level transitions play a key role. The work described here was done in collaboration with Isard Dunietz. 1 Introduction The time-evolution due to Bs 0 B0 s mixing is governed by the B s mass eigenstates Bs eavy and Bs ight which are characterized by their mass eigenvalues M (s), M (s) and decay widths Γ (s), Γ(s). Because o these mixing eects, oscillatory M s t terms with M s M (s) M (s) show up in the time-dependent transition rates 1 Γ(Bs 0 (t) ) and Γ(Bs(t) 0 ) describing decays o initially present Bs 0 and Bs 0 mesons into a inal state, respectively. The strength o the Bs 0 B0 s oscillations is measured by the mixing parameter x s M s /Γ s,whereγ s (Γ (s) +Γ(s) )/2. Within the Standard Model one expects 2 x s = O(20) implying very rapid Bs 0 B0 s oscillations which require an excellent vertex resolution system to keep track o the M s t terms. That is obviously a ormidable experimental task. owever, as pointed out by Dunietz 3,itmay not be necessary to trace the rapid M s t oscillations in order to obtain insights into the mechanism o CP violation. This remarkable eature is due to the expected sizable width dierence 4 Γ s Γ (s) Γ(s). The major contributions to Γ s, which may be as large as O(20%) o the average decay width Γ s, originate rom b cc s transitions into inal states that are common both to Bs 0 and B0 s. Because o this width dierence already untagged B s rates, which are deined by Γ[(t)] Γ(B 0 s (t) )+Γ(B 0 s(t) ), (1) may provide valuable inormation about the phase a Invited talk given at the XXVIII International Conerence on igh Energy Physics ICEP 96, Warsaw, Poland, July 25 31, 1996, to appear in the proceedings. structure o the observable ξ (s) ( ) =exp iθ (s) A(B 0 s ) M 12 A(Bs 0 ), (2) where Θ (s) M 12 is the weak Bs B 0 s 0 mixing phase 1. This can be seen nicely by writing Eq. (1) in a more explicit way as ollows: [( Γ[(t)] 1+ ξ (s) 2) ( e Γ(s) t +e Γ(s) t) 2Reξ (s) (e Γ(s) t e Γ(s) t)]. (3) In this expression the rapid oscillatory M s t terms,whichshowupinthetagged rates, cancel 3. Thereore it depends only on the two exponents e Γ(s) t and e Γ(s) t,whereγ (s) and Γ(s) can be determined e.g. rom the angular distribution 5 o the decay B s J/ψφ. From an experimental point o view such untagged analyses are clearly much more promising than tagged ones in respect o eiciency, acceptance and purity. 2 A Transparent Example In order to illustrate these untagged rates in more detail, let me discuss an estimate o the angle γ o the usual non-squashed unitarity triangle 6 o the Cabibbo Kobayashi Maskawa matrix 7 (CKM matrix) using untagged B s K + K and B s K 0 K 0 decays. This approach has been proposed very recently by Dunietz and mysel 8.Using the SU(2) isospin symmetry o strong interactions to relate the QCD penguin contributions to these decays (electroweak penguins are color-suppressed

2 in these modes and thus play a minor role), we obtain Γ[K + K (t)] P 2[ ( 1 2 r cos ρ cos γ + r 2 cos 2 γ ) e Γ(s) t + r 2 sin 2 γe Γ(s) t] (4) and Γ[K 0 K 0 (t)] P 2 e Γ(s) t, (5) where r r e iρ = T P ei(δ T δ P ). (6) ere P denotes 9 the b s QCD penguin amplitude, T is the color-allowed b ūu s tree amplitude, and δ P and δ T are the corresponding CP-conserving strong phases. In order to determine γ rom the untagged rates Eqs. (4) and (5) we need an additional input that is provided by the SU(3) lavor symmetry o strong interactions. I we neglect the color-suppressed current-current contributions to B + π + π 0 we ind 9 T λ K 2 A(B + π + π 0 ), (7) π where λ is the Wolenstein parameter 10, K and π are the K and π meson decay constants, respectively, and A(B + π + π 0 ) denotes the appropriately normalized B + π + π 0 decay amplitude. Since P is known rom B s K 0 K 0,thequantity r = T / P can be estimated with the help o Eq. (7) and allows the extraction o γ rom the part o Eq. (4) evolving with the exponent e Γ(s) t. 3 B s Decays into Admixtures o CP Eigenstates As we will see in a moment, one can even do better than in the previous section, i.e. without using an SU(3) lavor symmetry input, by considering the decays corresponding to B s KK where two vector mesons (or higher resonances) are present in the inal states An Extraction o γ using Untagged B s K + K and B s K 0 K 0 Decays The untagged angular distributions o these decays, which take the general orm 11 [(θ, φ, ψ; t)] = [ ] b (k) (t)+b (k) (t) g (k) (θ, φ, ψ), k (8) provide many more observables than the untagged modes B s K + K and B s K 0 K 0 discussed in Section 2. ere θ, φ and ψ are generic decay angles describing the kinematics o the decay products arising in the decay chain B[ s K ( πk) ] K ( πk). The observables b (k) (t)+b (k) (t) governing the time-evolution o the angular distribution Eq. (8) are given by real or imaginary parts o bilinear combinations o decay amplitudes that are o the ollowing structure: [ ] (K A (t) A (t) K ) e B s (t) (9) (K K ) e B s (t) + ( ) B s B s. In this expression and are labels that deine the relative polarizations o K and K in inal state conigurations ( K K ) (e.g. linear polarization states 12 {0,, }) with CP eigenvalues ηcp: (CP) ( K K ) = η CP ( K K ). (10) An analogous relation holds or. The observables o the angular distributions or B s K + K and B s K 0 K 0 are given explicitly in Re. 8.In the case o the latter decay the ormulae simpliy considerably since it is a penguin-induced b sd d mode and receives thereore no tree contributions. Using as in Section 2 the SU(2) isospin symmetry o strong interactions, the QCD penguin contributions o these decays can be related to each other. I one takes into account these relations and goes very careully through the observables o the angular distributions, one inds that they allow the extraction o the CKM angle γ without any additional theoretical input 8. In particluar no SU(3) symmetry arguments as in Section 2 are needed. The angular distributions provide moreover inormation about the hadronization dynamics o the corresponding decays, and the ormalism 8 developed or B s K + K applies also to B s ρ 0 φ i we perorm a suitable replacement o variables. Since that channel is expected to be dominated by electroweak penguins 13, it may allow interesing insights into the physics o these operators. 3.2 The Gold-plated Transitions to Extract η This subsection is devoted to an analysis 8 o the untagged decays B s Ds + Ds and B s J/ψφ, 2

3 which is the counterpart o the gold-plated mode B d J/ψK S to measure the angle β o the unitarity triangle. These decays are dominated by a single CKM amplitude. Consequently the hadronic uncertainties cancel in the quantity ξ (s) deined by Eq. (2), which takes in that particular case the orm ξ (s) =exp(iφ CKM ), (11) and the observables o the angular distributions simpliy considerably. A characteristic eature o these angular distributions is intererence between CP-even and CP-odd inal state conigurations leading to observables that are proportional to (e Γ(s) t e Γ(s) t) sin φ CKM. (12) ere the CP-violating weak phase is given by 2 φ CKM =2λ 2 η O(0.03), where the Wolenstein parameter η ixes the height o the unitarity triangle 6. The observables o the angular distributions 8 or both the color-allowed channel B s Ds + Ds and the color-suppressed transition B s J/ψφ each provide suicient inormation to determine the CP-violating weak phase φ CKM rom their untagged data samples thereby ixing the Wolenstein parameter η. The extraction o φ CKM is not as clean as that o β rom B d J/ψK S. This is due to the smallness o φ CKM with respect to β enhancing the importance o the unmixed amplitudes proportional to the CKM actor Vub V us which are similarly suppressed in both cases. 4 B s Decays caused by b cu s The B s decays discussed in this section are pure tree decays and probe the CKM angle γ in a clean way 14. There are by now well-known strategies onthemarketusingthetimeevolutionsosuch modes, e.g. B s D 0 φ 14,15 and B s D s ± K 16, to extract γ. owever, in these strategies tagging is essential and the rapid M s t oscillations have to be resolved which is an experimental challenge. The question what can be learned rom untagged data samples o these decays, where the M s t terms cancel, has been investigated by Dunietz in Re. 3. In the untagged case the determination o γ requires additional inputs: a measurement o the untagged B s DCPφ 0 rate in the case o the color-suppressed modes B s D 0 φ, and a theoretical input corresponding to the ratio o the unmixed rates Γ(Bs 0 Ds K + )/Γ(Bs 0 Ds π + )in the case o the color-allowed decays B s D s ± K. This ratio can be estimated with the help o the actorization hypothesis which may work reasonably well or these color-allowed channels. Interestingly the untagged data samples may exhibit CP-violating eects that are described by observables o the orm Γ[(t)] Γ[(t)] (e Γ(s) t e Γ(s) t) sin ϱ sin γ. (13) ere ϱ is a CP-conserving strong phase shit and γ is the usual angle o the unitarity triangle. Because o the sin ϱ actor, a non-trivial strong phase shit is essential in that case. Consequently the CP-violating observables Eq. (13) vanish within the actorization approximation predicting ϱ {0,π}. Since actorization may be a reasonable working assumption or the color-allowed modes B s D s ± K, the CP-violating eects in their untagged data samples are expected to be very small. On the other hand, the actorization hypothesis is very questionable or the color-suppressed decays B s D 0 φ and sizable CP violation may show up in the corresponding untagged rates 3. Concerning such CP-violating eects and the extraction o γ rom untagged rates, the decays B s Ds ± K and B s D 0 φ are expected to be more promising than the transitions discussed above. As was shown in Re. 17,thetimedependences o their untagged angular distributions allow a clean extraction o the CKM angle γ without any additional input. The inal state conigurations o these decays are not admixtures o CP eigenstates as in Section 3. They can, however, be classiied by their parity eigenvalues. A characteristic eature o the angular distributions are intererences between parity-even and parity-odd conigurations that may lead to potentially large CP-violating eects in the untagged data samples even when all strong phase shits vanish. An example o such an untagged CP-violatingobservable is the ollowing quantity 17 : Im {[ A (t) A (t) ]} +Im {[ A C (t) A C (t) ]} (e Γ(s) t e Γ(s) t) { R cos(δ ϑ ) 3

4 + R cos(δ ϑ ) } sin γ. (14) In that expression bilinear combinations o certain decay amplitudes (see Eq. (9)) show up, {0, } denotes a linear polarization state 12 and δ, ϑ are CP-conserving phase shits that are induced through strong inal state interaction eects. For the details concerning the observable Eq. (14) in particular the deinition o the relevant charge-conjugate amplitudes A C and the quantities R the reader is reerred to Re. 17. ere I would like to emphasize only that the strong phase shits enter in the orm o cosine terms. Thereore non-trivial strong phases are in contrast to Eq. (13) not essential or CP violation in the corresponding untagged data samples and one expects, even within the actorization approximation, which may apply to the color-allowed modes B s Ds ± K, potentially large eects. Since the sot photons in the decays Ds D s γ, D 0 D 0 γ are diicult to detect, higher resonances exhibiting signiicant all-charged inal states, e.g. D s1 (2536) + D + K 0, D 1 (2420) 0 D + π with D + D 0 π +,maybemorepromising or certain detector conigurations. A similar comment applies also to the mode B s Ds + D s discussed in Subsection Conclusions The oscillatory M s t terms arising rom Bs 0 Bs 0 mixing, which may be too rapid to be resolved with present vertex technology, cancel in untagged rates o B s decays that depend thereore only on the two exponents e Γ(s) t and e Γ(s) t. I the width dierence Γ s is sizable as is expected rom theoretical analyses untagged B s decays may allow the determination both o the CKM angle γ and o the Wolenstein parameter η and may urthermore provide valuable insights into the mechanism o CP violation and the hadronization dynamics o the corresponding decays. To this end certain angular distributions may play a key role. Compared to the tagged case, such untagged measurements are much more promising in view o eiciency, acceptance and purity. A lot o statistics is required, however, and the natural place or these experiments seems to be a hadron collider. Obviously the easibility o untagged strategies to extract CKM phases depends crucially on a sizable width dierence Γ s. Even i it should turn out to be too small or such untagged analyses, once Γ s 0 has been established experimentally, the ormulae developed in Res. 8,17 have also to be used to determine CKM phases correctly rom tagged measurements. In this sense we cannot lose and an exciting uture concerning B s decays may lie ahead o us! Acknowledgment I would like to thank Isi Dunietz or the most pleasant and enjoyable collaboration on the topics presented in this talk. Reerences 1. I. Dunietz and J.. Rosner, Phys. Rev. D34 (1986) 1404; I. Dunietz, Ph. D. thesis, Ann. Phys. 184 (1988) 350; M. Gronau, Phys. Rev. ett. 63 (1989) A. Ali and D. ondon, DESY , hepph/ ; A.J. Buras, Nucl. Instr. and Meth. in Phys. Res. A368 (1995) I. Dunietz, Phys. Rev. D52 (1995) For a very recent analysis o Γ s and other reerences see M. Beneke, G. Buchalla and I. Dunietz, hep-ph/ ; M. Beneke, these proceedings, hep-ph/ A.S. Dighe, I. Dunietz,.J. ipkin and J.. Rosner, Phys. ett. B369 (1996) Chau and W.-Y. Keung, Phys. Rev. ett. 53 (1984) 1802; C. Jarlskog and R. Stora, Phys. ett. B208 (1988) N. Cabibbo, Phys. Rev. ett. 10 (1963) 531; M. Kobayashi and T. Maskawa, Progr. Theor. Phys. 49 (1973) R. Fleischer and I. Dunietz, TTP96-07, hepph/ M. Gronau et al., Phys. Rev. D52 (1995) Wolenstein, Phys. Rev. ett. 51 (1983) This notation is due to Amol Dighe, private communication. 12. J.. Rosner, Phys. Rev. D42 (1990) R. Fleischer, Phys. ett. B332 (1994) 419; N.G. Deshpande, X.-G. e and J. Trampetic, Phys. ett. B345 (1995) 547; M. Gronau et al., Phys. Rev. D52 (1995) R. Aleksan et al., hep-ph/

5 15. M. Gronau and D. ondon, Phys. ett. B253 (1991) R. Aleksan, I. Dunietz and B. Kayser, Z. Phys. C54 (1992) R. Fleischer and I. Dunietz, TTP96-08, hepph/