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ψφ decays
6 August 1998
Physics Letters B 433 Ž1998. 147–149
Resolving a discrete ambiguity in the CKM angle b through Bu,d ™ Jrc K ) and Bs ™ Jrc f decays Amol S. Dighe a , Isard Dunietz b, Robert Fleischer
c
a
b
The Abdus Salam International Centre for Theoretical Physics, 34100 Trieste, Italy Theoretical Physics DiÕision, Fermi National Accelerator Laboratory, BataÕia, IL 60510, USA c Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Received 15 April 1998 Editor: R. Gatto
Abstract It is well known that sinŽ2 b ., where b is one of the angles of the unitarity triangle of the CKM matrix, can be determined in a theoretically clean way by measuring mixing-induced CP violation in the decay Bd ™ Jrc K S . Another clean extraction of this CKM angle is provided by the time-dependent angular distribution for the decay products of Bd ™ Jrc Ž™ lqly. K ) 0 Ž™ p 0 K S ., where we have more observables at our disposal than in the case of Bd ™ Jrc K S , so that in addition to sinŽ2 b . also cosŽ2 b . can be probed in a direct way. Unfortunately a sign ambiguity remains in cosŽ2 b .. If it could be resolved, a discrete ambiguity in the extraction of the CKM angle b could be resolved as well, which would allow a more incisive test of the CKM model of CP violation. This note shows that detailed time-dependent studies of Bu, d ™ Jrc K ) and Bs ™ Jrc f decay processes can determine the sign of cosŽ2 b ., thereby removing the corresponding ambiguity in the extraction of the CKM angle b . q 1998 Elsevier Science B.V. All rights reserved.
The conventional methods for determining the angles a , b and g of the usual unitarity triangle w1x of the Cabibbo–Kobayashi–Maskawa matrix ŽCKM matrix. w2x leave several discrete ambiguities w3x. This is also the case for the ‘‘gold-plated’’ mode Bd ™ Jrc K S . The mixing-induced CP asymmetry arising in this channel allows only a theoretically clean determination of sinŽ2 b ., so that a discrete four-fold ambiguity for the extracted value of b g w08,3608x remains. In the recent literature, several strategies were proposed to resolve ambiguities of this kind w4x. Another clean probe of the CKM angle b is provided by the observables of the angular distribu-
tions for the decay products of Bd ™ Jrc Ž™ lq ly . K ) 0 Ž™ p 0 K S . modes w5–7x. Such observables can in general be expressed in terms of decay amplitudes as < Af Ž t. <2, Im
Re
A)f˜ Ž t . A f Ž t . ,
A)f˜ Ž t . A f Ž t . ,
Ž 1.
where f and f˜ are labels for specific final-state configurations. The full three-angle distributions for tagged Bd Ž t . ™ Jrc K ) 0 Ž™ p 0 K S . decays are given in w6–8x. Throughout this note, by tagging we
0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 8 0 - 7
148
A.S. Dighe et al.r Physics Letters B 433 (1998) 147–149
mean making the distinction of initially, i.e. at t s 0, present unmixed Bd,0 s and Bd0, s mesons. Weighting functions have been derived to extract the corresponding observables in an efficient way from experimental data w6,7x. The time evolution of the interference terms in Ž1., i.e. of the real and imaginary parts of bilinear combinations of certain decay amplitudes, allows the determination w7x of sinŽ d 1,2 . and cosŽ d 1 y d 2 ., where d 1 and d 2 are CP-conserving strong phases, and of sinŽ2 b . and
phases dˆ1,2 are the flavour SUŽ3. counterparts of d 1,2 . In the strict SUŽ3. limit, we have dˆ1,2 s d 1,2 . The time evolution of these observables takes the following form w7x:
cos Ž d 1,2 . cos Ž 2 b . .
where terms of O Ž f 2 . have been neglected, GH , GL denote the decay widths of the Bs mass eigenstates, G' Ž GH q GL .r2, and k s 1,2. Consequently, the strong phases dˆ1,2 can be determined unambiguously by resolving the rapid D m s t oscillations in Ž3.. Comparing the resulting values for sinŽ dˆ1,2 . and cosŽ dˆ1 y dˆ2 . with their unhatted analogues, which can be determined from the Bu, d ™ Jrc K ) observables, we obtain valuable information on SUŽ3. breaking. In order to fix the sign of cosŽ2 b . with the help of Ž2., we just need the sign of cosŽ d 1 . or cosŽ d 2 ., which is provided by the sign of cosŽ dˆk . determined from the Bs Ž t . ™ Jrc f observables. The SUŽ3. flavour symmetry should work reasonably well to determine this sign, unless
Ž 2.
The CP-conserving observables < A f Ž t s 0.<, sinŽ d 1,2 . and cosŽ d 1 y d 2 . can be determined to a higher accuracy from the much larger data samples arising for B "™ Jrc K ) " transitions, and untagged Bd decays into Jrc K ) 0 Ž™ Kq py . and JrcK ) 0Ž™ Ky pq . states w6,7x. At first sight, one may think that sinŽ d 1,2 . and cosŽ d 1 y d 2 . extracted this way from the Bu, d ™ Jrc K ) angular distributions will allow the determination of cosŽ2 b . with the help of the terms given in Ž2.. A closer look shows, however, that this is unfortunately not the case, since we do not have sufficient information to fix the signs of cosŽ d 1,2 ., thereby leaving a sign ambiguity for cosŽ2 b .. The purpose of this letter is to point out that this ambiguity can be resolved with the help of tagged, time-dependent studies of Bs ™ Jrc f decays. The angular distributions are given in w7,8x, and weighting functions to extract the observables from experimental data can be found in w7x. An important feature of these observables is that they allow the determination of a CP-violating weak phase f w7,9x, which takes a very small value, of O Ž0.03., within the Standard Model, and represents a sensitive probe for new-physics contributions to Bs0 – Bs0 mixing. Provided there is a sizeable mass difference between the mass eigenstates BsH and BsL , this phase can even be extracted from untagged Bs data samples w10x, where the rapid D m s t oscillations cancel w11x. Another important feature is the fact that the Žy .
tagged, time-dependent B s Ž t . ™ Jr c Ž ™ q y. q y. Ž l l f ™ K K observables corresponding to the ‘‘Im’’ terms in Ž1. provide sufficient information to determine dˆ1,2 unambiguously, where the strong
ey Gt sin Ž dˆk y D m s t . Žy . 1
q
2
Ž eyG H t y ey G L t . cos Ž dˆk . f ,
Ž 3.
A.S. Dighe et al.r Physics Letters B 433 (1998) 147–149
in Bs Ž t . ™ Jrc f decays at planned experiments at the Tevatron and the LHC. Thus, in a not too distant future, the determination of dˆk and the resolution of the ambiguity Žrelated to the sign of cosŽ2 b .. in the extraction of the CKM angle b from Bd ™ Jrc K ) 0 Ž™ p 0 K S . decays may become feasible. Let us note that there remains a two-fold ambiguity for b in this approach, since we cannot decide whether b lies within the intervals w08,1808x or w1808,3608x. In each interval, b is, however, fixed unambiguously. Consequently, the original four-fold ambiguity arising in the extraction of b from sinŽ2 b . can be reduced to just a two-fold ambiguity. Usually it is argued that ´ K , which measures indirect CP violation in the kaon system, implies the former range w4x. While the Bd ™ Jrc K ) 0 Ž™ p 0 K S . mode is very accessible at B factories operating at the F Ž4S . resonance, detectors at hadron accelerators should study the feasibility of the p 0 reconstruction. The Bd ™ Jrc r 0 , Jrc v modes could be added to Bd ™ Jrc K ) 0 Ž™ p 0 K S . in order to resolve the b ambiguity. If penguin amplitudes are neglected, the time evolution of these decay modes also depends on the CKM angle b and, in the limit of the SUŽ3. flavour symmetry, their strong phases are equal to those of their SUŽ3. counterparts. In summary, traditional methods allow tests of the CKM picture of CP violation only up to discrete ambiguities. The resolution of these ambiguities would make such CKM tests significantly more powerful. In this letter, making use of the many observables that are available from angular correlations, we have proposed an approach to resolve a discrete ambiguity in the determination of the CKM angle b that may be simpler than strategies advocated earlier w4x. More generally, angular-correlation methods can also be formulated to remove discrete CKM ambiguities in b , 2 b q g s p q b y a and g from colour-allowed processes w14x.
149
Acknowledgements This work was supported in part by the Department of Energy, Contract No. DE-AC0276CHO3000. References w1x L.L. Chau, W.-Y. Keung, Phys. Rev. Lett. 53 Ž1984. 1802; J.D. Bjorken, private communication; C. Jarlskog, R. Stora, Phys. Lett. B 208 Ž1988. 268. w2x N. Cabibbo, Phys. Rev. Lett. 10 Ž1963. 531; M. Kobayashi, T. Maskawa, Progr. Theor. Phys. 49 Ž1973. 652. w3x For a recent review, see, for instance, A.J. Buras, R. Fleischer, University of Karlsruhe preprint TTP97-15, 1997, hepphr9704376, to appear in Heavy Flavours II, A.J. Buras, M. Lindner ŽEds.., World Scientific, Singapore, 1998. w4x Y. Grossman, Y. Nir, M.P. Worah, Phys. Lett. B 407 Ž1997. 307; Y. Grossman, H.R. Quinn, Phys. Rev. D 56 Ž1997. 7259; Y. Grossman, B. Kayser, Y. Nir, Phys. Lett. B 415 Ž1997. 90; J. Charles, A. Le Yaouanc, L. Oliver, O. Pene, ` J.-C. Raynal, LPTHE-Orsay 97r70, 1997, hep-phr9801363; L. Wolfenstein, Carnegie Mellon University preprint CMUHEP-98-01, 1998, hep-phr9801386; B. Kayser, talk given at the Moriond Workshop on Electroweak Interactions and Unified Theories, Les Arcs, France, March 1997, hepphr9709382, to appear in the Proceedings. w5x B. Kayser, M. Kuroda, R.D. Peccei, A.I. Sanda, Phys. Lett. B 237 Ž1990. 508. w6x I. Dunietz, H.R. Quinn, A. Snyder, W. Toki, H.J. Lipkin, Phys. Rev. D 43 Ž1991. 2193. w7x A.S. Dighe, I. Dunietz, R. Fleischer, preprint CERN-THr9885, FERMILAB-PUB-98r093-T, ICr98r25, 1998, hepphr9804253. w8x A.S. Dighe, I. Dunietz, H.J. Lipkin, J.L. Rosner, Phys. Lett. B 369 Ž1996. 144. w9x I. Dunietz, in: P. McBride, C. Shekhar Mishra ŽEds.., Proceedings of the Workshop on B Physics at Hadron Accelerators, Snowmass, Colorado, Fermilab-CONF-93r267, Batavia, 1993, p. 83. w10x R. Fleischer, I. Dunietz, Phys. Rev. D 55 Ž1997. 259. w11x I. Dunietz, Phys. Rev. D 52 Ž1995. 3048. w12x C.P. Jessop et al., CLEO Collaboration, Phys. Rev. Lett. 79 Ž1997. 4533. w13x M. Schmidt, S. Pappas, CDF Collaboration, private communication. w14x I. Dunietz, in progress.
Physics Letters B 433 Ž1998. 147–149
Resolving a discrete ambiguity in the CKM angle b through Bu,d ™ Jrc K ) and Bs ™ Jrc f decays Amol S. Dighe a , Isard Dunietz b, Robert Fleischer
c
a
b
The Abdus Salam International Centre for Theoretical Physics, 34100 Trieste, Italy Theoretical Physics DiÕision, Fermi National Accelerator Laboratory, BataÕia, IL 60510, USA c Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Received 15 April 1998 Editor: R. Gatto
Abstract It is well known that sinŽ2 b ., where b is one of the angles of the unitarity triangle of the CKM matrix, can be determined in a theoretically clean way by measuring mixing-induced CP violation in the decay Bd ™ Jrc K S . Another clean extraction of this CKM angle is provided by the time-dependent angular distribution for the decay products of Bd ™ Jrc Ž™ lqly. K ) 0 Ž™ p 0 K S ., where we have more observables at our disposal than in the case of Bd ™ Jrc K S , so that in addition to sinŽ2 b . also cosŽ2 b . can be probed in a direct way. Unfortunately a sign ambiguity remains in cosŽ2 b .. If it could be resolved, a discrete ambiguity in the extraction of the CKM angle b could be resolved as well, which would allow a more incisive test of the CKM model of CP violation. This note shows that detailed time-dependent studies of Bu, d ™ Jrc K ) and Bs ™ Jrc f decay processes can determine the sign of cosŽ2 b ., thereby removing the corresponding ambiguity in the extraction of the CKM angle b . q 1998 Elsevier Science B.V. All rights reserved.
The conventional methods for determining the angles a , b and g of the usual unitarity triangle w1x of the Cabibbo–Kobayashi–Maskawa matrix ŽCKM matrix. w2x leave several discrete ambiguities w3x. This is also the case for the ‘‘gold-plated’’ mode Bd ™ Jrc K S . The mixing-induced CP asymmetry arising in this channel allows only a theoretically clean determination of sinŽ2 b ., so that a discrete four-fold ambiguity for the extracted value of b g w08,3608x remains. In the recent literature, several strategies were proposed to resolve ambiguities of this kind w4x. Another clean probe of the CKM angle b is provided by the observables of the angular distribu-
tions for the decay products of Bd ™ Jrc Ž™ lq ly . K ) 0 Ž™ p 0 K S . modes w5–7x. Such observables can in general be expressed in terms of decay amplitudes as < Af Ž t. <2, Im
Re
A)f˜ Ž t . A f Ž t . ,
A)f˜ Ž t . A f Ž t . ,
Ž 1.
where f and f˜ are labels for specific final-state configurations. The full three-angle distributions for tagged Bd Ž t . ™ Jrc K ) 0 Ž™ p 0 K S . decays are given in w6–8x. Throughout this note, by tagging we
0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 8 0 - 7
148
A.S. Dighe et al.r Physics Letters B 433 (1998) 147–149
mean making the distinction of initially, i.e. at t s 0, present unmixed Bd,0 s and Bd0, s mesons. Weighting functions have been derived to extract the corresponding observables in an efficient way from experimental data w6,7x. The time evolution of the interference terms in Ž1., i.e. of the real and imaginary parts of bilinear combinations of certain decay amplitudes, allows the determination w7x of sinŽ d 1,2 . and cosŽ d 1 y d 2 ., where d 1 and d 2 are CP-conserving strong phases, and of sinŽ2 b . and
phases dˆ1,2 are the flavour SUŽ3. counterparts of d 1,2 . In the strict SUŽ3. limit, we have dˆ1,2 s d 1,2 . The time evolution of these observables takes the following form w7x:
cos Ž d 1,2 . cos Ž 2 b . .
where terms of O Ž f 2 . have been neglected, GH , GL denote the decay widths of the Bs mass eigenstates, G' Ž GH q GL .r2, and k s 1,2. Consequently, the strong phases dˆ1,2 can be determined unambiguously by resolving the rapid D m s t oscillations in Ž3.. Comparing the resulting values for sinŽ dˆ1,2 . and cosŽ dˆ1 y dˆ2 . with their unhatted analogues, which can be determined from the Bu, d ™ Jrc K ) observables, we obtain valuable information on SUŽ3. breaking. In order to fix the sign of cosŽ2 b . with the help of Ž2., we just need the sign of cosŽ d 1 . or cosŽ d 2 ., which is provided by the sign of cosŽ dˆk . determined from the Bs Ž t . ™ Jrc f observables. The SUŽ3. flavour symmetry should work reasonably well to determine this sign, unless
Ž 2.
The CP-conserving observables < A f Ž t s 0.<, sinŽ d 1,2 . and cosŽ d 1 y d 2 . can be determined to a higher accuracy from the much larger data samples arising for B "™ Jrc K ) " transitions, and untagged Bd decays into Jrc K ) 0 Ž™ Kq py . and JrcK ) 0Ž™ Ky pq . states w6,7x. At first sight, one may think that sinŽ d 1,2 . and cosŽ d 1 y d 2 . extracted this way from the Bu, d ™ Jrc K ) angular distributions will allow the determination of cosŽ2 b . with the help of the terms given in Ž2.. A closer look shows, however, that this is unfortunately not the case, since we do not have sufficient information to fix the signs of cosŽ d 1,2 ., thereby leaving a sign ambiguity for cosŽ2 b .. The purpose of this letter is to point out that this ambiguity can be resolved with the help of tagged, time-dependent studies of Bs ™ Jrc f decays. The angular distributions are given in w7,8x, and weighting functions to extract the observables from experimental data can be found in w7x. An important feature of these observables is that they allow the determination of a CP-violating weak phase f w7,9x, which takes a very small value, of O Ž0.03., within the Standard Model, and represents a sensitive probe for new-physics contributions to Bs0 – Bs0 mixing. Provided there is a sizeable mass difference between the mass eigenstates BsH and BsL , this phase can even be extracted from untagged Bs data samples w10x, where the rapid D m s t oscillations cancel w11x. Another important feature is the fact that the Žy .
tagged, time-dependent B s Ž t . ™ Jr c Ž ™ q y. q y. Ž l l f ™ K K observables corresponding to the ‘‘Im’’ terms in Ž1. provide sufficient information to determine dˆ1,2 unambiguously, where the strong
ey Gt sin Ž dˆk y D m s t . Žy . 1
q
2
Ž eyG H t y ey G L t . cos Ž dˆk . f ,
Ž 3.
A.S. Dighe et al.r Physics Letters B 433 (1998) 147–149
in Bs Ž t . ™ Jrc f decays at planned experiments at the Tevatron and the LHC. Thus, in a not too distant future, the determination of dˆk and the resolution of the ambiguity Žrelated to the sign of cosŽ2 b .. in the extraction of the CKM angle b from Bd ™ Jrc K ) 0 Ž™ p 0 K S . decays may become feasible. Let us note that there remains a two-fold ambiguity for b in this approach, since we cannot decide whether b lies within the intervals w08,1808x or w1808,3608x. In each interval, b is, however, fixed unambiguously. Consequently, the original four-fold ambiguity arising in the extraction of b from sinŽ2 b . can be reduced to just a two-fold ambiguity. Usually it is argued that ´ K , which measures indirect CP violation in the kaon system, implies the former range w4x. While the Bd ™ Jrc K ) 0 Ž™ p 0 K S . mode is very accessible at B factories operating at the F Ž4S . resonance, detectors at hadron accelerators should study the feasibility of the p 0 reconstruction. The Bd ™ Jrc r 0 , Jrc v modes could be added to Bd ™ Jrc K ) 0 Ž™ p 0 K S . in order to resolve the b ambiguity. If penguin amplitudes are neglected, the time evolution of these decay modes also depends on the CKM angle b and, in the limit of the SUŽ3. flavour symmetry, their strong phases are equal to those of their SUŽ3. counterparts. In summary, traditional methods allow tests of the CKM picture of CP violation only up to discrete ambiguities. The resolution of these ambiguities would make such CKM tests significantly more powerful. In this letter, making use of the many observables that are available from angular correlations, we have proposed an approach to resolve a discrete ambiguity in the determination of the CKM angle b that may be simpler than strategies advocated earlier w4x. More generally, angular-correlation methods can also be formulated to remove discrete CKM ambiguities in b , 2 b q g s p q b y a and g from colour-allowed processes w14x.
149
Acknowledgements This work was supported in part by the Department of Energy, Contract No. DE-AC0276CHO3000. References w1x L.L. Chau, W.-Y. Keung, Phys. Rev. Lett. 53 Ž1984. 1802; J.D. Bjorken, private communication; C. Jarlskog, R. Stora, Phys. Lett. B 208 Ž1988. 268. w2x N. Cabibbo, Phys. Rev. Lett. 10 Ž1963. 531; M. Kobayashi, T. Maskawa, Progr. Theor. Phys. 49 Ž1973. 652. w3x For a recent review, see, for instance, A.J. Buras, R. Fleischer, University of Karlsruhe preprint TTP97-15, 1997, hepphr9704376, to appear in Heavy Flavours II, A.J. Buras, M. Lindner ŽEds.., World Scientific, Singapore, 1998. w4x Y. Grossman, Y. Nir, M.P. Worah, Phys. Lett. B 407 Ž1997. 307; Y. Grossman, H.R. Quinn, Phys. Rev. D 56 Ž1997. 7259; Y. Grossman, B. Kayser, Y. Nir, Phys. Lett. B 415 Ž1997. 90; J. Charles, A. Le Yaouanc, L. Oliver, O. Pene, ` J.-C. Raynal, LPTHE-Orsay 97r70, 1997, hep-phr9801363; L. Wolfenstein, Carnegie Mellon University preprint CMUHEP-98-01, 1998, hep-phr9801386; B. Kayser, talk given at the Moriond Workshop on Electroweak Interactions and Unified Theories, Les Arcs, France, March 1997, hepphr9709382, to appear in the Proceedings. w5x B. Kayser, M. Kuroda, R.D. Peccei, A.I. Sanda, Phys. Lett. B 237 Ž1990. 508. w6x I. Dunietz, H.R. Quinn, A. Snyder, W. Toki, H.J. Lipkin, Phys. Rev. D 43 Ž1991. 2193. w7x A.S. Dighe, I. Dunietz, R. Fleischer, preprint CERN-THr9885, FERMILAB-PUB-98r093-T, ICr98r25, 1998, hepphr9804253. w8x A.S. Dighe, I. Dunietz, H.J. Lipkin, J.L. Rosner, Phys. Lett. B 369 Ž1996. 144. w9x I. Dunietz, in: P. McBride, C. Shekhar Mishra ŽEds.., Proceedings of the Workshop on B Physics at Hadron Accelerators, Snowmass, Colorado, Fermilab-CONF-93r267, Batavia, 1993, p. 83. w10x R. Fleischer, I. Dunietz, Phys. Rev. D 55 Ž1997. 259. w11x I. Dunietz, Phys. Rev. D 52 Ž1995. 3048. w12x C.P. Jessop et al., CLEO Collaboration, Phys. Rev. Lett. 79 Ž1997. 4533. w13x M. Schmidt, S. Pappas, CDF Collaboration, private communication. w14x I. Dunietz, in progress.