Gronau, Michael

London, D

Rosner, Jonathan L

CERN Document Server

CERN-TH.7407/94EFI-94-41UdeM-GPP-TH-94-08ASSUMPTIONS UNDERLYING DETERMINATIONOF A WEAK COUPLING PHASEMichael GronauTheoretical Physics Division, CERNCH-1211 Geneva 23, Switzerland1Jonathan L. RosnerEnrico Fermi Institute and Department of PhysicsUniversity of Chicago, Chicago, IL 60637andDavid LondonLaboratoire de Physique NucleaireUniversite de Montreal, Montreal, QC, Canada H3C 3J7ABSTRACTWe clarify the assumptions underlying the determination of theweak phase γ from a comparison of rates for charged B mesondecays to  and K.CERN-TH.7407/94August 19941Permanent address: Department of Physics, Technion { Israel Institute of Technology,Haifa 32000, Israel1In a recent paper [1] we showed that it is possible to obtain the weakphase γ by measurements of charged B-meson decay rates to +K0, 0K+,+0, and their charge-conjugate states. One of the ingredients in this anal-ysis was the observation that, in the limit in which annihilation diagrams areneglected, the decay B+ ! +K0 is pure penguin. It has subsequently beensuggested [2] that there is a hidden assumption in this result. What followsare some comments which explain the relation between our explicit assump-tion and the interpretation of Ref. [2]. We also wish to remind the readerabout previous discussions of the potential of this method for determiningthe weak phase γ, as a function of the strong phase dierence and of γ itself.A diagrammatic approach, equivalent to an SU(3) decomposition of am-plitudes, was adopted in Ref. [1]. Let q stand for d or s, and let unprimedand primed amplitudes correspond to q = d; s, respectively. The domi-nant amplitudes are expected to be (i) a color-favored \tree" contribution Tor T 0 with subprocess b ! quu, (ii) a color-suppressed tree contribution Cor C 0 with this same subprocess, and (iii) a penguin contribution P or P 0with subprocess b! q. (We omit reference to gluons or SU(3) singlet quark-antiquark pairs.) Other amplitudes, expected to be suppressed in comparisonwith these, are (iv) an annihilation subprocess A or A0 involving bu ! qu,(v) an exchange subprocess E or E0 involving bq ! uu, and (vi) a \penguinannihilation" subprocess PA or PA0 involving bq ! [SU(3) singlet system].The neglect of contributions (iv) { (vi) in comparison with (i) { (iii) wasnoted explicitly to be equivalent to the assumption that rescattering eectsare unimportant. For example, a nal state which can be reached through theannihilation diagram can also be reached through a tree diagram followed bya rescattering. Several tests of this hypothesis were proposed [1, 3]. One canexpect such an assumption to lead to relations between nal-state phases indierent decay channels. Indeed, one such phase relation was noted to existbetween B !  and B ! K [4]. Another example was given in Ref. [2].The fundamental process involved in the decays is b! s+ (light quark {antiquark pair). Let us denote a decay amplitude by A if it involves the lightquark-antiquark pair with isospin 1, and B if the pair has isospin zero. Wecan decompose amplitudes for B ! K charge states into isospin amplitudesA3=2; A1=2, and B1=2, where the subscript denotes the total isospin of the nalstate [5]. We also quote the decomposition into amplitudes associated withgraphs:A(B0 ! −K+) = A3=2 +A1=2 − B1=2 = −(T 0+ P 0) ; (1)p2A(B0 ! 0K0) = 2A3=2 − A1=2 +B1=2 = −C 0 + P 0 ; (2)A(B+ ! +K0) = A3=2 +A1=2 +B1=2 = P 0 +A0 ; (3)p2A(B+ ! 0K+) = 2A3=2 − A1=2− B1=2 = −(T 0+ C 0 + P 0 +A0) : (4)2Solving for the respective amplitudes, we ndA3=2 = −C0 + T 03; (5)A1=2 =2C 0 − T 0 + 3A06; (6)B1=2 = P 0 +T 0 +A02: (7)The point raised in Ref. [2] is that the I = 3=2 combinationA(+K0) +p2A(0K+) = 3A3=2 = −(T 0+ C 0) p2A^eiγei3 (8)and the I = 1=2 tree contribution to the combination[2A(+K0)−p2A(0K+)]tree = 3(A1=2 +B1=2)tree= +(T 0 + C 0)  p2C^eiγei1 (9)should have the same strong nal-state phases 1 = 3 if their sum,[3A(+K0)]tree =p2A^eiγei3 +p2C^eiγei1 (10)is to vanish.In the graphical description of Ref. [1], this is automatically the case, sincethe amplitude A3=2 and the tree contribution to the combination A1=2 +B1=2are both proportional to T 0 + C 0. Thus, the equivalence of the strong nal-state phases, 1 = 3, is not a hidden assumption, but is rather a directconsequence of our assumption that the annihilation diagrams are negligible.If the annihilation amplitude A0 is neglected, the only remaining contributionto Eq. (3) is then P 0, as stated.It is noted in Ref. [2] that the method could provide a powerful con-straint on the weak phase γ even if only an upper limit on CP violation isobtained from B ! 0K. Indeed, it was stressed in Ref. [1] that γ can bedetermined even without observing a CP asymmetry, similar to the methodproposed in Ref. [6]. Here too, the precision of determining this angle is high-est for γ = =2. The range of strong phases for which γ can be measuredwith a given precision is maximal for this value of the angle [7].We thank L. Wolfenstein for fruitful discussions. M. G. wishes to thankthe CERN Theory Division and J. R. wishes to thank the Fermilab TheoryGroup for their warm hospitality. This work was supported in part by theUnited States { Israel Binational Science Foundation under Research GrantAgreement 90-00483/3, by the German-Israeli Foundation for Scientic Re-search and Development, by the Fund for Promotion of Research at theTechnion, by the United States Department of Energy under Contract No.DE FG02 90ER40560, and by the N. S. E. R. C. of Canada and les Fonds F.C. A. R. du Quebec.3References[1] M. Gronau, J. L. Rosner, and D. London, Phys. Rev. Lett. 73, 21 (1994).[2] L. Wolfenstein, Phys. Rev. D 50, 1994, to be published.[3] M. Gronau, O. F. Hernandez, D. London, and J. L. Rosner, Phys. Rev. D50, Oct. 1, 1994.[4] M. Gronau et al., Ref. [3], end of Sec. III B.[5] Y. Nir and H. R. Quinn, Phys. Rev. Lett. 67, 541 (1991); M. Gronau,Phys. Lett. B 265, 389 (1991); L. Lavoura, Mod. Phys. Lett. A 7,1553 (1992); H. J. Lipkin, Y. Nir, H. R. Quinn, and A. E. Snyder,Phys. Rev. D 44, 1454 (1991).[6] M. Gronau and D. Wyler, Phys. Lett. B 265, 172 (1991).[7] S. L. Stone, in Beauty 93, Proceedings of the First International Work-shop on B Physics at Hadron Machines, Liblice Castle, Melnik, CzechRepublic, January 1993, ed. P. E. Schlein, Nucl. Instrum. Meth. 33, 15(1993).4

Assumptions underlying determination of a weak coupling phase

Abstract

Similar works

Full text

Available Versions

CERN Document Server