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Volume 96B, number 3,4 PHYSICS LETTERS 3 November 1980 NONLEPTONIC DECAYS OF CHARMED MESONS Hiroshi KONASHI Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan Received 10 June 1980 The experimental data on the relative rates in charmed meson decays are explained in terms of the Kobayashi-Maskawa mixing matrix of the six-quark model and the dipole type form factors for mesons. The recent measurement of charmed meson decays [I] has given rise to much interest. Because the re- sults, listed below, are markedly different from their previous theoretical expectations: r (D 0 ~ K-K+)/F(D 0 + K-Tr +) = (11.3 + 3.0)% (1) r(D -+ n - rr+)/P(D 0 -+ K -n +) = (3.3 + 1.5)% (2) F(D 0 -+ K0n0)/p(D0 -+ K-rr +) = 0.63 -+ 0.35 (3) B(D + -+ R.rr+)/B(D 0 ---* K-rr +) = 0.68 + 0.33 (4) r(D +) => (5.8 -+ 1.5) r(D). (5) A variety of proposals [2] have been made. In several of these, a strong "20-plet" dominance is assumed in the weak couplings in order to account for the above eqs. (3), (4) and (5). This assumption, however, does not seem to be explained by present theoretical under- standing of the electroweak and strong interactions. In this letter, we show the experimental results can be understood in a phenomenological way without the assumption of a strong "20-pier" dominance. The reasonable assumptions made are (i) that the electro- weak currents are given by the standard SU(2) X U(1) model of six quarks with the Kobayashi-Maskawa mix- ing matrix [3], and (ii) the dipole type form factors are valid for charmed meson decays as well as for pion decays [e.g. 4]. The effective nonleptonic lagrangian for c -+ ual~ is te f f (a /v2) 1 = Uub Uca [~- C_ {(fib)(~c) - (a-b) (~c)} (6) 1 + 5C+ ((nb)(Sc) + (ab)(Oc))l. We have suppressed the Lorentz (/~) and color (a) in- dices and U//stands for the mixing angles in the Kobayashi-Maskawa matrix. The coefficients C_ and C+ denote the short-distance renormalization effects of hard gluons for "20-plet" and "84-plet", respective- ly. Using reasonable parameters, one obtains C_ = 2.4 and C+ = 0.64 in the leading log limit [5]. According to Barger and Pacvasa, we neglect the quark-anti- quark pair creation [2]. Then, the matrix elements of the effective lagrangian between color-singlet hadronic states are factorized in the form, {M+M'lLeffl D) (7) = (G/x /~)Uub f*a+(M+ I (ub)l 0)(M'I (ac)l D), for D(c~) -+ M+(ul~) + M'(a?t), and (MOM'[Leff[ D) (8) = (G/~2) Uub fca - Volume 96B, number 3,4 PHYSICS LETTERS 3 November 1980 and q x (M'(p')[ Vx(O)l D(p)) (10) _ 1 1 (2a)3 X/4X/4X/4X/4X/4X/4X~00 P% f+(q2)(D2 - M'2) ' where we have expressed the mass of each particle by the particle's symbol. And we use the usual decay con- stants of kaons and pions, whose values are fTr_+=x/~fno=0.97a, f~=fKC3 = 1.23a. C 3 is the mixing angle; cos 03" Here, we note the fol- lowing relations hold in this case when the mixing an- gles are taken into account; fM(q 2) = fM(0)(1 -- qZ//.t2)-2, (11) fK(0) -- f~+ (0) -- X/~ Lo (0). Where ~2 is a normalization mass of the axial vector form factors and we use -1 (GeV/c2) 2 for numerical calculations. These equations reveal that SU(3) break- ing in fM(q2 ) at q2 = 0 is small. As a natural extension of this conception, we adopt the dipole type form fac- tors for D -+ M' transitions: f+(q2) = f+(0)(1 -- q2/u2)-2 (12) where/~+2 is a normalization mass of the vector form factors and we use ~0.71 (GeV/c2) 2 for numerical es- timates. Then, the ratios between charmed meson decay rates are given in the mode by ,22 2 2 2 P(D + K-K +) _ R(D 0, K - , K +) f~ f+(K )S1C 3 P(D -~ K -a +) R(D 0, K - , a +) f2+f+2(a2) C~ ' p(D 0 . rr -a +) y(D 0 -+ K -a +) _ S1C 2 _ R(D 0, ~r-, ~-+) (/9 2 ~,2)2 2 2 R(D 0, K - , a +) (/9 2 -K2) 2 (C1C'2C 3 +$2S3 ei8 )2' F(D o ~ ~0 fro) p(D 0 --* K-~r +) _ R(D 0, g0, a0) fK2X2 f+2(K 2) (D 2 _ a2)2 R(D 0 ,K ,a ) 2 2 2 2 _K2)2 ' f~+x+f+(a ) (D 2 and B(D +~g0a+ )_ R(D +,g0,1r +) 3' B(D 0 -+ K -a +) R(D 0 , K - , 7r +) (13) [X+4+f+(a2)(D2 K2)+x fKf+(K2)(D2_ a2)] 2 2 22 2 2 22 ;~X+f~,(a )(0 -X ) where R(x,y, z) = (x 4 +y4 +z 4 _ 2x2y2 _ 2y2z2 - 2z2x2)l/2/16ax3, T = r(D+)/r(DO) and C i = cos 0 i and S i = sin 0 i are the mixing angles. With the mea- sured quantities of cos 01 , f~, fK and the mass of each particle, and the assumptions made previously, we ob- tain the relative rates in charmed meson decays as fol- lows: F(D 0 -+ K-K+) /p(D 0 -+ K -v +) = 0.36C 2, p(D 0 ~ a-~+)/V(D 0 ~ K -a +) = 1.2 IS 1 C2/(C 1 C2C 3 + $2S3ei6 )] 2, P(D 0 -~ g0r0) /p(D0 -+ K -a +) = 0.43, B(D + -* ff0n+)/B(D0 -+ K -a +) = 0.42. Comparing the above predictions for P(D 0 ~ K - K+)/ P(D 0 --* K-Tr +) and P(D 0 -+ a-Tr+)/F'(D 0 -+ K- l r +) with the experimental results, we have the constraints on the mixing angles: 0.48 < I C31 < 0.63 and 0.12 < IS 1 C2/(C 1 C2C 3 + $2S3 ei~ )[ ~ 0.20. These constraints are not inconsistent with the previ- ous works [5]. And these predicted values agree with the experimental results quite well. In summary, we find that the strong constraints on the mixing angles of the Kobayashi-Maskawa matrix are obtained and the dipole type form factors are still useful for charmed meson decays. A more precise ex- perimental data such as semileptonic decays of charmed mesons are not only useful to determine the mixing matrix but also give the crucial test for the as- sumption of using the dipole type form factors in charmed meson decays. 379 Volume 96B, number 3,4 PHYSICS LETTERS 3 November 1980 References [1] G.S. Abrams et al., Phys. Rev. Lett. 43 (1979) 481; J. Kirkby, Proc. Intern. Symp. on Lepton and photon interactions at high energies (FNAL, Batavia, 1979); V. Luth, Proc. Intern. Syrup. on Lepton and photon interactions at high energies (FNAL, Batavia, 1979); G. Gidal, Proc. Intern. Conf. of High energy physics (Geneva, Switzerland, 1979). [2] V. Barger and S. Pakvasa, Phys. Rev. Lett. 43 (1979) 812; M. Suzuki, Phys. Rev. Lett. 43 (1979) 818, Phys. Lett. 85B (1979) 91; B. Guberina et al., Phys. Lett. 89B (1979) 111; D.G. Sutherland, Phys. Lett. 90B (1980) 173; I.I.Y. Bigi, Phys. Lett. 90B (1980) 177; [3] [4] [5] N. Deshpande, M. Gronau and D. Sutherland, Phys. Lett. 90B (1980) 431; H. Fritzsch and P. Minkowski, Phys. Lett. 90B (1980) 455; T. Hayashi et al., preprint RRK 80-2 (1980). S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in Elementary particle physics, ed. N. Svartholm (Stockholm, 1968) p. 367; M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. R. Marshak, Riazuddin and C. Ryan, Theory of weak interactions in particle physics (Wiley, New York, 1969). M.K. Gaillard and B.W. Lee, Phys. Rev. Lett. 33 (1974) 108; G. Altarelli and L. Maiani, Phys. Lett. 52B (1974) 351; J. Ellis, M.K. Gaillard and D.V. Nanopoulous, Nucl. Phys. B100 (1975) 313. 380