The nonleptonic decays B+ → D+D0 and c s B+ → D+D0 in a relativistic quark model. c s M.A. Ivanov,a J.G. K¨orner,b O.N. Pakhomovab,c aBogoliubov Laboratory of Theoretical Physics, 3 0 Joint Institute for Nuclear Research, 141980 Dubna, Russia 0 bInstitut fu¨r Physik, Johannes Gutenberg-Universita¨t, D-55099 Mainz, Germany 2 n cSamara State University, 443011 Samara, Russia a J 0 2 Abstract 2 v InthewakeofexploringCP-violationinthedecaysofB andBc mesons,weperform 1 the straightforward calculation of their nonleptonic decay rates within a relativistic 9 quark model. We confirm that the decays B D D0 and B D D0 are well 2 c s c s → → 2 suited to extract the Cabibbo-Kobayashi-Maskawa angle γ through the amplitude 1 relations because their decay widths are the same order of magnitude. In the b c 2 − sector the decays B DK and B DD lead to squashed triangles which are 0 c → → / therefore not so useful to determine the angle γ experimentally. We also determine h p the rates for other nonleptonic Bc-decays and compare our results with the results - of other studies. p e h Key words: Cabibbo-Kobayashi-Maskawa matrix elements; nonleptonic decays; : v bottom and bottom-charm mesons; relativistic quark model. i X PACS: 12.15.Hh,12.39.Ki,13.25.Hw r a Aswas pointedoutin[1]and[2,3]thedecays B+ D+D0(D0)arewellsuited c → s for an extraction of the CKM angle γ through amplitude relations. These decays are better suited for the extraction of γ than the similar decays of the B and B mesons because the triangles in latter decays are very squashed. u d The B meson has been observed by the CDF Collaboration [4] in the decay c B J/ψlν. One could expect around 5 1010 B events per year at LHC [5] c c → × which gives us hope to use the B decay modes for the studying CP-violation. c In the case of the B D D0(D0) decays the relevant amplitude relations c s → can be written in the form [2] √2A(B+ D+D0)=A(B+ D+D0)+A(B+ D+D0), c → s + c → s c → s Preprint submitted to Elsevier Science 1 February 2008 (1) √2A(B− D−D0)=A(B− D−D0)+A(B− D−D0), c → s + c → s c → s where D0 = D0 + D0 /√2 is a CP-even eigenstate. The diagrams de- | +i (cid:16)| i | i(cid:17) scribing the decays B+ D+D0 and B+ D+D0 are shown in Figs. 1 c → s c → s and 2, respectively. The color-enhanced amplitude of B+ D+D0 can be seen to be proportional to V† V 0.0029 exp(iγ). Atc t→he sams e time the ub · cs ≈ · decay amplitude for B+ D+D0 is proportional to V V 0.0088 but c → s bc · us ≈ color-suppressed. Simple estimates made in [2] give A(B+ D+D0) A(B− D−D0 c → s = c → s = (1). (2) (cid:12)(cid:12)A(B+ D+D0)(cid:12)(cid:12) (cid:12)(cid:12)A(B− D−D0)(cid:12)(cid:12) O (cid:12) c → s (cid:12) (cid:12) c → s (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) This implies that all sides of the amplitude triangles suggested in [2,6] have similar lengthsasshowninFig.3.Itallowsonetoextract themagnitudeofthe weak CKM-phase γ from the measurement of the B± D± +(D0,D0,D0) c → s + decay widths. The method [6] of the extraction of γ from Eq. (1) is based on the parametrization of the amplitudes as A(B+ D+D0)=A(B− D−D0) = A¯ eiδ¯, c → s c → s | | (3) A(B+ D+D0)= A eiγeiδ, A(B− D−D0)) = A e−iγeiδ, c → s | | c → s | | ¯ where δ and δ are the strong final state interaction phases. Introducing the notation A(B+ D+D0) A and A(B− D−D0) A Eq. (1) | c → s + | ≡ | +| | c → s + | ≡ | −| can be rewritten as A 2 + A 2= A 2 + A¯ 2 +2 A A¯ cosγ cos(δ¯ δ), + − | | | | | | | | | || | − (4) A 2 A 2=2 A A¯ sinγ sin(δ¯ δ). + − | | −| | | || | − The four solutions for sinγ are given by [6] 1 sinγ = Y Y Y Y , (5) 4 A A¯ (cid:26)±q ++ −− ±q +− −+(cid:27) | || | 2 2 where Y = A + A¯ 2 A 2 and Y = 2 A 2 A A¯ . Thus, ±+ ± ±− ± h| | | |i − | | | | −h| |−| |i the measurements of the rates of the six decays in Eq. (1) will determine the magnitude of γ with the four-fold ambiguity in Eq. (5). The way to resolve the ambiguity was discussed in [6]. 2 In contrast to B D D, the corresponding ratios for B KD and B c s c → → → DD are [2] A(B+ K+D0) A(B− K−D0 → = → = (0.1), (6) (cid:12)(cid:12)A(B+ K+D0)(cid:12)(cid:12) (cid:12)(cid:12)A(B− K−D0)(cid:12)(cid:12) O (cid:12) → (cid:12) (cid:12) → (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)A(B+ D+D0)(cid:12) (cid:12)A(B− D−D0 (cid:12) c → = c → = (0.1) (7) (cid:12)(cid:12)A(B+ D+D0)(cid:12)(cid:12) (cid:12)(cid:12)A(B− D−D0)(cid:12)(cid:12) O (cid:12) c → (cid:12) (cid:12) c → (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) which can be seen to lead to squashed triangles which are not very suited to measure γ. Some estimates of the branching ratios have been obtained before in [7]-[10] with widely divergent results. We employ here a relativistic quark model [11] to provide an independent evaluation of these branching ratios. This model is based on an effective interaction Lagrangian which describes the coupling between hadrons and their constituent quarks. For example, the coupling of the meson H to its constituent quarks q and q¯ is given by the 1 2 Lagrangian (x) = g H(x) dx dx F (x,x ,x )q¯(x )Γ λ q(x ). (8) int H 1 2 H 1 2 1 H H 2 L Z Z Here, λ and Γ are Gell-Mann and Dirac matrices which entail the flavor H H and spin quantum numbers of the meson field H(x). The shape of the vertex function F can in principle be found from the Bethe-Salpeter equation as H was done e.g. in [12]. However, we choose a phenomenological approach where the vertex functions are modelled by a simple form. The function F must be H invariant under the translation F (x+a,x +a,x +a) = F (x,x ,x ) and H 1 2 H 1 2 should decrease quite rapidly in the Euclidean momentum space. In our previous papers [13] we omit a possible dependence of the vertex func- tions on external momenta under calculation of the Feynman diagrams. This implies a dependence on how loop momenta are routed through the diagram at hand. In our last paper [14] and in the present calculation we employ a particular form of the vertex function given by m x +m x F (x,x ,x ) = δ x 1 1 2 2 Φ ((x x )2). (9) H 1 2 H 1 2 (cid:18) − m +m (cid:19) − 1 2 where m and m are the constituent quark masses. The vertex function 1 2 F (x,x ,x ) evidently satisfies the above translational invariance condition. H 1 2 We are able to make calculations explicitly without any assumptions concern- ing the choice of loop momenta. 3 The coupling constants g in Eq. (8) are determined by the so called compos- H iteness condition proposed in [15] and extensively used in [16]. The compos- iteness condition means that the renormalization constant of the meson field is set equal to zero 3g2 Z = 1 HΠ˜′ (m2 ) = 0 (10) H − 4π2 H H where Π˜′ is the derivative of the meson mass operator. In physical terms H the compositeness condition means that the meson is composed of a quark and antiquark system. For the pseudoscalar and vector mesons treated in this paper one has 1 d d4k Π˜′ (p2)= pα Φ˜2( k2) P 2p2 pα Z 4π2i P − tr γ5S (k +w p)γ5S (k w p) 1 21 2 12 (cid:20) 6 6 6 − 6 (cid:21) 1 pµpν 1 d d4k Π˜′ (p2)= gµν pα Φ˜2 ( k2) V 3(cid:20) − p2 (cid:21)2p2 pα Z 4π2i V − tr γνS (k +w p)γµS (k w p) 1 21 2 12 (cid:20) 6 6 6 − 6 (cid:21) where w = m /(m +m ). ij j i j The leptonic decay constant f is calculated from P 3g d4k P Φ˜ ( k2)tr OµS (k +w p)γ5S (k w p) = f pµ. 4π2 Z 4π2i P − (cid:20) 1 6 21 6 2 6 − 12 6 (cid:21) P 3g d4k 1 V Φ˜ ( k2)tr OµS (k +w p)γ ǫ S (k w p) = f ǫµ , 4π2 Z 4π2i V − (cid:20) 1 6 21 6 · V 2 6 − 12 6 (cid:21) m V V V The transition form factors P(p ) P(p )(V(p )) are calculated from the 1 2 2 → Feynman integrals 3g4Pπg2P′ Z 4dπ42kiΦ˜P(−(k +w13p1)2)Φ˜P′(−(k +w23p2)2) (11) tr S (k+ p )OµS (k+ p )γ5S (k)γ5 = F (q2)Pµ +F (q2)qµ, 2 2 1 1 3 + − · (cid:20) 6 6 6 6 6 (cid:21) 3g g d4k P V Φ˜ ( (k +w p )2)Φ˜ ( (k +w p )2) (12) 4π2 Z 4π2i P − 13 1 V − 23 2 (ǫ ) tr S (k+ p )OµS (k+ p )γ5S (k)γ ǫ = V ν 2 2 1 1 3 V · (cid:20) 6 6 6 6 6 · (cid:21) m +m P V 4 gµν PqA (q2)+PµPν A (q2)+qµPνA (q2)+iεµναβP q V(q2) , 0 + − α β n− o whereOµ = γµ(1 γ5).WeusethelocalquarkpropagatorsS (k) = 1/(m k) i i − 6 − 6 where m is the constituent quark mass. As discussed in [13,14], we assume i that m < m + m in order to avoid the appearance of imaginary parts in H 1 2 the physical amplitudes. This holds true for the light pseudoscalar mesons but is no longer true for the light vector mesons. We shall therefore employ iden- tical masses for the pseudoscalar mesons and the vector mesons in our matrix element calculations but use physical masses in the phase space calculation. This is quite a reliable approximation for the heavy mesons, e.g. D∗ and B∗ whose masses are almost the same as the D and B, respectively. However, for the light mesons this approximation is not so good since the K∗(892) has a mass much larger than the K(494). For this reason we exclude the light vec- tor mesons from our considerations. The fit values for the constituent quark masses are taken from our papers [13,14] and are given in Eq. (13). m m m m u s c b (13) 0.235 0.333 1.67 5.06 GeV We employ a Gaussian for the vertex functuon Φ˜ (k2) = exp( k2/Λ2 ) and H E − E H determine the size parameters Λ2 by a fit to the experimental data, when H available, or to lattice simulations for the leptonic decay constants. The nu- merical values for Λ are H Λ Λ Λ Λ Λ Λ Λ π K D Ds B Bs Bc (14) 1.00 1.60 1.70 1.70 2.00 2.00 2.05 GeV We have used the technique outlined in our previous papers [13,14] for the nu- merical evaluation of the Feynman integrals in Eqs. (11) and (12). The results of our numerical calculations are well represented by the parametrization F(0) F(s) = (15) 1 asˆ+bsˆ2 − withsˆ= q2/m2 .Using such aparametrizationfacilitatesfurther integrations. Bc The values of F(0), a and b are listed in Table 1. The calculated values of the leptonic decay constants are given in Eq. (16). They agree with the available 5 experimental data and the results of the lattice simulations. fK+ fD0 fD∗0 fDs fDs∗ fB fBc (16) 0.161 0.215 0.348 0.222 0.329 0.180 0.398 GeV Therelevant effective Hamiltonianforthedecays B D D0 andB D D0 c s c s → → is written as G H = F C (µ) V V† (¯bu) (c¯s) +V V† (¯bc) (u¯s) eff −√2 n 1 (cid:16) cs ub · V−A V−A us cb · V−A V−A(cid:17) † ¯ † ¯ + C (µ) V V (bs) (c¯u) +V V (bs) (u¯c) (17) 2 (cid:16) cs ub · V−A V−A us cb · V−A V−A(cid:17)o where V A refers to Oµ = γµ(1 γ5). We use the numerical values of the − − Wilsoncoefficientsattherenormalizationscaleµ = m givenbyC = 1.107 b,pole 1 and C = 0.248 as given in [17]. Note that we interchange the labeling 1 2 2 − ↔ of the coefficients to be consistent with the papers [7]-[10]. StraightforwardcalculationofthematrixelementsofthedecaysB D D0(D D0) c s s → by using the effective Hamiltonian in Eq.(17) reproduces the result of the fac- torization method. We have G A(B+ D+D0)= F V†V (18) c → s √2 · ub cs a fBcD(m2 )(m2 m2 )+fBcD(m2 )m2 f ×n 1 h + Ds Bc − D0 − Ds Dsi· Ds + a fBcDs(m2 )(m2 m2 )+fBcDs(m2 )m2 f , 2 h + D0 Bc − Ds − D0 D0i· D0o G A(B+ D+D0)= F V†V (19) c → s √2 · bc us a fBcDs(m2 )(m2 m2 )+fBcDs(m2 )m2 f × 2 h + D0 Bc − Ds − D0 D0i· D0 +annihilation channel where a = C + ξC and a = C + ξC with ξ = 1/N . As usual we put 1 1 2 2 2 1 c the QCD color factor ξ = 0 according to 1/N -expansion. Also we drop the c annihilation processes from the consideration. Note that the calculation of the matrix elements of the nonleptonic decays involving the vector D-mesons in the final states proceed in a similar way. We extend our analysis to the semileptonic and nonleptonic decays of B-meson. For numerical evaluation we have used the set of the parameters: m = 5.279 B+ 6 GeV, τ = 1.655 ps, m = 6.4 GeV, τ = 0.46 ps, a = 1.107, a = B+ Bc Bc 1|ξ=0 2|ξ=0 0.248 and − V V V V V V ud us ub cd cs bc | | | | | | | | | | | | (20) 0.98 0.22 0.003 0.22 0.98 0.040 First,to illustrate thequality ofour calculations, we list some branching ratios of the B-meson decays in Table 2 and compare them with the experimental data. The exclusive nonleptonic decay widths of the B and B mesons for c arbitrary values of a and a are listed in Table 3 whereas their branching 1 2 ratios for a = 1.107 and a = 0.248 are given in Table 4. One can see 1 2 − that as it was expected the magnitudes of the branching ratios of the decays B D D0 and B D D0 are very close to each other.It gives us hope that c s c s → → they can be measured in the forthcoming experiments. Finally, in Table 5 we compare our results with the results of other studies. One can see that there are quite large differences between the predictions of the different models. This work was completed while M.A.I. and O.N.P. visited the University of Mainz. M.A.I. appreciates the partial support by Graduiertenkolleg “Eichthe- orien”, the Russian Fund of Basic Research grant No. 01-02-17200 and the Heisenberg-Landau Program. O.N.P. thanks DAAD and Federal Russian Pro- gram ”Integracia” Grant No. 3057. References [1] M. Masetti, Phys. Lett. B 286, 160 (1992). [2] R. Fleischer and D. Wyler, Phys. Rev. D 62, 057503 (2000) [arXiv:hep-ph/0004010]. [3] R. Fleischer, “B physics and CP violation,” arXiv:hep-ph/0210323. [4] F. Abe et al. [CDF Collaboration], Phys. Rev. Lett. 81, 2432 (1998) [arXiv:hep-ex/9805034]; Phys. Rev. D 58, 112004 (1998) [arXiv:hep- ex/9804014]. [5] I. P. Gouz, V. V. Kiselev, A. K. Likhoded, V. I. Romanovsky and O. P. Yushchenko, “Prospects for the B/c studies at LHCb,” arXiv:hep- ph/0211432. [6] M. Gronau and D. Wyler, Phys. Lett. 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J. directC 4, 18 (2002) [arXiv:hep-ph/0205287]. [15] A. Salam, Nuovo Cim. 25, 224 (1962); S. Weinberg, Phys. Rev. 130, 776 (1963); K. Hayashi et al., Fort. der Phys. 15, 625 (1967). [16] G.V.EfimovandM.A.Ivanov,“TheQuarkConfinementModelOfHadrons,” Bristol, UK: IOP (1993) 177 p; Int. J. Mod. Phys. A 4, 2031 (1989). [17] A. Ali, E. Lunghi, C. Greub and G. Hiller, Phys. Rev. D 66, 034002 (2002) [arXiv:hep-ph/0112300]. [18] K. Hagiwara et al. [Particle Data Group Collaboration], Phys. Rev. D 66, 010001 (2002). Table 1 Form factors for B+ D0(D∗0) and B+ D+(D∗+) transitions. Form factors are c c s s → → approximated by the form F(q2) = F(0)/(1 asˆ+bsˆ2) with sˆ= q2/m2 . − Bc B+ D0(D∗0) B+ D+(D∗+) c c s s → → F(0) a b F(0) a b F 0.189 2.47 1.62 0.194 2.47 1.61 + F -0.194 2.43 1.54 -0.183 2.43 1.53 − A 0.284 1.30 0.15 0.312 1.40 0.16 0 A 0.158 2.15 1.15 0.168 2.21 1.19 + A -0.328 2.40 1.51 -0.329 2.41 1.51 − V 0.296 2.40 1.49 0.298 2.41 1.49 8 Table 2 Comparison of some branching ratios of the B-meson decays with the available experimental data. This work PDG [18] B+ D0e+ν 0.024 0.0215 0.0022 → ± B+ D∗0e+ν 0.056 0.053 0.008 → ± B+ K+D0 2.8 10−4 (2.9 0.8) 10−4 → · ± · B+ D+D0 0.013 0.013 0.004 s → ± B+ D+D∗0 0.008 0.012 0.005 s → ± B+ D∗+D0 0.019 0.009 0.004 s → ± B+ D∗+D∗0 0.046 0.027 0.010 s → ± Table 3 Exclusive nonleptonic decay widths of the B andB mesons in 10−15 GeV. c B+ K+D0 (0.364a +0.286a )2 B+ K+D0 0.00915a2 1 2 2 → → B+ K+D∗0 (0.342a +0.442a )2 B+ K+D∗0 0.0219a2 1 2 2 → → B+ D+D0 4.367a2 s 1 → B+ D+D∗0 2.707a2 s 1 → B+ D∗+D0 6.300a2 s 1 → B+ D∗+D∗0 14.84a2 s 1 → B+ D+D0 (0.0147a +0.0146a )2 B+ D+D0 0.753a2 c 1 2 c 2 → → B+ D+D∗0 (0.0107a +0.0234a )2 B+ D+D∗0 1.925a2 c 1 2 c 2 → → B+ D∗+D0 (0.0233a +0.0106a )2 B+ D∗+D0 0.399a2 c 1 2 c 2 → → B+ D∗+D∗0 (0.0235a +0.0235a )2 B+ D∗+D∗0 1.95a2 c 1 2 c 2 → → B+ D+D0 (0.0689a +0.0672a )2 B+ D+D0 0.0405a2 c s 1 2 c s 2 → → B+ D+D∗0 (0.0503a +0.106a )2 B+ D+D∗0 0.101a2 c s 1 2 c s 2 → → B+ D∗+D0 (0.101a +0.0498a )2 B+ D∗+D0 0.0222a2 c s 1 2 c s 2 → → B+ D∗+D∗0 (0.104a +0.110a )2 B+ D∗+D∗0 0.109a2 c s 1 2 c s 2 → → 9 Table 4 Branching ratios of some nonleptonic decay widths of the B andB mesons in cal- c culated for a = 1.107 and a = 0.248. 1 2 − B+ K+D0 2.76 10−4 B+ K+D0 1.41 10−6 → · → · B+ K+D∗0 1.82 10−4 B+ K+D∗0 3.38 10−6 → · → · B+ D+D0 1.11 10−7 B+ D+D0 3.24 10−5 c c → · → · B+ D+D∗0 0.25 10−7 B+ D+D∗0 8.28 10−5 c c → · → · B+ D∗+D0 3.76 10−7 B+ D∗+D0 1.71 10−5 c c → · → · B+ D∗+D∗0 2.84 10−7 B+ D∗+D∗0 8.38 10−5 c c → · → · B+ D+D0 2.48 10−6 B+ D+D0 1.74 10−6 c s c s → · → · B+ D+D∗0 0.60 10−6 B+ D+D∗0 4.34 10−6 c s c s → · → · B+ D∗+D0 6.88 10−6 B+ D∗+D0 0.95 10−6 c s c s → · → · B+ D∗+D∗0 5.41 10−6 B+ D∗+D∗0 4.69 10−6 c s c s → · → · Table 5 Exclusive nonleptonic decay widths of the B meson in units of 10−15 GeV. c Comparison with other studies. Process This paper [7] [8] [9] [10] [5] B+ D+D0 0.0405a2 0.0340a2 0.168a2 0.01a2 0.0415a2 0.176a2 c s 2 2 2 2 2 2 → B+ D+D∗0 0.101a2 0.0354a2 0.143a2 0.009a2 0.0495a2 0.260a2 c s 2 2 2 2 2 2 → B+ D∗+D0 0.0222a2 0.0334a2 0.0658a2 0.087a2 0.0201a2 0.166a2 c s 2 2 2 2 2 2 → B+ D∗+D∗0 0.109a2 0.0564a2 0.128a2 0.15a2 0.0597a2 0.951a2 c s 2 2 2 2 2 2 → 10