In consistency with the b --> c type of (quasi) two body decays, recently
observed two body decays of B mesons are studied in a hybrid perspective in
which their amplitude is given by a sum of factorizable and non-factorizable
ones, and a role of the latter in these decays are discussed.Comment: 7 page

A. Abada

B. Block

C. T. Sachrajda

C. W. Bauer

D. E. Groom

G. Buchalla

H. Georgi

J. D. Bjorken

K. Abe

K. Hagiwara

K. Terasaki

M. Bauer

M. Beneke

M. J. Dugan

M. Kobayashi

M. S. Nehring

N. Cabibbo

N. Isgur

P. L. Frabetti

S. Oneda

T. E. Coan

Y.-Y. Keum

English

arXiv

Crossref

Newly observed two-body decays of
                    B

Terasaki, K

Kyoto University Research Information Repository

TitleNewly observed two-body decays of B mesons in a hybridperspectiveAuthor(s)Terasaki, KCitationPHYSICAL REVIEW D (2003), 67(9)Issue Date2003-05-01URL http://hdl.handle.net/2433/50462RightCopyright 2003 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityNewly observed two-body decays of B mesons in a hybrid perspectiveK. TerasakiYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan~Received 28 October 2002; published 14 May 2003!In consistency with B¯ →D (*)p , J/cK¯ , and J/cp decays, recently observed B0→Ds1p2 and B¯ 0→Ds1K2 decays are studied in a hybrid perspective in which their amplitude is given by a sum of factorizableand nonfactorizable ones.DOI: 10.1103/PhysRevD.67.097501 PACS number~s!: 13.25.Hw~Quasi-!two-body decays of B mesons have been studiedextensively by using factorization @1,2#. However, recentlymeasured rates @3# for the color mismatched spectator ~CMS!decays, B¯ d0→D (*)0p0, are much larger than the expectationof factorization. It suggests that nonfactorizable contribu-tions can play an important role in these decays. In addition,very recently, B¯ 0→Ds1K2 and B0→Ds1p2 have been ob-served @4#. The rate for the former is again much larger thanthe expectation of factorization; i.e., it is expected to bestrongly suppressed ~the helicity suppression! since it is de-scribed by an annihilation diagram in the weak boson massmW→‘ limit. It means that the nonfactorizable contributionis dominant in this decay. The latter is a pure spectator decay,b¯→u¯1(cs¯), but does not satisfy the kinematical conditionof color transparency @5#, so that it is not very clear if thefactorization works well in this decay. Therefore, it is mean-ingful to study a possible role of nonfactorizable contribu-tions in the newly observed B¯ d0→D (*)0p0, B¯ 0→Ds1K2, andB0→Ds1p2 decays in consistency with the b→c type ofdecays, B¯ →D (*)p , J/cK¯ , and J/cp .We first review briefly our ~hybrid! perspective ~see Ref.@6# for more details!. Our starting point is to assume that theamplitude can be decomposed into a sum of factorizable andnonfactorizable ones (M FA and M NF , respectively!. M FA isestimated by using the factorization while M NF is assumed tobe dominated by dynamical contributions of various hadronstates and calculated by using a hard pion ~or kaon! approxi-mation in the infinite momentum frame ~IMF! @7,8# since, inthe existing theories such as QCD sum rule @9#, QCD factor-ization @10#, p QCD @11#, and soft-collinear effective theory@12# which treat hadronic weak interactions of heavy me-sons, it is still too complicated to account for dynamicalcontributions of various hadron states at the B meson massscale. In this approximation, M NF is given by a sum of thesurface term (M S) which is given by a sum of all possiblepole amplitudes and the equal-time commutator term (M ETC)which arises from the contribution of nonresonant ~multihad-ron! intermediate states @13#. Corresponding to the abovedecomposition of the amplitude, the effective weak Hamil-tonian, Hw.(GF /A2)$c1O11c2O2%1H.c. ~where c1 andc2 are the Wilson coefficients with QCD corrections @14#!, isdecomposed into a sum of the Bauer-Stech-Wirbel ~BSW!Hamiltonian @1#, Hw(BSW), and an extra term, H˜ w , i.e., Hw→Hw(BSW)1H˜ w , by using the Fierz reshuffling, whereHw(BSW) is given by a sum of products of colorless currentsand might provide the factorizable amplitude. However, the‘‘external’’ hadron states which sandwich Hw(BSW) might in-teract sometimes with each other through hadron dynamics~like a rescattering, etc.!. In this case, the corresponding partof the amplitude is nonfactorizable and should be included inM NF , so that the values of the coefficients, a1 and a2, inM FA arising from Hw(BSW) might not be the same as the origi-nal a1(BSW)5c11c2 /Nc and a2(BSW)5c21c1 /Nc in Hw(BSW),where Nc is the color degree of freedom. Since such a had-ron dynamics cannot be controlled by the perturbative QCD,we will treat a1 and a2 as adjustable parameters later. Theextra term H˜ w which is given by a color singlet sum ofcolored current products provides nonfactorizable amplitudesin the present perspective, although, in Ref. @2#, contribu-tions from H˜ w have been included in the factorized ampli-tudes by considering the effective colors.Explicit expression of factorized and nonfactorizable am-plitudes for the B¯ →D (*)p , J/cK¯ , and J/cp decays havealready been given in Ref. @6# in which M ETC and M S withcontributions of low lying meson poles are taken into ac-count. In the same way, we can calculate the amplitude forthe b¯→u¯1(cs¯) decays, B0→Ds(*)1p2. These amplitudes,however, include many parameters, i.e., form factors, decayconstants of heavy mesons, asymptotic matrix elements ofH˜ w ~matrix elements of H˜ w taken between single hadronstates with infinite momentum!, phases, d˜ I , of M ETC(I) (B¯→Dp), (I5 12 and 32 ), relative to M S and the relative phase(D) between M FA and M NF which has not been considered inour previous studies. The other parameters involved areknown or can be estimated by using related experimentaldata and asymptotic flavor symmetries @15#.To obtain improved values of the above amplitudes, weupdate values of parameters involved. Asymptotic matrix el-ements of axial charges are estimated as follows @8#, i.e.,u^r0uAp1up2&u.1.0 by using the partially conserved axial-vector current ~PCAC! and the measured decay rate G(r→pp)exp.150 MeV @16#. Here we take ^r0uAp1up2&51.0 and the other ones can be related to it by using relatedasymptotic flavor symmetries, for example,A2^D*1uAp1uD0&52^r0uAp1up2&, etc., as in our previ-ous study @6#. As the values of the Cabibbo-Kobayashi-Maskawa ~CKM! matrix elements @17# and the decay con-stants, we take Vcs.Vud.0.98, Vcd.20.22, Vcb.0.040,uVub /Vcbu.0.090, and f p.130.7 MeV, f K.160 MeVfrom Ref. @16#. The decay constant, f J/c.406 MeV, can bePHYSICAL REVIEW D 67, 097501 ~2003!0556-2821/2003/67~9!/097501~4!/$20.00 ©2003 The American Physical Society67 097501-1obtained from G(J/c→e1e2)exp55.2660.37 keV @16#.The updated values of the decay constants of heavy mesons,f D.0.226 GeV, f Ds.0.250 GeV, and f B.0.198 GeV, aretaken from the lattice QCD @18#, and f D*. f D and f Ds*. f Ds are assumed as expected by the heavy quark effectivetheory ~HQET! @19#. The form factors, F0(DB¯ )(mp2 ) andA0(D*B¯ )(mp2 ), are estimated by using the HQET and the dataon the semi-leptonic decays of B mesons @16# asF0(DB¯ )(mp2 ).0.74 and A0(D*B¯ )(mp2 ).0.65. The form factors,F0(pB¯ )(q2) and F1(pB¯ )(q2), are estimated by using extrapola-tion formulas based on the lattice QCD @20#. We here takeF0(pB)(mD2 ).0.28, F0(pB)(mDs2 ).0.32, F1(pB)(mD*2 ).0.34,F1(pB)(mc2 ).0.50, and F1(KB)(mc2 ).0.59. The annihilationamplitudes which contain F0(Dp)(mB2 ) and A0(D*p)(mB2 ) willbe small and neglected because of the helicity suppression.The asymptotic matrix element of H˜ w is parametrized by^D0uH˜ w(ud;cb)uB¯ d0&VcbVud f p 5BH31025 ~GeV!, ~1!where H˜ w(ud;cb) is a component of H˜ w which is given by asum of O˜ 1(ud;cb)5VudVcb$2(a(d¯ tau)L(c¯tab)L% and O˜ 2(ud;cb)5VudVcb$2(a(c¯tau)L(d¯ tab)L% with the color SUc(3) gen-erator ta. To evaluate the B¯ →D*p amplitudes, we assume^D*0uH˜ w(ud;cb)uB¯ d*0&5^D0uH˜ w(ud;cb)uB¯ d0& ~2!as expected by the HQET. All the other asymptotic matrixelements of H˜ w involved in the nonfactorizable amplitudesare combined with the ones in Eq. ~2!, i.e.,^J/cuH˜ w(cd;cb)uB¯ d*0&5S VcdVcs D ^J/cuH˜ w(cs;cb)uB¯ s*0&5S VcdVudD ^D (*)0uH˜ w(ud;cb)uB¯ d(*)0&52S VcdVcbVcsVub D ^Ds(*)1uH˜ w(cs;ub)uBu(*)1&,~3!by inserting commutation relations, @VK0,H˜ w(cs;cb)#5(Vcs /Vcd)H˜ w(cd;cb) , @VD0,H˜ w(cd;cb)#5(Vcd /Vud)H˜ w(ud;cb) ,@VD¯ 0,H˜ w(cs;cb)#5(Vcb /Vub)H˜ w(cs;ub) , between related asymp-totic states ~single hadron states with infinite momentum!and using asymptotic SU f(3) and SU f(4) relations,^B¯ s*0uVK0uB¯ d*0&521, ^D*0uVD0uJ/c&521, etc. To obtainthe last equality in Eq. ~3!, we have used the CP invariancewhich is always assumed in this Brief Report and^$qq¯ %0uO˜ 1u$qq¯ %0&50 from a quark counting @21#, whereO˜ 65O˜ 16O˜ 2. The $qq¯ %0’s denote the low lying mesons. Inthis way, we can obtain M FA and M NF in the second and thirdcolumns, respectively, of Table I, where we have neglectedsmall contributions of annihilation terms in M FA and excitedmeson poles in M NF .TABLE I. Factorized and nonfactorizable amplitudes for the B¯ →D (*)p , J/cK¯ , J/cp , and B0→Ds(*)1p2 decays. The CKM matrix elements are factored out.Decay AFA~31025 GeV! ANF~31025 GeV!B¯ 0→D1p2 1.94a1eiD2H 43 eid˜ 1/22 13 eid˜ 3/2J BHB¯ 0→D0p021.14S f D0.226 GeVD a2eiD 2H 2A23eid˜ 1/21A23eid˜ 3/2J BHB2→D0p21.94a1H 110.48S f Df p D S a2a1D J eiD eid˜3/2BHB¯ 0→D*1p2 21.68a1eiD 20.694BHB¯ 0→D*0p0 1.07S f D*0.226 GeVD a2eiD 0.983BHB2→D*0p221.68 a1H 110.52S f D*f p D S a2a1D J eiD 20.696BHB2→J/cK2 23.60a2eiD 20.548BHB¯ 0→J/cK¯ 0 23.60a2eiD 20.548BHB2→J/cp2 23.08a2eiD 20.692BHB¯ 0→J/cp0 2.18a2eiD 0.489BHB0→Ds1p2 1.95a1eiD eid˜ 1BHB0→Ds*1p2 1.54a1eiD 0.70BHBRIEF REPORTS PHYSICAL REVIEW D 67, 097501 ~2003!097501-2We now look for values of parameters, a1 , a2 , D , d˜ I ,(I51/2, and 3/2), and BH , which reproduce the measuredbranching ratios for the B¯ →D (*)p , J/cK¯ , and J/cp de-cays. a1 and a2 are treated as adjustable parameters withvalues around a1(BSW) and a2(BSW). The phase d˜ I is restrictedin the region ud˜ Iu,90 ° since resonant contributions havealready been extracted as pole amplitudes in M S while D andBH are treated as free parameters. The result is not verysensitive to d˜ I , and the coefficients, a1 and a2, favor valuesclose to the ones taken in Ref. @2# which is based on thefactorization. The above implies that the nonfactorizablecontribution is not very important in the color favored de-cays. We can reproduce the experimental data (Bexp) com-piled by the Particle Data Group 2002 @22# taking values ofparameters in the range 1.00&a1&1.13, 0.28&a2&0.31,24 °&uDu&32 °, ud1/2u&70 °, 10 °&ud3/2u&90 °, and 0.09&BH&0.25. To see more explicitly a role of the non-factorizable contribution, we list a typical result on thebranching ratios ~near the best fit to Bexp) for a151.08, a250.29, d˜ 150.0 °, d˜ 35690 °, D5628 °, and BH50.19 inTable II, where we have used t(B2)51.67310212 s andt(B¯ 0)51.54310212 s from Ref. @22#. BFA and Btot are givenby M FA and M tot5M FA1M NF , respectively. As seen inTable II, BFA in which M NF is discarded is hard to reproducethe data on the CMS decays, B¯ →D (*)0p0. If we add M NF ,however, we can get a much better fit to the data includingthe CMS decays. In the color favored B¯ →D (*)p decays,M NF is rather small ~but it can interfere efficiently with themain amplitude, M FA). In the B2→D0p2 decay, however, itis very small. In the B¯ →J/cK¯ and J/cp decays, the colorsuppression does not work so well that M NF is not dominantin contrast with the B¯ →D (*)0p0 although all of them are theCMS decays.Next, we study the B0→Ds(*)1p2 decays comparingwith the B2→D0p2 which has been studied above. Usingthe same values of parameters as the above, i.e., a151.08, a250.29, BH50.19, we obtainuM NF~B0→Ds1p2!u.0.09uM FA~B0→Ds1p2!u, ~4!uM NF~B0→Ds*1p2!u.0.08uM FA~B0→Ds*1p2!u, ~5!which imply that the factorization works considerably wellin these decays although they do not satisfy the condition ofthe color transparency. Neglecting the rather small M NF inthe B0→Ds1p2 and using the same values of parameters asthe above, we obtainuM ~B0→Ds1p2!u.0.074uM ~B2→D0p2!u, ~6!where we have used uVub /Vcbuexp.0.090 @16#. The mea-sured branching ratio for the B2→D0p2 decay @22# leads ustoB~B0→Ds1p2!.2.731025, ~7!which reproduces well the recent measurements @4#,B~B0→Ds1p2!BABAR5~3.162.0!31025,B~B0→Ds1p2!BELLE5~2.420.811.060.7!31025.In the same way, we obtain B(B0→Ds*1p2).1.731025,which is again compatible with the experimental upper limits@4#.In the B¯ 0→Ds1K2 decay, M FA is strongly suppressed be-cause of the helicity suppression, so that M NF dominates thedecay in the present perspective, i.e.,M ~B¯ 0→Ds1K2!.M NF~B¯ 0→Ds1K2!.2iVcbVudS f pf KD ^D0uH˜ wuB¯ d0&VcbVud f p eid˜ 1. ~8!The same value of parameters as the above leads toB~B¯ 0→Ds1K2!.2.831025, ~9!which should be compared with the measured values @4#B~B¯ 0→Ds1K2!BABAR5~3.262.0!31025,B~B¯ 0→Ds1K2!BELLE5~4.621.111.261.3!31025.In summary, we have studied the recently observed de-cays, B¯ →D (*)0p0, B0→Ds1p2, and B¯ 0→Ds1K2, in con-sistency with the b→c type of decays, B¯ →D (*)p , J/cK¯ ,and J/cp , providing their amplitude by a sum of factorizedand nonfactorizable ones. To study the nonfactorizable am-plitudes, we have used the asymptotic SU f(3) and SU f(4)symmetries which may be broken. The size of the symmetrybreaking can be estimated from the value of the form factor,f 1(0), in the matrix element of the related vector current,where f 1(0)51 in the symmetry limit. From the measuredvalues of the form factors, f 1(pD)(0)50.7160.06 @23# andu f 1(pD)(0)/ f 1(K¯ D)(0)u51.0060.13 @24#, the asymptoticSU f(4) symmetry seems to be broken to the extent of 30%TABLE II. A typical result on the branching ratios (31023) forB¯ →D (*)p , J/cK¯ , and J/cp decays, where the values of the pa-rameters involved are given in the text. BFA and Btot are given byM FA and M tot , respectively. Bexp are taken from Ref. @22#.Decays BFA Btot BexpB¯ 0→D1p2 4.0 3.1 3.060.4B¯ 0→D0p0 0.10 0.24 0.2760.06B2→D0p2 5.6 5.6 5.360.5B¯ 0→D*1p2 3.1 2.6 2.7660.21B¯ 0→D*0p0 0.09 0.22 0.2260.10B¯ 0→D*0p2 4.1 4.7 4.660.4B2→J/cK2 0.82 0.99 1.0160.05B¯ 0→J/cK¯ 0 0.75 0.91 0.8760.05B2→J/cp2 0.030 0.039 0.04260.007B¯ 0→J/cp0 0.014 0.018 0.02160.005BRIEF REPORTS PHYSICAL REVIEW D 67, 097501 ~2003!097501-3while the asymptotic SU f(3) still works well. However, sucha large symmetry breaking has not caused any serious prob-lem in the present study since M NF is much smaller than M FAexcept for some decays in which M FA is strongly suppressedand whose experimental errors are still large. For more pre-cise studies, of course, more detailed information of the sym-metry breaking will be needed.The amplitude with final state interactions has been in-cluded in the nonfactorizable one. For the color favored B¯→D (*)p decays, M NF has been rather small and, therefore,the final state interactions seem to be not very important ~butnot necessarily negligible! in these decays. In the B¯→D (*)0p0 which are the CMS decays, M NF has been domi-nant since M FA is suppressed because of the color suppres-sion. In the B¯ →J/cK¯ and J/cp , which also are the CMSdecays, however, the color suppression has not worked sowell that M NF has not been dominant in contrast with theB¯ →D (*)0p0 decays. In the B0→Ds(*)1p2 decays whichare the color favored b¯→u¯1(cs¯) type of spectator decays,M NF has been small. The values of parameters which repro-duce the measured branching ratios for the B¯ →D (*)p ,J/cK¯ , and J/cp decays have led to B(B0→Ds(*)1p2) con-sistent with the very recent measurements. It means that thefactorization works considerably well in these decays al-though they do not satisfy the condition of color transpar-ency. In the B¯ 0→Ds1K2 decay which is the annihilation de-cay, M NF has been dominant and reproduced the very recentmeasurements within their large errors. All the above suggestthat dynamical contributions of hadrons should be carefullytreated in hadronic weak decays of B mesons.In the CMS decays, B¯ 0→D (*)0p0, J/cK¯ , and J/cp , theannihilation decay, B¯ 0→Ds1K2, and the b¯→u¯1(cs¯) type ofspectator decay, B0→Ds1p2, both of the theoretical and ex-perimental ambiguities are still large although their measuredrates have been reproduced considerably well by taking ac-count of the nonfactorizable contributions. More theoreticaland experimental studies on these decays will be needed.The author would like to acknowledge Professor T. Onogifor discussions. He also thanks the Yukawa Institute for The-oretical Physics at Kyoto University. Discussions during theYITP workshop, YITP-W-02-08, ‘‘TEA ~Theoretical-Experimental-Astronomical Particle Physics! 02,’’ were use-ful to complete this work.@1# M. 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Newly observed two-body decays of B mesons in a hybrid perspective

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Newly observed two-body decays of B mesons in a hybrid perspective

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