Hadronic B decays involving even-parity charmed mesons

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<ul><li><p>PHYSICAL REVIEW D 68, 094005 ~2003!Hadronic B decays involving even-parity charmed mesons</p><p>Hai-Yang ChengInstitute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China</p><p>~Received 15 July 2003; published 7 November 2003!</p><p>HadronicB decays containing an even-parity charmed meson in the final state are studied. Specifically we</p><p>focus on the Cabibbo-allowed decaysBD** p(r), D** Ds(* ) , Ds** D (* ) and BsDs** p(r), whereD**denotes generically ap-wave charmed meson. TheBD** transition form factors are studied in the improvedversion of the Isgur-Scora-Grinstein-Wise quark model. We apply heavy quark effective theory and chiralsymmetry to study the strong decays ofp-wave charmed mesons and determine the magnitude of theD1</p><p>1/2-D13/2</p><p>mixing angle~the superscript standing for the total angular momentum of the light quark!. Except for the decayto D1(2427)</p><p>0p2 the predictions ofB(B2D** 0p2) agree with experiment. The sign of theD11/2-D13/2mixing angle is found to be positive in order to avoid a severe suppression of the production ofD1(2427)</p><p>0p2.The interference between color-allowed and color-suppressed tree amplitudes is expected to be destructive inthe decayB2D1(2427)0p2. Hence, an observation of the ratioD1(2427)0p2/D1(2427)1p2 can be usedto test the relative signs of various form factors as implied by heavy quark symmetry. Although the predictedB2D1(2420)0r2 at the level of 331023 exceeds the present upper limit, it leads to the ratioD1(2420)r</p><p>2/D1(2420)p2'2.6, as expected from the factorization approach and from the ratiof r / f p</p><p>'1.6. Therefore, it is crucial to have a measurement of this mode to test the factorization hypothesis. ForB</p><p>Ds** D decays, it is expected thatDs0* D*Ds1D as the decay constants of the multiplet (Ds0* ,Ds1) becomethe same in the heavy quark limit. The preliminary Belle observation of fairly less abundant production of</p><p>Ds0* D than Ds1D is thus a surprise. What is the cause for the discrepancy between theory and experimentremains unclear. Meanwhile, it is also important to measure theB decay intoDs1(2536)D</p><p>(* ) to see if it issuppressed relative toDs1(2463)D</p><p>(* ) to test the heavy quark symmetry relationf Ds1(2536)! f Ds1(2463). Under</p><p>the factorization hypothesis, the production ofDs2* D is prohibited as the tensor meson cannot be produced fromthe V2A current. Nevertheless, it can be induced via final-state interactions or nonfactorizable contributions</p><p>and hence an observation ofBDs2* D (* ) could imply the importance of final-state rescattering effects.DOI: 10.1103/PhysRevD.68.094005 PACS number~s!: 13.25.Hw, 12.39.St, 13.25.Ft, 14.40.LbivB</p><p>isurat</p><p>ti</p><p>1</p><p>2.46</p><p>d by</p><p>-</p><p>I. INTRODUCTION</p><p>Interest in even-parity charmed mesons has been revby the recent discovery of a new narrow resonance by Ba@1#. This state, which can be identified withJP501, is</p><p>lighter than most theoretical predictions for a 01 cs state@2#.Moreover, a renewed lattice calculation@3# yields a largermass than what is observed. This unexpected and surprdisparity between theory and experiment has sparked a flof many theory papers. For example, it has been advocthat this new state is a four-quark bound state1 @4,5# or aDKmolecular@9# or even aDp atom @10#. On the contrary, ithas been put forward that, based on heavy quark effectheory and chiral perturbation theory, the newly observ0556-2821/2003/68~9!/094005~15!/$20.00 68 0940edar</p><p>ingryed</p><p>ved</p><p>Ds(2317) is the 01 cs state and that there is a 11 chiral</p><p>partner with the same mass splitting with respect to the2</p><p>state as that between the 01 and 02 states@11,12#. The ex-istence of a new narrow resonance with a mass nearGeV which can be identified with the 11 state was firsthinted at by BaBar and has been observed and establisheCLEO @13# and Belle@14#.</p><p>Although theDs0* (2317) andDs1(2463) states were dis</p><p>covered in charm fragmentation ofe1e2cc, it will bemuch more difficult to measure the counterpart ofDs0* (2317)andDs1(2463) in the nonstrange charm sectornamely,D0*andD1owing to their large widths. Indeed, the broadD0*andD1 resonances were explored by Belle@14# in chargedBand</p><p>n</p><p>t</p><p>l</p><p>e</p><p>e</p><p>1The low-lying noncharm scalar mesons in the conventionalqq8 states are predicted by the quark potential model to lie in between 12 GeV, corresponding to the nonet statesf 0(1370),a0(1450),K0* (1430), andf 0(1500)/f 0(1710). The light scalar nonet formed bys(600),k(800), f 0(980), anda0(980) can be identified primarily as four-quark states@6#. It has been argued@7# that a strong attraction betwee</p><p>(qq)3* and (qq)3 @6,8#, where3* and3 here refer to color, and the absence of the orbital angular momentum barrier in thes-wave four-quark</p><p>state may explain why the scalar nonet formed by four-quark bound states is lighter than the conventionalqq nonet. By the same token, i</p><p>is likely that a scalarcnns four-quark state, wheren5u, d, will be lighter than the 01 p-wavecs state, where a typical potential mode</p><p>prediction gives 2487 MeV@2#. It has been suggested in@4# to search for exotic four-quarkcqqq charmed meson production inB decays.Particularly noteworthy are resonances in the doubly chargedDs</p><p>1p1 (D1K1) and wrong pairingD1K2 channels. However, contrary to th</p><p>case of scalar resonances, the 11 Ds(2463) state is unlikely a four-quark state as it is heavier than the axial-vector meson formed bycs. A</p><p>nonobservation of a heavier and broad 01 cs state will not support the four-quark interpretation ofDs(2317).2003 The American Physical Society05-1</p></li><li><p>op</p><p>th.toao</p><p>isenar</p><p>e</p><p>or</p><p>m</p><p>avfh</p><p>.</p><p>-s</p><p>e</p><p>. A</p><p>ld</p><p>u-</p><p>, 0ld-</p><p>he</p><p>yet</p><p>and</p><p>ts-</p><p>rk</p><p>edoupe</p><p>HAI-YANG CHENG PHYSICAL REVIEW D 68, 094005 ~2003!to D1p2p2 and D* 1p2p2 decays ~see Table I!. Thestudy of even-parity charmed meson production inB decays,which is the main object of this paper, also provides anportunity to test heavy quark effective theory.</p><p>This work is organized as follows. The masses and widof p-wave charmed mesons are summarized in Sec. IIorder to determine the mixing angle of the axial-veccharmed mesons, we apply heavy quark effective theorychiral symmetry to study their strong decays. The decay cstants ofp-wave charmed mesons andBD** form factorsare studied in Sec. III within the Isgur-Scora-Grinstein-Wquark model. The production ofp-wave charmed mesons iB decays is studied in detail in Sec. IV. Conclusionspresented in Sec. V.</p><p>II. MASS SPECTRUM AND DECAY WIDTH</p><p>In the quark model, the even-parity mesons are convtionally classified according to the quantum numbersJ,L,S:the scalar and tensor mesons correspond to2S11LJ5</p><p>3P0 and3P2, respectively, and there exit two different axial-vectmeson statesnamely,1P1 and</p><p>3P1which can undergomixing if the two constituent quarks do not have the samasses. For heavy mesons, the heavy quark spinSQ de-couples from the other degrees of freedom in the hequark limit, so thatSQ and the total angular momentum othe light quarkj are separately good quantum numbers. Ttotal angular momentumJ of the meson is given byJW5 jW</p><p>1SW Q with SW 5sW1SW Q being the total spin angular momentumConsequently, it is more natural to useLJ</p><p>j</p><p>5P23/2, P1</p><p>3/2, P11/2, and P0</p><p>1/2 to classify the first excitedheavy meson states whereL here is the orbital angular momentum of the light quark. It is obvious that the first and laof these states are3P2 and</p><p>3P0, while @16#</p><p>uP13/2&amp;5A2</p><p>3u1P1&amp;1A13u3P1&amp;,</p><p>uP11/2&amp;52A1</p><p>3u1P1&amp;1A23u3P1&amp;. ~2.1!</p><p>In the heavy quark limit, the physical eigenstates withJP</p><p>511 areP13/2 andP1</p><p>1/2 rather than3P1 and1P1.</p><p>The masses and decay widths of even-parity2 ~or p-wave!charmed mesonsDJ* andDsJ* are summarized in Table I. Wshall use 11 and 181 or D1 andD18 to distinguish betweentwo different physical axial-vector charmed meson stateswe shall see below, the physical 11 state is primarilyP1</p><p>1/2,while 181 is predominatelyP1</p><p>3/2. A similar broadD1 statenot listed in Table I was reported by CLEO@17# with M52461235</p><p>142 MeV andG52902831104 MeV. For the known nar-</p><p>2If the even-parity mesons are the bound states of four quathey are in an orbitals wave. In this case, one usesJP501 rather3P0 to denote scalar mesons, for example.09400-</p><p>sInrndn-</p><p>e</p><p>n-</p><p>e</p><p>y</p><p>e</p><p>t</p><p>s</p><p>row resonances, the Belle measurement of theD2*0 width</p><p>~see Table I! is substantially higher than the current woraverage of 2365 MeV @15#.</p><p>In the heavy quark limit, the states within the chiral doblets (01,11) with j 51/2 and (181,21) with j 53/2 aredegenerate. After spontaneous chiral symmetry breaking1</p><p>states acquire masses while 02 states become massless Gostone bosons. As shown in@11#, the fine splitting between 01</p><p>and 02 is proportional to the constituent quark mass. Thyperfine mass splittings of the fourp-wave charmed mesonstates arise from spin-orbit and tensor-force interactions@seeEq. ~2.21! below#, while the spin-spin interaction is solelresponsible for the hyperfine splitting within the multipl(02,12).</p><p>From Table I and the given masses of pseudoscalarvector charmed mesons in the PDG@15#, it is found empiri-cally that the hyperfine splittings within the chiral multiple(01,11), (181,21), and (02,12) are independent of the flavor of the light quark:</p><p>m~D2* !2m~D18!'m~Ds2* !2m~Ds18 !'37 MeV,</p><p>m~Ds1!2m~Ds0* !'m~Ds* !2m~Ds!5144 MeV,</p><p>m~D1!2m~D0* !'m~D* !2m~D !5143 MeV. ~2.2!</p><p>However, the fine splittings</p><p>m~Ds0* !2m~Ds!'m~Ds1!2m~Ds* !'350 MeV,</p><p>m~D0* !2m~D !'m~D1!2m~D* !'430 MeV ~2.3!</p><p>s,</p><p>TABLE I. The masses and decay widths of even-parity charmmesons. We follow the naming scheme of the Particle Data Gr@15# to add a superscript * to the states if the spin-parity is in thnormal sense, JP501,12,21, . . . . The fourp-wave charmedmeson states are thus denoted byD0* , D1 , D18 , and D2* . In theheavy quark limit,D1 hasj 51/2 andD18 hasj 53/2 with j being thetotal angular momentum of the light degrees of freedom.</p><p>State Mass~MeV! Width ~MeV! Ref.</p><p>D0* (2308)0 2308617615628 276621618660 @14#</p><p>D1(2427)0 2427626620615 384275</p><p>1107624670 @14#D18(2420)</p><p>0 2422.261.8 18.923.514.6 @15#</p><p>2421.461.560.460.8 23.762.70.264.0 @14#D18(2420)</p><p>6 242765 2868 @15#D2* (2460)</p><p>0 2458.962.0 2365 @15#2461.662.160.563.3 45.664.466.561.6 @14#</p><p>D2* (2460)6 245964 2527</p><p>18 @15#</p><p>Ds0* (2317) 2317.360.4 ,7 @1,13,14#</p><p>Ds1(2463) 2463.661.761.0 ,7 @13,14#</p><p>Ds18 (2536) 2535.3560.3460.5 ,2.3 @15#Ds2* (2573) 2572.461.5 1524</p><p>15 @15#5-2</p></li><li><p>rsun</p><p>Eqn-</p><p>el</p><p>ee1kth</p><p>ng</p><p>,e</p><p>o</p><p>ofivetrye</p><p>ark</p><p>in</p><p>he,e</p><p>gthe</p><p>up-ble</p><p>so</p><p>-</p><p>HADRONIC B DECAYS INVOLVING EVEN-PARITY . . . PHYSICAL REVIEW D68, 094005 ~2003!depend on the light quark flavor.3 Since the fine splittingbetween 01 and 02 or 11 and 12 should be heavy flavoindependent in the heavy quark limit, the experimental re~2.3! implies that the fine splitting is light quark mass depedent. Indeed, if the first line rather than the second line of~2.3! is employed as an input for the fine splittings of nostrange charmed mesons, one will predict@11#</p><p>M ~D0*6!52217 MeV, M ~D0*</p><p>0!52212 MeV,</p><p>M ~D16!52358 MeV, M ~D1</p><p>0!52355 MeV, ~2.6!</p><p>which are evidently smaller than what are measured by B@14#.</p><p>It is interesting to note that the mass difference betwstrange and nonstrange charmed mesons is of order 100MeV for 02, 12, 181, and 21 as expected from the quarmodel. As a consequence, the experimental factm(Ds0* )'m(D0* ) andm(Ds1);m(D1) is very surprising.</p><p>In the heavy quark limit, the physical mass eigenstatesD1and D18 can be identified withP1</p><p>1/2 and P13/2, respectively.</p><p>However, beyond the heavy quark limit, there is a mixibetweenP1</p><p>1/2 and P13/2, denoted byD1</p><p>1/2 and D13/2, respec-</p><p>tively,</p><p>D1~2427!5D11/2cosu1D1</p><p>3/2sinu,</p><p>D18~2420!52D11/2sinu1D1</p><p>3/2cosu. ~2.7!</p><p>Likewise for strange axial-vector charmed mesons,</p><p>Ds1~2463!5Ds11/2cosus1Ds1</p><p>3/2sinus ,</p><p>Ds18 ~2536!52Ds11/2sinus1Ds1</p><p>3/2cosus . ~2.8!</p><p>SinceD11/2 is much broader thanD1</p><p>3/2 as we shall see shortlythe decay width ofD18(2420) is sensitive to the mixing anglu. Our task is to determine theD1</p><p>1/2-D13/2 mixing angle from</p><p>the measured widths. In contrast, the present upper limitsthe widths ofDs1(2463) andDs18 (2536) do not allow us toget any constraints on the mixing angleus . Hence, we willturn to the quark potential model to extractus as will beshown below.</p><p>3Likewise, considering the spin-averaged masses of the dou(01,11) and (181,21),</p><p>m01~D ![1</p><p>4m~D0* !1</p><p>3</p><p>4m~D1!, m12~D ![</p><p>3</p><p>8m~D18!1</p><p>5</p><p>8m~D2* !,</p><p>~2.4!the hyperfine mass splittings</p><p>m12~D !2m01~D !'48 MeV, m12~Ds!2m01~Ds!'132 MeV~2.5!</p><p>also depend on the light quark flavor. Based on a quark-me</p><p>model, the spin-weighted massesm01(D)52165650 MeV @18#</p><p>and m01(Ds)52411625 MeV @19# were predicted, while experimentally they are very similar~see Table I!.09400lt-.</p><p>le</p><p>n10</p><p>at</p><p>n</p><p>It is suitable and convenient to study the strong decaysheavy mesons within the framework of heavy quark effecttheory in which heavy quark symmetry and chiral symmeare combined@20#. It is straightforward to generalize thformalism to heavy mesons inp-wave excited states@21#.The decayD0* undergoes ans-wave hadronic decay toDp,while D1</p><p>1/2 can decay intoD* by s-wave andd-wave pionemissions but only the former is allowed in the heavy qulimit mc`:</p><p>G~D0* Dp!5gD0* Dp</p><p>2 pc</p><p>8pmD02</p><p>,</p><p>G~D11/2D* p!5gD</p><p>11/2D* p</p><p>2 pc</p><p>8pmD11/2</p><p>2 , ~2.9!</p><p>wherepc is the c.m. momentum of the final-state particlestheB rest frame. The tensor mesonD2* decays intoD* or Dvia d-wave pion emission. In the heavy quark limit where ttotal angular momentumj of the light quark is conservedD1</p><p>3/2Dp is prohibited by heavy quark spin symmetry. Thexplicit expressions for the decay rates are@21#</p><p>G~D13/2D* p!5 1</p><p>6p</p><p>mD*mD</p><p>13/2</p><p>h82</p><p>Lx2</p><p>pc5</p><p>f p2 ,</p><p>G~D2* D* p!51</p><p>10p</p><p>mD*mD2</p><p>h82</p><p>Lx2</p><p>pc5</p><p>f p2 ,</p><p>G~D2* Dp!51</p><p>15p</p><p>mDmD2</p><p>h82</p><p>Lx2</p><p>pc5</p><p>f p2 , ~2.10!</p><p>where Lx is a chiral symmetry breaking scale,f p5132 MeV, andh8 is a heavy-flavor-independent couplinconstant. Thepc</p><p>5 dependence of the decay rate indicatesd-wave nature of pion emission. From Eq.~2.10! we obtain</p><p>G~D2*0D1p2!</p><p>G~D2*0D* 1p2! 5</p><p>2</p><p>3</p><p>mD</p><p>mD*S pc~D2* Dp!pc~D2* D* p! D</p><p>5</p><p>52.3,</p><p>~2.11!</p><p>in excellent agreement with the measured value of 2.360.6@15#.</p><p>Since the d-wave decay is severely phase-space spressed, it is evident thatD0* andD1 are very broad, of order250 MeV in their widths, whereasD18 and D2* are narrowwith widths of order 20 MeV.</p><p>The strong couplings appearing in Eq.~2.9! are given by</p><p>gD0* Dp</p><p>5AmD0mDmD0</p><p>2 2mD2</p><p>mD0</p><p>h</p><p>f p,</p><p>gD11/2D* p5AmD11/2mD*</p><p>mD11/2</p><p>22mD*</p><p>2</p><p>mD11/2</p><p>h</p><p>f p, ~2.12!</p><p>ts</p><p>n</p><p>5-3</p></li><li><p>on</p><p>o</p><p>s</p><p>end</p><p>ley</p><p>re</p><p>f</p><p>a</p><p>th</p><p>tryTs</p><p>ofen</p><p>ibu-</p><p>V</p><p>n-</p><p>to</p><p>ehe</p><p>theethe</p><p>.</p><p>HAI-YANG CHENG PHYSICAL REVIEW D 68, 094005 ~2003!with h being another heavy-flavor-independent coupling cstant in the effective Lagrangian@21#. It can be extractedfrom the measured width ofD0* (2308) ~see Table I! to be</p><p>h50.6560.12. ~2.13!</p><p>From the averaged width 29.464.2 MeV measured forD2*0</p><p>we obtain</p><p>h8</p><p>Lx50.6760.05 GeV21. ~2.14!</p><p>Substituting the couplingsh andh8 into Eqs.~2.9! and~2.10!leads to</p><p>G~D13/2D* p!510.561.5 MeV,</p><p>G~D11/2D* p!5248692 MeV,</p><p>~2.15!</p><p>where we have assumed thatD13/2 has a mass close t</p><p>D18(2420) andm(D11/2)'mD1(2427). Therefore,D1</p><p>3/2 ismuch narrower thanD1</p><p>1/2 owing to the phase-space suppresion for d waves. However, the physical stateD18(2420) canreceive ans-wave contribution as there is a mixing betweD1</p><p>1/2 and D13/2 beyond the heavy quark limit....</p></li></ul>