Weak hadronic decays of charmed mesons emitting pseudoscalar and axial-vector mesons

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<ul><li><p>Weak hadronic decays of charmed mesons emitting pseudoscalar and axial-vector mesons</p><p>Neelesh Sharma and R. C. VermaDepartment of Physics, Punjabi University, Patiala-147 002, India</p><p>(Received 31 March 2005; published 11 January 2007)</p><p>In this paper, we investigate phenomenologically several weak decays of charmed mesons emitting apseudoscalar meson and an axial-vector meson. Decay amplitudes are obtained using the factorizationscheme in the spectator quark model. Branching ratios for the Cabibbo angle-favored, Cabibbo angle-suppressed, and Cabibbo angle-doubly-suppressed decays are obtained and compared with availableexperimental results.</p><p>DOI: 10.1103/PhysRevD.75.014009 PACS numbers: 13.25.Ft, 14.40.Aq, 14.40.Ev, 14.40.Lb</p><p>I. INTRODUCTION</p><p>The spectator model using the factorization ansatz hasachieved substantial success [14] in explaining most ofthe exclusive two-body D-meson decays emitting s-wavemesons. The model involves the expansion of transitionamplitudes in terms of a few form factors, which provideessential information on the structure of the mesons.Therefore, it is natural to extend the phenomenologicalstudy to charmed meson decays emitting p-wave mesons[510]. Theoretically, these decays are expected to besuppressed, as there is less phase space available.However, the measured branching ratios for the observedmodes are found to be rather large.</p><p>In the present paper, we study two-body weak hadronicdecays emitting an axial-vector meson A1 and a pseu-doscalar meson P0 in the Cabibbo-favored mode, theCabibbo-suppressed mode, and the Cabibbo-doubly-suppressed mode. Using the factorization scheme to obtainthe decay amplitudes, we calculate the branching ratios ofthese decay modes.</p><p>The paper is organized as follows. In Sec. II, we discussthe axial-vector meson spectroscopy. Methodology forcalculating D ! PA decays is presented in Sec. II.Section IV gives numerical results, and discussions andconclusions are given in the last section.</p><p>II. AXIAL-VECTOR MESON SPECTROSCOPY</p><p>Experimentally, two types of axial-vector mesons,AJPC 1 and A0JPC 1, behave well with re-spect to the quark model q q expectations. The JPC 1nonet has two isoscalar states, besides the isovector non-strange a11:260 mesons and isospinor strange mesons.There exist three good 1 candidates, f11:282,f11:420, and f011:512, one more than the expectednumber. This indicates that one of the three mesons is anon-q q state. Particle Data Group [11] indicates thatf11:420 is a multiquark state in the form of a K K boundstate [12] or a K K deuteronlike state [13]. In the presentanalysis, we define mixing of the isoscalar states as</p><p> f11:285 12</p><p>p u u d d cosA ss sinA;</p><p>f011:512 12</p><p>p u u d d sinA ss cosA:(1)</p><p>Similarly, mixing of two isoscalar mesons, h11:170 andh011:380, is defined as</p><p> h11:170 12</p><p>p u u d d cosA0 ss sinA0 ;</p><p>h011:380 12</p><p>p u u d d sinA0 ss cosA0 :(2)</p><p>Proximity of a11:260 and f11:285 and, to a lesserextent, that of b11:235 and h11:170 indicates the idealmixing for both 1 and 1 nonets, i.e.,</p><p> A A0 0: (3)This is also supported by their decay patterns. f11:285decays predominantly to 4 and , while f011:512decays to K K. Similarly, h11:170 decays predomi-nantly to , and h011:380 decays to K K and KK states.</p><p>Experimentally, the isodoublet strange mesonsK11:270 and K11:400 are given by a mixture of 3P1and 1P1 states [14],</p><p> K1 K1A sin K1A0 cos;K1 K1A cos K1A0 sin;</p><p>(4)</p><p>where K1A and K1A0 are the strange partners of a11:260and b11:235, respectively. Particle Data Group [11] as-sumes maximal mixing ( 45). From the experimentalinformation on masses and partial rates of K11:270 andK11:400, Suzuki [15] found two possible solutions with atwofold ambiguity, 33 and 57.</p><p>For and 0 pseudoscalar states, we use</p><p> 0:547 12</p><p>p u u d d sinP ss cosP;</p><p>00:958 12</p><p>p u u d d cosP ss sinP;(5)</p><p>where PidealPphysical:Pphysical10</p><p>PHYSICAL REVIEW D 75, 014009 (2007)</p><p>1550-7998=2007=75(1)=014009(9) 014009-1 2007 The American Physical Society</p><p>http://dx.doi.org/10.1103/PhysRevD.75.014009</p></li><li><p>and 23 for quadratic and linear mass formulas [11],respectively.</p><p>III. METHODOLOGY</p><p>A. Weak Hamiltonian</p><p>The general current current weak Hamiltonians HWfor charmed changing modes are classified as follows:</p><p>(i) Cabibbo-favored C S 1 decays,</p><p> HW GF2</p><p>p VudVcsa udsc a2sd uc; (6a)</p><p>(ii) Cabibbo-suppressed C 1;S 0 decays,</p><p> HW GF2</p><p>p VudVcdaf ud dc usscg</p><p> a2f dd uc ss ucg; (6b)taking VusVcs VudVcd;</p><p>(iii) Cabibbo-doubly-suppressed C S 1decays,</p><p> HW GF2</p><p>p VusVcda us dc a2 ds uc: (6c)</p><p>Here, q q is shorthand for a color singlet combina-</p><p>tion q1 5q. The parameters a1 and a2 relateto the short-distance QCD Wilson coefficients. Forhadronic charmed decays, we use a1 1:26 anda2 0:51.</p><p>B. Decay amplitudes and rates</p><p>The decay rate formula for D ! PA decays is given by</p><p> D ! PA p3C</p><p>8m2AjAD ! PAj2; (7)</p><p>where pC is the magnitude of the three-momentum of afinal-state particle in the rest frame of the D meson and mAdenotes the mass of the axial-vector meson.</p><p>The factorization scheme expresses the decay ampli-tudes as a product of the matrix elements of weak currents(up to the scale factor of GF</p><p>2p CKM elements) as</p><p> AD ! PA hPjJj0ihAjJjDi hAjJj0ihPjJjDi;AD ! PA0 hPjJj0ihA0jJjDi hA0jJj0ihPjJjDi:</p><p>(8)</p><p>Using Lorentz invariance, matrix elements of the currentbetween meson states can be expressed [5,11] as</p><p> hPjJj0i ifPk; hAjJj0i 2 mAfA; hA0jJj0i 2 mA0fA0 ;hAPAjJjDPDi l 2 c2 PDPD PA c2 PDPD PA;hA0PA0 jJjDPDi r 2 s2 PDPD PA0 s2 PDPD PA0 ;</p><p>(9)</p><p>and</p><p> hPPPjJjDPDi PD PP</p><p>m2D m2Pq2</p><p>q</p><p>FDP1 q2</p><p>m2D m2Pq2</p><p>qFDP0 q2;</p><p>which yield</p><p> AD ! PA 2mAfAFD!P1 m2A fPFD!Am2P;AD ! PA0 2mA0fA0FD!P1 m2A0 fPFD!A</p><p>0 m2P;(10)</p><p>where</p><p> FD!Am2P l m2D m2Ac m2Pc;FD!A0 m2P r m2D m2A0 s m2Ps:</p><p>(11)</p><p>C. Decay constants and form factors</p><p>Decay constants of pseudoscalar mesons are wellknown. However, for axial-vector mesons, decay constantsfor JPC 1 mesons may vanish due to the C-paritybehavior. Under charge conjugation, the two types of axial-vector mesons transform as</p><p>Mab1 ! Mba1Mab1 ! Mba1</p><p>a; b 1; 2; 3</p><p>where Mab denotes meson 3 3 matrix elements in SU3flavor symmetry. Since the weak axial-vector current trans-forms as Aab ! Aba under charge conjugation, onlythe 1 state can be produced through the axial-vectorcurrent in the SU3 symmetry limit [15]. In this work, weuse the following values of decay constants [11,16,17] ofthe axial-vector 1 mesons and pseudoscalar 0 me-sons:</p><p>NEELESH SHARMA AND R. C. VERMA PHYSICAL REVIEW D 75, 014009 (2007)</p><p>014009-2</p></li><li><p> fa1 0:203 GeV; fK1A 0:175 GeV;ff1 ff01 0:221 GeV; f 0:133 GeV;fK 0:160 GeV;f 0:133 GeV; and f0 0:126 GeV:</p><p>For FDP form factors, experimental branching ratios[11] of semileptonic decays D ! K= l l providethe following values:</p><p> FDK1 0 FDK0 0 0:76;FD1 0 FD0 0 0:69; FD1 0 0:68;FDsK1 0 0:64; and FDs1 0 0:72</p><p>which match well with the form factors given by the Bauer-Stech-Wirbel (BSW) model [1]. Therefore, we take theBSW form factors for other FDP transitions. Momentumdependence of the form factors is taken as</p><p> Fq2 F01 q2=m2n(12)</p><p>with pole mass m given by vector meson masses. Theoriginal BSW model [1] assumes a monopole behavior</p><p>(n 1) for both the form factors, which is not consistentwith the heavy quark symmetry scaling relations for heavy-to-light transitions. However, in the modified BSW model[18], consistency with the heavy quark symmetry is re-stored by taking dipole behavior for q2 dependence for theform factor F1. Since the BSW model provides the formfactors only for P0 ! P0 or P0 ! V1 tran-sitions, we calculate form factors FDA and FDA</p><p>0using the</p><p>ISGW quark model [5].</p><p>IV. NUMERICAL RESULTS AND DISCUSSIONS</p><p>Sandwiching the weak Hamiltonian (6) between theinitial and the final states, the decay amplitudes for variousD ! PA decay modes are obtained by using (8) and (9).The decays can be categorized as follows:</p><p>(I) involving P0 ! P0 transitions only,(II) involving P0 ! A1 transitions only, and</p><p>(III) involving both P0 ! P0=A1 transitions.Their respective decay amplitudes are given in Tables I, II,and III. Branching ratios obtained for these categories aregiven in Tables IV, V, and VI, respectively. For theCabibbo-favored decays, though many decay channelsare available for D mesons, the experimental measure-</p><p>TABLE I. Decay amplitudes for D ! PA decays involving P0 ! P0 transitions.Decay Decay amplitude</p><p>(a) Cabibbo-favored decays GF2</p><p>p cos2cD0 ! Ka1 2a1ma1fa1FD!Km2a1 D0 ! 0 K01</p><p>2</p><p>psina2mK1fK1AF</p><p>D!m2K1 D0 ! 0 K01</p><p>2</p><p>pcosa2mK1fK1AF</p><p>D!m2K1 D0 ! K01</p><p>2</p><p>psin sinpa2mK1fK1AF</p><p>D!m2K1 Ds ! K K01 2 sina2mK1fK1AFDs!Km2K1 Ds ! K K01 2 cosa2mK1fK1AFDs!Km2K1 Ds ! a1 2 cospa1ma1fa1FDs!m2a1 (b) Cabibbo-suppressed decays GF</p><p>2p sinc cosc</p><p>D0 ! a1 2a1ma1fa1FD!m2a1 D0 ! KK1 2 sina1mK1fK1AFD!Km2K1 D0 ! 0f01 </p><p>2</p><p>pa2mf01ff01F</p><p>D!m2f01 D ! K0K1 2 sina1mK1fK1AFD!Km2K1 D ! f01 2a2mf01ff01FD!m2f01 Ds ! K0a1 2a1ma1fa1FDs!Km2a1 Ds ! Ka01</p><p>2</p><p>pa2ma1fa1F</p><p>Ds!Km2a1 Ds ! Kf1 </p><p>2</p><p>pa2mf1ff1F</p><p>Ds!Km2f1 (c) Cabibbo-doubly suppressed decays GF</p><p>2p sin2c</p><p>D0 ! 0K012</p><p>psina2mK1fK1AF</p><p>D!m2K1 D0 ! 0K01</p><p>2</p><p>pcosa2mK1fK1AF</p><p>D!m2K1 D0 ! K1 2 sina1mK1fK1AFD!m2K1 D0 ! K1 2 cosa1mK1fK1AFD!m2K1 D0 ! K01</p><p>2</p><p>psin sinpa2mK1fK1AF</p><p>D!m2K1 D ! K01 2 sina2mK1fK1AFD!m2K1 D ! K01 2 cosa2mK1fK1AFD!m2K1 D ! 0K1 </p><p>2</p><p>psina1mK1fK1AF</p><p>D!m2K1 D ! 0K1 </p><p>2</p><p>pcosa1mK1fK1AF</p><p>D!m2K1 D ! K1</p><p>2</p><p>psin sinpa1mK1fK1AF</p><p>D!m2K1 </p><p>WEAK HADRONIC DECAYS OF CHARMED MESONS . . . PHYSICAL REVIEW D 75, 014009 (2007)</p><p>014009-3</p></li><li><p>ments for branching ratios are available only for D0 !Ka1 , D</p><p> ! K0a1 , D0 ! K1 , D ! K01, and ex-perimental upper limits are available for D0 ! 0 K01,D0 ! 0 K01, D0 ! K0a01, D0 ! K1 , and D ! K01.</p><p>In the earlier work [6], BD0 ! Ka1 1:46% andBD ! K0a1 3:75% were obtained using monopoleq2 dependence of the FDP1 q2 form factor, which are muchless than the experimental values. Recently, Cheng [16] haspointed out that the q2 dependence of the form factorFDP1 q2 should have dipole form rather than the monopoleform in order to be consistent with the heavy quark sym-metry, which yields a higher branching ratio for D ! Ka1decays. In the present work, using the dipole q2 depen-dence, we calculate BD ! K0a1 9:45% whichagrees well with the experimental value 8:2 1:7%.The calculated branching ratio BD0 ! K0a01 0:004%is consistent with the experimental upper limit </p></li><li><p>AD0!Ka1 A Ka1 expiKa11=2 </p><p>1r</p><p>Ka1</p><p>2expi Ka1</p><p>;</p><p>AD0! K0a0112</p><p>p A Ka1 expi Ka11=2 </p><p>1r Ka1 expi Ka1;AD! K0a1 A</p><p>Ka13=2 expi</p><p>Ka13=2 ; (14)</p><p>where A Ka1 2=3A Ka11=2 , r Ka1 AKa13=2 =A</p><p>Ka11=2 , and the phase</p><p>difference</p><p> Ka1 Ka11=2 Ka13=2 :</p><p>Thus, elastic FSI yield the following branching ratios (%):</p><p> BD0 ! Ka1 1:431:195 cos Ka1;BD0 ! K0a01 1:421:003 cos Ka1:</p><p>(15)</p><p>It is obvious that any nonzero value of the phase difference Ka1 will enhance D0 ! K0a01 and deplete D0 ! Ka1 .Thus, we find that the elastic FSI works in the wrongdirection here. Besides the elastic FSI, these decays maybe affected by inelastic FSI involving quark exchangediagrams, because the produced quarks have enough timeto rearrange before combining to form the final-state had-</p><p>rons. Recently, Cheng [16] has shown that the FSI caninduce large long-distance W-exchange terms. Assumingthat the data analysis [4] for D ! K decay holds also forD ! Ka1, he has obtained BD0 ! Ka1 6:2% ingood agreement with the experimental value 7:21:1%. Note that the branching ratio of BD ! K0a1 9:45% remains unaffected by the W-exchange process andthe FSI effects.</p><p>For D ! K1= K1 decays modes, we have calculatedthe branching ratios for different choices of the K1A-K1A0mixing angle ( 33, 45, 57) using dipole q2 behaviorof the FDP1 q2 form factors. Theoretical values of branch-ing ratios of D0 ! K1 =0 K01, D ! K01, and D0 !0 K01 are consistent with experimental data for all themixing angles. However, the experimental upper limitBD ! K01 favors the choice of 33, which inturn implies BD ! K01 6:52%, BD0 !K1 0:17%, and BD0 ! 0 K01 0:69%. Wewish to remark here that 33 has been obtained earlierby Godfrey and Isgur [14] in a unified quark model analy-sis, and also favored by K11:400 production in decays[15]. This choice of the mixing angle yields the largestbranching ratio BD0 ! K1 0:16%, though it isstill lower than the observed value 1:14 0:31%.Similar to the D0 ! Ka1 decays, this decay mode isalso likely to have contribution from the W-annihilationand W-exchange processes. Including the factorizable con-</p><p>TABLE III. Decay amplitudes for D ! PA decays involving P0 ! P0=A1 transitions.Decay Amplitude</p><p>(a) Cabibbo-favored decaysD ! K0a1 2a1ma1fa1FD!Km2a1 a2fKFD!a1 m2K1 D ! K01 2 sina2mK1fK1AFD!m2K1 sina1fFD!K1A m2 cosa1fFD!K1A0 m2D ! K01 2 cosa2mK1fK1AFD!m2K1 cosa1fFD!K1A m2 sina1fFD!K1A0 m2(b) Cabibbo-suppressed decaysD0 ! 0a01 a2ma1fa1FD!m2a1 a2f2p FD!a1 m2D0 ! a01 sinpa2ma1fa1FD!m2a1 </p><p>cosp2</p><p>p sinp2 a2fFD!a1 m2D0 ! 0f1 a2mf1ff1FD!m2f1 </p><p>a2f2</p><p>p FD!f1 m2D0 ! f1 sinpa2mf1ff1FD!m2f1 </p><p>cosp2</p><p>p sinp2 a2fFD!f1 m2D ! a01</p><p>2</p><p>pa2fa1ma1F</p><p>D!m2a1 a1f2p FD!a1 m2D ! 0a1</p><p>2</p><p>pa1fa1ma1F</p><p>D!m2a1 a2f2p FD!a1 m2D ! a1 </p><p>2</p><p>psinpa1ma1fa1F</p><p>D!m2a1 cosp sinp</p><p>2p a2fFD!a1 m2</p><p>D ! f1 2</p><p>pa2mf1ff1F</p><p>D!m2f1 a1f</p><p>2p FD!f1 m2</p><p>Ds ! K1 2 cosp sina1mK1fK1AFDs!m2K1 cosp sinsinp sin</p><p>2p a2fFDs!K1A m2</p><p> cos cosp cos sinp2p a2fFDs!K1A0 m2Ds ! K1 2 cos cospa1mK1fK1AFDs!m2K1 cos cosp </p><p>cos sinp2</p><p>p a2fFDs!K1A m2cosp sin sin sinp2p a2fFDs!K1A0 m2(c) Cabibbo-doubly-suppressed decays</p><p>Ds ! KK01 2 sina2mK1fK1AFDs!Km2K1 sina1fKFDs!K1A m2K cosa1fKFDs!K1A0 m2KDs ! KK01 2 cosa2mK1fK1AFDs!Km2K1 cosa1fKFDs!K1A m2K sina1fKFDs!K1A0 m2KDs ! K0K1 2 sina1mK1fK1AFDs!Km2K1 sina2fKFDs!K1A m2K cosa2fKFDs!K1A0 m2KDs ! K0K1 2 cosa1mK1fK1AFDs!Km2K1 cosa2fKFDs!K1A m2K sina1fKFDs!K1A0 m2K</p><p>WEAK HADRONIC DECAYS OF CHARMED MESONS . . . PHYSICAL REVIEW D 75, 014009 (2007)</p><p>014009-5</p></li><li><p>tribution of such diagrams, the decay amplitudes of D ! K1 get modified to (leaving aside the scale factorGF2</p><p>p cos2C) </p><p>AD0 ! K1 a1fFD!K1m2 a2fDFK1!m2D;</p><p>AD0 ! 0 K01 12</p><p>p a22mK1fK1AFD!m2K1</p><p> a2fDFK1!m2D; (16)</p><p>where</p><p> FD!K1 sinFD!K1A cosFD!K1A0 ;FK1! sinFK1A! cosFK1A0!;</p><p>and</p><p> fK1 fK1A sin fK1A0 cos:</p><p>For fD 0:3 GeV and 33, we find that the experi-</p><p>mental value BD0 ! K1 1:14 0:31% requiresFK1!m2D 1:33 0:29 GeV. This in turn enhan-ces the branching ratio for D0 ! 0 K01 to 0:55 0:06%,which remains well below the experimental upper limit</p></li><li><p>V. CONCLUSIONS</p><p>In this paper, we have studied hadronic weak decays ofcharmed mesons into pseudoscalar and axial-vector me-sons in Cabibbo-favored, Cabibbo-suppressed, and</p><p>Cabibbo-doubly-suppressed channels. At present, experi-mental information is available only for D !Ka1= K1= K1 decay modes. We make the following</p><p>conclusions:(i) D decays are well understood with the dipole be-</p><p>TABLE V. Branching ratio for D, Ds meson decays involving P0 ! A1 transitions.Decays Br (%) 33 Br (%) 45 Br (%) 57 Experiment (%)(a) Cabibbo-favored decaysD0 ! K0a01 0.004 0.004 0.004 </p></li><li><p>havior for q2 dependence of the form factor FDP1 q2,which is justified in light of the heavy quark sym-metry arguments.</p><p>(ii) Available data for D ! K1= K1 decay modesfavor 33 for K1 K1 mixing.</p><p>(iii) The D0 decays seem to require siza...</p></li></ul>