Inclusive weak decays of charmed mesons

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<ul><li><p>IL NUOVO CIMENTO VOL. 99 A, N. 1 Gennaio 1988 </p><p>Inclusive Weak Decays of Charmed Mesons. </p><p>L. ANGELINI, L. NITTI and M. PELLICOR0 </p><p>Dipartimento di Fisica dell'Universitd - Bari, Italia Istituto Nazionale di Fisica Nucleare, Sezione di Bari - Bari, Italia </p><p>G. PREPARATA (*) </p><p>INFN-Laboratori Nazionali di Frascati - Frascati, Italia </p><p>(ricevuto l'l Giugno 1987) </p><p>Summary. m We show that a complete understanding of the inclusive weak decays of the charmed meson D can be achieved within the realistic approach to quark interactions provided by anisotropic chromodynamics (ACD). A possible qualitative explanation is offered for the observed different behaviours of the D O and F + mesons. We find evidence against the nonleptonic ,,enhancement~ predicted by asymptotic freedom. </p><p>PACS. 14.40.Jz. - Charmed and other heavy mesons and meson resonances. PACS. 13.25. - Hadronic decays of mesons. PACS. 12.90. - Miscellaneous theoretical ideas and models. </p><p>1. - In t roduct ion . </p><p>The physics of weak decays of heavy flavoured mesons has deservedly received a great deal of attention, both experimental and theoretical, because it offers a unique opportunity to test, in a well-defined and limited domain, our ideas about hadron structure and short-distance behaviour. </p><p>The standard theoretical analyses of ref. (1.2) have indeed concentrated their </p><p>(*) On leave of absence from: Dipartimento di Fisica, Universit~ di Bari, Italia. (1) j. ELLIS, M. K. GAILLARD and D. V. NANOPOULOS: Nucl. Phys. B, 100, 313 (1975); N. CABIBBO and L. MAIANI: Phys. Lett. B, 73, 418 (1978); B. FAKIROV and B. STECH: Nucl. Phys. B, 133, 315 (1978). (2) L.L. CHAN: Phys. Rep., 95, 1 (1983); B. STECH: Heidelberg Preprint HD-THEP-85-8 (1985). </p><p>45 </p></li><li><p>46 L. ANGELINI~ L. NI2~I, M. PELLICOR0 and G. PREPARATA </p><p>attention on the very relevant fact that according to our experience with precocious light-cone behaviour, already for charmed quarks (i.e. for a quark mass of the order of 2 GeV) one could use the ideas of asymptotic freedom (AF) to obtain very simple and testable predictions concerning a number of features of inclusive charmed particles' weak decays. In particular, in perturbative QCD it has been demonstrated (~) that to leading order in as, AF predicts the dominance of the spectator diagrams reported in fig. 1, which pictures the nonleptonic inclusive weak decay of charmed mesons as a ~-decay process (') c--. s + u + d, while the semi-leptonic decay is obtained by substituting the quark pair ud with the lepton pair ~ee. </p><p>u &amp; </p><p>C S </p><p>uobs </p><p>Fig. 1. - The spectator diagram for charmed-meson decay. </p><p>The immediate consequence of this extremely simple and appealing picture is that, independently of hadronic structure effects, the semi-electronic branching ratio turns out to be about 20%. A prediction that seems to work quite well for D-mesons but is definitely at variance with the observed D O decays (5). It must be recalled that before the discovery in 1979 (6) of the very different behaviour of D and D O mesons, later confirmed by the direct measurements of lifetimes, the observed average value of the semi-electronic branching ratio was about 10%, in remarkable agreement with a more refined analysis of the perturbative gluonic corrections to the simple diagrams of fig. 1. According to it the effective nonleptonic Hamiltonian differs from the simple W-exchange interaction in the following way(7): (Vcs, V,d are entries of the Kobayashi-Maskawa matrix(S)): </p><p>(1.1) He~ = G/V2vcsVud [cleF(1 - Ys) c~y,(1 - Ys) d + czar'(1 - ~,s) d~,,(1 - ~'s) c], </p><p>(2) N. CABIBBO, G. CORBO and L. MAIANI: Nucl. Phys. B, 155, 93 (1979). (4) In this paper we shall consider only the Cabibbo-allowed portion of the weak Hamiltonian. (~) MARK III COLLABORATION: Phys. Rw. Lett., 54, 1976 (1985). (~) DELCO COLLABORATION: Phys. Rev. Leg., 45, 329 (1980); E531 COLLABORATION: Phys. Rev. Left., 45, 1049 (1980); MARK II COLLABORATION: Phys. Rev. Lett., 45, 39 (1980). (7) For a very recent review see: A. J. PUPAS: in Proceedings of the 1985 EPS Conference on High Energy Physics, edited by L. NITTI and G. PREPARATA (Bali, 1985), p. 1037. (8) M. KOBAYASHI and T. MASKAWA: Prog. Theor. Phys., 49, 652 (1973). </p></li><li><p>INCLUSIVE WEAKS DECAYS OF CHARMED MESONS 47 </p><p>where C+ 4- C_ C+ - - C _ _ </p><p>(1.2) c l - ~ , c2 = 2 </p><p>and </p><p>(1.3) c+ j , d_ =-2d+ =8. </p><p>Setting the relevant mass scale of the decay 1.2GeV</p></li><li><p>48 L. ANGELINI, L. NITTI, M. PELLICORO and G. PREPARATA </p><p>failure of the AF ideas is no surprise to us for, after having developed in the last few years a succesful theoretical framework for computing hadronic properties completely independent of AF, we have now the proof that AF and perturbative QCD are inconsistent with the nonperturbative confined QCD (1~). </p><p>Thus in this paper we shall pursue our ideas unbiased by AF, and replace the effective weak nonleptonic Hamiltonian (1.1) by the simpler form </p><p>(1.5) He~ = G/V2 Vcs Vud ~r~(1 -- rs) CUr,(1 -- rs) d , </p><p>to which (1.1) reduces for C = 1. Anisotropic chromodynamics (ACD)(12), a gauge theory of confined coloured quarks proposed a few years ago, is the theoretical framework in which we shall carry out our computations. Without entering into the details of this approach, which have been reiteratively expounded in the literature (1~), we would like to recall that </p><p>i) ACD allows one to compute the meson spectrum and wave functions of all q~-mesons in terms of a minimal set of inputs (the string tension ~2, and the ,~constituent~ quark masses mq (,4); </p><p>ii) when looked upon inclusively (i.e. summing over the final states) and at high energy (E &gt;&gt; mq) the final quarks behave in ACD as free spin-l/2 fermions of masses m (15.~); </p><p>iii) the weak currents have accordingly simple (Dirac) matrix elements between high-energy quarks. </p><p>Note that, as stressed above, all gluonic effects, in particular those predicted by AF (,7), are neglected, for it is an important qualifying difference of ACD from perturbative QCD that such effects may be (almost) always overlooked (~). </p><p>The plan of our paper is as follows: in sect. 2 we analyse the semi-leptonic decays of charmed mesons, while the spectator diagrams for the nonleptonic decays are calculated in sect. 3. A discussion of the origin, and a possible </p><p>(11) G. PREPARATA: Essential quantum instability of the perturbative Yang-Mills vacuum, preprint BA-GT/85-18. (19 G. PREPARATA: Phys. Lett. B, 102, 327 (1981); 108, 187 (1982). (is) For a review of ACD, see G. PREPARATA in Fundamental Interactions Cargese 1981, edited by M. LEVY, J. L. BASDEVANT, D. SPEISER, J. WEYERS, M. JACOB and R. GASTMANS (New York, N.Y., 1982), p. 421. (14) j . L. BASDEVANT, P. COLANGELO, G. PREPARATA: Yuovo Cimento A, 71, 445 (1982). (15) p. CEA, G. NARDULLI, G. PREPARATA: Z. Phys. C, 16, 135 (1982). (is) L. ANGELINI, L. NITTI, M. PELLICORO and G. PREPARATA: Riv. Nuovo Cimento 6, no. 3 (1983). (lr) M. K. GAILLARD and B. W. LEE: Phys. Rev. Lett., 33, 108 (1974); G. ALTARELLI and L. MAIANI: Phys. Lett. B, 52, 351 (1974). (18) Except for the ~,third,~ jet in e+e - annihilation: L. ANGELINI, L. NYI'rI, M. PELLICORO, G. PREPARATA and G. VALENTI: Phys. Lett. B, 119, 456 (1982). </p></li><li><p>INCLUSIVE WEAKS DECAYS OF CHARMED MESONS 49 </p><p>computation of the difference between the nonleptonic decays of the different charmed mesons is taken up in sect. 4. Finally we devote the last section to the conclusions. </p><p>2. - The semi-leptonic decays. </p><p>The diagram describing the semi-leptonic process </p><p>(2.1) D, F--~e~e+ X </p><p>is reported in fig. 3. According to ACD, in order to completely evaluate this </p><p>v[ </p><p>el, us -~ </p><p>Fig. 3. - The spectator diagram describing the semi-leptonie decay of charmed mesons. </p><p>diagram we need to know the D, F wave functions, and perform the summation over the final states X. In fig. 4 we report the calculated momentum </p><p>1.6 </p><p>~. 1.2 </p><p>0.8 </p><p>0.4 </p><p>i </p><p>0 2.0 0.5 1.0 1.5 p ( GeV/c ) </p><p>2.4- </p><p>2.0 </p><p>Fig. 4. - The momentum distributions Ip~(p)f ~ of constituent quarks inside a D-meson (full line) and a F-meson (dashed line). </p><p>4 - Il Nuovo Cimento A. </p></li><li><p>50 L. ANGELINI, L. NITTI, M. PELLICORO and G. PREPARATA </p><p>distributions fp~(p)I 2 of the c-quark in the charmed mesons D and F, respectively. Note that in ACD (,4) the ,</p></li><li><p>INCLUSIVE WEAKS DECAYS OF CHARMED MESONS 51 </p><p>60 </p><p>50 </p><p>40 </p><p>30 </p><p>~ 20 </p><p>0 </p><p>-1(} 0 I , OL.6 [ 0.2 0.4 0.8 1.4 </p><p>p(GeV/c) </p><p>T ,V++ I I </p><p>1.0 1.2 </p><p>Fig. 6. - Our prediction for the leptonic spectrum of D-decay compared with experimental data (19). </p><p>(number of spin states) by phase space. In this way we obtain </p><p>B(D--) Ke ~e) (2.4a) - 1.4, </p><p>B(D--. K*e%e) D 1.5 + 0.8 (9) </p><p>exp. DO 0.8 ___ 0.5 ' </p><p>B(D--. K= e%) (2.4b) - 0.065, exp. = 0.07 (lo), </p><p>B(D--* (K + K*)e~) </p><p>again in agreement with data. </p><p>3. - The spectator d iagrams o f non lepton ic decays. </p><p>If in fig. 5 we replace the lepton pair ~ee with a quark pair ud we obtain the spectator diagrams of nonleptonic decays. By following the colour degree of freedom of the quarks we can separate two distinct contributions: the colour- enhanced diagram, where the ud -pair gives rise to a physical (colourless) hadron state, and the colour-suppressed diagram, where the u-quark combines with the spectator antiquark, while the d combines with the s-quark. The latter contribution is about a factor of 9 smaller than the former. The numerical evaluation of these contributions yields for the spectator nonleptonic widths </p><p>(3.1a) F~(D +') = 5.9- 1011 s- 1 , </p><p>(3. lb) F~(F ) = 7.4.1011 S- 1 </p></li><li><p>52 L. ANGELINI, L. NITTI, M. PELLICORO and G. PREPARATA </p><p>On the other hand, the measured nonleptonic widths are (20) </p><p>(3.2a) /'u(D ) = (7.3 + 2.8). 1011 s- 1, </p><p>(3.2b) F,e(D ) = (1.95 + 0.9). 10 TM s-1. </p><p>Thus we get good agreement only for the D+-decay, while D O is off about a factor of 2. In the next section we shall discuss the probable explanation for such discrepancy. </p><p>4. - The nonspectator contributions to the nonleptonic decays. </p><p>The contributions to the nonleptonic decay rate we wish to discuss in this section are of the general type depicted in fig. 7, where by emission and reabsorption of a W-boson the charmed meson turns itself into a pair q~ of light quarks. </p><p>W ~ </p><p>D~F @ ~ q , (- 'T:~ho~rons </p><p>{ _ ~ </p><p>q </p><p>Fig. 7. - The diagram contributing to the nonleptonic partial widths of D O and F mesons. </p><p>It is immediate to verify that this mechanism can contribute to D O and F but not to D+-decay, for there exists no s~ pair of positive charge. We have thus a natural explanation for the good agreement between (3.1a) and (3.2a). By inserting between the two weak-current operators a complete set of intermediate states and dominating them by qq-states (21) we realize at once that the diagrams in fig. 2 are obtained by keeping for D O the ACD q~-continuum and for F the vacuum. </p><p>In ACD the diagrams in fig. 2 can be easily evaluated from the detailed knowledge of Dad F wave functions and the result is </p><p>(4.1a) /'.s(D ) = (0.62- 101) s- 1 , </p><p>(4.1b) /'.~(F +) = (1.07" 1010) S -1 , </p><p>(~) We calculate the nonleptonic widths using the world averages of the lifetimes and of the semi-leptonic branching ratios (lo). (21) This dominance has been discussed and justified for exotic (in the sense of Regge theory) W + W--channels in K ~onleptonic decays in G. PREPARATA: Phys. Lett. B, 34, 412 (1971); G. NARDULLI, G. PREPARATA and D. ROTONDI: Phys. Rev. D, 27, 557 (1983). </p></li><li><p>INCLUSIVE WEAKS DECAYS OF CHARMED MESONS 53 </p><p>W- </p><p>D C ~ S </p><p>~) </p><p>W + </p><p>F c ~ - ~ u s U o1, </p><p>b~ ) </p><p>W &amp; </p><p>~)c x ~ u </p><p>b z ) </p><p>Fig. 8. - The relevant diagrams with no helicity suppression contributing to the nonleptonic partial widths o D o and F + mesons. </p><p>which are far smaller than required by the experimental values (3.2b) and basically in agreement with previous estimates (22). We believe that the most likely explanation of the missing nonleptonic rate in D O and F decay lies in contributions to the mechanism in fig. 7 which do not suffer from light-quark helicity suppression. The most likely candidates for such contributions are reported in fig. 8. The evaluation of these diagrams, although straightforward, is rather laborious. We hope to be able to report on our calculation soon. </p><p>Assuming that we have correctly isolated the nonspectator contributions to the D O and F nonleptonic decays, we can evaluate the charged multiplicity in D O decay, by treating the final light-quarks pair as a fire string (16) and using the EPOS decay program (=), already succesfully tested in e e--physics. We obtain for the average charged particles' multiplicities </p><p>(4.2a) D~ = 2.26, exp. 2.16+0.16(u) , </p><p>(4.2b) D = 2.8, exp. 2.46 + 0.14 (u), </p><p>(4.2c) F+ = 2.8 , </p><p>again in good agreement with experiment. </p><p>(22) V. BARGER, J. P. LEVEILLE, P. M. STEVENSON, R. J. N. PHILLIPS: Phys. Rev. Lett., 45, 83 (1980). (~) L. ANGELINI, a. NITTI, M. PELLICORO, G. PREPARATA and G. VALENTI: Comp. Phys. Comm., 34, 371 (1985). (u) MARK II COLLABORATION: Phy8. Rev. D, 24, 78 (1981). </p></li><li><p>54 L. ANGELINI, L. NITTI, M. PELLICORO and G. PREPARATA </p><p>We end this section by recalling that we have obtained a natural and complete explanation for the D+-decays, while we hope to have isolated the important (helicity-unsuppressed) contributions to the nonspectator nonleptonic diagrams. </p><p>5. - Conc lus ions . </p><p>The main achievement of this paper is, in our opinion, a full quantitative understanding of D decay, and a qualitative insight upon its difference with F and D O decays. In particular the almost perfect agreement found between our calculation of the semi-leptonic branching ratio of D and experiment leaves little or no room for the nonleptonic enhancement (1.4) predicted by AF. In addition we have correctly explained the lepton spectrum as well as the relative ratio between K, K* and higher resonances in semi-leptonic D decay. Finally, assuming that we have isolated the correct mechanism accounting for the missing nonleptonic rate in D O and F decay, we have computed the average charged multiplicities again in good agreement with the experimental information. </p><p>It is our feeling that, once freed from the constraints imposed by AF, the physics of the weak decays of charmed mesons resumes a simplicity and beauty that seemed definitely lost. </p><p> RIASSUNTO </p><p>Si mostra che una comprensione completa dei decadimenti deboli inclusivi dei mesoni D + pub essere conseguita nell'ambito delrapproccio realistico alle interazioni tra quark fornito daUa cromodinamica anisotropa (ACD). Si presenta una possibile spiegazione qualitativa per il diverso comportamento osservato nel caso dei mesoni D O ed F . 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