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a* IC/83/122 IMTERIIAL REPORT I. INTRODUCTION (Limited distribution) International Atomic Energy Agency In 1979, and Fayyazuddin ±' deduced the £1 = 4- rule and the

and (d/f)w ratio in ncn-leptonic weak decays cf non-churned harycms by considering Iducational Scientific and Cultural Organization weak- boson exchange graphs as the finalogue of gluon exchange. In tliis

IIONAL CENTRE FOB THEORETICAL PHYSICS paper we apply the same standard current algebra techniques, extended to SU(U), and obtain a good fit to the known non-leptonic ds_vay rates A _ •* pK , A , -»• An+ using the same scale parameter of Riazuduin and Fayyazuddin . We compare our analysis with model results and with a recent calculation, using the HIT Dag.

HON-LEPTOHIC WEAK DECAYS OF CHARMED BAKYC8S * II. COMBINING CURRENT ALGEBRA AND QUARK AMPLITUDES

F. Hussain** and H. Scadron*** The starting point of our analysis is the veak Hamiltonian density International Centre for Theoretical Physics, Trieste, .

with the hadronic part of the weak V-A left-handed SU(M quark current ' ' given by ABSTRACT

The non-leptonic decay rates A pK Ai pK and AA A are calculated +• using current algehra and an evaluation of the matrix element /B |H^' '\B. > using non-relativistic SU(6) wave functions. The results are found to he in (2) good agreement with experiment. The results are also compared with earlier _ t- o quark model and MIT bag model calculations. with G = 1.026 x 10 m~ and where u,d,s and c stand for the respective quark fields and 6 is the Cabibbo angle. As a first approximation we ignore ° 5) the short distance QCD effects so that the Cabibbo-enchanced charm changing effective Hamiltonian is

MIRAMARE - TRIESTE (3) August 1983 This obeys the selection rule AC = AS = AI = 1. Later we shall comment upon short distance modifications of ('}), * To be submitted for publication. 2) The matrix element for baryon decay processes can be written as •* Permanent address: Department of Physics, Quaid-i-Azam University, , . *•* Supported in part by the U.S. Department of Energy under Contract So, DE-ACO2-8OER1O663. M = - (h) Permanent address: Physics Department, University of Arizona, Tucson, Arizona 85721, USA. -2- III. CHARMED ** +I11 AND A +pK DECAYS where A and B are the (parity violating) s-wave and (parity conserving) p- wave amplitudes respectively. The three-hadron matrix elements of the weak Since there is experimental data on only two charmed hyperon decays Hemiltonian may be reduced to baryon 10 baryon .ransition elements of H,, by A+ -+ Air* and A+ -*• pK we shall give the details of the calculation and the applying standard soft-meson techniques , to give results for just these decays. The relevant formula for these two processes are: (5) 1. A+ •* An c , r In this case the current commutator terra vanishes so that Here Ql is the axial generator associated with the meson P1' and f is the 5 P corresponding pseudoscalar meson decay constant (f =93 MeV, f,, = 112 HeV). A '- A (9a) H (q) are possible pole term contributions and M (oj represent the factorization (quark decay diagram) contributions. The V-A structure of Hw leads to [Q^,K ] = -[Q^HU] so that i

(6) (9b) where QJ is the SU(lt) charge with the quantum numbers of the meson ?^ and 2. At - PK° operates on the baryon states to its left and right as an SU(1O generator.

The s-wave contributions of the -r baryon pole terms are suppressed, A -- A, (10a) whereas the commutator term contributes only to the s-wave amplitudes A so that A (10b) = " f (7a)

(7b) Here the masses of the baryons are denoted by the corresponding particle symbols, f and d denote the usual SU{3) antisymmetric and symmetric raeson- where B . is obtained from Fig.l as baryon coupling with f + d = 1 and f/d = 1/2. g = 13.1»5 is the pion-nucleon pole ° coupling constant,

C e> The task now is to compute ^B |H^' '| B^ . Riazuddin and (8) Fayyaauddin calculated these matrix elements for non-charmed hyperon decays ^« J using non-relativistic SU(2) quark wave functions for the baryons. We repeat the same calculation noting that the wave functions for A , E are obtained with ^n|H^*c' |i*^ • n^'c' . Here g is the strong coupling constant. 8) from the corresponding strange hyperons by making the replacement s •+ c. Note that because QJ is a generator of SV{h) and the 5- baryon states The quark scattering diagram is shown in Fig.2. we are considering are members of the 20 multiplet of SU(li), both amplitudes The effective Hy is the Fourier transform of the non-relativistic are described as a sum of terms involving transitions of the form {.B |HJ?"°*|B "} limit of Fig.2. This gives in leading order Finally, using SO(U), the factorization contributions can be specified in terms of the measured form factors of current transitions between knovn bsryons as has been discussed by Buxas 7) c ±-<5 &.*«,!( +- u.c J M.J = (11) (16)

where we have included the colour suppression factor 1/3. Thus the amplitude and KrL" ' = 0. Here a. and B, are operators which respectively transform for Fig.U Is a c-quark into an s-quark and a d-quark into a u-quark. Working out the spin and unitary spin components of the various constituent quark baryon transitions, 6C-I using the non-relativistic SU(6) wave functions , we find (17)

, P <-i The current matrix elements in (lit) and (17) have been related by Euras to (12) the measured form factors of current transitions involving known baryons. vhere is the transition matric element evaluated by Riazuddin 1) 2) and Fayyazuddin ' . Thus IV. RESULTS

Finally we get (13) I- is. Co«i. i) where a 1 GeV) ^ 0-5, I, A denote the masses of the respective baryons, m and m are the constituent masses of the non-strange and strange A = - «J? (18a) respectively with fi = 3^0 MeV and m = 510 MeV. The factorization or quark diagram contributions to A + Air and 6 -

(HO (18b)

To obtain the amplitude for Fig.lt it is convenient to perform the 2) Fierz transformation 3 , (15)

This equation is valid for V-A currents and Fermi statistics of the quarks. (19a) It transforms the Hamiltonian to a. form containing neutral V-A currents. Because the particles are colour singlets the amplitude corresponding to Fig. is obtainable from the effective Hamiltonian

-5- -6- (£1) B - The new effective Hamiltonian will change the quark diagram amplitude for A +*i (Fig.3) by the factor C + *• C and the amplitude corresponding to Fig.l* by the factor C, + 3C . Though these effects are appreciable they (19b) are not as significant as the effect of employing H of Eq.(21) in evaluating the matrix element Xl |H*^* " ]A \ . Usingithe values of C and £ 3* As described in Ref.T the q = 0 values of the form factors H C2 preferred by Korner et al., C^ = 1.315, CU = -O.585 we find that the and ICC are fixed from the vector and axial vector form factors of known matrix element ^Z 1HH.'C'|A "^ is enhanced by a factor of 1.9. The total baryons effect of including these short distance factors is to significantly increase the widths for both decay modes to r L (20) II P -l-il* 10 and

We take g. = 1.25** and — = — . Following Korner et al. we use the which are too large as compared to the experimental values. However we notice A 1* P 3* 2 9 invariant form factors H:: (q) and IC (q ) to continue from cf = that the ratio Of the decay widths still fits very well (Table II). o n = m" where m is the mass of the relevant pseudoscalar meson, i.e. pP P The conclusion we reach is that the unmodified Hamiltonian gives very m or We use the standard dipole form factor of the form good results whereas the QCD corrected effective Hamiltonian does not fit the experimental rates. The modified Hamiltonian also does not give as good results of charmed meson decays as obtained by ignoring short distance corrections " . - it" As suggested by Guberina et al. one possible solution would be to include with 2.1U GeV arm ,

The theoretical values of the partial -widths r for Air and pK"

are compared with the experimental values in Table I and with other calculations. ACKNOWLEDGMENTS We see that our model fits the known experimental results 'better than either the quark model calculations of Korner et al.jWhich seriously The authors are grateful to Professors Riazuddin and Fayyazuddin for overestimates the width for the pK° mode, or the MIT bag model calculation, very helpful discussions. They also thank Professor Ahdus Salam, the which seriously overestimates the Ait mode by a factor of 3-5. Our results International Atomic Energy Agency and UHESCO for hospitality at the are well within the experimental errors for 'both modes. International Centre for Theoretical Physics, Trieste.

However there is a caveat to these claims. We have not included possible short distance effects of strong interaction QCD. The appearance of a new term (sd) (uc) (neutral current) interaction is expected from the short-distance expansion of the W-boson exchange amplitude in an asymptotically 12) free of coloured quarks. One obtains the effective Kamiltonian

-7- REFERENCES Table I 1) Riazuddin and Fayyazuddin, Phys. Hev. Dl8, 3578 (1978); Phys. Rev. D19_,.l63O (1976). Current 10) 11) algebra 2) M.D. Scadron, Rep, Prog. Phys. W*_, 213 (1981). Experiment Quark model MIT bag Present caleul. 3) H. Cabibbo, Phys. Rev. Lett. 10, 531 (1963). too large by k) S. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2_, 1285 (1970). 0.6 a factor of 0.77 3-5 • 5) B.W. Lee and M.K. Gaillard, Phys. Rev, Lett. 33., 108 (197>O; G. Altarelli and L. Maiani, Phys. Lett. 52B, 351 (1971*). 1.00 8.9 1.1(0 1.61* pK° -0.78 6) H.E. Marshak, Riazuddin and C.P. Ryan, Theory of Weak Interactions in {Wiley, B.Y. 1969). 11 7) A.J. Buras, Nucl. Phys. B109_, 373 (1976). Partial widths for A+ + Air+ and pK in units of 10'

8} J.Fin.jord and F. Ravnaal, Phya. Lett. 58B, 61 (1975).

9) W. Thirring, Acta Phys. Aust. Suppl. II, 205 (1965).

10) J.G. Korner, G. Kramer and J. Willrodt, Z. Phys. C2, 117 (1979).

11) D. Ebert and W. Rallies, CERH preprint TH.3598. Table II 12) M.K. Gaillard, B.W. Lee and J.L. Rosner, Rev. Mod. Phys. V[_, 227 U9T5h

J. Ellis, M.K, Gaillard and D.V. Hanopoulos, Hucl. Phys. B100, 313 (1975); Without short- With short- G. Altarelli, U. Cabibbo and L. Maiani, Phys. Hev. Lett.35., 635 (1975). Experiment distance factors distance factors

13) M.D. Scadron, Univ. of Arizona preprint (1983). 1 0.1*7 0.56 lit) B. Guberina, D. Tadic and J. Trampetic, Z. Phys. C13., 251 (1982). 0.5 * r(PK°)

FIGURE CAPTIONS Ratio of partial widths R Fig.l Rapidly Varying baryon poles in Bi-»Bf P^

Fig. 2 V scattering of quarks in the jharm changing non-leptonic weak Hamiltonlan density.

Fie. 3 Quark diagram for A+ + Atr+ c

Fig.l* Quark diagram for A+ c

-9- -10- B

Flg.3 Flfl.1

**'*

u >p CoaO

Flg.2 FIO-4

-11- -12-