Volume 67B, number 3 PHYSICS LETTERS 11 April 1977

N O N L E P T O N I C D E C A Y S O F C H A R M E D F M E S O N

A N D T H E E X I S T E N C E O F (c, ~k) R D O U B L E T

Ken SASAKI Department of Physics, Kyoto University, Kyoto 606, Japan

Received 6 December 1976

It is shown that the analysis of decay modes of the charmed pseudoscalar meson F ± into 3n and K±K~ may pro- vide a sensitive test for the existence of a right-handed c and h quark doublet.

Recently there has been considerable work on the construction of new models of weak interactions, most of which have more than four quarks and involve right-handed currents. Those models with right-handed currents may involve a right-handed c and ~, quark doublet (c, ~')R [1]. It has been argued by Barger and Nanopoulos [2] that an upper experimental bound on the decay rate ff ~ vb- could establish whether a (c, ~')R doublet exists. In this paper we wish to show that the analysis of decay modes of the charmed pseu- doscalar meson F±(c~( or ;k~- state) into 3zr and K±K 0 may serve as a sensitive test for the existence of a (c, ~)R doublet.

For the general form of the hadronic IACI = IAS[ = 1 current jCS, we may write,

jCS - u = c ' r~ , (~+/37s)X. (1)

In the "standard" Weinberg-Salam-GIM (WSG1M) model (case A) [3], a = cos 0 C and/3 = - c o s 0 C, on the other hand in those models with a (c, X)R doublet (case B) [ l ] , a = 1 + c o s 0 C and/3 = 1 - c o s 0 C.

At the first glance there is a great difference between the two types of models in the prediction of the pure leptonic decay rate of the F meson, because the axial- vector part ofJ CS contributes to this process. Analo- gously to the 7r~2 and K~2 decays, we define fl-" by

(0 [jCS[ F +(q)) = i~¢fi fF q~/3. (2)

If we assumefF ~ f K , we obtain

F(F + ~ U + v ) / r ( K ÷ -+/a÷v) ~ (mF/mK)/32/sin 2 0 C.

(3)

Taking the predicted value for the mass of F [4], m F = 1.975 GeV, and with I-'(K + ~ O+v) ~ 0.51 × 108 sec -1 and tan20c = 0.056, we predict

F(F+ ~/a+v) =4 .0 × 109 sec -1 for case A (4)

= 3.1 × 106 sec- 1 for case B.

The ratio of the two predicted values in eq. (4) is cos20c/(1 - c o s 0 c ) 2 ~ 1.3 × 103. Although this re- sult may be important for the explanation o f dimuon production in neutrino-induced reactions, it is not decisive of the existence of a (c, ~')R doublet. Many other factors should also be considered for the analy- sis of dimuon events, such as leptonic and semilep- tonic decays of other charmed particles and their pro- duction rates, etc.

Let us now turn our attention to the nonleptonic decays of F, especially, to the decay mode F ~ 3n. We take the current × current form for the nonleptonic weak Hamiltonian. When we simply write a quark diagram for the process F ÷ ~ 3rr in fig. 1, we find that the two quark lines of initial particle are joined together and are disconnected with the quark lines of final particles except through the weak boson propa- gator. It should be emphasized that this kind of dia- grams in which the quark lines of initial hadron are disconnected with those of final hadrons first appeared

c C

FL-,3 Fig. I . Qnark diagram for the decay process o f F + ~ 3n.

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Volume 67B, number 3 PHYSICS LETTERS 11 April 1977

in the nonleptonic decay processes of F --* pions (and F* --* pions).

When we estimate the matrix element of (37r[H w IF +) by inserting intermediate states between the two cur- rents (only states with quantum numbers of vacuum are allowed), we expect that the summation over in- termediate states is saturated by vacuum state because of the unique feature of the quark diagram of the above process. Thus we expect

(3zrlHwIF +) ~ (01J,CSIF+). (5)

vector part of J r S contributes to the process The axial F + --* 3rr. Hence there will appear a large difference between case A and case B in the prediction of the decay rate for F ~ 3n.

The above statement can be supported by a simple dynamical model. The model, which we shall use later for the calculation of the relevant decay rates, is a generalization of the vector-meson-dominance model suggested by Sakurai.[5] to describe the nonleptonic decays of ordinary hadrons. Recently this model has been extensively used by Borchardt and Mathur for the study of weak decays of the lowest-lying charmed mesons [6].

From now we restrict the number of quark flavours to four, p, n, 2t and c. (Models of weak currents may have many quarks, then we consider their subgroup with four quarks.) Following ref. [6], we denote vec- tor and axial vector currents as

V ~ - ~ A ~ - - ,a = q~7,q , ,~ - q~u75 q , (6)

where a,/3 = 1 . . . . . 4, which refer to p, n, X and c, respectively. In the WSGIM model (case A), the weak Hamiltonian for nonleptonic decays contains a singlet, a 20-pier and a 84-plet SU(4) representations. We as- sume the 20-plet dominance in case A [7]. When a model also involves a (c, ~.)R doublet (case B), the weak Hamiltonian may further contain the 15-plet, 45-plet and 45*-plet representations. Indeed, the 20, 45, 45* and 84 contribute to our relevant IAC1 = )ASI = 1 decay processes in case B (note that the 15-plet does not contribute to these processes). The 45-plet and 84-plet contain the SU(3) decuplet and 27-plet representations, respectively, both of which induce the z&/= 3/2 transition in strangeness-changing decays of ordinary hadrons. Therefore we assume that the 20-plet also dominates in case B. Then, for both cases

A and B, the relevant IACI = IASI = 1 piece of the non- leptonic weak Hamiltonian may be written as

(7) + @ - v1}

Next, we adopt the phenomenological relations

-,,/~(mv/.rv)~,~, A '~ =v%"va,,~ (8) V~u/3 - 2 a u/3

where ¢~# and P~ are the physical vector-meson and pseudoscalar-meson fields, and m 2 / f v and fp are the corresponding couplings. For the three-meson strong vertex (one vector and two pseudoscalar mesons), we adopt the SU(4) invariant interaction Hamiltonian

H S = ig Tr(¢ue*~ue ) . (9)

We normalize all decays to the K 0 --* zr+n - decay rate, so we would not need the value of the strong coupling constant g.

The process F + -* pTr would proceed through the diagram in fig. 2. In terms of this generalized Sakurai model, the unique feature of the quark diagram of fig. 1 can be restated as follows: the weak vertex should appear before the strong vertex in the above process. Compare with the diagrams in figs. 3 and 4 which describe the decay processes F + --* K + ~0 and K 0 -~/.r+/r - -"

Calculations of the relevant amplitudes are now

. P

o , j / ', ,,~_ __' _.' ~" F [ D'" o

Fig. 2. F +'-, pn decay Fig. 3. F ÷'-* K+K. ° diagram, decay diagram.

7/ ' - 7/* i

I i

° t, ° ' , K - * 5 . . . . . K'" ,~. ~_ . . . . 7 / ' -

O - . . . . .

Fig. 4. K~ ~ rr+n - decay diagram.

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Volume 67B, number 3 PHYSICS LETTERS 11 April 1977

straightforward:

A (F ÷-, po.+) = - .4 p%o)

= G cOS O c/3flrfFgm2F _ m2 ff

A (F + -* K + ~0)

- - 2 p ( r t ) . e ( , o ) ,

G - 1 2 = - i ~ c°s 0C - ~ f K ~D,g(mF - m 2 ) ,

(lO)

(11)

A (K 0 -" lr+lr - )

1 2 m 2 ) , = - G cos 0 C sin 0 C fTr ~K* g(mK -

(12)

where e(p) is the polarization vector o f p . For numeri- cal values o f f p a n d f v , we follow ref. [ 6 ] ; f F = fK =fn (~93 MeV), andm2,/f2D , = m2K,/,f~, = rn2/.f~ (~2 .23 × 10 - 2 (GeV)2). We takemD ° = 2.0 ~eV-[4] . With F(K 0 -q" *r+n - ) = 7.7 × 109 ( s - l ) , we obtain the

predicted values for decay rates F ( F ÷ ~ pTr) and [ ' (F + K + ~0), which are exhibited in table 1. Case A and case B predict much different values for

the decay rate of F + ~ prr. This result comes from the fact that the value of/3 in eq. (1) differs largely between the two cases. On the other hand, the same partial decay rate is predicted for F + --' K+K 0, since t3 - a = - 2 cos 0 C in eq. (11) for both cases. The predicted ratio of lr '(F + ~ pTr) to I ' ( F + ~ K + ~0) is very much different in two cases:

F ( F + '* pn) = 8.3 Case A

r(F+ -* K+ r'°) (13)

= 6.3 X I0 -3 Case B .

How reliable is the above est imate? It is true that the SU(4) symmetry for hadronic vertices which we adopted in eq. (9) may be badly broken. But even when we allow such a large SU(4) breaking aLthe relevant strong vertices as, say, ¼ ~ g2D, FK ]g2omr ~ 4 (compare with the SU(3) breaking, for example,

2 4KK/gl~Tr ~ ~ 1.3), the distinction of the predicted ratio in case A and case B is still quite apparent.

From these analyses, it is concluded that the decay channel F-* ~ 3n wouid be observed easier than F ±

K ~ K 0 in case A but that a converse statement can

Table 1 Predictions for the decay rates of F ÷ -~ p*r and F + ~ K+K °, based on the generalized Sakurai model. Case A is the "stand- ard" Weinberg-Salam-GIM model, and case B is models with a (c, h) R doublet.

Case A Case B a t = C O S 0 C oc= I + C O S 0 C

/3= - C O S 0 C /3 = 1 - COS0 C

F(F +---, p%+) 3.8 x 1012 s -I 2.9 X 109 s -1 = F(F + ~ p÷~,O)

F(F÷-~K+K °) 9.2x 1011 s -1 9.2x 1011 s -1

be made in case B. In fact F ± -+ 37r would be much suppressed as compared with F ± -+ K±K 0 if a (c, X)R doublet may exist. Therefore the observation of the charmed F meson and the analysis of its decay modes into 3rr and K±K 0 will serve as an important test whether a (c, X) R doublet exists or not.

The author is very grateful to Dr. M. Kobayashi for instructive comments and discussions, and also for reading through the manuscript. He has also bene- fited from discussions with Professor S. Tanaka, Professor T. Muta, Drs. M. Bando, T. Uematsu and T. Kugo.

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[2J V. Barger and D.V. Nanopoulos, Phys. Lett. 63B (1976) 168.

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[4] A. De Rujula, H. Georgi and S.L. Glasbow, Phys. Rev. D12 (1975) 147.

[5] J.J. Sakurai, Phys. Rev. 156 (1967) 1508. [6] S.R. Borchardt and V.S. Mathur, Phys. Rev. Lett. 36

(1976) 1287; preprint UR-581. [7] Y. lwasaki, Phys. Rev. Lett. 34 (1975) 1407;

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