Volume 155B, number 1,2 PHYSICS LETTERS 16 May 1985

B R A N C H I N G R A T I O I N T O S P I N S I N G L E T A N D T R I P L E T M E S O N S IN S E M I L E P T O N I C DECAYS O F HEAVY F L A V O R E D M E S O N S

Mahlko S U Z U K I

Department of Physics and Lawrence Berkeley Laboratory, Untverslty of Cahforma, Berkeley, CA 94720, USA

Recewed 20 February 1985

The dlfferenual decay rates into spin slnglet and triplet of the Q'c 1 states m the sermleptomc heavy-flavored-meson decays Qc t --, Q'q + d + v are calculated relanwstacally m the spectator picture of heavy flavor decays They deterrmne the production rauo r = F((S = 1)+ d + v)/F((S = 0)+ d + v) xf final-state interactions cause no fl~ppmg between smglet and triplet, as is the case m the D and B decays We find that r vanes as a function of a = m(Q')/m(Q) from 2 at a = 0 to 3 at a =1.

Semileptonic decays have played an important role as the signature o f heavy flavored hadrons. In D and B decays, the distribution of hadronic invariant mass has been measured to be in general agreement with the theoretical expectations [1 ]. The purpose of this short paper is to show that even the spin structure of final hadrons can be predicted almost unambiguously in the conventional picture o f heavy flavor decays.

The process of interest is the sermleptonic decay of the heavy flavored meson Q~,

Q~I -+ Q'q + ~ + v, (1)

where Q is a heavy quark bound with a spectator quark ~l (fi or d) in a 1S 0 meson state. We compute here the energy spectra and the partial decay rates for the spin singlet and triplet Q'q final states in the spectator model o f the heavy quark decays. The invariant ha- dronic mass in the final state, given by ([p(Q') + p(fi or a)] 2)1/2, is bounded from above by 0.92 GeV and 2.03 GeV in D and B decays, respectively, when m b = 5 GeV, m e = 1.55 GeV, m s = 0.5 GeV, and mu, d = 0.3 GeV are chosen. These upper bounds are sub- ject to smearing when the Fermi motion is included in Q?t. Consequently, the c-quark in B decay must form D or D* since no other charmed meson states exist below 2.03 GeV. While the scale is different in D decay, the final s-quark in D decay must still form dominantly K or K* since there are no other con- spicuous strange meson states below 0.92 GeV and

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two-meson production sufers from a severe phase space suppression. Therefore, Q'~I is in S-wave. Then, conservation o f total angular momentum forbids final- state interactions from flipping the Q'~ spin between the singlet and triplet. This argument does not apply to T decay. The invariant hadronic mass of the final state in T decay extends up to ~6.1 GeV for m t = 40 GeV and m b = 5 GeV. Since several of the excited B meson states are expected to exast below 6.1 GeV, higher orbital angular momenta are important in the T decay. In order to identify the spin singlet and trip- let Q'~I productions with the 1Lj and 3Lj bottom- flavored-meson production, we have to introduce a dynamical assumption that spin flip is neghgible in the final-state interactions. This assumption is quite plau- sible; the long range forces responsible for hadroniza- tion are predominantly the spin-independent confining force and the coulombic force ofg luon exchange.

For the purpose o f covariant calculation, it is con- venient to quantize the spins of Q and ~ in the ini- tial state along the direction o f the final Q' momen- tum in the rest frame of Q?:I. (See fig. 1 .) We express our results in terms of the reduced rates, removing the multiplicative factor due to the quark mixing an- gles. In the approximation of no Fermi motion and no final-state spin flip for L :/: 0, we can show for the standard V - A interactions that the differentaal de- cay rates for Q?q ~ (Q'?q)s=0,1 + ~ + v with m R ~ 0 are given by

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Volume 155B, number 1,2 PHYSICS LETTERS 16 May 1985

~ . ~ i I /

i /

r ' 0

(a)

a:l/8 I - ; , ~

01 02 03 0.4 05 x = EQ,/M

Fig. 1 Spins in the semlleptomc Qq decay.

dPo[dx = (G2M5/24rr3) (1 + a) 2 (x 2 - a2)l/2(x - a),

(2) dPl/dX = (G2M5[24rr 3) (x 2 - a2)l/2

× [(1 - a ) 2 a + (5 - 2a + 5a2)x - 8x2] , (3)

where the subscripts 0 and 1 refer to singlet (S = 0) and triplet (S = 1), and x = ElM and a = m/M with E being the Q' energy and m (M) being the Q' (Q) mass. The bars above the rates are to remind that the quark mixing factors have been removed. The energy spec- tra are quite different between the singlet and triplet productions. The singlet production dPo/dx vanishes fast at the lower limit o f x (E = m) where the lepton pair £v must be emitted back to back; the helicities of £v violate the angular momentum conservation along the direction of the lepton momenta. At the upper limit o f x (= (1 + a2)/2) where £ and v are emitted in the same direction, the differential decay rates dPo/dx and dPl/dX become equal for any value ofa . We have plotted dPo,1/dx in fig. 2a for two val- ues o f a , a = 1/3 applicable to D and B decays and a = 1/8 applicable probably to T decay.

In the D and B decays, the entire allowed ranges in x correspond to the formation of K, K* and D, D*, respectively, as discussed previously. Integrating (2) and (3) from x = a to (1 + a2)/2, we obtain the K(D) and K*(D*) formation rates as

PO = (G2M5/576rr3) (1 + a) 2

× [ 1 - 3 a - 3 a 2 + 3 a 4 + 3 a 5 - a 6 - 1 2 a 3 1 n a ] , ( 4 )

3

L~ t -

z!

(b)

/

I I I I

. t , . t . = . , . ,

t*b b-c 0.5 C*S

a

Fig. 2 (a) Energy spectra of the fmal quark Q' when Q'q is m spin singlet and triplet, a = m/M. The vertical scales fora = 1/8 and fora = 1/3 are arbitrary. (b) The ratio I'1/I" o plotted against the parameter a (= m(Q')/M(Q)).

Pl = (G2M5/57&r3)[2 + a - 16a 2 + 9a 3 - 9a 5

+ 1 6 a 6 - a 7 - 2a8+ 12a3(1 - 4a + a2 ) lna ] . (5)

They add up to the well-known formula for the total rate,

['0 + Pl = (G2M5/192rt3)

X (1 - Ka 2 + 8a 6 - a 8 - 24a41n a).

The ratio Pl/l-'0 has been plotted in fig. 2b for the range of values allowed for a. The ratio increases monotonically from 2 at a = 0 to 3 at a = 1. As a -+ 1, the Gamow-Teller transition dominates in P l , whale the Fermi transition dominates in F 0 . In the limit of a ~ 1, the lepton pair £v is emitted nearly indepen- dently in the rest frame of Q~. Then, it is fairly easy to understand why P l /P0 approaches the value 3 It is interesting to compare our relativistic results with the existing estimates based on the nonrelativastic quark and/or a favor symmetry of one kind or an- other [2], which involve a large ambiguity especially in D decay.

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Volume 155B, number 1,2 PHYSICS LETTERS 16 May 1985

In T decay, the final Q'?:I covers the B meson states o f L ¢ 0, too. Therefore, the ratio r l / r 0 = 2 .3 - 2 .4 from fig. 2b represents the ratio for the sums of the decay rates

r 0 = r(T ~ B(O-)£v) + F(T ~ B**(1P1)~v ) + .... (6)

r 1 = P(T -~ B*(1- )£v) + F(T ~ B**(3Pj)~p) + . . . . (7)

I f one wishes to obtain the ratio of the rates into 1S 0 and 3S1, F(T -+ B*(1-)~v)/r(T ~ B(0- )~v) , one should integrate the two curves for a = 1/8 in fig. 2a over a small region above x = a. Since the singlet pro- duction is suppressed near x = a, I ' (T ~ B(O-)£v) will turn out to be insignificant as compared with F(T

B*(1 - )£v ) . Needless to say, this predict ion is inde- pendent of the assumption on the final-state interac- tions. This is one of the interesting predictions which result directly from the present analysis. In fact, fig. 2a will turn out to be very useful when the B me- son spectrum becomes known. The curves integrated over a given segment o f x represent the product ion rates of b?:l bound state(s) and/or resonance(s) whose masses fall in this segment o f x through the relation

m ( b ~ m 2 m 2 al/2 =( b +2mtmu,dx + u,d" ' (8)

up to small smearing due to the initial Fermi motion. In summary, the decay rates into the spin singlet

and tr iplet qua rk -an t iqua rk states in the semilep- tonic heavy-flavored-meson decays have been com- puted relativistically in the spectator quark model. They give directly the rates for the K and K* produc- t ion in D decay and for the D and D* product ion in

B decay. In T decay, the formulas will allow us to estimate the B and B* product ion once we know the P-wave B state masses. Otherwise, they give the sums

of the product ion rates for the B states up to around 6 GeV.

Further details such as the energy dependence of the longitudinal polarizations of the final K*, D*, and B* will be presented elsewhere together with tech- nical aspects o f the computat ion.

This work was supported by the Faculty Research Grant of the University of California, Berkeley, the US National Science Foundat ion Research Grant PHY-81-18547, and the US Department of Energy Research Contract DE-AC03-76SF-O0098.

References

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