Download pdf - Branching ratio into spin singlet and triplet mesons in semileptonic decays of heavy flavored mesons

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 hadronic 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 triplet 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 initial state along the direction o f the final Q' momentum 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 decay rates for Q?q ~ (Q'?q)s=0,1 + ~ + v with m R ~ 0 are given by

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


~ . ~ 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 spectra 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 values 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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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 production 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 independent of the assumption on the final-state interactions. 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 meson 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 semileptonic 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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