Volume 99B, number 3 PHYSICS LETTERS 19 February 1981

DOES THE ENHANCEMENT MECHANISM WORK IN NON-LEPTONIC DECAYS

OF HEAVY PSEUDOSCALAR MESONS WITH NEW FLAVORS (b AND t)?

Tsunehiro KOBAYASHI and Nobuo YAMAZAKI Institute of Physics, University of Tsukuba, lbaraki 305, Japan

Received 4 November 1980

We show that the enhancement mechanism proposed for non-leptonic decays of charmed mesons (D °, F ¢) is still im- portant for decays of pseudoscalar mesons with b flavor, while its effect is small for mesons with t flavor. For the semi- leptonic branching ratios of B- 0aft), B°(bd) and B°(b~) we predict BR(B- -, evX ~ 9%, BR(B ° --, evX) ~ BR(Bs°~ euX)

5%. We can also derive the enhancement of the branching ratios of uneharmed f'mal states in B decays, i.e. BR (B- ~ Xs~) 20%, BR(B ° ~ Xsa ) ~ 10%, where Xs~(sa) = KTr + K*rlr + ... + ~,KK, + ....

In the previous paper [ 1 ] it was pointed out that the unitarity diagrams represented by quark diagrams with non-exotic intermediate states are important to understand the large enhancement o f non-leptonic de- cays o f charmed mesons (D O and F +). This enhance'- ment mechanism also explains the enhancement of the A / = 1/2 amplitude of K-meson decays [1]. It may be interesting to study the same enhancement mechanism in decays of heavy pseudoscalar mesons composed of a heavy quark (b or t quark) and a light antiquark (fi, a or g quark). In this note we study the enhancement mechanism Of weak decays o f heavy pseudoscalar mesons [2] in terms of the model proposed in ref. [1].

We start with the six-quark model [3]. Consider- ing that the mixing angle for b ~ c is expected to be much larger than that o f b ~ u in the weak current for b decay, we may take the effective currents for b and t decays as follows:

]b = ObgiT#( 1-- ?5)bi, for the b quark,

ju t = 0tti3,t~(1 - 3"5)bi, for the t quark , (1)

where the i of %. is the color index and 0 b and 0 t stand for the mixing parameters defined in the six- quark model. We can estimate that I 0 b I will be at most comparable with sin 0 e (0 e = Cabibbo angle) and l0 t I is comparable with cos 0 c ~. 1. The unitarity diagrams with non-exotic intermediate states can be

drawn for the decays of B 0 (bd), B - (bfi), B 0 (bg) and T O (tO) as shown in fig. 1, whereas there is no such diagram for the decays of T + (td) and T~'(tg).

(B°-a) o at o ra

~] ~ ~ - ~]'I-~//~ ~I- (¢ rOss b W rm

terms)

(B°'-- b) (E~- b) S S

b ~ b b ~ b

a . o

(~-a)

b b+{cross ' terms) bL . . . . . . . . . . . b ~ l l l l l l l # " "

C ¢

( T ° - a ) a

o ~7:~-2'/)77~ 0 +t +(cross terms) t l - l l l / - J l l l l J l ~f111111111

b b

Fig. 1. Enhancement diagrams for decays of B and T mesons, where wavy lines denote W bosom.

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Volume 99B, number 3 PHYSICS LETTERS 19 February 1981

Following the formula derived in ref. [1] , we can write the non-leptonic decay widths for these diagrams as follows:

]"{qL 1 2 2 gG FIOJ[ cos20c

X MM4f2 M OSR (ql c12)[IR 12NR C ~ , (2)

with IIRI ~ 1/8~r2, where]" = b or t , M is the mass of the initial meson, M 9 the mass for the form factor de- fined by (1 -q2/M())- l , which is chosen to b e M 0

m b + m e for B mesons and M 0 ~ m t + m b for T mesons (mq = quark mass), as expected in the vector dominance model. In ( 2 ) f M = the meson decay con- stant, a ~ ( q l q2) = the s-wave meson -meson scatter- hag cross section with a non-exotic ql C:t2 intermediate state, N R = the number of possible scattering proces- ses and c~t = the short distance enhancement factor which should be determined in each diagram given in fig. 1. They are chosen as shown in table 1, and c~t = K 2 for all processes in fig. 1, where B e = (b~), T b = (tl0) with (spin)parity = 0 - and D*, F*, B c and T~

_1 are the vectormesons and K 1 - g [(1 + k)c+ + (1 - k)c_] is defined by

c = [1 + 1 ( 3 3 - 2Nf) a s ln(Mw2/U2)] 12/(33-2Nf)

and c+ = c 21/2 (Nf is the number of flavors and % is the running coupling constant of QCD). Following ref. [11, we take k = - 1 / 3 . Then K 1 ~ 1.52 (c_ = 1.92, c+ = 0.72) for B decays with mass M B = 5.3 GeV .1 a n d K 1 ~ 1.35 (c_ = 1.64, c+ = 0.78) for T-

,1 Recent observations [4] of T"' at center-of-mass energy ,~ 10.55 GeV in e÷e - collisions indicate that 5.14 GeV < M B < 5.29 GeV. Our results are quite insensitive to changing M B from 5.1 GeV to 5.3 GeV.

Table 1

Diagrams fM o ~(q 1 q2 ) NR

(B°-a) fn(P,Bc,B e) o~(cff) 64

(B°(-)-b) ]'F(F*) o~(sd(fi)) 16

(BO-a) fF(F*,Bc,B ~) a~(~) 64

(Bs °-b) IF(F*) o~(s~) 16

(T o -a) fn(p,Tb,T~) o]~(b(~) 64

decays with M T = 20 GeV, where Nf = 6 is used and M T = 20 GeV is taken as an example.

Now we may write the non-leptonic decay widths of the B and T mesons in terms of the sum of the con- tributions given in eq. (2) and the free quark decay contribution [2,5,6]. As was discussed in ref. [6], the contributions of the exchange diagram for B- and T- meson decays are negligibly small .2 . Hence the ratios of the non-leptonic decay widths to the leptonic ones are obtained as follows*3 :

FNL(B0)/FSL(B ~ evX) ~ I 'NL(B0)/PsL(Bs -~ evX)

6.4 + 24o~(qlC:12 ; m b ) , (3)

F N L ( B - ) / F s L ( B ~ evX) ~ 6.4 + 4.8 o~t(q 1 q2; mb)

for M B = 5.3 GeV and m b = 5 GeV and

PNL(T0)/FsL(T - evX) ~ 7.6 + 10o~(ql 7t2 ; m b ) ,

FNL(T +) /FsL(T r* evX)

FNL(Ts+)/PsL(Ts --* evX) ~ 7 .6 , (4)

for M T = m t = 20 GeV, where the value of o R should be taken in units of mb. In (3) and (4) we use the semi- leptonic decay formula

PSL = g(e) G210q 12m5 /(192n3) ,

where g(e) stands for the quark mass correction fac- tor [7]. The factor g(e) is estimated to be 0.52 for the decays of b ~ c + (dfi, eP e or/a~u), 0.19 for b ~ c + (sg), 0.13 for b ~ c + (rPr) and 0.6 for t -~b + any- thing, where m c = 1.5 GeV, m r = 1.784 GeV and m u

m d ~ mu -~ m e ~ 0 are used and all masses except the b-quark mass are neglected in t decays. In the de- rivation of (3) and (4) the following relation is also postulated as a first approximation:

fM 1 fM2 o~(qlct2)=f20~(rc+K-;sa)

for M1, M 2 = rr, p, D, D*, F, F*, B, B*, Be, B~, T b and T~. One should remember that the introduct ion of symmetry breaking, e.g., o(c~) ~ o(cfi) ~ a(s~ and

,2 The contribution of the exchange diagrams is estimated to be at most a few percent for B- decay and much smaller than one percent for T + decay.

,3 In the following equations contributions arising from Cabibbo-suppressed decays are included, the ratios of which to the allowed decays are tan 02c .

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Volume 99B, number 3 PHYSICS LETTERS 19 February 1981

sd), leads to a small reduction of the enhancement fac- tors for B 0 and B 0. A simple Regge pole analysis gives

o~t (mb) -~ 1.5 (1.865/5.3) ~ 0.53, for B mesons , (5)

o~t (rob) ~ 1.5(1.865/20) ~ 0.14, for T mesons,

where 1.5 mb at the D-meson mass is taken from ref. [1 ] . For the choice given in (5) we obtain the follow- ing semi-leptonic decay branching ratios , 4 .

BR(B 0 ~ evX) ~ BR(B 0 -~ evX) ~ 4 .7%,

BR(B- ~ evX) ~ 8 .9%, BR(T 0 -~ evX) ~. 8 .4%,

(6) BR(T + --* euX) ~ BR(T + --* evX) ~ 9 .5%.

For the ratios of lifetimes (r) we have

r B - / r B o ~, r B - / rBo s ~ 1 .9 ,

rT÷/rTO ~ 1.1 , rT*/rTs ~ 1 . (7)

The above values for B mesons are insensitive to chang- ing the b-quark mass from 4.6 GeV to 5 GeV.

For the decay o f B c [= (b~)] meson we must take account of another diagram with a (b~)-intermediate state representing the decays B e -~ (B or B*) + any- thing. As shown in fig. 2, two diagrams correspond- ing to the two decays ~ ~ a + (rid) and ~ -* g + (fis), interfere destructively. I f the cancellation between the two diagrams is almost complete, the Bc-decay is written in terms of the free b-quark decays * s and the enhanced diagrams similar to those in r iga and we ob- tain BR(B c ~ evX) ~ 7%. We, however, know that a symmetry breaking between d and s quarks exists in

.4 Using o Rs = 1 mb instead of 1.5 mb at the D-meson mass in (5), we obtain BR (B ° -~ evX) ~ BR(B ° -~ evX) ~- 5.8% and BR(B-~ evX) ~ 9.6%. We note thaf the ratio rD÷/rDO

10 for D-decays is maintained for the choice of o~ = 1 mb. .s The free decays of the ~ quark are suppressed by the Iactor

(me~rob) 5 ~ 1/400 in comparison with those of the b-quark.

0 fi

(-sin0cCOS0 c ) (sin~-o::~) Fig. 2. Enhancement diagrams contributing to the decays B e

(B or B*) + anything, where wavy lines denote W bosons.

the two diagrams, If the ratio sin 0e/ 10bl is large enough to compensate for the cancellation, we can ex- pect a large decay width for the decays B c -~ (B or B*) + anything. In this case the lifetime o f B~- can possibly be shorter than that expected from the evalu- ation via the b-quark decays. Observing those decays in experiments seems very interesting.

An interesting feature in our model is seen in de- cays via the diagram ( B 0 ( - ) - b ) . The matr ix elements for hadronic final states translated from the sd(fi) sys- tem without breaking the OZI rule, for instance g.rr, K*rr, K, mr . . . . . are predicted to be comparable with (or possibly a little bit smaller than) that of the de- cays via the diagram (B 0 - a ) . The branching ratios for those decay modes are expected to be , 6 .

B R ( B - -~ Xsfi) ~ 23%, BR(B° -* Xs~) ~ 12%, (8)

where Xs~ = KTr + KTrTr + KKI~ + . . . . The uncharmed final states, including D and Fmesons , are strongly sup- pressed by phase space. The enhancement o f these de. cay modes is remarkably different from the predict ion of the W-exchange dominance [8] and the 20 dimen- sional representation dominance [9] , where no such enhancement appears as we assume that the transition b ~ u is negligible ,7 . This is a very good test for our model.

Our model is also different from the model with a quark number selection rul e [10], because the con- tr ibution of the free quark decay diagrams forbidden by the selection rule is comparable with that of the diagrams allowed by the selection rule in B decays and the former is larger than the latter in T decays.

Finally we would like to stress the characteristic features of our model:

(i) The enhancement by non-exotic diagrams [1] and suppression by the exchange diagrams [6] de- creases as the meson masses increase, and the domi-

These ratios are sensitive to the estimation of oSR (cu- or ~ ) in (5). For the choice of o~(sfi) = 1 mb instead of 1.5 mb at the D-meson mass we obtain BR(B- ~ Xs~ )

16% and BR(B ° ~Xsa ) ~. 10%. The uncharmed hadronic states without e- and Y-quark components resulting from the free decay b ~ c + (cs) may be suppressed, because (i) this free decay is suppressed by the mass correction factor g(e ) and 0i) the uneharmed state production via the annihilation of c and ~ quarks will be suppressed by a jet-like structure expected in the free b-quark decays.

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Volume 99B, number 3 PHYSICS LETTERS 19 February 1981

nance of the free quark decay diagrams may be realiz- ed in T decays.

(ii) For B decays the enhancement mechanism still works and the enhancement of uncharmed hadronic final states is a good test for our model.

We may conclude that our model will be tested in weak decays of heavy mesons.

Note added. The CLEO collaboration has measured the following semi-leptonic branching ratios for B mesons: BR(B ~ fiX) ~. (7.5 + 3.1)%for muons and BR(B ~ eX) ~ (16 + 4 + 7)% and (18 -+ 9)% in indepen- dent searches for electrons. See CLEO collaboration, pre- print CorneU University, CLNS-80/464 (1980).] Our re- suits are consistent with these data within the errors. In order to test our model definitely, more precise data for B - and B 0 are expected.

References

[1] T. Kobayashi and N. Yamazaki, Phys. Rev. Lett. 45 (1980) 1070.

[2] The first discussion of B-decays was done by: J. Ellis, M.K. Gaillard, D.V. Nanopoulos and S. Rudaz, Nucl. Phys. B133 (1977) 285.

[3] M. Kobayashi and T. Masukawa, Prog. Theor. Phys. 49 (1973) 652.

[4] D. Andrews et al., Phys. Rev. Lett. 45 (1980) 219; G. Finocchiaro et al., Phys. Rev. Lett. 45 ,(1980) 222.

[5] M.K. Galliard, B.W. Lee and J.L. Rosnet, Rev. Mod. Phys. 47 (1975) 277; J. Ellis, M.K. GaiUard and D.V. Nanopoulous, Nuel. Phys. B100 (1975) 313; N. Cabibbo and L. Maiani, Phys. Lett. 73B (1978) 418; D. Fakirov and B. Stech, Nuel. Phys. B133 (1978) 315.

[6] T. Kobayashi and N. Yamazaki, Prog. Theor. Phys. 65 (1981) No. 2, to be published.

[7] R.E. Maxshak, Riazuddin and C.P. Ryan, Theory of weak interactions in particle physics (Wiley, New York); N. Cabibbo and L. Maiani, Phys. Lett. 79B (1978) 109.

[8] H. Fritzsch and P. Minkowski, Phys. Lett. 90B (1980) 455; W. Berrtreuther, O. Machtman and B. Stech, Z. Phys. C4 (1980) 257.

[9] Y. Koide, Phys. Rev. D20 (1979) 1739; B. Guberina, S. Nussinov, R.D. Peecei and R. Rilehel, Phys. Lett. 89B (1979) 111.

[10] T. Hayashi, M. Nakagawa, M. Nitto and S. Ogawa, Ptog. Theor. Phys. 49 (1973) 351; 52 (1974) 636; M. Matsuda, M. Nakagawa and S. Ogawa, Ptog. Theor. Phys. 64 (1980) 264; E. Ma, S. Pakvasa and W.A. Simmons, Z. Phys. C5 (1980) 309.

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