Two-body decays of Bs mesons

Physics Letters B 283 ( 1992 ) 434-438 North-Holland

PHYSICS LETTERS B

Two-body decays of Bs mesons J. B i j n e n s a n d F. H o o g e v e e n Theoo' Divtsion, CERN, CH- 1211 (ieneva 23, Swttzerland Received 3 April 1992

We have calculated the decay rates of the B~meson in a number of exclusive two-body decay channels using the Bauer-StechWirbel model for current matrix elements. The influence of the free parameters of the model on the predictions is studied. The total branching ratio of the B, into final states which only contain stable charged panicles is found to be about 10 -~.

1. Introduction

2. The B a u e r - S t e c h - W i r b e l model

It does not have to be emphasized that the study o f B-mesons is worthwhile. The observation o f mixing and the prospect o f observing CP-violation in the Bsystem are exciting and may give deep insight into the inner m e c h a n i s m o f the standard model. O f the pseudoscalar B-mesons only the B" and the B ° have been seen in fully reconstructed events [ 1,2 ]. So far the B~-meson has remained elusive. Although it would be highly surprising if this state did not exist, it is very i m p o r t a n t to actually find it. The B~ meson is expected to mix strongly with its antiparticle and m a n y proposed schemes [ 3 ] for detecting CP-violation in the B-system use this particle as initial state. In this paper we calculate a n u m b e r of branching ratios o f the B,-meson into two-body channels. To do so we employ the model o f Bauer, Stech and Wirbel [4,5] which these authors have used to calculate the two body decays o f D ' , D °, B" and B" mesons. The organization o f this paper is as follows. In section 2 we briefly review the BSW model and discuss the input p a r a m e t e r s o f the model. In section 3 we give the results o f our calculation together with a discussion o f the sensitivity o f our results with respect to variations in the input parameters. Finally in section 4 we discuss goldplated decay chains where the decay products at the end o f the decay chain arc all charged pions, kaons, muons or electrons. These can be used for the direct detection o f the B~.

With the current state o f the art it is impossible to calculate matrix element for processes involving strongly interacting particles from first principles using the standard model. Therefore one has to resort to models. The model used here is a model by Bauer. Stech and Wirbel [4,5]. This model can be used to calculate the decay o f a spin 0 meson in two mesons o f spin 0 or 1, as well as its semileptonic decay. The two main ingredients are factorization and an ansatz for the current matrix elements. The short distance effective action is given by

434

G~ ~;,r,= ~ to, ( p ) (z~d') (,¢'c) + c2(J~ ) ( S ' d ' ) (tk') ] +[(ad')~(es')]+

....

(2.1)

where the primes on the quark fields denote the interaction eigcnstates. The factorization assumption a m o u n t s to replacing the short distance effective action by the following product of meson currents Gf : [ a l (ad')H(ff'C)H + a 2 ( s ' d ' )H(/~(')H :/;"f = x 5. +...]:,

(2.2)

where the index H means that the quarks denoted inside the bracket arc to be contracted with one and the same meson. In this way the problem o f calculating decay matrix elements reduces to the calculation of current matrix elements between a one meson Mate

0370-2693/92/$ 05.00 ~) 1992 ELsevierScience Publishers B.V. All rights reserved.

Volume 283. number 3.4

PHYSICS LETTERS B

and the vacuum or between two one meson states. The former matrix elements are simply given by the corresponding decay constants:

(OlJ, lAO-- )=iJ'~p,.

(OIJ, I.41-- ) = G m , F , .

1I June 1992

I'(0)= m.~-ram m,-me

g(½(a,+a,).

2o.tjr.Oe

o~ +oG ~(b,+G))

,//(a,, ~,,).#(ae. t,,,i

x

(2.3)

,'t2(0)- m . , + m e . 4 j ( 0 ) for a spin zero and a spin one particle respectively. The latter kind o f matrix element can be expressed in terms of Lorentz-scalar formfactors:

(BO IJ, JAO:)= P.,+Pe +

q2

Fl(q 2)

,,,,~

-

(2.6)

,

exp(-ax2-bx)

I

(2.4)

g(a,b)= j" dx( l - x ) e x p ( - a x 2 - b x )

,

(2.7)

o

dient of the BSW model makes its appearance. These

1"o(0) =.4o(0) = "f( ½(a., +aB). ~(b, +be)) x."fl a.,, b, ).f(ae. b,)

X

-m~

(I

authors assume that using the wavefunctions of a relativistic harmonic oscillator model provides a reasonable approximation for the wavefunction of a physical meson. This amounts to taking the following expression for the formfactors at zero m o m e n t u m transfer:

2(.04(.0

"G +,n~,,

f(a, b ) = J d x x ( l - x )

where q=pj-pB, and e" is the polarization vector of the spin one meson. The q2 behaviour is assumed to be dominated by the nearest pole which has the correct quantum numbers. At this point the third ingre-

~l - F r o m

b,=-

I

m.,-mu ),t2(q: q 2 G, )

+ i e*~q q- 2muq, Ao(q 2 ) .

m

# ,

with similar expressions for ae, be. The functions f and g, given by

q- q, ) ..|l(q 2) +i(m-, +mu)( e * - ~*'ff

\ m .~ + m e

,n~

v j~,e l'(q 2)

I'I I 4 - F m t t

_ ie,. q(p.,,,- Pro,

.4o(0) .

[n this expression m.~m arc the masses of the mesons in the matrix element, mtL,m are the masses of the nonspectator quarks, and m2 is the spectator quark mass. The numbers a.j, b.t are given by a~-

2

2m~ D I t - - DT t~

(2.5 cont'd)

q, Fo(q z ) .

( B I : J./, J,-10' )

A,(O)=

t H . l - - D I /~

#

g(~(a, +al~), ~(b~ +bl~) ) ,/'f(a ,. b,)f(at~, bu)

(2.5)

can be readily expressed in terms of the error function. The parameters ~o.~and ~oe define the spatial extensions of the two mesons sandwiching the currents. (Small oJ means a large meson and vice versa). Final state interactions play an i m p o n a n t role in the analogous decays of the D-meson, but for the B~ meson they are expected to be small due to the large mass o f the b-quark. In the following we have neglected all final state interactions. Also neglected are the effects of weak annihilation contributions, penguin diagrams and long distance radiative corrections. Short distance Q C D radiative corrections are in principle incorporated in the coefficients a j, a2, but rather than calculating these numbers from first principles we are using the values which have been obtained from a comprehensive fit to nonstrange B-meson decays [ 6 ] a~=l.ll,

a2=0.21 .

(2.8)

The values of the decay constants used can be found in table 1, whereas the polcmasses are listed in table 2. Many of these pole masses correspond to unob435

Volume 283. number 3.4

PHYSICS LETTERS B

Table 1 The decay constants in MeV.

Meson

Deca~ constant (MeV)

~' 131.7 K.D,D~.q~ 160.6 q(uu) 94 q'(uu) 65

Meson

Decay constant (MeV)

p,a~,K*.D*.D~ m Q J/~

221 156 233 382

Table 2 The pole masses in MeV. ,/"

cb

ub

bs

0+ 0I~ 1

6800 6300 6730 6340

5780 5277.6 5710 5331.3

5890 5400 5820 5430

served states. In principle the same values are being used as those o f BSW, unless more precise information has become available in the mean time. As constituent quark masses we assume 350 MeV for the up and down quarks, 550 MeV for the strange quarks and 1.7 GeV for the charm quark. The b-quark mass was taken to be 4.9 GeV. The values o f the Kobayashi-Maskawa matrix elements were taken from reference [7], for l"ud, V~, l',.d and I~, whereas l'~,b and l'ob were taken to be 0.05 and 0.005 respectively. Finally for the B~ mass itself we adopted 5.4 GeV as a standard value. 3. Results Going through the formalism described above and assuming that the lifetime of the B~-meson is the same as the average lifetime o f the nonstrange B-mesons 11.8× I0- ~ s we have calculated the branching ratios for the B~ into a number o f two meson decay channels. The results o f our calculation can be found in table 3. There we list the branching ratio for the decay, the KM matrix elements which appear in the amplitude. All two-body decay channels in the list add up to a branching ratio of I 1.4%. The branching ratio simply scales quadratically with the KM matrix elements. The decays into a pair of charged particles 436

11 June 1992

scale as a~ and the decays into a pair of neutral particles scale like a~. In this calculation all effects of mixing a n d / o r CP violation are ignored. If the B~meson is mixing strongly, as it is expected to do, then the branching ratios for a mass eigenstate to decay in a particular channel is just the average o f the branching ratios o f this channel and its charge conjugate. The mass of the B~-meson is presently not known. For the numerical estimates in table 1 a value of 5.4 GeV has been used. Increasing this value to 5.5 GeV leads to an increase of all branching ratios considered. The largest increase is by 29% and occurs for the channel TI'J/~¢. whereas the smallest increase is 17% which occurs in the channel rl ~°. The changes in the decay channels with branching ratios over one promille are more or less uniformly distributed between + 2 7 % a n d + 18%. For every particle the parameter co is a free parameter defining the spatial extension of the meson. A value of 400 MeV gives a reasonable description of D and non-strange B-meson decays, and has bcen adopted here for all particles. Increasing the co of the B~-meson to 500 MeV while keeping the rest fixed has drastic consequences for some decay channels. The branching ratio of the oJ/~¢ channel increases by 39%. The other important decay channels increase by between 20% and 30%. The q-' dependence of the formfactors is modeled using the assumption that the nearest resonance with the correct quantum numbers dominates. Many of the resonances which are needed are themselves not yet detected however and consequently their mass is unknown. The values from table 2 arc used to produce the numbers in table 3. To see the effect of this uncertainty, we increased all pole masses by 10%. The resulting change in the branching ratios was very" small, most important channels decreased by a few percent. The oJ/~¢ channel was among a few exceptions and decreased by 13%. The overall error of any calculation is very hard to estimate. From the uncertainties in the input parameters discussed above we know that the inaccuracy can be at least as large as a factor of two for the decay Bs-.0J/tl/and as large as 1.5 for the important decay channels involving D-mesons. The relative importance of the different decay channels is however much less influenced by the uncertainty in the input parameters. The order o f importance of the largest

Lfir

~-0l X£1"O

~ I ~",1 q".,l I I '". I q".I I r I '".1
°~zb °~z'b °lt0 (.tO Ilia

,-01 X~9"O , - 0 1 X98"0 ~-01XT.I'O ~-01XZl '0

I '",! q".l I ~l'".l
~1 +,,I q,.1 I ~1 ~. .I ~ .I I

• he .b.b

01X£9"0

~1 .I ,11 Z '" ,-t ~" -I,I

.hb

I '",I ""A I

I ~".1%1 I ~lP~lq"ll ~1'".1 ~".11 :1~° 1~°.11 I '". 1 ""A I ~-I P:I"",I I ~11'°,l "°,I I I i,,, 1 ,a,,1 I :11'~.9
qnl

~1P". 1 q". I I p", I "". 1 r. I '~1 q".l

z I Pn,l qn I ~I ~"A ~.I : I ~'~.1'~".1 ~.I I:l~'| qn 1

I P".I 'a",l r. I ~",19",1 : I I">,l
~-01 xO'C'O ~-01X lIFO

~1 ~". 1 ~, 11

~-0I X81"O

:1 P~I q".| I ~1 ~".1 ~.1 I rl P~,l q~ I I z l Pn'I qn'l I ~ I '",I ~.1 l zl P".i qn.| I

~-01XLI'0

~eU rob.

rl ~",1 ~".11

oOo*~l__

zl ~",1 '~>,! I ~1 "~.1
~e,b.

~-01X~ZO

°dLt

z-()I X S I ' 0 ~-0l X£['O

od,h

- 01 X I~U'0

m tt

- 0 l Xi~Z'O -01 x 6 £ ' 0 -01 x 6 £ ' 0 -Ol x Zt~'0 - 0 1 X 6lr'O -.Ol xl29'0

~

°de ox) o(~_o~1 ~lz¢ . b.o,,)l

()d-~bl coo,,N o.Oh -G +,~I i%"A COo~I odo~I ~(>.~I -~1 +,)1 o.(] ,it -"A+~! -G+~I ~l-to~A -.O+)l -,~I,,~l -o)I +)1 oCIh

~-01 x6~'O ~-01X09"O -01 x 0 9 ' 0 -01 x g g o -01XL9"0 - 0 l XOL'O -01XOL'O -01X~L'O -01X9L'O -01X9L'0 -01XIl'0 -OlXEl'O -01 XLI'0 -01X81'O -0IX81"0 0 i X0Z'0 -OIXIUO

-~o31 o*(Io.~I oato.~l boo31 (1%)I bo>I OO l-t oCl .h

~. Ol X 1,6"0 9 -01 X ~ 1"0 9-01XSl'0 9-01XSl'O
o*Oo~l

- 01 X £ 9"0

o!lea Su!q3uealt

lauueqD

z lS~l "". I I I P". 1 "". 1 I

~1V".l q".l l zl ".1%1 I ~1P". 1%1 I el pn, l q:',| I z I ~1 "".1 I ~ 'i P:~.|q[ I I

zI

:1 <~1 <~.1 I ~ i, ~it, I tin. I I zl p" a~ p" .1 ",.ll z I ">A %1 I ~l ~';1 q';l [ l:l P~,| q'~.l I ~ I p",l q~A I I .~ .ll I P~,1 '~ ,I I zl ~",,1%1 I z:l P:',1 q". I I ~1 "".1 ~'.1 I :1 '".1 "~,11 '~l P",l q:',l I :1 ~.1%1 I rl " ,1 ",,1 I pn rl ,1 "~,11 ~1Pn.l q~.l I l:, q:~ ~1 ,1 .11 :1 '~.! ">,II :! pn| q~,ll rl ~".t ~ .tl pn q~ ~1 .1 ,11

zl Pn, I q':', I I ~I '~.l q~.ll luatuala x!alelAl

-01 × L U 0 -01XSU0 -01 × 0 £ 0 -01 x I£'0 -0I x££'0 -01 x6£'0 -01 × 61,0 .-01 x01"0 ,-01XZI'O ~-01Xl, l 0 ~-01XI71 "0 )--01XITI'O ,-01X£ZO ,-Of x £ £ 0 ,-01 x t:,£'0

.-01XL£O ,-01 X 6 £ ' 0 . - 0 1 X OirO .-01 x £1r'O ,-01 x ~t,'O .-01 x 9 F O ,-01XL~'O .-01 x 6 ~ ' 0 ,-OI XL9'O .-01 x £ 8 0 -01X~l'O ~-01 x £ 1 O t-Ol X~l'O c-OI X 6 1 O c-Of X £ U O ~-01 x o £ o ~-01XOlr'O ~-01X Zt~'O ~-0l X 9t~'0 c-Of X ILO -01X~L'O r-01X I I0 z-Of x £ 1 O r-01 x 1~£'0 ~-01 x l tr0 ~-01X ~ ' 0 ~-01X9L'O ~:-01 x trg'O ~-01X£lO ~-OIX~I'O -01Xl~l'O ~-01X91"O ~-01XZUO o!lea 8u!q3ueal[l

oG.b ooG0 o(210 c~Utt -,,,U +o~ ~G.b >bo>l . lt÷.~

~U +.'>1 _at ~ ~ / f ,>1 o,(30 2 0 +~1 _O+.~l -;U+>I _!e+>l

_d+>l .'Ix? 2e,,~l o(lc~.~A "hh Jh/f o.>l 'b,tt oClo~A o.(I()~1 tall h /k/f h - o Acl o.Oo*~1 -~.ka - (1)0

- . 0 ~Cl -.)I +'CI -.Cl +;Cl mlro 2U +;(1 _u ~Cl 2 u ~(I -~Cl2U

'e2G _dUo -d2cI - 2 0 +20 13uueq.3

•~00"0=anA pue ~0"0 =4~A 'set -01 × 8 ' I I = .az .iZ.0=z~ "11"l = 49 Sulttlnsse 'sXe3ap ~ll .~poq-o~Al aoj so!leJ ~u!q3ueafl £ alqel Z661 0unf I I

II S~131.1.3q SDISXI-Id

17'£ aaqtunu '£812 atunloA

Volume 283, number 3.4

PHYSICS LETTERS B

c h a n n e l s is not changed, oven by rather drastic changes in input parameters,

IlJune1992

Table 4 Thc contribution of several two-bodychannels to the branching ratio B~-,stablecharged particles only. Branchingratio

Channel

4. Goldplated decay chains Decay chains in which the stable particles at the end of the chain are all charged are especially suited for reconstructions of the B~ meson. So far not a single fully r e c o n s t r u c t e d B~ has been seen. The n u m e r ical values below have been taken from ref. [ 7 ]. First consider the decays of the D~" -meson. In 2.7% of the time it decays in Qn ± and 1.3% of its decays produce On -~n -~n ~ . O f these roughly half are followed by' the decay o--,K+K -. The decay channel KsK* followed by the decay of the Ks into two charged pions has a branching ratio of about 0.9%. Furthermore the decay channels K.*-~K + and K*+Ks have a probability of 2.6% and 1.6% respectively. The charged K* decays into a charged pion and Ks one third of the time, whereas the K *° decays into K+rt - two thirds of the time. Finally the direct decay n-~ n ± n : has a branching ratio of 1.2%. The decay mode K + K - r t + adds another 0,67%. Taking cve~'thing together wc notice that the D [ decays 6.7% of the time into a final state in which all the particles are charged. Thc a( meson decays with a branching ratio of 50% to n : n ~n ~ through port±. The J / ~ decays into e+c - or It+p - with a branching ratio o f 13.8%. Another interesting possibility is the ~ ' which dccays either in charged leptons or into J / ~ c n + n - followed by' J/0/ decaying into e+e - o r / a + p ~. These channels contribute 6. 1% to the all charged branching ratio of the W. The branching ratio B ~ - , ~ ' has been calculated using the same methods as in section 3. All together we find the branching ratios for goldplated decays as they are displayed in table 4. Adding eve~,thing together we note that roughly one promille of the B~ decays goes into a goldplated decay chain. Furthermore to all these decay channels one may add n + n - and or K ' K - pairs ad libitum (or at least upto the kinematical limit). This may

438

I)," a( D," n oJ/w D,+ D~D,' K O~ K +a~ K+n Ds+K*-

4 . 3 7 X 10 -4

3.7f)× 10 4 0.88× 10 -~ 0.34X 10 -'~ 0.28X 10-4 0.24X 10..4 0.19X 10-4 0.14× l0 "~ 0. l I X 10-4

"

10.3 X 10 - 4

s u m

bring a substantial improvement as a comparison with the D-system shows. F(D*_,~-~n+n,n

F ( D ÷ _,~(.~ n+ )

.)

=2.5.

F(D°--, K - n + n * n - ) =2.1. F( D ° -, K - n + )

(4.1)

Acknowledgement The authors like to thank A. Pich and P. Kluit for a n u m b e r of pleasant discussions.

References [ 1] ARGUS ('ollab.. H. Albrecht et al., Z. Phys. (" 48 (1990) 543. [2] CLEO Collab., D. Borleletto et al., Inclusive and exclusive decays of B-mesons to final states including charm and charmonium mesons, preprint CLNS-91-1102. [ 3 ] CP violation, ed. ('. Jarlskog (World Scientific, Singapore, 1989). [4J M. Bauer. B. Stech and M. Wirbel, Z. Phys. C 34 (1987) 103. [5]M. Wirbel. B. Stech and M. Bauer, Z. Phys. C 29 1f985) 037. [6] M. Neubert et al., Exclusive weak decays of B-mesons. preprtnt HD-THEP-91-28. [ 7 ] Particle Data Group, J.J. Hernandezet al., Reviewof particle properties, Phys. Lett. B 239 (1990) I.