Volume 229. numlx'r 4 PHYSICS LETTERS B 19 October 1989

SEMILEPTONIC DECAYS OF B MESONS INTO ,:v,

P. HEIL1GER and L.M. SEHGAL 111 Phystkah.sches Instttut. Technische tlnchschule Aachen, D-5 I00 Aachen, I"RG

Received 14 June 1989; revised manuscript received 21 June 1989

We discuss the characteristics of the decays B ,D'tv and B-, D*xv. with particular reference to probtng form factors that are inaccessible in the eve and pv. modes. Features dtscussed include the ~:c ratio, the x spectrum and the x polarization.

I . I n t r o d u c t i o n

Decays of B mesons into final statcs containing a zg, pair constitute a significant fraction of all semilcptonic B decays. These decays have an intrinsic interest, as they can probe certain form factors characterizing the matrix elements (DIJ,,I B), (D*IJ , I B). etc. which produce effects proportional to m~ that are inaccessible in the e~,, and 119, decay modes. In addition, the decays B-,rg, X followed by "t--.v,e~', or "t ,vdag~, act as sources of secondary lcptons that constitute a background to the spectrum of primary electrons and muons emitted in B decays.

2 . Q u a r k m o d e l g u i d e l i n e s

The decay widths associated with B-, X~v, for ~ = "t and ~ = e can be simply estimated on the basis of the quark transitions b--.c~vr or b--. u~v~:

192n'2 , [ \rob ) (m~ 'n~ ) ] F~= GFmu 2 /me m~ 21 IV~hl I | - - , - - , 0 +lV~bl , ,0 . (1)

In b FI| b

Hcrc l(x, y. 0) is a phase space factor given by [ 1 ]

l(x,y, 0) = ~fll - 2 ( x 2 + y 2 ) + (.r2-),2) z

× [ l -7 (x2+y2)+6x2y2-2(x2+y 2) 2-62c2y2(x2+),2) +(x2-y2 ) ( x4 -y '+5y2-5x2 ) ]

I - x 2 - y 2 + ~ / l - 2 ( x 2 + y 2) + ( x 2 - y 2) ~ + 12(x'+ya-2x4y 4) In

2x)'

+ 12(x4-y 4) In x 2 + y 2 - (x2-y2) 2_ (x2_y2)~/i _ 2 ( x 2 + y 2 ) + (x2_y2) 2 (2a)

2xy

Special cases of interest are

l(x,x, O ) = x / l - 4 x 2 ( I - 1 4 x 2 - 2 x 4- 12x 6) +24x4(I - x 4) In I + x / I - 4 x 2 (2b) I - x/'l- - 4.'¢ 2 '

l(x, O, O) =l(O,x, 0 )= I -8X~+8X6-X~-24X4 In x . (2c)

0370-2693/89/$ 03.50 ©Elscvicr Science Publishers B.V. ( North-Holland Physics Publishing Division )

409

Volume 229, number 4 PHYSICS LETTERS B 19 October 1989

Taking m~/mb.~ m,/m,..~ ~ and neglecting m~ or m~ we obtain

1', IV,.bl21(~, ~,O)+lV,,hl2l(O. ~,O) F~- IVcbl21(~,O,O)+IVo, 121(O,O,O) " (3 )

Using the fact that I V~b I 2/I V~v 12 < 0.04 [ 2 ] and thc phase space factors I( ~, [. 0) = 0. I 19, I(0, [, 0) = 1( [, 0, 0) =0 .447 , 1(0, 0, 0) = I,

FJF~ ~0.266. ( 4 )

The energy spectrum of the T given by the transition b ~c ' t -9 , is "

dE 2 s G~mu 2).1/2 6){4xyA+ d x - 96]r 3 II"~bl ( I , y , [ ( I - O ) 2 - y 2 ] B }

where

x=E:/mb, 6=(m,/mb) 2, y=l--2x+&, 2(x,y,z)=x2+y2+z2--2xy-2yz+2zx,

A=23/2 l'm~,)------"0 ' B=2Z'/2 l'm~,)-----"0 l + m ~ ' \mu)/ J"

ot

L-- C3

1,4

1.2

0 8

0.6

0 4

0,2

O. 0 0,4 0,8 1 2 1 6

L e p t o n S p e c t r u m

b --> c-ru

" This is similar to the electron spectrum in la decay with a masswe v,. See e.g. ref. [ 3 ].

Fig. 1. Momcntum spectrum of the x-lepton from the decay b-,cxv.

I 2,4

p , ( O e V )

4 1 0

Volume 229. number 4 PHYSICS LETTERS B

Assuming mJmh.~ m J m h = ~, the momentum spectrum of the T is shown in fig. I. Finally the longitudinal polarization of the x- in thc decay b- ,c~-9 , is

v l P t = - -

c l - ( m ~ / m b E , ) g ( E , ) / f ( E , ) '

where

( 2m . ) g ( E , ) = m h ( m b - E , ) 1 + rnb +rn, --2mhE, J f (E , )=O.5(rn~-2mbE,+rn~ - m ~ ) + g ( E , ) 2 2 '

v being the velocity of the x in the b rest frame.

19 October 1989

(6)

3. Exclusive decay B--, D,:v,

The matrix element of B-, Dl~v+ is characterized by two form factors defined by

( D(P + )1 V~, I B (p))> =f+ (q2) (p+p,) , , +f_ (q2) ( p _ p . ) , , (7)

where q = p - p ' . The contribution off_ to the decay rate is proportional to m~ and hence negligible for ~ =e or I.t, but significant for B ~ D w , . The double differential spectrum in the energies of the lepton and the D meson is [4]

d2F G~ 2( d/';~dED - 4n ~ IV~bl If+ 12,4+Ref*+J'-B+ I f -12C) ,

where

A = rnt)[ 2E~Ev - mr,( Wo - ED) + ~m~( IA;, - ED) -- rn~Ev] ,

B = m ~ [ E v - ~ ( W o - E D ) ] , C=~m~(Wo-Et~)

with

Wo=(rn~ + m ~ - m ~ ) / 2 m a , E ~ = r n B - E D - E ~ . (8)

For the form factors f , (q2) we have considered two hypotheses: (a) The parametrization of Bauer. Stech and Wirbel (BSW) according to which [5]

h, f ( q 2 ) = I -q2~ m2, , i=O, l , (9a)

wherefo andf~ arc related to f_, by

f ÷ ( q 2 ) = f , ( q 2 ) , f _ ( q 2 ) _ r n ~ - m ~ q2 ~ , ( q 2 ) - f l (q2) ] • (9b)

The pole residues are h~= h, =0.69, the pole positions being mo=6.8 GeV and m~ =6.34 GeV. (b) The parametrization of AItomari and Wolfenstein (AW) [6] in which the normalization off_+ at

q'= qm,~ = (rnB -roD) 2 is determined with the help of the quark model

m D ----4- m .

f : (q~,~)'~ 2 x / m ~ m , , (10)

and the q2 dependencc is given by a polc at a mass of 6.8 GeV.

411

Volume 229, number 4 PHYSICS LETTERS B 19 October 1989

Meson Spect rum

. - - .

,> ¢1

(D v

tm

b~

1 4

1.2

0 8

06

04

0.2

-B ~> DT~' a

I I

x

IiI

B$

, , ,

0 4 0 8 1 2 1 6 0,

0 2 24

Lepton Spect rum po(OeV)

,> 0 v d

c~

.

I 4

12

B ~ DT/J b

I

0 8

0 6

0 4

0 2

0 I I I I I 0 0 4 0 .8 1 2 1.6 2 2 4

p,(GeV)

Fig. 2. Characteristics o f decay B ~ Dtv, with form factors from AW [ 6 ] and BSW [ 5 ]: (a ) m o m e n t u m spectrum of the D meson, (b) m o m e n t u m spectrum of the "t lepton, ( c ) longitudinal polarization o f the '[ lepton, compared with the quark model result.

412

Volume 229. number 4 PHYSICS LETTERS B 19 October 1989

- 0 2

- 0 4

- 0 6

- 0 8

Longitud]nol Polorisotion

C

BSW

B ~ DTI.,

AW

QM -I

0 0 2 0 4 0 6 0 8 1 12 1 4

E,.(CeV)

Fig. 2 (cont inued) .

A comparison of the two parametrizations shows that f_ If÷ is much smallcr for BSW as compared to AW. Wc now summarize the main results concerning the decay B--,Dxv,, which may be obtaincd in direct analogy

with the well-known results for K~ decay [4 ]. (i) The ratio of B--,Dzv~ to B--,Dev¢ is

F,(B--,D) _ /0 "25 ( A W ) . (11) I'~(B--,D) (0.30 (BSW).

These numbers bracket the quark model value given in cq. (4). (ii) The momentum spectrum of the D meson, dcrived from eq. (8), is shown in fig. 2a for thc two choices

of form factors. The BSW spectrum is slightly softer. (iii) The momentum spectrum of the T is shown in fig. 2b and is practically indistinguishable for the AW and

BSW cases. (iv) The longitudinal polarization of the x is depicted in fig. 2c, whcrc the quark model result is also shown

for comparison. It may be recalled that for fixed q2, the polarization vector of the lepton is given by [4] P~=A/ IAI with

E ~ , - (12)

where

413

PHYSICS LETTERS B

Meson Spectrum

(_9 v

"ct db

L ED

L

1 4

~2

0 8

0 6

0 4

0.2

W'BIW

0.4 0 8 I 2 1 .6 2 . 0

B ~ D'Tz, a

Volume 229, number 4 19 October 1989

Lepton Spectrum

I 2.4

po'(GeV)

'> u

(-9 v & ED

L ED

u..

't b B ~> D'TI.'

12

1

I I I l

0 8

0 6

0.4

0 2

0 Q

BSW

0 4 0 8 I 2 ~ 6 2, 2 4

p , ( G e V )

Fig. 3. Characteristics of decay B , D*rv, with form factors from AW [6 ] and BSW [ 5 ]: (a) momentum spectrum of the D* meson, (b) momentum spectrum of the t lepton. (c) longitudinal polarization of the 't lepton, compared with the quark model result.

414

Volume 229, number 4 PHYSICS LETTERS B 19 October 1989

o

- 0 2

- 0 4

- 0 6

- 0 8

Longitudinol Polorisotion

C

t I I [ 0 2 0 4 0.6 0 8

AW

BSW

QM I I | -1

0 I. 1.2 1,4

~(OeV)

Fig. 3 ( con t inued) .

2m~ a , (~ ) = m e

[ E ~ + ~ ( ~ - 1 ) ( E o ~ , , -Eo) ] , a2( ~) =rn~ + rnBEq( ~ - l ) + I r n ~ ( ~ - - l ) 2 ,

"E"D max = rn~ + m ~ - r n ~

2mH

~= f_ (q2) , f÷(q2).

Thus for a fixed kinematical configuration, the direction of polarization dcpends only on the ratio 4.

4. Exclusive decay B ~ D*~v,

The matrix element ( D*IJ,~I B) is characterized by four form factors, defined by

(D*(~)~J~`~B)=f(q2)(~+a~(q2)(p~+p~)"~p~+a-(q2)(pn-p~)u~pB+ig~` 'p~(p~+pD)p(pu-pD)° . (13)

The quark model prescription of Altomari and Wolfcnstein yields, for q2 = q2 ....

, , I r a . ( ' • f ~ ~ ' , g~ 2-~n~Ni m B , a+~ ~ m ~ . . . . . 1+ rnr~ I - - mB , a _ = a + + x/ '~D-mn" (14)

The extrapolation to other q2 values follows a pole hypothesis with a pole mass of 6.8 GeV. The form factor a_ contributes to the decay rate in proportion to rn~. In the AW model a_ is close to zero, a feature which remains valid in the BSW ansatz. The double differential cross section in the lepton and D* energies is given in the appendix.

415

Volume 229. number 4 PHYSICS LETTERS B 19 October 1989

The principal results for the decay B--, D*xv, arc as follows: (i) The decay width of B ~ D*rv, compared to the electron mode is

F,(B--,D*) ~0.26 ( A W ) , Fe(B--,D*) - /.0.24 (BSW). (15)

(ii) The D* momentum spectrum is shown in fig. 3a and is essentially the same for the BSW and AW form factors.

(iii) The lepton momentum spectrum is exhibited in fig. 3b. Here we note a significant difference, the BSW prescription giving a harder spectrum.

(iv) The T lepton polarization as a function of the ~ energy, is shown in fig. 3c. (v) The decay width of B--,D~'o,~Tv compared to B~D~'~,~v (where the subscript denote longitudinal and

transverse polarization) is

F,( B--, D~'o.g) ~0.77 ( A W ) , F~(B--,D~'~.,)- (0.79 (BSW), (16)

whercas in the case of electron emission we get

F~(B--,Df'o,g) ~0.91 ( A W ) , F¢(B--,D,*,~,.)- ( I . 07 (BSW). (17)

5. Summao'

(i) Decays of the B meson into DrVT should occur with a probability of (25-30)% compared to the decay B --, Dev,.

( ii ) Decays into D*rv, should occur with a probability of ( 24-26 )% relative to B --, D*ev,. (iii) Decays with higher mass D states may be reasonably approximatcd by the quark model b--,c~v~, which

gives a x:e ratio of 27%. ( iv ) The excl u si ve branch i ng ratios B -4 D ( D* ) rv, depend on the choice of form factors. Th is dependence is

also present in the meson and lepton spectra as displayed in figs. 2 and 3. (v) The ratio D,o.,'tv/D,*~,,drv should bc about 0.8, somewhat smaller than thc ratio 0.9- I. 1 expectcd for the

cv channel.

Appendix

d ~ r G~I V~bl 2 - - - ( f 2/i, + 2mB./h + A 2 + 2m ~fa_ A 3 - 8fgA4

dEc dED. 16~3mB

+ a t AsB, +2m~a+ a_ A6B, + m~a'- ..17B, +8g2Aa).

where

A, = ~ (,n ~, +m~,. - m ~ ) - m,ED.

2 + m ~ {[ - ~(m~, +m~, . + m ~ ) +mB(E~ +ED.) l [ I (m~, - -m~ . +m~) --mBE~I},

416

Volume 229. number 4 PHYSICS LETTERS B

A2 = m ~ ( - 4 E ~ - 3 E D . +mB)+mB(4E~ED. + 4 E ~ +m~. ) -m~(E~ - ~ED. )

+ roD.1 tunED, m~(4E~--mB+2ED.)--m,~(4E~+4E~ED.+~m~)+m~ 3 E ~ - } m ~

I a~ = - m ~ + m~ (E~,. +E~) + m f,----: m~ED. [ m,,(t:~ + 2ED.) + ~ tm~ --,n~. + m~)] ,

.44=mm~ED.[ i 2 ~ (m~+m~.+m~)+rnn(2E~+ED.) ]_m~.mr~E~+~m 2 2 - u(mt~+m~. + m ~ ) - m 3 ( F ~ + E D . ) ,

A s = m ~ ( - 2 m ~ - 2 m ~ , . - 3 2 ~ m~ + 8mBE~ + 4mt~Er,. - 8E~ - 8E~ED.)

+ mB( 3m~ ED. + 4m~ E~) + ~m~.m~ - '~m~" ,

A~, = ~ ( 3 m ~ -m~, . + m~ ) -m . , ( 2E~ + E D . ) ,

A7 = ~ ( , n ~, + rn ~,. - m ~ . ) - mB/ : ' o . ,

= ~t-:~. m~,,7~, , . )I~ + A8 ( m 2 -2 ~ 2 - (m~,+mD. m~)--muEt~.]+2m~ED.E~[~(m~+m~>.+m~)--mB(E~+ED.)]

+ m ~ l - ~ 2 , (mn +mb. +m~)+mB(Ee +ED.)]2+m2D .mBEo2 2,

I B, = - m ~ + m~,-~, m~E~,..

19 October 1989

References

[ I ] J.L. Cortes. X.Y. Pham and A. Tounsi. Phys. Rev. D 25 (1982) 188. [ 2 ] Panicle Data Group. G . P Yost et al., Review ofparlicle properties, Ph~,s. Lett. B 204 ( 1988 ) I. [31 R.E. Shrock, Phys. Rcv. D 24 (1981) 1275. [4] See e.g.R.E. Marshak, Riazuddin and C.P. Ryan, Theory of weak interactions in particle physics ( Wiley-lnterscience, New York,

1969). [ 5 ] M. Bauer, B. Slech and M. Wirbcl, Z. Phys. C 34 ( 1987 ) 103. [6] T. Altomari and L. Wolfenstein, Phys. Rcv. Lctt. 58 (1987) 1583;

scc also B. Grinstein. N. Isgur, D. Scora and M. Wise, Phys. Rcv. D 39 (1989) 799.

4 1 7