Leptonic decay constants of heavy mesons in QCD

Physics Letters B 297 (1992) 181-185 North-Holland

PHYSICS LETTERS B

i

Leptonic decay constants of heavy mesons in QCD A.V. T k a b l a d z e 1 Institutefor High Energy Physics, 142 284 Protvino, Moscowregion, Russian Federation Received 16 June 1992; revised manuscript received 9 October 1992

A formula for calculating the axial decay constant of heavy-light quark mesons has been obtained by means of the QCD sum rules. The calculations are based on the use of the excited state mass spectrum of the corresponding system. The calculated values for the charmed mesons are re= 145 MeV and fF= 160 MeV. For B and B~ mesons, the dependencies of the decay constants on the b-quark mass are presented. At mb=4.55 GeV the results are fB= 165 MeV and fBs= 177 MeV.

1. Introduction The experimental values o f the axial decay constants o f h e a v y - l i g h t pseudoscalar mesons are not known so far. But we are to know t h e m to extract the SM parameters from the experimental data on the Bo/~o-mixing [ 1 ] and semileptonic decays o f B and D mesons. The search for new physics in heavy meson rare decays also requires that the hadronic matrix elements should be exactly n o r m a l i z e d (which is impossible without knowledge o f the decay constants) t2]. The leptonic decay constants o f heavy pseudoscalar mesons have been estimated within various theoretical models (lattice Q C D , Q C D sum rules, potential models, bag m o d e l s ) . The o b t a i n e d values differ a p p r o x i m a t e l y by a factor o f 2 (see table in ref. [ 3 ] ). Even the results one has got by means o f the Q C D sum rules are not the same. The constant fB= 190 ± 30 MeV [4] has been c o n f i r m e d by the data o f refs. [ 5,6 ]. However, t h e f o values are different in all three works (200 MeV [4], 160 MeV [6], 225 MeV [ 5 ] ) . The application o f the nonrelativistic sum rules leads to a much lower value o f f B = 9 0 MeV and f D = 160 MeV [7]. The Borel sum rules in ref. [8] gave f B = 130 MeV and f D = 170 MeV. The difference between the fB values from refs. [4,8 ] m a y be due to the different choices o f the b-quark mass. It is noteworthy that the results o b t a i n e d from the Borel sum Also at Kutaisi State University, Georgia. Elsevier Science Publishers B.V.

rules are strongly dependent on the heavy quark mass. In the heavy quark effective theory ( H Q E T ) two different versions o f the Q C D sum rules give different results for fB: 195 MeV [9] and 130 MeV [10]. Though there are some ambiguities in this a p p r o a c h connected with 1 ~me-corrections and the choice o f b-quark mass as well [ 10, l 1 ]. In ref. [ 12 ] the authors discussed a method, allowing one, within the Q C D sum rules, to d e t e r m i n e the electromagnetic widths o f the vector meson excited state using the d a t a on the massive spectrum o f these states. The proposed m e t h o d has been applied to determine the axial constant o f the Be-meson leptonic decay [ 13 ]. The Bc radial excitation masses have been borrowed from potential models. The advantages o f this m e t h o d against the Borel sum rules and the moment technique are that, first, we do not find strong dependencies on the heavy quark mass and, second, one may get rid of a p a r a m e t e r like the " Q C D continu u m " threshold not restricting by the lowest meson in the saturated physical part o f the sum rule. Usually, the appearance o f this p a r a m e t e r leads to additional uncertainties [ 5,6,14 ]. In the present paper, the a b o v e - m e n t i o n e d m e t h o d is exploited to establish the axial constants on B and D mesons (section 2). These constants are d e p e n d e n t on the masses o f the radial excitations o f B and D. Still, they can be found with good enough accuracy in the framework o f potential models. In section 3 we discuss the choice o f the p a r a m e t e r s and present numerical results. The conclusion s u m m a r i z e s our consideration and corn181

Volume 297, number 1,2

PHYSICS LETTERSB

24 December 1992

pares the obtained data with results of other works.

In this definitionf~= 132 MeV. An account of the power corrections gives us the following sum rules:

2. F o r m u l a for d e c a y c o n s t a n t s

M4 f~ k=l ( m e + m q ) 2 ( M 2 +Q 2)

Let us consider the two-point function associated with the axial current j = iQ~sq (Q and q meaning the heavy and light quarks, respectively):

1 [" Im II(s) as - n J ~s + Q d s + C l ( Q 2 ) - - n ( G G }

+ C2( Q 2 ) ( mqqq} + C3( Q2)mQg ( qauv • ½2aqG~u~}

H(q 2) = i ~ d4x exp(iqx)

+ C4( QZ)oq ( qq} z , X (OITUs(x)js(O))l O}.

(1)

Write for H(qZ) the dispersion relation 1 H(q 2) = ~

[

ds Im H(s)

.

3 (q2_ m 2e ) 2 {

q2

+In

q2

1+

q2

2 lnq2_

mQ

M4f2k e x p ( - M 2 r ) (me+mq) 2

k=l

The subtraction terms are left out from (2), since later, after the transition to the Borel sum rules, they are eliminated all the same. The I m H ( q 2) expression has the form [ 14,15 ] Im H(q 2) = 8 n

with Ci ( Q 2) being the relevant Wilson coefficients. The application of the Borel operator IS~(Q z) to eq. (5) results in

(2)

~ s--q

(mQ+mq) 2

m~

k, q2 ]

m 2

q2

+~ln~2

q --mQ

+ mq~2~ In q2 - rn 2e + _ rn~ _ m~

4 a ~ [ 9 + 2 / (rnZe'~

~-

q2--mzQ

+ln--

q l}

q2--m~

(5)

I m H ( s ) e x p ( - s z ) d s + C ~ ( z ) ( }i,

=-~

(6)

where £ T ( x ) = lim ~ 5 n,x~oo n/x=T

n.

-

and C~(z) =£~( Q2)Ci( Q2) ,

ln~

C'j = -

exp(-rn~z),

C~ = -

12n e x p ( - m ~ r ) ,

1

where

C 3' = - ½ z ( I - ~ 1m Q2r ) e x p ( - m ~ z ) ,

/(x)=-- i dyy-J !n(l-y) 0

C 4' = _ 8 n z ( 2

is the Spence function. Suppose that the physical part of the sum rules is saturated by an infnite number of narrow resonances with the masses Mk and the axial constantsf~. Then Im/-/(q2) is expressed like 1_im H ( s ) _

M~f~k (mQ+mq)

8 ( s - - M 2) .

(3)

, e r4 - ~ , ~, ,2o r - g m

2)

exp(-rn~z)

the C~(r) here are obtained in the limit mq-~0. Now for the exponential series in the left-hand side of (6) use the Euler-McLoren formula [ 16]: k=l

M~f~, e x p ( - M ~ r )

2

Here and in what follows me and mq stand for the masses of the heavy and light quarks. We define the leptonic decay constant f f o r the meson M in a standard way as

=

( m [ Q.yuysqlO ) = - ipufM .

Make again the Borel transformation £M~(r) in (6)

182

(4)

dMk-~-~kM~f~exp(-M~r) Mn n--I

+ ~ M k4f2k e x p ( - M 2 r ) + ....

(7)

k=0

and take into account ( 7 ) . Then forfk we get f ~ = 2 ( m e + m q ) 2 dMk 1 Im Mk3 dk n

II(M 2) .

(8)

At k = 1 this formula gives the constant values for lowest mesons. Here we have exploited the following p r o p e r l y o f the Borel operator: £dx) [x" exp(-bx) ]-, Jr+")(z-b)

.

In this case, the c o n t r i b u t i o n from the power corrections vanishes. The reason is the a p p e a r a n c e o f factors like J+ <") (me--Mk) 2 2 in the coefficient functions ( mQ~ Mk). The singular expression like J ~+n)( m ~ M 2) is due to our earlier transition to the limit mq---~O. In the case o f the Bc meson for C1(S)=L~(r)CI(z) " ~ ' we derive the expression C'(=

- ~ ~1 ( ~ + (mQ72mq)2)(6A,-15A2+4A3),

where (2N-3)!! /{, 1)N AN=-- 2(2N_2)!!v/~,'-- ~_ ,

v= 1--

24 December 1992

PHYSICS LE'I TERS B

Volume 297, number 1,2

4mqmQ s--(mQ--mq) 2"

At ma=O we have C ' { ( s ) = 0 , a n d no singularities arise. While calculating the electromagnetic widths of J/ gt and 7" mesons and the decay constant o f Be [ 12,13 ], we had an insufficient contribution o f the power corrections ( ~ 10% for J/q/and 7" and ~ 50/0 for Bc). But in our case these corrections yield no c o n t r i b u t i o n at all (at least, this c o n t r i b u t i o n may be neglected). It seems that the strong Q C D v a c u u m n o n p e r t u r b a t i v e effects, involved in hadron formation, are allowed for in the factor dMk/dk implicitly in terms o f the meson masses.

Let us calculate the factor dMk/dk from three points using simple rules o f numerical differentiation. Since the functional dependence o f the heavy meson mass on the principal q u a n t u m n u m b e r is rather smooth, three points must give a reliable result. This may be checked with the ~u and 7` systems, whose spectra are well known from experiment. At k = l, the difference between the derivatives in both cases (three and four points) is less than 1%. An argument in favour o f the three points is also a considerable loss o f accuracy in the calculations of the higher radial excitation mass. The values o f dMk/dk for different mesons, obtained by means o f the spectra [ 17 ], are presented in table I. As follows from eq. (8), fg only depends on the heavy quark mass and the strong interaction constant O~s(M2),unlike in the Borel sum rules [6,8], where the physical part I m / / ( q 2) is only saturated by the lowest meson, and there appears an additional parameter, i.e. the c o n t i n u u m threshold), The quark mass in ( 5 ), i.e. in ( 8 ), is renormalized on-shell. In the literature [4-6,8 ], one finds different values for the on-shell heavy quark masses: m e = 1.35-1.46 GeV,

mb = 4 . 5 - 4 . 8

GeV.

Figs. la, 1b show the leptonic decay constants o f the B and D mesons versus the heavy quark masses. The strange quark mass is m s = 150 MeV. One should note, however, that both for D and B mesons the as corrections in ( 7 ) are of the o r d e r o f 40% o f the total value (cf. refs. [6,8] ). To diminish the as contributions, one should pick up for the mass another n o r m a l i z a t i o n point p 2 = _ rn~ (instead o f pZ = m ~ ) [ 18,19 ]. The on-shell masses are related to the new masses through the formula

mQ(p2=m~)=mQ(p2=--m~)(l + 21n 2 ~ ) . Fig. 2 gives now axial constants versus the new masses mQ(p2=--m~). The mass range is in correspond-

3. On choosing the parameters and numerical results Table 1 We d e t e r m i n e d the factor dMk/dk from the data on the spectra o b t a i n e d within the potential m o d e l [ 17 ]. W i t h the precision o f d e t e r m i n a t i o n o f the excited state mass o f 50 MeV (which is the claim o f all the potential m o d e l s ) , the relative error for the D meson mass is ~ 2%, and for the B it is ~ 1%.

Meson

dM•/dkl

D F B B,

0.74 0.67 0.64 0.63

kffi 1

183

Volume 297, number 1,2

PHYSICS LETTERS B

24 December 1992

fB,Bs in MeV

fD,F in M e V 180

'

'

'

'

'

'

'

'

'

I

'

'

'

'

'

'

'

'

180

'

,

a

,

,

,

I

'

,

,

,

I

'

'

'

i

b

~-

160 160 140 140 120 120

......... 1.3

' ......... 1.4

i00

....

1.5

' .... 4.6

4.5

me in G e V

D .... 4.7

4.8

m~ in GeV

Fig. 1. The dependencies of the leptonic decay constants (a) of charmed mesons on the c-quark mass mc(p2=mZ~), (b) of beauty mesons on the b-quark mass mb(p2 = m 2). fD,F in M e V 180

....

, .........

_

~ , B s in M e V , ....

200

'

'

'

'

1

'

'

'

'

1

'

'

'

'

a

b 180

160

160 140 -

120 .... 1.15

~

, ......... 1.2

i .... 1.3

140 i

i

i

4.1

me in G e V

I

J

l

t

,

l

[

l

l

4.2

l

4.3

,

4.4

mb in G e V

Fig. 2. The dependencies of the leptonic decay constants (a) of charmed mesons on the c-quark mass me(p2=-m~), (b) of beauty mesons on the b-quark mass mb(p2= -mZ). e n c e w i t h the v a r i a t i o n s o f the on-shell masses in fig. 1. In this case the ors c o r r e c t i o n s for the D m e s o n s reach 25%, and for B m e s o n s ( 1 0 - 1 5 )% ( t h e as cont r i b u t i o n decreases w i t h the g r o w t h o f the b - q u a r k m a s s ) . As is clear f r o m fig. 2, the v a l u e s o f f o a n d fF, in fact, do not change, whereas those offn andfB, h a v e i n c r e a s e d by 10-15 MeV. N o t e , t h a t in o u r calculations we are n o t faced w i t h a strong d e p e n d e n c e on the h e a v y q u a r k mass as in the Borel s u m rules (especially in the case o f c h a r m e d p a r t i c l e s ) [6,8 ]. F o r the r u n n i n g c o n s t a n t as, we h a v e used the standard formula 184

12~ as= 33-2nf

1

ln(Q2/A 2) '

w i t h A = 130 + 30 MeV. T h e v a r i a t i o n o f A b e t w e e n 130 +_ 30 M e V alters the o b t a i n e d results by o n l y 2 - 3 MeV. T h e i n a c c u r a c y o f o u r e s t i m a t e s is d e t e r m i n e d m a i n l y by the precision o f the value o f the factor d M k / d k o f ~ 5%. T h e e r r o r is the s a m e in the s u b s e q u e n t orders o f p e r t u r b a t i o n theory. T h e r e f o r e , the r e l a t i v e e r r o r in o u r c a l c u l a t i o n s is ~ 10%.

Volume 297, number 1,2

PHYSICS LETTERS B

24 December 1992

4. Conclusion

Acknowledgement

F r o m the Q C D sum rules we have derived the formula calculating the decay constants of heavy pseudoscalar mesons by means of the excited state spectra of the corresponding mesons. These state spectra are determined in the framework of the potential model. The resultant formula f o r f d o e s not contain the Borel parameter or the " Q C D c o n t i n u u m " threshold involved in the Borel sum rules. This allows one to construct easily the dependence of f on the heavy quark mass (fig. 2). The obtained values for fD= 145 MeV and f v = 160 MeV practically do not change when the c-quark mass varies in reasonable limits. Provided the theoretical uncertainties are taken into account the value offD agrees with the data of refs. [ 6-8 ]. The leptonic decay constant for the F meson turns out to be lower than the predictions obtained within the Q C D sum rules [ 5,6,20 ]. The constantsfB andfB, are strongly dependent on the b-quark mass. At m b ( p 2 = - m 2) = 4 . 1 7 GeV, we have fB= 165 MeV, which agrees beautifully with the results of ref. [19], where the same b-quark mass is used. When mb(pZ=-m~,)=4.4 GeV ( m b ( P 2 = m~) = 4 . 8 G e V ) , t h e n f b = 132 MeV, which goes well with the value obtained in ref. [8 ]. This result is also in agreement with the recent calculations offs in the static quark approximation (.f~-= 130 MeV at m b= 4.8 GeV) [ 21 ]. The same v a ! ~ forfB is obtained from Q C D sum rules in H Q E T [10]. We may state that the ambiguity of the estimated value of f is due to the choice of the b-qUark mass. The saturation of the hadronic part of the Borel sum rules by the few lowest lying resonances leads to the f s and fD values; _~hich are not contradictory to the results presemed in this paper [ 22 ]. I n the limit of the SUr(3 ) symmetry, the axial constants for the D and F a n d B and Bs mesons coincide. The SUf(3 ) symmetry for f , and f~ is broken up to 20%. The values obtained for the fD and fF differ by 12%, while the difference between t h e f s a n d f s , values is 7%, Therefore, with the mass of the second quark growing, the relative difference between the leptonic decay constants of the strange and nonstrange mesons diminishes.

In conclusion I would like to thank V.G. Kartvelishvili, V.V. Kiselev and A.K. Likhoded for useful discussions.

References [ 1] J. Ellis,J.S. Hagelinand S. Rudaz, Phys. Lett. B 192 ( 1987) 201. [2 ] S. Bertolini,F. Borzumatiand A. Masiero, Phys. Lett. B 192 (1987) 437. [ 3 ] P. Colangelo,G. Nardulli and M. Pietroni, preprint BARI TH/90-79 (1990). [4] L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rep. 127 (1985) I. [5] C.A. Dominguez and N. Paver, Phys. Lett. B 197 (1987) 423;B 199 (1987) 596 (E). [6] S. Narison, Phys. Lett. B 198 (1987) 104. [7 ] A.R. Zhitnitsky, I.R. Zhitnitsky and V.L. Chernyak, Soy. J. Nucl. Phys. 38(5) (1983) 773. [ 8 ] T.M. Aliev and V.L. Eletskii, Soy. J. Nucl. Phys. 38 ( 1983) 936. [9] E. Baggan,P. Ball, V.M. Braun and H.G. Dosch, Phys. Lett. B278 (1992) 457. [ 10] N.F. Nasrallah, K. Schilcherand Y.L. Wu, Phys. Lett. B 261 (1991) 131. [ 11 ] D.J. Broadhurst and A.G. Grozin, Phys. Lett. B 274 ( 1992 ) 421. [ 12] B.V. Geshkenbein, Yad. Fiz. 42 ( 1985 ) 991 [ in russian ]; S.S. Grigoryan, preprint IHEP 88-7 (Serpukhov, 1988) [in russian ]. [ 13] V.V. Kiselevand A.V. Tkabladze, Sov. J. Nucl. Phys. 50(6) (1989) 1063. [ 14] L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Lett. B 104 (1981) 305. [ 15] D.J. Broadhurst, Phys. Lett. B 101 ( 1981 ) 423. [ 16] Handbook of mathematical functions, eds. M. Abramowitz and I.A. Stegun (Nauka, Moscow, 1979) [in russian]. [ 17] S. Godfrey and N. Isgur, Phys. Rev. D 32 (1985) 189; J.L. Basdevant, P. Colangelo and G. Preparata, Nuovo Cimento A 71 (1982) 445. [ 18] M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385. [19] L.R. Reinders, Phys. Rev. D 38 (1988) 947. [20] M.A. Shifman,Usp. Fiz. Nauk. 151 (1987) 193 [Soy. Phys. Usp. 30 (1987) 91]. [21 ] C.A. Dominguez and N. Paver, Phys. Lett. B 276 (1992) 197. [22] J. Liu and K.-T. Chao, University of Minnesota preprint TPI-MIMN-91/52-T ( 1991 ).

185