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Leptonic decay constants of charm and beauty mesons in QCD
Volume 197, number 3
PHYSICS LETTERS B
29 October 1987
L E P T O N I C DECAY C O N S T A N T S O F C H A R M A N D B E A U T Y M E S O N S I N Q C D C.A. D O M I N G U E Z 1.2 Deutsches Elektronen-Synchrotron, DESE D-2000 Hamburg 52, Fed. Rep. Germany
and N. PAVER 3 Dipartimento di Fisica Teorica, Universitd di Trieste, 1-34100 Trieste, Italy Scuola Internazionale Superiore di Studi Avanzati, 1-34100 Trieste, Italy and lstituto Nazionale di Fisica Nucleate, Sezione di Trieste, 1-34100 Trieste, Italy
Received 24 June 1987
Leptonic decay constants of heavy pseudoscalar mesons are estimated in QCD by means of Hilbert transform power-moment sum rules at Q2=0. The meson masses are also obtained in order to assess the reliability of these predictions. Our results are fD/f~ = 1.7 +_0.2, fv/f~ = 2.1 --+0. l, and fa/f~ = I. 1- 1.6. As a byproduct a bound on fas/fB is obtained.
An accurate knowledge o f the leptonic decay constants o f heavy pseudoscalar mesons is quite i m p o r t a n t as they play the key role o f absolute n o r m a l i z a t i o n s to a wide variety o f heavy flavour weak transitions. In the absence o f e x p e r i m e n t a l data, considerable effort has been d e v o t e d to the theoretical estimate o f FD, fv, fB, etc., in particular in the f r a m e w o r k o f Q C D sum rules [ 1-4] (for a review o f other estimates see e.g. refs. [ 5,6]). As is well known, this m e t h o d relates through dispersion relations low energy parameters, such as particle masses and coupling constants, to the short-distance o p e r a t o r p r o d u c t expansion ( O P E ) o f current correlators. This OPE is assumed to be valid in the presence o f n o n - p e r t u r b a t i v e effects which are p a r a m e t r i z e d by a set o f vacu u m expectation values o f quark and gluon fields. These v a c u u m condensates induce power corrections to a s y m p t o t i c freedom [ 7 ]. Although this m e t h o d seems well defined, there is some freedom in the choice o f the o p t i m a l weight in the dispersion relation a n d in the m o d e l used for parametrizing the spectral function. Also, the m e t h o d is sensitive to the n u m b e r o f loops considered in the calculation o f the a s y m p t o t i c f r e e d o m contribution, as well as to the n u m b e r a n d actual values o f the v a c u u m condensates included in the OPE. All this has resulted in a wide spectrum o f predictions for fD, fa, etc., e.g. fB = 6 0 - 2 0 0 MeV. Since on the one hand, better knowledge o f these constants would allow for m o r e accurate predictions in a n u m b e r o f semi-leptonic and non-leptonic transitions, a n d on the other h a n d there has been recent progress in the d e t e r m i n a t i o n o f v a c u u m condensates, we feel that a systematic reanalysis o f this p r o b l e m is needed. We a t t e m p t to do this here by choosing as a starting p o i n t the Hilbert transform power m o m e n t sum rules at Q 2 = 0. Aside from keeping the n u m b e r o f free p a r a m e t e r s to a m i n i m u m , these sum rules p r o v i d e a natural framework to study systems with a heavy quark, in the sense that at Q 2 = 0 n o n - p e r t u r b a t i v e effects are p a r a m e t r i z e d through the O P E as a series in inverse powers o f the heavy quark mass m o, i.e. the only large mass scale. To this extent, 1/mQ plays the unambiguous role o f the short-distance expansion parameter. The situation is not so transparent with Laplace Alexander von Humboldt Research Fellow. -" On leave of absence from Facultad de Fisica, Pontificia Universidad Catolica de Chile, Santiago, Chile. 3 Supported by MPI (Ministero della Pubblica Istruzione, Italy). 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
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transform sum rules as they involve an additional short-distance parameter, i.e. the Laplace variable a =-I/M z. Nevertheless, we shall also comment on this method at the end. In our analysis we incorporate the leading power corrections of dimension d~< 6 and consider perturbative effects up to order O (as). Regarding the spectral function parametrization, in addition to the lowest pseudoscalar meson pole we include continuum contributions and compute the corrections they induce at the one and two-loop level. Finally, to assess the reliability of our results for the leptonic decay constants we also estimate the masses of the heavy pseudoscalar mesons. We begin by defining the two-point function 9'5 (q) = i
J d4x exp (iqx) (0[T(0UA~,(x) 0"A,(0)) 10),
(1)
where
O~'A~,(x)= (mq + mo):Ct(x) i)'sQ(x):,
(2)
with q(x) ( Q ( x ) ) being a light (heavy) quark field and mq (mQ) its corresponding current mass. By selecting q = u , d, s and Q = c , b the axial-vector current divergences (2) will have the quantum numbers of D, F and B mesons. The function ~5 (q) satisfies the dispersion relation (Q2-= _ q2 > 0) oo
g s ( Q 2 ) = n1J f ds I_mys(s) (s+Q 2) +subtractions,
(3)
0
defined up to two subtractions, arising from external renormalization, which can be disposed of by taking at least two derivatives in (3). This procedure leads to the Hilbert transform or power moment sum rules, which at Q2=O become (__)n+l
("
d
)- - n + l g/,(Q2) Q2 o _ 1 ~f s___~5_ ds im q/5(s)"
(4)
Up to the actual values of the vacuum condensates the moments ~0(") can be computed in QCD and thus eq. (4) relates the resonance parameters in the spectral function (1/n)Im ~us(s) to the fundamental perturbative and non-perturbative QCD parameters in ~o("). Althoughthe quark and gluon condensates cannot yet be computed from first principles, they can be extracted from independent experimental data using QCD sum rules in other channels, e.g. the charmonium system, e+e - total cross sections, etc. The perturbative QCD expression for the spectral function at the two-loop level, as first obtained correctly in refs. [3,8], is given by 1 im gs(q2 )
rc
AF =
3(mq+mQ)20"~V[l+4°q ( 8rc2q2 ~ ] (7-- v2)
+ i=, ~ {(v+v-')[Li2(a,a2,-Li2(-aj)-lnailnfl~l+A,
lna,+Bilnfli})],
(5,
where Liz(X) = - i dt t -I ln(1 - t ) ,
(6)
0
3(3mq+mQ~ A, = ~ k mq +mQ /
424
19+2v2 +3v 4 32v
mq(mq--mQ)(l+v+2v/a,), c]2v(1 +v)
(7)
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PHYSICSLETTERSB
Bt=2+2(m2q-mS) qZv
'
mq ( l - v ) °tl=--mo i ~ p
' ]~1=
29 October 1987
x/1 +ott (1 +v) 2 4v '
(8)
and qZ=q2 (mq_mQ)2, v----(1 --4mqmQ/(12)1/2.The expressions for A2, B2, az andfl2 are obtained from (7), (8) by the interchange mq*-+mQ. In particular, in the limit mq--*0, well justified for up- and down-quarks as in the case of D and B~.d mesons, as exact integration of (5) leads to [ 3 ]
~ok'~b= ~
3 (~--~) "-1
B(n, 3)(1 +a°a~),
(9)
where B(x, y) is the beta function and a ° are the rational numbers 3rt o ~zz -~-a,- 6 -1
2 n+l
6 "~z[1 (~ n + ~ 4- ~=t 7 +
1 n
1 (n+l)
1 (~3))lJ -( n +- 2 ) +
.
(10)
Parametrizing non-perturbative corrections to asymptotic freedom through the OPE and keeping the leading vacuum condensates of dimension d~< 6, one has, always neglecting mq, m S 2 C4 04 > -~- m~ q2 3 C5(05> IP'5(qZ) IN'P-- mS--q 4 (mS--q2) +
(m~ _q2)2
(m S _q2)3
(m~ _ q2)4 ]
where the C. ( 0 . ) are defined as C 4 ( 0 4 > = ( (as/12=)G 2 -mQdlq ),
(12)
C5 (05) = (gsCliaG'i~q),
(13)
C 6 ( 0 6 ) =~Ogs ((cWuJ, q ) Z ~lTuaq/ • q
(14)
Computing the non-perturbative part of~0(") through (11 ) and adding it to (9) one finds using (4) that for the first few moments, with increasing n the perturbative piece of ~o(n)decreases in magnitude but at the expense of an increase in the non-perturbative contributions. As a compromise we then choose to consider only the first two moments; this will allow us to estimate fp as well as Mp. From eq. (11 ) one obtains ~(I) N.P.--
C4(04~ + 3C5(05) m~
4
1 (C4(04) N.o. m ~ \
m~
5 C6(O6> 3 m~ '
m~
3C5(05) +2
m~
11 C 6 ( O 6 ) ) 3 m~ "
(15)
(16)
Adding (15), (16) to (9) completes the theoretical calculation of the LHS of (4) in QCD. Turning to the hadronic spectral function appearing on the RHS of (4) we choose the following parametrization: lzt Im ~5(s) HAD=2fZM4~(s--M~') +O(S--So) 1 Im g/s(s) A.V.'
(17)
where So is the threshold for asymptotic freedom, the second term in (17) is given in (5), and ( 0 IA~ IP(p) > =ix/2fpp,,,
(18) 425
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defines the leptonic decay constant fe; with this normalization e.g. f~ = 93.2 MeV. Given the rather large masses of the D, F and B mesons we would expect the spectral function to be relatively smooth for s>M g, and thus the second term in (17), i.e. the perturbative QCD continuum, should be a reasonable parametrization of Im ~/5(S)HAD at such energies. In other words, we assume that potential radial excitations of the heavy pseudoscalar mesons do not show up as prominent narrow peaks in the spectral function. In this case the first two moments read
2f ~,
1
M2 - 8n 2 [ ( 1 - a , ) + a s ( 0 . 7 5 1 - b ~ ) ] +
C4(04)
3 C5(05)
m------~-Q + 4
2f~ M~( 1 M 2 - m~ 3-~2nz [ ( l - a 2 ) + o t s ( 1 . 7 0 6 - b 2 ) ] +
m~
C4(04)m~ +
5 C6(06)
3
23C5 (O5)rn~
m~
'
) 113 C6 (O6)m~ '
(19) (20)
where al, a2, bl, b2 are the continuum corrections which can be computed by integrating the second term in (17). Concerning the values of the QCD parameters entering the sum rules (19), (20) we use for the "on-shell" quark masses: mc(Q 2 = m 2) = 1.3 GeV, mb(Q z = m b2) =4.6 GeV, ots(m 2) =0.296, and as(rn 2) =0.214. For the vacuum condensates we take m c ( d l q ) = - 0 . 0 1 4 GeV 4 and m b ( ~ q ) = - - 0 . 0 5 5 GeV 4, which follow from the most recent determination of light quark masses and condensates [ 9 ]; ( otsG 2) / 12 n = 0.003 GeV 4 as extracted from e+e - data [ 10] and charmonium [ 11 ]. The quark-gluon condensate (gsCzlitrG'2q) = 2 M 2 (clq) seems to play a particularly important role in the QCD analysis of the baryon spectrum. The results obtained for the mass parameter M~ are, however, somewhat controversial. We take M 2 = (0.5 _+0.1 ) GeV 2 as a compromise between e.g. M~ -~ (0.1-0.4) GeV 2 [ 12] or M 2 -~ (0.6-1.0) GeV 2 [ 13], and M 2 ---0.3 GeV 2 [ 14], the latter value being suggested by charmonium sum rules. We have found that the higher value M 2 -- 1 GeV 2 [ 15] is in disagreement with the observed D- and F-meson masses. Such a sensitivity to the quark-gluon condensate is actually an interesting aspect of eqs. (19), (20). Finally, for C6(O6) we use C6(O6) -- - 9nots(dlq) 2, which takes into account a factor of five increase in the factorization estimate according to recent results from an e + e - analysis [ 10 ] as well as previous claims from other sources [ 12,15 ]. Starting with the D-meson and solving (19), (20) for values of the asymptotic freedom threshold within the wide range So = 2M 2 - 3 M ~ we obtain MD = 1.85 --+0.15 GeV,
(21)
to be compared with MDlexp = 1.87 GeV, and fD = 160-+ 15 MeV.
(22)
Continuum corrections in this channel are important, i.e. without them MD--~2-2.2 GeV, but yet not so large as to spoil the predictive power of the sum rules [ 16 ]. We wish to stress that So is a free parameter in the sumrule approach; the choice we made is an educated guess based on experience in other channels. The two important points are: (i) predictions should be reasonably stable under changes in So within a (hopefully) wide region, and (ii) obtaining a value for FD from the sum rules is not enough. Even with everything else fixedfD is clearly a function of So. Hence, some other quantity whose experimental value is well known, e.g. MD in our case, should be estimated simultaneously in order to assess the reliability of the prediction for fD- It is rewarding that our results above meet these two requirements. Considering next the F-meson, if we continue to neglect the light quark mass, ms in this case, we would obviously find MF=Mo, and fF=fD. Given the uncertainties in the values of the vacuum condensates and in So, which reflect themselves on the errors in (21 ), (22), we will be unable to predict the small SU (3) breaking mass splitting Mv-MD. Notice that the mass is obtained from the ratio of two sum rules. However, the situation is better for fF. For example, by retaining terms of order O (c) and O (e In e), with ~-= mJme, eq. (19) becomes 426
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2f2/M2 = (1/8n2){ 1 + 3e+ as[0.751 + (6In)e(1 + n 2 / 9 + l n e)] } + continuum+ non-perturbative.
29 October 1987 (23)
A comparison with the corresponding relation for f o leads to the approximate expression fv/fD ~-- ( MF/MD)(1 + 3 0 ~- 1.2.
(24)
We have solved the two sum rules in this channel by numerical integration of the imaginary part eq. (5) retaining ms but without approximations, i.e. without expanding in E as in eq. (23). Concerning the non-perturbative terms, at the present level of accuracy we have safely ignored the small SU(3) breaking in the quark condensates and used the same values of Cn(On) as before. In the broad duality region So ~ - 2 M ~ - 3 M 2 we obtain M y = 1.9+0.1 GeV,
(25)
as expected, and fv = 194_+ 12 MeV,
(26)
in nice agreement with the ~-expansion result eq. (24). Turning to the B-meson an inspection of the various contributions to ( 19 ), (20) shows that except for Ca (O4) the non-perturbative terms are unimportant. Continuum corrections in this channel are very large, a fact known from previous analyses [2], e.g. without them we would predict MB= 8 GeV as opposed to MBlexp= 5.27 GeV. Including these corrections and allowing for the asymptotic freedom threshold to vary in the range So 1 . 1 M ~ - 2 M ~ brings MB down to Ms--- (5.1-6.2) GeV. Given the large value of the B-meson mass it would be reasonable to expect a precocious onset of asymptotic freedom in this channel, e.g. So - (1.1-1.3)M 2,
(27)
which narrows down the resulting mass to MB =5.2+_0.2 GeV.
(28)
It should be emphasized, though, that (28) is not a genuine prediction. If we have not had beforehand information on MB and had allowed So to vary over a wider range the uncertainty would have been considerably larger. The above procedure is then to be understood more as a determination of So than of MB. However, since our purpose is to predict fB, assessing the reliability of this prediction through the resulting value of MB, we can use So in the range (27) to obtain fB = 104-150 MeV.
(29)
Using the experimental value of MB as input, the eigenvalue solution for So is: So= 1.2 ME, and f8 = 127 MeV. However, the result (29) provides a better feeling for the true uncertainties involved in this channel. Concerning the B,-meson, its decay constant is given by an expression analogous to eq. (23) except that now =- ms~rob "~ 0.04. Given the smallness of ~ and of the non-perturbative power corrections we expect the ratio fB,/f8 to be more accurately predicted than either fBs or fB separately. Our result for this ratio is fB,/fB > MB,/MB,
(30)
which should be useful in connection with oscillations in the B°-B°-meson system (for reviews see e.g. refs. [5,17]). The most severe source of uncertainty in the estimate offB, eq. (29), is its power dependence on So. This is a characteristic feature of the Hilbert transform sum rules (4), which has a bigger impact on fB than on fD orfv, eqs. ( 2 2 ) a n d (26). To minimize this sensitivity to some extent, we have selected a duality region for so by requiring that the ratio between eqs. (19) and (20) reproduces the experimental value of MB within some error, e.g. 5%. In principle, Laplace transform QCD sum rules are expected to be far less sensitive to So on account of their exponential weight. In the present application these sum rules are [4] 427
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so
2f2p-~ exp(-Mz/M2)= ,!odsexp(-s/M2) llm~us(S)QcD +exp(-m~/M z) C4(04)---~-~
1-2M2]C5(05>+- ~
2 - 2M 2
6M 4J
C6(06) ,
(31)
where
i dsexp(-s/M ) l~rl m
g/5(s) Q C D - -
m~
- M2(M 2+So) exp( -
3
8n 2
{m~[El(m~/M2)_El(So/M2)] +M2(M2+m~) exp(_m~/M2 )
S o [ M 2 ) - 2m~M2 [ exp(
-m~/M 2) - e x p (
- S o ~ M 2 ) ] + O(Ots)} ,
(32)
with P - - D , F, B, etc., and co
E l ( Z ) = f e x p , - t ) tit.
(33)
Earlier estimates based on these sum rules [4] have claimed somewhat smaller values offD and fB than the ones obtained here through Hilbert transforms. However, the point we wish to raise here is that although eq. (31 ) does exhibit a better (softer) behaviour on So, it has conceptual as well as numerical disadvantages. First, the power corrections in eq. (31 ) appear dominated by C4 ( 04 > on account of the smallness of C5 ( 05 > and C6 (O6 > rather than because of the usual Laplace suppression; notice that the terms multiplying C5 (O5 > and C6 ( O6 > are roughly comparable. This means that inside the "sum rule window" ( M 2-- 0.8-2.1 GeV 2 for P = D and M 2~-3.3-4.2 GeV 2 for P - - B ) the Laplace variable M E has lost the unambiguous character of the shortdistance expansion parameter it usually has in the familiar applications to light quark systems. A second cause of discomfort is the pronounced sensitivity of the perturbative contribution eq. (32) to the values of mQ and M E. In fact, the dependence on rnQ is in this case exponential. For instance, in the case o f f a a 4% increase in mb from rob----4.6 GeV to rob----4.8 GeV produces a change in the perturbative contribution of a factor of 2-3 for M2,-'3-5 GeV 2. On the other hand, for fixed So and mQ, eq. (32) changes by a factor of 8-16 for P = D and by a factor of 40 for P = B, inside the "sum rule window" in M E. The corresponding changes in the nonperturbative piece are roughly a factor of 3 and 20 (in the same direction), respectively. In the case offD these extreme variations are offset by corresponding large variations in the exponential on the LHS of eq. (31 ), so that in the end the result fOrfo appears somewhat stable. All things considered we findfo ~ 120-150 MeV inside the "window" M 2 ~ 1.5-3 GeV 2, and for So- (I.5-3)M2D, which is consistent with eq. (22). However, in view of the above remarks we would not attach much significance to this result. For fa the huge variations of the perturbative and non-perturbative contributions are not entirely offset by the variation of the exponential on the LHS of eq. (31 ), so that uncertainties are beyond the 100% level. We wish to point out that such a dangerous situation is not encountered in the usual applications of Laplace sum rules to light quark systems. Also, these problems do not affect the Hilbert transform power moments at Q2--0, which in our view, are more reliable to treat charm and beauty mesons as they lead to stable predictions. In conclusion, collecting our results and normalizing to f~ = 93.2 MeV we have found
fDIf~=l.7+_0.2, fvlf~=2.1+_O.1, fnlf~=l.l-l.6, fBs/fa>~Mas/MB.
(34)
The quality of these predictions is gauged by the results obtained for the meson masses in the same framework, i.e. eqs. (21), (25), and (28). As usual, however, the ultimate test will have to come from experiment. In this
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References [ 1 ] V.A. Novikov, M.A. Shifman, M.B. Voloshin, A.I. Vainshtein and V.I. Zakharov, in: Proc. Neutrino-78 Conf. (West Lafayette, IN, 1978); E.G. Floratos, S. Narison and E. de Rafael, Nucl. Phys. B 155 (1979) 115; V.S Mathur and T. Yamawaki, Phys. Lett. B 107 (1981 ) 127; E.V. Shuryak, Nuel. Phys. B 198 (1982) 83; S. Narison, Z. Phys. C 14 (1982) 263. [2] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. B 104 (1981) 305. [3] D.J. Broadhurst and S.C. Generalis, Open University Report No. OUT-4102-8/R (1982). [4] S.C. Generalis, Ph.D. Thesis, Open University Report No. OUT-4102-13 (1984); A.R. Zhitnitskii, I.R. Zhitnitskii and V.L. Chernyak, Sov. J. Nucl. Phys. 38 (1983) 775; T.M. Aliev and V.L. Eletsky, Soy. J. Nucl. Phys. 38 (1983) 936. [ 5 ] A. All, DESY Report No. DESY-86-041, and Proc. LEP-Physics-Jamboree, CERN (Yellow) Report No. CERN-86-02, Vol. 2 (CERN, Geneva, 1986). [6] J. K6rner, in: Proc. Intern. Symp. on Production and decay of heavy hadrons (Heidelberg, 1986). [7] M.A. Shifman, A.1. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385,448; L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Rev. 127 (1985) 1. [8] D.J. Broadhurst, Phys. Lett. B 101 (1981) 423. [9] C.A. Dominguez, M. Kremer, N.A. Papadopoulos and K. Schilcher, Z. Phys. C 27 (1985) 481; C.A. Dominguez and E. de Rafael, Ann. Phys. (NY) 174 (1987) 372. [ 10] R.A. Bertlmann, C.A. Dominguez, M. Loewe, M. Perrottet and E. de Rafael, DESY Report, to be published; see also R.A. Bertlmann, in: Proc. XVII Intern. Symp. on Multiparticle dynamics, eds. M. Markytan, W. Majerotto and J. Mac Naughton ( World Scientific, Singapore, 1987); C.A. Dominguez, DESY Report No. DESY-87-002 (1987), in: Proc. Intern. Workshop on Quarks, gluons and hadronic matter (Cape-Town, 1987), to be published. [ 11 ] J. Marrow and G. Shaw, Z. Phys. C 33 (1986) 237. [ 12] Y. Chung, H.G. Dosch, M. Kremer and D. Schall, Z. Phys. C 25 (1984) 151. [ 13] V.M. Belyaev and B.L. Ioffe, Sov. Phys. JETP 56 (1982) 493. [ 14] S.N. Nikolaev and A.V. Radyushkin, Soy. J. Nucl. Phys. 39 (1984) 91. [ 15 ] M. Kremer and G. Schierholz, DESY Report No. DESY-87-024 (1984); Phys. Lett. B 194 (1987) 283. [ 16] M. Kremer, N.A. Papadopoulos and K. Schilcher, Phys. Lett. B 143 (1984) 476. [ 17] T. Nakada, SIN Report No. PR-86-11 (1986). [ 18] G. Eilam, B. Margolis and R.R. Mendel, Phys. Rev. D 35 (1987) 1723.
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L E P T O N I C DECAY C O N S T A N T S O F C H A R M A N D B E A U T Y M E S O N S I N Q C D C.A. D O M I N G U E Z 1.2 Deutsches Elektronen-Synchrotron, DESE D-2000 Hamburg 52, Fed. Rep. Germany
and N. PAVER 3 Dipartimento di Fisica Teorica, Universitd di Trieste, 1-34100 Trieste, Italy Scuola Internazionale Superiore di Studi Avanzati, 1-34100 Trieste, Italy and lstituto Nazionale di Fisica Nucleate, Sezione di Trieste, 1-34100 Trieste, Italy
Received 24 June 1987
Leptonic decay constants of heavy pseudoscalar mesons are estimated in QCD by means of Hilbert transform power-moment sum rules at Q2=0. The meson masses are also obtained in order to assess the reliability of these predictions. Our results are fD/f~ = 1.7 +_0.2, fv/f~ = 2.1 --+0. l, and fa/f~ = I. 1- 1.6. As a byproduct a bound on fas/fB is obtained.
An accurate knowledge o f the leptonic decay constants o f heavy pseudoscalar mesons is quite i m p o r t a n t as they play the key role o f absolute n o r m a l i z a t i o n s to a wide variety o f heavy flavour weak transitions. In the absence o f e x p e r i m e n t a l data, considerable effort has been d e v o t e d to the theoretical estimate o f FD, fv, fB, etc., in particular in the f r a m e w o r k o f Q C D sum rules [ 1-4] (for a review o f other estimates see e.g. refs. [ 5,6]). As is well known, this m e t h o d relates through dispersion relations low energy parameters, such as particle masses and coupling constants, to the short-distance o p e r a t o r p r o d u c t expansion ( O P E ) o f current correlators. This OPE is assumed to be valid in the presence o f n o n - p e r t u r b a t i v e effects which are p a r a m e t r i z e d by a set o f vacu u m expectation values o f quark and gluon fields. These v a c u u m condensates induce power corrections to a s y m p t o t i c freedom [ 7 ]. Although this m e t h o d seems well defined, there is some freedom in the choice o f the o p t i m a l weight in the dispersion relation a n d in the m o d e l used for parametrizing the spectral function. Also, the m e t h o d is sensitive to the n u m b e r o f loops considered in the calculation o f the a s y m p t o t i c f r e e d o m contribution, as well as to the n u m b e r a n d actual values o f the v a c u u m condensates included in the OPE. All this has resulted in a wide spectrum o f predictions for fD, fa, etc., e.g. fB = 6 0 - 2 0 0 MeV. Since on the one hand, better knowledge o f these constants would allow for m o r e accurate predictions in a n u m b e r o f semi-leptonic and non-leptonic transitions, a n d on the other h a n d there has been recent progress in the d e t e r m i n a t i o n o f v a c u u m condensates, we feel that a systematic reanalysis o f this p r o b l e m is needed. We a t t e m p t to do this here by choosing as a starting p o i n t the Hilbert transform power m o m e n t sum rules at Q 2 = 0. Aside from keeping the n u m b e r o f free p a r a m e t e r s to a m i n i m u m , these sum rules p r o v i d e a natural framework to study systems with a heavy quark, in the sense that at Q 2 = 0 n o n - p e r t u r b a t i v e effects are p a r a m e t r i z e d through the O P E as a series in inverse powers o f the heavy quark mass m o, i.e. the only large mass scale. To this extent, 1/mQ plays the unambiguous role o f the short-distance expansion parameter. The situation is not so transparent with Laplace Alexander von Humboldt Research Fellow. -" On leave of absence from Facultad de Fisica, Pontificia Universidad Catolica de Chile, Santiago, Chile. 3 Supported by MPI (Ministero della Pubblica Istruzione, Italy). 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
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transform sum rules as they involve an additional short-distance parameter, i.e. the Laplace variable a =-I/M z. Nevertheless, we shall also comment on this method at the end. In our analysis we incorporate the leading power corrections of dimension d~< 6 and consider perturbative effects up to order O (as). Regarding the spectral function parametrization, in addition to the lowest pseudoscalar meson pole we include continuum contributions and compute the corrections they induce at the one and two-loop level. Finally, to assess the reliability of our results for the leptonic decay constants we also estimate the masses of the heavy pseudoscalar mesons. We begin by defining the two-point function 9'5 (q) = i
J d4x exp (iqx) (0[T(0UA~,(x) 0"A,(0)) 10),
(1)
where
O~'A~,(x)= (mq + mo):Ct(x) i)'sQ(x):,
(2)
with q(x) ( Q ( x ) ) being a light (heavy) quark field and mq (mQ) its corresponding current mass. By selecting q = u , d, s and Q = c , b the axial-vector current divergences (2) will have the quantum numbers of D, F and B mesons. The function ~5 (q) satisfies the dispersion relation (Q2-= _ q2 > 0) oo
g s ( Q 2 ) = n1J f ds I_mys(s) (s+Q 2) +subtractions,
(3)
0
defined up to two subtractions, arising from external renormalization, which can be disposed of by taking at least two derivatives in (3). This procedure leads to the Hilbert transform or power moment sum rules, which at Q2=O become (__)n+l
("
d
)- - n + l g/,(Q2) Q2 o _ 1 ~f s___~5_ ds im q/5(s)"
(4)
Up to the actual values of the vacuum condensates the moments ~0(") can be computed in QCD and thus eq. (4) relates the resonance parameters in the spectral function (1/n)Im ~us(s) to the fundamental perturbative and non-perturbative QCD parameters in ~o("). Althoughthe quark and gluon condensates cannot yet be computed from first principles, they can be extracted from independent experimental data using QCD sum rules in other channels, e.g. the charmonium system, e+e - total cross sections, etc. The perturbative QCD expression for the spectral function at the two-loop level, as first obtained correctly in refs. [3,8], is given by 1 im gs(q2 )
rc
AF =
3(mq+mQ)20"~V[l+4°q ( 8rc2q2 ~ ] (7-- v2)
+ i=, ~ {(v+v-')[Li2(a,a2,-Li2(-aj)-lnailnfl~l+A,
lna,+Bilnfli})],
(5,
where Liz(X) = - i dt t -I ln(1 - t ) ,
(6)
0
3(3mq+mQ~ A, = ~ k mq +mQ /
424
19+2v2 +3v 4 32v
mq(mq--mQ)(l+v+2v/a,), c]2v(1 +v)
(7)
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PHYSICSLETTERSB
Bt=2+2(m2q-mS) qZv
'
mq ( l - v ) °tl=--mo i ~ p
' ]~1=
29 October 1987
x/1 +ott (1 +v) 2 4v '
(8)
and qZ=q2 (mq_mQ)2, v----(1 --4mqmQ/(12)1/2.The expressions for A2, B2, az andfl2 are obtained from (7), (8) by the interchange mq*-+mQ. In particular, in the limit mq--*0, well justified for up- and down-quarks as in the case of D and B~.d mesons, as exact integration of (5) leads to [ 3 ]
~ok'~b= ~
3 (~--~) "-1
B(n, 3)(1 +a°a~),
(9)
where B(x, y) is the beta function and a ° are the rational numbers 3rt o ~zz -~-a,- 6 -1
2 n+l
6 "~z[1 (~ n + ~ 4- ~=t 7 +
1 n
1 (n+l)
1 (~3))lJ -( n +- 2 ) +
.
(10)
Parametrizing non-perturbative corrections to asymptotic freedom through the OPE and keeping the leading vacuum condensates of dimension d~< 6, one has, always neglecting mq, m S 2 C4 04 > -~- m~ q2 3 C5(05> IP'5(qZ) IN'P-- mS--q 4 (mS--q2) +
(m~ _q2)2
(m S _q2)3
(m~ _ q2)4 ]
where the C. ( 0 . ) are defined as C 4 ( 0 4 > = ( (as/12=)G 2 -mQdlq ),
(12)
C5 (05) = (gsCliaG'i~q),
(13)
C 6 ( 0 6 ) =~Ogs ((cWuJ, q ) Z ~lTuaq/ • q
(14)
Computing the non-perturbative part of~0(") through (11 ) and adding it to (9) one finds using (4) that for the first few moments, with increasing n the perturbative piece of ~o(n)decreases in magnitude but at the expense of an increase in the non-perturbative contributions. As a compromise we then choose to consider only the first two moments; this will allow us to estimate fp as well as Mp. From eq. (11 ) one obtains ~(I) N.P.--
C4(04~ + 3C5(05) m~
4
1 (C4(04) N.o. m ~ \
m~
5 C6(O6> 3 m~ '
m~
3C5(05) +2
m~
11 C 6 ( O 6 ) ) 3 m~ "
(15)
(16)
Adding (15), (16) to (9) completes the theoretical calculation of the LHS of (4) in QCD. Turning to the hadronic spectral function appearing on the RHS of (4) we choose the following parametrization: lzt Im ~5(s) HAD=2fZM4~(s--M~') +O(S--So) 1 Im g/s(s) A.V.'
(17)
where So is the threshold for asymptotic freedom, the second term in (17) is given in (5), and ( 0 IA~ IP(p) > =ix/2fpp,,,
(18) 425
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defines the leptonic decay constant fe; with this normalization e.g. f~ = 93.2 MeV. Given the rather large masses of the D, F and B mesons we would expect the spectral function to be relatively smooth for s>M g, and thus the second term in (17), i.e. the perturbative QCD continuum, should be a reasonable parametrization of Im ~/5(S)HAD at such energies. In other words, we assume that potential radial excitations of the heavy pseudoscalar mesons do not show up as prominent narrow peaks in the spectral function. In this case the first two moments read
2f ~,
1
M2 - 8n 2 [ ( 1 - a , ) + a s ( 0 . 7 5 1 - b ~ ) ] +
C4(04)
3 C5(05)
m------~-Q + 4
2f~ M~( 1 M 2 - m~ 3-~2nz [ ( l - a 2 ) + o t s ( 1 . 7 0 6 - b 2 ) ] +
m~
C4(04)m~ +
5 C6(06)
3
23C5 (O5)rn~
m~
'
) 113 C6 (O6)m~ '
(19) (20)
where al, a2, bl, b2 are the continuum corrections which can be computed by integrating the second term in (17). Concerning the values of the QCD parameters entering the sum rules (19), (20) we use for the "on-shell" quark masses: mc(Q 2 = m 2) = 1.3 GeV, mb(Q z = m b2) =4.6 GeV, ots(m 2) =0.296, and as(rn 2) =0.214. For the vacuum condensates we take m c ( d l q ) = - 0 . 0 1 4 GeV 4 and m b ( ~ q ) = - - 0 . 0 5 5 GeV 4, which follow from the most recent determination of light quark masses and condensates [ 9 ]; ( otsG 2) / 12 n = 0.003 GeV 4 as extracted from e+e - data [ 10] and charmonium [ 11 ]. The quark-gluon condensate (gsCzlitrG'2q) = 2 M 2 (clq) seems to play a particularly important role in the QCD analysis of the baryon spectrum. The results obtained for the mass parameter M~ are, however, somewhat controversial. We take M 2 = (0.5 _+0.1 ) GeV 2 as a compromise between e.g. M~ -~ (0.1-0.4) GeV 2 [ 12] or M 2 -~ (0.6-1.0) GeV 2 [ 13], and M 2 ---0.3 GeV 2 [ 14], the latter value being suggested by charmonium sum rules. We have found that the higher value M 2 -- 1 GeV 2 [ 15] is in disagreement with the observed D- and F-meson masses. Such a sensitivity to the quark-gluon condensate is actually an interesting aspect of eqs. (19), (20). Finally, for C6(O6) we use C6(O6) -- - 9nots(dlq) 2, which takes into account a factor of five increase in the factorization estimate according to recent results from an e + e - analysis [ 10 ] as well as previous claims from other sources [ 12,15 ]. Starting with the D-meson and solving (19), (20) for values of the asymptotic freedom threshold within the wide range So = 2M 2 - 3 M ~ we obtain MD = 1.85 --+0.15 GeV,
(21)
to be compared with MDlexp = 1.87 GeV, and fD = 160-+ 15 MeV.
(22)
Continuum corrections in this channel are important, i.e. without them MD--~2-2.2 GeV, but yet not so large as to spoil the predictive power of the sum rules [ 16 ]. We wish to stress that So is a free parameter in the sumrule approach; the choice we made is an educated guess based on experience in other channels. The two important points are: (i) predictions should be reasonably stable under changes in So within a (hopefully) wide region, and (ii) obtaining a value for FD from the sum rules is not enough. Even with everything else fixedfD is clearly a function of So. Hence, some other quantity whose experimental value is well known, e.g. MD in our case, should be estimated simultaneously in order to assess the reliability of the prediction for fD- It is rewarding that our results above meet these two requirements. Considering next the F-meson, if we continue to neglect the light quark mass, ms in this case, we would obviously find MF=Mo, and fF=fD. Given the uncertainties in the values of the vacuum condensates and in So, which reflect themselves on the errors in (21 ), (22), we will be unable to predict the small SU (3) breaking mass splitting Mv-MD. Notice that the mass is obtained from the ratio of two sum rules. However, the situation is better for fF. For example, by retaining terms of order O (c) and O (e In e), with ~-= mJme, eq. (19) becomes 426
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2f2/M2 = (1/8n2){ 1 + 3e+ as[0.751 + (6In)e(1 + n 2 / 9 + l n e)] } + continuum+ non-perturbative.
29 October 1987 (23)
A comparison with the corresponding relation for f o leads to the approximate expression fv/fD ~-- ( MF/MD)(1 + 3 0 ~- 1.2.
(24)
We have solved the two sum rules in this channel by numerical integration of the imaginary part eq. (5) retaining ms but without approximations, i.e. without expanding in E as in eq. (23). Concerning the non-perturbative terms, at the present level of accuracy we have safely ignored the small SU(3) breaking in the quark condensates and used the same values of Cn(On) as before. In the broad duality region So ~ - 2 M ~ - 3 M 2 we obtain M y = 1.9+0.1 GeV,
(25)
as expected, and fv = 194_+ 12 MeV,
(26)
in nice agreement with the ~-expansion result eq. (24). Turning to the B-meson an inspection of the various contributions to ( 19 ), (20) shows that except for Ca (O4) the non-perturbative terms are unimportant. Continuum corrections in this channel are very large, a fact known from previous analyses [2], e.g. without them we would predict MB= 8 GeV as opposed to MBlexp= 5.27 GeV. Including these corrections and allowing for the asymptotic freedom threshold to vary in the range So 1 . 1 M ~ - 2 M ~ brings MB down to Ms--- (5.1-6.2) GeV. Given the large value of the B-meson mass it would be reasonable to expect a precocious onset of asymptotic freedom in this channel, e.g. So - (1.1-1.3)M 2,
(27)
which narrows down the resulting mass to MB =5.2+_0.2 GeV.
(28)
It should be emphasized, though, that (28) is not a genuine prediction. If we have not had beforehand information on MB and had allowed So to vary over a wider range the uncertainty would have been considerably larger. The above procedure is then to be understood more as a determination of So than of MB. However, since our purpose is to predict fB, assessing the reliability of this prediction through the resulting value of MB, we can use So in the range (27) to obtain fB = 104-150 MeV.
(29)
Using the experimental value of MB as input, the eigenvalue solution for So is: So= 1.2 ME, and f8 = 127 MeV. However, the result (29) provides a better feeling for the true uncertainties involved in this channel. Concerning the B,-meson, its decay constant is given by an expression analogous to eq. (23) except that now =- ms~rob "~ 0.04. Given the smallness of ~ and of the non-perturbative power corrections we expect the ratio fB,/f8 to be more accurately predicted than either fBs or fB separately. Our result for this ratio is fB,/fB > MB,/MB,
(30)
which should be useful in connection with oscillations in the B°-B°-meson system (for reviews see e.g. refs. [5,17]). The most severe source of uncertainty in the estimate offB, eq. (29), is its power dependence on So. This is a characteristic feature of the Hilbert transform sum rules (4), which has a bigger impact on fB than on fD orfv, eqs. ( 2 2 ) a n d (26). To minimize this sensitivity to some extent, we have selected a duality region for so by requiring that the ratio between eqs. (19) and (20) reproduces the experimental value of MB within some error, e.g. 5%. In principle, Laplace transform QCD sum rules are expected to be far less sensitive to So on account of their exponential weight. In the present application these sum rules are [4] 427
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so
2f2p-~ exp(-Mz/M2)= ,!odsexp(-s/M2) llm~us(S)QcD +exp(-m~/M z) C4(04)---~-~
1-2M2]C5(05>+- ~
2 - 2M 2
6M 4J
C6(06) ,
(31)
where
i dsexp(-s/M ) l~rl m
g/5(s) Q C D - -
m~
- M2(M 2+So) exp( -
3
8n 2
{m~[El(m~/M2)_El(So/M2)] +M2(M2+m~) exp(_m~/M2 )
S o [ M 2 ) - 2m~M2 [ exp(
-m~/M 2) - e x p (
- S o ~ M 2 ) ] + O(Ots)} ,
(32)
with P - - D , F, B, etc., and co
E l ( Z ) = f e x p , - t ) tit.
(33)
Earlier estimates based on these sum rules [4] have claimed somewhat smaller values offD and fB than the ones obtained here through Hilbert transforms. However, the point we wish to raise here is that although eq. (31 ) does exhibit a better (softer) behaviour on So, it has conceptual as well as numerical disadvantages. First, the power corrections in eq. (31 ) appear dominated by C4 ( 04 > on account of the smallness of C5 ( 05 > and C6 (O6 > rather than because of the usual Laplace suppression; notice that the terms multiplying C5 (O5 > and C6 ( O6 > are roughly comparable. This means that inside the "sum rule window" ( M 2-- 0.8-2.1 GeV 2 for P = D and M 2~-3.3-4.2 GeV 2 for P - - B ) the Laplace variable M E has lost the unambiguous character of the shortdistance expansion parameter it usually has in the familiar applications to light quark systems. A second cause of discomfort is the pronounced sensitivity of the perturbative contribution eq. (32) to the values of mQ and M E. In fact, the dependence on rnQ is in this case exponential. For instance, in the case o f f a a 4% increase in mb from rob----4.6 GeV to rob----4.8 GeV produces a change in the perturbative contribution of a factor of 2-3 for M2,-'3-5 GeV 2. On the other hand, for fixed So and mQ, eq. (32) changes by a factor of 8-16 for P = D and by a factor of 40 for P = B, inside the "sum rule window" in M E. The corresponding changes in the nonperturbative piece are roughly a factor of 3 and 20 (in the same direction), respectively. In the case offD these extreme variations are offset by corresponding large variations in the exponential on the LHS of eq. (31 ), so that in the end the result fOrfo appears somewhat stable. All things considered we findfo ~ 120-150 MeV inside the "window" M 2 ~ 1.5-3 GeV 2, and for So- (I.5-3)M2D, which is consistent with eq. (22). However, in view of the above remarks we would not attach much significance to this result. For fa the huge variations of the perturbative and non-perturbative contributions are not entirely offset by the variation of the exponential on the LHS of eq. (31 ), so that uncertainties are beyond the 100% level. We wish to point out that such a dangerous situation is not encountered in the usual applications of Laplace sum rules to light quark systems. Also, these problems do not affect the Hilbert transform power moments at Q2--0, which in our view, are more reliable to treat charm and beauty mesons as they lead to stable predictions. In conclusion, collecting our results and normalizing to f~ = 93.2 MeV we have found
fDIf~=l.7+_0.2, fvlf~=2.1+_O.1, fnlf~=l.l-l.6, fBs/fa>~Mas/MB.
(34)
The quality of these predictions is gauged by the results obtained for the meson masses in the same framework, i.e. eqs. (21), (25), and (28). As usual, however, the ultimate test will have to come from experiment. In this
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References [ 1 ] V.A. Novikov, M.A. Shifman, M.B. Voloshin, A.I. Vainshtein and V.I. Zakharov, in: Proc. Neutrino-78 Conf. (West Lafayette, IN, 1978); E.G. Floratos, S. Narison and E. de Rafael, Nucl. Phys. B 155 (1979) 115; V.S Mathur and T. Yamawaki, Phys. Lett. B 107 (1981 ) 127; E.V. Shuryak, Nuel. Phys. B 198 (1982) 83; S. Narison, Z. Phys. C 14 (1982) 263. [2] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. B 104 (1981) 305. [3] D.J. Broadhurst and S.C. Generalis, Open University Report No. OUT-4102-8/R (1982). [4] S.C. Generalis, Ph.D. Thesis, Open University Report No. OUT-4102-13 (1984); A.R. Zhitnitskii, I.R. Zhitnitskii and V.L. Chernyak, Sov. J. Nucl. Phys. 38 (1983) 775; T.M. Aliev and V.L. Eletsky, Soy. J. Nucl. Phys. 38 (1983) 936. [ 5 ] A. All, DESY Report No. DESY-86-041, and Proc. LEP-Physics-Jamboree, CERN (Yellow) Report No. CERN-86-02, Vol. 2 (CERN, Geneva, 1986). [6] J. K6rner, in: Proc. Intern. Symp. on Production and decay of heavy hadrons (Heidelberg, 1986). [7] M.A. Shifman, A.1. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385,448; L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Rev. 127 (1985) 1. [8] D.J. Broadhurst, Phys. Lett. B 101 (1981) 423. [9] C.A. Dominguez, M. Kremer, N.A. Papadopoulos and K. Schilcher, Z. Phys. C 27 (1985) 481; C.A. Dominguez and E. de Rafael, Ann. Phys. (NY) 174 (1987) 372. [ 10] R.A. Bertlmann, C.A. Dominguez, M. Loewe, M. Perrottet and E. de Rafael, DESY Report, to be published; see also R.A. Bertlmann, in: Proc. XVII Intern. Symp. on Multiparticle dynamics, eds. M. Markytan, W. Majerotto and J. Mac Naughton ( World Scientific, Singapore, 1987); C.A. Dominguez, DESY Report No. DESY-87-002 (1987), in: Proc. Intern. Workshop on Quarks, gluons and hadronic matter (Cape-Town, 1987), to be published. [ 11 ] J. Marrow and G. Shaw, Z. Phys. C 33 (1986) 237. [ 12] Y. Chung, H.G. Dosch, M. Kremer and D. Schall, Z. Phys. C 25 (1984) 151. [ 13] V.M. Belyaev and B.L. Ioffe, Sov. Phys. JETP 56 (1982) 493. [ 14] S.N. Nikolaev and A.V. Radyushkin, Soy. J. Nucl. Phys. 39 (1984) 91. [ 15 ] M. Kremer and G. Schierholz, DESY Report No. DESY-87-024 (1984); Phys. Lett. B 194 (1987) 283. [ 16] M. Kremer, N.A. Papadopoulos and K. Schilcher, Phys. Lett. B 143 (1984) 476. [ 17] T. Nakada, SIN Report No. PR-86-11 (1986). [ 18] G. Eilam, B. Margolis and R.R. Mendel, Phys. Rev. D 35 (1987) 1723.
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