8 November 2001

Physics Letters B 520 (2001) 115123www.elsevier.com/locate/npe

c, b quark masses and fD(s), fB(s) decay constants frompseudoscalar sum rules in full QCD to order 2s

Stephan Narison a,b

a Laboratoire de Physique Mathmatique et Thorique, UM2, Place Eugne Bataillon, 34095 Montpellier Cedex 05, Franceb Center for Academic Excellence on Cosmology and Particle Astrophysics (CosPA), Department of Physics, National Taiwan University,

Taipei, 10617 Taiwan, ROC

Received 30 August 2001; accepted 10 September 2001Editor: G.F. Giudice

Abstract

The pseudoscalar sum rules of the heavylight quark systems are used for extracting simultaneously the c, b quark massesand the decay constants fD(s) , fB(s) of the D(s),B(s) mesons. To order

2s , one obtains the running quark masses: mc(mc) =

(1.10 0.04) GeV, mb(mb) = (4.05 0.06) GeV implying mb(MZ) = (2.65 0.05) GeV; the perturbative pole masses:Mc = (1.46 0.04) GeV, Mb = (4.69 0.06) GeV; and the decay constants: fD = (205 20) MeV, fB = (203 23) MeVand fDs = (235 24) MeV, fBs = (236 30) MeV, in the normalization where f = 130.56 MeV. The fitted values of thepole and running masses satisfy quite well their three-loop perturbative relation. The value fD fB confirms earlier findingsfrom the sum rule that the 1/

MP heavy quark symmetry scaling law is affected by large 1/MP corrections.

1. Introduction

One of the most important parameters of the stan-dard model is the quark masses. However, contrary tothe leptons, where the physical mass can be identifiedwith the pole of the propagator, the quark masses aredifficult to define because of confinement. Some at-tempts have been made in order to define the heavyquark pole mass within perturbation theory, where ithas been shown to be IR-finite [1] and independentof the choice of the regularization and renormalizationschemes used [2]. More recently, it has been noticed,in the limit of a large number of flavours, that the re-summation of perturbative series can induce a nonper-

E-mail address: narison@lpm.univ-montp2.fr (S. Narison).

turbative term, which can affect the truncated pertur-bative result, and can, then, limit the accuracy of thepole mass determination (for reviews see, e.g., [3,4]).One may bypass the previous problems, by working,at a given order of perturbative QCD, with the run-ning quark masses, which are treated like couplingconstants of the QCD Lagrangian (see, e.g., [5]), andwhere nonperturbative-like effect is expected to be ab-sent. It is also known that the decay constants fD,Bof the pseudoscalar D,B mesons play an essentialrle in the neutral DD and BB oscillations, andin D,B decays, and can directly be measured in theD,B ll leptonic decays. In addition, it is im-portant for your knowledge of heavy quark symme-try, how the value of these decay constants deviatefrom the nave 1/

MP scaling law, expected to oc-

cur when the pseudoscalar meson mass is infinitely

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116 S. Narison / Physics Letters B 520 (2001) 115123

large. A lot of efforts has been furnished in the lit-erature [6] for extracting directly from the data therunning masses of the light and heavy quarks and theheavy quark perturbative pole masses using the SVZQCD spectral sum rules (QSSR) [7] (for a completereview, see, e.g., [8]), whilst fD,B come from differ-ent forms of the pseudoscalar sum rules [815] sincethe pioneering work of [16]. In this Letter, I shall con-sider a direct extraction of the running charm and bot-tom quark masses from pseudoscalar two-point func-tion sum rules where the 2s correction has been re-cently obtained in [17] and where we shall also usethe observed values of the meson masses: MD =1.869 GeV and MB = 5.279 GeV. In the same time,we shall simultaneously estimate the decay constantsfD(s) , fB(s) of the D(s) and B(s) mesons. The simul-taneous extraction of the quark mass and fP togetherwith the extraction of fP using the running quark masshas been initiated in our previous work [13,14] to or-der s , and will be improved here to order 2s . Allprevious works [813] with the exception of the re-cent work in [15] have used, as input, the pole massvalue for extracting fP , where, as we have mentionedpreviously, the definition of the pole mass might beaffected by some nonperturbative contributions [3,4].Moreover, the extraction of the quark mass value fromthe pseudoscalar sum rule itself which is an improve-ment of our previous works [13,14] is not done in allprevious works.

2. The QCD spectral sum rules

We shall work with the pseudoscalar two-pointcorrelator:

(1)5(q2

) i d4x eiqx0|T Jq(x)J q (0)|0,built from the heavylight quark current: Jd(x) =(mQ + md)Q(i5)d, and which has the quantumnumbers of the D and B mesons. mQ is the heavyquark mass, and we shall neglect the d quark masshere. The corresponding Laplace transform sum rulesare:

L( )=t

dt et 1

Im5(t) and

(2)R( ) dd

logL( ),

where t is the hadronic threshold. The latter sumrule, or its slight modification, is useful, as it is equalto the resonance mass squared, in the simple dualityansatz parametrization of the spectral function:

1

Im5(t) 2f 2DM4D

(t M2D

)(3)+ QCD continuum(t tc),

where the QCD continuum comes from the disconti-nuity of the QCD diagrams, which is expected to give agood smearing of the different radial excitations. 1 Thedecay constant fD is analogous to f = 92.32 MeV.However, in order to avoid some confusion, and fora more direct comparison with the lattice results, weshall abandon our favorite normalization, and adopt inthe rest of the Letter, the one:

(4)fD

2fD,

a normalization where f = 130.56 MeV, and whichthe different experimental groups have also adopted;tc is the QCD continuum threshold, which is, like thesum rule variable , an (a priori) arbitrary parame-ter. In this Letter, we shall impose the tc and sta-bility criteria for extracting our optimal results. 2 TheQCD expression of the correlator is well-known totwo-loop accuracy (see, e.g., [8] and the explicit ex-pressions given in [9]), in terms of the perturbativepole massMQ, and including the nonperturbative con-densates of dimensions less than or equal to six. 3 Thesum rule reads:

L( )=M2Q{ M2Q

dt et 182

1 At the optimization scale, its effect is negligible, such that amore involved parametrization is not necessary.

2 The corresponding tc value very roughly indicates the positionof the next radial excitations.

3 A different expression of the coefficient of the quarkgluonmixed condensate is given in [18]. This change affects only slightlythe result. We shall include the negligible contribution from thedimension six four-quark condensates. Notice that there is somediscrepancy on the value of the four-quark coefficient in theliterature.

S. Narison / Physics Letters B 520 (2001) 115123 117

[

3t (1 x)2(

1 + 43

(s

)f (x)

)

+(s

)2R2s

]

(5)+ [pC4O4 +C6O6 ]eM2Q},

where R2s is the new 2s -term obtained semi-analytically in [17] and is available as a MATHEMAT-ICA package program Rvs.m. The other terms are:

x M2Q/t,

f (x)= 94

+ 2 Li2(x)+ logx log(1 x)

32

log(1/x 1) log(1 x)+ x log(1/x 1) (x/(1 x)) logx,

C4O4 = MQdd +sG

2/12,

(6)

C6O6 =M3Q

2

(1 1

2M2Q

)g

d

a

2Ga d

(

8

27

)(2 M

2Q

2 M

4Q

2

6

) s2.

The previous sum rules can be expressed in terms ofthe running mass mQ(), 4 through the perturbativethree-loop relation [1,2,19]:

Mpole

(7)

= m(p2)[1 + (43

+ ln p2

M2

)(s

)

+[KQ +

(221

24 13

36n

)lnp2

M2

+(

15

8 n

12

)ln2

p2

M2

](s

)2],

4 It is clear that, for the nonperturbative terms which are knownto leading order of perturbation theory, one can use either therunning or the pole mass. However, we shall see that this distinctiondoes not affect notably the present result.

where, in the RHS, Mpole M is the pole mass and:

(8)

KQ = 17.1514 1.04137n+ 43

i =Q

(r mi

MQ

).

For 0 r 1, (r) can be approximated, within anaccuracy of 1% by:

(9)(r) 2

8r 0.597r2 + 0.230r3.

Throughout this Letter we shall use the values of theparameters [8,20] given in Table 1.

We have used for the mixed condensate the parame-trization:

(10)g

d

a

2Ga d

=M20 dd,

and we deduce the value of the QCD scale from thevalue of s(MZ)= (0.1184 0.031) given in [6,21].

3. The D-meson channel

We study in Fig. 1, the prediction of fD from theLaplace sum rules L and the one of MD from ratioof moments R for given value of the running charmquark mass mc(mc). The influences of the choice ofthe continuum threshold tc and of the sum rule scale are shown in details. Our optimal results correspondto the case where both stability in and in tc arereached. However, for a more conservative estimateof the errors we allow deviations from the stabilitypoints, and we take:

tc (6 9.5) GeV2,(11) (1.2 0.2) GeV2,

and where the lowest value of tc corresponds to thebeginning of the -stability region. Values outside theabove ranges are not consistent with the stability crite-ria. One can inspect that the dominant nonperturbativecontribution is due to the dimension-fourMcdd lightquark condensate, and test that the OPE is not bro-ken by high dimension condensates at the optimiza-tion scale. However, the perturbative radiative correc-tions converge slowly, as the value of fD increases by12% after the inclusion of the s correction and thesum of the lowest order plus s -correction increasesby 21% after the inclusion of the 2s term, indicating

118 S. Narison / Physics Letters B 520 (2001) 115123

Fig. 1. Laplace sum rule analysis of the decay constant fD , the running mass mc(mc) and the pole massMc : (a) fD versus the sum rule scale at given mc(mc) and for different values of the continuum threshold tc ; (b) the same as (a) but forMD ; (c) and (d) effects of mc(mc) onMDand fD . The circle is the solution given by the data on MD ; (e) to (h) the same as (a) to (f) but for the pole mass.

S. Narison / Physics Letters B 520 (2001) 115123 119

Table 1Different sources of errors in the estimates of the decay constants (in MeV) and quark masses (in GeV). We have exagerately enlarged the errorbars of different input in order to have conservative errors

Sources |1fD |mc |1fD |Mc |1mc | |1Mc| |1fB |mb |1fB |Mb |1mb | |1Mb |4 = (325 43)MeV 7.4 6.2 0.03 0.03 5 = (225 30)MeV 3.6 3.0 0.02 0.02 =Mc,b 1/2(Mc,b mc,b) 9.3 14.0 Geom. estimate of 3s -term 7.9 8.2 1.7 2.0

D = (1.2 0.2) GeV2 1.1 1.1 0.01 0.01 B = (0.35 0.05) GeV2 8.2 5.7 0.02 0.026.0 tDc

[GeV2

] 9.5 2.1 2.8 0.01 0.01

36 tBc[GeV2

] 50 2.4 2.8 0.05 0.03

dd1/3(1 GeV)= (229 18)MeV 8.8 8.9 0.01 0.01 5.2 7.1 sG

2 = (0.07 0.03) GeV2 1.8 1.1 0.005 0.005 0.9 1.0 M20 = (0.8 0.1) GeV2 0.7 0.7 0.005 0.01 2.2 2.1 0.03 0.04s2 = (5.8 2.4) 104 GeV6 0.2 0.3 0.6 1.0 0.01 0.01From our estimate of mc,b orMc,b 10.9 9.6 15 13

Total 20 17 0.04 0.04 23 17 0.06 0.06

that the total amount of corrections of 21% is still areasonnable correction despite the slow convergenceof the perturbative series, which might be improvedusing, like in [10], a resummed series. However, asthe radiative corrections are both positive, we expectthat this slow convergence will not affect in a sensibleway the final estimate. In order to improve the pertur-bative contributions, we have estimated the possiblecontribution of the 3s -term by assuming that its coef-ficient is the geometric sum of the s and 2s contribu-tions. This effect is shown in Table 1, which is has aquite reasonnable value. A more precise answer on thehigher order perturbative contribution needs an evalu-ation of the 3s term which we hope to be available inthe near future. In doing the analysis, one can also no-tice that the relative size of the perturbative correctionsis smaller in the physical observable fD (perturba-tive + nonperturbative) than in the perturbative graphalone. This is due to fact that the rle of the con-densate is important at the optimization scale, whichthen decreases the relative weight of the perturbativeradiative corrections in the OPE. The estimate of MDfrom the ratio of moments R is less affected by radia-tive corrections, which tend to cancel each others due

to the form of the sum rule. The behaviour of the op-timized values of fD and MD versus different valuesof mc(mc) is given in Figs. 1(a) and (b). One can ex-plicitly see in Figs. 1(c) and (d) that both MD and fDare very sensitive to the change of mc(mc). This fea-ture allows to have a good determination of the quarkmass and then of fD , once the experimental value ofMD is used. Adding quadratically the different errorsgiven in Table 1, we deduce the final estimate:

mc(mc)= (1.10 0.04) GeV,(12)fD = (201 20)MeV,

where as mentioned previously, we have used the nor-malization f = 130.56 MeV. These optimal valuescorrespond to tc = 6.5 GeV2 and = 1.2 GeV2. Thevalue of tc roughly corresponds to a radial excitationwith a mass-splitting relative to the ground state massof aboutM which is phenomenologically acceptable.A similar analysis shown in Figs. 1(e)(h) is done forthe pole mass. The discussions presented previouslyapply also here, including the one of the radiative cor-rections. We quote the final result:

Mc = (1.47 0.04) GeV,

120 S. Narison / Physics Letters B 520 (2001) 115123

(13)fD = (208 17)MeV,where the error is slightly smaller here due to theabsence of the subtraction scale uncertainties. For fD ,we consider as a final estimate the mean value of thetwo predictions and taking the largest errors:

(14)fD = (204 20)MeV.Using our previous estimate of fDs /fD = 1.15 0.04[13], we can also deduce:

(15)fDs = (235 24)MeV.One can immediately compare the present predictionswith the one obtained to order s using the sameprocedure [14]:

mc(mc)= (1.08 0.11) GeV,(16)fD = (201 15)MeV,

where one can notice a good agreement between thes and 2s results, though the error in [14] is smalleras the effect of the subtraction point has not beentaken into account. However, the agreement seemsparadoxal in view of the fact that radiative correctionstend to increase the value of fD compared to thelower orders result. The different truncations of theexpression of s and of the relation between the poleand running mass also affect the absolute value offD , which tend to compensate the increase due tothe radiative corrections of the correlator. Therefore,a nave comparison becomes misleading. As one cansee in Table 1, the main source of errors is due to thevariations of the quark mass and to a lesser extent tothe ones of , and . The effect of tc on theresult is relatively small from the value tc larger than6 GeV2, where one starts to have a stability. Thisresult for the mass is also in agreement with the onefromMJ/ [6,22]:

(17)mc(mc)=(1.23+0.040.05

)GeV,

but lower than the one from [23] using nonrelativisticBalmer formula for the cc bound state. One can cross-check that the two values of mc(mc) and Mc give theratio:

(18)Mc/mc(mc) 1.33,which satisfies quite well the three-loop perturbativerelation Mc/mc(mc) = 1.33 obtained from the pre-vious Eq. (7). This could be a nontrivial result if

one has in mind that the quark pole mass definitioncan be affected by nonperturbative corrections notpresent in the standard SVZ-OPE. In particular, it maysignal that 1/q2 correction of the type discussed in[3,24], if present, will only affect weakly the stan-dard SVZ-phenomenology as observed explicitly inthe light quark, gluonia and hybrid channels [24]. Re-cent results to order 2s [12], using the analogous sumrule and using as input the pole mass value, givesfD = (195 20) MeV. The result is slightly lowerthan the result given here, though in agreement withinthe errors. We understand this slight difference as dueto the lower value of the QCD continuum thresholdused there, which corresponds to tc 5.6 GeV2 if oneuses our pole mass value Mc = 1.47 GeV. As shownin Fig. 1(e), this tc value is on the boarder of the stability region. It also indicates that the error intro-duced by the choice of tc could have been underesti-mated in this result. Fig. 1(g) also indicates that fD isquite sensitive in full QCD to the change of the Mcvalue, while the weaker dependence obtained in [12]is also related to the values of tc used there. Quenchedand unquenched lattice results for fD are compiled in[25]. The quenched results range from (192 18) to(221 17) MeV, in fair agreement with our resultswithin the errors. The two available unquenched re-sults [26,27] lead to the average:

f latD = (220 20)MeV,(19)f latDs = (254 29)MeV,

which is again in agreement within the errors with ourprevious estimate. Finally, one can also compare thevalue of fDs with the experimental value [29]:

(20)f expDs = (286 60)MeV,which agrees within 1 with our prediction. Improve-ments of our predictions for fDs need an estimate ofthe ratio fDs /fD to order

2s for the ms and ss cor-

rections.

4. The B-meson channel

We extend the previous analysis to the case of theB-meson, which again is an update of our previouswork in [8,9,14]. The analysis is still similar to the onedone in the D-channel, and is summarized in Fig. 2

S. Narison / Physics Letters B 520 (2001) 115123 121

Fig. 2. The same as Fig. 1 but for the b-quark and B-meson.

122 S. Narison / Physics Letters B 520 (2001) 115123

and in Table 1. Using the running b-quark mass, asa free parameter, we obtain at the optimization scale = 0.375 GeV2 and tc = 38 GeV2:mb(mb)= (4.05 0.06) GeV,

(21)fB = (205 23)MeV,while using the pole mass as a free parameter, we get:

Mb = (4.69 0.06) GeV,(22)fB = (200 17)MeV,

from which we deduce the average:

(23)fB = (203 23)MeV,where we have taken the largest errors. One can againcross-check that the two values of mb(mb) and Mblead to

(24)Mb/mb(mb)= 1.16,to be compared with 1.15 from the three-loop pertur-bative relation in Eq. (7), and might indirectly indicatethe smallness of the 1/q2 correction if any. Our re-sult of mb can be compared with our previous estimatefrom -systems [22]:

(25)mb(mb)= (4.23 0.05) GeV,and with similar values from recent estimates [6,30]. Our value of the perturbative pole mass is inagreement within the errors with the one in [6,22,30]but is again lower than the one in [23]. Our value offB is in fair agreement with the recent results fB =(206 20) MeV obtained in [12] using HQET sumrules and the one fB = (197 23) MeV obtained in[15] using the Laplace sum rule like in this work. Theslight difference is due to the different appreciationsof the continuum threshold tc and stability regionsin each papers. More specifically, errors related tothe choice of tc at their choice of lower -valuesappear to be underestimated in these works. At sucha choice of low -values, the ground state contributionto the sum rule is smaller than in the present analysis.Unquenched lattice results [27,28] give the meanvalue:

f latB = (198 37)MeV,(26)f latBs /f

latB = 1.17 0.03,

where the largest error has been taken. These valuesagree with our previous determination in Eq. (23) andwith our earlier estimate [13]:

(27)fBs /fB = 1.16 0.05,which has been confirmed from the recent analysisof [15]. Combining this SU(3) breaking ratio with ourestimate of fB , one obtains:

(28)fBs = (236 30)MeV.The extension of the previous analysis to the D andB channels is in progress.

5. Summary

We have updated our previous estimate of thequark masses and decay constants in [8,9,14] usingthe recent expression [17] of 2s corrections for theheavylight pseudoscalar correlators. Our results forthe masses in Eqs. (12), (13), (21), (22) and for thedecay constants in Eqs. (14), (15), (23), (28) confirmprevious estimates to two-loops. The results for therunning masses are:

mc(mc)= (1.10 0.04) GeV,mb(mb)= (4.23 0.05) GeV, Eqs. (12), (21),where mb(mb) runned until the Z-mass implies toorder 2s :

5

mb(Mb)= (2.65 0.05) GeV.The pole masses are:

Mc = (1.47 0.04) GeV,Mb = (4.69 0.06) GeV, Eqs. (13), (22).The decay constants are:

fD = (204 20)MeV,fB = (203 23)MeV, Eqs. (14), (23).Using our previous SU(3) breaking prediction onfPs /fP [13], one also deduces:

fDs = (235 24)MeV,fBs = (236 30)MeV, Eqs. (15), (28).

5 Motivated by a remark of Gordon Kane.

S. Narison / Physics Letters B 520 (2001) 115123 123

The three-loop corrections tend to push the valuesof the decay constants to higher values restoring theslight discrepancy between the sum rules and re-cent unquenched lattice values. The resulting equalityfD fB confirm earlier findings from the sum rule[9] indicating large corrections to the 1/

MP heavy

quark symmetry scaling law. Values of the quarkmasses obtained from the pseudoscalar sum rules arein agreement with the one from the quarkonia sumrules [6,22,30], but lower than the ones obtained in[23] from nonrelativistic Balmer formulae. The fittedvalues of the running and perturbative pole masses sat-isfy quite well their three-loop perturbative relation,which may indicate that 1/q2-like terms [3,24] havenegligible effect in this channel like in the case of thelight quark systems.

Acknowledgements

It is a pleasure to thank W.-Y. Pauchy Hwang forthe hospitality at CosPA-NTU (Taipei), where part ofthis work has been done, Abdesslam Arhrib for hishelp in running the three-loop package Rvs.m with hisnewer MATHEMATICA version, Matthias Steinhauseron the use of Rvs.m, Arifa Ali Khan for commentingthe CP-PAC results and Valya Zakharov for somecommunications.

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