arxiv:1511.05903v3 [hep-ph] 29 nov 2015

Nuclear and Particle Physics Proceedings 00 (2015) 1–12

Nuclear andParticle Physics

Proceedings

Decay Constants of Heavy-Light Mesons from QCDI,II

Stephan Narisona

aLaboratoire Particules et Univers de Montpellier, CNRS-IN2P3, Case 070, Place Eugene Bataillon, 34095 - Montpellier, France.

Abstract

We summarize our recently improved results for the pseudoscalar [1, 2], vector and Bc [3] meson decay constants from QCD spectralsum rules where N2LO ⊕ estimate of the N3LO PT ⊕ d ≤ 6 condensates have been included in the SVZ expansion. The “optimalresults” based on stability criteria with respect to the variations of the Laplace/Moments sum rule variables, QCD continuumthreshold and subtraction constant µ are compared with recent sum rules and lattice calculations. To understand the “apparenttension” between some recent results, we present in Section 8 a novel extraction of for fB∗/ fB from heavy quark effective theory(HQET) sum rules by including the normalization factor (Mb/MB)2 relating the pseudoscalar to the universal HQET correlatorsfor finite b-quark and B-meson masses. We obtain fB∗/ fB = 1.025(16) in good agreement with 1.016(16) from spectral sum rules(SR) in full QCD [3]. We complete the paper by including new improved estimates of the scalar, axial and B∗c meson decayconstants (Sections 11–13). For further phenomenological uses, we attempt to extract a Global Average of the sum rules and latticedeterminations which are summarized in Tables 2–6. We do not found any deviation of these SM results from the present data.

Keywords: QCD spectral sum rules, Heavy quark effective theory (HQET), Heavy-light mesons, Meson decay constants.

1. IntroductionThe decay constants fP/S , fV/A of (pseudo)scalar and (ax-ial)vector mesons are of prime interests for understanding therealizations of chiral symmetry in QCD and for controllingthe meson (semi-)leptonic decay widths, hadronic couplingsand form factors 1. In addition to the well-known values offπ=130.4(2) MeV and fK=156.1(9) MeV which control thelight flavour chiral symmetries [9], it is also desirable to extractthe ones of the heavy-light charm and bottom quark systemswith high-accuracy. This improvement program has been initi-ated by the recent predictions of fD(s) , fB(s) [1, 2] and pursued forfD∗(s)

, fB∗(s)and fBc in [3] from QCD spectral sum rules (QSSR)

[10] which are improved predictions of earlier estimates [15–27] 2 since the pioneering work of Novikov et al. (NSV2Z)[28]. Here, these decay constants are normalized as:

〈0|JP/S (x)|P〉 = fP/S M2P/S ; 〈0|JµV/A(x)|V/A〉 = fV/AMVV/Aεµ,(1)

where: εµ is the vector polarization; JP/S (x) ≡ (mq + / −MQ)q(iγ5/1)Q and JµV/A(x) ≡ q(γµ/γµγ5)Q are the local heavy-light pseudoscalar, scalar and vector currents; q ≡ d, c; Q ≡

IPlenary talk given at QCD15 (18th International QCD Conference and 30thAnniversary), 29 june-3 july 2015 (Montpellier-FRA) and at HEPMAD15 (7thInternational Conference), 17-22 september 2015 (Antananarivo-MGA).

IIEn hommage aux victimes des attentats du 13 Novembre 2015 a Paris-FRA.Email address: [email protected] (Stephan Narison)

1Some applications to D and B-decays can e.g be found in [5–9].2For reviews with complete references, see e.g: [1, 2, 4, 11–14, 20].

c, b; P ≡ D, B, Bc; S ≡ D∗0, B∗0, V ≡ D∗, B∗, A ≡ D1, B1 and

their s-quark analogue; the decay constants fP/S , fV/A are re-lated to the leptonic widths Γ[P/S → l+νl] and Γ[V/A → l+l−].The associated two-point correlators are:

ψP/S (q2) = i∫

d4x eiq.x〈0|T JP/S (x)JP/S (0)†|0〉 ,

ΠµνV/A(q2) = i

∫d4x eiq.x〈0|T JµV/A(x)JνV/A(0)†|0〉

= −

(gµν −

qµqν

q2

)ΠT

V/A(q2) +qµqνq2 ΠL

V/A(q2)(2)

where ΠTV/A(q2) has more power of q2 than the transverse two-

point function used in the current literature for mq = mQ inorder to avoid mass singularities at q2 = 0 when mq goes tozero. Π

µνV/A(q2) (vector/axial) and ψS/P(q2) (scalar/pseudoscalar)

correlators are related each other through the Ward identities:

qµqνΠµνV/A(q2) = ψS/P(q2) − ψS/P(0) , (3)

where the (non)perturbative parts of ψS/P(0) are known (see e.g.[11]) and play an important role for absorbing mass singular-ities which appear during the evaluation of the PT two-pointfunction. For extracting the decay constants, we shall workwith the well established (inverse) Laplace sum rules [we usethe terminology : (inverse) Laplace instead of Borel sum ruleas it has been demonstrated in [29] that its QCD radiative cor-

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rections satisfy the (inverse) Laplace transform properties] 3 :

LP/S ,V/A(τ, µ) =

∫ tc

(mq+MQ)2dt e−tτ 1

πIm

[ψP/S ,Π

TV/A

](t, µ) .(5)

For improving the extraction of the decay constants fD∗(s)and

fB∗(s), we shall also work with the ratio:

RV/A,P ≡LV,A(τV/A)LP(τP)

, (6)

in order to minimize the systematics of the approach, the ef-fects of heavy quark masses and the continuum threshold un-certainties which are one of the main sources of errors in thedeterminations of the individual decay constants. τV/A, τP de-note the values of the sum rule τ-variable at which each indi-vidual sum rule is optimized (minimum or inflexion point). Ingeneral, τV/A , τP as we shall see later on which requires somecare for a precise determination of the ratio of decay constants.This ratio of sum rule has lead to a successful prediction of theSU(3) breaking ratio fPs/ fP [32] such that, from it, one expectsto extract precise values of the ratio fV/A/ fP.

2. QCD expression of the two-point correlators

The QCD expression of the Laplace sum rule LP/S (τ, µ) in the(pseudo)scalar channel can be found in [1, 2, 22] for full QCDincluding N2LO perturbative QCD corrections and contribu-tions of non-perturbative condensates up to the complete d = 6dimension condensates 4. The one of the vector channel hasbeen given within the same approximation in [3]. Some com-ments are in order :• The expressions of NLO PT in [33, 34], of N2LO PT in

[35], of the non-perturbative in [16, 28] and the light quarkmass corrections in [16, 33, 36] have been used. The N3LOPT contributions have been estimated by assuming the geomet-ric growth of the series [37] which is dual to the effect of a1/q2 term [38, 39]. The PT expressions have been originallyobtained using a pole quark mass which is transformed into theMS -scheme by using the relation between the running mQ(µ)and pole mass MQ in the MS -scheme to order α2

s [40–44].• The LO contributions in αs up to the d = 4 gluon:

〈αsG2〉 ≡ 〈αsGaµνG

µνa 〉 , d = 5 mixed quark-gluon: 〈qGq〉 ≡

〈qgσµν(λa/2)Gaµνq〉 = M2

0〈qq〉 and d = 6 quark: 〈d jd〉 ≡〈dgγµDµGµν

λa2 d〉 = g2〈dγµ

λa2 d

∑q qγµ

λa2 q〉 ' − 16

9 (παs) ρ〈dd〉2,(after the use of the equation of motion) condensates, have beenobtained originally by NSV2Z [28] . ρ ' 3 − 4 measures thedeviation from the vacuum saturation estimate of the d = 6four-quark condensates [45–47].

3One can also work with moment sum rules:

M(n)qb (µ) =

∫ tc

(mq+Mb)2

dttn+2

1π

Imψqb(t, µ) , (4)

like in [1] or with τ-decay like finite energy sum rules [30] inspired from τ-decay [31] but these different sum rules give approximately the same results asthe one from (inverse) Laplace sum rules.

4Note an unfortunate missprint of 1/π in front of Imψ(t) in Ref. [1].

• The contribution of the d = 6 gluon condensates: 〈g3G3〉 ≡

〈g3 fabcGaµνG

bνρG

cρµ〉 , 〈 j

2〉 ≡ g2〈(DµGaνµ)2〉 = g4〈

(∑q qγν λ

a

2 q)2〉 '

− 643 (παs)2ρ〈dd〉2, after the use of the equation of motion, have

been deduced from [16] (Eqs. II.4.28 and Table II.8).• One can notice that the gluon condensate 〈αsG2〉 and 〈G3〉

contributions flip sign from the pseudoscalar to the vector cor-relators while there is an extra mQM2

0〈dd〉 term with a posi-tive contribution in the pseudoscalar channel from the mixedcondensate. Similar observation arises for the signs of the chi-ral quark and mixed quark gluon condensates from the pseu-doscalar to the scalar and from the vector to the axial correla-tors. We shall see in Fig. 3 that these different signs transforme.g the minimum in τ for the pseudoscalar into an inflexionpoint for the vector sum rule.• It is clear that, for some non-perturbative terms which are

known to leading order of perturbation theory, one can use ei-ther the running or the pole mass. However, we shall see thatthis distinction does not affect, in a visible way, the present re-sult, within the accuracy of our estimate, as the non-perturbativecontributions are relatively small though vital in the analysis.

Table 1: QCD input parameters: the original errors for 〈αsG2〉, 〈g3G3〉 andρ〈qq〉2 have been multiplied by 2-3 for a conservative estimate.

Parameters Values Ref.αs(Mτ) 0.325(8) [31, 45, 48, 49]mc(mc) 1261(12) MeV average [49–51]mb(mb) 4177(11) MeV average [49, 50]µq (253 ± 6) MeV [11, 30, 52, 53]M2

0 (0.8 ± 0.2) GeV2 [22, 46, 54]〈αsG2〉 (7 ± 3) × 10−2 GeV4 [45, 47, 50, 55–60]〈g3G3〉 (8.2 ± 2.0) GeV2 × 〈αsG2〉 [50]ραs〈qq〉2 (5.8 ± 1.8) × 10−4 GeV6 [45–47]ms (0.114 ± 0.006) GeV [11, 30, 52, 53, 61]κ ≡ 〈ss〉/〈dd〉 (0.74 ± 0.12) [11, 62]

3. QCD input parameters

The QCD parameters which shall appear in the following anal-ysis will be the charm and bottom quark masses mc,b (we shallneglect the light quark masses q ≡ u, d), the light quark conden-sate 〈qq〉, the gluon condensates 〈αsG2〉 and 〈g3G3〉 the mixedcondensate 〈qGq〉 defined previously and the four-quark con-densate ραs〈qq〉2. Their values are given in Table 1.• We shall work with the running light quark condensates

and masses. They read:

〈qq〉(τ) = −µ3q (−β1as)2/β1/C(as)

〈qGq〉(τ) = −M20 µ

3q (−β1as)1/3β1/C(as) ,

ms(τ) = ms/(−Log

√τΛ

)2/−β1C(as) , (7)

where β1 = −(1/2)(11 − 2n f /3) is the first coefficient of the βfunction for n f flavours; as ≡ αs(τ)/π; µq is the spontaneousRGI light quark condensate [64]. The QCD correction factorC(as) in the previous expressions is numerically [44]:

C(as) = 1 + 1.1755as + 1.5008a2s + ... : n f = 5 , (8)

which shows a good convergence.


• The value of the running 〈qq〉 condensate is deduced fromthe well-known GMOR relation: (mu +md)〈uu+ dd〉 = −m2

π f 2π ,

where fπ = 130.4(2) MeV [9] and the value of (mu + md)(2) =

(7.9 ± 0.6) MeV obtained in [30] which agrees with the PDGin [49] and lattice averages in [65]. Then, we deduce the RGIlight quark spontaneous mass µq given in Table 1.• For the heavy quarks, we shall use the running mass and

the corresponding value of αs evaluated at the scale µ.• To be conservative, we have enlarged the original errors of

some parameters (gluon condensate, SU(3) breaking,...) by afactor 2-3. We do not consider the lattice result [63] for κ whichis in conflict with SR one from various channels and ChPT.

We shall see that the effects of the gluon and four-quark con-densates on the values of the decay constants are small thoughthey play an crucial role in the stability analysis.

4. Parametrization of the spectral function

• Minimal Duality Ansatz (MDA)We shall use MDA for parametrizing the spectral function:

1π

ImψP/S (t) ' f 2P M4

P/S δ(t − M2P) + “QCD cont.”θ(t − tP

c ) ,

1π

ImΠTV (t) ' f 2

V M2Vδ(t − M2

V ) + “QCD cont.”θ(t − tVc ), (9)

where fP,V are the decay constants defined in Eq. (1) and thehigher states contributions are smeared by the “QCD contin-uum” coming from the discontinuity of the QCD diagrams andstarting from a constant threshold tP

c , tVc which is independent

on the subtraction point µ in this standard minimal model.• Test of the Minimal Duality Ansatz from J/ψ and Υ

The MDA presented in Eq. (9), when applied to the ρ-mesonreproduces within 15% accuracy the ratio Rdd measured fromthe total cross-section e+e− → I = 1 hadrons data (Fig. 5.6 of[4]), while in the case of charmonium, M2

ψ from ratio of mo-ments R(n)

cc evaluated at Q2 = 0 has been compared with the onefrom complete data where a remarkable agreement for highern ≥ 4 values (Fig. 9.1 of [4]) has been found.

Recent tests of MDA from the J/ψ and Υ systems have beendone in [1]. Taking (tψc )1/2 ' Mψ(2S ) − 0.15 GeV and (tΥc )1/2 '

MΥ(2S ) − 0.15 GeV, we show (for instance) the ratio betweenL

expQQ

and LdualQQ

in Fig. 1 for the J/π and Υ systems indicatingthat for heavy quark systems the role of the QCD continuum issmaller than in the case of light quarks while the exponentialweight suppresses efficiently the QCD continuum contributionand enhances the one of the lowest resonance. We have usedthe simplest QCD continuum for massless quarks [50] 5:

QCD cont. = (1 + as + 1.5a2s − 12.07a3

s)θ(t − tc).

One can see in Fig. 1 that the MDA, with a value of√

tc aroundthe one of the 1st radial excitation mass, describes well the com-plete data in the region of τ-stability of the sum rules [50]:

τψ ' (0.8 ∼ 1.4) GeV−2, τΥ ' (0.2 ∼ 0.4) GeV−2, (10)

5We have checked that the including mass corrections give the same results.

as we shall see later on. Though it is difficult to estimate withprecision the systematic error related to this simple model, thisfeature indicates the ability of the model for reproducing accu-rately the data. We expect that the same feature is reproducedfor the open-charm and beauty vector meson systems wherecomplete data are still lacking. Moreover, MDA has been alsoused in [66] in the context of large Nc QCD, where it providesa very good approximation to the observables one computes.

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r L

0.0 0.1 0.2 0.3 0.4 0.50.90

0.95

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r L

a) b)Figure 1: τ-behaviour of the ratio of Lexp

bb/Ldual

bbfor√

tc = MΥ(2S )-0.15 GeV. The reddashed curve corresponds to the strict equality for all values of τ : a) charmonium, b)bottomium channels.

a) b)Figure 2: a) τ-behaviour of R(τ) normalized to the ground state energy E0 for the har-monic oscillator. 2 and 4 indicate the number of terms in the approximate series; b) Com-mented regions for different values of τ.

5. Optimization and Stability criteria

• τ and tc-stabilitiesIn order to extract an optimal information for the lowest reso-nance parameters from this rather crude description of the spec-tral function and from the approximate QCD expression, oneoften applies the stability criteria at which an optimal result canbe extracted. This stability is signaled by the existence of astability plateau, an extremum or an inflexion point versus thechanges of the external sum rule variables τ and tc where the si-multaneous requirement on the resonance dominance over thecontinuum contribution and on the convergence of the OPE issatisfied. This optimization criterion demonstrated in series ofpapers by Bell-Bertmann [60] in the case of τ by taking the ex-amples of harmonic oscillator and charmonium sum rules andextended to the case of tc in [4, 11] gives a more precise mean-ing of the so-called “sum rule window” originally introducedby SVZ [10] and used in the sum rules literature.• µ subtraction point stability

We shall add to the previous well-known stability criteria, theone associated to the requirement of stability versus the varia-tion of the arbitrary subtraction constant µ often put by hand inthe current literature and which is often the source of large er-rors from the PT series in the sum rule analysis. The µ-stability


procedure has been applied recently in [1, 2, 53, 67] 6 whichgives a much better meaning on the choice of µ-value at whichthe observable is extracted, while the errors on the results havebeen reduced due to a better control of the µ region of variationwhich is not often the case in the literature.

6. The decay constants fD and fD∗

• Direct determinations from LP,V (τ, µ)We start by showing in Fig. 3, the τ-behaviour of the decay

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tcP@GeV2D, Μ = mc

a) b)Figure 3: a) τ-behaviour of fD from LP for different values of tc, at a given value of thesubtraction point µ = mc; b) the same as in a) but for fD∗ from LV .

constants fD∗ and fD at a given value of the subtraction pointµ = mc for different values of the continuum threshold tc. Wehave assumed that : (tD∗

c )1/2−(tDc )1/2 ' MD∗−MD = 140.6 MeV,

for the vector and pseudoscalar channels where the QCD ex-pressions are truncated at the same order of PT and NP series.At this value of µ, we deduce the optimal value versus τ (min-imum or inflexion point for τ ' 0.8 − 1 GeV−2) and tc (5.6-8GeV2). Next, we study the µ variation of these results whichwe show in Fig. 4. We deduce the mean result from [1]:

fD = 204(6) MeV , (11)

where the largest error for each data point comes from tc 7. Thevalue of fD∗ at the minimum in µ = (1.5 ± 0.1) GeV is [3]:

fD∗ = 253.5(11.5)tc (5.7)τ(13)svz(1)µ = 253.5(18.3) MeV, (12)

with : (13)svz = (0.5)αs (12.3)α3s(0.6)mc (3.8)〈dd〉

(1.8)〈αsG2〉(1.4)〈dGd〉(0.4)〈g3G3〉(0.4)〈dd〉2 , (13)

where the SVZ-OPE error is dominated by the estimate of theα3

s corrections (95%) and where again the error due to µ hasbeen mutiplied by a factor 2 for a more conservative error. Oneshould notice that the accurate value of fD comes from the meanof different data shown in Fig. 4, while the error from fD∗ istaken from the one at the minimum for µ ' 1.45 GeV.• The ratio fD∗/ fD and improved determination of fD∗

One can notice in Fig. 3 that working directly with the ratio inEq. (6) by taking the same value τV = τP is inaccurate as thetwo sum rules LV (τ) and LP(τ) are not optimized at the same

6Some alternative approaches for optimizing the PT series are in [68].7One should note that the extraction of the charm quark running mass by re-

quiring that the sum rules should reproduce the D-meson mass is obtained in thesame range of values of the previous set of stability parameters [1]. The sameresults are obtained for the other mesons. This fact increases the confidence onthe predictions of the unknown decay constants.

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0252

253

254

255

256

Μ @GeVD

f D*@M

eVD

a) b)

Figure 4: a) fD from LSR versus µ and for mc=1467 MeV. The filled (grey) region is theaverage with the corresponding averaged errors. The dashed horizontal lines are the oneswhere the errors come from the best determination; b) fD∗ for different values of µ.

value of τ (minimum for fD and inflexion point for fD∗ ). There-fore, for a given value of tc, we take separately the value of eachsum rule at the corresponding value of τ where they presentminimum and/or inflexion point and then take their ratio. Fora given µ, the optimal result corresponds to the mean obtainedin range of values of tc where one starts to have a τ-stability(tc ' 5.6 − 5.7 GeV2 for τ ' 0.6 GeV−2) and a tc-stability(tc ' 9.5 ∼ 10.5 GeV2 for τ ' 0.8 GeV−2). In Fig. 5, we lookfor the µ-stability of the previous optimal ratio fD∗/ fD in the set(τ, tc). We obtain at the minimum µ = (1.5 ± 0.1) GeV:

fD∗/ fD = 1.218(6)tc (27)τ(23)svz(4)µ ,= 1.218(36)with : (23)svz = (7)αs (2)α3

s(3)mc (0)〈dd〉(18)〈αsG2〉

(12)〈dGd〉(0)〈g3G3〉(1)〈dd〉2 . (14)

The error from µ has been multiplied by 2 to be conservative.The largest error comes from τ which is due to the inaccuratelocalization of the inflexion point. The one due to tc is smalleras expected in the ratio which is not the case for the direct ex-traction of the decay constants. The ones due to 〈αsG2〉 and〈dGd〉 are large due to the opposite sign of their contributionsin the vector and pseudoscalar channels which add when takingthe ratio. Using the value fD = 204(6) MeV in Eq. (11) and theratio in Eq. (14), we deduce the improved value:

fD∗ = 248.5(10.4) MeV , (15)

where the errors have been added quadratically. Taking themean of the two results in Eqs. (12) and (15), one obtains:

〈 fD∗〉 = 249.7(10.5)(1.2)syst = 250(11) MeV , (16)

where the 1st error comes from the most precise determinationand the 2nd one from the distance of the mean value to it.

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D

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0252

253

254

255

256

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a) b)

Figure 5: a) fD∗ / fD versus the subtraction point µ; b) The same as a) but for fD∗ .

• Upper bound on fD∗

We derive an upper bound on fD∗ by considering the positivity


of the QCD continuum contribution to the spectral function andby taking the limit where tc → ∞ in Eq. (5) which correspondsto a full saturation of the spectral function by the lowest groundstate contribution. The result of the analysis versus the changeof τ for a given value of µ = 1.5 GeV is given in Fig. 6a whereone can observe like in the previous analysis the presence of aτ-inflexion point. We also show the good convergence of the PTseries by comparing the result at N2LO and the one includingan estimate of the N3LO term based on the geometric growthof the PT coefficients. We show in Fig 6b the variation of theoptimal bound versus the subtraction point µ where we find aregion of µ stability from 1.5 to 2 GeV. We obtain:

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N3LO

N2LO

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a) b)

Figure 6: Upper bounds on fD∗ : a) at different values of τ for a given value µ =1.5GeV of the subtraction point µ. One can notice a good convergence of the PT series bycomparing the calculated N2LO and estimated N3LO terms; b) versus the values of thesubtraction point µ.

fD∗ ≤ 267(10)τ(14)svz(0)µ MeV ,

with : (14)svz = (1.4)αs (13)α3s(1.3)mc (3)〈dd〉

(3)〈αsG2〉(2.5)〈dGd〉(0)〈g3G3〉(0.7)〈dd〉2 . (17)

Alternatively, we combine the upper bound fD ≤ 218.4(1.4)MeV obtained in [1, 2] with the ratio in Eq. (14) and deduce:

fD∗ ≤ 266(8) MeV , (18)

where we have added the errors quadratically. The good agree-ment of the results in Eqs. (17) and (18) indicates the self-consistency of the approaches. This bound is relatively strongcompared to the estimate in Eq. (16). A comparison of our re-sults with the ones from some other sources is shown in Table 4.

7. The decay constants fB and fB∗ from SR in full QCD

• Direct estimate of fB and fB∗

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a) b)

Figure 7: a) τ-behaviour of fB from LP for different tc, at µ = mb; b) the same as in a)but for fB∗ from LV .

We extend the analysis to the case of the B∗ meson. We usethe set of parameters in Table 1. The τ-behaviours of fB and f ∗Bin Fig. 7 have a shape similar to the case of fD and fD∗ . We

estimate fB∗ from Fig. 7b where the τ-stability is reached fromtc = 34 GeV2 while the tc-one starts from tc = (55 − 60) GeV2.We show the µ-behaviour of the optimal result in Fig. 8. At theinflexion point µ = (5.5 ± 0.5) GeV, we deduce:

fB∗ = 239(38)tc (1)τ(2.7)svz(1.4)µ = 239(38) MeV,with : (2.7)svz = (0.7)αs (2)α3

s(0.4)mb (1.6)〈dd〉

(0.4)〈αsG2〉(0.3)〈dGd〉(0)〈g3G3〉(0)〈dd〉2 . (19)

• The ratio fB∗/ fB

We extract directly the ratio fB∗/ fB from the ratio of sum rules.We show in Fig. 9a its τ-behaviour for different values of tc forµ = 5.5 GeV from which we deduce as optimal value the meanof the τ-minima obtained from tc = 34 to 60 GeV2. We showin Fig. 9b the µ-behaviour of these optimal results where wefind a minimum in µ around 3.8-4.5 GeV and a slight inflexionpoint around 5.5 GeV. We consider as a final result the mean ofthe ones from these two regions of µ:

fB∗/ fB = 1.016(12)tc (1)τ(9)svz(6)µ = 1.016(16) , (20)

with:

(9)svz = (3)αs (6)α3s(3)mb (4)〈dd〉(3)〈αsG2〉(1)〈dGd〉(1)〈g3G3〉(1)〈dd〉2 (21)

Combining the results in Eq. (20) with the value fB = 206(7)MeV obtained in [1, 2], we deduce:

fB∗ = 209(8) MeV , (22)

which is more accurate than the direct determination in Eq. (19)and where the main error comes from the one of fB extracted in[1, 2]. We consider the result in Eq. (22) which is also the meanof the results in Eqs. (22) and (19) as our final determination.

4.5 5.0 5.5 6.0 6.5237

238

239

240

241

Μ @GeVD

f B*@M

eVD

Figure 8: Values of fB∗ at different values of the subtraction point µ.

• Upper bound on fB∗

ò

ç

à

0.00 0.05 0.10 0.15

0.96

0.98

1.00

1.02

1.04

Τ @GeV-2D

f B*�f

B

65

à 60

55

50

45

ç 40

ò 35

34

tc@GeV2D, Μ = 5.5 GeV

3 4 5 6 71.010

1.015

1.020

1.025

1.030

1.035

Μ @GeVD

f B*�f

B

a) b)

Figure 9: fB∗ / fB: a) τ-behaviour for different tc at µ = 5.5 GeV; b) µ-behaviour.

Like in the case of the D∗ meson, we extract directly an upperbound on fB∗ by using the positivity of the QCD continuum tothe spectral function. We show the τ- and the µ-behaviours ofthe optimal bound in Fig. 10. We deduce at µ = (6 ± 0.5) GeV:

fB∗ ≤ 295(14)τ(4)svz(10)µ = 295(18) MeV . (23)


0.05 0.10 0.15 0.20 0.25

0.2

0.4

0.6

0.8

1.0

Τ @GeV-2D

f B*

@GeV

D

ò N3LO

N2LO

Μ = 6 GeV

4 5 6 7 8

240

260

280

300

Μ @GeVD

f B*@M

eVD

a) b)

Figure 10: Upper bound of fB∗ from LP: a) τ-behaviour at µ = 6 GeV for N2LO andN3LO truncation of the PT series; b) µ-behaviour.

We consider the previous values of fD∗ and fB∗ as improvementof our earlier results in [4] and [18].

8. fB∗/ fB from HQET sum rules

This ratio is under a good control from heavy quark effectivetheory (HQET) sum rules. It reads in the large mass limit [25,26, 69]:

Rb ≡fB∗√

MB∗

fB√

MB= RPT +

RNPT

Mb,

RPT = 1 −23

as − 6.56a2s − 84.12a3

s = 0.892(13) ,

RNPT =23

Λ

(1 +

53

as

)− 4GΣ

(1 +

32

as

), (24)

with : as ≡ (αs/π)(Mb) = 0.070(1), Λ ≡ MB −

Mb = (499 ± 60) MeV, where we have used the pole massMb=4.810(60) MeV to order α2

s deduced from the running massmb(mb) = 4.177(11) MeV [50] and we have retained the largererror from PDG [49]. We have estimated the error in the PTseries by assuming a geometric growth of the coefficients [37].

GΣ(Mb) ≡ GΣ[αs(Mb)]−3/2β1 +827

Λ , (25)

where the invariant quantity: GΣ ' −(0.20 ± 0.01) GeV hasbeen extracted inside the t-stability region of Fig. 7 from [26].We obtain :

Rb = 0.938(13)PT (5)NP(6)Mb,Λ = 0.938(15) . (26)

To convert the above HQET result to the one of the full theory atfinite quark and meson masses, we have to include the normal-ization factor (MB/Mb)2 relating the pseudoscalar to the uni-versal HQET correlators according to the definition in Eq. (1).Then, we deduce (see e.g. [2, 26, 27]):

fB∗

fB=

(MB

Mb

) √MB

MB∗Rb = 1.025(13)Mb (15)Rb , (27)

in fair agreement with our result in Eq. (20) and the ones in[8, 26, 70] but higher than the ones in [71, 72]. To make a directcomparison of our results with the lattice calculations, it is de-sirable to have a direct lattice calculation from the pseudoscalarcorrelator built from the current in Eq. (1). The result of [72] isdifficult to interpret as they use a non-standard threshold of theQCD continuum. However, their requirement of maximal sta-bility for τ ≤ 0.15 GeV−2 is outside the “standard sum rule (SR)

window” obtained around 0.2-0.3 GeV−2 (minimum or inflex-ion point in our figures) where the lowest resonance dominatesthe sum rule. Taking literally their set (tc, τ)=( 33 GeV2, 0.05GeV−2) where their “duality” is obtained (Fig. 3 of [72]) intoour Fig. 9, one obtains: fB∗/ fB = .985 more comparable withtheir result 0.944 but meaningless from the SR point of view asit comes from a region dominated by the QCD continuum.

9. SU(3) breaking for fD∗s and fD∗s/ fD∗

We pursue the same analysis for studying the S U(3) break-ing for fD∗s and the ratio fD∗s/ fD∗ . We work with the completemassive (ms , 0) LO expression of the PT spectral function ob-tained in [64] and the massless (ms = 0) expression known toN2LO used in the previous sections. We include the NLO PTcorrections due to linear terms in ms obtained in [70]. We showthe τ-behaviour of the results in Fig. 11a for a given µ = 1.5GeV and different tc. We study their µ-dependence in Fig. 11bwhere a nice µ stability is reached for µ ' 1.4 − 1.5 GeV. We

ç

ò

ç

ò

0.0 0.2 0.4 0.6 0.8 1.00.1

0.2

0.3

0.4

0.5

Τ @GeV-2D

f Ds*

@GeV

D

10

ò 9

ç 8

7

6

ò 5.8

ç 5.7

tcD*

@GeV2D, Μ = 1.5 GeV

1.30 1.35 1.40 1.45 1.50 1.55 1.60270

271

272

273

274

Μ @GeVD

f D*s@M

eVD

a) b)

Figure 11: fD∗sfrom LV : a) τ-behaviour for different tc, at µ =1.5 GeV; b) µ-behaviour.

have used : (tD∗sc )1/2 − (tD∗

c )1/2 ≈ MD∗s −MD∗ = 102 MeV. Takingthe conservative result ranging from the beginning of τ-stability(tc ' 5.7 GeV2) until the beginning of tc-stability of about (9–10) GeV2, we obtain at µ=1.5 GeV:

fD∗s = 272(24)tc (2)τ(18)svz(2)µ = 272(30) MeV ,

with : (18)svz = (0)αs (14)α3s(1)mc (2)〈dd〉(1.5)〈αsG2〉

(0.8)〈dGd〉(0)〈g3G3〉(0)〈dd〉2 (0.3)ms (2)κ . (28)

Taking the PT linear term in ms at lowest order and tc = 7.4GeV2, we obtain fD∗s = 291 MeV in agreement with the one293 MeV of [70] obtained in this way. The inclusion of thecomplete LO term decreases this result by about 5 MeV whilethe inclusion of the NLO PT S U(3) breaking terms increasesthe result by about the same amount. Combining the result inEq. (28) with the one in Eq. (16), we deduce the ratio:

fD∗s/ fD∗ = 1.090(70) , (29)

where we have added the relative errors quadratically. Alterna-tively, we extract directly the previous ratio using the ratio ofsum rules. We show the results in Fig. 12a versus τ and for dif-ferent values of tc at µ = 1.5 GeV. τ-stabilities occur from τ ' 1to 1.5 GeV−2. We also show in Fig 12b the µ-behaviour of theresults where a good stability in µ is observed for µ ' (1.4−1.5)GeV in the same way as for fD∗s . We deduce:

fD∗s/ fD∗ = 1.073(1)tc (16)τ(2)µ(50)svz = 1.073(52) ,with : (50)svz = (1)αs (45)α3

s(0)mc (2)〈dd〉(16)〈αsG2〉

(3)〈dGd〉(1)〈g3G3〉(2)〈dd〉2 (4)ms (13)κ , (30)


ç

ò

ç

ò

0.0 0.5 1.0 1.51.04

1.05

1.06

1.07

1.08

1.09

1.10

Τ @GeV-2D

f Ds*

�fD

*

10

ò 9

ç 8

7

6

ò 5.8

ç 5.7

tcD*

@GeV2D, Μ = 1.5 GeV

1.30 1.35 1.40 1.45 1.50 1.55

1.068

1.070

1.072

1.074

Μ @GeVD

f D*s�f

D*

a) b)

Figure 12: fD∗s/ fD∗ : a) τ-behaviour for different tc and for µ = 1.5 GeV; b) µ-behaviour.

where, for asymmetric errors, we have taken the mean of thetwo extremal values. The error associated to τ take into ac-counts the fact that, for some values of tc, the τ-minima forfD∗ and fD∗s do not coıncide. Comparing Eqs. (29) and (30),one can see the advantage of a direct extraction from the ratioof moments due to the cancellation of systematic errors in theanalysis. Taking the mean of Eqs. (29) and (30), we deduce:

fD∗s/ fD∗ = 1.08(6)(1)syst , (31)

where the 1st error comes from the most precise determinationand the 2nd one from the distance of the mean value to thecentral value of this precise determination. Using Eq. (31), fD∗

in Eq. (16) and its upper bound in Eq. (18), we predict in MeV:

fD∗s = 270(19) , fD∗s ≤ 287(8.6)(16) = 287(18) . (32)

Future experimental measurements of fD∗ and fD∗s though mostprobably quite difficult should provide a decisive selection ofthese existing theoretical predictions.

10. SU(3) breaking for fB∗s and fB∗s/ fB∗

0.00 0.05 0.10 0.15 0.200.98

1.00

1.02

1.04

1.06

1.08

1.10

Τ @GeV-2D

f Bs*

�fB

*

65

à 60

55

50

45

ç 40

ò 35

34


4.5 5.0 5.5 6.0

1.045

1.050

1.055

1.060

Μ @GeVD

f B*s�f

B*

a) b)

Figure 13: fB∗s/ fB∗ : a) τ-and tc-behaviours at a given µ = 5 GeV; b) µ-behaviour.

We extend the analysis done for the D∗s to the case of the B∗s-meson. We show, in Fig. 13a, the τ-behaviour of the ratiofB∗s/ fB∗ at µ=5 GeV and for different values of tc where the τ-stability starts from tc = 40 GeV2 while the tc one is reached fortc ' (60 − 65) GeV2. Our optimal result is taken in this rangeof tc. We study the µ-behaviour in Fig. 13b where an inflexionpoint is obtained for µ = (5 ± .5) GeV. At this point, we obtain:

fB∗s/ fB∗ = 1.054(8)tc (0)τ(6)µ(4.6)svz = 1.054(11) ,with : (4.6)svz = (2)αs (2.5)α3

s(0)mb (2)〈dd〉(1.5)〈αsG2〉

(0)〈dGd〉(0)〈g3G3〉(0)〈dd〉2 (0)ms (2)κ . (33)

We show, in Fig. 14a, the τ-behaviour of the result for fB∗s atµ=5.5 GeV and for different values of tc. For fB∗s , τ-stabilitystarts from tc ' 34 GeV2 while tc-stability is reached from tc '(50 − 65) GeV2. We show in Fig. 14b the µ-behaviour of these

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.2

0.3

0.4

0.5

0.6

0.7

Τ @GeV-2D

f Bs*

@GeV

D

65

à 60

55

50

45

ç 40

ò 35

34


4.5 5.0 5.5 6.0 6.5 7.0 7.5250

255

260

265

270

275

280

285

Μ @GeVD

f B*s@M

eVD

a) b)

Figure 14: fB∗s: a) τ-behaviour for different tc at a given µ = 6 GeV; b) µ-behaviour.

optimal results. One can notice a slight inflexion point at µ=6GeV which is about the one (5.0-5.5) GeV obtained previouslyfor the ratio fB∗s/ fB∗ . At this value of µ, we obtain:

fB∗s = 271(39)tc (0)τ(3)svz(6)µ = 271(40) MeV ,

with : (3)svz = (1)αs (1.5)α3s(0.5)mb (2)〈dd〉(0.5)〈αsG2〉

(0.5)〈dGd〉(0)〈g3G3〉(0.5)〈dd〉2 (0)ms (1)κ . (34)

Combining this result with the one of fB∗ in Eq. (19) obtainedwithin the same approach and conditions, we deduce the ratio:

fB∗s/ fB∗ = 1.13(25) , (35)

where the large error is due to the determinations of each ab-solute values of the decay constants. We take as a final valueof the ratio fB∗s/ fB∗ the most precise determination in Eq. (33).Combining this result with the final value of fB∗ in Eq. (22) andwith the upper bound in Eq. (23), we deduce our final estimate:

fB∗s = 220(9) MeV , fB∗s ≤ 311(19) MeV . (36)

11. The scalar meson decay constants fD∗0(0s)

and fB∗0(0s)

• Due to large discrepancies of the fD∗0 values among differ-ent SR [73–75] and between lattice [76, 77] results (see Table3), we update our previous estimates in [4, 18, 22]. We usethe same QCD inputs (N2LO for PT within the MS -scheme⊕ d ≤ 6 condensates) and the same criteria as for fD,B [1] forobtaining the optimal results. The difference beween the scalarand pseudoscalar SR is the overall factor (MQ ± ms)2 and thesigns of the 〈qq〉 and mixed condensates effects for ms = 0.We show fD∗0 versus τ in Figs. 15a,b where the τ-stability about(0.4 − 0.6) GeV−2 of the SR for µ = τ−1/2 constrains tc inside(7 − 9) GeV2 (Fig. 15a) 8. In Fig. 15b, we show the SR for agiven µ = 2 GeV, and tc = (7−11) GeV2-stability. The effect ofthe truncation of the PT series is analyzed by using instead theinvariant quark mass in Eq. 7 and the QCD scale Λ. We show inFig. 15c the results versus µ and using either the running (blueopen circle data points) or the invariant (red triangle data) mass.Taking their average, we deduce the value in Table 2 where 95%of the error comes from the above range of tc-values. The errorbecomes 12 instead of 7 if one considers the most precise deter-mination. Upper bound is derived in Fig. 16 using the positivityof the spectral function. We extract from the SR at µ =

√1/τ

8Larger tc corresponds to an apprent small τ-stability outside the SR windowand leads to a larger fD∗0 (Fig. 15b).


the SU(3) breaking ratio fD∗0s/ fD∗0 with the result in Table 3. The

error of the ratio comes mainly from the S U(3) breaking terms.These results confirm and improve the previous one in [22].• Doing similar analysis for fB∗0 and fB∗0s

, we notice that onlythe SR subtracted at µ = τ−1/2 (Fig. 15d) exhibits τ-stabilityfor tc ' (35 − 38) GeV2. We use MB∗0 ' MB∗0s

= 5733 MeVassuming that MB∗0 −MB ' MD∗0 −MD = 448 MeV supported bythe SR estimate 413(210) MeV in [4, 18]. We deduce the resultsin Table 2, where like for fD∗0 , the errors come mainly from tcand the S U(3) breaking parameters. These results improve andconfirm previous estimates in [4, 18].• fD∗0 agrees with the (inaccurate) phenomenological value

206(120) MeV from B → D∗∗π [76]. The agreement for fB∗0but not fD∗0 with the SR one in [75] is puzzling and could bedue to a different treatment of the QCD input and evolutionparameters which are sensitive at low scale. A comparison withsome other determinations are given in Table 3. A future precisemeasurement of fD∗0 will select among the theoretical estimates.

ò

à

ç

0.0 0.2 0.4 0.6 0.80.10

0.15

0.20

0.25

0.30

Τ @GeV-2D

f D*

0@G

eVD

ç 1110

à 98

ò 7

tc@GeV2D

ò

à

ç

0.0 0.2 0.4 0.6 0.80.10

0.15

0.20

0.25

0.30

Τ @GeV-2D

f D*

0@G

eVD

ç 1110

à 98

ò 7

tc@GeV2D

a) b)

òò

òò ò ò

ç

ç

ç

ç

ç ç

1.0 1.5 2.0 2.5 3.0

160

180

200

220

240

260

280

Μ @GeVD

f D*

0@M

eVD

ò

à

0.00 0.05 0.10 0.15 0.20 0.25 0.300.1

0.2

0.3

0.4

0.5

Τ @GeV-2D

f B*

0@G

eVD

ç 55

50

à 45

40

ò 38

35

tc@GeV2D

c) d)

Figure 15: a) fD∗0versus τ and tc for µ = τ−1/2; b) The same as a) but for µ = 2 GeV; c)

Optimal value of fD∗0versus µ: blue open circles are data from SR with running mass. Red

triangles are from SR using invariant mass. The green open circle comes from the SR forµ = τ−1/2. The grey region is the average with averaged errors. The green dotted horizontallines correspond to the error coming from the best determination. ; d) fB∗0

versus τ and tcfor µ = 1/

√τ. A similar behaviour is obtained for fB∗0s

.

ò

ò

0.0 0.2 0.4 0.6 0.80.15

0.20

0.25

0.30

Τ @GeV-2D

f D*

0@G

eVD

ò Μ =1.8 GeV

çç

0.5 1.0 1.5 2.0 2.5 3.0 3.5210

220

230

240

250

260

270

Μ @GeVD

f D0

*@M

eVD

a) b)

Figure 16: Upper bound of fD∗0: a) versus τ for µ = 1.8 GeV; b) for different µ. The red

open cercle is the optimal value.

12. The axial meson decay constants fD1 and fB1

In the zero light quark mass limit, the axial-vector correlatorcan be deduced from the vector one by changing the sign of thechiral condensate contributions. Using the expression known toN2LO ⊕ d ≤ 6 condensates, we extract directly the ratio fD1/ fD

and fB1/ fB by taking each value of fD1 and fB1 at a τ-minimum

ò

ç

0.0 0.2 0.4 0.6 0.80.1

0.2

0.3

0.4

0.5

0.6

Τ @GeV-2D

f D1

@GeV

D

11

10

ç 9

ò 8

7

tc@GeV2D, Μ = mc

ò

ç

à

0.0 0.1 0.2 0.3

0.2

0.4

0.6

0.8

Τ @GeV-2D

f B1

@GeV

D

60

à 55

50

45

40

ç 38

ò 35

34

tc@GeV2D, Μ = mb

a) b)

ç ç

ç

çç

0.5 1.0 1.5 2.0 2.5 3.01.0

1.2

1.4

1.6

1.8

2.0

2.2

Μ @GeVD

f D1�f

D

ç

çç

ç

ç

ç

ç ç

3 4 5 6 7 81.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

Μ @GeVD

f B1�f

B

c) d)

Figure 17: a) fD1 versus τ and tc for µ = mc; b) The same as a) but for fB1 ; c) Optimalvalue of fD1 for different µ: the green dashed line is the average. The green region is thevalue if one takes the errors from the weighted average. ; d) The same as c) but for fB1 .The horizontal green lines correspond to the error from the best determination.

for a given (µ, tc) (Figs. 17a,b) and the corresponding value offD (Fig. 3) and fB (Fig. 7) at the same of set (µ, tc). We useMB1 − MB∗ ' MB∗0 − MB=448 MeV as indicated by the result417(212) MeV in [4, 18]. The optimal ratios obtained in thisway are given for different µ in Figs. 17c,d. The average givenin Table 4 comes from the dashed coloured regions.

à

ç

0.0 0.1 0.2 0.3 0.4 0.5

0.2

0.4

0.6

0.8

1.0

Τ @GeV-2D

f Bc@G

eVD

ç 70

60

50

à 44

42


Figure 18: τ-behaviour of fBc from LP for different tc, and for µ = 7.5 GeV.

4 6 8 10

360

380

400

420

440

460

480

500

Μ @GeVD

f Bc@M

eVD

6.0 6.5 7.0 7.5 8.0 8.5 9.0440

450

460

470

480

490

Μ @GeVD

f Bc@M

eVD

a) b)

Figure 19: fBc at different values of the subtraction point µ: a) estimate; b) upper bound

13. The decay constants fBc and fB∗c

We complete the analysis by the estimate and bound of the de-cay constant fBc of the Bc(6277) meson bc bound state and offB∗c of its vector partner B∗c, where the light quarks d, s are re-placed by the heavy quark c. Our analysis will be very similarto the one in [18, 78, 79] but we shall use the running c and bquark masses instead of the pole masses and we shall includeN2LO radiative corrections.• The dynamics of the Bc and B∗c is expected to be different

from the B and B∗ one because, by using the heavy quark massexpansion, the heavy quark 〈cc〉 and quark-gluon mixed 〈cGc〉condensates defined in Section 2 behave as [78]:

〈cc〉 = −1

12πmc〈αsG2〉 −

〈g3G3〉

1440π2m3c,


ç

ò

ç

0.00 0.05 0.10 0.15 0.200.3

0.4

0.5

0.6

0.7

0.8

0.9

Τ @GeV-2D

f Bc*

@MeV

D

ç 70

60

52

ò 50

ç 48

tc @GeV2D, Μ = 6 GeV

çç

3 4 5 6 7 8400

420

440

460

480

500

520

Μ @GeVD

f Bc*

@MeV

D

a) b)

Figure 20: fB∗c: a) versus τ for different tc and at a given µ; b) optimal values at τ ' 0.1

GeV−2 versus µ. he open red circle is the optimal value at the inflexion point.

0.00 0.05 0.10 0.15 0.20 0.250.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Τ @GeV-2D

f Bc*

@MeV

D ç Μ = 5.5 GeV

çç

3 4 5 6 7 8460

480

500

520

540

Μ @GeVD

f Bc*

@MeV

D

a) b)

Figure 21: fB∗cbound : a) versus τ for different tc and at a given µ; b) optimal values at

τ ' 0.14 GeV−2 versus µ. The open red circle is the optimal bound at the inflexion point.

〈cGc〉 =mc

π

(log

mc

µ

)〈αsG2〉 −

〈g3G3〉

48π2mc. (37)

These behaviours are in contrast with the ones of the light quark〈qq〉 and mixed quark-gluon 〈qGq〉 condensates [4, 11].• The complete expression of the perturbative NLO pseu-

doscalar spectral function has been obtained in [16] and explic-itly written in [78], where we transform the pole to the runningmasses. We add to this expression the N2LO result in [35] formc = 0. We consider as a source of errors the N3LO con-tribution from an assumed geometric growth of the PT series[37] which mimics the 1/q2-term parametrizing the large orderterms of PT series [38, 39].• The Wilson coefficients of the non-perturbative 〈αsG2〉 and

〈g3G3〉 contributions are also given in [78].• Like in the previous cases, we study the SR versus τ and

tc as shown in Fig. 18. One can note that the non-perturbativecontributions are small (about 1−2 MeV) indicating that the dy-namics of the Bc meson is dominated by the perturbative con-tributions. This feature might explain the success of the non-relativistic potential models for describing the Bc-like hadrons[78]. The optimal result is obtained from tc = 44 GeV2 (be-ginning of τ-stability) until tc = (50 − 60) GeV2 (beginning of

à

à

à

à

ò

òò

ç

ç ç

ç

æ

æ

0 1 2 3 4 5 6 7100

200

300

400

500

M @GeVD

f M@M

eVD

Figure 22: Behaviour of the meson decay constants versus the meson masses: greenopen circle: pseudoscalar π,D, B, Bc ; purple triangle: S U(3) breaking: K,Ds, Bs; bluebox: vector ρ,D∗, B∗, B∗c ; brown full circle: scalar D∗0, B

∗0; light green: axial A1,D1, B1.

Table 2: Pseudoscalar D(s), B(s) meson decay constants. NS2R refers to Non-Standard Sum Rules not using either the minimal QCD continuum ansatz northe exponential / moments weights. The errors of the Global Average arerescaled by 1.2 as in [9] for accounting slight tensions between different de-terminations.

Sources Refs.fD [MeV] fDs [MeV] fDs/ fD204(6) 246(6) 1.21(4) SR [1]203(23) 235(24) 1.15(4) SR [19, 20]≤ 218(2) ≤ 254(2) − SR [1]≤ 279 ≤ 321(12) − SR [20]≤ 230 ≤ 270 − SR [23]201(13) 238(18) 1.15(5) SR [70]208(10) 240(10) 1.15(6) SR [75]195(20) − − HQET-SR [27]204.0(4.6) 243.2(4.9) 1.170(23) SR Average209.2(3.3) 249.3(1.8) 1.187(2) Latt average 2 ⊕ 1 [9, 65]212.1(1.2) 248.8(1.3) 1.172(3) Latt average 2 ⊕ 1 ⊕ 1 [80, 81]211.8(1.1) 249.0(1.1) 1.173(3) Latt final average206(9) 245(16) 1.19(3) NS2R [82]177(21) 205(22) 1.16(3) NS2R [83]211.2(1.1) 249.1(1.1) 1.173(3) Global average203.7(4.7) 257.8(4.1) − Data [9]

fB [MeV] fBs [MeV] fBs/ fB206(7) 234(5) 1.14(3) SR [1]203(23) 236(30) 1.16(5) SR [19, 20]≤ 235(4) ≤ 251(6) − SR [1]197(23) 232(25) − SR [21]194(15) 231(16) 1.19(10) SR [75]207(13) 242(15) 1.17(4) SR [70]199(29) − − HQET-SR [2]206(20) − − HQET-SR [27]204.0(5.1) 234.5(4.4) 1.154(21) SR Average190(4) 227(2) 1.207(12) Latt average 2 ⊕ 1 [9, 65]186(4) 227(4) 1.216(8) Latt average 2 ⊕ 1 ⊕ 1 [84, 85]188(3) 227(2) 1.213(7) Latt final average186(14) 222(12) 1.19(5) NS2R [86]192(15) 228(20) 1.18(2) NS2R [87]191.9(2.5) 228.1(1.8) 1.204(6) Global average196(24) − − Data [9]

tc-stability). We show in Fig. 19 the µ-behaviour of the results,where there is an inflexion point for µ = (7.5±0.5) GeV for theestimate (Fig. 19a) and the upper bound (Fig. 19b). At theseoptimal points, we deduce:

fBc = 436(38)tc (2)αs (2)α3s(7)mc (6)µ = 436(40) MeV ,

fBc ≤ 466(9)αs (2)α3s(12)mc (8)µ = 466(16) MeV . (38)

We may consider these results as a confirmation and improve-ment of the earlier ones in [4, 18, 78, 79]. The results in Eq.(38) will restrict the wide range of fBc given in the current liter-ature and may be used for extracting the CKM angle Vcb fromthe Bc → τντ leptonic width.• We do a similar analysis for the B∗c vector meson where

we replace the strange by the charm quark mass in the B∗s SRexpression 9. We use the rescaled MB∗c = 6350(20) MeV frompotential model [78] using the experimental value of MBc . Thedifferent steps are shown in Fig. 21. We obtain the results inTable 6 for τ ' 0.15 GeV−2.

Summary and Conclusions

•We have reviewed our recent determinations of the heavy-light pseudoscalar [1, 2], vector [3] mesons decay constants

9A complete expression of the spectral function to NLO is given in [13] butit has singularity for mc → 0 and needs to be checked.


Table 3: Scalar D∗0(s), B∗0(s) meson decay constants. Values marked with † are

not included in the average. The one with ∗ has been rescaled by 1.5 for ac-counting a slight tension with our estimate.

Sources Refs.fD∗

0[MeV] fD∗

0s[MeV] fD∗

0s/ fD∗

0122(43)†–360(90)† − − Latt. [76, 77]128(13)†–373(19)† − − SR [73–75]217(24) 202(23) 0.93(2) SR [22]221(12) 202(19) 0.912(23) SR New≤ 243(8) ≤ 222(6) − SR New220(11) 202(15) 0.922(15) SR Average

fB∗0

[MeV] fB∗0s

[MeV] fB∗0s/ fB∗

0263(51) − − SR [4, 18]281(14) 274(20)∗ − SR [75]271(26) 233(21) 0.865(54) SR New≤ 302(8) ≤ 261(17) − SR New278(12) 255(15) − SR Average

Table 4: Vector D∗, B∗ and axial D1, B1 meson decay constants. The n f = 2Lattice and Global Average errors marked with ∗ is rescaled by 1.5 for account-ing slight tensions between different estimates.

Sources Ref.fD∗ [MeV] fD∗/ fD fB∗ [MeV] fB∗/ fB250(11) 1.218(36) 209(8) 1.016(16) SR [3]242(16) 1.20(8) 210(11) 1.02(5) SR [70]263(21) − 213(18) − SR [75]− − − 1.025(16) HQET-SR New≤ 266(8) − ≤ 295(18) − SR [3]≤ 297 − ≤ 261 − SR [70]250(8) 1.215(30) 210(6) 1.020(11) SR Average278(24)∗ 1.208(27) − 1.051(17) Latt. n f = 2 [8]− − 175(6) 0.941(26) Latt. NR [71]− − − 1.018(14) Latt Average252(22) 1.22(8) 182(14) 0.944(21) NS2R [72]253(7) 1.212(20) 192(6)∗ 1.008(12)∗ Global Average

fD1 [MeV] fD1/ fD fB1 [MeV] fB1/ fB363(11) 1.78(1) 385(18) 1.87(6) SR New≤ 388(5) − ≤ 440(16) − SR New333(20) − 335(18) − SR [75]356(16)∗ − 360(19)∗ SR Average

from the standard Laplace and Moment sum rules (SR) intro-duced by SVZ [10]. Motivated by different tensions in the lit-erature, we have completed the review by presenting new de-terminations of the vector fB∗/ fB, scalar fD∗0,B

∗0, axial fD1,B1 and

vector fB∗c meson decay constants. Our results based on stabilitycriteria are summarized in Tables 2–6 and compared with someother recent sum rules and lattice results. Our quoted errors aremainly due to the choice of the QCD continuum threshold andto the estimate of the α3

s PT series. This latter is not consideredin some other SR results.• Then, we have attempted to give a Global Average of sum

rules and lattice results (results from some other approaches cane.g be found in [91]), which can be used for further phenomeno-logical applications. To take into account the slight tensionsamong different determinations, we have rescaled some errors.• One can see in these Tables that there are fair agreements

among estimates from different standard sum rule (SR) ap-proaches. The slight difference is mainly due to a different ap-preciation of the optimal results from the choice of the QCDcontinuum threshold, subtraction constant µ and of the set ofinput QCD parameters. The quoted upper bounds come fromthe positivity of the spectral functions.

Table 5: S U(3) breakings for the vector D∗s and B∗s decay constants.

Sources Ref.fD∗s

[MeV] fD∗s/ fD∗ fB∗s

[MeV] fB∗s/ fB∗

270(29) 1.08(6) 220(14) 1.054(11) SR [3]293(17) 1.21(5) 210(11) 1.20(4) SR [70]308(21) − 255(19) − SR [75]≤ 287(18) − ≤ 317(17) − SR [3]≤ 347 − ≤ 296 − SR [70]290(11) 1.16(4) 221(7) 1.064(10) SR Average306(27) 1.21(6) 214(19) − NS2R [72]311(14)∗ 1.16(6) − − Latt. n f = 2 [8]274(6) − 213(7) − Latt. n f = 2 ⊕ 1 & NR [71, 90]282.5(4.8) 1.17(3) 216.9(4.6) 1.064(11) Global Average

Table 6: Pseudoscalar Bc and vector B∗c mesons decay constants. The n f = 2Lattice error marked by ∗ is rescaled by 1.5. The same for NS2R for accountingthe tension with other estimates.

Sources Ref.fBc [MeV] fB∗c

[MeV] fB∗c/ fBc

436(40) − − − SR [3]≤ 466(16) − − − SR [3]503(171) − − − Pot. Mod. [78]528(29)∗ − − − NS2R [89]427(6) − − − Latt. NR [88]434(23)∗ − 422(20)∗ 0.988(27) Latt. n f = 2 [71]− 453(20) 1.043(45) SR New− ≤ 505(6) − SR New432(6) − 438(14) 1.003(23) Global Average

• A slight tension exists between SR, NS2R and lattice re-sults for fB∗/ fB [72]. To give insight into this problem, we haverederived this ratio in Section 8 from HQET sum rules. Thereis also a tension for fBc from NS2R [89] (see comments in [3]).The N2SR results do not affect significantly the SR average andthey are considered in the Global Average.• Large tensions exist in the scalar sector among various SR

and between lattice results for fD∗0 . Our updated results givenin Table 3 using the running MS charm mass and SR stabilitycriteria confirm our previous findings [22].• The results indicate a good realization of heavy quark sym-

metry for the B and B∗ mesons ( fB ≈ fB∗ ) as expected fromHQET [25] but signal large charm quark mass and radiativeQCD corrections for the D and D∗ mesons ( fD ≈ fB, ..) whichare known since a long time (see e.g among others [17, 93]).• The S U(3) breaking is typically 20% for fDs/ fD ≈ fBs/ fB

and 10% for fD∗s/ fD∗ ≈ fB∗s/ fB∗ which are almost constant asthey behave like ms/ωc [32] where ωc ≡

√tc − Mb ≈ M′B −

MB is independent of Mb. The reverse effect for fD∗s/ fD∗ ≈

fB∗s/ fB∗ < 1 comes mainly from the overall (1 − ms/mQ) factornot compensated by MD∗0s/B∗0s

' MD∗0/B∗0 [32].• It is informative to show the behaviour of our predictions of

the pseudoscalar and vector meson decay constants versus thecorresponding meson masses in Fig. 22. We use fρ = (221.6 ±1.0) MeV from its electronic width [49] and fA1 = 207 MeVfrom SR analysis [4]. One can notice similar MQ behavioursof these couplings where the the 1/

√MQ HQET relations [25,

92] are not satisfied due to large αs and 1/MQ corrections asnoted earlier [17, 93]. One can notice small S U(3) but largeS U(4) breaking for Bc, B∗c. Chiral symmetries are badly brokenbetween the vector D∗, B∗ and axial D1, B1 meson couplings.• We do not found any deviation of these Standard Model

results from the present data.


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