$\overline m_c$ and $\overline m_b$ from $M_{Bc}$ and improved estimates of $f_{Bc}$ and $f_{Bc(2S)}$

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m_c and m_b from M _Bc and improved estimates of

f _Bc and f _Bc(2S)

Stephan Narison

To cite this version:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

m

_c

and

m

_b

from

M

_B

_c

and

improved

estimates

of

f

_B

_c

and

f

_B

_c

₍

_2S

₎

Stephan Narison

LaboratoireParticulesetUniversdeMontpellier,CNRS-IN2P3,Case070,PlaceEugèneBataillon,34095,Montpellier,France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received19December2019

Receivedinrevisedform11January2020 Accepted12January2020

Availableonline15January2020 Editor:J.Hisano

Keywords:

QCDspectralsumrules Perturbativeandnon-perturbative calculations

Hadronandquarkmasses Gluoncondensates

We extract(forthefirsttime) thecorrelated valuesofthe runningmassesmc andmb from MBc using QCDLaplacesumrules(LSR)withinstabilitycriteriawhereperturbative (PT) expressionsatN2LOand non-perturbative(NP)gluoncondensatesatLOareincluded.Allowingthevaluesofmc,b(mc,b)tomove

inside the enlarged rangeofrecent estimates fromcharmonium and bottomiumsum rules (Table 1) obtained using similar stability criteria, we deduce: mc(mc)=1286(16) MeV and mb(mb)=4202(8)

MeV. Combined with previous estimates (Table 2), we deduce a tentative QCD Spectral Sum Rules (QSSR)average: mc(mc)=1266(6) MeVand mb(mb)=4197(8) MeVwherethe errorscomefrom the

precisedeterminationsfrom J/ψ and ϒsumrules.Asaresult,wepresentanimprovedpredictionof

fBc=371(17)MeVandthetentativeupperbound fBc(2S)≤139(6)MeV,whichareusefulforafurther

analysisofBc-decays.

©2020TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Extractions of the perturbative (quark masses,

α

s) and

non-perturbative quark and gluon condensates QCD parameters are very important as they will serve as inputs in different phe-nomenological applications of the (non)-standard model. Lattice calculationsare an useful tool fora such project but alternative analyticalapproachesbasedonQCDfirstprinciples(Chiral pertur-bation, effective theory and QCD spectral sum rules (QSSR)) are usefulcomplementandindependentcheckoftheprevious numer-icalsimulationsastheygiveinsightsforabetterunderstandingof the(non)-perturbativephenomenainsidethehadron“blackbox”.

Inthis note,1 _{we shall use theLaplace version [2–7] of QSSR} introducedby Shifman-Vainshtein-Zakharov(SVZ)[2,3,8–21] fora newextractionoftherunningquark massesmc andmb fromthe

Bc

(

0−+

)

-mesonmass whichwe shall usefor improvingthe

pre-diction on its decay constant fBc done previously using similar

approachesin [22–27] and for deriving a tentative upper bound on fBc(2S).

2. TheQCDLaplacesumrules

•

TheQCDinterpolatingcurrent

We shall be concerned with the following QCD interpolating current:

E-mailaddress:snarison@yahoo.fr.

1 _{Somepreliminaryresultsofthisworkhasbeenpresentedin [1].}



0

|

J5

(

x

)

|

P

 =

fPM2P

:

J5

(

x

)

≡ (

mc

+

mb

)

c

¯

(

i

γ

5

)

b

,

(1)

where: J5

(

x

)

isthelocalheavy-lightpseudoscalarcurrent;mc,bare

renormalized mass ofthe QCD Lagrangian; fP isthe decay

con-stant related to theleptonic widths



[

P

→

l+

ν

l

]

andnormalised

as fπ

=

132 MeV.

•

Formofthesumrules

We shall work with the Finite Energy version of the QCD Laplacesumrules(LSR)andtheirratios:

L

c n

(

τ

,

μ

)

=

tc



(mc+mb)2 dt tne−tτ 1

π

Im

ψ

5

(

t

,

μ

) ,

R

c n

(

τ

)

=

L

c n+1

L

c n

,

(2)

where

τ

is the LSR variable,n is the degree of moments, tc is

the thresholdof the “QCD continuum” whichparametrizes, from the discontinuityof theFeynman diagrams, thespectral function Im

ψ

5

(

t

,

m2Q

,

μ

)

where

ψ

5

(

t

,

m2Q

,

μ

)

isthe(pseudo)scalar

correla-tor:

ψ

5

(

q2

)

=

i



d4x e−iqx



0

|

T

J5

(

x

) (

J5

(

0

))

†

|

0

.

(3)

3. QCDexpressionofthetwo-pointfunction

Usingthe SVZ[2] OperatorProductExpansion(OPE),the two-pointcorrelatorcanbewrittenintheform:

https://doi.org/10.1016/j.physletb.2020.135221

2 S. Narison / Physics Letters B 802 (2020) 135221

ψ

5

(

q2

)

=

∞



(mc+mb)2 dt t

−

q2 1

π

Im

ψ

5

(

t

,

μ

)

|

P T

+ 

α

sG2



CG2

(

q2

,

μ

)

+ 

g3G3



C_G3

(

q2

,

μ

)

+ · · · ,

(4)

where

μ

isthesubtractionscale; Im

ψ

5

(

t

,

μ

)

|

P T isthe

perturba-tivepartofthespectralfunction;C_G2 andC_G2 are(perturbatively)

calculableWilsoncoefficients;



α

sG2



and



g3G3



arethe

non-per-turbative gluon condensate of dimensionsd

=

4

,

6 contributions where: G2

≡

Ga_{μν G}μνa and g3G3

≡

g3fabcGaμν Gνρ,bGμ,

c

ρ .As

explic-itlyshowninRef. [23],CG2 andC_G2 includetheonesofthequark

andmixedquark-gluoncondensatethroughtherelation[2,28,29]:

 ¯

Q Q

 = −

1 12

π

mQ



α

sG2

 −



g3_G3

_

1440

π

2_m3 Q

,

 ¯

Q G Q

 =

mQ

π



logmQ

μ





α

sG2

 −



g3G3



48

π

2_m Q

,

(5)

fromtheheavyquarkmassexpansion.

•

q2

₌

_{0 behaviourofthecorrelator}

ToNLO,theperturbativepartof

ψ

5

(

0

)

reads[6,10,11,30]:

ψ

5

(

0

)

|

P T

=

3 4

π

2

(

mb

+

mc

)



m3_bZb

+

m3cZc



,

(6) with: Zi

=



1

−

logm 2 i

μ

2

 

1

+

10 3 as



+

2aslog2 m2_i

μ

2

,

(7)

wherei

≡

c

,

b;

μ

istheQCDsubtractionconstantandas

≡

α

s

/

π

is

theQCDcoupling.ThisPTcontributionwhichispresentherehas tobeaddedtothewell-knownnon-perturbativecontribution:

ψ

5

(

0

)

|

N P

= −(

mb

+

mc

)

¯

cc

+ ¯

bb

 ,

(8)

for absorbing logn

₍

₋

_m2

i

/

q2

)

mass singularities appearing during

theevaluationofthe PTtwo-point function,atechnicalpointnot oftencarefullydiscussedinsome papers.Workingwith

ψ

5

(

q2

)

is

safeas

ψ

5

(

0

)

,whichdisappears aftersuccessive derivatives,does

not affectthe pseudoscalar sumrule.This is not the caseofthe longitudinal part of the axial-vector two-point function



(_A0)

(

q2

)

builtfromtheaxial-vectorcurrent:

Jμ_A

≡ ¯

c

(

γ

μ

γ

5

)

b

,

(9)

whichisrelatedto

ψ

5

(

q2

)

throughtheWardidentity[6,10,11]:



(_A0)

(

q2

)

=

1 q2

ψ

5

(

q2

)

− ψ

5

(

0

)

,

(10)

andwhichisalsooften(uncorrectly)usedinliterature.

•

LOandNLOPerturbativecontributionatlargeq2

Thecompleteexpressions ofthePT spectralfunction hasbeen obtained to LO in [31], to NLO in [30] and explicitly written in [23].Itreads(i

≡

c

,

b): Im

ψ

5

(

t

)

|

P T

=

3

(

mb

+

mc

)

2 8

π

t q

¯

4_v



1

+

4 3



¯

α

s

π

 

3 8

(

7

−

v 2

₎

+



i=b,c

(

v

+

v−1

) (

L2

(

α

1

α

2

)

−

L2

(

−

α

i

)

−

log

α

1log

β

i

)

+

Ailog

α

i

+

Bilog

β

i



+

O

(

α

2 s

)



(11) where L2

(

x

)

= −

x



0 dy y log

(

1

−

y

)

(12) and Ai

=

3 4 3mi

+

mj mi

+

mj

−

19

+

2v2

+

3v4 32v

−

mi

(

mi

−

mj

)

¯

q2_v

₍

₁

₊

_v

₎



1

+

v

+

2v 1

+

α

i



;

Bi

=

2

+

2 m2_i

−

m2_j

¯

q2_v

;

(13)

α

i

=

mi mj 1

−

v 1

+

v

;

β

i

=



1

+

α

i

(

1

+

v

)

2 4v

¯

q2

=

t

− (

mb

−

mc

)

2

;

v

=



1

−

4mbmc

¯

q2

,

wheremiistheon-shell/polemass.

•

Higher OrdersPerturbativecontributionsatlargeq2

In the absence of a complete result to order

α

2

s, we shall

approximatively use the expression of the spectral function for

mc

=

0: Im

ψ

5

(

t

)

|

N2L OP T



m_b2t 8

π



¯

α

s

π



2 R2s

,

(14)

where R2s has been obtained semi-analytically in [32,33] and is

availableasaMathematicapackageprogramRvs.m.

Weexpectthatitisagoodapproximationbecauseweshallsee that theNLOcontributionsinduce(asexpected) smallcorrections intheratioofmomentsusedtodeterminemc,bduetothepartial

cancellationofthiscontribution.

We estimatethe accuracyofthisapproximationby comparing thisN2LO approximation withtheone obtainedassuming a geo-metricgrowthofthePTcoefficients[34].

We estimate the error dueto the truncation of the PT series fromtheN3LOcontributionestimated,asabove,fromageometric growth of the PT series which is expected to mimic the phe-nomenological1

/

q2 dimension-twocontributionparametrizingthe uncalculatedlargeordertermsofPTseries[35–38].

• 

α

sG2



contributionatlargeq2

We shall use the complete expression of the gluon conden-sate



α

sG2



contributiontothetwo-point correlatorgivenin[23],

whichagreeswithknownresultsformb

=

mc [10,11].TheWilson

coefficient reads: C_G2

(

q2

,

μ

)

=

1

π

∞



(mb+mc)2



−

mbmct



t

−

m2_b

−

mbmc

−

mc2



2

(

Q2

₊

_t

₎

_[

_t

₋₍

_m b

−

mc

)

2

]

3/2

−

√

mbmc



(

mb

+

mc

)

2

[

t

−(

mb

+

mc

)

2

]

16

[

Q2

_{+ (}

_m b

+

mc

)

2

]

2

−

1 16

[

Q2

₊₍

_m b

+

mc

)

2

]

×

(

mb

+

mc

)

2

+

(

5m2 b

+

18mbmc

+

5m2c

)

[

t

− (

mb

+

mc

)

2

]

8mbmc



×

dt

[

t

−(

mb

+

mc

)

2

]

5/2

,

(15)

• 

g3_G3

_

_{contributionatlargeq}2

A similar (but lengthy) expression of the



g3_G3

_

_condensate

contribution can also be obtained from [23], where it has been checked that it agrees with known result for mb

=

mc [29]. It

reads: C_G3

(

q2

)

=

1

π

∞



2 dt

[

t

−

2

_]

9/2



mbmct 6

(

t

+

Q2

₎

_[

_t

_{− (}

_m b

−

mc

)

2

]

7/2

×



3t4

−

2

(

3m2_b

+

2mbmc

+

3m2c

)

t3

+(

5m3_bmc

+

18m2bm2c

+

5mbm3c

)

t2

+2

(

3m6_b

+

m5_bmc

−

6m4bm2c

−

6m3bm3c

−

6mb2m4c

+

mbm5c

+

3m6c

)

t

−

3

(

m8_b

+

m7_bmc

−

m5bm3c

−

2mb4m4c

−

mb3m5c

+

mbm7c

+

mc8

)



−

√

mbmc



−

7

4

(

t

−

2

)

3 192

(

Q2

₊

2

₎

4

+



7

2

(

t

−

2

)

2 192

+

A

(

t

−

2

)

3 1536mbmc



2

(

Q2

₊

2

₎

3

+



−

7

4

(

t

−

2

₎

192

−

2A

(

t

−

2

₎

2 1536mbmc

+

B

(

t

−

2

₎

3 24576m2 bm2c



×

1

(

Q2

₊

2

₎

2

+



7

4 192

+

2A

(

t

−

2

)

1536mbmc

−

B

(

t

−

2

)

2 24576m_b2m2c

−

C

(

t

−

2

)

3 196608m3_bm3c



×

1 Q2

₊

2

 

,

with:

=

mb

+

mc A

=

51m_b2

+

166mbmc

+

51m2c B

=

31m4_b

−

836m3_bmc

−

1862mb2m2c

−

836mbm3c

+

31m4c C

=

277m4_b

+

596m_b3mc

−

514m2bm2c

+

596mbm3c

+

277m4c

.

(16)

•

FromOn-shelltotheM S-scheme

We transform the pole masses mQ to the running masses

mQ

(

μ

)

using theknown relationin the M S-scheme to order

α

2s

[39–47]: mQ

=

mQ

(

μ

)

1

+

4 3as

+ (

16

.

2163

−

1

.

0414nl

)

a 2 s

+

ln

μ

2 m2_Q



as

+ (

8

.

8472

−

0

.

3611nl

)

a2s



+

ln2

μ

2 m2_Q

(

1

.

7917

−

0

.

0833nl

)

a 2 s

...



,

(17)

fornl

=

3 lightflavours.Inthefollowing,weshallusenf

=

5 total

numberofflavoursforthenumericalvalueof

α

s. 4. QCDinputparameters

TheQCDparameterswhichshallappearinthefollowing anal-ysiswillbe the charmandbottom quark massesmc,b,the gluon

condensates



α

sG2



and



g3G3



.

Table 1

QCD inputparametersfromrecentQSSRanalysisbased onstability criteria.The valuesofmc(mc)andmb(mb)comefromrecentmomentsSRandtheirratios[48]

wheretheerrorshavebeenmultipliedbyafactor2tobeconservative. Parameters Values Sources Ref.

αs(MZ) 0.1181(16)(3) Mχ0c,b−Mη_c,b Ratios of LSR [49]

mc(mc) 1264(12)MeV J/ψfamily Mom. [48]

mb(mb) 4188(16)MeV ϒfamily Mom. [48]

αsG2 (6.35±0.35)×10−2GeV4 Hadrons QSSR average [49]

g3_G3_{ (8.2±2.0)GeV}2_{× α}

sG2 J/ψfamily Mom. [50,51]

Ratios of LSR [52]

•

QCDcoupling

α

s

WeshallusefromtheMχ0c

−

Mηc mass-splittingsumrule[49]:

α

s

(

2

.

85

)

=

0

.

262

(

9

)

;

α

s

(

Mτ

)

=

0

.

318

(

15

)

;

α

s

(

MZ

)

=

0

.

1183

(

19

)(

3

)

(18)

whichismoreprecisethantheonefromMχ0b

−

Mηb [49]:

α

s

(

9

.

50

)

=

0

.

180

(

8

)

;

α

s

(

Mτ

)

=

0

.

312

(

27

)

;

α

s

(

MZ

)

=

0

.

1175

(

32

)(

3

).

(19)

TheseleadtothemeanvaluequotedinTable1,whichisin com-pleteagreementwiththeworldaverage[53]:

α

s

(

MZ

)

=

0

.

1181

(

11

),

(20)

butwithalargererror.

•

c andb quarkmasses

For the c and b quarks, we shall use the recent

determina-tions [48] of therunning massesandthe corresponding value of

α

sevaluatedatthescale

μ

obtainedusingthesamesumrule

ap-proach from charmonium and bottomium systems. The range of valuesgiveninTable1enlargedbyafactor2are withinthePDG average[53].

•

Gluonandquark-gluonmixedcondensates

Forthe



α

sG2



condensate,weusetherecentestimateobtained

fromacorrelationwiththevaluesoftheheavyquark massesand

α

s whichcanbe compared withthe QSSRaveragefromdifferent

channels[49].

Theoneof



g3G3



comesfromaQSSRanalysisofcharmonium systems.TheirvaluesaregiveninTable1.

5. Parametrisationofthespectralfunction

–Inthepresentcase,wherenocompletedataontheBc

spec-tralfunctionareavailable,weusethedualityansatz:

Im

ψ

5

(

t

)



f2PM4P

δ(

t

−

M2P

)

+ (

t

−

tc

)

“Q C D continuum”

,

(21)

forparametrizingthespectralfunction. MP and fP arethelowest

groundstatemassandcouplinganalogueto fπ .The“QCD contin-uum”is theimaginarypart oftheQCD two-point function while

tc isitsthreshold.Withinasuchparametrization,oneobtains:

R

c

n

≡

R



M2P

,

(22)

indicating that the ratioof moments appears to be a usefultool forextractingthemassofthehadrongroundstate[10–14].

– This simple model has been tested in different channels where complete dataare available (charmonium, bottomium and

4 S. Narison / Physics Letters B 802 (2020) 135221

model,thesumrulereproduces well theone usingthecomplete data,whilethemassesofthelowestgroundstatemesons( J

/ψ,

ϒ

and

ρ

) havebeenpredictedwithagoodaccuracy.Intheextreme caseoftheGoldstonepion,the sumrule usingthespectral func-tionparametrizedbythissimplemodel[10,11] andthemore com-plete oneby ChPT[54] leadto similarvalues ofthesumoflight quark masses

(

mu

+

md

)

indicating the efficiency of this simple

parametrization.

–An eventual violationof thequark-hadron duality (DV)[55, 56] has been frequently tested in the accurate determination of

α

s

(

τ

)

fromhadronic

τ

-decaydata[56–58], whereitsquantitative

effectinthespectralfunctionwasfoundtobelessthan1%. Typi-cally,theDVhastheform:



Im

ψ

5

(

t

)

∼ (

mc

+

mb

)

2t e−κtsin

(

α

+ β

t

)θ (

t

−

tc

) ,

(23)

where

κ

,

α

,

β

are model-dependent fitted parameters but not basedfrom first principles. Withinthismodel, wherethe contri-bution isdoubly exponential suppressedin theLaplace sumrule analysis, we expect that in the stability regions where the QCD continuumcontributiontothesumruleisminimalandwherethe optimalresultsinthispaperwillbeextracted, suchduality viola-tionscanbesafelyneglected.

– Therefore,we (apriori) expect that one can extract witha goodaccuracythec andb runningquarkmassesandtheBc decay

constantwithintheapproach.Aneventualimprovementofthe re-sultscan bedone after amore completemeasurement ofthe Bc

pseudoscalar spectral function which is not an easy task though therecentdiscovery byCMS[59] ofthe Bc

(

2S

)

state at6872(1.5)

MeVisagoodstartingpointinthisdirection.

–Inthefollowing,inordertominimizetheeffectsofunknown higherradialexcitationssmearedbytheQCDcontinuumandsome eventual quark-duality violations,we shallwork withthe lowest ratioofmoments

R

c₀forextractingthequarkmassesandwiththe lowestmoment

L

c₀forestimatingthedecayconstant fBc.Moment

withnegative n willnot beconsideredduetotheir sensitivityon thenon-perturbativecontributionssuchas

ψ

5

(

0

)

.

6. Optimizationcriteria

For extractingthe optimal results fromthe analysis, we shall useoptimizationcriteria(minimumsensitivity)oftheobservables versus the variation of the external variables namelythe

τ

sum ruleparameter,theQCD continuumthresholdtc andthe

subtrac-tionpoint

μ

.

Results based on these criteria have led to successful predic-tions inthe current literature [10,11].

τ

-stability hasbeen intro-ducedand testedby Bell-Bertlmann usingthe toy model of har-monic oscillator [9] and applied successfully inthe heavy [4,5,9, 60–68] and light quarks systems [2,3,10–14,69]. It has been ex-tended later on to thetc-stability [10–13] andto the

μ

-stability

criteria[27,49,63,69,70].

Stability onthe numbern ofheavy quark moments have also beenused[48,50–52].

One cannotice inthe previous worksthat these criteriahave ledto moresolidtheoreticalbasisandnoticeableimprovedresults fromthesumruleanalysis.

7. mcandmbfromMBc

Inthefollowing,welookforthestabilityregionsoftheexternal parameters

τ

,

tcand

μ

whereweshallextractouroptimalresult.

•

τ

stability

Fig. 1. MBc asfunctionofτfordifferentvaluesoftc,forμ=7.5GeVandforgiven

valuesofmc,b(mc,b)inTable1.

Fig. 2. MBcasfunctionoftcforμ=7.5 GeVandfortherangeofτ-stabilityvalues.

Weusethecentralvaluesofmc,b(mc,b)giveninTable1.

Ina firststep,fixingthevalue of

μ

=

7

.

5 GeVwhichweshall justifylaterandwhichisthecentralvalueof

μ

= (

7

.

5

±

0

.

5

)

GeV obtainedin [27],weshowinFig.1thebehaviourofMBc for

differ-entvaluesoftcwherethecentralvaluesofmc

(

mc

)

=1264MeVand

mb

(

mb

)

=4188 MeV giveninTable1havebeen used.Weseethat

the inflexion points at

τ

 (

0

.

30

∼

0

.

32

)

GeV−2 appear fortc

≥

52 GeV2. The smallest value of

√

tc is around the Bc

(

2S

)

mass

of 6872(1.5) MeV recently discovered by CMS[59] but does not necessarilycoïncide withit asthe QCDcontinuum isexpectedto smear all higherstatescontributions to thespectral function. In-stead,itsvalueisrelatedbydualitytothegroundstateparameters asdiscussedin[71] fromaFESRanalysisofthe

ρ

-mesonchannel.

•

tcstability

WeshowinFig.2thebehaviourofMBc versustc whichisvery

stable. For definiteness, we take tc in the range 52 to 79 GeV2

wherewehaveaslightmaximumattc



60 GeV2.Atthisrangeof

tc values,onecaneasily checkthattheQCD continuum

contribu-tion tothesumruleis(almost) negligible.Tohavemoreinsights onthiscontribution,weshowinFig.3theratioofthecontinuum overthelowestgroundstatecontributionaspredictedbyQCD:

rc

≡



_∞ tc dte −tτ

_

_Im

_ψ

cont 5



tc (mc+mb)2dte −tτ

_

_Im

_ψ

Bc 5 (24)

whereonecanindeedseethattheQCDcontinuumtothemoment sumrule

L

c₀ isnegligibleinthisrangeoftc values.This

Fig. 3. Ratiorcofthecontinuumoverthelowestgroundstatecontributionas

func-tionoftc at thecorrespondingτ-stabilitypoints forμ=7.5 GeVandfor given

valuesofmc,b(mc,b)inTable1.Thedashed-dottedlineisthecontributionfora

VectorMesonDominance(VDM)assumptionofthespectralfunction.

Fig. 4. mc(mc)as functionofμ forτ0.32 GeV2 and forthe centralvalue of

mb(mb)giveninTable1.

Givene.g.the centralvalueofmb

(

mb

)

=

4188 MeVfromTable1

andusing

τ

= .

32 GeV−2 _andt

c

=

60 GeV2,weshowinFig.4the

correlated values of mc

(

mc

)

at different values of

μ

needed for

reproducingMBc.Weobtainaninflexionpointat:

μ

= (

7

.

5

±

0

.

1

)

GeV

,

(25)

which we shall use in the following. Thisvalue agrees with the one

μ

= (

7

.

5

±

0

.

5

)

GeVquotedin[27] usingdifferentways.

•

Extractingtheset

(

mc

,

mb

)

Inthefollowing,we studythecorrelationbetweenmc andmb

neededforreproducingtheexperimentalmass[53]:

Mexp_B

c

=

6275

.

6

(

1

.

1

)

MeV

,

(26)

fromthe ratio

R

c₀ of Laplace sum rules defined in Eqs. (2) and (22).

–Allowingmc

(

mc

)

tomoveintherange:

mc

(

mc

)

 (

1252

−

1282

)

MeV (27)

fromthe J

/ψ

and Mχ0c−Mηc mass-splitting sumrules, we show

in Fig. 5, the predictions for MBc as a function of mb

(

mb

)

. The

bandistheerrorinducedbythechoiceofthestabilitypoints

τ

=

(

0

.

30

−

0

.

32

)

GeV−2 whichisabout(12-13) MeV,whiletheerror duetosomeotherparametersarenegligible.Then,wededuce:

mb

(

mb

)

= (

4195

−

4245

)

MeV

,

(28)

whichleadstothecorrelatedsetofvaluesinunitsofMeV:

[

mb

(

mb

),

mc

(

mc

)

] = [

4195

,

1282

] ... [

4245

,

1252

] .

(29)

Fig. 5. MBc asfunctionofmb(mb)fordifferentvaluesofmc(mc),forμ=7.5 GeV

andfortherangeofτ-stabilityvaluesτ= (0.30−0.32)GeV−2_.

Thisresultshowsthatasmallvalueofmc iscorrelatedtoalarge

valueofmb andvice-versa.

– Scrutinizing Fig. 5, one can see that, atfixed mb, e.g. 4220

MeV, MBc increaseswiththemc valuesgiveninthelegend

(verti-cal line),asintuitivelyexpected. Ontheother,fixing thevalue of

mc attheoneinthelegendsay1282MeV,onecan see

(straight-linewithapositive slope)that MBc increaseswhenmb increases

onthemb-axisasalsointuitivelyexpected.

–Consideringthatthevaluesofmb

(

mb

)

areinsidetherange:

mb

(

mb

)

= (

4176

−

4209

)

MeV

,

(30)

allowedfromthe

ϒ

sumrulesasgiveninTable2,wecandeduce fromFig.5strongerconstraintsonmb

(

mb

)

:

mb

(

mb

)

= (

4195

−

4209

)

MeV

=

4202

(

7

)

MeV

,

(31)

wherethe errorissimilar to theaccurate value fromthe

ϒ

sum ruleinTable2.Thisisduetothesmallintersectionregionofthe resultsfromthe J

/ψ,

ϒ

andtheBc sumrules.Withthisrangeof

values,wededuce:

mc

(

mc

)

=

1286

(

8

)

f ig.4

(

14

)

τ

(

1

)

tc MeV

,

=

1286

(

16

)

MeV

,

(32)

wheretheerrorsduetosomeotherparametersandtothe trunca-tionofthePTseriesarenegligible.Thesusbscript f ig

.

4 indicates that theerrorcomesfromtheintersectionregion betweenthe

ϒ

andBc sumrulesinFig.5.

We consider thevalues inEqs. (31) and (32) as ourfinal de-terminationsofmb

(

mb

)

andmc

(

mc

)

fromMBc andcombined

con-straintsfromthe J

/ψ

and

ϒ

sumrules.

•

Comments

–Onecannoticethat theeffectofthePTradiativecorrections are quite smallin theratio ofmoments becausethe onesof the absolutemoments

L

0,1 tendtocompensate eachothers.Thisfact

can be checkedfrom a numericalparametrization ofthe LSR ra-tio.Attheoptimizationscale

τ



0

.

32 GeV−2and

μ

=

7

.

5 GeV,it reads(as

≡

α

s

/

π

):

√

R

0

|

N2L OP T



√

R

0

|

L OP T

(

1

−

0

.

16as

−

0

.

42a2s

) ,

(33)

whiletheLSRlowestmomentis:

√

L

0

|

N2L OP T



√

L

0

|

L OP T

(

1

+

6as

+

26

.

4a2s

) .

(34)

6 S. Narison / Physics Letters B 802 (2020) 135221

Table 2

Valuesofmc(mc)andmb(mb)comingfromourrecentQSSRanalysisbasedon

sta-bilitycriteria.Someotherdeterminationscanbefoundin[53]. Parameters Values [MeV] Sources Ref.

mc(mc) 1256(30) J/ψfamily Ratios of LSR17 [49]

1266(16) Mχ0c−Mηc Ratios of LSR [49]

1264(6) J/ψfamily MOM & Ratios of MOM [48] 1286(66) MD Ratios of LSR [70]

1286(16) MBc Ratios of LSR (This work)

1266(6) Average This work mb(mb) 4192(17) ϒfamily Ratios of LSR [49]

4188(8) ϒfamily MOM & Ratios of MOM [48] 4236(69) MB Ratios of MOM & of LSR [70]

4213(59) MB Ratio of HQET-LSR [63]

4202(7) MBc Ratios of LSR (This work)

4196(8) Average This work

theone 36a2_s whichone wouldobtain using a geometric growth oftheas PT coefficients[36].Therefore,theerrorinduced by the

differenceofthetwoN2LOapproximationsisnegligible.

–Thecontributionofthegluoncondensatesisalsosmallatthe optimizationscaleas



α

sG2



increasesMBc byabout5MeVwhile



g3_G3

_

_{decreases itby 1MeV. Thesecontributions are smalland}

alsoshow thegoodconvergenceofthe OPE.Then, itsinduces an increaseofabout6MeV inthequark massvaluesandintroduces anegligible errorof1MeV.However,thenon-perturbative contri-butionsareimportantforhavingthe

τ

-stabilityregion.

– As the QCD continuum contribution which is expected to smearall radial excitation contributions is negligible atthe opti-mizationregionduetotheexponentialdumpingfactorofthesum rule, we expect that some eventual DV discussed previously can be safely neglected dueto its doubly exponential suppression in theLSR analysis. We alsoexpect that the effectsof higherradial excitationscanbesimilarlyneglectedliketheoneoftheQCD con-tinuum.

8. ComparisonwithsomeotherQSSRdeterminations

Wecomparetheprevious resultswiththeonesinTable2 ob-tained from some other QSSR analysis using the same stability criteria. A tentative average of the central values and using the errorfromthemostprecise predictionsfrom J

/ψ

and

ϒ

families leadstotheaveragesquotedinTable2.

9. Revisiting fBc

Usingthepreviouscorrelatedvaluesof

(

mc

,

mb

)

,wereconsider

theestimateof fBc recentlydoneinRef. [27].

•

τ

andtcstabilities

Weshowthe

τ

-behaviourof fBc inFig.6fordifferentvaluesof tc for

μ

=

7

.

5 GeV andfor

[

mc

(

mc

),

mb

(

mb

)

]

= [

1264

,

4188

]

MeV

fromTable1.We start tohave

τ

-stability (minimum)fortheset [tc

,

τ

]=[47GeV2,0.22GeV−2]andtc-stabilityfortheset[60 GeV2,

(

0

.

30

−

0

.

32

)

GeV−2_{].Onecannoticethatthe}

_τ

_{-stabilitystarts}

ear-lierforsmallertc-value thantothecaseoftheratioofmoments

usedin thepreceding sections.Tobe conservative, weshall con-siderthevalueof fBc obtainedinsidethislargerrangeoftc-values

andtakeasafinal valueits mean.In thisrange,thevalue of fBc

increasesbyabout15MeV.

•

μ

stability

Westudytheinfluenceof

μ

on fBc inFig.7giventhevaluesof

τ

andmb,c.Weseeaclearstabilityfor

μ

= (

7

.

2

−

7

.

5

)

GeVwhich

isconsistent withtheone forMBc andwiththeone obtainedin

[27] indicatingtheself-consistency oftheanalysis.

Fig. 6. fBc versusτ givenμ=7.5 GeVand[mc(mc),mb(mb)]= [1264,4188]MeV

fromTable1.

Fig. 7. fBc as function ofμ for τ0.31 GeV−2 and for the central value of

mc,b(mc,b)giveninEqs. (31) and(32).

•

Higherorders(HO)PTcorrections

– The N2LO contribution increases the prediction fromLO

⊕

NLOby24MeV.Weestimatetheerrorinducedbyusingtheresult atmc

=

0 bycomparingitwiththeoneobtainedfromtheestimate

oftheN2LOcoefficient usingageometricgrowthofthePTseries (see Eq. (34)). The difference induces an error of

±

9

.

6a2s which

correspondsto

±

9 MeV.

– Weestimate the errorduetothe uncalculated higherorder (HO) partof thePT seriesfromthe N3LO contributionestimated byusingthegeometricgrowthofthecoefficientsgiveninthe nu-merical expression in Eq. (34) which is

±

158a3_s. Itintroduces an errorofabout10MeV.

•

Gluoncondensatecontributions

The inclusion of the



α

sG2



condensate increases the sum of

thePT contributionsby 3MeV,whiletheinclusionofthe



g3_G3

_

decreases the prediction by 1 MeV. These contributions and the inducederrorarenegligible.

•

Result

–TheresultoftheanalysisinunitsofMeVis:

fBc

=

368

(

1

)

τ

(

8

)

tc

(

7

)

mc

(

5

)

mb

(

1

)

αs

(

1

)

μ9

)

N2L O

(

10

)

H O

=

368

(

18

)

MeV

,

(35)

ifoneusesthemassvaluesobtainedinEqs. (31) and(32),whileit is:

fBc

=

381

(

1

)

τ

(

8

)

tc

(

3

)

mc

(

5

)

mb

(

1

)

αs

(

1

)

μ

(

9

)

N2L O

(

10

)

H O

ifone usesthetentativemassaverages giveninTable 2.We take asafinalresultthemeanofthetwodeterminations:

fBc

=

371

(

17

)

MeV

.

(37)

Thisresultimprovesthepreviousone fBc

=

436

(

40

)

MeVobtained

recentlyin Ref. [27] and the earlierresults in [22,24,25]. It con-firms and improves the one fBc

=

388

(

29

)

MeV averaged from

moments and LSR in [23] where the values of the pole masses havebeenused.However,itdisagreeswithsomeresultsincluding the lattice one reviewed in Table 3 of [27]. New estimates from thelattice approachisneededforclarifyingthe issue.Comments relatedtosomeofthepreviousworkshavebeenalreadyaddressed in [27] andcanbeconsultedthere.

10. Attemptedupperboundfor fBc(2S)

We attempt to give an upper bound to fBc(2S) by using a

“two resonances +QCD continuum”parametrization ofthe spec-tralfunction. However,we are aware onthe factthat dueto the exponential suppression of the Bc

(

2S

)

contribution compared to

Bc

(

1S

)

and of its eventual smaller couplingas expected for the

observed radial excitations in some other channels, we may not extractwithagoodprecisionthe Bc

(

2S

)

decayconstantfromthis

approach. Instead by using the positivity of the QCD continuum contributionfortc

≥

47 GeV2 justabove M2Bc(2S),one obtainsthe

semi-analyticexpressionfrom

L

c₀:

ρ

Bc

≡



fBc(2S) fBc



2



_M Bc(2S) MBc



4 e−  M2_Bc(2S)−M2_Bcτ

≤

3

.

6%

,

(38)

atthe

τ

-stability of about 0

.

22 GeV−2 as can be deduced from Fig.3.Usingthepreviousvalueof fBc inEq. (37),wededuce:

fBc(2S)

≤ (

139

±

6

)

MeV

,

(39)

indicatingthat the radial excitation couplesweaker to the corre-sponding quark current than the ground state meson. This fea-turehasbeenalreadyobservedexperimentallyinthecaseoflight (

π

,

ρ

)andheavy(

ψ,

ϒ

)mesons.

11. Summaryandconclusions

•

mcandmb

We have used QCD Laplace sum rules to estimate (for the firsttime) the correlated values ofmc

(

mc

)

and mb

(

mb

)

from the

Bc-mesonmassallowingthemtomoveinsidetheextended

(mul-tipliedbyafactor2:seeFig.5andTable1)rangeofvaluesallowed bycharmoniumandbottomiumsumrules.Thesevalues:

mb

(

mb

)

=

4202

(

7

)

MeV

(

Eq. (31)

),

mc

(

mc

)

=

1286

(

16

)

MeV

(

Eq. (32)

),

agreewithpreviousrecentonesfromcharmoniumandbottomium systemsquotedinTable2.Theerrorsaresimilartotheonesfrom

J

/ψ

and

ϒ

sum rules. They have been relatively reduced com-paredtotheonesfromthe D andB mesonmassesthankstothe extraconstraintson therange ofvariations ofmc,b

(

mc,b

)

usedin

Fig.5from J

/ψ

and

ϒ

sumrules.Usingthesevaluesandtheones fromrecentdifferentQSSRdeterminationscollectedinTable2,we deducetheQSSRaverage:

mc

(

mc

)

=

1266

(

6

)

MeV and mb

(

mb

)

=

4196

(

8

)

MeV

,

wheretheerrorcomesfromthemostaccuratedeterminations.

•

Decayconstants fBcandfBc(2S)

Using the new results in Eqs. (31) and (32), we improve our previous predictionsof fBc [22,23,27] whichbecomes more

accu-ratedueto the inclusionof HO PT correctionsandto theuse of modernvaluesoftheQCDinputparameters:

fBc

=

371

(

17

)

MeV

(

Eq. (37)

)

AnupperboundfortheBc

(

2S

)

decayconstantisalsoderived:

fBc

(

2S

)

≤ (

139

±

6

)

MeV

(

Eq. (39)

) .

Thesenewresultswillbe usefulforfurther phenomenological analysis.

Improvement ofour results requiresa complete evaluation of the spectral function at N2LO and a future measurement of the

Bc

(

2S

)

decayconstant.Weplantoextendtheanalysisinthis

pa-pertosomeother Bc-likemesonsinafuturework. References

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