Table Of ContentBottom Mass from Nonrelativistic Sum Rules at
NNLL∗
3
1
0
2 Maximilian Stahlhofen†
n DESYTheoryGroup,Notkestraße85,D-22607Hamburg,Germany
a E-mail: [email protected]
J
4
2
We report on a recent determination of the bottom quark mass from nonrelativistic (large-n) ϒ
] sumruleswithrenormalizationgroupimprovement(RGI)atnext-to-next-to-leadinglogarithmic
h
p (NNLL)order. Thecomparisontopreviousfixed-orderanalysesshowsthattheRGIcomputedin
- thevNRQCDframeworkleadstoasubstantialstabilizationofthetheoreticalsumrulemoments
p
e withrespecttoscalevariations. Asinglemomentfit(n=10)totheavailableexperimentaldata
h
yieldsM1S=4.755±0.057 ±0.009 ±0.003 GeVforthebottom1Smassandm (m )=
[ b pert αs exp b b
4.235±0.055 ±0.003 GeVforthebottomMSmass. Thequoteduncertaintiesrefertothe
3 pert exp
v perturbativeerrorandtheuncertaintiesassociatedwiththestrongcouplingandtheexperimental
3
input.
9
4
3
.
1
0
3
1
:
v
i
X
r
a
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗preprintDESY13-008,UWThPh-2013-3
†Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
BottomMassfromNonrelativisticSumRulesatNNLL MaximilianStahlhofen
1. Introduction
Determinationsofthebottomquarkmassm havebeenthesubjectofalargenumberofQCD
b
precisionstudiesinthepast. ForasummarywerefertoRef.[1]. Thebottommassisanimportant
parameterinnumeroustheoreticalpredictionsnotonlywithin,butalsobeyondthestandardmodel.
The data from e+e− collisions is the common experimental input in many determinations of
m ,becauseinparticulartheregionclosetothebb¯ thresholdandtheϒresonancesofthetotalcross
b
sectionareverysensitivetothebottommassparameter. Oneclassicapproachisbasedonthesum
rule[2]thatstatestheequalityoftheexperimentalmoment
(cid:90) ∞ ds
Pnexp = sn+1Rbb¯(s), (1.1)
0
where Rbb¯ = σ(e+e− → bb¯+X)/σpt is the measured inclusive (hadronic) bottom pair produc-
tioncrosssectionnormalizedbyσ =4πα2/3s,andthecorrespondingtheoreticalexpressionPth
pt n
obtained from an operator product expansion (OPE) in QCD. For not too large n nonperturbative
powercorrectionstoPtharesuppressedandthetheoreticalpredictionisdominatedbytheperturba-
n
tiveQCD(pQCD)resultforanexternalbottomquarkpair. Concerningtheappropriatetheoretical
formalism bottom mass determinations from the equation Pth(m )=Pexp differ depending on the
n b n
valuesforn. Wedistinguishtwoclasses.
For n< 3 the theoretical moment Pth is governed by fluctuations at the scale m . Therefore
∼ n b
higherordertermsintheOPEtypicallyscalelikepowersofΛ /m andtheconventionalpQCD
QCD b
resultisinprinciplesufficientforaprecisedeterminationofthebottommass. Recentlow-nanaly-
ses[3,4]employanapproximatefour-looppQCDcalculation. Ref.[5]usesarelatedvariantofthis
method,wheretheintegrationinEq.(1.1)isonlycarriedoutoverafiniterangeandacompensating
term (according to Cauchy’s theorem) is added to the theory prediction. Both low-n approaches
have the drawback that precise experimental data is currently only available in the region close to
the production threshold (and for the ϒ resonances) and this deficiency has to be compensated by
additional theory input in one way or the other1, see e.g. Refs. [6, 4] for discussions on the cor-
respondinguncertainties. Powercorrectionstotheadditionaltheorycontribution2 inthemoments
arecommonlyassumedtobesmall.
On the other hand large-n moments, where 4<n<10, receive only negligible contributions
∼ ∼
from the energy regions beyond threshold and are dominated by the experimentally well-known
ϒ-resonances and hence nonrelativistic bound state dynamics. The corresponding sum rules are
thereforeoftencallednonrelativisticorϒsumrules. Duetothenonrelativisticnatureofthelarge-n
√
momentsinadditiontothehardscalem thesoftscalem / nandtheultrasoftscalem /nemerge
b b b
as relevant short-distance scales and the convergence of the OPE requires the upper limit n<10.
∼
√
The hierarchy between these scales induces sizable terms ∝(α n)k, the so-called Coulomb sin-
s
gularities,andlargelogarithms∝(α ln(n))l intheperturbativeloopexpansion. Theresummation
s
1In fact the finite energy sum rule used in Ref. [5] is equivalent to an infinite energy sum rule, if in the range
abovethefiniteenergylimitthetheoryresultisusedforthetotalcrosssectionRbb¯ inEq.(1.1)andtheinfinitemoment
integrationconverges.
2relatedtothefactthattheenergyintegrationcontourneedstobedeformedontothepositiverealaxisclosetothe
finiteenergycutoff
2
BottomMassfromNonrelativisticSumRulesatNNLL MaximilianStahlhofen
of the Coulomb singular terms to all orders can be performed within the effective field theory
NRQCD [7, 8]. Extensions of this framework like the pNRQCD [9, 10] and the vNRQCD [11]
formalism also allow the systematic resummation of the logarithmic terms. The renormalization
group improved (RGI) result for the theoretical large-n moments is expressed as a simultaneous
√
expansioninα and1/ nandschematicallytakestheform
s
√ (cid:104) √ √ (cid:105)
P ∼ ∑(α n)k(α ln(n))l 1(LL);α ,1/ n(NLL);α2,α / n,1/n(NNLL);... (1.2)
n s s s s s
k,l
for the leading logarithmic (LL), next-to leading logarithmic (NLL) and next-to-next-to leading
logarithmic (NNLL) order. Prior to the work presented here the RGI bottom mass determination
fromlarge-nsumrulesreachedNLLandpartlyNNLLlevel[12,13],butdidnotincludetheNNLL
runningofthedominantheavyquarkpairproductioncurrent. Earlierfixed-orderanalyses[14,15,
16,17,18,19] up tonext-to-next-to-leading order (NNLO) only resumthe Coulomb singularities
and count (α ln(n))l as αl in Eq. (1.2). The convergence of the fixed-order expansion however
s s
turned out to be rather unsatisfactory, see Ref. [20] for a review. As we will show below RGI
computationsimprovetheconvergencepropertiessubstantiallyandallowforareliableandprecise
determinationofthebottomquarkmassfromnonrelativisticsumrules.
Foranydeterminationofthebottommassparameterwithaprecisionatthepercentlevelitis
mandatorytoadoptanappropriateshort-distancemass-schemeinordertoavoidO(Λ )infrared
QCD
renormalon ambiguities. Suitable mass schemes are the MS scheme for the low-n sum rules and
so-calledthresholdmass-schemes[21,22,23,24]forthelarge-nsumrules.
The present talk focuses on the determination of the 1S bottom mass [22] from RGI large-n
sumrulesandismostlybasedontherecentlypublishedRef.[25]. Inthisanalysissinglemoment
fits of the mass parameter are carried out including for the first time the almost complete NNLL
correction to the theoretical moments. The still missing contribution from the NNLL soft mix-
ing correction to the running of the heavy quark production current can be neglected under the
assumptionthatitssizeiscomparabletothealreadyknownsoftNNLLterms.
2. ExperimentalMoments
Thedominantcontribution(87%-98%forn=6-12)totheexperimentalmomentsinEq.(1.1)
for large n comes from the first four ϒ resonances, ϒ(1S)-ϒ(4S), which we construct from their
electromagneticdecaywidthsandmasses[1]usingthenarrowwidthapproximation. Forthecon-
tribution from the threshold region (5.7% - 1.4% for n=6-12) we use BABAR data [26] in the
√ √
energy range between s=10.62 and s=11.21 and follow the approach of Ref. [27]. Finally
the continuum region above 11.21 GeV contributes only a tiny fraction. It can be modelled by
the respective pQCD result [28] assigning a model uncertainty of 10% to the cross section with-
outintroducinganumericallyrelevanterrortotheexperimentalmoments. Weemphasizethatthis
continuumcontributionshouldberegardedasaroughestimateforthe(missing)experimentaldata
ratherthananadditionaltheoryinput. Theprecisenumbersfortherelevantexperimentalmoments
togetherwiththeirstatisticalandsystematicalerrorscanbefoundinRef.[25].
3
BottomMassfromNonrelativisticSumRulesatNNLL MaximilianStahlhofen
3. TheoreticalMomentsatNNLL
For details on the derivation of the theoretical moments according to the scheme in Eq. (1.2)
we refer to Refs. [14, 15, 18, 25]. The resummation of nonrelativistic logarithms follows the vN-
RQCDapproach[25]. Hereweshallonlydiscussthegeneralstructureandthelatestcomputational
progressconcerningtheRGIofthetheoreticallarge-nmoments. Thetheorypredictionforthenor-
malized total bb¯ pair production cross section Rbb¯(s) in the threshold region is due to the optical
theoremrelatedtononrelativisticcurrent-currentcorrelators,whichdescribetheproductionandan-
nihilationofaheavyquarkpair. Explicitresultsforthesecorrelatorscanbeadoptedfromprevious
worksontt¯thresholdproductionine+e− collisions[29,30,31,32]. Afterthemomentintegration
overstheresultforthen-thmomentthroughNNLLordercanbeexpressedas
√
3N Q2 π (cid:26) (cid:27)
Pth,NNLL= c b c (h,ν)2ρ (h,ν)+2c (h,ν)c (h,ν)ρ (h,ν) , (3.1)
n 4n+1(Mpole)2nn3/2 1 n,1 1 2 n,2
b
where the ρ arise from the integration of the nonrelativistic current correlators and the c are
n,i i
Wilson coefficients of the respective effective currents. The variables h and ν are introduced to
parametrizethematchingandrenormalizationscalesoftheeffectivetheory. Thenaturalchoiceis
√
h∼1,ν ∼1/ n. Theresidualdependenceofthebottommassfitontheseparametersisusedfor
theperturbativeerrorestimateinSec.4.
Equation(3.1)explicitlydependsonthebottompolemassMpole,whichwetranslatetothe1S
b
massM1S usingtherelation
b
Mpole=M1S{1+∆LL+∆NLL+[(∆LL)2+∆NNLL+∆NNLL]}. (3.2)
b b c m
The∆termsarelabeledaccordingtothenonrelativisticordercountingschemeinEq.(1.2). Explicit
expressionscanbefoundinRef.[30]. Forthefinaltheoreticalexpressionusedinthesinglemoment
fitsbelowweconsistentlyexpandouttheperturbativeseriesfortheWilsoncoefficientsc together
i
with the nonrelativistic expansion series for the ρ and Mpole in Eq. (3.1).3 The convergence
n,i b
propertiesofthisexpansionarediscussedindetailinRef.[25].
Apart from the NNLL correction to the renormalization group (RG) running of the Wilson
coefficient c associated with the dominant heavy quark production current all relevant contribu-
1
tions to Eq. (3.1) are known completely. Concerning the NNLL running of c all (“non-mixing”)
1
contributions from genuine vNRQCD three-loop diagrams were computed in Ref. [32]. The cor-
responding pNRQCD calculation is not available at present. The remaining NNLL (“mixing”)
contributions are generated by corrections to the vNRQCD four-quark operator (“potential”) co-
efficients appearing in the NLL anomalous dimension of c . Recent results for the ultrasoft NLL
1
runningofthesubleading(O(v)andO(v2))nonrelativisticquark-antiquarkpotentials[33,34,35]
completed the calculation of the ultrasoft part of the NNLL mixing contributions [34]. It is the
dominantNNLLmixingeffect[36,25]intherunningofc . Likewisetheultrasofttermsdominate
1
the NNLL non-mixing running [32]. Thus at present the only unknown piece in Pth,NNLL is the
n
NNLLsoftmixingcontributiontotheRGevolutionofc .
1
3WeareforcedtosimultaneouslyexpandouttheseriesforMbpole andtheρn,i inthewayexplainedinRef.[18]in
ordertoachieveapropercancellationoftheleadingrenormalon.
4
BottomMassfromNonrelativisticSumRulesatNNLL MaximilianStahlhofen
1.30
1.25
1.20
c (ν)
1 1.15
c (1)
1
1.10
1.05
1.00
0.3 0.4 0.5 0.6 0.7 0.8
Ν
Figure1: RGevolutionofthecurrentcoefficientc : NLL(blue)andapproximateNNLLresult(red)with
1
uncertaintyduetotheunknownNNLLsoftmixingcontribution(lightredband).
Figure 1 shows the dependence of the current coefficient c (ν)≡c (h=1,ν) on the renor-
1 1
malizationparameterν forthecompleteNLLresult(blue)andanapproximateNNLLresult(red),
whereallknownNNLLcontributions4 areadded. The(lightred)bandaroundtheNNLLcurveis
generated by varying all known soft NNLL contributions to that curve by a factor between 0 and
2. The uncertainty due to the unknown NNLL soft mixing terms estimated by this band is much
smallerthanthelargetotalNNLLcorrectionfromtherunningintherelevantrange0.3(cid:46)ν (cid:46)0.6
andcansafelybeneglectedinthefollowing[25].
The (known) leading nonperturbative power correction to the moment Pth,NNLL is associated
n
with the gluon condensate [14] and turns out to be completely negligible for our analysis [25].
Higherorderpowercorrectionsaresufficientlysuppressedforn(cid:46)10.
4. SingleMomentFits
ThevNRQCDexpressionforthetheoreticalmomentPth,NNLL exhibitsaresidualdependence
n
on the scale µ =hm , where the effective theory is matched to full QCD, as well as on the two
h b
correlated renormalization scales µ =hm ν (soft) and µ =hm ν2 (ultrasoft). Here and in the
S b U b
following m ≡ M1S. The three unphysical scales can be consistently parametrized by the two
b b
variableshandν. Inordertoestimatetheuncertaintiesfromhigherorderperturbativecorrections
wechoosetovarytheparametersforthebottommassfitswithintheh-ν regionaroundthedefault
√
valuesν =ν :=1/ n+0.2andh=1asdefinedinFig.2a. Theplotalsoshowsthecontoursof
∗
theresultforM1S fromtheequationPth(M1S)=Pexp,i.e. aNNLLsinglemomentfitforn=10.
b 10 b 10
Withtheseconventionsfortheh-ν scalingvariationswecannowgenerateerrorbandsaround
thedefaultfits(dashedlines)forM1S(α )asshowninFig.3. Thetwopanelsinthisfigurecompare
b s
thefitsofthebottommassasafunctionofthestrongcouplingα (M )usingtheoreticalmoments
s Z
calculatedinthefixed-order(a)andtheRGIapproach(b). Thefixed-ordermomentsareobtained
by switching off the all-order resummation of nonrelativistic logarithms in our RGI moments as
explainedinRef.[25]. Sinceatleadingorder(LO)theonlyrelevantphysicalscaleisthesoftscale
4These even include the first logarithm ∝α3lnν in the NNLL series of the soft mixing contribution [32]. The
s
subsetofspin-dependenttermsintheNNLLsoftmixingcontributionisalsoknown[37],buttinyandneglectedhere.
AllrelevantanalyticNNLLexpressionsforc aregiveninRef.[25].
1
5
BottomMassfromNonrelativisticSumRulesatNNLL MaximilianStahlhofen
a) b)
2.0 4.73
4.90
4.67 4.85
1.5
"
V
4.80
f 4.82 Ge
!
4.76 4.7 S 4.75
1.0 1
M
4.85 4.70
4.91
4.974.88
0.5 4.94 4.79 4.7 4.65
5 10 15 20
0.5 1.0 1.5 2.0
n
h
Figure2: Panela): Contourplotofthe1SbottommassdeterminedfromPth(m )=Pexp asafunctionof
10 b 10
theparametershand f ≡ν/ν . ThedifferentcontoursarelabeledbytherespectivemassvalueinGeV.The
∗
regionintheh-f planeboundedbythereddashedlinerepresentstheparameterspacewescantodetermine
the variation of the mass, which contributes to our perturbative error estimate. The region is defined by
0.75≤h≤1/0.75anddemandingthat 0.5µ∗ ≤µ ≤2µ∗, where µ∗ =m ν2. Theredpointinsidethis
U U U U b ∗
areaindicatesourdefaultvalues f =h=1forthemassdetermination. Panelb): 1Smassresults(dots)with
perturbativeerrorbarsfromsinglemomentfitsforn=4ton=20asexplainedinthetext.
theLOandLLbandsinFig.3agreeexactly. Comparingthenext-to-leading(NLO)withtheNLL
and in particular the NNLO with the NNLL results we however observe much larger scale vari-
ations of the fixed-order results. This clearly indicates a substantially improved precision related
to the resummation of logarithms in the RGI approach. As argued in Ref. [25] we believe that in
contrasttotheLLband,whichisgeneratedonlybysoftscalevariations,andthe(w.r.t. thedefault
fits) strongly asymmetric NLL error bands the NNLL mass range gives a reliable estimate of the
perturbative uncertainty. The observed bottom mass dependence on the input value for α (M ) is
s Z
rathermildandatleastintheinterval0.113≤α (M )≤0.120lineartoagoodapproximation[25].
s Z
A far more detailed analysis of the numerical results including plots of Fig. 3 b type for different
valuesofnaswellasmultiplemomentfitshasbeencarriedoutinRef.[25].
Thefinalresultforthe1SbottommassfromtheNNLLRGIsinglemomentanalysisforn=10
outlinedaboveis
M1S = 4.755±0.057 ±0.009 ±0.003 GeV, (4.1)
b pert αs exp
where we have used the current world average α (M ) = 0.1183±0.0010 for the strong cou-
s Z
pling[38]. ThecentralvalueinEq.(4.1)correspondstotheresultofthedefaultfit(h=1,ν =ν ).
∗
The quoted errors refer to the perturbative uncertainty, which we estimate by half the size of the
band from the scale variations, and the errors from the uncertainties of α and the experimental
s
datausedforthefit.
Figure2bcomparestheM1S resultsandtheirrespectiveperturbativeerrorbarsfromfitsusing
b
the moments n = 4 to n = 20. Within the errors all central values are in very good agreement.
Wehoweveremphasizethatfornconsiderablylargerthan10formallytheOPEforthetheoretical
moments breaks down due to possibly uncontrolled higher order power corrections, although the
6
BottomMassfromNonrelativisticSumRulesatNNLL MaximilianStahlhofen
a) n(cid:61)10 b) n(cid:61)10
5.0 5.0 5.0 5.0
NNLO NNLL
4.8 4.8 4.8 4.8
LO LL
V (cid:76) V (cid:76)
e 4.6 4.6 e 4.6 4.6
G G NLL
(cid:72) (cid:72)
1S 4.4 4.4 1S 4.4 4.4
M NLOM
4.2 4.2 4.2 4.2
4.0 4.0 4.0 4.0
0.116 0.117 0.118 0.119 0.120 0.116 0.117 0.118 0.119 0.120
Α M Α M
s Z s Z
(cid:72) (cid:76) (cid:72) (cid:76)
Figure3: Comparisonofthemassesobtainedfromthefixedorder(a)andRGIcalculation(b)ofthe10-th
moment,Pth(m )=Pexp. InpanelsaandbweshowthemassvalueswithLO,NLO,NNLOandLL,NLL,
10 b 10
NNLLaccuracy, respectively. Thedashedlinesdisplaytheresultsfromthefitswiththedefaultvaluesfor
theparametershandν.Thecorresponding(partlyoverlapping)errorbandsweregeneratedbyvaryinghand
ν withintheparameterspacedefinedinFig.2a. (Wealsoaddedthetinyexperimentalerrorinquadrature,
which is however hardly visible.) Concerning panel a, we note that for some low m values in the NLO
b
band and the associated values for h and ν the ultrasoft coupling α (µ ) reaches 0.65 causing numerical
S U
instabilities.
leading power correction still appears to be small as long as n (cid:46) 20. We therefore regard the
error bars for n>10 shown in Fig. 2 b as a confirmation of the perturbative stability of the RGI
vNRQCDcalculation,butdonotusethemforquotingourfinalerrors. Ontheotherhandtheerror
bars for n<10 increase for smaller n because the sensitivity of the theoretical moments on the
massdecreases[25],cf. Eq.(3.1).
Usingtherespective(fixed-order)relationstothepolemass,seeRefs.[18,19]fordetails,we
cantranslatethe1SmassresultinEq.(4.1)totheMS-massandobtain
m (m ) = 4.235±0.055 ±0.003 GeV, (4.2)
b b pert exp
wherewehaveaddedanadditionalconversionerrorof15MeVtotheperturbativeuncertainty[25].
Interestingly the α dependence of the original 1S mass result in Eq. (4.1) and the intrinsic α
s s
dependence of the 1S-MS conversion formula almost cancel exactly. The remaining α induced
s
erroristhereforenegligibleandnotquotedinEq.(4.2).
5. Summary
We have presented the determination of the 1S bottom mass from (single) large-n moment
fits with RGI at NNLL order as carried out in Ref. [25]. The main result is given in Eq. (4.1).
ConvertedtotheMSschemeourresult(Eq.(4.2))isconsistentwiththeNLLRGIlarge-nresultof
Ref.[12],butnotquitecompatiblewiththelatestresultsfromlow-nsumruledeterminations[3,5].
We however note that our calculation (like Ref. [12]) treats the charm quark as massless, while
previousfixed-order analyses[39,19]have shownthatfinite charmmasseffects areenhancedfor
largenandcauseasizablemassshiftbetween−20and−30MeV.Asimilareffectisexpectedin
theRGIanalysisandmighthelptoreconcilethediscrepancy.
7
BottomMassfromNonrelativisticSumRulesatNNLL MaximilianStahlhofen
Acknowledgments
IwouldliketothankAndréHoangforcommentsonthemanuscript. Thisworkwassupported
bytheDFGunderEmmy-NoetherGrantNo. TA867/1-1.
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