On the perturbative expansion of tau hadronic spectral function moments

0.25 MB·English

by  Diogo Boito

On the perturbative expansion of τ hadronic
spectral function moments
3
1
0 Diogo Boito∗
2
PhysikDepartmentT31,TechnischeUniversitätMünchen,

n James-Franck-Straße1,D-85748Garching,Germany
a
E-mail: [email protected]
J

4
1
In the determination of α from tau decays several different moments of the hadronic spectral
s

] functions have been used. In a recent work, we performed an analysis of their perturbative be-
h
p haviour under two different assumptions for the higher-order coefficients of the Adler function.
- We showed that the various moments can be divided in a small number of classes. We con-
p
e cludedthatsomeofthemomentscommonlyemployedinα extractionsshouldbeavoideddueto
s
h
theirbadperturbativebehaviour. Furthermore,forthemomentsthathaveagoodperturbativebe-
[

haviour,andunderreasonableassumptionsforthehigher-orderbehaviouroftheAdlerfunction,
1
v fixed-orderperturbationtheory(FOPT)providesthesuperiorframeworkfortherenormalization
8
groupimprovement. Herewediscussanextensionofthisanalysiswhereweconsiderthepertur-
0
0 bativeseriesforvaluesofthehadronicinvariantmasssquareds0≤m2τ. Ourconclusionsarenot
3
alteredwithinareasonables window.
. 0
1
0
3
1
:
v
i
X
r
a
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
†TUM-HEP-873/13
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Perturbativeexpansionofτ hadronicspectralfunctionmoments DiogoBoito
1. Introduction
Inthelast20years,hadronictaudecayshavebeenanimportantsourceofempiricalinforma-
tion on fundamental parameters of QCD. Notably, the strong coupling, α , can be extracted with
s
a good precision at relatively low energies, close to the edge of the validity of perturbative QCD.
AfterthemeasurementofthespectralfunctionsatLEP,otherparameterssuchasthestrangequark
mass, the CKM matrix element V , as well as non-perturbative condensates could be extracted
us
(seee.g.[1]). Theextractionoftheseparametersreliesonsumrules. Quark-hadrondualityandthe
opticaltheoremallowtoexpressthedecayrateasweightedintegralsofthevectorandaxial-vector
spectralfunctionsrunningoverthehadronicinvariantmasssquaredsfromthresholduptom2.
τ
Using the analytic properties of the quark-antiquark correlators, the theoretical counter-part
oftheexperimentalquantitiesareexpressedascontour-integralsinthecomplexenergyplanewith
fixed |s| = m2. However, in the theoretical description of τ decays two main obstacles remain.
τ
The first is referred to as duality violations (DVs). They are related to the break-down of local
quark-hadron duality in the vicinity of the Minkowski axis (where resonance effects become im-
portant). In the past, they have been neglected due to a fortuitous kinematical suppression of the
problematic region in the contour integration. Recently, thanks to the progress in modelling DVs
realistically[3,4],theyhavebeenincludedself-consistentlyinafull-fledgedα analysis[5,6]. The
s
second important obstacle is the prescription for the renormalization group (RG) improvement of
theperturbativeseries. Themostwidelyemployedprescriptionsarefixed-orderperturbationtheory
(FOPT) [7, 8] and contour improved perturbation theory (CIPT) [9, 10]. When used in practice,
theyleadtodifferentα results. Withtherecentlycomputedα4 correction[11],thedifferencebe-
s s
cameevenmorepronounced. Severalworkshavedealtwiththisdiscrepancy[8,12,13,14,15,16]
in the light of the α4 term. The conclusions in favour of FOPT or CIPT (or a third prescription)
s
arebasedon(implicitorexplicit)assumptionsontheyetunknownhigherorderα corrections. In
s
thiscontext,thegoalofRef.[8]wastoconstructaplausiblemodelforthehigher-ordercorrections
of the Adler function from the leading renormalon singularities of its Borel transform, using only
generalRGargumentstodescribethestructureofthesingularitiesintheBorelplane. Aftermatch-
ingthemodeltotheknowncoefficientsinQCD,themainconclusionofRef.[8]wasthatFOPTis
to be preferred over CIPT, since FOPT provides a closer approach to the Borel resummed results
—inthespiritofanasymptoticseries.
Thisconclusionwasbasedsolelyontheanalysisoftheweightw (x),obtainedfromthekine-
τ
matics of the decay. This is not entirely satisfactory since realistic determinations of α employ
s
(and often require) several different weight functions w(x). In fact, any analytical w(x) gives
i i
rise to a valid sum-rule that emphasises a given part of the spectral functions, as well as different
contributions in the theoretical description. In the literature, several weight-functions have been
employed and yet little attention has been paid to the moment dependence of the convergence
properties of the perturbative series. We have addressed this question in Ref. [17] and pursued
the FOPT/CIPT comparison for several weight functions. We showed that the different moments
employed in the literature can be divided in a small number of categories. The characteristics of
theirperturbativeseriescouldbelinkedtogenericfeaturesofthemomentweightfunctionandthe
dominantrenormalonsingularitiesoftheAdlerfunction. Weconcludedthatsomeofthemoments
currentlyemployedinsomeα extractionsshouldbeavoidedduetothepoorconvergenceoftheir
s
2
Perturbativeexpansionofτ hadronicspectralfunctionmoments DiogoBoito
perturbative expansions. Additionally, for all moments that display good perturbative behaviour
— and under reasonable assumptions for the higher-order corrections — FOPT provided the best
frameworktotheRGimprovement.
A point that was not discussed in [17] is the stability of the conclusions with respect to s
0
variations(s beingtheupperlimitofintegrationinthesum-rule,seeEq.(2.1)). Therelevanceof
0
thisissueliesinthefactthatseveralα analysesusesum-ruleswherethedataareintegratedupto
s
s <m2. Hereweshowthattheconclusionsof[17]remainvalidwhens isvariedawayfromm2.
0 τ 0 τ
2. Theoreticalframework,model,andresults
We work with generalized sum-rules, where the weight function in the integrals can be any
analytical function w(x) and the upper limit of integration is taken to be any point s ≤m2. The
i 0 τ
experimentalsideofthesum-rulesarethenwrittenasintegralsoverthespectralfunctionsas
Rwi (s ) = 12πS |V |2(cid:90) s0 ds(cid:18)1− s (cid:19)2(cid:18)1+2 s (cid:19)(cid:20)ImΠ(1+0)(s)− 2s ImΠ(0) (s)(cid:21).
τ,V/A 0 EW ud s s s V/A s +2s V/A
0 0 0 0 0
(2.1)
The two point functions are defined as Πµν (p) ≡ i(cid:82)dxeipx(cid:104)Ω|T{Jµ (x)Jν (0)†}|Ω(cid:105) and they
V/A V/A V/A
assume the usual decomposition into longitudinal and transversal components. TheV and A cur-
rentsaregivenbyJµ (x)=(u¯γµ(γ )d)(x).
V/A 5
Thetheoreticalcounter-partofEq.(2.1)isobtainedfromthecounter-clockwisecontourinte-
grationofthecorrelatorswith|s|=s . Thecontributionsonthetheorysidecanbeorganizedas
0
(cid:20) (cid:21)
N
Rwi (s ) = cS |V |2 δtree+δ(0)(s )+ ∑ δ(D) (s )+δDV (s ) . (2.2)
V/A 0 2 EW ud wi wi 0 wi,V/A 0 wi,V/A 0
D≥2
Theperturbativecontributioniscontainedinδtree andδ(0),ofwhichδ(0) containstheloopcorrec-
wi wi wi
tions. InthechirallimittheyarethesameforV andAcorrelators,andcorrespondtotheperturba-
(1+0)
tiveseriesofΠ (s). The quark-mass corrections, aswellascontributionsfromoperatorswith
V/A
D>2intheOPE,areencodedinthetermsδ(D) ;DVcontributionsarerepresentedbyδDV .
wi,V/A wi,V/A
Our focus is on the behaviour of the perturbative correction and it is convenient to write it in
termsoftheRGinvariantAdlerfunction,whoseexpansioninα canbewrittenas
s
d N ∞ n+1 (cid:18) −s(cid:19)k−1
D(1+0)(s) ≡ −s Π(1+0)(s)= c ∑an ∑kc log , (2.3)
ds 12π2 µ n,k µ2
n=0 k=1
where a =α(µ)/π. RG invariance implies that only the coefficients c are independent. The
µ n,1
otherc canbeexpressedintermsofc andβ-functioncoefficients. Theperturbativecontribu-
n,k n,1
tiontothetheorysideofthesum-rulesisthen
∞ n 1 (cid:73) dx (cid:18)−s x(cid:19)
δ(0) = ∑ ∑kc W(x)logk−1 0 an , (2.4)
wi n,k 2πi x i µ2 µ
n=1k=1
|x|=1
withx=s/s andW(x)=2(cid:82)1dzw(z). DuetotheRGinvarianceofD(1+0)(s)onehasthefreedom
0 i x i
of setting the scale µ. The FOPT prescription corresponds to the choice µ2 =s . In this case,
0
3
Perturbativeexpansionofτ hadronicspectralfunctionmoments DiogoBoito
the coupling a(s ) is taken out-side the integrals and one is left with the integration of powers of
0
log(−x). TheCIPTchoicecorrespondtoµ2=−s x,whichresumsthelogarithmsbut,inturn,the
0
integralsaredone(numerically)overtherunningcouplinga(−s x).
0
In order to compare FOPT and CIPT as well as understand the perturbative behaviour of
spectralfunctionmoments, onemusthaveanansatzfor theunknownhigher-orderAdlerfunction
coefficientsc . HerewefollowthemethodintroducedinRef.[8]whichmakesuseoftheavailable
n,1
knowledge of the renormalon structure of the Borel transformed Adler function. The idea is to
construct a realistic model for the Borel transform using the leading singularities. We work with
thefunctionD(cid:98)(s)anditsBoreltransform,B[D(cid:98)](t),definedas
12π2 ∞ √ ∞ tn
N DV(1+0)(s) ≡ 1+D(cid:98)(s) ≡ 1+∑rnαs( s)n+1, B[D(cid:98)](t) ≡ ∑rnn!. (2.5)
c n=0 n=0
TheoriginalseriescanbeunderstoodasanasymptoticexpansionoftheinverseofB[D(cid:98)](t),
(cid:90) ∞
D(cid:98)(α) ≡ dte−t/αB[D(cid:98)](t), (2.6)
0
when the integral exists. Singularities of B[D(cid:98)](t) on the positive real axis (infra red (IR) renor-
malons)giverisetofixed-signasymptoticseriesandobstructtheBorelsummation,Eq.(2.6). This
introducesanambiguityintheintegralthatisexpectedtobecancelledagainstexponentiallysmall
termsinα ,orpowercorrections(duetothelogarithmicrunningofthecoupling). Singularitieson
s
thenegativerealaxis(ultravioletrenormalons(UV))giverisetosign-alternatingseries.
General RG arguments and the structure of the OPE allow one to determine the position and
strengthoftherenormalonsingularitiesinthetplane,thoughnottheirresidues[18]. Thefixed-sign
natureoftheexactlyknowncoefficientsoftheAdlerfunctionsuggestthatatlowandintermediate
orders the series is dominated by IR singularities. The reference model (RM) of [8] contains the
firsttwoIRandtheleadingUVsingularities. TheBoreltransformisgivenby
B[D(cid:98)](u) = B[D(cid:98)UV](u)+B[D(cid:98)IR](u)+B[D(cid:98)IR](u)+dPO+dPOu. (2.7)
1 2 3 0 1
Thestructureofthebranch-cutsingularitiescanbefoundin[8]. Theresiduesandthecoefficients
dPO arefixedbymatchingtotheexactlyknownc toc (augmentedbyanestimateforc ).
0,1 1,1 4,1 5,1
Within this model, the conclusion of Ref. [8] in favour of FOPT has been corroborated and
extended in our recent work [17]. All moments that display a good perturbative behaviour favour
theFOPTprescriptionwithintheRM.Thisconclusioncanbetracedbacktothecontributionofthe
leadingIRsingularity, relatedtotheD=4correctionsintheOPE.Ifthissingularityisarbitrarily
suppressed,onegeneratesamodel—lessrealistic,inouropinion—inwhichCIPTisthepreferred
prescription. To realize this scenario in practice, and assess possible model dependencies in our
conclusions, weintroducedthefollowingalternativemodel(AM)wheretheleadingsingularityis
absentwhereasthesub-leadingoneatu=4isexplicitlytakenintoaccount:
B[D(cid:98)](u) = B[D(cid:98)UV](u)+B[D(cid:98)IR](u)+B[D(cid:98)IR](u)+dPO+dPOu. (2.8)
1 3 4 0 1
WithintheAM,momentswithgoodperturbativebehaviourfavourCIPT.
4
Perturbativeexpansionofτ hadronicspectralfunctionmoments DiogoBoito
1.30 1.30
Borel Amb. s =1.5 GeV2
1.20 1.20 BCoIrPeTl Asm0=b1.. 5s0 G=meVτ22
CIPT s0=2.5 GeV2
1.10 1.10 C0IPT s =mτ2
0
1.00 1.00
0.90 0.90
Borel Amb. s =1.5 GeV2
Borel Am0b. s =mτ2
0.80 FOPT s =1.50 GeV2 0.80
FOPT s0=2.5 GeV2
0.70 FO0PT s =mτ2 0.70
0
1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15
(a) w (x)=(1−x)2(1+2x),FOPT. (b) w (x)=(1−x)2(1+2x),CIPT.
τ τ
(0)
Figure1: Referencemodel. δ (s )orderbyorderinα normalisedtotheBorelsumforFOPT(left)and
wτ 0 s
CIPT(right)withthreevaluesofs : 1.5GeV2,2.5GeV2,andm2. BandsgivetheBorelambiguities.
0 τ
ThemodelsrepresenttwoquitedifferentsituationsregardingtheinterplayoftheAdlerfunc-
tion coefficients and the running coupling effects. In the RM, there are cancellations between the
contributionfromthehigh-ordercoefficientsc andtherunningcouplingeffects,atagivenorder
n,1
in α . In this case FOPT is superior since it treats these contributions on an equal footing, while
s
CIPT misses the cancellations due to the resummation of the running effects to all orders. On the
other hand, the AM represents a situation where the running effects are dominant and should be
resummed. Inthiscase,thehigh-ordercoefficientscanbeneglectedandCIPTisabetterprescrip-
tion. SincethereisnoknownmechanismthatwouldnaturallysuppresstheleadingIRsingularity
inQCD,webelievethescenarioofEq,(2.7)tobemorerealistic.
Usingthesetwomodels,wecomparedinRef.[17]theperturbativeseriesinFOPTandCIPT
generatedfrom17polynomialweight-functionsw(x). Weshowedthattheycanbedividedintoa
i
small number of categories regarding the behaviour of their perturbative series. Generic features
of thefunctions w(x)(such asstarting or not withthe unity), together withthe assumptions upon
i
the Adler function, suffice to determine whether they are suitable for α extractions and whether
s
FOPT or CIPT is more suitable for the RG improvement. We showed that some of the weight
functionsusedintheliterature,e.g. polynomialscontainingsolelypowersofxi withi≥2,should
beavoidedduetotheirbadperturbativebehaviour. Wealsoprovidedfurtherargumentsthatsupport
theplausibilityoftheRMof[8]andconcludedthatforwell-behavedmomentsFOPTispreferred.
Anaspectthatwasnotconsideredin[17]wasthes dependenceoftheseconclusions. Thisis
0
importantbecausesum-ruleswithdifferentvaluesofs ≤m2areusedinextractionsofα [5,6,19].
0 τ s
Here we show explicit results for the FOPT/CIPT comparison for two moments within the two
models given in Eqs. (2.7) and (2.8) and considering three values of s : 1.5 GeV2, 2.5 GeV2, and
0
m2. The interval [1.5 GeV2: m2] spans the values used in the fits of [5, 6]. Since we intent to
τ τ
comparetheperturbativeseriesatdifferentvaluesofs anormalisationprocedureisinorder. For
0
better comparison, we normalise the series generated for each value of s by its corresponding
0
Borelsum,Eq.(2.6). Hence,intheplots,meaningfulseriesshouldbeasymptotictotheunity.
Westartbyconsideringthecaseofmomentsthathavegoodperturbativebehaviourfors =m2.
0 τ
As a representative we choose to use the kinematic moment w . In Fig. 1, we consider the FOPT
τ
andCIPTserieswithintheRM.Ontheleft-handside, Fig.1(a), oneobservesthatthenormalised
5
Perturbativeexpansionofτ hadronicspectralfunctionmoments DiogoBoito
1.80 1.80
Borel Amb. s =1.5 GeV2 Borel Amb. s =1.5 GeV2
Borel Am0b. s =mτ2 Borel Am0b. s =mτ2
1.60 FOPT s =1.50 GeV2 1.60 CIPT s =1.50 GeV2
FOPT s0=2.5 GeV2 CIPT s0=2.5 GeV2
1.40 FO0PT s0=mτ2 1.40 C0IPT s0=mτ2
1.20 1.20
1.00 1.00
0.80 0.80
1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15
(a) w (x)=(1−x)2x3(1+2x),FOPT. (b) w (x)=(1−x)2x3(1+2x),CIPT.
17 17
(0)
Figure2: Referencemodel. δ (s )orderbyorderinα normalisedtotheBorelsumforFOPT(left)and
w17 0 s
CIPT(right)withthreevaluesofs : 1.5GeV2,2.5GeV2,andm2. BandsgivetheBorelambiguities.
0 τ
FOPT series still behaves as a good asymptotic series even for s significantly smaller than m2.
0 τ
As expected, for lower s , the larger values of α amplify the divergent behaviour above the 8th
0 s
order. Nevertheless, the first few terms of the series approach the Borel resummed value. Note
also that the Borel sum has a larger ambiguity for smaller s due to larger α . On the right-hand
0 s
side, in Fig. 1(b), one sees that the poor performance of CIPT is amplified by the larger values of
the coupling at lower s . That is, within the RM, CIPT is not a good approximation to the Borel
0
resummedvalues,andevenlesssoforsmallers .
0
We now turn to a moment with bad perturbative behaviour: w (x)=(1−x)2x3(1+2x) (to
17
employ the notation of [17]). In Ref. [17] we showed that moments starting with powers of x
(that do not contain the unity) tend to have bad perturbative behaviour and are largely dominated
by power corrections. In Fig. 2 we address the s dependence of this conclusion. On the left,
0
Fig.2(a)showsthatforhighervaluesofs FOPTcanapproachtheBorelresultonlyathighorders
0
(not available exactly). At low s the series displays a wild behaviour and cannot be considered a
0
good approximation to the Borel sum. In CIPT, Fig. 2(b), the bad behaviour already observed for
s =m2 isamplifiedatlowers . Theseriesareerraticandcannotbeconsidersuitableasymptotic
0 τ 0
approximationstotheBorelsum. Notethatthismoment,despiteofitsbadperturbativebehaviour,
entersseveraldeterminationsofα fromτ decays(e.g. Refs.[20,21]).
s
Wecanperformthesameanalysisinthealternativemodel,Eq.(2.8),whichreceivesnocontri-
butionfromtheleadingIRsingularity. InFig.3,weshowtheseriesnormalisedtotheirrespective
Borel resummed values within the AM for w (x). In this model, CIPT provides the better frame-
τ
workalsoforlowervaluesofs ,asshowninFig.3(b). Theseriesremainsverystablefors =2.5
0 0
GeV2 andstillapproachestheBorelresultwell. Fors =1.5GeV2 CIPTcanstillbeconsidereda
0
goodapproximationtakingintoaccounttheamplifiedBorelambiguity. TheoscillationsofFOPT,
already present at s =m2, are much amplified for lower s (see Fig. 3(a)). Within the AM, the
0 τ 0
FOPTseriesarenotagoodapproximationtotheBorelresummedvalues.
To conclude we examine the case of w in the context of the AM. The results are shown in
17
Fig.4. The badperturbative behaviourof FOPT andCIPT remainsfor all values ofs . Thisis an
0
indicationofthemodelindependenceoftheconclusionthatw (andanumberofothermoments
17
alsodiscussedinRef.[17])shouldbeavoidedindeterminationsofα .
s
6
Perturbativeexpansionofτ hadronicspectralfunctionmoments DiogoBoito
1.40 1.40
Borel Amb. s =1.5 GeV2 Borel Amb. s =1.5 GeV2
1.30 FBOorPeTl Asm0=b1.. 5s0 G=meVτ22 1.30 BCoIrPeTl Asm0=b1.. 5s0 G=meVτ22
FOPT s0=2.5 GeV2 CIPT s0=2.5 GeV2
1.20 FO0PT s =mτ2 1.20 C0IPT s =mτ2
0 0
1.10 1.10
1.00 1.00
0.90 0.90
0.80 0.80
1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15
(a) w (x)=(1−x)2(1+2x),FOPT. (b) w (x)=(1−x)2(1+2x),CIPT.
τ τ
(0)
Figure3: Alternativemodel. δ (s )orderbyorderinα normalisedtotheBorelsumforFOPT(left)and
wτ 0 s
CIPT(right)withthreevaluesofs : 1.5GeV2,2.5GeV2,andm2. BandsgivetheBorelambiguities.
0 τ
1.80 1.80
Borel Amb. s =1.5 GeV2 Borel Amb. s =1.5 GeV2
1.60 Borel Am0b. s =mτ2 1.60 Borel Am0b. s =mτ2
FOPT s =1.50 GeV2 CIPT s =1.50 GeV2
1.40 FOPT s0=2.5 GeV2 1.40 CIPT s0=2.5 GeV2
FO0PT s =mτ2 C0IPT s =mτ2
0 0
1.20 1.20
1.00 1.00
0.80 0.80
0.60 0.60
0.40 0.40
1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15
(a) w (x)=(1−x)2x3(1+2x),FOPT. (b) w (x)=(1−x)2x3(1+2x),CIPT.
17 17
(0)
Figure4: Alternativemodel. δ (s )orderbyorderinα normalisedtotheBorelsumforFOPT(left)and
w17 0 s
CIPT(right)withthreevaluesofs : 1.5GeV2,2.5GeV2,andm2. BandsgivetheBorelambiguities.
0 τ
3. Conclusions
Recently, we have analysed the perturbative behaviour of several moments often employed
in analyses of α from τ decays under different assumptions for the large-order behaviour of the
s
Adler function [17]. We have shown that some of these moments should be avoided due to their
badperturbativebehaviour. Furthermore,underreasonableassumptionsfortheBoreltransformed
Adlerfunction, weshowedthatFOPTprovidesthepreferredframeworkfortheRGimprovement
ofmomentsthatdisplaygoodperturbativebehaviour.
Here we showed, for the first time, that these conclusions are still valid if one considers the
perturbativeseriesgeneratedbyFOPTandCIPTfors ≤m2. Thisisarelevantquestion,sincein
0 τ
α extractionsoneoftenconsiderssum-ruleswiths ≤m2. Wehaveshownexplicitlytheresultsfor
s 0 τ
two representative moments previously investigated in Ref. [17] for s =m2. The s dependence
0 τ 0
analysis was also carried out for the remaining moments studied in [17] with similar conclusions;
theyarenotshownhereforthesakeofbrevity.
7
Perturbativeexpansionofτ hadronicspectralfunctionmoments DiogoBoito
Acknowledgements
The author wishes to thank Martin Beneke and Matthias Jamin for the careful reading of the
manuscript. ThisworkwassupportedbytheAlexandervonHumboldtFoundation.
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