Confronting Spectral Functions from e+e- Annihilation and tau Decays: Consequences for the Muon Magnetic Moment

17 Pages·0.4 MB·English

by  Andreas Hocker

1
Confronting Spectral Functions from e+e Annihilation and τ Decays:
−
3
Consequences for the Muon Magnetic Moment
0
0
2 A. Ho¨ckera∗,

n
aLaboratoire de l’Acc´el´erateurLin´eaire,
a
J IN2P3-CNRS et Universit´e de Paris-Sud, F-91898 Orsay,France

5 (e-mail: [email protected])
1
Vacuum polarization integrals involve the vector spectral functions which can be experimentally determined

1 fromtwosources: (i)e+e− annihilationcrosssectionsand(ii)hadronicτ decays. Recentlyresultswithcompara-
v bleprecision havebecome available from CMD-2 on one side, and ALEPH,CLEO and OPAL on theother. The
4
comparison of the respective spectral functions involves a correction from isospin-breaking effects, which is eval-
0
uated. Afterthecorrection it isfound thatthedominantππ spectralfunctionsdonotagreewithin experimental
1
and theoretical uncertainties. Some disagreement is also found for the 4π spectral functions. The consequences
1
ofthesediscrepanciesforvacuumpolarization calculationsarepresented,withtheemphasisonthemuonanoma-
0
3 lous magnetic moment. Adding quadratically experimental and theoretical uncertainties, we find that the full
0 StandardModelpredictionofaµ deviatesfromtherecentBNLmeasurementatthelevelof3.0(e+e−-based)and
/ 0.9 (τ-based) standard deviations.
h
p
-
p
e
1. INTRODUCTION vantintegralscanbeexpressedintermsofanex-
h
: perimentally determined spectral function which
v Hadronic vacuum polarization in the photon isproportionaltothecrosssectionfore+e− anni-
i propagator plays an important role in the preci-
X hilation into hadrons. The accuracyof the calcu-
siontestsoftheStandardModel. Thisisthecase
r lations has therefore followed the progress in the
a fortheevaluationoftheelectromagneticcoupling
quality of the corresponding data [3]. Because
attheZ massscale,α(M2),whichreceivesacon-
Z the latter wasnot alwayssuitable, it wasdeemed
tribution ∆α (M2) of the order of 2.8 10−2
had Z necessary to resort to other sources of informa-
that must be known to an accuracy of better
tion. One such possibility was the use [4] of the
than 1% so that it does not limit the accuracy
vector spectral functions derived from the study
on the indirect determination of the Higgs boson
of hadronic τ decays [5] for the energy range less
mass from the measurement of sin2θ . Another
W than 1.8GeV. Anotherone occurredwhen itwas
example is provided by the anomalous magnetic
realized in the study of τ decays [6] that pertur-
moment a =(g 2)/2 of the muon, where the
µ µ− bative QCD could be applied to energy scales as
hadronic vacuum polarization component is the
low as 1-2 GeV, thus offering a way to replace
leadingcontributorto the uncertaintyof the the- poor e+e− data in some energy regions by a reli-
oretical prediction.
ableandprecisetheoreticalprescription[7–9]. Fi-
Starting fromRefs. [1,2]there is a long history
nally,withoutanyfurthertheoreticalassumption,
of calculating the contributions from hadronic
it was proposed to use QCD sum rules [10,11] in
vacuum polarization in these processes. As they
ordertoimprovethe evaluationinenergyregions
cannot be obtained from first principles because
dominated by resonances where one has to rely
ofthelowenergyscaleinvolved,thecomputation
onexperimentaldata. Using these improvements
reliesonanalyticityandunitaritysothattherele-
the lowest-orderhadronic contribution to a was
µ
∗WorkdoneincollaborationwithM.Davier,S.Eidelman
andZ.Zhang
2
found to be [11] providing a better understanding of this
critical area when relating vector τ and
ahad,LO = (692.4 6.2) 10−10 . (1)
µ ± isovector e+e− spectral functions.
The complete theoretical prediction includes in
addition QED, weak and higher order hadronic 2. MUON MAGNETIC ANOMALY
contributions.
The anomalous magnetic moment of the muon ItisconvenienttoseparatetheStandardModel
is experimentally known to very high accuracy. predictionfortheanomalousmagneticmomentof
Combinedwiththeolder,lesspreciseresultsfrom the muon into its different contributions,
CERN [12],the measurements fromthe E821ex-
periment at BNL [13–15], including the most re- aSµM = aQµED+ahµad+awµeak , (3)
cent result [16], yield
with ahad = ahad,LO + ahad,HO + ahad,LBL ,
aeµxp = (11659203±8) 10−10 , (2) where aµQµED = (1µ1658470.6µ±0.3) 10−µ10 is the
andareaimingatanultimateprecisionof410−10 pure electromagnetic contribution (see [32,33]
and references therein), ahad,LO is the lowest-
in the future. The previous experimental re- µ
order contribution from hadronic vacuum polar-
sult [15] was found to deviate from the theo-
retical prediction by 2.6 σ, but a large part of ization, ahµad,HO =(−10.0±0.6)10−10 is the cor-
the discrepancy was originating from a sign mis- responding higher-order part [34,4], and awµeak =
take in the calculation of the small contribution (15.4 0.1 0.2) 10−10, where the first error
± ±
fromthe so-calledlight-by-light(LBL)scattering is the hadronic uncertainty and the second is
diagrams [17,18]. The new calculations of the due to the Higgs mass range, accounts for cor-
LBL contribution [19–21] have reduced the dis- rections due to exchange of the weakly interact-
crepancy to a nonsignificant 1.6 σ level. At any ing bosons up to two loops [35]. For the LBL
rate it is clear that the presently achieved exper- partweaddthevaluesforthepion-polecontribu-
imental accuracy already calls for a more precise tion[19–21]andthe otherterms[20,21]toobtain
evaluation of ahµad,LO. ahµad,LBL =(8.6±3.5) 10−10.
New experimental and theoretical develop- By virtue of the analyticity of the vacuum
ments have prompted the re-evaluation of the polarization correlator, the contribution of the
hadronic contributions presentedin Ref. [22]and hadronic vacuum polarization to aµ can be cal-
reported here: culated via the dispersion integral [36]
new, precise results have been obtained at ∞
• α2(0) K(s)
NovosibirskwiththeCMD-2detectorinthe ahad,LO = ds R(s) , (4)
µ 3π2 s
region dominated by the ρ resonance [23], Z
and more accurate R measurements have 4m2π
beenperformedinBeijingwiththeBESde- where K(s) is the QED kernel [37] strongly em-
tector in the 2-5 GeV energy range [24]. phasizingthelow-energyspectralfunctionsinthe
integral (4). In effect, about 91% of the total
new preliminary results are available from
• contribution to ahad,LO is accumulatedat center-
the final analysis of τ decays with ALEPH µ
of-mass energies √s below 1.8 GeV and 73% of
using the full statistics accumulated at
ahad,LO is covered by the two-pion final state
LEP1 [25]; also the information from the µ
which is dominated by the ρ(770) resonance. In
spectral functions measured by CLEO [26,
Eq. (4), R(s) R(0)(s) denotes the ratio of the
27] and OPAL [28] has been incorporated ≡
’bare’ cross section for e+e− annihilation into
in the analysis.
hadrons to the pointlike muon-pair cross section.
new results on the evaluation of isospin At low mass-squared, R(s) is taken from experi-
•
breaking have been produced [29–31], thus ment.
3
3. DATA FROM e+e− ANNIHILATION ALEPH [25,5], CLEO [26,27] and OPAL [28].
The branching fraction B for the τ
The exclusive low energy e+e− cross sections ν π−π0 (γ) decay mode is oπfπp0articular intere→st
τ
have been measured mainly by experiments run-
since it provides the normalization of the corre-
ning at e+e− colliders in Novosibirsk and Orsay.
sponding spectral function. The new value [25],
Due to the high hadron multiplicity at energies
B = (25.47 0.13) %, turns out to be larger
ππ0
above 2.5 GeV, the exclusive measurement of ±
∼ than the previously published one [42] based on
the respective hadronic final states is not practi-
the 1991-93 LEP1 statistics, (25.30 0.20) %.
cable. Consequently,the experiments atthe high ±
In the limit of isospin invariance, the corre-
energycollidershavemeasuredthetotalinclusive sponding e+e− isovector cross sections are cal-
cross section ratio R.
culated via the isospin rotations
The most precise data from CMD-2 on the
e+e− π+π− cross sections are now available in 4πα2
their→finalform[23]. Theydiffer fromthe prelim- σeI+=e1−→π+π− = s vπ−π0 , (5)
inary ones, releasedtwo years ago [38], mostly in
4πα2
tvhaerioturesatcmhaenngteosfrtehseulrteaddiaintivae croedrrueccttiioonnso.fTthhee σeI+=e1−→π+π−π+π− = 2· s vπ−3π0 , (6)
cross section by about 1% below the ρ peak and
4πα2
5% above. The overall systematic error of the fi- σeI+=e1−→π+π−π0π0 = s v2π−π+π0 (7)
nal data is quoted to be 0.6% and is dominated (cid:20)
by the uncertainties in the radiative corrections
(0.4%). Agreement is observed between CMD- −vπ−3π0 . (8)
(cid:21)
2 and the previous experiments within the much
The τ spectral function v (s) for a given vector
largeruncertainties(2-10%)quotedbythelatter. V
hadronic state V is defined by [43]
Large discrepancies are observed between the
different data sets for the e+e− π+π−π0π0 B(τ− ν V−) dN
→ v (s) → τ V
cross sections (see Fig. 5). These are probably V ∝ B(τ− ν e−ν¯ )N ds
τ e V
related to problems in the calculation of the de- →
−1
tection efficiency, since the efficiencies are small s 2 2s
1 1+ , (9)
in general(∼10−30%) and they are affected by × "(cid:18) − m2τ(cid:19) (cid:18) m2τ(cid:19)#
uncertainties in the decay dynamics that is as-
sumedinthe Monte Carlosimulation. Onecould where Vud = 0.9748 0.0010 (using [44]). The
| | ±
expect the more recentexperiments (CMD-2 [40] spectral functions are obtained from the cor-
andSND [41]) to be more reliable in this context responding invariant mass distributions, after
because of specific studies performed in order to subtracting out the non-τ background and the
identify the major decay processes involved. feedthroughfromother τ decaychannels,andaf-
Fore+e− π+π−π+π− theexperimentsagree ter a final unfolding from detector effects such as
reasonablyw→ell within their quoted uncertainties energy and angular resolutions, acceptance, cal-
(see Fig. 4 in Section 7). ibration and photon identification. Agreement
Adetailedcompilationandcompletereferences withinerrorsisobservedforthethreeinputspec-
of all the data used for this analysis are given in tralfunctions vπ−π0 (ALEPH,CLEO,OPAL),so
Ref. [22]. that we use their weighted average in the follow-
ing.
4. DATA FROM HADRONIC τ DECAYS 5. RADIATIVE CORRECTIONS FOR
e+e− DATA
Data from τ decays into two- and four-
pion final states τ− ν π−π0, τ− ν π−3π0 Radiative corrections applied to the measured
τ τ
→ →
and τ− ν 2π−π+π0, are available from e+e− cross sections are an important step in the
τ
→
4
experimental analyses. They involve the consid- Table 1
eration of several physical processes and lead to Expected sources of isospin symmetry breaking
between e+e− and τ spectral functions in the 2π
large corrections. We stress that the evaluation
of the integral in Eq. (4) requires the use of the and 4π channels, and the corresponding correc-
’bare’ hadronic cross section, so that the input tions to ahµad,LO as obtained from τ data.
data must be analyzed with care in this respect. Sources of Isospin ∆ahad,LO (10−10)
µ
Several steps are to be considered: Symmetry Breaking π+π−
Corrections are applied to the luminos-
• Short distance rad. corr. 12.1 0.3
ity determination, based on large-angle − ±
Long distance rad. corr. 1.0
Bhabha scattering and muon-pair produc- −
tion in the low-energy experiments, and mπ− 6=mπ0 (β in cross section) −7.0
small-angle Bhabha scattering at high en- mπ− 6=mπ0 (β in ρ width) +4.2
ergies. These processes are usually cor- mρ− 6=mρ0 0±2.0
ρ ω interference +3.5 0.6
rected for externalradiation,vertex correc- − ±
Electromagnetic decay modes 1.4 1.2
tions and vacuum polarization from lepton − ±
loops. Sum 13.8 2.4
− ±
Thehadroniccrosssectionsgivenbytheex-
•
perimentsareingeneralcorrectedforinitial
stateradiationandtheeffectofloopsatthe
In Eq. (4) one must incorporate in R(s)
electron vertex. •
the contributionsofallhadronicstatespro-
The vacuum polarization correction in duced at the energy √s. In particular, ra-
•
the photon propagator is a more delicate diative effects in the hadronic final state
point. The cross sections need to be must be considered, i.e., final states such
fully corrected for our use, i.e. σ = as V +γ (FSR) haveto be included. While
bare
σ (α(0)/α(s))2, where σ is the FSR has been added to the newest CMD-
dressed dressed
measuredcrosssectionalreadycorrectedfor 2 data [23], this is not the case for the
initial state radiation, and α(s) is obtained older data and thus has to be corrected
from resummation of the lowest-ordereval- for [22] “by hand”, using an analytical ex-
uation, giving α(s)=α(0)/(1 ∆α (s) pression computed in scalar QED (point-
lep
− −
∆α (s)). Whereas ∆α (s) can be cal- like pions) [45].
had lep
culated analytically, ∆α (s) is related by
had In summary, we correct each e+e− experimen-
analyticityandunitarity to a dispersionin-
tal result, but those from CMD-2 (ππ), by the
tegral, akin to Eq. (4). Since the hadronic
factor C C , where C accounts for
correctioninvolvestheknowledgeofR(s)at HVP · FSR HVP
hadronic vaccum polarization and C stands
allenergies,includingthosewherethemea- FSR
fortheFSRcorrection. Weassignuncertaintiesof
surements are made, the procedure has to
50% (vacuum polarization) and 100% (FSR cor-
be iterative, and requires experimental as
rections), which are considered to be fully corre-
well as theoretical information over a large
lated between all channels to which the correc-
energy range.
tions apply.
The new data from CMD-2 [23] are
explicitly corrected for both leptonic 6. ISOSPIN BREAKINGIN e+e− ANDτ
and hadronic vacuum polarization effects,
SPECTRAL FUNCTIONS
whereasthepreliminarydatafromthesame
experiment[38]anddatafromother exper- Therelationships(5),(6)and(7)betweene+e−
iments were not (see follow up of this dis- and τ spectral functions only hold in the limit of
cussion in Ref. [22]) exact isospin invariance. This is the Conserved
5
Vector Current (CVC) property of weak decays. its value differs from the one given in Ref. [46],
It follows from the factorization of strong inter- because subleading quark-level and hadron-level
actionphysics as produced throughthe γ and W contributions should not be added, as double
propagators out of the QCD vacuum. However, counting wouldoccur. The correctexpressionfor
we know that we must expect symmetry break- the π−π0 mode therefore reads
ingatsomelevelfromelectromagneticeffectsand
ShadG
even in QCD because of the up and down quark EW EM =(1.0233 0.0006) G , (11)
mass splitting. Since the normalization of the τ SEsuWb,lep ± · EM
spectralfunctionsisexperimentallyknownatthe
thesubleadinghadroniccorrectionsbeingnowin-
0.5%level,itisclearthatisospin-breakingeffects
corporated in the mass-squared-dependent G
EM
mustbe carefullyexaminedifonewantsthis pre-
factor. Equation (10) explicitly corrects for the
cision to be maintained in the vacuum polariza-
mass difference between neutral and charged pi-
tion integrals.
ons affecting the cross section and the width of
Because of the dominance of the ππ contribu-
the ρ.
tion in the energy range of interest for τ data,
The different contributions to the isospin-
wediscussmainlythischannel,followingourear-
breakingcorrectionsareshowninthe secondcol-
lier analysis [4]. The corrections on ahad,LO from
µ umnofTable1. Thedominantuncertaintystems
isospin breaking are given in Table 1. A more from the ρ±-ρ0 mass difference.
complete discussion, in particular with respect
Since the integral (4) requires as input the
to the corrections previously applied is given in
e+e− spectral function including FSR photon
Ref. [22].
emission,afinalcorrectionisnecessary. Itisiden-
The dominant contribution to the electroweak
ticaltothatappliedintheCMD-2analysis[23,45]
radiative corrections stems from the short dis-
(cf. Section 5). The total correction to the τ re-
tancecorrectiontotheeffectivefour-fermioncou- sult amounts to ( 9.3 2.4) 10−10.
pling τ− ν (du¯)− enhancing the τ amplitude − ±
→ τ There exists no comparable study of isospin
by the factor Shad =1.0194 [46]. This correction
EW breaking in the 4π channels. Only kinematic
leavesoutthepossibilityofsizeablecontributions
corrections resulting from the pion mass differ-
from virtual loops. This problem was studied in
ence have been considered so far [29], which we
Ref. [30] within a model based on Chiral Pertur-
have applied in this analysis. It creates shifts of
bationTheory. Inthiswaythecorrectlow-energy 0.710−10 ( 3.8%)and+0.110−10 (+1.1%)for
hadronic structure is implemented and a consis- −2π+2π− and−π+π−2π0, respectively.
tent frameworkhas been set up to calculate elec-
troweak and strong processes, such as the radia- 7. COMPARISON OF e+e− AND τ
tive corrections in the τ ν π−π0 decay. Their
→ τ SPECTRAL FUNCTIONS
new analysis [31] directly applies to the inclusive
radiative rate, τ ντπ−π0 (γ), as measured by The e+e− andthe isospin-breakingcorrectedτ
→
the experiments. The relation between the Born spectral functions can be directly compared for
level e+e− spectral function and the τ spectral the dominant 2π and 4π final states. For the 2π
function reads [31] channel,theρ-dominatedformfactorfallsoffvery
1 β3 F0 2 rapidly at high energy so that the comparison
vπ+π− = G β30 Fπ− vπ−π0(γ) , (10) can be performed in practice over the full energy
EM − (cid:12) π (cid:12) rangeofinterest. Thesituationisdifferentforthe
(cid:12) (cid:12)
where GEM GEM((cid:12)s) is(cid:12)the long-distance radia- 4π channels where the τ decay kinematics limits
≡ (cid:12) (cid:12)
tive correction involving both real photon emis- theexercisetoenergieslessthan 1.6GeV,with
∼
sion and virtual loops (the infrared divergence only limited statistics beyond.
cancelsinthe sum). Notethatthe short-distance Fig.3showsthecomparisonforthe2πspectral
S correction, discussed above, is already ap- functions. Visually,theagreementseemssatisfac-
EW
plied in the definition of v (cf. Eq. (9)), but tory, howeverthe large dynamicalrange involved
−
6
t Average TOF
preliminary
3 CMD-2 (02)
10
CMD
)
b
n OLYA (low)
(
n OLYA (high)
o 2
cti 10 DM1
e
S DM2
s
s
o
r
C 10
‹ 4m2 threshold
p
1
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
s (GeV2)
500 1500
400
) )
b b
n n
( (
n 300 n 1000
o o
cti cti
e e
S S
s 200 s
os 4m2p thresh. os
r fi r
C C
500
100
‹ s[ Low s expansion]
0
0.1 0.2 0.3 0.4 0.3 0.4 0.5 0.6 0.7 0.8
s (GeV2) s (GeV2)
Figure 3. Comparison of the π+π− spectral functions from e+e− and isospin-breaking corrected τ data,
expressed as e+e− cross sections. The band indicates the combined e+e− and τ result within 1σ errors.
It is given for illustration purpose only.
7
t Average CMD-2 t Average CMD-2
0.2 preliminary CMD 0.2 preliminary CMD
2|[t]Fp ODMLY1A 2|[t]Fp ODMLY1A
|)/ |)/
2|[t]Fp 0 2|[t]Fp 0
|– |–
]e ]e
e e
2|[ 2|[
Fp Fp
|( |(
-0.2 -0.2
0.25 0.5 0.75 1 0.5 0.55 0.6 0.65 0.7
s (GeV2) s (GeV2)
Figure1. Relative comparisonof theπ+π− spec- Figure 2. Relative comparison in the ρ region
tral functions from e+e− and isospin-breaking of the π+π− spectral functions from e+e− and
corrected τ data, expressed as a ratio to the τ isospin-breaking corrected τ data, expressed as a
spectral function. The band shows the uncer- ratio to the τ spectral function. The band shows
tainty on the latter. the uncertainty on the latter.
7.1. Branching Ratios in τ Decays and
CVC
A convenient way to assess the compatibil-
does not permit an accurate test. To do so, the ity between e+e− and τ spectral functions pro-
e+e− data are plotted as a point-by-point ratio ceeds withthe evaluationofτ decayfractionsus-
to the τ spectral function in Fig. 1, and enlarged ing the relevant e+e− spectral functions as in-
in Fig. 2, to better emphasize the region of the ρ put. All the isospin-breaking corrections dis-
peak. The e+e− data are significantly lower by cussed in Section 6 are included. The advan-
2-3% below the peak, the discrepancy increasing tage of this procedure is to allow a quantitative
to about 10% in the 0.9-1.0 GeV region. comparison using a single number. The weight-
The comparison for the 4π cross sections is ing of the spectral function is however differ-
given in Fig. 4 for the 2π+2π− channel and in ent from the vacuum polarization kernels. Us-
Fig.5 forπ+π−2π0. Thelatter suffersfromlarge ing the branching fraction B(τ− ν e−ν¯ ) =
τ e
→
differences between the results from the different (17.810 0.039)%, obtained assuming leptonic
±
e+e− experiments. The τ data, combining two universalityin the chargedweakcurrent[25],the
measured spectral functions according to Eq. (7) results for the main channels are given in Ta-
andcorrectedforisospinbreakingasdiscussedin ble 2. The errors quoted for the CVC values
Section6,liesomewhatinbetweenwithlargeun- are split into uncertainties from (i) the experi-
certainties above 1.4 GeV because of the lack of mental input (the e+e− annihilation cross sec-
statistics and a large feedthrough background in tions) and the numerical integration procedure,
theτ ν π−3π0mode. Inspiteofthesedifficul- (ii) the missing radiative corrections applied to
τ
→
ties the π−3π0 spectral function is in agreement the relevant e+e− data, and (iii) the isospin-
with e+e− data as can be seen in Fig. 4. It is breaking corrections when relating τ and e+e−
clear that intrinsic discrepancies exist among the spectral functions. The values for the τ branch-
e+e− experiments andthat a quantitativetest of ing ratios involve measurements [25,47,48] given
CVC in the π+π−2π0 channel is premature. without charged hadron identification, i.e., for
8
Table 2
Branching fractions of τ vector decays into 2 and 4 pions in the final state. Second column: world
average. Third column: inferred from e+e− spectral functions using the isospin relations (5-7) and
correcting for isospin breaking. The experimental error of the π+π− CVC value contains an absolute
procedural integration error of 0.08%. Experimental errors, including uncertainties on the integration
procedure, and theoretical (missing radiative corrections for e+e−, and isospin-breaking corrections and
V for τ) are shownseparately. Right column: differences between the direct measurements in τ decays
ud
and the CVC evaluations, where the separate errors have been added in quadrature.
Branching fractions (in %)
Mode
τ data e+e− via CVC ∆(τ e+e−)
−
τ− ν π−π0 25.46 0.12 23.98 0.25 0.11 0.12 +1.48 0.32
τ exp rad SU(2)
→ ± ± ± ± ±
0.30
τ− ντπ−3π0 1.01 0.08 1.09 0|.06exp 0.02{rzad 0.05SU(2}) 0.08 0.11
→ ± ± ± ± − ±
0.08
τ− ντ2π−π+π0 4.54 0.13 3.63 |0.19exp 0.04{rzad 0.09SU(2}) +0.91 0.25
→ ± ± ± ± ±
0.21
| {z }
the hπ0ν , h3π0ν and 3hπ0ν final states. The investigated.
τ τ τ
corresponding channels with charged kaons have
beenmeasured[49,50]andtheircontributionscan 8. SPECIFIC CONTRIBUTIONS
be subtracted out in order to obtain the pure pi-
Insomeenergyregionswheredatainformation
onic modes. As expected from the preceding dis-
is scarce and reliable theoretical predictions are
cussion, a large discrepancy is observed for the
τ ν π−π0 branchingratio,withadifferenceof available, we use analytical contributions to ex-
τ
→ tend the experimental integral. Also, the treat-
( 1.48 0.12 0.25 0.11 0.12 )%,
τ ee rad SU(2)
− ± ± ± ± mentofnarrowresonancesinvolvesaspecificpro-
where the uncertainties are from the τ branching
ratio, e+e− cross sections, e+e− missing radia- cedure.
tive corrections and isospin-breaking corrections 8.1. The π+π− Threshold Region
(including the uncertainty on V ), respectively.
ud To overcomethe lackofprecise dataatthresh-
Adding all errors in quadrature, the effect repre-
old energies and to benefit from the analyticity
sentsa4.6σ discrepancy. Sincethedisagreement
property of the pion form factor, a third order
between e+e− and τ spectral functions is more
expansion in s is used. The pion form factor
pronounced at energies above 750 MeV, we ex- F0 is connected with the π+π− cross section via
π
pect a smaller discrepancy in the calculation of F0 2 =(3s/πα2β3)σ . Theexpansionforsmall
ahad,LO becauseofthe steeply fallingkernelK(s) | π| 0 ππ
µ s reads
in this case.
1
The situation in the 4π channels is different. F0 = 1+ r2 s+c s2+c s3+O(s4) , (12)
Agreementisobservedfortheπ−3π0modewithin π 6h iπ 1 2
an accuracy of 11%, however the comparison is where we use r2 = (0.439 0.008) fm2 [51],
not satisfactory for the 2π−π+π0 mode. In the h iπ ±
andthetwoparametersc arefittedtothedata
1,2
latter case, the relative difference is very large,
in the range [2m , 0.6 GeV]. The results of the
π
(22 6)%, compared to any reasonable level of
± fits are explicitly quoted in Ref. [22]. We show
isospin symmetry breaking. As such, it rather
the functions obtained in Fig. 6. Good agree-
points to experimental problems that have to be
ment is observed in the low energy region where
9
45
40 ALEPHpreliminary 50 CALMEDP-H2 preliminary
CMD-2 (low)
OLYA
35 CMD-2 (high)
ND 40 ND
nb) 30 CMD nb) SND
ction ( 25 ODMLY1A (low) ction ( 30 DMM3N2
Se 20 DM1 (high) Se
Cross 15 DM2 Cross 20
10
10
5
0 0
1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4
s (GeV2) s (GeV2)
Figure 4. Comparison of the 2π+2π− spectral Figure 5. Comparison of the π+π−2π0 spectral
functions from e+e− and isospin-breaking cor- functions from e+e− and isospin-breaking cor-
rected τ data, expressed as e+e− cross sections. rected τ data, expressed as e+e− cross sections.
the expansion should be reliable. Since the fits mationoftheintegralwhendealingwithstrongly
incorporate unquestionable constraints from first concave functions such as the tails of Breit-
principles, we have chosento use this parameter- Wigner resonance curves.
ization for evaluating the integrals in the range We therefore perform a phenomenological fit
up to 0.5 GeV. of a BW resonance plus two Gaussians (only
one Gaussian is necessary for the ω) to account
8.2. Integration over the ω and φ Reso- forcontributionsotherthanthe singleresonance.
nances Bothfitsresultinsatisfactoryχ2 values. Wehave
In the regions around the ω and φ resonances accounted for the systematics due to the arbi-
wehaveassumedinthe precedingworksthat the trariness in the choice of the parametrization by
cross section of the π+π−π0 production on the varying the functions and parameters used. The
one hand, and the π+π−π0, K+K− as well as resulting effects are numerically small compared
K0K0 productionontheotherhandissaturated to the experimental errors.
S L
by the corresponding resonance production. In a Since the experiments quote the cross sec-
datadrivenapproachitishowevermorecarefulto tion results without correcting for leptonic and
directly integrate the measurement points with- hadronicvacuumpolarizationinthephotonprop-
outintroducingpriorassumptionsontheunderly- agator, we perform the correction here. The cor-
ing process dynamics [52]. Possible non-resonant rectionofhadronicvacuum polarizationbeing it-
contributionsandinterferenceeffectsarethusac- erative and thus only approximative, we assign
counted for. half of the total vacuum polarization correction
Notwithstanding,astraightforwardtrapezoidal as generous systematic errors (cf. Section 5). In
integration buries the danger of a bias: with in- spite of that, the evaluation of ahad,LO is dom-
µ
sufficient scan density, the linear interpolation of inated by the experimental uncertainties. Since
the measurements leads to a significant overesti- the trapezoidal rule is biased, we choose the re-
10
12 calculation can be found in our earlier publica-
Low s exp (O|s4|) tions [7,11,22]and in the references therein.
10 Fit to t data ( ) AtestoftheQCDpredictioncanbeperformed
Fit to e+e– data ( ) intheenergyrangebetween1.8and3.7GeV.The
8 contributiontoahad,LO inthisregioniscomputed
µ
2|F(s)p 6 tbherleoswh opl+dp– tpoarbeed(w33it.h87t±he0r.4e6su)l1t0,−(3140.9usin1g.8Q)C10D−,1t0ofbroemcotmhe-
| ±
data. The two values agree within the 5% accu-
4
racy of the measurements.
2
int. limit fit limit
9. RESULTS
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
s (GeV2) 9.1. Lowest Order Hadronic Contributions
We use the trapezoidal rule to integrate the
Figure 6. Fit of the pion form factor from 4m2 experimental data points (with the exception of
π
to 0.35 GeV2 using a third-order Taylor expan- thenarrowresonances). Correlationsbetweenthe
sion with the constraints at s = 0 and the mea- measurementsaswellasamongexperimentshave
sured pion r.m.s. charge radius from space-like beentakenintoaccount[22]. Beforeaddingupall
data [51]. The result of the fit is integrated only the contributions to ahµad,LO, we shall summarize
up to 0.25 GeV2. the procedure. Onthe one hand,thee+e−-based
evaluation is done in three pieces: the sum of
exclusive channels below 2 GeV, the R measure-
ments in the 2-5 GeV range and the QCD pre-
sultsbasedontheBWfitsforthefinalevaluation
diction for R above. Major contributions stem
of ahad,LO.
µ from the 2π (73%) and the two 4π (4.5%) chan-
nels. On the other hand, in the τ-based evalua-
8.3. Narrow cc and bb Resonances
tion,thelatterthreecontributionsaretakenfrom
The contributions from the narrow J/ψ reso-
τ dataupto1.6GeVandcomplementedbye+e−
nances are computed using a relativistic Breit-
data above, because the τ spectral functions run
Wignerparametrizationfortheirline shape. The
out of precision near the kinematic limit of the τ
physical values for the resonance parametersand
mass. Thus,fornearly77%ofahad,LO(contribut-
their errors are taken from the latest compila- µ
ing 80%ofthe totalerror-squared),two indepen-
tion in Ref. [44]. Vacuum polarization effects are
dent evaluations (e+e− and τ) are produced, the
already included in the quoted leptonic widths.
remainder being computed from e+e− data and
The total parametrization errors are then calcu-
QCD alone.
lated by Gaussian error propagation. This inte-
Fig.7givesapanoramicviewofthee+e− data
gration procedure is not followed for the ψ(3S)
in the relevant energy range. The shaded band
statewhichisalreadyincludedintheRmeasure-
below 2 GeV represents the sum of the exclu-
ments, andfor the Υ resonanceswhichare repre-
sive channels considered in the analysis. It turns
sented in an average sense (global quark-hadron
out to be smaller than our previous estimate [4],
duality) by the bb QCD contribution, discussed
essentially because more complete data sets are
next.
usedandnew informationonthe dynamicscould
8.4. QCD Prediction at High Energy be incorporatedin the isospin constraints for the
Since the emphasis in this paper is on a com- missing channels. The QCD prediction is indi-
plete and critical evaluation of spectral functions cated by the cross-hatched band. It is used in
from low-energy data, we have adopted the con- this analysisonly for energiesabove5 GeV. Note
servativechoiceofusingtheQCDpredictiononly thattheQCDbandisplottedtakingintoaccount
above an energy of 5 GeV. The details of the the thresholds for open flavourB states, in order