Measurement of branching fractions and spectral functions in $\tau$ decays

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functions in τ decays

M. Davier

To cite this version:

Measurement of Bran hing Fra tions and Spe tral Fun tions in  De ays Mi helDavier

a a

Laboratoiredel'A elerateurLineaire,IN2P3/CNRS-UniversitedeParis-Sud, BP34,91898Orsay,Fran e

FullLEP-Idata olle tedbytheALEPHdete torduring1991-1995runningareanalyzedinordertomeasure the de aybran hingfra tionsandthehadroni spe tralfun tions.Theanalysisfollowstheglobalmethodused inthepublishedstudybasedon1991-1993 data,withseveralimprovements,espe ially on erningthetreatment ofphotonsand

0

's. Extensivesystemati studiesareperformed,inordertomat hthelargestatisti softhedata sample orrespondingto327148measuredandidentiedde ays.Preliminaryvaluesforthebran hingfra tions areobtainedforthe2leptoni hannelsand11hadroni hannelsdenedbytheirrespe tivenumbersof harged parti lesand 

0

's. Using previouslypublishedALEPH results onnal stateswith harged andneutral kaons, orre tionsareappliedsothatbran hingratiosforex lusivenalstateswithoutkaonsarederived. Somephysi s impli ationsofthe resultsare given,inparti ular on erning universality inthe leptoni hargedweak urrent, isospininvarian eina1 de ays,andthe separationof ve torandaxial-ve tor omponentsofthetotal hadroni rate. Spe tralfun tionsareobtainedforthemainex lusive hannelsandthetotalin lusiverate,withseparation oftheve torandaxial-ve torstates.

1. Introdu tion

A ompleteandnalanalysisof de aysis pre-sentedusingaglobalmethod. All datare orded at LEP-I with the ALEPH dete tor are used, thusprovidinganupdateofthosepreviousresults whi h were based on partial data sets. The in- reasein statisti s|thefullsample orresponds to about 2.5 times the luminosity used in the last published global analysis [1,2℄| not only allows for a redu tion of the dominant statisti- al error but, moreimportantly, provides away to better study and orre t systemati biases. Several improvements of the method have been introdu ed in order to a hieve a better ontrol overthe most relevant systemati un ertainties: simulation-independent measurement of the  sele tion eÆ ien y, improved photon identi a-tion espe ially at low energy where the separa-tion between photons from  de ays and fake photonsfrom u tuationsin hadroni or ele tro-magneti showers is deli ate, a new method to orre t the Monte Carlo simulation for the rate offakephotons,andstri ter riteriafor hannels withlowbran hingfra tions. For onsisten yand in order to maximally prot from the improved

analysis alldatasetsre ordedfrom1991to1995 have been repro essed. The resultspresentedin thispaperthussupersedethosealreadypublished in Ref. [3,1,2℄. Only the measurements on nal states ontainingkaons,whi hwerealreadybased onthefullstatisti s,remainun hanged[6{9℄.

2. Experimentalmethod

2.1. The data and simulated samples A detailed des riptionof theALEPHdete tor anbefoundelsewhere[10,11℄.

Tau-pair events are simulated by means of a MonteCarloprogramwhi hin ludesinitialstate radiation omputeduptoorder

2

and exponen-tiated, andnalstateradiative orre tionsto or-der [12,13℄. The simulationof the subsequent  de aysalsoin ludessinglephotonradiationfor the de ayswithup to three hadronsin thenal state.

The data used in this analysis have been re ordedatLEPIin1991-1995,foratotal

2.2. Sele tion of  events

The prin ipal hara teristi s of  events are lowmultipli ity,ba k-to-ba ktopologyand miss-ingenergy. Ea heventisdividedintotwo hemi-spheres by an energy ow algorithm [11℄ whi h al ulates allthevisible energyavoiding double- ounting between the TPC and the alorime-ter information. Cuts on the kinemati vari-ables of the two hemispheres are used to veto Bhabha,muon-pair, hadroni Z and -indu ed events[14,1,2℄.

Weusethe'break-mix'methodintrodu edfor thedetermination of the rossse tion [14℄ to measure the eÆ ien y of all the sele tion uts. For every ut, one hemisphere of the event is hosenjudi iouslysothatit isunbiasedwith re-spe t to the ut under study and free of non- ba kgrounds. Thispro eduresele tstheopposite hemisphereasanunbiased de aywhi histhen stored away. Pairs of sele ted hemispheres are ombined to onstru t a  event sample built ompletely from data. This sample is used to measuretheeÆ ien yofthegiven ut.

ThemeasuredeÆ ien ies arefoundto bevery lose to those obtained by the simulation, devi-ations being at most at the few per mille level. This situation stems from the fa ts that the  de aydynami sis |apartfrom smallbran hing ratio hannels| very well known, the sele tion eÆ ien iesarelargeandthesimulationofthe de-te torisadequate.Theoverallsele tioneÆ ien y of  events is 78.8 %. This value in reases to 91.9 % when the  angular distribution is re-stri ted to thedete tor polar a eptan e,giving a better indi ation for theeÆ ien y of the uts designedto ex ludenon- ba kgrounds. In ad-dition,whenexpressedrelativelytoea h de ay, thesele tioneÆ ien iesareweaklydependenton thenal state,with atotal relativespan ofonly 10%forthe13 onsideredde aytopologies. 2.3. Estimation ofnon- ba kgrounds

A newmethod |alreadyimplementedforthe measurementofthe polarization[15℄|hasbeen developed to dire tly measure in the nal data samples the ontributions from the major non- ba kgrounds. The pro edure doesnot require

simulation of these hannels, but only a quali-tative des ription of the distribution of the dis- riminatingvariables. Thebasi ideais toapply uts on the data in order to redu e as mu h as possible the population while keeping ahigh eÆ ien yfortheba kgroundsour eunderstudy, i.e. the reverse of what is done in the  sele -tion. Thenon- ba kgroundsinea h hannelare listedinTable1. Theyamounttoatotalfra tion of(1:230:04)% inthefulldatasample. 2.4. Charged parti leidenti ation

Chargedparti leidenti ationisa hievedwith a likelihood method in orporating the informa-tion from the relevant dete tors. In this way, ea h harged parti le is assigned a set of prob-abilitiesfromwhi haparti letypeis hosen. No attemptismadeinthisanalysistoseparatekaons frompionsinthehadronsamplesin enalstates ontainingkaonshavebeenpreviouslystudied[6{ 8℄. Theperforman eoftheparti leidenti ation has beenstudied in detail using ontrol samples of Bhabha events,  pairs, -indu ed lepton pairs and hadronsfrom 

0

-tagged de aysover thefullangularandmomentum range[1℄. 2.5. Photonidenti ation

Thehigh ollimationof  de aysatLEP ener-giesquiteoftenmakesphotonre onstru tion dif- ult,sin ethesephotonsare losetooneanother or lose to the showers generated by harged hadrons. Of parti ular relevan e is the reje -tion of fake photons whi h may o ur be ause of hadroni intera tions, u tuations of ele tro-magneti showers, or the overlapping of several showers. These problems rea h a tolerablelevel thanks to thene granularityofECAL, in both transverse and longitudinal dire tions, but nev-ertheless theyrequirethedevelopmentofproper andreliablemethodsinorderto orre tlyidentify photon andidates.

A likelihood method is used for dis riminat-ing between genuine and fake photons. For ev-ery luster above a threshold of 300 MeV, an estimator P

is dened, P

dis- orre ted throughdetailed omparisons between dataandMonteCarlo. Majorimprovementswere introdu edatthisstageintheanalysis ompared to the previous one [2℄, mainly in the hoi e of variables and in the use of energy-dependent referen e distributions. Better photon energy alibration is a hieved ompared to the previ-ousanalysis. Photon onversionsarealso re on-stru ted.

2.6.  0

re onstru tion Threedierentkindsof

0

'sarere onstru ted: resolved

0

from two-photonpairing, unresolved 

0

from merged lusters, and residual  0

from the remaining singlephotons after removing ra-diative,bremsstrahlungandfakephotonswitha likelihoodmethod.

Afterthepairingandthe lustermoment anal-ysis, all the remaining photons inside a one of 30

Æ

around the jet axis are onsidered. Radia-tiveandbremsstrahlungphotonsaresele ted us-ingalikelihood methodandarekeptseparatein the  de ay lassi ation dis ussed below. The identi ation of nal state radiative photons in hadroni hannelsisadiÆ ulttaskandandtoa largeextenttheanalysisreliesonthedes ription of radiation at thegenerator level in theMonte Carlo[16℄.

2.7. De ay lassi ation

Ea h de ayis lassiedtopologi ally a ord-ing to the number of hadroni harged tra ks, their parti le identi ation and the number of 

0

's re onstru ted. While for 1-prong and 5-prong hannelstheexa tmultipli ityisrequired, thetra knumber in 3-prong hannels is allowed to be2,3or4,in orderto redu e systemati ef-fe tsduetotra kingandse ondaryintera tions. The denition of the leptoni hannels requires an identiedele tron ormuon withany number ofphotons. Hemispheresreje tedbyspe i uts areputinaspe ial lass,labelled14,whi hdoes not orrespond to a nominal physi al  de ay mode. Thenumbersof's lassiedinea hofthe onsideredde ay hannelsarelistedinTable1.

The KORAL07 generator [12℄ in the Monte Carlosimulationin orporatesallthede aymodes onsidered in this analysis, ex ept for the h4

0

Table1

Numbers of re onstru ted and estimated non- ba kground events in 1991-1993 and 1994-1995 datasets inthedierent onsideredtopologies.

lass 91-93 94-95 e 22405 59846 33100 74558  22235 40945 32145 38040 h 15126 9311 22429 10013 h 0 32282 14122 49008 17826 h2 0 12907 449 18317 8116 h3 0 2681 267 3411 359 h4 0 458 123 499 195 3h 11610 8720 17315 12930 3h 0 6467 9723 9734 16539 3h2 0 1091 277 1460 3610 3h3 0 124 134 150 257 5h 60 31 105 72 5h 0 36 165 59 216 14 4834 24938 7100 30352 sum 132316 181586 194832 2224107

de ay hannelforwhi haseparategenerationwas done using aphase spa e model. The omplete behaviourbetweenthegeneratedde aysandtheir re onstru ted ounterpartsusingthede ay lassi- ationisembodiedintheeÆ ien ymatrix. This matrix"

ji

givestheprobabilityofa de ay gen-erated in lass j to be re onstru ted in lass i. Obtainedinitiallyusingthesimulatedsamples,it is orre tedforee tswheredataandsimulation anpossiblydier,su hasparti leidenti ation and photon identi ation. Forthe latter ee t, a detailed pro edure has been developed in or-der to orre t thesimulation offakephotonsby omparisonwithdata.

3. Results

where n obs i

isthe observed events numberin re- onstru ted lassi,n

bk g i

thenon- ba kgroundin re onstru ted lass i, "

ji

the eÆ ien y of a pro-du ed lassj eventre onstru tedas lass i, and N

prod j

theprodu edeventsnumberof lassj. The lassnumbersi, j runfrom1to14, thelast one orrespondingtothereje ted andidates.

The analysis assumes a standard de ay de-s ription. One ould imagine unknown de ay modes not in ludedin the simulation, but sin e largedete tioneÆ ien iesarea hievedin the sele tion whi h is therefore robust, it would be diÆ ult for these de ays to pass unnoti ed. An independentmeasurementofthebran hingratio forundete tedde aymodes,usingadire tsear h withaone-sided tag,wasdoneinALEPH[17℄, limiting this bran hing ratio to less than 0.11% at 95%CL. This result justies the assumption thatthesumofthebran hingratiosforvisible de aysisequaltoone.

Thebran hingratiosareobtainedandlistedin Table 2. The mostimportant systemati un er-tainties originatefrom photonidenti ation and 

0

re onstru tion,andse ondaryintera tions. 3.2. From re onstru ted lasses to

ex lu-sivemodes

The 13 lasses orresponding to major  de- aymodesstill ontainthe ontributionsfrom -nalstatesinvolvingkaons. Thelatterare oming fromCabibbo-suppressed de aysormodeswith aKK pair,both hara terized bysmall bran h-ing ratios ompared to the nonstrange modes without kaons. Complete analyses of  de ays involvingneutralor hargedkaonshavebeen per-formed byALEPH onthefullLEP I data[6{8℄. TheyaresummarizedandmeasurementswithK

0 S andK

0 L

are ombinedinRef.[9℄.

The de aysinvolving or! mesonsalso re-quire spe ial attention in this analysis be ause single photons in their ele tromagneti de ay modes aretreatedas

0

andidates. Corre tions are applied, based on spe i measurements by ALEPH [18℄ and CLEO [20℄ of  de ay modes ontainingthosemesons. Thisbookkeepingtakes into a ount all the major de ay modes of the onsidered mesons [19℄, in luding the

isospin-+

smaller ontributions 3(;!) have been identi-ed and measured by CLEO[21℄. Even though the orre tionsfromthese hannelsareverysmall theyhavebeenin ludedforthesakeof omplete-ness. Finally, another verysmall orre tion has beenappliedtotakeintoa ountthea

1

radiative de ayinto [22℄.

3.3. Overall onsisten ytest

Reje ted hemispheresbe auseof harged par-ti le identi ation uts (2 GeV minimum mo-mentum and ECAL ra kvetoforsome1-prong modes, stri t denition of higher multipli ity hannels)arepla edin lass 14. Asalready em-phasized, this sample does not orrespond to a nominal de aymodeandshouldbeexplainedby all other measured fra tions in the other lasses and the eÆ ien y matrix. Thus the determina-tion of a hypotheti al signal in this lass is a measure of the level of onsisten y a hieved in theanalysis.

ForthisdeterminationtheeÆ ien yofthe pos-sible signalin lass 14is taken tobe100%. The results, already shown in Table2 separately for the 1991-1993and 1994-1995data sets, are on-sistentandare ombinedtogiveB

14

=(0:066 0:042 %. This valueis onsistent with zeroand provides a good he k of the overall pro edure at the 0.1% level for bran hing ratios. It oin- ides, approximately and a identally, with the limit a hieved of 0.11% at 95% CL in a dire t sear h for `invisible' de ays not sele ted in the 13- hannel lassi ation. In the following it is assumedthat all de aymodeshavebeen prop-erly onsideredatthe0.1%pre isionlevelandno physi s ontributionbeyondstandard de aysis further allowed. Thus the quantity B

14

is now onstrainedtobezero.

It anbefurthernoti edthatthisanalysis pro-vides abran hingratioin the33

0

lass whi h is onsistent with zero (see Table 2). The re-sult is therefore given asan upper limit at 95% CL, B

33

0<4:910 4

, onsistentwiththe mea-surementmadebyCLEO [23℄yieldingB

33 0 = (2:20:5)10

4

. Thenalstateisdominatedby and!resonan es[23℄andusingother hannels al-lowsonetoobtainalowerlimitforthisbran hing

Table2

Bran hingratios(%)forthere onstru tedtopologies(quasi-ex lusivemodes)from1991-1993and 1994-1995datasets;thersterrorisstatisti al andthese ondissystemati .

lass 91-93 94-95 e 17.8590.1120.058 17.7990.0930.045  17.3560.1070.055 17.2730.0870.039 h 12.2380.1050.104 12.0580.0880.083 h 0 26.1320.1500.104 26.3250.1230.090 h2 0 9.6800.1390.124 9.6630.1070.105 h3 0 1.1280.1100.086 1.2290.0890.068 h4 0 0.2270.0560.047 0.1630.0500.040 3h 9.9310.0970.072 9.7690.0800.059 3h 0 4.7770.0930.074 4.9650.0770.066 3h2 0 0.5170.0630.050 0.5510.0500.038 3h3 0 0.0160.0290.020 -0.0210.0230.019 5h 0.0980.0140.006 0.0980.0110.004 5h 0 0.0220.0100.009 0.0280.0080.007 14 0.0170.0430.042 0.0990.0350.037 (31)10 4

isusedasinputforthis hanneland the global analysis is performed in terms of the remaining12dened hannelswhi hareretted.

3.4. Final ombinedresults

The twosets of results are onsistent and are nally ombinedinTable3.

Thebran hingratiosobtainedforthedierent hannels are orrelatedwith ea h other. On one hand the statisti al u tuations in thedata and theMonteCarlosamplesaredrivenbythe multi-nomial distribution of the orrespondingevents, produ ing well-understood orrelations. Onthe otherhandthesystemati ee tsalsoindu e im-portant and nontrivial orrelations between the dierent hannels. All the systemati studies were done keeping tra k of the orrelated vari-ations in the nal bran hing ratio results, thus allowingaproperpropagationoferrors.

The present results are onsistent with those of the previously published ALEPH analysis [1, 2℄. The leptoni bran hing ratios also agree within errorswith the resultsof anindependent ALEPHanalysiswhi hdoesnotrelyonaglobal method[24℄.

4. Dis ussion ofthe results

4.1. Comparisonwith otherexperiments Figs. 1, 2, 3, 4 show that the results of this analysis are in good agreementwith those from othermostpre iseexperiments. Inallthese ases, ALEPHa hievesthebestpre ision.

4.2. Universality in the leptoni harged urrent

4.2.1. -e universality from the leptoni bran hingratios

InthestandardV-Atheorywithleptoni ou-pling g

l

at the Wl l

Table3

Combinedresultsfortheex lusivebran hing ra-tios (B)formodeswithoutkaons. The ontribu-tionsfrom hannelswith  and! aregiven sep-arately,thelatteronlyfortheele tromagneti ! de ays. All resultsarefrom this analysis,unless expli itlystated. Thevalueslabelled*and**,*** are taken from ALEPH [18℄ and CLEO [21,20℄, respe tively. mode B stat  syst [%℄ e 17.8370.0720.036  17.3190.0700.032  10.8280.0700.078   0 25.4710.0970.085  2 0 9.2390.0860.090  3 0 0.9770.0690.058  4 0 0.1120.0370.035    + 9.0410.0600.076    +  0 4.5900.0570.064    + 2 0 0.3920.0300.035    + 3 0 (estim.) 0.0130.0000.010 3 2 + 0.0720.0090.012 3 2 +  0 0.0140.0070.006   0   0.1800.0400.020  2 0   0.0150.0040.003    +   0.0240.0030.004 a 1 (! )(estim.) 0.0400.0000.020  !(! 0 ; +  )  0.2530.0050.017   0 !(! 0 ; +  ) ; 0.0480.0060.007  2 0 !(! 0 ; +  )  0.0020.0010.001    + !(! 0 ; +  )  0.0010.0010.001 f(x) = 1 8x+8x 3 x 4 12x 2 lnx (4) Numeri ally, the W propagator orre tion and the radiative orre tions are small: Æ

 W = 1+ 2:9
10 4 andÆ  =1+43:2
10 4 .

Takingtheratioofthetwoleptoni bran hing fra tions, a dire t test of -e universality is ob-tained. Themeasuredratio

B  B

e

=0:97090:00600:0029 (5) agrees with the expe tation whi h is equal to 0.97257 when universality holds. Alternatively themeasurementsyieldtheratioof ouplings

g  g =0:99910:0033 (6)

17

18

19

20

B(e

νν

–

) (%)

ALEPH 89-90

ARGUS

ALEPH 91-93

CLEO 97

DELPHI 91-95

OPAL 91-95

L3 91-95

ALEPH 91-95 Prel.

average

18.09

±

0.45

±

0.45

17.5

±

0.3

±

0.5

17.79

±

0.12

±

0.06

17.76

±

0.06

±

0.17

17.877

±

0.109

±

0.11

17.81

±

0.09

±

0.06

17.806

±

0.104

±

0.076

17.837

±

0.072

±

0.036

17.823

±

0.051

Figure 1. Comparison of ALEPH measurement with publishedpre ise resultsfrom other experi-mentsfor !e.

whi his onsistentwithone.

This result is in agreementwith the best test of -e universality of the W ouplings obtained in the omparison ofthe ratesfor  !

 and  !e

e

de ays,wheretheresultsofthetwomost a urate experiments [26,27℄ anbeaveragedto yield

g ge

= 1:00120:0016. The results have omparable pre ision, but it should be pointed out that theyarein fa t omplementary. The resultgivenhereprobesthe ouplingto a trans-verseW(heli ity1)whilethede aysmeasure the ouplingto alongitudinal W (heli ity0). It is indeed on eivablethat either approa h ould be sensitiveto dierentnonstandard orre tions to universality.

4.2.2. Testsof - and -euniversality Comparing the rates for ( ! 

radia-17

18

19

B(

µνν

–

) (%)

ALEPH 89-90

ARGUS

OPAL 90-92

ALEPH 91-93

CLEO

DELPHI 91-95

L3 91-95

ALEPH 91-95 Prel.

average

17.35

±

0.41

±

0.37

17.4

±

0.3

±

0.5

17.36

±

0.27

±

0

17.31

±

0.11

±

0.05

17.37

±

0.08

±

0.18

17.325

±

0.095

±

0.077

17.342

±

0.11

±

0.067

17.319

±

0.07

±

0.032

17.331

±

0.054

Figure 2. Comparison of ALEPH measurement withpublished pre iseresultsfrom other experi-mentsfor !.

tive orre tions,oneobtains  g  g   2 =      m  m   5 B e f( m 2 e m 2  ) f( m 2 e m 2  )
W
(7)  g  g e  2 =      m  m   5 B  f( m 2 e m 2  ) f( m 2  m 2  )
W
(8) wheref( m 2 e m 2  )=0:9998,
W = Æ  W Æ  W =1 2:9
10 4 ,
= Æ  Æ  =1+8:5
10 5 , and  l isthe leptonl lifetime.

FromthepresentmeasurementsofB e ,B  ,the  mass [19℄, m  = (1777:03 +0:30 0:26 ) MeV (domi-natedbytheBESresult[28℄),the lifetime[19℄, 

=(290:61:1)fsandtheotherquantitiesfrom Ref.[19℄,universality anbetested:

g  g = 1:00090:00230:00190:0004 (9)

11.5

12

12.5

B(h

ν

) (%)

CLEO 97

OPAL 91-95

ALEPH 91-95 Prel.

average

11.52

±

0.05

±

0.12

11.98

±

0.13

±

0.16

11.524

±

0.07

±

0.078

11.584

±

0.076

Figure 3. Comparison of ALEPH measurement with publishedpre ise resultsfrom other experi-mentsfor !h (sum of andK).

g  g

e

= 1:00010:00220:00190:0004;(10) where theerrorsarefrom the orresponding lep-toni bran hingratioandthe lifetimeandmass, respe tively.

4.2.3. - universality from the pioni bran hingratio

The measurementofB 

also permits an inde-pendenttestof-universalitythroughthe rela-tion  g  g   2 = B  B !      2m  m 2  m 3   1 m 2  =m 2  1 m 2  =m 2  ! 2 Æ = ; (11)

where the radiative orre tion [29℄ amounts to Æ

=

=1:00160:0014Usingtheworld-averaged valuesforthe and (

 and 



25

26

27

28

B(h

π

o

ν

) (%)

CLEO 94

OPAL 91-95

ALEPH 91-95 Prel.

average

25.87

±

0.12

±

0.42

25.89

±

0.17

±

0.29

25.924

±

0.097

±

0.085

25.916

±

0.116

Figure 4. Comparison of ALEPH measurement withpublished pre iseresultsfrom other experi-mentsfor !h 0  (sumof 0  andK 0 ).

the bran hing ratio for the de ay  !  [19℄, thepresentresultforB

 ,oneobtains g  g  =0:99620:00480:00190:0002; (12) omparing the measured value (B



= 10:823 0:104)%to theexpe tedoneassuming universal-ity(10:9100:042)%. ThequotederrorsinEq.12 arefrom the pionmodebran hingratioand the  lifetime andmass,respe tively.

Thetwodeterminationsof g g obtainedfromB e andB 

are onsistentwithea hotherand anbe ombinedtoyield g  g  =1:00000:00210:00190:0004; (13) where the errorsare from the ele tron and pion bran hing ratioand the lifetime and mass, re-spe tively. Universalityof the  and  harged- urrent ouplings holds at the 0.29% level with

17.4

17.6

17.8

18

18.2

B

_e

(%)

ALEPH Preliminary

B

_e

ALEPH

B

_µ

ALEPH

B

_π

ALEPH

τ

_τ

WA

average

Figure 5. The measuredvalue forB e

ompared topredi tionsfromothermeasurementsassuming leptoni universality. Theverti albandgivesthe averageofalldeterminations.

mination of B e

andB 

, and theworld-averaged valueforthe lifetime.

The onsisten yofthepresentbran hingratio measurements with leptoni universality is dis-playedin Fig.5where the resultfor B

e

is om-paredto omputedvaluesofB

e usingasinputB  (assuming e  universality),   and   (  universality),andB  and  (  universality). All valuesare onsistentandyieldtheaverage B universality e =(17:8100:039)% : (14) 4.2.4. a 1 de ays to 3 and 2 0

Withthelevelofpre isionrea heditis interest-ingto omparetheratesinthe3and2

0 han-nels whi h are ompletely dominated by the a

1 resonan e. Thedominant  intermediate state leadstoequalrates,butasmallisospin-breaking ee tisexpe tedfromdierent hargedand

nel.

A re ent CLEO partial-wave analysis of the 2

0

nal state [30℄ has shown that the sit-uation is in fa t mu h more ompli ated with many intermediate states, in parti ular involv-ing isos alars, amounting to about 20% of the total rate and produ ing strong interferen e ef-fe ts. A good des ription of the a

1

de ays was a hieved in the CLEO study, whi h an be ap-pliedtothe3nal state,predi ting[30℄ aratio oftherates 3/2

0

equalto 0.985. Thisvalue, whi h in ludes known isospin-breakingfrom the pion masses,turns outto bein good agreement withthemeasuredvaluefromthisanalysiswhi h showstheexpe tedtrend

B 3 B 2 0 =0:9790:018: (15) 4.2.5. The  0

bran hing ratio in the on-text ofa

had  The

0

nalstateisdominatedbythe reso-nan e. Itsmassdistribution|orbetterthe or-responding spe tral fun tion| is a basi ingre-dient of va uum polarization al ulations, su h asthat used for omputing the hadroni ontri-butiontotheanomalousmagneti momentofthe muona

had 

. Inthis asethe ontributionis dom-inant(71%)andtherefore ontrolsthenal pre i-sionoftheresult. ItwasproposedinRef.[31℄ to usethespe tralfun tionsobtainedfromthe mea-surementofhadroni de aysinordertoimprove thepre isionof thepredi tionfora

had 

. The al- ulationwaslaterimprovedwiththehelpofQCD onstraintsforenergiesabovethe mass[32℄and evenbelow[33℄.

The normalization of the spe tral fun tion is provided by the bran hing fra tion B

 0. The presentworldaverageis ompletelydominatedby thepublished ALEPHresult[2℄. Thenewresult givenhereislargerby0.68%,thusone anexpe t aslightlylarger ontributionto a

had 

.

A newevaluation [34℄ wasavailable,usingthe preliminary spe tral fun tions from the present analysis, the published CLEO results [35℄ and new results from e

+

e annihilation from CMD-2 [36℄. Revision of the CMD-2 results [37℄ prompted a re-evaluation [38℄, whi h revealed a

+

fun tions. Whereasthe estimateleadstoa pre-di tion onthemuon magneti momentin agree-ment with the latest most pre ise measurement from the BNL experiment E-821 [39℄, the pre-di ted value using only e

+

e data lies 2.4 stan-darddeviationsbelowthemeasurement. Inview of this situation,it isimportantto he kall the ingredients,inparti ularthedeterminationofthe bran hingratioB

 0.

Asmostofthesystemati un ertaintyin B 

0 omes from =

0

re onstru tion, it is interest-ing to ross he k the results in the 'adja ent' hadroni modes, i.e. the  and 2

0

hannels. Thisispossibleifuniversalityintheweak harged urrent is assumed, leading to an absolute pre-di tion of B



using as input the  lifetime (see Se tion 4.2.2), and by omputing B

2 0 from the measurementof B

3

whi h is essentially un- orrelated with B



0 (see Se tion 4.2.4). The two omparisons,B  B uni  =( 0:080:11)%, B 2 0 B iso 2 0 =(0:060:16)%,donotpointto anysystemati biasinthedeterminationofB

 0 within thequotedun ertainty.

4.2.6. Separation of ve tor and axial-ve tor ontributions

Fromthe ompleteanalysisofthe bran hing ratios presented in this paper, it is possible to determine the nonstrange ve tor (V) and axial-ve tor(A) ontributions to thetotal  hadroni width, onveniently expressed in terms of their ratiostotheleptoni width, alledR

;V andR

;A , respe tively. The determination of the strange ounterpartR

;S

isalreadypublished[9℄. The ratio R

for the total hadroni width is al ulatedfrom thedifferen e oftheratioof the totalhadroni widthandele troni bran hing ra-tio, R  = 1 B e B  B e = 1 B e 1:97257 = 3:6420:012: (16) takingforB( !e  e  

)thevalueobtainedin Se tion 4.2 assuminguniversalityin theleptoni weak urrent. UsingtheALEPHmeasurementof thestrangewidthratio[9℄,veryslightlymodied

byCLEO[40℄ R

;S

=0:16030:0064; (17) thefollowingresultisobtainedforthenonstrange omponent

R ;V+A

=3:482  0:014: (18) SeparationofVandA omponentsinhadroni nal states with only pions is straightforward. Isospininvarian erelatesthespin-parity ofsu h systems to their number of pions: G-parity =1 (evennumber) orrespondstove torstates,while G=-1(oddnumber)tagsaxial-ve torstates. This propertypla es astrongrequirementonthe eÆ- ien yof

0

re onstru tion,a onstraintthatwas stronglyintegratedinthisanalysis.

ModeswithaKKpairarenotingeneral eigen-statesof G-parityand ontribute tobothV and A hannels. While the de ay to K K

0

is pure ve tor, the situation in the KK mode is not lear and a onservative axial-ve torfra tion of 0:750:25isassumed. Forthede aysintoKK no informationis available in this respe t anda fra tion0:50:5istaken.

The total nonstrange ve tor and axial-ve tor ontributionsobtainedinthisanalysisare: R ;V = 1:7870:0110:007; (19) R ;A = 1:6950:0110:007; (20) where the se ond errorsre e t theun ertainties in theV=A separationin the hannels withKK pairs. Taking areofthe orrelationsbetweenthe respe tive un ertainties, one obtains the dier-en ebetweentheve torandaxial-ve tor ompo-nents,whi hisphysi allyrelatedtotheamountof nonperturbative QCD ontributions in the non-strangehadroni  de aywidth:

R ;V A

=0:0920:0180:014; (21) whereagainthese onderrorhasthesame mean-ingasin Eqs.(19)and(20). Theratio

R ;V A R

;V+A

=0:0260:007 (22) is a measure of the relative importan e of

non-4.3. Summaryofbran hingfra tions The resultspresentedhere are ombinedwith previously published ALEPH results on nal stateswithkaonsin Table4.

5. Spe tral fun tions

Thehadroni massspe train thedierent ex- lusive hannels iare orre tedbythe appropri-ate kinemati fa tor and normalized by B

i =B

e in 140mass bins. The orre ted spe traare un-foldedfromdete toree tsbyaregularized inver-sionofthe140  140dete torresponsematrix[41℄. Studies are performedusing the fullanalysis for every sour e of systemati un ertainty. Corre-sponding ovarian ematri esarebuiltfor and 

0

re onstru tion,energy alibrationand resolu-tion for harged parti lesand 

0

's, tra kingand se ondaryintera tions, and theunfolding pro e-dure. Spe tral fun tions are thus obtained for theleading hannels: 

0 ,2 0 ,3 0 ,4 0 ,3, 3 0 ,32 0 ,and5.

The spe tral fun tion for the  0

hannel is giveninFig.6,whileFig.7showsthe onsisten y betweentheunfoldedmassspe trainthe3and 2

0

hannels.

The in lusive ve tor (V)and axial-ve tor(A) spe tralfun tionsare determined(Fig.8). They are respe tively dominated by the low-lying  and a

1

resonan es and are shown to approa h thesmoothQCDpredi tionat largermasses, al-thoughtheyare learlynot`asymptoti 'atthe mass. The onvergen etotheperturbativeQCD regimeismu hbetterrealizedfortheV+A spe -tralfun tion,asshowninFig.9,whiletheV A part in Fig. 10 undergoes large damped os illa-tionsaroundzero.

The QCD analysis of the nal spe tral fun -tionsisinprogress.

6. Con lusions

Table4

AsummarylistofALEPHbran hingratios(%). The labels V, A and S refer to the nonstrange ve torand axialve tor,and strange omponents, respe tively. The! de aymodesmarked(*)are ele tromagneti ( 0 ; +  ). mode B tot [%℄ ALEPHprelim. e 17.8370.080  17.3190.077  10.8280.105 A   0 25.4710.129 V  2 0 9.2390.124 A  3 0 0.9770.090 V  4 0 0.1120.051 A    + 9.0410.097 A    +  0 4.5900.086 V    + 2 0 0.3920.046 A    + 3 0 0.0130.010 V 3 2 + 0.0720.015 A 3 2 +  0 0.0140.009 V   0  0.1800.045 V (3)  0.0390.007 A a 1 (! ) 0.0400.020 A  ! (*) 0.2530.018 V   0 ! (*) 0.0480.009 A (3) ! (*) 0.0030.003 V K K 0 0.1630.027 V K  0 K 0 0.1450.027 (7525)%A  K 0 K 0 0.1530.035 (7525)%A K K +  0.1630.027 (7525)%A (KK) 0.050.02 (5050)%A K 0.6960.029 S K  0 0.4440.035 S K 0  0.9170.052 S K 2 0 0.0560.025 S K  +  0.2140.047 S K 0   0 0.3270.051 S (K3) 0.0760.044 S K  0.0290.014 S

10

-1

1

10

0

1

2

3

Mass

2

(GeV/c

2

)

2

|F

π

|

2

ALEPH 91-95

Gounaris-Sakurai Fit

10

15

20

25

30

35

40

45

50

0.4

0.5

0.6

0.7

0.8

spe tralfun tionttedusing the Gounaris-Sakurai parametrization with on-tibutions fromthe,', and" resonan es.

0

0.05

0.1

0.15

0.2

0.25

x 10

-2

0

1

2

3

Mass

2

(GeV/c

2

)

2

(1/N)dN/0.025 (GeV/c

2

)

2

τ

−

_→

_π

₂

_π

0

_ν

τ

τ

−

_→

_π

−

_π

+

_π

−

_ν

τ

Figure7. Comparisonofthe orre tedmass spe -traforthe3 and2

0

0

0.5

1

1.5

2

2.5

3

0

1

2

3

Mass

2

(GeV/c

2

)

2

v

1

ALEPH 91-95

τ

−

→

_(V

−

_,I=1)

ν

τ

Perturbative QCD (massless)

Parton model prediction

ππ

0

π

3

π

0

,3

ππ

0

,6

π

(MC)

ωπ

,

ηππ

0

,KK

0

(MC)

π

KK-bar(MC)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

1

2

3

Mass

2

(GeV/c

2

)

2

a

1

ALEPH 91-95

τ

−

→

_(A

−

_,I=1)

ν

τ

Perturbative QCD (massless)

Parton model prediction

π

2

π

0

,3

π

π

4

π

0

,3

π

2

π

0

,5

π

π

KK-bar(MC)

Figure 8. The ve tor and axial-ve tor spe -tral fun tions, with ontributions from ex lusive hannels indi ated (shapes of ontributions la-beled \MC" taken from the simulation). The QCD predi tion is obtained with 

s (M 2 Z ) = 0:120.

0

0.5

1

1.5

2

2.5

3

0

1

2

3

Mass

2

(GeV/c

2

)

2

v

1

+a

1

ALEPH 91-95

τ

−

_→

_(V,A,I=1)

_ν

τ

Perturbative QCD (massless)

Parton model prediction

Figure 9. The in lusive V +A spe tral fun -tion and predi tionsfrom theparton modeland frommasslessperturbativeQCDusing

s (M 2 Z )= 0:120.

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0

1

2

3

Mass

2

(GeV/c

2

)

2

v

1

-a

1

ALEPH 91-95

τ

−

_→

_(V,A,I=1)

_ν

τ

Perturbative QCD/Parton model

multipli ityupto4 0

'sinthenalstate. Major improvementsareintrodu edwithrespe ttothe published analysisandabetterunderstanding is a hieved,in parti ularintheseparationbetween genuine andfakephotons. As modeswith kaons (K  ,K 0 S ,andK 0 L

)havealreadybeenstudiedand published withthefullstatisti s,thenonstrange modes withoutkaonsareemphasized. Taken to-gether these resultsprovide a omplete des rip-tionofthe de aymodesupto6hadronsinthe nalstate.

Themeasuredbran hingratiovaluesare inter-nally onsistentandagreewithknown onstraints fromothermeasurementsintheframeworkofthe StandardModel. Thepre isionrea hedand the ompleteness of the results are for the moment unique. Morespe i ally, theresultsonthe lep-toni and pioni fra tions lead to powerfultests of universalityin the hargedleptoni weak ur-rent, showing that the e   ouplings are equal within 2-3 per mille. The bran hing ra-tio of  ! 

 

0

, whi h is of parti ular inter-est tothea uratedetermination ofva uum po-larizationee ts, is determined with apre ision of 0.5% to be (25:470:13) %. Also the ratio of a

1

bran hing fra tions into 2 0

and3 nal statesis measuredtobe0:9790:018,in agree-mentwithexpe tationfrompartialwaveanalyses ofthese de ays. Separatingnonstrangehadroni hannels into ve tor (V) and axial-ve tor (A) omponents and normalizing to the ele troni width yields the ratios R

;V = 1:7870:013, R ;A = 1:6950:013, R ;V+A = 3:4820:014 andR ;V A =0:0920:023.

The spe tral fun tions for the main hadroni modeshavebeenextra tedfromthemass distri-butions. The separated ve tor and axial-ve tor omponentsare the basi input for QCD analy-sesandva uum polarization al ulations.

A knowledgements

I would like to thank the organizers of the Tau04 Workshop for their hospitality and their eÆ ient running of the meeting. Many thanks are due to my olleagues ChangzhengYuan and Zhiqing Zhang for their major ontributions to

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