Dispersive analysis of $τ^-\toπ^-π^0ν_τ$ Belle data

0.18 MB·

by  Daniel Gómez Dumm

NuclearPhysicsB
Proceedings
Supplement
NuclearPhysicsBProceedingsSupplement00(2013)1–3
Dispersive analysis of τ− → π−π0ν Belle data
τ
DanielGo´mezDumma,PabloRoigb
aIFLP,CONICET−Dpto.deF´ısica,UniversidadNacionaldeLaPlata,C.C.67,1900LaPlata,Argentina.
bGrupdeF´ısicaTeo`rica,InstitutdeF´ısicad’AltesEnergies,UniversitatAuto`nomadeBarcelona,E-08193Bellaterra,Barcelona,Spain.
3
1
0
2 Abstract

n
WeanalyseBelledataonthedecayτ− →π−π0ν usingadispersiverepresentationofthevectorformfactorwhichis
a τ
J consistentwithchiralsymmetryandpreservesanalyticityandunitarityexactly.Wefittheunknowntheoreticalparam-
0 etersfromthedata,determiningthevaluesoftherelatedlow-energyobservables(cid:68)r2(cid:69)π andcπ. Theimplementation
3 ofisospinbreakingeffectsisalsodiscussed[1]. V V

] Keywords: Taudecays,dispersionrelations,chiralLagrangian,hadronicformfactors,mesonresonances
h
p
-
p 1. Introduction light-flavoured QCD allows to build an effective field
e theory, known as Chiral Perturbation Theory (χPT)
h Thevectorformfactorofthepion,Fπ(s),encodesall
[ unknown strong interaction dynamics iVn τ− → π−π0ντ [lo7w, 8-e,n9e,rg1y0]p,awrthoicfhthiessapbelcetrtuomd,esbcurtibfaeilasccautrlaatregleyrtihne-

1 decays[2]. Itisdefinedas variant masses [11]. In this region, new particles (ρ,
v √ K(cid:63), a , ...) are excited and their momenta and masses
7 <π−(p )π0(p )|d¯γµu|0>= 2(p −p )µFπ(s),(1) 1
π− π0 π− π0 V are large enough to prevent their use in the expansion
6
1 with s=(pπ− + pπ0)2. Thisformfactorisnotonlyrele- parametersoftheeffectivetheory. Thereforethesenew
7 vantfortheunderstandingofthehadronizationofQCD activedegreesoffreedomhavetobeincludedintheac-
1. currents at low energies (see [3]), but also represents a tion, andanewexpansionparameterisneeded. Inthis
0 crucialingredientfortheevaluationoftheleadingorder respect,theinverseofthenumberofcoloursoftheQCD
3 (LO) hadronic contribution to the anomalous magnetic gaugegrouphasproventobeausefulquantitytobuild
1
moment of the muon, which provides a very stringent theexpansionupon[12,13,14]. Amodelizationofthis
:
v probe of new physics [4]. In addition, from the high- ideainthemesonsectorforthreeflavoursisprovidedby
Xi energyperspective,theππchannelis(togetherwiththe ResonanceChiralTheory(RχT)[15,16,17,18],which
three pion channel) essential to follow the spin in the recoverstheχPTresultsatnext-to-leadingorder(NLO).
r
a Higgs di-tau decay channels at LHC [5], and thus to Sinceitisknownthatthelightestresonancesdominate
determine its spin and CP properties with the help of the dynamics, the infinite tower of states predicted in
TAUOLA[6]. the NC → ∞ limit can be restricted to the first excita-
tions,takingintoaccountasmanystatesasrequiredby
thedata. Inaddition,itisseenthattheinclusionofres-
2. Theoreticalsetting
onancewidthsisessentialtodescribetheobservedphe-
In τ− → π−π0ν decays the electroweak part of nomenology,althoughwidthsariseatNLOinthe1/NC
τ
expansion. In principle, one can take into account this
the process is theoretically under control, while the
effectbycomputingoff-shellwidthsconsistentlywithin
hadronization of the quark currents is more involved
RχT [19, 20]. On the other hand, as in any effective
since in the spanned energy region QCD is essentially
theory,thesymmetrypropertiesdeterminetheoperators
non perturbative. The approximate chiral symmetry of
/NuclearPhysicsBProceedingsSupplement00(2013)1–3 2
allowedintheLagrangian,butleavethecorresponding M2
= (cid:110) (cid:104) V (cid:105)(cid:111) .(5)
couplingconstantsunknown. However,theQCDshort- M2 1+ s (cid:60)e A (s)+ 1A (s) −s−iM Γ (s)
distance behaviour of the Green functions and related V 96π2F2 π 2 K V V
formfactors[15,16,17,18,21,22,23,24,25,26]pro- Thustherelevant(I = 1, J = 1)phaseshiftistakento
videasetofrelationsamongthesecoefficientsthatren- be
dersRχTmorepredictive.
(cid:61)mFπ(s)(0)
tanδ1(s)= V . (6)
1 (cid:60)eFπ(s)(0)
V
3. Vectorformfactorofthepionandfitstodata
Thisisnowusedasinputforathree-subtracteddisper-
Different approaches have been developed to deal sionrelationfortheformfactor. Inthiswayonegets
wpuitthatitohnedoifveFrsVπe(se)neartgyNNreLgiOmeins.χFPoTr s[2<7,M2ρ28,,th2e9c,o3m0-] FVπ(s) = exp(cid:26)α1s+ α22s2
pcarobvileitsyuusepfutol.1InGoerVd,eurntoitaernizlaartgioenthteecdhonmiqauienso[f31a,pp3l2i]- +s3 (cid:90) ∞ ds(cid:48) δ11(s(cid:48))  . (7)
π (s(cid:48))3(s(cid:48)−s−i(cid:15))
and the Omne`s solution to the dispersion relation have 4m2π
been employed [33, 34, 35]. Finally, in order to reach
Kinematical isospin corrections can be easily in-
energiesuptoMτ,theinclusionoftheρ(cid:48)resonance[36] cluded in Eq. (7) by considering different masses for
andevenatowerofresonances,inspiredintheN →∞
C thechargedandneutralpionsandkaons. Inaddition,at
limit[37,38], havebeenproposed. FromtheRχTLa-
the same order one should also take into account elec-
grangian, including only the ρ(770) multiplet one ob-
tromagnetic corrections [2, 40], which enter through a
tains local term felm and a global factorG (s) [41]. Thus
local EM
F G s weconsiderthreepossibleexpressionsfortheformfac-
Fπ(s)=1+ V V , (2)
V F2 M2 −s tortoperformourfitstoBelledata:
V
where F is the pion decay constant in the chiral limit, • FitI,correspondingtoFVπ(s)fromEq.(7).
M = M , and F andG measurethestrengthofthe
V ρ V V • FitII,sameasIbutwiththeinclusionofkinemat-
ρππandρVµ couplings, respectively, andVµ standsfor icalcorrections(m (cid:44)m ,m (cid:44)m ).
π± π0 K± K0
the quark vector current. If the vanishing of the form
factoratlargeenergiesisrequired,therelationF G = • Fit III, including kinematical and electromagnetic
V V
F2isobtained,yieldingFπ(s)= MV2 . Nowonecando corrections.
V M2−s
V
better[33]andmatchthisexpressiontotheNLOresult Ourfittingparametersare M ,F,α andα . Itisfound
ρ 1 2
inχPT.Finalstateinteractionsareincludedthroughthe thatwithouttheinclusionofadditionalresonances,one
so-called chiral loop functions AP(s,µ2 = Mρ2). Then, canobtaingoodfitstoBelledatafors(cid:46)1.5GeV2.The
the unitarity and analyticity constraints determine the bestfitresultstothefirst30points(centralvalueofthe
Omne`sexponentiationofthefullloopfunction,leading bin corresponding to 1.525 GeV2, with 0.05 GeV2 bin
to width)areshowninTable1,wherewehaveconsidered
M2 (cid:40) −s (cid:34) 1 (cid:35)(cid:41) the 1/NππdNππ/ds distribution measured by Belle (this
FVπ(s)= M2 −V sexp 96π2F2 Aπ(s)+ 2AK(s) . (3) includes error correlations). These fits show, firstly,
V that the dispersive description of the form factor can
Hereonecannotsimplyincludetheresonancewidthby indeed successfully account for the experimental data
replacingM2−sbyM2−s−iM Γ (s)inthepropagator, up to s (cid:46) 1.5 GeV2, and secondly, that the approach
V V V V
sincethiswoulddoublecount(cid:61)m[A (s)]andanalytic- employed by the Belle Collaboration (named here as
P
itywouldbeviolatedatNNLOinthechiralexpansion. II) is indeed an adequate one, as it yields the lowest
Wefollowinsteadaproceduresimilartothatproposed χ2/ndf valuesaccordingtoourfits. Noticethat, given
inRef.[39]fortheKπvectorformfactor,inwhichuni- thelowenergythresholdforthisdecay,thesubtraction
tarity and analyticity are satisfied exactly. The starting constantsarefixedatarelativelylowenergyscale,and
pointisaformfactorasinEq.(2),wheretheloopfunc- thedispersiverepresentationturnsouttobeinsensitive
tionsareresummedintothedenominator: to the dynamics at large energies. In order to get a re-
sultfortheformfactorthatcanbevalidupto s = M2,
M2 τ
FVπ(s)(0) = M2 (cid:110)1+ s (cid:104)A (Vs)+ 1A (s)(cid:105)(cid:111)−s (4) ttheremeexdpiaretesseionnerginiesEqto. a(7p)hceannombeeneo.lgo.gimcaaltlcyheaddeaqtuainte-
V 96π2F2 π 2 K
/NuclearPhysicsBProceedingsSupplement00(2013)1–3 3
function[1]likethatgiveninRef.[42](includedinthe on this topic with M. Davier, H. Hayashii, G. Lo´pez
newversionofTAUOLA[43]),ortheGounaris-Sakurai Castro and A. Pich are acknowledged. We thank
parametrization[44]usedbyBelle[45]. Hayashii Hisaki for providing us with Belle correla-
In the table, the errors quoted in single brackets are tion matrix. This work has been partially funded by
thoseresultingonlyfromthefit,i.e.neglectingthesys- CONICET (Argentina) under grant # PIP 02495, and
tematic errors arising from our theoretical approach. byANPCyT(Argentina)undergrant#PICT2011-0113
Thesearee.g.givenbytheenergyrangetobefitted,the and by the Spanish grants FPA2007-60323, FPA2011-
numberofsubtractionsandthevalueoftheupperinte- 25948andbytheSpanishConsoliderIngenio2010Pro-
gration limit in the dispersive integral. In order to es- grammeCPAN(CSD2007-00042).
timatetheassociatedtotalerrorswehaveextendedour
fit up to energies in the range [1.325,1.525] GeV2, we
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Parameter FitValue(I) FitValue(II) FitValue(III)
M 0.8431(5)((17)) 0.8280(4)((14)) 0.8276(4)((21))
ρ
F 0.0901(2)((5)) 0.0902(2)((4)) 0.0906(2)((4))
π
α 1.87(1)((3)) 1.84(1)((3)) 1.81(1)((2))
1
α 4.29(1)((7)) 4.34(1)((7)) 4.40(1)((6))
2
χ2/ndf 1.37 1.39 1.56
Γ (M2) 0.207(1)((3)) 0.194(1)((3)) 0.192(1)((4))
ρ ρ
Table1: FitresultstotheBelle1/NππdNππ/dsdistribution,includingcorrelationsbetweenerrors. Theerrorsinsingleanddouble
bracketscorrespondtothosearisingonlyfromthefitandthoseobtainedafterconsideringtheoreticalsystematics, respectively.
EnergyunitsaregiveninGeVpowers.Γ (M2)isobtainedusingthefittedvaluesofM andF andisgivenonlyforreference.
ρ ρ ρ π
Determination (cid:68)r2(cid:69)π (GeV−2) cπ (GeV−4)
V V
Ourfit 10.86(14) 3.84(3)
BijnensandTalavera[30] 11.22(41) 3.85(60)
PichandPortole´s[34] 11.04(30) 3.79(4)
Table2:Low-energyobservablesofthevectorpionformfactoruptothequadraticterm.Wegivetheresultsfromourfit,theO(p6)
χPTanalysisinRef.[30]andthedispersiveanalysisinRef.[34].