Towards a determination of the tau lepton dipole moments

0.18 MB·English

by  Matteo Fael

NuclearPhysicsB
Proceedings
Supplement
NuclearPhysicsBProceedingsSupplement00(2013)1–4
Towards a determination of the tau lepton dipole moments
M.Faela,b,L.Mercollic,M.Passerad
aDipartimentodiFisicaeAstronomia,Universita`diPadova,35131Padova,Italy
bInstitutfu¨rTheoretischePhysik,Universita¨tZu¨rich,CH-8057,Zu¨rich,Switzerland
cPrincetonUniversity,DepartmentofAstrophysicalSciences,Princeton,NJ,08544,USA
dIstitutoNazionaleFisicaNucleare,35131Padova,Italy
3
1
0
2

n
Abstract
a
J The τ anomalous magnetic moment a and electric dipole moment d have not yet been observed. The present
τ τ
2 bounds on their values are of order 10−2 and 10−17e cm, respectively. We propose to measure a with a precision
τ
2 ofO(10−3)orbetterandimprovetheexistinglimitsond usingpreciseτ− → l−ν ν¯ γ (l = eorµ)datafromhigh-
τ τ l
luminosityBfactories. Adetailedfeasibilitystudyofthismethodisunderway.
]
h
p
p- 1. Introduction O(10−6);severaltheoriesexistwherethisna¨ıvescaling
e isviolatedandmuchlargereffectsareexpected[3].
Theveryshortlifetimeoftheτlepton(2.9×10−13s)
h Ever since the very first discovery of CP violation
[ makes it very difficult to measure its electric and mag- in the 1960s there have been ongoing efforts to mea-
netic dipole moments. The present resolution on its
1 surefundamentalelectricdipolemoments(EDMs). In-
anomalousmagneticmomenta isonlyofO(10−2)[1],
v τ deed,EDMsviolateparityandtimereversal;ifCPT is
2 more than an order of magnitude larger than its Stan-
agoodsymmetry,T violationimpliesCPviolationand
0 dardModel(SM)prediction α (cid:39) 0.001. Infact,while
2π viceversa. TheSMpredictionsforleptonEDMsarefar
3 theSMvalueofa isknownwithatinyuncertaintyof
5 5×10−8 [2],thissτhortlifetimehassofarpreventedthe too small to be seen by projected future experiments.
. Hence,theobservationofanon-vanishingleptonEDM
1 determinationofa measuringtheτspinprecessionina
τ would be evidence for a CP-violating new physics ef-
0 magneticfield,likeintheelectronandmuong−2exper-
3 fect[4]. Theexperimentaldeterminationorastringent
iments. Instead, experiments focused on various high-
1 boundontheτleptonEDMd posesthesamedifficul-
precision measurements of τ pair production in high- τ
: tiesthatoneencountersinthecaseofa :theτ’slifetime
v energy processes, comparing the measured cross sec- τ
i is very short. Nonetheless, a CP violation signature
X tions with the SM predictions. As these processes in- arising from d was searched for in the e+e− → τ+τ−
r volve off-shell photons or taus in the ττ¯γ vertices, the reactionreachiτngasensitivitytod of10−17ecm[5].
a measuredquantityisnotdirectlya . τ
τ After a brief review of the present experimental and
A precise measurement of a would offer an excel-
τ theoretical status of the τ dipole moments, we discuss
lent opportunity to unveil new physics effects. Indeed,
thepossibilitytoprobethemviaprecisemeasurements
in a large class of theories beyond the SM, new con-
ofthedecaysτ− →l−ν ν¯ γ(l=eorµ).
tributionstotheanomalousmagneticmomentofalep- τ l
ton l of mass m scale with m2. Therefore, given the
l l
large factor m2/m2 ∼ 283, the g−2 of the τ is much 2. Theanomalousmagneticmomentofthetau
τ µ
more sensitive than the muon one to electroweak and
newphysicsloopeffectsthatgivecontributionspropor- TheSMpredictionfora isgivenbythesumofQED,
τ
tionaltom2. Inthesescenarios,thepresentdiscrepancy electroweak (EW) and hadronic terms. The QED con-
l
inthemuong−2suggestsanew-physicseffectina of tribution has been computed up to three loops: aQED =
τ τ
/NuclearPhysicsBProceedingsSupplement00(2013)1–4 2
117324(2) × 10−8 (see [6] and references therein), the non-SM contribution to a . We refer to [10] for a
τ
where the uncertainty π2ln2(m /m )(α/π)4 ∼ 2×10−8 concise review of older, less stringent limits on a , de-
τ e τ
has been assigned for uncalculated four-loop contribu- rived from data samples of e+e− → τ+τ− (for photon
tions. The errors due to the uncertainties of the O(α2) squared four-momentum q2 up to (37GeV)2) [11] and
(5×10−10)andO(α3)terms(3×10−11),aswellasthat Z0 → τ+τ−γ (where q2 = 0 but the radiating τ is not
induced by the uncertainty of α (8×10−13), are negli- on-shell)[12, 13,14, 15]. Other indirectbounds on a
τ
gible. Thesumoftheone-andtwo-loopEWcontribu- werestudiedin[16].
tionsisaEW = 47.4(5)×10−8 (see[7,2]andreferences Comparing Eqs. (1) and (2) (their difference is
τ
therein).Theuncertaintyencompassestheestimateder- roughlyonestandarddeviation),itisclearthatthesen-
rors induced by hadronic loop effects, neglected two- sitivity of the best existing measurements is still more
loopbosonictermsandthemissingthree-loopcontribu- than one order of magnitude worse than needed. The
tion.Italsoincludesthetinyerrorsduetotheuncertain- possibility to improve these limits is certainly not ex-
tiesinM andm . cluded. Future high-luminosity B factories such as
top τ
Similarly to the case of the muon g−2, the leading- Super-KEKB [17] offer new opportunities to improve
order hadronic contribution to a is obtained via a dis- the determination of the τ magnetic properties. The
τ
persionintegralofthetotalhadroniccrosssectionofthe authors of [18] proposed to determine the Pauli form
e+e− annihilation (the role of low energies is very im- factor F (q2) ofthe τ via τ+τ− production in e+e− col-
2
portant, althoughnotasmuchasfora ). Theresultof lisions at the Υ resonances with sensitivities possibly
µ
thelatestevaluation,usingthewholebulkofexperimen- down to O(10−5) or even better. In [19] the reanaly-
taldatabelow12GeV,isaHLO = 337.5(3.7)×10−8 [2]. sisofvariousmeasurementsofthecrosssectionofthe
τ
The hadronic higher-order (α3) contribution aHHO can processe+e− → τ+τ−,thetransverseτpolarizationand
τ
be divided into two parts: aHHO = aHHO(vp)+aHHO(lbl). asymmetryatLEPandSLD,aswellasthedecaywidth
τ τ τ
The first one, the O(α3) contribution of diagrams con- Γ(W → τν ) at LEP and Tevatron allowed the authors
τ
taining hadronic self-energy insertions in the photon tosetamodel-independentlimitonnewphysicscontri-
propagators, is aHHO(vp) = 7.6(2) × 10−8 [8]. Note butions,
τ
that na¨ıvely rescaling the corresponding muon g−2 re- −0.007<aNP <0.005, (3)
τ
sult by a factor m2/m2 leads to the incorrect estimate
τ µ a bound stronger than that in Eq. (2). This analysis,
aHHO(vp) ∼ −28×10−8 (even the sign is wrong!). Es-
τ like earlier ones, was performed without radiative cor-
timates of the light-by-light contribution aHHO(lbl) ob-
τ rections, but the authors checked that the inclusion of
tained rescaling the corresponding one for the muon initial-stateradiationdidnotaffectsignificantlytheob-
g−2 by a factor m2/m2 fall short of what is needed –
τ µ tainedbounds.
this scaling is not justified. The parton-level estimate
of [2] is aHHO(lbl) = 5(3) × 10−8, a value much lower
τ
3. TheτleptonEDM
than those obtained by na¨ıve rescaling. Adding up the
abovecontributionsoneobtainstheSMprediction[2]
In the SM, the only sources of CP violation are the
aSM =aQED+aEW+aHLO+aHHO =117721(5)×10−8. (1) CKM-phase and a possible θ-term in QCD. A funda-
τ τ τ τ τ mental EDM can only be generated at the three-loop
Errorswereaddedinquadrature. level [20], yielding a SM contribution which is far be-
The present PDG limit on the τ lepton g−2 was low experimental capabilities. Models for physics be-
derived in 2004 by the DELPHI collaboration from yondtheSMgenerallyinducelargecontributionstolep-
e+e− → e+e−τ+τ− total cross section measurements at tonandneutronEDMs,andalthoughtherehasbeenno
√
sbetween183and208GeVatLEP2(thestudyofa experimentalevidenceforanEDMsofar,thereiscon-
τ
via this channel was proposed in [9]): −0.052 < a < siderable hope to gain new insights into the nature of
τ
0.013 at 95% confidence level [1]. Reference [1] also CPviolationthroughthiskindofexperiments.
quotestheresultintheform: Thecurrent95%confidencelevellimitsontheEDM
oftheτleptonaregivenby
a =−0.018(17). (2)
τ
−2.2<Re(d )<4.5 (10−17 ecm),
τ
(4)
This limit was obtained comparing the experimentally −2.5<Im(d )<0.8 (10−17 ecm);
τ
measuredvaluesofthecrosssectiontotheSMvalues,
assuming that possible deviations were due to non-SM they were obtained by the Belle collaboration [5] fol-
values of a . Therefore, Eq. (2) is actually a limit on lowingtheanalysisof[21]fortheimpactofaneffective
τ
/NuclearPhysicsBProceedingsSupplement00(2013)1–4 3
operator for the τ EDM in the process e+e− → τ+τ−. whichcanbedeterminedasfollows.
Comparedtoa ,theexperimentalsensitivitytod isnot Theleading-order(LO)predictionoftheLagrangian
τ τ
significantlyhighersinceinnaturalunitstheboundson L +L forthedifferentialdecayrateofapolar-
QED Fermi
d quoted above are of the order of 10−3 GeV−1. Fur- izedτ−intol−+ν¯ +ν +γ(l=eorµ)is,intheτlepton
τ l τ
thermore,asdiscussedpreviouslyinthecaseofa ,the restframe,
τ
analysisof[5]islimitedtotheLOprecisionofthecal-
culationof[21].Therefore,webelievethatthereiscon- dΓ6 αm5G2 √ (cid:34)
0 = τ F y x2−4r2 G (x,y,c)
siderableroomforanimprovementofthesebounds. dxdydΩ dΩ (4π)8 0
l γ
√ (cid:35)
+ x2−4r2J (x,y,c)(cid:126)n·pˆ +yK (x,y,c)(cid:126)n·pˆ , (12)
4. Radiativeleptonicτdecays 0 l 0 γ
Weproposetomeasurethedipolemomentsoftheτ where α = 1/137.035999174(35) [25] is the fine-
leptonthroughitsradiativeleptonicdecays[22,23,24] structureconstant,G =1.663788(7)×10−5GeV−2[26]
F
istheFermicouplingconstant,m =1.77682(16)GeV
τ
τ− →l−ντν¯lγ, withl=eorµ. (5) [27],r=ml/mτ,andthekinematicvariables
Theauthorsof[19]and[21]haveappliedeffectiveLa- 2E 2E
x= l, y= γ, c≡cosθ (13)
grangian techniques to study aτ and dτ. Our strategy mτ mτ
is similar: we describe radiative τ decays through an
effective Lagrangian Leff which contains the QED La- arethenormalizedenergiesoftheleptonl−andthepho-
grangianfortheleptons,theeffectiveFermiLagrangian, ton, which are respectively emitted at solid angles Ωl
andtheeffectiveoperatorsthatcontributetotheanoma- and Ωγ, and θ is the angle between their momenta (cid:126)pl
lousmagneticmomentandEDMoftheτlepton,i.e. and(cid:126)pγ. Also,n=(0,(cid:126)n)istheτpolarizationvectorwith
n2 = −1 and n· p = 0. The functionsG , J and K
τ 0 0 0
Leff =LQED+LFermi+ca4eΛ Oa−cd2iΛ Od, (6) weTrehecoompepruatteodrOin[g2e8n,e2r9a,te3s0a,d3d1i]t.ionalcontributionsto
a
thedifferentialdecayrateinEq.(12). Theycanbesum-
whereO aregivenby
a,d marisedintheshift[24]
O = τ¯σ τFµν,
a µν (7) G0(x,y,c) → G0(x,y,c) + a˜τGa(x,y,c), (14)
O = τ¯σ γ τFµν.
d µν 5
andsimilarlyfor J and K . TheoperatorO produces
The scale Λ represents the scale where any kind of 0 0 d
theadditionalterm
physicswhichisnotdescribedbyLeff generatesacon- √ (cid:16) (cid:17)
tribution to the τ’s electric ormagnetic dipole moment d˜ m y x2−4r2pˆ · pˆ ×(cid:126)n G (x,y,z) (15)
τ τ l γ d
and is therefore larger than the electroweak scale, i.e.
Λ > M . For simplicity we assume the scale Λ to be insidethesquarebracketsofEq.(12)[23]. Tinyterms
Z
equal for both operators O , knowing that in fact the ofO(a˜2)andO(d˜2)wereneglected.
a,d
scalefortheEDMismuchhigherthanthatfortheg−2. Ourgoalistoprovideamethodtodetermineaτ with
The effective Lagrangian Leff in Eq. (6) gives the fol- aprecisionofO(10−3)orbetter. Thiscallsforananal-
lowingpredictionsfortheτdipolemoments: ogousprecisiononthetheoreticalside. Forthisreason,
wehavecorrectedthedecayrateinEq.(12)toinclude
α m
a = +c τ +··· (8) radiative corrections at next-to-leading order (NLO) in
τ 2π a Λ
QED[22,23,31,32].ThecomparisonofthisNLOpre-
1
d = c +··· (9) diction,modifiedbytheadditionaltermsinEq.(14)for
τ d Λ
G (andsimilarlyfor J and K ),tosufficientlyprecise
0 0 0
where the dots indicate higher-order contributions not dataallowstodeterminea˜ (andtherebya viaEq.(8))
τ τ
relevantforourdiscussion(note,inEq.(9),thatd has possiblydowntothelevelofO(10−4).Similarly,includ-
τ
noQEDcontribution). Wethendefinetheparameters ing the additional term in Eq. (15), one can also deter-
m mined˜τwhich,atthislevelofprecision,coincideswith
a˜τ ≡ ca Λτ, (10) dτ(seeEq.(9)). Thisanalysisisinprogress[23].
The contributions from the two effective operators
1
d˜τ ≡ cd Λ, (11) Oa,d to the electromagnetic form factors are the same
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dellaRicercaScientificaundertheprogramPRIN2010-
11, and by the European Programmes UNILHC (con-
tract PITN-GA-2009-237920) and INVISIBLES (con-
tractPITN-GA-2011-289442).