Update of g-2 of the muon and Delta alpha

0.22 MB·English

by  T. Teubner

CPC(HEP&NP),2009,33(X):1—6 Chinese Physics C Vol.33,No.X, Xxx,2009
2 ∆
Update of g of the muon and α
−
T. Teubner1 K. Hagiwara2 R. Liao1 A.D. Martin3 Daisuke Nomura2
1(Department ofMathematical Sciences,UniversityofLiverpool,LiverpoolL693BX,U.K.)
0 2(TheoryCenter,KEK,Tsukuba,Ibaraki305-0801, Japan)
1 3(Department ofPhysicsandInstituteforParticlePhysicsPhenomenology, UniversityofDurham,DurhamDH13LE,U.K.)
0
2 Abstract WeupdateourStandardModelpredictionsforg−2ofthemuonandforthehadroniccontributions

n totherunningoftheQEDcoupling,∆α(5)(M2). Particularemphasisisputonrecentchangesinthehadronic
had Z
a contributionsfrom new data in the 2π channel and from the energy region just below 2 GeV.
J
9 Key words Anomalous magnetic moment, muon, runningcoupling, hadronic contributions
2
PACS 13.40.Em, 14.60.Ef, 12.15.Lk

]
h
p 1 Introduction completelydominatedby the hadroniccontributions,
-
p ∆α(5)(q2). Recall α(M2) is the least well known of
e The anomalous magnetic moment of the muon, thehsaedt of precision obseZrvables [G ,M ,α(M2)], so
h a =(g 2)/2, has been the subject of wide interest F Z Z
[ µ − its error is a limiting factor in the electroweak fits of
and detailed research. The discrepancy between its
the SM as performed e.g. by the LEP Electroweak
1 experimental value as measured by BNL [1] and its
Working Group.
v
prediction within the Standard Model (SM) is one of
1
0 thefew–ifnottheonly–experimentalsignofphysics
2 Standard Model prediction of g 2
4 beyond the SM (apart from neutrino mixing). This −
5
has triggereda lot of intense scrutiny of both the ex- Theanomalousmagneticmomentofthemuonre-
.
1 perimental determinationand the theoreticalevalua- ceives contributions from all sectors of the SM. The
0
tion of a . The BNL experiment E821 has achieved QED and electroweak (EW) corrections can be cal-
0 µ
1 animpressiveprecisionof0.5ppm[1],andfurtherim- culated within perturbation theory and are, through
: provements may only be reached with the planned many impressive works, well under control. In the
v
i experiments at Fermilab and J-PARC, see the pre- compilation for aSM presented here we use aQED =
X µ µ
sentations [2, 3]. As discussed below, the SM predic- 116584718.08(15)10−11[6,7]andaEW=(154 2)10−11
r · µ ± ·
a tionreliesheavilyonthe experimentalinformationof [8], as e.g. reviewed in [9]. The hadronic contribu-
the measuredhadroniccrosssections atlow energies. tions can not be reliably calculated in perturbative
During the last 15 years,the SM prediction has gone QCD(pQCD)astheloop-integralsaredominatedby
through several phases of improvement and consol- low momentum transfer, i.e. the non-perturbative
idation and has now, for the first time, reached an region of QCD. They are typically divided into the
accuracy even slightly better than the experimental leading (LO) and higher-order (HO) vacuum polari-
value. Thisismainlyduetothebigeffortstomeasure sation(VP),andtheso-calledlight-by-lightcontribu-
the hadronic cross sections with increasing accuracy tions, which are also subleading: ahad=ahad,LOVP+
µ µ
and the progress of various groups working on the ahad,HOVP+ahad,l−by−l. While the VP induced cor-
µ µ
data-drivenevaluationof the hadronic contributions, rections can be calculated with methods based on
which are in fairly good agreement (though further dispersion relations and using experimentally mea-
improvementsareforeseenaswillbediscussedbriefly sured hadronic cross sections as input, the light-by-
in Section 4). In this article we will concentrate on light scattering contributions can be estimated only
themainchangesinthehadroniccontributionstoa , usingmodels.∗ Theresultsfromdifferentgroupsvary
µ
updating the works [4, 5]. considerably, both w.r.t. the mean value and the er-
Similarly to g 2, the theoretical uncertainties ror. Forareviewpresentedatthisconferencesee[10].
−
in the running of the QED coupling, α(q2), are IntheSMpredictionofg 2presentedhereweusethe
−
1)E-mail:[email protected]
∗Firstprinciplecalculations withinlatticegaugefieldtheoryareunderwaybutverydifficultandatanearlystage.
(cid:13)c2009ChinesePhysicalSocietyandtheInstituteofHighEnergyPhysicsoftheChineseAcademyofSciencesandtheInstitute
No.X T.Teubneretal: Updateofg−2ofthemuonand∆α 2
value ahad,l−by−l=(10.5 2.6) 10−10, which has been of the 2π channel presented at the PhiPsi09 confer-
µ ± ·
obtained in [11] as a combination of results based on ence but not yet available for public use). The grey
different models.† In the following we will discuss in bandshowstheresultofourdatacombinationinthis
more detail recent changes in the VP contributions. channel including also older data, but excluding the
KLOE data, see the discussion below.
2.1 Hadronic VP contributions
1400
TheLOhadronicVPcontributionsarecalculated Fit (w/o KLOE)
Mean of fit (w/o KLOE)
Mean of fit (all sets)
using the dispersion integral 1200 CKMLDO-E2 ((0068))
SND (04, re-anal.)
1 ∞ CMD-2 (02, re-anal.)
ahad,LOVP= dsσ0 (s)K(s), (1) 1000
wtiohnergeivKin(µgs)h=ighm3es2µst·(w04e.πi4g3.h.Zt.m1t)o2πilsowaehksatndoewnenrgkieersn√els,fuanncd- +-+-σππ(ee -> ) [nb] 680000
σ0 (s) is the hadroniccrosssectionfor e+e− γ∗
had → → 400
hadrons(+γ). The superscript 0 indicates that the
‘undressed’ cross section must be used, i.e. the cross 200
section without VP effects in the virtualphoton, but
0
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
including final state radiation (FSR) of photons. To Energy [GeV]
arrive at the best compilation for σ , at low ener- Fig. 1. Most important datainthe2π channel
had
gies (√s<2 GeV) about 24 hadronic channels (ex- and fits as described in thetext.
clusive final states) have to be summed, and in each
250
channel the data from different experiments have to Mean ofF fiitt ((ww//oo KKLLOOEE))
Mean of fit (all sets)
be combined. At intermediate energies σhad is mea- SND (0C4M, rDe--2a n(0a6l.))
200
suredinclusively. PerturbativeQCDcanonlybeused
away from resonances, and (most) data driven anal-
ytmhseernesstuhwsoel.dr.p.tQ.FCorarDddioaetntailvyilesfcooorfrrteehnceetridgoainetssaaainnbdpovuteth,etthhdeeaitoraptecrneoambt¯b-- +-+-σππ(ee -> ) [nb] 110500
bination through a non-linear χ2 fit, as used in the
min
work reported here, see [4, 5]. Note that there are 50
uncertainties w.r.t. the correct application of radia-
tive corrections (undressing of VP, possible addition 0
0.3 0.35 0.4 0.45 0.5 0.55
of neglected photon FSR) to the data, especially in Energy [GeV]
the case of older data sets. In our analysis this leads Fig. 2. Low energy region close to the2π threshold.
tothe assignmentofaseparateerrorduetoradiative
corrections, δahad,VP+FSR 1.8 10−10, which alone is
µ ≃ · The lowest energy region close to the 2π thresh-
about ten times bigger than the uncertainty of the
old is displayed in Fig. 2. The recent CMD-2 data
electroweak contributions aEW.
µ represented by (red) triangles and error bars clearly
2.1.1 Recent changes in the 2π channel demonstrate the improvement from this single data
Morethan70%ofahad is coming fromthe ρ 2π set alone, with the fit in this region being dominated
µ →
channel. Following older experiments, the cross sec- by these data. Figure 3 shows an enlargement of the
tione+e− π+π− hasbeenmeasuredinrecentyears ρ-ωinterferenceregionwhichisnowverywellmapped
→
with increasing accuracy by the Novosibirsk experi- out by the consistent data sets. However, Fig. 3 also
ments CMD-2 and SND, see e.g. [14] for a short re- shows that the KLOE data are undershooting the
viewoftheirresults. Figure1displaystheimpressive combination of the other data in this region, while
agreement of the most recent data sets from CMD-2 typically being higher than other data at lower ener-
and SND, together with recentdata from KLOE[15] gies as canbe seen by the (green) solidline in Fig. 1.
obtained with the method of Radiative Return (see This apparent difference in shape is highlighted in
[16] for a detailed review of the method and its ap- Fig. 4, which shows the normalised difference of the
plication, and [17] for the very new KLOE analysis KLOEcrosssectionandthecombinationoftheother
†Notethatthisisslightlydifferentfrom(thoughcertainlycompatiblewith)thevalueahµad,l−by−l=(116±40)·10−11 asobtained
in[12,13]anddiscussedin[10].
No.X T.Teubneretal: Updateofg−2ofthemuonand∆α 3
2π data by the square markers, while the band dis- the non-linear χ2 fit.‡ Note that if we calculate
min
plays the size of the error of the compilation with- the contribution to g 2 from the KLOE data alone,
−
out KLOE. (The bands displayed in the figures are we obtain aππ,KLOE =(384.16 3.47) 10−10, in per-
µ ± ·
obtained from the diagonal elements of the fit’s full fect agreement with the result of the integral over
covariance matrix.) our compilation of all 2π data without KLOE (but
in the range of the KLOE data), for which we get
1500
1400 MeaMne oafnF f iiott fC(( wwfKiMt//L oo(DOa KK-lEl2 LLs ((OOe00tEE68s))))) uaµπseπ,twh/eouptrKoLcOeEdu=re(3a8d4o.p1t2e±d2a.l5re1a)d·y10i−n1[04.]Wwheetrheeeraerfloireer
SND (04, re-anal.)
1300 CMD-2 (02, re-anal.) KLOE data (now superseded by [15]) were included
1200 after integration, by calculating a weighted average
+-+-σππ(ee -> ) [nb] 11010000 wKnaiLtthOioEnth.,eanindtengortalpoevreforrmthiengdaatapocionmt-pbiyla-ptiooinntwciothmobuit-
900
BaBar has also published their first measurement
800 ofthe2π channelbasedonRadiativeReturn[18](see
700 also[19]),findingsomediscrepancieswithKLOE.We
600 havenotincludedthenewBaBardataintheanalysis
0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82
Energy [GeV] presented here as they were not available for public
Fig. 3. ρ-ω interference region in the2π channel.
use at the time of the conference, but see [20, 21] for
an analysis which includes them.
0.15
FitK wL/Oo EK LdOatEa 2.1.2 Energy region below 2 GeV
0.1 Important changes in the data input have also
happened in the regionbetween 1.4 and 2 GeV. This
0.05 region is particularly difficult, as a growing number
σσσ( - )/KLOEFitFit 0 ocgefososmdibulaletcica-hunardadcrhyoa.ns Hteoxocwblueesviievnrec,lufitdhneeadsletsoetnaoetebrsgtiaebisnecaσorhmeadeasbwoaitvche-
-0.05
the reach of the Novosibirsk machine§, and the qual-
ity of the available data was not very good. Alter-
-0.1
natively, one can rely on inclusive R measurements,
-0.15 but for this only rather old and not very precise
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Energy (GeV) data are available. This situation has changed with
Fig. 4. Normalised difference of the KLOE
BaBar measuring, through Radiative Return, many
data [15] and the data compilation excluding
channels with higher accuracy than earlier experi-
KLOE; the band represents the error of the
ments. These include new data for 2π+2π− [23],
compilation.
K+K−π0, K0πK [24], 2π+2π−π0, K+K−π+π−π0,
S
One should keep in mind that the KLOE data 2π+2π−η [25], 2π+2π−2π0 [26] used for our updated
are obtained via the method of Radiative Return analysis. Figures 5 – 7 exemplify the influence of the
at fixed collider centre-of-mass energy, whereas the new BaBar data. The new data are not always in
other data are measured via the traditional method good agreement with other sets; in such cases the fit
of energy scan by adjusting the e+e− beam ener- has a poor quality and we scale up the error of the
gies. Hence Monte Carlo simulation tools including channel’s contribution by χ2 /d.o.f. (e.g. in the
min
radiative corrections and also the systematic effects 2π+2π−2π0 channel χ2 /dp.o.f.=2.7.)
min
arecompletely differentbetweenthe two approaches. 2.1.3 Sum rule analysis
Up to now it is not clear what causes the differ- Notethatinpreviousg 2analyses[4,5]wefound
−
ence in the shapes of the 2π data, and more stud- a discrepancy between using the available inclusive
ies are underway to clarify the situation. Unfortu- data in the region 1.4 to 2 GeV and adding the ex-
nately this difference in shape prevents us from in- clusive channels. The inclusive data were lower than
cluding the KLOE data in a straightforward way in
‡Thefitallowsareadjustmentoftheoverallnormalisationofthedatasetswithintheirsystematicerrors. IncludingtheKLOE
datawouldleadtoabadχ2 /d.o.f.andunnaturalnormalisationeffects pullingthefitupward,see[4]foradetaileddiscussion.
min
§Thiswillchangewiththecurrentupgrade,see[22].
No.X T.Teubneretal: Updateofg−2ofthemuonand∆α 4
the sum of the exclusive channels, though they were over the exclusive for energies √s = 1.43...2 GeV.
found to be similar in shape. Since then the situation has changed: the hadronic
datahaschangedslightly,beingloweratlowenergies
7
Fit (w/o BaBar) and also at energies above 2 GeV. Also, with the in-
Fit (all sets)
BaBar (08)
6 DDMM12 ((8921)) clusionoftherecentBaBardatathesumoftheexclu-
sive channels in the region 1.4 to 2 GeV has become
5
slightlylowerandmoreaccuratethanbefore. Ourup-
+-0σπ(ee -> KK) [nb]s 34 dftehareteencdotrssruuemmsporrunuldleeisnagbnaasuslyemsdisrouinslepsiuQnmtCemDgraaarlirsseeodmveiarndteFhietgod.am8t;aadtbcifhy-
fitting for α as a free parameter (see [5] for details).
2 s
It is clear that the new sum over exclusive channels
1 is more accurate than the old inclusive data and also
morecompatiblewiththepredictionsbasedonpQCD
0
1.4 1.5 1.6 1.7 1.8 1.9 2
Energy [GeV] with a world average value of αs. We therefore are
Fig. 5. K0Kπ channel with improvement due nowcombining the resultsfromthe inclusive andthe
S
to recent BaBar data [24]. sum over exclusive data in this energy region.¶
45
Fit w/o BaBar
Fit all sets
40 BaDBMa2r ((9005)) (m,n,√s–0) data (fraction (%))
DM1 (82)
M3N (79) (0,0,3.7) inclusive (13.0)
35 (1,0,3.7) inclusive (20.4)
(1,1,3.7) inclusive (13.1)
+-+-+-σππππ(ee -> ) [nb] 12235050 ((((((((20112011,,,,,,,,00010001,,,,,,,,33333222.))))...7555)))) iiiiiiiinnnnnnnncccccccclllllllluuuuuuuussssssssiiiiiiiivvvvvvvveeeeeeee ((((((((2122223429667903........79942068))))))))
(2,0,2.5) inclusive (24.2)
10
(0,0,3.7) exclusive
5 (1,0,3.7) exclusive
(1,1,3.7) exclusive
(2,0,3.7) exclusive
0
1.4 1.5 1.6 1.7 1.8 1.9 2 (0,0,3) exclusive
Energy [GeV] (1,0,3) exclusive
(1,1,3) exclusive
Fig. 6. 2π+2π− channel. (2,0,3) exclusive
(0,0,2.5) exclusive
(1,0,2.5) exclusive
(1,1,2.5) exclusive
14
Fit (w/o BaBar) (2,0,2.5) exclusive
Fit (all sets)
BaBar (06)
12 MDM3N2 ((7896)) αS(M Z2) 0.1 0.12 0.14
Fig.8. Resultsfordifferentsum-rules[5]trans-
10
+-+-+-00σππππππ(ee -> ) [nb] 468 2.1.I4n ltaaOhdtetedhdwietoiirnorltcndohaatavonpegrrteaehsgdeeiacnoitmifdoαnprsoeo(srMfutaαlZ2tns)t;f.otcrhheaahµbnaagdn,eVdsPsdhioswcusssed
above, compared to [4], we have also included new
2 datainotherchannels: K+K− fromCMD-2 [27]and
SND[28],K0K0 fromSND[29],π+π−π0 fromCMD-
0 1.4 1.5 1.6 1.7 1.8 1.9 2 S L
Energy [GeV] 2 [30], ωπ0 from KLOE [31], and inclusive R data at
Fig. 7. 2π+2π−2π0 channel. higher energies above 2 GeV from BES [32, 33] and
CLEO [34]. Figure 9 shows the recent BES data to-
In [5] we performed a QCD sum rule analysis, con- gether with the fit of all inclusive data in this region
cluding that the inclusive data are more compatible and the prediction from pQCD. While the contribu-
with pQCD and the world average of α , and there- tiontog 2issignificantlysmallerthanpreviously,it
s −
forechosetousetheinclusivedatainsteadofthesum is obvious that pQCD, which is in perfect agreement
¶Note that the sum over exclusive channels stills requires, due to the lack of experimental information, the use of iso-spin
relationsforunknownchannels, whichinturnresultsinalargeerrorfromthepoorlyknownKK¯ππ channel.
No.X T.Teubneretal: Updateofg−2ofthemuonand∆α 5
with the three latest BESII (09) data points [33], is corresponds to a 4σ discrepancy. The situation is
still somewhat lowerthan the fit for energiesbelow 3 displayedin Fig. 10 where we also show our previous
GeV. result and the most recent results from Jegerlehner
and Nyffeler [13] andthree different predictions from
4.5 Inclusive Davier et al. [20, 21, 35]: While their e+e− based re-
Inclusive mean
pQCD
pQCD mean sultwithout the new BaBar2π (labelled‘w/o BaBar
4 BEBSE SII ((0019))
BES (99) (09)’ in the figure) data agrees very well with our
evaluation, the result including these new data lead
3.5
toashiftupwards,butstillcompatiblewiththeother
R(s) 3 e+e− basedresult. The thirdresultemploys,inaddi-
tion to e+e− data, also the use of τ spectral function
2.5 data from ALEPH, OPAL, CLEO and BELLE.
2
1.5
2 2.5 3 3.5 4 HMNT (06)
√s [GeV]
Fig.9. RecentdatafromBEStogetherwiththe JN (09)
fit of all inclusive data compared to pQCD in
Davier et al, τ (09)
the energy region above2 GeV.
Davier et al, e+e–w/o BaBar (09)
w/ BaBar (09)
Numerically the changes to ahad,LOVP from the
µ HLMNT (09)
different energy regions amount to (units of 10−10,
comparedtoresultsfrom[4]): 0.76(0.32 1.43GeV, experiment
− −
low energy exclusive channels), +2.10 (1.43 2 GeV,
− BNL
results from inclusive and sum over exclusive data
BNL (new from shift in λ)
combined), 1.35 (2 11.09 GeV, higher energy in-
− −
clusive data). This accidentally leads to a near per- 170 180 190 200 210
fect cancellation of the shifts, with the total result aµSM × 1010 – 11659000
ahad,LOVP=(689.4 3.6 1.8 ) 10−10. The first
µ ± exp± rad · Fig. 10. Comparison of recent predictions for
error is coming from the statistical and systematic
g−2compared to theBNL measurement.
error of the data, whereas the second error is our
estimate of the uncertainty in the radiative correc-
To translate the charged current induced hadronic
tions as mentioned above. However, compared to
τ decay data into the required spectral functions
our earlier result from [4], ahad,LOVP(HMNT06) =
µ
requires the application of isospin breaking correc-
(689.4 4.2 1.8 )10−10,thereisafurtherreduc-
± exp± rad · tions, which can only be predicted in a model de-
tion in the error.
pendent way. Earlier τ based results from Davier et
The higher-order VP induced contributions can
al. were incompatible with e+e− based results, but
also be calculated using a dispersion integral, see [5]
with a re-evaluation of the isospin breaking correc-
for details. Our updated result is nearly unchanged
tions [21, 35, 36] they find the result displayed in
andreadsahad,HOVP=( 9.79 0.06 0.03 )10−10.
µ − ± exp± rad · Fig. 10 (labelled ‘τ (09)’) which is now marginally
2.2 SM predictions compared to the BNL consistent. These findings were discussed controver-
measurement siallyatthePhiPsi09conference;Benayounpresented
analternativeapproachbasedonHidden LocalSym-
Combiningthe QED,EWandhadroniccontribu-
metry anddynamical (ρ,ω,φ) mixing. With this and
tions as discussedabove,we arriveatour SM predic-
with a global fit he gets consistency of the τ spec-
tionoftheanomalousmagneticmomentofthemuon,
tral function with the e+e− data and an improved
aSM(HLMNT09)=(116 591 773 48) 10−11. (2) 2π contribution to g 2 [37]. Keeping in mind that
µ ± · −
otherstudiesobtainedaverylargeuncertaintyofthe
DuetoasmallshiftofCodata’smuontoprotonmag- isospin breaking corrections (see e.g. [38]) we believe
netic ratioλthe experimentalvaluefora fromBNL that, until this issue is better understood, the pre-
µ
is now aEXP=116592089(63) 10−11 [2]. This results dictions of g 2 based on e+e− data alone are more
µ · −
in a difference aEXP aSM=(31.6 7.9) 10−10 which reliable.
µ − µ ± ·
No.X T.Teubneretal: Updateofg−2ofthemuonand∆α 6
3 Running QED coupling α(q2) cies are reported by other groups, depending on the
data used and the details of the analyses. However,
Basedonthe same data compilation,we alsopre-
for all e+e− based analyses the discrepancy is about
dict the hadronic contributions to the running of the
3 4σ and standing all scrutiny.
QED coupling, α(q2)=α/(1−∆αlep−∆αhad). The −For the future, further improvements of the SM
hadronic VP is important for many studies where
prediction will be possible. The method of Radiative
high accuracy radiative corrections are required, as
Return has proven extremely powerful and will be
is the case forthe hadroniccontributionsto g 2,see
− leading to many more results. In addition to the al-
[16] for a very recent review. Of particular impor-
readyreportednew 2π datafromKLOE[17],further
tance as input in EW precision fits is the quantity
∆α(5)(M2), the hadronic contributions to the run- analysesareongoing,andthereareexcitingprospects
had Z withKLOE2[39]. Thereisalsoarichprogrammego-
ning α from five-flavours (the contribution from the
ingonatBELLE,andthepossibilityofSuperBELLE
top quark is usually added using pQCD). Our up-
dated value is ∆α(5)(M2)=0.02760 0.00015. This at the horizon. CMD-3 and SND at VEPP-2000
had Z ± currently commissioned in Novosibirsk are aiming at
isslightlyhigherandsignificantlymoreaccuratethan
largely improving the exclusive measurements in the
thenumberfromBurkhardtandPietrzykusedasde-
regionbelow 2 GeV, whereas BESIII at BEPCII will
fault by the LEP EW Working Group, andleads e.g.
cover higher energies. With all these developments
to a lower preferred Higgs mass and a lower upper
the error of the SM prediction of g 2 is expected to
limit from the famous ‘Blue Band Plot’. −
shrinkevenfurther,sothatinthefuturethelight-by-
light contributions may eventually become the limit-
4 Summary and outlook
ing factor. Obviously there is an extremely strong
We have given an update of the SM prediction case for new the g 2 experiments as planned by the
−
of g 2, emphasising recent developments for the g 2 Collaboration for Fermilab [2] and at J-PARC
− −
hadronic contributions. With the e+e− based anal- [3]. With this, g 2 will become even more powerful
−
ysis presented here we obtain a 4σ discrepancy be- in establishing and constraining physics beyond the
tween the experimentally measured value of a and Standard Model.
µ
its SM prediction. Slightly less significant discrepan-
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