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M. Davier
To cite this version:
The Hadroni Contribution to (g 2) Mi helDavier a E-mail: davierlal.in2p3.fr a
Laboratoiredel'A elerateurLineaire,IN2P3/CNRS-UniversitedeParis-Sud,
BP34,91898Orsay,Fran e
Theevaluationof thehadroni ontribution tothe muon magneti anomaly a is reviewed,in luding a new
estimate using pre ise results on the
+
spe tral fun tionfrom the KLOE Collaboration. It is found that
theKLOEdata on rmtosomeextentthepreviouse
+
e annihilationdatainthis hannel,anda entuatethe
disagreement withthe isospin-breaking- orre ted spe tralfun tion from !
0
de ays. Corre ting for
theempiri aldi eren einthemassofthe hargedandtheneutrallo allyimproves,butdoesnotresolvethis
dis repan y. Apreliminaryreevaluation(in ludingtheKLOEdata)ofthee
+
e -basedStandardModelpredi tion
ofa
resultsinadeviationof2.7standarddeviationsfromtheBNLmeasurement.
1. Introdu tion
Hadroni va uum polarization (HVP) in the
photon propagator plays an important role in
manypre isiontestsoftheStandardModel. This
isthe aseforthemuonanomalousmagneti
mo-menta
(g
2)=2,wheretheHVP omponent
is the leading ontributor to the un ertainty of
the Standard Model predi tion. The HVP
on-tribution is omputed by means of a dispersion
relationasanintegraloverexperimentally
deter-minedspe tralfun tions.Itisthepropertyofthis
dispersionrelationthat the spe tralfun tion
providesthemajorpartofthetotalHVP
ontri-bution,sothattheexperimentale ortfo useson
this hannel.
Spe tral fun tions are dire tly obtained from
the ross se tions of e
+
e annihilation into
hadrons. The a ura y of the al ulations has
therefore followed the progress in the quality of
the orresponding data[1℄. Be ause the latter
were not always suitable, it was deemed
ne es-sary to resort to other sour es of information.
One su h possibility was the use of the ve tor
spe tral fun tions[2℄ derived from the study of
hadroni de ays[3℄ for the energy range less
than m
' 1:8GeV=
2
. For this purpose, the
isospinrotationthat leadsfromthe harged to
theneutrale
+
e nalstatehastobethoroughly
orre tedforisospin-breakinge e ts.
Also,itwasdemonstratedthatessentially
per-turbativeQCD ouldbeapplied toenergys ales
aslowas1{2GeV[4,5℄,thuso eringawayto
re-pla epoore
+
e datainsomeenergyregionsbya
reliableandpre isetheoreti alpres ription[6-11℄.
Detailed reanalyses in luding all available
ex-perimentaldatahavebeenpublishedin
Refs.[12-14℄ (see also the preliminary results given in
Refs.[15,16℄), taking advantage of pre ise
re-sults in the hannel fromthe CMD-2
experi-ment[17℄ and fromthe ALEPHanalysis of
de- ays[18℄, and bene ting from a more omplete
treatment of isospin-breaking orre tions[19,20℄.
It was found that the e
+
e and the
isospin-breaking- orre ted spe tral fun tions do not
agree within their respe tive un ertainties, thus
leadingtoin onsistentpredi tionsforthe
lowest-orderhadroni ontributiontoa
. Thedominant
ontribution to the dis repan y stems from the
hannelwithadi eren eof( 11:96:4
exp 2:4 rad 2:6 SU(2) (7:3 total ))10 10 , andamore
signi ant energy-dependent deviation. When
omparedtotheworldaverageofthemuon
mag-neti anomaly measurements, dominated by the
resultsfromtheBNLexperiment[21℄,
a = (11659208:05:8)10 10 ; (1) therespe tivee +
e and -basedpredi tions
devia-23
24
25
26
27
B(
τ
–
→
ν
τ
π
–
π
o
) (in %)
CLEO
OPAL
L3
ALEPH
preliminary
τ
Average
e
+
e
–
CVC
25.42
±
0.12
±
0.42
25.44
±
0.17
±
0.29
25.44
±
0.16
±
0.10
25.47
±
0.10
±
0.09
25.46
±
0.10
24.52
±
0.32
Figure 1. The measured bran hing ratios for
!
0
ompared to the predi tion from
the e
+
e !
+
spe tral fun tion applying the
isospin-breaking orre tion fa tors dis ussed in
Ref.[13℄. Themeasuredbran hingratiosarefrom
ALEPH [18℄, CLEO [22℄ and OPAL [23℄. The
L3andOPALresultsareobtainedfromtheir h
0
bran hing ratio,redu edbythesmallK
0
ontri-butionmeasuredbyALEPH[24℄ andCLEO[25℄.
errorsinquadrature.
Theproblembetween ande
+
e dataismore
noti ablewhen omparingthe !
0
bran h-ing fra tion with the predi tion obtained from
integrating the orresponding
isospin-breaking- orre tede
+
e spe tralfun tion. Here,the
fun -tionundertheintegrandislesssele tivethanitis
the asefortheHVP ontributiontoa
,leading
to a dis repan y of 2.9 standard deviations[12℄,
asshowninFig.1.
This summer, new data on the spe tral
fun tion in the mass region between 0:60 and
0:97GeV=
2
werepresentedbytheKLOE
Collab-oration[26℄, using the|forthe purpose of
pre i-sion measurements|innovativete hnique of the
radiativereturn[27℄. The statisti al pre ision of
these data by far outperforms the Novosibirsk
sample,butthesystemati errorsareabouttwi e
aslargeasthoseobtainedbyCMD-2. Newdata
the BABAR Collaboration[28℄ on the
-nalstate. Theyunveilalarger rossse tionsand
a resonant peak at around 1:6GeV=
2
that was
missed by the previous DM2 measurement[29℄.
TheBABARdataarenot(yet)usedinthe
prelim-inary reevaluation of thelowest-orderHVP
on-tribution given here. The orre tion to a
had;LO
when using the BABAR data for this mode [30℄
is of theorder of +110
10
. Also, preliminary
BABAR resultsare available onthe2
+
2 nal
statewhi hareoverallmorepre isethanexisting
data [30℄. They arenot in luded in thepresent
evaluationandtheire e twouldbeto hangethe
hadroni ontributionbyabout 110
10 .
2. Muon Magneti Anomaly
Itis onvenienttoseparatetheStandardModel
(SM) predi tionfortheanomalousmagneti
mo-mentofthemuonintoitsdi erent ontributions,
a SM = a QED +a had +a weak ; (2) with a had = a had;LO +a had;HO +a had;LBL ; (3) andwherea QED =(11658472:00:2)10 10 is
thepureele tromagneti ontribution(see[31,32℄
and referen es therein), a
had;LO is the lowest-order HVP ontribution, a had;HO = ( 10:0 0:6) 10 10
is the orresponding higher-order
part[33,2℄, and a weak = (15:4 0:1 0:2) 10 10
, where the rst error is the hadroni
un- ertainty and the se ond is due to the Higgs
mass range, a ounts for orre tions due to
ex- hangeoftheweaklyintera tingbosonsuptotwo
loops[34℄. Forthelight-by-light(LBL)s attering
part, a
had;LBL
, we use the value (12:03:5)
10 10
fromthelatestevaluation[35℄,slightly
or-re tedforthemissing ontributionfrom(mainly)
thepionbox.
Owing to unitarity and to the analyti ity of
the va uum-polarizationfun tion, thelowest
or-derHVP ontributiontoa
anbe omputedvia
thedispersionintegral[36℄
where K(s) is a well-known QED kernel, and
R (0)
(s) denotes the ratio of the \bare" ross
se tion for e
+
e annihilation into hadrons to
the pointlike muon-pair ross se tion. The
bare ross se tion is de ned as the measured
rossse tion orre tedfor initial-stateradiation,
ele tron-vertex loop ontributions and
va uum-polarization e e ts in the photon propagator.
However,photonradiationinthe nalstateis
in- ludedinthebare rossse tionde nedhere. The
reasonforusingthebare(i.e.,lowestorder) ross
se tion is that a full treatment of higher orders
is anyhowneeded at thelevel of a
, sothat the
useofthe\dressed" rossse tionwouldentailthe
riskof double- ountingsomeof thehigher-order
ontributions.
The fun tion K(s) 1=s in Eq. (4) gives a
strong weight to the low-energy part of the
in-tegral. About 91% of the total ontribution to
a had;LO
isa umulatedat enter-of-massenergies
p
sbelow1:8GeVand 73%of a
had;LO
is overed
bythe nal state,whi h isdominatedbythe
(770)resonan e.
3. The Input Data
Adetailed ompilation ofall theexperimental
datausedintheevaluationofthedispersion
inte-gral(4)isprovidedinRefs.[13,12℄. Alsodis ussed
thereinisthe orre tivetreatmentofradiative
ef-fe tsappliedtosomeofthemeasurements. The
spe tralfun tionisobtainedbyaveragingthe
re-sults from ALEPH[3℄, CLEO[37℄andOPAL[38℄,
whi hexhibitsatisfa torymutualagreement.
A omparison of the e
+
e !
+
data
andthe orresponding spe tralfun tion,
repre-sentedasapoint-by-pointratiotothe spe tral
fun tion is givenin Fig.2. Several observations
anbemade.
Asigni antdis repan y,mainlyabovethe
peak is found between and the e
+ e
datafromCMD-2aswellasolderdatafrom
OLYA.
Overall,the KLOEdata on rm thetrend
exhibitedbytheothere
+
e data.
Some disagreement between KLOE and
data arelarge), on the peak (KLOE
be-low CMD-2) as well as on the high mass
side(KLOEdataarelow).
At this stage, the spe tralfun tion has not
been orre ted for a possible
0
mass and
width splitting[41,40℄. In ontrastto earlier
ex-perimental[3℄ and theoreti al results[39℄, a
om-bined pion form fa tor t[40℄ to the newpre ise
data on e
+
e and spe tral fun tions leads to
m m 0 = (2:30:8)MeV= 2 , while no
sig-ni antwidth splittingisobservedwithinthe t
errorof1:7MeV=
2 .
Note that ifthemassdi eren eisto betaken
asanexperimentalfa t,alargerwidthdi eren e
wouldbeexpe ted. Usinga hiralmodelofthe
resonan e[42,19,20℄,onehas
0 = m 0 m 3 0 3 + EM (5) where EM
isthewidthdi eren efrom
ele tro-magneti de ays. Thisleadstoatotalwidth
dif-feren e of (2:10:5)MeV=
2
that is marginally
onsistent with the observed value[40℄. is
ob-servedwithin the terrorof1:7MeV=
2 .
Considering the mass splitting in the
isospin-breaking orre tion of the spe tral fun tion
tendstolo allyimprove(thoughnotrestore)the
agreement between and CMD-2 data, leaving
an overallnormalization dis repan y. In reasing
the
0
width splitting by+3MeV=
2
im-provestheagreementbetween andKLOEdata
inthepeakregion,whileit annot orre tthe
dis- repan iesin thetails. Note thata orre tionof
the mass splitting alone would in rease the
dis- repan y betweenthe and e
+ e -based results fora had;LO .
Duringthepreviousevaluationsofa
had;LO
,the
results using respe tively the and e
+
e data
werequotedindividually,butonthesamefooting
sin e the e
+
e -based evaluation wasdominated
by the data from a single experiment(CMD-2).
The on rmation of this dis repan y by KLOE
dis redits the -based result for the use in the
dispersion integral until a better understanding
of thedynami al originof the observed e e t is
-0.3
-0.2
-0.1
0
0.1
0.2
0.2
0.4
0.6
0.8
1
1.2
KLOE
CMD-2
CMD
OLYA
DM1
τ
Average
preliminary
s (GeV
2
)
(
|
F
π
|
2
[
ee
]
–
|
F
π
|
2
[
τ
]
)
/
|
F
π
|
2
[
τ
]
-0.3
-0.2
-0.1
0
0.1
0.2
0.5
0.55
0.6
0.65
0.7
KLOE
CMD-2
CMD
OLYA
DM1
τ
Average
preliminary
s (GeV
2
)
(
|
F
π
|
2
[
ee
]
–
|
F
π
|
2
[
τ
]
)
/
|
F
π
|
2
[
τ
]
Figure2. Relative omparisonof the
+
spe tralfun tionsfrom e
+
e -annihilationdataand
isospin-breaking- orre ted data,expressedasaratiotothe spe tralfun tion. Theshadedbandindi atesthe
errorsofthe data. Thee
+
e dataarefromKLOE[26℄,CMD-2[17℄,CMD,OLYAandDM1(referen es
giveninRef.[12℄). Therighthand plotemphasizestheregionofthepeak.
tan e.
4. Results
Thein lusion ofthe KLOE data de reases
the ontributionfromthismodefrom[12℄(450:2
4:91:6 rad )10 10 to(448:34:11:6 rad )10 10
forthe energyinterval between0.5and 1:8GeV.
Notethat theadditionalsystemati errordue to
radiative e e ts originates from the energy
re-gionsnot overedbythere entKLOEand
CMD-2measurements,whereafulltreatmentof
radia-tive orre tionsis applied. Thepreliminary
esti-mateoftheintegral(4)givenbelowin ludes one
additionalimprovementwithrespe t toRef.[12℄:
perturbative QCDis used insteadof
experimen-tal data in the region between1:8 and 3:7GeV,
wherenon-perturbative ontributionstointegrals
over di erently weighed spe tral fun tions were
foundto besmall[7℄. This resultsin aredu tion
ofa
had;LO
by 110
10
. Allother ontributions
to the dispersionintegralare equalto those
de- nedin Ref.[12℄. + hadroni ontributionis a had;LO =(693:45:33:5 rad )10 10 ; (6)
where the se ond error is due to our treatment
of (potentially) missing radiative orre tions in
the older data[13℄. Adding to this the QED,
higher-order hadroni , light-by-light s attering,
and weak ontributions given in Se tion 2, one
nds a SM = (11659182:86:3 had;LO+HO 3:5 had;LBL 0:3 QED+EW )10 10 : (7)
Thisvalue anbe omparedtothepresent
mea-surement(1);addingallerrorsinquadrature,the
di eren ebetweenexperimentandtheoryis
a exp a SM =(25:29:2)10 10 ; (8)
whi h orresponds to 2.7 \standard deviations"
(tobeinterpretedwith areduetothedominan e
of systemati errors in the SM predi tion). A
graphi al omparison of the result(7) with
140
150
160
170
180
190
200
210
a
µ
– 11
659
000 (10
–10
)
BNL-E821 04
DEHZ 03 (e
+
e
–
-based)
DEHZ 03 (
τ
-based)
HMNT 03 (e
+
e
–
-based)
J 03 (e
+
e
–
-based)
TY 04 (e
+
e
–
-based)
DEHZ 04 (e
+
e
–
-based)
BNL-E821 04
180.9
±
8.0
195.6
±
6.8
176.3
±
7.4
179.4
±
9.3 (preliminary)
180.6
±
5.9 (preliminary)
182.8
±
7.2 (preliminary)
208
±
5.8
Figure 3. Comparison of theresult (7)with the
BNL measurement[21℄. Also given are our
pre-vious estimates[12℄, where the triangle with the
dotted error barindi ates the -based result, as
well as the estimates from Refs.[14-16℄, not yet
in ludingtheKLOEdata.
5. Con lusion and Perspe tive
In spite of the new and pre ise data on the
two-pion spe tral fun tion from the KLOE
Col-laboration, the lowest order hadroni
va uum-polarization ontributionremainsthemost
riti- al omponentintheStandardModelpredi tion
ofa
. The entral pie e ofinformationprovided
by the present KLOE data is that they on rm
thedis repan ybetweenthe dataande
+
e
an-nihilationobservedinthis hannel[12℄.Thissaid,
wepointoutthat therealsoo ursdisagreement
betweenKLOEand CMD-2 data in someofthe
energyregions.
Anempiri alisospin-breaking orre tionofthe
resonan elineshape(massandwidth)improves
but doesnotrestore the agreement betweenthe
two data sets. It is a onsequen e of this
on- rmation that, until the CVC puzzle is solved,
onlye
+
e datashouldbeusedfortheevaluation
of thedispersionintegral. Doingso,and
in lud-ing the KLOE data, we nd that the Standard
Model predi tion of a di ers from the
experi-mental valueby2.7standarddeviations.
We are looking forward to the forth oming
results on the low- and high-energy two-pion
spe tral fun tion from the CMD-2
Collabora-tion. Thesedata willhelpto signi antlyredu e
the systemati un ertaintydue to the orre tive
treatment of radiative e e ts, often omitted by
partbythepreviousexperiments.
The initial-state-radiation program of the
BABAR ollaborationhasalreadyproved its
per-forman eby publishingthe spe tralfun tion for
+
0
(andsoonfor2
+
2 ),whileresultsfor
thetwo-pion nalstateareexpe ted.
A knowledgements
I thankAndreasHo ker,SimonEidelmanand
Zhiqing Zhangfor the fruitful ollaborationand
BillMar ianoformany larifyingdis ussions.
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