SCANNING OF THE e +
e !
+
CROSS-SECTION BELOW 1 GeV
AT DANE BY RADIATIVE EVENTS
M. I. Konhatnij, N. P. Merenkov
*
NationalSieneCentreKharkovInstituteofPhysisandTehnology
61108,Kharkov,Ukraine
Submitted21February2002
Thetaggedphotoneventsforthemeasurementofthee +
e ! +
totalross-setionbytheradiativereturn
methodatDANEis disussed. Theeetsausedbythenotexatlyhead-onollisionofbeamsandbythe
QEDradiative orretions are investigated. Theessential partonsists ofthe analysis ofthe event seletion
rulesthatensuretherejetionofthe3-pionhadronistateandtakethemainpropertiesofthemultiplepurpose
KLOEdetetorinto aount. Thestudyofthenon-head-oneetis performedintheBornapproximationby
integratingoverthetaggedphotonangles,whereastheradiativeorretionsarealulatednegletingthiseet.
Togetherwith the quasireal eletronapproah, this allows us toderive analytial formulasforthe orretion
tothe ross-setionoftheinitial-stateradiativeproess. Somenumerialalulationsillustrateouranalytial
results.
PACS:12.20.-m,13.40.-f,13.60.-Hb,13.88.+e
1. INTRODUCTION
The reent measurement of the muon anomalous
magneti moment a
= (g 2)
=2 performed in the
BrookhavenE861experimentwiththeeletroweakpre-
ision[1℄hasboostedtheinterestinarenewedtheoret-
ialalulationofthis quantity[2℄. Thereportednew
worldaverage has shown the disrepany of 2.6 stan-
dard deviations with respet to the theoretial value
basedonthe StandardModel alulation[3℄, and this
mayopen awindowinto possiblenew physisbeyond
the Standard Model. On the other hand, theonlu-
sion about a signiant disrepany between the re-
ported data and the Standard Model predition may
besomewhatpremature.
Theoretial estimations of a
inlude several on-
tributions involvingthenonperturbativehadronise-
toroftheStandardModel: vauumpolarization,light-
by-light sattering, and higher-order eletroweak or-
retions. Hadronieets in thetwo-loop eletroweak
ontributionaresmall,oftheorderoftheexperimental
error,andtheassoiatedtheoretialunertaintyanbe
broughtunder safeontrol[4℄.
Thesituationwithhadronieetsin thelight-by-
*
E-mail:merenkovkipt.kharkov.ua
lightsatteringradiallyhangedinreentmonths(af-
ter the E861 data were reported) due to works ited
in Ref. [5℄. In these works, the authors used the de-
sriptionofthe 0
transitionformfator basedon
alarge-N
C
expansionandshort-distanepropertiesof
QCD to alulate the pseudosalar hannel ontribu-
tion (in the
system). The orresponding result
disagreesby only its overall sign with the latest pre-
vious alulationsof two dierent groups[6, 7℄. It is
interesting to note that the result in Ref. [5℄ fored
both these groups to arefully hek their programs,
and they reently found their own (dierent) soures
ofthewrongsignforthepseudosalarhannel[8,9℄.
Themainingredientofthetheoretialpreditionof
a
,whihisresponsibleforthebulkofthetheoretial
error,istheontributionofthehadronvauumpolar-
ization. The problem is that it annot be omputed
analytially beause perturbative QCD loses its pre-
ditionpoweratlowand intermediateenergies,where
on the other hand, the orresponding eet is maxi-
mum. But this ontribution an be alulated from
thedata onthetotalhadroniross-setion
h forthe
proess e +
e ! hadrons using a dispersion relation
[10℄. Beause the existing data about
h
ome from
dierentsouresand donot alwaysmeettherequired
auray,theyaresupplementedwithatheoretialin-
put. Therefore, dierent estimationsgive dierentre-
sults, whih either strengthen the dierene between
theoretial and experimental values of a
or make it
onlymarginal[1114℄.
The ross-setion
h
also plays an important role
in the evolution of the running eletromagneti ou-
pling
QED
from low to high energies. This means
thattheinterpretationofmeasurementsathigh-energy
eletronpositron and eletronproton olliders de-
pendsonthepreiseknowledgeof
h
,withoneperent
aurayorevenbetter.
The updated hadroni light-by-light ontribution
[5, 8,9℄ dereasesthe disrepanybetween thetheory
and theexperimentfor a
to 1.5 standarddeviations,
andthedisagreementbetweentheStandardModeland
thereportedexperimentalvaluebeomesnotsosharp.
Nevertheless,whenthefullsetofdataattheBNLol-
laborationisanalyzed,theexperimentalerrorbarsare
expetedto derease bytheadditionalfator threeat
least, and this hallenges a new test of the Standard
Model. Thehigh-preisiondataabout
h
willplaythe
keyrolein thistest.
Wenotethatthedatareentlyderivedinthediret
sanning of
h
by CMD-2 [15℄ and BES II [16℄ ol-
laborationswereinludedin anewanalysis [17℄. This
signiantly redues the error in the hadroni ontri-
bution to the shiftof
QED
but doesnot removethe
disrepany in a
. Therefore, there existsan eminent
physial reason for new measurements to aumulate
high-preisiondataabout
h
atthetotalenter-of-mass
frameenergiesbelow1GeV.
Theoldideatousetheinitial-stateradiativeevents
in theeletronpositronannihilationproess
e (p
1 )+e
+
(p
2
)!(k)+hadrons(q) (1)
for thesanningof thetotalhadroniross-setion
h
hasbeomequite attrativereently[1822℄. Thisra-
diativereturnapproahallowsperformingthesanning
measurements at the aelerators running at a xed
energy, and this irumstane is the main advantage
ompared to thetraditionaldiret sanning. Therea-
sonisthatthemostimportantphysialparameters,the
luminosityandthebeamenergy,remainthesamedur-
ingtheentiresanningatxed-energy olliders. They
mustthereforebedeterminedonlyone,whihanbe
done with a very high auray. The drawbak is of
ourse a loss in the event number, and it is obvious
thatonlyhigh-luminosityaeleratorsanbeompeti-
tivewhentheradiativereturnmethodis used.
It is a general opinion that the high-luminosity
DANE mahineoperating in theresonaneregion
withmultiple-purpose detetorKLOEistheidealol-
lidertosan
h (q
2
)withtheenter-of-massenergy p
q 2
varyingfrom thethresholdto 1GeV just byradiative
events. Itisnowwellunderstoodthatinthisenergyre-
gion,thetotalhadroniross-setionismainlyfullled
by the ontribution of the resonane. This in turn
impliesthat the dominanthadroni nalstate is that
ofthehargedpion pair +
, and theKLOEdete-
torallowsmeasuringboththephotonenergydeposited
in alorimeters and the 3-momenta of pions running
throughthedrifthamber[20;23℄.
Suh a wide range of experimental possibilities of
the KLOE detetor an provide a realization of two
approahestosanningthe +
hannelontribution
to
h (q
2
): with tagged photonevents [19;20;22℄ and
withoutphoton tagging [22;24;25℄. The last method
has some advantages beause it allows inluding the
events with ollinear initial-state radiative photons,
whih leadstotheinreaseof theross-setionbythe
energylogarithm enhanementfator[26℄.
On theother hand,therst data aboutthe tradi-
tionaltaggedphotonsanningof
h (q
2
)atDANEare
reported [26℄ (we also note that large radiative event
rateswereobservedbytheBaBarCollaboration[27℄).
To extrat
h (q
2
) at dierent squared di-pion invari-
natmasseswith one perentauray, onemustpre-
iselyanalyzetheinitial-stateradiativeeventsandtake
thenal-stateradiativeeventsandtheinitial-statera-
diative interferene as a bakground. Moreover, the
radiative orretions must be alulated for all these
ontributions [21℄. For a realisti experimental event
seletion,this task anusually besolved by meansof
MonteCarloeventgenerators. Butforsomeidealon-
ditions,analytialalulationsmaybeperformedwith
highauray,andthisis averyimportanttestofthe
requiredoneperentaurayproduedbytheMonte
Carlogenerators.
The high-preision analysis of the initial-state ra-
diativeeventsisthemainattributeoftheradiativere-
turnmethod. Theorrespondingradiationorretions
were onsidered in a number of papers by both the
MonteCarlogenerators[19;22℄andanalytialalula-
tions[18;22;24℄. Inpresentpaper,wederiveanalytial
formulasfordierent ontributionsto theinitial-state
radiative ross-setion inluding the rst-order radia-
tive orretions. We use the samerules for the event
seletionasgivenin[24℄.Theseseletionrulesaremax-
imumdrawneartheexperimental ones [20;23℄exept
thedi-pionangular phasespae, forwhihweusethe
entire 4 openingangle. Thisisnottheasewiththe
realisti measurements at DANE beause there ex-
istsaso-alledblind zone in theKLOEdetetor with
the opening angle about 15 Æ
along the eletron and
positronbeamdiretions.Anypartileinsidethisblind
zoneannotbedetetedeitherbytheKLOEalorime-
tersorbytheKLOEdrifthamber.
InSe. 2,webriey reall the seletionrules used
hereandanalyzetheBornross-setionbynumerially
integrating the suiently ompliatedanalytial for-
mulasgivenin[24℄,whihtakethenon-head-onbeam
ollision into aount. In Se. 3, theontributions to
radiative orretions due to the virtualand soft pho-
ton emission are obtained by analytially integrating
over the angular phase spae of the photon that de-
posits its energy in alorimeters. InSe. 4, theeet
ofanadditionalhardphotonemissioninside theblind
zone is investigated and the orresponding ontribu-
tion to radiative orretions is derived. To perform
the analytial alulation, we use the quasireal ele-
tronapproximation[28℄forboththedierentialross-
setionandunderlyingkinematis. InSe.5,thetotal
radiativeorretionis derivedandsomenumeriales-
timates aregiven. Theeliminationoftheauxiliaryin-
fraredparameterisdemonstratedand thedependene
oftheradiativeorretionstotheinitial-stateradiative
ross-setiononthe squareddi-pioninvariantmassq 2
and physial parametersdening theseletion rulesis
investigated. WebrieysummarizeourresultsinCon-
lusion. IntheAppendies,wegivesomeformulasthat
areusefulinourintermediatealulations.
2. SELECTION RULESANDTHE
INITIAL-STATE BORNRADIATIVE
CROSS-SECTION
As mentioned above, the multiple purpose KLOE
detetor allows independently measuring the photon
energywithtwoalorimeters QCALand EMCALand
the 3-momenta of the harged pions with the drift
hamber. The seletion rules that we onsider here
an be formulated as follows: any event is inluded
if only one hard photon with the energy ! > !
m ,
!
m
= 50 MeV, hits the alorimeters and if the dif-
ferene between the lost energy and the lost 3-
momentummodulusjKjdoesnotexeedasmallvalue
E; 1;whereE is thebeamenergy. Thelost en-
ergyisdenedasthedierenebetweenthetotalinitial
energy andthe sumofthe hargedpion energies, and
thelost3-momentumisdened similarly.
Therst ruleimplies that in addition to onehard
photon, onlysoftphotonsthat annot bereordedby
the KLOE detetor anhit thealorimeters. Thera-
diationoftheadditionalhardphotonisallowedinside
the blind zone. The seond rule ensures the removal
of the 3-pion hadroni state arising due to possible
!
+
0
and ! ! +
0
deays. Theneutral
pionquiklydeaysintotwoquanta;oneofthesehas
timetolightinthealorimeters,whereastheotheran
yawayintotheblindzone. Itiseasytosee thatthis
ruledoesnotallowthelostinvariantmasstobegreater
than2E 2
,andthe3-pionstateisthereforeforbidden
if < 0:035:Thus, the following event seletionuts
areimposed[20℄:
jKjE; !>!
m
;
=0:02; !
m
=50MeV:
(2)
Therstinequalityin(2) isimportantinalulat-
ingtheontributionausedbytheemissionoftwohard
photons(one inside the alorimeters and the other in
theblind zone)beauseitonlyaetsthephasespae
oftwohardphotons. AttheBornlevel(withonlyone
photoninsidethealorimeters),wemustthereforetake
theseondrestritionin (2)into aountbyintrodu-
tionthetrivialfuntion. Thedierentialdistribution
overthedi-pioninvariantmassanbewrittenas[24℄
d B
dq 2
=
2 2
(q 2
)
(S q 2
)dosd'
4S(2E jP
jsinos') 2
(S+T
1 )
2
+(S+T
2 )
2
T
1 T
2
(! !
m ); (3)
!=
S q 2
2(2E jP
jsinos')
;
where and '(!) are the polar and azimuthal an-
gles (energy) of the photon deteted by the KLOE
alorimeters. The approximation used hereis valid if
E 2
2
0
m
2
; where 2
0
is the opening angle of the
blind zone and m is the eletron mass. The ross-
setion(q 2
)oftheproesse +
e ! +
isexpressed
throughthepioneletromagnetiformfatorF
(q
2
)as
(q 2
)=
2
jF
(q
2
)j 2
3q 2
1 4m
2
q 2
3=2
;
where m
is the pion mass. The invariants entering
Eq.(3)aregivenby
T
1
= 2p
1
k= !(2E 2P
z
os jP
jsinos');
P
z
=E
1 jP
j
2
8E 2
;
T
2
= 2p
2
k= !(2E+2P
z
os jP
jsinos');
S=2p
1 p
2
=4E 2
jP
j
2
; q 2
=S+T
1 +T
2 :
In writingthese expressions, wetook into aount
that at DANE, theeletron andpositron beamsex-
erisenotexatlyahead-onollisionattheinteration
1 2
3
50 100 200 250
z
0:2 0:4 0:6 0:8 1:0
0 150
2
3 1
a b
50 100 200 250
z 150
0
0:901 0:902 0:903
Fig.1. TheBornross-setion d
dz
=
2 (q
2
)
oftheinitial-stateradiativeproesse +
e !+ +
atdierentlimiting
anglesoftheblindzoneversusvariablez:Figure1aorrespondstotheontributionoftheregionD>1denedbyEq.(4).
Thesumofontributionsoftheregions1>D>s
0 ands
0
>D> 1isshownin Fig.1b(seeRef.[24℄, Eqs.(19)(21),
fortheorrespondinganalytialformulas). Itdependsontheminimumenergyofthetaggedphoton!
m
,whihwehoose
as50MeV.0=5 Æ
(1),10 Æ
(2),15 Æ
(3)
point,but there exists asmall rossingangle between
themthatisequaltojP
j=E,wherejP
j=12:5MeV.
Beause ofanonzerorossingangle, theenergyofthe
tagged photonbeomes dependent on its angular po-
sition, whihompliates theexatanalytialalula-
tions. Thus,thequestionarisesastothemagnitudeof
theorrespondingeet. Asshownin [24℄,thereexist
threeregions,
D>1; 1>D>sin
0
; sin
0
>D> 1;
where
D=
4E(E !
m ) q
2
jP
j
2
2!
m jP
j
;
andtheformoftheinitial-stateradiativeross-setion
is dierent in eah region. The analytialexpressions
for the distribution overthe di-pioninvariantmass is
simpleinthersttworegions,butitseemsthatonlya
numerialintegrationwithrespettothephotonpolar
angleispossibleinthethird region. Wealsonotethat
in the limitingaseasjP
j!0, onlytherst region
anourwiththeobviousrestrition
4E(E !
m )>q
2
:
The results of our alulations of the Born ross-
setion are shown in Figs. 1 and 2. It follows from
Fig.1thattheontributionoftherstregion(D>1)
dominates in a wide interval of the di-pion invariant
masses. Withintheapproximation
4E 2
(1
0 )P
2
;
R Φ
1
3 2
0.2 0.4 0.6 0.8 1.0
0
z 0.50
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
Fig.2. Theeetofanon-head-onollisionoftheele-
tronandpositron beams atDANEis shownforthe
ontributionoftheregionD>1intermsoftheratio
denedby Eq.(6)atdierent
0
=5 Æ
(1), 10 Æ
(2),
15 Æ
(3). Theapproximation4E 2
(1
0 )jP
j
2
is
usedinalulatingthisratio. ThequantityRisgiven
perent
theorrespondingross-setionanbewrittenas[24℄
d B
0
dq 2
=
2 (q
2
)
2E 2
F
0
(D 1);
F
0
=
S
S q 2
1+ S q
2
2S
2ln 1+
0
1
0 +
jP
j
2
2E 2
ln 1+
0
1
0 +
0
s 2
0
S q 2
S
0
1+ jP
j
2
2E 2
(3
2
0 )
; (4)
where
0
=os
0 , s
0
=sin
0
; and thefuntion de-
nesthemaximumpossiblevalueofq 2
:
Theform ofF
0
andD in Eq.(4) is validifall po-
laranglesbetween
0
and
0
arepermittedforthe
taggedphoton. Iflarge-anglephotonsradiatedbetween
l
and
l
arenotreorded,wemustwrite
d B
l
dq 2
=
2 (q
2
)
2E 2
(F
0 F
l )(D
l
1); (5)
where F
l
anbederived from F
0
by simply replaing
0 with
l and
D
l
=
4E(E !
m ) q
2
jP
j
2
2!
m jP
js
l
; s
l
=sin
l :
InSe.3,weonsiderthis asewith
l
=40 Æ
andall
itthemodiedEMCALsetup.
ToindiatetheeetofnonzeroP
intherstre-
gion,weshowinFig.2theratio
R
= F
0
F(z;
0 )
F(z;
0 )
; F(z;
0 )=F
0 (P
=0)=
=
1+z 2
1 z ln
1+
0
1
0
(1 z)
0
; z= q
2
4E 2
: (6)
ForthemodiedEMCALsetup,theorrespondingra-
tio isgivenby
R l
= F
0 F
l
[F(z;
0
) F(z;
l )℄
F(z;
0
) F(z;
l )
: (7)
AsanbeseenfromFig.2,this eetdoesnotexeed
vepermilleat
0
=5 Æ
anddereasesastheangle
0
grows.
The ontribution of the third region (sin
0
>
> D > 1) is negligible everywhere, and the seond
region (1>D >sin
0
) ontributesonlyinside avery
narrowintervaloftheorder210 3
nearthemaximum
possible di-pion invariant mass squared(see Fig.1b).
Wetherefore onludethat for restritions(2)onsid-
ered here, the eet of non-head-on ollisions on the
event seletion at DANE is about several per mille
in themostimportantresonaneregionandisunder
ontrol where it annot be negleted. In this region,
theorrespondingross-setionanthereforebegiven
byEq.(4)with theF(z;
0
)insteadofF
0
withthere-
quiredauray.
3. VIRTUALANDSOFTCORRECTIONS
High-preisiontheoretialpreditionsareneessary
in order to reah the auray of one perent in the
measurementofthepion ontributionto thehadroni
ross-setion at DANE by radiative events. These
preditionsmustat leastinlude therst-order radia-
tive orretions that aount for the virtual and real
softphotonontributionintheoverallphasespaeand
anadditionalontributiondueto ahardphotonemis-
sioninsidetheblindzone. Inalulatingradiativeor-
retions,wenegletP
at theverybeginningand set
s;t
1 , andt
2
equalto S;T
1 ,and T
2 atP
=0;respe-
tively.
To alulate the virtual and soft orretions, we
start from the orresponding expression derived in
Ref.[24℄(Eq.(30)),performthetrivialazimuthalangle
integration,andwritetheresultintheonvenientform
d V+S
dq 2
=
2
2
(q 2
)
(1 z)dos
2s
(q
2
t
1 )
2
+(q 2
t
2 )
2
t
1 t
2
+T
(1 x
m
z); (8)
where
T= 3
2 T
g 1
8q 2
T
11 (q
2
t
1 )
2
+T
22 (q
2
t
2 )
2
+
+(T
12 +T
21 )(sq
2
t
1 t
2 )
; s=4E 2
;
=4(L
s
1)ln+3L
q +
2 2
3 9
2 +
+4
ln
1
ln
1+
0
1
0 +ln
2
1 ln
1+
1
1
1
;
L
s
=ln s
m 2
; L
q
=ln q
2
m 2
; x
m
=
!
m
E :
The aboveexpressionfor ontainsthree softpa-
rameters ,
1
, and
2
: The rst restrits the soft
photonenergyinsidetheblindzonewiththevalueE:
Itisauxiliaryandanelswhentheontributionaused
bythehardphotonemissionisadded(seeSe.5). The
parameters
1 and
2
are physial. Theyare dened
bythesensitivity
1
E of theQCAL alorimeter that
surrounds the blind zone and overs polar angles of
thedetetedphotonfrom
0 to
1
=20 Æ
with respet
toboththeeletronandthepositronbeamdiretions
(
1
=os
1
)andbythesensitivity
2
EoftheEMCAL
alorimeter that oversthe photonangles between
1
and
1
. Theaseofaslightlymodiedgeometryof
EMCAL(withthepolaranglesfrom
l
=40 Æ
to
l
notovered[20℄)isonsideredinAppendix A.Theo-
eients T
g and T
ik
in theright-hand side of Eq. (5)
arealulatedasfuntions oftheinvariantss, t
1 , and
t
2
inRef.[24℄ (seealso[29;22℄).
Ouraimistoanalytiallyintegratedierentialdis-
tribution (8) withrespet to the tagged photonpolar
angle. Theintegrationofthetermontainingistriv-
ialandyields
d S+V
dq 2
=
2
2
(q 2
)
2E 2
F(z;
0
)(1 x
m
z): (9)
Toperformtheremainingintegrations,itis onve-
nienttorepresentthequantityT as
T =L 2
q L
2
s +
2s
q 2
s 2L
qs +
2s(s 2q 2
)
(q 2
s) 2
+1
L
q +
+ 2(2q
2
s)s
(q 2
s) 2
L
s +L
1q
4s 2
(q 2
s)t
2 2(q
4
+s 2
)
(q 2
s)t
1 +2
+
+ln t
1
m 2
sq 2
(s+t
2 )
2 +
q 2
+3s
s+t
2
1 2(L
q L
s )
1
2s 2
t
2 +
q 4
+s 2
t
1
1
q 2
s
+
+ 2(q
4
+3s 2
)
6L
qs +3(L
2
s L
2
q )+
2
3(q 4
2sq 2
s 2
)
6(q 2
s)t
1
sq 2
(s+t
1 )
2 (q
2
+3s)
s+t
1
L
q
2
3 +
+ s
s+t
1 +
1
2 +(t
1
$t
2
); (10)
where
L
qs
=Li
2
1 q
2
s
; L
1q
=Li
2
1 t
1
q 2
:
TheSpenefuntionL
1q
hasanonzeroimaginarypart,
butwemusttakeintoaountonlyitsrealpartinour
alulations,
ReL
1q
=
2
6 ln
t
1
q 2
ln
1 t
1
q 2
Li
2
t
1
q 2
:
We also notethat the oeientsT
g andT
i;k
ontain
the termsinvolvingt 2
1;2 and t
3
1;2
[22;24;29℄,but these
vanishin thequantityT.
Integrating thepiee of ross-setion(5) that on-
tainsthequantityT withrespettothetaggedphoton
polarangles,weobtain
d S+V
T
dq 2
=
2
2
(q 2
)
4E 2
(1 z)F
T (z;
0
)(1 x
m
z); (11)
where the funtion F
T (z;
m
) is given in Appendix A
for arbitrary values of the limiting angle
m
. Here,
we onsider the ase where
m
=
0
and use the ap-
proximation1
0
1, whih issuiently good for
0 10
Æ
(preisely this ase is suitable for the blind
zoneoftheKLOEdetetor),
(1 z)F
T (z;
0
!1)= 4ln
3
z
3(1 z) +
+ 2(1+z
2
)
1 z ln
1 z
z ln
2
z (3 z)ln (1 z)
2
z
lnz+
+
4(1 2z)
1 z
lnz+4(1 z)ln(1 z)
2
3 z 2(1+z)ln 1 z
z
Li
2
(1 z)+
+ 8
1 z
Li
3
(1 z)+z 2
Li
3
z 1
z
5+z
2(1+z 2
)
1 z
lnzln 2
1
0
2 +
+
1 z 2
1 z
2(1+z 2
)
1 z
ln (1 z)
2
z
lnz+2Li
2 (1 z)
ln
1
0
2
: (12)
Thetotalvirtualandsoft orretionisthe sumof(9)
and(11).
For the modied form of the EMCAL alorime-
ter, the expressions for and F(z;
0
) in Eq. (9) and
F
T (z;
0
)inEq.(11)mustbehangedas
!+4ln
2 ln
1+
l
1
l
;
F(z;
0
)!F(z;
0
) F(z;
l );
F
T (z;
0 )!F
T (z;
0 ) F
T (z;
l );
(13)
wheretheexpressiongiveninAppendixAmustbeused
forthefuntion F
T (z;
l )at
m
=
l :
In alulating the virtual and soft orretions, we
negletedtermsoftheorder
i
; i=1;2ompared to
unity. Thisaurayimpliesthesamerelationbetween
thetaggedphotonenergy!andthesquareddi-pionin-
variantmassq 2
asin theBorn approximation,
!=E(1 z) (14)
(provided that P
= 0); and it sues to guarantee
theone perentpreision.
4. HARDPHOTON EMISSION INSIDETHE
BLIND ZONE
Seletion rules (2) used here permit the radiation
ofanadditionalinvisible photoninsidetheblindzone.
Fortheevents
e (p
1 )+e
+
(p
2
)!(k
1 )+(k
2 )+
+
(q); (15)
one photonwith the4-momentum k
2
hitsthe photon
detetor andtheotherphoton(withthe4-momentum
k
1
) is ollinear and esapes it. It is obvious that re-
lation (14)betweenthetagged photonenergyandthe
squareddi-pioninvariantmassisviolatedin thisase.
Toalulatetheorrespondingontributionintora-
diativeorretionsanalytially,westartwithusingthe
quasireal eletronapproximationfor boththe form of
theross-setionandtheunderlyingkinematis. Phys-
ially, this implies that weneglet terms of the order
1
0
2
0
=2andm 2
=E 2
2
0
omparedtounity. Were-
allthat 2
0
=20:02fortheKLOEdetetor. Inaor-
dane with the quasireal eletron approximation, the
dierentialross-setionofproess(15)anbewritten
inthesameformasfortheinlusive(untaggedphoton)
eventseletion[25℄,
d H
=2
2
2
(q 2
)
4E 2
x P(x;L
0 )dx
(q 2
xu
1 )
2
+(q 2
u
2 )
2
xu
1 u
2
!
2 d!
2 d
2 (!
2
!
m
); (16)
P(x;L
0 )=
1+x 2
1 x L
0 2x
1 x
;
L
0
=ln E
2
2
0
m 2
;
2
=os
2
;
x=1
!
1
E
; u
1
= 2k
2 p
1
; u
2
= 2k
2 p
2
;
where !
2 (
2
) is the energy (the polar angle) of the
tagged photon and !
1
is the energy of the invisible
ollinearphoton.
The fator (=2)P(x;L
0
)dx desribes the radia-
tion probability of the ollinear photon by the initial
eletron, the fator2 aounts forthe sameontribu-
tion aused bythe initial positron ollinear radiation,
and the rest is in fat theross-setion of proess (1)
with thereduedeletron4-momentum (p
1
!xp
1 )at
P
=0:
Ouraim is now to derivethe dierential distribu-
tionoverthesquareddi-pioninvariantmassq 2
;andit
is onvenientto usethe relationbetweenq 2
and
2 in
ordertoavoidtheintegrationover
2
intheright-hand
sideofEq.(16). Todisentangletheseletionrulesand
obtain the integration region, it is also useful to in-
trodue thetotal photonenergy =!
1 +!
2
instead
of!
2 ,
q 2
=4E(E )+2!
1
!
2
(1
2 );
d
2
! dq
2
2!
1
!
2
; d!
2
!d
2 :
(17)
Takingintoaountthatin termsofnewvariables,
u
1
= 4E
2
z
!
1
; u
2
= 4E
!
1 [!
1 ( !
1
) E
z
℄;
z
= E(1 z);
(18)
weanrewriteEq.(16)as
d H
dq 2
=2
2
2
(q 2
)
4E 2
M(z;L
0
;!
1
;)
d!
1 d
E
z
( !
1
!
m
); (19)
where
M(z;L
0
;!
1
;)= L
0 4(L
0
1)E
z
! 2
1
zE 2
L
0
(E !
1 )
2 +
[2z (1+z)L
0
℄E
E !
1
+
+ [2(1+z
2
)L
0
4z (1 z) 2
℄E 2
(1 z)( 2!
1 )E
!
1
( !
1
) E
z
:
(20)
Wenote that although theterm ontaining ! 2
1 in the
denominatoranbenegletedin theaseoftheinlu-
siveeventseletion[25℄,it isnowimportant andeven
ontributestotheanellationoftheauxiliaryinfrared
parameter:
Wenowndtheintegrationregionforthevariables
!
1
and determined by restritions (2) and by the
inequalities
m
<
2
<
m
; E<!
1
< Ex
m
(21)
limitingthepossibleanglesforthetaggednonollinear
photonandtheenergiesof theinvisible ollinearpho-
ton. Here,wedonotrequire1
m
tobesmall,bearing
inmindafurtherappliationtothemodiedEMCAL
setup. Therstrestritionin(2)denesthemaximum
valueof ; the minimum value of anbe obtained
fromrelation(17)at !
1
=E and
2
=
m ,
max
=E(1 z)
1+
2
;
min
=E(1 z)
1+
(1
m )
2
:
(22)
Theondition
2
>
m
impliesthat
! <!
1
<! +
;
!
=
2
"
1 s
1
8E
z
2
(1+
m )
#
:
(23)
Finally, theinequality
2
<
m
an be formulated
asfollows: ifthevaluesofaresuhas
<
;
=E(1 z)
1+
(1
m
)(1 z)
8
;
then
!
1
>!
+
or !
1
<! ;
!
=
2
"
1 s
1
8E
z
2
(1
m )
#
:
(24)
Theonsistentombinationofthesetofinequalities
(21)(24) for!
1
and denes the integration region.
Ingeneral,thisregiondependsonthedi-pioninvariant
massthroughz;itisshowninFig.3,whereweusethe
notation
=E(1 z)(1+);
x
=E(1 z)
1+
(1
0 )x
m y)
2(1 z)
;
y=1 z x
m :
(25)
This region diers from the orresponding region for
theinlusiveeventseletion[25℄.
Wenote that in the alulation,weontrolled our
analytial integration bymeans of the numerialone.
Thisallowedustoonludethat intheasewhere
z<z
=1 4
1
m
;
the ontribution of the topregion in Fig. 3a issmall,
and weexluded it from onsideration. Although the
regions in Figs. 3b and are dierent, the respetive
ontributions toross-setion(19)havethesameana-
lytialform.
The list of the integrals that are reguired in both
ases,z<z
andz>z
,isgiveninAppendixB.Using
these integrals, wewrite the ontributionof theaddi-
tional hard photon emission inside blind zone to the
radiativeorretionsas
d H
dq 2
=
2
2
(q 2
)
2E 2
[G
ln+P(z;L
0 )G
p +L
0 G
1 +G
0
℄: (26)
If thetaggedphoton isdetetedin theangular region
0
> >
0
; the oeientsG
i
(i =;p;1;0) are
givenby
G
=4(L
0 1)
1 z+
1+z 2
1 z l
0
; l
0
=ln
2
0
4
; (27)
G
p
= ln 2
(1 z)
2(1 z x
m )x
m 2ln
1 z
x
m
ln
4(1 z)x
m
2
4Li
2
x
m
1 z
+ 2
2l
0 ln
(1 z)(1 z x
m )
2x
m
; (28)
G
1
=x
m
2+
1+ 1
z+x
m
ln 1 z
x
m
2(1 z)
2ln
2 +3
+
+(2z+x
m )
1
z+x
m 1
l
0 +ln
2(1 z x
m )
(1+z)
1
2 ln
2
z lnzln x
m
1 z
ln (z+x
m )
1 l
0 ln
2x
m
(1 z)
+Li
2
(1 z)+
+Li
2
x
m
z
Li
2 (z+x
m )+
2
6
; (29)
G
0
=
2
3
(2 3z)+4(1 z)
1+ln
2
+
+2z
1
2 ln
2
z lnzln x
m
1 z
+ln(z+x
m )
l
0 +ln
2x
m
(1 z)
+
+Li
2
(1 z)+Li
2
x
m
z
Li
2 (z+x
m )
i
+
+(1 z)
ln 2
2(1 z x
m )x
m
(1 z) +
+ln 1 z
x
m ln
4(1 z)x
m
2
+2l
0 ln
(1 z x
m )
2
+
+2Li
2
x
m
1 z
+Li
2
(1 z)
2(1 z x
m )x
m
; (30)
wherewepasstothelimit1
0
1andtakeintoa-
ountthatonlytheregionsinFigs.3bandontribute
inthislimitingase.
Todesribe theontributioninto radiativeorre-
tionsausedbythedoublehardphotonemissionwith
thetagged photonin therange
0
> >
l ,
l
> >
0
; whih orresponds to the modied
EMCALsetup,wemustevaluatethedierene
d H
dq 2
(
m
=
0 )
d H
dq 2
(
m
=
l
): (31)
Wehave
00 00 00 00 00 00 11 11 11 11 11 11
000000 000000 000000 000000 000000 000000 000000 000000
111111 111111 111111 111111 111111 111111 111111 111111
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000
111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111
00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111
x
Exm
!
l
max min
!
l=
!
+
! l
=
E x
m
! l
=
!l
=
!
!l
=
!
E a
Exm x
E
!
l
max min
! l
=
E x
m
! l
=
! l
=
!
!l
=
!
!
l
=
!
+ b
Exm x
E
!
l
max min
! l
=
E x
m
! l
=
! l
=
!
! l
=
!
Fig.3. Theintegrationregionfortheontributionofthehardphotonemissioninsidetheblindzoneintheregionsz<z(3a),
z<z<1 2xm (3b),and1 2xm<z<1 xm(3).WeneglettheontributionofthetoppieeinFig.3a,whihis
justiedbynumerialontrol
d H
M
dq 2
=
2
2
(q 2
)
2E 2
(G
G
l
)ln+
+P(z;L
0 )(G
p G
l
p )+L
0 (G
1 G
l
1 )+G
0 G
l
; (32)
wheretheexpressionsforthefuntionsG l
i
aregivenin
Appendix C.
Inthis ase, there also exists an additional region
wheretheradiationofthehardphotonatlargeangles
an ontribute. It overs polar angles from
l up to
l
: We take only one piee of the orresponding
ontributionintoaount(theonethatisproportional
to ln)andwriteitas[24℄
d L
M
dq 2
= d
B
l
dq 2
2 4ln
1
ln
1+
l
1
l
; (33)
where d B
l
=dq 2
is dened by Eq. (5). The remaining
ontribution issmallbeauseofrestrition(2)and we
expetthat itisparameteriallyequalto =2l
0 rel-
ativetotheBornross-setion.
5. THETOTALRADIATIVECORRECTION
The total radiative orretion to the ross-setion
oftheinitial-stateradiationproess(1)withthe +
hadroninalsateisdenedbythesumoftheontri-
butionsausedbythevirtualandrealsoftphotonemis-
sionand by theradiationofthehardollinearphoton
insidetheblindzoneoftheKLOEdetetor. Inalulat-
ing theradiativeorretion,wesupposethat P
=0,
beausetheorrespondingeet dueto thenon-head-
on ollision of beams annot be greaterthan 10 3
at
theradiativeorretionlevel. Itiseasytoseethatthe
auxiliaryinfraredut-oparametervanishesforboth
formsoftheEMCALalorimeter.If
0
>>
0 it
entersthissumintheombination
2
2
(q 2
)
2E 2
ln
4(L
s
1)+4ln 1
0
1+
0
F(z;
0 )+G
; (34)
wheretheexpressioninsidetheurlybraketsvanishes
inthelimitingasewhere1
0
1,whihwasusedin
alulatingG
: We anthereforewrite the analytial
expressionforthederivedradiativeorretionas
d RC
dq 2
= d
B
dq 2
Æ RC
; Æ RC
=
2 V
F(z;
0 )
; (35)
V =F~ (z;
0 )+
1
2
(1 z)F
T (z;
0 )+
+P(z;L
0 )G
p +L
0 G
1 +G
0
; (36)
where d B
=dq 2
is dened by Eq. (4) at P
= 0, ~
is without the terms ontainingln, and the limit
1
0
1mustbetakenforthefuntionsF(z;
0 )and
F
T (z;
0 ):
ForthemodiedEMCAL alorimeter,the expres-
sionin theurlybraketsin (34)isreplaedby
4
L
s
1+ln (1
0 )(1+
l )
(1+
0 )(1
l )
ln 1+
l
1
l L
0 +1
[F(z;
0
) F(z;
l
)℄; (37)
where theterms in thesquare brakets orrespond to
theontributionofthevirtualandsoftphotonemission
andtheremainingtermsareausedbythelarge-angle
(largerthan
l
)andsmall-angle(smallerthan
0 )hard
photonradiation.
Thetotalradiativeorretionanthenbewrittenas
1
2
a
0 0.2 0.4 0.6 0.8 1.0
z δ RC
− 0.28
− 0.26
− 0.24
− 0.22
− 0.18
− 0.16
− 0.20
1
2
b
0 0.2 0.4 0.6 0.8 1.0
z δ RC
− 0.16
− 0.18
− 0.20
− 0.22
− 0.24
− 0.26
− 0.14
Fig.4. Trendsinthez dependeneofthequantityÆ RC
denedbyEq.(35)underthevariationofthephysialparameters
0 andx
m : a
0
=10 Æ
,x
m
=0:098 (1);
0
=5 Æ
,x
m
=0:098 (2); b
0
=7:5 Æ
,x
m
=0:039 (1);
0
=7:5 Æ
,
xm=0:098 (2). Allurvesarealulatedat=0:02,1=0:002,and2=0:01
2
1
a
0 0:2 0:4 0:6 0:8 1:0
z Æ
RC
0:16
0:18
0:20
0:22
0:24
0:26 0:14
2
1
b
0 0:2 0:4 0:6 0:8 1:0
z Æ
RC
0:26 0:24 0:22 0:20 0:18 0:16 0:14 0:12
Fig.5. Inueneofthephysialparameters,1,and2onthezdependeneofÆ RC
:Theminimumenergyofthetagged
photonis50MeV(xm=0:098);a0=7:5 Æ
,=0:02,2=0:01,1=0:002(1)0.01(2);b0=7:5 Æ
,=0:03,
2=0:015,1=0:002(1) 0.015(2)
Æ RC
l
=
2
V
M
[F(z;
0
) F(z;
l )℄
; (38)
V
M
=~
M [F(z;
0
) F(z;
l )℄+
1 z
2
[F
T (z;
0 ) F
T (z;
l
)℄+P(z;L
0 ) G
p G
l
p
+
+L
0 G
1 G
l
1
+G
0 G
l
0
;
~
M
=+4~ ln
2 ln
1
l
1+
l : (39)
Toidentify trendsin thebehaviorof theradiative
orretion,westudyitsdependeneonthephysialpa-
rameters that dene event seletion rules(2), namely
andx
m
,andthedependeneontheopeningangleof
theblind zone
0
and the respetivesensitiveness
1
and
2
oftheQCALand EMCALalorimeters.
TheresultsforÆ RC
givenbyEq.(35)areshownin
Figs.4and5.Aswasexpeted,theradiativeorretion
islargeandnegativebeausethepositiveontribution
aused by the real photon radiation annot ompen-
sate the negativeone-loop orretion. This eet in-
tensies beause the rst inequality in (2) dereases
thephasespaeoftheadditionalinvisiblerealphoton.
Theabsolutevalueofradiativeorretiondependsonz
andhangesfrom14%nearthe +
pairprodution
thresholdto25%atthemaximumpossiblesquareddi-
pioninvariantmass. In themoreinterestingregionof
the resonane(0:5<z <0:7), it amountsto about
2 1
a
0 0:2 0:4 0:6 0:8 1:0
z 0:14
0:16
0:18
0:20
0:22
0:24
0:26
0:28 Æ
RC
l
2 1
b
0 0:2 0:4 0:6 0:8 1:0
z 0:12
0:14
0:16
0:18
0:20
0:22
0:24
0:26
0:28 Æ
RC
l
Fig.6. The z dependeneof Æ RC
l
dened by Eq. (38) at dierentvalues of
0 and x
m
: a
l
= 40 Æ
, x
m
= 0:098,
0
=10 Æ
(1),5 Æ
(2);b
0
=7:5 Æ
,
l
=40 Æ
,x
m
=0:039,(1),0.098(2)
1420%.
Themain peuliarities in thebehaviour of thera-
diativeorretionarerelatedtothehangeofthepos-
itiveontribution aused bythe radiation of anaddi-
tional invisible hard photon. If the limiting angle
0
dereases, theabsolutevalueinreasesbeausethein-
visiblephotonangularphasespaeisthenompressed.
Conversely, thederease of theminimal energyof the
tagged photon leads to an expansion of the energy
phasespae oftheinvisiblephotonat axed valueof
(seeFig.3),andtherefore,toadereaseoftheabso-
lute value. A similar eet ours asthe parameter
grows. Butthetotalenergyofboththetagged and
the invisible photonsthen inreases,and the absolute
valuedereasesasin thepreviousase.
The hange of the parameters
1
and
2 aets
the energy phase spae of an additional real invisible
softphotoninsidetheKLOEalorimeters. Ifthesepa-
rametersareinreased, theorrespondingphasespae
expandsandtheabsolutevaluedereases.
Thetotalrst-orderradiativeorretionÆ RC
l
forthe
modiedEMCAL setup is shown in Fig. 6. Near the
threshold,itissomewhatsmallerthanÆ RC
intheabso-
lutevalue,butitgrowsmorerapidlywiththeinrease
ofz:
Ouralulationsarerestritedbyonsidering only
therst-orderorretiontotheBornross-setion. But
alarge valueoftheabsolute valuerequires evaluating
the eets of higher-order QED orretions to larify
the questionof whether ourapproximationsuesto
provide the one per ent auray even in the region
of theresonane. We hopeto alulate theseeets
elsewhere.
6. CONCLUSIONS
The radiativereturn method with tagged photons
oersa uniqueopportunity for ameasurementof the
totalhadroniross-setion(e +
e !hadrons)overa
widerangeofenergies. Thedereaseoftheeventnum-
ber iseasily ompensated by ahigh luminosity ofthe
neweletronpositronolliders. Of apartiular inter-
estaretheexperimentaleortsatlowandintermediate
energiesbeausethey aremandatory forthefuture of
theeletroweakpreisionphysis.
Suess of the preision studies of the hadroni
ross-setion through the measurement of radiative
eventsrelies onthe mathinglevelofreliabilityofthe
theoretial expetation. The prinipal problem is the
analysisofradiativeorretionsto theinitial-statera-
diative ross-setion at realisti onditions as regards
theevent seletion. Inthe presentwork, we havede-
velopedtheapproahproposed inRef. [24℄forahigh-
preisionanalytial alulation of the e +
e !
+
hannel ontribution to the hadroni ross-setion.
Thishannel dominatesintherange below1GeV be-
ause of the radiativereturn on the resonane and
theorrespondingontribution anbe measuredwith
a high-preision at the DANE aelerator with the
multiple-purposeKLOEdetetor[26℄.
Ouralulations inlude theanalysis ofthe eets
relatedtonon-head-onollisionofbeamsandtherst-
orderradiativeorretion. Wehavedemonstratedthat
attheBornlevel,thenon-head-oneetsdonotexeed
several per mille. To derive the radiative orretion,
wenegletedthiseetsandalsoappliedthequasi-real
eletronmethod [28℄to desribeeventswithtwohard
photons,onetaggedbytheKLOEalorimetersandthe
otherinvisibleinsidetheblindzone. Thisapproahhas
allowedus to analytially disentanglerealisti restri-
tionsrelatedtotheeventseletionrulesandtheKLOE
detetor geometry. Therst-orderQEDradiativeor-
retionobtainedinthiswayisnegativeandlargeinthe
absolute value. Weinvestigatedthemaintrendsin its
behaviour at the variationof the physial parameters
thatdeneexperimentalrestritionsoneventseletion;
weonludethat thehigher-orderorretionsmustbe
evaluatedinordertoensuretheoneperentauray
requiredforthetheoretialpreditions.
APPENDIXA
Here,wegivetheexatresultof theanalytialan-
gular integrationof the quantityT (seeEq. (8)) with
respettothetaggedphotonpolaranglesatarbitrary
valuesofthelimitingangle
m
andthesquareddi-pion
invariantmass,
m
Z
m
Td=2F
T (z;
m
); (A.1)
(1 z)F
T (z;
m )= 2
1+z 2
1 z lnzln
2
1
m
2 +T
1 (z;
m )ln
1
m
2 +T
0 (z;
m );
where
T
0 (z;
m )=
4
3(1 z) ln
3
z+2
m
z 5
2 +
(1 z)[ 4zlnz+((1 2
m )(1 z
2
)+8z)ln(1 z)℄
[1+
m
+(1
m
)z℄[1
m
+(1+
m )z℄
+
+
3 6z+z 2
1 z lnz
+2[2
m (1 z)℄
1+ln 1+
m
2
ln
1
(1
m
)(1 z)
2
+
+Li
2
(1+
m
)(1 z)
2z
2[2+
m
(1 z)℄
ln
1
(1+
m
)(1 z)
2
ln 1 z
z +Li
2
(1
m
)(1 z)
2z
+
+4(1 z)
m Li
2
1 z
z
+ 4(1+z
2
)
1 z ln
1 z
z
Li
2
(1
m
)(1 z)
2z
Li
2
(1+
m
)(1 z)
2z
+
+ 8
1 z
2lnzLi
2
1
m
2
+ln 1 z
z
Li
2
(1
m
)(1 z)
2
Li
2
(1+
m
)(1 z)
2
z 2
Li
3
(1
m
)(1 z)
2z
Li
3
(1+
m
)(1 z)
2z
Li
3
1
m
(1+
m )z
+Li
3
(1
m )z
1+
m
Li
3
(1
m
)(1 z)
2
+Li
3
(1+
m
)(1 z)
2
+
+ 5+z
2
4
lnzln 2
1+
m
2
+ 2
1 z ln
1+
m
2
2z 2
ln 2
z (3 z)(1 z)lnz+
1+2z z 2
2
+2(1+z 2
)
lnzln(1 z) Li
2
(1+
m
)(1 z)
2z
+
+4Li
2
(1
m
)(1 z)
2
+2(1 z 2
)Li
2
1 z
z
+(1 z) 2
(1+
m )
3+
m
+(1
m )z
1+
m
+(1
m )z
; (A.2)
T
1 (z;
m )=
2
1 z
2z 2
ln 2
z+(3 z)(1 z)
lnz
1+2z z 2
2
2(1+z 2
)
lnzln(1 z) Li
2
(1
m
)(1 z)
2z
4Li
2
(1+
m
)(1 z)
2
2(1 z 2
)Li
2
1 z
z
(1 z)[2+
m
(1 z)℄ln
1
(1+
m
)(1 z)
2
(1 z) 2
(1
m )
3
m
+(1+
m )z
1
m
+(1+
m )z
: (A.3)
WehereusethestandardnotationfortheSpenefun-
tions
Li
2 (x)=
1
Z
0 dt
t
ln(1 xt); Li
3 (x)=
1
Z
0 dt
t
lntln(1 xt):
If weassume that
m
=
0 , 1
0
1, the resultin
Eq.(12)isreovered.
APPENDIXB
Intheasewherez>z
,theintegrationregionfor
the double hard photonemission is shown in Figs.3b
and and the orresponding dierential ross-setion
is dened by Eq. (19). The list of the neessary in-
tegrals is dened by expansion (20) of the quantity
M(z;L
0
;!
1
;)andisgivenby
I
1
= Z
d!
1 d
E
z
=y
2 ln
y(1
m )
x
m ln
1 z
x
m
;
y=1 z x
m
;
(B.1)
I
2
= Z
d!
1 d
(E !
1 )
z
=
2
6 +
1
2 ln
2
z+
+ln(z+x
m )ln
(1
m )x
m
(1 z)
+lnzln 1 z
x
m +
+Li
2
x
m
z
+Li
2
(1 z) Li
2 (z+x
m
); (B.2)
I
3
= Z
E d!
1 d
(E !
1 )
2
z
= x
m
z(z+x
m )
ln 1 z
x
m
1+z
z
ln(z+x
m )
y
z+x
m ln
(1
m )y
; (B.3) I
4
= Z
( 2!
1 )d!
1 d
[!
1
( !
1
) E
z
℄
z
=
=
2
6 1
2 ln
2 1+
m
2
ln
1
m
1+
m ln
x
m y
2(1 z) 2
1
2 ln
2 2x
m y
(1 z) Li
2
1
m
2
Li
2
(1 z)
2x
m y
; (B.4)
I
5
= Z
(1 z)E d!
1 d
[!
1
( !
1
) E
z
℄
z
=
=
2
2 4y
1 z 2Li
2
x
m
1 z
2(1+
m )
1
m
ln 1+
m
2 ln
2
1
m x
m
1 z
+
+ln
1+
m
1
m +
x
m
1 z
+
+ln
1
m
1+
m ln
2 2
x
m
y(1 z) 1
2 ln
2 2y
3
2 ln
2 1 z
x
m +ln
1 x
x
m ln
y
2x 2
m +
+Li
2
y(1
m )
2(1 z)
Li
2
y(1
m )
(1+
m
)(1 z)
+
+Li
2
x
m (1
m )
(1+
m )(1 z)
Li
2
x
m (1
m )
2(1 z)
; (B.5)
I
6
= Z
d!
1 d
! 2
1
=
(1 z)
2
2(1
m
ln) (1+
m )ln
1+
m
(1
m )
lny+ x
m
1 z
: (B.6)
Inalulatingtheseintegrals,wenegletedtermsof
theorderx 2
m
and(1
m )x
m
omparedtounity;these
termsareofthesameorder astheparameter:
Intheaseswhere
z<1 4
1
m
;
wemustintegrateovertheregionshowninFig.3a. As
mentionedabove, theontribution of the toppiee of
thisregion,where
Ex
m
>!
1
>! +
;
is small (about 12%) ompared to the bottom
oneand anbe negleted. This approximationis suf-
ienttoprovidetheoneperentaurayofradiative
orretion. Weuse thenotationJ
i
, similarly to I
i ,to
labelseparateintegralsoverthisregion,
J
1
=I
1 +L
K 2(K
+
K );
K
= 1
2 1
s
1
4
(1
m
)(1 z)
!
;
L
K
=ln K
+
K
;
(B.7)
J
2
=I
2 lnzL
K +Li
2
((1 z)K ) Li
2
((1 z)K
+ )
Li
2
(1 z)K
z
+Li
2
(1 z)K
+
z
; (B.8)
J
3
=I
3 +
1 z
z L
K
1+z
z ln
z+(1 z)K
+
z+(1 z)K
; (B.9)
J
4
=I
4 +ln
1
m
2
ln 1
m
2 1
2 ln
1
m
1+
m
; (B.10)
J
5
=2ln
1
m
1+
m ln+
2
6 +2Li
2
(K ) 4K +
+ 1
2 ln
2
1
m
1+
m
2ln(1 z)ln
1
m
1+
m
1
2 ln
2
(1
m )
2(1+
m
)(1 z)
1
2 ln
(1+
m )
2
4(1
m
)(1 z) L
K
+lnK lnK
+ +
+ 2(1+
m )
1
m ln
2(1+
m
+(1
m )K )
(1+
m )(1+
m
+(1
m )K
+ )
+
+Li
2
(1
m )K
+
1+
m
Li
2
(1
m )K
1+
m
+Li
2
(1
m )K
2
Li
2
(1
m )K
+
2
; (B.11)
J
6
=I
6 +
(1 z)(1
m )
2
ln
(1 z)(1
m )
x
m
1 z
2lnK
+ 2(K
+ K )
: (B.12)
APPENDIXC
InthisAppendix,wegivetheanalytialformofthe
funtionsG l
i
forarbitraryvaluesofthelimitingtagged
photonangle
l
: Theonlyonditionon
l
usedin Ap-
pendix Bisthat
(1
l )x
m 1:
Thisrestrits
l
bythevaluesabout45 Æ
: Withtheex-
eptionofG l
,thefuntionsG l
i
aredierentforz>z
andz<z
:Intherstase,wehave
G l
(z>z
)=4(L
0 1)
l
(1 z)+ 1+z
2
1 z ln
1
l
1+
l
ln; (C.1)
G l
p (z>z
)= 2ln
1
l
1+
l ln
(1 z)y
2x
m ln
2 x
m
2(1 z)y 2ln
1 z
x
m ln
(1 z)x
m
y 2
+ 2
8y
1 z
4(1+
l )
1
l ln
(1+
l )[(1
l
)(1 z)+x
m (1
l )℄
2[2(1 z) x
m (1
l )℄
4Li
2
x
m
1 z)
+2Li
2
(1
l )y
2(1 z)
2Li
2
(1
l )x
m
2(1 z)
+2Li
2
(1
l )x
m
(1+
l
)(1 z)
2Li
2
(1
l )y
(1+
l
)(1 z)
; (C.2)
G l
1 (z>z
)=x
m
2(2
l )+
1+z+x
m
z+x
m ln
1 z
x
m
+2(1 z)
(1
l
)lny (1+
l )ln
1+
l
3
+
+
y(2z+x
m )
z+x
m ln
(1
l )y
+(1+z)
ln 1 z
x
m ln
z+x
m
z
+ln (z+x
m )
1 ln
1
l
+
+Li
2
z 1
z
Li
2
x
m
z
+Li
2 (z+x
m )
2
6
; (C.3)
G l
0 (z>z
)=
2
3
(3z 2) 2x
m
(3
l ) 2z
ln 1 z
x
m ln
z+x
m
z
ln(z+x
m )ln
1
l
Li
2
z 1
z
+Li
2
x
m
z
+Li
2 (z+x
m )
+(1 z)
8+ 1
2 ln
2 1+
l
2 ln
2
1
l
1+
l +
+ln 2
(1
l )
2(1+
l )y
+4yln
1
l
1+
l +2ln
1 z
x
m ln
1 z
y
2(1
l
)lny+Li
2
1
l
2
+2Li
2
x
m
1 z
+
+Li
2
(1 z)
2yx
m
+Li
2
(1
l )x
m
2(1 z)
Li
2
(1
l )y
2(1 z)
Li
2
(1
l )x
m
(1+
l
)(1 z)
+
+Li
2
(1
l )y
(1+
l
)(1 z)
+2 1+
l
1
l
ln (1+
l )[(1
l
)(1 z)+x
m (1
l )℄
2[2(1 z) x
m (1
l )℄
ln 1+
l
: (C.4)
Intheasewherez<z
, theorrespondingfuntions G l
i
aregivenby
G l
p (z<z
)=
2
3
8K ln
2
2(1 z) 2ln
1
l
1+
l ln
(1 z)
2
+2lnK
+ lnK
ln
(1+
l )
2
4(1
l
)(1 z) L
K +
4(1+
l )
1
l ln
2[1+
l
+(1
l )K ℄
(1+
l )[1+
l
+(1
l )K
+
℄ +2Li
2
(1
l )K
2
4Li
2
(K ) 2Li
2
(1
l )K
+
2
+2Li
2
(1
l )K
+
1+
l
2Li
2
(1
l )K
1+
l
; (C.5)
G l
1 (z<z
)=2x
m (3 2
l )+
2 y
1 y
x
m ln
1 z
x
m +yln
y(1
l )
+
+2(1 z)
(3 2
l )(K
+
K ) 3 (2
l )L
K
+(1
l
)lny+(1+
l )ln
1+
l
+
+(1+z)
2
6 +
1+ln
(1 z)
(1
l )x
m
ln(z+x
m )+lnz
L
k ln
1 z
x
m
+ln
z+(1 z)K
+
z+(1 z)K +
+Li
2
z 1
z
+Li
2 (z+x
m ) Li
2
x
m
z
+
+Li
2
(z 1)K
z
Li
2
(z 1)K
z
+Li
2
((1 z)K
+ ) Li
2
((1 z)K )
; (C.6)
G l
0 (z<z
)= 4x
m (1
l
)+(1 z)
6
2
3
2(3 2
l )(K
+
K )+ 1
2 ln
2 (1 z)
2
yx
m
2(1
l
)lny 2(1+
l )ln
1+
l
+ln
2
1
l
2
2ln
1
l
1+
l ln
1
l
2yx
m +ln
2
2(1 z)(1+
l )
(1
l )
+
+
2(1
l )+
1
2 ln
(1+
l )
2
4(1
l
)(1 z)
L
K lnK
+
lnK ln (1 z)
2
yx
m ln
2(1+
l
)(1 z)
(1
l )
2Li
2
(K ) Li
2
(1
l )K
2
+Li
2
(1
l )K
+
2
+Li
2
(1
l )K
(1+
l )
Li
2
(1
l )K
+
1+
l
+
+Li
2
1
l
2
+Li
2
(1 z)
2yx
m
+ 2(1+
l )
1
l ln
(1+
l )[1+
l
+(1
l )K
+
℄
2[1+
l
+(1
l )K ℄
+
+2z
2
6 +ln
(1
l )x
m
(1 z)
ln(z+x
m ) lnz
L
K ln
1 z
x
m
+
+Li
2
x
m
z
Li
2
z 1
z
Li
2 (z+x
m )+Li
2
((1 z)K ) Li
2
((1 z)K
+ )+
+Li
2
(z 1)K
+
z
Li
2
(z 1)K
+
z
: (C.7)
REFERENCES
1. R. M. Carey et al., (g-2) Collaboration, Phys. Rev.
Lett. 82,1632 (1999);H. M. Brownetal., (g-2)Col-
laboration, Phys. Rev. D 62, 0901101 (2000); Phys.
Rev.Lett.86,2227(2000);B.LeeRobertsetal.,(g-2)
Collaboration,E-printarhiveshep-ex/0111046.
2. A. H. H®ker, E-print arhives hep-ph/0111243;
J.F.deTroónizandF.J.Yndurain,E-printarhives
hep-ph/0111258.
3. V.M. HuglesandT.Kinoshita, Rev.Mod.Phys.71,
133(1999).
4. S. Peris, M. Perrottet and E. de Rafael, Phys.Lett.
B 355, 523 (1995); A. Czarneski, B. Krause, and
W. J. Mariano, Phys. Rev. D 52, R2619 (1995);
E. A. Kuraev, T. V. Kuhto, and A. Shiller, Yad.
Fiz. 51, 1631 (1990); E. A. Kuraev, T. V. Kuhto,
Z. K.Silagadze, and A. Shiller, Preprint, INP 90-66,
Novosibirsk(1990).
5. M. Kneht and A. Nyeler, E-print arhives
hep-ph/0111058;M.Kneht,A.Nyeler,M.Perrottet,
andE.deRafael,E-printarhiveshep-ph/0111059.
6. J.Bijnens,E.Pallante,andJ.Prades,Phys.Rev.Lett.
75,3784 (1995),Nul.Phys.B474,379(1996).
7. M. Hayakawa, T.Kinoshita, and A. I. Sanda, Phys.
Rev. Lett. 75, 790 (1995), Phys. Rev. D 54, 3137
(1996); M. Hayakawa and T. Kinoshita, Phys. Rev.
D57,465(1998).
8. J.Bijnens,E.Pallante,andJ.Prades,PreprintLUTP
01-37,E-printarhiveshep-ph/0112255.
9. M. Hayakawa and T. Kinoshita, Preprint
KEK-TH-993,E-printarhiveshep-ph/0112102.
10. N. Cabibbo and R. Gatto, Phys. Rev. 124, 1577
(1961); M. Gourdinand E. de Rafael, Nul.Phys.B
10,667(1967).
11. A. Charneki and W. J. Mariano, E-print arhives
hep-ph/0102122; W.J.MarhianoandB.L.Roberts,
E-print arhives hep-ph/0105056; K. Melnikov,
SLUC-PUB-8844, E-print arhives hep-ph/0105267;
F. J. Yndurain, E-print arhives hep-ph/0103212;
F.Jegerlehner,E-printarhiveshep-ph/0001386.
12. M. Davier and H. H®ker, Phys. Lett. B 419, 419
(1998),435,427(1998).
13. S.Narison,Phys.Lett.B513,53(2001);V.Cirigliano,
G. Eker, and H. Neufeld, Phys. Lett. B 513, 361
(2001).
14. J. F. Troóniz and F. J. Yndurain, E-print arhives
hep-ph/0106025; J. Prades, E-print arhives hep-
ph/0108192.
15. R. R. Akhmetshin et al., E-print arhives
hep-ex/9904027; Nul. Phys. A 675, 424 (2000);
Phys.Lett.B475,190(2000).
16. D. Kong, E-print arhiveshep-ph/9903521: J. Z. Bai
et al., BES Collaboration, Phys. Rev. Lett. 84, 594
(2000);E-printarhiveshep-ph/0102003.
17. F. Jegerlehner, E-print arhives hep-ph/0104304,
E-printarhiveshep-ph/0105283.
18. A.B.Arbuzovetal.,JHEP12(1998)009;M.Konhat-
nijandN.P.Merenkov,JETPLett.69,811(1999).
19. S.Binner, J. H. K¶hn,andK. Melnikov, Phys.Lett.
B 459,279(1999); K.Melnikov etal., Phys.Lett.B
477,114(2000); H.Czy¹andJ.H.K¶hn,Eur.Phys.
J.C18,497(2001).
20. S.Spagnolo,Eur.Phys.J.C6,637(1999);G.Cataldi
etal.,KLOEMEMO195,August13,1999.
21. A. Hoefer, J. Gluza, F. Jegerlehner, E-print arhives
hep-ph/0107154.
22. G. Rodrigo et al., Eur. Phys. J. C 22, 81 (2001),
E-printarhiveshep-ph/0106132;G.Rodrigo, E-print
arhives hep-ph/0111151; G. Rodrigo et al., E-print
arhiveshep-ph/0112184.
23. G. Cataldi etal., Physis and Detetors at DANE,
569(1999).
24. V.A.Khozeetal.,Eur.Phys.J.C18,481(2001).
25. N. P. Merenkov and O.N. Shekhovtsova, Pis'maZh.
Exs. Teor. Fiz. 74, 69 (2001); V. A. Khoze et al.,
E-print arhives hep-ph/0202021 (submitted to Eur.
Phys.J.C).
26. M. Adinol et al., KLOE Collaboration, E-print
arhiveshep-ex/0006036;A.Aloisioetal.,KLOECol-
laboration,E-printarhiveshep-ex/0107023;A.Denig
(on behalf of KLOECollaboration), E-print arhives
hep-ex/0106100.
27. E. P. Solodov (BABAR Collaboration), E-print
arhiveshep-ex/0107027.
28. V.N.Baier,V.S.Fadin,andV.A.Khoze,Nul.Phys.
B65,381(1973).
29. E. A.Kuraev,N. P.Merenkov, andV.S.Fadin,Sov.
J.Nul.Phys.45,486(1987).