The radiative return at phi- and B-factories: FSR for muon pair production at next-to-leading order

0.64 MB·English

by  Henryk Czyz

EPJ manuscript No.
(will be inserted by the editor)
CERN-PH-TH/2004-061
TTP04-07
Φ B
Radiative return at - and -factories:
5 FSR for muon pair production at next-to-leading order ⋆
0
0
2
Henryk Czyz˙1 a, Agnieszka Grzelin´ska1 b, Johann H. Ku¨hn2 c, and Germ´an Rodrigo3,4 d

n
a 1 Instituteof Physics, Universityof Silesia, PL-40007 Katowice, Poland.
J 2 Institut fu¨r Theoretische Teilchenphysik, Universit¨at Karlsruhe, D-76128 Karlsruhe, Germany.
7 3 Department of Physics, Theory Division, CERN, CH-1211 Geneva23, Switzerland.
2 4 Institutode F´ısica Corpuscular, E-46071 Valencia, Spain.

2
April 7, 2004
v
8
Abstract. Muon pair production through the radiative return is of importance for a measurement of the
7
0 hadronicproductioncrosssectionintwoways:itprovidesanindependentcalibrationanditmaygiveriseto
4 animportantbackgroundforameasurementofthepionformfactor.WiththismotivationtheMonteCarlo
0 event generator PHOKHARA is extended to include next-to-leading order radiative corrections to the
4 reaction e+e− µ+µ−γ. Furthermore, virtual ISR corrections to FSR from pions are introduced, which
→
0 extends the applicability of the generator into a new kinematical regime. Finally, the effect of photon
/ vacuum polarization is introduced into this newversion of thegenerator.
h
p
-
p 1 Introduction The generator was subsequently extended to simulate
e
the production of other hadronic final states: four pions
h
in [3,4] and nucleon pairs in [5], as well as muons [6].
: The high luminosity of currently operating Φ- and B-
v The influence of collinear lepton pair radiation on these
meson factories offers the unique opportunity to measure
i measurements was investigated in [7].
X the famous ratio R σ(e+e− hadrons)/σ over a
≡ → point An important step forward was the evaluation of the
r wide range of energies in one single experiment, through
a complete one-loop corrections to ISR [8,9] and the con-
theradiativereturn.Thispossibilitywassuggestedalong
struction of the event generator PHOKHARA [6], which
time agoforpion-pairproduction[1].Inviewofthecapa-
included these virtual corrections and the corresponding
bilities of the currently operating machines, the idea was
radiationoftwophotons.PHOKHARA,likeEVA,canbe
revived in [2], where a Monte Carlo event generator EVA
easily used to simulate reactions with arbitrary hadronic
wasconstructed,whichsimulatestheproductionofapion
final states, once a specific model has been chosenfor the
pair plus one hard photon from initial- and final-state ra-
relevantmatrix element of the hadronic current;it is cur-
diation (ISR and FSR). The program includes additional
rently available for 2π, 4π, pp¯and nn¯ production [6,4,5]
collinear radiation from the incoming electrons through
and of course for muon pairs. A number of cross section
structure function techniques and was used to demon-
measurements based on the radiative return, which have
strate the feasibility of such a measurement in ongoing
madeuseofsimulationsbasedonEVAandPHOKHARA,
experiments.
have been presented recently: a precise measurement of
⋆ Work supported in part by BMBF under grant number thepionformfactormadebytheKLOECollaboration[10,
11,12,13]; preliminary results on four-prong final states
05HT9VKB0, EC 5th Framework Programme under contract
have been obtained by the BABAR Collaboration [14].
HPRN-CT-2002-00311 (EURIDICE network), TARI project
HPRI-CT-1999-00088, Polish State Committee for Scientific Extensive discussions of theoretical and experimental as-
Research (KBN) under contract 2 P03B 017 24, BFM2002- pects of the radiative return, including various topics not
00568, Generalitat Valenciana under grant GRUPOS03/013, covered by the present paper, can be found in the liter-
and MCyT undergrant FPA-2001-3031. ature[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30];
a e-mail: [email protected] considerationsrelatedtotheradiativereturncanbefound
b e-mail: [email protected] in [31,32]; perspectives of its use outside hadronic cross
c e-mail: [email protected] section measurements and the long term potential of the
d e-mail: [email protected] method are outlined in [33].
2 H. Czyz˙ et al.: Radiative return at Φ- and B-factories: FSR for muon pair production at next-to-leadingorder
Considering measurements with a precision close to
one percent, FSR starts to become an issue, in particu- e+ (cid:22)+
lar the contribution from the two-step process e+e−
γγ∗( π+π−γ) [18]. FSR has to be modeled separate→ly
→
for every different mode. In leading order (LO), it is now
included in PHOKHARA for the π+π− and the µ+µ− e(cid:0) (cid:22)(cid:0)
modes, in next-to-leading order (NLO), i.e. for the two- (a)
stepprocesswithonehardphotonfromISRandrealplus
virtual radiation from the final state for π+π− only.
It is the purpose of the present paper to extend the
same approach to final states with muon pairs. This is
motivated, on the one hand, by the fact that this reac-
tion is an important background for the pion form factor (b) (
)
measurement, on the other hand, that it may be used for
a measurement of the ratio between hadronic final states Fig. 1. NLO corrections to the reaction e+e− µ+µ−γ: (a)
→
and muon pairs of the same invariant mass. This might double real emission, (b) virtual corrections to final-state ver-
lead to a direct determination of the R-ratio, with the tex, and (c) virtual corrections to initial-state vertex. Only
cancellation of many uncertainties that arise in the abso- representativeFeynman diagrams are depicted.
lute cross section determination.
Although,technicallyspeaking,thetreatmentofmuons
is quite similar to that of pions, a number of physics as-
ducedinSection4,whichalsocontainsadiscussionofthe
pects are quite different. First of all, in contrast to radia-
influence of this additional term on present experimental
tionfrommuons,theamplitudeforphotonradiationfrom
studies. The implementation of vacuum polarization and
pions can only be parametrized by a model form factor,
its effect onthe crosssectionandasymmetries willbe the
and the model dependence becomes particularly relevant
subject of Section 5. Section 6 contains a brief summary
for energetic photons. Then, at larger cms energies, rele-
and our conclusions.
vant e.g. to the B-factories, the leading order FSR with
only one photon in the final state is strongly suppressed
inthepioncase,asaconsequenceofthe pionformfactor,
whereasthe point-likebehaviourofthe muonleadstoim-
2 FSR for muons at next-to-leading order and
portant contributions from ISR as well as FSR in leading
order. its implementation in PHOKHARA
Inordertoarriveatafullresultformuonpairproduc-
tion at NLO, corrections with combined virtual emission
fromtheinitialstateandrealhardemissionfromthefinal The most relevant contributions to FSR at NLO, which
state(Fig.1c)mustbeincluded,andcombinedwiththose aredepictedschematicallyinFig.1,consistofdoublereal
from“simultaneous”softrealISRandhardFSR(Fig.1a). photon emission diagrams, where one photon is emitted
To offer a uniform program, the same contributions will off the initial-state leptons and the other is emitted from
alsobeincludedforpionpairproduction.However,wewill the final state (Fig. 1a), final-state vertex corrections to
demonstrate that their effect is small already for KLOE single initial-state photon emission (Fig. 1b), and initial-
energies, in particular for the standard KLOE cuts, and state vertex corrections to single final-state photon emis-
completely irrelevant for higher energies. sion (Fig. 1c).
Finally we study the influence of the leptonic and ha- The amplitude for the process depicted in Fig. 1a
dronicvacuumpolarizationontheradiativereturn,where
we adopt the parametrization suggested in [34]. For the e+(p1,λe+) + e−(p2,λe−)
→
measurementoftheintegratedcrosssectionwithcutssup- µ+(q1,λµ+) + µ−(q2,λµ−) + γ(k1,λ1) + γ(k2,λ2) ,
pressing FSR, vacuum polarization just leads to a multi-
(1)
plicativefactorthatcanbe takenintoaccountbycorrect-
ingtheresultattheendoftheexperimentalanalysis.The
forward–backward or the charge asymmetry, however, is is given by
affected by vacuum polarization effects in a non-trivial
manner. (4πα)3/2
Let us briefly outline the content of this paper. The MIHFSNLO(λe+,λe−,λµ+,λµ−,λ1,λ2)=− Qˆ2
amplitudesforrealandvirtualradiationinNLOformuon (cid:26)
pairproduction,whicharethemainadditionalingredients vI†(p1,λe+) A(λµ+,λµ−,λ1,λ2) uI(p2,λe−)
inthenewprogram,willbeintroducedinSection2.Tests
of the technical precision of the program and some of its +vI†I(p1,λe+) B(λµ+,λµ−,λ1,λ2) uII(p2,λe−)
physicsresultswillbepresentedinSection3.TheISRcor- (cid:27)
rectiontorealhardFSRfortheπ+π− modewillbeintro- + k1 k2,λ1 λ2 , (2)
↔ ↔
(cid:0) (cid:1)
H. Czyz˙ et al.: Radiative return at Φ- and B-factories: FSR for muon pair production at next-to-leadingorder 3
where the index H indicates that both photons are hard, only, and
1+β2
A(λµ+,λµ−,λ1,λ2)= ηV+S(s′,Ecut)= 2 µ log(t)+1 log(2w′)
2 − 2β
ε∗(k1,λ1)−k1+−2ε∗(k1,λ1)·p1 D−(λ2,λµ+,λµ−) log(t) 5 (cid:20) µ (cid:21) 1 β2
(cid:0)+ D−(λ2,λµ+,λµ−)(cid:0)22εk∗12(·kkp11,1·λp12)(cid:1)·p2−k1+ε∗(k1,λ1)−(cid:1) , − 1s+′βµβµ2(cid:20)22sL′i−2 7m22µβµ+3m2µxπµ2(cid:21)−l2og+(tl)olgo(cid:18)g −14+µβ(cid:19)µ .
(3) − β 1+β − 2 − 2
µ (cid:20) (cid:18) µ(cid:19) (cid:18) (cid:19)(cid:21)
(10)
and
Ecut is the maximal energy of the soft photon in the s′
2
B(λµ+,λµ−,λ1,λ2)= rest frame, and
ε∗(k1,λ1)+k1−−2ε∗(k1,λ1)·p1 D+(λ2,λµ+,λµ−) β = 1 4m2/s′ , t= 1−βµ ,
(cid:0) 2k1·p1 (cid:1) µ − µ 1+βµ
+ D+(λ2,λµ+,λµ−) 2ε∗(k1,λ1)·p2−k1−ε∗(k1,λ1)+ , w′ =qEcut/√s′ , 1 = s′ +1 . (11)
2k p 2 x 2m2
(cid:0) 1· 2 (cid:1) µ µ
(4)
The virtual (Fig. 1c) plus soft photon corrections to the
with initial-state vertex can be written as [36]
α
Qˆ =p1+p2−k1 =q1+q2+k2 , s′ =Qˆ2 . (5) dσIVF+SSNLO,i = π δV+S(s,Eγmin) dσF(0S)R(s) , (12)
Dµ is the current describing the µ+µ−γ final state [4], where dσ(0) is the leading order e+e− µ+µ−γ cross
FSR →
which we report below for completeness section,withthephotonemittedoffthefinalleptonsonly,
and, for m2 s:
e ≪
Dµ(λ2,λµ+,λµ−)=ie u†I(q2,λµ−)A˜µ(λ2)vI(q1,λµ+) δV+S =2 (L 1)log(2w)+ 3L 1+ζ(2) . (13)
(cid:26) − 4 −
(cid:26) (cid:27)
+u†II(q2,λµ−)B˜µ(λ2)vII(q1,λµ+) , Emin is the maximal energy of the soft photon (or the
(cid:27) γ
(6) minimal energy of the hard photon) in the s rest frame,
while
with
s
L=log and w =Emin/√s . (14)
A˜µ(λ )= 2q2·ε∗(k2,λ2)+ε∗(k2,λ2)−k2+ σµ− (cid:18)m2e(cid:19) γ
2
(cid:0) 2k2·q2 (cid:1) To match hard, soft and virtual radiation smoothly, the
σµ− 2q ε∗(k ,λ )+k+ε∗(k ,λ )− energy cutoff (Ecut) has to be transformed from the rest
1· 2 2 2 2 2 , (7) frameoftheµ+µ2−γsystem(withthephotonemittedfrom
− 2k q
(cid:0) 2· 1 (cid:1) the finalstate)to the laboratoryframe (e+e− cms frame)
(Emin). In fact it is necessary to recalculate the soft pho-
γ
2q ε∗(k ,λ )+ε∗(k ,λ )+k− σµ+ ton contribution, as the cut on Q2 = (q1+q2)2 depends
B˜µ(λ )= 2· 2 2 2 2 2 in the latter case on the angle between the two emitted
2
2k q
(cid:0) 2· 2 (cid:1) photons. Now
σµ+ 2q ε∗(k ,λ )+k−ε∗(k ,λ )+
1· 2 2 2 2 2 , (8) 1+β2
− (cid:0) 2k2·q1 (cid:1) ηV+S(s′,E2cut)=−2 2β µ log(t)+1
(cid:20) µ (cid:21)
manadtrσicµe±s a=nd(Ia,±±=σi)a,µσw±hefroerσain,yifo=ur1-,v2e,c3to,raareµ.the Pauli log(2w)+1+ s′ log s 2+log 1−βµ2
µ × s′ s s′ − 4
The virtual (Fig. 1b) plus soft photon corrections to (cid:20) − (cid:18) (cid:19)(cid:21) (cid:18) (cid:19)
log(t) 5 1+β2 2β
the final-state vertex can be written as [35] s′ 7m2 +3m2x µ 2Li µ
− s′β 2 − µ µ µ − β 2 1+β
µ (cid:20) (cid:21) µ (cid:20) (cid:18) µ(cid:19)
dσV+S = α ηV+S(s′,Ecut) dσ(0) (s′) , (9) π2 1+βµ
IFSNLO,f π 2 ISR log(t)log . (15)
− 2 − 2
(cid:18) (cid:19)(cid:21)
where dσ(0) is the leading order e+e− µ+µ−γ cross UsingEqs.(2)to(15)theimplementationofFSRincom-
ISR →
section, with the photon emitted off the initial leptons bination with ISR is straightforward.
4 H. Czyz˙ et al.: Radiative return at Φ- and B-factories: FSR for muon pair production at next-to-leadingorder
Table 1. Total cross section (nb), corresponding to the Q2 e+e− µ+µ−γ(γ)
distributions from Fig. 2, for the process e+e− µ+µ−γ at 0.002 → a
NLO,for different values of thesoft photon cuto→ff. dσ(w=10−5)/dσ(w=10−4) 1
0.0015 dQ2 dQ2 −
w √s= 1.02 GeV 10.52 GeV
10−4 28.237(1) 0.19854(7) 0.001
10−5 28.236(1) 0.19883(14)
0.0005
0
We have not included diagrams where two photons
0.0005
are emitted from the final state, neither final-state ver- −
tex corrections with associated real radiation from the fi- 0.001 √s=1.02GeV
nal state [37]. This constitutes a radiative correction to −
0 < θ < 180
FSR and will give non negligible contributions only for 0.0015 ◦ γ ◦
those cases, where at least one photon is collinear with − 0◦ < θµ± < 180◦
one of the muons. Box diagrams with associated real ra- 0.002
− 0 0.2 0.4 0.6 0.8 1
diation from the initial- or the final-state leptons, as well
Q2(GeV2)
as pentagon diagrams, are also neglected. As long as one
e+e µ+µ γ(γ)
considers charge symmetric observables only, their con- − −
0.0025 →
tribution is neither divergent in the soft nor the collinear b
limitandthusoforderα/π withoutanyenhancementfac- 0.002 dσ(w=10−5)/dσ(w=10−4) 1
dQ2 dQ2 −
tor. We want to stress that we have included only C-even
0.0015
gauge invariant sets of diagrams (see Fig. 9 of Ref. [18]
for a graphical representation of the equivalent set of di- 0.001
agrams in pion pair production). The box and pentagon 0.0005
diagrams that we have neglected are related to ISR-FSR
0
interferences, and therefore are suppressed after suitable
kinematicalcutsusedwithintheradiativereturnmethod. 0.0005
−
One could also neglect some small contributions from the
0.001
diagrams taken into account, but they are kept for the − √s =10.52GeV
sake of completeness, as the full 1-loop radiative correc- −0.0015 0◦ < θγ < 180◦
tionswillbeconsiderinafuturepublication.Atthispoint
0.002 0 < θ < 180
thesesmallcontributionswillbecomerelevantallowingac- − ◦ µ± ◦
curatecalculationsnotrestrictedtoradiativereturnphys- 0.0025
− 0 1 2 3 4 5 6 7
ical configurations. Q2(GeV2)
Fig. 2. Comparison of the Q2 distribution for two values of
thesoftphotoncutoff(w=10−4 vs.10−5)for√s=1.02GeV
3 Tests of the Monte Carlo program and
(a)and√s=10.52GeV(b).Oneofthephotonswasrequired
discussion of physical results tohaveenergy>10MeV(for√s=1.02GeV)and>100MeV
(for √s=10.52 GeV). One sigma statistical errors are shown
A number of tests were performed to ensure the tech- for theratio of the two distributions.
nical precision of the new version of PHOKHARA. The
square of the matrix element, summed over polarizations
of the final particles and averaged over polarizations of
the same separationparameter w was chosen for ISR and
the initial particles, was calculated with FORM [38], us-
FSR corrections. Choosing w = 10−4 or less, the result
ingthe standardtracemethod.Externalgaugeinvariance
becomesindependentofw,uptoexpectedsystematicdif-
wascheckedanalyticallywhenusingthetracemethodand
ferences of order 10−4. These systematic differences are
numericallyforthe amplitude calculatedwith the helicity
visible already in Fig. 2a, where χ2/d.o.f = 30/14, while
amplitude method. The two results for the square of the
in Fig. 2b statistical errors are about 0.1%, and χ2/d.o.f
matrixelementsummedoverpolarizationswerecompared
= 3.5/7. The tests prove that the analytical formula de-
numerically. The code based on the result from the trace
scribing soft photon emission as well as the Monte Carlo
methodwaswritteninquadrupleprecisionto reducecan-
integrationinthesoftphotonregionarewellimplemented
cellations.Thecodebasedontheresultobtainedwiththe
in the program.
helicity amplitude method uses double precision for real
andcomplexnumbersandisnowincorporatedinthecode For the purpose of further tests we have also calcu-
ofPHOKHARA4.0.Agreementof13significantdigits(or latedthe integratedcontributionofhardphotonemission
better) was found between the two codes. The sensitivity fromthe finalstate to the ISR spectrum.Integratingover
of the integrated cross section to the choice of the cutoff all angles and energies, from Ecut (defined in the s′ rest
2
w can be deduced from Fig. 2 and Table 1. For simplicity frame) to the kinematical limit of the final-state photon,
H. Czyz˙ et al.: Radiative return at Φ- and B-factories: FSR for muon pair production at next-to-leadingorder 5
we find that this contribution can be written as e+e− µ+µ−γ
→
α
dσIHFSNLO,f = π ηH(s′,E2cut) dσI(S0R) (s′) , (16) 0.0008 σ(PHOKHARA) 1 w = 10−4
σ(analytical) −
0.0006
where
1+β2 t 0.0004
ηH(s′,Ecut)= µ Li 1 m Li (t t)
2 − β 2 − t − 2 m
µ (cid:20) (cid:18) (cid:19) 0.0002
log2(t) 1+β2 t
µ m
+ζ(2)+ log(tm t) log 0
2 − − β t
(cid:21) (cid:20) µ (cid:18) (cid:19)
2 + log(tm) 1−βµ2 1+ xµ(2−βm2 ) −0.0002
− β 1 β2 (1 β2 ) FSRatLO
(cid:21) µ (cid:26) − m(cid:20) − m (cid:21) 0.0004
4t(β2 β2 ) log(t ) −
+(1+β2 ) log µ− m + m
m (1 β2)(1 β2 ) 2 0.0006
(cid:20) (cid:18) − µ − m (cid:19) (cid:21) − 1 10 100
1 17 β2 2x xµ 33 17β2 √s(GeV)
− 8 − µ− µ − 4β β − µ
(cid:20) (cid:21)(cid:27) m µ(cid:26) Fig. 3. Comparison between theLO FSRcross section calcu-
+ 1 27+11β2 6(1−βµ2) lated analytically (Eq. (20)) and calculated by PHOKHARA
1−βm2 (cid:20)− µ− 1−βm2 (cid:21)(cid:27) sfotartaistfiixcaedl evrarolures.of w=Eγmin/√s. Errors represent one sigma
2log(1 t t) , (17)
m
− −
t, x and β are defined in Eq. (11), and
µ µ
For single-photonemissionfromthe finalstate andno
4m2 1 β further photon radiation, a formula similar to Eq. (16)
µ m
β 1 , t = − . holds:
m ≡s − Q2m m 1+βm α
σH = ηH(s,Emin) σ(0) (s) , (20)
Here Q2 is the maximum value of Q2 FSR π γ e+e−→µ+µ−
m
with ηH defined in Eq. (17), Emin defined now in the s
Q2 =s′ 2Ecut√s′ . γ
m − 2 restframeandσ(0) the lowestordercrosssection
e+e−→µ+µ−
For small Ecut, w = Ecut/√s′ 1, the function ηH of the process e+e− µ+µ−.
2 2 ≪ →
reduces to The results of the tests are collected in Figs. 3 and 4
proving the excellent technical precision of this part of
1+β2
ηH(s′,Ecut) log(2w) 2+ µ log(t) PHOKHARA,wellbelow0.05%.Resultsofsimilartestsat
2 ≃ β NLOarecollectedinFig.5,whereresultsofPHOKHARA
(cid:20) µ (cid:21)
1+β2 1+β are compared with the analytical result of Eq. (19). The
+ µ Li (t2) ζ(2) 2log(t)log µ latterchecksthetechnicalprecisionoftheimplementation
2
β − − 2
µ (cid:20) (cid:18) (cid:19)(cid:21) of double-photon emission, where one photon is emitted
log(t) 4β2 from the initial state and the other from the final state,
+ 5+β2 +6x 2log µ
8β − µ µ − 1 β2 together with the corresponding virtual and soft correc-
µ (cid:20) (cid:21) (cid:18) − µ(cid:19) tionstothe final-statevertex.Anagreementof1%onthe
3 11
+ x + . (18) function η means in fact an agreement better than 10−4
µ
2 4
on the cross section thanks to the additional factor α/π
(compare Eq. (9) and Eq. (20)).
Adding virtual, soft (Eq. (10)) and hard (Eq. (18)) cor-
The relative size the NLO FSR contributions to the
rections, the familiar correction factor [39]
e+e− µ+µ−γ(γ)crosssectionintroducedinPHOKHA-
→
1+β2 RA depends on the event selection used. In the follow-
η(s′)= µ 4Li2(t)+2Li2( t) ing,we will indicate some ofits characteristicfeatures. In
β −
µ (cid:20) Fig.6a the Q2 differentialcrosssectionis shownwith two
log(t)log (1+βµ)3 +3log 1−βµ2 log(β ) peaks at low (‘soft’ muon pair production) and large Q2
− 8β2 4β − µ (soft photon emission) values. In Fig. 6b the correspond-
(cid:18) µ (cid:19)(cid:21) (cid:18) µ (cid:19)
ing relative NLO FSR contributions are shown. They are
2 3β 3log(t)
+ µ(5 3β2) (33+22β2 7β4) Q2-dependent and might be as big as a few per cent. As
βµ(3−βµ2)(cid:20) 8 − µ − 48 µ− µ (cid:21) seen by comparison of Figs. 6b and 6c the relative size of
(19) thiscontributiondoesdependontheeventselectionused.
At KLOE, the process e+e− µ+µ−γ(γ) is a back-
is recovered. ground to the measurement of th→e e+e− π+π− cross
→
6 H. Czyz˙ et al.: Radiative return at Φ- and B-factories: FSR for muon pair production at next-to-leadingorder
e+e µ+µ γ
− − + (cid:0) + (cid:0)
0.001 → e e ! (cid:22) (cid:22) (cid:13)((cid:13))
0:02
σ(PHOKHARA) 1
σ(analytical) − √s=1.02GeV 0:015 (cid:17)(P(cid:17)(HanOaKlyHti
AaRl)A) (cid:0) 1
0.0005
0:01
0:005
0
0
(cid:0)0:005
0.0005
− (cid:0)0:01 p = 1.02 GeV
s
FSRatLO (cid:0)0:015
0.001
− 1e 06 1e 05 0.0001 0.001 0.01 0.1 1 (cid:0)0:02
− − Emin(GeV) 0 0:2 0:4 0:6 0:8 1
γ 0 2
s(GeV )
Fig. 5. Comparison between analytical function η (Eq. (19))
e+e− µ+µ−γ and η obtained from PHOKHARAfor √s= 1.02 GeV.
0.001 →
σ(PHOKHARA) 1
σ(analytical) −
emitted from the final state, are expected to be smaller,
0.0005 but will be relevant anyhow if the precision requirement
for the background estimate is of the order of a few per
cent.
The process e+e− µ+µ−γ(γ) is even more impor-
0
tant for BABAR, wher→e it is used as Q2-dependent lumi-
nositymonitoring.Asaresulttheaccuracyrequirementis
higher. In Fig. 8 some features of the newly implemented
0.0005 corrections are shown in the interesting Q2 range, from
−
the point of view of using the radiative return method
√s=1.02GeV FSRatLO formeasurementsofthe hadroniccrosssection.InFig.8a
theQ2 differentialcrosssectionisgiven,includingandex-
0.001
− 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 cluding NLO FSR corrections. As seen from Fig. 8b the
Emin(GeV) relative size of those corrections is quite small, of the or-
γ
der of 1%, if no angular cuts are applied; but, as shown
Fig. 4. Comparison between theLO FSR cross section calcu-
in Fig. 8c, their relevance depends on the event selection
lated analytically (Eq. (20)) and calculated by PHOKHARA
and the corrections can be as big as 10% for some event
forafixedvalueof√s=1.02GeV.Errorsrepresentonesigma
selection and Q2 range.
statistical errors.
4 ISR corrections to FSR real hard emission
section by the radiative return method, due to possible for two-pion final state
pion–muon misidentification. The angular cuts used by
now by KLOE diminish the low Q2 part of the cross sec- Initial-state vertex corrections to FSR for the reaction
tion, while the track mass cut, which is specifically intro- e+e− π+π−γ (Fig. 9a) were not included in version
→
duced to constrain the final state to configurations with 3.0 of PHOKHARA, and therefore, for consistency, dou-
π+π− and one photon only (see Ref. [40]), does not allow ble photon radiation (Fig. 9b) was restricted to emission
foreventswithµ+µ− andonephotononly(seeFig.7afor ofhardISR.Inthepresentversionoftheeventgenerator,
correspondingcrosssections).Ifonly angularcuts areap- PHOKHARA 4.0, these corrections [36] have been added
plied, the newly introduced corrections are at the level of and the generation of double photon emission is uncon-
a few per cent (Fig. 7b). However,if in additionthe track strained, provided that one of the photons is hard. The
mass cut is applied, leaving only events with two hard virtual and soft corrections to the initial-state vertex are
photons, the contribution of one-photon ISR plus one- identicaltothemuoncaseandEq.(12)holds,withdσ(0)
FSR
photonFSRtothetwo-photonsampleisupto35%ofthe nowbeingtheleadingordere+e− π+π−γ crosssection,
→
two-photon ISR sample (Fig. 7c), thus this contribution and the photon being emitted off the final pions only.
is indispensable for a reliable background estimate. The We report here also formulae analogous to Eqs. (10)
left-over two-hard-photon corrections, with both photons and (15), as we have found representations simpler than
H. Czyz˙ et al.: Radiative return at Φ- and B-factories: FSR for muon pair production at next-to-leadingorder 7
+ (cid:0) + (cid:0) + (cid:0) + (cid:0)
e e !(cid:22) (cid:22) (cid:13)((cid:13)) a e e !(cid:22) (cid:22) (cid:13)((cid:13))
110 IFSNLO 35 noM cut IFSLO
IFSLO IFSNLO
100 p 1.02GeV trM cut ISR
2 ) 90 s = a 2 ) 30 tr IFSR
GeV 80 0Æ < (cid:18)(cid:13) < 180Æ GeV 25 p 1.02GeV
Æ Æ =
d(cid:27) (nb=2dQ 7600 0 < (cid:18)(cid:22)(cid:6) < 180 d(cid:27) (nb2dQ 2105 (cid:18)(p~(cid:22)+ +Æsp~=(cid:22)(cid:0)) < 15Æ oÆr > 165Æ
50 < (cid:18)(cid:22)(cid:6) < 130
50
10
40
30 5
20
0
10 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1
0 0:2 0:4 0:6 0:8 1 Q2(GeV2)
2 2
Q (GeV ) + (cid:0) + (cid:0)
e e !(cid:22) (cid:22) (cid:13)((cid:13))
+ (cid:0) + (cid:0) 0:02 b
e e !(cid:22) (cid:22) (cid:13)((cid:13))
0:03
(cid:27)(IFSLO) (cid:0)12dQ 00::00210 b LO)d(cid:27)(IFSLO) =2dQ (cid:0)00::00110
d(cid:27)(IFSNLO)d =2dQ (cid:0)(cid:0)(cid:0)000:::000123 ps = 1.02GeV d(cid:27)(IFSN 2dQ (cid:0)(cid:0)(cid:0)000:::000234 ps =1.02GeÆVor Æ
0Æ < (cid:18)(cid:13) < 180Æ (cid:18)(p~(cid:22)+ +p~(cid:22)(cid:0)) < 15 > 165
(cid:0)0:04 (cid:0)0:05 Æ Æ
Æ Æ 50 < (cid:18)(cid:22)(cid:6) < 130
0 < (cid:18)(cid:22)(cid:6) < 180
(cid:0)0:05 (cid:0)0:06
0:3 0:4 0:5 0:6 0:7 0:8 0:9 1
2 2
(cid:0)0:06 Q (GeV )
0 0:2 0:4 0:6 0:8 1 + (cid:0) + (cid:0)
Q2(GeV2) 0:35 c e e !(cid:22) (cid:22) (cid:13)(cid:13)
+ (cid:0) + (cid:0)
e e !(cid:22) (cid:22) (cid:13)((cid:13))
0:02
p = 1.02 GeV R) 0:3
(cid:0)1 0:01 s or c (cid:27)(IS 2dQ
d(cid:27)(IFSLO) 2dQ 0 (cid:18)(cid:13)4<0Æ1<5Æ(cid:18)(cid:22)(cid:6)(cid:18)<(cid:13) >14106Æ5Æ dd(cid:27)(IFSR) =2dQ 0:02:52 Mtr cut
=
O) (cid:0)0:01
L
SN 2Q 0:15 p 1.02GeV
d(cid:27)(IF d (cid:0)0:02 0:1 (cid:18)(p~(cid:22)+ +s =~p(cid:22)(cid:0)) < 15Æ or> 165Æ
Æ Æ
(cid:0)0:03 50 < (cid:18)(cid:22)(cid:6) < 130
0:05
(cid:0)0:04 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1
2 2
Q (GeV )
Fig. 7. Thesizeof theNLOFSRcorrections (IFSNLO)com-
(cid:0)0:05
0 0:2 0:4 0:6 0:8 1 paredwiththesum ofNLOISRandLOISRandFSRcontri-
Q2(GeV2) butions(IFSLO)tothee+e− µ+µ−γ(γ)crosssection:a)Q2
→
differentialcrosssectionswithangularcutsusedbyKLOE;b)
Fig. 6. Thesize oftheNLOFSRcorrections (IFSNLO)com-
relative difference of the cross sections with angular cuts used
pared with the sum of NLO ISR and LO ISR and FSR con-
byKLOE(notrackmasscut);c)relativedifferenceofthecross
tributions(IFSLO)tothee+e− µ+µ−γ(γ)crosssection:a)
Q2 differential cross sections wit→h no angular cuts; b) relative sections with angular cuts used by KLOE (with track mass
cut).Thetrackmasscutallowsonlyfortwo-photoneventsand
difference of the cross sections with no angular cuts; c) rela-
here IFSR stands for two- and one-photon ISR + one-photon
tivedifferenceofthecrosssectionswithangularcutschosento
FSR,while ISR stands for two-photon ISR only.
suppress FSR at LO.
8 H. Czyz˙ et al.: Radiative return at Φ- and B-factories: FSR for muon pair production at next-to-leadingorder
+ (cid:0) + (cid:0)
0:35 a e e !(cid:22) (cid:22) (cid:13)((cid:13)I)FSNLO e+ (cid:25)+
1
IFSLO
2
0:3
)
2 V e(cid:0) (cid:25)(cid:0)
e 0:25 p 10.52GeV (a) (b)
G
s =
=
(nb 0:2 0Æ < (cid:18)(cid:13) < 180Æ Fig. 9. Virtual (a) and real (b) radiative corrections to real
d(cid:27) 2dQ 0:15 0Æ < (cid:18)(cid:22)(cid:6) < 180Æ FSRemission from two-pion finalstate.
0:1
0:05 e+e(cid:0) !(cid:25)+(cid:25)(cid:0)(cid:13)((cid:13))
0
0
0 1 2 3 4 5 6 7 8 9
2 2 (cid:0)0:005
Q (GeV ) 1
+ (cid:0) + (cid:0) (cid:0)
e e !(cid:22) (cid:22) (cid:13)((cid:13))
0:014 b 3:0) 2Q (cid:0)0:01
p = 10.52GeV (cid:27)( d
1 s d =
d(cid:27)(IFSLO) (cid:0)2dQ 0:00:1021 d(cid:27)(4:0) 2(cid:0)dQ (cid:0)00:0:0125 p 1.02 GeV
O) = 0:008 s =
L
SN 2Q (cid:0)0:025 0Æ < (cid:18)(cid:13) < 180Æ
F d
d(cid:27)(I 0:006 0Æ < (cid:18)(cid:25)(cid:6) < 180Æ
(cid:0)0:03
0:004
0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1
Æ Æ Q2(GeV2)
0:002 0 < (cid:18)(cid:13) < 180 Fig. 10. Comparison between PHOKHARA 3.0 and
Æ Æ
0 < (cid:18)(cid:22)(cid:6) < 180 PHOKHARA4.0for√s=1.02GeV.Thepionandphoton(s)
0 angles are not restricted.
0 1 2 3 4 5 6 7 8 9
2 2
Q (GeV )
+ (cid:0) + (cid:0)
e e !(cid:22) (cid:22) (cid:13)((cid:13))
0:11 c
1 0:1
(cid:0)
(cid:27)(IFSLO) 2dQ 00::0098 p20sÆ=<1(cid:18)0(cid:13).5<2G16e0VÆ rptieroaendvsi,owusitlyhsroefptoprhteodtoinne[n1e8r]g.yTchuetvEi2rcututainl tphluess′sroefsttcforarrmece-,
d 0:07
LO) = 0:06 30Æ < (cid:18)(cid:22)(cid:6) < 150Æ
SN 2Q
F d 0:05
d(cid:27)(I 0:04 ηV+S(s′,Ecut)= 2 1+βπ2 log(t )+1 log(2w)
2 − 2β π
0:03 (cid:20) π (cid:21)
2+β2 1 β2
0:02 π log(t ) 2+log − π
π
− β − 4
0:01 π (cid:18) (cid:19)
1+β2 1+β
0 π 2log(tπ)log π
0 1 2 3 24 52 6 7 8 9 − 2βπ (cid:26)− (cid:18) 2 (cid:19)
Q (GeV ) 2β
Fig. 8. The size of the next-to-leading FSR corrections (IF- +4Li2 π π2 , (21)
1+β −
SNLO)comparedwiththesumofNLOISRandLOFSRcon- (cid:18) π(cid:19) (cid:27)
tributions (IFSLO) to the e+e− µ+µ−γ(γ) cross section:
a) Q2 differential cross sections w→ith no angular cuts at B-
factories; b) relative difference of the cross sections with no
angular cuts; c) relative difference of the cross sections with where w = Ecut/√s′, β = 1 4m2/s′ and t = (1
angularcutsresemblingtheBABARdetector(ine+e−cms).If β )/(1+β ),2while, withπ a soft−photoπn energy cπut Em−in
twophotonsareemitted,bothhard(withenergy>100MeV) π π p γ
photonsare required to bewithin indicated angular cuts.
H. Czyz˙ et al.: Radiative return at Φ- and B-factories: FSR for muon pair production at next-to-leadingorder 9
+ (cid:0) + (cid:0) posed they contribute up to 2–3% to the Q2 differential
e e !(cid:25) (cid:25) (cid:13)((cid:13))
0:002 cross section, as shown in Fig. 10. However, for KLOE
cuts used for low-angle measurements of the pion form
0:0015 p 1.02GeV factor, the additional IFSNLO corrections, not included
1 s = a in PHOKHARA 3.0, are well below 0.1%, as anticipated
(cid:0) 0:001
in[18].ItisshowninFig.11foreventselectionswithand
(cid:27)(3:0) 2dQ 0:0005 without the track mass cut. It is worth mentioning that
d = the IFSNLO corrections change also the ratio
d(cid:27)(4:0) 2dQ 0 R(Q2)= 4(1+ 2Qm22µ)βµ ddQσπ2 , (23)
(cid:0)0:0005 βπ3|Fπ(Q2)|2 ddQσµ2
(cid:0)0:001
Æ Æ which is expected to be 1 for ISR only, when no cuts are
50 < (cid:18)(cid:25)(cid:6) < 130
or applied.FromFigs.12and13itis clearlyvisiblethatone
(cid:0)0:0015 Æ Æ
(cid:18)(p~(cid:25)+ +p~(cid:25)(cid:0)) < 15 > 165 has to take the IFSNLO corrections into account and use
(cid:0)0:002 a Monte Carlo event generator, when extracting the pion
0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 formfactorfromtheratioofpionandmuoncrosssections.
2 2
Q (GeV ) ThisistrueatbothΦ-andB-factories.Asimilarsituation
+ (cid:0) + (cid:0)
e e !(cid:25) (cid:25) (cid:13)((cid:13)) can be expected for other hadronic final states.
0:002
0:0015 M cut p 1.02GeV
5 Vacuum polarization implementation
1 tr s = b
(cid:0) 0:001
(cid:27)(3:0) 2dQ 0:0005 PreHctOioKnHsfAorRaAll4fi.n0ailnhcaluddroensiaclsstoatveascuanudmmpuoolanrpizraotdiounctcioonr-,
d
= withtheactualimplementationofRef.[34].Ascanbean-
(cid:27)(4:0) 2dQ 0 ticipatedfromFig.1,thevacuumpolarizationcorrections
d
arecalculatedatdifferentscales fordifferent types ofdia-
(cid:0)0:0005 grams(Qˆ2,Q2,s,etc.).Intheenergyrangethatisimpor-
tant for measurements of the components of the hadronic
(cid:0)0:001
Æ Æ crosssectionthroughtheradiativereturnmethod,thevac-
50 < (cid:18)(cid:25)(cid:6) < 130
(cid:0)0:0015 Æ or Æ uumpolarizationcorrectionshaveanon-trivialbehaviour
(cid:18)(p~(cid:25)+ +p~(cid:25)(cid:0)) < 15 > 165 (see Fig. 14) due to the hadronic contributions. For event
(cid:0)0:002 selections, for which only ISR corrections are important,
0:3 0:4 0:5 0:6 0:7 0:8 0:9 1
2 2 the vacuum polarizationcontributionis just a multiplica-
Q (GeV )
tive factor depending on Q2, so that one can correct the
Fig. 11. Comparison between PHOKHARA 3.0 and
eventgeneratorresultsevenaftergeneration.However,for
PHOKHARA 4.0 for √s = 1.02 GeV: a) the pion polar an-
◦ ◦ eventselectionswhere othercontributionsarealsoimpor-
gles are restricted to 50 < θπ± < 130 and angular cuts are
tant, event generation with complete implementation of
imposed on the missing momentum; b) the same as (a) with
additional track mass cut. vacuum polarization might become necessary. An exam-
ple is shown in Fig. 15, where the vacuum polarization
contributions do change the forward–backward asymme-
in the e+e− cms frame, they read tryofthepionpolarangledistribution.Theeffect,ofupto
5%, is mainly caused by the choice of s in the close vicin-
1+β2 ity of the Φ resonance, which is nothing but the KLOE
ηV+S(s,s′,Emin)= 2 π log(t )+1
γ − 2β π case. As seen in Fig. 14 the vacuum polarization around
(cid:20) π (cid:21)
s′ s the Φ is much larger than at lower energies. As a result,
log(2w)+1+ log thevacuumpolarizationcorrectiontotheISR–FSRinter-
× s′ s s′
(cid:20) − (cid:18) (cid:19)(cid:21) ference, which contributes to the numerator of the asym-
2+β2 1 β2 metry, is larger than the corresponding corrections to its
π log(t ) 2+log − π
− β π − 4 denominator,which is dominated by ISR. Therefore,vac-
π (cid:18) (cid:19)
uumpolarizationcorrectionsdonotdropintheratio.This
1+β2 1+β
π log(t )log π effecthastobe takenintoaccount,whenstudiesofmodel
π
− 2β − 2
π (cid:26) (cid:18) (cid:19) dependence of FSR are performed.
2β
+4Li π π2 , (22)
2
1+β −
(cid:18) π(cid:19) (cid:27) 6 Summary and Conclusions
where w =Emin/√s.
γ
ThenumericalimportanceofthecorrectionsfromFig.9 Precisionmeasurementsofthehadroniccrosssectionthrough
depends on the event selection. If no angularcuts are im- theradiativereturntoanaccuracyofabout1%requirethe
10 H. Czyz˙ et al.: Radiative return at Φ- and B-factories: FSR for muon pair production at next-to-leadingorder
1:04
1:8 p =10.52GeV
(cid:27)(cid:22) s
=d(cid:27)(cid:22) 1:6 (cid:27)=d(cid:25) 1:03 0Æ < (cid:18)(cid:13) < 180Æ
(cid:25) d
32)(cid:12)=((cid:12)jFj)d(cid:27)(cid:22)(cid:25)(cid:25) 11::42 ps =1.02GeV 232Q)(cid:12)=((cid:12)jFj)(cid:22)(cid:25)(cid:25) 11::00211 0Æ < (cid:18)(cid:22)(cid:6);(cid:25)(cid:6) < 180Æ
22m=Q(cid:22) 1 50Æ < (cid:18)(cid:22)(cid:6);(cid:25)(cid:6) < 130Æ 2+2m=(cid:22) 0:99
4(1+2 0:8 (cid:18)(p~(cid:22)+;(cid:25)+ +p~(cid:22)(cid:0);(cid:25)o(cid:0)r)><11655ÆÆ 4(1 0:98
0:6
0:97
0:4 0 0:2 0:4 0:6 0:8 1 1:2 1:4 1:6
0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 Q2(GeV2)
2 2
Q (GeV ) 1:06
p =10.52GeV
1:2 d(cid:27)(cid:22) 1:04 s
= Æ Æ
(cid:27)(cid:22) (cid:27)(cid:25) 20 < (cid:18)(cid:13) < 160
2Fj)d(cid:27)=d(cid:25)(cid:25) 1:11:15 p0Æs<=(cid:18)1(cid:13).0<2G18e0VÆ 32=((cid:12)jFj)d(cid:25)(cid:25) 1:021 30Æ < (cid:18)(cid:22)(cid:6);(cid:25)(cid:6) < 150Æ
3=((cid:12)j(cid:25) 1:05 0Æ < (cid:18)(cid:22)(cid:6);(cid:25)(cid:6) < 180Æ 2Q)(cid:12)(cid:22) 0:98
2Q)(cid:12)(cid:22) 1 22m=(cid:22) 0:96
2m=(cid:22) 1+ 0:94
+2 0:95 4(
1 0:92
4(
0:9
0:9
0 0:2 0:4 0:6 0:8 1 1:2 1:4 1:6
2 2
0:85 Q (GeV )
0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 Fig. 13. TheratioRofEq.(23)for√s=10.52 GeVandtwo
2 2
Q (GeV ) different eventselections.
1:8
(cid:22)
(cid:27)
d 1:6
=(cid:25) 0.4
(cid:27)
d
) 1:4 0.35
2j(cid:25)
F
3j(cid:25) 0.3
(cid:12) 1:2
=((cid:22) 0.25
(cid:12)
)
2Q 1 p = 1.02GeV 0.2
2m=(cid:22) s or (cid:1)(cid:11)
2 Æ Æ 0.15
+ 0:8 (cid:18)(cid:13) < 15 (cid:18)(cid:13) > 165
4(1 40Æ < (cid:18)(cid:22)(cid:6);(cid:25)(cid:6) < 140Æ 0.1
0:6
0.05
0
0:4
0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1
2 2 -0.05
Q (GeV )
Fig. 12. TheratioR ofEq.(23)for√s=1.02GeV,forthree -0.1
different event selections. 0 2 p4 6 8 10
2
Q (GeV)
Fig. 14. Theenergydependenceof∆αcalculatedbythepro-
gram of Ref. [34].