Erratum to: Precise measurement of Γ(K→eν(γ))/Γ(K→μν(γ)) and study of K→eνγ
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arXiv:0907.3594v2 [hepex] 22 Jul 2009
EPJ manuscript No.
(will be inserted by the editor)
Precise measurement of Γ(K → eν(γ))/Γ(K → µν(γ)) and
study of K → eνγ
The KLOE Collaboration
F. Ambrosino3,4, A. Antonelli1, M. Antonelli1, F. Archilli8,9, P. Beltrame2, G. Bencivenni1, C. Bini6,7, C. Bloise1,
S. Bocchetta10,11, F. Bossi1, P. Branchini11, G. Capon1, D. Capriotti10T. Capussela1, F. Ceradini10,11, P. Ciambrone1,
E. De Lucia1, A. De Santis6,7, P. De Simone1, G. De Zorzi6,7, A. Denig2, A. Di Domenico6,7, C. Di Donato4, B. Di
Micco10,11, M. Dreucci1, G. Felici1, S. Fiore6,7, P. Franzini6,7, C. Gatti1, P. Gauzzi6,7, S. Giovannella1, E. Graziani11,
M. Jacewicz1, V. Kulikov13, G. Lanfranchi1, J. LeeFranzini1,12, M. Martini1,5, P. Massarotti3,4, S. Meola3,4,
S. Miscetti1, M. Moulson1, S. M¨ uller2, F. Murtas1, M. Napolitano3,4, F. Nguyen10,11, M. Palutan1, A. Passeri11,
V. Patera1,5, P. Santangelo1, B. Sciascia1, A. Sibidanov1, T. Spadaro1, M. Testa1, L. Tortora11, P. Valente7,
G. Venanzoni1, and R. Versaci1,5
1Laboratori Nazionali di Frascati dell’INFN, Frascati, Italy
2Institut f¨ ur Kernphysik, Johannes Gutenberg  Universit¨ at Mainz, Germany
3Dipartimento di Scienze Fisiche dell’Universit` a “Federico II”, Napoli, Italy
4INFN Sezione di Napoli, Napoli, Italy
5Dipartimento di Energetica dell’Universit` a “La Sapienza”, Roma, Italy
6Dipartimento di Fisica dell’Universit` a “La Sapienza”, Roma, Italy
7INFN Sezione di Roma, Roma, Italy
8Dipartimento di Fisica dell’Universit` a “Tor Vergata”, Roma, Italy
9INFN Sezione di Roma Tor Vergata, Roma, Italy
10Dipartimento di Fisica dell’Universit` a “Roma Tre”, Roma, Italy
11INFN Sezione di Roma Tre, Roma, Italy
12Physics Department, State University of New York, Stony Brook, USA
13Institute for Theoretical and Experimental Physics, Moscow, Russia
Received: date / Revised version: date
Abstract. We present a precise measurement of the ratio RK = Γ(K → eν(γ))/Γ(K → µν(γ)) and a
study of the radiative process K → eνγ, performed with the KLOE detector. The results are based on
data collected at the Frascati e+e−collider DAΦNE for an integrated luminosity of 2.2 fb−1. We find
RK = (2.493 ± 0.025stat ± 0.019syst) × 10−5, in agreement with the Standard Model expectation. This
result is used to improve constraints on parameters of the Minimal Supersymmetric Standard Model with
lepton flavor violation. We also measured the differential decay rate dΓ(K → eνγ)/dEγ for photon energies
10 < Eγ < 250 MeV. Results are compared with predictions from theory.
PACS. 13.20.Eb Decays of K mesons
1 Introduction
The decay K±→ e±ν is strongly suppressed, ∼few×10−5,
because of conservation of angular momentum and the
vector structure of the charged weak current. It there
fore offers the possibility of detecting minute contributions
from physics beyond the Standard Model (SM). This is
particularly true of the ratio RK = Γ(K → eν)/Γ(K →
µν) which, in the SM, is calculable without hadronic un
certainties [1,2]. Physics beyond the SM, for example mul
Correspondence to: Mario.Antonelli@lnf.infn.it,
Tommaso.Spadaro@lnf.infn.it
tiHiggs effects inducing an effective pseudoscalar interac
tion, can change the value of RK. It has been shown in Ref.
3 that deviations of RKof up to a few percent are possible
in minimal supersymmetric extensions of the SM (MSSM)
with non vanishing eτ scalar lepton mixing. To obtain ac
curate predictions, the radiative process K → eνγ (Ke2γ)
must be included. In Ke2γ, photons can be produced via
internalbremsstrahlung (IB) or directemission (DE), the
latter being dependent on the hadronic structure. Inter
ference among the two processes is negligible [4]. The DE
contribution to the total width is approximately equal to
that of IB [4] but is presently known with a 15% fractional
accuracy [5].
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2The KLOE Collaboration: Precise measurement of Γ(K → eν(γ))/Γ(K → µν(γ)) and study of K → eνγ
RK is defined to be inclusive of IB, ignoring how
ever DE contributions. A recent calculation [2], which in
cludes order e2p4corrections in chiral perturbation theory
(χPT), gives:
RK= (2.477 ± 0.001) × 10−5.
RKis not directly measurable, since IB cannot be distin
guished from DE on an eventbyevent basis. Therefore,
in order to compare data with the SM prediction at the
percent level or better, one has to be careful with the DE
part.1
DE can proceed through vector and axialvector tran
sitions, with effective coupling V and A, respectively:
(1)
d2Γ(Ke2γ,DE)
dxdy
?(V + A)2fDE+(x,y) + (V − A)2fDE−(x,y)?,
where GF is the Fermi coupling, θCis the Cabibbo angle
[6], x = 2Eγ/MK, y = 2Ee/MK are the dimensionless
photon and electron energies in the kaon rest frame (both
lying between 0 and 1), and
=G2
FsinθC2αemM5
64π2
K
×
(2)
fDE+(x,y) = (x + y − 1)2(1 − x),
fDE−(x,y) = (1 − y)2(1 − x).
Terms proportional to (me/MK)2are neglected. The pho
ton energy spectrum in the CM is shown in Fig. 1 with
its IB, DE+, and DE−contributions.2The DE terms are
evaluated with constant V, A coupling and calculated in
χPT at O(p4) [4]. We define the rate R10as:
(3)
??
??
??
??
??
??
??
??
?
??? ????????????
E?(MeV)
??
DE?
DE+
(d d
?? E°)
10
(2 MeV)
£
5
/?¹2
??
Fig. 1. CM photon spectrum for Ke2γ decay. Inner brem
sstrahlung (IB) and positive and negative helicity direct emis
sion (DE+and DE−) contributions are also shown.
R10= Γ(K → eν(γ), Eγ< 10 MeV)/Γ(K → µν).
1
The same arguments apply in principle to Γ(K → µν).
However, there is no helicity suppression in this case. IB must
be included and DE can be safely neglected.
2“+” and “−” refer to the photon helicity.
(4)
Evaluating the IB spectrum to O(αem) with resummation
of leading logarithms, R10 includes 93.57 ± 0.07% of the
IB,
R10= RK× (0.9357± 0.0007).
The DE contribution in this range is expected to be neg
ligible. R10 is measured without photon detection. Some
small contribution of DE is present in the selected sample.
In particular, DE decays have some overlap with the IB
emission at high pe. We have also measured the differential
width
dRγ
dEγ
Γ(K → µν)
for Eγ> 10 MeV and pe> 200 MeV requiring photon
detection, both to test χPT predictions for the DE terms
and to reduce possible systematic uncertainties on the R10
measurement.
(5)
=
1dΓ(K → eνγ)
dEγ
,(6)
2 DAΦNE and KLOE
DAΦNE, the Frascati φ factory, is an e+e−collider oper
ated at a total energy√s = mφ∼1.02 GeV. φ mesons are
produced, essentially at rest, with a visible cross section
of ∼ 3.1 µb and decay into K+K−pairs with a BR of
∼ 49%. During 20012005 KLOE collected an integrated
luminosity of about 2.2 fb−1, corresponding to ∼3.3 billion
of K+K−pairs. Kaons have a momentum of ∼100 MeV
corresponding to a velocity βK∼0.2. The mean kaon path
is λK∼90 cm. Observation of a K±meson signals, or tags,
the presence of a K∓meson. Kaon production and decay
are studied with the KLOE detector, consisting essentially
of a drift chamber, DC, surrounded by an electromagnetic
calorimeter, EMC. A superconducting coil provides a 0.52
T magnetic field.
The DC, see Ref. 7, is a cylinder of 4 m in diameter
and 3.3 m in length. It contains 12,582 drift cells arranged
in 58 stereo layers uniformly filling the sensitive volume.
The momentum resolution for tracks at large polar angle
is σ(p⊥/p⊥)≤0.4%.
The EMC is a lead/scintillatingfiber sampling calori
meter [8] consisting of a barrel and two endcaps, with good
energy resolution, σE/E ∼ 5.7%/?E(GeV), and excellent
time resolution, σT = 54 ps/?E(GeV)⊕140 ps. The EMC
provides also particle identification, based on the pattern
of energy deposits in the EMC cells. An example of the
difference between electron and muon patterns is shown
in Fig. 2.
The trigger [9] uses both EMC and DC information.
Two energy deposits above threshold (E > 50 for barrel
and > 150 MeV for endcaps) are required for the EMC
trigger. The DC trigger is based on wire hit multiplicity.
The logical OR of EMC and DC triggers is used for the
measurement presented. The trigger efficiency is evaluated
from data.
Cosmicray rejection is performed by the trigger hard
ware. Residual cosmic ray and machine background events
are removed by an offline software filter using calorimeter
information before track reconstruction.
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The KLOE Collaboration: Precise measurement of Γ(K → eν(γ))/Γ(K → µν(γ)) and study of K → eνγ3
?
020406080100 120140
e
E
(MeV)
Fig. 2. Energy deposit pattern in the EMC cells for a 200 MeV
electron (left) and a muon (right) from two KL → πℓν events.
The detector response is obtained by means of the
KLOE Monte Carlo (MC) simulation program Geanfi,
Ref. 10. Changes in machine parameters and background
conditions are simulated on a runbyrun basis in order to
properly take into account the induced effects.
The MC samples used for this analysis correspond to
integrated luminosities of 4.4 fb−1for the main K±decay
modes and of 500 fb−1for decays with BR’s less than 10−4.
The effects of initial and finalstate radiation are included
in the simulation at the event generator level [10,11]. For
Ke2γevents, the IB component is described at O(e2) inclu
ding resummation of leading logarithms [11], while the DE
component is described with χPT at O(e2p4) [4]. Unless
otherwise specified, when comparing data with simulation
we rescale MC samples to an integrated luminosity of 2.2
fb−1, assume the SM value for RK, and use the theoretical
prediction for the DE/IB fraction.
3 Selection of leptonic kaon decays
K±decays are signaled by the observation of two tracks
with the following conditions. One track must originate
at the interaction point (IP) and have momentum in the
interval {70, 130} MeV, consistent with being a kaon from
φdecay. The second track must originate at the end of the
previous track and have momentum larger than that of the
kaon, with the same charge. The second track is taken as a
decay product of the kaon. The point of closest approach
of the two tracks is taken as the kaon decay point D and
must satisfy 40< rD<150 cm, zD <80 cm. The geomet
rical acceptance with these conditions is ∼56%, while the
decay point reconstruction efficiency is ∼51%. From the
measured kaon and decay particle momenta, pKand pd,
we compute the squared mass m2
decay K → ℓν assuming zero missing mass:
m2
ℓof the lepton for the
ℓ= (EK− pK− pd)2− p2
ℓis shown in Fig. 3, upper curve,
from MC simulation. The muon peak is quite evident,
higher masses corresponding to non leptonic and semilep
tonic decays. No signal of the K → eν (Ke2) decay is
visible. The very large background around zero mass is
the tail of the K → µν (Kµ2) peak, due to poor measure
ments of pK, pd or the decay angle, αKd. The expected
d.(7)
The distribution of m2
signal from Ke2γis also shown in Fig. 3, lower curves, sep
arately for Eγ>10 and <10 MeV. The expected number
of Ke2 decays in the sample is ∼30,000. A background
rejection of at least 1000 is necessary, to obtain a 1% pre
cision measurement of Γ(Ke2), with an efficiency of ∼30%.
m2`
(MeV )
2
Events/(1360 MeV )
2
10
10
10
10
10
2
3
6
7
8
010,000 20,000 30,000 40,000 50,000
Fig. 3. MC distribution of m2
Ke2γ with Eγ < 10 MeV (> 10 MeV) is shown by the dashed
(dotted) lines.
ℓ, solid line. The contribution of
The kinematics of the twobody decay φ → K+K−
provides an additional measurement of pK. The kaon mo
mentum at the IP is computed from its direction at the
IP and the known value of the φ 4momentum.3The com
puted value is extrapolated to the decay point D, account
ing for K energy losses in the material traversed. These
are relevant, since the kaon velocity is ∼ 0.2. The mate
rial amount traversed has been determined to within 1%,
thus reducing its contribution to the momentum resolu
tion to below 0.5 MeV. The total resolution of the mea
surement is ∼ 1 MeV, comparable with that from track
reconstruction. We require the two pK determinations to
agree within 5 MeV.
Further cuts are applied to the daughter track. Reso
lution of track parameters is improved by rejecting badly
reconstructed tracks, i.e., with χ2(track fit)/ndf > 7.5.
Events with poorly determined decay angles are mostly
due to tracks with improper leftright assignment in the
reconstruction of the DC hits. This happens often when a
large majority of the hits associated to the daughter track
are on a single stereo view. These events are removed by
a cut on the the fractional difference of the number of hits
on each stereo view.
Finally, using the expected errors on pK and pdfrom
tracking, we compute event by event the error on m2
The distribution of δm2
ℓdepends slightly on the opening
angle αKd, which in turn has different distribution for Ke2
and Kµ2. Events with large value of δm2
ℓ, δm2
ℓ.
ℓare rejected:
3The average value of φ 4momentum is determined on a
runbyrun basis from Bhabha events, while eventbyevent
fluctuations are dominated by the beam energy spread.
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4The KLOE Collaboration: Precise measurement of Γ(K → eν(γ))/Γ(K → µν(γ)) and study of K → eνγ
δm2
equalize the losses due to this cut for Ke2and Kµ2.
The effect of quality cuts on m2
Fig. 4. The background in the Ke2signal region is effec
tively reduced by more than one order of magnitude with
an efficiency of ∼70% for both Ke2and Kµ2.
ℓ< δmax, with δmax defined as a function of αKd, to
ℓresolution is shown in
m2(MeV )
`
2
Events/(1360 MeV )
2
102
103
104
105
106
107
108
?50000 5000 10000 1500020000
Fig. 4. m2
cuts for MC Kµ2 (upper plots) and Ke2 with Eγ < 10 MeV
(lower plots). Black dots represent data after quality cuts.
ℓspectrum before (dashed) and after (solid) quality
Information from the EMC is also used to improve
background rejection. To this purpose, we extrapolate the
secondary track to the EMC surface and associate it to a
nearby EMC cluster. This requirement produces a signal
loss of about 8%.
Energy distribution and position along the shower axis
of all cells associated to the cluster allow for e/µ par
ticle identification. For electrons, the cluster energy Ecl
is a measurement of the particle momentum pd, so that
Ecl/pdpeaks around 1, while for muons Ecl/pdis on av
erage smaller than 1. Moreover, electron clusters can also
be distinguished from µ (or π) clusters by exploiting the
granularityof the EMC: electrons shower and deposit their
energy mainly in the first plane of EMC, while muons be
have like minimum ionizing particles in the first plane and
deposit a sizeable fraction of their kinetic energy from the
third plane onward, when they are slowed down to rest
(Bragg’s peak), see Fig. 2.
All useful information about shower profile and total
energy deposition are combined with a 1225201 struc
ture neural network trained on KL→ πℓν and Kµ2data,
taking into account variations of the EMC response with
momentum and impact angle on the calorimeter. The dis
tribution of the neural network output, NN, for a sample
of KL → πeν events is shown in Fig. 5, for data and
MC. Additional separation has been obtained using time
of flight information. The data distribution of NN as func
tion of m2
ℓis shown in Fig. 6. A clear K → eν signal can
be seen at m2
ℓ∼ 0 and NN ∼ 1.
14000
12000
10000
800
600
400
200
0
0.00.4 0.60.81.01.2 0.2
counts
NN
Fig. 5. Neuralnetwork output, NN, for electrons of a KL →
πeν sample from data (black) and MC (red).
?8000 ?6000 ?4000 ?200002000 4000
(MeV )
6000
m22
`
1.0
0.7
0.9
0.6
0.8
NN
103
102
10
counts/350 MeV /0.02
2
Fig. 6. Data density in the NN, m2
ℓplane.
Some 32% of the events with a K decay in the fidu
cial volume, have a reconstructed kink matching the re
quired quality criteria and an EMC cluster associated to
the lepton track; this holds for both Ke2and Kµ2. In the
selected sample, the contamination from K decays other
than Kℓ2is negligible, as evaluated from MC. R10, Eq. 4,
is obtained without requiring the presence of the radiated
photon. The number of K → eν(γ), is determined with
a binned likelihood fit to the twodimensional NNvs m2
distribution. Distribution shapes for signal and Kµ2back
ground are taken from MC; the normalization factors for
the two components are the only fit parameters. The fit
has been performed in the region −3700 < m2
MeV2and NN > 0.86. The fit region accepts ∼ 90%
of K → eν(γ) events with Eγ < 10 MeV, as evaluated
from MC. A small fraction of fitted K → eν(γ) events
have Eγ > 10 MeV: the value of this “contamination”,
fDE, is fixed in the fit to the expectation from simulation,
fDE= 10.2%. A systematic error related to this assump
tion is discussed in Sect. 5.
ℓ
ℓ< 6100
We count 7064±102 K+→ e+ν(γ) events and 6750±
101 K−→ e−¯ ν(γ), 89.8% of which have Eγ < 10 MeV.
The signaltobackground correlation is ∼ 20% and the
Page 5
The KLOE Collaboration: Precise measurement of Γ(K → eν(γ))/Γ(K → µν(γ)) and study of K → eνγ5
m`
22
(MeV )
Events/700 MeV2
0
2000
4000
?20000200040006000
0
5000
10000
m`
22
(MeV )
?200002000 40006000
Events/700 MeV2
Fig. 7. Fit projections onto the m2
and Kµ2 background (dotted line). The contribution from Ke2 events with Eγ > 10 MeV is visible in the left panel (dashed
line).
ℓaxis for NN > 0.98 (left) and NN < 0.98 (right), for data (black dots), MC fit (solid line),
χ2/ndf is 113/112 for K+and 140/112 for K−.4Fig. 7
shows the sum of fit results for K+and K−projected onto
the m2
ℓaxis in a signal (NN > 0.98) and a background
(NN < 0.98) region. The residual contribution of Ke2γ
events with Eγ> 10 MeV is also shown.
The number of Kµ2 events is obtained from a fit to
the m2
ℓdistribution. The fraction of background events
under the muon peak is estimated from MC to be less than
one per mil. We count 2.878 × 108(2.742 × 108) K+→
µ+ν(γ) (K−→ µ−¯ ν(γ)) events. The difference between
K+and K−counts is due to K−nuclear interactions in
the material traversed.
3.1 Ke2γ event counting
In order to study Ke2γ decays, we apply the same selec
tion criteria as for Ke2, but a tighter PID cut, NN > 0.98.
We also require one and only one photon in time with
the K decay. Photons are identified by selecting a clus
ter with energy greater than 20 MeV. This requirement
reduces machine background and suppresses most of the
IB events, leaving a sample dominated by direct emission
process (DE). Moreover, the difference between the pho
ton and the electron measured time of flight has to lie
within two standard deviations from its expected value.
The fraction of signal events satisfying all of these ad
ditional requests is ∼ 25%. The m2
selected events is shown in Fig. 8 for data and MC. Ke2γ
decays with pe> 200 MeV and pe< 200 MeV are shown
separately. The highmomentum component is dominated
by DE+process, DE−accounting for 2% only (Eq. 2),
and is the only relevant for the systematic related to the
R10 measurement: high pe values correspond to low val
ues of m2
ℓwhere the Ke2signal lies. The lowmomentum
component, with contributions from both DE+and DE−
processes, is completely overwhelmed by Ke3events with
one undetected photon from π0decay.
ℓdistribution for the
4The χ2/ndf of the K−fit improves to 114/98 for a fit range
NN > 0.88, with negligible difference in the measured value for
R10.
Events/4000 MeV2
10
102
103
104
0 10000 20000 30000 40000 50000
m2
`
(MeV )
2
Fig. 8. m2
line) for events with a detected photon. MC Ke2γ events with
pe< 200 MeV (gray), pe> 200 MeV (dashed) and Ke3 events
(dotdashed) are shown separately.
ℓdistribution for data (black dots) and MC (solid
Further rejection of Ke3events is provided by kinemat
ics. The photon energy in the laboratoty frame, Eγ(lab),
can be calculated for Ke2γdecays from the measured pho
ton direction, the kaon momentum pK and the electron
momentum pe, with a resolution of ∼ 12 MeV. The reso
lution on ∆E = Eγ(lab)−Eγ, EMCis that of the calorime
ter, σ ∼ 30 MeV for Eγ(lab) = 200 MeV. The number of
Ke2γ events is found from a binned likelihood fit in the
∆E/σm2
ℓplane. This provides a better signal to noise
figure, compared to using cuts on ∆E and m2
tion shapes for signal and Kµ2and Ke3backgrounds are
taken from MC. The amounts of the three components are
the fit parameters.
For the measurement of the differential width, Eq. 6,
we boost Eγ(lab) to the kaon rest frame (Eγ) and perform
independent fits for five Eγbins between 10 MeV and the
kinematic limit, as defined in Table 1. For each Eγ bin,
we are able to extract the number of Ke2γ events with
pe> 200 MeV. Because of limited statistics, the counting
is done combining the kaon charges. Results are listed in
Table 1. The total Ke2γ count, with Eγ > 10 MeV and
ℓ. Distribu
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6The KLOE Collaboration: Precise measurement of Γ(K → eν(γ))/Γ(K → µν(γ)) and study of K → eνγ
Eγ (MeV)
Signal counts
χ2/ndf
10 to 50
55±16
80/66
50 to 100
219±24
141/105
100 to 150
463±32
87/106
150 to 200
494±38
100/106
200 to 250
253±26
116/102
Table 1. Fit results for the number of Ke2γ events with pe > 200 MeV, in five Eγ energy bins.
pe> 200 MeV, is 1484±63 events. Fig. 9 shows the sum
of the fit results on all of the Eγbins, projected onto the
∆E/σ axis for the signal region (top), defined as m2
8000 MeV2or 14000 < m2
background region (bottom). In the latter, Kµ2dominate
the region 8000 < m2
the region above 20000 MeV2(see Fig. 8).
ℓ<
ℓ< 20000 MeV2, and for the
ℓ< 14000 MeV2, while Ke3dominate
0
100
200
Events
?5
?2.502.55
¢ /
E ¾
0
200
400
600
?5
?2.502.55
¢ /
E ¾
Events
Fig. 9. Ke2γ fit projections onto the ∆E/σ(∆E) axis for the
signal region (as defined in the text, top) and background re
gion (bottom) for data (black dots), MC fit (solid line), Kµ2
(dashed line) and Ke3 (gray line). All Eγ bins are added.
4 Efficiency
The ratios of Ke2to Kµ2and Ke2γto Kµ2efficiencies are
evaluated with MC and corrected for possible differences
between data and MC, using control samples. We evaluate
dataMC corrections separately for each of the following
analysis steps: decay point reconstruction (kink), quality
cuts, clustercharged particle association; for Ke2γevents,
the efficiency for selection of a photon cluster is added,
too. For each step, the correction is defined as the ratio of
data and MC efficiencies measured on the control sample,
each folded with the proper kinematic spectrum of Ke2
(or Kµ2) events.
Decay point reconstruction efficiencies are evaluated
using pure samples of Kµ2and Ke3; these are tagged by
the identification of the twobody decay, Kµ2or K → ππ0
(Kπ2), of the other kaon [12] and selected with tagging and
EMC information only, without using tracking.
A 99.5% pure K±
plus one and only one EMC cluster with energy E > 90
MeV, not due to the tagging kaon decay products. The
properties of the selected Kµ2 event are evaluated using
time and position of the cluster and the kaon momentum
obtained from the tagging (with 1% resolution). The muon
momentum and the decay point position are determined
a priori, without using the kaon and electron tracks, with
a resolution of about 5 MeV and about 2 cm, respectively.
The tracking efficiency is determined as a function of the
decay point position and the decay angle, by counting the
number of events in which a kink is reconstructed out of
the number of Kµ2candidate events.
K±
tecting the photons from π0decay with time of flights
consistent with a single point in the tagged kaon track ob
tained from the tagging kaon. Second, a third cluster with
energy, time, and position consistent with the expectation
from a Ke3decay is selected. The electron momentum and
the kaon decay point are determined a priori with a reso
lution of ∼20 MeV (dominated by the measurement of π0
momentum) and ∼2 cm, respectively.
The corrections to MC efficiencies range between 0.90
and 0.99 depending on the decay point position and on
the decay angle. The simulation is less accurate in case of
overlap between lepton and kaon tracks, and with decays
occurring close to the inner border of the fiducial volume.
Samples of KL(e3), KL(µ3), and Kµ2 decays with a
purity of 99.5%, 95.4%, and 100.0% respectively, are used
to evaluate lepton cluster efficiencies. These samples are
selected using tagging and DC information only, without
using calorimeter, see Refs. 13–15. The efficiency has been
evaluated as a function of the particle momentum sepa
rately for barrel and endcap. The correction to MC ef
ficiencies ranges between 0.98 and 1.01 depending on the
momentum and on the point of impact on the calorimeter.
The singlephoton detection efficiency for data and MC
is evaluated as a function of photon energy using Kπ2
events, in which one of the two photons from π0decay is
identified, allowing an a priori determination of the po
sition and of the energy of the second one. The average
correction factor to MC efficiency is ∼0.98.
The trigger efficiency has been evaluated solely from
data. The probabilities ǫTRG
trigger condition to be satisfied in a DCtriggered (EMC
triggered) event are evaluated in Ke2enriched and Kµ2
pure samples. The efficiency for the logical OR of the EMC
and DC trigger conditions is given by ǫTRG
ǫTRG
DC, and it is ∼ 0.99 for both Ke2 and Kµ2,
µ2sample is obtained with K∓tagging
e3decays are selected in K∓tagged events first de
EMC(ǫTRG
DC) for the EMC (DC)
EMC+ ǫTRG
DC
−
EMC× ǫTRG
Page 7
The KLOE Collaboration: Precise measurement of Γ(K → eν(γ))/Γ(K → µν(γ)) and study of K → eνγ7
with a ratio ǫTRG(Ke2)/ǫTRG(Kµ2) = 0.9988(5). A possi
ble bias on the previous result due to correlation between
EMC and DC triggers is also taken into account, which is
evaluated to be 0.997(1) using MC simulation.
The event losses induced by the cosmic veto applied at
the trigger level and by the background rejection filter ap
plied offline (FILFO) are evaluated from samples of down
scaled events, in which the veto conditions are registered
but not enforced. The ratio of Ke2to Kµ2efficiencies are
1.0013(2) and 0.999(4) for cosmic veto and FILFO, respec
tively. The statistical error due to the FILFO correction
is 0.4%, and dominates the total uncertainty in trigger,
cosmic veto, and FILFO corrections.
5 Systematic errors
The absolute values of all of the systematic uncertainties
on R10and Rγ, the integral of Eq. 6 for Eγ> 10 MeV, are
listed in Table 2; as a comparison, the statistical uncer
tainty is reported as well. All of the sources of systematic
error are discussed below.
δ(R10)×105
0.024
δ(Rγ)×105
0.066Statistical error
Systematic error
Counting:fit
DE
kink
trigger
e,µ cluster
γ cluster
0.007
0.005
0.014
0.009
0.005

0.019
0.004

0.009
0.006
0.003
0.003
0.013
Efficiency:
Total systematic error
Table 2. Summary of statistical and systematic uncertainties
on the measurements of R10 and Rγ.
To minimize possible biases on Ke2event counting due
to the limited knowledge of the momentum resolution, we
used Kµ2data to carefully tune the MC response on the
tails of the m2
ℓdistribution. This has been performed in
sidebands of the NN variable, to avoid bias due to the
presence of Ke2signal. Similarly, for the NN distribution
the EMC response in the MC has been tuned at the level of
single cell, using Kℓ3data control samples. Residual differ
ences between data and MC Ke2and Kµ2NN shapes have
been corrected by using the same control samples. Finally,
to evaluate the systematic error associated with these pro
cedures, we studied the variation of the results with dif
ferent choices of fit range, corresponding to a change of
overall purity from ∼ 75% to ∼ 10%, for K → eν(γ) with
Eγ< 10 MeV, and from ∼ 31% to ∼ 10%, for K → eν(γ)
with Eγ > 10 MeV and pe > 200 MeV. The results are
stable within statistical fluctuations. A systematic uncer
tainty of ∼ 0.3% for both R10 and dRγ/dEγ, indepen
dently on Eγ, is derived by scaling the uncorrelated errors
so that the reduced χ2value equals unity (see also Table
2).
Ke2event counting is also affected by the uncertainty
on fDE, the fraction of Ke2events in the fit region which
are due to DE process. This error has been evaluated by
repeating the measurement of R10with values of fDEvar
ied within its uncertainty, which is ∼ 4% according to our
measurement of the Ke2γdifferential spectrum (Sects. 3.1
and 6). Since the m2
ℓdistributions for Ke2γwith Eγ< 10
MeV and with Eγ > 10 MeV overlap only partially, the
associated fractional variation on R10is reduced: the final
error due to DE uncertainty is 0.2% (Table 2).
Different contributions to the systematic uncertainty
on ǫe2/ǫµ2are listed in Table 2. These errors are domina
ted by the statistics of the control samples used to correct
the MC evaluations. In addition, we studied the variation
of each correction with modified controlsample selection
criteria. We found neglible contributions in all cases but
for the kink and quality cuts corrections, for which the
bias due to the controlsample selection and the statistics
contribute at the same level.
The total systematic error is ∼ 0.8% for both R10and
Rγ measurements, to be compared with statistical accu
racies at the level of ∼ 1% and ∼ 4%, respectively. As
a further crosscheck on the results, and particularly on
the criteria adopted to obtain the data/MC corrections,
we measured with the same analysis method the ratio
Rℓ3= Γ(Ke3)/Γ(Kµ3). We found Rℓ3= 1.507 ± 0.005stat
and Rℓ3 = 1.510 ± 0.006stat for K+and K−. These re
sults agree within the quoted accuracy with the value ex
pected from the worldaverage formfactor slope measure
ments [16], Rℓ3= 1.506± 0.003.
6 Results and interpretation
6.1 RK and leptonflavor violation
The number of K → eν(γ) events with Eγ < 10 MeV,
the number of K → µν(γ) events, the ratio of Ke2 to
Kµ2efficiencies and the measurement of R10are given in
Table 3 for K+, K−and both charges combined. K+and
K−results are consistent within the statistical error. The
systematic uncertainty is common to both charges.
To compare the R10 measurement with the inclusive
RK prediction from SM, we take into account the accep
tance of the 10 MeV cut for IB, Eq. 5. We obtain:
RK= (2.493 ± 0.025stat± 0.019syst) × 10−5,
in agreement with SM prediction of Eq. 1. In the frame
work of MSSM with leptonflavor violating (LFV) cou
plings, RK can be used to set constraints in the space of
relevant parameters, using the following expression [3]:
(8)
RK= RSM
K ×
?
1 +
?m4
m4
K
H
??m2
τ
m2
e
???∆31
R
??2tan6β
Ris the effective
?
,(9)
where MHis the chargedHiggs mass, ∆31
eτ coupling constant depending on MSSM parameters,
and tanβ is the ratio of the two Higgs superfields vacuum
expectation values. The regions excluded at 95% C.L. in
the plane MH–tanβ are shown in Fig. 10 for different
values of the effective LFV coupling ∆31
R.
Page 8
8The KLOE Collaboration: Precise measurement of Γ(K → eν(γ))/Γ(K → µν(γ)) and study of K → eνγ
N(Ke2)N(Kµ2)
2.878 × 108
2.742 × 108
5.620 × 108
ǫe2/ǫµ2
R10
K+
K−
K±
6348 ± 92 ± 23
6064 ± 91 ± 22
12412 ± 129 ± 45
0.944 ± 0.003 ± 0.007
0.949 ± 0.002 ± 0.007
0.947 ± 0.002 ± 0.007
(2.336 ± 0.033 ± 0.019) × 10−5
(2.330 ± 0.035 ± 0.019) × 10−5
(2.333 ± 0.024 ± 0.019) × 10−5
Table 3. Number of Ke2 and Kµ2 events, efficiency ratios and results for R10 for K+, K−, and both charges combined; first
error is statistical, second one is systematic.
20
40
60
80
200 400 600800
RK=(2.493§0.032) 10
£
?5
∆13= 103
∆13= 5 104
∆13= 104
charged Higgs mass (GeV )
2
tan( )
¯
Fig. 10. Excluded regions at 95% C.L. in the plane MH–tanβ
for ∆31
R= 10−4,5 × 10−3,10−3.
6.2 Measurement of dRγ/dEγ
Results on the differential spectrum are given in Table 4.
For each Eγbin we measure ∆Rγ, the integral of dRγ/dEγ
over the bin width. In Fig. 11 top, our measurements are
Eγ (MeV)
10 to 50
50 to 100
100 to 150
150 to 200
200 to 250
ǫ(e2)/ǫ(µ2)
0.104±0.003
0.192±0.001
0.184±0.001
0.183±0.001
0.174±0.002
∆Rγ (10−6)
0.94±0.30±0.03
2.03±0.22±0.02
4.47±0.30±0.03
4.81±0.37±0.04
2.58±0.26±0.03
Table 4. dRγ/dEγ results. Most of the efficiency ratio error
is common to all energy bins.
compared to the prediction from χPT at O(p4) [4] and
from the Light Front Quark model (LFQ) of Ref. 17. In
tegrating over Eγfrom 10 MeV to 250 MeV, we obtain:
Rγ= (1.483 ± 0.066stat± 0.013syst) × 10−5,
in agreement with the prediction Rγ= 1.447×10−5, which
is obtained using the values for the effective couplings (V
and A) from O(e2p4) χPT [4] and using worldaverage
values for all of the other relevant parameters. The Rγ
prediction includes a 1.32(1)% contribution from IB. This
result confirms within a 4% error the amount of DE com
ponent in our MC.
(10)
0100200300
0
0.1
0.2
0.3
0.4
0.5
E°
(MeV)
LFQM
Âe p
PT()
O
2 4
O(
Fit
)
e p
2 6
PT
Â
E°
(MeV)
0
0.1
0.2
0.3
0.4
0.5
¢R°
(10
)
?5
¢R°
(10
)
?5
Fig. 11. ∆Rγ = [1/Γ(Kµ2)] × [dΓ(Ke2γ/dEγ] vs Eγ. On top
data (black dots) are compared to χPT predictions at O(e2p4)
and to the LFQ model, see text. At the bottom data are fitted
to χPT at O(e2p6). The IB contribution is shown (red line).
The comparison of the measured spectrum with the
χPT prediction shown in Fig. 11 top suggests the presence
of a form factor, giving a dependence of the effective cou
plings on the transeferred momentum, W2= M2
as predicted by χPT at O(e2p6) [17]. The formfactor pa
rameters are obtained by fitting the measured Eγ distri
bution with the theoretical differential decay width given
in Eq. 2, with the vector effective coupling expanded at
first order in x: V = V0(1 + λ(1 − x)). The axial effec
tive coupling A is assumed to be independent on W as
predicted by χPT at O(e2p6) [17]. The small contribution
from DE−transition to our selected events does not allow
a fit to the related V − A component. Therefore, in the
fit V0− A is kept fixed at the expectation from χPT at
O(e2p4), while V0+A and λ are the free parameters. The
result of this fit is shown in Fig. 11 bottom. We obtain:
K(1 − x),
V0+ A = 0.125 ± 0.007stat± 0.001syst,
λ = 0.38 ± 0.20stat± 0.02syst,
with a correlation of 0.93 and a χ2/ndof = 1.97/3. Our fit
confirms at ∼ 2σ the presence of a slope in the vector form
factor, in agreement with the value from χPT at O(e2p6),
λ ∼ 0.4.
Page 9
The KLOE Collaboration: Precise measurement of Γ(K → eν(γ))/Γ(K → µν(γ)) and study of K → eνγ9
7 Conclusions
We have performed a comprehensive study of the process
Ke2γ. We have measured the ratio of Ke2γand Kµ2widths
for photon energies smaller than 10 MeV, without photon
detection requirement. We find:
R10= (2.333 ± 0.024stat± 0.019stat) × 10−5.
From this result we derive the inclusive ratio RK to be
compared with the SM prediction:
(11)
RK= (2.493 ± 0.025stat± 0.019syst) × 10−5,
in excellent agreement with the SM prediction
(12)
RK= (2.477 ± 0.001) × 10−5.
Our result improves the accuracy with which RKis known
by a factor of 5 with respect to the present world average
and allows severe constraints to be set on new physics
contributions in the MSSM with lepton flavor violating
couplings as shown in Fig. 10.
To obtain the value of RK from the measurement of
R10knowledge of radiative effects is required for both in
ner bremsstrahlung and direct emission. The latter is im
portant for the helicity suppressed K → eν decay but is
not precisely known nor the differential width has ever
been measured. We have therefore measured the differen
tial decay width for Ke2γas a function of Eγ, normalized
to Kµ2, in the momentum region pe > 200 MeV, in the
kaon rest frame. Our result for the direct emission width
is in agreement with the expectation from χPT and gives
an indication of the presence of O(e2p6) contributions.
(13)
Acknowledgements We thank the DAΦNE team for
their efforts in maintaining low background running con
ditions and their collaboration during all datataking. We
want to thank our technical staff: G. F. Fortugno and
F. Sborzacchi for their dedicated work to ensure an effi
cient operation of the KLOE Computing Center; M. Anelli
for his continuous support to the gas system and the safety
of the detector; A. Balla, M. Gatta, G. Corradi and G. Pa
palino for the maintenance of the electronics; M. Santoni,
G. Paoluzzi and R. Rosellini for the general support to
the detector; C. Piscitelli for his help during major main
tenance periods. This work was supported in part by EU
RODAPHNE, contract FMRXCT980169; by the Ger
man Federal Ministry of Education and Research (BMBF)
contract 06KA957; by the German Research Foundation
(DFG),’Emmy Noether Programme’, contracts DE839/1
4.
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