Determination of $\pi\pi$ scattering lengths from measurement of $\pi^+\pi^-$ atom lifetime

Abstract

The DIRAC experiment at CERN has achieved a sizeable production of
$\pi^+\pi^-$ atoms and has significantly improved the precision on its lifetime
determination. From a sample of 21227 atomic pairs, a 4% measurement of the
S-wave $\pi\pi$ scattering length difference $|a_0-a_2| =
(.0.2533^{+0.0080}_{-0.0078}|_\mathrm{stat}.{}^{+0.0078}_{-0.0073}|_\mathrm{syst})M_{\pi^+}^{-1}$
has been attained, providing an important test of Chiral Perturbation Theory.

Full text

Page 1

Determination of ππ scattering lengths from measurement of π+π−atom lifetime
B.Adevaa, L.Afanasyevb, M.Benayounc, A.Benellid, Z.Berkae, V.Brekhovskikhf, G.Caragheorgheopolg, T.Cechake, M.Chibah,
P.V.Chliapnikovf, C.Ciocarlang, S.Constantinescug, S.Costantinii, C.Curceanu (Petrascu)g, P.Doskarovae, D.Dreossik, D.Drijardl,∗,
A.Dudarevb, M.Ferro-Luzzil, J.L.Fungueiri˜ no Pazosa, M.Gallas Torreiral,a, J.Gerndte, P.Gianottij, D.Goldini, F.Gomeza, A.Gorinf,
O.Gorchakovb, C.Guaraldoj, M.Gugiug, M.Hansroull, Z.Honsm, R.Hoseke, M.Iliescuj,g, V.Karpukhinb, J.Klusone, M.Kobayashin,
P.Kokkaso, V.Komarovb, V.Kruglovb, L.Kruglovab, A.Kulikovb, A.Kuptsovb, K.I.Kurodab, A.Lambertop, A.Lanarol,q, V.Lapshinf,
R.Lednickyr, P.Lerustec, P.Levi Sandrij, A.Lopez Agueraa, V.Lucherinij, T.Makis, I.Manuilovf, J.Marint, J.L.Narjouxc,
L.Nemenovb,l, M.Nikitinb, T.Nunez Pardoa, K.Okadau, V.Olchevskiib, A.Pazosa, M.Pentiag, A.Penzov, J.M.Perreaul, M.Ploa,
T.Pontag, G.F.Rappazzop, A.Riazantsevf, J.M.Rodrigueza, A.Rodriguez Fernandeza, A.Romero Vidalj, V.M.Ronjinf, V.Rykalinf,
J.Saboridoa, C.Santamarinaa, J.Schacherw, C.Schuetzi, A.Sidorovf, J.Smolike, F.Takeutchiu, A.Tarasovb, L.Tauscheri, M.J.Tobara,
T.Trojeke, S.Trusovx, V.Utkinb, O.V´ azquez Docea, S.Vlachosi, O.Voskresenskayab, T.Vrbae, C.Willmottt, V.Yazkovx,
Y.Yoshimuran, M.Zhabitskyb, P.Zrelovb
aSantiago de Compostela University, Spain
bJINR Dubna, Russia
cLPNHE des Universites Paris VI/VII, IN2P3-CNRS, France
dZurich University, Switzerland
eCzech Technical University in Prague, Prague, Czech Republic
fIHEP Protvino, Russia
gIFIN-HH, National Institute for Physics and Nuclear Engineering, Bucharest, Romania
hTokyo Metropolitan University, Japan
iBasel University, Switzerland
jINFN, Laboratori Nazionali di Frascati, Frascati, Italy
kINFN, Sezione di Trieste and Trieste University, Trieste, Italy
lCERN, Geneva, Switzerland
mNuclear Physics Institute ASCR, Rez, Czech Republic
nKEK, Tsukuba, Japan
oIoannina University, Ioannina, Greece
pINFN, Sezione di Trieste and Messina University, Messina, Italy
qUniversity of Wisconsin, Madison, USA
rInstitute of Physics ASCR, Prague, Czech Republic
sUOEH-Kyushu, Japan
tCIEMAT, Madrid, Spain1
uKyoto Sangyo University, Kyoto, Japan
vINFN, Sezione di Trieste, Trieste, Italy
wBern University, Switzerland
xSkobeltsin Institute for Nuclear Physics of Moscow State University, Moscow, Russia
Abstract
The DIRAC experiment at CERN has achieved a sizeable production of π+π−atoms and has significantly improved the precision on
its lifetime determination. From a sample of 21227 atomic pairs, a 4% measurement of the S-wave ππ scattering length difference
|a0− a2| =
?
0.2533+0.0080
−0.0078
???stat
+0.0078
−0.0073
???syst
?
M−1
π+ has been attained, providing an important test of Chiral Perturbation Theory.
Keywords:
DIRAC experiment, elementary atom, pionium atom, pion scattering
1. Introduction
Pionium (A2π) is the π+π−hydrogen-like atom, with 378 fm
Bohr radius, which decays predominantly into π0π0[1]. The al-
ternativeγγ decayaccountsforonly∼ 0.4%ofthetotalrate[2].
∗Corresponding author
Email address: Daniel.Drijard@cern.ch (D.Drijard)
1Associated with the university of Santiago de Compostela for technical
support in the GEM/MSGC detector
Its ground-state lifetime is governed by the ππ S-wave scatter-
ing lengths aI, with total isospin I = 0,2 [1, 3]:
Γ2π0 =2
9α3p?(a0− a2)2(1 + δ)M2
?
in the atom rest frame, α is the fine-structure constant, and δ =
(5.8±1.2)·10−2is a correction of order α due to QED and QCD
[3] which ensures a 1% accuracy of equation (1). The value of
π+,
(1)
where p?=
M2
π+− M2
π0− (1/4)α2M2
π+ is the π0momentum
Preprint submitted to Physics Letters B October 5, 2011
arXiv:1109.0569v2 [hep-ex] 3 Oct 2011

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a0and a2can be rigorously calculated in Chiral Perturbation
Theory (ChPT) [4, 5], predicting a0−a2= (0.265±0.004)M−1
and the A2πlifetime τ = (2.9 ± 0.1) · 10−15s [6]. The mea-
surement of Γ2π0 provides an important test of the theory since
a0− a2is sensitive to the quark condensate defining the spon-
taneous chiral symmetry breaking in QCD [7]. The method re-
ported in this article implies observation of the pionium state
through its ionization into two pions. Given its large Bohr
radius, this is directly sensitive to ππ scattering at threshold,
Mππ ∼ 2Mπ+, and thus delivers a precision test of the theory
without requiring threshold extrapolation, as for semileptonic
Ke4 decays [8], or substantial theoretical input as for K → 3π
decays [9].
π+
2. Pionium formation and decay
In collisions with target nuclei, protons can produce pairs
of oppositely charged pions. Final-state Coulomb interaction
leads to an enhancement of π+π−pairs at low relative c.m. mo-
mentum (Q) and to the formation of A2πbound states or pio-
nium. These atoms may either directly decay, or evolve by ex-
citation (de-excitation) to different quantum states. They would
finally decay or be broken up (be ionized) by the electric field of
the target atoms. In the case of decay, the most probable chan-
nel is π0π0and the next channel is γγ with a small branching ra-
tio of 0.36%. In the case of breakup, characteristic atomic pion
pairs emerge [10]. These have a very low Q (< 3 MeV/c) and
very small opening angle in the laboratory frame (< 3 mrad).
A high-resolution magnetic spectrometer (∆p/p ∼ 3 · 10−3) is
used [11] to split the pairs and measure their relative momen-
tum with sufficient precision to detect the pionium signal. This
signal lays above a continuum background from free (unbound)
Coulomb pairs produced in semi-inclusive proton-nucleus in-
teractions. Other background sources are non-Coulomb pairs
where one or both pions originate from a long-lived source
(η,η?,Λ,...) and accidental coincidences from different proton-
nucleus interactions.
The first observation of A2π was performed in the early
1990s [12]. Later, the DIRAC experiment at CERN was able
to produce and detect ∼ 6000 atomic pairs and perform a first
measurement of the pionium lifetime [13]. We now present fi-
nal results from the analysis of ∼ 1.5·109events recorded from
2001 to 2003. Compared to the results in [13], this analysis has
reduced systematic errors and improved track reconstruction,
mostly due to the use of the GEM-MSGC detector [11] infor-
mation, which leads to a larger signal yield. The present data
come from collisions of 20 and 24 GeV/c protons, delivered by
the CERN PS, impinging on a thin Ni target foil of 94 or 98 µm
thickness for different run periods.
3. Pionium detection and signal analysis
Low relative-momentum prompt and accidental π+π−pairs
are produced at the target and selected by the multi-level trig-
ger when their time difference, recorded by the two spectrom-
eter arms, is |∆t| < 30 ns. A suitable choice of the target ma-
terial and thickness provides the appropriate balance between
the A2πbreakup and annihilation yields, with reduced multiple-
scattering [14, 15]. For a thin Ni target, of order ∼ 10−3X0,
the relative c.m. momentum Q of the atomic pairs is less than
∼ 3 MeV/c and their number is ∼ 10% of the total number of
free pairs in the same Q region [16]. The experiment is thus
designed for maximal signal sensitivity in a very reduced re-
gion of the π+π−phase space. This is done by selective trigger-
ing and by exploiting the high resolution of the spectrometer
and background rejection capabilities. The longitudinal (QL)
and transverse (QT) components of? Q, defined with respect to
the direction of the total laboratory momentum of the pair, are
measured with precisions 0.55 MeV/c and 0.10 MeV/c, respec-
tively.
The double differential spectrum of prompt π+π−pairs Npr
(defined by |∆t| < 0.5 ns), composed of atomic nA, Coulomb
NC, non-Coulomb NnC, and accidental Naccpairs, can be χ2-
analysed in the (QT, QL) plane by minimizing the expression
[Mij− Fij
[Mij+ (σij
χ2=
?
ij
A− Fij
A)2+ (σij
B]2
B)2]
.
(2)
Here
M(QT,QL) =
?
d2Npr
dQTdQL
?
∆QT∆QL,
(3)
and the sum in (2) runs over a two-dimensional grid of |QL| <
15 MeV/c and |QT| < 5 MeV/c, with bin centres located at val-
ues (Qi
The FAand FBfunctions describe the A2πsignal and the NC+
NnC+ Naccthree-fold background, respectively; σAand σBare
their statistical errors. The analysis is based on the parametriza-
tion of FAand FBand the precise Monte Carlo simulation of the
detector response.
The FAsignal has been simulated [17, 18] according to an
accurate model of A2πproduction, propagation [14], and inter-
action with the target medium [15, 19–21].
In the background FB, the NnCand the Naccdouble differ-
ential spectra were parametrized according to two-body phase
space and Lorentz boosted to the laboratory frame using the
observed pion pair spectra [17]. The spectrum of NCpairs is
enhanced at low? Q with Q defined at the point of production, by
the Coulomb interaction according to the Gamow–Sommerfeld
factor
2πMπα/Q
1 − exp(−2πMπα/Q).
The finite size of the production source and final-state interac-
tion effects have been calculated [22, 23] and applied to sim-
ulated atomic and Coulomb pairs. An additional momentum-
dependent correction has been applied to the simulated NC
spectrum to take into account a small (< 0.5%) contamination,
measured by time-of-flight [24], due to misidentified K+K−
pairs. Small admixtures of misidentified p¯ p and residual con-
tamination from e+e−pairs have been measured and produce
no effect on the final result.
The fraction of accidental pairs in FB was measured by
time-of-flight to be ωacc ? 12.5%, averaged over the pair mo-
mentum and the different data sets.
T, Qj
L) and uniform bin size ∆QT = ∆QL= 0.5 MeV/c.
AC(Q) =
(4)
2

Page 3

0
1000
2000
3000
4000
5000
6000
7000
02468 10 12
|QL|(MeV/c)
14
Entries/0.25MeV/c
MC (nA)
QT < 2MeV/c
0
2500
5000
7500
10000
12500
15000
17500
20000
22500
02468 10 12
|QL|(MeV/c)
14
Entries/0.25MeV/c
MC (NC+NnC+Nacc)
MC (NnC+Nacc)
QT < 2MeV/c
Figure 1: |QL| fit projections of the π+π−spectrum from data (dots) and sim-
ulation (MC lines). The top plot shows the experimental spectrum compared
with the simulated background components (no pionium signal), with (solid
line) and without (dotted line) Coulomb pairs (NC). The bottom plot shows
the experimental |QL| spectrum after background subtraction and the simulated
pionium spectrum.
The experimental resolutions on the momentum and open-
ing angle must be accurately simulated in order to extract the
narrow pionium signal. Multiple-scattering in the target and
the spectrometer is the primary source of uncertainty on the
QT measurement. In order to achieve the desired QT reso-
lution, the scattering angle must be known with ∼ 1% preci-
sion, which is beyond the currently available GEANT descrip-
tion [25]. An improved multiple-scattering description was im-
plemented based on dedicated measurements of the average
scattering angle off material samples [26]. A cross-check with
the standard GEANT description was made by comparing the
momentum evolution of the measured distance between π+and
π−at the target [27].
The QLresolution was checked using Λ decays with small
opening angle. The widths of reconstructed real and simulated
Λ → pπ−were compared. A 3.4% relative difference was ob-
served and attributed to residual fringing magnetic field effects,
multiplescatteringinthedownstreamvacuumchannelexitwin-
dow, and to a small misalignment between the spectrometer
arms. Such effects have been altogether absorbed into an addi-
tional Gaussian smearing term, of width 0.66·10−3, convoluted
with the simulated momentum resolution function.
The only free parameters in (2) are the number of detected
0
5000
10000
15000
20000
25000
30000
35000
012345
QT(MeV/c)
Entries/0.25MeV/c
|QL| < 2 MeV/c
0
500
1000
1500
2000
2500
3000
3500
012345
QT(MeV/c)
Entries/0.25MeV/c
|QL| < 2 MeV/c
0
200
400
600
800
1000
1200
1400
1600
1800
x 102
012345
QT(MeV/c)
Entries/0.25MeV/c
2 < |QL| < 15 MeV/c
Figure 2: QT fit projections of the π+π−spectrum from data (dots) and sim-
ulation (line). The left plots show the comparison between the experimental
spectra and the full simulated background. The plots correspond to different
QLregions: top left plot in the A2πsignal region (low |QL|) and bottom left
plot away from it (higher |QL|). The right plot shows the QT spectrum after
background subtraction and the simulated pionium spectrum.
(MeV/c)
L
Q
-20
-15
-10
-5
0
5
10
15
20
(MeV/c)
Q
-8
-6
-4
-2
0
2
4
6
8
)
2
Entries/(0.5x0.5(MeV/c)
0
200
400
600
800
1000
1200
1400
1600
Figure 3: Coulomb subtracted two-pion correlation function measured in the
(Q⊥,QL) plane, showing the pionium signal. Q⊥is the signed projection of? Q
into a generic transverse axis (azimuthal invariance is ensured by the absence
of beam and target polarization)
3

Page 4

atomic pairs (nrec
pairs (Nrec
dimensional space |QL| < 15 MeV/c, QT< 5 MeV/c, for values
of the total pair momentum p between 2.6 and 6.8 GeV/c [28].
A constraint on the total number of reconstructed prompt pairs
is applied such that Npr(1 − ωacc) = Nrec
In Figs. 1 and 2, the |QL| and QTprojections of the experi-
mental prompt π+π−spectrum are shown in comparison to the
fitted simulated background spectrum (FA= 0). After subtrac-
tion of the FBbackground, the experimental A2πsignal emerges
at small values of |QL| (Fig. 1) and QT (Fig. 2) and can be
compared with the simulated FAsignal. As expected, multiple-
scattering in the target and upstream detectors broadens the QT
signal shape. This is clearly shown in the 2-dimensional plot of
Fig. 3. The overall agreement between the best-fit experimen-
tal and simulated spectra is excellent, over the entire QT,QL
domain.
A) and the fraction of non-Coulomb/Coulomb
nC/Nrec
The minimization is performed in two-
C).
C+ Nrec
nC+ nrec
A.
4. Pionium breakup probability
The pionium breakup probability, Pbr, is defined as the ra-
tio nA/NAbetween the number nAof observed pairs from pi-
onium ionization caused by target atoms and the total number
NAof pionium atoms formed by final-state interaction. The lat-
ter can be inferred by quantum mechanics from the number of
Coulomb-interacting pairs measured at low Q according to the
expression [10]
?∞
where Ω is the domain of integration |QL| < 2 MeV/c and
QT< 5 MeV/c, yielding Kth= 0.1301. Differences in detector
acceptance and reconstruction efficiency for nAand NCpairs,
?Aand ?Crespectively, are taken into account by correcting the
theoretical factor Kthas
Kexp(Ω) = Kth(Ω)?A(Ω)
?C(Ω).
Those differences arise mainly from the lesser resolution of the
upstream detectors for identifying close tracks at very low QT.
This occurs more frequently for atomic pairs than for Coulomb
pairs.
The breakup probability is thus determined as
nrec
Nrec
NA(Ω)
NC(Ω)=(2πMπα)3
π
·
n=11/n3
ΩAC(Q)d3Q
?
= Kth(Ω),
(5)
(6)
Pbr=nA
NA
=
A(Ω)
C(Ω)·
1
Kexp(Ω).
(7)
The momentum-dependent Kexpfactor (6) has been cal-
culated from fully reconstructed Monte Carlo atomic and
Coulomb pairs. Using (6) and (7), 35 independent Pbr val-
ues are obtained for the five independent data sets and for
seven 600 MeV/c wide bins of the A2πmomentum from 2.6 to
6.8GeV/c, byappropriatelyfoldingthemomentumdependence
of Kexp.
In Table 1 the fitted yields are given for the different
momentum-averaged data sets.
atomic pairs have been detected. The reported Pbrvalues are
Overall, more than 2 · 104
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
3 3.54 4.55 5.566.5
p(GeV/c)
Pbr
Figure 4: The dependence of the measured Pbr, averaged over all data sets,
from the pionium laboratory momentum and the Monte Carlo prediction cor-
responding to the ground-state lifetime of 3.15 · 10−15s obtained from the best
fit.
only indicative of the amount of variation expected with respect
to the different experimental conditions, and they are not used
in the final momentum-dependent fit.
A slight increase of the measured Pbrwith increasing pio-
nium momentum is observed in Fig. 4 (data points), which is a
consequence of the longer decay path, and hence the greater
breakup yield, expected at higher atom momenta. The con-
tinuous curve represents the predicted evolution of Pbr with
pionium laboratory momentum, for the value of the pionium
ground-state lifetime τ = 3.15·10−15s obtained from this anal-
ysis.
The dependence of the A2πbreakup probability on the spe-
cific choice of the integration domain Ω has been verified. The
measured Pbr, averaged over the data sets, is indeed very stable
versus variations of the |QL|, QTintegration limits as shown in
Fig. 5.
0.35
0.4
0.45
0.5
0.55
0.6
0.25 0.50.751 1.251.5 1.7522.25
|QL,cut|(MeV/c)
Pbr
0.35
0.4
0.45
0.5
0.55
0.6
0.51 1.52 2.53 3.54 4.55
QT,cut(MeV/c)
Pbr
Figure 5: Stability of the average Pbrwith respect to variation of the: (top) |QL|
(for QT< 5 MeV/c ) and (bottom) QT(for |QL| < 2 MeV/c) integration limits,
in 0.5 MeV/c bins.
4

Page 5

Table 1: Fit results for QT< 5MeV/c and |QL| < 15MeV/c.
NC
6020±216
9321±274
5886±210
21227±407
Ni,
94 µm, 24 GeV/c
98 µm, 24 GeV/c
98 µm, 20 GeV/c
combined samples
pbeam
χ2/ndf
2127/2079
4288/4149
4257/4144
nA
NnC
45624±4501
93148±5754
60867±4397
199639±8526
Nacc
63212±208
98499±255
59392±144
221103±359
Pbr
0.441±0.018
0.452±0.015
0.472±0.020
546003±4549
828554±5811
496820±4441
1871377±8613
5. Results and systematic errors
A detailed assessment of the systematic errors affecting the
Pbrmeasurement has been carried out, considering all known
sources of uncertainty in the simulation and in the theoretical
calculations. The largest systematic error comes from a ∼ 1%
uncertainty in the multiple-scattering angle inside the Ni tar-
get foil which induces a ±0.0077 error on Pbr. The momentum
smearing correction can increase Pbrby ∼ 2% and thus pro-
duce a ±0.0026 systematic error. The double-track resolution
at small angles can change Pbrby 1.1% and generate a sys-
tematic error of ±0.0014. The admixture of K+K−changes Pbr
by ∼ 1%. The uncertainty on such contamination is 15% and
produces a systematic error of ±0.0011 on Pbr. The finite-size
correction to the point-like approximation creates a maximum
0.8% variation of the simulated yield of Coulomb pairs and a
systematic error of ±0.0011 on Pbr. The influence of the final-
state strong interaction on the τ dependence of Pbris negligi-
ble [18, 22]. The trigger response efficiency was measured us-
ing minimum-bias events and accidental pairs from calibration
runs. The efficiency is high and quite uniform in the selected
QT, QLdomain and it drops by ∼ 2% per MeV/c at |QL| > 15
MeV/c. The simulated and experimental trigger efficiencies
agree to better than 0.5%, in the same |QL| range. This max-
imum deviation increases the breakup probability by ∼ 3% and
thus produces a systematic error of ±0.0004. Background hits
in the upstream spectrometer region, generated by beam and
secondary interactions in the target region, are the source of a
±0.0001 systematic error on Pbr. The effect of the lower purity
of the 94 µm Ni target foil compared to the 98 µm is an under-
estimation of Pbrby ∼ 1.1%. This corresponds to a systematic
error of ±0.0013 for the corresponding data set.
The dependence of Pbron the atom lifetime τ, its momen-
tum, and the target parameters has been extensively studied for
several target materials, both by exactly solving the system of
transport equations [14, 18] describing the A2πexcitation/de-
excitation, breakup and annihilation, and by simulating [15]
the A2πpropagation in the target foil. The precision reached
by these calculations is at the level of 1% [29], which is re-
flected in a ±0.0042 systematic error on Pbrfor a lifetime τ =
3.15 · 10−15s. The result of these calculations defines three
functions Pbr(τ, p), one for each of the combinations of target
thickness and beam momentum. The functions Pbr(τ, p) are
further convoluted with the experimental momentum spectra
of Coulomb pairs inside the seven (600 MeV/c wide) mom-
entum slices of the pionium laboratory momentum, from 2.6
to 6.8 GeV/c. This approach ensures that within each slice the
non-linear dependence of Pbr(τ) on the laboratory momentum
is negligible.
Coulomb pairs, which have a momentum spectrum similar
to that of atomic pairs, are taken from prompt pairs in the? Q
region away from the A2πsignal, after subtraction of the non-
Coulomb contribution. The values of the systematic errors are
summarized in Table 2.
Table 2: Summary of systematic errors on Pbr.
source
multiple scattering
momentum smearing
double-track resolution
K+K−and p¯ p
trigger simulation
background hits
target impurity
finite size
calculation of Pbr(τ)
Overall error
σ
±0.0077
±0.0026
±0.0014
±0.0011
±0.0004
±0.0001
±0.0013
±0.0011
±0.0042
±0.0094
6. Conclusions
Finally, the Pbrmeasurements, obtained for the different ex-
perimental conditionsand A2πmomentum ranges, and theirpre-
dicted Pbr(τ, p) values (see Fig.6), were used in a maximum
likelihood fit of the lifetime τ [30]. Both statistical and system-
atic uncertainties were taken into account in the maximization
procedure.
, s
τ
2 2.53 3.54
-15
10
×
br
0.50
P
0.38
0.40
0.42
0.44
0.46
0.48
m 24GeV/c
m 24GeV/c
µ
µ
Ni 98
Ni 94
Figure 6: Function Pbr(τ) corresponding to the dependence on pionium lifetime
of the breakup probability for different targets.
5

Page 6

Our final measurement of the ground-state A2πlifetime yi-
elds τ =
3.15+0.20
−0.19
Taking into account A2π → γγ and using formula (1), we
obtain the ππ scattering length difference
?
where the systematic error includes the 0.6% uncertainty in-
duced by the theoretical uncertainty on the correction δ.
In conclusion, we have measured the ground-state lifetime
of pionium with a total uncertainty of ∼ 9%. This represents the
most accurate lifetime measurement ever obtained and has al-
lowed us to determine the scattering length difference |a0− a2|
with a ∼ 4% accuracy. Our result is in agreement with val-
ues of the scattering lengths obtained from Ke4[8] and K3π[9]
decay measurements using a completely different experimental
approach.
?
???stat
+0.20
−0.18
???syst
?
× 10−15s.
|a0− a2| =
0.2533+0.0080
−0.0078
???stat
+0.0078
−0.0073
???syst
?
M−1
π+,
(8)
7. Acknowledgments
We are indebted to CERN for continuous support and
the PS team for the excellent performance of the accelerator.
We acknowledge the computing help from CESGA (Spain).
This work was funded by CERN, INFN (Italy), INCITE and
MICINN (Spain), IFIN-HH (Romania), the Ministry of Edu-
cation and Science and RFBR grant 01-02-17756-a (Russia),
the Grant-in-Aid from JSPS and Sentanken-grant from Kyoto
Sangyo University (Japan).
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