Observation of transverse polarization asymmetries of charged pion pairs in e+e- annihilation near √s = 10.58 GeV.

Abstract

The interference fragmentation function translates the fragmentation of a quark with a transverse projection of the spin into an azimuthal asymmetry of two final-state hadrons. In e(+)e(-) annihilation the product of two interference fragmentation functions is measured. We report nonzero asymmetries for pairs of charge-ordered π(+)π(-) pairs, which indicate a significant interference fragmentation function in this channel. The results are obtained from a 672 fb(-1) data sample that contains 711 × 10(6) π(+)π(-) pairs and was collected at and near the Υ(4S) resonance, with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider.

Full text

Page 1

arXiv:1104.2425v1 [hep-ex] 13 Apr 2011
Belle DRAFT 2011-5 KEK Preprint 2010-53
Observation of the interference fragmentation function for charged pion pairs in e+e−
annihilation near√s = 10.58 GeV.
A. Vossen,9,50R. Seidl,34I. Adachi,8H. Aihara,43T. Aushev,20,13V. Balagura,13W. Bartel,3M. Bischofberger,24
A. Bondar,1,31M. Braˇ cko,22,14T. E. Browder,7M.-C. Chang,2A. Chen,25P. Chen,27B. G. Cheon,6K. Cho,17
Y. Choi,38S. Eidelman,1,31M. Feindt,16V. Gaur,40N. Gabyshev,1,31A. Garmash,1,31B. Golob,21,14
M. Grosse Perdekamp,9,34J. Haba,8K. Hayasaka,23Y. Horii,42Y. Hoshi,41W.-S. Hou,27H. J. Hyun,19
K. Inami,23A. Ishikawa,35M. Iwabuchi,49Y. Iwasaki,8T. Iwashita,24N. J. Joshi,40H. Kichimi,8H. O. Kim,19
M. J. Kim,19B. R. Ko,18T. Kumita,45J. S. Lange,4M. J. Lee,37S.-H. Lee,18M. Leitgab,9,34Y. Li,47C. Liu,36
D. Liventsev,13R. Louvot,20S. McOnie,39H. Miyata,29Y. Miyazaki,23R. Mizuk,13G. B. Mohanty,40E. Nakano,32
S. Nishida,8O. Nitoh,46A. Ogawa,34T. Ohshima,23S. Okuno,15G. Pakhlova,13H. Park,19H. K. Park,19
M. Petriˇ c,14L. E. Piilonen,47S. Ryu,37H. Sahoo,7Y. Sakai,8O. Schneider,20C. Schwanda,11O. Seon,23
M. Shapkin,12V. Shebalin,1,31T.-A. Shibata,33,44J.-G. Shiu,27P. Smerkol,14Y.-S. Sohn,49E. Solovieva,13
S. Staniˇ c,30M. Stariˇ c,14M. Sumihama,33,5T. Sumiyoshi,45Y. Teramoto,32M. Uchida,33,44S. Uehara,8
T. Uglov,13Y. Unno,6S. Uno,8G. Varner,7A. Vinokurova,1,31C. H. Wang,26M.-Z. Wang,27P. Wang,10
Y. Watanabe,15E. Won,18B. D. Yabsley,39Y. Yamashita,28V. Zhilich,1,31P. Zhou,48and V. Zhulanov1,31
(The Belle Collaboration)
1Budker Institute of Nuclear Physics, Novosibirsk
2Department of Physics, Fu Jen Catholic University, Taipei
3Deutsches Elektronen–Synchrotron, Hamburg
4Justus-Liebig-Universit¨ at Gießen, Gießen
5Gifu University, Gifu
6Hanyang University, Seoul
7University of Hawaii, Honolulu, Hawaii 96822
8High Energy Accelerator Research Organization (KEK), Tsukuba
9University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
10Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
11Institute of High Energy Physics, Vienna
12Institute of High Energy Physics, Protvino
13Institute for Theoretical and Experimental Physics, Moscow
14J. Stefan Institute, Ljubljana
15Kanagawa University, Yokohama
16Institut f¨ ur Experimentelle Kernphysik, Karlsruher Institut f¨ ur Technologie, Karlsruhe
17Korea Institute of Science and Technology Information, Daejeon
18Korea University, Seoul
19Kyungpook National University, Taegu
20´Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), Lausanne
21Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana
22University of Maribor, Maribor
23Nagoya University, Nagoya
24Nara Women’s University, Nara
25National Central University, Chung-li
26National United University, Miao Li
27Department of Physics, National Taiwan University, Taipei
28Nippon Dental University, Niigata
29Niigata University, Niigata
30University of Nova Gorica, Nova Gorica
31Novosibirsk State University, Novosibirsk
32Osaka City University, Osaka
33Research Center for Nuclear Physics, Osaka
34RIKEN BNL Research Center, Upton, New York 11973
35Saga University, Saga
36University of Science and Technology of China, Hefei
37Seoul National University, Seoul
38Sungkyunkwan University, Suwon

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39School of Physics, University of Sydney, NSW 2006
40Tata Institute of Fundamental Research, Mumbai
41Tohoku Gakuin University, Tagajo
42Tohoku University, Sendai
43Department of Physics, University of Tokyo, Tokyo
44Tokyo Institute of Technology, Tokyo
45Tokyo Metropolitan University, Tokyo
46Tokyo University of Agriculture and Technology, Tokyo
47CNP, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
48Wayne State University, Detroit, Michigan 48202
49Yonsei University, Seoul
50Indiana University, Bloomington, Indiana 47408
The interference fragmentation function translates the fragmentation of a quark with a transverse
projection of the spin into an azimuthal asymmetry of two final-state hadrons. In e+e−annihilation
the product of two interference fragmentation functions is measured. We report nonzero asymme-
tries for pairs of charge-ordered π+π−pairs, which indicate a significant interference fragmentation
function in this channel. The results are obtained from a 672 fb−1data sample that contains
711 × 106π+π−pairs and was collected at and near the Υ(4S) resonance, with the Belle detector
at the KEKB asymmetric-energy e+e−collider.
PACS numbers: 13.88.+e,13.66.-a,14.65.-q,14.20.-c
This paper addresses the extraction of the interference
fragmentation function (IFF) in e+e−annihilation near
10.58 GeV center-of-mass energy.
gested by Collins [1], is sensitive to the transverse polar-
ization of the fragmenting quark and thus can be used
as a quark polarimeter. The previous measurement of
the Collins fragmentation function [2, 3] with the Belle
detector allowed the first global analysis of transversity
[4] to be performed using data from HERMES [5] and
COMPASS [6]. Knowledge of the IFF will allow com-
plementary access to transversity and a comparison to
the Lattice QCD calculations [7]. Moreover, by detect-
ing a second hadron, the sensitivity to the quark spin
survives integration over transverse momenta. Thus, un-
like the Collins effect, collinear models can be used for
factorization and the QCD evolution of the fragmenta-
tion function is known [8]. Like the Collins function, the
IFF is chiral-odd and can be used to extract transversity
from asymmetries measured in polarized semi-inclusive
deep inelastic scattering (SIDIS) [9, 10] or proton-proton
scattering [11].
The quantity sensitive to the transverse polarization of
quarks is a cosine modulation of the azimuthal angle φ
of the plane spanned by the momenta of the two hadrons
h1, h2around the fragmenting quark direction with re-
spect to the transverse quark spin. However, while the
quark spin is unknown in unpolarized e+e−scattering,
The IFF, first sug-
the two primordial quarks appear in two back-to-back
jets. The kinematics of the process is shown in Fig. 1.
Thus, instead of measuring the azimuthal angle between
the spin vector and the vector R = Ph1−Ph2describing
the two-hadron-plane, one measures an azimuthal cor-
relation of two hadron pairs detected in opposite hemi-
spheres α = 1,2. The angles φ1 and φ2 are defined in
the center-of-mass system (CMS) between Rα and the
event plane spanned by the electron-positron axis ˆ z and
the thrust axis ˆ n [12]. They can be expressed in terms
of measured quantities as:
φ{1,2} = sgn[ˆ n · (ˆ z × ˆ n × (ˆ n × R1,2)}]
× arccos
?ˆ z × ˆ n
|ˆ z × ˆ n|·
ˆ n × R1,2
|ˆ n × R1,2|
?
.
(1)
As in the Collins analysis, a second method can be ap-
plied, which does not directly depend on the thrust axis
to calculate the angles, but defines the reference axis via
the momentum of the second hadron pair P2and corre-
sponding angles φ1Rand φ2R. Using either set of angles,
φ1,φ2 or φ1R,φ2R, one can obtain a cos(φ1(R)+ φ2(R))
modulation proportional to the interference fragmenta-
tion functions normalized by the corresponding unpolar-
ized di-hadron fragmentation functions. The amplitude
of this modulation in e+e−annihilation is according to
Boer [13]:

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a12R(z1,z1,m2
1,m2
2) ∝
1
2
sin2θ
1 + cos2θ

?
q,q
e2
qz2
1z2
2H< )q
1 (z1,m2
1) H< )q
1 (z2,m2
2)


×

?
q,q
e2
qz2
1z2
2Dq
1(z1,m2
1) Dq
1(z2,m2
2)


−1
,
(2)
and a similar formula for the cos(φ1+ φ2) modulation
amplitude a12. The interference fragmentation function
H< ),q
1
of a quark q ( and charge eq) , and its polarization-
independent counterpart Dq
1, depend on the fractional
energy zα
α and on its invariant mass mα. The CMS energy is
denoted by√s and the polar angle θ is defined between
the lepton axis and the reference axis in the CMS. As the
polar angular dependence is a clear indication of initial
transverse quark polarization, its asymmetry dependence
was studied.
CMS
= 2Eα/√s of the hadron pair in hemisphere
Ph1
R1Ph1+ Ph2
π − φ1
Ph3
φ2− π
Thrust axis ˆ n
e−
e+
Ph2
Ph4
FIG. 1: Azimuthal angle definitions for φ1 and φ2 as defined
relative to the thrust axis in the CMS.
Collins and Ladinsky[14] used the linear sigma model
to make the first predictions for π-π correlations. An-
other approach makes use of a partial wave analysis to
arrive at predictions for H∢
1, which receives essential con-
tributions from the interference of meson pairs (pions and
kaons) in relative S- and P-wave states [15, 16]. A strong
dependence on the invariant mass of the hadron pair is
predicted. Predictions for spin effects that can be ob-
served at the B-factories can be found in papers by Jaffe,
Jin and Tang [17] and Radici, Jakob and Bianconi [18],
with the latter being recently extended to e+e−anni-
hilation [19] at Belle energies. Jaffe and collaborators
estimate the final-state interactions of the meson pairs
from meson-meson phase shift data in [20], where it is
observed that S- and P-wave production channels inter-
fere strongly in the mass region around the ρ, the K∗and
the φ meson resonances, and give rise to a sign change of
the IFF.
The present analysis is based on a data sample of 672
fb−1, collected with the Belle detector at the KEKB
asymmetric-energy e+e−(3.5 on 8 GeV) collider [21]
operating at the Υ(4S) resonance and 60 MeV below.
The Belle detector is a large-solid-angle magnetic spec-
trometer that consists of a silicon vertex detector (SVD),
a 50-layer central drift chamber (CDC), an array of
aerogel threshold Cherenkov counters (ACC), a barrel-
like arrangement of time-of-flight scintillation counters
(TOF), and an electromagnetic calorimeter (ECL) com-
prised of CsI(Tl) crystals located inside a superconduct-
ing solenoid coil that provides a 1.5 T magnetic field.
An iron flux-return yoke located outside of the coil is in-
strumented to detect K0
Lmesons and to identify muons
(KLM). The detector is described in detail elsewhere [22].
Two inner detector configurations were used. A 2.0 cm
radius beampipe and a 3-layer silicon vertex detector
were used for the first sample of 157 fb−1, while a 1.5 cm
radius beampipe, a 4-layer silicon detector and a small-
cell inner drift chamber were used to record the remaining
516 fb−1[23].
The most important selection criterion is the event
shape variable thrust, T, the maximum of which defines
the thrust axis ˆ n :
T
max
=
?
h|PCMS
?
h
· ˆ n|
|
h|PCMS
h
. (3)
The sum extends over all detected particles, and PCMS
denotes their momenta in the CMS. The deviation of
the reconstructed thrust axis from the generated quark-
antiquark pair axis for light quarks is 135 mrad with an
RMS of 90 mrad, as obtained from the simulated sam-
ple of events. This value is compatible with those cited
earlier in the Collins analysis [2]. Since the two pairs
of hadrons should appear in a two-jet topology, events
are selected with a thrust value larger than 0.8. The
contamination from B decays in this event sample is
around 2% [3]. As the hadron pairs are sampled only
in the barrel region of the detector, one has to ensure
that for those pairs all possible azimuthal angles around
the thrust axis lie also within this acceptance. For this
purpose only events with a thrust axis pointing into the
h

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central detector are considered with the z component of
the thrust unit vector |ˆ nz| < 0.75. In order to obtain a
reliable thrust axis and to reduce the contribution from
e+e−→ τ+τ−events, the reconstructed energy of an
event is required to be above 7 GeV. Tracks are required
to lie in the central part of the detector acceptance corre-
sponding to −0.6 < cos(θLAB) < 0.9, where θLABis the
polar angle in the laboratory frame. This corresponds
to a nearly symmetric track selection in the CMS frame,
with the polar angle range −0.79 < cos(θCMS) < 0.74.
All tracks are required to originate from a region around
the reconstructed interaction point, which is defined by
the requirements dr < 2 cm and |dz| < 4 cm, where dr
and dz are the distance of closest approach to the in-
teraction point in the plane perpendicular to the beam
direction and along the direction of the beams.
tracks are required to have a minimal fractional energy
z =2Eh
√s> 0.1. The fractional energy zαof each hadron
pair is thus at least 0.2. The effect of the minimal hadron
energy requirement on the decay angular distribution will
be discussed later. Pions were selected among the recon-
structed charged tracks by vetoing identified muons, elec-
trons and protons, and requiring a kaon - pion particle
identification likelihood to be larger than 0.7 [24]. With
these requirements the rate of fake pions in the selected
sample is below 5%. The overall fraction of misiden-
tified pions, obtained from simulated data, is added as
a relative systematic uncertainty of the final measured
asymmetries and is correlated between the bins defined
below.
In addition to θLAB, other polar angles in this anal-
ysis are the polar angle of the thrust axis in the CMS
θt = acos(ˆ nz) and the decay angles of a hadron pair
in their respective center-of-mass systems θ1d,2ddefined
with respect to the first (i.e., positive) hadron.
lowest-order interference fragmentation term has a sinθd
distribution.
Any combination of two charged pions with opposite
charge is combined in a pair if the two hadrons are in the
same hemisphere. For the analysis we select two pion
pairs belonging to opposite hemispheres.
the requirement of an opening angle relative to the thrust
axis cosψ = |(ˆ n·Ph)|/|Ph| > 0.8 selects only tracks that
have at least a certain fraction of their momentum along
the thrust axis. After these selection criteria, the total
data sample contains 711×106π+π−pairs (1.58 di-pion
pairs per event). Throughout this paper the order of the
pion pairs used for calculating R1,2 is always π+π−in
both hemispheres. The data is binned in either 8 × 8
m1,m2 bins between 0.25 GeV/c2and 2 GeV/c2or in
9×9 z1,z2bins between 0.2 and 1.0. The first method of
assessing the interference fragmentation function is based
on measuring a cos(φ1+ φ2) modulation of two hadron
pair yields (N(φ1+φ2)) on top of the flat distribution due
to the unpolarized part of the fragmentation functions.
The unpolarized part is given by the average bin content
All
The
In addition,
?N12?. The normalized distribution is then defined as
R12:=N(φ1+ φ2)
?N12?
. (4)
The two-pion pair yields N(φ1(R)+ φ2(R)) are obtained
for each kinematic bin in 16 equidistant bins of the az-
imuthal angles.The normalized azimuthal di-hadron
yields, R12(R)can be parameterized as:
R12(R)= a12(R)cos(φ1(R)+ φ2(R)) + b12(R)+
c12(R)sin(φ1(R)+ φ2(R)) + d12(R)cos2(φ1(R)+ φ2(R))(5)
where the parameter b12(R)should be unity due to the
normalization. The parameter a12(R)is the amplitude
proportional to the interference fragmentation functions.
The normalized distribution is fit to equation (5) with
a12(R), b12(R), c12(R)and d12(R)as free parameters. The
reduced χ2values of the individual fits over all run ranges
and bins are well described by a χ2distribution with a
mean value close to unity.
The PYTHIA event generator [25] used in this analysis
does not contain the spin effects related to the IFF, and
thus all asymmetries are expected to vanish. A check
can be performed for the kinematic effects that could
mimic the spin-induced asymmetries. For this purpose
light quark (uds) events and charm quark events have
been generated, which were tracked through the detec-
tor in a GEANT [26] simulation and then fully recon-
structed. Asymmetries were evaluated at the generated
4-momentum level, as well as for reconstructed events.
The results of this analysis are summarized in Table I,
where effects of a finite detector acceptance are clearly
visible. They can be significantly reduced via the open-
ing angle selection. The sum of the absolute value of
the reconstructed asymmetries and their statistical un-
certainties in the simulated sample were assigned as bin-
by-bin systematic uncertainties of the data asymmetries.
They represent the largest systematic uncertainty, which
are up to several % in the lowest statistics bins.
TABLE I: MC results in % averaged over all z bins for gener-
ated uds events (uds gen), within the geometrical acceptance
(uds gen. acc.) as well as reconstructed uds and charm events.
Samplez1,z2-Asymmetries
?a12?
No opening angle cut
−0.089 ± 0.008 −0.108 ± 0.008
uds gen. acc. −0.488 ± 0.011 −0.490 ± 0.011
uds rec.−0.401 ± 0.007 −0.428 ± 0.007
charm rec.−0.446 ± 0.041 −0.388 ± 0.044
With opening angle cut of 0.8
uds gen.−0.038 ± 0.013 −0.035 ± 0.013
uds gen. acc. −0.112 ± 0.016 −0.113 ± 0.016
uds rec.0.020 ± 0.010
charm rec.0.006 ± 0.040
?a12R?
uds gen.
0.006 ± 0.010
0.027 ± 0.040

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5
Mixed events:
tion between the hadron pairs on the quark and the anti-
quark side of an event, taking one hadron pair of another
event should destroy this correlation and the asymme-
tries obtained for such a mixed-event data sample should
vanish unless detector effects introduce artificial asym-
metries. Two ways of extracting event-mixed asymme-
tries were applied: using a hadron pair of a first event in
combination with a pair of a second event, and taking the
axis information either from the first or the second event.
The values from data are (−0.019±0.017)% for a12and
(−0.012 ± 0.017)% for a12R. These values are included
as absolute systematic uncertainties in the results.
Higher harmonics:
The higher-order terms in Eq. (5)
are needed to reproduce the azimuthal variations well.
Generally these different harmonics are orthogonal and
should not interfere with each other, but a limited ac-
ceptance can introduce other asymmetries. The small
differences in a12(R)of up to 1% between either fitting
the first two terms or all are assigned as a bin-by-bin
systematic uncertainty.
Weighted MC asymmetries:
were introduced into the MC generator for hadron pairs
around the quark-antiquark axis and then reconstructed
to test the validity of the reconstruction method. The a12
asymmetries, which directly depend on using the thrust
axis as a proxy for the quark-antiquark axis, are recon-
structed to (92 ± 1)% of the generated value, and the
a12R asymmetries to (99 ± 1)%. Corresponding correc-
tion factors are applied to the measured asymmetries and
the uncertainties were assigned as a systematic error.
Process contributions:
The thrust selection alone al-
ready reduces the background from Υ(4S) decays to a
negligible level. The charm contribution, however, has
nearly the same thrust distribution as that for light
quarks. On the other hand, since pions from charmed
mesons are the product of a decay chain, the fractional
energies fall off more rapidly than for light quarks. There-
fore the relative charm contribution also falls off from
nearly 50% at lowest z bins to a few % at high z. The
charm contribution in the mass bins first falls as can
be seen in Fig. 2 but then increases again for invariant
masses around 1 GeV/c2.
There is a small contribution from τ pairs rising to
several % at high z. When analyzing a τ enhanced data
sample without the minimal energy requirement one finds
asymmetries of a12 = (−1.31 ± 0.13)% averaged over
the whole kinematic range. This asymmetry can be ex-
plained by the sizeable residual contribution from con-
tinuum events in the τ enhanced data. The relative con-
tributions from τ pair events multiplied by their average
asymmetry are added as systematic error, which is, how-
ever, negligibly small.
Correlation studies:
In order to exclude possible ef-
fects of correlations between different kinematic and az-
imuthal bins, MC studies have been performed, which
As the asymmetry requires a correla-
Artificial asymmetries
bin number
0 10 2030 4050 60
relative process contributions
0
0.2
0.4
0.6
0.8
1
0.25
< m
0.40
<
1
0.40
< m
0.50
<
1
0.50
< m
0.62
<
1
0.62
< m
0.77
<
1
0.77
< m
0.90
<
1
0.90
< m
1.10
<
1
1.10
< m
1.50
<
1
1.50
< m
2.00
<
1
±
B
charm
ττ
B
uds
0
B
,
0
FIG. 2: Relative contributions of various processes for pion
pairs as a function of the 8×8 m1,m2bin number. The closed
circles denote light quark-antiquark pair events, inverted tri-
angles – charm events, triangles – charged B meson pairs,
open circles – neutral B meson pairs and squares – τ pairs.
TABLE II: Integrated asymmetries for the two reconstruction
methods and their average kinematics.
?z1?,?z2?
?m1?,?m2?
?sin2θt/(1 + cos2θt)?
?sinθ1d?,?sinθ2d?
?cosθ1d?,?cosθ2d?
a12
a12R
0.4313
0.6186 GeV/c2
0.7636
0.9246
0.0013
−0.0196 ± 0.0002(stat.) ± 0.0022(syst.)
−0.0179 ± 0.0002(stat.) ± 0.0021(syst.)
did not find any such effects.
Inverted thrust selection:
was also analyzed to test whether the azimuthal correla-
tion of the two hadron pairs decreases. On average the
asymmetries were 45% smaller.
Results:
The results can be seen in Fig. 3 as a func-
tion of the fractional energies and in Fig. 4 as a func-
tion of the di-pion invariant masses. One sees a large
asymmetry rising with increasing fractional energy and
invariant mass with an indication of leveling off at the
highest invariant masses. At higher masses or fractional
energies an asymmetry of up to 10% corresponds to in-
terference fragmentation functions of more than 30% the
size of the corresponding unpolarized two-hadron frag-
mentation function. The results averaged over all kine-
matic bins are summarized in Table II. The a12Rresults
show similar dependencies and magnitudes.
Summary:
Large azimuthal asymmetries for two
π+π−pairs in opposite hemispheres were extracted from
a 672 fb−1data sample. The asymmetries monotonically
decrease as a function of z1,2and m1,2and no sign change
is observed in contrast to [17]. The interference fragmen-
tation function can be extracted from those asymmetries
The inverse thrust selection

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12
a
-0.14

12
a
-0.04
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
< 0.28
2
0.20 < z < 0.35
2
0.28 < z < 0.42
2
0.35 < z
-0.14

12
a
-0.04
-0.12
-0.1
-0.08
-0.06
-0.02
0
0.02
0.04
< 0.50
2
0.42 < z < 0.57
2
0.50 < z < 0.65
2
0.57 < z
1
z
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.14
-0.12
-0.1
-0.08
-0.06
-0.02
0
0.02
0.04
< 0.72
2
0.65 < z
1
z
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
< 0.82
2
0.72 < z
1
z
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
< 1.00
2
0.82 < z
FIG. 3: a12 modulations for the 9×9 z1,z2 binning as a func-
tion of z1for the z2bins. The shaded (green) areas correspond
to the systematic uncertainties.

12
-0.02
a
-0.16

12
a
-0.04
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
0
0.02
2
< 0.40 GeV/c
2
< m
2
0.25 GeV/c
2
< 0.50 GeV/c
2
< m
2
0.40 GeV/c
2
< 0.62 GeV/c
2
< m
2
0.50 GeV/c
-0.16

12
a
-0.04
-0.14
-0.12
-0.1
-0.08
-0.06
-0.02
0
0.02
2
< 0.77 GeV/c
2
< m
2
0.62 GeV/c
2
< 0.90 GeV/c
2
< m
2
0.77 GeV/c
2
< 1.10 GeV/c
2
< m
2
0.90 GeV/c
]
2
[GeV/c
1
m
0.4 0.6 0.81 1.2 1.4 1.6
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.02
0
0.02
2
< 1.50 GeV/c
2
< m
2
1.10 GeV/c
]
2
[GeV/c
1
m
0.4 0.6 0.811.2 1.4 1.6
2
< 2.00 GeV/c
2
< m
2
1.50 GeV/c
FIG. 4: a12 modulations for the 8 × 8 m1,m2 binning as a
function of m2 for the m1 bins. The shaded (green) areas
correspond to the systematic uncertainties.
and used in a global fit to the SIDIS data to obtain the
transversity distribution function.
We thank the KEKB group for excellent operation of
the accelerator, the KEK cryogenics group for efficient
solenoid operations, and the KEK computer group and
the NII for valuable computing and SINET3 network sup-
port. We acknowledge support from MEXT, JSPS and
Nagoya’s TLPRC (Japan); ARC and DIISR (Australia);
NSFC (China); DST (India); MEST, KOSEF, KRF (Ko-
rea); MNiSW (Poland); MES and RFAAE (Russia);
ARRS (Slovenia); SNSF (Switzerland); NSC and MOE
(Taiwan); NSF and DOE (USA).
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