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arXiv:1006.4221v1 [hep-ex] 22 Jun 2010
Effectsoftransversityindeep-inelasticscatteringbypolarizedprotons
Hermes Collaboration
A. Airapetianℓ,o, N. Akopovz, Z. Akopove, E.C. Aschenauerf,1, W. Augustyniaky, R. Avakianz,
A. Avetissianz, E. Avetisyane, A. Bacchetta2, S. Belostotskir, N. Bianchij, H.P. Blokq,x, A. Borissove,
J. Bowlesm, I. Brodskyℓ, V. Bryzgalovs, J. Burnsm, M. Capiluppii, G.P. Capitanij, E. Cisbaniu,
G. Ciulloi, M. Contalbrigoi, P.F. Dalpiazi, W. Deconincke,3, R. De Leob, L. De Nardok,e,
E. De Sanctisj, M. Diefenthalern,h, P. Di Nezzaj, M. D¨ urenℓ, M. Ehrenfriedℓ, G. Elbakianz,
F. Ellinghausd,4, U. Elschenbroichk, R. Fabbrif, A. Fantonij, L. Felawkav, S. Frullaniu, D. Gabbertk,f,
G. Gapienkos, V. Gapienkos, F. Garibaldiu, V. Gharibyanz, F. Giordanoe,i, S. Gliskeo,
M. Golembiovskayaf, C. Hadjidakisj, M. Hartige,5, D. Haschj, G. Hillm, A. Hillenbrandf, M. Hoekm,
Y. Hollere, I. Hristovaf, Y. Imazuw, A. Ivanilovs, H.E. Jacksona, H.S. Jok, S. Joostenn,k, R. Kaiserm,
G. Karyanz, T. Kerim,ℓ, E. Kinneyd, A. Kisselevr, N. Kobayashiw, V. Korotkovs, V. Kozlovp,
P. Kravchenkor, L. Lagambab, R. Lambn, L. Lapik´ asq, I. Lehmannm, P. Lenisai, L.A. Linden-Levyn,
A. L´ opez Ruizk, W. Lorenzono, X.-G. Luf, X.-R. Luw, B.-Q. Mac, D. Mahonm, N.C.R. Makinsn,
S.I. Manaenkovr, L. Manfr´ eu, Y. Maoc, B. Marianskiy, A. Mart´ ınez de la Ossad, H. Marukyanz,
C.A. Millerv, Y. Miyachiw,6, A. Movsisyanz, M. Murraym, A. Mussgillere,h, E. Nappib, Y. Naryshkinr,
A. Nassh, M. Negodaevf, W.-D. Nowakf, L.L. Pappalardoi, R. Perez-Benitoℓ, N. Pickerth,
M. Raithelh, P.E. Reimera, A.R. Reolonj, C. Riedlf, K. Rithh, G. Rosnerm, A. Rostomyane,
J. Rubinn, D. Ryckboschk, Y. Salomatins, F. Sanftlw, A. Sch¨ afert, G. Schnellf,k, B. Seitzm,
T.-A. Shibataw, V. Shutovg, M. Stancarii, M. Staterai, E. Steffensh, J.J.M. Steijgerq, H. Stenzelℓ,
J. Stewartf, F. Stinzingh, S. Taroianz, A. Terkulovp, A. Trzcinskiy, M. Tytgatk, P.B. van der Natq,
Y. Van Haarlemk,7, C. Van Hulsek, D. Veretennikovr, V. Vikhrovr, I. Vilardib, C. Vogelh, S. Wangc,
S. Yaschenkof,h, H. Yec, Z. Yee, S. Yenv, W. Yuℓ, D. Zeilerh, B. Zihlmanne, P. Zupranskiy
aPhysics Division, Argonne National Laboratory, Argonne, Illinois 60439-4843, USA
bIstituto Nazionale di Fisica Nucleare, Sezione di Bari, 70124 Bari, Italy
cSchool of Physics, Peking University, Beijing 100871, China
dNuclear Physics Laboratory, University of Colorado, Boulder, Colorado 80309-0390, USA
eDESY, 22603 Hamburg, Germany
fDESY, 15738 Zeuthen, Germany
gJoint Institute for Nuclear Research, 141980 Dubna, Russia
hPhysikalisches Institut, Universit¨ at Erlangen-N¨ urnberg, 91058 Erlangen, Germany
iIstituto Nazionale di Fisica Nucleare, Sezione di Ferrara and Dipartimento di Fisica, Universit` a di Ferrara, 44100 Ferrara, Italy
jIstituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, 00044 Frascati, Italy
kDepartment of Subatomic and Radiation Physics, University of Gent, 9000 Gent, Belgium
ℓII. Physikalisches Institut, Universit¨ at Gießen, 35392 Gießen, Germany
mDepartment of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
nDepartment of Physics, University of Illinois, Urbana, Illinois 61801-3080, USA
oRandall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
pLebedev Physical Institute, 117924 Moscow, Russia
qNational Institute for Subatomic Physics (Nikhef), 1009 DB Amsterdam, The Netherlands
rPetersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300 Russia
sInstitute for High Energy Physics, Protvino, Moscow region, 142281 Russia
tInstitut f¨ ur Theoretische Physik, Universit¨ at Regensburg, 93040 Regensburg, Germany
uIstituto Nazionale di Fisica Nucleare, Sezione Roma 1, Gruppo Sanit` a and Physics Laboratory, Istituto Superiore di Sanit` a, 00161 Roma,
Italy
vTRIUMF, Vancouver, British Columbia V6T 2A3, Canada
Preprint submitted to Physics Letters B 23 June 2010
Page 2
wDepartment of Physics, Tokyo Institute of Technology, Tokyo 152, Japan
xDepartment of Physics and Astronomy, VU University, 1081 HV Amsterdam, The Netherlands
yAndrzej Soltan Institute for Nuclear Studies, 00-689 Warsaw, Poland
zYerevan Physics Institute, 375036 Yerevan, Armenia
Abstract
Single-spin asymmetries for pions and charged kaons are measured in semi-inclusive deep-inelastic scattering of positrons and
electrons off a transversely nuclear-polarized hydrogen target. The dependence of the cross section on the azimuthal angles of the
target polarization (φS) and the produced hadron (φ) is found to have a substantial sin(φ + φS) modulation for the production of
π+, π−and K+. This Fourier component can be interpreted in terms of non-zero transversity distribution functions and non-zero
favored and disfavored Collins fragmentation functions with opposite sign. For π0and K−production the amplitude of this Fourier
component is consistent with zero.
Key words: semi-inclusive DIS, single-spin asymmetries, polarized structure functions, transversity, Collins function
PACS: 13.60.-r, 13.88.+e, 14.20.Dh, 14.65.-q
Most of our knowledge about the internal structure of
nucleons comes from deep-inelastic scattering (DIS) exper-
iments. At the energies of current fixed-target experiments,
the dominant process in DIS of charged leptons by nucle-
ons is the exchange of a single space-like photon with a
squared four-momentum −Q2much larger than the typi-
cal hadronic scale, usually set to be the squared mass M2
of the nucleon. The cross section for this lepton scattering
process can be decomposed in a model-independent way in
terms of structure functions. Factorization theorems based
on quantum chromodynamics (QCD) provide an interpre-
tation of these structure functions in terms of parton dis-
tribution functions (PDFs), which ultimately reveal crucial
aspects of the dynamics of confined quarks and gluons.
Polarized inclusive DIS on nucleons, lN → l′X (where X
denotes the undetected final state), neglecting weak boson
exchange can be described by four structure functions (see,
e.g, Refs. [1,2]). They can be interpreted using collinear
factorization theorems (see, e.g, Ref. [3,4] and references
therein). Three of the structure functions contain contri-
butions at leading order in an expansion in M/Q (twist
expansion). These contributions include the leading-twist
(twist-2) quark distribution functions fq
(for simplicity, the dependence on Q2has been dropped).
The variable x represents the fraction of the nucleon mo-
1(x) and gq
1(x) [2]
1Now at: Brookhaven National Laboratory, Upton, NY 11772-5000,
USA
2Address: Dipartimento di Fisica Nucleare e Teorica, Universit` a di
Pavia and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia,
via Bassi 6, 27100 Pavia, Italy
3Now at: Massachusetts Institute of Technology, Cambridge, MA
02139, USA
4Now at: Institut f¨ ur Physik, Universit¨ at Mainz, 55128 Mainz, Ger-
many
5Now at: Institut f¨ ur Kernphysik, Universit¨ at Frankfurt a.M., 60438
Frankfurt a.M., Germany
6Now at: Department of Physics, Yamagata University, Kojirakawa-
cho 1-4-12, Yamagata 990-8560, Japan
7Now at: Carnegie Mellon University, Pittsburgh, PA 15213, USA
mentum carried by the parton in a frame where the nucleon
moves infinitely fast in the direction opposite to the probe.
The hard probe defines a specific direction (q in Fig. 1),
usually denoted as longitudinal, and the transverse plane
perpendicular to it. In a parton-model picture, fq
scribes the number density of quarks of flavor q in a fast-
moving nucleon without regard to their polarization. The
PDF gq
1(x) describes the difference between the number
densities of quarks with helicity equal or opposite to that of
the nucleon if the nucleon is longitudinally polarized. The
integrals over x of fq
and axial charge of the nucleon, respectively.
There is a third leading-twist PDF, the function hq
called the transversity distribution (see Ref. [5] for a re-
view on the subject). Its integral over x is related to the
tensor charge of the nucleon [6]. It can be interpreted as
the difference between the densities of quarks with trans-
verse(Pauli-Lubanski) polarization parallel or anti-parallel
to the transverse polarization of the nucleon [7]. In contrast
to fq
1(x), due to helicity conservation, there ex-
ist no gluon analog of hq
Therefore, hq
1(x) cannot mix with gluons under QCD evo-
lution.
The transversity distribution does not appear in any
structure function in inclusive DIS because it is odd under
inversion of the quark chirality. It must be combined with
another chiral-odd nonperturbative partner to appear in
a cross section for hard processes involving only QED or
QCD, as such interactions preserve chirality. For this rea-
son, in spite of decades of inclusive DIS studies, no exper-
imental information on the transversity distribution was
available until recently. In lepton-nucleon scattering, the
transversity distribution can be accessed experimentally
only in semi-inclusive DIS with a transversely polarized
target, where it can appear in combination with, e.g., the
1(x) de-
1(x) and gq
1(x) are related to the vector
1(x)8,
1(x) and gq
1(x) in the case of spin-1
2targets.
8In literature, the distribution functions fq
are also denoted as q(x), ∆q(x), and δq(x), respectively.
1(x), gq
1(x), and hq
1(x)
2
Page 3
chiral-odd Collins fragmentation function [8]. This Letter
presents a measurement of the associated signal.
In semi-inclusive DIS, lN → l′hX, where a hadron h is
detected in the final state in coincidence with the scattered
lepton,the crosssectiondependson,amongothervariables,
the hadron transverse momentum and its azimuthal orien-
tation with respect to the lepton scattering plane about the
virtual-photon direction. If the target is polarized and the
polarization of the final state is not measured, the semi-
inclusive DIS cross section can be decomposed in terms of
18 semi-inclusive structure functions (see, e.g, Ref. [9]).
When the transverse momentum of the produced hadron
is small compared to the hard scale Q, semi-inclusive DIS
can be described using transverse-momentum-dependent
factorization [10,11]. The semi-inclusive structure func-
tions can be interpreted in terms of convolutions involv-
ing transverse-momentum-dependent parton distribution
and fragmentation functions [12]. The former encode in-
formation about the distribution of partons in a three-
dimensional momentum space, and the latter describe the
hadronization process in a three-dimensional momentum
space. Hence, the study of semi-inclusive DIS not only
opens the way to the measurement of transversity, but
also probes new dimensions of the structure of the nu-
cleon and of the hadronization process, thus offering new
perspectives to our understanding of QCD.
When performing a twist expansion, eight semi-inclusive
structure functions contain contributions at leading order,
related to the eight leading-twist transverse-momentum-
dependent PDFs [9]. One of these structure functions is
interpreted as the convolution of the transversity distri-
bution function hq
T) (not integrated over the trans-
verse momentum) and the Collins fragmentation function
H⊥q→h
1
(z,k2
to the correlation between the transverse polarization of
the fragmenting quark and kT [8]. Here, z in the target-
rest frame denotes the fraction of the virtual photon energy
carried by the produced hadron h, pTdenotes the trans-
verse momentum of the quark with respect to the parent
nucleon direction, and kTdenotes the transverse momen-
tum of the fragmenting quark with respect to the direc-
tion of the produced hadron. This structure function mani-
fests itself as a sin(φ+φS) modulation in the semi-inclusive
DIS cross section with a transversely polarized target. Its
Fourier amplitude, henceforth named Collins amplitude, is
denoted as 2?sin(φ+φS)?
UT, where φ (φS) represents the
azimuthal angle of the hadron momentum (of the trans-
verse component of the target spin) with respect to the
lepton scattering plane and about the virtual-photon direc-
tion, in accordance with the Trento Conventions [13] (see
Fig. 1). The subscript UT denotes unpolarized beam and
target polarization transverse with respect to the virtual-
photon direction. Other azimuthal modulations have dif-
ferent origins and involve other distribution and fragmen-
tation functions. They can be disentangled through their
specific dependence on the two azimuthal angles φ and φS
1(x,p2
T), which acts as a polarimeter being sensitive
h
k′
k
ST
Ph
Ph⊥
q
φ
φS
Fig. 1. The definition of the azimuthal angles φ and φS relative to
the lepton scattering plane.
(see, e.g, Refs. [9,14,15]). Results on, e.g., the sin(φ − φS)
modulation of this data set were reported in Ref. [16].
Non-zero Collins amplitudes were previously published
for charged pions from a hydrogen target [17], based on
a small subset (about 10%) of the data reported here,
consisting of about 8.76 million DIS events. Collins am-
plitudes for unidentified hadrons were measured on pro-
tons [18] and for pions and kaons, albeit consistent with
zero, on deuterons [19–21] by the Compass collaboration.
In Refs. [22,23] the first joint extraction of the transversity
distribution function and the Collins fragmentation func-
tion was carried out, under simplifying assumptions, using
preliminaryresultsfromasubsetofthepresentdataincom-
bination with the deuteron data from the Compass collab-
oration[19–21]and e+e−annihilationdata from the Belle
collaboration [24,25]. Recently, significant amplitudes for
two-hadron production in semi-inclusive DIS, which con-
stitutes an independent process to probe transversity, were
measured at the Hermes experiment [26] providing ad-
ditional evidence for a non-zero transversity distribution
function.
In this Letter, in addition to much improved statistical
precision on the charged pion results, the Collins ampli-
tudes for identified K+, K−, and π0are presented for the
first time for a proton target. The data reported here were
recorded during the 2002–2005 running period of the Her-
mes experiment with a transversely nuclear-polarized hy-
drogen target stored in an open-ended target cell internal
to the 27.6GeV Hera polarized positron/electron storage
ring at Desy. The two beam helicity states are almost per-
fectly balanced in the present data, and no measurable con-
tribution arising from the residual net beam polarization
to the amplitudes extracted was observed. The target cell
was fed by an atomic-beam source [27], which uses Stern–
Gerlach separation combined with radio-frequency transi-
tions of hyperfine states. The target cell was immersed in
a transversely oriented magnetic holding field. The effects
of this magnetic field were taken into account in the recon-
struction of the vertex positions and the scattering angles
of charged particles. The nuclear polarization of the atoms
was flipped at 1–3 minutes time intervals, while both the
polarization and the atomic fraction inside the target cell
were continuously measured [28]. The average magnitude
of the proton-polarization component perpendicular to the
beam direction was 0.725±0.053.Scattered leptons and co-
3
Page 4
incident hadrons were detected by the Hermes spectrome-
ter[29].Leptonswereidentifiedwithanefficiencyexceeding
98% and a hadron contamination of less than 1%. Charged
hadrons detected within the momentum range 2–15 GeV
were identified using a dual-radiator RICH by means of
a hadron-identification algorithm that takes into account
the event topology. The detection of the neutral pions is
based on the measurements of photon pairs in the electro-
magnetic calorimeter. These were accepted only if Eγ> 1
GeV and 0.10 GeV < Mγγ < 0.17 GeV, where Eγ and
Mγγdenote the photon energy and the photon-pair invari-
ant mass, respectively. The combinatorial background was
evaluated in the side-bands 0.06 GeV < Mγγ< 0.10 GeV
and 0.17 GeV < Mγγ< 0.21 GeV.
Events were selected according to the kinematic require-
ments W2> 10GeV2, 0.023 < x < 0.4, 0.1 < y < 0.95,
and Q2> 1GeV2, where W2≡ (P + q)2, Q2≡ −q2≡
−(k − k′)2, y ≡ (P · q)/(P · k), and x ≡ Q2/(2P · q) are
the conventional DIS kinematic variables with P, k and k′
representing the four-momenta of the initial state target
proton, incident and outgoing lepton, respectively. In or-
der to minimize target fragmentation effects as well as to
exclude kinematic regions where contributions from exclu-
sive channels become sizable, coincident hadrons were only
included if 0.2 < z < 0.7, where z ≡ (P · Ph)/(P · q) and
Phis the four-momentum of the produced hadron.
The cross section for semi-inclusive production of
hadrons using an unpolarized lepton beam and a target
polarized transversely with respect to the virtual pho-
ton direction includes a polarization-averaged part and
a polarization-dependent part. The former contains two
cosine modulations and the latter contains a total of five
sine modulations [9,14,15]:
dσh(φ,φS) = dσh
UU
?
1 +
2
?
n=1
2?cos(nφ)?h
UUcos(nφ)
+ |ST|
5
?
i=1
2?sinΦi?h
UTsinΦi
?
,
(1)
whereSTdenotesthetransverse(with respecttothevirtual
photon direction) component of the target-protonpolariza-
tion vector and Φ = [φ+φS,φ−φS,φS,2φ−φS,3φ−φS].
The dependence of the cross section and of the azimuthal
amplitudes on x, y, z, and Ph⊥has been suppressed. The
subscript UU denotes unpolarized beam and unpolarized
target, and dσh
UUrepresents the cross section averagedover
φ and over beam and target polarizations.
The Collins amplitude 2?sin(φ+φS)?
preted in the parton model as [14]
h
UTcan be inter-
2?sin(φ+φS)?
h
UT(x,y,z,Ph⊥)
C?−
=
(1 − y)
(1 − y + y2/2)
Ph⊥·kT
|Ph⊥| Mhhq
C?fq
1(x,p2
T)Dq→h
T)H⊥q→h
1
(z,k2
(z,k2
T)?
1(x,p2
1
T)?
,
(2)
where Ph⊥≡ |Ph−(Ph·q)q
of the produced hadron, and Dq→h
averagedquark fragmentation function. The notation C de-
notes the convolution [9]
|q|2
| is the transverse momentum
1
is the polarization-
C?...?= x
?
q
e2
q
?
d2pTd2kTδ(2)
?
pT− kT−Ph⊥
z
??...?,
(3)
where the sum runs over the quark flavors q, and eqare the
quark electric charges in units of the elementary charge.
Expressions similar to Eq. (2) hold for the other azimuthal
modulations in Eq. (1) [9]. Note that, as the quark fla-
vors enter the cross section with the square of their electric
charge, the u-quarks provide the dominant contribution to
the production of, e.g., π+/K+for proton targets (com-
monly denoted as “u-quark dominance”).
Experimentally, the Fourier amplitudes of the yields for
opposite transverse target-spin states were extracted using
a maximum-likelihood fit alternately binned in x, z, and
Ph⊥, but unbinned in φ and φS. This is equivalent to a
Fourier decomposition of the asymmetry
Ah
UT(φ,φS) ≡
1
|ST|
dσh(φ,φS) − dσh(φ,φS+ π)
dσh(φ,φS) + dσh(φ,φS+ π), (4)
for perfectly balanced target polarization and in the limit
of very small φ and φS bins. The asymmetry amplitudes
for neutral pions were corrected for the effects of the
combinatorial background evaluated in the side-bands
of the photon-pair invariant mass spectrum. In addition
to the five sine terms in Eq. (1), the fit also included a
sin(2φ + φS)term,arisingfromthesmallbutnon-vanishing
target-polarization component that is longitudinal to the
virtual-photon direction when the target is polarized per-
pendicular to the beam direction [30]. In order to avoid
cross contamination arising from the limited spectrometer
acceptance, the six amplitudes were extracted simultane-
ously. The fit did not include the cos(nφ) modulations of
Eq. (1). As a consequence, one cannot expect a priori that
the Fourier amplitudes extracted are identical to those of
Eq. (1). However, in the following they will be considered
to be equivalent because inclusion in the fit of estimates
[31] for the cos(φ) and cos(2φ) amplitudes of the unpo-
larized cross section resulted in negligible effects on the
extracted amplitudes.
The extracted Collins amplitudes are shown in Fig. 2 as
a function of x, z, or Ph⊥. They are positive for π+and K+,
negative for π−, and consistent with zero for π0and K−
at a confidence level of at least 95% based on a Student’s
t-test including the systematic uncertainties. Note that the
x, z, and Ph⊥dependences in Fig. 2 are three projections
of the same data and are thus fully correlated.
A scale uncertainty of 7.3% on the extracted amplitudes,
notshowninFig.2,arisesfromtheaccuracyinthemeasure-
ment of the target polarization. Effects from acceptance,
smearingdue to detector resolution, higher order QED pro-
cesses and hadron identification procedure based on the
RICH are not corrected for in the data. Rather, the size
4
Page 5
0
0.05
2 〈sin(φ+φS)〉UT
π
π+
-0.1
0
π0
-0.05
0
π-
0
0.1
2 〈sin(φ+φS)〉UT
K
K+
-0.1
0
0.1
10
-1
x
K-
0.40.6
z
0.51
Ph⊥ [GeV]
Fig. 2. Collins amplitudes for pions and charged kaons as a function
of x, z, or Ph⊥. The systematic uncertainty is given as a band at the
bottom of each panel. In addition there is a 7.3% scale uncertainty
from the accuracy in the measurement of the target polarization.
of all these effects was estimated using a Pythia6 Monte
Carlo simulation [32] tuned to Hermes hadron multiplicity
data and exclusive vector-meson production data [33–35]
and including a full simulation of the Hermes spectrom-
eter. A polarization state was assigned to each generated
event using a model that reflects the (transversetarget) po-
larization dependent part of the cross section (see Eq. (1)).
This model was obtained through a fully differential (i.e
differential in the four relevant kinematic variables x, Q2,
z, and Ph⊥) 2ndorder polynomial fit [36,37] of real data.
The asymmetry amplitudes, extracted from the simulated
data by means of the same analysis procedure used for the
real data, were then compared with the model, evaluated
in each bin at the mean kinematics, to obtain an estimate
of the global impact of the effects listed above. The result
was included in the systematic uncertainty and constitutes
the largest contribution. It accounts for effects of nonlin-
earity of the model, as it includes the difference in each bin
between the average model and the model evaluated at the
average kinematics. The impact on the extracted ampli-
tudes of contributions [30] from the non-vanishing longitu-
dinal target-spin component was estimated based on previ-
ous measurements of single-spin asymmetries for longitu-
dinally polarized protons [38,39]. The resulting relatively
small effect was included in the systematic uncertainty.
A Monte Carlo simulation was used to estimate the frac-
tion of pions and kaons originating from the decay of ex-
clusively produced vector mesons, updating previous re-
sults reported in Ref. [40]. For charged pions, this fraction
is dominated by the decay of ρ0mesons and, in the kine-
matic region covered by the present analysis, is of the or-
der of 6-7%. The vector-meson fractions for neutral pions
and charged kaons are of the order of 2-3%. The z and Ph⊥
dependences of the fraction of pions and kaons stemming
from the decay of exclusively produced vector mesons are
shown in [16] for the two kinematic regions Q2< 4 GeV2
and Q2> 4 GeV2(the x dependence was not reported due
to the strong correlation between x and Q2in the data).
They exhibit maxima at high z and low Ph⊥. These con-
tributions are considered part of the signal and were not
used to correct the pion and kaon yields analysed in the
present work. However, this information can be useful for
the interpretation of the results.
In general, the non-vanishing amplitudes shown in Fig. 2
increase in magnitude with x. This is consistent with the
expectation that transversitymainly receives contributions
fromthe valencequarks.A nonnegligiblecontributionfrom
the sea quarks cannot be excluded, but is not expected to
be large due to the fact that transversity cannot be gener-
ated in gluon splitting. The amplitudes are also found to
increase with z, in qualitative agreement with the results
for the Collins fragmentation function from the Belle ex-
periment [24,25]. The results of Fig. 2 also show that the
π−amplitude is of opposite sign to that of π+and larger in
magnitude. A possible explanation is dominance of u fla-
vor among struck quarks, in conjunction with a substantial
magnitude with opposite signof the disfavoredCollinsfrag-
mentation function describing, e.g, the fragmentation of u
quarks into π−mesons, as already suggested in Ref. [17].
Opposite signs for the favored and disfavored Collins frag-
mentation functions are not in contradiction to the Belle
results [24,25] and are supported by the combined fits re-
ported in [22]. They can be understood in light of the
string model of fragmentation [41] (and also of the Sch¨ afer–
Teryaev sum rule [42]). If a favored pion is created at the
stringendbythe firstbreak,adisfavoredpionfromthe next
break is likely to inherit transverse momentum in the op-
posite direction. The string fragmentation model, the base
of the successful and widespread Jetset generator [43],
predicts such a Ph⊥ strong negative correlation between
favored and disfavored pions.
Under the assumption of isospin symmetry, the fragmen-
tation functions for neutral pions are assumed equal to the
average of those for charged pions. Factorization of the
semi-inclusive cross section results in the following isospin
relation for the Collins amplitudes for pions:
?sin(φ + φS)?π+
UT+ C?sin(φ + φS)?π−
− (1 + C)?sin(φ + φS)?π0
UT
UT= 0 ,
(5)
5
Page 6
whereC denotestheratioofthepolarization-averagedcross
sections for semi-inclusive charged-pion production (C ≡
dσπ−
UU). The extracted pion amplitudes areconsistent
with Eq. (5).
The Fourier amplitudes for K+are found to be larger
than those for π+at a confidence level of at least 90%
(99%) based on a Student’s t-test including (not including)
the systematic uncertainties. On the other hand, the am-
plitudes for π−and K−exhibit a very different behavior,
the former being significantly negative, while the latter is
consistent with zero in the whole kinematic range. Here,
however, one should keep in mind that, in contrast to π−,
a K−has no valence quarks in common with the target
proton and sea quark transversity is expected to be small.
In interpreting the various features of the extracted am-
plitudes, and in particular the differences between those of
pions and kaons, the largely unknown role of several con-
curring factors should be considered. Among these are, e.g,
(i) the role of sea quarks in conjunction with possibly large
fragmentationfunctions; (ii) the variouscontributionsfrom
decay of semi-inclusively produced vector-mesons which,
based on a Monte Carlo simulation, are mainly ρ and ω
mesons producing pions (up to 37% and 10%, respectively),
and K∗and φ mesons producing kaons (up to 41% and
3.5%, respectively); (iii) the kT dependences of the frag-
mentation functions, which can be different for different
hadrons and can have an effect on the extracted amplitudes
through the convolution of Eqs. (2) and (3).
Up to this point, the discussion is based on Eq. (2) and is
thus valid up to twist-3. It is therefore interestingto investi-
gate the possible presence of twist-4 contributions. To this
end, the Q2dependence of the extracted amplitudes was
studied in more detail. To minimize effects arising from the
strong correlation between x and Q2in the data, the events
in each x bin were divided into two sub-bins, with Q2below
and above the mean value ?Q2(xi)? for the original bin (see
Fig. 3). However, due to the limited statistics it was not
possible to significantly constrain the twist-4 contributions
by fitting the data in Fig. 3 with various Q2dependences
(including the appropriate y-dependent prefactor of Eq. 2).
In summary, non-zero Collins amplitudes in semi-
inclusive DIS were measured for charged pions and posi-
tively charged kaons. These amplitudes can be interpreted
as due to the transverse polarization of quarks in the tar-
get, revealed by its influence on the fragmentation of the
struck quark. They thus support the existence of non-zero
transversity distribution functions in the proton and also
the existence of non-zero Collins fragmentation functions.
In particular, by comparing the Collins amplitudes of π+
and π−, it appears that fragmentation that is disfavored
in terms of quark flavor has an unexpected importance,
and enters with a sign opposite to that of the favored one.
In contrast to the expectation that the π+and the K+
Collins amplitudes should have similar magnitudes, based
on the common u-quark dominance, the amplitude for K+
is found to be significantly larger than that for π+. This
UU/dσπ+
-0.1
0
0.1
Q2 < 〈Q2(xi)〉
Q2 > 〈Q2(xi)〉
2 〈sin(φ+φS)〉UT
π+
1
10
10
-1
xx
〈Q2〉 [GeV2]
〈Q2〉 [GeV2]
π-
10
-1
x x
Fig. 3. Collins amplitudes for charged pions as functions of x. The
Q2range for each i-bin in x was divided into the two regions above
and below the average Q2of that bin (?Q2(xi)?). The bottom panels
show the x-dependence of the average Q2.
could be an indication of, e.g, an unanticipated behavior of
the Collins fragmentation functions possibly in conjunction
with a non negligible role of the sea quarks in the nucleon.
Collins amplitudes consistent with zero are measured for
π0and K−. These data should considerably improve the
precision of transversity extractions from future global fits.
We gratefully acknowledge the Desy management for its
supportandthe staffatDesyandthecollaboratinginstitu-
tionsfortheirsignificanteffort.Thisworkwassupportedby
theMinistryofEconomyandtheMinistryofEducationand
Scienceof Armenia;the FWO-FlandersandIWT, Belgium;
the Natural Sciences and Engineering Research Council of
Canada;theNationalNaturalScienceFoundationofChina;
the Alexander von Humboldt Stiftung, the German Bun-
desministerium f¨ ur Bildung und Forschung (BMBF), and
the Deutsche Forschungsgemeinschaft (DFG); the Italian
Istituto Nazionale di Fisica Nucleare (INFN); the MEXT,
JSPS, and G-COE of Japan; the Dutch Foundation for
Fundamenteel Onderzoek der Materie (FOM); the Russian
Academy of Science and the Russian Federal Agency for
Science and Innovations; the U.K. Engineering and Physi-
cal Sciences Research Council, the Science and Technology
Facilities Council, and the Scottish Universities Physics Al-
liance; the U.S. Department of Energy (DOE) and the Na-
tional Science Foundation (NSF); and the European Com-
munity Research Infrastructure Integrating Activity under
the FP7 ”Study of strongly interacting matter (Hadron-
Physics2, Grant Agreement number 227431)”.
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and simulationof the
DESY-THESIS-
transversityandtransverse-
SumrulesfortheT-odd
7
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