Observation of the naive-T-odd Sivers effect in deep-inelastic scattering.

Abstract

Azimuthal single-spin asymmetries of leptoproduced pions and charged kaons were measured on a transversely polarized hydrogen target. Evidence for a naive-T-odd, transverse-momentum-dependent parton distribution function is deduced from nonvanishing Sivers effects for pi(+), pi(0), and K(+/-), as well as in the difference of the pi(+) and pi(-) cross sections.

Full text

Page 1

arXiv:0906.3918v2 [hep-ex] 18 Dec 2009
Observation of the Naive-T-odd Sivers Effect in Deep-Inelastic Scattering
A. Airapetian,12,15N. Akopov,26Z. Akopov,26E.C. Aschenauer,6W. Augustyniak,25A. Avetissian,26E. Avetisyan,5
A. Bacchetta,5B. Ball,15N. Bianchi,10H.P. Blok,17,24H. B¨ ottcher,6C. Bonomo,9A. Borissov,5V. Bryzgalov,19
J. Burns,13M. Capiluppi,9G.P. Capitani,10E. Cisbani,21G. Ciullo,9M. Contalbrigo,9P.F. Dalpiaz,9
W. Deconinck,5,15R. De Leo,2L. De Nardo,15,5E. De Sanctis,10M. Diefenthaler,14,8P. Di Nezza,10J. Dreschler,17
M. D¨ uren,12M. Ehrenfried,12G. Elbakian,26F. Ellinghaus,4U. Elschenbroich,11R. Fabbri,6A. Fantoni,10
L. Felawka,22S. Frullani,21D. Gabbert,6G. Gapienko,19V. Gapienko,19F. Garibaldi,21V. Gharibyan,26
F. Giordano,5,9S. Gliske,15C. Hadjidakis,10M. Hartig,5D. Hasch,10G. Hill,13A. Hillenbrand,6M. Hoek,13
Y. Holler,5I. Hristova,6Y. Imazu,23A. Ivanilov,19H.E. Jackson,1H.S. Jo,11S. Joosten,14,11R. Kaiser,13
T. Keri,13,12E. Kinney,4A. Kisselev,18V. Korotkov,19V. Kozlov,16P. Kravchenko,18L. Lagamba,2R. Lamb,14
L. Lapik´ as,17I. Lehmann,13P. Lenisa,9L.A. Linden-Levy,14A. L´ opez Ruiz,11W. Lorenzon,15X.-G. Lu,6
X.-R. Lu,23B.-Q. Ma,3D. Mahon,13N.C.R. Makins,14S.I. Manaenkov,18L. Manfr´ e,21Y. Mao,3B. Marianski,25
A. Martinez de la Ossa,4H. Marukyan,26C.A. Miller,22Y. Miyachi,23A. Movsisyan,26M. Murray,13
A. Mussgiller,5,8E. Nappi,2Y. Naryshkin,18A. Nass,8M. Negodaev,6W.-D. Nowak,6L.L. Pappalardo,9
R. Perez-Benito,12P.E. Reimer,1A.R. Reolon,10C. Riedl,6K. Rith,8G. Rosner,13A. Rostomyan,5J. Rubin,14
D. Ryckbosch,11Y. Salomatin,19F. Sanftl,20A. Sch¨ afer,20G. Schnell,6,11K.P. Sch¨ uler,5B. Seitz,13
T.-A. Shibata,23V. Shutov,7M. Stancari,9M. Statera,9J.J.M. Steijger,17H. Stenzel,12J. Stewart,6F. Stinzing,8
S. Taroian,26A. Terkulov,16A. Trzcinski,25M. Tytgat,11A. Vandenbroucke,11P.B. van der Nat,17
Y. Van Haarlem,11C. Van Hulse,11M. Varanda,5D. Veretennikov,18V. Vikhrov,18I. Vilardi,2C. Vogel,8
S. Wang,3S. Yaschenko,6,8H. Ye,3Z. Ye,5S. Yen,22W. Yu,12D. Zeiler,8B. Zihlmann,5and P. Zupranski25
(The Hermes Collaboration)
1Physics Division, Argonne National Laboratory, Argonne, Illinois 60439-4843, USA
2Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70124 Bari, Italy
3School of Physics, Peking University, Beijing 100871, China
4Nuclear Physics Laboratory, University of Colorado, Boulder, Colorado 80309-0390, USA
5DESY, 22603 Hamburg, Germany
6DESY, 15738 Zeuthen, Germany
7Joint Institute for Nuclear Research, 141980 Dubna, Russia
8Physikalisches Institut, Universit¨ at Erlangen-N¨ urnberg, 91058 Erlangen, Germany
9Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara and
Dipartimento di Fisica, Universit` a di Ferrara, 44100 Ferrara, Italy
10Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, 00044 Frascati, Italy
11Department of Subatomic and Radiation Physics, University of Gent, 9000 Gent, Belgium
12Physikalisches Institut, Universit¨ at Gießen, 35392 Gießen, Germany
13Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
14Department of Physics, University of Illinois, Urbana, Illinois 61801-3080, USA
15Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
16Lebedev Physical Institute, 117924 Moscow, Russia
17National Institute for Subatomic Physics (Nikhef), 1009 DB Amsterdam, The Netherlands
18Petersburg Nuclear Physics Institute, Gatchina, Leningrad region 188300, Russia
19Institute for High Energy Physics, Protvino, Moscow region 142281, Russia
20Institut f¨ ur Theoretische Physik, Universit¨ at Regensburg, 93040 Regensburg, Germany
21Istituto Nazionale di Fisica Nucleare, Sezione Roma 1, Gruppo Sanit` a
and Physics Laboratory, Istituto Superiore di Sanit` a, 00161 Roma, Italy
22TRIUMF, Vancouver, British Columbia V6T 2A3, Canada
23Department of Physics, Tokyo Institute of Technology, Tokyo 152, Japan
24Department of Physics and Astronomy, Vrije Universiteit, 1081 HV Amsterdam, The Netherlands
25Andrzej Soltan Institute for Nuclear Studies, 00-689 Warsaw, Poland
26Yerevan Physics Institute, 375036 Yerevan, Armenia
(Dated: December 18, 2009)
Azimuthal single-spin asymmetries of lepto-produced pions and charged kaons were measured
on a transversely polarized hydrogen target. Evidence for a naive-T-odd, transverse-momentum-
dependent parton distribution function is deduced from non-vanishing Sivers effects for π+, π0, and
K±, as well as in the difference of the π+and π−cross sections.
PACS numbers: 13.60.-r, 13.88.+e, 14.20.Dh, 14.65.-q

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The ongoing experimental effort in spin-dependent
high-energy scattering and attendant theoretical work
continue to indicate that the spins of the quarks and glu-
ons are not sufficient to explain the nucleon spin [1]. The
investigation of the only remaining contribution, that of
orbital angular momentum of the constituents, is clearly
essential. Transverse-momentum-dependent parton dis-
tribution functions are recognized as a tool to study spin-
orbit correlations, hence providing experimental observ-
ables for studying orbital angular momentum. One par-
ticular example is the Sivers function f⊥
the correlation between the momentum direction of the
struck quark and the spin of its parent nucleon. This
correlation is commonly defined as the Sivers effect. A
non-vanishing f⊥
1Tcontributes to, e.g., single-spin asym-
metries (SSAs) in semi-inclusive deep-inelastic scattering
(DIS) off transversely polarized protons, ep↑→ e′hX,
where h is a hadron detected in coincidence with the
scattered lepton e′.
For a long time, transverse SSAs had been assumed
to be negligible in hard scattering processes. They are
odd under naive time reversal, i.e., time reversal of three-
momenta and angular momenta, and thus require inter-
ference of amplitudes with different helicities and phases.
In QED and perturbative QCD, these ingredients are
suppressed [3, 4]. Therefore, in semi-inclusive DIS they
must be ascribed to the non-perturbative parts in the
cross section, i.e., to specific parton distribution and
fragmentation functions, commonly categorized as being
naive-T-odd. The idea of a naive-T-odd quark distribu-
tion function goes back to an interpretation [2] of large
left-right asymmetries observed in pion production in the
collision of unpolarized with transversely polarized nucle-
ons [5]. It was argued that such asymmetries could be
attributed to a left-right asymmetry in the distribution
of unpolarized quarks in transversely polarized nucleons,
i.e., an asymmetry that exists before the pion is formed
in the fragmentation process, and that does not vanish at
high energies. A decade after an initial proof [6] that this
distribution function, now termed the Sivers function,
must vanish because of time-reversal invariance of QCD,
it was realized through the pioneering work in Ref. [7]
and subsequently in Refs. [8, 9, 10] that this proof ap-
plies only to transverse-momentum-integrated distribu-
tion functions.A gauge link, previously neglected in
the definition of gauge-invariant distribution functions,
invalidates the original proof for the case of transverse-
momentum-dependent distribution functions. The gauge
link provides the phase for the interference (required for
naive-T-oddness), and can be interpreted as an interac-
tion of the struck quark with the color field of the target
remnant [11].
The inclusion of the gauge link has profound conse-
quences on factorization proofs and on the concept of
universality, which are of fundamental relevance for high-
energy hadronic physics. A direct QCD prediction is a
1T[2], describing
Sivers effect in the Drell–Yan process that has the oppo-
site sign compared to the one in semi-inclusive DIS [8].
For hadron production in proton-proton collisions the sit-
uation is more intricate [12], leading to a violation of
standard factorization and universality, even for the case
of unpolarized collisions [13]. Therefore, the study of the
Sivers effect in semi-inclusive DIS and other processes
is of utmost importance for our understanding of high-
energy scattering involving hadrons.
The Sivers effect has been related to the orbital motion
of quarks inside a transversely polarized nucleon since the
seminal work in Ref. [2]. In the calculation of Ref. [7], it
became clear that orbital angular momentum of quarks
is needed for a non-vanishing Sivers effect as it arises
through overlap integrals of wave-function components
with different orbital angular momenta.
quantitative relation has yet been found between f⊥
the orbital angular momentum of quarks. One faces a
similar quandary with the anomalous magnetic moment
κ of the nucleon: it also requires wave function compo-
nents with non-vanishing quark orbital angular momen-
tum without constraining the net orbital angular momen-
tum [14]. Indeed, f⊥
1Tinvolves overlap integrals between
the same wave function components that also appear in
the expressions for κ as well as for the total angular mo-
mentum in the Ji relation [15] for the nucleon-spin de-
composition [7, 14].
An interesting link between κ and f⊥
in Ref. [16]: the sign of the quark-flavor contribution to
κ determines the sign of f⊥
1Tfor that quark flavor. If the
final-state interactions are attractive, as one would as-
sume for the confining color force, a positive flavor con-
tribution to κ leads to a negative f⊥
angle definitions follow the Trento Conventions [17].)
In semi-inclusive DIS, f⊥
1Tleads to SSAs in the dis-
tribution of hadrons in the azimuthal angle about the
virtual-photon direction. In general, azimuthal SSAs
provide important information not only about the Sivers
function but also about other distribution and fragmen-
tation functions.For example, transversity [18], de-
scribing the distribution of transversely polarized quarks
in transversely polarized nucleons, combined with the
naive-T-odd Collins fragmentation function [6], also leads
to SSAs. The keys to extracting different combinations
of the various distribution and fragmentation functions
are their different dependences on the two azimuthal an-
gles φ and φS of the hadron momentum Phand of the
transverse component ST of the target-proton spin, re-
spectively, about the virtual-photon direction (cf. [17]).
The Sivers effect manifests itself as a sin(φ − φS) modu-
lation in the azimuthal distribution [19].
In this Letter clear evidence for a non-vanishing Sivers
function is reported. The sin(φ − φS) modulations in
semi-inclusive DIS are measured for pions and charged
kaons, as well as in the difference between the π+and
π−cross sections, providing sensitivity to f⊥
However, no
1Tand
1Twas suggested
1T. (The sign and
1Tfor both

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3
valence and sea quarks.
The data reported here were recorded during the 2002–
2005 running period of the Hermes experiment using
a transversely nuclear-polarized hydrogen gas target in-
ternal to the 27.6GeV Hera lepton (e+or e−) storage
ring at Desy. The open-ended target cell was fed by an
atomic-beam source [20] based on Stern–Gerlach sepa-
ration combined with radio-frequency transitions of hy-
perfine states. The nuclear spin direction was flipped
at 1–3min time intervals, while both nuclear polariza-
tion and the atomic fraction inside the target cell were
continuously measured [21]. The average magnitude of
the proton-polarization component perpendicular to the
lepton-beam direction was 0.725 ± 0.053.
Scattered leptons and coincident hadrons were de-
tected by the Hermes spectrometer [22]. Leptons were
identified with an efficiency exceeding 98% and a hadron
contamination of less than 1%. Charged hadrons with
momentum 2GeV < |Ph| < 15GeV were identified
using a dual-radiator ring-imagingˇCerenkov detector
(RICH) [23]. For this a hadron-identification algorithm
was employed that takes into account the topology of
the whole event, in contrast to the track-level algorithm
in previous analyses [24].
ject to the requirements Q2> 1GeV2, W2> 10GeV2,
0.1 < y < 0.95, and 0.023 < x < 0.4, where Q2≡ −q2≡
−(k − k′)2, W2≡ (P + q)2, y ≡ (P · q)/(P · k), and
x ≡ Q2/(2P · q). Here, P, k, and k′represent the four-
momenta of the target proton, the incident lepton, and
the scattered lepton, respectively. Coincident hadrons
were accepted if 0.2 < z < 0.7, where z ≡ (P ·Ph)/(P ·q).
The cross section for semi-inclusive production of
hadrons using an unpolarized lepton beam on a trans-
versely polarized target can be written as [19, 25, 26]
Events were selected sub-
σ(φ,φS) = σUU{1 + 2?cosφ?UUcosφ + 2?cos2φ?UUcos2φ
+ |ST|[2?sin(φ−φS)?UTsin(φ − φS) + ...]}, (1)
where
polarization-dependent
only the sin(φ − φS) modulation (the Sivers term), is
written out explicitly. Here, the subscript UT denotes
unpolarized beam and transverse target polarization
(with respect to the virtual-photon direction), while σUU
represents the φ-independent part of the polarization-
independent cross section. The sin(φ − φS) amplitude
can be interpreted in the quark-parton model as [19]
fivesinemodulations
part,
contribute
but, for
tothe
convenience,
2?sin(φ − φS)?UT= −
?
qe2
?
qf⊥,q
qe2
1T(x,p2
qfq
T) ⊗WDq
T) ⊗ Dq
1(z,K2
1(z,K2
T)
1(x,p2
T)
,
(2)
where the sums run over the quark flavors, the eqare the
quark charges, and f1and D1are the spin-independent
quark distribution and fragmentation functions, respec-
tively. The symbol ⊗ (⊗W) represents a (weighted) con-
volution integral over intrinsic and fragmentation trans-
verse momenta pTand KT, respectively.
The amplitudes of the five sine modulations in Eq. (1)
were extracted simultaneously to avoid cross contamina-
tion. For this a maximum-likelihood fit was used [27],
with the data alternately binned in x, z, and Ph⊥ ≡
|Ph−(Ph·q)q
|q|2
|, but unbinned in φ and φS. A sixth term,
arising from the small but non-vanishing target-spin com-
ponent that is longitudinal to the virtual-photon direc-
tion when the target is polarized perpendicular to the
beam direction [28], was also included in the fit.
A scale uncertainty of 7.3% on the extracted Sivers
amplitudes arises from the accuracy of the target-
polarization determination.
timates [29] for the cosφ and cos2φ amplitudes of the
unpolarized cross section had negligible effects on the
amplitudes extracted. Possible contributions [28] to the
amplitudes from the non-vanishing longitudinal target-
spin component were estimated based on measurements
of SSAs on longitudinally polarized protons [30, 31] and
included in the systematic uncertainty. Effects from the
hadron identification using the RICH, the geometric ac-
ceptance, smearing due to detector resolution, and radia-
tive effects are not corrected for in the data. Rather, the
size of all these effects was estimated using a simulation
tuned to the data, which involved a fully differential poly-
nomial fit to the measured azimuthal amplitudes [32].
The result was included in the systematic uncertainty
and constitutes the largest contribution.
Based on a Pythia6 Monte Carlo simulation [33]
tuned to Hermes data, the fraction of charged pions
(kaons) stemming from the decay of exclusive vector-
meson channels was estimated to be about 6–7% (2–3%).
Among the contributions of all the vector mesons to the
pion samples, that of the ρ0is dominant. A different
observable, for which the contributions from exclusive ρ0
mesons cancels, is the pion-difference asymmetry
Inclusion in the fit of es-
Aπ+−π−
UT
(φ,φS) ≡
1
|ST|
(σπ+
(σπ+
U↑−σπ−
U↑−σπ−
U↑) − (σπ+
U↑) + (σπ+
U↓−σπ−
U↓−σπ−
U↓)
U↓), (3)
the SSA in the difference in the π+and π−cross sec-
tions for opposite target-spin states ↑,↓.
this asymmetry helps to isolate the valence-quark Sivers
functions:under some assumptions, such as charge-
conjugation and isospin symmetry among pion fragmen-
tation functions, one can deduce from Eq. (2) that this
SSA stems mainly from the difference (f⊥,dv
in the Sivers functions for valence down and up quarks.
The resulting Sivers amplitudes for pions, charged
kaons, and for the pion-difference asymmetry are shown
in Fig. 1 as functions of x, z, or Ph⊥. They are positive
and increase with increasing z, except for π−, for which
they are consistent with zero. In the case of π+, K+,
and the pion-difference asymmetry, the data suggest a
saturation of the amplitudes for Ph⊥? 0.4 GeV and are
consistent with the predicted linear decrease in the limit
of Ph⊥going to zero.
In addition,
1T
− 4f⊥,uv
1T
)

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4
0
0.05
0.1
2 〈sin(φ-φS)〉UT
π+
-0.1
0
0.1
2 〈sin(φ-φS)〉UT
π0
-0.05
0
0.05
2 〈sin(φ-φS)〉UT
π-
0
0.1
0.2
2 〈sin(φ-φS)〉UT
K+
-0.1
0
0.1
2 〈sin(φ-φS)〉UT
K-
0
0.25
10
-1
x
2 〈sin(φ-φS)〉UT
π+ − π-
0.40.6
z
0.51
Ph⊥ [GeV]
FIG. 1: Sivers amplitudes for pions, charged kaons, and the
pion-difference asymmetry (as denoted in the panels) as func-
tions 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 target-polarization measurement.
In order to further examine the influence of exclusive
vector-meson decay and other possible
contributions, several studies were performed. Raising
the lower limit of Q2to 4 GeV2eliminates a large part
of the vector-meson contribution. Because of strong cor-
relations between x and Q2in the data, this is presented
only for the z and Ph⊥dependences. No influence of the
vector-meson fraction on the asymmetries is visible as
shown in Fig. 2. For the x dependence shown in Fig. 3,
each bin was divided into two Q2regions below and above
the corresponding average Q2(?Q2(xi)?) for that x bin.
While the averages of the kinematics integrated over in
those x bins do not differ significantly, the ?Q2? values
for the two Q2ranges change by a factor of about 1.7.
The asymmetries do not change by as much as would
1
Q2-suppressed
-0.1
0
0.1
0.2
2 〈sin(φ-φS)〉UT
π+
Q2 < 4 GeV2
Q2 > 4 GeV2
0
0.1
0.40.6
z
Nh
VM / Nh
0.5
Ph⊥ [GeV]
1
K+
0.40.6
z
0.5
Ph⊥ [GeV]
1
FIG. 2: Sivers amplitudes for π+(left) and K+(right) as
functions of z or Ph⊥, compared for two different ranges in Q2
(high-Q2points are slightly shifted horizontally). The corre-
sponding fraction of pions and kaons stemming from exclusive
vector mesons, extracted from a Monte Carlo simulation, is
provided in the bottom panels.
0
0.1
Q2 < 〈Q2(xi)〉
Q2 > 〈Q2(xi)〉
2 〈sin(φ-φS)〉UT
π+
1
10
10
-1
xx
〈Q2〉 [GeV2]
〈Q2〉 [GeV2]
K+
10
-1
x x
FIG. 3: Sivers amplitudes for π+(left) and K+(right) as
functions of x. The Q2range for each bin was divided into
the two regions above and below ?Q2(xi)? of that bin. In the
bottom the average Q2values are given for the two Q2ranges.
have been expected for a sizable
bution, e.g., the one from longitudinal photons to the
spin-(in)dependent cross section. However, while the π+
asymmetries for the two Q2regions are fully consistent,
there is a hint of systematically smaller K+asymmetries
in the large-Q2region.
An interesting facet of the data is the difference in the
π+and K+amplitudes shown in Fig. 4. On the basis
of u-quark dominance, i.e., the dominant contribution to
π+and K+production from scattering off u-quarks, one
might naively expect that the π+and K+amplitudes
should be similar.The difference in the π+and K+
amplitudes may thus point to a significant role of other
quark flavors, e.g., sea quarks. Strictly speaking, even in
the case of scattering solely off u-quarks, the fragmenta-
tion function D1, contained in both the numerator and
denominator in Eq. (2), does not cancel in general as it
appears in convolution integrals. This can lead not only
1
Q2-suppressed contri-

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5
-0.05
0
0.05
0.1
0.15
10
-1
x
2 〈sin(φ-φS)〉K
UT - 2 〈sin(φ-φS)〉π
UT
+
+
10
-1
x
Q2 < 〈Q2(xi)〉
10
-1
x
Q2 > 〈Q2(xi)〉
FIG. 4: Difference of Sivers amplitudes for K+and π+as
functions of x for all Q2(left), and separated into ”low-” and
”high-Q2” regions as done for Fig. 3.
to additional z-dependences, but also to a difference in
size of the Sivers amplitude for π+and K+. Higher-twist
effects in kaon production might also contribute to the
difference observed: in the low-Q2region, where higher-
twist should be more pronounced, the π+and K+am-
plitudes disagree at the confidence level of at least 90%,
based on a Student’s t-test, while being statistically con-
sistent in the high-Q2region.
As scattering off u-quarks dominates in these data, the
positive Sivers amplitudes for π+and K±suggest a large
and negative Sivers function for u-quarks. This is sup-
ported by the positive amplitudes of the difference asym-
metry, which is dominated by the contribution from va-
lence u-quarks. The vanishing amplitudes for π−require
cancelation effects, e.g., from a d-quark Sivers function
opposite in sign to the u-quark Sivers function. In combi-
nation with deuteron data from the Compass collabora-
tion [34], a large positive d-quark Sivers function can be
deduced [35]. These fits have yet to be updated with the
final results presented here, as well as with preliminary
proton data from Compass [36].
In summary, non-zero Sivers amplitudes in semi-
inclusive DIS were measured for production of π+, π0,
and K±, as well as for the pion-difference asymme-
try. They can be explained by the non-vanishing naive-
T-odd, transverse-momentum-dependent Sivers distribu-
tion function.This function also plays an important
role in transverse single-spin asymmetries in pp collisions,
and is linked to orbital angular momentum of quarks in-
side the nucleon. Although no quantitative conclusion
about their orbital angular momentum can be inferred,
the Sivers function provides important constraints on the
nucleon wave function and thus indirectly on the total
quark orbital angular momentum. For instance, in the
approach of Ref. [11], the measured positive Sivers asym-
metries for π+and K+mesons correspond to a positive
contribution of u-quarks to the orbital angular momen-
tum, under the assumption that the production of π+
and K+mesons is dominated by scattering off u-quarks.
We gratefully acknowledge the Desy management for
its support, the staff at Desy and the collaborating insti-
tutions for their significant effort, and our national fund-
ing agencies and the EU RII3-CT-2004-506078 program
for financial support.
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