# Effects of transversity in deep-inelastic scattering by polarized protons

**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 (phi_S)and the produced hadron (phi) is found to have a substantial sin(phi+phi_S) modulation for the production of pi+, pi- 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 pi0 and K- production the amplitude of this Fourier component is consistent with zero. Comment: 7 pages, 3 figures

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**ABSTRACT:**Recent progress in the field of spin physics of high energy particle interactions is reviewed with particular emphasis on the spin structure functions as measured in polarized deep inelastic lepton-nucleon scattering (DIS). New measurements are presented to obtain more direct information on the composition of the nucleon angular momentum, with results from semi-inclusive DIS accessing flavour-separated parton distribution functions (PDF) and with first data from hard exclusive reactions which may be interpreted in terms of recently developed generalizations of parton distribution functions (GPD). Finally, experimental prospects are outlined which will lead to a further development of the virtues of QCD phenomenology of the spin structure of the nucleon.International Journal of Modern Physics A 01/2002; 17(23). · 1.13 Impact Factor - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We summarize the standard factorization theorems for hard processes in QCD, and describe their proofs. Comment: 100 pages, 27 figures. Authors' affiliations updated compared with published version09/2004; - SourceAvailable from: R. L. Jaffe[Show abstract] [Hide abstract]

**ABSTRACT:**We study the chiral-odd spin structure functions of the nucleon, {ital h}{sub 1}({ital x}) and {ital h}{sub 2}({ital x}), their physical significance, sum rules, and model estimates. We show that they can be measured in the Drell-Yan process with polarized beams at order {ital Q}{sup 0} and {ital Q}{sup {minus}1}, respectively.Physical Review Letters 08/1991; 67(5):552-555. · 7.73 Impact Factor

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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 B23 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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andsimulationof the

DESY-THESIS-

transversityandtransverse-

SumrulesfortheT-odd

7

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