The spin structure function <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml

HAL Id: hal-01466197

https://hal.archives-ouvertes.fr/hal-01466197

Submitted on 15 Feb 2017

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

The spin structure function g1p of the proton and a test

of the Bjorken sum rule

C. Adolph, R. Akhunzyanov, M. Alexeev, G. Alexeev, A. Amoroso, V.

Andrieux, V. Anosov, A. Austregesilo, C. Azevedo, B. Badelek, et al.

To cite this version:

C. Adolph, R. Akhunzyanov, M. Alexeev, G. Alexeev, A. Amoroso, et al.. The spin structure function

g1p of the proton and a test of the Bjorken sum rule. Modern Physics Letters B, World Scientific

Publishing, 2016, 753, pp.18 - 28. �10.1016/j.physletb.2015.11.064�. �hal-01466197�

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

The

spin

structure

function

g

₁

p

of

the

proton

and

a

test

of

the

Bjorken

sum

rule

C. Adolph

i

,

R. Akhunzyanov

h

, M.G. Alexeev

ab

, G.D. Alexeev

h

,

A. Amoroso

ab

,

ac

,

V. Andrieux

v

,

V. Anosov

h

,

A. Austregesilo

q

,

C. Azevedo

b

,

B. Badełek

af

,

F. Balestra

ab

,

ac

,

J. Barth

e

, G. Baum

1

,

R. Beck

d

,

Y. Bedfer

v

,

k

,

J. Bernhard

n

,

k

,

K. Bicker

q

,

k

,

E.R. Bielert

k

,

R. Birsa

z

,

J. Bisplinghoff

d

, M. Bodlak

s

,

M. Boer

v

,

P. Bordalo

m

,

2

,

F. Bradamante

y

,

z

,

C. Braun

i

, A. Bressan

y

,

z

,

∗

,

M. Büchele

j

, E. Burtin

v

,

L. Capozza

v

,

3

, W.-C. Chang

w

,

M. Chiosso

ab

,

ac

,

I. Choi

ad

,

S.U. Chung

q

,

4

, A. Cicuttin

aa

,

z

,

M.L. Crespo

aa

,

z

, Q. Curiel

v

,

S. Dalla Torre

z

,

S.S. Dasgupta

g

,

S. Dasgupta

y

,

z

, O.Yu. Denisov

ac

,

L. Dhara

g

, S.V. Donskov

u

,

N. Doshita

ah

,

V. Duic

y

,

M. Dziewiecki

ag

, A. Efremov

h

,

P.D. Eversheim

d

,

W. Eyrich

i

,

A. Ferrero

v

,

M. Finger

s

,

M. Finger jr.

s

,

H. Fischer

j

, C. Franco

m

,

N. du Fresne von Hohenesche

n

,

J.M. Friedrich

q

,

V. Frolov

h

,

k

,

E. Fuchey

v

,

F. Gautheron

c

,

O.P. Gavrichtchouk

h

,

S. Gerassimov

p

,

q

,

F. Giordano

ad

, I. Gnesi

ab

,

ac

,

M. Gorzellik

j

,

S. Grabmüller

q

,

A. Grasso

ab

,

ac

, M. Grosse-Perdekamp

ad

,

B. Grube

q

, T. Grussenmeyer

j

,

A. Guskov

h

,

F. Haas

q

,

D. Hahne

e

,

D. von Harrach

n

,

R. Hashimoto

ah

,

F.H. Heinsius

j

,

F. Herrmann

j

, F. Hinterberger

d

,

N. Horikawa

r

,

6

,

N. d’Hose

v

,

C. -Yu Hsieh

w

, S. Huber

q

,

S. Ishimoto

ah

,

7

,

A. Ivanov

h

, Yu. Ivanshin

h

,

T. Iwata

ah

,

R. Jahn

d

, V. Jary

t

,

P. Jörg

j

,

R. Joosten

d

,

E. Kabuß

n

, B. Ketzer

q

,

8

_{, G.V. Khaustov}

u

_,

_{Yu.A. Khokhlov}

u

,

9

_,

_{Yu. Kisselev}

h

_,

F. Klein

e

, K. Klimaszewski

ae

,

J.H. Koivuniemi

c

,

V.N. Kolosov

u

, K. Kondo

ah

,

K. Königsmann

j

, I. Konorov

p

,

q

, V.F. Konstantinov

u

,

A.M. Kotzinian

ab

,

ac

,

O. Kouznetsov

h

,

*

Corresponding authors.

E-mailaddresses:Andrea.Bressan@cern.ch(A. Bressan), Fabienne.Kunne@cern.ch(F. Kunne).

1 _{Retired from Universität Bielefeld, Fakultät für Physik, 33501 Bielefeld, Germany.} 2 _{Also at Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal.}

3 _{Present address: Universität Mainz, Helmholtz-Institut für Strahlen- und Kernphysik, 55099 Mainz, Germany.}

4 _{Also at Department of Physics, Pusan National University, Busan 609-735, Republic of Korea and at Physics Department, Brookhaven National Laboratory, Upton, NY 11973,}

USA.

5 _{Supported by the DFG Research Training Group Programme 1102 “Physics at Hadron Accelerators”.} 6 _{Also at Chubu University, Kasugai, Aichi, 487-8501 Japan.}

7 _{Also at KEK, 1-1 Oho, Tsukuba, Ibaraki, 305-0801 Japan.}

8 _{Present address: Universität Bonn, Helmholtz-Institut für Strahlen- und Kernphysik, 53115 Bonn, Germany.} 9 _{Also at Moscow Institute of Physics and Technology, Moscow Region, 141700, Russia.}

10 _{Present address: RWTH Aachen University, III. Physikalisches Institut, 52056 Aachen, Germany.} 11 _{Present address: Uppsala University, Box 516, SE-75120 Uppsala, Sweden.}

12 _{Supported by the German Bundesministerium für Bildung und Forschung.} 13 _{Supported by Czech Republic MEYS Grant LG13031.}

14 _{Supported by SAIL (CSR), Govt. of India.} 15 _{Supported by CERN-RFBR Grant 12-02-91500.}

16 _{Supported by the Portuguese FCT – Fundação para a Ciência e Tecnologia, COMPETE and QREN, Grants CERN/FP/109323/2009, CERN/FP/116376/2010 and CERN/FP/123600/}

2011.

17 _{Supported by the MEXT and the JSPS under the Grants Nos. 18002006, 20540299 and 18540281; Daiko Foundation and Yamada Foundation.} 18 _{Supported by the DFG cluster of excellence ‘Origin and Structure of the Universe’ (www.universe-cluster.de).}

19 _{Supported by EU FP7 (HadronPhysics3, Grant Agreement number 283286).}

20 _{Supported by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities.} 21 _{Supported by the Polish NCN Grant DEC-2011/01/M/ST2/02350.}

22 _Deceased.

http://dx.doi.org/10.1016/j.physletb.2015.11.064

0370-2693/©2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by

M. Krämer

q

, P. Kremser

j

,

F. Krinner

q

,

Z.V. Kroumchtein

h

,

N. Kuchinski

h

,

F. Kunne

v

,

∗

,

K. Kurek

ae

, R.P. Kurjata

ag

, A.A. Lednev

u

, A. Lehmann

i

,

M. Levillain

v

,

S. Levorato

z

,

J. Lichtenstadt

x

,

R. Longo

ab

,

ac

,

A. Maggiora

ac

,

A. Magnon

v

,

N. Makins

ad

,

N. Makke

y

,

z

,

G.K. Mallot

k

,

C. Marchand

v

,

A. Martin

y

,

z

,

J. Marzec

ag

,

J. Matousek

s

,

H. Matsuda

ah

,

T. Matsuda

o

,

G. Meshcheryakov

h

,

W. Meyer

c

, T. Michigami

ah

, Yu.V. Mikhailov

u

,

Y. Miyachi

ah

,

A. Nagaytsev

h

,

T. Nagel

q

,

F. Nerling

n

, D. Neyret

v

, V.I. Nikolaenko

u

,

J. Novy

t

,

k

,

W.-D. Nowak

j

, A.S. Nunes

m

,

A.G. Olshevsky

h

,

I. Orlov

h

,

M. Ostrick

n

,

D. Panzieri

a

,

ac

,

B. Parsamyan

ab

,

ac

, S. Paul

q

,

J.-C. Peng

ad

,

F. Pereira

b

,

M. Pesek

s

,

D.V. Peshekhonov

h

,

S. Platchkov

v

,

J. Pochodzalla

n

,

V.A. Polyakov

u

,

J. Pretz

e

,

10

,

M. Quaresma

m

, C. Quintans

m

, S. Ramos

m

,

2

, C. Regali

j

,

G. Reicherz

c

,

C. Riedl

ad

,

E. Rocco

k

,

N.S. Rossiyskaya

h

,

D.I. Ryabchikov

u

,

A. Rychter

ag

,

V.D. Samoylenko

u

,

A. Sandacz

ae

,

C. Santos

z

, S. Sarkar

g

,

I.A. Savin

h

,

G. Sbrizzai

y

,

z

,

P. Schiavon

y

,

z

,

K. Schmidt

j

,

5

,

H. Schmieden

e

,

K. Schönning

k

,

11

,

S. Schopferer

j

,

A. Selyunin

h

, O.Yu. Shevchenko

h

,

22

,

L. Silva

m

,

L. Sinha

g

,

S. Sirtl

j

,

M. Slunecka

h

,

F. Sozzi

z

, A. Srnka

f

,

M. Stolarski

m

,

M. Sulc

l

,

H. Suzuki

ah

,

6

, A. Szabelski

ae

,

T. Szameitat

j

,

5

,

P. Sznajder

ae

,

S. Takekawa

ab

,

ac

,

J. ter Wolbeek

j

,

5

, S. Tessaro

z

,

F. Tessarotto

z

, F. Thibaud

v

,

F. Tosello

ac

,

V. Tskhay

p

,

S. Uhl

q

,

J. Veloso

b

, M. Virius

t

,

T. Weisrock

n

,

M. Wilfert

n

, R. Windmolders

e

,

K. Zaremba

ag

,

M. Zavertyaev

p

,

E. Zemlyanichkina

h

, M. Ziembicki

ag

,

A. Zink

i

a_{UniversityofEasternPiedmont,15100Alessandria,Italy}

b_{UniversityofAveiro,DepartmentofPhysics,3810-193Aveiro,Portugal}

c_{UniversitätBochum,InstitutfürExperimentalphysik,44780Bochum,Germany}12,19

d_{UniversitätBonn,Helmholtz-InstitutfürStrahlenundKernphysik,53115Bonn,Germany}12

e_{UniversitätBonn,PhysikalischesInstitut,53115Bonn,Germany}12

f_{InstituteofScientificInstruments,ASCR,61264Brno,CzechRepublic}13

g_{MatrivaniInstituteofExperimentalResearch&Education,Calcutta-700030,India}14

h_{JointInstituteforNuclearResearch,141980Dubna,Moscowregion,Russia}15

i_{UniversitätErlangen–Nürnberg,PhysikalischesInstitut,91054Erlangen,Germany}12

j_{UniversitätFreiburg,PhysikalischesInstitut,79104Freiburg,Germany}12,19

k_{CERN,1211Geneva23,Switzerland}

l_{TechnicalUniversityinLiberec,46117Liberec,CzechRepublic}13

m_{LIP,1000-149Lisbon,Portugal}16

n_{UniversitätMainz,InstitutfürKernphysik,55099Mainz,Germany}12

o_{UniversityofMiyazaki,Miyazaki889-2192,Japan}17

p_{LebedevPhysicalInstitute,119991Moscow,Russia}

q_{TechnischeUniversitätMünchen,PhysikDepartment,85748Garching,Germany}12,18

r_{NagoyaUniversity,464Nagoya,Japan}17

s_{CharlesUniversityinPrague,FacultyofMathematicsandPhysics,18000Prague,CzechRepublic}13

t_{CzechTechnicalUniversityinPrague,16636Prague,CzechRepublic}13

u_{StateScientificCenterInstituteforHighEnergyPhysicsofNationalResearchCenter‘KurchatovInstitute’,142281Protvino,Russia} v_{CEAIRFU/SPhNSaclay,91191Gif-sur-Yvette,France}19

w_{AcademiaSinica,InstituteofPhysics,Taipei,11529Taiwan}

x_{TelAvivUniversity,SchoolofPhysicsandAstronomy,69978TelAviv,Israel}20

y_{UniversityofTrieste,DepartmentofPhysics,34127Trieste,Italy} z_{TriesteSectionofINFN,34127Trieste,Italy}

aa

AbdusSalamICTP,34151Trieste,Italy

ab_{UniversityofTurin,DepartmentofPhysics,10125Turin,Italy} ac_{TorinoSectionofINFN,10125Turin,Italy}

ad_{UniversityofIllinoisatUrbana–Champaign,DepartmentofPhysics,Urbana,IL61801-3080,USA} ae_{NationalCentreforNuclearResearch,00-681Warsaw,Poland}21

af_{UniversityofWarsaw,FacultyofPhysics,02-093Warsaw,Poland}21

ag_{WarsawUniversityofTechnology,InstituteofRadioelectronics,00-665Warsaw,Poland}21

ah_{YamagataUniversity,Yamagata,992-8510Japan}17

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received 23 April 2015

Received in revised form 18 November 2015 Accepted 23 November 2015

Available online 27 November 2015 Editor: M. Doser

New results for the double spin asymmetry Ap₁ and the proton longitudinal spin structure function

gp₁ are presented.TheywereobtainedbytheCOMPASSCollaborationusingpolarised200GeV muons scatteredoff alongitudinally polarisedNH3 target.The data werecollectedin2011and complement

those recorded in 2007 at 160 GeV,in particular atlower values of x. They improve the statistical precision of gp₁(x) by about afactor oftwo in the region x0.02. A next-to-leading orderQCD fit tothe g1 worlddata isperformed. Itleadstoanewdetermination ofthe quarkspincontribution to

the nucleonspin, ,ranging from0.26 to 0.36,and toa re-evaluationof the firstmoment of g₁p. The uncertainty of is mostlydue to the large uncertainty in the present determinations ofthe gluonhelicitydistribution.AnewevaluationoftheBjorkensumrulebasedontheCOMPASSresults for

thenon-singletstructurefunctiongNS

1 (x,Q2) yieldsasratiooftheaxialandvectorcouplingconstants

|gA/gV|=1.22±0.05(stat.)±0.10(syst.),whichvalidatesthesumruletoanaccuracyofabout9%.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Thedeterminationofthelongitudinalspinstructureofthe nu-cleonbecameoneoftheimportantissuesinparticlephysicsafter the surprisingEMC resultthat the quark contribution tothe nu-cleonspinisverysmallorevenvanishing[1].Thepresent knowl-edgeonthelongitudinalspinstructurefunctionoftheproton,gp₁, originatesfrommeasurements of theasymmetry Ap₁ inpolarised leptonnucleon scattering.In alltheseexperiments,longitudinally polarisedhigh-energyleptonswerescatteredofflongitudinally po-larisednucleonornucleartargets.AtSLACandJLabelectronbeams wereused,electronandpositronbeamsatDESYandmuonbeams atCERN. Detailson the performance oftheseexperiments anda collectionoftheirresultscanbefounde.g.inRef.[2].

InthisLetter,wereportonnewresultsfromtheCOMPASS ex-periment atCERN. By measuring Ap₁, we obtain results on gp₁ in the deep inelastic scattering (DIS) region. They cover the range from1

(

GeV

/

c

)

2 to190

(

GeV

/

c

)

2 inthephoton virtuality Q2 _and from0.0025to0.7intheBjorkenscalingvariablex.Thenewdata, whichwerecollectedin2011atabeamenergyof200 GeV, com-plement earlierdata taken in 2007at 160 GeV that covered the range 0

.

004

<

x

<

0

.

7 [3]. In the newly explored low-x region, ourresultssignificantlyimprovethestatisticalprecisionofg₁p and thereby allow usto decreasethe low-x extrapolation uncertainty inthedeterminationoffirstmoments.

Inthefollowingsection,theCOMPASSexperimentisbriefly de-scribed.ThedataselectionprocedureispresentedinSection3and themethodofasymmetry calculationinSection 4.The resultson

Ap₁

(

x

,

Q2

₎

_{and g}p

1

(

x

,

Q2

)

are given in Section 5. A new next-to-leadingorder(NLO)QCDfittotheexistingnucleong1 datainthe region Q2

_>

₁

₍

_GeV

_/

_c

₎

2 _{isdescribed inSection 6.Section7deals} withthedeterminationoffirstmomentsof g₁p andtheevaluation oftheBjorkensumruleusingCOMPASSdataonly.Conclusionsare giveninSection8.

2. Experimentalsetup

The measurements were performedwith the COMPASS setup at the M2 beam line of the CERN SPS. The data presented in thisLettercorrespondtoan integratedluminosityof0.52 fb−1.A beam of positive muons was used with an intensity of 107 s−1 in a 10 s long spill every 40 s. The nominal beam momentum was 200 GeV/c with a spread of 5%. The beam was naturally polarised with a polarisation PB

≈

0

.

8, which is known with a precision of 0

.

04. Momentum and trajectory of each incoming particle were measured in a set of scintillator hodoscopes, scin-tillatingfibreandsilicondetectors.Thebeamwasimpingingona solid-stateammonia(NH3) targetthat provideslongitudinally po-larisedprotons.The threeprotons inammoniawere polarisedup to

|

PT

|

≈

0

.

9 by dynamic nuclear polarisation with microwaves. Forthispurpose,thetargetwas placedinsidealarge-aperture su-perconductingsolenoidwithafieldof2.5 Tandcooled to60 mK byamixtureofliquid3_Heand4_{He.Thetargetmaterialwas} con-tainedinthreecylindricalcellswithadiameterof4 cm,whichhad their axesalong thebeamlineandwere separatedby adistance of5 cm. The outer cells with a lengthof 30 cm were oppositely polarised to the central one, which was 60 cm long. In orderto compensate foracceptancedifferencesbetweenthecells, the po-larisationwasregularlyreversed byrotationofthemagneticfield

direction. In order to guard against unknown systematic effects, thedirectionofthepolarisationrelativetothemagneticfieldwas reversedonceduringthedatatakingperiodbyexchangingthe mi-crowave frequenciesappliedtothecells.Ten NMRcoils surround-ing the target material allowed fora measurement of PT with a precision of 0.032 for both signs of the polarisation. The typical dilution due to unpolarisable material in the target amounts to about 0.15.

The experimental setup allowed forthemeasurement of scat-teredmuonsandproducedhadrons.Theseparticlesweredetected in atwo-stage, open forward spectrometerwithlarge acceptance in momentum andangle.Each spectrometer stage consistedof a dipole magnet surrounded by tracking detectors. Scintillating fi-bre detectors and micropattern gaseous detectors were used in the beamregionandclose tothe beam,while multiwire propor-tional chambers, drift chambers andstraw detectors covered the largeouterareas.Scatteredmuonswereidentifiedinsetsof drift-tubeplaneslocatedbehindironandconcreteabsorbersinthefirst and second stages. Particle identification with the ring imaging Cerenkovdetectororcalorimetersisnotusedinthismeasurement. The‘inclusivetriggers’werebasedonacombinationofhodoscope signalsforthescatteredmuons, whilefor‘semi-inclusivetriggers’ an energy deposit of hadron tracks in one of the calorimeters was required,optionally incoincidence with an inclusive trigger. A detaileddescriptionofthe experimentalsetup can befound in Ref.[4].

3. Dataselection

Theselectedeventsarerequiredtocontainareconstructed in-coming muon, a scattered muon and an interaction vertex. The measured incident muon momentum has to be in the range 185 GeV

/

c

<

pB

<

215 GeV

/

c.Inorder toequalise thebeam flux throughalltargetcells,theextrapolatedbeamtrackisrequiredto pass all ofthem. The measured longitudinalposition of the ver-tex allows usto identify the target cell in which the scattering occurred. The radialdistanceof thevertexfromthebeamaxisis requiredtobelessthan1

.

9 cm,by whichthecontributionof un-polarisedmaterialisminimised.Allphysicstriggers,inclusiveand semi-inclusive ones, are included inthis analysis. Inorder to be attributedtothescatteredmuon,atrackisrequiredtopassmore than30 radiationlengthsofmaterialandithastopointtothe ho-doscopes thathavetriggered theevent.InordertoselecttheDIS region,onlyeventswithphotonvirtualityQ2

_>

₁

₍

_GeV

_/

_c

₎

2_are se-lected.Inaddition,therelativemuonenergytransfer, y,isrequired to bebetween0

.

1 and0

.

9.Here, thelower limitremovesevents thataredifficulttoreconstruct,whiletheupperlimitremovesthe regionthatisdominatedbyradiativeevents.Thesekinematic con-straintsleadtotherange0

.

0025

<

x

<

0

.

7 andtoaminimummass squaredofthehadronicfinalstate, W2_,of12

₍

_GeV

_/

_c2

₎

2_.Afterall selections, thefinal sample consistsof77 million events.The se-lected sample is dominated by inclusive triggers that contribute 84% to the total number of triggers. The semi-inclusive triggers mainly contribute to the high-x region, where they amount to abouthalfofthetriggers.Inthehigh- Q2_{regionthesemi-inclusive} triggersdominate.

4. Asymmetrycalculation

Theasymmetrybetweenthecrosssectionsforantiparallel(

↑↓

) andparallel(

↑↑

)orientationsofthelongitudinalspinsofincoming muonandtargetprotoniswrittenas

Ap_LL

=

σ

↑↓

₋

_σ

↑↑

σ

↑↓

+

σ

↑↑

.

(1)

Thisasymmetryisrelatedtothe longitudinalandtransversespin asymmetriesAp₁andAp₂,respectively,forvirtual-photonabsorption bytheproton: Ap_LL

=

D

(

Ap₁

+

η

Ap₂

) .

(2) Thefactors

η

=

γ

(

1

−

y

−

γ

2_y2

_/

₄

₋

_y2_m2

_/

_Q2

₎

(

1

+

γ

2_y

_/

₂

₎₍

₁

₋

_y

_/

₂

₎

₋

_y2_m2

_/

_Q2 (3) and D

=

y

((

1

+

γ

2_y

_/

₂

₎₍

₂

₋

_y

₎

₋

_{2 y}2_m2

_/

_Q2

₎

y2

₍

₁

₋

_2m2

_/

_Q2

₎₍

₁

₊

_γ

2

₎

₊

₂

₍

₁

₊

_R

₎₍

₁

₋

_y

₋

_γ

2_y2

_/

₄

₎

(4)

dependontheeventkinematics,with

γ

=

2Mx

/



Q2_{,m themuon} andM theprotonmass.Thevirtual-photondepolarisationfactorD

dependsalsoon the ratio R

=

σ

L

/

σ

T, where

σL

(

σ

T) is thecross section for the absorption of a longitudinally (transversely) po-larisedvirtual photon by a proton.The asymmetry Ap₁ is defined as

Ap₁

=

σ

1/2

−

σ

3/2

σ

1/2

+

σ

3/2

,

(5)

where

σ1

/2

(

σ

3/2

)

istheabsorptioncrosssection ofatransversely polarised virtual photon by a proton with total spin projection 1 2



3 2



inthephotondirection. Sinceboth

η

and Ap₂ [5]aresmall intheCOMPASSkinematicregion, Ap₁



Ap_LL

/

D andthe longitudi-nalspinstructurefunction[6]isgivenby

g₁p

=

F p 2 2x

(

1

+

R

)

A p 1

,

(6)

where Fp₂ denotesthe spin-independentstructure functionofthe proton.

Thenumberofevents,Ni,collectedfromeachtargetcellbefore andafterreversalofthetarget polarisationisrelatedtothe spin-independentcrosssection

σ

=

σ

1/2

+

σ

3/2 andtothe asymmetry

Ap₁ as

Ni

=

ai

φ

ini

σ

(

1

+

PBPTf D Ap₁

),

i

=

o1

,

c1

,

o2

,

c2

.

(7) Here,aiistheacceptance,

φ

itheincomingmuonflux,nithe num-ber of target nucleons and f the dilution factor, while PB and

PT were already introduced in Section 2. Events fromthe outer target cell are summed, thus the four relations of Eq. (7) corre-spondingto the two sets of target cells (outer, o and central, c)

andthetwo spin orientations(1 and2) resultin a second-order equationinAp₁ fortheratio

(

No1Nc2

)/(

Nc1No2

)

.Fluxesand accep-tancescancel inthisequation, if theratio ofacceptances forthe two setsof cells is thesame before andafterthe magnetic field rotation[7].Theasymmetriesarecalculatedseparatelyforeachof thosesub-samples. Eachperiod before andaftersuch rotationof themagneticfieldisconsideredasonesub-sampleandthe asym-metriesarecalculatedseparatelyforeachofthesesub-samples.In ordertominimisethestatisticaluncertainty,allquantitiesusedin

Table 1

Contributions to the systematic uncertainty on Ap1with multiplicative (top) and

ad-ditive (bottom) components.

Beam polarisation PB/PB 5% Target polarisation PT/PT 3.5% Depolarisation factor D(R)/D(R) 2.0–3.0% Dilution factor f/f 2% Total Amult 1 0.07 A p 1

False asymmetry Afalse1 <0.84·σstat

Transverse asymmetry η·Ap2 <10−2

Radiative corrections ARC

1 10−4–10−3

theasymmetry calculationare evaluated eventby eventwiththe weightfactor[7,8]

w

=

PBf D

.

(8)

Thepolarisationoftheincomingmuonsasafunctionofthebeam momentumisobtainedfromaparametrisationbasedona Monte Carlosimulation ofthebeam line. Theeffectivedilution factor f

is givenby the ratioofthe total crosssection for muonson po-larisable protons to the one on all nucleiin the target, whereby their measured composition is takeninto account. It is modified by acorrectionfactorthat accountsforthedilutiondueto radia-tive events on unpolarised protons [9]. The target polarisation is not includedinthe eventweight,becauseit maychangeintime andgeneratefalseasymmetries.Theobtainedasymmetriesare cor-rected forspin-dependent radiative effects according to Ref. [10]

andforthe14NpolarisationasdescribedinRefs.[3,11].Ithasbeen checkedinthesamebinningasfortheasymmetry determination that theuseofsemi-inclusivetriggersdoesnot biasthe determi-nation of Ap₁. The final value of Ap₁ is obtainedas the weighted averageoftheresultsfromthesub-samples.

Systematicuncertaintiesarecalculatedtakingintoaccount mul-tiplicativeandadditive contributions to Ap₁. Multiplicative contri-butionsoriginate fromtheuncertaintiesofthetargetpolarisation, thebeampolarisation,thedepolarisationfactor(mainlyduetothe uncertaintyof R) andthedilutionfactor.Whenaddedin quadra-ture, these uncertainties result in a total uncertainty



Amult

1 of 0

.

07 Ap₁. They are shown in Table 1, which also shows the ad-ditive contributions. The largest additive contribution to the sys-tematic uncertainty is the one from possible false asymmetries. Theirsizeisestimatedwithtwo differentapproaches. Inthefirst approach, the central target cell is artificially divided into two consecutive 30 cm long parts. Calculating the asymmetry using the two outer cells or the two central parts, the physics asym-metry cancels and thus two independent false asymmetries are formed. Both arefound to be consistentwith zero.Thistest was done usingthesamesub-samplesasforthephysics asymmetries determination. In order to check for a false asymmetry due to time-dependent effects,the asymmetries Ap₁ obtainedfromthese sub-samplesarecomparedbyusingthemethodof“pulls”[12].No significantbroadeningofpulldistributionsisobserved.Thesepulls areusedtosetanupperlimitonthesystematicuncertaintydueto false asymmetries Afalse₁ .Dependingon thex-bin, valuesbetween 0

.

4

·

σ

stat and0

.

84

·

σ

stat areobtained. Furtheradditivecorrections originatefromneglecting Ap₂ andfromtheuncertaintyinthe cor-rection ARC₁ totheasymmetry Ap₁,whichisduetospin-dependent radiative effects. The totalsystematic uncertaintyis givenby the quadraticsumofthecontributionsinTable 1.

5. Resultson

A

p₁and

g

p₁

The dataare analysedinterms of A₁p and gp₁ asafunction of

x and Q2_{.The x dependence of A}p

bin is shown in Fig. 1 together with the previous COMPASS re-sultsobtainedat160 GeV [3]andwithresultsfromother exper-iments[1,13–16] includingthose by SMCat 190 GeV [17],while thelatestresultsfromJLab[18] werenotincludedbecauseofthe

W2

>

12

(

GeV

/

c2

)

2 cut. The bandsat the bottom representthe systematic uncertainties of the COMPASS results as discussed in

Fig. 1. The

asymmetry

Ap₁as a function of x at

the measured values of

Q2_as

ob-tained from the COMPASS data at 200 GeV. The new data are compared to the COMPASS results obtained at 160 GeV[3]and to the other world data (EMC [1],

CLAS [13], HERMES [14], E143 [15], E155 [16], SMC [17]). The bands at the bottom

indicate the systematic uncertainties of the COMPASS data at 160 GeV (upper band) and 200 GeV (lower band). (Coloured version online.)

Section4.Thenewdataimprovethestatisticalprecisionatleastby a factoroftwo inthelow-x region,whichiscovered bytheSMC and COMPASS measurements only. The good agreement between all experimental results reflects the weak Q2 dependenceof Ap₁. Thisis alsoillustratedinFig. 2,whichshows Ap₁ asa functionof

Q2 insixteenintervalsofx fortheCOMPASSdatasetsat160 GeV and200 GeV.Innoneofthex bins, asignificant Q2 dependence isobserved.ThenumericalvaluesofAp₁

(

x

)

andAp₁

(

x

,

Q2

)

obtained at200 GeV aregiveninAppendix AinTables 8 and 9.

The longitudinal spin structure function gp₁ is calculatedfrom

A₁p usingEq.(6),theF₂pparametrisationfromRef.[17]andthe ra-tio R from Ref. [19].The new resultsare shownin Fig. 3 atthe measuredvaluesofQ2 incomparisonwiththepreviousCOMPASS results obtained at 160 GeV and with SMC results at 190 GeV. The systematic uncertainty of g₁p is calculated using the contri-butions from Table 1includingin additionan uncertainty for F₂p

of 2–3%[17].Comparedtothe SMCexperiment, thepresent sys-tematicuncertaintiesarelargerduetoamorerealisticestimateof falseasymmetries,whichisbasedonrealevents.

The world data on g₁p as a function of Q2 _{for various x are} showninFig. 4.Thedatacoverabouttwodecadesinx andin Q2

formostofthex range,exceptforx

<

0

.

02,wheretheQ2rangeis muchmorelimited.Thenewdataimprovethekinematiccoverage in theregion ofhigh Q2 _{andlow x values, which givesa better} lever arm forthedetermination ofquark andgluon polarisations fromtheDGLAPevolutionequations.Inaddition,theextensionof measurementstolowervaluesofx isimportanttobetterconstrain thevalueofthefirstmomentof gp₁.

Fig. 2. The

asymmetry

Ap₁ as a function of Q2_{in bins of x obtained}

_{from the 200 GeV (red squares) and 160 GeV (blue circles) COMPASS data. The band at the bottom}

Fig. 3. The

spin-dependent structure function

xgp1at the measured values of Q2as

a function of x.

The COMPASS data at 200 GeV (red squares) are compared to the

results at 160 GeV (blue circles) and to the SMC results at 190 GeV (green crosses) for Q2

>1 (GeV/c)2. The bands from top to bottom indicate the systematic

un-certainties for SMC 190 GeV, COMPASS 200 GeV and COMPASS 160 GeV. (Coloured version online.)

Fig. 4. World

data on the spin-dependent structure function

g₁p as a function of

Q2_{for various values of x with}

_{all COMPASS data in red (full circles: 160 GeV, full}

squares: 200 GeV). The lines represent the Q2_{dependence for each value of x,}

_as

determined from a NLO QCD fit (see Section6). The dashed ranges represent the

region with W2_{<10 (GeV/c}2₎2_{. Note that the data of the individual x bins}

_are

staggered for clarity by adding 12.1–0.7i, i=0 . . .17. (Coloured version online.)

6. NLOQCDfitof

g

1worlddata

WeperformedanewNLOQCDfitofthespin-dependent struc-turefunction g1 in theDISregion, Q2

>

1

(

GeV

/

c

)

2,considering allavailableproton,deuteronand3_{Hedata.Thefitisperformedin} theMS renormalisation andfactorisation scheme.Forthe fit,the sameprogramisusedasinRef.[20],whichwasderivedfrom pro-gram2inRef.[17].Theregion W2

<

10

(

GeV

/

c2

)

2 isexcluded as itwasinrecentanalyses[21].Notethattheimpactofhigher-twist

effectswhenusingasmallerW2_{cutisconsideredinRef.[22].The} totalnumberofdatapointsusedinthefitis495(seeTable 2),the numberofCOMPASSdatapointsis138.

The neutron structure function gn₁ is extracted from the 3He data,whilethenucleonstructurefunction gN

1 isobtainedas

gN₁

(

x

,

Q2

)

=

1 1

−

1

.

5

ω

D

g₁d

(

x

,

Q2

),

(9)

where

ωD

is a correction for the D-wave state in the deuteron,

ω

D

=

0

.

05

±

0

.

01 [27], andthedeuteron structurefunction gd1 is givenpernucleon.Thequarksingletdistribution



qS

(

x

)

,thequark non-singletdistributions



q3

(

x

)

and



q8

(

x

)

,aswell asthegluon helicity distribution



g

(

x

)

,which appearin theNLO expressions for g₁p, gn

1 and gN1 (seee.g. Ref.[17]),areparametrisedata refer-encescale Q₀2 asfollows:



fk

(

x

)

=

η

k xαk

₍

₁

−

_x

₎

βk

₍

₁

+

_γ

kx

)



1 0xαk

(

1

−

x

)

βk

(

1

+

γ

kx

)

dx

.

(10)

Here,



fk

(

x

)

(k

=

S

,

3

,

8

,

g)represents



qS

(

x

)

,



q3

(

x

)

,



q8

(

x

)

and



g

(

x

)

and

η

k is the first moment of



fk

(

x

)

at the reference scale.Themomentsof



q3 and



q8 are fixedatanyscaleby the baryondecayconstants(F

+

D)and(3F

−

D),respectively, assum-ingSU

(

2

)

f andSU

(

3

)

fflavoursymmetries.Theimpactofreleasing these conditions is investigated and included in the systematic uncertainty.Thecoefficients

γ

k arefixedtozeroforthetwo non-singletdistributionsastheyarepoorlyconstrainedandnotneeded to describe the data. The exponent

β

g, which is not well deter-mined fromthedata,is fixedto 3

.

0225 [28] andthe uncertainty fromtheintroducedbiasisincludedinthefinal uncertainty.This leaves 11 free parameters in the fitted parton distributions. The expressionfor

χ

2 _{ofthefitconsistsofthreeterms,}

χ

2

=

Nexp



n=1

⎡

⎢

⎣

N



datan i=1

⎛

⎝

gfit1

−

N

ngdata_1,i

N

n

σ

i

⎞

⎠

2

+



1

−

N

n

δ

N

n



2

⎤

⎥

⎦ +

χ

_positivity2

.

(11)

Onlystatisticaluncertainties ofthedataaretakenintoaccount in

σ

i. The normalisation factors

N

n ofeach data set n areallowed to vary taking into account the normalisation uncertainties

δN

n. Ifthelatterare unavailable,theyare estimatedasquadraticsums oftheuncertainties ofthebeamandtargetpolarisations.The fit-ted normalisations are found to be consistent with unity, except fortheE155protondatawherethenormalisationishigher,albeit compatiblewiththevaluequotedinRef.[16].

In order to keep the parameters within their physical ranges, the polarised PDFs are calculated at every iteration of the fit andrequiredtosatisfy thepositivityconditions

|

q

(

x

)

+ ¯

q

(

x

)

|

≤

q

(

x

)

+ ¯

q

(

x

)

and

|

g

(

x

)

|

≤

g

(

x

)

at Q2

=

1

(

GeV

/

c

)

2 [29,30],which is accomplished by the

χ

2

positivity term in Eq. (11). This proce-dure leads to asymmetric values of the parameter uncertainties when the fitted value is close to the allowed limit. The unpo-larised PDFs andthe corresponding value of the strong coupling constant

α

s

(

Q2

)

are takenfromthe MSTW parametrisation[28]. Theimpact ofthechoiceofPDFsisevaluatedby usingtheMRST distributions[31]forcomparison.

Inordertoinvestigatethesensitivityoftheparametrisationof thepolarisedPDFstothefunctionalforms,thefitisperformedfor severalsetsoffunctionalshapes.Theseshapesdoordonotinclude the

γS

and

γg

parametersofEq.(10)andaredefinedatreference scales rangingfrom1

(

GeV

/

c

)

2 to63

(

GeV

/

c

)

2.Itisobserved[8]

Fig. 5. Results

of the QCD fits to

g1world data at Q2=3 (GeV/c)2for the two sets of functional shapes as discussed in the text. Top: singlet xqS(x)and gluon distribution

xg(x). Bottom: distributions of x[q(x)+ ¯q(x)]for different flavours (u, d ands).

Continuous lines correspond to the fit with

γS=0, long dashed lines to the one with γS=0. The dark bands represent the statistical uncertainties, only. The light bands, which overlay the dark ones, represent the total systematic and statistical uncertainties

added in quadrature. (Coloured version online.)

that mainlytwo sets offunctionalshapesare neededtospan al-mostentirelythe rangeofthe possible



qS

₍

_x

₎

_and

_

_g

₍

_x

₎

distri-butions allowed by the data. These two sets of functional forms yieldtwoextremesolutionsfor



g

(

x

)

.For

γg

=

γ

S

=

0 (

γ

g

=

0 and

γ

S

=

0) anegative (positive) solutionfor



g

(

x

)

isobtained. Both solutions areparametrised at Q2

0

=

1

(

GeV

/

c

)

2 andleadto simi-larvaluesofthereduced

χ

2_{ofthefitsofabout1.05/d.o.f.Changes} inthefitresultthatoriginate fromusingother(converging) func-tionalformsareincludedinthesystematicuncertainty.

TheobtaineddistributionsarepresentedinFig. 5.Thedark er-ror bands seen in this figure stem from generating several sets of g1 pseudo-data, which are obtainedby randomising the mea-sured g1 valuesusingtheirstatisticaluncertaintiesaccordingtoa normaldistribution.Thiscorresponds toaone-standard-deviation accuracy of the extracted parton distributions. A thorough anal-ysis of systematic uncertainties of the fitting procedure is per-formed.The most importantsource isthe freedom inthe choice of the functional forms for



qS

(

x

)

and



g

(

x

)

. Further uncer-tainties arise from the uncertainty in the value of

α

s

(

Q2

)

and from effects of SU

(

2

)

f and SU

(

3

)

f symmetry breaking. The to-talsystematic andstatistical uncertaintiesarerepresented bythe light bands overlaying the dark ones in Fig. 5. For both sets of functional forms discussed above,



s

(

x

)

+ ¯

s

(

x

)

stays negative. It is differentfromzero for x



0

.

001 as are



d

(

x

)

+ ¯

d

(

x

)

and



u

(

x

)

+ ¯

u

(

x

)

.Thesingletdistribution



qS

₍

_x

₎

_{iscompatiblewith} zeroforx



0

.

07.

The inclusion of systematic uncertainties in the fit leads to much larger spreads in the first moments ascompared to those obtainedby only propagating statisticaluncertainties. Theresults

forthe firstmoments aregiven inTable 3.In thistable,



de-notes the first moment of the singlet distribution. Notethat the first moments of



u

+ ¯

u,



d

+ ¯

d and



s

+ ¯

s are not inde-pendent, sincethe first moments ofthe non-singletdistributions arefixedbythedecayconstantsFandDateveryvalueofQ2.The largeuncertaintyin



g

(

x

)

,whichismainlyduetothefreedomin thechoiceofitsfunctionalform,doeshowevernotallowto deter-mine thefirstmomentof



g

(

x

)

fromtheavailableinclusivedata only.

The fitted gp₁ and gd₁ distributions at Q2

=

3

(

GeV

/

c

)

2 are showninFig. 6together withthedataevolvedtothesamescale. The two curvescorrespond to the two extremefunctional forms discussed above, which lead to either a positive or a negative



g

(

x

)

. The dark bands represent the statistical uncertainties as-sociated with each curve andthe light bands representthe total systematic and statistical uncertainties added in quadrature.The valuesfor gp₁ are positiveinthewhole measuredregion downto

x

=

0

.

0025,whilegd

1 isconsistentwithzeroatlow x.

7. Firstmomentsof

g

1andBjorkensumrulefromCOMPASSdata The new data on gp₁ together with the new QCD fit allow a more precise determination of the first moments

1

(

Q2

)

=



1

0 g1

(

x

,

Q2

)

dx oftheproton,neutronandnon-singletspin struc-turefunctionsusingCOMPASSdataonly.Thelatteroneisdefined as

g₁NS

(

x

,

Q2

)

=

gp₁

(

x

,

Q2

)

−

g₁n

(

x

,

Q2

)

Table 2

List of experimental data sets used in this analysis. For each set the number of points, the χ2_{contribution and the fitted normalisation factor is given for the two functional}

shapes discussed in the text, which lead to either a positive or a negative function g(x).

Experiment Function extracted Number of points χ2 _{Normalisation}

g(x) >0 g(x) <0 g(x) >0 g(x) <0 EMC[1] Ap1 10 5.2 4.7 1.03±0.07 1.02±0.07 E142[23] An 1 6 1.1 1.1 1.01±0.07 0.99±0.07 E143[15] gd 1/F d 1 54 61.4 59.0 0.99±0.04 1.01±0.04 E143[15] gp1/F p 1 54 47.4 49.1 1.05±0.02 1.08±0.02 E154[24] An 1 11 5.9 7.4 1.06±0.04 1.07±0.04 E155[25] gd 1/F d 1 22 18.8 18.0 1.00±0.04 1.00±0.04 E155[16] gp1/F p 1 21 50.0 49.7 1.16±0.02 1.16±0.02 SMC[17] Ap1 59 55.4 55.4 1.02±0.03 1.01±0.03 SMC[17] Ad 1 65 59.3 61.5 1.00±0.04 1.00±0.04 HERMES[14] Ad 1 24 28.1 27.0 0.98±0.04 1.01±0.04 HERMES[14] Ap1 24 14.0 16.2 1.08±0.03 1.10±0.03 HERMES[26] An 1 7 1.6 1.2 1.01±0.07 1.00±0.07 COMPASS 160 GeV[20] gd 1 43 33.1 37.7 0.97±0.05 0.95±0.05 COMPASS 160 GeV[3] Ap1 44 50.8 49.1 1.00±0.03 0.99±0.03

COMPASS 200 GeV (this work) Ap1 51 43.6 43.2 1.03±0.03 1.02±0.03

Fig. 6. Results

of the QCD fits to

g1p(left) and gd1(right) world data at Q2=3 (GeV/c)2as functions of x.

The curves correspond to the two sets of functional shapes as

discussed in the text. The dark bands represent the statistical uncertainties associated with each curve and the light bands, which overlay the dark ones, represent the total systematic and statistical uncertainties added in quadrature. (Coloured version online.)

Theintegral

NS₁

(

Q2

₎

_{atagivenvalue of Q}2 _{isconnected tothe} ratiogA

/

gVoftheaxialandvectorcouplingconstantsviathe fun-damentalBjorkensumrule[32]

NS₁

(

Q2

)

=

1



0 g₁NS

(

x

,

Q2

)

dx

=

1 6





gA gV





CNS₁

(

Q2

) ,

(13)

whereCNS₁

(

Q2

)

isthenon-singletcoefficientfunctionthatisgiven uptothird orderin

α

s

(

Q2

)

inperturbative QCDinRef. [33].The calculationuptothefourthorderisavailableinRef.[34].

Duetosmalldifferencesinthekinematics ofthedatasets, all points of the three COMPASS g1 data sets (Table 2) are evolved to the Q2 value of the 160 GeV proton data. A weighted aver-age of the 160 GeV and 200 GeV proton data is performed and thepoints atdifferentvaluesof Q2 _{andthe samevalue ofx are} merged.

Forthedeterminationof

p₁and

₁d,thevaluesofgp₁ andgd₁are evolvedto Q2

₌

_{3 (GeV}

_/

_c)2 _{andtheintegralsarecalculatedinthe} measuredrangesofx.Inordertoobtainthefullmoments,theQCD fitisusedtoevaluatetheextrapolationtox

=

1 andx

=

0 (see Ta-ble 4). The moment

n₁ is calculated using gn

1

=

2gN1

−

g p 1. The

Table 3

Value ranges of first moments of quark distributions, as obtained from the QCD fit when taking into account both statistical and systematic uncertain-ties, as detailed in the text.

First moment Value range at Q2_=3(GeV/c)2  [0.26,0.36]

u+ ¯u [0.82,0.85]

d+ ¯d [−0.45,−0.42]

s+ ¯s [−0.11,−0.08]

systematicuncertainties ofthemomentsincludetheuncertainties of PB, PT, f , D and F2. The uncertainties dueto the dominant additive systematicuncertainties forthe spin structure functions cancel to a large extent in the calculation of the first moments andarethusnottakenintoaccount.Inaddition,theuncertainties fromtheQCDevolutionandthosefromtheextrapolationare ob-tainedusingtheuncertaintiesgiveninSection6.Thefullmoments are given in Table 5. Note that also

N₁ is updated compared to Ref.[20]usingthenewQCDfit.

For the evaluation of the Bjorken sum rule, the procedure is slightly modified. Before evolving from the measured Q2 _to

Q2

₌

_{3 (GeV}

_/

_c)2_,gNS

Table 4

Contribution to the first moments of g1at Q2=3 (GeV/c)2. Limits in parentheses

are applied for the calculation of N

1. The uncertainties of the extrapolations are

negligible. x range p1 1N 0–0.0025 (0.004) 0.002 0.000 0.0025 (0.004)–0.7 0.134±0.003 0.047±0.003 0.7–1.0 0.003 0.001 Table 5

First moments of g1at Q2=3 (GeV/c)2using COMPASS data only. 1 δ stat1 δ syst 1 δ evol 1 Proton 0.139 ±0.003 ±0.009 ±0.005 Nucleon 0.049 ±0.003 ±0.004 ±0.004 Neutron −0.041 ±0.006 ±0.011 ±0.005 Table 6 Results of the fit of q3(x)at Q2 0=1 (GeV/c)2. Param. Value η3 1.24±0.06 α3 −0.11±0.08 β3 2.2+₋00..54 χ2_{/NDF 7.9/13} Table 7 First moment NS

1 at Q2=3 (GeV/c)2 from the

COM-PASS data with statistical uncertainties. Contributions from the unmeasured regions are estimated from the NLO fit to gNS1 . The statistical uncertainty is determined

using the error band shown in Fig. 7.

x range NS 1 0–0.0025 0.006±0.001 0.0025–0.7 0.170±0.008 0.7–1.0 0.005±0.002 0–1 0.181±0.008

iscalculatedfromtheprotonanddeuteron g1 data.23Sincethere isnomeasuredCOMPASSvalueofgd

1 correspondingtothenewg p 1 pointatx

=

0

.

0036,thevalueof gd

1 fromtheNLOQCDfitisused inthiscase.Thefitofg₁NS isperformedwiththesameprogramas discussed inthe previous section butfittingonly thenon-singlet distribution



q3

(

x

,

Q2

)

.Theparametersofthisfitaregivenin

Ta-ble 6 and a comparison of the fitted distribution with the data points is shown in Fig. 7. The statistical error band is obtained withthe samemethod asdescribed inthe previous section. The systematicuncertaintiesofthefitare muchsmallerthanthe sta-tisticalones.Theadditionalnormalisationuncertaintyisabout 8%.

Theintegral ofgNS

1 inthemeasuredrangeof0

.

0025

<

x

<

0

.

7 iscalculatedusingthedatapoints.The contributionfromthe un-measuredregion isextractedagainfromthe fit.Thevarious con-tributionsarelistedinTable 7andthedependenceof

NS₁ onthe lowerlimit oftheintegral isshowninFig. 8.The contributionof themeasured x rangetotheintegral correspondsto93

.

8% ofthe fullfirst moment,whilethe extrapolationto0 and1 amountsto 3

.

6% and 2

.

6%, respectively. Comparedto the previous result[3], thecontributionoftheextrapolationtox

=

0 isnowbyaboutone thirdsmallerthanbeforeduetothelarger x rangeofthepresent data.Thevalueoftheintegralforthefullx rangeis

NS₁

=

0

.

181

±

0

.

008

(

stat.

)

±

0

.

014

(

syst.

).

(14)

23 _{The results for g}NS

1 as well as for A p 1and g

p

1are available at HEPDATA [35].

Fig. 7. Values

of

xgNS

1 (x)at Q2=3 (GeV/c)2compared to the non-singlet NLO QCD

fit using COMPASS data only. The error bars are statistical. The open square at low-est x is

obtained with

gd

1 taken from the NLO QCD fit. The band around the curve

represents the statistical uncertainty of the NS fit, the band at the bottom the sys-tematic uncertainty of the data points. (Coloured version online.)

Fig. 8. Values

of

x1ming

NS

1 dx as a function of xmin. The open circle at x=0.7 is

obtained from the fit. The arrow on the left side shows the value for the full range, 0 ≤x≤1.

The total uncertainty of

NS₁ is dominated by the systematic uncertainty, which is calculated using the same contributions as used for the values in Table 5. The largest contribution stems fromthe uncertaintyofthebeampolarisation (5%);other contri-butions originatefromuncertaintiesinthecombinedprotondata, i.e.those oftarget polarisation,dilutionfactor anddepolarisation factor. Theuncertainties in thedeuteron datahavea smaller im-pact as the first moment of gd₁ is smaller than that of the pro-ton. The uncertainty due to the evolution to a common Q2 _is foundtobe negligiblewhenvarying Q₀2 between1

(

GeV

/

c

)

2 and 10

(

GeV

/

c

)

2.Theoverallresultagreeswellwithourearlierresult

NS₁

=

0

.

190

±

0

.

009

±

0

.

015 inRef.[3].

The result for

NS₁ is used to evaluate the Bjorken sum rule withEq.(13).Usingthecoefficient function CNS₁

(

Q2

)

atNLOand

α

s

=

0

.

337 at Q2

=

3

(

GeV

/

c

)

2,oneobtains

|

gA

/

gV| =1

.

22

±

0

.

05

(

stat.

)

±

0

.

10

(

syst.

).

(15) The comparison of the value of

|

gA

/

gV

|

from the present analysis and the one obtainedfrom neutron

β

decay,

|

gA

/

gV

|

=

1

.

2701

±

0

.

002[36],providesavalidationoftheBjorkensumrule withanaccuracyof9%.Notethatthecontributionof



g cancelsin Eq.(12)andhencedoesnotenter theBjorken sum.Higher-order perturbative corrections are expected to increase slightly the re-sult.ByusingthecoefficientfunctionCNS₁ atNNLOinsteadofNLO,

|

gA

/

gV

|

isfound tobe 1.25, closerto values stemming from the neutronweakdecay.

8. Conclusions

The COMPASS Collaboration performed new measurements of thelongitudinaldoublespinasymmetry Ap₁

(

x

,

Q2

)

andthe longi-tudinalspinstructurefunctiong₁p

(

x

,

Q2

)

oftheprotonintherange 0

.

0025

<

x

<

0

.

7 and in theDIS region, 1

<

Q2

_<

₁₉₀

₍

_GeV

_/

_c

₎

2_, thusextendingthepreviouslycoveredkinematicrange[3]towards largevaluesofQ2_{andsmallvaluesofx.Thenewdataimprovethe} statisticalprecisionofg₁p

(

x

)

byaboutafactoroftwoforx



0

.

02.

The world data for g₁p, g₁d and g₁n were used to perform a NLO QCD analysis, including a detailed investigation of system-aticeffects.Thisanalysisthusupdatesandsupersedestheprevious COMPASSQCDanalysis[20].Itwas foundthatthecontributionof quarkstothenucleonspin,



,liesintheinterval0.26and0.36 at Q2

=

3

(

GeV

/

c

)

2, wherethe interval limits reflectmainly the largeuncertaintyinthedeterminationofthegluoncontribution.

When combined withthe previously published results onthe deuteron [20], the new gp₁ data provide a new determination of the non-singlet spin structure function g₁NS and a new evalua-tion of the Bjorken sum rule, which is validated to an accuracy ofabout 9%.

Acknowledgements

We gratefully acknowledge the support ofthe CERN manage-ment andstaff andthe skill andeffort of the technicians ofour collaboratinginstitutes.Thisworkwasmadepossiblebythe finan-cialsupportofourfundingagencies.

Appendix A. Asymmetryresults

AsymmetryresultsaregiveninTables 8 and9.

Table 8

Values of Ap₁and gp₁as a function of x at

the measured values of

Q2_{. The first uncertainty is statistical, the second one is systematic.}

x range x y Q2_((GeV/c)2_{) A}p 1 g p 1 0.003–0.004 0.0036 0.800 1.10 0.020±0.017±0.007 0.60±0.51±0.22 0.004–0.005 0.0045 0.726 1.23 0.017±0.012±0.005 0.43±0.31±0.13 0.005–0.006 0.0055 0.677 1.39 0.020±0.012±0.005 0.44±0.26±0.11 0.006–0.008 0.0070 0.629 1.61 0.0244±0.0093±0.0041 0.43±0.16±0.08 0.008–0.010 0.0090 0.584 1.91 0.019±0.010±0.006 0.27±0.15±0.09 0.010–0.014 0.0119 0.550 2.33 0.0431±0.0086±0.0045 0.51±0.10±0.06 0.014–0.020 0.0167 0.518 3.03 0.0719±0.0091±0.0060 0.642±0.081±0.061 0.020–0.030 0.0244 0.492 4.11 0.0788±0.0097±0.0065 0.514±0.063±0.048 0.030–0.040 0.0346 0.477 5.60 0.088±0.013±0.010 0.424±0.063±0.054 0.040–0.060 0.0488 0.464 7.64 0.114±0.013±0.009 0.401±0.044±0.036 0.060–0.100 0.0768 0.450 11.7 0.166±0.014±0.013 0.376±0.031±0.033 0.100–0.150 0.122 0.432 18.0 0.264±0.019±0.019 0.372±0.027±0.029 0.150–0.200 0.172 0.416 24.8 0.318±0.027±0.024 0.298±0.025±0.024 0.200–0.250 0.223 0.404 31.3 0.337±0.036±0.030 0.224±0.024±0.021 0.250–0.350 0.292 0.389 39.5 0.389±0.037±0.029 0.166±0.016±0.013 0.350–0.500 0.407 0.366 52.0 0.484±0.055±0.051 0.095±0.011±0.010 0.500–0.700 0.570 0.339 67.4 0.73±0.11±0.09 0.0396±0.0058±0.0053 Table 9 Values of Ap1and g p

1as a function of x at

the measured values of

Q2. The first uncertainty is statistical, the second one is systematic.

x range x y Q2_((GeV/c)2_{) A}p 1 g p 1 0.003–0.004 0.0035 0.771 1.03 0.059±0.029±0.014 1.79±0.87±0.45 0.0036 0.798 1.10 −0.004±0.027±0.012 −0.12±0.81±0.37 0.0038 0.840 1.22 0.002±0.032±0.012 0.05±0.98±0.37 0.004–0.005 0.0044 0.641 1.07 0.006±0.021±0.008 0.15±0.50±0.19 0.0045 0.730 1.24 0.021±0.020±0.008 0.53±0.51±0.20 0.0046 0.817 1.44 0.023±0.022±0.011 0.60±0.59±0.28 0.005–0.006 0.0055 0.540 1.11 0.009±0.024±0.011 0.18±0.46±0.21 0.0055 0.661 1.36 0.026±0.020±0.008 0.56±0.42±0.17 0.0056 0.795 1.68 0.022±0.020±0.008 0.51±0.47±0.18 0.006–0.008 0.0069 0.442 1.14 0.033±0.020±0.009 0.50±0.32±0.14 0.0069 0.580 1.50 0.041±0.015±0.007 0.71±0.27±0.12 0.0071 0.757 2.02 0.006±0.014±0.007 0.12±0.27±0.13 0.008–0.010 0.0089 0.349 1.17 0.007±0.027±0.013 0.08±0.32±0.16 0.0089 0.483 1.62 0.029±0.018±0.007 0.40±0.25±0.10 0.0090 0.710 2.41 0.015±0.014±0.006 0.24±0.23±0.09 0.010–0.014 0.0116 0.278 1.21 0.044±0.026±0.013 0.41±0.24±0.12 0.0117 0.401 1.75 0.040±0.017±0.011 0.42±0.18±0.11 0.0120 0.656 2.92 0.044±0.011±0.005 0.56±0.14±0.07 0.014–0.020 0.0164 0.206 1.26 0.087±0.034±0.015 0.58±0.22±0.11 0.0165 0.313 1.92 0.100±0.020±0.011 0.77±0.16±0.09 0.0168 0.605 3.74 0.063±0.011±0.006 0.60±0.10±0.06 0.020–0.030 0.0239 0.177 1.55 0.072±0.030±0.016 0.36±0.15±0.08 0.0240 0.280 2.49 0.079±0.025±0.011 0.45±0.14±0.07 0.0246 0.575 5.16 0.079±0.011±0.008 0.545±0.077±0.061 0.030–0.040 0.0341 0.173 2.18 0.103±0.035±0.016 0.39±0.13±0.06 0.0343 0.272 3.50 0.099±0.041±0.018 0.43±0.18±0.08 0.0347 0.559 7.07 0.083±0.015±0.013 0.421±0.075±0.066