Final COMPASS results on the deuteron spin-dependent structure function g(1)(d) and the Bjorken sum rule

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Final

COMPASS

results

on

the

deuteron

spin-dependent

structure

function

g

₁

d

and

the

Bjorken

sum

rule

C. Adolph

h

,

M. Aghasyan

y

,

R. Akhunzyanov

g

,

M.G. Alexeev

aa

,

G.D. Alexeev

g

,

A. Amoroso

aa

,

ab

,

V. Andrieux

ac

,

u

,

N.V. Anfimov

g

,

V. Anosov

g

,

K. Augsten

g

,

s

,

W. Augustyniak

ad

,

A. Austregesilo

p

,

C.D.R. Azevedo

a

,

B. Badełek

ae

,

F. Balestra

aa

,

ab

,

M. Ball

c

,

J. Barth

d

,

R. Beck

c

,

Y. Bedfer

u

,

J. Bernhard

m

,

j

,

K. Bicker

p

,

j

,

E.R. Bielert

j

,

R. Birsa

y

,

M. Bodlak

r

,

P. Bordalo

l

,

1

,

F. Bradamante

x

,

y

,

C. Braun

h

,

A. Bressan

x

,

y

,

M. Büchele

i

,

W.-C. Chang

v

,

C. Chatterjee

f

,

M. Chiosso

aa

,

ab

,

I. Choi

ac

,

S.-U. Chung

p

,

2

,

A. Cicuttin

z

,

y

,

M.L. Crespo

z

,

y

,

Q. Curiel

u

,

S. Dalla Torre

y

,

S.S. Dasgupta

f

,

S. Dasgupta

x

,

y

,

O.Yu. Denisov

ab

,

∗

,

L. Dhara

f

,

S.V. Donskov

t

,

N. Doshita

ag

,

Ch. Dreisbach

p

,

V. Duic

x

,

W. Dünnweber

3

,

M. Dziewiecki

af

,

A. Efremov

g

,

P.D. Eversheim

c

,

W. Eyrich

h

,

M. Faessler

3

,

A. Ferrero

u

,

M. Finger

r

,

M. Finger jr.

r

,

H. Fischer

i

,

C. Franco

l

,

N. du Fresne von Hohenesche

m

,

J.M. Friedrich

p

,

V. Frolov

g

,

j

,

E. Fuchey

u

,

F. Gautheron

b

,

O.P. Gavrichtchouk

g

,

S. Gerassimov

o

,

p

,

J. Giarra

m

,

F. Giordano

ac

,

I. Gnesi

aa

,

ab

,

M. Gorzellik

i

,

4

,

S. Grabmüller

p

,

A. Grasso

aa

,

ab

,

M. Grosse Perdekamp

ac

,

B. Grube

p

,

T. Grussenmeyer

i

,

A. Guskov

g

,

F. Haas

p

,

D. Hahne

d

,

G. Hamar

x

,

y

,

D. von Harrach

m

,

F.H. Heinsius

i

,

R. Heitz

ac

,

F. Herrmann

i

,

N. Horikawa

q

,

5

,

N. d’Hose

u

,

C.-Y. Hsieh

v

,

6

,

S. Huber

p

,

S. Ishimoto

ag

,

7

_,

_{A. Ivanov}

aa

,

ab

_,

_{Yu. Ivanshin}

g

_,

_{T. Iwata}

ag

_,

_{V. Jary}

s

_,

_{R. Joosten}

c

_,

P. Jörg

i

,

E. Kabuß

m

A. Kerbizi

x

,

y

,

B. Ketzer

c

,

G.V. Khaustov

t

,

Yu.A. Khokhlov

t

,

8

,

9

,

Yu. Kisselev

g

,

F. Klein

d

,

K. Klimaszewski

ad

,

J.H. Koivuniemi

b

,

V.N. Kolosov

t

,

K. Kondo

ag

,

K. Königsmann

i

,

I. Konorov

o

,

p

,

V.F. Konstantinov

t

,

A.M. Kotzinian

aa

,

ab

,

O.M. Kouznetsov

g

,

M. Krämer

p

,

P. Kremser

i

,

F. Krinner

p

,

Z.V. Kroumchtein

g

,

23

,

Y. Kulinich

ac

,

F. Kunne

u

,

K. Kurek

ad

,

R.P. Kurjata

af

,

A.A. Lednev

t

,

23

,

A. Lehmann

h

,

M. Levillain

u

,

S. Levorato

y

,

Y.-S. Lian

v

,

10

,

J. Lichtenstadt

w

,

R. Longo

aa

,

ab

,

A. Maggiora

ab

,

A. Magnon

ac

,

N. Makins

ac

,

N. Makke

x

,

y

_,

_{G.K. Mallot}

j

,

∗

_,

_{B. Marianski}

ad

_,

_{A. Martin}

x

,

y

_,

_{J. Marzec}

af

_,

_{J. Matoušek}

r

,

y

_,

H. Matsuda

ag

,

T. Matsuda

n

,

G.V. Meshcheryakov

g

,

M. Meyer

ac

,

u

,

W. Meyer

b

,

Yu.V. Mikhailov

t

,

M. Mikhasenko

c

,

E. Mitrofanov

g

,

N. Mitrofanov

g

,

Y. Miyachi

ag

,

A. Nagaytsev

g

,

F. Nerling

m

,

D. Neyret

u

,

J. Nový

s

,

j

,

W.-D. Nowak

m

,

G. Nukazuka

ag

,

A.S. Nunes

l

,

A.G. Olshevsky

g

,

I. Orlov

g

,

M. Ostrick

m

,

D. Panzieri

ab

,

11

,

B. Parsamyan

aa

,

ab

,

S. Paul

p

,

J.-C. Peng

ac

,

F. Pereira

a

,

M. Pešek

r

,

D.V. Peshekhonov

g

,

N. Pierre

m

,

u

,

S. Platchkov

u

,

J. Pochodzalla

m

,

V.A. Polyakov

t

,

J. Pretz

d

,

12

,

M. Quaresma

l

,

C. Quintans

l

,

S. Ramos

l

,

1

_,

_{C. Regali}

i

_,

_{G. Reicherz}

b

_,

_{C. Riedl}

ac

_,

_{M. Roskot}

r

_,

_{N.S. Rossiyskaya}

g

_,

D.I. Ryabchikov

t

,

9

,

A. Rybnikov

g

,

A. Rychter

af

,

R. Salac

s

,

V.D. Samoylenko

t

,

A. Sandacz

ad

,

C. Santos

y

,

S. Sarkar

f

,

I.A. Savin

g

,

T. Sawada

v

,

G. Sbrizzai

x

,

y

,

P. Schiavon

x

,

y

,

K. Schmidt

i

,

4

,

H. Schmieden

d

,

K. Schönning

j

,

13

,

E. Seder

u

,

y

,

A. Selyunin

g

,

L. Silva

l

,

L. Sinha

f

,

S. Sirtl

i

,

M. Slunecka

g

,

J. Smolik

g

,

A. Srnka

e

,

D. Steffen

j

,

p

,

M. Stolarski

l

,

O. Subrt

j

,

s

,

M. Sulc

k

,

H. Suzuki

ag

,

5

,

A. Szabelski

ad

,

y

,

T. Szameitat

i

,

4

,

P. Sznajder

ad

,

S. Takekawa

aa

,

ab

,

*

Correspondingauthors.

E-mailaddresses:oleg.denisov@cern.ch(O.Yu. Denisov),gerhard.mallot@cern.ch(G.K. Mallot),malte.christian.wilfert@cern.ch(M. Wilfert).

http://dx.doi.org/10.1016/j.physletb.2017.03.018

0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

M. Tasevsky

g

,

S. Tessaro

y

,

F. Tessarotto

y

,

F. Thibaud

u

,

A. Thiel

c

,

F. Tosello

ab

,

V. Tskhay

o

,

S. Uhl

p

,

A. Vauth

j

,

J. Veloso

a

,

M. Virius

s

,

J. Vondra

s

,

S. Wallner

p

,

T. Weisrock

m

,

M. Wilfert

m

,

∗

,

R. Windmolders

d

,

J. ter Wolbeek

i

,

4

,

K. Zaremba

af

,

P. Zavada

g

,

M. Zavertyaev

o

,

E. Zemlyanichkina

g

,

N. Zhuravlev

g

,

M. Ziembicki

af

,

A. Zink

h

,

a_{UniversityofAveiro,Dept.ofPhysics,3810-193Aveiro,Portugal}

b_{UniversitätBochum,InstitutfürExperimentalphysik,44780Bochum,Germany}14,15

c_{UniversitätBonn,Helmholtz-InstitutfürStrahlen- undKernphysik,53115Bonn,Germany}14

d_{UniversitätBonn,PhysikalischesInstitut,53115Bonn,Germany}14

e_{InstituteofScientificInstruments,ASCR,61264Brno,CzechRepublic}16

f_{MatrivaniInstituteofExperimentalResearch&Education,Calcutta-700 030,India}17

g_{JointInstituteforNuclearResearch,141980Dubna,Moscowregion,Russia}18

h_{UniversitätErlangen–Nürnberg,PhysikalischesInstitut,91054Erlangen,Germany}14

i_{UniversitätFreiburg,PhysikalischesInstitut,79104Freiburg,Germany}14,15

j_{CERN,1211Geneva23,Switzerland}

k_{TechnicalUniversityinLiberec,46117Liberec,CzechRepublic}16

l_{LIP,1000-149Lisbon,Portugal}19

m_{UniversitätMainz,InstitutfürKernphysik,55099Mainz,Germany}14

n_{UniversityofMiyazaki,Miyazaki889-2192,Japan}20

o_{LebedevPhysicalInstitute,119991Moscow,Russia}

p_{TechnischeUniversitätMünchen,PhysikDept.,85748Garching,Germany}14,3

q_{NagoyaUniversity,464Nagoya,Japan}20

r_{CharlesUniversityinPrague,FacultyofMathematicsandPhysics,18000Prague,CzechRepublic}16

s_{CzechTechnicalUniversityinPrague,16636Prague,CzechRepublic}16

t_{StateScientificCenterInstituteforHighEnergyPhysicsofNationalResearchCenter‘KurchatovInstitute’,142281Protvino,Russia} u_{IRFU,CEA,UniversitéParis-Saclay,91191Gif-sur-Yvette,France}15

v_{AcademiaSinica,InstituteofPhysics,Taipei11529,Taiwan}

w_{TelAvivUniversity,SchoolofPhysicsandAstronomy,69978TelAviv,Israel}21

x_{UniversityofTrieste,Dept.ofPhysics,34127Trieste,Italy} y_{TriesteSectionofINFN,34127Trieste,Italy}

z_{AbdusSalamICTP,34151Trieste,Italy}

aa_{UniversityofTurin,Dept.ofPhysics,10125Turin,Italy} ab_{TorinoSectionofINFN,10125Turin,Italy}

ac_{UniversityofIllinoisatUrbana-Champaign,Dept.ofPhysics,Urbana,IL61801-3080,USA} ad_{NationalCentreforNuclearResearch,00-681Warsaw,Poland}22

ae_{UniversityofWarsaw,FacultyofPhysics,02-093Warsaw,Poland}22

af_{WarsawUniversityofTechnology,InstituteofRadioelectronics,00-665Warsaw,Poland}22

ag_{YamagataUniversity,Yamagata992-8510,Japan}20

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received15December2016 Receivedinrevisedform3March2017 Accepted10March2017

Availableonline16March2017 Editor: M.Doser

Keywords:

COMPASS

Deepinelasticscattering Spin

Structurefunction Partonhelicitydistributions

Finalresultsarepresentedfromtheinclusivemeasurementofdeep-inelasticpolarised-muonscattering onlongitudinallypolariseddeuteronsusinga6_{LiDtarget.Thedataweretakenat160 GeVbeamenergy} andtheresultsareshownforthekinematicrange1(GeV/c)2_<Q2_<100(GeV/c)2_{inphotonvirtuality,} 0.004<x<0.7 in the Bjorkenscaling variable and W>4GeV/c2 _{inthe mass ofthe hadronic final} state.Thedeuterondouble-spinasymmetryAd₁andthedeuteronlongitudinal-spinstructurefunctiongd₁ are presented inbinsofx and Q2_{.Towardslowestaccessiblevalues ofx, g}d

1 decreasesand becomes consistent with zero within uncertainties. The presented final gd

1 values together with the recently published final gp₁ values ofCOMPASSare used to againevaluate the Bjorken sum ruleand perform theQCDfittotheg1worlddataatnext-to-leadingorderofthestrongcouplingconstant.Inbothcases, changesincentralvaluesoftheresultingnumbersarewellwithinstatisticaluncertainties.The flavour-singletaxialchargea0,whichisidentifiedintheMS renormalisationschemewiththetotalcontribution of quark helicities to the nucleonspin, is extracted atnext-to-leading orderaccuracy from only the COMPASSdeuterondata:a0(Q2=3(GeV/c)2)

=

0.32

±

0.02stat±0.04syst±0.05evol.Togetherwiththe recentresultsontheprotonspinstructurefunctiong₁p,theresultsongd

1constitutetheCOMPASSlegacy onthemeasurementsofg1throughinclusivespin-dependentdeepinelasticscattering.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1 _{AlsoatInstitutoSuperiorTécnico,UniversidadedeLisboa,Lisbon,Portugal.}

2 _{AlsoatDept.ofPhysics,PusanNationalUniversity,Busan609-735,RepublicofKoreaandatPhysicsDept.,BrookhavenNationalLaboratory,Upton,NY11973,USA.} 3 _{SupportedbytheDFGclusterofexcellence‘OriginandStructureoftheUniverse’(www.universe-cluster.de).}

4 _{SupportedbytheDFGResearchTrainingGroupProgrammes1102and2044.} 5 _{AlsoatChubuUniversity,Kasugai,Aichi487-8501,Japan.}

6 _{AlsoatDept.ofPhysics,NationalCentralUniversity,300JhongdaRoad,Jhongli32001,Taiwan.} 7 _{AlsoatKEK,1-1Oho,Tsukuba,Ibaraki305-0801,Japan.}

8 _{AlsoatMoscowInstituteofPhysicsandTechnology,MoscowRegion,141700,Russia.} 9 _{SupportedbyPresidentialgrantNSh-999.2014.2.}

1. Introduction

About a quarter of a century ago, measurements of the spin-dependent structure function gp₁ by EMC[1]led to the very sur-prising result that the quark spin contribution to the nucleon spin of 1/2 might be very small or even vanishing, albeit with large experimental uncertainties. This result initiated enormous experi-mental and theoretical activities to study the spin structure of the nucleon. In subsequent measurements by SMC[2], an upgraded ap-paratus was used to confirm with better precision that only about one third of the spin of the nucleon is made up by quark spins. This observation is supported by recent lattice QCD simulations

[3].

In the last two decades, several new experiments were set up at various laboratories to study the longitudinal spin structure of the nucleon in even more detail. These experiments included COM-PASS at CERN using the CERN SPS muon beam line at energies 160 GeV and 200 GeV, HERMES at DESY using the 27

.

5 GeV elec-tron beam of HERA, many experiments at the 6 GeV electron beam of Jefferson Laboratory, as well as PHENIX and STAR at the proton-proton collider RHIC with a centre of mass energy of 270 GeV. Except for the latter two, in all other experiments the longitudinal spin structure of the nucleon was studied by inclusive measure-ments of spin-dependent deep-inelastic lepton-nucleon scattering (DIS) using longitudinally polarised beams and targets, in partic-ular by measuring double-spin cross-section asymmetries. More details can be found in recent reviews, see e.g. Ref.[4].

The measured value of the parton helicity contribution to the proton spin is very sensitive to the minimal experimental acces-sible value of the Bjorken-x variable. Therefore measurements at low x are

crucial to understand the spin structure of the nucleon.

According to theoretical expectations, new contributions to the DGLAP QCD evolution, e.g. double logarithmic terms [5,6], may be important in this region. Perturbative QCD is considered to be ap-plicable for values of Q2 _{as low as 1}

₍

_GeV

_/

_c

₎

2_{. At COMPASS, using}

a 160 GeV muon beam, this corresponds to a minimal value of x

equal to 0.004.

In this Letter, results are presented on the longitudinal double-spin asymmetry Ad

1 and the longitudinal spin structure function

gd

1 of the deuteron. They are obtained from data taken in 2006

with the CERN 160 GeV longitudinally polarised muon beam and a longitudinally polarised 6_{LiD target. These results are described}

and compared to those published earlier [7] for the 2002–2004 data. The analysis of the combined 2002–2006 data yields the fi-nal COMPASS results on Ad₁ and gd₁. Moreover, the combined data set analysed in this work extends to high Q2_{values that were}

for-11 _{AlsoatUniversityofEasternPiedmont,15100Alessandria,Italy.}

12 _{Presentaddress: RWTH Aachen University, III.Physikalisches Institut, 52056} Aachen,Germany.

13 _{Presentaddress:UppsalaUniversity,Box516,75120Uppsala,Sweden.} 14 _{SupportedbytheGermanBundesministeriumfürBildungundForschung.} 15 _{SupportedbyEUFP7(HadronPhysics3,GrantAgreementnumber283286).} 16 _{SupportedbyCzechRepublicMEYSGrantLG13031.}

17 _{SupportedbySAIL(CSR),Govt.ofIndia.} 18 _{SupportedbyCERN-RFBRGrant12-02-91500.}

19 _{Supported bythe Portuguese FCT – Fundação para a Ciência e Tecnologia,} COMPETEandQREN,GrantsCERN/FP109323/2009,116376/2010,123600/2011and CERN/FIS-NUC/0017/2015.

20 _{Supportedbythe MEXTand the JSPS under the Grants No.18002006, No.} 20540299andNo.18540281;DaikoFoundationandYamadaFoundation.

21 _{SupportedbytheIsraelAcademyofSciencesandHumanities.} 22 _{SupportedbythePolishNCNGrant2015/18/M/ST2/00550.} 23 _Deceased.

merly only reached by SMC, thereby improving considerably the statistical accuracy. Together with the results on the proton spin structure function gp₁ [8,9], the results for gd₁ constitute the COM-PASS legacy on the measurements of g1 through inclusive DIS.

The Letter is organised as follows. Experimental set-up and data analysis are described in Sect.2. The physics context of the analysis and details on the calculation of asymmetries are given in Sect.3. In Sect.4, the results are presented and interpreted. Summary and conclusions are given in Sect.5. The reader is referred to Ref.[10]

for a detailed description of the analysis. 2. Experimentalset-upanddataanalysis

The COMPASS spectrometer used in 2002–2004 and the up-grades of the polarised target solenoid and the RICH detector per-formance in 2005 are described in detail in Ref.[12]. In 2006 the target material was 6_{LiD contained in three cells instead of two.}

They were located along the beam one after the other and had a diameter of 3 cm. The two outermost cells had a length of 30 cm and the central cell was 60 cm long. The deuteron polarisation in

6_{LiD was P}

T

≈

0

.

52, and the direction of the target polarisation in

the outer cells was opposite to that of the central one. The po-larisation direction was inverted on a regular basis by rotating the direction of the target solenoid magnetic field. During the data tak-ing, the direction of the polarisation with respect to the solenoid field was inverted by repolarisation in opposite directions keeping the solenoid field unchanged. The tertiary M2 beam of the CERN SPS delivered a naturally polarised muon beam with a polarisation of PB

≈

0

.

8. The nominal momentum was 160 GeV

/

c with a

mo-mentum spread of 5%. Momentum and trajectory of each beam particle were measured by sets of scintillator hodoscopes, scin-tillating fibre and silicon detectors. The particles produced in an interaction were detected in a two-stage open forward spectrom-eter with large momentum and angular acceptance. Each stage contained a dipole magnet complemented with various tracking detectors (scintillating fibre detectors, micropattern gaseous detec-tors, multiwire proportional chambers, drift chambers, straw de-tectors), as well as hadron and electromagnetic calorimeters. In the first stage, a RICH detector was available for hadron identifi-cation. Scattered muons were detected by drift tube planes and multiwire proportional chambers located behind iron and con-crete absorbers. Two types of triggers were used in this analysis. The “inclusive” trigger was based on a signal from a combina-tion of hodoscope signals from the scattered muon. The “semi-inclusive” triggers required an energy deposition in one of the calorimeters with an optional coincidence with the inclusive trig-ger.

Events with a reconstructed interaction point in one of the three target cells are selected requiring at least a reconstructed incoming muon and a scattered muon. The measured momentum of the incident muon has to be in the range 140 GeV

/

c

<

pB

<

180 GeV

/

c,

and the extrapolated beam track has to cross all target

cells to equalise the flux through them. The amount of unpolarised material surrounding the polarised material is minimised by a ra-dial cut on the vertex position of r

<

1

.

4 cm. The scattered muon is identified by requiring that it has passed more than 30 radia-tion lengths and points to the hodoscope that triggered the event. In addition, kinematic constraints on the scattering process are ap-plied. A photon virtuality of Q2

_>

₁

₍

_GeV

_/

_c

₎

2 _{is required and the}

relative virtual-photon energy has to be in the range 0

.

1

<

y

<

0

.

9. Here, the lower limit removes events that are difficult to recon-struct, and the upper limit removes events, the kinematics of which are dominated by radiative effects. These selection criteria

lead to the kinematic range 0

.

004

<

x

<

0

.

7 and to a minimal mass of the hadronic final state of W

>

4 GeV

/

c2. The final sample con-sists of 46 million events.

3. Asymmetrycalculation

The longitudinal double-spin asymmetry for one-photon ex-change in inclusive DIS on the deuteron, μd →

μ

X, is defined as a function of x and Q2 as follows:

Ad₁

=

σ

T 0

−

σ

2T

2

σ

T

.

(1)

Here,

σ

T

J is the γ∗-deuteron absorption cross section for total spin projection J in the direction of the virtual photon γ∗ and

σ

T

_{= (}

_σ

T

0

+

σ

1T

+

σ

2T

)/

3 the deuteron photoabsorption cross

sec-tion for transverse virtual photons. This asymmetry is derived from the asymmetry between the cross sections for parallel and an-tiparallel oriented longitudinal spins of beam particle and target nucleon24_: Ad_LL

=

σ

↑↓

₋

_σ

↑↑

σ

↑↓

+

σ

↑↑

=

D

(

A d 1

+

η

Ad2

).

(2)

Here, also the contribution from the transverse spin asymmetry Ad₂

is taken into account. The factors

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)

depend only on the kinematics of the process, with γ

=

2Mx

/



Q2_{. The symbols m and M denote}

_the

_{mass of the muon and}

the nucleon, respectively. The factor R in the depolarisation fac-tor D represents

the ratio of the cross sections for the absorption

of a longitudinally and a transversely polarised photon by a nu-cleon. In COMPASS kinematics, the factor ηand the asymmetry A2

are both small, and hence the contribution ηA2is neglected in the

calculation of A1and g1.

For the calculation of the asymmetry, the number of events in each target cell for both polarisation directions can be expressed as

Ni

=

ai

φ

ini

σ

(

1

+

PBPTf D Ad1

) ,

i

=

o1

,

c1

,

o2

,

c2

.

(5)

Here, ai is the acceptance,

φi

the incoming flux, ni the number of target nucleons, σ the spin-averaged cross section and f the di-lution factor. There are four equations describing the two solenoid field directions (1, 2) for the combined outer cells (o) and the cen-tral cell (c). They are combined into one second-order equation in A1for the ratio

(

No1Nc2

)/(

No2Nc1

)

, where acceptance and flux 24 _{Whileforaspin-1/2 targetthefirstequalityinEq.(2)isstrict,foraspin-1} targetthereisanextracontributioninthedenominatoroftheasymmetryAd

LL= σ↑↓−σ↑↑

σ↑↓+σ↑↑+σ↑0, whichis connectedto the structurefunction b1. This functionis expectedtobesmall[13],asalsoconfirmedbyameasurement[14],andhence neglectedhere.

Table 1

SummaryforthesystematicuncertaintyofA1.

Beam polarisation PB/PB 5% Target polarisation PT/PT 5% Depolarisation factor D(R)/D(R) 2–3% Dilution factor f/f 2–3% Total Amult 1 0.08·Ad1 False asymmetry Afalse <0.75· Astat1 Transverse asymmetry η·Ad

2 <10−4

Rad. corrections ARC ₁₀−5_–10−3

Fig. 1. ComparisonbetweentheresultsonAd

1obtainedfromthe2006datasetand thepreviousresultsfromCOMPASS.

cancel. The asymmetry is calculated for periods of stable data tak-ing, which are combined using the weighted mean. In the asym-metry calculation, each event is used with a weight factor in order to minimise the statistical uncertainty:

w

=

PBf D

.

(6)

Systematic uncertainties are calculated taking into account mul-tiplicative and additive contributions. The multiplicative contri-bution



Amult

1 comprises the uncertainties on beam and target

polarisations and the uncertainties on depolarisation and dilution factors. The size of each of these contributions is shown in the up-per part of Table 1. The lower part of the table shows the additive contributions from i) possible false asymmetries, ii) the neglect of the transverse asymmetry A2 and iii) the uncertainty on

spin-dependent radiative corrections. False asymmetries are investigated using two methods. In one method, possible false asymmetries are studied by calculating the asymmetry between cells with the same polarisation direction, i.e. between both outermost target cells and for the two halves of the central cell. Both asymmetries are found to be consistent with zero. In the other method, “pulls” [15] are used to check for time-dependent effects. Here the asymmetry is calculated for each subsample and compared to the final asymme-try. No significant broadening is observed in these distributions. The statistical limitation of this method leads to an uncertainty between 38% and 75% of the statistical uncertainty. This represents the largest additive contribution.

4. Results

The double-spin asymmetry Ad₁ and the spin-dependent struc-ture function gd₁ are calculated in bins of x and Q2. In Fig. 1 the results in bins of x obtained

from the 2006 data set are compared

to the results from the 2002–2004 data[7], which demonstrates the good agreement between both data sets (the χ2 _probability

is 63%). The 2006 data increase the statistics of the 2002–2004 data by approximately 50%. The results from both data sets are combined using the weighted mean. In Fig. 2, the combined COM-PASS results on Ad₁ are compared to the world data on Ad₁ at the measured values of Q2_{. All data sets agree well with one another.}

Fig. 2. ComparisonbetweenthecombinedCOMPASSresultsonAd

1andtheworld data(CLAS[16],HERMES[17],SMC[2],E155[18]andE143[19]).Alldatapoints areshownattheirmeasuredQ2_values.

The data confirm the well-known weak Q2 _{dependence of the}

asymmetry. This is also illustrated in Fig. 3, which shows the Q2

dependence of the COMPASS data for each x bin.

No clear

depen-dence on Q2 is visible in any x bin.The numerical values of the combined data for Ad₁

(

x

)

and Ad₁

(

x

,

Q2

)

are given in Tables A.1 and A.2of the appendix.

The spin-dependent structure function g₁dis calculated from the asymmetry Ad₁using g₁d

(

x

,

Q2

)

=

F d 2

(

x

,

Q2

)

2x

(

1

+

R

(

x

,

Q2

₎₎

A d 1

(

x

,

Q2

) .

(7)

The parametrisation of the unpolarised structure function F₂d is taken from Ref. [2] and the parametrisation of the ratio R is

taken from Ref.[20]. The x dependence

of the

structure function is shown in Fig. 4 together with the results from SMC [2] that were obtained at a higher beam energy of 190 GeV. In the figure, the two COMPASS data points at lowest x are

obtained as averages

from the four lowest x bins

used

in this analysis. The systematic uncertainties are shown by bands at the bottom. The COMPASS data do not support large negative values of the structure func-tion at low x,

an indication of which may be seen in the SMC data.

Instead, g₁d is compatible with zero for x decreasingtowards the lower limit of the measured range.

The new results on the spin-dependent structure function gd₁, which are shown in Fig. 5 together with the world data in bins of x and Q2_{, constitute the final COMPASS results and hence}

su-Fig. 3. Results on Ad

Fig. 4. ComparisonbetweenSMC[2]andcombinedCOMPASSresultson gd 1.The systematicuncertaintyisillustratedbythebandsatthebottom.Alldatapointsare shownattheirmeasuredQ2_values.

Fig. 5. Worlddata onthespin-dependentstructurefunctiongd

1 asafunctionof Q2_{forvariousvaluesofx withthecombinedCOMPASSdataasfilledcircles.The} linesrepresentthe Q2 _{dependenceforeach valueofx asdetermined fromthe} updatedNLOQCDfittotheworlddata.Thedashedpartsrepresenttheregionwith

W2_<10(GeV/c2₎2_.

persede the ones published in Ref.[7]. They improve the statistical precision of the combined world data on gd

1, in particular at low x

where SMC is the only other experiment that contributes.

The NLO QCD fit on the g1 world data described in detail in

Ref.[9]is repeated using the updated results for gd₁. The fit results are shown as curves in Fig. 5 for the various x bins. Compared to the previous analysis, the changes in central values of resulting parameters are of the order of statistical uncertainties. The param-eters of the QCD fit are available together with the deuteron results on HepData[21].

The presented final gd₁ values together with the final COMPASS results on gp₁ [8,9]are used to re-evaluate the Bjorken sum rule as described in the same reference. The results

Table 2

ContributionstoN

1 atQ2=3(GeV/c)2 with statistical uncertainties from the COMPASSdata. x range N 1 0÷0.004 −0.0015÷0.001 0.004÷0.7 0.045±0.002 0.7÷1.0 0.001



₁NS

=

0

.

192

±

0

.

007stat

±

0

.

015syst and

|

gA

/

gV

| =

1

.

29

±

0

.

05stat

±

0

.

10syst (8)

agree within statistical errors with the previously published ones.

The new combined data are also used to update the results for the first moment of the spin-dependent structure function of the nucleon,



N₁

(

Q2

₎

₌



1

0 g1d

(

x

,

Q2

)/(

1 −1

.

5

ω

D

)

dx. Here ωD

=

0

.

05 ±0

.

02[22]is the correction for the D-state admixture in the deuteron. The first moment is calculated by evolving the values of g₁d to the common value Q2

₌

₃

₍

_GeV

_/

_c

₎

2_{. From these values}

the contribution to the first moment from the measured x range

is calculated. The contributions from the unmeasured regions are estimated using the parameterisations of parton distributions from our NLO QCD analysis [9]. For the region 0

<

x

<

0

.

004 the contri-bution lies between −0

.

0015 and 0.001. Such a small contribution can be expected as gd

1is consistent with zero for x

<

0

.

02. It is also

consistent with zero at scales Q2

<

1

(

GeV

/

c

)

2 where the mea-surements extend to even lower values of x [11]. For the region 0

.

7

<

x

<

1 the contribution to the first moment of gd

1 is as small

and equal to 0.001. The contributions from the different x ranges

are shown in Table 2. The updated value of the first moment from COMPASS data alone is:



₁N

(

Q2

=

3

(

GeV

/

c

)

2

)

=

0

.

046

±

0

.

002stat

±

0

.

004syst

±

0

.

005evol

.

(9)

The systematic uncertainty contains the bin-to-bin correlated uncertainties of PB, PT, f , D,

ω

D and F2. The uncertainties of PB

and PT dominate, while the impact of possible false asymmetries

largely cancels in the discussed integral. An uncertainty of 100% is used for the contribution from the large-x extrapolation, while for the low-x extrapolation the uncertainty is taken as half of the full range, i.e. 0.00125. Both are included in the evolution uncer-tainty.

All presently available experimental information supports the observation that gd₁ vanishes when x decreasesdown to the low-est accessible values. As can be seen in Fig. 6, the first moment of gd

1 measured from only the COMPASS deuteron data approaches

its asymptotic value already in the experimentally accessible region for Q2

₌

₃

₍

_GeV

_/

_c

₎

2_{. It can hence be used for physics}

interpreta-tion without using proton data and without invoking the Bjorken sum rule.

The structure function g₁das physical observable is factorisation-scheme independent, whereas its representation as convolution(s) of quark, anti-quark, and gluon helicity distributions with respec-tive Wilson coefficient functions[23,24]involves a possible scheme dependence. In the ‘modified minimal subtraction’ (MS) factori-sation scheme [25], the first moment of the gluon coefficient function vanishes, and hence the first moment



d₁ does not de-pend on the gluon helicity distribution. This allows for the direct determination of the flavour-singlet axial charge a0from the

COM-PASS



₁d result using only the axial charge a8 as an additional

a0

(

Q2

)

=

1



CMS_S

(

Q2

₎



9



₁N

−

1 4a8



C MS NS

(

Q2

)



(10) Here,



CMS

S

(

Q2

)

and



CNSMS

(

Q2

)

are the singlet and non-singlet

coefficient functions, which are calculated in perturbative QCD in Ref. [26,27]. In the MS factorisation scheme, a0 is

identi-fied with the total quark contribution to the nucleon spin: a0 MS

=



= (

u

+ ¯

u

)

+ (

d

+ ¯

d

)

+ (

s

+ ¯

s

)

. Here,



(−)

f is the first moment of helicity distribution of flavour- f quarks. Assum-ing SU(3) flavour symmetry, the value a8

=

0

.

585 ±0

.

025 [28]

is used. With αs

=

0

.

337 ±0

.

012 for Q2

=

3

(

GeV

/

c

)

2 and the corresponding NLO value for



C_SMS

(

Q2

)

= 

C_NSMS

(

Q2

)

=

0

.

893 the flavour-singlet axial charge is obtained using



N₁ as obtained in Eq.(9):

a0

(

Q2

=

3

(

GeV

/

c

)

2

)

=

0

.

32

±

0

.

02stat

±

0

.

04syst

±

0

.

05evol

.

(11)

The largest contribution to the total uncertainty originates from the uncertainties in the evolution of gd₁ to a common value of

Q2. This is due to the large uncertainty of the polarised gluon

Fig. 6. Values of_x1

ming

d

1/(1−1.5ωD)dx asafunctionofxmin.Theopencircleat x=0.7 isobtainedfromthefit.Thearrowontheleftsideshowsthevalueforthe fullrange,0≤x≤1.

distribution obtained in the fits. The extrapolation uncertainty is given by the extrapolation uncertainty of



₁N as explained above, and is included in the evolution uncertainty. This inde-pendent result on a0 is consistent with the value of a0

ob-tained from the COMPASS NLO QCD fit [9] of the world data. Note the remarkably good statistical and systematic accuracy of this result obtained from only the COMPASS deuteron data when comparing to the corresponding accuracy of the fit result

[9].

5. Summaryandconclusions

We have presented new results on the longitudinal spin struc-ture function gd₁ from data taken in 2006 and we have combined this data with our previous measurements. All data were taken us-ing the 160 GeV CERN muon beam and a longitudinally polarised

6_{LiD target. The results cover the kinematic range 0}

_.

₀₀₄

_<

_x

_<

₀

_.

_7,

1

(

GeV

/

c

)

2

<

Q2

_<

₁₀₀

₍

_GeV

_/

_c

₎

2 _{and W}

_>

_{4 GeV}

_/

_c2_{. The}

double-spin asymmetry is studied in bins of x and Q2. The combined results for gd

1 at low x

(

x

<

0

.

03

)

improve considerably the

pre-cision compared to the only existing result in this region, which originates from SMC. Now, gd

1 appears consistent with zero at the

presently lowest accessible values of x.The combined set of data was included in our NLO QCD fit to the g₁p, gd

1 and gn1 world

data. In addition, a re-evaluation of the Bjorken sum rule was per-formed using only COMPASS results. Both for the QCD NLO fit and the Bjorken sum rule, the new values stay compatible with the published ones within statistical uncertainties. The analysis of the COMPASS deuteron data alone leads to a NLO determination of the flavour-singlet axial charge a0

=

0

.

32

±

0

.

02stat

±

0

.

04syst

±

0

.

05evol

at Q2

₌

₃

₍

_GeV

_/

_c

₎

2 _{from the first moment of g}d

1. Together with

the results on the proton spin structure function gp₁ [8,9], the re-sults for gd₁ constitute the COMPASS legacy on the measurements of g1.

Acknowledgements

We gratefully acknowledge the support of the CERN manage-ment and staff and the skill and effort of the technicians of our collaborating institutes. This work was made possible by the finan-cial support of our funding agencies.

Appendix A. Appendix

The results for Ad₁and gd₁ are given in Tables A.1 andA.2.

Table A.1 Valuesfor Ad

1andgd1 asafunctionofx atthemeasuredvaluesofQ2forthecombined2002–2006data.Thefirst uncertaintyisstatistical,thesecondoneissystematic.

x range x Q2_((GeV/c)2_{) A}d 1 gd1 0.004–0.005 0.0046 1.10 −0.0054±0.0074±0.0048 −0.13±0.17±0.11 0.005–0.006 0.0055 1.22 0.0003±0.0058±0.0043 0.00±0.12±0.09 0.006–0.008 0.0070 1.39 −0.0011±0.0042±0.0023 −0.016±0.071±0.040 0.008–0.010 0.0090 1.62 −0.0087±0.0049±0.0031 −0.121±0.064±0.038 0.010–0.020 0.0141 2.19 −0.0011±0.0032±0.0024 −0.010±0.027±0.019 0.020–0.030 0.0244 3.29 0.0075±0.0048±0.0034 0.043±0.028±0.018 0.030–0.040 0.0346 4.43 0.0095±0.0064±0.0042 0.043±0.028±0.018 0.040–0.060 0.0487 6.06 0.0159±0.0063±0.0044 0.051±0.021±0.014 0.060–0.100 0.0766 9.00 0.0527±0.0070±0.0072 0.111±0.015±0.015 0.100–0.150 0.121 13.5 0.095±0.010±0.011 0.123±0.013±0.014 0.150–0.200 0.171 18.6 0.121±0.015±0.016 0.101±0.013±0.014 0.200–0.250 0.222 23.8 0.160±0.022±0.020 0.0744±0.0096±0.0096 0.250–0.350 0.290 31.1 0.190±0.023±0.022 0.076±0.010±0.009 0.350–0.500 0.405 43.9 0.317±0.037±0.036 0.0576±0.0069±0.0067 0.500–0.700 0.567 60.8 0.494±0.082±0.084 0.0254±0.0042±0.0045

Table A.2 ValuesforAd

1andgd1asafunctionofx andQ2forthecombined2002–2006data.Thefirstuncertaintyisstatistical, thesecondoneissystematic.

x range x Q2_((GeV/c)2_{) A}d 1 g d 1 0.004–0.005 0.0045 1.03 0.005±0.013±0.010 0.12±0.30±0.23 0.004–0.005 0.0046 1.09 −0.001±0.013±0.008 −0.02±0.29±0.19 0.004–0.005 0.0047 1.20 −0.023±0.013±0.008 −0.54±0.30±0.19 0.005–0.006 0.0055 1.07 −0.008±0.010±0.007 −0.15±0.20±0.12 0.005–0.006 0.0055 1.21 0.003±0.010±0.008 0.06±0.21±0.16 0.005–0.006 0.0056 1.39 0.004±0.011±0.006 0.08±0.22±0.14 0.006–0.008 0.0069 1.13 −0.0058±0.0075±0.0042 −0.09±0.11±0.06 0.006–0.008 0.0069 1.39 0.0011±0.0075±0.0043 0.02±0.12±0.07 0.006–0.008 0.0072 1.70 0.0007±0.0075±0.0043 0.01±0.13±0.07 0.008–0.010 0.0089 1.22 −0.0070±0.0084±0.0055 −0.08±0.10±0.07 0.008–0.010 0.0089 1.65 0.0021±0.0083±0.0052 0.03±0.11±0.07 0.008–0.010 0.0091 2.11 −0.0245±0.0083±0.0059 −0.36±0.12±0.09 0.010–0.020 0.0132 1.44 −0.0090±0.0051±0.0034 −0.076±0.043±0.029 0.010–0.020 0.0135 2.23 0.0028±0.0051±0.0033 0.027±0.050±0.032 0.010–0.020 0.0156 3.24 0.0009±0.0051±0.0034 0.009±0.049±0.033 0.020–0.030 0.0239 1.95 0.0198±0.0082±0.0062 0.101±0.042±0.032 0.020–0.030 0.0240 3.53 −0.0083±0.0082±0.0069 −0.051±0.050±0.042 0.020–0.030 0.0253 5.22 0.0075±0.0082±0.0056 0.048±0.053±0.037 0.030–0.040 0.0342 2.51 0.014±0.011±0.008 0.052±0.043±0.029 0.030–0.040 0.0344 4.82 0.007±0.011±0.009 0.033±0.051±0.043 0.030–0.040 0.0352 7.24 0.006±0.011±0.008 0.029±0.054±0.038 0.040–0.060 0.0477 3.38 0.005±0.011±0.009 0.014±0.032±0.025 0.040–0.060 0.0482 6.43 0.012±0.011±0.007 0.040±0.036±0.023 0.040–0.060 0.0502 10.1 0.021±0.011±0.007 0.072±0.037±0.025 0.060–0.100 0.0744 4.93 0.034±0.012±0.009 0.067±0.024±0.019 0.060–0.100 0.0757 9.28 0.052±0.012±0.012 0.111±0.026±0.025 0.060–0.100 0.0796 15.6 0.065±0.012±0.010 0.140±0.026±0.022 0.100–0.150 0.119 6.99 0.058±0.017±0.014 0.072±0.022±0.017 0.100–0.150 0.120 13.8 0.070±0.017±0.014 0.092±0.023±0.019 0.100–0.150 0.124 24.2 0.148±0.017±0.019 0.191±0.023±0.025 0.150–0.200 0.171 9.06 0.099±0.026±0.019 0.082±0.022±0.016 0.150–0.200 0.171 19.2 0.119±0.026±0.021 0.101±0.022±0.018 0.150–0.200 0.174 33.9 0.127±0.026±0.022 0.106±0.022±0.018 0.200–0.250 0.221 11.2 0.150±0.037±0.028 0.087±0.022±0.017 0.200–0.250 0.221 25.2 0.171±0.037±0.029 0.100±0.021±0.017 0.200–0.250 0.224 43.5 0.151±0.037±0.032 0.085±0.021±0.018 0.250–0.350 0.287 14.3 0.187±0.040±0.032 0.071±0.015±0.012 0.250–0.350 0.288 33.4 0.187±0.040±0.032 0.068±0.015±0.012 0.250–0.350 0.295 56.2 0.185±0.040±0.033 0.062±0.014±0.011 0.350–0.500 0.400 20.0 0.396±0.065±0.056 0.070±0.012±0.010 0.350–0.500 0.402 46.4 0.266±0.066±0.051 0.043±0.011±0.008 0.350–0.500 0.411 74.1 0.288±0.063±0.050 0.041±0.009±0.007 0.500–0.700 0.569 62.1 0.501±0.082±0.084 0.0204±0.0033±0.0035 References

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