Interplay among transversity induced asymmetries in hadron leptoproduction

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Physics

Letters

B

www.elsevier.com/locate/physletb

Interplay

among

transversity

induced

asymmetries

in

hadron

leptoproduction

C. Adolph

i

,

R. Akhunzyanov

h

,

M.G. Alexeev

ab

,

G.D. Alexeev

h

,

A. Amoroso

ab

,

ac

,

V. Andrieux

v

,

V. Anosov

h

,

W. Augustyniak

ae

,

A. Austregesilo

q

,

C.D.R. Azevedo

b

,

B. Badełek

af

,

F. Balestra

ab

,

ac

,

J. Barth

e

,

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

,

1

,

F. Bradamante

y

,

z

,

∗

,

C. Braun

i

,

A. Bressan

y

,

z

,

M. Büchele

j

,

E. Burtin

v

,

W.-C. Chang

w

,

M. Chiosso

ab

,

ac

,

I. Choi

ad

,

S.-U. Chung

q

,

2

,

A. Cicuttin

aa

,

z

,

M.L. Crespo

aa

,

z

,

Q. Curiel

v

,

N. d’Hose

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

,

C. Elia

y

,

z

,

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

,

3

,

C.-Y. Hsieh

w

,

S. Huber

q

,

S. Ishimoto

ah

,

4

,

A. Ivanov

ab

,

ac

,

Yu. Ivanshin

h

,

T. Iwata

ah

,

R. Jahn

d

,

V. Jary

t

,

P. Jörg

j

,

R. Joosten

d

,

E. Kabuß

n

,

B. Ketzer

q

,

5

,

G.V. Khaustov

u

,

Yu.A. Khokhlov

u

,

6

,

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

,

*

Correspondingauthor.

E-mailaddress:Franco.Bradamante@cern.ch(F. Bradamante).

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

2 _{AlsoatDepartmentofPhysics,PusanNationalUniversity,Busan609-735,RepublicofKoreaandatPhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,} USA.

3 _{AlsoatChubuUniversity,Kasugai,Aichi487-8501,Japan.} 4 _{AlsoatKEK,1-1Oho,Tsukuba,Ibaraki305-0801,Japan.}

5 _{Presentaddress:UniversitätBonn,Helmholtz-InstitutfürStrahlen- undKernphysik,53115Bonn,Germany.} 6 _{AlsoatMoscowInstituteofPhysicsandTechnology,MoscowRegion141700,Russia.}

7 _{Presentaddress:RWTHAachenUniversity,III.PhysikalischesInstitut,52056Aachen,Germany.} 8 _{SupportedbytheDFGResearchTrainingGroupProgramme1102“PhysicsatHadronAccelerators”.} 9 _{Presentaddress:UppsalaUniversity,Box516,SE-75120Uppsala,Sweden.}

10 _{SupportedbytheGermanBundesministeriumfürBildungundForschung.} 11 _{SupportedbyEUFP7(HadronPhysics3,GrantAgreementnumber283286).} 12 _{SupportedbyCzechRepublicMEYSGrantLG13031.}

13 _{SupportedbySAIL(CSR),Govt.ofIndia.} 14 _{SupportedbyCERN-RFBRGrant12-02-91500.}

15 _{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.

16 _{SupportedbytheMEXTandtheJSPSundertheGrantsNo.18002006,No.20540299andNo.18540281;DaikoFoundationandYamadaFoundation.} 17 _{SupportedbytheDFGclusterofexcellence‘OriginandStructureoftheUniverse’(www.universe-cluster.de).}

18 _{SupportedbytheIsraelScienceFoundation,foundedbytheIsraelAcademyofSciencesandHumanities.} 19 _{SupportedbythePolishNCNGrantDEC-2011/01/M/ST2/02350.}

20 _Deceased.

http://dx.doi.org/10.1016/j.physletb.2015.12.042

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

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

,

B. Marianski

ae

,

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

,

P. Montuenga

ad

,

A. Nagaytsev

h

,

F. Nerling

n

,

D. Neyret

v

,

V.I. Nikolaenko

u

,

J. Nový

t

,

k

,

W.-D. Nowak

j

,

G. Nukazuka

ah

,

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

,

G. Pesaro

y

,

z

,

M. Pesek

s

,

D.V. Peshekhonov

h

,

S. Platchkov

v

,

J. Pochodzalla

n

,

V.A. Polyakov

u

,

J. Pretz

e

,

7

,

M. Quaresma

m

,

C. Quintans

m

,

S. Ramos

m

,

1

,

C. Regali

j

,

G. Reicherz

c

,

C. Riedl

ad

,

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

,

8

,

H. Schmieden

e

,

K. Schönning

k

,

9

,

S. Schopferer

j

,

A. Selyunin

h

,

O.Yu. Shevchenko

h

,

20

,

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

,

3

,

A. Szabelski

ae

,

T. Szameitat

j

,

8

,

P. Sznajder

ae

,

S. Takekawa

ab

,

ac

,

J. ter Wolbeek

j

,

8

,

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

,

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}10,11 d_{UniversitätBonn,Helmholtz-InstitutfürStrahlen- undKernphysik,53115Bonn,Germany}10 e_{UniversitätBonn,PhysikalischesInstitut,53115Bonn,Germany}10

f_{InstituteofScientificInstruments,ASCR,61264Brno,CzechRepublic}12

g_{MatrivaniInstituteofExperimentalResearch&Education,Calcutta-700030,India}13 h_{JointInstituteforNuclearResearch,141980Dubna,Moscowregion,Russia}14 i_{UniversitätErlangen–Nürnberg,PhysikalischesInstitut,91054Erlangen,Germany}10 j_{UniversitätFreiburg,PhysikalischesInstitut,79104Freiburg,Germany}10,11 k_{CERN,1211Geneva23,Switzerland}

l_{TechnicalUniversityinLiberec,46117Liberec,CzechRepublic}12 m_{LIP,1000-149Lisbon,Portugal}15

n_{UniversitätMainz,InstitutfürKernphysik,55099Mainz,Germany}10 o_{UniversityofMiyazaki,Miyazaki889-2192,Japan}16

p_{LebedevPhysicalInstitute,119991Moscow,Russia}

q_{TechnischeUniversitätMünchen,PhysikDepartment,85748Garching,Germany}10,17 r_{NagoyaUniversity,464Nagoya,Japan}16

s_{CharlesUniversityinPrague,FacultyofMathematicsandPhysics,18000Prague,CzechRepublic}12 t_{CzechTechnicalUniversityinPrague,16636Prague,CzechRepublic}12

u_{StateScientificCenterInstituteforHighEnergyPhysicsofNationalResearchCenter‘KurchatovInstitute’,142281Protvino,Russia} v_{CEAIRFU/SPhNSaclay,91191Gif-sur-Yvette,France}11

w_{AcademiaSinica,InstituteofPhysics,Taipei11529,Taiwan}

x_{TelAvivUniversity,SchoolofPhysicsandAstronomy,69978TelAviv,Israel}18 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}19

af_{UniversityofWarsaw,FacultyofPhysics,02-093Warsaw,Poland}19

ag_{WarsawUniversityofTechnology,InstituteofRadioelectronics,00-665Warsaw,Poland}19 ah_{YamagataUniversity,Yamagata992-8510,Japan}16

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Articlehistory:

Received27October2015

Receivedinrevisedform24November2015 Accepted15December2015

Availableonline17December2015 Editor:M.Doser

Inthefragmentationofatransverselypolarizedquarkseveralleft–rightasymmetriesarepossibleforthe hadronsinthejet.Whenonlyoneunpolarizedhadronisselected,itexhibitsanazimuthalmodulation knownastheCollinseffect.Whenapairofoppositelychargedhadronsisobserved,threeasymmetries canbeconsidered,adi-hadronasymmetryandtwosinglehadronasymmetries.Inleptondeepinelastic scattering ontransversely polarizednucleons all theseasymmetriesare coupledwiththe transversity distribution. Fromthe highstatistics COMPASSdata onoppositelycharged hadron-pairproductionwe have investigated for the first time the dependence of these three asymmetries on the difference of the azimuthal angles of the two hadrons. The similarity of transversity induced single and di-hadron asymmetries is discussed. A new analysis of the data allows quantitative relationships to be

establishedamongthem,providingforthefirsttimestrongexperimentalindicationthattheunderlying fragmentationmechanismsarealldrivenbyacommonphysicalprocess.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The description of the partonic structure of the nucleon at leading twist in the collinear case requires the knowledge of three parton distribution functions (PDFs), the number, helicity andtransversityfunctions.Verymuchlikethehelicitydistribution, which gives the longitudinal polarization of a quark in a longi-tudinallypolarizednucleon,thetransversity distributiongivesthe transversepolarization of aquark in atransversely polarized nu-cleon.Its firstmoment,thetensorcharge,isafundamental prop-ertyofthe nucleon.Whilethe numberandthehelicityPDFscan beobtainedfromcross-sectional measurementsofunpolarizedor doublypolarized lepton–nucleondeeply inelasticscattering(DIS), respectively,thetransversitydistributionischiral-oddandassuch canbe measured onlyif foldedwithanotherchiral-odd quantity. As suggested more than 20 years ago [1,2], it can be accessed in semi-inclusive DIS (SIDIS) off transversely polarized nucleons fromaleft–rightasymmetryofthehadronsproducedinthestruck

quark fragmentation with respect to the plane defined by the

quarkmomentumandspin directions.Recently,boththeHERMES

and the COMPASS experiments have provided unambiguous

evi-dencethat transversity isdifferentfromzeroby measuring SIDIS offtransverselypolarizedprotons[3].Twodifferentprocesseshave beenaddressed.Inthefirstprocess,atargetspinazimuthal asym-metry in single-hadron production is measured, referred to as Collinsasymmetry [2].It dependsontheconvolution of

transver-sity and a hadron transverse-momentum dependent chiral-odd

fragmentationfunction (FF),theCollins function,which describes thecorrelationbetweenthehadrontransversemomentumandthe transversepolarizationofthefragmentingquark.Thesecond pro-cessistheproductionoftwooppositely chargedhadrons[1,4–6]. Inthiscasetheso-calleddi-hadrontargetspinazimuthal asymme-tryoriginatesfromthe couplingoftransversity toadi-hadronFF, alsoreferred to as interferenceFF, in principleindependent from the Collins function. In both cases, measurements of the corre-spondingazimuthalasymmetriesofthehadronsproducedine+e−

annihilation [7–9] provided independent information on the two typesofFFs,allowing forfirstextractionsoftransversityfromthe SIDISande+e−data[10–13].

ThehighprecisionCOMPASSmeasurementsontransversely po-larizedprotons[14,15]showedthatinthex-Bjorkenregion,where theCollinsasymmetryisdifferentfromzeroandsizable,the pos-itiveandnegativehadronasymmetriesexhibitamirrorsymmetry andthedi-hadronasymmetryisveryclosetoandsomewhatlarger thantheCollinsasymmetryforpositivehadrons.Thesefacts have beeninterpreted asexperimental evidenceof acloserelationship betweentheCollins andthe di-hadronasymmetries,hinting ata common physics origin of the two FFs [15–18], assuggested in the3P0recursivestringfragmentationmodel[19,20]and,forlarge

invariant mass of the hadron pair, in Ref. [21]. The interpreta-tionisalsosupported bycalculationswitha specificMonteCarlo model[22].

In order to better investigate the relationship between the Collinsasymmetry andthedi-hadron asymmetrythe correlations betweentheazimuthalanglesofthefinalstate hadronsproduced intheSIDISprocess

μ

p

→

μ

h+h−X havebeenstudiedusingthe COMPASS data. These correlations play an important role in the understandingofthehadronizationmechanismandinsofarhave beenstudiedonlyinunpolarizedSIDIS[23].Inthisarticleforthe

Fig. 1. DefinitionoftheCollinsangleC ofahadronh.ThevectorspT,s ands arethehadrontransversemomentumandthespinoftheinitialandstruckquarks respectively.

first time the results forSIDIS off transversely polarized protons arepresented.Theinvestigationhasproceededthroughthree ma-jorsteps:

i) the Collins asymmetries for positive and negative hadrons

have been compared with the corresponding asymmetries

measured in the SIDISprocess

μ

p

→

μ

h+h−X , i.e.when in thefinalstateatleasttwooppositelychargedhadronsare de-tected(2hsample);

ii) usingthe2hsampletheasymmetriesofh+andh−havebeen

measured and their relation has been investigated as func-tionof

φ,

thedifferenceoftheazimuthal anglesofthetwo hadrons;

iii) thedihadronasymmetryhasbeenmeasuredasfunctionof

φ

and, using a new generalexpression, compared withthe h+

andh−asymmetries.Theintegratedvaluesofthethree asym-metrieshavealsobeencompared.

2. TheCOMPASSexperimentanddataselection

COMPASS is a fixed-target experiment at the CERN SPS tak-ing datasince2002[24].The presentresultshavebeenextracted fromthe datacollected in2010witha160 GeV/c

μ

+ beamand atransverselypolarizedproton(NH3)target,alreadyusedto

mea-surethetransversespinasymmetries[14,25,15].Theyrefertothe 2hsample,i.e.SIDISeventsinwhichatleastonepositiveandone negativehadronhavebeendetected.

TheselectionoftheDISeventsandofthehadronsisdescribed indetailinRef.[15].Standardcutsareappliedonthephoton vir-tuality ( Q2

_>

_{1 GeV}2_/c2_{),on thefractional energytransferto the}

virtualphoton(0.1

<

y

<

0.9),andontheinvariantmassofthe fi-nal hadronicstate (W

>

5 GeV/c2_{).Specific tothisanalysisisthe}

requirementthat each hadronmusthavea fractionofthevirtual photonenergyz1,2

>

0.1,wherethesubscript1referstothe

pos-itive hadron andsubscript 2to thenegative hadron. Aminimum value of0.1 GeV/c for the hadrontransverse momenta pT 1,2

en-suresgoodresolutionintheazimuthal angles.Asshownin

Fig. 1

thevirtualphotondirectionisthez axis ofthecoordinatesystem while the x axisis directed along the lepton transverse momen-tum.Thedirectionofthe y axisischosentohavearight-handed coordinate system. Transverse componentsof vectorsare defined withrespecttoz axis.

The 2h sample consists of33 million h+h− pairs, tobe com-pared with the 85 million h+ or 71 million h− of the standard eventsample (1hsample)ofthe previousanalysis [15],whereat least one hadron (either positive or negative) per event was re-quired.

Fig. 2. (Coloronline.)ComparisonoftheCLasymmetriesforh1(fullredcircles)and h2(fullblacktriangles)inlp→lh+h−X withthestandardCollinsasymmetriesin lp→lh±X forz>0.1 (opencirclesandtriangles)measuredasfunctionofx. Table 1

Integrated values of the Collins and CL asymmetries for positive and negative hadronsintheregionx>0.032.

Collins asymmetry Collins-like asymmetry h1 −0.017±0.002 −0.018±0.003 h2 0.018±0.002 0.020±0.003

3. Comparisonof1hand2hsampleasymmetries

Foreach hadron the Collins angle



C i, i

=

1,2, is defined as

usual as



C i

= φ

i

+ φ

S

−

π

, where

φ

i is the azimuthal angle

of the hadron transverse momentum,

φ

S is the azimuthal angle

of the transversenucleonspin,and

π

− φ

S istheazimuthalangle

ofthespin



softhestruckquark[26],asshownin

Fig. 1

.Allthe azimuthalanglesaremeasuredaroundthez axis.Forthepositive and negative hadrons in the 2h sample, the amplitudes AsinC 1

CL1

andAsinC 2

CL2 ofthesin



C 1,2 modulationsinthecross-sectionhave

beenextractedwiththesamemethodasofRef. [15]andlabeled “CL” (Collins-like) to distinguish them from the standard Collins asymmetries,whicharedefinedinthe1hsample.

Withintheaccuracy ofthemeasurementsthe CLasymmetries turnout to be the sameasthe standard Collins asymmetries,as can be seen in Fig. 2. The integrated values of the Collins and CLasymmetries are givenin Table 1for x

>

0.032 that is inthe

x-regionwheretheyaresizable.Takingintoaccountstatistical cor-relationthedifferencebetweentheCollinsandtheCLasymmetries islessthanastandarddeviationforbothh1 andh2.This

observa-tionimpliesthattheCollinsasymmetrydoesnotdependon addi-tionalobservedhadronsintheevent.Asanimportantresultofthe firststepofthisinvestigationthe2hsamplecanbeusedtostudy themirrorsymmetryandtoinvestigatetheinterplaybetweenthe Collinssingle-hadronasymmetryandthedi-hadronasymmetry,as described in the following. All the results of the following work areobtainedinthekinematicalregionx

>

0.032,whichistheone wheretheCollinsandthedi-hadronasymmetriesarelargest.

4.

φ

dependenceoftheCLasymmetriesofpositiveand negativehadrons

Theazimuthal correlationsbetween

φ

1 and

φ

2 intransversely

polarizedSIDIShadbeeninvestigatedbymeasuringthe asymme-triesas a function of

|φ|

[17], where

φ

= φ

1

− φ

2. The final

resultsasfunctionof

φ

areshownin

Fig. 3

.Thetwoasymmetries looklikeevenfunctionsof

φ,

arecompatiblewithzerowhen

φ

tends to zero, and increase in magnitudeas

φ

increases.Very muchasin

Fig. 2

themirrorsymmetrybetweenpositiveand neg-ativehadronsisa strikingfeatureofthedata.Theoverall picture agreeswiththeexpectationfromthe3_P

0recursivestring

fragmen-tationmodelofRefs.[19,20],whichpredictsamaximumvaluefor

φ



π

.

Fig. 3. (Coloronline.)TheAsinC 1

CL1 (redcircles)andthe A

sinC 2

CL2 (blacktriangles)vs.

φ.Superimposedarethefittingfunctionsdescribedinthetext.

The framework to access the

φ

dependenceof CL

asymme-trieswasproposedinRef.[27].Afterintegrationoverx,Q2_,z 1,z2,

p2_{T 1}andp2_{T 2} thecross-sectionfortheSIDISprocesslN

→

lh+h−X

canbewrittenas d

σ

h1h2 d

φ

1d

φ

2d

φ

S

=

σ

U

+

ST



σ

C 1

(



pT 1

× 

q

)

· 

s

|

pT 1

× 

q

| |

s

|

+

σ

C 2

(

p



T 2

× 

q

)

· 

s

|

pT 2

× 

q

| |

s

|



=

σ

U

+

ST[

σ

C 1sin



C 1

+

σ

C 2sin



C 2]

,

(1)

wheretheunpolarized

σ

U andthepolarized

σ

C 1 and

σ

C 2structure

functions(SFs)mightdependon cos

φ.

Toaccesstheazimuthal correlationsofthepolarizedSFsEq.(1)isrewrittenintermsof

φ

1

and

φ,

oralternativelyintermsof

φ

2 and

φ:

d

σ

h1h2 d

φ

1d

φ

d

φ

S

=

σ

U

+

ST



σ

C 1

+

σ

C 2cos

φ



sin



C 1

−

σ

C 2sin

φ

cos



C 1



,

d

σ

h1h2 d

φ

2d

φ

d

φ

S

=

σ

U

+

ST



σ

C 2

+

σ

C 1cos

φ



sin



C 2

+

σ

C 1sin

φ

cos



C 2



.

(2)

Withthechangeofvariablesaboveanewmodulation,ofthetype cos



C 1,2,appearsinthecrosssection,whichcanthenbe

rewrit-ten in terms of the sine and cosine modulations of the Collins angle of either the positive or the negative hadron. The explicit expressionsforthefourasymmetriesare:

AsinC 1 CL1

=

1 DN N

σ

C 1

+

σ

C 2cos

φ

σ

U

,

AcosC 1 CL1

= −

1 DN N

σ

C 2sin

φ

σ

U

,

AsinC 2 CL2

=

1 DN N

σ

C 2

+

σ

C 1cos

φ

σ

U

,

AcosC 2 CL2

=

1 DN N

σ

C 1sin

φ

σ

U

,

(3)

where DN N isthemeantransverse-spin-transfercoefficient, equal

to0.87forthesedata.

Fig. 4

showsthemeasuredvaluesofthenew asymmetries AcosC 1,2

CL1,2 .Itisthefirsttimethattheyaremeasured.

Theyhavealmost equalvalues forpositiveandnegative hadrons, seemto beoddfunctionsof

φ

,andaverage tozerowhen inte-grating over

φ.

Notethat the data arein very good agreement withEq.(3)if

(

σ

C 1

/

σ

U

)

= −(

σ

C 2

/

σ

U

)

=

const.

Thequantities

σ

C 1

/

σ

U and

σ

C 2

/

σ

U,whichinprinciplecanstill

befunctionsof

φ,

canbeobtainedfromthemeasured asymme-triesusing

σ

C 1

σ

U

=

DN N



AsinC 1 CL1

+

A cosC 1 CL1 cot

φ



,

Fig. 4. (Coloronline.)TheAcosC 1

CL1 (redcircles)andtheA

cosC 2

CL2 (blacktriangles)vs.

φ.Superimposedarethefittingfunctionsdescribedinthetext.

Fig. 5. (Coloronline.)ThemeasuredvaluesoftheratiosσC 1/σU andσC 2/σU.The linesgivethefittedmeanvalues,−0.015±0.004 forh1and0.022±0.004 forh2. Table 2

Fittedvaluesofthea parameterforthetwoCLasymmetriesforpositiveand nega-tivehadrons. AsinC CL A cosC CL h1 0.014±0.003 0.025±0.005 h2 0.016±0.003 0.017±0.005

σ

C 2

σ

U

=

DN N



AsinC 2 CL2

−

A cosC 2 CL2 cot

φ



.

(4)

The values ofthe ratios

σ

C 1

/

σ

U and

σ

C 2

/

σ

U extracted fromthe

measured asymmetries are given in Fig. 5. Within the statistical uncertaintytheyareconstant,hintingatsimilarazimuthal correla-tionsinpolarizedandunpolarizedSFs. Moreover,they arealmost equalinabsolutevalueandofoppositesign.Assuming

(

σ

C 1

/

σ

U

)

=

−(

σ

C 2

/

σ

U

)

=

const.,themeasuredasymmetriescanbefittedwith

thesimplefunctions

±

a

(1

−

cosφ)inthecaseofthesine asym-metries,andasin

φ

forthecosineasymmetries.Theresultsofthe fitsforpositive(negative)hadronsarethedashedred(dot-dashed black)curvesshownin

Figs. 3 and

4.Theagreementwiththe mea-surementsisverygoodandthefourvaluesfortheconstantsa are

compatible,ascanbeseenin

Table 2

.

Asaconclusionofthissecondstepoftheinvestigation,theh1

andh2 CLasymmetriesasfunctionsof

φ

agreewiththe

expec-tationfromthe3_P

0recursivestringfragmentationmodelandwith

the calculationsof the

φ

dependence obtained in Ref. [27]. As intheone-hadronsample amirrorsymmetryforthepositiveand negative hadron sine asymmetries isobserved in the 2h sample, whichisaconsequenceoftheexperimentallyestablishedrelation

σ

C 1

= −

σ

C 2.

Theseresultsallow aquantitativerelationbetweentheh1 and

h2CLasymmetriesandthedi-hadronasymmetrytobederived,as

describedinthefollowing.

5. ComparisonofCLanddi-hadronasymmetries

The thirdand last stepof thisinvestigation hasbeen the for-malderivationofaconnectionbetweentheCLandthedi-hadron

Fig. 6. (Coloronline.) Asin2h,S

CL2h vs.φandthecorrespondingfit(blackfullcurve). Thedashedredanddot-dashedblackcurvesarethefitstoAsinC 1

CL1 and A

sinC 2

CL2 fromFig. 3.

asymmetries and the comparisonwith the experimental data. In the standard analysis, the

φ

integrated di-hadron asymmetry is measured from the amplitude of the sine modulation of the angle



R S

= φ

R

+ φ

S

−

π

, where

φ

R is the azimuthal angle of

the relative hadron momentum R



=



z2



p1

−

z1



p2

/

[z1

+

z2]

=:

ξ

2



p1

− ξ

1



p2. In the present analysis, the azimuthal angle

φ

2h of

thevector



RN

= ˆ

pT 1

− ˆ

pT 2 isevaluatedforeachpairofoppositely

chargedhadrons,withthehatindicatingunitvectors.Asdiscussed in Ref. [15], the azimuthal angle

φ

R is strongly correlated with

φ

2h

= [φ

1

+ φ

2

+

π

sgn(φ)

]/

2,wheresgn isthesignumfunction.

Also,introducingtheangle



2h,S

= φ

2h

+φ

S

−

π

,whichisakindof

meanoftheCollinsangleofthepositiveandnegativehadrons af-tercorrectingfora

π

phasedifference,itwasshown[15]thatthe di-hadronasymmetrymeasuredfromtheamplitudeofsin



2h,S is

essentially identical to the standard di-hadron asymmetry.In or-der to establish a connection betweenthe di-hadron asymmetry and the CL asymmetries



2h,S will be used rather than



R S in

the following. Starting fromthe general expression for the cross section given in Eq. (1), changing variables from

φ

1 and

φ

2 to

φ

and

φ

2h,andusingthe relationssin



2h,S

= ( ˆ

RN

× ˆ

q

)

· ˆ

s and

cos



2h,S

= −

sgn(φ)( ˆPN

× ˆ

q

)

· ˆ

s,where



PN

= ˆ

pT 1

+ ˆ

pT 2,Eq.(1)

canberewrittenas:

d

σ

h1h2 d

φ

d

φ

2hd

φ

S

=

σ

U

+

ST 1 2



σ

C 1

−

σ

C 2



×



2

(

1

−

cos

φ)

sin



2h,S

−

sgn

(φ)

σ

C 1

+

σ

C 2



×



2

(

1

+

cos

φ)

cos



2h,S

,

(5) whichsimplifiesto d

σ

h1h2 d

φ

2hd

φ

d

φ

S

=

σ

U

+

ST

·

σ

C 1

·



2

(

1

−

cos

φ)

·

sin



2h,S (6)

using theexperimental result

σ

C 2

= −

σ

C 1.This last cross-section

impliesasinemodulationwithamplitude

Asin2h,S CL2h

=

1 DN N

σ

C 1

σ

U

·



2

(

1

−

cos

φ).

(7)

At variancewiththesingle hadroncase, no Acos2h,S

CL2h asymmetry

ispresentinEq.(6)andthemeasuredvaluesareindeed compati-blewithzero.In

Fig. 6

the Asin2h,S

CL2h asymmetryisshowntogether

with the curve

−

c

√

2(1

−

cosφ) with c

=

0.017

±

0.002 (black solid line)asobtainedby thefit.Thedashed redanddot-dashed black curvesare thefitted curvesa

(1

−

cosφ) of

Fig. 3

.As can be seenthefitisgood,andthevalue ofc iscompatiblewiththe corresponding values of

Table 2

,in agreement withthe fact that

σ

C i

/

σ

U isthesameforthethreeasymmetries.Evaluatingtheratio

amplitudesone gets a value of 1.4

±

0.2 which agrees withthe value 4/

π

evaluated fromEqs. (7)and(3) andwithour original observationthatthedi-hadronasymmetryissomewhatlargerthan theCollinsasymmetryforpositivehadrons.

6. Conclusions

We have shown that in SIDIS hadron-pair production the

x-dependentCollins-likesinglehadronasymmetriesofthepositive

and negative hadrons are compatible with the standard Collins asymmetries and are mirror symmetric. Also, the Collins-like asymmetriesexhibita

±

a

(1

−

cos

φ)

dependenceon

φ,

which wehave derived fromthegeneralexpression forthe two-hadron cross-sectionandis aconsequenceof theexperimentally verified similar

φ

dependence of the unpolarized and polarized struc-turefunctionsandthemirrorsymmetryofthepolarizedstructure functions.

Mostimportant,forthefirst time ithas beenshownthat the amplitude of the di-hadron asymmetry asa function of

φ

has a very simple relation to that of the single hadron asymmetries inthe2hsample,namelyitcanbewrittenas

−

a

√

2(1

−

cosφ), wheretheconstanta isthesameasthatwhichappearsinthe ex-pressionsfortheCollins-likeasymmetries.Afterintegrationon

φ,

thedi-hadron asymmetry is largerthan the single hadron asym-metries by a factor 4/

π

, in good agreement with the measured values.

In conclusion, we have shown that the integrated values of Collinsasymmetriesinthe1hsamplearethesameasthe Collins-likeasymmetriesof2hsamplewhichinturnarerelatedwiththe integratedvaluesofdi-hadronasymmetry.Thisgivesanindication thatboththesinglehadron anddi-hadrontransverse-spin depen-dentfragmentation functionsare driven by the sameelementary mechanism.Asaconsequenceofthisimportantconclusionwecan add that the extraction of transversity distribution using the

di-hadron asymmetry in SIDIS does not represent an independent

measurementwithrespecttothe extractionswhichare basedon theCollinsasymmetry.

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