\PHnumber2015–199 \PHdateSeptember 20, 2019
\CollaborationThe COMPASS Collaboration \ShortAuthorThe COMPASS Collaboration
In the fragmentation of a transversely polarized quark several leftright asymmetries are possible for the hadrons in the jet. When only one unpolarized hadron is selected, it exhibits an azimuthal modulation known as Collins effect. When a pair of oppositely charged hadrons is observed, three asymmetries can be considered, a dihadron asymmetry and two single hadron asymmetries. In lepton deep inelastic scattering on transversely polarized nucleons all these asymmetries are coupled with the transversity distribution. From the high statistics COMPASS data on oppositely charged hadronpair production we 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 dihadron asymmetries is discussed. A new analysis of the data allows to establish quantitative relationships among them, providing for the first time strong experimental indication that the underlying fragmentation mechanisms are all driven by a common physical process.
\Submitted(to be submitted to Phys. Lett. B)
The COMPASS Collaboration
C. Adolph\Irefnerlangen, R. Akhunzyanov\Irefndubna, M.G. Alexeev\Irefnturin_u, G.D. Alexeev\Irefndubna, A. Amoroso\Irefnnturin_uturin_i, V. Andrieux\Irefnsaclay, V. Anosov\Irefndubna, W. Augustyniak\Irefnwarsaw, A. Austregesilo\Irefnmunichtu, C.D.R. Azevedo\Irefnaveiro, B. Badełek\Irefnwarsawu, F. Balestra\Irefnnturin_uturin_i, J. Barth\Irefnbonnpi, R. Beck\Irefnbonniskp, Y. Bedfer\Irefnnsaclaycern, J. Bernhard\Irefnnmainzcern, K. Bicker\Irefnnmunichtucern, E. R. Bielert\Irefncern, R. Birsa\Irefntriest_i, J. Bisplinghoff\Irefnbonniskp, M. Bodlak\Irefnpraguecu, M. Boer\Irefnsaclay, P. Bordalo\Irefnlisbon\Arefa, F. Bradamante\Irefnntriest_utriest_i, C. Braun\Irefnerlangen, A. Bressan\Irefnntriest_utriest_i, M. Büchele\Irefnfreiburg, E. Burtin\Irefnsaclay, W.C. Chang\Irefntaipei, M. Chiosso\Irefnnturin_uturin_i, I. Choi\Irefnillinois, S.U. Chung\Irefnmunichtu\Arefb, A. Cicuttin\Irefnntriest_ictptriest_i, M.L. Crespo\Irefnntriest_ictptriest_i, Q. Curiel\Irefnsaclay, N. d’Hose\Irefnsaclay, S. Dalla Torre\Irefntriest_i, S.S. Dasgupta\Irefncalcutta, S. Dasgupta\Irefnntriest_utriest_i, O.Yu. Denisov\Irefnturin_i, L. Dhara\Irefncalcutta, S.V. Donskov\Irefnprotvino, N. Doshita\Irefnyamagata, V. Duic\Irefntriest_u, M. Dziewiecki\Irefnwarsawtu, A. Efremov\Irefndubna, C. Elia\Irefnntriest_utriest_i, P.D. Eversheim\Irefnbonniskp, W. Eyrich\Irefnerlangen, A. Ferrero\Irefnsaclay, M. Finger\Irefnpraguecu, M. Finger jr.\Irefnpraguecu, H. Fischer\Irefnfreiburg, C. Franco\Irefnlisbon, N. du Fresne von Hohenesche\Irefnmainz, J.M. Friedrich\Irefnmunichtu, V. Frolov\Irefnndubnacern, E. Fuchey\Irefnsaclay, F. Gautheron\Irefnbochum, O.P. Gavrichtchouk\Irefndubna, S. Gerassimov\Irefnnmoscowlpimunichtu, F. Giordano\Irefnillinois, I. Gnesi\Irefnnturin_uturin_i, M. Gorzellik\Irefnfreiburg, S. Grabmüller\Irefnmunichtu, A. Grasso\Irefnnturin_uturin_i, M. GrossePerdekamp\Irefnillinois, B. Grube\Irefnmunichtu, T. Grussenmeyer\Irefnfreiburg, A. Guskov\Irefndubna, F. Haas\Irefnmunichtu, D. Hahne\Irefnbonnpi, D. von Harrach\Irefnmainz, R. Hashimoto\Irefnyamagata, F.H. Heinsius\Irefnfreiburg, F. Herrmann\Irefnfreiburg, F. Hinterberger\Irefnbonniskp, N. Horikawa\Irefnnagoya\Arefd, C.Yu Hsieh\Irefntaipei, S. Huber\Irefnmunichtu, S. Ishimoto\Irefnyamagata\Arefe, A. Ivanov\Irefndubna, Yu. Ivanshin\Irefndubna, T. Iwata\Irefnyamagata, R. Jahn\Irefnbonniskp, V. Jary\Irefnpraguectu, P. Jörg\Irefnfreiburg, R. Joosten\Irefnbonniskp, E. Kabuß\Irefnmainz, B. Ketzer\Irefnmunichtu\Areff, G.V. Khaustov\Irefnprotvino, Yu.A. Khokhlov\Irefnprotvino\Arefg, Yu. Kisselev\Irefndubna, F. Klein\Irefnbonnpi, K. Klimaszewski\Irefnwarsaw, J.H. Koivuniemi\Irefnbochum, V.N. Kolosov\Irefnprotvino, K. Kondo\Irefnyamagata, K. Königsmann\Irefnfreiburg, I. Konorov\Irefnnmoscowlpimunichtu, V.F. Konstantinov\Irefnprotvino, A.M. Kotzinian\Irefnnturin_uturin_i, O. Kouznetsov\Irefndubna, M. Krämer\Irefnmunichtu, P. Kremser\Irefnfreiburg, F. Krinner\Irefnmunichtu, Z.V. Kroumchtein\Irefndubna, N. Kuchinski\Irefndubna, F. Kunne\Irefnsaclay, K. Kurek\Irefnwarsaw, R.P. Kurjata\Irefnwarsawtu, A.A. Lednev\Irefnprotvino, A. Lehmann\Irefnerlangen, M. Levillain\Irefnsaclay, S. Levorato\Irefntriest_i, J. Lichtenstadt\Irefntelaviv, R. Longo\Irefnnturin_uturin_i, A. Maggiora\Irefnturin_i, A. Magnon\Irefnsaclay, N. Makins\Irefnillinois, N. Makke\Irefnntriest_utriest_i, G.K. Mallot\Irefncern, C. Marchand\Irefnsaclay, B. Marianski\Irefnwarsaw, A. Martin\Irefnntriest_utriest_i, J. Marzec\Irefnwarsawtu, J. Matousek\Irefnpraguecu, H. Matsuda\Irefnyamagata, T. Matsuda\Irefnmiyazaki, G. Meshcheryakov\Irefndubna, W. Meyer\Irefnbochum, T. Michigami\Irefnyamagata, Yu.V. Mikhailov\Irefnprotvino, Y. Miyachi\Irefnyamagata, P. Montuenga\Irefnillinois, A. Nagaytsev\Irefndubna, F. Nerling\Irefnmainz, D. Neyret\Irefnsaclay, V.I. Nikolaenko\Irefnprotvino, J. Nový\Irefnnpraguectucern, W.D. Nowak\Irefnfreiburg, G. Nukazuka\Irefnyamagata, A.S. Nunes\Irefnlisbon, A.G. Olshevsky\Irefndubna, I. Orlov\Irefndubna, M. Ostrick\Irefnmainz, D. Panzieri\Irefnnturin_pturin_i, B. Parsamyan\Irefnnturin_uturin_i, S. Paul\Irefnmunichtu, J.C. Peng\Irefnillinois, F. Pereira\Irefnaveiro, G. Pesaro\Irefnntriest_utriest_i, M. Pesek\Irefnpraguecu, D.V. Peshekhonov\Irefndubna, S. Platchkov\Irefnsaclay, J. Pochodzalla\Irefnmainz, V.A. Polyakov\Irefnprotvino, J. Pretz\Irefnbonnpi\Arefh, M. Quaresma\Irefnlisbon, C. Quintans\Irefnlisbon, S. Ramos\Irefnlisbon\Arefa, C. Regali\Irefnfreiburg, G. Reicherz\Irefnbochum, C. Riedl\Irefnillinois, N.S. Rossiyskaya\Irefndubna, D.I. Ryabchikov\Irefnprotvino, A. Rychter\Irefnwarsawtu, V.D. Samoylenko\Irefnprotvino, A. Sandacz\Irefnwarsaw, C. Santos\Irefntriest_i, S. Sarkar\Irefncalcutta, I.A. Savin\Irefndubna, G. Sbrizzai\Irefnntriest_utriest_i, P. Schiavon\Irefnntriest_utriest_i, K. Schmidt\Irefnfreiburg\Arefc, H. Schmieden\Irefnbonnpi, K. Schönning\Irefncern\Arefi, S. Schopferer\Irefnfreiburg, A. Selyunin\Irefndubna, O.Yu. Shevchenko\Irefndubna\Deceased, L. Silva\Irefnlisbon, L. Sinha\Irefncalcutta, S. Sirtl\Irefnfreiburg, M. Slunecka\Irefndubna, F. Sozzi\Irefntriest_i, A. Srnka\Irefnbrno, M. Stolarski\Irefnlisbon, M. Sulc\Irefnliberec, H. Suzuki\Irefnyamagata\Arefd, A. Szabelski\Irefnwarsaw, T. Szameitat\Irefnfreiburg\Arefc, P. Sznajder\Irefnwarsaw, S. Takekawa\Irefnnturin_uturin_i, J. ter Wolbeek\Irefnfreiburg\Arefc, S. Tessaro\Irefntriest_i, F. Tessarotto\Irefntriest_i, F. Thibaud\Irefnsaclay, F. Tosello\Irefnturin_i, V. Tskhay\Irefnmoscowlpi, S. Uhl\Irefnmunichtu, J. Veloso\Irefnaveiro, M. Virius\Irefnpraguectu, T. Weisrock\Irefnmainz, M. Wilfert\Irefnmainz, K. Zaremba\Irefnwarsawtu, M. Zavertyaev\Irefnmoscowlpi, E. Zemlyanichkina\Irefndubna, M. Ziembicki\Irefnwarsawtu and A. Zink\Irefnerlangen
turin_pUniversity of Eastern Piedmont, 15100 Alessandria, Italy
aveiroUniversity of Aveiro, Department of Physics, 3810193 Aveiro, Portugal
bochumUniversität Bochum, Institut für Experimentalphysik, 44780 Bochum, Germany\Arefsl\Arefss
bonniskpUniversität Bonn, HelmholtzInstitut für Strahlen und Kernphysik, 53115 Bonn, Germany\Arefsl
bonnpiUniversität Bonn, Physikalisches Institut, 53115 Bonn, Germany\Arefsl
brnoInstitute of Scientific Instruments, AS CR, 61264 Brno, Czech Republic\Arefsm
calcuttaMatrivani Institute of Experimental Research & Education, Calcutta700 030, India\Arefsn
dubnaJoint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia\Arefso
erlangenUniversität Erlangen–Nürnberg, Physikalisches Institut, 91054 Erlangen, Germany\Arefsl
freiburgUniversität Freiburg, Physikalisches Institut, 79104 Freiburg, Germany\Arefsl\Arefss
cernCERN, 1211 Geneva 23, Switzerland
liberecTechnical University in Liberec, 46117 Liberec, Czech Republic\Arefsm
lisbonLIP, 1000149 Lisbon, Portugal\Arefsp
mainzUniversität Mainz, Institut für Kernphysik, 55099 Mainz, Germany\Arefsl
miyazakiUniversity of Miyazaki, Miyazaki 8892192, Japan\Arefsq
moscowlpiLebedev Physical Institute, 119991 Moscow, Russia
munichtuTechnische Universität München, Physik Department, 85748 Garching, Germany\Arefsl\Arefsr
nagoyaNagoya University, 464 Nagoya, Japan\Arefsq
praguecuCharles University in Prague, Faculty of Mathematics and Physics, 18000 Prague, Czech Republic\Arefsm
praguectuCzech Technical University in Prague, 16636 Prague, Czech Republic\Arefsm
protvinoState Scientific Center Institute for High Energy Physics of National Research Center ‘Kurchatov Institute’, 142281 Protvino, Russia
saclayCEA IRFU/SPhN Saclay, 91191 GifsurYvette, France\Arefss
taipeiAcademia Sinica, Institute of Physics, Taipei, 11529 Taiwan
telavivTel Aviv University, School of Physics and Astronomy, 69978 Tel Aviv, Israel\Arefst
triest_uUniversity of Trieste, Department of Physics, 34127 Trieste, Italy
triest_iTrieste Section of INFN, 34127 Trieste, Italy
triest_ictpAbdus Salam ICTP, 34151 Trieste, Italy
turin_uUniversity of Turin, Department of Physics, 10125 Turin, Italy
turin_iTorino Section of INFN, 10125 Turin, Italy
illinoisUniversity of Illinois at UrbanaChampaign, Department of Physics, Urbana, IL 618013080, U.S.A.
warsawNational Centre for Nuclear Research, 00681 Warsaw, Poland\Arefsu
warsawuUniversity of Warsaw, Faculty of Physics, 02093 Warsaw, Poland\Arefsu
warsawtuWarsaw University of Technology, Institute of Radioelectronics, 00665 Warsaw, Poland\Arefsu
yamagataYamagata University, Yamagata, 9928510 Japan\Arefsq {Authlist}
aAlso at Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal
bAlso at Department of Physics, Pusan National University, Busan 609735, Republic of Korea and at Physics Department, Brookhaven National Laboratory, Upton, NY 11973, U.S.A.
cSupported by the DFG Research Training Group Programme 1102 “Physics at Hadron Accelerators”
dAlso at Chubu University, Kasugai, Aichi, 4878501 Japan\Arefsq
eAlso at KEK, 11 Oho, Tsukuba, Ibaraki, 3050801 Japan
fPresent address: Universität Bonn, HelmholtzInstitut für Strahlen und Kernphysik, 53115 Bonn, Germany
gAlso at Moscow Institute of Physics and Technology, Moscow Region, 141700, Russia
hPresent address: RWTH Aachen University, III. Physikalisches Institut, 52056 Aachen, Germany
iPresent address: Uppsala University, Box 516, SE75120 Uppsala, Sweden
lSupported by the German Bundesministerium für Bildung und Forschung
mSupported by Czech Republic MEYS Grant LG13031
nSupported by SAIL (CSR), Govt. of India
oSupported by CERNRFBR Grant 120291500
pSupported 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
qSupported by the MEXT and the JSPS under the Grants No.18002006, No.20540299 and No.18540281; Daiko Foundation and Yamada Foundation
rSupported by the DFG cluster of excellence ‘Origin and Structure of the Universe’ (www.universecluster.de)
sSupported by EU FP7 (HadronPhysics3, Grant Agreement number 283286)
tSupported by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities
uSupported by the Polish NCN Grant DEC2011/01/M/ST2/02350
Deceased
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 and transversity functions. Very much like the helicity distribution, which gives the longitudinal polarization of a quark in a longitudinally polarized nucleon, the transversity distribution gives the transverse polarization of a quark in a transversely polarized nucleon. Its first moment, the tensor charge, is a fundamental property of the nucleon. While the number and the helicity PDFs can be obtained from crosssection measurements of unpolarized or doubly polarized leptonnucleon deeply inelastic scattering (DIS), respectively, the transversity distribution is chiralodd and as such can be measured only if folded with another chiralodd quantity. As suggested more than 20 years ago [1, 2], it can be accessed in semiinclusive DIS (SIDIS) off transversely polarized nucleons from a leftright asymmetry of the hadrons produced in the struck quark fragmentation with respect to the plane defined by the quark momentum and spin directions. Recently, both the HERMES and the COMPASS experiments have provided unambiguous evidence that transversity is different from zero by measuring SIDIS off transversely polarized protons [3]. Two different processes have been addressed. In the first process, a target spin azimuthal asymmetry in singlehadron production is measured, the socalled Collins asymmetry [2]. It depends on the convolution of transversity and a hadron transversemomentum dependent chiralodd fragmentation function (FF), the Collins function, which describes the correlation between the hadron transverse momentum and the transverse polarization of the fragmenting quark. The second process is the production of two oppositely charged hadrons [1, 4, 5, 6]. In this case the socalled dihadron target spin azimuthal asymmetry originates from the coupling of transversity to a dihadron FF, also referred as interference FF, in principle independent from the Collins function. In both cases, measurements of the corresponding azimuthal asymmetries of the hadrons produced in annihilation [7, 8, 9] provided independent information on the two types of FFs, allowing for first extractions of transversity from the SIDIS and data [10, 11, 12, 13].
The high precision COMPASS measurements on transversely polarized protons [14, 15] showed that in the Bjorken region, where the Collins asymmetry is different from zero and sizable, the positive and negative hadron asymmetries exhibit a mirror symmetry and the dihadron asymmetry is very close to and somewhat larger than the Collins asymmetry for positive hadrons. These facts have been interpreted as experimental evidence of a close relationship between the Collins and the dihadron asymmetries, hinting at a common physics origin of the two FFs [15, 16, 17, 18], as suggested in the P recursive string fragmentation model [19, 20] and, for large invariant mass of the hadron pair, in Ref. [21]. The interpretation is also supported by calculations with a specific Monte Carlo model [22].
In order to better investigate the relationship between the Collins asymmetry and the dihadron asymmetry the correlations between the azimuthal angles of the final state hadrons produced in the SIDIS process have been studied using the COMPASS data. These correlations play an important role in the understanding of the hadronization mechanism and in so far have been studied only in unpolarized SIDIS [23]. In this article for the first time the results for SIDIS off transversely polarized protons are presented. The investigation has proceeded through three major steps:

the Collins asymmetries for positive and negative hadrons have been compared with the corresponding asymmetries measured in the SIDIS process , i.e. when in the final state at least two oppositely charged hadrons are detected (2h sample);

using the 2h sample the asymmetries of and have been measured and their relation has been investigated as function of , the difference of the azimuthal angles of the two hadrons;

the dihadron asymmetry has been measured as function of and, using a new general expression, compared with the and asymmetries. The integrated values of the three asymmetries have also been compared.
2 The COMPASS experiment and data selection
COMPASS is a fixedtarget experiment at the CERN SPS taking data since 2002 [24]. The present results have been extracted from the data collected in 2010 with a 160 GeV/c beam and a transversely polarized proton (NH) target, already used to measure the transverse spin asymmetries [14, 25, 15]. They refer to the 2h sample, i.e. SIDIS events in which at least one positive and one negative hadron have been detected.
The selection of the DIS events and of the hadrons is described in detail in Ref. [15]. Standard cuts are applied on the photon virtuality ( GeV/c), on the fractional energy transfer to the virtual photon (), and on the invariant mass of the final hadronic state ( GeV/c). Specific to this analysis is the requirement that each hadron must have a fraction of the virtual photon energy , where the subscript 1 refers to the positive hadron and subscript 2 to the negative hadron. A minimum value of 0.1 GeV/c for the hadron transverse momenta ensures good resolution in the azimuthal angles. As shown in Fig. 1 the virtual photon direction is the z axis of the coordinate system while the x axis is directed along the lepton transverse momentum. The direction of the y axis is chosen to have a righthanded coordinate system. Transverse components of vectors are defined with respect to z axis.
The 2h sample consists of 33 million pairs, to be compared with the 85 million or 71 million of the standard event sample (1h sample) of the previous analysis [15], where at least one hadron (either positive or negative) per event was required.
3 Comparison of 1h and 2h sample asymmetries
For each hadron the Collins angle is defined as usual as , where is the azimuthal angle of the hadron transverse momentum, is the azimuthal angle of the transverse nucleon spin, and is the azimuthal angle of the spin of the struck quark [26], as shown in Fig. 1. All the azimuthal angles are measured around the axis. For the positive and negative hadrons in the 2h sample, the amplitudes and of the modulations in the crosssection have been extracted with the same method as of Ref. [15] and labeled “CL” (Collinslike) to distinguish them from the standard Collins asymmetries, which are defined in the 1h sample.
Within the accuracy of the measurements the CL asymmetries turn out to be the same as the standard Collins asymmetries (Fig. 2 and Table 1), implying that the Collins asymmetry does not depend on additional observed hadrons in the event. As an important result of the first step of this investigation the 2h sample can be used to study the mirror symmetry and to investigate the interplay between the Collins singlehadron asymmetry and the dihadron asymmetry, as described in the following. All the results of the following work are obtained in the kinematical region , which is the one where the Collins and the dihadron asymmetries are largest.
Collins Asymmetry  Collinslike Asymmetry  

4 dependence of the CL asymmetries of positive and negative hadrons
The azimuthal correlations between and in transversely polarized SIDIS had been investigated by measuring the asymmetries as functions of [17], where . The final results as function of are shown in Fig. 3. The two asymmetries look like even functions of , are compatible with zero when tends to zero, and increase in magnitude as increases.
Very much as in Fig. 2 the mirror symmetry between positive and negative hadrons is a striking feature of the data. The overall picture agrees with the expectation from the P recursive string fragmentation model of Refs. [19, 20], which predicts a maximum value for .
The framework to access the dependence of CL asymmetries was proposed in Ref. [27]. After integration over , , , , and the crosssection for the SIDIS process can be written as
where the unpolarized and the polarized and structure functions (SFs) might depend on . To access the azimuthal correlations of the polarized SFs Eq. (4) is rewritten in terms of and , or alternatively in terms of and :
(2) 
With the change of variables above a new modulation, of the type , appears in the cross section, which can then be rewritten in terms of the sine and cosine modulations of the Collins angle of either the positive or the negative hadron. The explicit expressions for the four asymmetries are:
(3) 
where is the mean transversespintransfer coefficient, equal to 0.87 for these data. Figure 4 shows the measured values of the new asymmetries . It is the first time that they are measured. They have rather similar values for positive and negative hadrons, seem to be odd functions of , and average to zero when integrating over . Note that the data are in very good agreement with Eq. (4) if .
The quantities and , which in principle can still be functions of , can be obtained from the measured asymmetries using
(4) 
The values of the ratios and extracted from the measured asymmetries are given in Fig. 5. Within the statistical uncertainty they are constant, hinting at similar azimuthal correlations in polarized and unpolarized SFs. Moreover, they are almost equal in absolute value and of opposite sign.
Assuming , the measured asymmetries can be fitted with the simple functions in the case of the sine asymmetries, and for the cosine asymmetries. The results of the fits for positive (negative) hadrons are the dashed red (dotdashed black) curves shown in Fig. 3 and 4. The agreement with the measurements is very good and the four values for the constants are well compatible, as can be seen in Table 2.
0.014 0.003  0.025 0.005  
0.016 0.003  0.017 0.005 
As a conclusion of this second step of the investigation, the and CL asymmetries as functions of agree with the expectation from the P recursive string fragmentation model and with the calculations of the dependence obtained in Ref. [27]. As in the onehadron sample a mirror symmetry for the positive and negative hadron sine asymmetries is observed in the 2h sample, which is a consequence of the experimentally established relation .
These results allow to derive a quantitative relation between the and CL asymmetries and the dihadron asymmetry, as described in the following.
5 Comparison of CL and dihadron asymmetries
The third and last step of this investigation has been the formal derivation of a connection between the CL and the dihadron asymmetries and the comparison with the experimental data. In the standard analysis, the integrated dihadron asymmetry is measured from the amplitude of the sine modulation of the angle , where is the azimuthal angle of the relative hadron momentum . In the present analysis, the azimuthal angle of the vector is evaluated for each pair of oppositely charged hadrons, with the hat indicating unit vectors. As discussed in Ref. [15], the azimuthal angle is strongly correlated with , where is the signum function. Also, introducing the angle , which is a kind of mean of the Collins angle of the positive and negative hadrons after correcting for a phase difference, it was shown [15] that the dihadron asymmetry measured from the amplitude of is essentially identical to the standard dihadron asymmetry. In order to establish a connection between the dihadron asymmetry and the CL asymmetries will be used rather than in the following. Starting from the general expression for the cross section given in Eq. (4), changing variables from and to and , and using the relations and , where , Eq. (4) can be rewritten as:
(5)  
which simplifies to
(6) 
using the experimental result . This last crosssection implies a sine modulation with amplitude
(7) 
At variance with the single hadron case, no asymmetry is present in Eq. (6) and the measured values are indeed compatible with zero. In Figure 6 the asymmetry is shown together with the curve with (black solid line) as obtained by the fit. The dashed red and dotdashed black curves are the fitted curves of Fig. 3. As can be seen the fit is good, and the value of is well compatible with the corresponding values of Table 2, in agreement with the fact that is the same for the three asymmetries. Evaluating the ratio of the integrals of the dihadron amplitudes over the onehadron amplitudes one gets a value of which agrees with the value evaluated from Eqs. (7) and (4) and with our original observation that the dihadron asymmetry is somewhat larger than the Collins asymmetry for positive hadrons.
6 Conclusions
We have shown that in SIDIS hadronpair production the dependent Collinslike single hadron asymmetries of the positive and negative hadrons are well compatible with the standard Collins asymmetries and are mirror symmetric. Also, the Collinslike asymmetries exhibit a dependence on , which we have derived from the general expression for the twohadron crosssection and is a consequence of the experimentally verified similar dependence of the unpolarized and polarized structure functions and the mirror symmetry of the last ones.
Most important, for the first time it has been shown that the amplitude of the dihadron asymmetry as a function of has a very simple relation to that of the single hadron asymmetries in the 2h sample, namely it can be written as a , where the constant is the same as that which appears in the expressions for the Collinslike asymmetries. After integration on , the dihadron asymmetry has to be larger than the single hadron asymmetries by a factor , in good agreement with the measured values.
In conclusion, we have shown that the integrated values of Collins asymmetries in the 1h sample are the same as the Collinslike asymmetries of 2h sample which in turn are related with the integrated values of dihadron asymmetry. This gives indication that both the single hadron and dihadron transversespin dependent fragmentation functions are driven by the same elementary mechanism. As a consequence of this important conclusion we can add that the extraction of transversity distribution using the dihadron asymmetry in SIDIS does not represent an independent measurement with respect to the extractions which are based on the Collins asymmetry.
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