Collins and Sivers transverse-spin asymmetries in inclusive muoproduction of $\rho^0$ mesons

The production of vector mesons in deep inelastic scattering is an interesting yet scarsely explored channel to study the transverse spin structure of the nucleon and the related phenomena. The COMPASS collaboration has performed the first measurement of the Collins and Sivers asymmetries for inclusively produced $\rho^0$ mesons. The analysis is based on the data set collected in deep inelastic scattering in $2010$ using a $160\,\,\rm{GeV}/c$ $\mu^+$ beam impinging on a transversely polarized $\rm{NH}_3$ target. The $\rho^{0}$ mesons are selected from oppositely charged hadron pairs, and the asymmetries are extracted as a function of the Bjorken-$x$ variable, the transverse momentum of the pair and the fraction of the energy $z$ carried by the pair. Indications for positive Collins and Sivers asymmetries are observed.


Introduction
In semi-inclusive deep inelastic scattering (SIDIS) → ′ ℎ , a high-energy lepton scatters off a target nucleon , and in the final state the scattered lepton ′ is observed in coincidence with at least one hadron ℎ produced in the current fragmentation region. This process presently is the main tool to study the 3-dimensional partonic structure of the nucleon, i.e. the transverse spin and transverse momentum distributions of partons, and the possible correlations between their spin, their motion, and the spin of the nucleon. In the present quantum chromodynamics (QCD) framework, such information is encoded in the Transverse Momentum Dependent Parton Distribution Functions (TMD PDFs). In SIDIS, the above mentioned correlations induce azimuthal asymmetries in the angular distributions of the produced hadrons, which are interpreted in terms of convolutions of TMD PDFs and Transverse Momentum Dependent Fragmentation Functions (TMD FFs) 1.
Among the accessible asymmetries, the transverse single spin asymmetries (TSAs), which arise for a transversely polarized target nucleon, have been extensively studied in recent years. In particular, the Collins asymmetry [2] arises from the convolution between the chiral-odd transversity PDF ℎ 1 [3] and the chiral-odd and T-odd Collins FF ⊥ ℎ 1 [2]. The transversity PDF, which is the difference between the number density of partons with transverse polarisation parallel and antiparallel to the transverse polarisation of the parent nucleon, is the least well known among the three collinear PDFs needed for the complete characterization of the nucleon structure at leading order. The Collins TMD FF describes the correlation between the transverse polarisation of a fragmenting quark and the transverse momentum of the produced hadron, and probes the quark-spin dependence of the fragmentation process.
Another important TSA is the so-called Sivers asymmetry, which is interpreted as the convolution of the Sivers function ⊥ 1T [4], the TMD PDF that describes the transverse momentum distribution of an unpolarized quark in a transversely polarized nucleon, and the spin-averaged TMD FF ℎ 1 that describes the fragmentation of an unpolarized quark into an unpolarized hadron.
Collins and Sivers asymmetries have been measured since 2005, in particular in SIDIS off protons, deuterons or neutrons for unidentified charged hadrons and for identified pions, kaons and protons by the HERMES [5], COMPASS [6,7,8] and JLab [9] experiments. Phenomenological analyses of the Collins asymmetries and the corresponding asymmetries measured in + − annihilation to hadrons [10,11,12] have led to the extraction of both the transversity PDF and the Collins FF [13,14]. Similarly, from the HERMES and COMPASS measurements of the Sivers asymmetry, the Sivers function was extracted by several authors [15,16,17]. For a recent review on transverse spin effects on semi-inclusive processes see, e.g. Ref. [18].
Relevant information on nucleon structure and the fragmentation process can also be obtained from measurements of TSAs in inclusive production of vector mesons in DIS [19]. The Collins asymmetry for vector mesons couples the transversity PDF to the Collins FF ⊥VM 1 , which describes the fragmentation of a transversely polarized quark into a vector meson. The investigation of this channel could shed new light on the still poorly known quark-spin dependence of the fragmentation process. Model predictions based on the recursive string+ 3 0 model of polarized quark fragmentation [20,21] suggest that the Collins asymmetry for mesons has the opposite sign compared to that of positive pions. Depending on the choice of the parameters, the Collins asymmetry for vector meson production can be as large as for positive pions.
1For TMD PDFs and TMD FFs we use the notation of Ref. [1].
Up to now, transverse spin asymmetries for vector meson production have not been measured, neither in SIDIS nor in + − annihilation into hadrons. The low statistics of the produced vector mesons and the high combinatorial background make the measurement of these asymmetries very challenging.
In this article we present the first measurement of the Collins and Sivers asymmetries for 0 mesons produced in DIS off a transversely polarized proton target. The analysis is performed on the COMPASS data collected in 2010. The same data have already been used for many published results, e.g., the Collins and Sivers asymmetries for unidentified charged hadrons [6,7], pions and kaons [8], and dihadron production asymmetries [22]. The final data set used for the analysis described in this paper consists of 2.6 × 10 6 0 mesons.
The article is organized as follows. The formalism of vector meson production in SIDIS is introduced in Sec. 2. Section 3 describes the experimental apparatus and the data set used for this analysis. The method used for the extraction of TSAs is explained in Sec. 4. The extraction of the 0 signal is described in Sec. 5 and the results for Collins and Sivers 0 asymmetries are given in Sec. 6. In Sec. 7 conclusions are drawn.

Theoretical formalism
The kinematics for the production of a vector meson (VM) in SIDIS off transversely polarised protons in the one-photon exchange approximation is schematically shown in Fig. 1. The process is represented in the !" Fig. 1: Kinematics of the SIDIS process for 0 meson production in the gamma-nucleon reference system. gamma-nucleon system (GNS), namely in the reference system where the momentum of the exchanged virtual photon defines theẑ axis and thex-ẑ plane is the lepton scattering plane with thex axis along the transverse component of the lepton momenta [23].
If the polarisation of the vector meson is not considered, at leading twist TMD PDFs and TMD FFs and leading order order QCD, the differential cross section of the process → ′ 0 is [24]: Here, is the Bjorken variable, is the fraction of the initial lepton energy loss in the target rest frame, and 2 is the virtuality of the photon. The variable is the fraction of the energy of the virtual photon carried by the produced VM in the target rest frame and T is the modulus of its transverse momentum P T in the GNS. The variables VM and S are the azimuthal angle of P T where indicates a TMD PDF, indicates a TMD FF, and is a weight factor depending on the intrinsic quark transverse momentum k T in the GNS and on the transverse momentum p ⊥ of the VM with respect to the direction of the scattered quark.
The expression for the Collins asymmetry can be obtained from the ratio between the transverse-spin-dependent and the spin-averaged terms of the cross section in Eq. (1), and is given by The functions VM 1 and ⊥VM 1 describe the fragmentation of an unpolarized and a transversely polarized quark into a vector meson, respectively.
The Sivers asymmetry reads [24] sin Siv UT The expected number of vector mesons is where the other possible modulations are neglected. Here, is the dilution factor that takes into account the fraction of polarisable protons in the target, and t is the average transverse polarisation of the polarisable protons in the target.
For the specific case of 0 meson production considered in this work, we use all oppositely charged hadron pairs in the event. The four-momentum of the 0 candidate is given by h 1 + h 2 , where h 1 and h 2 are the momenta of the positive and the negative hadron of the pair, respectively. The fractional energy of the 0 candidate is given by = h 1 + h 2 and its transverse momentum in the GNS is given by P T = P h 1 T + P h 2 T , where we have indicated by h i and P h i T ( = 1, 2) the fraction of the virtual photon energy carried by the hadron ℎ in the target rest system, and the transverse momentum of ℎ in the GNS, respectively. In the following we indicate with hh the azimuthal angle of P T , and with hh = √︁ ( h 1 + h 2 ) 2 the invariant mass of the pair. As it will be discussed in Sec. 5, the data set of 0 candidates contains a sizeable combinatorial background due to non-resonant hadron pairs.

Experimental apparatus and data sample
The COMPASS experiment, a fixed target experiment located at the M2 beamline of the CERN SPS, is in operation since 2002. A detailed description of the apparatus can be found in Ref. [25,6,8]. The data used in this analysis were collected in 2010 using a 160 GeV/c + beam and a transversely polarized NH 3 target. The target consisted of three cylindrical cells with neighbouring cells polarized in opposite directions in order to collect data simultaneously for both target spin orientations. The average polarisation of the hydrogen nuclei was ⟨ t ⟩ ≃ 0.8. The average dilution factor was ⟨ ⟩ ≃ 0.15, with a slight dependence on and constant in and hT [7]. The data taking was divided in twelve periods of about ten days. In order to compensate for acceptance effects the polarisation was reversed in the middle of each period.
Only events with an incoming and an outgoing muon track and at least two produced charged hadrons are considered. Equal flux along the target cells is assured by requiring the extrapolated incoming muon to cross all three target cells. We only consider hadrons coming from the muon production vertex, thus tracks from mesons produced in weak decays do not enter this analysis. In order to ensure the deep inelastic regime, we require 2 > 1.0 (GeV/c) 2 and the invariant mass of the final hadronic system > 5 GeV/c 2 . The Bjorken variable ranges between 0.003 and 0.7. The selection > 0.1 removes events with poorly reconstructed virtual photon energy and < 0.9 removes events with large radiative effects. For hadrons, we require ℎ > 0.1 to ensure the current fragmentation regime and ℎT > 0.1 GeV/c to ensure good resolution in the respective azimuthal angle.
The dihadron samples with pairs ℎ + ℎ − , ℎ + ℎ + , ℎ − ℎ − of charged hadrons are selected as described in the following. In order to avoid the contribution from non-SIDIS diffractive events, exclusively produced ℎ + ℎ − pairs are rejected by requiring a missing energy miss > 3.0 GeV, where miss = ( 2 X − 2 )/2 , and 2 X = ( + p − ) 2 where , p and are the four momenta of the exchanged photon, the target proton and the hadron pair respectively. The requirements < 0.95, 0.1 GeV/c < P T < 4.0 GeV/c and 0.35 GeV/c 2 < M hh < 3.0 GeV/c 2 are also applied in order to define the kinematic range of the analysis. The selection > 0.3 is applied in order to enhance the fraction of 0 mesons. No further selection improving the signal over background ratio could be found.
The selected dihadron samples consist of about 3.4 × 10 7 ℎ + ℎ − pairs, 1.1 × 10 7 ℎ + ℎ + pairs and 0.7 × 10 7 ℎ − ℎ − pairs. The distributions of , and T for the selected ℎ + ℎ − sample are shown as filled histograms in the left, middle and right panels of Fig. 2, respectively. The empty histogram in the middle panel of the same figure shows the distribution without the requirements on the hadron pairs. The exclusive peak at = 1, rejected by the requirement on the missing energy, is clearly visible.
The invariant mass distribution for the ℎ + ℎ − sample is shown in Fig. 3, where the empty (filled) histogram is the invariant mass distribution before (after) the application of the cuts on the hadron pair variables. The peak corresponding to the 0 (770) invariant mass is clearly visible as well as the broad structures corresponding to the 0 (980) and 2 (1240) mesons. As the requirement of the missing energy rejects exclusive events, it also reduces the significance of the 0 peak.

Method for the extraction of the transverse single spin asymmetries
The invariant mass range is divided in the four regions that are defined in Tab. 1 and shown in Fig. 3 as separated by the vertical lines. Region II covers the 0 invariant mass peak and will be referred to as the " 0 region" in the following. Regions I and III are dominated by the combinatorial background and in the following will be referred to as the "side band" regions. Region IV has a comparable number of pairs as region III and is included to study the invariant mass dependence of TSAs for the combinatorial background, although it is not used in the extraction of the 0 asymmetries.
The extraction of the 0 TSAs proceeds using the following steps. First, the fraction of 0 mesons in the 0 region is evaluated. Then, the transverse single spin asymmetry sin X UT of the ℎ + ℎ − pairs in the 0 region is extracted. The angle X indicates either the Collins angle Coll or the Sivers angle Siv . The asymmetries are measured for all kinematic bins by using the unbinned maximum likelihood method [22] and the fit function ( Coll , Siv ) = 0 × (1 + NN t C sin Coll + t S sin Siv ) in agreement with the r.h.s. of Eq. (5). The parameters 0 , C and S are determined from the fitting procedure and are used to calculate the asymmetries sin Coll UT = C and sin Siv UT = S . The transverse single spin asymmetry sin X UT,bg of the background is evaluated as the mean of the asymmetries in regions I and III using the same procedure. In order to obtain the asymmetry for 0 mesons, the background transverse single spin asymmetry is subtracted from the asymmetry in the 0 region according to

Estimation of the 0 signal
In order to determine the fraction of 0 mesons in the invariant mass region II, it is necessary to evaluate the contribution of the combinatorial background. The shape of the background distribution in the 0 region is taken from the sum of the invariant mass distributions of ℎ + ℎ + and ℎ − ℎ − pairs. As normalisation the ratio between the number of ℎ + ℎ − pairs and the number of like-sign pairs in the invariant mass interval 0.46 GeV/c 2 < M hh < 0.56 GeV/c 2 is used, where the ratio between the invariant mass distributions of opposite and like-sign pairs is constant. For M hh < 0.50 GeV/c 2 the background subtraction procedure yields negative counts, meaning that the estimated background is larger than the ℎ + ℎ − distribution. The distribution of like-sign pairs is in fact not expected to reproduce the full invariant mass spectrum of ℎ + ℎ − pairs because of the different resonant structures of like-sign and unlike-sign pairs. However it can be used to estimate the background in the 0 region. The estimated invariant mass distribution of the background is shown in the top panel of Fig. 4 by the red points. It is subtracted from the ℎ + ℎ − distribution (continuous histogram) to obtain the difference of the extracted 0 distribution can be successfully described by a Breit-Wigner function provides confirmation that the form of the combinatorial background in region II and its normalisation is evaluated correctly.
The fraction of 0 mesons in region II is calculated by dividing the number of 0 mesons by the total number of ℎ + ℎ − pairs in the same region. The contamination from the decay of 0 mesons, estimated to be about 4%, is neglected. Also, since the 0 distribution is described by a width fixed to the PDG value, possible interference effects with the (782) decays are neglected. The total number of 0 mesons is estimated to be 2.6 × 10 6 and the average signal fraction is ⟨ ⟩ = 0.18. This procedure is applied to all , and T bins, and the values of are shown in Fig. 5. We find to be almost constant and about 0.18 as a function of and it increases with T and . The high value (about 0.38) in the last bin can be understood in terms of the string fragmentation model, where heavier resonances are produced mostly with large fractional energies [26].
As consistency check we compared the counts of 0 mesons obtained by summing separately over the , and T bins, which results in similar values that differ by less than 2% with respect to the integrated value. As a further check, the measured 0 distribution is compared for each kinematic bin to that expected by using a Breit-Wigner function with mass peak and width fixed to the PDG values and the normalisation fitted to the measured 0 distribution in the 0 region. The largest differences on the 0 counts are found to be less than 7.5% and located in the first two bins. These differences are taken into account in the evaluation of the systematic uncertainty of the final asymmetries. Moreover, it is checked that nearly the same background in the 0 region can be obtained with an alternative method that combines the invariant mass distributions simulated with the PYTHIA 8 event generator [27] for the different background components (resonant and non-resonant contributions) and Breit-Wigner functions with PDG parameters for the 0 , 0 and 2 resonances to fit the total ℎ + ℎ − distribution up to M hh = 1.4 GeV/c 2 . The differences between the two methods are small and are taken into account in the systematic uncertainty of the measured asymmetries for 0 mesons.

Results for Collins and Sivers asymmetries
The Collins asymmetry sin Coll UT for all selected ℎ + ℎ − pairs is shown in Fig. 6 as a function    The background asymmetry is evaluated taking the average of the asymmetries in regions I and III. According to Eq. (6), the background asymmetry is first rescaled by the factor 1 − and then subtracted from the Collins asymmetry in the 0 region. The contribution of the background asymmetry in the 0 region is shown in the top panel of Fig. 7 as a function of , and T . It has mostly negative values, as expected from Fig. 6, although the uncertainties are large. The final Collins asymmetry for 0 mesons is shown in the bottom panel of Fig. 7 and the corresponding values are given in Tab. 2. Given the large uncertainties, no clear trend can be seen as a function of and . As a function of T the asymmetry increases for T < 0.5 GeV/c , as suggested by the simulations carried out with the recursive string+ 3 0 model in Ref. [21], using the scenario where the production of vector mesons with longitudinal polarization in the GNS is favoured.
The systematic uncertainty on the extracted transverse single spin asymmetries for 0 mesons is estimated to be about 0.6 times the statistical one. As described above, this estimate accounts for the systematic uncertainty on The average value of the Collins asymmetry for 0 mesons is shown in Fig. 8. The asymmetry is positive with a significance of 2.3 standard deviations, evaluated taking into account both statistical and systematic uncertainties, and is in agreement with the model predictions [20]. Comparing the average value of the measured 0 Collins asymmetry with that of the simulated one [21], shown as the open point in Fig. 8, consistency is found within about one standard deviation.
The same procedure is repeated for the Sivers asymmetry. The asymmetry sin Siv UT is shown as a function of , , T and M hh in Fig. 9 for the different invariant mass regions. The Sivers asymmetry in the 0 region exhibits positive values and a clear trend with . Contrary to the case of the Collins asymmetry, the Sivers asymmetry is positive and significant also in the side-bands. This indicates that the contribution of the background to the asymmetry is large in the 0 region, as shown in the top panel of Fig. 10. The background-subtracted final Sivers asymmetry for 0 mesons is shown in the bottom panel in Fig. 10 and the corresponding values are given in Tab. 2. The shown uncertainties are the statistical ones. As for the Collins asymmetry, the systematic uncertainty is evaluated to be 0.6 times the statistical one. The average value of the asymmetry is shown in Fig. 8. It is found to be positive with a significance of 1.8 standard deviations. A positive Sivers asymmetry for 0 mesons is expected because, by momentum conservation in the hard scattering, the Sivers function induces a modulation on the direction of the struck quark which propagates to all the hadrons produced in the fragmentation process. The Sivers asymmetry for 0 mesons is thus naively expected to be similar to the average value of the Sivers asymmetries for positive and negative pions, which is positive [8]. Given the large uncertainties, no clear trends as a function of the kinematic variables can be seen for the 0 asymmetry.

Conclusions
The COMPASS Collaboration has performed the first measurement of the Collins and Sivers transverse single spin asymmetries for 0 mesons produced in DIS off transversely polarized protons. The full data set of SIDIS events collected by COMPASS in 2010 was analysed. An indication for a positive Collins asymmetry is found. The result is in agreement with the expectation from the recursive string+ 3 0 model of the polarized quark fragmentation process. Also an indication for a positive Sivers asymmetry is found, in agreement with the parton model. These measurements are complementary to the single hadron asymmetries, since they give new input to studying and understanding of the hadronisation process. This work shows that the measurement of TSAs for inclusive vector meson production in DIS is feasible and could be done with higher precision at future facilities.