Collins and Sivers asymmetries in muonproduction of pions and kaons off transversely polarised protons

Measurements of the Collins and Sivers asymmetries for charged pions and charged and neutral kaons produced in semi-inclusive deep-inelastic scattering of high energy muons off transversely polarised protons are presented. The results were obtained using all the available COMPASS proton data, which were taken in the years 2007 and 2010. The Collins asymmetries exhibit in the valence region a non-zero signal for pions and there are hints of non-zero signal also for kaons. The Sivers asymmetries are found to be positive for positive pions and kaons and compatible with zero otherwise.


Introduction
The description of the nucleon spin structure is still one of the open issues in hadron physics.In the last decades major progress in this field has been made by an interplay between new experimental results and the development of non-collinear QCD.The first information on transverse spin and transverse momentum effects become available recently.Presently, the complete description of quarks in the nucleon includes all possible correlations between quark spin, quark transverse momentum and nucleon spin [1,2].At leading twist, these correlations are described for each quark flavour by eight transverse momentum dependent (TMD) parton distribution functions (PDFs).After integration over transverse momentum, only three of them survive, namely the number density, the helicity and the transversity PDFs.One way to access experimentally these TMD PDFs is via semi-inclusive lepton-nucleon deep inelastic scattering (SIDIS), i.e. by studying deep-inelastic scattering (DIS) with detection of at least one of the produced hadrons.When the target nucleon is transversely polarised, the SIDIS cross section exhibits different azimuthal modulations [3] in different combinations of the two angles φ S and φ h .These are the azimuthal angles of the initial nucleon transverse spin vector and of the produced hadron momentum in a reference system, in which the z-axis is the virtual photon direction and the x − z plane is the lepton plane according to Ref. [4].The amplitudes of the modulations in the cross section (the so-called transverse spin asymmetries) are proportional to convolutions of TMD PDFs with TMD fragmentation functions.The two most thoroughly studied transverse spin asymmetries are the Collins and Sivers asymmetries.The Collins asymmetries allow access to the transversity PDFs coupled to the Collins fragmentation functions [5].The Sivers asymmetries give access to the Sivers PDFs [6], which describe the correlations between quark transverse momentum and nucleon spin.A Sivers PDF always appears in combination with an 'ordinary' (unpolarised) fragmentation function that describes the fragmentation of a quark into a hadron.
In this Paper, we present results on the Collins and Sivers asymmetries for pions and kaons produced on transversely polarised protons in a NH 3 target.These measurements are in line with the set of measurements done by the COMPASS Collaboration in the last years.Results on polarised deuterons were obtained for unidentified hadrons [7], pions and kaons [8], and on polarised protons for charged unidentified hadrons [9][10][11].The results presented in this Letter are extracted from all available COMPASS data taken in 2007 and 2010 using transversely polarised protons.For the measurements in 2007 and 2010, a similar spectrometer configuration was used.As compared to the measurements on transversely polarised deuterons, the measurements on transversely polarised protons benefit from a major upgrade of the apparatus performed in 2005.Of particular relevance for these measurements is the upgrade of the RICH detector [12], which led to improved efficiency and purity for the samples of identified particles, and the use of a new target solenoid magnet with a polar angle acceptance of 180 mrad as compared to the 70 mrad of the magnet used until 2005.Measurements of these asymmetries by the HERMES experiment exist [13,14] in a different kinematic range.Comparison with these results are also presented in the Paper.

Apparatus and data selection
The COMPASS spectrometer [15] is in operation in the North Area of CERN since 2002.The µ + beam provided by the M2 beam line had a momentum of 160 GeV/c, a momentum spread ∆p/p = ±5%, and a longitudinal polarisation of −80% that originated from the π-decay mechanism.The mean beam intensity was about 2.3 × 10 7 µ + /s and 4 × 10 7 µ + /s with spill lengths of 4.8 s and 10 s in 2007 and 2010, respectively.The target consisted of three cylindrical cells of 4 cm diameter, each separated by gaps of 5 cm.The length of the central cell was 60 cm and that of the two outer ones 30 cm.For the measurement of transverse spin effects, the target material was polarised along the vertical direction.In order to reduce systematic effects, neighbouring cells were polarised in opposite directions, which allows for simultaneous data taking with both target spin directions.To further minimise systematic effects, the polarisation of each cell was reversed every 4-5 days.During the 2007 data taking, a total amount of 12 × 10 9 events (440 TB) was recorded in six periods, each consisting of two sub-periods of data taking with opposite polarisation.In 2010 about 37 × 10 9 events (1.9 PB) were recorded over twelve periods.
Only events with a photon virtuality Q 2 > 1 (GeV/c) 2 and a mass of the hadronic final state W > 5 GeV/c 2 have been used to ensure the kinematic region of DIS.The upper limit on the fractional energy of the virtual photon (y) was set to 0.9 to reduce uncertainties due to electromagnetic radiative corrections and contamination from pion decay.A lower limit on y is required to ensure a good resolution in this variable.In the standard analysis this limit has been set to 0.1.A complementary sample with 0.05 < y < 0.1 was also studied, mainly to address the Q 2 dependence of the asymmetries.The Bjorken variable x covers the range from 0.003 to 0.7.A minimum value of 0.1 GeV/c for the hadron transverse momentum p h T with respect to the virtual photon direction was required to ensure good resolution in the measured azimuthal angle.A minimum value for the relative hadron energy z with respect to the virtual photon energy is needed to select hadrons from the current fragmentation region.This value has been set to 0.2 for the standard sample, while a complementary lower-z region (0.1 < z < 0.2) was also studied.
The stability of the apparatus during data taking is crucial.Therefore, various tests were performed using the 2007 and 2010 data, as described in [9][10][11].As a result from these quality tests, only four periods of data taking in 2007 were used for the analysis of the Sivers asymmetries, while for the Collins asymmetry all six periods were used.This can be understood as the Sivers asymmetry is very sensitive to instabilities in the spectrometer acceptance, because it represents the amplitude of the modulation that depends on the azimuthal angle of the hadron transverse momentum with respect to the target spin vector, which is aligned along a fixed direction.Due to improved detector stability, all periods of 2010 could be used for the extraction of both asymmetries.

Charged pions and kaons
The RICH detector information was used to identify charged hadrons as kaons and pions.The pattern of the detected photons in the detector was analysed taking into account the predicted path of the charged particle to compute likelihood values [16] for each reconstructed track entering the RICH acceptance.The likelihoods L were computed for different mass hypotheses (L M , with M = π, K, p, e) and for the hypothesis of absence of signal, namely the so called background hypothesis (L back ).A mass value is attributed to a track if the likelihood for the corresponding mass hypothesis is the largest.In addition, cuts on the ratio of the largest likelihood value to the second largest one were added to improve the figure of merit given by the product of the identification efficiency and sample purity, as defined below.Pions and kaons were identified in the momentum range between the Cherenkov threshold (about 2.6 GeV/c for pions, 9 GeV/c for kaons) and 50 GeV/c.A specific cut on the ratio L K /L back was applied to minimise the contamination of protons in the kaon sample in the momentum range between the kaon threshold and the proton one (about 18 GeV/c).In Fig. 1 the distributions in momentum (p), z and Q 2 for identified pions and kaons are shown and compared to those for unidentified hadrons.The mean values of the p h T , z and Q 2 distributions as a function of x are shown in Fig. 2.
Both the sample purity and the identification efficiency were measured and found to be compatible in the two data taking years as well as in the different periods of each year.The particle identification efficiencies and misidentification probabilities were determined using samples of pions from the K 0 decay and of kaons from the φ decay.The efficiencies are about 97% for pions and 94% for kaons.These values start to decrease for momenta about 30 GeV/c and reach values of 60% in the larger momentum region.In order to achieve high values of the sample purity, the misidentification probabilities between  pions and kaons were kept as low as a few percent even at the largest momentum values.The purity is defined as the fraction of K (π) inside the identified K (π) sample and depends also on the different population of the various particle types.It was evaluated from the particle identification efficiencies and misidentification probabilities and the number of identified kaons and pions, since the proton contribution is very small as already mentioned.The average purity values for pions are above 99%.The kaon purity is shown in Fig. 3 as a function of x, z, and p h T ; it is about 94% with a mostly mild dependence on the variables.The strongest dependence is visible in the large z region for the negative kaon sample, which is due to the increasing ratio of pions to kaons.The resulting statistics for charged pions and kaons after all cuts are shown in Table 1.
The K 0 identification is based on the detection of two oppositely charged tracks coming from a secondary vertex, for which the 2-pion invariant mass lies in the window m K 0 ± 20 MeV/c 2 .A separation between the primary and the secondary vertex of at least 10 cm was required.Furthermore, the angle between the reconstructed momentum vector of the track pair and the vector connecting the primary and secondary vertices was required to be smaller than 10 mrad.On the left side of Fig. 4, the Armenteros-Podolansky plot of the hadron pair is shown, in which the transverse momentum p T of one hadron with respect to the sum of hadron momenta is shown as a function of the difference of the longitudinal momenta over their sum, (p L1 − p L2 )/(p L1 + p L2 ).The K 0 band is clearly visible as well as the Λ and Λ bands.In order to exclude the contamination by the Λ/ Λ signal, the p T region between 80 MeV/c and 110 MeV/c was excluded.The background from e + e − pairs was suppressed by a lower cut on p T at 40 MeV/c.For the detected K 0 s, the difference between their mass value and the PDG [17] value is shown in the right panel of Fig. 4, where the vertical lines at ±20 MeV/c 2 enclose the K 0 s used for further analysis.The same cuts on z and p T as for the charged hadron samples were applied to the neutral kaons.The resulting statistics for K 0 are given in Table 1.

Results
In order to extract the transverse spin asymmetries from the data, the same procedure as described in Refs.[10,11] was used.The asymmetries were evaluated in bins of the kinematic variables x, z, or p h T , using the same binning as in our previous analyses [8].All the numerical results are available on HEPDATA.The (φ S ,φ h ) distributions from the different target cells and sub-periods were fitted using an extended maximum likelihood estimator [9], and the eight transverse spin asymmetries expected in the SIDIS process were extracted simultaneously.
The resulting sin(φ h + φ s ) and sin(φ h − φ s ) modulations yield the Collins and Sivers asymmetries, respectively, after division by i) the target material dilution factor, ii) the average target proton polarisation, and for the Collins asymmetry iii) the transverse spin transfer coefficient.The dilution factor of the ammonia target, taking also into account the electromagnetic radiative corrections, was evaluated in each x bin [10]: it increases with x from 0.14 to 0.17.As a function of z and p h T the dilution factor turns out to be almost constant with an average value of 0.15.
A non-flat azimuthal acceptance introduces correlations between the various modulations resulting from the fit.The correlation coefficients for Collins vs. Sivers asymmetries are found to be small and below 0.2 for all bins.Moreover, the asymmetries measured along different projections of the (x, z, p h T ) phase space are statistically correlated, because the overall sample of events is the same.In the case of COMPASS, these correlation coefficients for the Collins and for the Sivers asymmetries are all smaller than about 0.3, but non-negligible, so that they should be taken into account in any global fit.They are slightly different for kaons and pions due to the different kinematic coverage of the two samples.
In order to estimate the systematic uncertainties, several tests were performed based on our previous work for the charged hadrons [9][10][11].The effect of changes in the azimuthal acceptance between the data sets used to extract the asymmetries was quantified building false asymmetries, namely assuming a wrong polarisation direction in the target cells.The Collins and Sivers asymmetries were extracted splitting the data according to the scattered muon direction (up and down, left and right), and the statistical compatibility of the results was checked.No such false asymmetry was observed within the accuracy of the measurement and the point-to-point systematic uncertainties were evaluated from these tests as a fraction of the statistical error.For 2010, this fraction is 0.6 and for 2007 it ranges between 0.5 and 0.7.The systematic uncertainty due to particle misidentification is very small and included in these fractions.All results are subject to a 3% scale uncertainty that results from the uncertainties in the target polarisation and dilution factor.
For both years of data taking, the asymmetries were evaluated in each period and their compatibility was checked.While for 2010 this test shows good agreement among the results of the different periods, for 2007 it introduces an additional source of systematic uncertainties for the Sivers asymmetries.Very much like in the case of unidentified hadrons, an additional absolute uncertainty of ±0.012 was assigned to the π + Sivers asymmetry.This value is taken to be half of the difference between the mean asymmetries evaluated using the data from the beginning and the end of the 2007 data taking.Figure 5 shows the Collins and Sivers asymmetries for pions as a function of x from the two data taking years, obtained as weighted mean of the asymmetries from the different periods.The two measurements are in good agreement.The substantial improvement of the statistical precision of the 2010 data with respect to the 2007 data amounts to a factor of 1.6 for the Collins asymmetry and of 1.9 for the Sivers asymmetry.The final results were obtained combining the two samples, taking into account the different statistical and systematic uncertainties.The resulting systematic uncertainties are about 0.6 of the statistical ones for all the particle types.
The Collins asymmetries as a function of x, z, or p h T measured by COMPASS for pions and kaons on transversely polarised protons are shown in Fig. 6.The pion asymmetries are very similar to the unidentified hadron asymmetries [10]: at small x they are compatible with zero, while in the valence region they show an increasing signal, which has opposite sign for π + and π − .This naively indicates that the unfavoured and favoured Collins fragmentation functions have opposite sign.The results for charged kaons, although with larger statistical uncertainties, show a similar trend: in particular the K + asymmetry has a negative trend with increasing x, and the K − one is positive on average.The Collins asymmetry for neutral kaons shows a positive trend with increasing z.The average asymmetry is positive but compatible with zero within the statistical uncertainty.In order to investigate in more detail the behaviour of the asymmetries as a function of z and p h T , the asymmetries for charged hadrons were evaluated in a region where the signal is different from zero, namely x > 0.032.The results are shown in Fig. 7 for pions and kaons.They are in good agreement with the other existing measurements on a proton target from the HERMES experiment [13].This is a non-obvious result, as in the last x bins the COMPASS Q 2 values are larger by a factor 3-4 than the HERMES ones.The weak Q 2 dependence of the Collins asymmetry is also supported by a recent global fit [18] of the HERMES pion results, the COMPASS preliminary pion asymmetries from the 2010 data, and the Belle [19] e + e − → π + π − asymmetries, which is able to provide a good description of all the data sets.A comparison between the final results of this paper and the fit is shown in Fig. 8.The Collins asymmetry for charged hadrons was further investigated by extending the standard kinematic ranges in z and y.Compared to the above presented results, the asymmetries extracted in the low-z region (0.1 < z < 0.2) gave no indication for a substantial z-dependence, neither for pions nor for kaons.Similarly, in the low-y region (0.05 < y < 0.1) the pion asymmetries do not exhibit any special behaviour, while the kaon ones suffer from too low statistics.
The Sivers asymmetries measured by COMPASS for pions and kaons on transversely polarised protons are shown in Fig. 9. Also in this case, the pion asymmetries are very similar to the unidentified hadron asymmetries [11].The asymmetries for negative pions and kaons, as well as for neutral kaons are compatible with zero, while for positive pions and kaons there is a clear evidence for a positive signal extending over the full measured x region and increasing with z.Very intriguing is the fact that the K + signal is larger than the π + one, which indicates a possibly not negligible role of sea quarks [20][21][22].This is well visible in Fig. 10, where the two asymmetries are directly compared, and from the mean values in the x > 0.032 region, which are respectively 0.027 ± 0.005 and 0.043 ± 0.014.Unlike the case of the Collins asymmetry, the Sivers asymmetry measured by COMPASS at large x for positive pions and kaons is smaller than the one from HERMES [14].This difference is well visible also in the z and p h T variables when selecting the x > 0.032 region of the COMPASS data, as shown in Fig. 11.Several fits, which include the recently revisited Q 2 evolution, were performed using HERMES asymmetries [14], Fig. 8: Comparison between the Collins asymmetries for pions and one of the fits in [18] (fit with standard parameterisation and fit of A 12 Belle asymmetries [19]).The preliminary asymmetries from 2010 data are included in the fit.
COMPASS asymmetries on deuteron [8] and for unidentified hadrons on proton [11].In Fig. 12, the results of some of these fits [23][24][25], which employ Q 2 TMD evolutions, are shown to well reproduce the COMPASS results.It will be interesting to see the results of such fits when the results presented in this Letter will be included.More information on the Q 2 evolution is provided by the study of the Sivers asymmetries in the low-y region between 0.05 and 0.1.The pion asymmetries in this region are compared in the left panel of Fig. 13 to the asymmetries obtained in the standard y range and with the cut x > 0.032.The mean Q 2 values of these two samples are respectively 3.5 (GeV/c) 2 and 1.8 (GeV/c) 2 .As for unidentified hadrons, there is an indication for an increase of the π + asymmetries at low-y.The dependence of the Sivers asymmetries with z is further investigated considering the z region between 0.1 and 0.2, where the asymmetries show smaller values.The comparison of the pion asymmetries as a function of x for the two separated z ranges are shown in the right panel of Fig. 13.For negative pions, a positive signal shows up in the low-z region, which is not observed for larger values of z.
In summary, using the high statistics data collected in 2007 and 2010, COMPASS has measured the Collins and Sivers asymmetries in muonproduction of charged pions and charged and neutral kaons produced off transversely polarised protons.The high energy muon beam allowed the measurement of a broad kinematic range in x and Q 2 .The x, z and p T dependences of the asymmetries were studied.Further investigations extending the range in z and y were also performed.The Collins asymmetries are definitely different from zero for pions and there are hints of a non-zero signal also for kaons, although in this case the statistical significance is marginal.The Sivers asymmetries are positive for positive pions and kaons, although different in size.This result is of particular interest since it can be used to access the sea quark Sivers PDFs.The results presented in this paper provide an important input for the global analyses.Together with other measurements covering complementary kinematic ranges, they allow the study of the Q 2 dependence of the asymmetries and the quantitative extraction of the Collins FF and of the transversity and Sivers PDFs.This information is crucial for the predictions for future Drell-Yan measurements and for measurements at future high-energy electron-ion colliders.Fig. 12: Comparison between the Sivers asymmetries for pions and existing global fits [23][24][25], in which the COMPASS results for the unidentified hadrons on protons [11] are included.

Fig. 3 :
Fig.3: Purity of the identified positive and negative kaons as a function of x, z, p h T .

Fig. 4 :
Fig. 4: Left: Armenteros-Podolansky plot of the hadron pair.Right: Difference of the invariant mass of the hadron pair and the PDG value of the K 0 mass.The mass range used for the analysis is shaded.

Fig. 5 :Fig. 6 :Fig. 7 :
Fig. 5: Left: comparison between the Collins asymmetries for pions as a function of x, extracted from 2007 and 2010 data taking.Right: the same comparison for the Sivers asymmetries.

Fig. 9 :
Fig.9: The Sivers asymmetries for charged pions (top), charged kaons (middle) and neutral kaons (bottom) as a function of x, z and p h T .

Fig. 10 :
Fig. 10: The Sivers asymmetries for positive pions and kaons, as a function of x.

Fig. 11 :
Fig. 11: The Sivers asymmetries for positive pions (top) and kaons (bottom) on proton as a function of x, z and p h T , requiring x > 0.032.The asymmetries are compared to HERMES results.

Table 1 :
Final statistics for 2007 and 2010 for identified charged pions and kaons and neutral kaons.