Contribution of exclusive diffractive processes to the measured azimuthal asymmetries in SIDIS

Hadron leptoproduction in Semi-Inclusive measurements of Deep-Inelastic Scattering (SIDIS) on unpolarised nucleons allows one to get information on the intrinsic transverse momentum of quarks in a nucleon and on the Boer-Mulders function through the measurement of azimuthal modulations in the cross section. These modulations were recently measured by the HERMES experiment at DESY on proton and deuteron targets, and by the COMPASS experiment using the CERN SPS muon beam and a $^6$LiD target. In both cases, the amplitudes of the $\cos\phi_h$ and $\cos 2\phi_h$ modulations show strong kinematic dependences for both positive and negative hadrons. It has been known since some time that the measured final-state hadrons in those SIDIS experiments receive a contribution from exclusive diffractive production of vector mesons, particularly important at large values of $z$, the fraction of the virtual photon energy carried by the hadron. In previous measurements of azimuthal asymmetries this contribution was not taken into account, because it was not known that it could distort the azimuthal modulations. Presently, a method to evaluate the contribution of the exclusive reactions to the azimuthal asymmetries measured by COMPASS has been developed. The subtraction of this contribution results in a better understanding of the kinematic effects, and the remaining non-zero $\cos 2\phi_h$ modulation gives indication for a non-zero Boer-Mulders effect.

Hadron leptoproduction in Semi-Inclusive measurements of Deep-Inelastic Scattering (SIDIS) on unpolarised nucleons allows one to get information on the intrinsic transverse momentum of quarks in a nucleon and on the Boer-Mulders function through the measurement of azimuthal modulations in the cross section. These modulations were recently measured by the HERMES experiment at DESY on proton and deuteron targets, and by the COMPASS experiment using the CERN SPS muon beam and a 6 LiD target. In both cases, the amplitudes of the cos φ h and cos 2φ h modulations show strong kinematic dependences for both positive and negative hadrons. It has been known since some time that the measured final-state hadrons in those SIDIS experiments receive a contribution from exclusive diffractive production of vector mesons, particularly important at large values of z, the fraction of the virtual photon energy carried by the hadron. In previous measurements of azimuthal asymmetries this contribution was not taken into account, because it was not known that it could distort the azimuthal modulations. Presently, a method to evaluate the contribution of the exclusive reactions to the azimuthal asymmetries measured by COMPASS has been developed. The subtraction of this contribution results in a better understanding of the kinematic effects, and the remaining non-zero cos 2φ h modulation gives indication for a non-zero Boer-Mulders effect.
(to be submitted as a Rapid Communication to Nuclear Physics B)

Introduction
The azimuthal asymmetries in Semi-Inclusive measurements of Deep-Inelastic Scattering (SIDIS) on unpolarised nucleons are a powerful tool to access the quark intrinsic transverse momentum k T and the Boer-Mulders [1] Transverse Momentum Dependent Parton Distribution Function (TMD PDF) h ⊥ 1 . The target spin-averaged differential SIDIS cross section for the production of a hadron h is given in the one-photon exchange approximation [2] where φ h is the azimuthal angle of the hadron with respect to the lepton scattering plane, in a reference system in which the z-axis is the virtual photon direction and the x-axis is defined by the scattered lepton transverse momentum. The transverse momentum p h T of the hadron is the component of p h orthogonal to the z-axis and z is the fraction of the available energy carried by the hadron. The quantity x is the Bjorken variable, y is the fractional energy of the virtual photon, σ 0 is the φ h -independent part of the cross section, λ is the longitudinal polarisation of the incident lepton, and ε 1 , ε 2 and ε 3 are kinematic factors depending on y. The amplitudes A XU f (φ h ) are referred to as azimuthal asymmetries in the following. The superscripts UU and LU refer to unpolarised beam and target, and to longitudinally polarised beam and unpolarised target, respectively. In particular, within the pQCD factorized approach [2], the twist-3 azimuthal asymmetry A UU cos φ h gives a direct access to k 2 T through the Cahn effect [4], which is expected to be the main contributor to A UU cos φ h . The twist-2 part of the asymmetry A UU cos 2φ h gives access to the Boer-Mulders TMD PDF. Measurements of the "unpolarised" SIDIS azimuthal asymmetries were recently performed by the HERMES Collaboration for charged hadrons, pions and kaons using both proton and deuteron targets [5], and by the COMPASS Collaboration for charged hadrons using a deuteron ( 6 LiD) target [3]. They all show strong dependences on the kinematic variables. Several phenomenological analyses (for more details see Ref. [6]) did not succeed either in reproducing the data or in extracting the Boer-Mulders PDF. As a result the present knowledge of the quark intrinsic transverse momentum has very large uncertainties and a possible non-zero Boer-Mulders function in the SIDIS cross section has still to be demonstrated.
Looking at the COMPASS results, a few aspects for the A UU cos φ h asymmetry are particularly intriguing. This asymmetry is expected to be mainly due to the kinematic Cahn effect and should be negative, with absolute value increasing almost linearly with z and p h T and proportional to k 2 T (see e.g. Ref. [6]). The trend of the data is, however, quite different. The measured z dependence of the integrated asymmetry 2 shows a strong increase of absolute value starting at z 0.5. Moreover, looking at the three-dimensional result 3 , at high z the p h T dependence is the opposite of the expected one, and the x dependence changes behaviour from low to high z.
These observations suggest that another mechanism, different from the TMD parton model, is at work in hadron production at large z. As a matter of fact it is known that the charged hadron SIDIS sample at large z and at small p h T contains a non-negligible contribution of hadrons from the decay of vector mesons (VM) produced in exclusive diffractive processes. This contribution was indeed taken into account in the measurements of hadron multiplicities [7][8][9][10][11]. Now, for the first time, we have investigated the effect of this VM contribution on the azimuthal asymmetries. We have measured azimuthal asymmetries for h + and h − originating from the decay of exclusively produced VMs (referred to in the following as "exclusive-VM hadrons"), and found them to be large. Since they do not have an interpretation in the framework of the parton model TMD formalism, we have subtracted this contribution from the published COMPASS asymmetries. This correction considerably improves the agreement with the expectations for A UU cos φ h and has also a noticeable effect for A UU cos 2φ h . The paper is organized as follows: in Section 2 the measurement of the azimuthal modulations for exclusive-VM hadrons is described. In Section 3 we present the calculation of the fraction of exclusive-VM hadrons in the measured hadron sample. In Section 4 we describe the procedure used to subtract the exclusive-VM hadron contribution to the azimuthal asymmetries published by COMPASS, and give the final results.

Azimuthal modulations of exclusive-VM hadron
In order to evaluate the contribution of exclusive-VM hadrons to the published azimuthal asymmetries [3] obtained from the COMPASS data collected in 2004, we have analysed the 2006 COMPASS data, which were recently used to measure the hadron multiplicities in SIDIS [8][9][10][11], and for which all the necessary simulated data are available. The experimental conditions of the two data sets are very similar, since the same target material ( 6 LiD) was used, once limiting the spectrometer acceptance to the same restricted kinematic region investigated in Ref. [3].
The azimuthal modulations of the exclusive-VM hadrons are measured selecting DIS events as in Ref. [3], i.e. by using: where Q 2 is the exchanged photon virtuality and W the final state hadronic mass. The events are then selected requiring in the final state, in addition to the scattered muon, only two oppositely charged hadrons with z > 0.1. The fraction of the final-state energy that is carried by the hadron pair, z t , is shown in the left panel of Fig. 1. Hadron pairs originating from exclusively produced vector mesons appear as the sharp peak at z t 1 and are selected by requiring z t > 0.95. Contributions from other processes, which appear as background to this peak, are neglected in the present analysis.
The z distribution for the positive hadrons of the selected pairs is shown in the right panel of the same figure. Most of the hadrons come from ρ 0 decays. The broad structure at 0.4 < z < 0.6 is due to hadrons from φ meson decays, whose contribution is less than 10% of that of the ρ 0 .
The |φ h | distribution of the exclusive-VM hadrons shows large modulations, as can be seen in the  left panel of Fig. 2 for positive hadrons. Furthermore the |φ h | distribution strongly depends on z, as can be seen from the right panel in Fig. 2, again for h + . From that 2-dimensional distribution one notices that the amplitude of the cos φ h modulation changes sign with z. The same properties are observed also for h − .
The acceptance-corrected azimuthal modulations of the positive and negative exclusive-VM hadrons are fitted in each x, z and p h T bin of Ref. [3] with the function where the amplitudes a 0 , a 1 and a 2 are free parameters. The sin φ h modulation is not included because parallel studies on exclusive vector-meson production in COMPASS do not exhibit such a modulation [12]. a UU,excl cos φ h amplitude is very large in absolute value at large and small z, and changes sign at z 0.5. The a UU,excl cos 2φ h amplitudes are smaller but still non-negligible.
It should be noted that the results of the present analysis refer to a 6 LiD target and COMPASS kinematics. The observed azimuthal asymmetries for exclusive-VM hadrons depend on the angular distributions for ρ 0 decay and production, which are determined by Spin Density Matrix Elements (SDMEs). The SDMEs depend on ρ 0 transverse momentum [13] and on the mechanism of its production. In particular, for coherent production on the target nuclei, which dominates at small p h T , one may expect different angular distributions (different SDMEs) than those for the production on a single free or quasi-free nucleon.

Fraction of exclusive-VM hadrons in the SIDIS sample
For a quantitative estimate of the exclusive-VM hadron contribution to the unpolarised azimuthal asymmetries, it is necessary to determine the number N excl h of exclusive-VM hadrons relative to the total number of hadrons N tot h , i.e. the ratio r = N excl h /N tot h . Here we use a parameterisation obtained from previous works [8][9][10], which was based on a combined use of HEPGEN [14] and LEPTO [15] Monte Carlo generators. The former one is used to model differential cross sections of various hard processes of exclusive leptoproduction of single mesons or photons at COMPASS kinematics. For the determination of r, only exclusive ρ 0 production, which gives the main contribution to the exclusive-VM hadrons, is taken into account in the present study. By doing this, we might underestimate r, but only in the bins at lowest p h T and z 0.5, where it could be larger by at most a factor 1.2.
Since the binning in Ref. [8][9][10] is different from that in Ref. [3], we had to parameterise r as a function of x, z and p h T . The estimated values of r in all the kinematic bins are shown in Fig. 4 and are assumed to be the same for positive and negative hadrons. As one can see, the fraction of pions coming from the decay of exclusively produced ρ 0 is very large at large z and small p h T , where it reaches 50%, and diminishes for decreasing z and increasing p h T . The overall systematic uncertainty on r is estimated to be approximately 30% and is mainly due to the uncertainty on the knowledge of the diffractive cross section [8][9][10].

Results for the unpolarised SIDIS azimuthal asymmetries
The exclusive-VM hadron contributions to the published azimuthal asymmetries r a UU,excl cos φ h and r a UU,excl cos 2φ h are calculated in each x, z and p h T bin of Ref. [3]. The results for the smallest p h T bin, i.e. 0.1 GeV/c < p h T < 0.3 GeV/c, are shown for h + and h − in the second column of Fig. 3. As can be seen, the contribution of exclusive-VM hadrons is clearly different from zero and reaches values up to 20% at large z in this low p h T range. The contribution to the cos 2φ h modulation is smaller but still non-negligible, in particular if compared to the measured values of the asymmetries.

: SIDIS A UU
cos φ h asymmetry on 6 LiD for h + (red circles) and h − (black circles) after subtracting from the published asymmetry [3] the contribution of exclusive-VM hadrons, as function of x, in z and p h T bins. Inner error bars denote statistical uncertainties, outer ones statistical and systematic uncertainties added in quadrature. using where A UU, publ cos iφ h are the published values. A similar expression is used to obtain A UU cos 2φ h .
The resulting A UU cos φ h azimuthal asymmetries are shown in the third column of Fig. 3, again for the smallest p h T bin. After subtraction, the x dependence of the asymmetry becomes weaker, and in particular only a few positive values that are hard to be described by the Cahn effect remain. The last column of the figure shows the comparison between the asymmetries as published and after subtracting the contribution of exclusive VMs for h + . One can also see that the contribution of exclusive-VM hadrons is sizable at all z.
The results for A UU cos φ h and A UU cos 2φ h for positive and negative hadrons are shown in all x, z and p h T bins in Fig. 5 and 6, respectively. The inner error bars correspond to the statistical uncertainties only, while the outer bars represent the total uncertainties. The increase in the statistical uncertainties is due to the low statistics of the exclusive-VM hadrons. The systematic uncertainties have been evaluated by adding in quadrature the uncertainties of the published results (estimated to be of the same order of the statistical ones) and those due to the subtraction procedure. For the last ones the dominant contribution is that of the poor knowledge of r, which can cause an uncertainty at most as large as the statistical one, apart from a few bins at the highest zand lowest x-values. The total uncertainties are evaluated by adding in quadrature the statistical and the systematic uncertainties. The numerical values of the asymmetries are available on HepData [16].
In spite of the large uncertainties we consider this work as a major step forward in understanding the 3D structure of the nucleon. To give an idea of the impact, in Fig. 7 we compare A UU cos φ h with a simple Monte Carlo simulation for the Cahn effect. We have used the Monte Carlo code of Refs. [17,18], describing the fragmentation of polarised quarks, which was modified to include the Cahn effect. This is achieved by modulating the fragmenting quark direction according to the lepton-quark hard cross section calculated for a non-zero k T [4]. The p h 2 T dependence on z is built in and a suitable dependence of k 2 T on x has been used to reproduce the values of A UU cos φ h at z 0.5. The agreement is satisfactory and the trends are similar over all bins, except for the two bins at p h T > 0.5 GeV/c and z > 0.7. The same Monte Carlo simulation is also used to investigate the twist-4 cos 2φ h azimuthal modulations generated by the Cahn effect. The resulting amplitudes A UU cos 2φ h turn out to be compatible with zero. Other contributions, which are not generated by Boer-Mulders and Collins effect, appear also at twist-4 or higher orders. Although these contributions are not very well known, they should be suppressed as 1/Q 2 , thus it is most likely that the non-zero A UU cos 2φ h values of Fig. 6 are an indication of a non-zero Boer-Mulders PDF. Specifically, the corrected data for A UU cos 2φ h for positive hadrons still show a strong z dependence in the highest p h T -bin, with a significance above 5 σ . The phenomenological study of this effect is, however, beyond the scope of the present paper.

Conclusions
The COMPASS Collaboration has measured the azimuthal modulations of positive and negative hadrons from the decay of exclusive vector mesons produced in the scattering of 160 GeV/c muons on a 6 LiD target. The amplitudes of the modulations are found to be large and of the same sign for positive and negative hadrons. These hadrons constitute a contamination to the SIDIS hadron sample. Their contribution to the previously published COMPASS A UU, publ cos φ h and A UU, publ cos 2φ h unpolarised azimuthal asymmetries is estimated quantitatively and shown to be non-negligible over all the explored kinematic region and in particular at large z. After subtracting their cos φ h amplitudes, the A UU cos φ h asymmetries turn out to be in reasonable agreement over most of the explored kinematic region with a Monte Carlo simulation implementing the Cahn effect, except for a very few bins at large z and large p h T . The experimental determination of this important correction to already published data, which so far was never evaluated, is expected to have significant impact onto phenomenological analyses. When implemented, it could hopefully allow for a successful disentangling of the various contributions to the data and for a first extraction of the Boer-Mulders function.