Hadron Transverse Momentum Distributions in Muon Deep Inelastic Scattering at 160 GeV/$c$

Multiplicities of charged hadrons produced in deep inelastic muon scattering off a $^6$LiD target have been measured as a function of the DIS variables $x_{Bj}$, $Q^2$, $W^2$ and the final state hadron variables $p_T$ and $z$. The $p_T^2$ distributions are fitted with a single exponential function at low values of $p_T^2$ to determine the dependence of $\langle p_T^2 \rangle$ on $x_{Bj}$, $Q^2$, $W^2$ and $z$. The $z$-dependence of $\langle p_T^2 \rangle$ is shown to be a potential tool to extract the average intrinsic transverse momentum squared of partons, $\langle k_{\perp}^2 \rangle$, as a function of $x_{Bj}$ and $Q^2$ in a leading order QCD parton model.


Introduction
Semi-Inclusive measurements of Deep Inelastic Scattering (SIDIS) of leptons off nucleons provide information about the partonic structure of the nucleon and the hadronization of partons, and hence offer a wide testing ground of Quantum Chromodynamics (QCD). Subject of the present study are the transverse momentum distributions of charged hadrons produced in the current fragmentation region in lepton-nucleon scattering off unpolarised nucleons. The hadron transverse momentum p T is defined with respect to the virtual photon direction. The following standard notations are used: and for the incoming and outgoing lepton, N for the target nucleon, h for the outgoing hadron and X for the unobserved particles in the final state; l, l , P, and p denote the 4-momenta of , , N, and h. The general expression for the differential SIDIS cross section describing the reaction + N → + h + X in the one-photon approximation is [1,2]: Here, λ is the helicity of the incoming lepton and the standard SIDIS variables are used: the 4-momentum transfer q = (l − l ), the photon virtuality Q 2 = −q 2 , the Bjorken scaling variable x B j = Q 2 /2P · q, the hadron fractional energy z = P · p/P · q and the azimuthal angle φ h of the transverse momentum of the hadron with respect to the lepton scattering plane around the virtual photon direction. After integration over φ h , the cross section does not depend on the initial lepton polarisation λ . The hadron multiplicity per interaction is defined as the ratio of the differential SIDIS cross section over the differential DIS cross section d 2 σ DIS (x B j , Q 2 )/dx B j dQ 2 . Thus, the differential hadron multiplicity, d 2 n h /dzd p 2 T , depends on four variables, x B j , Q 2 , z, p 2 T : Within a pQCD Leading Order (LO) parton model the shape of the p 2 T distributions depends on the intrinsic transverse momentum k ⊥ of the partons and the transverse momentum of the hadrons p ⊥ acquired during parton fragmentation. The amount of the contributions of k ⊥ and p ⊥ may depend on the hadron type, parton flavour, and on kinematic variables such as x B j , Q 2 and z. Already in the 1970s, SIDIS was understood as a tool to access the intrinsic transverse momentum of the partons (see e.g. [3] and references therein). The connection between the intrinsic transverse momenta of the parton k ⊥ and that of the hadron p ⊥ and the measured transverse momentum p T of the produced hadron is illustrated in Fig. 1, assuming single photon exchange and leading order pQCD.
During the last three decades significant efforts, both in experimental and theoretical studies of (polarised) SIDIS, have been undertaken. Currently this process is considered to be one of the most promising to study the hadronization process and also the spin-dependent three-dimensional structure of the nucleon (see, e.g. [4]). Recently, a complete QCD treatment of transverse momentum and spin-dependent SIDIS was presented in Ref. [5] where factorization was derived in terms of well defined unintegrated or Transverse Momentum Dependent parton distribution and fragmentation functions (TMDs) with individual hard scale evolution properties. This formalism has been applied in Ref. [6] to obtain the Q 2 evolution of unpolarised TMDs; a mandatory information needed for a correct comparison of data measured in experiments at different hard scales [4].
Hadron leptoproduction has been studied by many experiments. Some recent examples are: JLab [7], HERMES [8] and E665 [9]. Earlier, EMC [10] covered most of the kinematic range of COMPASS. However, COMPASS has collected much more data in this range and the statistical errors of the present  Fig. 1: Sketch showing the kinematic variables for the absorption of a virtual photon by a parton with intrinsic transverse momentum k ⊥ and the subsequent hadronization. The transverse momentum of the observed hadron is denoted by p T when defined with respect to the virtual photon direction in the photon nucleon center of mass system and by p ⊥ when defined with respect to the scattered parton direction.
analysis are therefore significantly smaller, although only part of the available data has been used. The results presented here are obtained from data taken during the year 2004. More details of the analysis are described in Ref. [11].

Experiment, Data Selection and Acceptance
The COMPASS experiment is installed on the M2 beam line of the CERN SPS [12]. Polarised 160 GeV/c muons with an intensity of 2 × 10 8 µ/spill (one spill of 4.8 s length per 16.8 s) and a polarisation of 80% are scattered off a longitudinally polarised 6 LiD target. In 2004 the target consisted of two cells with opposite polarisation which was reversed every 8 hours. It has been verified that summing up the data from both cells yields a data sample with vanishing polarisation for the present analysis. The COMPASS detector is a large acceptance two-stage spectrometer which covers the kinematic range from quasi-real photoproduction to DIS. Both stages are equipped with hadron calorimeters and use absorber walls for muon identification. Charged particles emerging from the primary interaction vertex in the forward direction are identified as muons if they traverse at least 30 radiation length, otherwise they are identified as hadrons. The selection requires reconstructed trajectories in the detectors situated upstream and downstream of the first magnet. This ensures that the track momentum and sign of charge are well defined by bending in the magnetic field. The COMPASS ability to separate pions, kaons and protons with a Ring Imaging Cherenkov detector was not used in this analysis. Muon interactions with Q 2 > 1.0 (GeV/c) 2 and 0.1 < y < 0.9 are selected, where y = ν/E µ , and ν = E µ − E µ is the difference between the laboratory energies of the incoming and outgoing muon µ and µ . With the above selection, the hadronic energy squared W 2 = 2Mν + M 2 − Q 2 is > 25 GeV/c, above the nucleon resonance region. Here, M is the nucleon mass. The total number of inclusive events selected for this analysis is 45.8 × 10 6 , corresponding to an integrated luminosity of 775 pb −1 . The events are sampled into 23 intervals in Q 2 from 1 to 10 (GeV/c) 2 and x B j from 0.004 to 0.12, as shown in Fig. 2. The ranges and average values of Q 2 and x B j are shown in Tab. 1. Each of these (x B j , Q 2 ) intervals is further subdivided into 8 intervals in z from 0.2 to 0.8.
In order to correct for event losses caused by the non uniform acceptance of the COMPASS spectrometer, a full Monte Carlo (MC) simulation has been performed. The events were generated with LEPTO [13], passed through the spectrometer with a GEANT [14] based simulation program and reconstructed with the reconstruction software as the real data events.
The SIDIS acceptances A (+,−) SIDIS for detecting, together with the scattered muon, a positive (h + ) or negative hadrons (h − ) respectively factorize in an inclusive muon acceptance A incl (Q 2 , y) and a positive or negative hadron acceptance A h (+,−) ( lab p T , lab η). These acceptances depend on the spectrometer charac-  teristics, making the use of variables defined in the laboratory frame preferable; therefore, the transverse momentum lab p T , the polar angle lab θ , and the pseudorapidity lab η = − ln(tan lab θ 2 ) of the hadron are defined with respect to the direction of the incoming muon. The choice of lab θ is particularly convenient to exhibit the acceptance cut due to the aperture limit of the polarised target magnet at lab θ = 70 mrad for the upstream edge of the target. The factorization of hadron and muon acceptances implies that the differential multiplicities only depend on A h (+,−) since A incl cancels, see Eq. 2. Figure 3 shows the hadron acceptances A h − and A h + used in the analysis.
The four-dimensional acceptance used in the present analysis is integrated over the azimuthal angle of the hadrons, i.e. does not take into account the azimuthal modulations in the cross section [2]. The systematic effect on the extracted p 2 T have been investigated and found to be negligible.

Results
The differential multiplicities d 2 n h± /dzd p 2 T in bins of (Q 2 , x B j ) are defined in the introduction in terms of the semi-inclusive and inclusive differential cross sections. They are obtained as the acceptance corrected number of hadrons ∆ 4 N h± in 8 × 40 (z, p 2 T ) bins and 23 (∆x B j , ∆Q 2 ) bins, divided by the number of muon interactions in the same (∆x B j , ∆Q 2 ) bins: .
The distributions for two selected (Q 2 , x B j ) bins are shown in Fig. 4 for all z intervals. The full data set, including more p 2 T bins, is available on HEPDATA [15]. As can be seen from Eq. 3 the uncertainty of the integrated luminosity cancels and the only contributions to systematic uncertainties of the multiplicities come from the hadron acceptance and the assumption of factorization of hadron and muon acceptance. The total systematic uncertainty due to acceptance has been estimated to be 5% [11]. Only statistical errors are shown in the figures.
It is interesting to compare the values and W 2 -dependence of p 2 T obtained from the fit at small p T with the values and W 2 -dependence of p 2 T all . The W 2 -dependence of p 2 T , obtained from the fit in the bin 0.5 < z < 0.6 is shown in Fig. 8, that one of p 2 T all in Fig. 9. In addition to the data points, Fig. 9 shows lines, which represent fits of the data points assuming a linear function of lnW 2 . Because of the Q 2 -dependence, the last points are somewhat below the fit. The authors of Ref. [18] first suggested that p 2 T all should depend linearly on the µN center of mass energy squared s. They have verified their prediction with results from three fixed target experiments: JLab, HERMES and COMPASS, see Fig. 10. Fig. 10a shows the p 2 T distribution of charged hadrons with 0.5 < z < 0.6 and integrated over Q 2 and x B j , measured by COMPASS, which was used to determine the acceptance corrected p 2 T all . Fig. 10b taken from Ref. [18] shows the dependence of p 2 T all on s. Their value for COMPASS, represented by the black dots, was not corrected for acceptance. The new, acceptance corrected COMPASS value p 2 T all added to Fig. 10b (red dot) is shown in a recent paper [19], and used to quantify the p T broadening [20] in a model to determine the Sivers and Boer-Mulders asymmetries at COMPASS and HERMES. The result of the model of Pasquini and Schweitzer was closer to the COMPASS data when p T broadening is included. The authors of Ref. [18] also note that p 2 T all may depend linearly on W 2 rather than s.  convoluted with two unintegrated soft universal functions: f q (x B j , k ⊥ ), the parton distribution function of quark of flavor q and D h q (z, p ⊥ ), the fragmentation function defined as the number density of hadron h resulting from the fragmentation of a quark of flavor q. With the further assumption that both f q (x B j , k ⊥ ) and D h q (z, p ⊥ ) follow Gaussian distributions with respect to the transverse momentum variables k ⊥ and p ⊥ , respectively, the cross section can be approximated [16] at first order in O(k ⊥ /Q) by: where all the parameters describing the transverse momentum dependence of TMDs for a given quark flavor q are contained in p 2 T q , through the relation: Here again, integration over the azimuthal angle has been performed. In Ref. [16] it was assumed that p 2 ⊥ and k 2 ⊥ in Eq. 5 are constants and independent of the quark flavor. In general, they may both depend on Q 2 and the active quark flavor q while p 2 ⊥ can depend further on z and the produced hadron type, and k 2 ⊥ may depend on x B j .  T vs W 2 for 0.5 < z < 0.6 and for a low (left) and a high (right) Q 2 interval, from a fit over 0.1 < p T GeV/c < 0.85. This is to be compared with Fig. 9 where p 2 T all is plotted. The average Q 2 for each W 2 bin are indicated.  T distribution of charged hadrons with 0.5 < z < 0.6 used to determine the acceptance corrected p 2 T all (left). The s-dependence of p 2 T all from Ref. [18] (right). The red star labeled COMPASS is the value from this analysis, the black dot labeled COMPASS (Schw.) is the value used in Ref. [18], obtained from data not corrected for acceptance.   The observed dependence of the fitted p 2 T on z 2 is shown for two (Q 2 , x B j ) intervals in Fig. 12. The relation between p 2 T and z 2 is certainly not linear as in Eq. 5. It should be noted that the non linear behaviour of the z 2 -dependence of p 2 T was reproduced qualitatively in a recent paper [22] by imposing kinematical constraints to the model leading to Eq. 4. A more general ansatz for the contributions of the intrinsic transverse momenta p ⊥ and k ⊥ to the measured hadron transverse momentum p T is where p 2 ⊥ (z) is a function of z and should be taken from other measurements. The dependence of k ⊥ is still the same as in Eq. 6, with a constant average k 2 ⊥ . The knowledge of p 2 ⊥ (z) could be taken from DIS event generators which are supposed to incorporate all known properties of jet fragmentation. In Fig. 13 the measured values of p 2 T are compared with those of a simulation using the event generator LEPTO 1 . Two cases were simulated in the MC: interactions without intrinsic transverse parton momenta k 2 ⊥ = 0 (open squares) and interactions with k 2 ⊥ = 0.25 (GeV/c) 2 (open crosses). For k 2 ⊥ = 0.25 (GeV/c) 2 , the agreement between p 2 T from simulated events and from data (full squares) is striking for lower values of Q 2 , apart from the highest z 2 bins. For values of Q 2 larger than 4 (GeV/c) 2 , the data are significantly above the simulation. The significant differences between positive and negative hadrons at larger z values are not reproduced by the MC simulation. This comparison suggests that k 2 ⊥ can be extracted from the data, provided a detailed tuning of the jet fragmentation parameters would be performed. In addition, it should be noted that the present event selection includes all semi-inclusive events. Thus events which are not due to DIS, i.e. the absorption of the virtual photon by a quark with subsequent quark fragmentation, but to other mechanisms like diffractive vector meson production are included in the sample. The treatment of this kind of background to DIS requires special efforts, in order to extract k 2 ⊥ .

Conclusion
The main result of this study is the measurement of differential multiplicities of charged hadrons produced in unpolarised SIDIS of muons off a 6 LiD target. The acceptance corrected multiplicities in a 4-dimensional (z, p 2 T , x B j , Q 2 ) phase space are given in Ref. [15] separately for positively and negatively charged hadrons.