Physics at a Fixed-Target Experiment Using the LHC BeamsView this Special Issue
Research Article | Open Access
L. Massacrier, B. Trzeciak, F. Fleuret, C. Hadjidakis, D. Kikola, J. P. Lansberg, H.-S. Shao, "Feasibility Studies for Quarkonium Production at a Fixed-Target Experiment Using the LHC Proton and Lead Beams (AFTER@LHC)", Advances in High Energy Physics, vol. 2015, Article ID 986348, 15 pages, 2015. https://doi.org/10.1155/2015/986348
Feasibility Studies for Quarkonium Production at a Fixed-Target Experiment Using the LHC Proton and Lead Beams (AFTER@LHC)
Being used in the fixed-target mode, the multi-TeV LHC proton and lead beams allow for studies of heavy-flavour hadroproduction with unprecedented precision at backward rapidities, far negative Feynman-, using conventional detection techniques. At the nominal LHC energies, quarkonia can be studied in detail in , , and collisions at GeV and in Pb + and Pb + collisions at GeV with luminosities roughly equivalent to that of the collider mode that is up to 20 fb−1 yr−1 in and collisions, up to 0.6 fb−1 yr−1 in collisions, and up to 10 nb−1 yr−1 in Pb + collisions. In this paper, we assess the feasibility of such studies by performing fast simulations using the performance of a LHCb-like detector.
Since its start-up, the large hadron collider (LHC), the most energetic hadron collider ever built so far, has already made the demonstration of its outstanding capabilities. These can greatly be complemented by the addition of a fixed-target physics program. Its multi-TeV beams indeed allow one to study , , and collisions at a center-of-mass (c.m.s.) energy GeV as well as and collisions at GeV, with the high precision typical of the fixed-target mode. In this context, the proposal of a fixed-target experiment at the LHC , referred to as AFTER@LHC, has been promoted  in order to complement the existing collider experiments such as the Relativistic Heavy Ion Collider (RHIC) or the future Electron-Ion Collider (EIC) project in a similar energy range. The idea underlying the AFTER@LHC proposal is a multipurpose detector allowing for the study of a multitude of probes.
Various technological ways to perform fixed-target experiment at the LHC exist. On the one hand, the beam can be extracted by means of a bent crystal. This technology [2, 3] is currently developed as a smart beam-collimation solution and is studied by the UA9/LUA9 collaboration, respectively, at SPS and LHC. A bent crystal installed in the halo of the LHC beam would deflect the particles of the halo onto a target, with a flux of proton/s without any impact on the LHC performances [3–5].
On the other hand, the LHC beam can go through an internal-gas-target system in an existing (or new) LHC experiment. Such a system is already tested at low gas pressure by the LHCb collaboration to monitor the luminosity of the beam [6–8]. Data were taken at a center-of-mass energy of (54) GeV with () collisions during pilot runs in 2012 and 2013. Although this system, called SMOG, was tested during only few hours in a row, no decrease of the LHC performances was observed.
In the bent crystal case, the luminosity achievable with AFTER@LHC would surpass that of RHIC by 3 orders of magnitudes . We have reported in Table 1 the instantaneous and yearly integrated luminosities expected with the proton and Pb beams on various target species of various thicknesses, for the bent crystal as well as internal-gas-target options. Integrated luminosities as large as 20 fb−1 can be delivered during a one-year run of collisions with a bent crystal. Besides, it is worth mentioning that both technologies allow one to polarise the target, which is an important requirement to lead an extensive spin physics programme [1, 11].
Overall, thanks to the large luminosity expected, AFTER@LHC would become a quarkonium , prompt photon, and heavy-flavour observatory [1, 13] in and collisions where, by instrumenting the target-rapidity region, gluon and heavy-quark distributions of the proton, the neutron and the nuclei can be accessed at large and even at larger than unity in the nuclear case . In addition, the fixed-target mode allows for single-target-spin-asymmetry measurements over the full backward rapidity domain up to [15, 16]. Also, the versatility in the target choices offer a unique opportunity to study the nuclear matter versus the hot and dense matter formed in heavy ion collisions which can be studied during the one-month lead run. In the latter case, modern detection technology (such as high granularity calorimeter) should allow for extensive studies of quarkonium excited states, from to and resonances thanks to the boost of the fixed-target mode .
In this paper, we report on a feasibility study of quarkonium production at a fixed-target experiment using LHC beams. In Section 2, we outline the simulation framework which was used. In Section 3, we describe how a fast simulation of a detector response has been implemented, following a LHCb-like detector setup. In Section 4, we present the charmonium and bottomonium family studies performed with the simulations at GeV. In Section 5, we present multiplicity studies in and collisions as well as the expected nuclear modification factors for and in collisions at GeV. Finally in Section 6 some prospects for measurement at GeV are given. Section 7 gathers our conclusions.
2. Simulation Inputs
In order to get the most realistic minimum bias simulations at AFTER@LHC energy for quarkonium studies in the dimuon decay channels, we have simulated the quarkonium signal and all the background sources separately to have under control the transverse momentum and rapidity input distributions as well as the normalisation of the different sources.
The simulation has been performed for collisions at GeV. On the one hand, the quarkonium signal and the correlated background (Drell-Yan, , ) were simulated with HELAC-ONIA  which produces outputs following the format of Les Houches Event Files . The outputs were then processed with PYTHIA (PYTHIA 8.185 ) to perform the hadronisation, the initial/final-state radiations, and the decay of the resonances. On the other hand, the uncorrelated background was obtained from minimum bias collisions generated with PYTHIA.
The relative normalisation of the signal and background sources was performed according to the production cross section of the process (taking into account initial phase space cuts, if any). Values of the cross section and the number of simulated events , not to be confused with the expected events for a specific luminosity, are reported in Table 2. The cross section values are integrated over rapidity and .
2.1. Signal and Correlated Background
2.1.1. Quarkonium Signal
, , , , and were simulated in a data-driven way. The amplitude of (where is the quarkonium) is expressed in an empirical functional form : where with the partonic center-of-mass energy and the mass of the quarkonium taken from the PDG table .
The parameters , , , and were determined by fitting the differential cross section to the experimental data. The dedicated codes used to perform the fit and to generate unweighted events for quarkonium production have been implemented in HELAC-ONIA  and we used MSTW2008NLO PDF set  provided in LHAPDF5  and the factorisation scale . In order to constrain the nontrivial energy dependence of quarkonium production, we used the differential measurements of charmonium production performed by the PHENIX collaboration at RHIC, in collisions at GeV  to predict the corresponding yields at GeV. Given the lack of such measurements for at RHIC, we performed a combined fit to CDF , ATLAS , CMS , and LHCb [10, 28] data on production. The values of the fitted parameters are listed in Table 3. For illustration, the comparison between fits and the selected experimental data is shown in Figure 1.
In order to increase the statistics of the simulated data sample, the decay of the quarkonium in PYTHIA is forced into the dimuon decay channel. The simulated yields are then weighted by the cross section for this process multiplied by the Branching Ratio (BR).
2.1.2. Open Charm
Open charm production was simulated with the process in HELAC-ONIA. In order to avoid the huge theoretical uncertainties in the state-of-the-art perturbative calculations, open charm yields at GeV are also computed in a data-driven way following the method described in the previous section. Similarly, the matrix element of is determined using (1). The parameters are obtained from a fit to the -differential cross section measured by the STAR experiment  in collisions at GeV (see Figure 3). We obtained , , and GeV/c when by using CTEQ6L1  and by fixing the quark mass to GeV/c2 and the factorisation scale to . The of the fit is equal to 4.39 with 10 experimental data points. The tuned result is shown in Figure 2. The evolution of the cross section with the energy down to GeV is then given by HELAC-ONIA.
After embedding the Les Houches Event File into PYTHIA, muons from the underlying PYTHIA event can be produced on top of muons from the initial pair. The combination of those additional muons with a muon from the initial pair is not included in our definition of open charm correlated background. We have however checked that this contribution is negligible. In order to increase the statistics, , , , and were forced to decay into muons and only those decay muons were considered as correlated background. pairs coming from all possible combinations, , , , , , and , are considered. The simulated events are weighted by the production cross section times the pair Branching Ratio times the fraction of quark fragmenting to , , , or . This fraction is obtained from PYTHIA and found to be 95%.
2.1.3. Open Beauty
The theoretical uncertainty on open beauty production is relatively smaller than the one on open charm production. We therefore calculated open beauty production yields with a Leading Order (LO) matrix element which was normalised to the Next-To-Leading-Order (NLO) factor. The NLO cross section with the same setup was calculated by MADGRAPH5_AMC@NLO . We used CTEQ6L1 (CTEQ6M) for the LO (NLO) calculation. The factor is found to be 1.83. The renormalisation and factorisation scale is with the mass of the quark taken as GeV/c2. We have adopted a similar definition for the open beauty correlated background as the one of open charm (see the previous section).
Drell-Yan (DY) correlated background was simulated with the process at LO where is a pair of the same flavour light quarks. The LO calculation was done with the CTEQ6L1 pdf set and the renormalisation and factorisation scale was set to . In order to have enough statistics in the and mass window, a phase space cut requesting that the invariant mass of the dimuons () is greater than 2.5 GeV/c2 was applied. For the simulation of the DY background under the family peaks, a phase space cut GeV/c2 was applied. The DY cross section obtained with HELAC-ONIA at GeV is compared to the existing E866 data at the same energy . A factor 1.2 is needed to match the data and therefore it was also applied at GeV. Such a factor is known to approximately account for the higher-order QCD corrections.
2.2. Uncorrelated Background
The uncorrelated background was obtained from a minimum bias PYTHIA simulation at GeV using the process SoftQCD:nonDiffractive with the MRSTMCal.LHgrid LHAPDF (6.1.4) set . By comparing our simulation of open charm with a low statistic pure minimum bias PYTHIA simulation, we have checked that the contribution of dimuons originating from a muon from charm/beauty and a muon from / is negligible. The dominant source of uncorrelated opposite-sign muon pairs is the simultaneous semimuonic decay of uncorrelated and/or . In order to avoid possible double counting of signal and correlated background processes, the following hard processes have been switched off from the minimum bias simulations: HardQCD:hardccbar, HardQCD:hardbbbar, WeakSingleBoson:ffbar2gmZ (in order to avoid Drell-Yan pair production.), Charmonium:all and Bottomonium:all,.
3. Fast Simulation of the Response of a LHCb-Like Detector
The HELAC-ONIA and PYTHIA generators provide the opposite-sign muon pairs from quarkonium decays, correlated and uncorrelated backgrounds sources, as defined in the previous section. In order to account for the detector resolution and the particle identification capabilities of a given detector and to investigate the feasibility of the quarkonium studies in collisions at GeV, the detector response needs to be simulated. For this purpose, we have chosen a detector setup similar to the LHCb detector . A forward detector is very well suited as a fixed-target experiment setup as well, with a good tracking and particle identification capabilities.
According to LHCb analysis cuts, muons in our simulations are required to have their transverse momentum satisfying GeV/c  and their pseudorapidity in the laboratory frame satisfying . The cut range corresponds to the LHCb detector coverage. Since the momentum resolution reported by LHCb is (0.6)% for a momentum of 3 (100) GeV/c , we consider a momentum resolution of 0.5%. The single identification efficiency is taken to be 98%, which is an average efficiency obtained by LHCb for muons coming from decays, for GeV/c and GeV/c . These cuts and the abovementioned detector response on the muons are applied to simulate the quarkonium states and all the background sources.
In the case of uncorrelated background, as discussed in Section 2, most of the originate from or decays. If a or decays to a before 12 m along the axis, the is rejected by the tracking system and it is not considered in the simulation. 12 m corresponds to the distance where the calorimeters, followed by the muon stations, are placed in the LHCb detector setup. If the is produced beyond 12 m or if a is misidentified with in the muon stations, a misidentification probability is applied. The misidentification probabilities depend on the total particle momentum and were reported by the LHCb collaboration in . These probabilities are parameterised with the following functions: and , and they are shown in Figures 4(a) and 4(b), for and , respectively. Based on the single identification efficiency , the dimuon, , efficiency is calculated as a product of the single efficiencies: . For muons coming from or decays, misidentification probabilities are used: or , respectively, for and , and for prompt muons .
The pair efficiency is extracted in each kinematic phase space point and is shown as a function of the dimuon invariant mass, transverse momentum, and rapidity in Figure 5. This efficiency is used to correct dimuon spectra obtained with the uncorrelated background PYTHIA simulations.
4. Quarkonium Production Studies in H Collisions at GeV
In this section, we show results on the quarkonium production studies in the dimuon decay channels, with the dominant background sources. Simulations have been performed for a 7 TeV proton beam on a hydrogen target (), which gives GeV. We consider an integrated luminosity of 10 fb−1 which is expected to be obtained after half of a LHC year with the crystal mode, as described in Section 1 and Table 1.
4.1. Background Studies
These simulations allow us to quantify the background sources in the quarkonium studies in the dimuon decay channel, which could potentially make the quarkonium signal extraction more difficult or even prevent from obtaining a clear signal. In particular, this may be critical for the excited states. We present here simulations of invariant mass of opposite-sign muon pairs, , from the quarkonia and from the dominant background sources, in two mass ranges; see Figure 6. The first range corresponds to the and invariant mass windows and the second one to the mass range of the , , and . The invariant mass distributions are integrated over the whole transverse momentum and rapidity ranges. The plots show the simulated quarkonium signals and the background, separately from the different sources, and the black solid line is a sum of all contributions. The background sources correspond to an integrated luminosity of 10 fb−1.
In the and invariant mass window, the dominant background source is from uncorrelated pairs, mostly from and decays. The contributions from Drell-Yan and continuum are very small. In the case of states, the Drell-Yan contribution is the dominant one. Under the peak, the contribution from the continuum is negligible, and it is not considered here.
The significance (, where S is the number of signal counts and is the number of background counts, in the invariant mass range ) and the signal to background ratio () of each quarkonium state are given in the following:(i) 102, ,(ii), ,(iii), ,(iv), ,(v), ,for , , , , and , respectively.
Transverse momentum and rapidity distributions for the quarkonium signals and for each background source were also studied. As an example, the and distributions in the mass range, GeV/c2 (corresponding to ), are shown in Figure 7. It is visible that the distributions for the and different backgrounds differ. In more backward or forward rapidity regions, the signal to background ratio increases. This can also be seen in Figure 8, where the dimuon invariant mass distributions in and mass window are shown in three rapidity ranges. In terms of transverse momentum, one can obtain a very clean signal when going to higher . Above ~4 GeV/c, the uncorrelated background starts to vanish. Since , , and Drell-Yan simulations are LO simulations, the spectra of these correlated background sources are not shown here.
4.2. Quarkonium Simulations
We have also studied the and rapidity coverage reach of the quarkonium signals. The transverse momentum distributions are shown in Figure 9(a), for , , , , and , from the top to the bottom distribution. Similarly, Figure 9(b) shows the rapidity distribution for each quarkonium state. With an integrated luminosity of 10 fb−1 the quarkonium studies can be carried out in a wide rapidity and range. It should be possible to study signals up to GeV/c, and and could be studied even up to GeV/c. All the quarkonium states can be measured down to GeV/c.
This study is limited by the rapidity range of , in the laboratory frame, due to the pseudorapidity cuts on the decay . The red -axis on the top of Figure 9(b) denotes the rapidity in the center-of-mass frame. The rapidity shift for a 7 TeV proton beam on a fixed-target is −4.8; that is, . and signals can be studied in the whole mentioned rapidity range, while the lowest rapidity reach for is ~2.5–3.
5. Quarkonium Measurements in Collisions at GeV and Pb + H Collisions at GeV
5.1. Multiplicity in Proton-Nucleus Collisions
In proton-nucleus collisions, the high track multiplicity may induce a high detector occupancy and lead to a reduction of the detector capabilities. Since LHCb has successfully measured the and production in + Pb collisions at TeV [39, 40], one would expect a good capability of such detector under similar particle multiplicity environment. In the following, the charged particle multiplicity has been generated with the EPOS generator [41, 42] in different configurations: collisions at TeV in collider mode (the lead goes in the detector direction), collisions at GeV, and collisions at GeV in fixed-target mode. The charged particle multiplicity is dominated by the multiplicity. By comparing these three distributions as a function of the pseudorapidity of the particle in the laboratory frame as shown in Figure 10, one can conclude that the charged particle multiplicity in a fixed-target mode never exceeds the one obtained in collisions at TeV in the collider mode in the full pseudorapidity range: a detector with the LHCb capabilities will be able to run in such conditions.
5.2. Prospects for the Measurements of the Nuclear Modification Factors for and in Pb Collisions at GeV
To illustrate the potential offered by AFTER@LHC in collisions at GeV, we have evaluated, in this section, the impact of the nuclear modification of the gluon densities in nucleons within large nucleus, generically referred to as gluon shadowing, and its uncertainty as encoded in the nuclear PDF set EPS09. For that, we have used the probabilistic Glauber Monte-Carlo framework, JIN [43, 44], which allows us to encode different mechanisms for the partonic production and to interface these production processes with different cold nuclear matter effects, such as the aforementioned shadowing, in order to get the production cross sections for proton-nucleus and nucleus-nucleus collisions. JIN also straightforwardly computes any nuclear modification factor for minimum bias collisions or in specific centrality classes. In the case of proton-nucleus () collisions, it is the ratio of the yield per inelastic collision in collisions to the yield in collisions at the same energy multiplied by the average number of binary collisions in a typical collision, : In the presence of a net nuclear effect, is defined such that it differs from unity. In the simplest case of minimum bias collisions, one should have
As in , we have used the central curve of EPS09 as well as four specific extreme curves (minimal/maximal shadowing, minimal/maximal EMC effect), which reproduce the envelope of the gluon nPDF uncertainty encoded in EPS09 LO .
In addition to the modification of the partonic densities, quarkonium production in collisions can be affected by other effects, for instance, by the nuclear absorption which depends much on the nature of the object traversing the nuclear medium. If the meson is already formed, it may be affected more than a smaller preresonant pair. To discuss such an effect, it is useful to introduce the concept of the formation time, , based on the Heisenberg uncertainty principle and the time, in the rest frame of the meson, to discriminate between two states, for instance, the and the . In fact, one finds [45, 47] that such a time is similar for the charmonium and bottomonium states and is on the order of 0.3–0.4 fm. Obviously, this time has to be boosted in the frame where the nuclear matter sits. For smaller than the nucleus radius, the quarkonium is formed before escaping it. In the fixed-target mode with a proton beam and a nuclear target, the boost factor is simply . We therefore obtain as in Table 4.
One sees that looking at quarkonium production in collisions at different backward rapidities allows one to look at quarkonia traversing the nuclear matter at very different stages of their evolution. This effect could theoretically be studied by giving an ad hoc rapidity dependence to the effective absorption cross section, . This is left for a future study since, here, we wish to consider only the nPDF effects and the expected statistics. Other effects to be considered are the coherent energy loss  (expected to grow in the forward region) and the rescattering by comovers  (expected to grow with the multiplicity along the direction).
Since we wish to assess the discriminating power of the possible data to be taken with AFTER@LHC, we attribute to the EPS09 central values statistical uncertainties which directly follow from the differential yields, respectively, expected in and collisions. For that we take an integrated luminosity of 10 fb−1 for the runs and 100 pb−1 for the runs, in accordance with the luminosities discussed above (see Table 1). As this stage, we do not consider additional systematical uncertainties. This simplifying assumption could be lifted in a more detailed study which would also take into account a possible detector acceptance (and related efficiencies) as done in the previous section. In particular, we do not expect that the rapidity region for would be easily accessible.
In Figure 11, we show the rapidity dependence of for and its dependence near . The only million of to be collected per year allows for the measurement of a with a much better precision than the gluon nPDF, nearly up to . In addition, one notes that the nuclear modification factor is certainly measurable up to GeV/c.
In Figure 12, we also show the rapidity dependence of for and its dependence near . In both cases, the luminosity to be taken in a year at AFTER@LHC yields to statistical uncertainties which are largely negligible as compared to the nPDF uncertainties; the statistical uncertainties are not even visible in Figure 12. We expect this to hold also for the although its yields are down by a factor of 100.
As aforementioned, the nPDFs do not account for all the expected nuclear matter effects. However, it is clear that combining the measurements of , , and for (as a LHCb-like detector would do) will allow one to pin down the existence of a possible gluon EMC and antishadowing effect. We also stress that the complications induced by a rapidity dependence of could be avoided by the parallel measurement of for nonprompt which can only be sensitive to the energy loss since the quark decay (weakly) into the , way outside the nucleus. Figure 13 shows that the trend is similar compared to . Measuring the dependence of for prompt and should also avoid the sensitivity on formation time effects.
6. Prospects of Pb Measurements at GeV
The charged particle multiplicity has been generated with the EPOS generator [41, 42] in different configurations: at TeV in collider mode, , , and at GeV in fixed-target mode. By comparing these three distributions in the pseudorapidity of the particle in the laboratory frame as shown in Figure 14, one can conclude that the charged particle multiplicity in a fixed-target mode never exceeds the one obtained in collisions at TeV obtained in a collider mode in the full pseudorapidity range: a detector with the ALICE MFT+Muon detector  capability will be able to run in such conditions. Detailed studies are needed to evaluate up to which multiplicity a detector such as LHCb would be able to take good quality data.
In summary, we have shown that in a fixed-target mode with an integrated luminosity of 10 fb−1, using 7 TeV LHC proton beam on a hydrogen target, and with a detector setup and performances similar to the LHCb detector, quarkonium studies in the dimuon decay channel can be performed over a wide transverse momentum range and rapidity in the center-of-mass from ~−2.8 for and and ~−2 for states to ~0. We have performed simulations of the dominant background sources contributing to the invariant mass spectrum. The uncorrelated background was obtained using PYTHIA generator and dimuons from correlated background sources, , , and Drell-Yan, were simulated using both HELAC-ONIA and PYTHIA generators. The estimated background level allows for , , , , and measurements in the dimuon decay channel with good signal to background ratios.
These simulations set the stage for further ones including, on the one hand, the detection of photon from wave or decay or from the production of a pair, whose studies at low transverse momentum can provide important insight on the gluon transverse dynamics [51–54], and, on the other hand, the large combinatorial background typical of and collisions in which the study of excited quarkonium at AFTER@LHC energies is of paramount importance [1, 13]. We note that the DELPHES  framework seems particularly well suited to account for the photon detectability in such prospective studies.
Along our investigations, we have also noted that the main source of dimuons around the masses is from the Drell-Yan process (see Figure 6(b)). This gives us great confidence that the corresponding cross section can easily be extracted in this mass region in collisions, a fortiori with a vertex detector allowing for tagging the heavy-flavour muons. We therefore consider that the single-spin asymmetries for Drell-Yan pair production can indeed be extracted using a light polarised target. Motivations for such studies are discussed in [15, 16, 56]. Quarkonium polarisation measurements are of course also possible, given the large statistical samples.
As regards the case of collisions, we have had a first look at the charged particle multiplicities as a function of the laboratory pseudorapidity. We have found out that, for all the possible fixed-target modes, + Pb, Pb + H, and Pb + Pb, these are smaller than the ones reached in the collider modes, where the LHCb was used ( + Pb and Pb + at 5 TeV). We therefore believe that a detector with similar characteristics as compared to LHCb can very well be used in the fixed-target mode (our observation is obviously supported by the preliminary analysis of the LHCb-SMOG data taken during the pilot run of beam (Pb beam) on a Neon gas target from 2012 (2013) at a c.m.s energy of GeV (54 GeV) ).
In view of the above, we have evaluated the impact, and its uncertainty, on the nuclear modification of the gluon densities on prompt and nonprompt and in the form of . We have found that the measurements at backward rapidities allow one to search for the gluon antishadowing, the gluon EMC effect, and even the Fermi motion effect on the gluons with the unheard statistical precisions. The statistics are large enough to perform such measurement with the and probably also with and allowing for thorough investigations of the formation time effect of the meson propagating in the nuclear matter. Overall, our results confirm the great potential of AFTER@LHC for heavy-quark and quarkonium physics.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank C. Da Silva, D. d’Enterria, E. G. Ferreiro, R. Mikkelsen, S. Porteboeuf-Houssais, A. Rakotozafindrabe, P. Robbe, M. Selvaggi, M. Schmelling, P. Skands, and Z. Yang for the important and stimulating discussions. This research was supported in part by the ERC Grant 291377 “LHCtheory: Theoretical predictions and analyses of LHC physics: advancing the precision frontier,” by the COPIN-IN2P3 Agreement, by the French P2I0 Excellence Laboratory, by the French CNRS via the grants PICS-06149 Torino-IPNO and FCPPL-Quarkonium4AFTER & PEPS4AFTER2, by the European Social Fund within the framework of realizing the project “Support of intersectoral mobility and quality enhancement of research teams at Czech Technical University in Prague,” CZ.1.07/2.3.00/30.0034, by Grant Agency of the Czech Republic, Grant no. 13-20841S, and by the Foundation for Polish Science Grant HOMING PLUS/2013-7/8.
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Copyright © 2015 L. Massacrier et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.