We update the study of the total Psi and Upsilon production cross section in proton-proton collisions at RHIC energies using the QCD-based Color-Singlet (CS) Model, including next-to-leading order partonic matrix elements. We also include charm-quark initiated processes which appear at leading order in alpha_s, but which have so far been overlooked in such studies. Contrary to earlier claims, we show that the CS yield is consistent with measurements over a broad range of J/Psi rapidities. We also find that charm-quark initiated processes, including both intrinsic and sea-like charm components, typically contribute at least 20% of the direct J/Psi yield, improving the agreement with data both for the integrated cross section and its rapidity dependence. The key signature for such processes is the observation of a charm-quark jet opposite in azimuthal angle phi to the detected J/Psi. Our results have impact on the proper interpretation of heavy-quarkonium production in heavy-ion collisions and its use as a probe for the quark-gluon plasma.
We discuss the production of $D$-mesons and $J/ψ$ in high multiplicity proton-proton and proton-nucleus collisions within the Color-Glass-Condensate (CGC) framework. We demonstrate that the modification of the LHC data on $D$ and $J/ψ$ yields in high multiplicity events relative to minimum bias events arise from a significant enhancement of the gluon saturation scales of the corresponding rare parton configurations in the colliding protons and nuclei. For a given event multiplicity, we predict these relative yields to be energy independent from $\sqrt{s}=200$ GeV at RHIC to the highest LHC ene…
Recently we have proved the NRQCD factorization in heavy quarkonium production at high energy colliders at all orders in coupling constant. In this paper we extend this to non-equilibrium QCD and prove the NRQCD factorization in heavy quarkonium production at all orders in coupling constant in non-equilibrium QCD. This proof is necessary to study heavy quarkonium production from quark gluon plasma at RHIC and LHC.
Recently we have proved NRQCD color octet factorization of S-wave and P-wave heavy quarkonia production at all orders in coupling constant in QCD in vacuum at high energy colliders in Eur. Phys. J. C76 (2016) 448 and in arXiv:1807.04722 [hep-ph] respectively. In this paper we extend this to prove NRQCD color octet factorization of P-wave heavy quarkonium production in non-equilibrium QCD at RHIC and LHC at all orders in coupling constant. This proof is necessary to study the quark-gluon plasma at RHIC and LHC.
A model for the production of quarkonium states in the midrapidity region at RHIC and LHC energy range is presented which explores well understood properties of QCD only. An increase of the quarkonium hadronisation time with the initial energy leads to a gradual change of the most important phenomena from fixed target- to collider-energies. We evaluate nuclear effects in the quarkonium production due to medium modification of the momentum distribution of the heavy quarks produced in the hard interactions, i.e. due to the broadening of the transverse momentum distribution. Other nuclear effects…
We briefly review the existing psi(2S) data taken at RHIC, the Tevatron and the LHC. We systematically compare them with colour-singlet-model predictions as a function of the center-of-mass energy, of the quarkonium rapidity and of the quarkonium transverse momentum. The overall agreement is good except for large transverse momenta. This points at the existence of large NNLO corrections or points at colour-octet dominance.
The double spin asymmetry for exclusive $J/ψ$ pair production in the polarized $p-p$ collisions at RHIC is investigated. Our study shows that the asymmetry in this process is measurable at RHIC-SPIN experiments in the near future, and, hence it can be used to determine the gluon distribution function $ΔG (x)$ in the polarized nucleon.
At RHIC the cross section for c-cbar production will be large enough such that approximately 10 pairs will be produced in each central collision. If a region of deconfined quarks and gluons is subsequently formed, one would expect that the mobility of the charm quarks will enable them to form J/Psi through "off-diagonal" combinations, involving a quark and an antiquark which were originally produced in separate incoherent interactions. We present model estimates of this effect, which indicate that the signal for deconfinement at RHIC may possibly be J/Psi enhancement rather than suppression.
We present a brief overview of the most relevant current issues related to quarkonium production in high energy proton-proton and proton-nucleus collisions along with some perspectives. After reviewing recent experimental and theoretical results on quarkonium production in pp and pA collisions, we discuss the emerging field of polarisation studies. Thereafter, we report on issues related to heavy-quark production, both in pp and pA collisions, complemented by AA collisions. To put the work in a broader perspective, we emphasize the need for new observables to investigate quarkonium production …
The photonuclear production of vector mesons in ultraperipheral heavy ion collisions is investigated within the collinear approach using different parameterizations for the nuclear gluon distribution. The integrated cross section and the rapidity distribution for the $AA \to V AA$ ($V = J/Ψ, Υ$) process are computed for energies of RHIC and LHC. A comparison with the recent PHENIX data on coherent production of $J/Ψ$ mesons is also presented. We demonstrate that the study of the exclusive quarkonium photoproduction can be used to constrain the nuclear effects in the gluon distribution.
I give a brief overview of the recent theoretical progress in the study of quarkonium production in proton-nucleus collisions in view of the recent LHC and RHIC results. A special emphasis is put on the excited states such as the psi', Upsilon(2S) and Upsilon(3S).
The production process of quarkonia in proton-proton (pp) collision is a very good probe of the parton structure of the proton. Recent experimental data of the production of J/psi+vector boson or quarkonium pairs at the LHC and Tevatron suggest the relevance of double parton scatterings (DPS). We discuss here the single parton scattering (SPS) contribution to the J/psi +Z, J/psi +W, and J/psi +J/psi productions in hadron collisions. By revisiting the computations of the SPS contributions to the J/psi+Z and J/psi +W productions, we demonstrate that the ATLAS data in fact show evidence for DPS.
I present a short review of recent studies of quarkonium production in proton-proton and proton-nucleus collisions at collider energies in Small-$x$ formalism.
The study of a quasi back-to-back isolated pair made of a heavy quarkonium, such as a J/psi or a Upsilon, and a photon produced in proton-proton collisions at the LHC, is probably the optimal way to get right away a first experimental determination of two gluon transverse-momentum-dependent distributions (TMDs) in an unpolarized proton, f1^g and h1^perp,g the latter giving the distribution of linearly polarized gluons. To substantiate this, we calculate the transverse-momentum-dependent effects that arise in the process under study and discuss the feasibility of their measurements.
Quarkonium production in heavy-ion collisions is a well-known signature of the formation of a plasma of quarks and gluons (QGP). After thirty years from the first measurements at SPS energies, a large wealth of results is now accessible from high-energy experiments at RHIC and LHC, and these new data are contributing to sharpen the picture of the quarkonium behaviour in A-A collisions. In this paper, an overview of the main results on both charmonium and bottomonium production in p-A and A-A collisions is presented, focussing on the most recent achievements from the RHIC and LHC experiments.
This report reviews the study of open heavy-flavour and quarkonium production in high-energy hadronic collisions, as tools to investigate fundamental aspects of Quantum Chromodynamics, from the proton and nucleus structure at high energy to deconfinement and the properties of the Quark-Gluon Plasma. Emphasis is given to the lessons learnt from LHC Run 1 results, which are reviewed in a global picture with the results from SPS and RHIC at lower energies, as well as to the questions to be addressed in the future. The report covers heavy flavour and quarkonium production in proton-proton, proton-…
We study the production of heavy quarkonium states in high energy proton-nucleus collisions. Following earlier work of Blaizot, Fujii, Gelis, and Venugopalan, we systematically include both small $x$ evolution and multiple scattering effects on heavy quark pair production within the Color Glass Condensate (CGC) framework. We obtain for the first time expressions in the Non-Relativistic QCD (NRQCD) factorization formalism for heavy quarkonium differential cross sections as a function of transverse momentum and rapidity. We observe that the production of color singlet heavy quark pairs is sensit…
The objective of this paper is to assess the current theoretical understanding of the extensive set of quarkonium observables (for both charmonia and bottomonia) that have been attained in ultrarelativistic heavy-ion collisions over two orders of magnitude in center-of-mass energy. We briefly lay out and compare the currently employed theoretical frameworks and their underlying transport coefficients, and then analyze excitation functions of quarkonium yields to characterize the nature of the varying production mechanisms. We argue that an overall coherent picture of suppression and regenerati…
We study inclusive heavy quarkonium production with definite polarizations in polarized proton-proton collisions using the non-relativistic QCD color-octet mechanism. We present results for rapidity distributions of cross sections and spin asymmetries for the production of J/psi and psi' with specific polarizations in polarized p-p collisions at \sqrt s = 200 GeV and 500 GeV at the RHIC within the PHENIX detector acceptance range.
Quarkonium production in proton-proton (pp) collision provides interesting means to study the parton content and their correlations in the proton. Recent LHC and Tevatron data of J/psi + Z, J/psi + W and J/psi +J/psi production suggest the relevance of double parton scatterings (DPSs) as opposed to single parton scatterings (SPSs). In this proceedings contribution, we review the corresponding SPS contributions and discuss their upper limits set up by quark-hadron duality. These allow us to perform an improved extraction of the DPS yields and of their implications.
Quarkonium production in proton-proton collision is interesting in profiling the partons inside the nucleon. Recently, the impact of double parton scatterings (DPSs) was suggested by experimental data of associated quarkonium production (J/psi+Z, J/psi+W, and J/psi+J/psi) at the LHC and Tevatron, in addition to single parton scatterings (SPSs). In this proceedings contribution, we review the extraction of the effective parameter of the DPS through the evaluation of the SPS contributions under quark-hadron duality.
We present a phenomenological study of the $x_F$ dependence of quarkonium production in high energy proton-nucleus collisions. The \xf~dependence of comover contributions is introduced to account for the observed quarkonium suppression at low $x_F$. Combining comover contributions, nuclear shadowing effect, energy loss mechanism and nuclear absorption together we reproduce the overall \xf~dependence of E772/E789 data.
We study inclusive $χ_{cJ}$ production with definite polarizations in polarized proton-proton collisions at $\sqrt{s}$ = 200 GeV and 500 GeV at RHIC by using non-relativistic QCD (NRQCD) color-octet mechanism. We present results of rapidity distribution of $χ_{c0}$, $χ_{c1}$ and $χ_{c2}$ production with specific polarizations in polarized p-p collisions at RHIC within the PHENIX detector acceptance range. We also present the corresponding results for the spin asymmetries.
We propose a simple and model-independent procedure to account for the impact of the nuclear modification of the gluon density as encoded in nuclear collinear PDF sets on two-to-two partonic hard processes in proton-nucleus collisions. This applies to a good approximation to quarkonium, D and B meson production, generically referred to H. Our procedure consists in parametrising the square of the parton scattering amplitude, A_{gg -> H X} and constraining it from the proton-proton data. Doing so, we have been able to compute the corresponding nuclear modification factors for J/psi, Upsilon and …
We anticipate new features of quarkonium production in heavy ion collisions at RHIC and LHC energies which differ from a straightforward extrapolation of results at CERN SPS energy. General arguments indicate that one may expect quarkonium formation rates to increase more rapidly with energy and centrality than the production rate of the heavy quarks which they contain. This is due to new formation mechanisms in which independently-produced quarks and antiquarks form a bound quarkonium state. This mechanism will depend quadratically on the total number of initially-produced heavy quark pairs, …
The near threshold photo or electroproduction of heavy vector quarkonium off the proton is studied in quantum chromodynamics. Similar to the high-energy limit, the production amplitude can be factorized in terms of gluonic Generalized Parton Distributions and the quarkonium distribution amplitude. At the threshold, the threshold kinematics has a large skewness parameter $ξ$, leading to the dominance of the spin-2 contribution over higher-spin twist-2 operators. Thus threshold production data are useful to extract the gluonic gravitational form factors, allowing studying the gluonic contributio…
Quarkonium has long been proposed as one of the golden probes to identify the phase transition from confined hadronic matter to the deconfined quark-gluon plasma in heavy-ion collisions. Since then, we have achieved a better understanding, not only about the propagation of quarkonium in-medium, but also about its production. Recent theoretical developments in our comprehension of the quarkonium production mechanism in proton-proton collisions and on the propagation of quarkonia in proton-nucleus and nucleus-nucleus collisions are reviewed and discussed.
We show that the production of J/psi or Upsilon pairs in unpolarised pp collisions is currently the best process to measure the momentum-distribution of linearly-polarised gluons inside unpolarised protons through the study of azimuthal asymmetries. Not only the short-distance coefficients for such reactions induce the largest possible cos 4 phi modulations, but analysed data are already available. Among the various final states previously studied in unpolarised pp collisions within the TMD factorisation approach, di-J/psi production exhibits by far the largest asymmetries, up to 50% in the re…
We summarise the perspectives on heavy-quarkonium production at the LHC, both for proton-proton and heavy-ion runs, as emanating from the round table held at the HLPW 2008 Conference. The main topics are: present experimental and theoretical knowledge, experimental capabilities, open questions, recent theoretical advances and potentialities linked to some new observables.
Quarkonium production in proton-nucleus collisions is a powerful tool to disentangle cold nuclear matter effects. A model based on coherent energy loss is able to explain the available quarkonium suppression data in a broad range of rapidities, from fixed-target to collider energies, suggesting cold energy loss to be the dominant effect in quarkonium suppression in p-A collisions. This could be further tested in a high-energy fixed-target experiment using a proton or nucleus beam. The nuclear modification factors of J/$ψ$ and $Υ$ as a function of rapidity are computed in p-A collisions at $\sq…
We investigate the inclusive production of prompt J/psi mesons in polarized hadron-hadron and photon-hadron collisions in the factorization formalism of NRQCD. Numerical results are presented for BNL RHIC-Spin and the approved SLAC fixed-target experiment E161 to assess the feasibility to access the spin-dependent parton distributions in the polarized proton and photon. We point out that data on J/psi production taken by the PHENIX Collaboration in unpolarized proton-proton collisions at RHIC tend to favor the NRQCD factorization hypothesis, while they significantly overshoot the theoretical p…
We study the Central Exclusive Production (CEP) of chi_cJ and eta_c mesons at RHIC in proton-proton collisions. We consider the chi_cJ --> J/psi + gamma decay channels and, recalling that the J_z=0 suppression of the J=1,2 states can be compensated by their larger branchings to J/psi + gamma, present predictions of rates and distributions for chi_c(0,1,2) production. Particular attention is paid to the impact of p_t cuts applied to the outgoing protons, which can influence the relative rates significantly. The distribution in the azimuthal angle difference between the outgoing protons and the …
We outline the case for heavy-ion-physics studies using the multi-TeV lead LHC beams in the fixed-target mode. After a brief contextual reminder, we detail the possible contributions of AFTER@LHC to heavy-ion physics with a specific emphasis on quarkonia. We then present performance simulations for a selection of observables. These show that $Υ(nS)$, $J/ψ$ and $ψ(2S)$ production in heavy-ion collisions can be studied in new energy and rapidity domains with the LHCb and ALICE detectors. We also discuss the relevance to analyse the Drell-Yan pair production in asymmetric nucleus-nucleus collisio…
The contrast between model predictions for the transverse momentum spectra of J/Psi observed in Au-Au collisions at RHIC is extended to include effects of nuclear absorption. We find that the difference between initial production and recombination is enhanced in the most central collisions. Models utilizing a combination of these sources may eventually be able to place constraints on their relative magnitudes.
We perform a study of $W$-boson production in polarized proton-proton collisions through next-to-next-to-leading order (NNLO) in perturbative QCD. This calculation is required to extend the extraction of polarized parton distribution functions to NNLO accuracy. We present differential distributions at $\sqrt{s}=510$ GeV, relevant for comparison to measurements from the Relativistic Heavy Ion Collider (RHIC). The NNLO QCD corrections significantly reduce the scale dependence of the cross section. We compare the longitudinal single-spin asymmetries as a function of lepton pseudorapidity to RHIC …
The production of quarkonium in heavy ion collisions is studied at RHIC and LHC energies. General arguments indicate that, due to initial production of multiple quark pairs in each central collision, the final population of quarkonium may exhibit significant enhancements over straightforward extrapolation of behavior at SPS energy. Explicit calculations based on both a statistical hadronization picture and a kinetic formation mechanism in a deconfined state verify these general expectations. Such enhancements will alter the nature of how quarkonium yields may be used as a signature of deconfin…
The effect of parton energy loss in cold nuclear matter on the suppression of quarkonia (J/psi, Upsilon) in heavy-ion collisions is investigated, by extrapolating a model based on coherent radiative energy loss recently shown to describe successfully J/psi and Upsilon suppression in proton-nucleus collisions. Model predictions in heavy-ion collisions at RHIC (Au-Au, Cu-Cu, and Cu-Au) and LHC (Pb-Pb) show a sizable suppression arising from the sole effect of energy loss in cold matter. This effect should thus be considered in order to get a reliable baseline for cold nuclear matter effects in q…
The sensitivity of proton-induced $J/ψ$ and $Υ$ production to the possible $\bar u/\bar d$ asymmetry in the nucleon is studied. The ratio of the quarkonium production cross sections at large $x_F$ on hydrogen over deuterium targets, $σ(p+p) / σ(p+d)$, is shown to be sensitive to this asymmetry. Predictions of various theoretical models for this ratio are presented.
The recent results on net-proton and net-charge multiplicity fluctuations from the beam energy scan program at RHIC have drawn much attention to explore the critical point in the QCD phase diagram. Experimentally measured protons contain contribution from various processes such as secondaries from higher mass resonance decay, production process, and protons from the baryon stopping. Further, these contributions also fluctuate from event to event and can contaminate the dynamical fluctuations due to the critical point. We present the contribution of stopped proton and produced proton fluctuatio…
The anomalous behavior of the spin-spin correlation at large momentum transfer in pp elastic scattering is described in terms of the degrees of freedom associated with the onset of the charm threshold. A non-perturbative analysis based on the symmetries of QCD is used to extract the relevant dynamics of the charmonium-proton interaction. The enhancement to pp amplitudes and their phase follow from analyticity and unitarity, giving a plausible explanation of the spin anomaly. The interaction between charmonium and light quarks in nuclei may form a distinct kind of nuclear matter, nuclear bound …
We examine carefully bottomonia hadroproduction in proton colliders, especially focusing on the LHC, as a way of probing the gluon density in protons. To this end we develop some previous work, getting quantitative predictions and concluding that our proposal can be useful to perform consistency checks of the parameterization sets of different parton distribution functions.
We summarize the theoretical description of charmonium and bottonium production in hadronic collisions and compare it to the available data from hadron-nucleon interactions. With the parameters of the theory established by these data, we obtain predictions for quarkonium production at RHIC and LHC energies.
The strong interaction between heavy quarks and the quark gluon plasma makes the open and hidden charm hadrons be sensitive probes of the deconfinement phase transition in high energy nuclear collisions. Both the cold and hot nuclear matter effects change with the colliding energy and significantly influence the heavy quark and charmonium yield and their transverse momentum distributions. The ratio of averaged quarkonium transverse momentum square and the elliptic flow reveal the nature of the QCD medium created in heavy ion collisions at SPS, RHIC and LHC energies.
A dense parton system is expected to be formed in the early stage of relativistic heavy-ion collisions at RHIC energies and above. The probability of a quark gluon plasma production and the resulting strength of its signatures depends strongly on the initial conditions associated to the distributions of partons in the nuclear wave functions. At very high energies, the growth of parton distributions should saturate, possibly forming a Color Glass Condensate, which is characterized by a bulk momentum scale $Q_s$. As a direct consequence, the possible signatures of the QGP should be $Q_s$-depende…
In this paper we give estimates for the proton--proton cross sections into pairs of quarkonium states $J/ψ$, $ψ(2S)$, $Υ(1S)$ and $Υ(2S)$ at the scheduled AFTER@LHC energy of $115$GeV. The estimates are based on the intrinsic heavy flavor mechanism which is observable for high values of $x_F$, a range outside the dominance of single parton and double parton scattering.
The transverse momentum spectra of $J/ψ$, $ψ(2S)$, and $Υ(nS, n=1,2,3)$ produced in proton-proton ($p$+$p$), proton-antiproton ($p$+$\bar{p}$), proton-lead ($p$+Pb), gold-gold (Au+Au), and lead-lead (Pb+Pb) collisions over a wide energy range are analyzed by the (two-component) Erlang distribution, the Hagedorn function (the inverse power-law), and the Tsallis-Levy function. The initial temperature is obtained from the color string percolation model due to the fit by the (two-component) Erlang distribution in the framework of multisource thermal model. The excitation functions of some paramete…
Nucleon electromagnetic form factors are considered in the framework of the generalized Skyrme model with dilaton-quarkonium field. In our recent publication we have got big discrepancies between calculated form factors and dipole approximation formula. Here we have reasonably good accordance between them in finite impulse region after vector meson dominance have been taken into account. Omega and Rho -meson have been included into only hadron structure of the photon.
We discuss the photoproduction of $Υ$ and $J/ψ$ at high energy $\bar{p}p$, $pp$ and heavy ion colliders. We predict large rates in $\bar{p}p$ interactions at the Fermilab Tevatron %and in heavy-ion interactions at the CERN LHC. These reactions can be and in $pp$ and heavy-ion interactions at the CERN LHC. The $J/ψ$ is also produced copiously at RHIC. These reactions can be used to study the gluon distribution in protons and heavy nuclei. We also show that the different CP symmetries of the initial states lead to large differences in the transverse momentum spectra of mesons
We calculate the yields of quarkonia in heavy ion collisions at RHIC and the LHC as a function of their transverse momentum. Based upon non-relativistic quantum chromodynamics, our results include both color-singlet and color-octet contributions and feed-down effects from excited states. In reactions with ultra-relativistic nuclei, we focus on the consistent implementation of dynamically calculated nuclear matter effects, such as coherent power corrections, cold nuclear matter energy loss, and the Cronin effect in the initial state. In the final state, we consider radiative energy loss for the…
We present the study of accessing unpolarized and linearly polarized gluon TMDs in J/ψ and Υ(1S) production in unpolarized proton-proton collision at LHC, RHIC and AFTER energies. Non- relativistic QCD based color octet model (COM) is used for estimating quarkonium production rates within transverse momentum dependent factorization formalism. A comparison is drawn between the experimental data and the transverse momentum distribution of quarkonium obtained in COM and color evaporation model.
Polarization measurements are the best instrument to understand how quark and antiquark combine into the different quarkonium states, but no model has so far succeeded in explaining the measured J/psi and Upsilon polarizations. On the other hand, the experimental data in proton-antiproton and proton-nucleus collisions are inconsistent, incomplete and ambiguous. New analyses will have to properly address often underestimated issues: the existence of azimuthal anisotropies, the dependence on the reference frame, the influence of the experimental acceptance on the comparison with other measuremen…
Recently we have proved the factorization of NRQCD S-wave heavy quarkonium production at all orders in coupling constant. In this paper we extend this to prove the factorization of infrared divergences in $χ_{cJ}$ production from color singlet $c{\bar c}$ pair in non-equilibrium QCD at RHIC and LHC at all orders in coupling constant. This can be relevant to study the quark-gluon plasma at RHIC and LHC.
We present additions to the MARTINI event generator for examining heavy quarks and quarkonia in heavy-ion collisions. All stages of a heavy-ion collision affect the observables associated with heavy quarks: the initial phase space of the heavy quarks are sampled with PYTHIA8.1, the heavy quarks are evolved using Langevin dynamics and a 3+1-dimensional hydrodynamical description of the heavy-ion collision, and are fragmented and hadronized using a modified version of the color evaporation model that takes into account non-trivial evolution in position space, as well as the possibility of recomb…
The experimental data from the RHIC and LHC experiments of invariant pT spectra in A+A and p + p collisions are analysed with Tsallis distributions in different approaches. The information about the freeze-out surface in terms of freeze-out volume, temperature, chemical potential and radial flow velocity for different particle species are obtained. Further, these parameters are studied as a function of the mass of the secondary particles. A mass-dependent differential freeze-out is observed which does not seem to distinguish between particles and their antiparticles. Further a mass-hierarchy i…
At high temperatures, strongly interacting matter becomes a plasma of deconfined quarks and gluons. In statistical QCD, deconfinement and the properties of the resulting quark-gluon plasma can be investigated by studying the in-medium behaviour of heavy quark bound states. In high energy nuclear interactions, quarkonia probe different aspects of the medium formed in the collision. We survey the results of recent charmonium production studies in SPS and RHIC experiments.
We propose a novel approach to compute the cross section of near-threshold $J/ψ$ and $Υ$ production in electron-proton scattering at large photon virtualities $Q^2$ based on an operator product expansion. We show that the process can be used to extract the gluon part of the D-term gravitational form factor of the proton. At the subleading level, it is also sensitive to the trace anomaly effect of QCD.
In this work we investigate the coherent photoproduction of psi(1S), psi(2S) and Upsilon (1S) states in the proton-nucleus collisions in the LHC energies. Predictions for the rapidity distributions are presented using the color dipole formalism and including saturation effects that are expected to be relevant at high energies. Calculations are done at the energy 5.02 TeV and also for the next LHC run at 8.8 TeV in proton-lead mode. Discussion is performed on the main theoretical uncertainties associated to the calculations.
