1 Introduction

Measurements of charm hadron production at the Large Hadron Collider (LHC) represent a unique opportunity to study the properties of quantum chromodynamics (QCD) in hadron collisions [29,30,31]—provide a better description of the charm baryon production cross sections. In this paper, we use Pythia 8.303, including relevant recent additions, to provide the shape of charm hadron distributions, in order to extrapolate to unmeasured regions.

While total charm cross-section measurements exist at lower energies [32,33,34], the total charm cross sections have been measured only in specific kinematic regions at the LHC. A recent phenomenological work uses the FONLL [25, 26] framework to extrapolate the \(D^{0}\) cross section at \(\sqrt{s}= 5.02~\)TeV measured by the ALICE and LHCb collaborations to the full phase space and derives an estimation of the total charm production cross section [35]. In this article, we review the existing measurements of the \(D^{0}, D^{+}, D^{+}_{s}\), \(\Lambda _{c}^{+}\), \(\** _{c}^{0}\) production cross sections as a function of \(p_{\textrm{T}}\) and y performed by the ALICE, CMS and LHCb collaborations at \(\sqrt{s}= 5.02~\)TeV  [4,5,6,7,8,9,10,11,12] and combine them into a total charm cross section, extrapolated to the full phase space. The choice of the centre-of-mass energy was driven by the abundance of available measurements and by the need of providing a complete reference for existing and future heavy-ion measurements of charm production performed at the same nucleon-nucleon energy. The available data on charm baryon production are used for the first time to provide an estimation of the total cross section. The extrapolation procedure is discussed in detail, including the assumptions made. The possible \(1\%\) intrinsic charm contribution in the proton, whose existence has not been confirmed by experimental data, is expected to appear at large rapidity beyond the coverage of LHC experiments and is not considered in the extrapolation.

The article is organised as follows. In Sect. 2, an overview of the different LHC experiments is presented, together with a description of the datasets used for the extrapolation. The extrapolation procedure is described in Sect. 3, and results are presented in Sect. 4.

2 Experiments and data samples

The collision data used in this article have been delivered by the LHC operating at the CERN laboratory and collected by the ALICE [36], CMS [37] and LHCb [38] collaborations. The results are based on recent measurements published by the ALICE [7,8,9,10,11,12], CMS [5, 6] and LHCb [4] collaborations on the open charm production. A summary of the results used is given in Table 1, and an overview consisting of D-meson cross sections as function of \(p_{\textrm{T}}\) is reported in Fig. 1. The integrated luminosities of the data samples are 19.3 nb\(^{-1}\), 27.4 pb\(^{-1}\) and 8.6 pb\(^{-1}\) for the ALICE, CMS and LHCb analyses, respectively.

Fig. 1
figure 1

Compilation of the measured double-differential cross sections of \(D^{0}\) (top), \(D^{+}\) (middle) and \(D^{+}_{s}\) (bottom) mesons as functions of \(p_{\textrm{T}}\) from the ALICE [7, 11], CMS [5] and LHCb [4] experiments

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Table 1 The summary of the considered data sets
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In the following, we briefly summarise the methodology used by each experiment in their own publications.

2.1 ALICE

The ALICE collaboration measured the cross section of prompt \(D^{0}, D^{+}, D^{*+}, D^{+}_{s}\), \(\Lambda _{c}^{+}\) and \(\** _{c}^{0}\) hadrons and their respective charge conjugates in the range of rapidity \(|y|<\) 0.5, using a data sample of about 990 million Minimum Bias (MB) events, which were collected during the 2017 pp run. This corresponds to an integrated luminosity of about 19.3 nb\(^{-1}\) . The charm hadrons were reconstructed via the hadronic decay channels \(D^{0} \rightarrow K^{-}\pi ^{+}\), \(D^{+} \rightarrow K^{+} K^{-}\pi ^{+}\), \(D^{*+} \rightarrow D^{0}\pi ^{+}\) with \(D^{0} \rightarrow K^{-}\pi ^{+}\), \(D_{s}^{+} \rightarrow \phi \pi ^{+} \rightarrow K^{+}K^{-}\pi ^{+}\), \(\Lambda _{c}^{+} \rightarrow pK^{-}\pi ^{+}\) and \(\Lambda _{c}^{+} \rightarrow pK^{0}_{\textrm{S}}\) \(\rightarrow p\pi ^+\pi ^-\) and the semileptonic decay channel \(\** _{c}^{0} \rightarrow {\textrm{e}}^+\** ^{-}\nu _{\textrm{e}}\) [39].

The prompt candidates are separated from the non-prompt coming from b-hadron decays either using predictions from FONLL, or by a data-driven approach based on the impact parameter distributions of the candidates. A different analysis technique is applied for the prompt \(D^{0}\) reconstruction, which is mainly based on particle identification and combinatorial background subtraction. It allows for extending \(D^{0}\) measurement down to \(p_{\textrm{T}} =0\).

The \(p_{\textrm{T}}\)-differential cross sections of prompt hadrons were obtained as

$$\begin{aligned} \left. \frac{\textrm{d}\sigma }{{\textrm{d}p}_{\textrm{T}}}\right| _{\scriptscriptstyle |y_{\textrm{lab}}|<0.5}=\frac{f_{\mathrm{{prompt}}}\cdot \left. N_{\textrm{raw}}\right| _{|y_{\textrm{lab}}|<y_{\mathrm{\scriptscriptstyle fid}}}}{2 \cdot \alpha _{\mathrm{\scriptscriptstyle y} }\cdot \Delta p_{\textrm{T}} \cdot ({\textrm{Acc}} \times \epsilon )_{\textrm{prompt}}\cdot \mathrm{{BR}}\cdot \mathcal {L}}. \end{aligned}$$
(1)

Here, \(N_{\textrm{raw}}\) is the raw yield extracted in a given \(p_{\textrm{T}}\) interval (of width \(\Delta p_{\textrm{T}}\)), \(({\textrm{Acc}}\times \epsilon )_{\textrm{prompt}}\) is the geometrical acceptance multiplied by the reconstruction and selection efficiency of prompt hadrons, and \(f_{\textrm{prompt}}\) is the fraction of prompt hadrons in the uncorrected yield. The factor \(\alpha _{\mathrm{\scriptscriptstyle y}}\) accounts for the \(p_{\textrm{T}}\)-dependent fiducial acceptance selection on the candidates. A factor 2 was added to the denominator to take into account that anti-particles are counted in the raw yield, but the corrected yields are given for only one particle type. Finally, \(\mathrm {\mathcal {L}} = \mathrm N^{MB}_{ev}/\sigma _{MB}\), where \(\mathrm N^{MB}_{ev}\) is the number of analysed MB events and \(\mathrm \sigma _{MB}=\) 50.9 mb is the inelastic cross section for the MB trigger condition.

A detailed description of the reconstruction and selection of the various charm hadrons, including the efficiency estimation and the systematic uncertainty evaluation, can be found in Refs. [7, 10,11,12].

2.2 CMS

The CMS collaboration measured charm cross sections for both prompt [5] and non-prompt [40] \(D^{0}\) production in both pp and heavy-ion collisions through the \(D^{0} \rightarrow K^{-}\pi ^{+}\) decay. The prompt cross section for pp collisions uses data samples with luminosity up to 27.4 pb\(^{-1}\) from the run at \(\sqrt{s}=5.02\) TeV recorded in 2015. The \(D^{0}\) rapidity was limited to \(|y|<1\) in the analysis to profit from the best possible tracking resolution, and from a dedicated trigger for the high-\(p_{\textrm{T}}\) part of the measurement which allowed the extension of the measurement up to \(p_{\textrm{T}} =100\) \({\textrm{GeV}}/c\) [5]. The high background due to the absence of dedicated pion and kaon identification limited the lower momentum of the measurement to transverse momenta above 2 \({\textrm{GeV}}/c\), in a sample of highly prescaled MB triggered events. The cross sections originally quoted for the sum of \(D^{0}\) and \({\overline{D}{}^0}\) were divided by 2 for the purpose of this work, in order to make them consistent with the ALICE convention.

In a separate result [6], \(\Lambda _{c}^{+} \rightarrow pK^{-}\pi ^{+}\) final states were measured in the same rapidity range \(|y|<1\) on the same data set. This measurement was limited to the transverse momentum region \(5<p_T<20\) \({\textrm{GeV}}/c\).

The CMS results overlap largely with the ALICE measurements which exhibit higher precision at low \(p_{\textrm{T}}\). Since the total cross section is driven by the lower \(p_{\textrm{T}}\) spectrum, the CMS measurements are not included in the final combination but used for systematic studies.

2.3 LHCb

The LHCb collaboration measured the cross section of prompt \(D^{0}\), \(D^{+}\), \(D^{+}_{s}\) and \(D^{*+}\) mesons at centre-of-mass energy \(\sqrt{s}=5.02\) TeV, using a data sample with an integrated luminosity of \(8.60\pm 0.33\) pb\(^{-1}\) recorded in 2015. The prompt component is separated from the non-prompt coming from b-hadron decays by using the high resolution of the vertex locator (VELO) detector. Thanks to the unique LHCb coverage in the forward rapidity region, the measurement is performed in the \(p_{\textrm{T}}\) range of \(0<p_{\textrm{T}} <10\) \({\textrm{GeV}}/c\) for \(D^{0}\) and \(D^{+}\) mesons, and \(1<p_{\textrm{T}} <10\) \({\textrm{GeV}}/c\) for \(D^{+}_{s}\) and \(D^{*+}\) mesons. The measured rapidity range covers the region \(2.0<y<4.5\). The final states \(D^{0} \rightarrow K^{-}\pi ^{+}\), \(D^+ \rightarrow K^- \pi ^+ \pi ^+\), \(D_{s}^{+} \rightarrow \phi \pi ^{+} \rightarrow K^{+}K^{-}\pi ^{+}\) and \(D^{*+} \rightarrow (D^0\rightarrow K^- \pi ^+) \pi ^+\) and their charge conjugated are reconstructed. A detailed description of the reconstruction, selection of signals and efficiency determination is discussed in Ref. [4]. The double-differential cross section is reported as a function of \(p_{\textrm{T}}\) and y:

$$\begin{aligned} \frac{{\textrm{d}}^2\sigma }{{\textrm{d}p}_{\textrm{T}}{\textrm{d}}y}=\frac{1}{\Delta p_{\textrm{T}}\Delta y}\times \frac{N_{\textrm{D}}}{\epsilon _D \times {\textrm{BR}} \times \mathcal {L}} \end{aligned}$$
(2)

where \(\Delta p_{\textrm{T}}=1\) \({\textrm{GeV}}/c\) and \(\Delta y=0.5\) are the widths of \(p_{\textrm{T}}\) and y bins, \(N_{\textrm{D}}\) is the measured signal yield of the D-meson plus the charge-conjugated yields, \(\epsilon _D\) is the total efficiency for the D-meson, \({\textrm{BR}}\) is the branching ratio of the decay, and \(\mathcal {L}\) is the total integrated luminosity.

3 Extrapolation procedure

As stated in the introduction, the ultimate goal of this paper is to obtain the total \(c\overline{c}\) cross section by extrapolating LHC measurements using Pythia for the estimation in the unmeasured regions of phase space. In this section, we will first provide details on the simulation by introducing the relevant parts of Pythia in Sect. 3.1. We then proceed explaining the estimation of the only remaining free parameter, namely the kinematic charm mass, in Sect. 3.2. Finally, in Sect. 3.3 we explain the extrapolation procedure itself, with an estimation of extrapolation uncertainty in Sect. 3.4.

3.1 Pythia

The Monte Carlo event generator Pythia [27] is one of the standard tools for generating simulated collision events at most of the collider experiments. In this section, the main model ingredients relevant for charm hadron production are introduced.

When simulating charmed hadron production down to very low \(p_{\textrm{T}}\), both perturbative and non-perturbative aspects need to be considered [29, 50]. The junctions have colour flows connecting multiple quarks in three-quark vertices. As such, they carry an intrinsic baryon (or anti-baryon) number, and will hadronise as baryons. The highest multiplet has a higher string tension than a normal triplet string, and when it hadronises, the overall suppression of strange quark production, will decrease. In the case of charm production, it means, to first order, that relatively more \(\Lambda _{c}^{+}\) baryons will be produced, at the expense of D-mesons, and that relatively more \(D^{+}_{s}\) mesons will be produced, at the expense of \(D^{+}\). Since this analysis uses Pythia to extrapolate measured spectra of hadrons to unmeasured regions, it is important to consider these effects. All the models introduce several new parameters, all of which are estimated in the light-flavour sector. Parameters, which have previously been shown to provide a good description of the multi-strange baryons in the light sector, are used here as well and are summarised in Table 2 in Appendix 1.

For this study, the most relevant parameter available for tuning is the charm quark mass, as long as the agreement with other data sets is not compromised, as will be explained in the following.

3.2 Estimating the kinematic charm mass

As explained in Sect. 3.1, the only remaining parameter left influencing the cross section of charmed hadrons is the charm quark mass. The charm mass enters in both the perturbative matrix elements and in the phase space selection, and the resulting kinematic charm mass is therefore not required to be identical to the current quark mass, though it should not deviate too far from it. In ref. [44], a value of \(m_{c}=1.5\) GeV/\(c^2\) was chosen, loosely based on data from WA82 [51], E769 [52] and E791 [53]. We updated this value using the large amount of recent data on D-meson cross sections from the LHC (Fig. 1).

In Pythia simulations, a variation of the charm quark mass \(m_{c}\) is performed from \(m_{c}=1.1\) GeV/\(c^2\) to 1.9 GeV/\(c^2\) with steps of \(\Delta m_{c}=0.1\) GeV/\(c^2\) (see Figs. 6, 7 and 8 in Appendix 2). The \(\chi ^2\) values are computed between the data and Pythia simulations for each \(m_{c}\) value. All available \(D^{0}\), \(D^{+}\) and \(D^{+}_{s}\) data points with \(p_{\textrm{T}} <6\,{\textrm{GeV}}/c\) are used in the calculation to reduce statistical fluctuations in the data and Pythia simulation at higher \(p_{\textrm{T}}\), yielding a total of 4 data points from CMS data, 28 from ALICE and 82 from LHCb. For the systematic uncertainties in these data points, the uncertainties between experiments are uncorrelated except for those from the branching ratios. Within each experiment, the CMS points are assumed to be uncorrelated, while both the ALICE and the LHCb uncertainties are partially correlated. As the correlation between systematic uncertainties in these measurements is not available, two extreme scenarios are considered: a) All experimental uncertainties between data points are uncorrelated, and b) all systematic uncertainties between ALICE and LHCb data points are fully correlated. Hence, all the data points are shifted up and down by \(1\sigma\) from their nominal values, and the corresponding \(\chi ^2\) values are calculated. The resulting \(\chi ^2\) versus the charm quark mass \(m_{c}\) is shown in Fig. 2, where the black points denote the uncorrelated scenario, and the red and blue show the fully correlated scenario with the upper and lower limit, respectively. Fifth-order polynomial functions are used to fit the kinematic charm mass. The minimum \(\chi ^2\) corresponds to the \(m_{c}\) value that agrees best with the data, which is found to be \(m_{c} =\) 1.29, 1.45 and 1.57 \({\textrm{GeV}}/c^2\) for the red, black and blue points, respectively. The differences between these \(m_{c}\) values are considered as the systematic uncertainty originated from the data uncertainty. The minimum \(\chi ^2\) values around 500 indicate that Pythia does not reproduce the D-meson data perfectly.

Fig. 2
figure 2

Fit of the kinematic charm mass. The global \(\chi ^2\) between Pythia and all available \({\textrm{D}}\)-meson data, fitted with a fifth order polynomial, is shown

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The resulting value (\(\pm 1\sigma\)) for the kinematic charm mass, which will be used in the following, is:

$$\begin{aligned} m_{c} = 1.45^{+0.12}_{-0.16} {\textrm{GeV}}/c^2. \end{aligned}$$
(3)

Figures 3 and 4 show the measured cross sections for \(D^{0}\) and \(\Lambda _{c}^{+}\) compared to the Pythia simulations with the bands corresponding to \(1\sigma\) variations around the kinematic charm mass. The same comparison is shown in Appendix 2 for \(D^{+}\) (Fig. 9), \(D^{+}_{s}\) (Fig. 10) and \(\** _{c}^{0}\) (Fig. 11).

We recommend to use this value of kinematic charm mass in future charm hadron studies at LHC, along with the model parameters given in Table 2 in Appendix 1.

Fig. 3
figure 3

Measured \(D^{0}\) cross section as a function of rapidity compared with Pythia simulations with the bands corresponding to a 1\(\sigma\) variation around the optimum

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Fig. 4
figure 4

Measured \(\Lambda _{c}^{+}\) cross section as a function of rapidity compared with Pythia simulations with the bands corresponding to a 1\(\sigma\) variation around the optimum

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3.3 Extrapolation to total \(c\overline{c}\) cross section

The total charm cross section in pp collisions at \(\sqrt{s}=5.02\) TeV is derived by extrapolating the available \(p_{\textrm{T}}\)- and y-differential cross sections of charm hadrons to the range \(0<p_{\textrm{T}} <36\,{\textrm{GeV}}/c\) and \(|y|<8\). The contribution to the total cross section above these bounds is considered to be negligible with respect to that at lower \(p_{\textrm{T}}\), as the \({\textrm{d}}\sigma /{\textrm{d}}y\) value for charm from FONLL pQCD calculations [25] falls to 0 for \(|y|<8\), and similarly less than \(0.01\%\) of the total \(p_{\textrm{T}}\)-differential cross section lies above \(p_{\textrm{T}} =36\, {\textrm{GeV}}/c\).

The measured charm hadron production cross sections are integrated in their visible ranges with a consistent treatment of correlations in their systematic uncertainties as described in their respective publications. When combining the measurements together, the uncertainties are assumed to be uncorrelated between experiments, apart from the branching ratio uncertainty, which is treated as fully correlated.

The extrapolation is performed separately for each hadron species, and in two steps, assuming that the \(p_{\textrm{T}}\) and y dependences are factorisable. First, an extrapolation in \(p_{\textrm{T}}\) (where needed) is performed within each experiment’s rapidity range (\(|y| < 0.5\) for ALICE; \(2.0<y<4.5\) for LHCb). The extrapolation factor is determined by dividing the integrated Pythia spectrum for \(p_{\textrm{T}} < 36\,{\textrm{GeV}}/c\) by that within the visible \(p_{\textrm{T}}\) range. Then, it is multiplied by the integrated visible production cross section. The statistical and systematic uncertainties on the integrated measurements are scaled by the same factor. The respective \(p_{\textrm{T}}\) extrapolation factors for each hadron species and experiment are detailed in Table 3.

The extrapolation in rapidity is performed using a similar method, by taking the ratio of the integrated Pythia spectra in \(|y|< 8\) over the visible rapidity range for \(p_{\textrm{T}} <36\) \({\textrm{GeV}}/c\). For the D-meson species, where measurements are available for both ALICE and LHCb, the visible range comprises the rapidity acceptances of both detectors together; for the charm baryon species where only ALICE measurements exist, the visible range was taken as \(|y|<0.5\). The rapidity interpolation factors are detailed in Table 4.

The total charm production cross section, \(\sigma _{c\overline{c}}\), is calculated from the measured ground-state hadron species as

$$\begin{aligned} \sigma _{c\overline{c}} = \sigma (D^{0})+\sigma (D^{+}) + \sigma (D^{+}_{s}) + \sigma (\Lambda _{c}^{+}) + 2 \times \sigma (\** _{c}^{0}). \end{aligned}$$
(4)

As no measured cross section is available for \(\** _{c}^{+}\) at this collision energy, a factor 2 is applied on the \(\** _{c}^{0}\) cross section in the calculation. This is motivated by the assumption that due to isospin symmetry the production rates of \(\** _{c}^{+}\) and \(\** _{c}^{0}\) are equivalent, and so enter into the total charm production cross section equally. At very forward rapidity, the possibility of an enhanced production of \(\** _{c}^{+}\) over \(\** _{c}^{0}\) is not considered due to lack of experimental data. The contribution of the \(\Omega _{\textrm{c}}\) baryon is considered to be small with respect to the other hadron species and so is not added to the central value. However, to account for the possible case that the \(\Omega _{\textrm{c}}\) enters significantly into the total charm cross section, an upper systematic uncertainty is assigned based on the \(\** _{c}^{0,+}\) cross section, under the extreme assumption of \(\Omega _{\textrm{c}}/\** _{c}^{0} =1\).

An extrapolation uncertainty is assigned to the total cross section by varying the bare charm mass used as an input parameter of Pythia as discussed above and recalculating the central value of the extrapolated cross section, resulting in an extrapolation uncertainty of approximately 2%. As the rapidity dependence of \(\** _{c}^{0,+}\)-baryon production is not well studied yet in pp collisions, an additional version of the extrapolation was performed, where the Monash tune of Pythia was used instead of the enhanced colour reconnection (CR) mode discussed above. This tune of Pythia predicts a significantly smaller charm baryon-to-meson ratio than the CR mode 2 for both \(\Lambda _{c}^{+}\) and \(\** _{c}^{0,+}\) baryons as measured by the ALICE collaboration at mid-rapidity. After extrapolating with this model, the relative contribution of \(\** _{c}^{0,+}\) baryons is reduced by approximately 7%, with corresponding increases in the relative abundances of \(D^{0}\), \(D^{+}\) and \(\Lambda _{c}^{+}\). The resulting total charm production cross section is lower by 8% with respect to the central prediction; this value is assigned as a lower bound on the extrapolation uncertainty in the final result.

3.4 Alternative extrapolation

An alternative, model-independent extrapolation method is employed to cross-check the result by replacing the Pythia spectrum with numerical functions fitted to data. The two-step extrapolation procedure remains unchanged. First, in each visible y interval, a Tsallis function [54] is used to fit the measured \(p_{\textrm{T}}\)-differential cross sections. The extrapolation factor is determined as the ratio of the integral of the Tsallis function in the range \(0<p_{\textrm{T}} <36\) \({\textrm{GeV}}/c\) to that within the visible \(p_{\textrm{T}}\) range. The integrated visible cross section is then multiplied by the factor. This step is performed for \(D^{0}\), \(D^{+}\) and \(D^{+}_{s}\) mesons in the y intervals of \(0<y<0.5\), \(2<y<2.5\), \(2.5<y<3\), \(3<y<3.5\), \(3.5<y<4\) and \(4<y<4.5\), and for \(\Lambda _{c}^{+}\) and \(\** _{c}^{0}\) baryons in \(0<y<0.5\). Next, for the extrapolation in rapidity, a Gaussian function is used to fit the extrapolated cross section in \(0<p_{\textrm{T}} <36\) \({\textrm{GeV}}/c\) as a function of y for each D-meson species. The extrapolation factor in rapidity is determined similarly, using \(|y|<8\) as the full rapidity range and the sum of the ALICE and LHCb acceptance as the visible region. The final extrapolated total cross section is compared to that extrapolated with Pythia, showing a \(\sim 4\%\) variation in the final value. This variation is added in quadrature as an extra contribution to the extrapolation uncertainties in the result.

4 Results and comparison to models

The data from the various experiments compared with Pythia with the dedicated tuning discussed in Sect. 3.1 are reported in Figs.  3, 4, 9, 10, and 11 as a function of rapidity and \(p_{\textrm{T}}\).

The total charm production cross section in pp collisions at \(\sqrt{s}=5.02~{\textrm{TeV}}\), as obtained from Eq. 4, is

$$\begin{aligned} \sigma _{c\overline{c}} = 8.34 \pm 0.22\mathrm { (stat.) } {}^{+0.37}_{-0.37} \mathrm { (syst.) } {}^{+0.36}_{-0.46} \mathrm { (extr.) } {}^{+0.68}_{-0} (\Omega _c) \,{\text {mb}}. \end{aligned}$$
(5)

The uncertainty sources are as described in Sect. 3. The value is also compared with model calculations from the next-to-leading order MNR pQCD framework [55] in Fig. 5 and with the charm cross sections measured by other experiments in pp and p–A collisions at different centre-of-mass energies [32,33,34]. It should be noted that the other experimental points shown, in particular those reported by the LHC collaborations at \(\sqrt{s}=2.76\,{\textrm{TeV}}\) and \(7\,{\textrm{TeV}}\), consider only the contributions of D-mesons scaled by the estimated fragmentation fractions from \(\mathrm e^+e^-\) collisions and do not include measurements of charm baryon production.

The uncertainties on the NLO (MNR) calculation are defined by summing in quadrature the contributions from a) variations in the factorisation and renormalisation scales \(\mu _{\textrm{f}}\) and \(\mu _{\textrm{r}}\) between 0.5 and 2, with the constraint \(0.5< \mu _{\textrm{f}}/\mu _{\textrm{r}} < 2\); b) variations in the charm mass between \(m_c=1.2\) and \(1.8\,\) \({\textrm{GeV}}/c\); and c) the uncertainties related to the applied PDF set (CTEQ6.6 [56]). The value of the \(c\overline{c}\) cross section is more than double the central value of the predictions, but is fully consistent within the theoretical uncertainties. It should be noted that the central value of the NLO calculations assumes a larger charm mass \((m_c = 1.5\,{\textrm{GeV}}/c)\) than the one determined above from fits to the experimental data, which strongly contributes to the lower predicted central value of the cross section.

Fig. 5
figure 5

The total charm production cross section measured in p–A and pp collisions by experiments [32,33,34] as a function of collision energy. For the measurement reported from this paper, the experimental uncertainties are shown as error bars and the extrapolation-related uncertainties are shown as a box. The measurements are compared with the NLO (MNR) predictions represented by the solid line [55]. The dashed lines denote the systematic uncertainties on the calculations (see text for more details)

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5 Conclusions

The total \(c\overline{c}\) cross section has been extrapolated using the open-charm hadron production measurements from the three LHC experiments ALICE, CMS and LHCb collected from pp collisions at \(\sqrt{s} =5.02\) TeV. The results have been presented as a function of the rapidity of the charm mesons and have, for the case of differential results, been compared to Pythia, including recent improvements in hadronisation models, and for the case of the integrated result, with a state-of-the-art fixed order calculation.

The main result, shown in Fig. 5, represents the most comprehensive measurement to-date of the total \(c\overline{c}\) cross section in pp collisions. We note that along with an increase in precision with respect to the earlier results due both to the increase in experimental precision for the LHC experiments between Run 1 and 2 of the LHC and the reduced reliance on model-dependent extrapolations in rapidity, the measured \(c\overline{c}\) cross section is now exactly at the edge of the uncertainty estimate of the NLO calculation.

The extrapolation procedure has been extensively described, being based on extracted shapes from Pythia, followed by a data-driven approach. We believe that this novel extrapolation procedure, and the slight update in the values of the parameter used in the shapes extraction, can be adopted in the future for this type of extrapolation at different centre-of-mass energies, as the ones previously and currently being measured at the LHC as well as at other accelerators.