We update predictions for lepton fluxes from the hadroproduction of charm quarks in the scattering of primary cosmic rays with the Earth’s atmosphere. The calculation of charm-pair hadroproduction applies the latest results from perturbative QCD through next-to-next-to-leading order and modern parton distributions, together with estimates on various sources of uncertainties. Our predictions for the lepton fluxes turn out to be compatible, within the uncertainty band, with recent results in the literature. However, by taking into account contributions neglected in previous works, our total uncertainties are much larger. The predictions are crucial for the interpretation of results from neutrino experiments like IceCube, when disentangling signals of neutrinos of astrophysical origin from the atmospheric background.

Article funded by SCOAP3

10^7$\,GeV) than pions and kaons.\footnote{\looseness=-1 More precisely, the critical energies in vertical direction for the charmed hadrons considered in this work amount to: $E_{D^0}^{crit}$ = 9.71 $\cdot 10^7$\,GeV,$\,\,\,$ $E_{D^+}^{crit}$ = 3.84 $\cdot 10^7$\,GeV,$\,\,\,$ $E_{D_s^+}^{crit}$ = 8.40 $\cdot 10^7$\,GeV,$\,\,\,$ $E_{\Lambda_c}^{crit}$ = 24.4 $\cdot 10^7$\,GeV.} These immediately decaying particles ($\tau \sim 10^{-12}$~s) give rise to the so called prompt flux, that is the object of study of this paper. In case of hadrons decaying into leptons, the flux of leptons coming from low energy hadrons, i.e., from hadrons with $E \ll E_{\rm crit}$, can be approximated by \begin{equation} \phi_{h\rightarrow l}^{\rm low} = Z_{h\,l}^{\rm low}\, \frac{Z_{p\,h}}{1-Z_{p\,p}}\, \phi^0_p \, , \label{philow} \end{equation} whereas the flux of leptons from hadrons with $E \gg E_{\rm crit}$ is approximated by \begin{equation} \phi_{h \rightarrow l}^{\rm high} = Z_{h\,l}^{\rm high}\, \frac{Z_{p\,h}}{1-Z_{p\,p}}\, \frac{E_{\rm crit,h}}{E_h} \, \frac{\ln(\Lambda_h/\Lambda_p)}{1-\frac{\Lambda_p}{\Lambda_h}} \, f(\theta) \, \phi^0_p \, , \label{phihigh} \end{equation} with $\Lambda_j (E_j)$ defined as $\Lambda_j (E_j) = \lambda_j (E_j)/(1-Z_{jj} (E_j))$. In the approximated solutions to the cascade equations outlined above, an energy dependence is understood in all fluxes $\phi$, all $Z$-moments and all attenuation lengths $\Lambda$. Note that the low energy lepton flux is isotropic, whereas the high energy lepton flux is characterized by an angular dependence $f(\theta)\sim 1/\cos(\theta)$ for $\theta$ $< 60^{\,\mathrm{o}}$, where $\theta$ is the angle with respect to the zenith, and by a more complex angular dependence close to the horizon. The solution in the intermediate energy range $E \sim E_{\rm crit}$ is obtained by the geometric approximation \begin{equation} \phi_{h \rightarrow l} (E_l) = \frac{\phi_{h \rightarrow l}^{\rm low} (E_l) \phi_{h \rightarrow l}^{\rm high} (E_l)}{(\phi_{h \rightarrow l}^{\rm low}(E_l) + \phi_{h \rightarrow l}^{\rm high}(E_l))} \, , \label{geometric} \end{equation} whose quality and validity depend on the particular shapes of $\phi_{h \rightarrow l}^{\rm low}$ and $\phi_{h \rightarrow l}^{\rm high}$ as a function of the lepton energy, see, e.g., figure~\ref{fig:interpola} in section~\ref{sec:predictions} below, for an example of this interpolation. In this way one gets the contribution $\phi_{h \rightarrow l}$ to the lepton flux from each hadron species $h$. Summation over all hadron species finally provides the total lepton flux $\phi_l$ for each lepton species $l$, that is $\phi_l = \sum_h \phi_{h \rightarrow l}$. Alternatively, the system of differential equations in eq.~(\ref{cascade}) can also be solved numerically. ]]>

5 \cdot 10^6 \, \mathrm{GeV} \, . \end{eqnarray} This is a reference spectrum used in earlier works on prompt lepton fluxes, and, although recent measurements have shown that it basically overestimates nucleon fluxes at the highest energies, we consider it for reference and comparison with older works~\cite{Pasquali-ml-1998ji, Enberg-ml-2008te, Bhattacharya-ml-2015jpa}. \item[2)] Gaisser 2012 (variant 1 and 2)~\cite{Gaisser-ml-2012zz}: \\ The first variant of the Gaisser spectrum, fitting available experimental data of different origin to an analytic expression with a number of parameters, is based on the hypothesis that three populations, one including CR particles accelerated by SuperNova remnants in our galaxy, a second one still of galactic origin but with an higher energy, and a third one of particles accelerated at extra-galactic sources, contribute to the measured CR spectrum. The three populations are characterized by different rigidities,\footnote{The rigidity of each population multiplied by the atomic number of each nuclear group, determines the characteristic energy where the corresponding all-particle CR spectrum exponentially cuts off. The larger the rigidity is, more extended is the spectrum at high energy.} they all include protons and nuclear groups (He, CNO, Mg-Si, Fe) with different spectral indices. The second variant of the Gaisser spectrum provides a special treatment of the third population, which is assumed to be composed of protons only, with large rigidity. \item[3)] Gaisser 2014 (variant 1 and 2)~\cite{Stanev-ml-2014mla, Gaisser-ml-2013bla}: \\ This uses the same functional form as in Gaisser 2012, but with updated parameters for an alternative fit of experimental data. In particular, the first variant of the spectrum involves three populations, two of galactic and one of extra-galactic origin, including the p, He, C, O, Fe nuclear groups, with dif\-fe\-rent rigidities with respect to the Gaisser 2012 case. The second variant differs from the first one because it includes an additional component from heavier nuclei, plus a fourth population, characterized by extra-galactic protons only, with large rigidity. This affects the ultra-high-energy part of the spectrum and improves the agreement with Auger data on cosmic ray composition at high-energy~\cite{Kampert-ml-2012mx}. \end{itemize} The all nucleon spectra corresponding to these cases are shown in figure~\ref{crspettri}. The effect of the different options on the shape of lepton fluxes is extensively discussed in section~\ref{sec:uncQCD}. \begin{figure} \centering \includegraphics[width=0.75\textwidth]{./disegni/crspectra_30.pdf} ]]>

E_{{\rm lab},\, D^0}$ contribute to the $Z$-moment at any given fixed energy $E_{{\rm lab},\, D^0}$. Although, in line of principle, $E_{\rm lab}$ can be very large, in practice it turns out that the largest contribution comes from values of $E_{\rm lab}$ within the range $E_{{\rm lab},\, D^0}$ $<$ $E_{\rm lab}$ $<$ $(100-1000) \times E_{{\rm lab},\, D^0}$, due to the fact that the distribution in $x_E = E_{{\rm lab},\, D^0}/E_{\rm lab}$ is rapidly suppressed for large $x_E$. As a consequence, for energies as those probed by IceCube, the contributions to the $Z$-moments come mainly from regions with a center-of-mass energy $\sqrt{S}$ not too high with respect to the energy range reached and probed so far at the LHC, where perturbative QCD in the standard formalism of collinear factorization has been tested to work. Any deviations from this formalism which may exist at the highest energies, e.g., in the form of non-linear effects (like gluon recombination as opposed to gluon splittings) or due to the dominance of large logarithms $\ln(S/m^2)$ subject to resummation on the basis of a different factorization formalism ($k_T$ factorization)~\cite{Ball-ml-2001pq}, are, thus, expected to have only a small impact on the $Z$-moments we are interested in for the aim of understanding the IceCube results. \begin{figure} \centering \includegraphics[width=0.75\textwidth]{./disegni/Z_pallhad_spettro0_pair1.pdf} ]]>

10^3$\,GeV, our central predictions are included in the uncertainty band of the latter, whereas the central predictions from the dipole model are included in our uncertainty band. In order to infer a value for the transition energy $E_{\rm trans}$ where the prompt neutrino flux overcomes the conventional one, in figure~\ref{fig:comp2} we compare our prompt lepton flux with the conventional neutrino flux originally computed in ref.~\cite{Honda-ml-2006qj} for a power-law CR primary spectrum and rescaled to one variant of the Gaisser spectra in ref.~\cite{Bhattacharya-ml-2015jpa}. We obtain $E_{\rm trans}$~=~$6.0^{+ 12}_{- 3}$~$\cdot 10^5$\,GeV. Interestingly, the central value lies well within the interval (4~$\cdot 10^5$~$-$~$10^6$)~GeV where the IceCube experiment did not observe any event after the full 988-day analysis~\cite{Aartsen-ml-2014gkd}. In fact, the IceCube collaboration has reported an excess of neutrinos in the diffuse flux, all lying in the neutrino energy regions [0.3~$-$~4]~$\cdot\,10^5$\,GeV and [1 $-$ 2] $\cdot\,10^6$\,GeV. According to our predictions, the ``empty'' region of IceCube corresponds to the ``conventional-prompt'' transition region, i.e., the region where the contributions of conventional neutrinos and prompt neutrinos to the total neutrino flux become of the same order of magnitude. We thus believe that the ``empty'' region seen by IceCube so far, should not be empty, but actually dominated by prompt neutrinos. However, the IceCube error bars in the ``empty'' region are still quite large, and we stress that the accumulation of more statistics is necessary before judgment can be made, whether this lack of signal is just an artifact due to poor statistics or due to some other technical issue, or instead has a real physical interpretation. \looseness=-1 At higher energies, on the other hand, the total observed neutrino flux $E_\nu^2 \, \phi(E_\nu)$ for $E_\nu$ in the [1 $-$ 2] $\cdot\,10^6$\,GeV energy interval appears slightly suppressed with respect to that in the [2 $-$ 3] $\cdot\,10^5$\,GeV bins. However, looking at our central prompt flux distributions and summing them with the distributions for the conventional flux, as a first rough estimate it turns out that we would expect a much larger suppression in the [1 $-$ 2] $\cdot\,10^6$\,GeV region with respect to the [2 $-$ 3] $\cdot\,10^5$\,GeV one, disfavoring the interpretation that the events seen by IceCube in the [1 $-$ 2] $\cdot\,10^6$\,GeV window are just due to a prompt neutrino component.\footnote{This interpretation is also disfavoured by IceCube observations of the arrival directions of the events with $E$ $>$ 6$\cdot 10^4$\,GeV, in presence of a $\mu$ veto (see figure~3 of ref.~\cite{Aartsen-ml-2014gkd}).} The difference between the IceCube (signal + background) observed total yield at high-energy and the yield for prompt neutrinos as predicted by our calculation is slightly reduced if we observe that our predictions have a sizable uncertainty band, meaning that even the shapes of the distributions can change in a non-negligible way when a higher-order calculation in QCD will be available. Furthermore, using as input primary CR fluxes including a population of extra-galactic protons with very-high rigidity (i.e.\ variants 2 of Gaisser spectra, instead of variants 1 which have a mixed extra-galactic component with a lower global rigidity for the extra-galactic population) also reduces the difference between neutrino predictions and the observed flux. In fact, the latter give rise to neutrino spectra which are less severely suppressed at the highest energies than those from models with extragalactic mixed components, as is evident when comparing, e.g., the left and right panels of figure~\ref{fig:comp2}, obtained with the variants 1 and 2 of the Gaisser 2014 spectrum, respectively. In order to go beyond these purely qualitative considerations and to draw more definite quantitative conclusions, one should definitely wait for more experimental statistics and also insert our fluxes into the specific experimental analysis software. In any case, we would like to emphasize that the transition region for the prompt \mbox{($\nu_{\mu}$ + $\bar{\nu}_\mu$)-}flux in our calculation turns out to be also a transition region for uncertainties, i.e., the QCD uncertainties dominate the total uncertainties at energies below the transition region whereas the astrophysical ones start to give a progressively sizable contribution above it, pointing to the importance and necessity of pursuing further studies of cosmic ray composition at the highest energies~\cite{Aloisio-ml-2013hya}, possibly accompanied by future measurements independent of Monte Carlo simulations of hadronic-interactions at the highest energies. ]]>