Prompt neutrinos from atmospheric c-cbar and b-bbar production and the gluon at very small x

TL;DR

The paper addresses predicting atmospheric prompt neutrinos from charm production in the forward, very-small-x regime and motivates extrapolating the proton gluon distribution to x ~ 10^-9 to enable flux predictions up to 10^9 GeV. It compares DGLAP-only, unified DGLAP/BFKL, and saturation-based extrapolations, identifies the GBW saturation approach as the most reliable at high energy, and extends the calculation to proton–air collisions to derive νμ and ντ fluxes, including b-bbar contributions. It finds that GBW-based predictions yield constrained fluxes with uncertainties around a factor of three, with ντ from charm dominating the atmospheric flux above ~10^4 GeV and beauty decays providing a significant additional contribution; these results have important implications for neutrino astronomy and cosmic ray physics, offering a robust atmospheric background model and a parameterization for high-energy charm production in p–air collisions.

Abstract

We improve the accuracy of the extrapolation of the gluon distribution of the proton to very small x, and show that the charm production cross section, needed to calculate the cosmic ray-induced `atmospheric' flux of ultrahigh energy prompt muon and tau neutrinos, may be predicted within perturbative QCD to within about a factor of three. We follow the sequence of interactions and decays in order to calculate the neutrino fluxes as a function of energy up to 10^9 GeV. We also compute the prompt neutrino tau flux from b-bbar production, hadronization and decay. New cosmic sources of neutrinos will be indicated if more prompt neutrinos are observed than predicted. If fewer neutrinos are observed than predicted, then constraints will be imposed on the nuclear composition of cosmic rays. The advantages of studying tau neutrinos are emphasized. We provide a simple parameterization of the prediction for the inclusive cross section for c quark production in high energy proton--air collisions.

Figures (11)

• Figure 1: The lowest-order diagram for $c\bar{c}$ production in high energy $pp$ collisions. The 'small $x$' gluon has typical values $x_2\simeq M_{c\bar{c}}^2/2sx_F$, where $x_F\sim 0.1$.
• Figure 2: The differential cross section $x_Fd\sigma/dx_F$ for charm production in $pp$ collisions, (\ref{['eq:dsigmabydx_F']}), at three different laboratory energies $E$, for three different ways of extrapolating the gluon to very small $x$. The models are (i) a double-leading-log DGLAP extrapolation for $x<10^{-5}$ (MRST), (ii) a unified DGLAP/BFKL approach with an $x^{-\lambda}$ extrapolation for $x<10^{-7}$ (KMS), and (iii) an extrapolation with saturation effects included (GBW). Plot (d) compares the GBW prediction at the three energies.
• Figure 3: The energy dependence of the relevant '$Z$ moment', (\ref{['eq:momentZ_c']}), of charm production in proton--nucleon collisions, as a function of the energy of the produced charm quark. The models are as in Fig. 2. The reason why the KMS prediction falls below the other predictions at the smaller values of $E_c$ is explained in Section \ref{['sec:predictions']}. The $\gamma=2.02$ moments are shown for illustration; in the calculation of the neutrino fluxes, the differential cross section $x_F d\sigma /dx_F$ is convoluted with the observed primary cosmic ray flux.
• Figure 4: The energy dependence of the relevant '$Z$ moment', $\sigma Z_c\equiv\int x^{2.02}(d\sigma^c/dx)dx$, of charm production in $p$-air collisions, as a function of the energy of the produced charm quark. The dashed curve corresponds to the GBW model (extended to include rescattering of the $c\bar{c}$ pair within the air nucleus). The upper dotted ($B$), lower dotted ($g_{3P}$) and continuous ($g_{3P}+B$) curves respectively include the growth of the proton radius, triple-Pomeron effects and the combination of these two effects. The dot-dashed curve is the scaling prediction of (\ref{['eq:lowerlimit']}), but with $\sigma_{\rm inel}$ corresponding to $p$--air collisions. The $\gamma=2.02$ moments are shown for illustration; in the calculation of the neutrino fluxes, the differential cross section $x_F d\sigma /dx_F$ is convoluted with the observed primary cosmic ray flux.
• Figure 5: The flux of prompt muon neutrinos at ground level, weighted by $E^3$, for different choices of the $c \rightarrow$ charmed hadron fragmentation function, $dn^{c \rightarrow i}/dx$. The curves correspond in descending order to assuming (i) no fragmentation $dn/dx\propto\delta(1-x)$, (ii,$\,$iii) $\delta(x_D-0.75x_c)$ and $\delta(x_{\Lambda_c} - \frac{1}{2}(x_c+1))$ for $x_c>0.1$, or $\delta(x_{\Lambda_c} - 1.47x_c)$ and (iv) a Peterson et al. fragmentation function PET with $\varepsilon_c = 0.043$RPP2002. In each case charm production is calculated using the GBW solid curve of Fig. 4.