Prompt neutrino fluxes from atmospheric charm

We calculate the prompt neutrino flux from atmospheric charm production by cosmic rays, using the dipole picture in a perturbative QCD framework, which incorporates the parton saturation effects present at high energies. We compare our results with the next-to-leading-order perturbative QCD result and find that saturation effects are large for neutrino energies above ${10}^6\mathrm{GeV}$, leading to a substantial suppression of the prompt neutrino flux. We comment on the range of prompt neutrino fluxes due to theoretical uncertainties.

Figure 1

The perturbative diagrams giving rise to scattering with a gluon of the ${\gamma }^{*}\to q\overline{q}$ fluctuation in deep inelastic scattering.

Figure 2

The perturbative diagrams giving rise to the scattering of a gluon with the $g\to q\overline{q}$ pair fluctuation in hadronic collisions.

Figure 3

Charm quark $x_F$ distribution in proton-air collisions at $E_p={10}^9\mathrm{GeV}$, calculated in the dipole model described in the text, with the standard choices of $m_c=1.3\mathrm{GeV}$, factorization scale ${\mu }_F=2m_c$, and the MRST2001 PDFs [59].

Figure 4

The NLO QCD $pA\to c\overline{c}X$ differential cross section as a function of Feynman $x_F$ for PRS [14] compared to the dipole model (DM) result for incident proton energies of ${10}^3$, ${10}^6$, ${10}^9\mathrm{GeV}$. The thicker lines are PRS and the thinner lines with the same color are DM at the same energy.

Figure 5

Prompt and conventional ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ fluxes in the vertical direction. The shaded band is the theoretical uncertainty band for the prompt flux calculated in this paper with the dipole model. The dashed line shows the conventional flux from Gaisser and Honda (GH) [11] and the dotted line is the conventional flux calculated in Ref. 15 (TIG).

Figure 6

Prompt muon neutrino fluxes obtained in perturbative QCD. The shaded area represents the theoretical uncertainty in the prompt neutrino flux evaluated in this paper, and the solid line in the band is our standard result. The dashed curve is the NLO perturbative QCD calculation of Ref. 14 (PRS), modified here to include fragmentation; the dotted curve is the saturation model result of Ref. 16 (MRS); and the dash-dotted curve is the LO perturbative QCD calculation of Ref. 15 (TIG).

Figure 7

The effect of fragmentation on predicted ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ fluxes. The solid lines are DM and NLO QCD (the latter multiplied by 2 to separate the lines) with Kniehl-Kramer fragmentation. The dashed lines are without fragmentation.

Figure 8

Theoretical uncertainty band (blue, dark band), compared to the uncertainty band from Ref. 6 (magenta, light band) with the overlapping region shown as the middle (light blue) band. The flux is scaled by $E$, in units of $1/{\mathrm{km}}^2\mathrm{yr}\mathrm{sr}$.

Figure 9

Prompt (solid line) and conventional (dashed lines) fluxes of ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$, ${\nu }_e+{\overline{\nu }}_e$, and ${\mu }^{+}+{\mu }^{-}$. The conventional fluxes are from Ref. 15. The three prompt fluxes are approximately equal and are therefore here represented by the ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux.

Figure 10

Prompt ${\nu }_{\tau }+{\overline{\nu }}_{\tau }$ flux (solid line) compared with the prompt ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux (dashed line).

Figure 11

Dependence on zenith angle of prompt and conventional ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ fluxes. The solid and dashed lines are DM with fragmentation in the vertical and horizontal directions, respectively. The dotted and dash-dotted lines are the Gaisser-Honda conventional fluxes [11] in the vertical and horizontal directions.