Interpretation of measurements of the muon charge ratio in the TeV range depends on the spectra of protons and neutrons in the primary cosmic radiation and on the inclusive cross sections for production of $\pi^\pm$ and $K^\pm$ in the atmosphere. Recent measurements of the spectra of cosmic-ray nuclei are used here to estimate separately the energy spectra of protons and neutrons and hence to calculate the charge separated hadronic cascade in the atmosphere. From the corresponding production spectra of $\mu^+$ and $\mu^-$ the $\mu^+/\mu^-$ ratio is calculated and compared to recent measurements. The comparison leads to a determination of the relative contribution of kaons and pions. Implications for the spectra of $\nu_\mu$ and $\bar{\nu}_\mu$ are discussed.
The muon charge ratio in the TeV range has been measured by MI-NOS [1,2] and more recently by OPERA [3]. Both analyses use an analytic approximation as a framework for making an inference about the separate contributions of the pion and kaon channels to the charge asymmetry. In this paper a more detailed derivation of the muon charge ratio is used for the analysis. The muon charge ratio is expressed in terms of the spectrumweighted moments for production of π ± and K ± by protons and neutrons in the primary cosmic radiation, following the analysis of Lipari [4]. The analysis here accounts for the special contribution of associated production of charged, positive kaons. This analysis also accounts for the effect of the energy dependence of the composition of the primary cosmic-ray nuclei. Measurements from ATIC [5] and CREAM [6,7] indicate that the spectra of helium and heavier nuclei become somewhat harder than the spectrum of protons above several hundred GeV. This feature for helium was recently confirmed by PAMELA [8].
Because muon neutrinos are produced together with muons in the processes π ± → µ ± + ν µ (ν µ ) and
these results also apply to ν µ and νµ . In the TeV range and above the contribution of muon decay to the intensity of muon neutrinos is negligible. For reasons of kinematics, kaons are relatively more important for neutrinos at high energy than for muons. An additional goal of this paper is to draw attention to the implications of the muon results for atmospheric neutrinos in the TeV energy range and beyond.
The excess of µ + in atmospheric muons can be traced to the excess of protons over neutrons in the primary cosmic-ray beam coupled with the steepness of the cosmic-ray spectrum, which emphasizes the forward fragmentation region in interactions of the incident cosmic-ray nucleons. The classic derivation of the muon charge ratio [9] considers muon production primarily through the channel p → π ± + anything. The atmospheric cascade equation for the intensity of nucleons as a function of slant depth X in the atmosphere is solved separately for N = n + p and ∆ = pn subject to the appropriate boundary conditions. For the total intensity of nucleons as a function of slant depth X (g/cm 2 )
where the nucleon attenuation length is Λ N = λ/(1 -Z N N ) and λ is the interaction length of nucleons in the atmosphere. The corresponding result for ∆
where
The Z-factors (like Z N N = Z pp + Z pn ) are spectrum-weighted moments of the inclusive cross sections for the corresponding hadronic process. For example, a particularly important moment for this paper is
for the process p + air → K + + Λ + anything.
The normalized inclusive cross section is weighted by x γ where γ is the integral spectral index for a power-law spectrum and x = E K /E p . Feynman scaling is assumed in these approximate formulas, so the parameters may vary slowly with energy, especially near threshold. However, the scaling approximation is relatively good because the moment weights the forward fragmentation region.
The next step is to solve the coupled equations for the production of charged pions by nucleons separately for Π + (X) + Π -(X) and for ∆ π = Π + (X) -Π -(X). The solutions are then convolved with the probability per g/cm 2 for decay to obtain the corresponding production spectra of muons and neutrinos. The decay kinematic factors are
for muons and (1 -r π ) γ (γ + 1) and
for neutrinos. In each of Eqs. 7 and 8 the first expression is a low-energy limit and the second a high energy limit, where low and high are with respect to the critical energy ǫ π . The ratio r π = m 2 µ /m 2 π = 0.5731. The forms for two-body decay of charged kaons are the same with r K = 0.0458.
The production spectra are then integrated over slant depth through the atmosphere to obtain the corresponding contributions to the lepton fluxes. Finally, the low and high-energy forms are combined into a single approximate expression.
For example, for the flux of ν µ + νµ the expression is
Here φ N (E ν ) = dN/d ln(E ν ) is the primary spectrum of nucleons (N) evaluated at the energy of the neutrino. The three terms in brackets correspond to production from leptonic and semi-leptonic decays of pions, kaons and charmed hadrons respectively. The term for prompt neutrinos from decay of charm has been included in Eq. 9 (see Ref. [10]) but will not be discussed further here. The numerator of each term of Eq. 9 has the form
with i = π ± , K, charm and BR iν is the branching ratio for i → ν. The first Z-factor in the numerator is the spectrum weighted moment of the cross section for a nucleon (N) to produce a secondary hadron i from a target nucleus in the atmosphere, defined as in Eq. 5. The second Z-factor is the corresponding moment of the decay distribution for i → ν + X, which is written explicitly in Eq. 8. The second term in each denominator is the ratio of the low-energy to the high-energy form of the decay distribution [11]. The forms for muo
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