Atmospheric neutrinos [82,83]

The theoretical flux calculations were reviewed by Stanev [84], Battistoni [85], and Lipari [82]. The largest uncertainties are in the primary flux and in the pion yield in $p$-nucleon scattering. The agreement between the Bartol and Honda flux calculations was somewhat accidental, since the Bartol group used a lower flux and higher $\pi$ yield, while Honda assumed the reverse. Recent new measurements of the primary flux by AMS [86] and BESS [87] have clarified the situation up to a few hundred GeV (the range relevant for fully contained and for partially contained or stopping events), indicating that the old (high) ``Weber data'' should be discarded. There is still a need for new measurements at middle energy (up to a few TeV), relevant to throughgoing muons. There has been some theoretical work on the pion yields, but major progress is expected from the HARP cross section measurements at CERN [88] (which are also critical to a possible $\nu$ factory).

In addition to the Bartol and Honda calculations, the newer FLUKA group [85] has explored theoretically-motivated (as opposed to semi-empirical) cross sections. Moreover, they have done a new 3-dimensional calculation. This yields somewhat higher fluxes in the horizontal ( $\cos \theta \sim 0$) direction than the older one-dimensional codes, leading to slightly higher $\Delta m^2$. Many smaller effects, including geometry, geomagnetic effects, seasonal variations, etc., have been carefully considered.

The conclusion is that the flux is well understood. There are no large problems, and the predicted $\nu_\mu/\nu_e$ ratio and zenith distributions are under control. All of the groups are working on refined calculations.

Ronga [34] reviewed the data from Soudan 2 and MACRO. Soudan 2 obtains a $\nu_\mu/\nu_e$ ratio of 0.68(11)(6) compared to expectations, and their up/down asymmetry is becoming significant. MACRO excludes pure $\nu_\mu \mbox{$\rightarrow$}\mbox{$\nu_s$}$ at 98%, mainly from the ratio of vertical to horizontal upward throughgoing events. (There are matter effects for $\mbox{$\nu_\mu$}\mbox{$\rightarrow$}\mbox{$\nu_s$}$, or for $\mbox{$\nu_\mu$}\mbox{$\rightarrow$}\mbox{$\nu_e$}$, but not for $\mbox{$\nu_\mu$}\mbox{$\rightarrow$}\mbox{$\nu_\tau$}$.)

Kajita [19] summarized the current status of Superkamiokande data:

• The zenith angle distributions are now very precise and beautiful. They are consistent with oscillations. However, they do not have the sensitivity to resolve oscillation wiggles, so neutrino decay scenarios cannot be excluded.
• Analysis of the data using the new 3d flux calculations will increase $\Delta m^2$ slightly from the value ( $\sim 3.2 \mbox{$\times$}10^{-3}$ eV$^2$) obtained using a 1d flux. The precise number is under investigation. The higher $\Delta m^2$ is good news for long baseline experiments.
• Pure $\nu_\mu \mbox{$\rightarrow$}\mbox{$\nu_s$}$ is excluded at 99% by a combination of upward throughgoing, high energy PC, and NC-enhanced multi-ring events. However, significant admixtures of $\nu_s$, such as $\nu_\tau + \mbox{$\nu_s$}$, in the final state cannot be excluded.
• SuperK has fit to a transition probability $\propto \sin^2 (\alpha LE^n)$, with $\alpha$ and $n$ free. They obtain $n = -1.0 \pm 0.14$, consistent with oscillations ($n=-1$), but excluding CPT violation ($n=0$), or violations of Lorentz invariance or the equivalence principle ($n=+1$).

In the future there will be refined flux calculations. It is possible that SuperK will be able to observe $\tau$ appearance at the 2-3$\sigma$ level, further constraining the sterile neutrino scenarios. Atmospheric neutrinos may be studied in new generation experiments, including MONOLITH [41] and ICARUS [44] at Gran Sasso and the ANTARES underwater experiment [89]. MONOLITH should have excellent capabilities for observing an oscillation dip, as well as matter effects [90]. Finally, terrestrial long baseline experiments K2K (KEK to Kamiokande) [42], MINOS (Fermilab-Soudan) [43], and CNGS (Cern-Gran Sasso) [44,45,91] will be able to study the atmospheric neutrino parameter range in much more detail.