Their paper looks at the IceCube results and finds confirmation--but they actually beg the question a bit--they assume the particles are taus and then see if this agrees with the Standard Model, rather than assuming the Standard Model and trying to estimate whether these are taus.
And though they wrote about supersymmetric s-tau ("stau") particles as a Beyond-The-Standard-Model alternative, these don't act like the usual predictions for staus. (There's some flexibility. Actually, alot. Actually there are a ridiculous number of unknown parameters in supersymmetry, so maybe they're right after all..)
Halzen said he was surprised that an acquaintance's paper refuting ANITA's first premise wasn't out already--perhaps it wasn't as easy as he thought.
ANITA looks for radio bursts in the atmosphere resulting from showers of particles . It can't see very many radio bursts coming down at it directly, but the radio waves of downgoing showers reflect off the Antarctic ice, so its coverage is actually quite wide. Thing is, a shower developing in the Earth's atmosphere will have a particular polarization. When it reflects off the ice, that switches. So if they see an upcoming shower's radio waves with the wrong polarization (and they see 2 such), they conclude that it wasn't a reflection--the parent particle was really coming up through the Earth. And the angles they see those coming from imply that the chord through the Earth is thousands of miles. And the energy of the shower they infer means the original particle that entered the Earth on the other side must have been mighty energetic: 100EeV or thereabouts. It would have interacted along the way, and the granddaughter particle that interacted for the last time in the crust would have had only about 10EeV left. That's still a lot, btw. And the probability that it could happen as described is, as noted above, very very low.
But... What if the ice isn't nice and homogeneous? Can you get an effective reflection that doesn't reverse the polarization of the radio waves generated by the shower? Then you just have an "ordinary" 10EeV cosmic ray shower. That would be my first guess, and that's Halzen's friend's estimation (so far).
Experimental glitches and surprises happen.
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