On Wednesday, Sam Ting gave the talk about AMS-02 at CERN. If you missed the talk, you may watch the 85-minute recorded video here:
Recent results from the AMS experiment (CERN web, thanks to Joseph S.)What I want to focus on are the slides 82-85, and especially 85, around 0:48:00-0:49:00. They show some particular events they have seen. The first three slides show a \(1\GeV\) electron and a positron with the same energy; a \(10\GeV\) electron and a positron of the same energy; a \(100\GeV\) electron and a positron of the same energy.
It could be a foolish IQ test – checking whether you know the geometric series – to ask you what is the next slide. Well, it could be a \(1\TeV\) electron and a positron of the same energy. Except that it's not. ;-)
Instead, the slide shows a \(982\GeV\) electron and a \(636\GeV\) positron; the latter figure is significantly lower than \(1\TeV\). Clearly, the number of electrons and positrons with similarly high energies that they could see was already pretty low. It seems extraordinarily natural to me to assume that \(982\GeV\) and \(636\GeV\) are the highest-energy electron and the highest-energy positron they have recorded so far: why wouldn't they boast about their biggest fish?
The text below is based on this assumption.
Sam Ting only showed the bins and the numbers of electron and positron events up to the highest \(260-350\GeV\) bin but we were not told the key information what was seen above this cutoff. Ting justified this silence by the data's not having a high enough confidence level, and so on. Still, there must exist some particular electron and positron events and it must be possible to count them.
We could have been told this raw data. We were not. I am convinced that he is trying to save some gunpowder for the AMS briefings in the following years – the spectrometer should operate for decades. At the same time, I have many reasons to think that if we were told something about the events above \(350\GeV\), we would see that their case for dark matter is much stronger than it looks from the officially published data.
Now, you should look at Figure 2a of the Cholis, Finkbeiner, Goodenough, Weiner 2008 paper. It shows how the positron fraction should behave for various neutralino masses in the most convincing models of WIMP, those based on neutralinos in supersymmetric theories.
The graph shows that if the known backgrounds were the only thing that contributes, the positron fraction should steadily decrease from \(0.05\) to \(0.01\) as you go from energies \(10\GeV\) to \(1,000\GeV\) on the log scale. Instead, AMS (and previously PAMELA and Fermi) saw the positron fraction increasing from \(0.05\) to \(0.15\) as you go from \(10\GeV\), a local minimum of the positron fraction, to \(350\GeV\).
This observed positron fraction is already much higher than the background – it's the "positron excess" that suggests new physics. However, pulsars could still be the mundane explanation of this excess. If the origin of this excess is truly new particle physics such as WIMPs, the drop of the positron fraction above \(350\GeV\) that we haven't seen should be steep.
The only new information I will use are the energies of the highest-energy events. The number of events of these high energies is pretty low. I was thinking how to approach the problem: Monte Carlo tests of various adjusted background-only and background-plus-signal distributions etc. Well, I would have to define lots of contrived fitting functions and do lots of complicated operations with them which would be a lot of work, especially given the fact that the final conclusions are very fuzzy and have an unimpressive statistical significance. At the end, I decided for a very simple strategy: to reasonably extend Table 1 of the AMS paper to the higher energies.
In the table below, I took the number of positrons in various bins and the positron fraction from Table 1 of the AMS paper. The total number of positrons+electrons in the first data column was calculated by a simple division. Now, the extra bins above \(350\GeV\) were added and the last bin in the electron+positron and positron columns that contained the highest-energy events of the two types was marked as \(1\). It could have been higher but both numbers have a comparable chance to be higher and the conclusions wouldn't dramatically change.
With the bins I chose, I interpolated the numbers in both columns (electron+positron, positron) by geometric series. And then I could calculate the positron fractions in the missing columns. The result looks like this:
| Energy/\({\rm GeV}\) | \(N(e^\pm)\) | \(N(e^+)\) | fraction |
| 100-115 | 2719 | 304 | 0.11 |
| 116-132 | 1953 | 223 | 0.11 |
| 133-151 | 1284 | 156 | 0.11 |
| 152-173 | 1056 | 144 | 0.12 |
| 174-206 | 902 | 134 | 0.14 |
| 207-260 | 660 | 101 | 0.15 |
| 261-350 | 465 | 72 | 0.15 |
| 351-450 | 194 | 17 | 0.09 |
| 451-550 | 81 | 4 | 0.05 |
| 551-650 | 34 | 1 | 0.03 |
| 651-750 | 14 | 0 | 0 |
| 751-850 | 6 | 0 | 0 |
| 851-950 | 2-3 | 0 | 0 |
| 951+ | 1 | 0 | 0 |
To make the story even shorter, my point is that the positron fraction must decrease to a very small value such as \(0.03\) in the bin around \(636\GeV\), the highest observed energy of a positron, because the number of electrons+positrons in that bin is still very high over there. It has to drop from the known value \(465\) in the \(260-350\GeV\) bin to \(1\) in the bin above \(950\GeV\). Because the number of electrons is behaving rather smoothly, I interpolated by some kind of geometric series to get \(34\) electrons+positrons in the \(551-650\GeV\) bin. I surely don't claim this number to be precise but I do think it is a good estimate and the number of positrons must be significantly larger than \(10\) in that bin, otherwise the graph of the number of electrons would be too bumpy which is unlikely.
The decrease from \(0.15\) below \(350\GeV\) to less than \(0.03\) above \(650\GeV\) may surely be classified as abrupt and it does suggest a WIMP (neutralino) mass of order \(350-650\GeV\). In fact, I think that there is a reason why AMS hasn't added another column above \(350\GeV\): the number of positron events was already very low over there, producing a positron fraction significantly smaller than \(0.15\). If the positron fraction were still close to \(0.15\) e.g. in \(350-500\GeV\), I am inclined to believe that they would still have added this extra bin.
If this extra reasoning is on the right track, the neutralino mass could be closer to the lower value, \(m_{\tilde \chi}\sim 350\GeV\). In fact, because all values between \(260\) and \(350\GeV\) were clumped together by AMS, it's equally plausible that the neutralino mass could be between \(260-350\GeV\) but closer to \(350\GeV\) because it seems that this "whole" bin is still behaving in the way that maximizes the positron fraction. Well, because the maximum of the positron fraction could actually be a bit higher than \(0.15\), perhaps close to \(0.20\), the \(260-350\GeV\) bin could be close to \(0.15\) because of the average of \(0.20\) in the lower half and \(0.10\) in the upper half. By these fuzzy ideas, I want to suggest that \(300\GeV\) is still plausible.
Again, I totally agree that even if AMS has the data sketched above, the data only support the conclusions I am trying to make at a rather low confidence level and it is very correct that AMS aren't trying to make bombshell announcements yet because they can't have the sufficient certainty.
On the other hand, it seems extraordinarily likely to me that Sam Ting must feel a bit strange because the picture he is presenting is significantly different from – weaker than – the picture he actually believes to be most likely based on the complete data he hasn't shown to us. I've tried to fill the gap above. The numbers above should make it a bit more explicit why I think that all the bloggers who say that AMS doesn't possess any hints are wrong: Katie Mack, Matthew Francis, Matthew Strassler, and probably many others.
If you look at the 2008 paper by Hooper, Blasi, Serpico, Table 1 shows you some positron fractions predicted by the pulsars, too. The decrease of the positron fraction is also rather steep, although not as steep as for WIMPs (a drop of the positron fraction to 1/5 of the maximum value just by doubling energy above the value producing the maximum seems unlikely with pulsars), but note that the positron fraction seems to be maximized for \(E\sim 100\GeV\), well below the apparent maximum near \(350\GeV\) suggested by the AMS data (with some extra reading in between the lines). That could be the real reasons why the pulsar explanation may suck although some people tried to argue that the models of pulsars could be adjusted and put on steroids to increase the maximum energy, too.
Note that in my "average" estimate of the high-energy particles, there were about \(310-311\) electrons and \(22\) positrons censored, producing \(0.07\) for the fraction above \(350\GeV\). The actual number of positrons and the fraction could be higher or lower. If the drop is truly steep, then the actual number of censored positrons is much lower than \(22\).
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