Here's what it says in the paper:
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Possible associations of transients with nuclear weapons testing might be considered for two reasons. From 1951 until the launch of Sputnik in 1957, at least 124 above-ground nuclear tests were conducted by the United States (U.S.), Soviet Union, and Great Britain. In some circumstances, nuclear radiation is known to cause a visible glow (i.e., Cherenkov radiation)5.
Cherenkov radiation is fairly well understood (Wikipedia,
Cherenkov radiation).
It is caused by charged particles travelling through a
dielectric medium faster than light can propagate through that medium (although the speed of those particles is always less than
c, the speed of light in a vacuum), and is often observed in nuclear reactors where the fuel rods are underwater.
I'm not sure there is a plausible mechanism whereby a nuclear test could generate visible Cherenkov radiation hundreds or thousands of miles away, excepting perhaps a piece of highly energetic fallout landing in a suitable medium, but as
@jdog has pointed out, fallout takes time to travel. Even if this were to occur, it is hard to imagine a situation where a sufficiently large amount of sufficiently energetic
airborne fallout, presumably in the form of a "cloud" of radioactive debris, would end up in a suitable medium in the atmosphere (areas of high humidity??) and be bright enough to be observed hundreds or thousands of miles away from the site of detonation, or having travelled hundreds or thousands of miles from the point of detonation. Like most airborne contaminants, fallout disperses, and radioactive decay is a thing.
There are no records whatsoever (AFAIK) of the blue glow of Cherenkov radiation being directly observed in the "normal" environment, even during/ after a nuclear test. We know nuclear tests were closely monitored, and often the detonation sites were surveyed.
The paper continues
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Consistent with this concept, glowing "fireballs" in the sky were reported in multiple instances to occur shortly after nuclear tests in locations where significant nuclear fallout was expected6,7
As
@jdog points out, the first in-text reference ([6]) that supposedly supports this claim is for Hastings' self-published
UFOs & nukes: extraordinary encounters at nuclear weapons sites 2nd ed., 2017.
The second ([7]) is for Knuth, K.H.
et al., "The new science of unidentified aerospace-undersea phenomena (UAP)", a pre-print (i.e. yet to be published in a peer-reviewed journal); 2025; Villarroel, Garry P. Nolan and the Tedesco brothers feature in a long list of co-authors
(link
https://www.arxiv.org/pdf/2502.06794; PDF attached below).
The relevant part of this paper is pages 58-65, "UAP and Nuclear Weapons", which in its first para. states
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Robert Hastings has been studying UFO incursions at nuclear sites for more than fifty years (since 1973), having interviewed over 150 former and retired military personnel involved in such cases...
...so the first ref. is for Hasting's self-published book, the second ref. is for a piece of work whose relevant part refers to Hasting's self-published book.
(Even allowing for the conventions used in citing references, I'm unhappy that Villarroel doesn't mention that she is a co-author of a paper that she refers to, but in fairness that might not be unusual. Doesn't mean we have to like it- I feel it provides a layer of opacity to the casual reader, who might not realise that the author, in this case Villarroel, is citing something they worked on, not an independent confirmatory source).
None of the examples/ descriptions given in the "UAP and Nuclear Weapons" section of the Knuth
et al. paper describe the steady blue glow of Cherenkov radiation:
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'Balls of Fire'... Brown quickly found the object, a bright ball of fire... It is suggested that the 'green fireballs' which appeared over Sandia in late 1948... two members of a patrol station southeast of Killeen base witnessed a small inch and a half diameter (4 cm) blinking violet light hovering about six feet off the ground and only about ten to twelve feet away... Five minutes later, four Army men located about two miles away observed a small bright light with a two to four inch metallic cone attached.
The violet light made me pause for thought, maybe an incorrect colour term had been used for a blueish glowing light; but barring an (unrecorded) accident involving the dispersal of fissile material, which caused no illness and required no clean-up that we're aware of, this seems extremely unlikely (and the problem of what dielectric medium might be present in the air to allow for Cherenkov radiation to occur remains).
The 1967 Malmstrom Air Force Base incident (Wikipedia
https://en.wikipedia.org/wiki/Malmstrom_UFO_incident) is mentioned by Knuth
et al. as "...the most famous of these events"; the claimed sightings are referred to as "incursions" indicating that the authors take it for granted that a physical object or objects were in fact present. Robert Salas, a 26-year old USAF lieutenant at the time of the incident, claimed over 18 years later to have seen a
red UFO (which, he claimed without any supporting evidence, incapacitated a Security Policeman). But the Knuth
et al. work doesn't describe Salas' description, which very clearly is not of Cherenkov radiation! Nor do Knuth
et al. refer to the USAF report on the incident (which was of interest to the USAF because it involved an electrical/ electronics malfunction which affected the ability to launch a small number of ICBMs for some hours, although of course there was no attempt or intention to
actually launch them).
The report, quoted on Wikipedia, stated
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Rumors of Unidentified Flying Objects (UFO) around the area of Echo Flight during the time of the fault were disproven. A Mobile Strike Team, which had checked all November Flight's [launch facilities] on the morning of 16 March 67, were question[ed] and stated that no unusual activity or sightings were observed.
The references used by Villarroel concerning
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are not "Consistent with this concept" [of Cherenkov radiation].
Citing questionable sources which provide evidence of questionable relevance (i.e.
not describing Cherenkov radiation) Villarroel forms a tentative, and frankly implausible, hypothesis linking unproven claims of "fireballs" (etc. etc.), nuclear weapons, Cherenkov radiation and transients.
Having constructed what might be thought of as a distinctly weedy straw man, Villarroel goes on to write
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We also considered a very different potential reason for links between nuclear testing and transients.
...and goes on to advance her aliens in geosynchronous orbits theory.
It's almost, "Here's a theory where I link a known physical phenomenon with unusual sightings, which lack evidential support, which do not describe that physical phenomenon. Still reading? Here's an even
more interesting theory..."

Sort of paving the way from highly questionable science to pseudoscience.