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Apparently, this discrepancy is one of the "worst predictions" in the history of science. Clearly the vacuum energy calculation depends on many approximations and it is not clear how it should be tied to the cosmological constant. Some articles claim that the two differ by 120 orders of magnitude, however it is not clear for me who was the first to publish this number or where it comes from. I have checked reviews like:

So who was the first to publish such a large difference of 120 or 122? Did the author specify values for both the vacuum energy calculation and measurements of the cosmological constant?

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    $\begingroup$ The discrepancy is between the values found from gravity/cosmology and QFT. As far as I know, the first to become aware of this was Pauli, see e.g. this lecture by Straumann. $\endgroup$
    – Tom Heinzl
    Jun 26 at 11:40
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    $\begingroup$ Another early work is by Zeldovich (1968), entitled "The cosmological constant and the theory of elementary particles". It has been republished as a "Golden Oldie" in 2008. Pauli's contribution is mentioned by the historical editor of the republication as their first reference. $\endgroup$
    – Tom Heinzl
    Jun 26 at 13:16
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    $\begingroup$ The measured value of the cosmological constant is roughly $\Lambda \approx 10^{-44} \mbox{GeV}^4$. In QFT, it is given by a divergent integral that has to be cut off, typically at some mass value, $M$, yielding the estimate $\Lambda \sim M^4$. Pauli chose the mass associated with the classical electron radius, implying $\Lambda \sim 10^{-5} \mbox{GeV}^4$ (off by 39 orders of magnitude). Choosing $M$ to be the Planck mass ($10^{19}$ GeV), one finds the 120 orders of magnitude discrepancy, $\Lambda \sim 10^{76} \mbox{GeV}^4$. So your question is: who first used the Planck mass as a cutoff? $\endgroup$
    – Tom Heinzl
    Jun 27 at 10:08
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    $\begingroup$ @TomHeinzl yes, who was the first to set that cutoff $\endgroup$
    – Mauricio
    Jun 27 at 11:00
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    $\begingroup$ @njuffa from time to time for specific bits of information I've emailed directly to the reference librarians of various libraries around the world, and invariably they have taken some time to access various things and share electronic copies with me. It's not a way to research a topic, but very often you can get copies of hard-to-find things that way. You may also consider searching a database of Masters and Ph.D. theses on the topic of vacuum energy and its predictions by narrowing your search to text of theses. Most universities make them accessible. It's slow and painstaking, but sometimes $\endgroup$
    – uhoh
    Jul 7 at 22:12

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