Why is the square of the neutrino mass negative?

In arXiv:hep-ph/0009291 this is explained by giving the example of:

$$m^2_{\nu_e}= -2.5 \pm 3.3 \text{eV}^2 \tag{1}$$

"Thee negative value of the neutrino mass-square simply means:" $$E^2/c^2 -p^2=m^2_{\nu_e} c^2 < 0$$ "The right-hand side in Eq. (3) can be rewritten as ($-m_s^2 c^2$), then $m_s$ has a positive value."

What is the meaning of $m_s$? This isn't explained in this article, and it makes no sense to me that the value for the square of a mass could be negative.

A similar question has been posted in Negative Neutrino Mass squared and in Negative Mass Square I do not understand the answers given.

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    $\begingroup$ Rather than posting a new question, I recommend adding comments to the linked questions & the answers there. Try to ask for specific clarifications, as "I do not understand the answers" leaves us floundering as how to help you. $\endgroup$ Commented Sep 9, 2020 at 12:19
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    $\begingroup$ Does this answer your question? Negative Mass Square $\endgroup$ Commented Sep 9, 2020 at 12:19
  • $\begingroup$ The only reason I didn't do that is because when I have done that before, I got little to no feedback, whilst when posting it again, I tend to get answers from a different POV that help me understand. That link was posted in my question and I did not understand the explanation. $\endgroup$ Commented Sep 9, 2020 at 12:29
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    $\begingroup$ You are referring to >20 year old data. The difficult-to-interpret parameter fits have not disappeared to date but they are much milder. Of course that fit parameter is not supposed to make sense as a fundamental physics quantity. Are you familiar with the fine mechanics of measurement? $\endgroup$ Commented Sep 9, 2020 at 15:42

1 Answer 1


The mass squared of the electron neutrino obviously cannot be negative as then the neutrino would have an imaginary mass, so the obvious conclusion is that there is some unknown (probably systematic) error in the experiment. The latest measurements I am aware of are the results from the KATRIN experiment, and while they do still give a negative value $m^2 = -1.0_{-1.1}^{+0.9}~\text{eV}^2$ this is not significantly different from $0$.

The paper you link is basically claiming the electron neutrino does in effect have an imaginary mass because it travels faster than light. This is not an explanation that most of us find convincing, especially since experimental measurements of the electron neutrino speed from the supernova 1987A give a result consistent with the speed of light, not faster than light.

If you are interested to read a detailed discussion of the subject I recommend the paper Neutrino mass limit from tritium beta decay by E. W. Otten, C. Weinheimer. This gives a very detailed discussion of the tritium experiment and possible sources of error, though without coming to any firm conclusion. Note that this paper predated the KATRIN results, though it does discuss the KATRIN experiment.

  • $\begingroup$ I am trying to cite a good source that will give me the upper boundary of the one or even all of the three neutrinos masses. So, does this make it less credible than the 1987 source and therefore should not be included in my work? $\endgroup$ Commented Sep 9, 2020 at 16:37
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    $\begingroup$ @user7077252 Cite the KATRIN result. The Tsao Chang, Guangjiong Ni paper is highly speculative and I would not cite it in anything I wanted to be taken seriously. $\endgroup$ Commented Sep 9, 2020 at 17:18

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