In a hypothetical neutrinoless beta decay, how the energy spectrum will look like?

Question

If we consider a hypothetical situation in which $\beta^-$ deacay is happening without emitting any anti-neutrino, to be precise, a neutrinoless beta decay is happening. So, in this case, how the energy spectrum will look like?

I know the energy spectrum in case when the beta decay takes place along with emitting an anti-neutrino. But I do not have any knowledge what will the spectrum look like when the anti-neutrino is not being emitted. If anyone can provide the spectrum along with explanation, that will be a great help. I am attaching the spectrum when anti-neutrino is emitted.

Answer 1

You would see a monoenergetic line. The fact that this isn't the case led Wolfgang Pauli to postulate the existence of the neutrino.

If the beta electron and the nucleus were all there was, then momentum conservation would dictate that after the decay, the two would fly apart in opposite directions. The entire decay energy (called the Q-value) of the decay would be split between the electron and the recoiling atom. Because of the huge mass difference (an electron weights 511keV$/c^2$, your example Bismuth-210 nucleus some 210GeV$/c^2$) that means that the electron flies away with approximately the Q-value of the decay.

By the way, experiments that search for neutrino-less double-beta decay do look for such a line in the combined energy of the two electrons coming out of the decay.

Answer 2

It's hard to say what your hypothetical situation would look like because your hypothetical situation would break most of the laws of quantum mechanics. It would be a very different universe than what we have today (if there even would be a universe).

Instead of thinking of a hypothetical situation that could never occur, it might be more instructive to compare the beta decay to an alpha decay. In an alpha decay, there is no additional particle created, so the emitted alpha particle energy is a discrete value (not a distribution). The change in binding energy is distributed between the alpha particle and the final nucleus, and the distribution is determined by their relative masses.