# Hydrogen and the neutron

I have a question about Hydrogen and the neutron.

So a neutral hydrogen atom has structure ${}^1H\equiv[p^+ e^-]$, a bound proton and electron. From a quantum theoretical viewpoint, one would write down an infinite number of interactions to describe this bound state. But I was thinking doesn't that suggest the following is possible?

My worry is of course the neutral current interaction in the centre - two things come to mind; the GIM mechanism suppressing flavour changing neutral currents (e.g., the instance where $\nu_e$ oscillates into $\nu_\mu$), and the neutron to proton ratio of hydrogen. If this occurs, the neutrino could eject from the diagram, leaving a lone neutron, which could decay $n\rightarrow p^+ e^- \bar{\nu}_e$. So why does this not occur? Many thanks for your thoughts!

One further thought - if the neutrino is Majorana, does this also enhance this hydrogen decay effect? (ref https://en.wikipedia.org/wiki/Bethe%E2%80%93Salpeter_equation)

• You mention that the free neutron decays into a proton, electron and neutrino. What does this imply about the relative masses of these particles? – By Symmetry Aug 10 at 10:24
• @BySymmetry He's not asking about a decay process, but about an interaction. – rob Aug 10 at 10:26
• @BySymmetry even though $m_n > m_p + m_e$, that does not mean to say these interaction cannot occur (Heisenberg uncertainty allows us to borrow the energy, off shell); again like I say, all sorts of effects could occur at that central propagator - neutrino oscillation, neutron neutrino scattering. But ultimately Hydrogen is stable, why if these processes are accessible? – MKF Aug 10 at 10:35

All of the interactions you have drawn do in fact contribute to the interaction between the electron and the proton. This is the sense in which it is a weak "force." The size of the neutral current interaction between the proton and the electron has recently been measured, with preliminary and final analyses published in 2013 and 2018. (I'm an author on both papers.)

The neutral current interaction,

$${\rm pe} \to {\rm p}Z\rm e \to pe,$$

has the same structure as the electromagnetic force mediated by the photon, but different behavior under parity symmetry and with a Yukawa potential

$$U_\text{weak} \sim \frac{q^\mathrm{weak}_\mathrm e q^\mathrm{weak}_\mathrm p}{r} \cdot e^{ - m_Z r}$$

rather than the $1/r$ Coulomb potential of electromagnetism. The mass of the $Z$ boson makes this interaction vanish exponentially with a distance scale $r_Z = \hbar c / m_Zc^2 \approx 0.0022\,\rm fm$, three orders of magnitude smaller than the size of the proton. So for the hydrogen atom bound states, there is a small correction to the energy from the part of the wavefunction where the electron overlaps with the proton. This correction would be "large" in $s$-wave states compared to $p$-wave states, since their overlap with the nucleus is different, but not large enough to contribute to the current uncertainty of the Rydberg constant, $E_1 = -13.6\,\mathrm{eV} \times (1 \pm 290\,\mathrm{ppb})$. That few hundred parts per billion is the size of the weak-interaction asymmetry in the papers I linked above, where the interaction energy was $1\,\rm GeV$ and the short-range weak interaction is less inaccessible.

The charged-current force between a proton and an electron would have its largest contribution from the loop diagram

$${\rm pe}\to {\rm n}W{\rm e} \to {\rm n}\nu \to {\rm p}W\nu \to {\rm pe}$$

and its permutations. This force has a shorter effective range because now you have to borrow enough energy for two heavy virtual bosons. That makes it even weaker at long range than the weak interaction from the single heavy boson. I can't remember how much weaker, but it wasn't a consideration at all in the experiment we did. (We did have about five years where the theory community had a disagreement about the $\gamma Z$ box diagram.)

As far as beta decay from the virtual $\rm n\nu$ state in the charged-current box diagram that doesn't matter, that would look from the outside like

$$\rm pe \to pe\bar\nu\nu$$

which violates conservation of energy unless the proton-electron system releases some binding energy. A fun calculation would be predict the branching ratio for $\bar\nu\nu$ emission from one of the Lyman or Balmer transitions.

In fact, that's a really interesting calculation. Suppose that the branching ratio for $\rm H^*\to H\bar\nu\nu$ is any tiny fraction you like: $10^{-20}$ or $10^{-50}$ or whatever. Then the photospheres of stars are diffuse emitters of few-eV neutrinos, which would effectively be a continuous source of quasi-relativistic dark matter. I think the evidence is that most dark matter is "cold," or nonrelativistic, but there's a good paper in an estimate of how rapidly these Lyman neutrinos would accumulate around a galaxy.

• Thanks @rob for the great answer - physically it makes sense that it had to be protected; but as you say, this must mean it can occur! Of course the Yukawa potential suppresses the interaction compared to the typical QED interaction and the neutrino cross sections are generally much smaller than those in QED ($o(10^{-40})$ or less often), so they rarely happen. As a secondary point then, it must be possible to take the Hydrogen atom and access these suppressed weak interactions by placing it into a thermal bath? I wonder if it would mimic the binding energy case you discussed above (i.e. EFT) – MKF Aug 10 at 12:44