To what degree of precision are atoms electrically neutral?

Question

It is said that if, say, the electric charge is not a Lorentz invariant, neutral atoms are no longer neutral, which is not experimentally valid. I want to know to what degree of precision atoms are measured to be electrically neutral and what would happen if, say, we assume that an oxygen atom has a superfluous charge of $10^{-30}$ C or $10^{-50}$ C.

Answer 1

See

Bressi, G., et al. "Testing the neutrality of matter by acoustic means in a spherical resonator," Physical Review A 83, (2011): 052101 (DOI: http://dx.doi.org/10.1103/PhysRevA.83.052101)

as one of the more recent experiments on this topic. The hypothesis has been tested since the 1920s on different materials using $4$ basic methods, and results using one specific methods are reported above. A difference between the proton and electron charges would produce sound in a SF$_6$ gas trapped in a spherical resonator.

Apparently water has not been tested explicitly but other systems, including high-$Z$ systems where relativistic effects could be important, have been investigated. In no case is there evidence that “neutral” matter is not in fact neutral: recent measurements have fractional uncertainties in the range of one part in $10^{-21}$, so roughly $10^{-40}$ C.

(There is also cosmological evidence but I cannot find the reference to this right now.)

Answer 2

The best bounds I am aware of come from cosmology. (Unfortunately, I know from experience that this is not something that the Particle Data Group tracks very assiduously.) If there were charge imbalances in the early universe (before recombination, when it was an opaque plasma), that would leave a strong impact on the cosmic microwave background. This is discussed in this 2005 paper [C. Caprini, P. G. Ferreira, JCAP 02, 006 (2005)] (arXiv version).

The bounds turn out to somewhat model dependent, but the general scale quoted in the paper for the electron-proton charge difference is $|q_{e-p}|\lesssim 10^{-26}e$, or about $10^{-45}$ C. My own interest in this comes from the related problem of bounds on the photon charge, which are even more model dependent—as discussed, for example, here and here.