How accurately is it known that protons have the same charge as electrons? Recently, I claimed that the proton and electron have the same charge, and that this has been experimentally verified up to $1$ part in $10^{26}$. This belief was naively based on CODATA's 2014 recommendation for the value of an elementary charge $e$[1]:
$$e=1.6021766208(98) \cdot 10^{-19}\ \mathrm{C}$$
where brackets signify standard uncertainty (assuming normal distribution, $\approx 0.68$ probability, coverage factor $k=1$). After delving more into the literature, I concluded that CODATA does not actually in a direct sense compare the charges of protons and electrons; rather it is based on an indirect approach.[2]
$$e= \sqrt{\frac{2h\alpha}{\mu_0c}}$$
Here $\alpha$ is the fine-structure constant.
Question: how accurately is proton–electron charge equality established? The ideal answer includes (if possible)


*

*experimental values of the charge of the electron and proton;

*uncertainty, including assumptions;

*if pertinent, was the charge of the neutron assumed to be zero or was it estimated?

*reference to primary literature


Piccard and Kessler (1925) verified that the charges match up to $5$ parts in $10^{21}$; Hillas and Cranshaw (1959) noted that the first pair had made unwarrated assumptions for the charge of a neutron. The latter pair showed that a proton's charge is no more than $5\cdot10^{-20}$ times different from an electron's.[3] King (1960) achieved the result $2.5 \pm 1.5 \cdot 10^{-20}$ from hydrogen atoms.[4]
Has this been improved upon?

[1] Mohr, P. J.; Newell, D. B.; Taylor, B. N. 'CODATA Recommended Values of the Fundamental Physical Constants: 2014'. Reviews of Modern Physics, 2016, 88 (3). DOI: 10.1103/RevModPhys.88.035009, freely available on arXiv here, also archived link.
[2] Mohr, P. J.; Taylor, B. N.; Newell, D. B. 'CODATA Recommended Values of the Fundamental Physical Constants: 2006'. Rev. Mod. Phys., 2008, 80 (2), 633–730. DOI: 10.1103/RevModPhys.80.633.
 DOI: 10.1103//RevModPhys.80.633, freely available on arXiv here, also archived link.
[3] M. Hillas, A.; E. Cranshaw, T. A. 'Comparison of the Charges of the Electron, Proton and Neutron'. Nature, 1959, 184, 892–893. DOI: 10.1038/184892a0.
[4] King, J. G. 'Search for a Small Charge Carried by Molecules'. Phys. Rev. Lett., 1960, 5 (12), 562–565. DOI: 10.1103/PhysRevLett.5.562.
 A: Since asking the question yesterday, I was either made aware of or found three more topical papers.

1. Marinelli, M.; Morpurgo, G. 'Searches of Fractionally Charged Particles in Matter with the Magnetic Levitation Technique'. Physics Reports, 1982, 85 (4), 161–258. DOI: 10.1016/0370-1573(82)90053-9.

Since this is quite long, I had trouble finding one concrete value, but it did lead me to reference 2. In reference 2, Marinelli–Morpungo's result is quoted as
$$\frac{q_\mathrm{p}+q_\mathrm{e}}{e}=(0.8\pm0.8)\cdot 10^{-21}.$$
Thank you, Ben Crowell, for pointing me to this study in your comment.


2. Unnikrishnan, C. S.; Gillies, G. T. 'The Electrical Neutrality of Atoms and of Bulk Matter'. Metrologia, 2004, 41 (5), S125. DOI: 10.1088/0026-1394/41/5/S03.

This pair gives a large table summarising results of proton–electron charge asymmetry.
$\hspace{1cm}$
The limit remains at $10^{-21}e$ as before. However, they propose a new
experiment which could push the limit to $10^{-26}$ in the future.

We have been considering using a torsion balance for a sensitive search for fractional charges and free quarks. The possibility is experimentally attractive since a carefully constructed torsion balance can achieve a charge sensitivity of about $10^{-2}e$ in a moderate electric field. Such an experiment is suitable to probe the hypothetical charge asymmetry between the electron and the proton with very high sensitivity. [---]
Thus, the experiment will be able to probe a charge asymmetry of $10^{-24}e$. Once this is established, the mass element can be changed to a heavier one with heavier wires or balls, each of the order of $1\ \mathrm{g}$, with a total mass of about $100\ \mathrm{g}$. Then, the total number of atoms is about $10^{24}$ with $3\cdot 10^{25 }$ electron–proton pairs. With fractional charge sensitivity, a torsion balance experiment can reach a charge asymmetry sensitivity of an impressive $10^{-26}e$.



3. Bressi, G.; Carugno, G.; Della Valle, F.; Galeazzi, G.; Ruoso, G.; Sartori, G. 'Testing the Neutrality of Matter by Acoustic Means in a Spherical Resonator'. Phys. Rev. A, 2011, 83 (5), 052101. DOI: 10.1103/PhysRevA.83.052101.

The result by Dylla and King (1973) is criticised: Bressi et al. claim the former pair's result is actually no better than $10^{-19}e$. Bressi et al's own result is (assuming charge conservation in $\beta^-\!$-decay)
$$\frac{q_\mathrm{p}+q_\mathrm{e}}{e}=(-0.1\pm1.1)\cdot 10^{-21}.$$
This is also their limit on the maximum charge of the neutron.
