# 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.

• Measuring a charge of subatomic particles means applying a classical concept to the quantum world. In quantum mechanics charge is a quantum number, not an arbitrary value like mass. What is the objective of this research? To prove quantum mechanics wrong in one part in $10^{26}$? Really? – safesphere Jan 13 '18 at 20:40
• @safesphere The reason is actually rather down-to-earth. I initially made the claim in this answer, and am currently attempting to perform a fact check. I appreciate your feedback. Is the question of an experimental upper bound for electron-proton charge asymmetry invalid? – Linear Christmas Jan 13 '18 at 20:50
• @safesphere You're completely in the woods with that. Measuring if the difference in the value of charge associate with one class of particle and the value help by another class is perfectly well defined within quantum mechanics. You seem to be assuming that there is only one quantization scale for charge (which may or may not be true) but that doesn't cause trouble with comparing the values that result from measurement of charge. – dmckee Jan 13 '18 at 20:58
• @safesphere So would I, because this measurement has been done. In other words this is not merely a valid question, but a useful question. – dmckee Jan 13 '18 at 21:13
• See Marinelli and Morpugo, PLB 137 (1984) 439, which gives a limit of about 10^-21. There are separate measurements of neutrality of the neutron, described in arxiv.org/pdf/hep-ph/9209259.pdf – Ben Crowell Jan 14 '18 at 1:10

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.

• Caveat: This answer was written based on a brief browsing, not a thorough analysis. I assumed all uncertainties are quoted as standard uncertainties as in the OP. Feel free to edit or point out any inconsistencies. If you add any new papers or if your edit is significant, you may also turn this into a community wiki. – Linear Christmas Jan 14 '18 at 14:41
• Proposed experiment with potential to achieve $10^{-28}e$ precision: Arvanitaki, Asimina, et al. "How to test atom and neutron neutrality with atom interferometry." Physical review letters 100.12 (2008): 120407. arXiv:0711.4636 – A.V.S. Jan 14 '18 at 14:59