# Should the electrical charge of $\pi^+$ (positive pion) be considered exact?

According to https://en.wikipedia.org/wiki/Elementary_charge, IIUC, electrical charge of an electron should be considered an exact value of 1.602176634 coulombs instead of an experimental value with limited precision:

From the 2019 redefinition of SI base units, that took effect on 20 May 2019, its value is exactly $$1.602176634×10^{−19}$$ C[1] by definition of the coulomb.

Also see "SI Brochure (2019)", p. 127.

It seems the standard reference of particle physics data usually only tells a particle's electrical charge by denoting them as $$X^+$$ (for a particle with a charge same as electron but positive), $$X^-$$ (for a particle with a charge same as electron), etc. For example, $$\pi^+$$: http://pdglive.lbl.gov/Particle.action?node=S008. Is this right? Then should these particles' charges now also be considered exact? If they should rather be considered approximated values, how to determine the value along with the precision and uncertainty?

• What does IIUC mean? – Kyle Kanos Jun 21 '19 at 23:26
• All charges of free elementary particles are integer multiples of the elementary charge of the electron, exactly, while confined particles such as quarks may have integer multiples of 1/3 of this elementary charge. This is a well-tested experimental result (through neutrality and charge conservation) and there are several theories attempting to explain it. Why would pions be special? – Cosmas Zachos Jun 22 '19 at 0:18
• The elementary charge is the ability of a particle to produce and absorb photons. It is either there or not, true or false, one or zero (or minus one). It is not an imprecise number. A fractional charge of quarks always adds up to a whole number. It simply means that two or three quarks have the ability to produce and absorb photons collectively. A free fractional charge does not exist. – safesphere Jun 22 '19 at 3:19
• Also conservation of charge is a law. – anna v Jun 22 '19 at 5:28
• @KyleKanos if I understand correctly, if I understand correctly. – AccidentalFourierTransform Jul 21 '19 at 20:13

Strictly speaking, only the charge of the electron is exact, and the charges of all other particles are determined theoretically or experimentally in relation to the electron charge. I am not aware of any sensible physics theory where the pion and electron charges differ, but more importantly the experimental constraints are extremely good.

Most obviously, one observed decay mode of negative pions is into an electron and an electron anti-neutrino, so if we accept that electric charge is conserved then any difference between the pion and electron charges must be equal to the charge of the neutrino. Experimental limits on neutrino electric charges range from $$2\times 10^{-8}\,e$$ for lab measurements down to $$2\times 10^{-15}\,e$$ from astrophysical observations. (Also note that if there was an otherwise invisible particle carrying away the tiny electric charge, then the decay would be a 3-body instead of 2-body decay, and the observed momentum distribution of the decay electrons would be very different than what is observed.)

The common pion proton interaction $$\pi^- p \rightarrow \pi^0 n$$ gives even better limits, although there are more steps in the reasoning. The experimental limit on any electron-proton charge magnitude difference is less than $$1\times 10^{-21}\,e$$, and the measured charge of the neutron is $$-0.2\pm0.8\times 10^{-21}\,e$$. If the neutral pion had an electric charge, then there would have to be a two photon state with an electric charge. Limits on the photon electric charge are really, really good ($$< 1\times10^{-46}\;\mathrm{or}\;<1\times10^{-35}$$e, depending on assumptions), and there are very strong theoretical arguments that the charge must be zero. So the observation of $$\pi^- p \rightarrow \pi^0 n$$ interactions, along with the above constraints on the proton, neutron, and neutral pion charges means that the charge difference between the electron and the negative pion must be $$< 1\times 10^{-21}\,e$$.

Similar strength limits can be established for all charged hadrons. e.g. The observation of charmed meson $$D^+\rightarrow \pi^+\pi^0$$ decays means the $$D^+$$ has the same charge as the $$\pi+$$.

Better limits can be established for charged leptons. The existence of $$\mu^-\rightarrow e^- \bar \nu_e \nu_\mu$$ and $$\tau^-\rightarrow e^- \bar \nu_e \nu_\tau$$ decays means that the $$\mu-e$$ and $$\tau-e$$ charge differences must be equal to the $$\nu_e-\nu_\mu$$ and $$\nu_e-\nu_\tau$$ charge differences. This must be zero since observed neutrino oscillations would be impossible (by electric charge conservation) if different neutrino flavours had different electric charges.

Getting back to your question of precision and uncertainty, if one wants to be pedantic the measured value of the pion charge is $$1.000000000000000000000\pm0.000000000000000000001\,e$$, but it would almost always be silly to write this out. In practice, treat the pion charge as if it were exact and equal to the defined charge of the electron.

• A follow-up question is that if one really wants to be pedantic and determine the experimental value (with precision and uncertainty) of electric charge for all particles, except electron which has a defined value and neutron which has its experimental value explicitly listed in PDG, then for each particle, one need to identify the decay mode giving the most tight constraint then do an analysis like the one you did for $\pi^-$? Or is it always $X.000000000000000000000 \pm 0.000000000000000000001~e$? – Computist Jul 17 '19 at 20:18
• I have expanded my answer above a bit to point out that the experimental value for all charged hadrons will be very similar, since their charges can all be connected via decays and interactions to the electron and pion charges. I would again emphasize that you should just assume all charges are exactly quantized in units of the electron charge unless you have some good reason not to. e.g. If you are interested in Standard Model extensions that could allow neutrinos or dark matter to have tiny electric charges, see arxiv.org/abs/1403.6344, arxiv.org/abs/1802.10094. – David Bailey Jul 21 '19 at 20:17