# Proton-to-electron mass ratio

In physics, the proton-to-electron mass ratio, μ or β, is simply the rest mass of the proton divided by that of the electron. Because this is a ratio of like-dimensioned physical quantity, it is a dimensionless quantity, a function of the dimensionless physical constants, and has numerical value independent of the system of units, namely:

μ = mp/me = 1836.15267389(17).[1]

The number enclosed in parentheses is the measurement uncertainty on the last two digits. The value of μ is known to about 0.4 parts per billion.

$\ \mu={m_p\over m_e} = {\alpha^2\over \pi r_pR_H} = 1836.15267\dots$

$\ m_p=$ mass of proton

$\ m_e=$ mass of electron

$\ \alpha=$ fine structure constant

$\ r_p=$ muonic hydrogen proton radius (from 2010 & 2013 proton radius experiments)

$\ R_H =$ Rydberg constant

This equation defines the proton to electron mass ratio and relates six fundamental physical constants.

## Discussion

μ is an important fundamental physical constant because:

$\alpha_s=-\frac{2\pi}{\beta_0 \ln(E/\Lambda_{QCD})}$
where β0 = −11 + 2n/3, with n being the number of flavors of quarks.

## Does μ vary over time?

Astrophysicists have tried to find evidence that μ has changed over the history of the universe. (The same question has also been asked of the fine structure constant.) One interesting cause of such change would be change over time in the strength of the strong force.

Astronomical searches for time-varying μ have typically examined the Lyman series and Werner transitions of molecular hydrogen which, given a sufficiently large redshift, occur in the optical region and so can be observed with ground-based spectrographs.

If μ were to change, then the change in the wavelength λi of each rest frame wavelength can be parameterised as:

$\ \lambda_i=\lambda_0[1+K_i(\Delta\mu/\mu)],$

where Δμ/μ is the proportional change in μ and Ki is a constant which must be calculated within a theoretical (or semi-empirical) framework.

Reinhold et al. (2006) reported a potential 4 standard deviation variation in μ by analysing the molecular hydrogen absorption spectra of quasars Q0405-443 and Q0347-373. They found that Δμ/μ = (2.4 ± 0.6)×105. King et al. (2008) reanalysed the spectral data of Reinhold et al. and collected new data on another quasar, Q0528-250. They estimated that Δμ/μ = (2.6 ± 3.0)×106, different from the estimates of Reinhold et al. (2006).

Murphy et al. (2008) used the inversion transition of ammonia to conclude that |Δμ/μ| < 1.8×106 at redshift z = 0.68.

Bagdonaite et al. (2013) used methanol transitions in the spiral lens galaxy PKS 1830-211 to find (∆μ/μ) = (0.0 ± 1.0) × 10−7 at z = 0.89, a stringent limit at this redshift.[2][3]

Note that any comparison between values of Δμ/μ at substantially different redshifts will need a particular model to govern the evolution of Δμ/μ. That is, results consistent with zero change at lower redshifts do not rule out significant change at higher redshifts.