# Proton-to-electron mass ratio

In physics, the proton-to-electron mass ratio (symbol μ or β) is the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms), a dimensionless quantity, namely:
    μ = m$_p$/⁠m$_e$ = 1836.152673426(32).
The number in parentheses is the measurement uncertainty on the last two digits, corresponding to a relative standard uncertainty of 1.7×10$^{−11}$.

## Discussion

μ is an important fundamental physical constant because:
* Baryonic matter consists of quarks and particles made from quarks, like protons and neutrons. Free neutrons have a half-life of 613.9 seconds. Electrons and protons appear to be stable, to the best of current knowledge. (Theories of proton decay predict that the proton has a half life on the order of at least 10$^{32}$ years. To date, there is no experimental evidence of proton decay.);
* Because they are stable, are components of all normal atoms, and determine their chemical properties, the proton is the most prevalent baryon, while the electron is the most prevalent lepton;
* The proton mass m$_p$ is composed primarily of gluons, and of the quarks (the up quark and down quark) making up the proton. Hence m$_p$, and therefore the ratio μ, are easily measurable consequences of the strong force. In fact, in the chiral limit, m$_p$ is proportional to the QCD energy scale, Λ$_{QCD}$. At a given energy scale, the strong coupling constant α$_s$ is related to the QCD scale (and thus μ) as
       $\alpha _{s}=-{\frac {2\pi }{\beta _{0}\ln(E/\Lambda _{\rm {QCD}})}}$
    where β$_0$ = −11 + 2n/3, with n being the number of flavors of quarks.

## Variation of μ 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 λ$_i$ of each rest frame wavelength can be parameterised as:
    $\ \lambda _{i}=\lambda _{0}\left[1+K_{i}{\frac {\Delta \mu }{\mu }}\right],$
where Δμ/μ is the proportional change in μ and K$_i$ 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)×10$^{−5}$. 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)×10$^{−6}$, different from the estimates of Reinhold et al. (2006).
Murphy et al. (2008) used the inversion transition of ammonia to conclude that |Δμ/μ| < 1.8×10$^{−6}$ at redshift z = 0.68. Kanekar (2011) used deeper observations of the inversion transitions of ammonia in the same system at z = 0.68 towards 0218+357 to obtain |Δμ/μ| < 3×10$^{−7}$.
Bagdonaite et al. (2013) used methanol transitions in the spiral lensing galaxy PKS 1830-211 to find ∆μ/μ = (0.0 ± 1.0) × 10$^{−7}$ at z = 0.89. Kanekar et al. (2015) used near-simultaneous observations of multiple methanol transitions in the same lens, to find ∆μ/μ < 1.1 × 10$^{−7}$ at z = 0.89. Using three methanol lines with similar frequencies to reduce systematic effects, Kanekar et al. (2015) obtained ∆μ/μ < 4 × 10$^{−7}$.
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.