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 = 1836.15267343(11).^[1]

The number in parentheses is the measurement uncertainty on the last two digits, corresponding to a relative standard uncertainty of 6.0×10⁻¹¹.^[1]

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 10³² 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

${\displaystyle \alpha _{s}=-{\frac {2\pi }{\beta _{0}\ln(E/\Lambda _{\rm {QCD)))))}$

where β₀ = −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:

${\displaystyle \ \lambda _{i}=\lambda _{0}\left[1+K_{i}{\frac {\Delta \mu }{\mu ))\right],}$

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 lens galaxy PKS 1830-211 to find ∆μ/μ = (0.0 ± 1.0) × 10⁻⁷ at z = 0.89.^[2][3] 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.