Dimensionless physical constant

Fine-structure constant

One of the dimensionless fundamental constants is the fine-structure constant:

${\displaystyle \alpha ={\frac {e^{2}}{4\pi \varepsilon _{0}\ \hbar c}}={\frac {e^{2}}{2\varepsilon _{0}hc}}=}$ 0.0072973525643(11),

where e is the elementary charge, ħ is the reduced Planck constant, c is the speed of light in vacuum, and ε₀ is the permittivity of free space. The fine-structure constant is fixed to the strength of the electromagnetic force. At low energies, α ≈ ⁠1/137⁠, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ ⁠1/127⁠. There is no accepted theory explaining the value of α.^[11]

Standard Model

The original Standard Model of particle physics from the 1970s contained 19 fundamental dimensionless constants describing the masses of the particles and the strengths of the electroweak and strong forces. In the 1990s, neutrinos were discovered to have nonzero mass, and a quantity called the vacuum angle was found to be indistinguishable from zero.^{[12]: 293–296}

The complete Standard Model requires 25 fundamental dimensionless constants.^[4] At present, their numerical values are not understood in terms of any widely accepted theory and are determined only from measurement. These 25 constants are:

• the fine structure constant;
• the strong coupling constant;
• fifteen masses of the fundamental particles (relative to the Planck mass m_P = 1.22089(6)×10¹⁹ GeV/c²), namely:
  • six quarks
  • six leptons
  • the Higgs boson
  • the W boson
  • the Z boson
• four parameters of the Cabibbo–Kobayashi–Maskawa matrix, describing how quarks oscillate between different forms;
• four parameters of the Pontecorvo–Maki–Nakagawa–Sakata matrix, which does the same thing for neutrinos.

Gravitational coupling constant

A gravitational coupling constant is a constant characterizing the gravitational attraction between a given pair of elementary particles. The electron mass is typically used, and the associated constant typically denoted α_G. It is a dimensionless quantity, with the result that its numerical value does not vary with the choice of units of measurement, only with the choice of particle.

The gravitational coupling constant, α_G, can be defined in terms of the gravitational attraction between two elementary particles:^[13]

$${\displaystyle \alpha _{\mathrm {G} }={\frac {Gm_{\mathrm {p} }^{2}}{\hbar c}}}$$

where:

• G is the gravitational constant;
• m_p is the particle rest mass;
• c is the speed of light in vacuum;
• ħ is the reduced Planck constant.

For a proton the value of the dimensionless constant is approximately 5×10⁻³⁹.^{[13][14]: 25}

In natural units where c = ħ = 1, the expression becomes^[15]: 1153 $${\displaystyle \alpha _{\mathrm {G} }=Gm_{\mathrm {p} }^{2}\ .}$$

In 1961, a dimensionless gravitational coupling constant was used by Robert H. Dicke.^[13] Dicke's work has been described as perhaps the first explicit discussion of the weak anthropic principle. This principle has various forms. The basic idea is that the constants in physical laws are limited to values which can give rise to physicists. Dicke reasoned that the gravitational coupling must be an extremely small number to ensure that expansion of the universe could continue against the force of gravity caused by all the matter in the universe.^[16]: 7 Dicke's analysis countered an earlier hypothesis by Paul Dirac, suggesting that the gravitational coupling constant depends on cosmic density. In a Big Bang model of the universe, the density varies strongly with time. Dicke noted that sufficient time must elapse for stars to form and this requires a very small gravitational coupling constant.^[15]: 1150

Cosmological constants

The cosmological constant, which can be thought of as the density of dark energy in the universe, is a fundamental constant in physical cosmology that has a dimensionless value of approximately 10⁻¹²².^[17] Other dimensionless constants are the measure of homogeneity in the universe, denoted by Q, which is explained below by Martin Rees, the baryon mass per photon, the cold dark matter mass per photon and the neutrino mass per photon.^[18]

Barrow and Tipler

Barrow and Tipler (1986) anchor their broad-ranging discussion of astrophysics, cosmology, quantum physics, teleology, and the anthropic principle in the fine-structure constant, the proton-to-electron mass ratio (which they, along with Barrow (2002), call β), and the coupling constants for the strong force and gravitation.

Martin Rees's 'six numbers'

Martin Rees, in his book Just Six Numbers,^[19] mulls over the following six dimensionless constants, whose values he deems fundamental to present-day physical theory and the known structure of the universe:

• N ≈ 10³⁶: the ratio of the electrostatic and the gravitational forces between two protons. This ratio is denoted α/α_G in Barrow and Tipler (1986). N governs the relative importance of gravity and electrostatic attraction/repulsion in explaining the properties of baryonic matter;^[20]
• ε ≈ 0.007: The fraction of the mass of four protons that is released as energy when fused into a helium nucleus. ε governs the energy output of stars, and is determined by the coupling constant for the strong force;^[20]: 53
• Ω ≈ 0.3: the ratio of the actual density of the universe to the critical (minimum) density required for the universe to eventually collapse under its gravity. Ω determines the ultimate fate of the universe. If Ω ≥ 1, the universe may experience a Big Crunch. If Ω < 1, the universe may expand forever.^[20]
• λ ≈ 0.7: The ratio of the energy density of the universe, due to the cosmological constant, to the critical density of the universe. Others denote this ratio by ${\displaystyle \Omega _{\Lambda }}$;^[20]: 110
• Q ≈ 10⁻⁵: The energy required to break up and disperse an instance of the largest known structures in the universe, namely a galactic cluster or supercluster, expressed as a fraction of the energy equivalent to the rest mass m of that structure, namely mc²;^[20]: 118
• D = 3: the number of macroscopic spatial dimensions.

Any plausible fundamental physical theory must be consistent with these six constants, and must either derive their values from the mathematics of the theory, or accept their values as empirical.