In physics, a dimensionless physical constant is a physical constant that is dimensionless, i.e. a pure number having no units attached and having a numerical value that is independent of whatever system of units may be used.^{[1]}^{: 525 } For example, if one considers one particular airfoil, the Reynolds number value of the laminar–turbulent transition is one relevant dimensionless physical constant of the problem. However, it is strictly related to the particular problem: for example, it is related to the airfoil being considered and also to the type of fluid in which it moves.
On the other hand, the term fundamental physical constant is used to refer to some universal dimensionless constants. Perhaps the best-known example is the fine-structure constant, α, which has an approximate value of 1⁄137.036.^{[2]}^{: 367 } The correct use of the term fundamental physical constant should be restricted to the dimensionless universal physical constants that currently cannot be derived from any other source.^{[3]}^{[4]}^{[5]}^{[6]}^{[7]} This precise definition is the one that will be followed here.
However, the term fundamental physical constant has been sometimes used to refer to certain universal dimensioned physical constants, such as the speed of light c, vacuum permittivity ε_{0}, Planck constant h, and the gravitational constant G, that appear in the most basic theories of physics.^{[8]}^{[9]}^{[10]}^{[11]} NIST^{[8]} and CODATA^{[12]} sometimes used the term in this way in the past.
There is no exhaustive list of such constants but it does make sense to ask about the minimal number of fundamental constants necessary to determine a given physical theory. Thus, the Standard Model requires 25 physical constants, about half of them the masses of fundamental particles (which become "dimensionless" when expressed relative to the Planck mass or, alternatively, as coupling strength with the Higgs field along with the gravitational constant).^{[13]}^{: 58–61 }
Fundamental physical constants cannot be derived and have to be measured. Developments in physics may lead to either a reduction or an extension of their number: discovery of new particles, or new relationships between physical phenomena, would introduce new constants, while the development of a more fundamental theory might allow the derivation of several constants from a more fundamental constant.
A long-sought goal of theoretical physics is to find first principles (theory of everything) from which all of the fundamental dimensionless constants can be calculated and compared to the measured values.
The large number of fundamental constants required in the Standard Model has been regarded as unsatisfactory since the theory's formulation in the 1970s. The desire for a theory that would allow the calculation of particle masses is a core motivation for the search for "Physics beyond the Standard Model".
In the 1920s and 1930s, Arthur Eddington embarked upon extensive mathematical investigation into the relations between the fundamental quantities in basic physical theories, later used as part of his effort to construct an overarching theory unifying quantum mechanics and cosmological physics. For example, he speculated on the potential consequences of the ratio of the electron radius to its mass. Most notably, in a 1929 paper he set out an argument based on the Pauli exclusion principle and the Dirac equation that fixed the value of the reciprocal of the fine-structure constant as 𝛼^{−1} = 16 + 1⁄2 × 16 × (16 − 1) = 136. When its value was discovered to be closer to 137, he changed his argument to match that value. His ideas were not widely accepted, and subsequent experiments have shown that they were wrong (for example, none of the measurements of the fine-structure constant suggest an integer value; in 2018 it was measured at α = 1/137.035999046(27)).^{[14]}
Though his derivations and equations were unfounded, Eddington was the first physicist to recognize the significance of universal dimensionless constants, now considered among the most critical components of major physical theories such as the Standard Model and ΛCDM cosmology.^{[15]}^{: 82 } He was also the first to argue for the importance of the cosmological constant Λ itself, considering it vital for explaining the expansion of the universe, at a time when most physicists (including its discoverer, Albert Einstein) considered it an outright mistake or mathematical artifact and assumed a value of zero: this at least proved prescient, and a significant positive Λ features prominently in ΛCDM.
Eddington may have been the first to attempt in vain to derive the basic dimensionless constants from fundamental theories and equations, but he was certainly not the last. Many others would subsequently undertake similar endeavors, and efforts occasionally continue even today. None have yet produced convincing results or gained wide acceptance among theoretical physicists.^{[16]}^{[17]}
An empirical relation between the masses of the electron, muon and tau has been discovered by physicist Yoshio Koide, but this formula remains unexplained.^{[18]} A recent theoretical derivation of mass ratios of charged elementary fermions explains why the Koide ratio for electron, muon and tau is not exactly 2/3 but close to it.^{[19]}
Dimensionless fundamental physical constants include:
One of the dimensionless fundamental constants is the fine-structure constant:
where e is the elementary charge, ħ is the reduced Planck's constant, c is the speed of light in a vacuum, and ε_{0} 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; Richard Feynman elaborates:
There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!
— Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 978-0-691-08388-9.
A theoretical derivation of the fine structure constant, based on unification in a pre-spacetime, pre-quantum theory in eight octonionic dimensions, has recently been given by Singh.^{[20]} This article derives the following expression
which agrees with the measured value to 2 parts in ten million. The match is claimed to be exact if a so-called Karolyhazy correction is accounted for, and a specific energy scale for the electro-weak symmetry breaking scale is assumed.
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.
The complete standard model requires 25 fundamental dimensionless constants (Baez, 2011). 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:
A possible theoretical explanation for the mass ratios of charged elementary fermions has been provided by Bhatt et al.^{[19]} in the context of a left-right symmetric extension of the standard model, based on the exceptional Jordan algebra of the octonions.
Dimensionless constants of the Standard Model | ||||
---|---|---|---|---|
Symbol | Description | Dimensionless value | Alternative value representation | |
m_{u} / m_{P} | up quark mass | 1.4 × 10^{−22} – 2.7 × 10^{−22} | 1.7–3.3 MeV/c^{2} | |
m_{d} / m_{P} | down quark mass | 3.4 × 10^{−22} – 4.8 × 10^{−22} | 4.1–5.8 MeV/c^{2} | |
m_{c} / m_{P} | charm quark mass | 1.04 × 10^{−19} | 1.27 GeV/c^{2} | |
m_{s} / m_{P} | strange quark mass | 8.27 × 10^{−21} | 101 MeV/c^{2} | |
m_{t} / m_{P} | top quark mass | 1.41 × 10^{−17} | 172.0 GeV/c^{2} | |
m_{b} / m_{P} | bottom quark mass | 3.43 × 10^{−19} | 4.19 GeV/c^{2} | |
θ_{12,CKM} | CKM 12-mixing angle | 0.23 | 13.1° | |
θ_{23,CKM} | CKM 23-mixing angle | 0.042 | 2.4° | |
θ_{13,CKM} | CKM 13-mixing angle | 0.0035 | 0.2° | |
δ_{CKM} | CKM CP-violating Phase | 0.995 | 57° | |
m_{e} / m_{P} | electron mass | 4.18546 × 10^{−23} | 511 keV/c^{2} | |
m_{νe} / m_{P} | electron neutrino mass | below 1.6 × 10^{−28} | below 2 eV/c^{2} | |
m_{μ} / m_{P} | Muon mass | 8.65418 × 10^{−21} | 105.7 MeV/c^{2} | |
m_{νμ} / m_{P} | muon neutrino mass | below 1.6 × 10^{−28} | below 2 eV/c^{2} | |
m_{τ} / m_{P} | tau mass | 1.45535 × 10^{−19} | 1.78 GeV/c^{2} | |
m_{ντ} / m_{P} | tau neutrino mass | below 1.6 × 10^{−28} | below 2 eV/c^{2} | |
θ_{12,PMNS} | PMNS 12-mixing angle | 0.5973±0.0175 | 34.22°±1° | |
θ_{23,PMNS} | PMNS 23-mixing angle | 0.785±0.12 | 45°±7.1° | |
θ_{13,PMNS} | PMNS 13-mixing angle | ≈0.077 | ≈4.4° | |
δ_{PMNS} | PMNS CP-violating Phase | Unknown | ||
α | fine-structure constant | 0.00729735 | 1/137.036 | |
α_{s} | strong coupling constant | ≈1 | ≈1 | |
m_{W±} / m_{P} | W boson mass | (6.5841 ± 0.0012) × 10^{−18} | (80.385 ± 0.015) GeV/c^{2} | |
m_{Z0} / m_{P} | Z boson mass | (7.46888 ± 0.00016) × 10^{−18} | (91.1876 ± 0.002) GeV/c^{2} | |
m_{H} / m_{P} | Higgs boson mass | ≈1.02 × 10^{−17} | (125.09 ± 0.24) GeV/c^{2} |
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}.^{[21]} 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.^{[22]}
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, in his book Just Six Numbers,^{[23]} 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 and ε govern the fundamental interactions of physics. The other constants (D excepted) govern the size, age, and expansion of the universe. These five constants must be estimated empirically. D, on the other hand, is necessarily a nonzero natural number and does not have an uncertainty. Hence most physicists would not deem it a dimensionless physical constant of the sort discussed in this entry.
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.
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