Dimensionless physical constant

In physics, a dimensionless physical constant, sometimes called a fundamental physical constant, is a physical constant that is dimensionless. It has no units attached and has a numerical value that is independent of the system of units used. Perhaps the best-known example is the fine-structure constant, α, which has approximate value of 1⁄137.036.

The term fundamental physical constant has also been used to refer to universal but dimensioned physical constants such as the speed of light c, vacuum permittivity ε₀, Planck's constant h, and the gravitational constant G. However, the numerical values of these constants are not fundamental, since they depend on the units used to express them. Increasingly, physicists reserve the use of the term fundamental physical constant for dimensionless physical constants that cannot be derived from any other source.

Unlike mathematical constants, the values of the dimensionless fundamental physical constants cannot be calculated; they are determined only by physical measurement. This is one of the unsolved problems of physics. A related question is why some of these constants apparently are set within a narrow range that allows for life to emerge in the Universe.

Introduction

The numerical values of dimensional physical constants depend on the units used to express them. So one can define a basis set of units so that selected dimensional physical constants have numerical value 1. The basis set may consist of units of time, length, mass, charge, and temperature, or an equivalent set. A choice of units is called a system of units.

For example, the International System of Units (SI) is the most generally used system of units today. The numerical values of dimensional physical constants are arbitrary and have no natural significance. (An exception is the vacuum permeability constant µ₀, whose numerical value of 4π×10⁻⁷ is a mathematical constant determined by the definition of the ampere in the SI system.)

The Planck units constitute another system of units. It is a system of natural units chosen so that the numerical values of the vacuum speed of light, the universal gravitational constant, and the constants of Planck, Coulomb, and Boltzmann are unity. These five dimensional physical constants then disappear from equations of physical laws, but as this occurs merely from the choice of units, these constants are considered not fundamental in an operationally distinguishable sense.

In contrast, the numerical values of dimensionless physical constants are independent of the units used. These constants cannot be eliminated by any choice of a system of units. Such constants include:

• α, the fine structure constant, the coupling constant for the electromagnetic interaction (≈1/137.036). Also the square of the electron charge, expressed in Planck units, which defines the scale of charge of elementary particles with charge.
• μ or β, the proton-to-electron mass ratio, the rest mass of the proton divided by that of the electron (≈1836.15). More generally, the ratio of the rest masses of any pair of elementary particles.
• α_s, the coupling constant for the strong force (≈1)
• α_G, the gravitational coupling constant (≈10⁻⁴⁵) which is the square of the electron mass, expressed in Planck units. This defines the scale of the masses of elementary particles and has also been used to express the relative strength of gravitation.

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

$\alpha ={\frac {e^{2}}{\hbar c\ 4\pi \varepsilon _{0}}}\approx {\frac {1}{137.03599908}},$

where e is the elementary charge, ħ is the reduced Planck's constant, c is the speed of light in a 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; 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 Feynman, Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 0-691-08388-6.

The analog of the fine structure constant for gravitation is the gravitational coupling constant. This constant requires the arbitrary choice of a pair of objects having mass. The electron and proton are natural choices because they are stable, and their properties are well measured and well understood. If α_G is calculated from the masses of two protons, its value is ≈10⁻³⁸.

The list of dimensionless physical constants increases in length whenever experiments measure new relationships between physical phenomena. The list of fundamental dimensionless constants, however, decreases when advances in physics show how some previously known constant can be computed in terms of others. A long-sought goal of theoretical physics is to find first principles from which all of the fundamental dimensionless constants can be calculated and compared to the measured values. A successful "Theory of Everything" would allow such a calculation, but so far, this goal has remained elusive.

Constants in the standard model and in cosmology

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:

• 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 CKM matrix, describing how quarks oscillate between different forms;
• four parameters of the Pontecorvo–Maki–Nakagawa–Sakata matrix, which does the same thing for neutrinos.

Dimensionless constants of the Standard Model
Symbol	Description	Dimensionless value	Alternative value representation
mu / mP	Up quark mass	1.4×10−22 – 2.7×10−22	1.7 – 3.3 MeV/c2
md / mP	Down quark mass	3.4×10−22 – 4.8×10−22	4.1 – 5.8 MeV/c2
mc / mP	Charm quark mass	1.04×10−19	1.27 GeV/c2
ms / mP	Strange quark mass	8.27×10−21	101 MeV/c2
mt / mP	Top quark mass	1.41×10−17	172.0 GeV/c2
mb / mP	Bottom quark mass	3.43×10−19	4.19 GeV/c2
θ12,CKM	CKM 12-mixing angle	0.23	13.1°
θ23,CKM	CKM 23-mixing angle	0.042	2.4°
θ13,CKM	CKM 13-mixing angle	0.0035	0.2°
δCKM	CKM CP-violating Phase	0.995	57°
me / mP	Electron mass	4.18546×10−23	511 keV/c2
mνe / mP	Electron neutrino mass	below 1.6×10−28	below 2 eV/c2
mμ / mP	Muon mass	8.65418×10−21	105.7 MeV/c2
mνμ / mP	Muon neutrino mass	below 1.6×10−28	below 2 eV/c2
mτ / mP	Tau mass	1.45535×10−19	1.78 GeV/c2
mντ / mP	Tau neutrino mass	below 1.6×10−28	below 2 eV/c2
θ12,PMNS	PMNS 12-mixing angle	0.5973±0.0175	34.22±1°
θ23,PMNS	PMNS 23-mixing angle	0.785±0.12	45±7.1°
θ13,PMNS	PMNS 13-mixing angle	≈0.077	≈4.4°
δPMNS	PMNS CP-violating Phase	Unknown
α	fine structure constant	0.00729735	1/137.036
αs	strong coupling constant	≈1	≈1
mW± / mP	W boson mass	6.5841±0.0012×10−18	80.385±0.015 GeV/c2
mZ0 / mP	Z boson mass	7.46888±0.00016×10−18	91.1876±0.002 GeV/c2
mH / mP	Higgs boson mass	≈1.02×10−17	125.09±0.24 GeV/c2

One constant is required for cosmology:

• the cosmological constant (in terms of Planck units) of Einstein's equations for general relativity, having a value of approximately 10⁻¹²².

Thus, currently there are 26 known fundamental dimensionless physical constants. However, this number may not be the final one. For example, if neutrinos turn out to be Majorana fermions, the Maki-Nakagawa-Sakata matrix has two additional parameters. Secondly, if dark matter is discovered, or if the description of dark energy requires more than the cosmological constant, further fundamental constants will be needed.

Well-known subsets

Certain dimensionless constants are discussed more frequently than others.

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, 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 fine structure constant (the dimensionless coupling constant for electromagnetism) to the gravitational coupling constant, the latter defined using two protons. In Barrow and Tipler (1986) and elsewhere in Wikipedia, this ratio is denoted α/α_G. N governs the relative importance of gravity and electrostatic attraction/repulsion in explaining the properties of baryonic matter;
• 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;
• 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 will experience a Big Crunch. If Ω < 1, the universe will expand forever;
• 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 $\Omega _{\Lambda }$;
• 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;
• D = 3: the number of macroscopic spatial dimensions.

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 cannot be measured. 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.

Calculation attempts

No formulae for the fundamental physical constants are known to this day.

The mathematician Simon Plouffe has made an extensive search of computer databases of mathematical formulae, seeking formulae for the mass ratios of the fundamental particles.

One example of numerology is by the astrophysicist Arthur Eddington. He set out alleged mathematical reasons why the reciprocal of the fine structure constant had to be exactly 136. When its value was discovered to be closer to 137, he changed his argument to match that value. Experiments have since shown that Eddington was wrong; to six significant digits, the reciprocal of the fine-structure constant is 137.036.

An empirical relation between the masses of the electron, muon and tau has been discovered by physicist Yoshio Koide, but this formula remains unexplained.

See also

• Cabibbo–Kobayashi–Maskawa matrix (Cabibbo angle)
• Coupling constant
• Dimensionless quantity
• Fine-structure constant
• Gravitational coupling constant
• Neutrino oscillation
• Physical cosmology
• Standard Model
• Weinberg angle
• Fine-tuned Universe
• Koide formula

Bibliography

• Martin Rees, 1999. Just Six Numbers: The Deep Forces that Shape the Universe. London: Weidenfeld & Nicolson. ISBN 0-7538-1022-0

External articles

General

• James R Johnson, 2015, "Discovering Nature's Hidden Relationships, an Unattainable Goal?"
• John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.
• Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle (1st ed.). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.
• Michio Kaku, 1994. Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford University Press.
• Fundamental Physical Constants from NIST
• Values of fundamental constants. CODATA, 2002.
• John Baez, 2002, "How Many Fundamental Constants Are There?"
• Plouffe. Simon, 2004, "A search for a mathematical expression for mass ratios using a large database."

Articles on variance of the fundamental constants

• John Bahcall, Charles Steinhardt, and David Schlegel, 2004, "Does the fine-structure constant vary with cosmological epoch?" Astrophys. J. 600: 520.
• John D. Barrow and Webb, J. K., "Inconstant Constants – Do the inner workings of nature change with time?" Scientific American (June 2005).
• Michael Duff, 2002 "Comment on time-variation of fundamental constants."
• Marion, H., et al. 2003, "A search for variations of fundamental constants using atomic fountain clocks," Phys.Rev.Lett. 90: 150801.
• Martins, J.A.P. et al., 2004, "WMAP constraints on varying α and the promise of reionization," Phys.Lett. B585: 29–34.
• Olive, K.A., et al., 2002, "Constraints on the variations of the fundamental couplings," Phys.Rev. D66: 045022.
• Uzan, J-P, 2003, "The fundamental constants and their variation: observational status and theoretical motivations," Rev.Mod.Phys. 75: 403.
• Webb, J.K. et al., 2001, "Further evidence for cosmological evolution of the fine-structure constant," Phys. Rev. Lett. 87: 091301.