# Dimensionless physical constant

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Dimensionless physical constant

In physics, a dimensionless physical constant (sometimes fundamental physical constant) is a universal physical constant that is dimensionless - having no unit attached, so its numerical value is the same under all possible systems of units. The best known example is the fine structure constant α, with the approximate value 1/137.036.

However, the term fundamental physical constant has also been used (as by NIST) to refer to universal but dimensional physical constants such as the speed of light c, vacuum permittivity ε0, Planck's constant h, or the gravitational constant G.

## Introduction

The numerical values of dimensional physical constants depends on the units used. The reason is that the numerical values of a selected basis set of dimensional physical constants can be normalized to 1 by a choice of units. The basis set consists of time, length, mass, charge, and temperature, or an equivalent set. A choice of units is called a system of units. The SI, the international system of units, is such a system of units. As another example, one system of units appears when the numerical values of the speed of light in a vacuum, the universal gravitational constant, and the constants of Planck, Coulomb, and Boltzmann, are all set to 1; this system of units is called the system of natural units or Planck units.

In contrast, the numerical values of dimensionless physical constants are independent on the units used. Such constants include:

At the present time, the values of the dimensionless physical constants cannot be calculated; they are determined only by physical measurement. This is one of the unsolved problems of physics.

The best known of the dimensionless 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 determines the strength of the electromagnetic force. Note that 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.

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 αG is calculated from 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. The reduction of chemistry to physics was a big step in this direction, since the properties of atoms and molecules can now be calculated from the Standard Model. 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 theta angle was found to be indistinguishable from zero.

The complete standard model requires 25 fundamental dimensionless constants (Baez, 2002). Their numerical values are, at present, not understood. These 25 constants are:

• the fine structure constant;
• the strong coupling constant;
• the masses of the fundamental particles (relative to the Planck mass), namely the six quarks, the six leptons, the Higgs boson, the W boson, and the Z boson;
• four parameters of the CKM matrix, describing how quarks oscillate between different forms;
• four parameters of the Maki-Nakagawa-Sakata matrix, which does the same thing for neutrinos.

One constant is required for cosmology:

Thus, currently there are 26 known fundamental dimensionless physical constants. However, this number may not be the final one. First, one of the mentioned constants, the Higgs boson mass, is unknown, as the Higgs boson has not yet been discovered. Secondly, if neutrinos turn out to be Majorana fermions, the Maki-Nakagawa-Sakata matrix has two additional parameters. Finally, 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/αG. 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 ΩΛ;
• 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. There are also compelling physical and mathematical reasons why D = 3.

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.

## Variation of the constants

The question whether the fundamental dimensionless constants depend on space and time is being extensively researched. Despite several claims, no confirmed variation of the constants has been detected.[citation needed]

## 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. However, such studies of fundamental constants often drift into numerology.

One well-known 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.

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