In physics, a dimensionless physical constant,
sometimes called a fundamental 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. Perhaps
the best-known example is the fine-structure
constant, α, which has an approximate value
of ​1⁄137.036.
The term fundamental physical constant is
normally used to refer to the dimensionless
constants, but has also been used (primarily
by NIST and CODATA) to refer to certain universal
dimensioned physical constants, such as the
speed of light c, vacuum permittivity ε0,
Planck constant h, and the gravitational constant
G, that appear in the most basic theories
of physics. Other physicists do not recognize
this usage, and reserve the use of the term
fundamental physical constant solely for dimensionless
universal physical constants that currently
cannot be derived from any other source. This
narrower usage will be followed here.
== Characteristics ==
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 coupling
constant).
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".
== History ==
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 + ½ × 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)).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.
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.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.
An empirical relation between the masses of
the electron, muon and tau has been discovered
by physicist Yoshio Koide, but this formula
remains unexplained.
== Examples ==
Dimensionless fundamental physical constants
include:
α, the fine structure constant, the coupling
constant for the electromagnetic interaction
(≈​1⁄137). 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). More generally,
the ratio of the rest masses of any pair of
elementary particles.
αs, the coupling constant for the strong
force (≈1)
αG, 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 the ratio of αG to the other coupling
constants has also been used to express the
strength of gravitation relative to the other
interactions.
=== Fine structure constant ===
One of the dimensionless fundamental constants
is the fine structure constant:
α
=
e
2
ℏ
c
4
π
ε
0
≈
1
137.03599908
,
{\displaystyle \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!
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 the masses of two protons,
its value is ≈10−38.
=== 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.
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 mP = 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.
=== 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. 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.
=== 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
Ω
Λ
{\displaystyle \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.
== See also ==
Cabibbo–Kobayashi–Maskawa matrix (Cabibbo
angle)
Coupling constant
Dimensionless numbers in fluid mechanics
Dimensionless quantity
Dirac large numbers hypothesis
Fine-structure constant
Gravitational coupling constant
Neutrino oscillation
Physical cosmology
Standard Model
Weinberg angle
Fine-tuned Universe
Koide formula
