Editing Dimensionless physical constant (section)

== Examples ==
Dimensionless fundamental physical constants include:
* ''α'', the [[fine-structure constant]], (≈&nbsp;{{sfrac|1|137}}). This is also the square of the [[elementary charge|electron charge]], expressed in [[Planck units]], which defines the scale of charge of [[elementary particle]]s with charge. The electron charge is the [[coupling constant]] for the [[electromagnetic interaction]].
* ''μ'' or ''β'', the [[proton-to-electron mass ratio]] (≈&nbsp;{{physconst|mp/me|round=0|ref=no}}), the [[rest mass]] of the [[proton]] divided by that of the [[electron]]. More generally, the ratio of the [[rest mass]]es of any pair of [[elementary particle]]s.
* ''α''<sub>s</sub>, the [[coupling constant]] for the [[strong force]] (≈&nbsp;1)

=== Fine-structure constant ===
One of the dimensionless fundamental constants is the [[fine-structure constant]]:
: <math> \alpha = \frac{e^2}{4 \pi \varepsilon_0 \ \hbar c}= \frac{e^2}{2 \varepsilon_0 h c} = </math> {{physconst|alpha|ref=no}},
where ''e'' is the [[elementary charge]], ''ħ'' is the reduced [[Planck constant]], ''c'' is the [[speed of light]] in vacuum, and ''ε''<sub>0</sub> is the [[permittivity of free space]]. The fine-structure constant is fixed to the strength of the [[electromagnetic force]]. At low energies, ''α'' ≈ {{sfrac|1|137}}, whereas at the scale of the [[Z boson]], about {{val|90|ul=GeV}}, one measures ''α'' ≈ {{sfrac|1|127}}. There is no accepted theory explaining the value of ''α''; [[Richard Feynman]] elaborates:

{{quote
 | There is a most profound and beautiful question associated with the observed coupling constant, ''e''{{snd}} 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!
 | {{Cite book
    |author=Richard P. Feynman
    |author-link=Richard Feynman
    |year=1985
    |title=QED: The Strange Theory of Light and Matter
    |publisher=[[Princeton University Press]]
    |page=129
    |isbn=978-0-691-08388-9
    |title-link=QED: The Strange Theory of Light and Matter
   }}
}}

=== Standard Model ===
The original [[Standard Model]] of [[particle physics]] from the 1970s contained 19 fundamental dimensionless constants describing the [[mass]]es of the particles and the strengths of the [[electroweak]] and [[strong force]]s. In the 1990s, [[neutrino]]s were discovered to have nonzero mass, and a quantity called the [[theta vacuum|vacuum angle]] was found to be indistinguishable from zero.<ref>Quint, W., & M. Vogel, ''Fundamental Physics in Particle Traps'' (Berlin/Heidelberg: Springer, 2014), [https://books.google.com/books?id=NJ25BQAAQBAJ&newbks=1&newbks_redir=0&lpg=PP1&hl=cs&pg=PA292&redir_esc=y#v=onepage&q&f=false pp. 293–296].</ref>{{rp|293–296}}

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 [[coupling constant|strong coupling constant]];
* fifteen [[mass]]es of the [[fundamental particle]]s (relative to the [[Planck mass]] ''m''<sub>P</sub> = {{val|1.22089|(6)|e=19|u=GeV/c2}}), namely:
** six [[quark]]s
** six [[lepton]]s
** the [[Higgs boson]]
** the [[W boson]]
** the [[Z boson]]
* four parameters of the [[Cabibbo–Kobayashi–Maskawa matrix]], describing how [[quark]]s oscillate between different forms;
* four parameters of the [[Pontecorvo–Maki–Nakagawa–Sakata matrix]], which does the same thing for [[neutrino]]s.

{| class="wikitable collapsible collapsed"
! colspan="4" | Dimensionless constants of the Standard Model
|-
! Symbol
! Description
! Dimensionless value
! Alternative value representation
|-
| ''m''<sub>u</sub> / ''m''<sub>P</sub>
| [[Up quark|up quark mass]]
| {{val|1.4e-22}} – {{val|2.7e-22}}
| 1.7–3.3 MeV/''c''<sup>2</sup>
|-
| ''m''<sub>d</sub> / ''m''<sub>P</sub>
| [[Down quark|down quark mass]]
| {{val|3.4e-22}} – {{val|4.8e-22}}
| 4.1–5.8 MeV/''c''<sup>2</sup>
|-
| ''m''<sub>c</sub> / ''m''<sub>P</sub>
| [[Charm quark|charm quark mass]]
| {{val|1.04431|0.0204768|0.0286675|e=-19}}
| {{Val|1.275|0.025|0.035|u=GeV/''c''<sup>2</sup>}}
|-
| ''m''<sub>s</sub> / ''m''<sub>P</sub>
| [[Strange quark|strange quark mass]]
| {{val|8.27e-21}}
| {{Val|95|9|3|u=MeV/''c''<sup>2</sup>}}
|-
| ''m''<sub>t</sub> / ''m''<sub>P</sub>
| [[Top quark|top quark mass]]
| {{val|1.415|0.00245721|e=-17}}
| {{val|172.76|0.3|u=GeV/''c''<sup>2</sup>}}
|-
| ''m''<sub>b</sub> / ''m''<sub>P</sub>
| [[Bottom quark|bottom quark mass]]
| {{val|3.43e-19}}
| 4.19 GeV/''c''<sup>2</sup>
|-
| ''θ''<sub>12,CKM</sub>
| [[Cabibbo–Kobayashi–Maskawa matrix|CKM 12-mixing angle]]
| {{val|0.22759|0.000873}}
| {{val|13.04|0.05|u=°}}
|-
| ''θ''<sub>23,CKM</sub>
| [[Cabibbo–Kobayashi–Maskawa matrix|CKM 23-mixing angle]]
| {{val|0.04154|0.00105}}
| {{val|2.38|0.06|u=°}}
|-
| ''θ''<sub>13,CKM</sub>
| [[Cabibbo–Kobayashi–Maskawa matrix|CKM 13-mixing angle]]
| {{val|0.003508|0.000192}}
| {{val|0.201|0.011|u=°}}
|-
| ''δ''<sub>13,CKM</sub>
| [[Cabibbo–Kobayashi–Maskawa matrix|CKM]] [[CP violation|CP-violating phase]]
| {{val|1.201|0.0785}}
| {{val|68.8|4.5|u=°}}
|-
| ''m''<sub>e</sub> / ''m''<sub>P</sub>
| electron mass
| {{val|4.18546e-23}}
| {{physconst|mec2_MeV|round=3|ref=no}}/''c''<sup>2</sup>
|-
| ''m''<sub>ν<sub>e</sub></sub> / ''m''<sub>P</sub>
| electron neutrino mass
| below {{val|9e-30}}
| below 0.11 eV/''c''<sup>2</sup>
|-
| ''m''<sub>μ</sub> / ''m''<sub>P</sub>
| muon mass
| {{val|8.65418e-21}}
| {{physconst|mmuc2_MeV|round=1|ref=no}}/''c''<sup>2</sup>
|-
| ''m''<sub>ν<sub>μ</sub></sub> / ''m''<sub>P</sub>
| muon neutrino mass
| below {{val|1.6e-28}}
| below 2 eV/''c''<sup>2</sup>
|-
| ''m''<sub>τ</sub> / ''m''<sub>P</sub>
| tau mass
| {{val|1.45535e-19}}
| 1.78 GeV/''c''<sup>2</sup>
|-
| ''m''<sub>ν<sub>τ</sub></sub> / ''m''<sub>P</sub>
| tau neutrino mass
| below {{val|1.6e-28}}
| below 2 eV/''c''<sup>2</sup>
|-
| ''θ''<sub>12,PMNS</sub>
| [[Pontecorvo–Maki–Nakagawa–Sakata matrix|PMNS 12-mixing angle]]
| {{val|0.58364|0.0122}}
| {{val|33.44|0.77|0.74|u=°}}
|-
| ''θ''<sub>23,PMNS</sub>
| [[Pontecorvo–Maki–Nakagawa–Sakata matrix|PMNS 23-mixing angle]]
| {{val|0.8587|0.0175|0.0227}}
| {{val|49.2|1.0|1.3|u=°}}
|-
| ''θ''<sub>13,PMNS</sub>
| [[Pontecorvo–Maki–Nakagawa–Sakata matrix|PMNS 13-mixing angle]]
| {{val|0.1496|0.00227|0.00209}}
| {{val|8.57|0.13|0.12|u=°}}
|-
| ''δ''<sub>Cp,PMNS</sub>
| [[Pontecorvo–Maki–Nakagawa–Sakata matrix|PMNS]] [[CP violation|CP-violating phase]]
| 2.95 ≤ ''δ'' ≤ 4.294
| 169° ≤ ''δ'' ≤ 246°
|-
| ''α''
| [[fine-structure constant]]
| {{physconst|alpha|round=8|ref=no}}
| 1 / {{physconst|alphainv|round=3|ref=no}}
|-
| ''α''<sub>s</sub>
| [[Coupling constant|strong coupling constant]]
| ≈ 1
| ≈ 1
|-
| ''m''<sub>W<sup>±</sup></sub> / ''m''<sub>P</sub>
| W boson mass
| {{val|6.5841|0.0012|e=-18}}
| {{val|80.385|0.015|u=GeV/''c''<sup>2</sup>}}
|-
| ''m''<sub>Z<sup>0</sup></sub> / ''m''<sub>P</sub>
| Z boson mass
| {{val|7.46888|0.00016|e=-18}}
| {{val|91.1876|0.002|u=GeV/''c''<sup>2</sup>}}
|-
| ''m''<sub>H</sub> / ''m''<sub>P</sub>
| Higgs boson mass
| ≈ {{val|1.02e-17}}
| {{val|125.09|0.24|u=GeV/''c''<sup>2</sup>}}
|}

=== 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<sup>−122</sup>.<ref>[[Robert Jaffe (physicist)|Jaffe, R. L.]], & Taylor, W., ''The Physics of Energy'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=drZDDwAAQBAJ&pg=PA419&redir_esc=y#v=onepage&q&f=false p. 419].</ref> 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.<ref name="Tegmark2014">{{cite book |first=Max |last=Tegmark |date=2014 |title=Our Mathematical Universe: My Quest for the Ultimate Nature of Reality |publisher=Knopf Doubleday Publishing Group |isbn=9780307599803 |page=[https://archive.org/details/ourmathematicalu0000tegm/page/252 252] |title-link=Our Mathematical Universe: My Quest for the Ultimate Nature of Reality }}</ref>

=== 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 constant]]s for the [[strong force]] and [[gravitation]].

=== Martin Rees's 'six numbers' ===
[[Martin Rees, Baron Rees of Ludlow|Martin Rees]], in his book ''Just Six Numbers'',<ref>Radford, T., [https://www.theguardian.com/science/2012/jun/08/just-six-numbers-martin-rees-review "''Just Six Numbers: The Deep Forces that Shape the Universe'' by Martin Rees—review"], ''[[The Guardian]]'', 8 June 2012.</ref> 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'' ≈ 10<sup>36</sup>: the ratio of the electrostatic and the gravitational forces between two [[proton]]s.  This ratio is denoted ''α''/''α''<sub>G</sub> in Barrow and Tipler (1986). ''N'' governs the relative importance of gravity and electrostatic attraction/repulsion in explaining the properties of [[baryonic matter]];<ref name="Rees, M. 2000, p">Rees, M. (2000)</ref>
* ''ε'' ≈ 0.007: The fraction of the mass of four [[proton]]s that is released as energy when [[nuclear fusion|fused]] into a [[helium]] nucleus. ''ε'' governs the [[Proton–proton chain reaction#Energy release|energy output of stars]], and is determined by the [[coupling constant]] for the [[strong force]];<ref>Rees, M. (2000), p. 53.</ref>
* Ω ≈ 0.3: the [[Friedmann equations#Density parameter|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 {{nowrap|Ω ≥ 1}}, the universe may experience a [[Big Crunch]]. If {{nowrap|Ω < 1}}, the universe may expand forever;<ref name="Rees, M. 2000, p"/>
* ''λ'' ≈ 0.7: The ratio of the energy density of the universe, due to the [[cosmological constant]], to the [[Critical density (cosmology)|critical density]] of the universe. Others denote this ratio by <math>\Omega_{\Lambda}</math>;<ref>Rees, M. (2000), p. 110.</ref>
* ''Q'' ≈ 10<sup>−5</sup>: 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 ''mc''<sup>2</sup>;<ref>Rees, M. (2000), p. 118.</ref>
* ''D'' = 3: the number of macroscopic spatial [[dimension]]s.

''N'' and ''ε'' govern the [[fundamental interaction]]s of physics. The other constants (''D'' excepted) govern the [[size of the universe|size]], [[age of the universe|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.