View source for Dimensionless physical constant

{{Short description|Physical constant with no units}}
{{Use dmy dates|date=October 2024}}
In physics, a '''dimensionless physical constant''' is a [[physical constant]] that is [[dimensionless quantity|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.<ref>Stroke, H. H., ed., ''The Physical Review: The First Hundred Years'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 1995), [https://books.google.com/books?id=3U2HSMHsouMC&pg=PA525&redir_esc=y#v=onepage&q&f=false p. 525].</ref>

The concept should not be confused with ''[[dimensionless number]]s'', that are not universally constant, and remain constant only for a particular phenomenon. In [[aerodynamics]] for example, if one considers one particular [[airfoil]], the [[Reynolds number]] value of the [[laminar–turbulent transition]] is one relevant dimensionless number 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.

The term '''fundamental physical constant''' is sometimes used to refer to some {{em|universal}} dimensionless constants. Perhaps the best-known example is the [[fine-structure constant]], ''α'', which has an approximate value of {{sfrac|1|{{physconst|alphainv|round=3|ref=no}}}}.<ref>Vértes, A., Nagy, S., Klencsár, Z., Lovas, R. G., & Rösch, F., eds., ''Handbook of Nuclear Chemistry'', (Berlin/Heidelberg: Springer, 2011), [https://books.google.com/books?id=NQyF6KaUScQC&pg=PA367&redir_esc=y#v=onepage&q&f=false p. 367].</ref>

== Terminology ==
It has been argued the term ''fundamental physical constant'' should be restricted to the dimensionless universal physical constants that currently cannot be derived from any other source;<ref>{{Cite web|url=http://math.ucr.edu/home/baez/constants.html|title=How Many Fundamental Constants Are There?|last=Baez|first=John|date=22 April 2011|website=math.ucr.edu|access-date=2018-04-13}}</ref><ref>{{Cite arXiv|last=Rich|first=James|date=2 April 2013|title=Dimensionless constants and cosmological measurements|eprint=1304.0577|class=astro-ph.CO}}</ref><ref name="hep-th1412.2040">{{cite journal|author=Michael Duff |title=How fundamental are fundamental constants?|journal=Contemporary Physics |arxiv=1412.2040|year=2014|volume=56 |issue=1 |pages=35–47 |doi=10.1080/00107514.2014.980093|bibcode=2015ConPh..56...35D |s2cid=118347723 |author-link=Michael Duff (physicist)}}</ref><ref>{{cite arXiv |last1=Duff |first1=M. J. |date=13 August 2002 |title=Comment on time-variation of fundamental constants |eprint=hep-th/0208093}}</ref><ref>{{cite journal |last1=Duff |first1=M. J. |last2=Okun |first2=L. B. |last3=Veneziano |first3=G. |title=Trialogue on the number of fundamental constants |journal=[[Journal of High Energy Physics]] |date=2002 |volume=2002 |issue= 3|pages=023 |arxiv=physics/0110060 |bibcode=2002JHEP...03..023D |doi=10.1088/1126-6708/2002/03/023|s2cid=15806354 }}</ref> this stricter definition is followed here.

However, the term ''fundamental physical constant'' has also been used occasionally to refer to certain universal dimensioned [[physical constant]]s, such as the [[speed of light]] ''c'', [[vacuum permittivity]] ''ε''<sub>0</sub>, [[Planck constant]] ''h'', and the [[Newtonian constant of gravitation]] ''G'', that appear in the most basic theories of physics.<ref name=":0" /><ref>http://physics.nist.gov/cuu/Constants/ NIST</ref><ref>{{Cite encyclopedia|url=https://www.britannica.com/science/physical-constant|title=Physical constant|encyclopedia=Encyclopedia Britannica|access-date=2018-04-13|language=en}}</ref><ref>{{Cite journal|last=Karshenboim|first=Savely G.|date=August 2005|title=Fundamental Physical Constants: Looking from Different Angles|journal=[[Canadian Journal of Physics]]|volume=83|issue=8|pages=767–811|doi=10.1139/p05-047|issn=0008-4204|arxiv=physics/0506173|bibcode=2005CaJPh..83..767K|s2cid=475086}}</ref> [[National Institute of Standards and Technology|NIST]]<ref name=":0">{{Cite web|url=https://physics.nist.gov/cuu/Constants/introduction.html|title=Introduction to the Fundamental Physical Constants|website=physics.nist.gov|access-date=2018-04-13}}</ref> and [[Committee on Data for Science and Technology|CODATA]]<ref>{{Cite journal|last1=Mohr|first1=Peter J.|last2=Newell|first2=David B.|last3=Taylor|first3=Barry N.|date=26 September 2016|title=CODATA Recommended Values of the Fundamental Physical Constants: 2014|journal=Reviews of Modern Physics|volume=88|issue=3|pages=035009|doi=10.1103/RevModPhys.88.035009|issn=0034-6861|arxiv=1507.07956|bibcode=2016RvMP...88c5009M|s2cid=1115862}}
</ref> sometimes used the term in this less strict manner.

== 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 are the [[mass]]es of [[fundamental particle]]s, which become "dimensionless" when expressed relative to the [[Planck mass]] or, alternatively, as coupling strength with the Higgs field along with the [[gravitational constant]].<ref>Kuntz, I., ''Gravitational Theories Beyond General Relativity'', (Berlin/Heidelberg: Springer, 2019), [https://books.google.com/books?id=xrWZDwAAQBAJ&pg=PA58&redir_esc=y#v=onepage&q&f=false pp. 58–61].</ref>

Fundamental physical constants cannot be derived and have to be [[metrology|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 [[Quantum cosmology|overarching theory unifying quantum mechanics and cosmological physics]]. For example, he speculated on the potential consequences of the ratio of the [[Classical electron radius|electron radius]] to its [[Electron rest mass|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<sup>−1</sup> = 16 + {{sfrac|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; the modern [[CODATA]] value is {{physconst|alphainv|symbol=yes|after=.}}

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 [[Lambda-CDM model|ΛCDM cosmology]].<ref>[[:fr:Dina K. Prialnik|Prialnik, D. K.]], ''An Introduction to the Theory of Stellar Structure and Evolution'' ([[Cambridge]]: [[Cambridge University Press]], 2000), [https://books.google.com/books?id=TGyzlVbgkiMC&pg=PA82&redir_esc=y#v=onepage&q&f=false p. 82].</ref> 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.<ref>{{Cite arXiv|last=Kragh|first=Helge|author-link=Helge Kragh|date=14 October 2015|title=On Arthur Eddington's Theory of Everything|eprint=1510.04046|class=physics.hist-ph}}</ref><ref>{{Cite journal|last=Gamow|first=G.|date=1 February 1968|title=Numerology of the Constants of Nature|journal=Proceedings of the National Academy of Sciences| language=en|volume=59| issue=2|pages=313–318| doi=10.1073/pnas.59.2.313| issn=0027-8424|pmid=16591598| pmc=224670|bibcode=1968PNAS...59..313G|doi-access=free}}</ref>

An [[Koide formula|empirical relation]] between the masses of the electron, muon and tau has been discovered by physicist [[Yoshio Koide]], but this formula remains unexplained.<ref>{{cite arXiv |last1=Rivero |first1=A. |last2=Gsponer |first2=A. |title=The strange formula of Dr. Koide |date=2 February 2008 |page=4 |eprint=hep-ph/0505220 }}</ref>

== 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.{{cn|date=June 2022}}

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.

== See also ==
* [[Cabibbo–Kobayashi–Maskawa matrix]] ([[Cabibbo angle]])
* [[Dimensionless numbers in fluid mechanics]]
* [[Dirac large numbers hypothesis]]
* [[Neutrino oscillation]]
* [[Physical cosmology]]
* [[Standard Model]]
* [[Weinberg angle]]
* [[Fine-tuned universe]]
* [[Koide formula]]

== References ==
{{reflist|30em}}

== Bibliography ==
* [[Martin Rees]], 1999. ''Just Six Numbers: The Deep Forces that Shape the Universe''. London: [[Weidenfeld & Nicolson]]. {{ISBN|0-7538-1022-0}}
* Josef Kuneš, 2012. [https://books.google.com/books?id=_jqUZIUXZBsC ''Dimensionless Physical Quantities in Science and Engineering'']. [[Amsterdam]]: [[Elsevier]]. {{ISBN|978-0-12-416013-2}}

== External articles ==
; General :
* {{citation |author=John Barrow |author-link=John D. Barrow |year=2002 |title=The Constants of Nature; From Alpha to Omega{{snd}} The Numbers that Encode the Deepest Secrets of the Universe |publisher=Pantheon Books |isbn=0-375-42221-8 }}
* {{BarrowTipler1986}}
* {{citation |author=Michio Kaku |author-link=Michio Kaku |year=1994 |title=Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension |title-link=Hyperspace (book) |publisher=[[Oxford University Press]] |bibcode=1994hsot.book.....K }}
* [https://physics.nist.gov/cuu/Constants/ Fundamental Physical Constants from NIST]
* [https://physics.nist.gov/cuu/Constants/Table/allascii.txt Values of fundamental constants.] [[CODATA]], 2002.
* [[John Baez]], 2002, "How Many Fundamental Constants Are There?"
* Simon Plouffe, 2004, "A search for a mathematical expression for mass ratios using a large database. {{Webarchive|url=https://web.archive.org/web/20070219210717/http://www.lacim.uqam.ca/%7Eplouffe/Search.htm |date=2007-02-19 }}"

; Articles on variance of the fundamental constants :
* {{cite journal | last1=Bahcall | first1=John N. |author-link=John N. Bahcall| last2=Steinhardt | first2=Charles L. | last3=Schlegel | first3=David | title=Does the Fine-Structure Constant Vary with Cosmological Epoch? | journal=The Astrophysical Journal | volume=600 | issue=2 | date=10 January 2004 | issn=0004-637X | doi=10.1086/379971 | pages=520–543|arxiv=astro-ph/0301507| bibcode=2004ApJ...600..520B | s2cid=8875571 }}
* [[John D. Barrow]] and Webb, J. K., "Inconstant Constants – Do the inner workings of nature change with time?" ''Scientific American'' (June 2005).
* [[Michael Duff (physicist)|Michael Duff]], 2002 "Comment on time-variation of fundamental constants."
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[[Category:Fundamental constants| ]]
[[Category:Dimensionless constants]]