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| In [[physics]], a '''dimensionless physical constant''', sometimes called '''fundamental physical constant''', is a [[physical constant]] that is [[dimensionless quantity|dimensionless]] – having no units attached, having a numerical value that is the same under all possible [[systems of units]]. A common example is the [[fine-structure constant]] α, with approximate value {{physconst|alpha|round=auto|after=.}}
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| The term "fundamental physical constant" has also been used (as by [http://physics.nist.gov/cuu/Constants/ NIST]) to refer to universal but dimensional [[physical constant]]s such as the [[speed of light]] ''c'', [[vacuum permittivity]] ''ε''<sub>0</sub>, [[Planck's constant]] ''h'', or the [[gravitational constant]] ''G''. However, the numerical value of these constants are not fundamental, since they are dependent on the units used to express them. Increasingly, physicists reserve the use of the term "fundamental physical constant" to refer only to dimensionless physical constants that cannot be derived from any other source and can only be measured from nature.<ref>[[John C. Baez]] (2011) "[http://math.ucr.edu/home/baez/constants.html How Many Fundamental Constants Are There?]"</ref>
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| ==Introduction==
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| The numerical values of dimensional physical constants are ''dependent'' on the units used to express these physical constants. As such it is possible to define a basis set of units so that selected ''dimensional'' physical constants are normalized to 1 solely because of the choice of units. The basis set may consist of [[time]], [[length]], [[mass]], [[electric charge|charge]], and [[temperature]], or an equivalent set. A choice of units is called a [[system of units]].
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| For example, the SI, the [[international system of units]], is such a system of units solely defined as convenient to human use and the numerical values of dimensional physical constants have no natural significance, only in a manner that relates to the [[anthropocentrism|human experience]]. As another example, a system of [[natural units]] called [[Planck units]] are defined so that the numerical values of the [[speed of light]] (in a vacuum), the [[universal gravitational constant]], and the constants of [[Planck constant|Planck]], [[Coulomb constant|Coulomb]], and [[Boltzmann constant|Boltzmann]], are all set to 1. Because, merely from the choice of units, these five dimensional physical constants disappear from equations of physical law, they are considered not fundamental in an operationally distinguishable sense.<ref>[[Michael Duff (physicist)|Michael Duff]] (2002) "[http://arxiv.org/abs/hep-th/0208093 Comment on time-variation of fundamental constants.]"</ref><ref>[[Michael Duff (physicist)|Michael Duff]], O. Okun and [[Gabriele Veneziano]] (2002) "[http://arxiv.org/abs/physics/0110060 Trialogue on the number of fundamental constants,]" ''Journal of High Energy Physics'' '''3''': 023.</ref>
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| 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:
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| *α, the [[fine structure constant]], the [[coupling constant]] for the [[electromagnetic interaction]] (≈1/137.036). 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.
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| *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 [[rest mass]]es of all [[elementary particles]] relative to that of the [[electron]].
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| *α<sub>s</sub>, the [[coupling constant]] for the [[strong force]] (≈1)
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| *α<sub>G</sub>, the [[gravitational coupling constant]] (≈10<sup>−38</sup>) which is the square of the [[electron mass]], expressed in [[Planck units]]. This defines the scale of the masses of [[elementary particle]]s and has also been used to express the relative strength of [[gravitation]].
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| Unlike [[mathematical constant]]s, the values of the dimensionless fundamental physical constants cannot be calculated; they are determined only by [[metrology|physical measurement]]. This is one of the [[Unsolved problems in physics|unsolved problems of physics]].
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| One of the dimensionless fundamental constants is the [[fine structure constant]]:
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| :<math> \alpha = \frac{e^2}{\hbar c \ 4 \pi \varepsilon_0} \approx \frac{1}{137.03599908},</math>
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| where ''e'' is the [[elementary charge]], ''ħ'' is the reduced [[Planck's constant]], ''c'' is the [[speed of light]] in a 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 ≈ 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:
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| {{quote|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]]|{{Cite book
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| |author=Richard P. Feynman
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| |year=1985
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| |title=[[QED: The Strange Theory of Light and Matter]]
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| |publisher=[[Princeton University Press]]
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| |page=129
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| |isbn=0-691-08388-6
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| |postscript=<!--None-->
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| }}}}
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| 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<sub>G</sub> is calculated from two protons, its value is ≈10<sup>−38</sup>.
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| 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.
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| ==Constants in the standard model and in cosmology==
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| 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 [[vacuum angle]] was found to be indistinguishable from zero.
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| The complete [[standard model]] requires 25 fundamental dimensionless constants ([http://math.ucr.edu/home/baez/constants.html Baez, 2002]). At present, their numerical values are not understood in terms of any widely accepted theory and are determined only from measurement. Based on Mp=1.22089(6)E19 GeV/c2, these 25 constants are:
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| *the [[fine structure constant]];
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| *the [[coupling constant|strong coupling constant]];
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| *fifteen [[mass]]es of the [[fundamental particle]]s (relative to the [[Planck mass]]), namely:
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| ** six [[quark]]s
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| ** six [[lepton]]s
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| ** the [[Higgs boson]]
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| ** the [[W boson]]
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| ** the [[Z boson]]
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| *four parameters of the [[CKM matrix]], describing how [[quark]]s oscillate between different forms;
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| *four parameters of the [[Pontecorvo–Maki–Nakagawa–Sakata matrix]], which does the same thing for [[neutrino]]s.
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| {| class="wikitable collapsible collapsed"
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| !colspan="5"|Dimensionless constants of the Standard Model
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| |-
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| ! Symbol
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| ! Description
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| ! Dimensionless value
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| ! Alternative value representation
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| |-
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| |''m''<sub>u</sub> / ''m''<sub>P</sub>
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| |Up quark mass
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| |1.4×10<sup>−22</sup> – 2.7×10<sup>−22</sup>
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| |1.7 – 3.3 MeV/c<sup>2</sup>
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| |-
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| |''m''<sub>d</sub> / ''m''<sub>P</sub>
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| |Down quark mass
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| |3.4×10<sup>−22</sup> – 4.8×10<sup>−22</sup>
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| |4.1 – 5.8 MeV/c<sup>2</sup>
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| |-
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| |''m''<sub>c</sub> / ''m''<sub>P</sub>
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| |Charm quark mass
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| |1.04×10<sup>−19</sup>
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| |1.27 GeV/c<sup>2</sup>
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| |-
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| |''m''<sub>s</sub> / ''m''<sub>P</sub>
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| |Strange quark mass
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| |8.27×10<sup>−21</sup>
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| |101 MeV/c<sup>2</sup>
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| |-
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| |''m''<sub>t</sub> / ''m''<sub>P</sub>
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| |Top quark mass
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| |1.41×10<sup>−17</sup>
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| |172.0 GeV/c<sup>2</sup>
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| |-
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| |''m''<sub>b</sub> / ''m''<sub>P</sub>
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| |Bottom quark mass
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| |3.43×10<sup>−19</sup>
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| |4.19 GeV/c<sup>2</sup>
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| |-
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| |''θ''<sub>12,CKM</sub>
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| |CKM 12-mixing angle
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| |0.23
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| |13.1°
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| |-
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| |''θ''<sub>23,CKM</sub>
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| |CKM 23-mixing angle
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| |0.042
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| |2.4°
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| |-
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| |''θ''<sub>13,CKM</sub>
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| |CKM 13-mixing angle
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| |0.0035
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| |0.2°
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| |-
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| |''δ''<sub>CKM</sub>
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| |CKM [[CP violation|CP-violating]] Phase
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| |0.995
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| |57°
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| |-
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| |''m''<sub>e</sub> / ''m''<sub>P</sub>
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| |Electron mass
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| |4.18546×10<sup>−23</sup>
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| |511 keV/c<sup>2</sup>
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| |-
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| |''m''<sub>ν<sub>e</sub></sub> / ''m''<sub>P</sub>
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| |Electron neutrino mass
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| |Unknown
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| |-
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| |''m''<sub>μ</sub> / ''m''<sub>P</sub>
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| |Muon mass
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| |8.65418×10<sup>-21<sup>
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| |105.7 MeV/c<sup>2</sup>
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| |-
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| |''m''<sub>ν<sub>μ</sub></sub> / ''m''<sub>P</sub>
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| |Muon neutrino mass
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| |Unknown
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| |-
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| |''m''<sub>τ</sub> / ''m''<sub>P</sub>
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| |Tau mass
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| |1.45535×10<sup>-19<sup>
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| |1.78 GeV/c<sup>2</sup>
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| |-
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| |''m''<sub>ν<sub>τ</sub></sub> / ''m''<sub>P</sub>
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| |Tau neutrino mass
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| |Unknown
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| |-
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| |''θ''<sub>12,PMNS</sub>
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| |PMNS 12-mixing angle
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| |{{val|0.5973|0.0175}}
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| |{{val|34.22|1|s=°}}
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| |-
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| |''θ''<sub>23.PMNS</sub>
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| |PMNS 23-mixing angle
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| |{{val|0.785|0.12}}
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| |{{val|45|7.1|s=°}}
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| |-
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| |''θ''<sub>13,PMNS</sub>
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| |PMNS 13-mixing angle
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| |≈0.077
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| |≈4.4°
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| |-
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| |''δ''<sub>PMNS</sub>
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| |PMNS [[CP violation|CP-violating]] Phase
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| |Unknown
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| |-
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| |α
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| |fine structure constant
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| |0.00729735
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| |1/137.036
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| |-
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| |α<sub>s</sub>
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| |strong coupling constant
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| |≈1
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| |≈1
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| |-
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| |''m''<sub>W<sup>±</sup></sub> / ''m''<sub>P</sub>
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| |W boson mass
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| |6.5841±0.0012×10<sup>−18</sup>
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| |80.385±0.015 GeV/c<sup>2</sup>
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| |-
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| |''m''<sub>Z<sup>0</sup></sub> / ''m''<sub>P</sub>
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| |Z boson mass
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| |7.46888±0.00016×10<sup>−18</sup>
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| |91.1876±0.002 GeV/c<sup>2</sup>
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| |-
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| |''m''<sub>H</sub> / ''m''<sub>P</sub>
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| |Higgs boson mass
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| |≈1.02×10<sup>−17</sup>
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| |≈125 GeV/c<sup>2</sup> (tentative)
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| |}
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| One constant is required for [[cosmology]]:
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| *the [[cosmological constant]] (in terms of [[Planck units]]) of [[Einstein field equation|Einstein's equations]] for [[general relativity]], having a value of approximately 10<sup>−122</sup>.
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| 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 fermion]]s, 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.
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| ==Well-known subsets==
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| Certain dimensionless constants are discussed more frequently than others.
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| ===Barrow and Tipler===
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| 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 [[gravitational coupling constant|gravitation]].
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| ===Martin Rees's Six Numbers===
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| [[Martin Rees, Baron Rees of Ludlow|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:
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| *''N''≈10<sup>36</sup>: the ratio of the [[fine structure constant]] (the dimensionless [[coupling constant]] for [[electromagnetism]]) to the [[gravitational coupling constant]], the latter defined using two [[proton]]s. In Barrow and Tipler (1986) and elsewhere in Wikipedia, this ratio is denoted α/α<sub>G</sub>. ''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), p. .</ref>
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| *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>
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| *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;<ref name="Rees, M. 2000, p"/>
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| *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>
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| *''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>
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| *''D'' = 3: the number of macroscopic spatial [[dimension]]s.
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| ''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 cannot be measured. Hence most physicists would not deem it a dimensionless physical constant of the sort discussed in this entry. There are also [[Spacetime#Privileged character of 3.2B1 spacetime|compelling physical and mathematical reasons why ''D'' = 3]].
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| 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.
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| ==Variation of the constants==
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| 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|reason=Please give a source for this assertion|date=June 2011}}
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| ==Calculation attempts==
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| No formulae for the fundamental physical constants are known to this day.
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| 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]].
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| 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.
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| 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.
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| ==See also==
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| *[[Cabibbo–Kobayashi–Maskawa matrix]] ([[Cabibbo angle]])
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| *[[coupling constant]]
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| *[[dimensionless quantities]]
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| *[[Fine-structure constant]]
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| *[[gravitational coupling constant]]
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| *[[Neutrino oscillation]]
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| *[[Physical cosmology]]
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| *[[Standard Model]]
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| *[[Weinberg angle]]
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| *[[Fine-tuned Universe]]
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| *[[Koide formula]]
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| ==References==
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| {{reflist|30em}}
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| ==Bibliography==
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| *[[Martin Rees]], 1999. ''Just Six Numbers: The Deep Forces that Shape the Universe''. London: Weidenfeld & Nicolson. ISBN 0-7538-1022-0
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| ==External articles==
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| ;General
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| *[[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.
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| *{{BarrowTipler1986}}
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| *[[Michio Kaku]], 1994. ''[[Hyperspace (book)|Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension]]''. [[Oxford University Press]].
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| *[http://physics.nist.gov/cuu/Constants/ Fundamental Physical Constants from NIST]
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| *[http://physics.nist.gov/cuu/Constants/Table/allascii.txt Values of fundamental constants.] [[CODATA]], 2002.
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| *[[John Baez]], 2002, "[http://math.ucr.edu/home/baez/constants.html How Many Fundamental Constants Are There?]"
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| *Plouffe. Simon, 2004, "[http://www.lacim.uqam.ca/%7Eplouffe/Search.htm A search for a mathematical expression for mass ratios using a large database.]"
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| ;Do the fundamental constants vary?
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| *[[John Bahcall]], Charles Steinhardt, and David Schlegel, 2004, "[http://www.arXiv.org/abs/astro-ph/0301507 Does the fine-structure constant vary with cosmological epoch?]" ''Astrophys. J. 600'': 520.
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| *[[John D. Barrow]] and Webb, J. K., "[http://www.sciam.com/article.cfm?chanID=sa006&articleID=0005BFE6-2965-128A-A96583414B7F0000 Inconstant Constants – Do the inner workings of nature change with time?]" ''Scientific American'' (June 2005).
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| *[[Michael Duff (physicist)|Michael Duff]], 2002 "[http://arxiv.org/abs/hep-th/0208093 Comment on time-variation of fundamental constants.]"
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| *Marion, H., et al. 2003, "[http://arXiv.org/abs/physics/0212112 A search for variations of fundamental constants using atomic fountain clocks,]" ''Phys.Rev.Lett. 90'': 150801.
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| *Martins, J.A.P. et al., 2004, "[http://arXiv.org/abs/astro-ph/0302295C WMAP constraints on varying α and the promise of reionization,]" ''Phys.Lett. B585'': 29–34.
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| *Olive, K.A., et al., 2002, "[http://arXiv.org/abs/hep-ph/0205269 Constraints on the variations of the fundamental couplings,]" ''Phys.Rev. D66'': 045022.
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| *Uzan, J-P, 2003, "[http://arXiv.org/abs/hep-ph/0205340 The fundamental constants and their variation: observational status and theoretical motivations,]" ''Rev.Mod.Phys. 75'': 403.
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| *Webb, J.K. et al., 2001, "Further evidence for cosmological evolution of the fine-structure constant," ''Phys. Rev. Lett. 87'': 091301.
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| [[Category:Fundamental constants| ]]
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