Constants and Scales

Status: Stable scaffold — quantitative claims pinned to CODATA/SI conventions; interpretive claims tagged. Last updated: 2026-06-08

This page catalogs the dimensionful and dimensionless parameters of fundamental physics, the unit conventions that distinguish physical content from bookkeeping, the characteristic scales (especially the Planck scale), the free-parameter count of the Standard Model + ΛCDM concordance cosmology, and the deepest fine-tuning puzzles (hierarchy, cosmological constant). It is the quantitative companion to THEORY_MAP.md, UNIFICATION_LANDSCAPE.md, and the ASSUMPTIONS_LEDGER.md. For the meaning of epistemic tags see EPISTEMICS.md.

1. Dimensionful constants: physics versus convention

The "fundamental constants" most often quoted are:

Symbol	Name	Role
$c$	speed of light	limiting speed; spacetime metric signature scale
$\hbar$	reduced Planck constant	quantum of action
$G$	Newton's gravitational constant	strength of gravity
$k_B$	Boltzmann constant	energy-per-temperature conversion
$e$	elementary charge	quantum of electric charge

A crucial and frequently muddled point: only dimensionless ratios carry frame- and unit-independent physical content ESTABLISHED. The numerical value of a dimensionful constant depends on the chosen unit system and can be altered (even set to 1) by a change of units without changing any predicted experimental outcome. The cleanest articulation is due to Duff, Okun, and Veneziano, who debated how many dimensionful constants are "truly fundamental" and concluded that the number of operationally meaningful constants is a matter of convention; the physically invariant content lives entirely in dimensionless combinations CONTESTED in emphasis, but the operational core is ESTABLISHED. See GLOSSARY.md for "dimensionless constant."

Concretely:

$c$ and $\hbar$ and $k_B$ are best understood as unit-conversion factors: $c$ converts seconds to meters (relating the time and space components of one four-vector), $\hbar$ converts energy to frequency, and $k_B$ converts temperature to energy INFERENCE — widely accepted; this is the logic behind natural units. Setting $c=\hbar=k_B=1$ removes them entirely without loss of physics.
$G$ sets the scale at which gravity becomes strong relative to quantum effects; its dimensionful value defines the Planck scale (§3). Whether $G$ is "more physical" than $c$ or $\hbar$ is itself partly conventional, but it cannot be removed simultaneously with $c$ and $\hbar$ — it instead fixes the unit of mass/length/time once those are chosen.
$e$ appears physically only through the dimensionless fine-structure constant $\alpha$ (§6).

This is why one can build a system (Planck units, §3) in which $c=\hbar=G=k_B=1$ : dimensionful constants are partly the residue of historically chosen units, and partly markers of where one regime of physics crosses into another.

Caveat to flag ESTABLISHED: a variation of a single dimensionful constant (e.g. "did $c$ change?") is not, by itself, an operationally well-posed question without specifying which dimensionless ratio is held fixed. Only variation of dimensionless constants is observable (§7). This is the "constants of nature are truly constant" assumption examined in the ASSUMPTIONS_LEDGER.md.

2. The 2019 SI redefinition

Since 20 May 2019, the SI base units are defined by fixing exact numerical values of seven defining constants rather than by physical artifacts ESTABLISHED. The relevant fixed values are:

\begin{aligned} \Delta\nu_{\mathrm{Cs}} &= 9\,192\,631\,770\ \text{Hz} \quad (\text{Cs-133 hyperfine transition}),\\ c &= 299\,792\,458\ \text{m/s},\\ h &= 6.626\,070\,15\times10^{-34}\ \text{J·s},\\ e &= 1.602\,176\,634\times10^{-19}\ \text{C},\\ k_B &= 1.380\,649\times10^{-23}\ \text{J/K},\\ N_A &= 6.022\,140\,76\times10^{23}\ \text{mol}^{-1},\\ K_{\mathrm{cd}} &= 683\ \text{lm/W}. \end{aligned}

These are now exact by definition (zero uncertainty), which inverts the historical relationship: the kilogram is defined via the fixed $h$ (and the Kibble balance / Avogadro sphere realize it), not the other way around ESTABLISHED.

Interpretive note INFERENCE: this codifies the §1 lesson institutionally. Fixing $c$ , $h$ , $e$ , $k_B$ turns them into pure conventions of the unit system; what remains experimentally determined are the dimensionless constants and the values of constants — such as $G$ and the fine-structure constant $\alpha$ — that the SI does not fix. In particular $G$ remains the least precisely measured fundamental constant (relative uncertainty $\sim10^{-5}$ , with persistent inter-experiment discrepancies) ESTABLISHED, and $\alpha$ is measured, not fixed.

3. Natural units and the Planck scale

Natural (particle-physics) units set $c=\hbar=1$ , so masses, momenta, energies, and inverse lengths/times are all measured in energy units (typically GeV). Useful conversions ESTABLISHED:

\hbar c \approx 0.1973\ \text{GeV·fm}, \qquad (\hbar c)^2 \approx 0.3894\ \text{GeV}^2\,\text{mb}.

Planck units further set $G=k_B=1$ , defining scales from $c,\hbar,G$ alone:

\ell_P=\sqrt{\frac{\hbar G}{c^3}}\approx 1.616\times10^{-35}\ \text{m},\qquad t_P=\sqrt{\frac{\hbar G}{c^5}}\approx 5.39\times10^{-44}\ \text{s},

M_P=\sqrt{\frac{\hbar c}{G}}\approx 2.176\times10^{-8}\ \text{kg}\approx 1.22\times10^{19}\ \text{GeV}/c^2,

T_P=\frac{M_P c^2}{k_B}\approx 1.42\times10^{32}\ \text{K}.

(Conventions vary: the reduced Planck mass $\overline{M}_P = M_P/\sqrt{8\pi}\approx 2.435\times10^{18}$ GeV is standard in cosmology and quantum gravity, because $8\pi G$ appears in the Einstein equations. Always check which is meant — see CONSTANTS_AND_SCALES.md §3 here as the canonical reference.)

What the Planck scale means. INFERENCE — strong consensus The Planck energy $M_P c^2\sim1.22\times10^{19}$ GeV is the scale at which the dimensionless gravitational coupling of a process, $\sim (E/M_P c^2)^2$ , becomes $O(1)$ : graviton exchange is no longer perturbatively small, and the effective-field-theory treatment of general relativity as a quantum field theory loses predictivity (an infinite tower of counterterms with uncontrolled coefficients). This is the clearest internal signal that quantum field theory on a fixed background must be replaced; see domains/quantum-field-theory.md and domains/general-relativity.md.

Critical caveat — what the Planck scale does NOT establish ESTABLISHED that this is the correct caution:

It is not a proven minimum length. The frequent claim " $\ell_P$ is the shortest measurable distance" is a heuristic/conjecture, motivated by gedanken arguments (a probe localizing to $\ell_P$ has energy $\sim M_P$ , forming a black hole) and realized in some quantum-gravity frameworks (generalized uncertainty principles, string theory's minimal length, loop quantum gravity's discrete area/volume spectra). But no framework with this feature is experimentally confirmed, and a fundamental minimum length is in tension with exact Lorentz invariance unless implemented carefully (e.g. via deformed/doubly-special relativity, which is SPECULATIVE). Tag: SPECULATIVE / OPEN.
The Planck scale marks where naive perturbative gravity-as-EFT fails, not necessarily where new physics first appears. New physics may intervene far below $M_P$ (a GUT scale $\sim10^{16}$ GeV, a seesaw scale $\sim10^{9\text{–}15}$ GeV, TeV-scale states), or — in scenarios like large extra dimensions or asymptotic safety — the effective scale of strong gravity could differ from $M_P$ SPECULATIVE. See OPEN_PROBLEMS.md.

4. Free parameters of the Standard Model

The Standard Model is a renormalizable QFT (see domains/particle-physics.md) whose Lagrangian, once the gauge group $SU(3)_c\times SU(2)_L\times U(1)_Y$ and the fermion representations are fixed, contains a finite set of numbers determined only by experiment. The commonly cited count is ~19 free parameters for the minimal SM (massless neutrinos), grouped as follows ESTABLISHED as a counting fact, modulo convention:

Group	Count	Parameters
Gauge couplings	3	$g_s$ , $g$ , $g'$ (equivalently $\alpha_s,\ \alpha_{\mathrm{em}},\ \sin^2\theta_W$ )
Charged-fermion masses / Yukawas	9	6 quark + 3 charged-lepton masses
Quark mixing (CKM)	4	3 angles + 1 CP-violating phase
Higgs sector	2	vev $v\approx246$ GeV (or $\mu^2$ ) and quartic $\lambda$ (equivalently $m_h, v$ )
QCD vacuum angle	1	$\bar\theta_{\mathrm{QCD}}$
Total (minimal SM)	19

Convention warning ESTABLISHED: the exact integer depends on bookkeeping. Some authors quote 18 (omitting $\bar\theta$ , since it is observationally consistent with zero — see strong-CP, §8 / OPEN_PROBLEMS.md); some quote $G$ as a 20th if gravity is appended. Treat "19" as a convention, not a theorem.

Neutrino sector. Oscillations prove neutrinos are massive (the one laboratory-confirmed breakdown of the minimal SM — see GAPS_AND_CONTRADICTIONS.md), adding parameters:

Dirac neutrinos: $+3$ masses $+3$ PMNS angles $+1$ Dirac phase $=+7$ → ~26.
Majorana neutrinos: as above $+2$ additional Majorana phases $=+9$ → ~28.

Whether neutrinos are Dirac or Majorana is OPEN, decided in principle by neutrinoless double-beta decay (see domains/particle-physics.md). So "the SM has ~26–28 parameters including neutrinos" is the honest statement [ESTABLISHED as counting; the physics of the neutrino sector is partly OPEN].

Cosmology. The base ΛCDM concordance model adds 6 parameters (§5), giving the often-quoted grand total of ~26–31 for "SM + ΛCDM" depending on neutrino assumptions and on whether one counts $G$ and overlapping inputs. ESTABLISHED as a rough convention; the precise number is not canonical.

The brute existence of these ~19–31 inputs — and especially the wild hierarchies among them (the top/electron Yukawa ratio is $\sim10^5$ ; see §6) — is the strongest sense in which the SM is descriptive, not explanatory. See OPEN_PROBLEMS.md (the flavor puzzle) and HYPOTHESES.md.

5. ΛCDM cosmological parameters

The base six-parameter ΛCDM model (see domains/cosmology.md) is conventionally specified as ESTABLISHED as the standard parametrization:

Parameter	Meaning
$\Omega_b h^2$	physical baryon density
$\Omega_c h^2$	physical cold dark matter density
$100\,\theta_{MC}$ (or $H_0$ )	acoustic angular scale / expansion rate
$\tau$	reionization optical depth
$A_s$ (or $\ln(10^{10}A_s)$ )	primordial scalar amplitude
$n_s$	scalar spectral index

Representative values (round numbers; consult primary sources for precision and error bars — see BIBLIOGRAPHY.md): the scalar tilt is measured to be slightly red, $n_s\approx0.965$ ESTABLISHED; the universe is spatially flat to sub-percent level ESTABLISHED; and $H_0\approx67\text{–}73$ km/s/Mpc depending on probe — the unresolved Hubble tension at the $\sim4$ – $6\sigma$ level CONTESTED. Spatial curvature $\Omega_k$ , the dark-energy equation of state $w$ , neutrino mass $\sum m_\nu$ , and extra relativistic species $N_{\mathrm{eff}}$ are held fixed in the base model but are standard extensions (each a probe of physics beyond base ΛCDM). $N_{\mathrm{eff}}\approx3.044$ is the SM prediction for three neutrino species including non-instantaneous decoupling ESTABLISHED. See GAPS_AND_CONTRADICTIONS.md for the $H_0$ and $S_8$ tensions.

6. Important dimensionless numbers

These are the genuinely physical, unit-independent constants ESTABLISHED values; explanations as noted:

Fine-structure constant:

\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}\approx\frac{1}{137.036}\approx7.297\times10^{-3}.

It is the coupling of QED at low (zero-momentum) energy. It runs: at the $Z$ pole $\alpha(M_Z)\approx 1/128$ ESTABLISHED. The integer "137" is famously suggestive but has no known fundamental significance — attempts to derive it (Eddington and others) are regarded as numerology CONTESTED → effectively OPEN, with strong consensus that no accepted derivation exists. The value of $\alpha$ is not explained by the SM; it is an input (§4).

Proton-to-electron mass ratio:

\mu \equiv \frac{m_p}{m_e}\approx 1836.15.

Most of the proton mass is QCD binding/gluon-field energy, not the sum of quark rest masses ESTABLISHED; thus $\mu$ is largely a statement about $\Lambda_{\mathrm{QCD}}$ relative to the electron Yukawa, intertwining the strong sector with electroweak symmetry breaking.

Strong coupling: $\alpha_s(M_Z)\approx0.118$ ESTABLISHED, running to small values at high energy (asymptotic freedom) and to strong coupling near $\Lambda_{\mathrm{QCD}}\sim200$ MeV.
Weak mixing angle: $\sin^2\theta_W\approx0.231$ (scheme- and scale-dependent) ESTABLISHED.
Gravitational coupling of two protons: $\alpha_G=Gm_p^2/(\hbar c)\approx5.9\times10^{-39}$ ESTABLISHED — quantifying how absurdly weak gravity is at particle masses, and the numerical core of the hierarchy problem (§7).

These dimensionless numbers are the real content; everything in §1–§3 is, by contrast, partly convention. See UNIFYING_PRINCIPLES.md.

7. The hierarchy / naturalness problem

Statement ESTABLISHED as a technical fact; CONTESTED as a "problem": in the SM, the Higgs mass-squared receives radiative corrections that, in a cutoff regularization, scale quadratically with the cutoff,

m_h^2 = (m_h^2)_{\text{bare}} + \delta m_h^2, \qquad \delta m_h^2 \sim \frac{\Lambda^2}{16\pi^2}.

If $\Lambda\sim M_P$ , reproducing the observed $m_h\approx125$ GeV requires the bare term and the correction to cancel to $\sim1$ part in $10^{34}$ . Equivalently, the electroweak scale $v\approx246$ GeV sits $\sim10^{16}$ below $M_P$ with no symmetry protecting a scalar mass (unlike fermion masses, protected by chiral symmetry, or gauge-boson masses, protected by gauge symmetry).

Epistemic status CONTESTED: this is not a logical inconsistency and not a domain-of-validity failure — the SM is perfectly predictive (the cutoff sensitivity is absorbed into a counterterm). It is a naturalness expectation: that dimensionless ratios should be $O(1)$ absent a protecting mechanism. The LHC's exclusion of the simplest TeV-scale solutions (low-energy SUSY, technicolor, simplest composite Higgs) has intensified debate over whether naturalness is the correct guide at all, versus environmental/anthropic selection CONTESTED/OPEN. See HYPOTHESES.md, OPEN_PROBLEMS.md, and the EFT-decoupling entry of the ASSUMPTIONS_LEDGER.md.

8. The cosmological-constant problem

The observed dark-energy density is $\rho_\Lambda^{\mathrm{obs}}\sim10^{-47}\ \mathrm{GeV}^4$ (equivalently a mass scale $\sim 2\times10^{-3}$ eV) ESTABLISHED. A naive QFT estimate of the vacuum (zero-point) energy density, cut off at a scale $M_{\mathrm{cut}}$ , gives $\rho_\Lambda^{\mathrm{QFT}}\sim M_{\mathrm{cut}}^4$ . The disagreement is the cosmological-constant problem — widely called the worst quantitative mismatch in physics ESTABLISHED that it is a severe puzzle.

The famous " $10^{120}$ " — read carefully ESTABLISHED, and the nuance is the important part:

With $M_{\mathrm{cut}}\sim M_P$ : $\rho_\Lambda^{\mathrm{QFT}}\sim M_P^4 \sim 10^{74}\ \mathrm{GeV}^4$ , so the ratio $\rho^{\mathrm{QFT}}/\rho^{\mathrm{obs}}\sim10^{120}$ . Hence the headline number.
With $M_{\mathrm{cut}}$ at the electroweak/TeV scale: $\rho_\Lambda^{\mathrm{QFT}}\sim(10^3\ \mathrm{GeV})^4=10^{12}\ \mathrm{GeV}^4$ , giving a ratio $\sim10^{59\text{–}60}$ .
Even the QCD chiral condensate alone contributes $\sim(\Lambda_{\mathrm{QCD}})^4\sim 10^{-3}\ \mathrm{GeV}^4$ , already $\sim10^{44}$ times too large.

The key conceptual point INFERENCE — broad consensus: the " $10^{120}$ " is the ratio of a cutoff-dependent, regularization-dependent estimate to the observed value. It is not a clean prediction. The naive sum-of-zero-point-energies is not Lorentz-invariant as written (a momentum cutoff breaks Lorentz invariance), and a proper treatment (dimensional regularization) makes the vacuum energy proportional to the masses of the fields, not to a quartic cutoff — changing the numerology though not the qualitative disaster. The robust statement is therefore: no symmetry or mechanism is known that cancels the large vacuum-energy contributions while leaving the tiny observed remainder, and the contributions that survive any reasonable accounting (e.g. the electroweak and QCD phase-transition condensates) already overshoot by dozens of orders of magnitude ESTABLISHED. The problem genuinely lives at the QFT/gravity interface: it is about how vacuum energy gravitates, and may signal that the naive QFT vacuum-energy object is simply the wrong thing to feed into Einstein's equations INFERENCE/OPEN. See domains/cosmology.md, domains/quantum-field-theory.md, and GAPS_AND_CONTRADICTIONS.md.

Proposed resolutions — supersymmetry (would cancel boson/fermion zero-point energies if unbroken; broken SUSY leaves a residue still far too large), the string landscape + anthropic selection (Weinberg's anthropic upper bound on $\Lambda$ famously preceded the discovery of acceleration), sequestering, degravitation, modified gravity — are all SPECULATIVE; none is established. See UNIFICATION_LANDSCAPE.md.

9. Do the constants vary?

Only variation of dimensionless constants is physically meaningful (§1). The most-constrained probes are $\alpha$ and $\mu=m_p/m_e$ :

The Oklo natural fission reactor ( $\sim2$ Gyr ago) bounds $|\Delta\alpha/\alpha|\lesssim 10^{-7}$ over that interval ESTABLISHED — order of magnitude; exact bound is analysis-dependent.
Atomic-clock comparisons bound present-day drift $|\dot\alpha/\alpha|$ at roughly $\lesssim 10^{-17}\,\text{yr}^{-1}$ [ESTABLISHED — order of magnitude; consult primary metrology literature in BIBLIOGRAPHY.md for current best values].
Quasar absorption spectra probe $\alpha$ and $\mu$ at cosmological lookback. Claims of a nonzero spatial/temporal variation of $\alpha$ (the "Webb dipole") have appeared but are not confirmed and are widely regarded as not yet established, with systematics a serious concern CONTESTED.
BBN and CMB consistency constrain combinations of constants in the early universe ESTABLISHED as constraints; INFERENCE in interpretation.

Bottom line: there is no confirmed evidence that any fundamental dimensionless constant varies ESTABLISHED — i.e. the null result is robust, but small variation is not excluded and is a generic prediction of theories with light dynamical scalars (quintessence, dilatons, the string landscape) INFERENCE/OPEN. This is the empirical face of the "constants are constant" entry in the ASSUMPTIONS_LEDGER.md. See also OPEN_PROBLEMS.md.

10. Anthropic reasoning (clearly flagged)

Several constants appear "fine-tuned" for complexity/life: the smallness of $\Lambda$ (§8), the proximity of the up/down quark mass difference and electromagnetic effects that make the proton lighter than the neutron, the strength of the strong force relative to electromagnetism (Hoyle-state carbon production), etc.

Status: SPECULATIVE. Anthropic reasoning — that observed values are conditioned on the existence of observers, drawn from an ensemble (multiverse / string landscape / eternal inflation) — is principled and has one notable predictive success (Weinberg's anthropic upper bound on $\Lambda$ , derived before the 1998 acceleration discovery, was of the right order). But it is not established physics: it depends on (i) the actual existence of an ensemble with varying constants SPECULATIVE, (ii) a well-defined probability measure over that ensemble — the unresolved measure problem of eternal inflation, which currently undermines predictivity (Boltzmann-brain and youngness pathologies) OPEN — and (iii) contested assumptions about what counts as "observers." Whether anthropic explanation is even within empirical science is itself CONTESTED. It must never be asserted as fact. See domains/cosmology.md, UNIFICATION_LANDSCAPE.md, and EPISTEMICS.md.

References

See BIBLIOGRAPHY.md for canonical sources: CODATA recommended values and the BIPM SI Brochure (9th ed., 2019 redefinition); Particle Data Group Review of Particle Physics (parameter values, $\alpha$ , $\alpha_s$ , masses); Planck collaboration cosmological-parameter papers (ΛCDM); Weinberg's review of the cosmological constant problem; and the Duff–Okun–Veneziano "trialogue" on the number of fundamental constants. Specific numerical precision and current best bounds should always be taken from these primary sources rather than from this page.