The Free Parameters of Physics
This appendix catalogs the complete set of dimensionless (or, where noted, dimensionful-but-irreducible) parameters that the evidence-based physics of this treatise — the Standard Model of particle physics and the base \(\Lambda\)CDM cosmological model — treats as free: numbers no confirmed theory yet predicts from anything more basic. The mathematical constants \(\pi\) and \(\ee\) (Section 7.9.1) and the dimensionful SI constants \(c\), \(\hbar\), \(G\), \(\epsilon_0\), and the fine-structure constant \(\alpha\) are catalogued separately in Physical Constants and SI Units, since they are either proven mathematical facts or exactly-defined/precisely-measured physical constants rather than free parameters in the sense used here.
Every entry below states, where one exists, how the quantity is constructed from more basic measured inputs; where no such construction exists, the entry says so plainly — the quantity is itself one of the truly free numbers of physics. Entries marked (open) are ones whose value, nature, or theoretical status is not yet settled by current evidence; each says explicitly what is and is not yet known, in keeping with this treatise's evidence-only scope (Epistemology and the Scientific Method). None of the speculative theoretical programs sometimes proposed to explain these numbers — supersymmetry, extra dimensions, or similar — is invoked here as an explanation, since none is yet supported by direct evidence; where relevant this is stated explicitly, and the broader question is returned to in What We Observe but Do Not Understand.
Electroweak and Higgs
The parameters catalogued here fix the electroweak sector of the Standard Model: the strength and mixing of the \(SU(2)_L\times U(1)_Y\) gauge interactions, the scale of spontaneous symmetry breaking, and the mass and self-coupling of the Higgs boson that breaks it. The fine-structure constant \(\alpha\), the third traditional input of electroweak fits alongside \(G_F\) and \(m_Z\), is already catalogued in Appendix B (docs/latex/appendices/B-constants-units.tex) and is not repeated here. All numerical values below are taken from the Particle Data Group's 2024 Review of Particle Physics. Because several of these quantities (the weak mixing angle, the gauge boson masses) are related by tree-level and loop-corrected Standard Model relations, they overdetermine the electroweak Lagrangian and their mutual consistency is itself a stringent test of the theory.
| Quantity | Symbol | Value |
|---|---|---|
| Fermi coupling constant | $G_F$ | $1.1663787(6)\times 10^{-5}\,/\mathrm{GeV}^{2}$ |
| Higgs vacuum expectation value | $v$ | $246.21965(6)\,\mathrm{GeV}$ |
| Weak mixing angle | $\sin^2\hat\theta_W(m_Z)$ | $0.23122(4)$ |
| W boson mass | $m_W$ | $80.3692(133)\,\mathrm{GeV}$ |
| Z boson mass | $m_Z$ | $91.1876(21)\,\mathrm{GeV}$ |
| Higgs boson mass | $m_H$ | $125.20(11)\,\mathrm{GeV}$ |
| Higgs quartic self-coupling | $\lambda$ | $0.1293(2)$ |
Fermi coupling constant
\(G_F\): \(1.1663787(6)\times 10^{-5}\,/\mathrm{GeV}^{2}\).
measured directly (muon lifetime: \(1/\tau_\mu = \dfrac{G_F^2 m_\mu^5}{192\pi^3}(1+\Delta q)\), \(\Delta q\) the known QED radiative correction).
The Fermi coupling constant \(G_F\) parametrizes the strength of charged-current weak interactions at energies far below the \(W\)-boson mass, in the effective four-fermion contact-interaction language that predates the discovery of the \(W\) and \(Z\) bosons. It is extracted with extraordinary precision from the muon lifetime, measured most accurately by the MuLan experiment at PSI, combined with QED radiative corrections computed to several loop orders. Because \(G_F\) is known to a relative precision of order \(5\times10^{-7}\), it – rather than \(v\) directly – is used as one of the primary electroweak input parameters, alongside \(\alpha\) (Appendix B) and \(m_Z\), in global electroweak fits. Through \(v=(\sqrt2 G_F)^{-1/2}\) it fixes the overall scale of electroweak symmetry breaking. [Navas:2024]
Higgs vacuum expectation value
\(v\): \(246.21965(6)\,\mathrm{GeV}\).
\(v \equiv (\sqrt{2}\,G_F)^{-1/2}\).
The vacuum expectation value \(v\) of the neutral component of the Higgs doublet sets the scale at which \(SU(2)_L\times U(1)_Y\) electroweak symmetry breaks spontaneously to the electromagnetic \(U(1)_{\mathrm{em}}\), and through the Yukawa couplings \(y_f=\sqrt2\,m_f/v\) it fixes the overall normalization of every fermion mass in the Standard Model. It is not measured directly but derived from the far more precisely known Fermi constant via \(v=(\sqrt2 G_F)^{-1/2}\), since \(G_F\) and \(v\) describe the same low-energy strength of the charged-current weak interaction. Its numerical value, close to \(246\,\mathrm{GeV}\), together with the Higgs mass \(m_H\) fixes the Higgs quartic self-coupling \(\lambda\) (see below). [Navas:2024]
Weak mixing angle
\(\sin^2\hat\theta_W(m_Z)\): \(0.23122(4)\).
\(\overline{\mathrm{MS}}\) scheme: \(\sin^2\hat\theta_W(\mu=m_Z) \equiv \dfrac{\hat g'^2}{\hat g^2+\hat g'^2}\Big|_{\overline{\mathrm{MS}}}\); on-shell scheme: \(\sin^2\theta_W \equiv 1-m_W^2/m_Z^2\).
The weak mixing (Weinberg) angle \(\theta_W\) rotates the neutral \(SU(2)_L\times U(1)_Y\) gauge fields \(W^3_\mu, B_\mu\) into the physical \(Z\) boson and the photon, and its sine-squared is the single additional number, beyond \(\alpha\) and \(G_F\) (or \(m_Z\)), needed to fix all electroweak neutral-current couplings at tree level. Because \(\sin^2\theta_W\) is a ratio of couplings, its numerical value is renormalization-scheme dependent: the value quoted here is the \(\overline{\mathrm{MS}}\) value evaluated at \(\mu=m_Z\), obtained from global electroweak fits to \(Z\)-pole asymmetries and low-energy neutral-current data. The alternative on-shell definition, \(\sin^2\theta_W\equiv1-m_W^2/m_Z^2\), gives a numerically different value – using the PDG-2024 \(m_W\) and \(m_Z\) listed below, \(1-m_W^2/m_Z^2\approx0.2232\) – because it absorbs electroweak radiative corrections differently order by order. [Navas:2024]
W boson mass
\(m_W\): \(80.3692(133)\,\mathrm{GeV}\).
\(m_W = \tfrac12 g v\), fixing the \(SU(2)_L\) gauge coupling \(g\) once \(v\) is known; measured directly.
The \(W^\pm\) boson mass is one of the two masses generated by the Higgs mechanism through the covariant derivative of the Higgs doublet, \(m_W=\tfrac12 gv\), so its measurement, combined with the independently fixed \(v\), determines the \(SU(2)_L\) gauge coupling \(g\). It is measured kinematically from the transverse-mass and transverse-momentum distributions of leptons in \(W\to\ell\nu\) decays at hadron colliders (Tevatron, LHC) and, less precisely, from \(e^+e^-\to W^+W^-\) threshold scans at LEP2. The value quoted is the PDG-2024 world average; the 2022 CDF-II measurement (\(80.4335(94)\,\mathrm{GeV}\)) lies in tension of several standard deviations with this average and with the Standard Model electroweak fit, a discrepancy still under active experimental scrutiny following subsequent 2024 measurements (including a CMS result and a revised ATLAS analysis) that favor the lower, SM-consistent value. [Navas:2024]
Z boson mass
\(m_Z\): \(91.1876(21)\,\mathrm{GeV}\).
\(m_Z = \tfrac12 v\sqrt{g^2+g'^2}\), fixing the combination \(g^2+g'^2\) once \(v\) is known; measured directly.
The \(Z\) boson mass is fixed, like \(m_W\), by electroweak symmetry breaking, \(m_Z=\tfrac12 v\sqrt{g^2+g'^2}\), so its measurement fixes the combination of \(SU(2)_L\) and \(U(1)_Y\) gauge couplings \(g^2+g'^2\) once \(v\) is known from \(G_F\). It is by far the most precisely known electroweak boson mass, extracted from the \(e^+e^-\to Z\) resonance line shape scanned at LEP1 and at the SLC, where it appears as the peak of the total hadronic and leptonic cross sections. Its relative precision of about \(2\times10^{-5}\) makes it, together with \(\alpha\) and \(G_F\), one of the three canonical input parameters of the electroweak Lagrangian in most global-fit conventions. [Navas:2024]
Higgs boson mass
\(m_H\): \(125.20(11)\,\mathrm{GeV}\).
measured directly (resonance peak in \(H\to\gamma\gamma\) and \(H\to ZZ^*\to4\ell\)); fixes the quartic self-coupling via \(m_H^2=2\lambda v^2\).
The Higgs boson mass, established in 2012 by ATLAS and CMS through the \(H\to\gamma\gamma\) and \(H\to ZZ^*\to4\ell\) decay channels, is measured directly as the invariant-mass peak of the decay products; unlike the fermion and gauge-boson masses, it is not predicted from a more fundamental parameter within the Standard Model but is itself a genuinely free input, fixing the strength of the Higgs field's quartic self-interaction \(\lambda\) through \(m_H^2=2\lambda v^2\). The value quoted is the PDG-2024 world average combining the high-resolution di-photon and four-lepton mass measurements of both experiments. Once \(m_H\) and \(v\) are fixed, every other tree-level Higgs property – total width, branching ratios, and the self-coupling \(\lambda\) – follows with no further free parameters, which is precisely what ongoing LHC Higgs-coupling measurements continue to test. [Navas:2024]
Higgs quartic self-coupling
\(\lambda\): \(0.1293(2)\).
\(\lambda = m_H^2/(2v^2)\).
The Higgs quartic self-coupling \(\lambda\) multiplies the \(\lambda(\Phi^\dagger\Phi)^2\) term of the Higgs potential and, together with \(v\), sets both the shape of the electroweak vacuum and the physical Higgs boson mass, \(m_H^2=2\lambda v^2\). Unlike \(m_H\), \(\lambda\) is not itself measured to comparable precision: its value quoted here is inferred algebraically from the two directly measured quantities \(m_H\) and \(v\), giving \(\lambda\approx0.129\). A genuinely independent, direct determination of \(\lambda\) requires observing Higgs pair production (\(pp\to HH\)) and extracting the Higgs trilinear self-coupling, a measurement that remains statistics-limited at the LHC and has so far only placed loose bounds around the Standard Model expectation. The measured smallness of \(\lambda\) relative to the top-quark Yukawa coupling squared is also central to the (unresolved) question of the stability of the electroweak vacuum under renormalization-group running to high energy scales. [Navas:2024]
Quantum Chromodynamics
Quantum chromodynamics (QCD), the SU(3)\(_c\) gauge sector of the Standard Model, is characterized by a running coupling \(\alpha_s\) and six quark mass parameters whose numerical values are, unlike the electroweak sector, deeply entangled with the choice of renormalization scheme and scale: light-quark masses are conventionally quoted in \(\overline{\text{MS}}\) at a fixed scale of \(2\,\mathrm{GeV}\), the charm and bottom masses in \(\overline{\text{MS}}\) at their own mass scale, and the top mass – uniquely among quarks, since it decays before hadronizing – as an approximate pole mass reconstructed from collider kinematics. This scheme-dependence is not a bookkeeping nuisance but reflects the underlying fact that confined color charges have no scheme-independent, asymptotic-state definition of mass in the way leptons do. The sector also harbors one of the cleanest open problems in physics: the QCD vacuum angle \(\bar\theta_{\rm QCD}\), which the symmetries of the Standard Model permit to take any value of order unity but which experiment constrains to be smaller than one part in \(10^{10}\), with no confirmed dynamical explanation.
| Quantity | Symbol | Value |
|---|---|---|
| Strong coupling constant | $\alpha_s(M_Z)$ | $0.1180(9)$ |
| Up-quark mass | $m_u$ | $2.16^{+0.49}_{-0.26}\,\mathrm{MeV}$ |
| Down-quark mass | $m_d$ | $4.67^{+0.48}_{-0.17}\,\mathrm{MeV}$ |
| Strange-quark mass | $m_s$ | $93.4^{+8.6}_{-3.4}\,\mathrm{MeV}$ |
| Charm-quark mass | $m_c(m_c)$ | $1.27(2)\,\mathrm{GeV}$ |
| Bottom-quark mass | $m_b(m_b)$ | $4.18^{+0.03}_{-0.02}\,\mathrm{GeV}$ |
| Top-quark mass | $m_t$ | $172.57(29)\,\mathrm{GeV}$ |
| QCD vacuum angle (strong CP problem) (open) | $\bar\theta_{\rm QCD}$ | $10^{-10}$ (see text) |
Strong coupling constant
\(\alpha_s(M_Z)\): \(0.1180(9)\).
\(\alpha_s \equiv g_s^2/(4\pi)\), where \(g_s\) is the QCD gauge coupling; the value is scale-dependent (renormalization-group running governed by the QCD \(\beta\)-function) and is conventionally quoted at \(\mu = M_Z\), the world average from lattice QCD, \(\tau\)-decay, deep-inelastic scattering, event-shape, and electroweak precision-fit determinations.
The strong coupling \(\alpha_s(\mu)\) measures the strength of the color force at renormalization scale \(\mu\); unlike the electromagnetic fine-structure constant \(\alpha\), which is essentially scale-independent at laboratory energies, \(\alpha_s\) runs rapidly and grows large at low energy, producing confinement, and shrinks logarithmically at high energy, producing asymptotic freedom – the property for which the 2004 Nobel Prize was awarded. By convention it is quoted at the reference scale \(\mu = M_Z \approx 91.19\,\mathrm{GeV}\), where the PDG world average combines lattice-QCD sum rules, \(\tau\)-lepton hadronic decays, deep-inelastic scattering structure functions, \(e^+e^-\) jet event shapes, and the running of \(\alpha_s\) inferred from \(Z\)-pole electroweak fits. Its value sets the size of perturbative QCD corrections throughout the treatise (hadron collider cross sections, jet physics, hadronic decay widths) and, via dimensional transmutation, fixes the intrinsic QCD scale \(\Lambda_{\rm QCD} \sim 200\,\mathrm{MeV}\) at which the coupling formally diverges and confinement sets in. [Navas:2024]
Up-quark mass
\(m_u\): \(2.16^{+0.49}_{-0.26}\,\mathrm{MeV}\).
measured directly, as a Lagrangian mass parameter (coefficient of \(\bar{u}u\)) in the \(\overline{\text{MS}}\) renormalization scheme at \(\mu = 2\,\mathrm{GeV}\); extracted from lattice QCD combined with QCD sum rules, chiral perturbation theory mass ratios, and hadron spectroscopy.
The up-quark mass is a free parameter of the Standard Model Lagrangian, fixed by the Yukawa coupling of \(u\) to the Higgs field, and is never observed as a free-particle mass because of color confinement. It is instead extracted as a running \(\overline{\text{MS}}\)-scheme mass parameter, conventionally quoted at \(\mu = 2\,\mathrm{GeV}\), from lattice QCD combined with chiral perturbation theory constraints on light-quark mass ratios and QCD sum rules applied to hadron spectra. This scheme dependence is physically essential: quoting “the” up-quark mass without specifying scheme and scale is meaningless, since the value shifts under renormalization-group running exactly as \(\alpha_s\) does. The smallness and asymmetric uncertainty of \(m_u\) relative to \(\Lambda_{\rm QCD}\) (light-quark masses are a small perturbation on hadron masses, most of which arises from gluon binding energy) also underlies the near-exact isospin symmetry of nuclear physics. [Navas:2024]
Down-quark mass
\(m_d\): \(4.67^{+0.48}_{-0.17}\,\mathrm{MeV}\).
measured directly, as a Lagrangian mass parameter in the \(\overline{\text{MS}}\) scheme at \(\mu = 2\,\mathrm{GeV}\); extracted from lattice QCD, chiral perturbation theory, and QCD sum rules, jointly with \(m_u\) and \(m_s\).
Like \(m_u\), the down-quark mass is a free Standard-Model parameter quoted in the \(\overline{\text{MS}}\) scheme at \(\mu = 2\,\mathrm{GeV}\), obtained jointly with the other light-quark masses from lattice QCD and chiral perturbation theory fits to pion, kaon, and eta meson masses and decay constants. That \(m_d > m_u\) by roughly a factor of two is the dominant source of the proton-neutron mass difference (partially offset by electromagnetic self-energy), and is responsible for the neutron being heavier than the proton – a fact of considerable cosmological importance, since it permits free-neutron beta decay and shapes primordial nucleosynthesis. As with \(m_u\), no experiment isolates a free down quark; all determinations are indirect, via QCD-mediated observables. [Navas:2024]
Strange-quark mass
\(m_s\): \(93.4^{+8.6}_{-3.4}\,\mathrm{MeV}\).
measured directly, as a Lagrangian mass parameter in the \(\overline{\text{MS}}\) scheme at \(\mu = 2\,\mathrm{GeV}\); extracted principally from lattice QCD and QCD sum rules applied to kaon and \(\eta\)/\(\eta'\) physics.
The strange-quark mass, quoted in the \(\overline{\text{MS}}\) scheme at the same reference scale \(\mu = 2\,\mathrm{GeV}\) used for \(m_u\) and \(m_d\), is roughly twenty times larger than the down-quark mass and is the dominant explicit breaking term for flavor SU(3) symmetry among the light hadrons. It is determined mainly from lattice QCD (moments of correlators, kaon masses and decay constants) and from QCD sum rules constrained by \(\tau\)-decay and \(e^+e^-\) data; its uncertainty, while still asymmetric and scheme-dependent, has shrunk considerably as lattice methods have matured. Together, \(m_u\), \(m_d\), \(m_s\) set the size of chiral symmetry breaking effects that organize the light hadron spectrum (pions and kaons as pseudo-Goldstone bosons of spontaneously broken chiral symmetry). [Navas:2024]
Charm-quark mass
\(m_c(m_c)\): \(1.27(2)\,\mathrm{GeV}\).
measured directly, as a Lagrangian mass parameter in the \(\overline{\text{MS}}\) scheme evaluated at its own mass scale, \(\mu = m_c\) (the conventional choice for heavy quarks, distinct from the fixed \(\mu = 2\,\mathrm{GeV}\) used for \(u,d,s\)); extracted from charmonium sum rules and lattice QCD.
Unlike the light quarks, the charm-quark mass is conventionally quoted in the \(\overline{\text{MS}}\) scheme evaluated at its own mass, \(\mu = m_c(m_c)\), rather than at the fixed \(2\,\mathrm{GeV}\) reference used for \(u,d,s\) – a scheme choice that avoids large logarithms in the perturbative relation between the running mass and observables near the charm threshold. The dominant determinations come from QCD sum rules applied to \(e^+e^- \to\) charm-quark-pair cross-section data and charmonium spectroscopy, cross-checked by lattice QCD. Because \(m_c\) sits close to \(\Lambda_{\rm QCD}\) times a moderate factor, charm physics occupies an intermediate regime where both perturbative QCD and nonperturbative (lattice, effective field theory) methods are needed and mutually validated. [Navas:2024]
Bottom-quark mass
\(m_b(m_b)\): \(4.18^{+0.03}_{-0.02}\,\mathrm{GeV}\).
measured directly, as a Lagrangian mass parameter in the \(\overline{\text{MS}}\) scheme evaluated at its own mass scale, \(\mu = m_b\); extracted from bottomonium sum rules, lattice QCD, and inclusive/exclusive \(B\)-meson semileptonic decay spectra.
The bottom-quark mass, like the charm mass, is quoted in the \(\overline{\text{MS}}\) scheme at its own scale \(\mu = m_b(m_b)\), which keeps the perturbative series for mass-dependent observables well behaved. It is determined from a combination of bottomonium (\(\Upsilon\)-system) sum rules, lattice QCD, and QCD-based analyses of inclusive and exclusive semileptonic \(B\)-meson decay spectra, all of which now agree to sub-percent precision. The bottom mass is a key input to CKM-matrix determinations from \(B\)-physics (Part on flavor physics) and to Higgs-boson branching-ratio predictions, since \(H \to b\bar{b}\) is the dominant Higgs decay channel and its rate depends quadratically on \(m_b\). [Navas:2024]
Top-quark mass
\(m_t\): \(172.57(29)\,\mathrm{GeV}\).
measured directly, as a pole mass (not an \(\overline{\text{MS}}\) running mass, unlike the other five quarks) from kinematic reconstruction of top-antitop pair-production events at the Tevatron and LHC.
The top quark is the only quark reported as a pole mass rather than an \(\overline{\text{MS}}\) running mass, because it decays via the electroweak interaction (\(t \to W^+ b\)) before it can hadronize – it is the one quark whose mass can be reconstructed directly from the kinematics of its (quasi-free) decay products, bypassing the scheme ambiguities inherent to confined quarks. The world average combines direct kinematic reconstructions from ATLAS, CMS, and the Tevatron experiments (CDF, D0). It should be noted that the “pole mass” extracted this way carries a residual, formally ill-defined nonperturbative ambiguity of order \(\Lambda_{\rm QCD}\) (the renormalon ambiguity), so the quoted value is understood as an operationally defined Monte-Carlo mass closely related to, but not rigorously identical to, the theoretical pole mass; conversion to a well-defined short-distance mass scheme (e.g. \(\overline{\text{MS}}\)) shifts the central value down by roughly \(9\,\mathrm{GeV}\) to \(10\,\mathrm{GeV}\) (e.g. \(\overline{m}_t(\overline{m}_t) \approx 163\,\mathrm{GeV}\) versus the pole-like value above). At \(\sim 173\,\mathrm{GeV}\), comparable to the electroweak scale, the top mass gives the largest Yukawa coupling in the Standard Model (\(y_t \approx 1\)) and dominates radiative corrections to the Higgs mass and to electroweak precision observables. [Navas:2024]
QCD vacuum angle (strong CP problem) (open)
\(\bar\theta_{\rm QCD}\): \(\abs{\bar\theta_{\rm QCD}} \lesssim 10^{-10}\) (90\% C.L., from the neutron electric dipole moment bound).
appears as the coefficient of the CP-violating topological term \(\mathcal{L}_\theta = \theta_{\rm QCD}\,\dfrac{g_s^2}{32\pi^2}\,G^a_{\mu\nu}\tilde{G}^{a\,\mu\nu}\) in the QCD Lagrangian; the physically observable combination \(\bar\theta_{\rm QCD} = \theta_{\rm QCD} + \arg\det(M_u M_d M_s\cdots)\) absorbs an equal, in-principle-independent phase from the quark mass matrix, and is bounded from above by the non-observation of a neutron electric dipole moment.
The QCD Lagrangian admits a gauge-invariant, Lorentz-invariant, renormalizable term \(\theta_{\rm QCD}\,(g_s^2/32\pi^2)\,G\tilde{G}\) built from the gluon field strength and its dual, which violates parity and time-reversal (hence CP, by CPT) and is entirely permitted by the symmetries of the Standard Model; the physical angle \(\bar\theta_{\rm QCD}\) also receives a contribution from the phase of the determinant of the quark mass matrix, so it is not removable by a field redefinition once electroweak symmetry breaking generates quark masses. A nonzero \(\bar\theta_{\rm QCD}\) of order unity is the generic theoretical expectation, since it is an allowed, unsuppressed term with no symmetry reason to vanish, and it would produce an observably large neutron electric dipole moment \(d_n \sim \bar\theta_{\rm QCD} \times e\,10^{-16}\,\mathrm{cm}\). Experimentally, the most sensitive constraint comes from ultracold-neutron electric-dipole-moment experiments, whose current upper bound on \(\abs{d_n}\) (of order \(e\,10^{-26}\,\mathrm{cm}\)) translates into \(\abs{\bar\theta_{\rm QCD}} \lesssim 10^{-10}\), a suppression by roughly ten orders of magnitude below its theoretically natural size. This unexplained fine-tuning – why a free, dimensionless parameter of the strong interaction sits so close to zero when nothing in the Standard Model forces it to – constitutes the strong CP problem, one of the cleanest and longest-standing naturalness puzzles in particle physics; it remains open within the evidence-based scope of this treatise; proposed dynamical solutions (e.g. a spontaneously broken Peccei–Quinn symmetry and its associated axion) are not yet supported by direct experimental evidence and are noted here only for completeness, not asserted. [Navas:2024]
Charged Lepton Masses and Quark Mixing (CKM)
The charged-lepton sector of the Standard Model exhibits a mass hierarchy spanning more than three orders of magnitude between the electron, muon, and tau, with no accepted dynamical explanation for the specific ratios observed – this is the long-standing "flavor puzzle." In the quark sector, the mismatch between the mass eigenbasis and the weak-interaction eigenbasis of the up- and down-type quarks is encoded in the unitary Cabibbo–Kobayashi–Maskawa (CKM) matrix \(V_{\mathrm{CKM}}\), conventionally written in the standard parametrization as a product of three Euler-like rotations through mixing angles \(\theta_{12}, \theta_{13}, \theta_{23}\) and a single physical complex phase \(\delta_{13}\) that is the sole source of CP violation in the quark sector at tree level in the SM. All four parameters, together with the nine \(|V_{ij}|\) moduli they generate, are extracted from a global fit to semileptonic decays, meson mixing, and CP-violating observables (e.g., \(\epsilon_K\), \(\sin 2\beta\)) tabulated by the Particle Data Group. The Jarlskog invariant \(J\) packages \(\theta_{12},\theta_{13},\theta_{23},\delta_{13}\) into the unique rephasing- and basis-independent scalar measure of CP violation, and its smallness relative to the maximal value permitted by unitarity (\(J_{\max}=1/(6\sqrt3)\approx0.096\)) is what makes CP-violating effects in flavor-conserving SM processes generically tiny.
| Quantity | Symbol | Value |
|---|---|---|
| Muon-to-electron mass ratio | $m_\mu/m_e$ | $206.7682830(46)$ |
| Tau-to-electron mass ratio | $m_\tau/m_e$ | $3477.23(23)$ |
| Cabibbo angle | $\theta_{12}$ | $13.02(4)\,^\circ$ |
| CKM mixing angle (13) | $\theta_{13}$ | $0.212(6)\,^\circ$ |
| CKM mixing angle (23) | $\theta_{23}$ | $2.40(5)\,^\circ$ |
| CKM CP-violating phase | $\delta_{13}$ | $1.144(27)\,\mathrm{rad}$ |
| Jarlskog invariant | $J$ | $3.08(15)\times 10^{-5}$ |
Muon-to-electron mass ratio
\(m_\mu/m_e\): \(206.7682830(46)\).
\(m_\mu/m_e\)(ratio of the independently measured muon and electron pole masses).
The muon-to-electron mass ratio is one of the most precisely known numbers in particle physics, determined from the independently measured muon mass (itself pinned down via muonium hyperfine spectroscopy and muon spin rotation in magnetic fields) and the electron mass from Penning-trap \(g\)-factor measurements. Both leptons couple identically to the electroweak gauge fields and differ only through their Yukawa couplings to the Higgs field, so this ratio is, in the Standard Model, simply the ratio of two a priori independent, unrelated Yukawa parameters. No symmetry or dynamical principle within the SM fixes or explains why \(m_\mu/m_e\approx207\) rather than some other value; this is the leptonic instance of the flavor puzzle. The ratio enters directly in tests of lepton universality, muonium spectroscopy, and the theoretical prediction of the muon's anomalous magnetic moment. [Navas:2024]
Tau-to-electron mass ratio
\(m_\tau/m_e\): \(3477.23(23)\).
\(m_\tau/m_e\)(ratio of the measured tau and electron pole masses).
The tau lepton mass, and hence this ratio, is measured principally from the tau pair production threshold scan in \(e^+e^-\) collisions (BES, Belle) via precise reconstruction of the threshold cross-section shape, giving a much larger relative uncertainty than the muon case because the tau's short lifetime prevents the ultra-precise spectroscopic techniques available for the muon and electron. As with \(m_\mu/m_e\), the Standard Model provides three independent Yukawa couplings for the charged leptons with no built-in relation between them, so the value \(m_\tau/m_e\approx3477\) is an unexplained input rather than a derived consequence of any known symmetry. Together with \(m_\mu/m_e\), this ratio is a central piece of evidence for the flavor puzzle: three generations of otherwise identical particles (same electroweak quantum numbers) with masses set by numerically unrelated, hierarchically spread coupling constants that the SM does not predict. [Navas:2024]
Cabibbo angle
\(\theta_{12}\): \(13.02(4)\,^\circ\).
\(\sin\theta_{12} = \abs{V_{us}}/\sqrt{\abs{V_{ud}}^2+\abs{V_{us}}^2}\)(standard parametrization of \(V_{\mathrm{CKM}}\)).
The Cabibbo angle is the largest of the three CKM mixing angles, first introduced by Cabibbo in 1963 to explain the suppression of strangeness-changing weak decays relative to strangeness-conserving ones, well before the third quark generation was known to exist. It is extracted from a combination of superallowed nuclear \(\beta\) decays and pion/kaon semileptonic decays (fixing \(\abs{V_{ud}}\) and \(\abs{V_{us}}\)) together with the CKM unitarity constraint. Its near-maximal size relative to \(\theta_{13}\) and \(\theta_{23}\) reflects the empirical fact that quark mixing is strongly hierarchical, concentrated between the first two generations, with the corresponding value \(\sin\theta_{12}\approx0.225\) often denoted the Wolfenstein parameter \(\lambda\). [Navas:2024]
CKM mixing angle (13)
\(\theta_{13}\): \(0.212(6)\,^\circ\).
\(\sin\theta_{13} = \abs{V_{ub}}\)(standard parametrization of \(V_{\mathrm{CKM}}\)).
This is the smallest of the three CKM mixing angles, governing the direct coupling between the first- and third-generation quarks. It is fixed essentially by \(\abs{V_{ub}}\), determined from semileptonic \(b\to u\ell\nu\) transitions in both inclusive and exclusive (\(B\to\pi\ell\nu\)) decay channels; a persistent few-percent-level tension between the inclusive and exclusive extractions of \(\abs{V_{ub}}\) remains one of the more actively studied discrepancies in flavor physics. The extreme smallness of \(\theta_{13}\) compared to \(\theta_{12}\) is part of the same unexplained mixing hierarchy that characterizes the quark sector. [Navas:2024]
CKM mixing angle (23)
\(\theta_{23}\): \(2.40(5)\,^\circ\).
\(\sin\theta_{23} = \abs{V_{cb}}/\sqrt{\abs{V_{cb}}^2+\abs{V_{tb}}^2}\approx\abs{V_{cb}}\)(standard parametrization of \(V_{\mathrm{CKM}}\)).
This angle governs mixing between the second and third quark generations and is fixed principally by \(\abs{V_{cb}}\), measured from semileptonic \(B\to D^{(*)}\ell\nu\) exclusive decays and from inclusive \(b\to c\ell\nu\) moment analyses. As with \(\theta_{13}\), mild tension persists between inclusive and exclusive determinations, currently at roughly the \(1\)–\(2\sigma\) level and under continued scrutiny with improved lattice QCD form factors. Its intermediate size, between the large \(\theta_{12}\) and the tiny \(\theta_{13}\), completes the hierarchical pattern of quark mixing angles that has no accepted first-principles explanation within the Standard Model. [Navas:2024]
CKM CP-violating phase
\(\delta_{13}\): \(1.144(27)\,\mathrm{rad}\).
measured directly, as the complex phase of \(V_{\mathrm{CKM}}\) in the standard (Chau–Keung) parametrization, from a global fit to CP-violating and CP-conserving flavor observables.
The phase \(\delta_{13}\) is the single irreducible complex parameter of the CKM matrix once unphysical quark-field rephasings are removed, and it is the origin of all CP violation observed in flavor-changing weak processes within the Standard Model, first evidenced in neutral kaon decays and later confirmed in the \(B\)-meson system by the BaBar and Belle experiments. It is extracted from a global fit combining CP-violating observables, most importantly the indirect CP violation parameter \(\epsilon_K\) in the kaon system and the mixing-induced CP asymmetry \(\sin2\beta\) measured in \(B^0\to J/\psi K_S^0\) decays, together with the CP-conserving \(\abs{V_{ij}}\) measurements that fix the angles. That a single phase in a \(3\times3\) unitary mixing matrix accounts for essentially all quark-sector CP violation measured to date is itself a strong, quantitatively successful prediction of the three-generation Standard Model, dating to Kobayashi and Maskawa's 1973 proposal. [Navas:2024]
Jarlskog invariant
\(J\): \(3.08(15)\times 10^{-5}\).
The Jarlskog invariant \(J\) is the unique combination of CKM angles and phase that is invariant under rephasing of the quark fields, and it appears as an overall multiplicative factor in every CP-violating asymmetry computable in the Standard Model quark sector, making it the single basis-independent scalar measure of how much CP violation the CKM mechanism can generate. Its value is obtained by propagating the fitted \(\theta_{12},\theta_{13},\theta_{23},\delta_{13}\) (or equivalently the fitted \(V_{ij}\) moduli and phase convention) through the defining combination above. That \(J\sim3\times10^{-5}\) is many orders of magnitude below both the naive \(\mathcal{O}(1)\) scale set by the largest mixing angle and the unitarity bound \(J_{\max}=1/(6\sqrt3)\approx0.096\) is the quantitative statement of why CP-violating effects, while firmly established, are numerically small throughout the flavor-conserving sector of the Standard Model – and why the observed matter-antimatter asymmetry of the universe cannot be explained by SM CP violation alone. [Navas:2024]
Neutrino Oscillations and the PMNS Matrix
Neutrino flavour oscillation, observed in solar, atmospheric, reactor, and accelerator beams, established that neutrinos are massive and that the weak-interaction flavour eigenstates \(\nu_e,\nu_\mu,\nu_\tau\) are related to the mass eigenstates \(\nu_1,\nu_2,\nu_3\) by a unitary mixing matrix, the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix \(U\), in direct analogy with the CKM matrix of the quark sector. In the standard three-flavour parametrization, \(U\) is written as a product of three rotations through the angles \(\theta_{12}\), \(\theta_{13}\), \(\theta_{23}\), one Dirac CP-violating phase \(\delta_{CP}\), and (if neutrinos are Majorana particles) two additional Majorana phases that do not affect oscillation probabilities. Oscillation probabilities depend on the mixing angles and on the two independent mass-squared splittings \(\Delta m^2_{21}\) and \(\Delta m^2_{32}\) (equivalently \(\Delta m^2_{31}=\Delta m^2_{32}+\Delta m^2_{21}\)), never on the absolute neutrino masses themselves, which remain a separate and still-open experimental question tied to the equally open question of whether the neutrino mass term is of Dirac or Majorana type. Values below are global-fit results as compiled in the Particle Data Group review of neutrino mixing.
| Quantity | Symbol | Value |
|---|---|---|
| Solar mixing angle | $\theta_{12}$ | $33.41(75)^\circ$ |
| Reactor mixing angle | $\theta_{13}$ | $8.54(14)^\circ$ |
| Atmospheric mixing angle | $\theta_{23}$ | $49.1(12)^\circ$ |
| Solar mass-squared splitting | $\Delta m^2_{21}$ | $7.53(18)\times 10^{-5}\,\mathrm{eV}^{2}$ |
| Atmospheric mass-squared splitting | $\Delta m^2_{32}$ | $2.453(34)\times 10^{-3}\,\mathrm{eV}^{2}$ |
| PMNS CP-violating phase (open) | $\delta_{CP}$ | $197(50)^\circ$ |
| Absolute neutrino mass scale and Dirac/Majorana nature (open) | $m_i,\ \textstyle\sum_i m_i$ | $0.12\,\mathrm{eV}$ (see text) |
Solar mixing angle
\(\theta_{12}\): \(33.41(75)^\circ\).
\(\theta_{12}=\arcsin\sqrt{\sin^2\theta_{12}},\quad \sin^2\theta_{12}=0.303(11)\) measured directly.
The solar mixing angle governs the \(\nu_2\)–\(\nu_1\) admixture and, together with \(\Delta m^2_{21}\), sets the survival probability \(P(\nu_e\to\nu_e)\) observed in solar-neutrino experiments (Homestake, GALLEX/SAGE, Super-Kamiokande, SNO) and in the long-baseline reactor experiment KamLAND, which probes the same oscillation parameters in vacuum rather than through solar matter effects. Because \(\theta_{12}\) is unambiguously bounded away from both \(0\) and \(45^\circ\), it is often called the best-measured of the three angles once matter (MSW) effects inside the Sun – confirmed decisively by SNO's neutral-current measurement – are accounted for. The value quoted is the current global-fit best fit to the combined solar and KamLAND data set; it is essentially identical for the normal and inverted mass orderings, since ordering sensitivity enters mainly through \(\theta_{13}\) and \(\Delta m^2_{31}\). [Navas:2024]
Reactor mixing angle
\(\theta_{13}\): \(8.54(14)^\circ\).
\(\theta_{13}=\arcsin\sqrt{\sin^2\theta_{13}},\quad \sin^2\theta_{13}=0.0220(7)\) measured directly.
\(\theta_{13}\) is the smallest of the three PMNS angles and was the last to be measured, established at high significance in 2012 by the short-baseline reactor experiments Daya Bay, RENO, and Double Chooz through the disappearance \(P(\bar\nu_e\to\bar\nu_e)\simeq 1-\sin^2(2\theta_{13})\sin^2(\Delta m^2_{31}L/4E)\) at \(\mathcal{O}(1\,\mathrm{km})\) baselines. Although numerically small, a nonzero \(\theta_{13}\) is what makes the CP-violating phase \(\delta_{CP}\) observable in \(\nu_\mu\to\nu_e\) appearance channels and what allows matter-effect sensitivity to the neutrino mass ordering in long-baseline accelerator experiments; its discovery reopened the experimental programme now pursued by T2K, NOvA, JUNO, DUNE, and Hyper-Kamiokande. The value is now known to about three percent precision and is essentially ordering-independent at this level. [Navas:2024]
Atmospheric mixing angle
\(\theta_{23}\): \(49.1(12)^\circ\).
\(\theta_{23}=\arcsin\sqrt{\sin^2\theta_{23}},\quad \sin^2\theta_{23}=0.572(20)\) measured (normal-ordering best fit).
The atmospheric angle controls \(\nu_\mu\) disappearance in atmospheric neutrinos (Super-Kamiokande, IceCube DeepCore) and in accelerator long-baseline experiments (T2K, NOvA, MINOS), and is close to, but not proven exactly equal to, the maximal-mixing value \(45^\circ\). Current global fits place the best fit in the upper octant (\(\theta_{23}>45^\circ\), equivalently \(\sin^2\theta_{23}>0.5\)) as quoted above, with a statistically nearly degenerate solution in the lower octant near \(\sin^2\theta_{23}\approx0.45\); resolving this octant ambiguity, which affects both \(\delta_{CP}\) extraction and mass-ordering sensitivity, is a stated goal of NOvA, T2K-II, DUNE, and Hyper-Kamiokande. The value is only mildly ordering-dependent and is the least precisely known of the three angles, at roughly the four percent level. [Navas:2024]
Solar mass-squared splitting
\(\Delta m^2_{21}\): \(7.53(18)\times 10^{-5}\,\mathrm{eV}^{2}\).
\(\Delta m^2_{21}\equiv m_2^2-m_1^2>0\) (sign fixed by the MSW matter resonance in the Sun); measured directly.
\(\Delta m^2_{21}\) is the smaller of the two independent squared-mass splittings and, together with \(\theta_{12}\), fixes the long oscillation length responsible for solar-neutrino conversion and for the KamLAND reactor-antineutrino oscillation pattern at \(\mathcal{O}(100\,\mathrm{km})\) baselines. Unlike the atmospheric splitting, its sign is unambiguously determined (positive, i.e. \(m_2>m_1\)) because solar matter effects (the MSW mechanism) break the vacuum degeneracy between the two sign choices; this is the only mass-ordering information oscillation experiments obtain without appeal to \(\theta_{13}\)-driven matter effects on the atmospheric splitting. The quoted global-fit value combines solar data with the reactor measurement from KamLAND and is known to about \(2\,\mathrm{\%}\) precision. [Navas:2024]
Atmospheric mass-squared splitting
\(\Delta m^2_{32}\): \(2.453(34)\times 10^{-3}\,\mathrm{eV}^{2}\).
\(\Delta m^2_{32}\equiv m_3^2-m_2^2\); magnitude measured directly, value quoted for Normal Ordering \((m_3>m_2>m_1)\); equivalently \(\Delta m^2_{31}=\Delta m^2_{32}+\Delta m^2_{21}\approx2.528\times 10^{-3}\,\mathrm{eV}^{2}\).
\(\Delta m^2_{32}\) sets the fast oscillation frequency probed by atmospheric neutrinos and by accelerator long-baseline experiments and, together with \(\theta_{23}\), dominates \(\nu_\mu\) disappearance in that regime. Its magnitude is known to about \(1.5\,\mathrm{\%}\), essentially identically whichever mass ordering is assumed, but its sign is not yet established at high statistical significance from oscillation data alone: this sign is precisely the neutrino mass-ordering (normal-vs-inverted) question. Combined global fits including reactor, atmospheric, and accelerator data presently favour Normal Ordering, the convention used above, at roughly the \(2\)–\(3\sigma\) level; JUNO's precision reactor measurement, together with continued running of atmospheric (IceCube-Upgrade, Super-K/Hyper-K) and accelerator (DUNE, T2K-II/NOvA) experiments, is expected to resolve the ordering within the coming decade. [Navas:2024]
PMNS CP-violating phase (open)
\(\delta_{CP}\): \(197(50)^\circ\).
measured via the CP-asymmetry between \(\nu_\mu\to\nu_e\) and \(\bar\nu_\mu\to\bar\nu_e\) appearance probabilities in long-baseline accelerator experiments, combined with the reactor constraint on \(\theta_{13}\).
The Dirac CP phase \(\delta_{CP}\) is the PMNS analogue of the CKM phase and, if nonzero and different from \(\pi\), allows leptonic CP violation in oscillations; it enters the appearance probabilities \(P(\nu_\mu\to\nu_e)\) and \(P(\bar\nu_\mu\to\bar\nu_e)\) measured by T2K and NOvA through terms proportional to \(\sin\delta_{CP}\sin\theta_{13}\). This is genuinely one of the least well-determined parameters in the Standard Model sector: current global fits give a best fit near \(\delta_{CP}\approx197^\circ\), closer to the CP-conserving value \(\pi\) (\(180^\circ\)) than to maximal CP violation (\(-\pi/2\), i.e. \(270^\circ\)); consistent with this, the CP-conserving values (\(0\) or \(\pi\)) are disfavoured only around the
level, so this does not constitute a discovery of leptonic CP violation, and the allowed range widens substantially – covering most of the \([0,2\pi)\) interval – once the mass-ordering degeneracy is marginalized over; T2K and NOvA show some mutual tension in \(\delta_{CP}\) depending on the assumed ordering. Next-generation long-baseline experiments, principally DUNE and Hyper-Kamiokande, are designed to reach
sensitivity to CP violation over a substantial fraction of the \(\delta_{CP}\) range; until such a measurement, the true value – and even the sign of CP violation – remains an open experimental question. [Navas:2024]
Absolute neutrino mass scale and Dirac/Majorana nature (open)
\(m_i,\ \textstyle\sum_i m_i\): \(\sum_i m_i < 0.12\,\mathrm{eV}\ \text{(95\% CL, cosmological bound, model-dependent)}\).
not determined by oscillations, which fix only \(\Delta m^2_{21}\) and \(\Delta m^2_{32}\); bounded independently by \(\beta\)-decay kinematics (KATRIN), cosmology (CMB\(+\)BAO), and \(0\nu\beta\beta\) searches.
Oscillation phases depend only on \(\Delta m^2_{ij}L/E\), so oscillation data are completely insensitive to the overall additive mass scale: they fix the two independent squared-mass splittings but not any individual \(m_i\), nor even the lightest mass \(m_{\rm lightest}\), which could in principle be zero. Three independent, complementary methods bound the absolute scale: direct kinematic spectroscopy of the tritium \(\beta\)-decay endpoint (KATRIN gives \(m_\beta\lesssim0.45\,\mathrm{eV}\) at \(90\,\mathrm{\%}\) CL from its early data run, with a design sensitivity down to about \(0.2\,\mathrm{eV}\)); cosmological structure formation and CMB lensing, which constrain the sum \(\sum_i m_i\) under \(\Lambda\)CDM assumptions (Planck 2018 combined with baryon acoustic oscillation data gives the bound quoted above); and searches for neutrinoless double beta decay (\(0\nu\beta\beta\)), which additionally probe whether the neutrino mass term is of Majorana type. Whether neutrinos are Dirac or Majorana fermions is a genuinely open question with no experimental resolution to date: \(0\nu\beta\beta\) would violate lepton number by two units and can occur only if neutrinos are massive Majorana particles, but current searches (KamLAND-Zen, GERDA, and successors) report no signal, placing only an upper bound on the effective Majorana mass parameter \(m_{\beta\beta}\) at the few-tens-of-\(\mathrm{meV}\) level (with sizeable nuclear-matrix-element uncertainty); a positive detection remains one of the primary open experimental goals of the field, bearing directly on the origin of neutrino mass (e.g. seesaw-type mechanisms) and on leptogenesis as a possible explanation of the cosmic baryon asymmetry. [Navas:2024] [Aghanim:2020]
Lambda-CDM Cosmological Parameters
The base \(\Lambda\)CDM model describes the observed universe with six free parameters fit jointly to the cosmic microwave background (and, for some derived quantities, external distance-ladder and large-scale-structure data): the physical baryon and cold-dark-matter densities \(\Omega_b h^2\) and \(\Omega_c h^2\), the amplitude and tilt of the primordial scalar power spectrum \(A_s\) and \(n_s\), the reionization optical depth \(\tau\), and the acoustic angular scale (from which \(H_0\) and \(\Omega_\Lambda\) are derived under the flatness assumption). All values below are taken from the Planck 2018 final data release, specifically the TT,TE,EE+lowE+lensing combination reported in Planck Collaboration VI, unless otherwise noted. The sector also includes the baryon-to-photon ratio \(\eta\), a derived cross-check quantity linking the CMB-era baryon density to the independent, much earlier Big Bang nucleosynthesis epoch, and it flags explicitly the Hubble tension, the unresolved several-sigma discrepancy between CMB-inferred and locally measured values of \(H_0\) that remains one of the most actively debated open problems in the field.
| Quantity | Symbol | Value |
|---|---|---|
| Baryon density parameter | $\Omega_b h^2$ | $0.02237(15)$ |
| Cold dark matter density parameter | $\Omega_c h^2$ | $0.1200(12)$ |
| Dark energy density parameter | $\Omega_\Lambda$ | $0.6847(73)$ |
| Hubble constant (open) | $H_0$ | $67.36(54)\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ (see text) |
| Scalar spectral index | $n_s$ | $0.9649(42)$ |
| Amplitude of the primordial scalar power spectrum | $A_s$ | $2.101\times 10^{-9}$ (see text) |
| Optical depth to reionization | $\tau$ | $0.0544(73)$ |
| Baryon-to-photon ratio | $\eta$ | $6.12(4)\times 10^{-10}$ |
Baryon density parameter
\(\Omega_b h^2\): \(0.02237(15)\).
\(\Omega_b h^2 \equiv (\rho_b/\rho_{\mathrm{crit},0})\,h^2\); measured directly from the relative heights of the odd\(/\)even acoustic peaks in the CMB temperature and polarization power spectra.
The physical baryon density \(\Omega_b h^2\) fixes the ratio of baryonic (ordinary) matter to the critical density, scaled by \(h^2\) to remove the degeneracy with \(H_0\) that would otherwise plague a bare \(\Omega_b\). It is pinned down by the CMB because baryons load the photon-baryon plasma before recombination, compressing odd acoustic peaks and suppressing even ones relative to a baryon-free plasma; the Planck 2018 TT,TE,EE+lowE+lensing fit gives \(\Omega_b h^2 = 0.02237(15)\), in excellent agreement with the independent value inferred from primordial deuterium abundances via Big Bang nucleosynthesis. This parameter is one of the six free numbers of the base \(\Lambda\)CDM model and is measured to better than one percent precision. [Aghanim:2020]
Cold dark matter density parameter
\(\Omega_c h^2\): \(0.1200(12)\).
\(\Omega_c h^2 \equiv (\rho_c/\rho_{\mathrm{crit},0})\,h^2\); measured directly, chiefly from the amplitude of the third acoustic peak and the overall peak-height envelope of the CMB power spectrum.
\(\Omega_c h^2\) is the physical density of cold (pressureless, collisionless) dark matter, the dominant matter component of the universe. In base \(\Lambda\)CDM all non-baryonic matter is assumed cold, so this parameter is equivalently quoted as the total non-baryonic dark-matter density \(\Omega_{\mathrm{dm}} h^2\); some analyses distinguish \(\Omega_c\) (cold) from a possible hot/warm component, but Planck finds no evidence requiring one. The Planck 2018 fit gives \(\Omega_c h^2 = 0.1200(12)\), roughly \(5.4\) times the baryon density, driven observationally by the enhanced gravitational driving of the third and higher acoustic peaks and by the low-\(\ell\) lensing potential power spectrum. Combined with \(\Omega_b h^2\) this yields a present-day total matter density \(\Omega_m = 0.3153(73)\). [Aghanim:2020]
Dark energy density parameter
\(\Omega_\Lambda\): \(0.6847(73)\).
\(\Omega_\Lambda \equiv \rho_\Lambda/\rho_{\mathrm{crit},0}\); in a spatially flat universe,\(\Omega_\Lambda = 1 - \Omega_m - \Omega_r\), derived from the CMB angular-diameter-distance constraint combined with the measured matter density.
\(\Omega_\Lambda\) is the fractional contribution of the cosmological constant (or, phenomenologically, dark energy with equation of state consistent with \(w=-1\)) to the total energy density today. It is not measured by a single spectral feature but emerges from the combination of the CMB's precise measurement of the angular acoustic scale \(\theta_*\) (which fixes the comoving distance to last scattering) together with the flatness prior and the measured matter density, and is corroborated independently by baryon acoustic oscillation (BAO) distance measurements and Type Ia supernova Hubble-diagram fits. Planck 2018 (TT,TE,EE+lowE+lensing) gives \(\Omega_\Lambda = 0.6847(73)\), confirming that dark energy dominates the present energy budget by roughly a factor of two over matter. Its microphysical origin — vacuum energy, a slowly rolling field, or a modification of gravity — remains unidentified, though its phenomenological value is measured to sub-percent precision. [Aghanim:2020]
Hubble constant (open)
\(H_0\): \(67.36(54)\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) (Planck 2018, TT,TE,EE+lowE+lensing, CMB-inferred within base \(\Lambda\)CDM); local distance-ladder determinations give values near \(73.0(1.0)\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\) [Riess:2022], in tension with the Planck value at the \(4\)–\(5\sigma\) level.
\(H_0 \equiv \dot a/a\big|_{t_0}\), the present-day cosmic expansion rate; measured either model-dependently from the CMB acoustic scale within an assumed \(\Lambda\)CDM background, or directly and locally from the distance-redshift relation of the cosmic distance ladder.
\(H_0\) sets the present expansion rate of the universe and, through \(1/H_0\), the characteristic age and size scale of the observable universe. Two broad classes of measurement exist: CMB/early-universe methods, which infer \(H_0\) indirectly by fitting the full \(\Lambda\)CDM model (including \(\Omega_b h^2\), \(\Omega_c h^2\), and the acoustic scale \(\theta_*\)) to the Planck power spectra and obtain \(H_0 = 67.36(54)\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\); and late-universe, model-independent distance-ladder methods (Cepheid-calibrated Type Ia supernovae, and independently the tip-of-the-red-giant-branch and Mira-variable calibrations), which cluster around \(73\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}\). The discrepancy between these two families of measurement — the Hubble tension — now exceeds \(4\)–\(5\sigma\) in the highest-precision comparisons and has resisted resolution by known systematics on either side; it is one of the most actively studied open problems in cosmology, with proposed resolutions ranging from unrecognized systematics to genuine early- or late-time extensions of \(\Lambda\)CDM (e.g. early dark energy, modified recombination physics), none yet confirmed. This entry reports the Planck (CMB) value as the primary \(\Lambda\)CDM-model parameter but flags the tension explicitly rather than presenting a single settled number. [Aghanim:2020] [Riess:2022]
Scalar spectral index
\(n_s\): \(0.9649(42)\).
power-law index of the primordial scalar (curvature) power spectrum,\(\mathcal{P}_{\mathcal{R}}(k) \propto k^{n_s - 1}\); measured directly from the scale dependence (tilt) of the CMB TT, TE, and EE power spectra across the observed multipole range.
\(n_s\) characterizes the near-scale-invariant tilt of the spectrum of primordial curvature perturbations laid down (in the standard picture) during an early inflationary epoch: \(n_s = 1\) would correspond to exact scale invariance (Harrison-Zel'dovich), while the measured value slightly below unity indicates a small red tilt. Planck 2018 measures \(n_s = 0.9649(42)\) from the shape of the CMB angular power spectrum, excluding exact scale invariance at high significance (\(n_s = 1\) is disfavored by more than \(8\sigma\)). This is one of the most precisely determined \(\Lambda\)CDM parameters and is a key discriminant among inflationary models, though the underlying inflaton dynamics (if any) remain outside the evidence-based scope of this treatise. [Aghanim:2020]
Amplitude of the primordial scalar power spectrum
\(A_s\): \(2.101\times 10^{-9}\) at pivot scale \(k_0 = 0.05\,/\mathrm{Mpc}\), conventionally reported as \(\ln(10^{10}A_s) = 3.044(14)\); the corresponding late-time, nonlinearly evolved amplitude is \(\sigma_8 = 0.8111(60)\).
\(A_s \equiv \mathcal{P}_{\mathcal{R}}(k_0)\), the amplitude of the dimensionless primordial curvature power spectrum at the pivot scale; measured directly from the overall (multipole-averaged) amplitude of the CMB temperature and polarization power spectra, degenerate with \(\tau\) through the combination \(A_s e^{-2\tau}\).
\(A_s\) fixes the overall normalization of the primordial curvature perturbations from which all later structure grows; because CMB anisotropies are damped by rescattering after reionization, the direct CMB observable is the combination \(A_s e^{-2\tau}\), so an independent handle on \(\tau\) (from large-scale polarization) is needed to break the degeneracy. Planck 2018 reports \(\ln(10^{10}A_s) = 3.044(14)\), i.e. \(A_s \approx 2.101\times 10^{-9}\). For late-time, nonlinear structure the amplitude is more commonly quoted as \(\sigma_8\), the rms mass-density fluctuation in spheres of radius \(8\,\mathrm{Mpc}\) obtained by evolving \(A_s\) forward through the transfer function and linear growth factor; Planck finds \(\sigma_8 = 0.8111(60)\). This treatise uses \(A_s\) (with \(\ln(10^{10}A_s)\)) as the primary \(\Lambda\)CDM parameter and cites \(\sigma_8\) as its late-time, structure-formation-friendly equivalent. [Aghanim:2020]
Optical depth to reionization
\(\tau\): \(0.0544(73)\).
\(\tau \equiv \int_0^{t_0} n_e(t)\,\sigma_T\, c\) , dt, the Thomson-scattering optical depth integrated from the observer back through the reionized intergalactic medium; measured directly from the suppression and large-angle bump of the CMB E-mode polarization power spectrum at low multipoles.
\(\tau\) measures the integrated column of free electrons that CMB photons scatter from after the universe reionizes (following the formation of the first stars and galaxies), which damps the temperature and E-mode power spectra on scales inside the horizon at reionization and generates a distinctive large-angle polarization bump. Planck 2018's low-\(\ell\) EE likelihood gives \(\tau = 0.0544(73)\), corresponding to an inferred midpoint reionization redshift around \(z_{\rm re}\sim 7\)–\(9\) under simple reionization histories, consistent with the epoch inferred from quasar Gunn-Peterson trough studies. Because \(\tau\) is degenerate with \(A_s\) in the small-scale CMB spectra (only the combination \(A_s e^{-2\tau}\) is well constrained there), its independent determination from large-angle polarization is essential to breaking that degeneracy and obtaining \(A_s\) itself. [Aghanim:2020]
Baryon-to-photon ratio
\(\eta\): \(6.12(4)\times 10^{-10}\).
\(\eta \equiv n_b/n_\gamma \approx 273.9\,\Omega_b h^2 \times 10^{-10}\), derived by combining the CMB-measured baryon density \(\Omega_b h^2\) with the photon number density fixed by the measured CMB blackbody temperature.
The baryon-to-photon ratio \(\eta\) (equivalently often quoted as \(\eta_{10} \equiv 10^{10}\eta\)) measures the tiny excess of baryons over antibaryons that survived annihilation in the early universe, expressed relative to the number density of CMB photons; it is the single parameter controlling the light-element yields (D, \(^3\)He, \(^4\)He, \(^7\)Li) of Big Bang nucleosynthesis. It is not measured independently in this catalog but is derived as a cross-check quantity from the CMB-fit baryon density \(\Omega_b h^2 = 0.02237(15)\) combined with the precisely known photon number density, giving \(\eta = 6.12(4)\times 10^{-10}\). The remarkable agreement between this CMB-derived value and the value independently required to match observed primordial deuterium and helium abundances is one of the strongest quantitative successes of standard Big Bang cosmology, cross-validating two entirely different physical epochs (nucleosynthesis at \(t\sim1\,\mathrm{s}\)–\(200\,\mathrm{s}\) versus recombination at \(t\sim380000\,\mathrm{yr}\)). [Aghanim:2020]
Gravitational Coupling, Planck Units, and the Hierarchy Problem
This sector collects the scales at which the treatise's two most disparate confirmed frameworks – quantum mechanics, governed by \(\hbar\), and general relativity, governed by \(G\) and \(c\) – become comparably important, together with the dimensionless measures of just how far current particle-physics scales sit from that regime. The Planck length, mass, and time are not free parameters but are fixed by dimensional analysis alone: the Buckingham \(\pi\) theorem (Theorem 2.3), proved in Section 2.2.1 from the rank-nullity theorem applied to the matrix of dimensional exponents, shows that \(\{\hbar,G,c\}\) admit no dimensionless combination among themselves, which is precisely the condition that makes the length, mass, and time built from them unique up to an unresolvable \(O(1)\) prefactor. The gravitational coupling constant \(\alpha_G\) then expresses, in the same dimensionless language as the fine-structure constant, how weak gravity is between two protons compared with the other measured interactions. The final three entries are mass ratios between confirmed particle-physics scales and the Planck mass; their extreme smallness is directly observed and uncontroversial, but the absence of any experimentally confirmed dynamical principle that explains why the scales are so separated – especially for the Higgs vacuum expectation value, whose radiative stability is not protected by any confirmed symmetry – constitutes the hierarchy (naturalness) problem, recorded here strictly as an open observational puzzle per this treatise's evidence-only scope.
| Quantity | Symbol | Value |
|---|---|---|
| Planck length | $\ell_P$ | $1.616255(18)\times 10^{-35}\,\mathrm{m}$ |
| Planck mass | $M_P$ | $2.176434(24)\times 10^{-8}\,\mathrm{kg}$ |
| Planck time | $t_P$ | $5.391247(60)\times 10^{-44}\,\mathrm{s}$ |
| Gravitational coupling constant | $\alpha_G$ | $5.91\times 10^{-39}$ |
| Electron-to-Planck mass ratio (open) | $m_e/M_P$ | $4.19\times 10^{-23}$ |
| Proton-to-Planck mass ratio (open) | $m_p/M_P$ | $7.69\times 10^{-20}$ |
| Higgs vacuum expectation value to Planck mass ratio (open) | $v/M_P$ | $2.01\times 10^{-17}$ |
Planck length
\(\ell_P\): \(1.616255(18)\times 10^{-35}\,\mathrm{m}\).
\(\ell_P = \sqrt{\dfrac{\hbar G}{c^3}}\).
The Planck length is the unique length scale constructible from the reduced Planck constant \(\hbar\), the Newtonian gravitational constant \(G\), and the speed of light \(c\) – the three constants that respectively govern quantum mechanics, gravitation, and relativity. Its dimensional exponents are fixed, not merely suggested, by the Buckingham \(\pi\) theorem (Theorem 2.3): among the three quantities \(\{\hbar, G, c\}\), whose dimension exponents span the full \(r=3\)-dimensional space of mechanical base dimensions \(\{\mathsf{M},\mathsf{L},\mathsf{T}\}\), the theorem's exponent matrix \(M\) has trivial kernel (\(n-r=3-3=0\)), so no dimensionless combination of \(\hbar\), \(G\), \(c\) alone exists; augmenting \(M\) with the target dimension \(\mathsf{L}\) then yields a \(3\times3\) linear system in the exponents of \(\hbar^a G^b c^d\) with a unique solution \(a=\tfrac12,\ b=\tfrac12,\ d=-\tfrac32\), i.e. \(\ell_P=\sqrt{\hbar G/c^3}\). The same rank-nullity argument that proves the theorem shows explicitly what it cannot deliver: any dimensionless numerical prefactor multiplying \(\ell_P\) is invisible to dimensional analysis and is fixed here by convention to \(1\). Numerically, using the CODATA 2018 values of \(\hbar\), \(G\), \(c\) tabulated in Physical Constants and SI Units, \(\ell_P \approx 1.6\times 10^{-35}\,\mathrm{m}\), some fifteen orders of magnitude below the smallest length probed directly by any collider experiment to date. [Tiesinga:2021]
Planck mass
\(M_P\): \(2.176434(24)\times 10^{-8}\,\mathrm{kg}\).
\(M_P = \sqrt{\dfrac{\hbar c}{G}}\).
By the same application of Theorem 2.3 used for the Planck length – the exponent system for a quantity of dimension \(\mathsf{M}\) built from \(\{\hbar,G,c\}\) has a unique solution because the three quantities admit no dimensionless combination among themselves – the Planck mass is fixed up to the same unresolvable \(O(1)\) prefactor as \(M_P=\sqrt{\hbar c/G}\). Equivalently \(M_P = \hbar/(\ell_P c)\). Its numerical value, about \(22\,\mu\mathrm{g}\), is unremarkable on laboratory scales – comparable to a fine dust grain – which is itself notable: the mass scale conventionally associated with quantum-gravitational effects becoming significant is macroscopic in mass even though the associated length (\(\ell_P\)) and energy (\(M_Pc^2\approx1.22\times 10^{19}\,\mathrm{GeV}\)) are wildly inaccessible to any existing or foreseeable particle experiment. [Tiesinga:2021]
Planck time
\(t_P\): \(5.391247(60)\times 10^{-44}\,\mathrm{s}\).
\(t_P = \sqrt{\dfrac{\hbar G}{c^5}} = \ell_P/c\).
Completing the triad, the Planck time is the unique time scale formed from \(\{\hbar,G,c\}\), fixed by exactly the same rank-3 exponent argument furnished by Theorem 2.3: \(t_P=\sqrt{\hbar G/c^5}=\ell_P/c\), again with a convention-fixed prefactor of \(1\) that dimensional analysis alone cannot supply. Numerically \(t_P\approx5.4\times 10^{-44}\,\mathrm{s}\), conventionally quoted as the timescale below which no confirmed physical theory – classical general relativity least of all, since it treats spacetime as a smooth continuum – is expected to remain a reliable description; no experiment has probed timescales anywhere near this regime. [Tiesinga:2021]
Gravitational coupling constant
\(\alpha_G\): \(5.91\times 10^{-39}\).
\(\alpha_G \equiv \dfrac{G m_p^2}{\hbar c}\).
In direct structural analogy with the electromagnetic fine-structure constant \(\alpha = e^2/(4\pi\epsilon_0\hbar c)\), which measures the strength of the electromagnetic coupling between two elementary charges in dimensionless form, the gravitational coupling constant \(\alpha_G\) measures the strength of the Newtonian gravitational attraction between two protons in the same dimensionless units. It is built entirely from constants already tabulated in Physical Constants and SI Units: \(G\), the proton mass \(m_p\), \(\hbar\), and \(c\). Its value, \(\alpha_G\approx5.9\times 10^{-39}\), is smaller than \(\alpha\) by roughly thirty-six orders of magnitude, making \(\alpha_G\) the quantitative, dimensionless measure of the empirical fact that gravity is by far the weakest of the interactions accessible to laboratory and particle-physics measurement. [Tiesinga:2021]
Electron-to-Planck mass ratio (open)
\(m_e/M_P\): \(4.19\times 10^{-23}\).
\(m_e/M_P,\quad m_e\) measured,\(\ M_P=\sqrt{\hbar c/G}\).
The electron mass, measured directly, is smaller than the Planck mass by a factor of about \(4.2\times 10^{-23}\). For a chiral fermion such as the electron, standard effective-field-theory power counting makes this smallness technically natural: radiative corrections to a fermion mass are proportional to the mass itself (protected by the approximate chiral symmetry restored as \(m_e\to0\)) rather than being pulled up quadratically toward the ultraviolet cutoff, so no fine-tuning is strictly required to keep \(m_e\) small under quantum corrections. The ratio is nonetheless recorded here as part of the same hierarchy-of-scales puzzle: no confirmed dynamical principle explains why the Yukawa coupling that ultimately sets \(m_e\) (via the Higgs mechanism, Part XI — Quantum Field Theory and the Standard Model) takes the minuscule value it does rather than some other value of order unity; that the coupling itself is unexplained – not that it is radiatively unstable – is the open question here. [Tiesinga:2021]
Proton-to-Planck mass ratio (open)
\(m_p/M_P\): \(7.69\times 10^{-20}\).
\(m_p/M_P,\quad m_p\) measured,\(\ M_P=\sqrt{\hbar c/G}\).
The proton mass is smaller than the Planck mass by a factor of about \(7.7\times 10^{-20}\), some three orders of magnitude larger a ratio than the electron's because most of the proton's mass arises not from the Higgs mechanism but from the strong-interaction confinement scale \(\Lambda_{\mathrm{QCD}}\) (Part XI — Quantum Field Theory and the Standard Model), with the current-quark Higgs-generated masses contributing only a small fraction. The ratio is nonetheless part of the same observed hierarchy: no confirmed dynamical mechanism explains why \(\Lambda_{\mathrm{QCD}}\), and hence the mass of ordinary baryonic matter, sits so far below the Planck scale rather than comparably close to it, even though the running of the strong coupling (asymptotic freedom, itself experimentally confirmed) makes the smallness only logarithmically – not quadratically – sensitive to the ultraviolet completion, unlike the scalar case below. [Tiesinga:2021]
Higgs vacuum expectation value to Planck mass ratio (open)
\(v/M_P\): \(2.01\times 10^{-17}\).
\(v/(M_P c^2),\quad v \approx 246\,\mathrm{GeV}\)(measured),\(\ M_Pc^2 \approx 1.22\times 10^{19}\,\mathrm{GeV}\).
The Higgs vacuum expectation value \(v\approx246\,\mathrm{GeV}\), which sets the electroweak scale and, through it, the masses of the \(W\) and \(Z\) bosons and (via Yukawa couplings) the charged fermions, is smaller than the Planck energy \(M_Pc^2\approx1.22\times 10^{19}\,\mathrm{GeV}\) by a factor of about \(2.0\times 10^{-17}\). This is the ratio at the heart of the hierarchy (naturalness) problem: unlike a fermion mass, the Higgs boson is a fundamental scalar, and nothing within the confirmed Standard Model protects its mass-squared parameter from receiving radiative corrections that scale quadratically with whatever ultraviolet cutoff the theory is embedded in – if that cutoff is the Planck scale, the corrections are technically expected to be some seventeen orders of magnitude larger than the observed electroweak scale itself, requiring an extreme and unexplained cancellation to keep \(v\) as small as it is measured to be. No experimentally confirmed symmetry or dynamical mechanism protecting the electroweak scale from such corrections has been observed to date (in particular, no signal of any such protecting structure has been seen at the LHC), so the smallness of \(v/M_P\) remains, strictly on the evidence, an open naturalness puzzle rather than a solved problem. [Navas:2024] [Tiesinga:2021]