Pion · R.A.V.E.N. Living Encyclopedia

The pion (π) is a meson of spin 0 and negative parity, forming an isospin triplet whose members have rest energies 139.570 39(18) MeV for π^± and 134.9768(5) MeV for π⁰ (Navas et al. 2024). It is a quark–antiquark bound state of quantum chromodynamics with valence content ud̄, ūd and the neutral combination of uū and dd̄, and it is the pseudo-Goldstone boson of the spontaneously broken chiral symmetry of the light-quark sector, so that its mass squared is proportional to the quark masses (Gell-Mann et al. 1968). As a spin-0 excitation it is described by a single-component field obeying the Klein–Gordon equation, and in a pionic atom it occupies a stationary state whose squared modulus gives the density of the state. The charged states carry electric charge ±e and source a Coulomb field, while the neutral state is uncharged and couples to two photons through the axial anomaly (Adler 1969; Bell & Jackiw 1969). The pion participates in the strong, electromagnetic and weak interactions. Elastic scattering from atomic electrons gives a root-mean-square charge radius of 0.659(4) fm for the charged pion (Navas et al. 2024), the average being dominated by the measurement of Amendolia et al. (1986), and the decay constant that sets the leptonic decay rate is f_π = 130.2(1.2) MeV from lattice quantum chromodynamics (Aoki et al. 2022; Navas et al. 2024). The charged pion decays by the weak interaction with a mean life of 2.6033(5) × 10⁻⁸ s, and the neutral pion decays electromagnetically with a mean life of 8.43(13) × 10⁻¹⁷ s (Navas et al. 2024). Pion exchange carries the long-range part of the nucleon–nucleon interaction, and pions produced when cosmic rays strike interstellar gas decay to the gamma rays observed from supernova remnants (Ackermann et al. 2013). The charged pion was identified in photographic emulsions exposed to cosmic rays by Lattes et al. (1947), and the neutral pion in the two-photon decays of mesons photoproduced at a synchrotron by Steinberger et al. (1950).

02 · Equations

Governing equations

The pion is the pseudo-Goldstone boson of the spontaneously broken chiral symmetry of quantum chromodynamics, and its low-energy behaviour is described by chiral perturbation theory. Relations are written in SI units or in natural units where stated; numerical values are from the Particle Data Group review (Navas et al. 2024).

Master · Chiral Lagrangian

Leading-order chiral Lagrangian

$$\mathcal L_2=\frac{f^2}{4}\,\mathrm{Tr}\!\left(\partial_\mu U\,\partial^\mu U^\dagger\right)+\frac{f^2}{4}\,\mathrm{Tr}\!\left(\chi U^\dagger+U\chi^\dagger\right),\qquad U=\exp\!\left(\frac{i\sqrt2\,\Phi}{f}\right)$$

Φ is the matrix of pion fields and χ carries the quark masses. The constant f in this form is the one in which f = f_π/√2 = 92.1 MeV (Gasser & Leutwyler 1984; Navas et al. 2024); the remaining relations on this page use f_π = 130.2(1.2) MeV. The mass term, the pion–pion scattering amplitudes and the low-energy constants of the relations below follow from this Lagrangian as an expansion in momenta and quark masses (Gasser & Leutwyler 1984).

01 · Relativistic wave equation

Klein–Gordon equation

$$\begin{gathered}\left(\Box+\frac{m_\pi^2c^2}{\hbar^2}\right)\phi=0,\\ E^2=p^2c^2+m_\pi^2c^4\end{gathered}$$

A spin-0 field has a single component, so no spin axis is defined. The parity of the pion is negative, which makes the field pseudoscalar and fixes the selection rules in its production and decay (Navas et al. 2024).

02 · Nuclear force

Yukawa potential and range

$$\begin{gathered}V(r)=-\frac{g^2}{4\pi}\,\frac{e^{-r/\lambda_\pi}}{r},\\ \lambda_\pi=\frac{\hbar}{m_\pi c}=1.4138\ \mathrm{fm}\end{gathered}$$

Exchange of a massive field gives a potential with an exponential cut-off at the reduced Compton wavelength of that field. For the pion that length is λ_π = 1.4138 fm (computed from Navas et al. 2024), and it sets the range of the one-pion-exchange part of the nucleon–nucleon interaction (Yukawa 1935).

03 · Mass

Gell-Mann–Oakes–Renner relation

$$m_\pi^2f_\pi^2=-\left(m_u+m_d\right)\langle\bar qq\rangle$$

The pion mass squared is proportional to the sum of the up and down quark masses and to the quark condensate, so it vanishes in the chiral limit (Gell-Mann et al. 1968). In the normalisation f_π = 130.2 MeV used here the condensate is ⟨q̄q⟩ = ⟨ūu + d̄d⟩ (Navas et al. 2024). The relation follows from the pion being the Goldstone boson of the spontaneously broken axial symmetry.

04 · Leptonic decay

Decay width and helicity suppression

$$\begin{gathered}\Gamma(\pi\to\ell\nu)=\frac{G_F^2}{8\pi}f_\pi^2\left|V_{ud}\right|^2m_\ell^2m_\pi\left(1-\frac{m_\ell^2}{m_\pi^2}\right)^{2},\\ \frac{\Gamma(e\nu)}{\Gamma(\mu\nu)}=1.230(4)\times10^{-4}\end{gathered}$$

The rate carries a factor m_ℓ², a consequence of angular-momentum conservation for a spin-0 parent decaying through a left-handed current, and it suppresses the electron channel. The measured ratio is 1.230(4) × 10⁻⁴ (Navas et al. 2024), in agreement with the calculation including radiative corrections (Cirigliano & Rosell 2007). The decay constant in this normalisation is f_π = 130.2(1.2) MeV (Navas et al. 2024).

05 · Anomaly

Two-photon decay of the neutral pion

$$\begin{gathered}\Gamma\!\left(\pi^0\to\gamma\gamma\right)=\frac{\alpha^2m_{\pi^0}^3c^4}{32\pi^3f_\pi^2\hbar},\\ \Gamma_{\rm LO}=7.79\ \mathrm{eV}\end{gathered}$$

The two-photon width is fixed at leading order by the axial anomaly, with no free parameter (Adler 1969; Bell & Jackiw 1969). The leading-order value in the normalisation f_π = 130.2 MeV is 7.79 eV (computed from Navas et al. 2024), and the measured width is 7.80(12) eV (Larin et al. 2020).

06 · Size

Electromagnetic form factor

$$\begin{gathered}F_\pi(Q^2)\simeq\frac{1}{1+Q^2/m_\rho^2},\\ \left\langle r^2\right\rangle=-6\hbar^2\left.\frac{dF_\pi}{dQ^2}\right|_{Q^2=0}\end{gathered}$$

Elastic scattering of pions from atomic electrons measures the space-like form factor, and its slope at zero momentum transfer gives a root-mean-square charge radius of 0.659(4) fm (Amendolia et al. 1986; Navas et al. 2024). The pole form with the ρ mass describes the data at low momentum transfer.

07 · Scattering

Pion–nucleon resonance

$$\begin{gathered}\sigma=\frac{8\pi}{k^2}\,\frac{\Gamma^2/4}{\left(\sqrt s-M_\Delta\right)^2+\Gamma^2/4},\\ M_\Delta=1232\ \mathrm{MeV},\quad \Gamma=117\ \mathrm{MeV}\end{gathered}$$

Elastic π⁺p scattering proceeds through the Δ(1232) resonance of isospin 3/2 and spin-parity 3/2⁺, whose Breit–Wigner mass is 1232 MeV and width 117 MeV (Navas et al. 2024). The prefactor 8π/k² is the unitarity bound on a single elastic channel of that spin.

08 · Flight

Decay length in a beam

$$\begin{gathered}L=\beta\gamma\,c\tau,\qquad c\tau=7.8045\ \mathrm{m},\\ r=\frac{p}{eB}\end{gathered}$$

The mean flight distance of a charged pion is βγcτ with cτ = 7.8045 m (computed from Navas et al. 2024), which sets the length of a pion beam line, and the second relation gives the bend radius in a uniform transverse field. For the neutral pion cτ = 25.3 nm (computed from Navas et al. 2024).

03 · Numbers

Measured properties

Values from the Particle Data Group review (Navas et al. 2024), with the exact 2019 SI definition of the elementary charge and constants from the CODATA 2022 adjustment (Mohr et al. 2025). Parenthesised digits give the standard uncertainty in the final digits shown.

±1.602 176 634 × 10⁻¹⁹ C

Charge of π^± · exact · SI 2019

139.570 39 MeV

Rest energy m(π^±)c² · Navas et al. 2024

0.659(4) fm

Charge radius · Navas et al. 2024

B(π⁰ → 3γ) < 3.1 × 10⁻⁸

C-invariance limit · 90% CL · McDonough et al. 1988

Quantity	Value	Status	Meaning & convention
Electric charge	±1.602 176 634 × 10⁻¹⁹ C	exact · SI 2019	The charged states carry exactly ±e; the neutral state carries no charge (Navas et al. 2024).
Mass, π^±	139.570 39(18) MeV	measured · 1.3 × 10⁻⁶	Equivalently 2.488 07 × 10⁻²⁸ kg (computed from Navas et al. 2024).
Mass, π⁰	134.9768(5) MeV	measured · 3.7 × 10⁻⁶	The neutral member of the isospin triplet (Navas et al. 2024).
Mass splitting	4.5936(5) MeV	derived · m(π^±)−m(π⁰)	Predominantly electromagnetic in origin (Navas et al. 2024).
Spin and parity	J^P = 0⁻	exact · representation	A pseudoscalar meson; no spin axis is defined for a spin-0 field (Navas et al. 2024).
Isospin	I = 1, I₃ = +1, 0, −1	exact · flavour assignment	The three charge states form one isospin triplet (Navas et al. 2024).
Valence content	ud̄ · (uū − dd̄)/√2 · ūd	exact · flavour assignment	The assignments for π⁺, π⁰ and π⁻ respectively (Navas et al. 2024).
Mean life, π^±	2.6033(5) × 10⁻⁸ s	measured · 1.9 × 10⁻⁴	Decay by the weak interaction, to μ⁺ν_μ in 99.987 70(4) % of decays (Navas et al. 2024).
Mean life, π⁰	8.43(13) × 10⁻¹⁷ s	measured · 1.5 × 10⁻²	Decay by the electromagnetic interaction, to two photons in 98.823(34) % of decays (Navas et al. 2024).
Decay length cτ, π^±	7.8045 m	derived · cτ	Multiplied by βγ it gives the mean flight distance in a beam (computed from Navas et al. 2024).
Charge radius, π^±	0.659(4) fm	measured · 6.1 × 10⁻³	Root-mean-square radius from the space-like electromagnetic form factor; the average is dominated by Amendolia et al. (1986) (Navas et al. 2024).
Decay constant f_π	130.2(1.2) MeV	lattice · 9.2 × 10⁻³	Lattice average in the normalisation in which the leptonic width carries f_π², used throughout this entry (Aoki et al. 2022; Navas et al. 2024); the value from Γ(π → μν) with \|V_ud\| from nuclear β decay is 130.56(14) MeV (Navas et al. 2024).
Compton wavelength, π^±	1.4138 fm	derived · ℏ/m_πc	Reduced wavelength; it sets the range of one-pion exchange (computed from Navas et al. 2024).
Two-photon width, π⁰	7.80(12) eV	measured · 1.5 × 10⁻²	Fixed at leading order by the axial anomaly (Larin et al. 2020).

Sources · PDG 2024 π^± listing S008 and π⁰ listing S009 (pdglive.lbl.gov) · Larin et al. Science 368, 506 (2020) · Amendolia et al. Nucl. Phys. B 277, 168 (1986) · Hennebach et al. Eur. Phys. J. A 50, 190 (2014) · NIST CODATA 2022 (physics.nist.gov/constants)

04 · Situations

Representative configurations

Six configurations computed from Particle Data Group values: pionic hydrogen, momentum selection of a pion beam in a magnetic field, pion–nucleon scattering through the Δ(1232), charged-pion decay to a muon, the emulsion tracks of Lattes et al. (1947), and neutral-pion gamma rays from a supernova remnant. Each scene states any scale factor applied to the rendering.

05 · Latest research

Recent literature

Preprints and papers retrieved at page load, ordered by submission date. arXiv: categories hep-ex, hep-ph, nucl-ex, nucl-th and hep-lat, abstracts matching the pion decay constant, the pion form factor, chiral perturbation theory, pion–nucleon scattering, pionic atoms, the neutral pion or pion photoproduction. INSPIRE-HEP: the same terms in titles of high-energy-physics records. Dates are arXiv submission dates and INSPIRE record dates; no publisher issue dates are used.

Querying arXiv · INSPIRE-HEP …

Date	Source	Title and authors	Reference
Retrieving …

06 · References

References

Declared representations

The valence pair is illustrativeThe two points inside the hero are the valence quark and antiquark oscillating within the charge radius; their positions are illustrative and their colours cycle to indicate colour exchange.

The charge radius has no edgeThe drawn shell marks the root-mean-square charge radius 0.659(4) fm (Navas et al. 2024); the pion has no surface, and its charge density falls off smoothly.

Spin 0 means no axisNothing rotates and no spin axis is drawn, because a spin-0 field carries no intrinsic angular momentum. The negative parity is a quantum number and not a shape (Navas et al. 2024).

Declared scalingWhere a physical rate or length is below display resolution (the decay time, the beam-line geometry, the remnant scale), the applied scale factor is stated in the scene.

Sources

H. Yukawa (1935). On the interaction of elementary particles I. Proc. Phys.-Math. Soc. Japan 17, 48.

C. M. G. Lattes, H. Muirhead, G. P. S. Occhialini & C. F. Powell (1947). Processes involving charged mesons. Nature 159, 694.

J. Steinberger, W. K. H. Panofsky & J. Steller (1950). Evidence for the production of neutral mesons by photons. Phys. Rev. 78, 802.

M. Gell-Mann, R. J. Oakes & B. Renner (1968). Behavior of current divergences under SU(3) × SU(3). Phys. Rev. 175, 2195.

S. L. Adler (1969). Axial-vector vertex in spinor electrodynamics. Phys. Rev. 177, 2426.

J. S. Bell & R. Jackiw (1969). A PCAC puzzle: π⁰ → γγ in the σ-model. Nuovo Cimento A 60, 47.

J. Gasser & H. Leutwyler (1984). Chiral perturbation theory to one loop. Ann. Phys. 158, 142.

S. R. Amendolia et al., NA7 (1986). A measurement of the space-like pion electromagnetic form factor. Nucl. Phys. B 277, 168.

R. McDonough et al. (1988). New searches for the C-noninvariant decay π⁰ → 3γ and the rare decay π⁰ → 4γ. Phys. Rev. D 38, 2121.

V. Cirigliano & I. Rosell (2007). Two-loop effective theory analysis of π (K) → eν̄_e[γ] branching ratios. Phys. Rev. Lett. 99, 231801.

M. Ackermann et al., Fermi-LAT (2013). Detection of the characteristic pion-decay signature in supernova remnants. Science 339, 807.

M. Hennebach et al. (2014). Hadronic shift in pionic hydrogen. Eur. Phys. J. A 50, 190.

I. Larin et al., PrimEx-II (2020). Precision measurement of the neutral pion lifetime. Science 368, 506.

S. Aoki et al., Flavour Lattice Averaging Group (2022). FLAG review 2021. Eur. Phys. J. C 82, 869.

S. Navas et al., Particle Data Group (2024). Review of particle physics. Phys. Rev. D 110, 030001 · PDG π^± listing.

P. J. Mohr et al. (2025). CODATA recommended values of the fundamental physical constants: 2022. Rev. Mod. Phys. 97, 025002 · physics.nist.gov/constants.