Pion

Possible applications

The use of pions in medical radiation therapy, such as for cancer, was explored at a number of research institutions, including the Los Alamos National Laboratory's Meson Physics Facility, which treated 228 patients between 1974 and 1981 in New Mexico, and the TRIUMF laboratory in Vancouver, British Columbia.

Theoretical overview

In the standard understanding of the strong force interaction as defined by quantum chromodynamics, pions are loosely portrayed as Goldstone bosons of spontaneously broken chiral symmetry. That explains why the masses of the three kinds of pions are considerably less than that of the other mesons, such as the scalar or vector mesons. If their current quarks were massless particles, it could make the chiral symmetry exact and thus the Goldstone theorem would dictate that all pions have a zero mass.

In fact, it was shown by Gell-Mann, Oakes and Renner (GMOR) that the square of the pion mass is proportional to the sum of the quark masses times the quark condensate: $${\displaystyle M_{\pi }^2=(m_{\text{u}}+m_{\text{d}})B+\mathcal{O}(m^2),}$$ with B the quark condensate: $${\displaystyle B={\left|\frac{⟨0|\overline{\mathrm{u}}\mathrm{u}|0⟩}{f_{\pi }^2}\right|}_{m_{\text{q}}\to 0}}$$ This is often known as the GMOR relation and it explicitly shows that ${\displaystyle M_{\pi }=0}$ in the massless quark limit. The same result also follows from light-front holography.

Empirically, since the light quarks actually have minuscule nonzero masses, the pions also have nonzero rest masses. However, those masses are almost an order of magnitude smaller than that of the nucleons, roughly ${\displaystyle m_{\pi }\approx \frac{\sqrt{v{m}_{\text{q}}}}{f_{\pi }}\approx \sqrt{m_{\text{q}}}}$ 45 MeV, where m_q are the relevant current quark masses, around 5–10 MeV/c².

The pion is one of the particles that mediate the residual strong interaction between a pair of nucleons. This interaction is attractive: it pulls the nucleons together. Written in a non-relativistic form, it is called the Yukawa potential. The pion, being spinless, has kinematics described by the Klein–Gordon equation. In the terms of quantum field theory, the effective field theory Lagrangian describing the pion-nucleon interaction is called the Yukawa interaction.

The nearly identical masses of π^±
and π⁰
indicate that there must be a symmetry at play: this symmetry is called the SU(2) flavour symmetry or isospin. The reason that there are three pions, π⁺
, π⁻
and π⁰
, is that these are understood to belong to the triplet representation or the adjoint representation 3 of SU(2). By contrast, the up and down quarks transform according to the fundamental representation 2 of SU(2), whereas the anti-quarks transform according to the conjugate representation 2*.

With the addition of the strange quark, the pions participate in a larger, SU(3), flavour symmetry, in the adjoint representation, 8, of SU(3). The other members of this octet are the four kaons and the eta meson.

Pions are pseudoscalars under a parity transformation. Pion currents thus couple to the axial vector current and so participate in the chiral anomaly.

Basic properties

Pions, which are mesons with zero spin, are composed of first-generation quarks. In the quark model, an up quark and an anti-down quark make up a π⁺
, whereas a down quark and an anti-up quark make up the π⁻
, and these are the antiparticles of one another. The neutral pion π⁰
is a combination of an up quark with an anti-up quark, or a down quark with an anti-down quark. The two combinations have identical quantum numbers, and hence they are only found in superpositions. The lowest-energy superposition of these is the π⁰
, which is its own antiparticle. Together, the pions form a triplet of isospin. Each pion has overall isospin (I = 1) and third-component isospin equal to its charge (I_z = +1, 0, −1).

Charged pion decays

The π^±
mesons have a mass of 139.6 MeV/c² and a mean lifetime of 2.6033×10⁻⁸ s. They decay due to the weak interaction. The primary decay mode of a pion, with a branching fraction of 0.999877, is a leptonic decay into a muon and a muon neutrino: $${\displaystyle \begin{array}{cc}{\pi }^{+}& ⟶{\mu }^{+}+{\nu }_{\mu }\\ {\pi }^{-}& ⟶{\mu }^{-}+{\overline{\nu }}_{\mu }\end{array}}$$

The second most common decay mode of a pion, with a branching fraction of 0.000123, is also a leptonic decay into an electron and the corresponding electron antineutrino. This "electronic mode" was discovered at CERN in 1958: $${\displaystyle \begin{array}{cc}{\pi }^{+}& ⟶{\mathrm{e}}^{+}+{\nu }_e\\ {\pi }^{-}& ⟶{\mathrm{e}}^{-}+{\overline{\nu }}_e\end{array}}$$

The suppression of the electronic decay mode with respect to the muonic one is given approximately (up to a few percent effect of the radiative corrections) by the ratio of the half-widths of the pion–electron and the pion–muon decay reactions, $${\displaystyle R_{\pi }={\left(\frac{m_e}{m_{\mu }}\right)}^2{\left(\frac{m_{\pi }^2-m_e^2}{m_{\pi }^2-m_{\mu }^2}\right)}^2=1.283\times {10}^{-4}}$$ and is a spin effect known as helicity suppression.

Its mechanism is as follows: The negative pion has spin zero; therefore the lepton and the antineutrino must be emitted with opposite spins (and opposite linear momenta) to preserve net zero spin (and conserve linear momentum). However, because the weak interaction is sensitive only to the left chirality component of fields, the antineutrino has always left chirality, which means it is right-handed, since for massless anti-particles the helicity is opposite to the chirality. This implies that the lepton must be emitted with spin in the direction of its linear momentum (i.e., also right-handed). If, however, leptons were massless, they would only interact with the pion in the left-handed form (because for massless particles helicity is the same as chirality) and this decay mode would be prohibited. Therefore, suppression of the electron decay channel comes from the fact that the electron's mass is much smaller than the muon's. The electron is relatively massless compared with the muon, and thus the electronic mode is greatly suppressed relative to the muonic one, virtually prohibited.

Although this explanation suggests that parity violation is causing the helicity suppression, the fundamental reason lies in the vector-nature of the interaction which dictates a different handedness for the neutrino and the charged lepton. Thus, even a parity conserving interaction would yield the same suppression.

Measurements of the above ratio have been considered for decades to be a test of lepton universality. Experimentally, this ratio is 1.233(2)×10⁻⁴.

Beyond the purely leptonic decays of pions, some structure-dependent radiative leptonic decays (that is, decay to the usual leptons plus a gamma ray) have also been observed.

Also observed, for charged pions only, is the very rare "pion beta decay" (with branching fraction of about 10⁻⁸) into a neutral pion, an electron and an electron antineutrino (or for positive pions, a neutral pion, a positron, and electron neutrino). $${\displaystyle \begin{array}{cc}{\pi }^{+}& ⟶{\pi }^0+{\mathrm{e}}^{+}+{\nu }_e\\ {\pi }^{-}& ⟶{\pi }^0+{\mathrm{e}}^{-}+{\overline{\nu }}_e\end{array}}$$

The rate at which pions decay is a prominent quantity in many sub-fields of particle physics, such as chiral perturbation theory. This rate is parametrized by the pion decay constant (f_π), related to the wave function overlap of the quark and antiquark, which is about 130 MeV.

Neutral pion decays

The π⁰
meson has a mass of 135.0 MeV/c² and a mean lifetime of 8.5×10⁻¹⁷ s. It decays via the electromagnetic force, which explains why its mean lifetime is much smaller than that of the charged pion (which can only decay via the weak force).

The dominant π⁰
decay mode, with a branching ratio of BR_γγ = 0.98823, is into two photons: $${\displaystyle {\pi }^0⟶2\gamma }$$

The decay π⁰
→ 3 γ (as well as decays into any odd number of photons) is forbidden by the C-symmetry of the electromagnetic interaction: The intrinsic C-parity of the π⁰
is +1, while the C-parity of a system of n photons is (−1)ⁿ.

The second largest π⁰
decay mode (BR_γee = 0.01174) is the Dalitz decay (named after Richard Dalitz), which is a two-photon decay with an internal photon conversion resulting in a photon and an electron-positron pair in the final state: $${\displaystyle {\pi }^0⟶\gamma +{\mathrm{e}}^{-}+{\mathrm{e}}^{+}}$$

The third largest established decay mode (BR_2e2e = 3.34×10⁻⁵) is the double-Dalitz decay, with both photons undergoing internal conversion which leads to further suppression of the rate: $${\displaystyle {\pi }^0⟶2{\mathrm{e}}^{-}+2{\mathrm{e}}^{+}}$$

The fourth largest established decay mode is the loop-induced and therefore suppressed (and additionally helicity-suppressed) leptonic decay mode (BR_ee = 6.46×10⁻⁸): $${\displaystyle {\pi }^0⟶{\mathrm{e}}^{-}+{\mathrm{e}}^{+}}$$

The neutral pion has also been observed to decay into positronium with a branching fraction on the order of 10⁻⁹. No other decay modes have been established experimentally. The branching fractions above are the PDG central values, and their uncertainties are omitted, but available in the cited publication.

^[a] ^ The quark composition of the π⁰
is not exactly divided between up and down quarks, due to complications from non-zero quark masses.