Eta and eta prime mesons

The eta (eta) and eta prime meson (eta prime) are isosinglet mesons made of a mixture of up, down and strange quarks and their antiquarks. The charmed eta meson (charmed eta) and bottom eta meson (bottom eta) are similar forms of quarkonium; they have the same spin and parity as the (light) eta defined,  but are made of charm quarks and bottom quarks respectively. The top quark is too heavy to form a similar meson, due to its very fast decay.

General
The eta was discovered in pion–nucleon collisions at the Bevatron in 1961 by Aihud Pevsner et al. at a time when the proposal of the Eightfold Way was leading to predictions and discoveries of new particles from symmetry considerations.

The difference between the mass of the $eta$ and that of the eta prime is larger than the quark model can naturally explain. This "$eta–eta prime$ puzzle" can be resolved  by the 't Hooft instanton mechanism, whose $547.862 MeV/c2$ realization is also known as the Witten–Veneziano mechanism. Specifically, in QCD, the higher mass of the $eta prime$ is very significant, since it is associated with the axial U$957.78 MeV/c2$(1)  classical symmetry, which is explicitly broken through the chiral anomaly upon quantization; thus, although the "protected" $eta$ mass is small, the $eta prime$ is not.

Quark composition
The $Eta$ particles belong to the "pseudo-scalar" nonet of mesons which have spin $5 s$ = 0 and negative parity, and $eta$ and $eta prime$ have zero total isospin, $3.2 s$, and zero strangeness, and hypercharge. Each quark which appears in an $eta$ particle is accompanied by its antiquark, hence all the main quantum numbers are zero, and the particle overall is "flavourless".

The basic SU(3) symmetry theory of quarks for the three lightest quarks, which only takes into account the strong force, predicts corresponding particles
 * $$ \mathrm{\eta}_1 = \frac{1}{\sqrt 3} \left( \mathrm{ u\bar{u} + d\bar{d} + s\bar{s} } \right) ~,$$

and
 * $$\mathrm{\eta}_8 = \frac{1}{\sqrt 6} \left( \mathrm{ u\bar{u} + d\bar{d} - 2s\bar{s} } \right) ~.$$

The subscripts are labels that refer to the fact that $η$$0 e$ belongs to a singlet (which is fully antisymmetrical) and $η$$1⁄N$ is part of an octet. However, the electroweak interaction – which can transform one flavour of quark into another – causes a small but significant amount of "mixing" of the eigenstates (with mixing angle $A$$J$ = −11.5°), so that the actual quark composition is a linear combination of these formulae. That is:


 * $$\left(\begin{array}{cc}

\cos\theta_\mathrm{P} & - \sin\theta_\mathrm{P} \\ \sin\theta_\mathrm{P} & \cos\theta_\mathrm{P} \end{array}\right) \left(\begin{array}{c} \mathrm{\eta}_8 \\ \mathrm{\eta}_1 \end{array}\right) = \left(\begin{array}{c} \mathrm{\eta} \\ \mathrm{\eta'} \end{array}\right) ~. $$

The unsubscripted name $eta$ refers to the real particle which is actually observed and which is close to the $η$$I$. The $eta prime$ is the observed particle close to $η$$1$.

The $eta$ and $eta prime$ particles are closely related to the better-known neutral pion $pion0,$ where
 * $$\mathrm{\pi}^0 = \frac{1}{\sqrt 2} \left( \mathrm{ u\bar{u} - d\bar{d} } \right) ~.$$

In fact, $pion0,$ $η$$8$, and $η$$θ$ are three mutually orthogonal, linear combinations of the quark pairs up quarkup antiquark, down quarkdown antiquark, and strange quarkstrange antiquark; they are at the centre of the pseudo-scalar nonet of mesons with all the main quantum numbers equal to zero.

η′ meson
The η′ meson (eta prime) is a flavor SU(3) singlet, unlike the eta. It is a different superposition of the same quarks as the eta meson (eta), as described above, and it has a higher mass, a different decay state, and a shorter lifetime.

Fundamentally, it results from the direct sum decomposition of the approximate SU(3) flavor symmetry among the 3 lightest quarks, $$\mathbb{3} \times \bar{\mathbb{3}} = \mathbb{1} + \mathbb{8}$$, where 1 corresponds to η1 before s light quark mixing yields eta prime.