Eta and eta prime mesons

The eta (
η
) and eta prime meson (
η′
) are isosinglet mesons made of a mixture of up, down and strange quarks and their antiquarks. The charmed eta meson (
η
_c) and bottom eta meson (
η
_b) are similar forms of quarkonium; they have the same spin and parity as the (light)
η
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.

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.^[2]

The difference between the mass of the
η
and that of the
η′
is larger than the quark model can naturally explain. This "
η
–
η′
puzzle" can be resolved^[3][4][5] by the 't Hooft instanton mechanism,^[6] whose ⁠1/ N ⁠ realization is also known as the Witten–Veneziano mechanism.^[7][8] Specifically, in QCD, the higher mass of the
η′
is very significant, since it is associated with the axial U_A(1) classical symmetry, which is explicitly broken through the chiral anomaly upon quantization; thus, although the "protected"
η
mass is small, the
η′
is not.

The
η
particles belong to the "pseudo-scalar" nonet of mesons which have spin J = 0 and negative parity,^[9][10] and
η
and
η′
have zero total isospin, I, and zero strangeness, and hypercharge. Each quark which appears in an
η
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

${\displaystyle \mathrm {\eta } _{1}={\frac {1}{\sqrt {3}}}\left(\mathrm {u{\bar {u}}+d{\bar {d}}+s{\bar {s}}} \right)~,}$

and

${\displaystyle \mathrm {\eta } _{8}={\frac {1}{\sqrt {6}}}\left(\mathrm {u{\bar {u}}+d{\bar {d}}-2s{\bar {s}}} \right)~.}$

The subscripts are labels that refer to the fact that η₁ belongs to a singlet (which is fully antisymmetrical) and η₈ 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 θ_P = −11.5°),^[11] so that the actual quark composition is a linear combination of these formulae. That is:

${\displaystyle \left({\begin{array}{cc}\cos \theta _{\mathrm {P} }&-\sin \theta _{\mathrm {P} }\\\sin \theta _{\mathrm {P} }&~~\cos \theta _{\mathrm {P} }\end{array}}\right)\left({\begin{array}{c}\mathrm {\eta } _{8}\\\mathrm {\eta } _{1}\end{array}}\right)=\left({\begin{array}{c}\mathrm {\eta } \\\mathrm {\eta '} \end{array}}\right)~.}$

The unsubscripted name
η
refers to the real particle which is actually observed and which is close to the η₈. The
η′
is the observed particle close to η₁.^[10]

The
η
and
η′
particles are closely related to the better-known neutral pion
π⁰
, where

${\displaystyle \mathrm {\pi } ^{0}={\frac {1}{\sqrt {2}}}\left(\mathrm {u{\bar {u}}-d{\bar {d}}} \right)~.}$

In fact,
π⁰
, η₁, and η₈ are three mutually orthogonal, linear combinations of the quark pairs
u

u
,
d

d
, and
s

s
; they are at the centre of the pseudo-scalar nonet of mesons^[9][10] with all the main quantum numbers equal to zero.

The η′ meson (
η′
) is a flavor SU(3) singlet, unlike the
η
. It is a different superposition of the same quarks as the eta meson (
η
), 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, ${\displaystyle \mathbb {3} \times {\bar {\mathbb {3} }}=\mathbb {1} +\mathbb {8} }$, where 1 corresponds to η₁ before s light quark mixing yields
η′
.

• List of mesons
• Quarkonium
• Special unitary group

• Eta and Eta' meson summaries at the Particle Data Group