Eta and eta prime mesons

Eta and eta prime mesons

Composition	η  : ≈ ${\textstyle \mathrm {\frac {1}{\sqrt {6}}} \left(u{\bar {u}}+d{\bar {d}}-2s{\bar {s}}\right)}$ η′  : ≈ ${\textstyle \mathrm {\frac {1}{\sqrt {3}}} \left(u{\bar {u}}+d{\bar {d}}+s{\bar {s}}\right)}$
Statistics	Bosonic
Family	Mesons
Interactions	Strong, Weak, Gravitation, Electromagnetic
Symbol	η , η′
Antiparticle	Self
Discovered	Aihud Pevsner et al. (1961)
Types	2
Mass	η  : 547.862±0.018 MeV/c^2[1] η′  : 957.78±0.06 MeV/c^2[1]
Mean lifetime	η : (5.0±0.3)×10^−19 s, η′ : (3.2±0.2)×10^−21 s
Decays into	η  : γ + γ or π^0 + π^0 + π^0 or π^+ + π^0 + π^− η′  : π^+ + π^− + η or ( ρ^0 + γ ) / ( π^+ + π^− + γ ) or π^0 + π^0 + γ
Electric charge	0 e
Spin	0
Isospin	0
Hypercharge	0
Parity	-1
C parity	+1

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.

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.^[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.

Quark composition

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.

η′ meson

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′
.

See also

• List of mesons
• Quarkonium
• Special unitary group

References

1. Light Unflavored Mesons as appearing in Olive, K. A.; et al. (PDG) (2014). "Review of Particle Physics". Chinese Physics C. 38 (9): 090001. arXiv:1412.1408. Bibcode:2014ChPhC..38i0001O. doi:10.1088/1674-1137/38/9/090001. S2CID 118395784.
2. Kupść, A. (2007). "What is interesting in
   η
   and
   η′
   Meson Decays?". AIP Conference Proceedings. 950: 165–179. arXiv:0709.0603. Bibcode:2007AIPC..950..165K. doi:10.1063/1.2819029. S2CID 15930194.
3. Del Debbio, L.; Giusti, L.; Pica, C. (2005). "Topological Susceptibility in SU(3) Gauge Theory". Physical Review Letters. 94 (3): 032003. arXiv:hep-th/0407052. Bibcode:2005PhRvL..94c2003D. doi:10.1103/PhysRevLett.94.032003. PMID 15698253. S2CID 930312.
4. Lüscher, M.; Palombi, F. (2010). "Universality of the topological susceptibility in the SU(3) gauge theory". Journal of High Energy Physics. 2010 (9): 110. arXiv:1008.0732. Bibcode:2010JHEP...09..110L. doi:10.1007/JHEP09(2010)110. S2CID 119213800.
5. Cè, M.; Consonni, C.; Engel, G.; Giusti, L. (2014). Testing the Witten–Veneziano mechanism with the Yang–Mills gradient flow on the lattice. 32nd International Symposium on Lattice Field Theory. arXiv:1410.8358. Bibcode:2014arXiv1410.8358C.
6. 't Hooft, G. (1976). "Symmetry Breaking through Bell-Jackiw Anomalies". Physical Review Letters. 37 (1): 8–11. Bibcode:1976PhRvL..37....8T. doi:10.1103/PhysRevLett.37.8.
7. Witten, E. (1979). "Current algebra theorems for the U(1) "Goldstone boson"". Nuclear Physics B. 156 (2): 269–283. Bibcode:1979NuPhB.156..269W. doi:10.1016/0550-3213(79)90031-2.
8. Veneziano, G. (1979). "U(1) without instantons". Nuclear Physics B. 159 (1–2): 213–224. Bibcode:1979NuPhB.159..213V. doi:10.1016/0550-3213(79)90332-8.
9. The Wikipedia meson article describes the SU(3) pseudo-scalar nonet of mesons including
   η
   and
   η′
   .
10. Jones, H. F. (1998). Groups, Representations and Physics. IOP Publishing. ISBN 978-0-7503-0504-4. Page 150 describes the SU(3) pseudo-scalar nonet of mesons including
    η
    and
    η′
    . Page 154 defines η₁ and η₈ and explains the mixing (leading to
    η
    and
    η′
    ).
11. Quark Model Review as appearing in Beringer, J.; et al. (PDG) (2012). "Review of Particle Physics" (PDF). Physical Review D. 86 (1): 010001. Bibcode:2012PhRvD..86a0001B. doi:10.1103/PhysRevD.86.010001.

External links

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