Eta and eta prime mesons

Eta and eta prime mesons
Composition	; ηいーた; : ≈ ; ; ηいーた′; : ≈ ;
Statistics	Bosonic
Family	Mesons
Interactions	Strong, Weak, Gravitation, Electromagnetic
Symbol	; ηいーた; , ; ηいーた′;
Antiparticle	Self
Discovered	Aihud Pevsner et al. (1961)
Types	2
Mass	; ηいーた; : 547.862±0.018 MeV/c2; ; ηいーた′; : 957.78±0.06 MeV/c2
Mean lifetime	; ηいーた; : (5.0±0.3)×10−19 s, ; ηいーた′; : (3.2±0.2)×10−21 s
Decays into	; ηいーた; :; ; γがんま; + ; γがんま; or ; ; πぱい0; + ; πぱい0; + ; πぱい0; or ; ; πぱい+; + ; πぱい0; + ; πぱい−; ; ; ηいーた′; :; ; πぱい+; + ; πぱい−; + ; ηいーた; or; (; ρろー0; + ; γがんま; ) / (; πぱい+; + ; πぱい−; + ; γがんま; ) or; ; πぱい0; + ; πぱい0; + ; γがんま; ;
Electric charge	0 e
Spin	0
Isospin	0
Hypercharge	0
Parity	-1
C parity	+1

The eta (
ηいーた
) and eta prime meson (
ηいーた′
) are isosinglet mesons made of a mixture of up, down and strange quarks and their antiquarks. The charmed eta meson (
ηいーた
_c) and bottom eta meson (
ηいーた
_b) are similar forms of quarkonium; they have the same spin and parity as the (light)
ηいーた
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 $η いーた$ and that of the
ηいーた′
is larger than the quark model can naturally explain. This " $η いーた - η いーた'$ 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 $η いーた'$ 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" $η いーた$ mass is small, the $η いーた'$ is not.

Quark composition

The $η いーた$ particles belong to the "pseudo-scalar" nonet of mesons which have spin $J$ = 0 and negative parity,^[9]^[10] and $η いーた$ and $η いーた'$ have zero total isospin, $I$ , and zero strangeness, and hypercharge. Each quark which appears in an $η いーた$ 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 $η いーた$ ₁ belongs to a singlet (which is fully antisymmetrical) and $η いーた$ ₈ 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 $θ しーた$ _P = −11.5°),^[11] 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 $η いーた$ refers to the real particle which is actually observed and which is close to the $η いーた$ ₈. The $η いーた'$ is the observed particle close to $η いーた$ ₁.^[10]

The $η いーた$ and $η いーた'$ particles are closely related to the better-known neutral pion $π ぱい 0,$ where

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

In fact, $π ぱい 0,$ $η いーた$ ₁, and $η いーた$ ₈ 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いーた′ meson

The ηいーた′ meson (
ηいーた′
) is a flavor SU(3) singlet, unlike the
ηいーた
. It is a different superposition of the same quarks as the eta meson (
ηいーた
), 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 ηいーた₁ before s light quark mixing yields
ηいーた′
.

References

^ ^a ^b 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.
^ Kupść, A. (2007). "What is interesting in
ηいーた
and
ηいーた′
Meson Decays?". AIP Conference Proceedings. 950: 165–179. arXiv:0709.0603. Bibcode:2007AIPC..950..165K. doi:10.1063/1.2819029. S2CID 15930194.
^ 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.
^ 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.
^ 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.
^ '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.
^ 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.
^ 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.
^ ^a ^b The Wikipedia meson article describes the SU(3) pseudo-scalar nonet of mesons including
ηいーた
and
ηいーた′
.
^ ^a ^b ^c 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いーた
and
ηいーた′
. Page 154 defines ηいーた₁ and ηいーた₈ and explains the mixing (leading to
ηいーた
and
ηいーた′
).
^ 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

[PDG14-1] 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.

[Kupsc-2] Kupść, A. (2007). "What is interesting in
ηいーた
and
ηいーた′
Meson Decays?". AIP Conference Proceedings. 950: 165–179. arXiv:0709.0603. Bibcode:2007AIPC..950..165K. doi:10.1063/1.2819029. S2CID 15930194.

[DelDebbio-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.

[Luscher-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.

[Ce-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.

[Hooft-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.

[Witten-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.

[Veneziano-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.

[WPMeson-9] The Wikipedia meson article describes the SU(3) pseudo-scalar nonet of mesons including
ηいーた
and
ηいーた′
.

[Jones-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いーた
and
ηいーた′
. Page 154 defines ηいーた₁ and ηいーた₈ and explains the mixing (leading to
ηいーた
and
ηいーた′
).

[PDG-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.

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