# Meson

### Parity

{{#invoke:main|main}}

If the universe were reflected in a mirror, most of the laws of physics would be identical—things would behave the same way regardless of what we call "left" and what we call "right". This concept of mirror reflection is called parity (P). Gravity, the electromagnetic force, and the strong interaction all behave in the same way regardless of whether or not the universe is reflected in a mirror, and thus are said to conserve parity (P-symmetry). However, the weak interaction does distinguish "left" from "right", a phenomenon called parity violation (P-violation).

Based on this, one might think that if the wavefunction for each particle (more precisely, the quantum field for each particle type) were simultaneously mirror-reversed, then the new set of wavefunctions would perfectly satisfy the laws of physics (apart from the weak interaction). It turns out that this is not quite true: In order for the equations to be satisfied, the wavefunctions of certain types of particles have to be multiplied by −1, in addition to being mirror-reversed. Such particle types are said to have negative or odd parity (P = −1, or alternatively P = –), whereas the other particles are said to have positive or even parity (P = +1, or alternatively P = +).

For mesons, the parity is related to the orbital angular momentum by the relation:[8]

${\displaystyle P=\left(-1\right)^{L+1}}$

where the L is a result of the parity of the corresponding spherical harmonic of the wavefunction. The '+1' in the exponent comes from the fact that, according to the Dirac equation, a quark and an antiquark have opposite intrinsic parities. Therefore the intrinsic parity of a meson is the product of the intrinsic parities of the quark (+1) and antiquark (−1). As these are different, their product is −1, and so it contributes a +1 in the exponent.

As a consequence, mesons with no orbital angular momentum (L = 0) all have odd parity (P = −1).

### C-parity

{{#invoke:main|main}}

C-parity is only defined for mesons that are their own antiparticle (i.e. neutral mesons). It represents whether or not the wavefunction of the meson remains the same under the interchange of their quark with their antiquark.[9] If

${\displaystyle |q{\bar {q}}\rangle =|{\bar {q}}q\rangle }$

then, the meson is "C even" (C = +1). On the other hand, if

${\displaystyle |q{\bar {q}}\rangle =-|{\bar {q}}q\rangle }$

then the meson is "C odd" (C = −1).

C-parity rarely is studied on its own, but the combination of C- and P-parity into CP-parity. CP-parity was thought to be conserved, but was later found to be violated in weak interactions.[10][11][12]

### G-parity

{{#invoke:main|main}}

G parity is a generalization of the C-parity. Instead of simply comparing the wavefunction after exchanging quarks and antiquarks, it compares the wavefunction after exchanging the meson for the corresponding antimeson, regardless of quark content.[13] In the case of neutral meson, G-parity is equivalent to C-parity because neutral mesons are their own antiparticles.

If

${\displaystyle |q_{1}{\bar {q_{2}}}\rangle =|{\bar {q_{1}}}q_{2}\rangle }$

then, the meson is "G even" (G = +1). On the other hand, if

${\displaystyle |q_{1}{\bar {q_{2}}}\rangle =-|{\bar {q_{1}}}q_{2}\rangle }$

then the meson is "G odd" (G = −1).

### Isospin and charge

{{#invoke:main|main}}

Combinations of one u, d or s quarks and one u, d, or s antiquark in JP = 0 configuration form a nonet.
Combinations of one u, d or s quarks and one u, d, or s antiquark in JP = 1 configuration also form a nonet.

The concept of isospin was first proposed by Werner Heisenberg in 1932 to explain the similarities between protons and neutrons under the strong interaction.[14] Although they had different electric charges, their masses were so similar that physicists believed they were actually the same particle. The different electric charges were explained as being the result of some unknown excitation similar to spin. This unknown excitation was later dubbed isospin by Eugene Wigner in 1937.[15] When the first mesons were discovered, they too were seen through the eyes of isospin. The three pions were believed to be the same particle, but in different isospin states.

This belief lasted until Murray Gell-Mann proposed the quark model in 1964 (containing originally only the u, d, and s quarks).[16] The success of the isospin model is now understood to be the result of the similar masses of the u and d quarks. Because the u and d quarks have similar masses, particles made of the same number of them also have similar masses. The exact specific u and d quark composition determines the charge, because u quarks carry charge +Template:Frac whereas d quarks carry charge −Template:Frac. For example the three pions all have different charges (Template:SubatomicParticle (Template:SubatomicParticleTemplate:SubatomicParticle), Template:SubatomicParticle (a quantum superposition of Template:SubatomicParticleTemplate:SubatomicParticle and Template:SubatomicParticleTemplate:SubatomicParticle states), Template:SubatomicParticle (Template:SubatomicParticleTemplate:SubatomicParticle)), but have similar masses (~Template:Val) as they are each made of a same number of total of up and down quarks and antiquarks. Under the isospin model, they were considered to be a single particle in different charged states.

The mathematics of isospin was modeled after that of spin. Isospin projections varied in increments of 1 just like those of spin, and to each projection was associated a "charged state". Because the "pion particle" had three "charged states", it was said to be of isospin I = 1. Its "charged states" Template:SubatomicParticle, Template:SubatomicParticle, and Template:SubatomicParticle, corresponded to the isospin projections I3 = +1, I3 = 0, and I3 = −1 respectively. Another example is the "rho particle", also with three charged states. Its "charged states" Template:SubatomicParticle, Template:SubatomicParticle, and Template:SubatomicParticle, corresponded to the isospin projections I3 = +1, I3 = 0, and I3 = −1 respectively. It was later noted that the isospin projections were related to the up and down quark content of particles by the relation

${\displaystyle I_{3}={\frac {1}{2}}[(n_{u}-n_{\bar {u}})-(n_{d}-n_{\bar {d}})],}$

where the n's are the number of up and down quarks and antiquarks.

In the "isospin picture", the three pions and three rhos were thought to be the different states of two particles. However in the quark model, the rhos are excited states of pions. Isospin, although conveying an inaccurate picture of things, is still used to classify hadrons, leading to unnatural and often confusing nomenclature. Because mesons are hadrons, the isospin classification is also used, with I3 = +Template:Frac for up quarks and down antiquarks, and I3 = −Template:Frac for up antiquarks and down quarks.

### Flavour quantum numbers

{{#invoke:main|main}}

The strangeness quantum number S (not to be confused with spin) was noticed to go up and down along with particle mass. The higher the mass, the lower the strangeness (the more s quarks). Particles could be described with isospin projections (related to charge) and strangeness (mass) (see the uds nonet figures). As other quarks were discovered, new quantum numbers were made to have similar description of udc and udb nonets. Because only the u and d mass are similar, this description of particle mass and charge in terms of isospin and flavour quantum numbers only works well for the nonets made of one u, one d and one other quark and breaks down for the other nonets (for example ucb nonet). If the quarks all had the same mass, their behaviour would be called symmetric, because they would all behave in exactly the same way with respect to the strong interaction. Because quarks do not have the same mass, they do not interact in the same way (exactly like an electron placed in an electric field will accelerate more than a proton placed in the same field because of its lighter mass), and the symmetry is said to be broken.

It was noted that charge (Q) was related to the isospin projection (I3), the baryon number (B) and flavour quantum numbers (S, C, B′, T) by the Gell-Mann–Nishijima formula:[17]

${\displaystyle Q=I_{3}+{\frac {1}{2}}(B+S+C+B^{\prime }+T),}$

where S, C, B′, and T represent the strangeness, charm, bottomness and topness flavour quantum numbers respectively. They are related to the number of strange, charm, bottom, and top quarks and antiquark according to the relations:

${\displaystyle S=-(n_{s}-n_{\bar {s}})}$
${\displaystyle C=+(n_{c}-n_{\bar {c}})}$
${\displaystyle B^{\prime }=-(n_{b}-n_{\bar {b}})}$
${\displaystyle T=+(n_{t}-n_{\bar {t}}),}$

meaning that the Gell-Mann–Nishijima formula is equivalent to the expression of charge in terms of quark content:

${\displaystyle Q={\frac {2}{3}}[(n_{u}-n_{\bar {u}})+(n_{c}-n_{\bar {c}})+(n_{t}-n_{\bar {t}})]-{\frac {1}{3}}[(n_{d}-n_{\bar {d}})+(n_{s}-n_{\bar {s}})+(n_{b}-n_{\bar {b}})].}$

## Classification

Mesons are classified into groups according to their isospin (I), total angular momentum (J), parity (P), G-parity (G) or C-parity (C) when applicable, and quark (q) content. The rules for classification are defined by the Particle Data Group, and are rather convoluted.[18] The rules are presented below, in table form for simplicity.

### Types of meson

Mesons are classified into types according to their spin configurations. Some specific configurations are given special names based on the mathematical properties of their spin configuration.

Types of mesons[19]
Type S L P J JP
Pseudoscalar meson 0 0 0 0
Pseudovector meson 1 1 + 1 1+
Vector meson 1 0 1 1
Scalar meson 1 1 + 0 0+
Tensor meson 1 1 + 2 2+

### Nomenclature

#### Flavourless mesons

Flavourless mesons are mesons made of pair of quark and antiquarks of the same flavour (all their flavour quantum numbers are zero: S = 0, C = 0, B = 0, T = 0).[20] The rules for flavourless mesons are:[18]

^ The C parity is only relevant to neutral mesons.
†† ^ For JPC=1−−, the ψ is called the Template:SubatomicParticle

• When the spectroscopic state of the meson is known, it is added in parentheses.
• When the spectroscopic state is unknown, mass (in MeV/c2) is added in parentheses.
• When the meson is in its ground state, nothing is added in parentheses.

#### Flavoured mesons

Flavoured mesons are mesons made of pair of quark and antiquarks of different flavours. The rules are simpler in this case: the main symbol depends on the heavier quark, the superscript depends on the charge, and the subscript (if any) depends on the lighter quark. In table form, they are:[18]

• If JP is in the "normal series" (i.e., JP = 0+, 1, 2+, 3, ...), a superscript ∗ is added.
• If the meson is not pseudoscalar (JP = 0) or vector (JP = 1), J is added as a subscript.
• When the spectroscopic state of the meson is known, it is added in parentheses.
• When the spectroscopic state is unknown, mass (in MeV/c2) is added in parentheses.
• When the meson is in its ground state, nothing is added in parentheses.

## Exotic mesons

{{#invoke:main|main}} These is experimental evidence for particles which are hadrons (i.e., are composed of quarks) and are color-neutral with zero baryon number, and thus by conventional definition are mesons. Yet, these particles do not consist of a single quark-antiquark pair, as all the other conventional mesons discussed above do. A tentative category for these particles is exotic mesons.

There are at least five exotic meson resonances that have been experimentally confirmed to exist by two or more independent experiments. The most statistically significant of these is the Z(4430), discovered by Belle in 2007 and confirmed by LHCb in 2014. It is a candidate for being a tetraquark: a particle composed of two quarks and two antiquarks.[21] See the main article above for other particle resonances which are candidates for being exotic mesons.

## List

{{#invoke:main|main}}

## Notes

1. J.J. Aubert et al. (1974)
2. J.E. Augustin et al. (1974)
3. S.W. Herb et al. (1977)
4. The Noble Foundation (1949) Nobel Prize in Physics 1949 – Presentation Speech
5. H. Yukawa (1935)
6. G. Gamow (1961)
7. J. Steinberger (1998)
8. C. Amsler et al. (2008): Quark Model
9. M.S. Sozzi (2008b)
10. J.W. Cronin (1980)
11. V.L. Fitch (1980)
12. M.S. Sozzi (2008c)
13. K. Gottfried, V.F. Weisskopf (1986)
14. W. Heisenberg (1932)
15. E. Wigner (1937)
16. M. Gell-Mann (1964)
17. S.S.M Wong (1998)
18. C. Amsler et al. (2008): Naming scheme for hadrons
19. W.E. Burcham, M. Jobes (1995)
20. For the purpose of nomenclature, the isospin projection I3 isn't considered a flavour quantum number. This means that the charged pion-like mesons (π±, a±, b±, and ρ± mesons) follow the rules of flavourless mesons, even if they aren't truly "flavourless".
21. LHCb collaborators (2014): Observation of the resonant character of the Z(4430)− state

## References

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

|CitationClass=journal }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

• {{#invoke:Citation/CS1|citation

|CitationClass=journal }} Template:Refend