Meson

Spin (quantum number S) is a vector quantity that represents the "intrinsic" angular momentum of a particle. It comes in increments of 1/2 ħ. The ħ is often dropped because it is the "fundamental" unit of spin, and it is implied that "spin 1" means "spin 1 ħ". (In some systems of natural units, ħ is chosen to be 1, and therefore does not appear in equations.)

Quarks are fermions—specifically in this case, particles having spin 1/2 (S = 1/2). Because spin projections vary in increments of 1 (that is 1 ħ), a single quark has a spin vector of length 1/2, and has two spin projections (S_z = +1/2 and S_z = −+1/2). Two quarks can have their spins aligned, in which case the two spin vectors add to make a vector of length S = 1 and three spin projections (S_z = +1, S_z = 0, and S_z = −1), called the spin-1 triplet. If two quarks have unaligned spins, the spin vectors add up to make a vector of length S = 0 and only one spin projection (S_z = 0), called the spin-0 singlet. Because mesons are made of one quark and one antiquark, they can be found in triplet and singlet spin states. The latter are called scalar mesons or pseudoscalar mesons, depending on their parity (see below).

There is another quantity of quantized angular momentum, called the orbital angular momentum (quantum number L), that is the angular momentum due to quarks orbiting each other, and comes in increments of 1 ħ. The total angular momentum (quantum number J) of a particle is the combination of intrinsic angular momentum (spin) and orbital angular momentum. It can take any value from J = |L − S| to J = |L + S|, in increments of 1.

Particle physicists are most interested in mesons with no orbital angular momentum (L = 0), therefore the two groups of mesons most studied are the S = 1; L = 0 and S = 0; L = 0, which corresponds to J = 1 and J = 0, although they are not the only ones. It is also possible to obtain J = 1 particles from S = 0 and L = 1. How to distinguish between the S = 1, L = 0 and S = 0, L = 1 mesons is an active area of research in meson spectroscopy.[10]

If the universe were reflected in a mirror, most 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, parity is related to the orbital angular momentum by the relation:[11]

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

where the L is a result of the parity of the corresponding spherical harmonic of the wavefunction. The "+1" 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 the "+1" that appears in the exponent.

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

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.[12] 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 more commonly in combination with P-parity into CP-parity. CP-parity was thought to be conserved, but was later found to be violated in weak interactions.[13][14][15]

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.[16]

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).

Combinations of one u, d or s quarks and one u, d, or s antiquark in J^P = 0⁻ configuration form a nonet.

Combinations of one u, d or s quarks and one u, d, or s antiquark in J^P = 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.[17] Although they had different electric charges, their masses were so similar that physicists believed that 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.[18] When the first mesons were discovered, they too were seen through the eyes of isospin and so 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).[19] 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 ++2/3 whereas d quarks carry charge −+1/3. For example, the three pions all have different charges (
π⁺
(
u

d
),
π⁰
(a quantum superposition of
u

u
and
d

d
states),
π⁻
(
d

u
)), but have similar masses (c. 140 MeV/c²) as they are each composed of a same total number of up and down quarks and antiquarks. Under the isospin model, they were considered 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"
π⁺
,
π⁰
, and
π⁻
, 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"
ρ⁺
,
ρ⁰
, and
ρ⁻
, 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

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

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 = +1/2 for up quarks and down antiquarks, and I₃ = −1/2 for up antiquarks and down quarks.

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. However, as 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:[20]

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

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:

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

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

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

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.

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).[23] The rules for flavourless mesons are:[21]

Nomenclature of flavourless mesons

q q content	JPC†	I
		0−+, 2−+, 4−+, ...	1+−, 3+−, 5+−, ...	1−−, 2−−, 3−−, ...	0++, 1++, 2++, ...
u d ${\displaystyle \mathrm {\tfrac {u{\bar {u}}-d{\bar {d}}}{\sqrt {2}}} }$ d u	1	π+ π0 π−	b+ b0 b−	ρ+ ρ0 ρ−	a+ a0 a−
Mix of u u , d d , s s	0	η η′	h h′	ω ϕ	f f′
c c	0	η c	hc	ψ††	χc
b b	0	η b	hb	ϒ	χb
t t	0	η t	ht	θ	χt

^† ^ The C parity is only relevant to neutral mesons.
^†† ^ For J^PC=1⁻⁻, the ψ is called the
J/ψ

In addition:

• 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 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:[21]

In addition:

• If J^P is in the "normal series" (i.e., J^P = 0⁺, 1⁻, 2⁺, 3⁻, ...), a superscript ∗ is added.
• If the meson is not pseudoscalar (J^P = 0⁻) or vector (J^P = 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.

Particle name	Particle symbol	Antiparticle symbol	Quark content	Rest mass (MeV/c2)	IG	JPC	S	C	B'	Mean lifetime (s)	Commonly decays to (>5% of decays)
Pion[25]	π+	π−	u d	139.57018±0.00035	1−	0−	0	0	0	(2.6033±0.0005)×10−8	μ+ + ν μ
Pion[26]	π0	Self	${\displaystyle \mathrm {\tfrac {u{\bar {u}}-d{\bar {d}}}{\sqrt {2}}} \,}$[a]	134.9766±0.0006	1−	0−+	0	0	0	(8.4±0.6)×10−17	γ + γ
Eta meson[27]	η	Self	${\displaystyle \mathrm {\tfrac {u{\bar {u}}+d{\bar {d}}-2s{\bar {s}}}{\sqrt {6}}} \,}$[a]	547.853±0.024	0+	0−+	0	0	0	(5.0±0.3)×10−19[b]	γ + γ or π0 + π0 + π0 or π+ + π0 + π−
Eta prime meson[28]	η′ (958)	Self	${\displaystyle \mathrm {\tfrac {u{\bar {u}}+d{\bar {d}}+s{\bar {s}}}{\sqrt {3}}} \,}$[a]	957.66±0.24	0+	0−+	0	0	0	(3.2±0.2)×10−21[b]	π+ + π− + η or ( ρ0 + γ ) / ( π+ + π− + γ ) or π0 + π0 + η
Charmed eta meson[29]	η c(1S)	Self	c c	2980.3±1.2	0+	0−+	0	0	0	(2.5±0.3)×10−23[b]	See η c decay modes
Bottom eta meson[30]	η b(1S)	Self	b b	9300±40	0+	0−+	0	0	0	Unknown	See η b decay modes
Kaon[31]	K+	K−	u s	493.677±0.016	​1⁄2	0−	1	0	0	(1.2380±0.0021)×10−8	μ+ + ν μ or π+ + π0 or π0 + e+ + ν e or π+ + π0
Kaon[32]	K0	K0	d s	497.614±0.024	​1⁄2	0−	1	0	0	[c]	[c]
K-Short[33]	K0 S	Self	${\displaystyle \mathrm {\tfrac {d{\bar {s}}-s{\bar {d}}}{\sqrt {2}}} \,}$[e]	497.614±0.024[d]	​1⁄2	0−	(*)	0	0	(8.953±0.005)×10−11	π+ + π− or π0 + π0
K-Long[34]	K0 L	Self	${\displaystyle \mathrm {\tfrac {d{\bar {s}}+s{\bar {d}}}{\sqrt {2}}} \,}$[e]	497.614±0.024[d]	​1⁄2	0−	(*)	0	0	(5.116±0.020)×10−8	π± + e∓ + ν e or π± + μ∓ + ν μ or π0 + π0 + π0 or π+ + π0 + π−
D meson[35]	D+	D−	c d	1869.62±0.20	​1⁄2	0−	0	+1	0	(1.040±0.007)×10−12	See D+ decay modes
D meson[36]	D0	D0	c u	1864.84±0.17	​1⁄2	0−	0	+1	0	(4.101±0.015)×10−13	See D0 decay modes
strange D meson[37]	D+ s	D− s	c s	1968.49±0.34	0	0−	+1	+1	0	(5.00±0.07)×10−13	See D+ s decay modes
B meson[38]	B+	B−	u b	5279.15±0.31	​1⁄2	0−	0	0	+1	(1.638±0.011)×10−12	See B+ decay modes
B meson[39]	B0	B0	d b	5279.53±33	​1⁄2	0−	0	0	+1	(1.530±0.009)×10−12	See B0 decay modes
Strange B meson[40]	B0 s	B0 s	s b	5366.3±0.6	0	0−	−1	0	+1	1.470+0.026 −0.027×10−12	See B0 s decay modes
Charmed B meson[41]	B+ c	B− c	c b	6276±4	0	0−	0	+1	+1	(4.6±0.7)×10−13	See B+ c decay modes

^[a] ^ Makeup inexact due to non-zero quark masses.
^[b] ^ PDG reports the resonance width (Γ). Here the conversion τ = ​^ħ⁄_Γ is given instead.
^[c] ^ Strong eigenstate. No definite lifetime (see kaon notes below)
^[d] ^ The mass of the
K⁰
_L and
K⁰
_S are given as that of the
K⁰
. However, it is known that a difference between the masses of the
K⁰
_L and
K⁰
_S on the order of 2.2×10⁻¹¹ MeV/c² exists.[34]
^[e] ^ Weak eigenstate. Makeup is missing small CP–violating term (see notes on neutral kaons below).

Particle name	Particle symbol	Antiparticle symbol	Quark content	Rest mass (MeV/c2)	IG	JPC	S	C	B'	Mean lifetime (s)	Commonly decays to (>5% of decays)
Charged rho meson[42]	ρ+ (770)	ρ− (770)	u d	775.4±0.4	1+	1−	0	0	0	~4.5×10−24[f][g]	π± + π0
Neutral rho meson[42]	ρ0 (770)	Self	${\displaystyle \mathrm {\tfrac {u{\bar {u}}-d{\bar {d}}}{\sqrt {2}}} }$	775.49±0.34	1+	1−−	0	0	0	~4.5×10−24[f][g]	π+ + π−
Omega meson[43]	ω (782)	Self	${\displaystyle \mathrm {\tfrac {u{\bar {u}}+d{\bar {d}}}{\sqrt {2}}} }$	782.65±0.12	0−	1−−	0	0	0	(7.75±0.07)×10−23[f]	π+ + π0 + π− or π0 + γ
Phi meson[44]	ϕ (1020)	Self	s s	1019.445±0.020	0−	1−−	0	0	0	(1.55±0.01)×10−22[f]	K+ + K− or K0 S + K0 L or ( ρ + π ) / ( π+ + π0 + π− )
J/Psi[45]	J/ψ	Self	c c	3096.916±0.011	0−	1−−	0	0	0	(7.1±0.2)×10−21[f]	See J/ψ (1S) decay modes
Upsilon meson[46]	ϒ (1S)	Self	b b	9460.30±0.26	0−	1−−	0	0	0	(1.22±0.03)×10−20[f]	See ϒ (1S) decay modes
Kaon[47]	K∗+	K∗−	u s	891.66±0.026	​1⁄2	1−	1	0	0	~7.35×10−20[f][g]	See K∗ (892) decay modes
Kaon[47]	K∗0	K∗0	d s	896.00±0.025	​1⁄2	1−	1	0	0	(7.346±0.002)×10−20[f]	See K∗ (892) decay modes
D meson[48]	D∗+ (2010)	D∗− (2010)	c d	2010.27±0.17	​1⁄2	1−	0	+1	0	(6.9±1.9)×10−21[f]	D0 + π+ or D+ + π0
D meson[49]	D∗0 (2007)	D∗0 (2007)	c u	2006.97±0.19	​1⁄2	1−	0	+1	0	>3.1×10−22[f]	D0 + π0 or D0 + γ
strange D meson[50]	D∗+ s	D∗− s	c s	2112.3±0.5	0	1−	+1	+1	0	>3.4×10−22[f]	D∗+ + γ or D∗+ + π0
B meson[51]	B∗+	B∗−	u b	5325.1±0.5	​1⁄2	1−	0	0	+1	Unknown	B+ + γ
B meson[51]	B∗0	B∗0	d b	5325.1±0.5	​1⁄2	1−	0	0	+1	Unknown	B0 + γ
Strange B meson[52]	B∗0 s	B∗0 s	s b	5412.8±1.3	0	1−	−1	0	+1	Unknown	B0 s+ γ
Charmed B meson†	B∗+ c	B∗− c	c b	Unknown	0	1−	0	+1	+1	Unknown	Unknown

^[f] ^ PDG reports the resonance width (Γ). Here the conversion τ = ​^ħ⁄_Γ is given instead.
^[g] ^ The exact value depends on the method used. See the given reference for detail.

There are two complications with neutral kaons:[13]

• Due to neutral kaon mixing, the
  K⁰
  _S and
  K⁰
  _L are not eigenstates of strangeness. However, they are eigenstates of the weak force, which determines how they decay, so these are the particles with definite lifetime.
• The linear combinations given in the table for the
  K⁰
  _S and
  K⁰
  _L are not exactly correct, since there is a small correction due to CP violation. See CP violation in kaons.

Note that these issues also exist in principle for other neutral flavored mesons; however, the weak eigenstates are considered separate particles only for kaons because of their dramatically different lifetimes.[13]

• What Happened to the Antimatter? Fermilab's DZero Experiment Finds Clues in Quick-Change Meson
• CDF experiment's definitive observation of matter-antimatter oscillations in the Bs meson