Spin, orbital angular momentum, and total angular momentum

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.^{[upper-alpha 1]}

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, either ( S_z = +1/2 or 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 , with three possible spin projections ( S_z = +1, S_z = 0, and S_z = −1), and their combination is called a vector meson or spin-1 triplet. If two quarks have oppositely aligned spins, the spin vectors add up to make a vector of length S = 0, and only one spin projection ( S_z = 0 ), called a scalar meson or spin-0 singlet. Because mesons are made of one quark and one antiquark, they are 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 also comes in increments of 1 ħ. The total angular momentum (quantum number J) of a particle is the combination of the two intrinsic angular momentums (spin) and the orbital angular momentum. It can take any value from J = |L − S| up 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.^[12]

P-parity

P-parity is left-right parity, or spatial parity, and was the first of several "parities" discovered, and so is often called just “parity”. 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:^[13][14]

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

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

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.^[15] 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 originally thought to be conserved, but was later found to be violated on rare occasions in weak interactions.^[16][17][18]

G-parity

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.^[19]

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

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

Combinations of one u, d, or s quark and one u, d, or s antiquark in J^P = 1⁻ configuration also form a nonet.

Flavour quantum numbers

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 (more negative) 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:^[23]

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

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:

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

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

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

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.

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).^{[lower-roman 1]} The rules for flavourless mesons are:^[24]

Nomenclature of flavourless mesons

q q content	I	JPC [lower-roman 2]
		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[lower-roman 3]	χc
b b	0	η b	hb	ϒ	χb
t t	0	η t	ht	θ	χt

1. ↑ For the purpose of nomenclature, the isospin projection I₃ is treated as if it were not 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".
2. ↑ C-parity is only relevant for neutral mesons.
3. ↑ For the special case 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.

Pseudoscalar mesons

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[27]	π+	π−	u d	139.57018±0.00035	1−	0−	0	0	0	(2.6033±0.0005)×10−8	μ+ + ν μ
Pion[28]	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[29]	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[30]	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[31]	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[32]	η b(1S)	Self	b b	9300±40	0+	0−+	0	0	0	Unknown	See η b decay modes
Kaon[33]	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[34]	K0	K0	d s	497.614±0.024	1⁄2	0−	1	0	0	[c]	[c]
K-Short[35]	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[36]	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[37]	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[38]	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[39]	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[40]	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[41]	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[42]	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[43]	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.^[36]
[e] ^ Weak eigenstate. Makeup is missing small CP–violating term (see notes on neutral kaons below).

Vector mesons

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[44]	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[44]	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[45]	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[46]	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[47]	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[48]	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[49]	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[49]	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[50]	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[51]	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[52]	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[53]	B∗+	B∗−	u b	5325.1±0.5	1⁄2	1−	0	0	+1	Unknown	B+ + γ
B meson[53]	B∗0	B∗0	d b	5325.1±0.5	1⁄2	1−	0	0	+1	Unknown	B0 + γ
Strange B meson[54]	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.

Notes on neutral kaons

There are two complications with neutral kaons:^[55]

• 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.^[55]