GlueX Physics Quark Model

GlueX Physics

Mesons in the Quark Model

Conserved Quantities in the Strong Interaction

To do this, I want to start with the very basics of the strong interaction, namely the conserved quantities, J, P, C, ···. With this, I want to look at spectroscopy within one specific model, the constituent quark model. This model is by no means perfect. It provides no explanation for confinement, and the role of gluons is not obvious. It also makes no absolute mass predictions, and no absolute rate predictions for decays. However it does make a rather large number of very good predictions. It also provides a very natural framework within which to classify mesons. It provides a natural handle to address issues such as structure and decays, and even makes some rather nice predictions for relative decay rates.

The strong interaction conserves a number of quantities, some of which are listed here.
B Baryon number.
Q Electric charge.
J Angular momentum.
S Strangeness.
I Strong isospin.
P Parity.
C Charge conjugation.
G G–parity.
Those that are used will be explained as we go along. However, a number of these are carried by the quarks themselves. In table 1 are given the quantum numbers of the three lightest quarks.

quark	B	Q	J	S	I	I_z
u	1/3	2/3	1/2	0	1/2	+1/2
d	1/3	-1/3	1/2	0	1/2	-1/2
s	1/3	-1/3	1/2	-1	0	0

Table 1: Quantum numbers of the quarks. B is baryon number, Q is electric charge, J is the spin, S is strangeness, I is the strong isospin and I_z is the projection of I along the quantization axis, (usually defined as z).

Building Mesons

In the constituent quark model, we treat a meson as a bound quark-antiquark pair, $q{\bar q}$ , and then draw an analogy to the positronium system, e⁺e⁻ to understand what we are seeing. In this picture the $q$ and the ${\bar q}$ both have spin $1$ . These can combine to either total spin $S=0$ , or total spin $S=1$ .

(S=0)	(S=1)
(1/(√ 2))[ ↑₁ ↓₂ - ↑₂↓₁ ]	↑₁↑₂ (1/(√ 2))[ ↑₁ ↓₂ + ↑₂↓₁ ] ↓₁↓₂

In addition to the total spin, we can have orbital angular momentum L between the $q{\bar q}$ pair. Then, the L and S can combine to total angular momentum J = L ⊕ S, where J =| L−S |,| L−S+1 |,···,| L+S |. The states can be written in spectroscopic notation as 2S+1LJ, and are shown for positronium in table 2. Using the quarks as given in table 1, we are then able to use L, S and J to construct the J^PC quantum numbers of the mesons.

State	S	L	J	P	C	J^PC	Mesons	Name
¹S₀	0	0	0	-	+	0^-+	π η η^' K	Pseudoscalar
³S₁	1	0	1	-	-	1^--	ρ ω φ K^*	Vector
¹P₁	0	1	1	+	-	1^+-	b₁ h₁ h₁^' K₁	Pseudo Vector
³P₀	1	1	0	+	+	0⁺⁺	a₀ f₀ f₀^' K^*₀	Scalar
³P₁	1	1	1	+	+	1⁺⁺	a₁ f₁ f₁^' K₁	Axial Vector
³P₂	1	1	2	+	+	2⁺⁺	a₂ f₂ f₂^' K^*₂	Tensor

Table 2: The positronium states as a function of L, S and J. These then correspond to the named mesons of the specified J^PC.

Parity

Let us start with parity, P. Mathematically,

parity is a reflection operator, and if the wave functions are eigenstates of the parity operator, then

P(ψ( ${\vec r}$ )) = ψ( $-{\vec r}$ ) = η_P ψ( ${\vec r}$ ).

Since applying parity twice should return us to the original state, the eigenvalues of parity, η_P can only be ±1. We can normally separate ψ into a radial and an angular piece,

ψ( ${\vec r}$ ) = R(r)Y_lm(θ, φ).

In this case, the operation of parity leaves R unchanged, but transforms the angular piece to Y_lm(π − θ, φ + π), and it can be shown that:

Y_lm(π − θ, φ + π) = (−1)^lY_lm(θ, φ).

Finally, fermions and antifermions have intrinsic opposite parity. This leads to the parity of a meson being:

P( $q{\bar q}$ ) = (−1)^L+1 (1)

In considering that parity is conserved in a reaction, we consider the decay A → B+C, where there is orbital angular momentum l between B and C. Parity conservation says that

P(A) = P(B) · P(C) · (−1)l. (2)

C-Parity

The next quantum number is charge conjugation, C, which represents a trans- formation of the particle into its antiparticle. This reverses several properties of the particle such as charge and magnetic moment. Clearly, in order for a particle to be an eigenstate of the C operator, it must be electrically neutral. If we consider the π◦, then:

C|π◦ >=η_C |π◦ >

where η_C = ±1. If we imagine a meson built from a quark and its antiquark, say $u{\bar u}$ , with some total wave function of both its position and spin, Ψ.

Ψ( ${\vec r}$ , ${\vec s}$ ) = R(r)Ylm(θ, φ)χ( ${\vec s}$ )

The charge conjugation operator acting on this state reverses the meaning of u and ${\bar u}$ . This has the effect of mapping ${\vec r}$ which points to the quark into $-{\vec r}$ so that it continues to point at the quark. Under the same arguments that we used in parity, this leads to a factor (−1)^L+1. This also flips the spin wave functions, leading to a factor of (−1) for the S=0 case and a factor of (+1) for the S=1 case. This is a factor of (−1)^S+1, which when combined with the L factor leads to:

C( $q{\bar q}$ ) = (−1)^L+S (3)

G-Parity

Clearly charged particles cannot be eigenstates of C,

C | π⁺ >= η | π⁻ >.

However, if we were to apply the C operator followed by a rotation in isospin,

R = exp(iπ I₂ )

such that

| I, I_z > → | I, -I_z > ,

then charged particles could be eigenstates of this operator. We define the G parity operator as G = CR, and from this it is easy to show that for a $q{\bar q}$ system,

G = C · (−1)^I.

These then lead to the following formulas.

J=L⊕S (4)
P = (−1)^L+1 (5)
C = (−1)^L+S (6)
G = (−1)^L+S+I (7)

Quantum Numbers of Mesons

Using these relationships to build up possible J^PC’s for mesons, we find that the following numbers are allowed:

0⁻⁺, 0⁺⁺, 1⁻⁻, 1⁺⁻, 1⁻⁻, 2⁻⁻, 2⁻⁺, 2⁺⁺, 3⁻⁻, 3⁺⁻,

3⁻⁻, · · ·

and looking carefully at these, we find that there is a sequence of J^PC’s which are not allowed for a simple $q{\bar q}$ system.

0⁻⁻, 0⁺⁻, 1⁻⁺, 2⁺⁻, 3⁻⁺, · · ·

These latter quantum numbers are known as explicitly exotic quantum numbers. If a state with these quantum numbers is found, we know that it must be something other than a normal, $q{\bar q}$ meson. Following the positronium analogy as in table 2, we can now assign the J^P(C) quantum numbers to the listed atomic states. In the case of mesons, we have three quarks, u, d and s which can be combined with three antiquarks. This leads to nine possible $q{\bar q}$ combinations with the same J^P(C), rather than the one positronium state.

SU(3)

If we now assume that the three quarks are flavor symmetric, then we can use the SU(3)–flavor group to build up the nominal nine mesons, (a nonet).

3⊗3=1⊕8

The nine members of the nonet are going to be broken into two groups, eight members of an octet, | 8 > and a single member of a singlet | 1 >. Under the SU(3) flavor assumption, all the members of the octet have the same basic coupling constants to similar reactions, while the singlet member could have a different coupling. The nominal $q{\bar q}$ combinations for the pseudoscalar mesons are shown below. The three π’s are isospin I = 1, while the K’s are all isospin 1/2. The | 1 > and | 8 > states are isospin 0.

	K⁰		K⁺
π^-		π⁰,η,η^'		π⁺
	${\bar K}$ ⁰		K⁺

In fact, the SU(3) quark content of the states is given as

	( $d{\bar s}$ )		( $u{\bar s}$
( $d{\bar u}$ )		(1/√ 2)( $u{\bar u}-d{\bar d}$ ) (1/√ 6)( $u{\bar u}+d{\bar d}-2s{\bar s}$ )		( $u{\bar d}$ )
	( $s{\bar d}$ )		( $s{\bar u}$ )
		(1/√ 3)( $u{\bar u}+d{\bar d}+s{\bar s}$ )