05399 - Quark Model (from Particle Data Group, 2010) [Nakamoura].pdf

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14. Quark model
1
14. QUARK MODEL
Revised September 2009 by C. Amsler (University of Z¨rich), T. DeGrand (University of Colorado,
u
Boulder), and B. Krusche (University of Basel).
14.1. Quantum numbers of the quarks
Quarks are strongly interacting fermions with spin 1/2 and, by convention, positive parity.
Antiquarks have negative parity. Quarks have the additive baryon number 1/3, antiquarks -1/3.
Table 14.1 gives the other additive quantum numbers (flavors) for the three generations of quarks.
They are related to the charge
Q
(in units of the elementary charge
e)
through the generalized
Gell-Mann-Nishijima formula
B
+
S
+
C
+
B
+
T
,
(14.1)
Q
=
I
z
+
2
where
B
is the baryon number. The convention is that the
flavor
of a quark (I
z
,
S, C, B,
or
T)
has
the same sign as its
charge
Q.
With this convention, any flavor carried by a charged meson has the
same sign as its charge,
e.g.,
the strangeness of the
K
+
is +1, the bottomness of the
B
+
is +1,
−
and the charm and strangeness of the
D
s
are each
−1.
Antiquarks have the opposite flavor signs.
Table 14.1:
Additive quantum numbers of the quarks.
Property
Quark
d
−
1
3
1
2
u
+
2
3
1
2
s
−
1
3
c
+
2
3
b
−
1
3
t
+
2
3
Q
– electric charge
I
– isospin
I
z
– isospin
z-component
S
– strangeness
C
– charm
B
– bottomness
T
– topness
0
0
−1
0
0
0
0
0
0
+1
0
0
0
0
0
0
−1
0
0
0
0
0
0
+1
−
1
2
+
1
2
0
0
0
0
0
0
0
0
14.2. Mesons
Mesons have baryon number
B
= 0. In the quark model, they are
qq
bound states of quarks
q
and antiquarks
q
(the flavors of
q
and
q
may be different). If the orbital angular momentum of
the
qq
state is , then the parity
P
is (−1)
+1
. The meson spin
J
is given by the usual relation
| −
s|
≤
J
≤ |
+
s|,
where
s
is 0 (antiparallel quark spins) or 1 (parallel quark spins). The charge
¯
conjugation, or
C-parity C
= (−1)
+s
, is defined only for the
q q
states made of quarks and their
own antiquarks. The
C-parity
can be generalized to the
G-parity G
= (−1)
I+
+s
for mesons
¯
u
made of quarks and their own antiquarks (isospin
I
z
= 0), and for the charged
ud
and
d¯
states
(isospin
I
= 1).
K. Nakamura
et al.,
JPG
37,
075021 (2010) (http://pdg.lbl.gov)
July 30, 2010
14:36
2
14. Quark model
The mesons are classified in
J
P C
multiplets. The = 0 states are the pseudoscalars (0
−+
)
and the vectors (1
−−
). The orbital excitations = 1 are the scalars (0
++
), the axial vectors
(1
++
) and (1
+−
), and the tensors (2
++
). Assignments for many of the known mesons are given
in Tables 14.2 and 14.3. Radial excitations are denoted by the principal quantum number
n.
The
very short lifetime of the
t
quark makes it likely that bound-state hadrons containing
t
quarks
and/or antiquarks do not exist.
States in the natural spin-parity series
P
= (−1)
J
must, according to the above, have
s
= 1
and hence,
CP
= +1. Thus, mesons with natural spin-parity and
CP
=
−1
(0
+−
, 1
−+
, 2
+−
,
¯
3
−+
,
etc.)
are forbidden in the
q q
model. The
J
P C
= 0
−−
state is forbidden as well. Mesons
with such
exotic
quantum numbers may exist, but would lie outside the
q q
model (see section
¯
below on exotic mesons).
Following SU(3), the nine possible
q q
combinations containing the light
u, d,
and
s
quarks are
¯
grouped into an octet and a singlet of light quark mesons:
3
⊗
3
=
8
⊕
1
.
(14.2)
A fourth quark such as charm
c
can be included by extending SU(3) to SU(4). However, SU(4)
is badly broken owing to the much heavier
c
quark. Nevertheless, in an SU(4) classification, the
sixteen mesons are grouped into a 15-plet and a singlet:
4
⊗
4
=
15
⊕
1
.
(14.3)
The
weight diagrams
for the ground-state pseudoscalar (0
−+
) and vector (1
−−
) mesons are
depicted in Fig. 14.1. The light quark mesons are members of nonets building the middle plane in
Fig. 14.1(a) and (b).
Isoscalar states with the same
J
P C
will mix, but mixing between the two light quark isoscalar
mesons, and the much heavier charmonium or bottomonium states, are generally assumed to be
negligible. In the following, we shall use the generic names
a
for the
I
= 1,
K
for the
I
= 1/2,
and
f
and
f
for the
I
= 0 members of the light quark nonets. Thus, the physical isoscalars are
mixtures of the SU(3) wave function
ψ
8
and
ψ
1
:
f
=
ψ
8
cos
θ
−
ψ
1
sin
θ ,
f
=
ψ
8
sin
θ
+
ψ
1
cos
θ ,
where
θ
is the nonet mixing angle and
1
¯
u
s
ψ
8
=
√
(u¯ +
dd
−
2s¯)
,
6
1
¯
u
s
ψ
1
=
√
(u¯ +
dd
+
s¯
)
.
3
(14.6)
(14.4)
(14.5)
(14.7)
The mixing angle has to be determined experimentally.
¯
These mixing relations are often rewritten to exhibit the
u¯
+
dd
and
s¯
components which
u
s
√
decouple for the “ideal” mixing angle
θ
i
, such that tan
θ
i
= 1/ 2 (or
θ
i
=35.3
◦
). Defining
α
=
θ
+ 54.7
◦
, one obtains the physical isoscalar in the flavor basis
1
¯
u
s
f
=
√
(u¯ +
dd)
cos
α
−
s¯
sin
α ,
2
July 30, 2010
14:36
(14.8)
14. Quark model
3
D
0
−
−
−
cs cd
cu
D
s
+
(a)
π
ds
;;;;;;;;;;;;;
–
;;;;;;;;;;;;;
−
;;;;;;;;;;;;;
−
;;;;;;;;;;;;;
;;;;;;;;;
K
0
;;;;;;;;;;;;;
;;;;;;;;;;;;;
du
;;;;;;;;;
;;;;;;;;;
;;;;;;;;;
−
;;;;;;;;;
su
;;;;;;;;;
;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;
K
–
−
;;;;;;;;;
π
0
η
us
;;;;;;;;;
+
;;;;;;;;;
;;;;;;;;;
−
;;;;;;;;;
;;;;;;;;;
;;;;;;;;;;;;;;
π
ud
η
c
η
′
sd
;;;;;;;;;;;;;;
−
;;;;;;;;;;;;;;
;;;;;;;;;;;;;;
D
+
+
K
;;;;;;;;;
C
Y
I
−
–
dc
−
sc
D
K
0
−
uc
D
0
D
*
0
−
−
−
cs cd
cu
D
s
−
D
s*
+
D
*
+
*
+
K
;;;;;;;;;
(b)
ρ
−
K
*
0
;;;;;;;;;;;;;
K
*
−
;;;;;;;;;;;;;
−
−
;;;;;;;;;
;;;;;;;;;;;;;
ds
us
;;;;;;;;;
0
;;;;;;;;;;;;;
;;;;;;;;;
;;;;;;;;;;;;;
;;;;;;;;;
−
−
;;;;;;;;;;;;;
;;;;;;;;;
;;;;;;;;;
du
;;;;;;;;;
;;;;;;;;;
;;;;;;;;;;;;;;
ud
;;;;;;;;;
−
;;;;;;;;;;;;;;
−
;;;;;;;;;
;;;;;;;;;;;;;;
su
;;;;;;;;;
;;;;;;;;;;;;;;
sd
;;;;;;;;;
;;;;;;;;;;;;;;
;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;
ρ
ω
J
ψ
φ
/
ρ
+
−
−
*
−
dc sc
D
K
*
0
−
uc
D
*
0
Figure 14.1:
SU(4) weight diagram showing the 16-plets for the pseudoscalar (a) and
vector mesons (b) made of the
u, d, s,
and
c
quarks as a function of isospin
I,
charm
C,
and
hypercharge
Y
=
S+B
−
C
. The nonets of light mesons occupy the central planes to which
3
the
c¯
states have been added.
c
and its orthogonal partner
f
(replace
α
by
α
– 90
◦
). Thus for ideal mixing (α
i
= 90
◦
), the
f
¯
becomes pure
s¯
and the
f
pure
u¯
+
dd.
The mixing angle
θ
can be derived from the mass
s
u
relation
4m
K
−
m
a
−
3m
f
√
,
(14.9)
tan
θ
=
2 2(m
a
−
m
K
)
which also determines its sign or, alternatively, from
4m
K
−
m
a
−
3m
f
.
(14.10)
tan
2
θ
=
−4m
K
+
m
a
+ 3m
f
Eliminating
θ
from these equations leads to the sum rule [1]
(m
f
+
m
f
)(4m
K
−
m
a
)
−
3m
f
m
f
= 8m
2
−
8m
K
m
a
+ 3m
2
.
a
K
(14.11)
*
D
s
−
This relation is verified for the ground-state vector mesons. We identify the
φ(1020)
with the
f
and the
ω(783)
with the
f
. Thus
φ(1020)
=
ψ
8
cos
θ
V
−
ψ
1
sin
θ
V
,
July 30, 2010
14:36
(14.12)
4
14. Quark model
ω(782)
=
ψ
8
sin
θ
V
+
ψ
1
cos
θ
V
,
(14.13)
with the vector mixing angle
θ
V
= 35
◦
from Eq. (14.9), very close to ideal mixing. Thus
φ(1020)
is nearly pure
s¯
. For ideal mixing, Eq. (14.9) and Eq. (14.10) lead to the relations
s
m
K
=
m
f
+
m
f
2
, m
a
=
m
f
,
(14.14)
which are satisfied for the vector mesons. However, for the pseudoscalar (and scalar mesons),
Eq. (14.11) is satisfied only approximately. Then Eq. (14.9) and Eq. (14.10) lead to somewhat
different values for the mixing angle. Identifying the
η
with the
f
one gets
η
=
ψ
8
cos
θ
P
−
ψ
1
sin
θ
P
,
η
=
ψ
8
sin
θ
P
+
ψ
1
cos
θ
P
.
(14.15)
(14.16)
Following chiral perturbation theory, the meson masses in the mass formulae (Eq. (14.9) and
Eq. (14.10)) should be replaced by their squares. Table 14.2 lists the mixing angle
θ
lin
from
Eq. (14.10) and the corresponding
θ
quad
obtained by replacing the meson masses by their squares
throughout.
The pseudoscalar mixing angle
θ
P
can also be measured by comparing the partial widths for
radiative
J/ψ
decay into a vector and a pseudoscalar [2], radiative
φ(1020)
decay into
η
and
η
[3], or
pp
annihilation at rest into a pair of vector and pseudoscalar or into two pseudoscalars
¯
[4,5]. One obtains a mixing angle between –10
◦
and –20
◦
.
The nonet mixing angles can be measured in
γγ
collisions,
e.g.,
for the 0
−+
, 0
++
, and 2
++
nonets. In the quark model, the amplitude for the coupling of neutral mesons to two photons
is proportional to
i
Q
2
, where
Q
i
is the charge of the
i-th
quark. The 2γ partial width of an
i
isoscalar meson with mass
m
is then given in terms of the mixing angle
α
by
Γ
2γ
=
C(5
cos
α
−
√
2 sin
α)
2
m
3
,
(14.17)
for
f
and
f
(α
→
α
– 90
◦
). The coupling
C
may depend on the meson mass. It is often assumed
to be a constant in the nonet. For the isovector
a,
one then finds Γ
2γ
= 9
C m
3
. Thus the
members of an ideally mixed nonet couple to 2γ with partial widths in the ratios
f
:
f
:
a
=
25 : 2 : 9. For tensor mesons, one finds from the ratios of the measured 2γ partial widths for
the
f
2
(1270) and
f
2
(1525) mesons a mixing angle
α
T
of (81± 1)
◦
, or
θ
T
= (27
±
1)
◦
, in accord
with the linear mass formula. For the pseudoscalars, one finds from the ratios of partial widths
Γ(η
→
2γ)/Γ(η
→
2γ) a mixing angle
θ
P
= (–18
±
2)
◦
, while the ratio Γ(η
→
2γ)/Γ(π
0
→
2γ)
leads to
∼
–24
◦
. SU(3) breaking effects for pseudoscalars are discussed in Ref. 6.
The partial width for the decay of a scalar or a tensor meson into a pair of pseudoscalar mesons
is model-dependent. Following Ref. 7,
Γ =
C
×
γ
2
× |F
(q)|
2
×
q .
(14.18)
July 30, 2010
14:36
14. Quark model
5
Table 14.2:
Suggested
qq
quark-model assignments for some of the observed light mesons. Mesons in bold face are included in the Meson
Summary Table. The wave functions
f
and
f
are given in the text. The singlet-octet mixing angles from the quadratic and linear mass
formulae are also given for the well established nonets. The classification of the 0
++
mesons is tentative and the mixing angle uncertain
due to large uncertainties in some of the masses. Also, the
f
0
(1710) and
f
0
(1370) are expected to mix with the
f
0
(1500). The latter is
not in this table as it is hard to accommodate in the scalar nonet. The light scalars
a
0
(980),
f
0
(980), and
f
0
(600) are often considered as
meson-meson resonances or four-quark states, and are therefore not included in the table. See the “Note on Scalar Mesons” in the Meson
Listings for details and alternative schemes.
n
2s+1
J
J
PC
ud, ud,
I
=1
1
√
(dd
−
uu)
2
I
=
us, ds; ds,
−us
K
K
∗
(892)
K
1B
†
∗
K
0
(1430)
1
2
I
=0
f
η
φ(1020)
h
1
(1380)
f
0
(1710)
f
1
(1420)
f
2
(1525)
η
2
(1870)
I
=0
f
η
(958)
ω(782)
h
1
(1170)
f
0
(1370)
f
1
(1285)
f
2
(1270)
η
2
(1645)
ω(1650)
θ
quad
[
◦
]
θ
lin
[
◦
]
1
1
S
0
1
3
S
1
1
1
P
1
1
3
P
0
1
3
P
1
1
3
P
2
1
1
D
2
1
3
D
1
1
3
D
2
1
3
D
3
1
3
F
4
1
3
G
5
1
3
H
6
2
1
S
0
2
3
S
1
†
0
−+
1
−−
1
+−
0
++
1
++
2
++
2
−+
1
−−
2
−−
3
−−
4
++
5
−−
6
++
0
−+
1
−−
π
ρ(770)
b
1
(1235)
a
0
(1450)
a
1
(1260)
a
2
(1320)
π
2
(1670)
ρ(1700)
−11.5
38.7
−24.6
36.0
K
1A
†
∗
K
2
(1430)
29.6
28.0
K
2
(1770)
†
K
∗
(1680)
K
2
(1820)
ρ
3
(1690)
a
4
(2040)
ρ
5
(2350)
a
6
(2450)
π(1300)
ρ(1450)
∗
K
3
(1780)
∗
K
4
(2045)
φ
3
(1850)
ω
3
(1670)
f
4
(2050)
32.0
31.0
f
6
(2510)
K(1460)
K
∗
(1410)
η(1475)
φ(1680)
η(1295)
ω(1420)
The 1
+±
and 2
−±
isospin
1
states mix. In particular, the
K
1A
and
K
1B
are nearly equal (45
◦
) mixtures of the
K
1
(1270) and
K
1
(1400).
2
The physical vector mesons listed under 1
3
D
1
and 2
3
S
1
may be mixtures of 1
3
D
1
and 2
3
S
1
, or even have hybrid components.
July 30, 2010
14:36

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