15. Quark Model

1

15. Quark Model
Revised August 2023 by C. Amsler (Stefan Meyer Inst.), V. Crede (Florida State U.) and T.
DeGrand (Colorado U., Boulder).

15.1
15.2
15.3
15.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quantum numbers of the quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exotic mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4.1 Tetraquarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4.2 Glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4.3 Hybrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.5 Baryons: qqq states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.5.1 Light baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.5.2 Charmed and bottom baryons . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.6 Magnetic moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.7 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.8 Lattice Calculations of Hadronic Spectroscopy . . . . . . . . . . . . . . . . . . . .
15.8.1 Spectroscopy of low-lying states . . . . . . . . . . . . . . . . . . . . . . . . . .
15.8.2 Excited state spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.8.3 Electromagnetic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2
3
9
10
11
12
13
13
19
23
25
26
27
27
34

15.1 Introduction
Quantum chromodynamics (QCD) is the theory of strong interactions. QCD is a quantum field
theory with an SU (Nc ) local “color” gauge symmetry with Nc = 3 colors and a collection of Nf“flavors” of colored fermions, the quarks. It involves a set of Nc2 − 1 = 8 non-Abelian gauge fields,
the gluons. QCD is believed to confine, that is, its physical states are color singlets with internal
quark and gluon degrees of freedom. This review is concerned with the description of the properties(masses and matrix elements for couplings to electromagnetism and the weak intractions) of the
low lying bound states of QCD. The shorthand expression for describing this physics is called the“quark model”.
The spectrum of strongly interacting particles consists of a tower of many states, which can beeither bosons (labelled as “mesons”) or fermions (labelled as “baryons”). The spectrum of baryons
and mesons exhibits a high degree of regularity. The organizational principle which best categorizes
this regularity is encoded in the quark model. All descriptions of strongly interacting states use thelanguage of the quark model. At the same time, the language is not precise. The quark model exists
on many levels: at the simplest level, it is an almost dynamics-free picture of strongly interacting
particles as bound states of quarks and antiquarks. As one refines the description, the quark model
can become a framework with more detailed descriptions of dynamics. At its most fundamental
level, it might be a description of QCD. In effective field theories of QCD at low energies the hadron
spectrum is not simply given by a series of states but is the manifestation of a complex dynamics
involving various types of states [1].
At its heart, the quark model assumes that mesons are bound states of quark - antiquark pairs,
and baryons are bound states of three quarks. These are the minimal particle content states which
can be color singlets in an SU (3) gauge theory. This approach cannot be justified directly fromS. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:19am

2

15. Quark Model

QCD; however, there is indirect evidence that this description has some fundamental validity fromthe version of QCD where the number of colors Nc is taken to infinity [2–4]. In that limit, mesons
are dominantly narrow (width proportional to 1/Nc ) bound states of a quark - antiquark pair, andbaryons have a mass which scales as Nc .
A better justification is that this approach works. Indeed, the quark model is much older than
QCD as a theory of the strong interactions (1973-1974, see the article on “50 Years of Quantum
Chromodynamics” [5]). In fact, the principal issue (circa 1963-64) in strong interaction physics
before QCD was to justify the success of the quark model in systemizing the properties of mesons
and baryons in terms of some more fundamental dynamics (QCD).
Today one knows that this is not the whole story. There are experimentally observed states
which either cannot be described, or have an uncomfortable description, as minimal quark number
states. Some of them have “exotic” (non-q̄q or qqq) quantum numbers. Given the successes of
the quark model, these are classified as “tetraquarks” (qq q̄ q̄), “pentaquarks” (qqqq q̄) or “glueballs”,
bound states of gluons, the gluonic degrees of freedom in QCD. Of course, such labels are imprecise:bound states with the same overall quantum numbers can mix, regardless of their internal degrees
of freedom.
This review has several parts. We start by describing the properties of strongly interacting
particles in terms of the properties of states made of a minimal number of quark fields which canbe coupled into a color singlet – two fields (a quark and an antiquark) for the mesons, and threequarks, for a baryon. Quarks come in six flavors. We describe the properties of mesons as q̄q
systems and baryons as qqq systems. Along the way we discuss hadronic bound states which do
not fit into this classification.
Finally, at the end of this review, we present results from lattice simulations of QCD, a direct
approach to the solution of QCD from its Lagrangian, without reference to models. Lattice simulations interact with the quark model in (at least) two ways: first, the interpolating fields which
are used in lattice simulations are usually directly based on quark model constructions. That is
the simplest way to create states with the desired quantum numbers, which can then be processedby the lattice calculation. The second way that lattice calculations interact with the quark model
comes when one wishes to put the lattice calculations into some context: without the quark model,
there are simply the results from the lattice calculations, and the results from experiment, and noway to understand why they are similar or different. The quark model is the framework which isalmost universally used to generate that context. Of course, that statement is equally valid when
one tries to systemize actual experimental data: the context is always some variation of a quark
model.

15.2 Quantum numbers of the quarks
As gluons carry no intrinsic quantum numbers beyond color charge, and because color is believed
to be permanently confined, the quantum numbers of strongly interacting particles are given by
the quantum numbers of their constituent quarks and antiquarks.
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 15.1 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 generalizedGell-Mann-Nishijima formula
B+ S + C + B + T,(15.1)
2
where B is the baryon number. The convention is that the quark flavor ( I z , S , C , B , or T ) has
the same sign as its charge Q . Antiquarks have the opposite flavor signs. With this convention,
Q = Iz+

31st May, 2024

15. Quark Model3

any flavor carried by a charged meson has the same sign as its charge, e.g., the strangeness of theK + (cs̄) is +1, the bottomness of the B + (ub̄) is +1, and the charm and strangeness of the Ds−(sc̄) are each −1.
Table 15.1: Quark quantum numbers.

Q – electric charge
I – isospin
I z – isospin z-component
S – strangenessC – charm
B – bottomness
T – topness

d

u

s

c

b

t

− 13
1
2
− 12

+ 32
1
2
+ 21

− 13

+ 23

− 13

+ 23

0
0
0
0

0 0 0 00 0 0 00 −1 0 0 0
0 0 +1 0 0
0 0 0 −1 0
0 0 0 0 +1

The hypercharge is defined as
Y =B+ S −C − B + T.
3

(15.2)

Thus Y is equal to 13 for the u and d quarks, – 23 for the s quark, and 0 for all other quarks. Moredetails and derivations on the quark structure of mesons and baryons can be found e.g. in Ref. [6].
The naming scheme for hadrons has been updated by the Particle Data Group in 2023, to
include the heavy “exotic” states discovered recently, the tetraquark (qq q̄ q̄) and pentaquark (qqqq q̄)candidates (see “Naming Scheme for Hadrons” in this Review).

15.3 Mesons
Mesons have baryon number B = 0. In the quark model, they are q q̄ 0 bound states of quarks q
and antiquarks q̄ 0 (the flavors of q and q 0 may be different).
If the orbital angular momentum of the q q̄ 0 state is `, then the parity P is (−1)`+1 . The meson
spin J is given by the usual relation |` − s| ≤ J ≤ |` + s|, where s = 0 (antiparallel quark spins) or
s = 1 (parallel quark spins). The charge conjugation, or C-parity C = (−1)`+s , is defined only forstates made of quarks and their own antiquarks. The C-parity can be generalized to the G-parity
G = (−1)I+`+s for mesons made of quarks and their own antiquarks (isospin I z = 0), and for the
charged ud¯ and dū states (isospin I = 1).
The mesons are classified in J P C multiplets. The ` = 0 states are the pseudoscalars (0−+ ) andthe vectors (1−− ). The orbital excitations ` = 1 are the scalars (0++ ), the axial vectors (1++ )and (1+− ) aka pseudovectors, and the tensors (2++ ). Assignments for many of the known mesons
are given in Tables 15.2, 15.3 and 15.4. Radial excitations are denoted by the principal quantum
number n. The very short lifetime of the t quark (→ W + b) 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 q̄ 0 model. The J P C = 0−− state is forbidden as well. Mesons with suchexotic quantum numbers may exist, but would lie outside the q q̄ 0 model (see section 15.4 below on
exotic mesons).
Following SU(3), the nine possible q q̄ 0 combinations containing the light u, d, and s quarks aregrouped into an octet and a singlet of light quark mesons:
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15. Quark Model

4

Table 15.2: Suggested q q̄ quark-model assignments for the lightest
mesons made of u, d and s quarks. Mesons in boldface are included in
the Summary Table. The wave functions f and f 0 are given in the text
(Eq. (15.9)) and the singlet-octet mixing angles in Table 15.5 below for
the well established nonets. The classification of the 0++ mesons is controversial: (i) the scalars a0 (980), K0∗ (700), f0 (980) and f0 (500), omitted
from the table, are often considered to be four-quark states, but are also
proposed for the ground state scalar nonet (see the chapter “Scalar Mesons
below 1 GeV” in this Review); (ii) there are three isoscalar 0++ mesons,f0 (1370), f0 (1500) and f0 (1710), both f0 (1500) and f0 (1710) being proposed as glueballs. The three states are expected to mix. The isoscalarassignments in the 21 S0 (0−+ ) nonet are also tentative. Details and alternative schemes can be found in “Spectroscopy of Light Meson Resonances”
in this Review.
a The 1+± and 2−± isospin 1 states mix. In particular, the K
1A and K1B
2
are nearly equal mixtures of the K1 (1270) and K1 (1400) [7]. The 2−±
mixed partner of the K2 (1770) is the established K2 (1820).
b The physical vector mesons are mixtures of 13 D and 23 S (for a discus1
1
sion see [8]).
c This state has also been proposed as a tetraquark state [9].
d The η(1475) and η(1405) (not shown) may be manifestations of a single
state [10].
e This state has also been proposed as the ground state tensor glueball [11, 12].
n2s+1 `J J P C

11 S0
13 S1
13 P0
11 P1
13 P1
13 P2
13 D1
11 D2
13 D3
13 F4
1 3 G5
21 S0
23 S1
23 P1
23 P2
21 D2
31 S0

I =1
I = 12¯ ūd,ud,us̄, ds̄;1¯¯ ūs√ (dd − uū) ds,
20−+ π
1−− ρ(770)
0++ a0 (1450)
1+− b1 (1235)
1++ a1 (1260)
2++ a2 (1320)
1−− ρ(1700)
2−+ π2 (1670)
3−− ρ3 (1690)
4++ a4 (1970)5−− ρ5 (2350)
0−+ π(1300)
1−− ρ(1450)
1++ a1 (1640)
2++ a2 (1700)
2−+ π2 (1880)0−+ π(1800)

I =0
f0

I =0
f

K
η
η 0 (958)∗
K (892)φ(1020) ω(782)K0∗ (1430) f0 (1370, 1500, 1710)
K1B a
h1 (1415) h1 (1170)
K1A a
f1 (1420) f1 (1285)
K2∗ (1430) f20 (1525) f2 (1270)
K ∗ (1680)b φ(2170)c ω(1650)
K2 (1770)a η2 (1870) η2 (1645)
K3∗ (1780) φ3 (1850) ω3 (1670)
K4∗ (2045) f4 (2300)
f4 (2050)
K5∗ (2380)
K(1460)η(1475)d η(1295)
K ∗ (1410)b φ(1680) ω(1420)
K1 (1650)K2∗ (1980) f2 (1950)e f2 (1640)
K(1830)31st May, 2024

η(1760)15. Quark Model

5

Table 15.3: cc̄ quark-model assignments for the charmonium mesons withestablished J P C and their corresponding open charm mesons. Mesons inboldface are included in the Meson Summary Table. The open flavor statesin the 1+− and 1++ rows are mixtures of the 1+± states.
a In unitarized chiral perturbation theory this meson splits into two statesat 2105 and 2451 MeV [1, 13].
b The masses are considerably smaller than most theoretical predictions.
They have also been considered as tetraquark or D(∗) K molecular states.
c This meson splits into two states at 2247 and 2555 MeV [1].
d Mixtures of the 1 3 D and 2 3 S states.
11
n2s+1 `J J P C

1 1 S01 3 S1
1 3 P01 3 P11 1 P1
1 3 P22 1 S02 3 S11 3 D11 3 D22 3 PJ
3 3 S12 3 D14 3 S11 3 D3

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

I = 21I =0¯
cū, cd;cs̄;
c̄u, c̄dc̄s±ηc (1S)
D
Ds
J/ψ(1S) D ∗Ds∗±∗ (2317)±b
χc0 (1P ) D0∗ (2300)a Ds0cχc1 (1P ) D1 (2430) Ds1 (2460)±bhc (1P )D1 (2420) Ds1 (2536)±∗ (2573)±χc2 (1P ) D2∗ (2460) Ds20ηc (2S)D0 (2550)
Ds0 (2590)+∗ (2700)±d
ψ(2S)D1∗ (2600)0 Ds1∗
0∗ψ(3770) D1 (2760)
Ds1 (2860)±d0ψ2 (3823) D2 (2740)χc0 (3860)χc2 (3930)
ψ(4040)ψ(4160)ψ(4415)∗ (2860)±
ψ3 (3842) D3∗ (2750) Ds3
I =0
cc̄

3⊗3=8⊕1 .

(15.3)

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, thesixteen mesons are grouped into a 15-plet and a singlet:
4 ⊗ 4 = 15 ⊕ 1 .

(15.4)

The weight diagrams for the ground-state pseudoscalar (0−+ ) and vector (1−− ) mesons are
depicted in Fig. 15.1. The light quark mesons are members of nonets building the middle plane in
Fig. 15.1(a) and (b).
Isoscalar states with the same J P C mix, but mixing between the two light quark isoscalarmesons, and the much heavier charmonium and 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, f
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15. Quark Model

6

Table 15.4: bb̄ quark-model assignments for the bottomonium mesons
with established J P C and their corresponding open bottom mesons.
n2s+1 `J

1 1 S01 3 S1
1 3 P01 3 P11 1 P1
1 3 P22 1 S02 3 S11 3 D22 3 PJ
2 1 P13 3 S13 3 PJ
4 3 S1

JPC

I =0
bb̄

I = 12¯bū, bd;b̄u, b̄d
B
B∗

I =0
bs̄;
b̄s
0
Bs
Bs∗

I =0
bc̄;
b̄c
±
Bc

0−+
ηb (1S)1−−Υ (1S)++0χb0 (1P )1++χb1 (1P )1+−
hb (1P )B1 (5721) Bs1 (5830)0∗ (5840)02++
χb2 (1P )B2∗ (5747) Bs2−+0ηb (2S)Bc (2S)±1−−Υ (2S)2−−
Υ2 (1D)0, 1, 2++ χb0,1,2 (2P )1+−
hb (2P )1−−Υ (3S)
0, 1, 2++ χb1,2 (3P )1−−Υ (4S)

and f 0 for the I = 0 members of the light quark nonets. Thus, the physical isoscalars are mixturesof the SU(3) wave function ψ8 and ψ1 :f 0 = ψ8 cos θ − ψ1 sin θ ,(15.5)

f = ψ8 sin θ + ψ1 cos θ ,(15.6)

where θ is the nonet mixing angle and1ψ8 = √ (uū + dd¯ − 2ss̄) ,6(15.7)

1
ψ1 = √ (uū + dd¯ + ss̄) .
3(15.8)The mixing relations are often rewritten to exhibit the√uū + dd¯ and ss̄ components which decouple
for the “ideal” mixing angle θi , such that tan θi = 1/ 2 (or θi = 35.3◦ ). Defining α = θ + 54.7◦ ,one obtains the physical isoscalar state in the flavor basis
1¯ cos α − ss̄ sin α ,f 0 = √ (uū + dd)
2(15.9)and its orthogonal partner f (replace α by α –90◦ ). Thus for ideal mixing (αi = 90◦ ), the f 0¯ The mixing angle θ can be derived by diagonalizing the
becomes pure ss̄ and the f pure uū + dd.
mass matrix!
m8 m81.
(15.10)m18 m1
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7

15. Quark ModelZ

Figure 15.1: 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 Iz , charm C , and hypercharge
Y =B+ S − C
3 . The nonets of light mesons occupy the central planes to which the cc̄ states
have been added.
The mass eigenvalues are mf 0 and mf . The mixing angle is given bytan θ =

m8 − mf 0.
m81(15.11)

Calculating m8 and m81 from the wave functions Eq. (15.7) and Eq. (15.8), and expressing thequark masses as a function of the I = 1/2 and I = 1 meson masses, one obtains
tan θ =

4mK − ma − 3mf 0√,
2 2(ma − mK )(15.12)

which also determines the sign of θ. Alternatively, one can express the mixing angle as a functionof all nonet masses. The octet mass is given by
m8 = mf 0 cos2 θ + mf sin2 θ
whencetan2 θ =4mK − ma − 3mf 0.
−4mK + ma + 3mf(15.13)(15.14)

Eliminating θ from Eq. (15.12) and Eq. (15.14) leads to the sum rule [14](mf + mf 0 )(4mK − ma ) − 3mf mf 0 = 8m2K − 8mK ma + 3m2a .
31st May, 2024

(15.15)

15. Quark Model

8

This relation is verified for the ground-state vector mesons. We identify the φ(1020) with the f 0
and the ω(783) with the f . Thus
φ(1020) = ψ8 cos θV − ψ1 sin θV ,(15.16)

ω(782) = ψ8 sin θV + ψ1 cos θV ,(15.17)

with the vector mixing angle θV = 36.5◦ from Eq. (15.14), very close to ideal mixing. Thus φ(1020)
is nearly pure ss̄. For ideal mixing, Eq. (15.12) and Eq. (15.14) lead to the relations
mK =mf + mf 0, ma = mf ,2

(15.18)which are satisfied for the vector mesons.
The situation for the pseudoscalar and scalar mesons is not so clear cut, either theoreticallyor experimentally. For the pseudoscalars, the mixing angle is small. This can be understoodqualitatively via gluon-line counting of the mixing process. The size of the mixing process between
the nonstrange and strange mass bases scales as αs2 , not αs3 , because of two rather than three gluon
exchange as it does for the vector mesons. It may also be that the lightest isoscalar pseudoscalars
mix more strongly with excited states or with states of substantial non-q̄q content, as will be
discussed below.
In fact a large mixing from hadronic loops is expected for scalar mesons, no matter what modelis assumed for q q̄ pair production [15]. A variety of analysis methods lead to similar results: First,
for these states, Eq. (15.15) is satisfied only approximately. Then Eq. (15.12) and Eq. (15.14) leadto somewhat different values for the mixing angle. Identifying the η with the f 0 one gets
η = ψ8 cos θP − ψ1 sin θP ,(15.19)0(15.20)η = ψ8 sin θP + ψ1 cos θP .

Following chiral perturbation theory, the meson masses in the mass formulae (Eq. (15.12) andEq. (15.14)) might be replaced by their squares. Table 15.5 lists the mixing angle θlin from
Eq. (15.14) (using the neutral members of the nonets) and the corresponding θquad obtained byreplacing the meson masses by their squares throughout. The mixing angles in the 1−− , 2++ and¯ mixing is predicted from hadronic loops
3−− nonets are not far from ideal, while larger ss̄-(uū + dd)++−++±
in the 0 , 0
and 1
nonets [15].
Table 15.5: Singlet-octet mixing angles for the well established nonets
from the linear mass formula Eq. (15.14) and its quadratic version in which
the masses are squared. The 1++ and 1+− nonet mixing angles dependon the mixing angle θK1 between K1A and K1B . The recommended values
are ∼23◦ and ∼28◦ for 1++ and 1+− , respectively, with θK 1 ∼35◦ [7].
n2s+1 `J J P C
11 S0
13 S1
13 P2
13 D3

θquad
θlin
◦
[ ]
[◦ ]
−+
0
−11.3 −24.5−−
1
39.2 36.5
2++ 29.6 28.0
3−− 31.8 30.8

The pseudoscalar mixing angle θP can also be measured by comparing the partial widths forradiative J/ψ decay into a vector and a pseudoscalar [16], radiative φ(1020) decay into η and η 0 [17],31st May, 2024

915. Quark Modelradiative decays between pseudoscalar and vector mesons [18], or p̄p annihilation at rest into a pairof vector and pseudoscalar or into two pseudoscalars [19, 20]. One obtains a mixing angle between–10◦ and –20◦ . More recently, a lattice QCD simulation, Ref. [21], has successfully reproduced themasses of the η and η 0 , and as a byproduct find a mixing angle θlin = −14.1(2.8)◦ . We return tothis point in Sec. 15.8.
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 isP
proportional to i Q2i , where Qi is the charge of the i-th quark. The 2γ partial width of an isoscalarmeson with mass m is then given in terms of the mixing angle α by√(15.21)Γ2γ = C(5 cos α − 2 sin α)2 m3 ,for f 0 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 finds Γ2γ = 9 C m3 . Thus the members of
an ideally mixed nonet couple to 2γ with partial widths in the ratios f : f 0 : a = 25 : 2 : 9. For
tensor mesons, one finds from the ratios of the measured 2γ partial widths for the f2 (1270) andf20 (1525) mesons a mixing angle αT of (81± 1)◦ , or θT = (27 ± 1)◦ , in accord with the linear mass
formula. For the pseudoscalars, one finds from the ratios of partial widths Γ (η 0 → 2γ)/Γ (η → 2γ)
a mixing angle θP = (–18 ± 2)◦ , while the ratio Γ (η 0 → 2γ)/Γ (π 0 → 2γ) leads to ∼ –24 ◦ . SU(3)breaking effects for pseudoscalars are discussed in [22].
The partial width Γ for the decay of a scalar or a tensor meson into a pair of pseudoscalarmesons is model-dependent. Following Ref. [23],Γ = C × γ 2 × |F (q)|2 × q ,(15.22)where C is a nonet constant, q the momentum of the decay products, F (q) a form factor, and γ 2
the SU(3) coupling. Details and explicit expressions for γ 2 and F (q) are given in “Spectroscopy ofLight Meson Resonances”. The decay of a q q̄ meson into a pair of mesons involves the creation ofa q q̄ pair, and SU(3) symmetry assumes that the matrix elements for the creation of ss̄, uū, anddd¯ pairs are equal. An excellent fit to the tensor meson decay widths is obtained assuming SU(3)symmetry and a pseudoscalar mixing angle θP ' –17 ◦ [23].
The analysis of resonances is complicated by the presence of thresholds such as K K̄ in a0 (980) →ηπ or f0 (980) → ππ, which affect the resonance masses and widths. A particularly nasty kinematiceffect is the triangle singularity in which one of the primary decay daughters in turn decays and isemitted backwards, catching up and scattering with the second primary product. This mechanism
generates fake peaks in the final state and has been proposed as alternative explanation for several
meson (and baryon) signals. Prominent examples are the η(1475) → K ∗ (→ Kπ)K̄ and η(1410) →a0 (980)(→ ηπ)π signals which could be due to one single state [10] (as shown in Table 15.2).
Furthermore, to parametrize resonances, many data analyses resort to Breit-Wigner amplitudes
which are not suitable to describe broad interfering mesons with the same quantum numbers, as
they violate unitarity. This is in particular the case in the mass range above 1500 MeV whereradial and orbital excitations, e.g. of broad scalar and tensor mesons, start to accumulate. A
better approach is to determine the T-matrix poles in the complex plane, where resonances areusually located on the second Riemann sheet.

15.4 Exotic mesons
There are two classes of colorless exotic mesons allowed in QCD: multiquark states (section
15.4.1) and mesons with active gluons, bound states of gluons (the glueballs, section 15.4.2) andq q̄ states with “valence” gluons (the hybrids, section 15.4.3). The first class contains the simplest
31st May, 2024

10

15. Quark Modelsystem consisting of two quarks and two antiquarks (tetraquarks). Compact tetraquarks (also called“diquonium” in the older literature) are made of diquarks and antidiquarks. Hadroquarkonia consistof a pair of heavy quark and antiquark in a compact core surrounded by a light-quark cloud. In
hadronic “molecules” the building blocks are color-neutral hadrons bound by meson exchanges.
15.4.1 Tetraquarks
The existence of a light nonet composed of four quarks (tetraquarks) with masses below 1 GeV
was suggested a long time ago [24,25]. Coupling two triplets of light quarks u, d, and s, one obtains
nine states, of which the six symmetric (uu, dd, ss, ud + du, us + su, ds + sd) form the six
dimensional representation 6, while the three antisymmetric (ud − du, us − su, ds − sd) form thethree dimensional representation 3 of SU(3):3 ⊗ 3 = 6 ⊕ 3̄ .

(15.23)Hence for tetraquarks one gets the reduction
3⊗3⊗3⊗3=6⊕3⊗6⊗3=3⊗3⊕6⊗6⊕6⊗3⊕3⊗6= 9 ⊕ 36 ⊕ 18 ⊕ 18.

(15.24)

Combining with spin and color and requiring antisymmetry for diquarks and antidiquarks, one
finds for ground states (zero angular momenta) that the most deeply bound tetraquarks (and hence
the lightest ones) form a nonet and are scalar mesons (see also [6]). The masses are estimated to be
below 900 MeV. The strange quark determines the mass splittings and one obtains a mass inverted
spectrum with a light isosinglet (f0 (500)), a medium heavy isodoublet (K0∗ (700)) and a heavy
isotriplet (a0 (980)) + isosinglet (f0 (980)). In alternative schemes these states build the lightest q q̄scalar nonet (for details see “Scalar mesons below 1 GeV” in this “Review”).
A plethora of new heavy multiquark candidates have been reported in the charmonium spectrum(Fig. 15.2). Details and references can be found in the review “Heavy non-q q̄ Mesons”, see also
Refs. [26–28].
The most prominent one is the χc1 (3872) (was X(3872)), first observed by BELLE in 2003 in
B-decays in the final state J/ψ π + π − [29] and firmly established by many experiments (Babar,
BESIII, CDF, D0, LHCb) in several production modes (e+ e− , Λb decay, p̄p, pp). This narrow state(< 1 MeV) decays mainly into D0 D̄0∗ and is right at the D0 D̄0∗ threshold. Its composition isbelieved to be mostly molecular, bound by pion exchange. Its existence was already predicted in1991 [30]. Many non-cc̄ candidates have been reported since then above the open charm threshold,with quantum numbers compatible with cc̄, but which are not expected in the pure charmonium
spectrum. In addition, one observes isovector (charged) mesons decaying into cc states plus acharged pion or kaon. (They are listed in this Review under “Other mesons”.) For example, the
1+− Tcc̄1 (3900)+ (was Zc (3900)+ ) and Tcc̄s1 (4000)+ (was Zcs (4000)− ) cannot be pure charmonia,
but are compatible with cc̄ud¯ and cc̄sū, respectively. They could belong to the same nonet, hence
identifying the other multiplet members might shed light on the structure of such states.
Even more remarkable are the observations by LHCb of a doubly charmed state Tcc (3875)+¯ [31] at the D0 D∗ threshold and of a Tccc̄c̄ (6900) [32] decaying into a pair of J/ψ(1S).
(ccūd))0Similar states begin to emerge also in the bottomonium spectrum, such as the 1+− Tbb̄1 (10610)+¯ Many of these states may be(was Zb (10610)+ ) and Tbb̄1 (10650)+ (was Zb (10650)+ ), both bb̄ud.
(∗)
hadronic modules made of charge conjugated pairs of mesons such as D(∗) , Ds or their excitations,and their B and B ∗ counterparts. They could also be mimicked by kinematical effects such as
triangle singularities.
31st May, 2024

15. Quark Model

11
4.8

4.6

4700 4685

4660
4500

4.4

4.2

4430

4S
4360

4274
4200,4220

4230

2D

m [GeV]

4140
4000

3S

4.0

2P

3915
3.8

1D

3900

3872

DD

2S
3.6

1P

3.4

3.2

1S
3.0

0-+ 1-- 0++ 1++ 2++ 1+- 1--

2--

3--

1

3

3

S0

3

S1

3

P0

3

P1

3

P2

1

P1

3

D1

D2

D3

Figure 15.2: Mass spectrum with known I G (J P C ) observed in the charmonium region. The DD̄
line shows the open charm threshold. The well-known cc̄ mesons are shown in black. (The 11 D2
ηc2 , expected around 3800 MeV, has not been seen yet.) The established additional observed states
are shown in red, and the states needing confirmation (i.e. omitted from the Summary Tables) arelabeled in green. The blue color denotes exotic isovectors.

15.4.2 Glueballs
QCD predicts the existence of extra isoscalar mesons which cannot be addressed by the quark
model. In the pure gauge theory they contain only gluons. The ground state glueball is predictedby lattice gauge theories to be 0++ , the first excited state 2++ . Errors on the mass predictions arelarge. From Ref. [33] one obtains 1750 (50) (80) MeV for the mass of the lightest 0++ glueball fromquenched QCD. As an example for the glueball mass spectrum, we show in Fig. 15.3 a calculationfrom [34]. A mass of 1710 MeV is predicted for the ground state, also with an error of about 100
MeV. Earlier work by other groups produced masses at 1650 MeV [35] and 1550 MeV [36] (seealso [37]. The first excited state has a mass of about 2.4 GeV, and the lightest glueball with exotic
quantum numbers (2+− ) has a mass of about 4 GeV.
These calculations are made in the so-called “quenched approximation” which neglects q q̄ loops.
However, both glue and q q̄ states couple to singlet scalar mesons. Therefore glueballs will mix withnearby q q̄ states of the same quantum numbers. The first results from lattice calculations, which
31st May, 2024

15. Quark Model

12

include these effects, indicate that the mass shifts are small. We return to a discussion of this point
in Sec. 15.8, see also Fig. 15.15.
The existence of three singlet scalar mesons around 1.5 GeV suggests additional degrees of
freedom such as glue, since only two mesons are predicted in this mass range (as indicated in Table
15.2). The f0 (1500) [23, 38] or, alternatively, the f0 (1710) [35], have been proposed as candidates
for the scalar glueball, both states having considerable mixing also with the f0 (1370). Following
Ref. [39] the glue is distributed among the 0++ isoscalars q q̄ mesons around 2 GeV. Other mixing
schemes, in particular with the f0 (500) and the f0 (980), have also been proposed [40]. According toa holographic model of low-energy QCD scalar glueballs decay strongly into kaons and η mesons, ingood agreement with data on the f0 (1710) [41]. Details can be found in the review “Spectroscopyof Light Meson Resonances” and in Ref. [42]. See also the review “Scalar Mesons below 1 GeV” inthis Review.

10
3

++

r0 MG

8
6

2

++

4

0

++

2

-+

0

-+

0

+-

2

+-

3

+-

1

+-

5
--

3-2-1

4

3

MG (GeV)

12

2

1

2
0

Reprinted with permission from Y. Chen et al, Phys. Rev. D73, 014516 (2006).
Copyright (2006) by the American Physical Society.

++

-+

+-

0

--

Figure 15.3: Predicted glueball spectrum from the lattice in quenched approximation (from [34]).
15.4.3 Hybrids
Mesons made of q q̄ pairs bound by excited gluons (q q̄g) are also predicted. Early model estimatesplaced them in the 1.9 GeV mass region, according to gluon flux tube models [43]. In the bag model
there are four nonets, among them an exotic 1−+ , around or above 1.4 GeV [44, 45].
Lattice QCD calculations show that the lightest hybrid is an exotic 1−+ state, at a mass around
2 GeV [46–48]. Fig. 15.13 will show an example of the spectroscopy from one group’s [48] simulation.
There are so far two candidates for exotic states with quantum numbers 1−+ , the π1 (1400) andπ1 (1600), which could be hybrids or four-quark states. However, a recent combined re-analysis ofπ1 production in diffractive π − p interaction and low energy p̄p annihilation leads to a single poleat ∼1560 MeV with a width of ∼390 MeV, although a two-pole scenario cannot be excluded [49].
Recently BESIII has reported evidence for the existence of a 1−+ η1 (1855) → ηη 0 [50] which could
be one of the isoscalar partners of the π1 (1600) in the 1−+ hybrid nonet [51]. (For details andreferences, see the review “Spectroscopy of Light Meson Resonances” in this Review.)
31st May, 2024

15. Quark Model

13

15.5 Baryons: qqq states
Baryons are fermions with baryon number B = 1, i.e., in the most general case, they arecomposed of three quarks plus any number of quark - antiquark pairs. Until recently, all establishedbaryons were consistent with 3-quark (qqq) configurations, which we mainly discuss in this section.
However, in 2015 the LHCb collaboration published first evidence for charmed ‘pentaquark’ statesof minimal quark content cc̄uud at invariant masses close to 4.4 GeV [52]. More refined LHCb
experiments have revealed evidence for four such states called Pcc̄ (4312)+ , Pcc̄ (4380)+ , Pcc̄ (4440)+ ,and Pcc̄ (4457)+ [53]. These states are located close to the thresholds of the production of ordinary
baryon-meson pairs like Σc+ D̄0 and Σc+ D̄?0 and are consistent with the predictions in terms ofmolecular-like states [54, 55]. A nice overview on the discussion of pentaquark and tetraquark
states is given in Ref. [56]. See also the article on “Pentaquarks” in this Review.
The color part of the baryon wave function is an SU(3) singlet, a completely antisymmetric stateof the three colors. Since the quarks are fermions, the wave function must be antisymmetric underinterchange of any two equal-mass quarks (up and down quarks in the limit of isospin symmetry).
Thus, it can be written as| qqq iA = | color iA × | space, spin, flavor iS ,(15.25)where the subscripts S and A indicate symmetry or antisymmetry under interchange of any two
equal-mass quarks. Note the contrast with the state function for the three nucleons in 3 H or 3 He:| N N N iA = | space, spin, isospin iA .

(15.26)This difference has major implications for the internal structure, magnetic moments, etc. (For anice discussion, see [57].)
15.5.1 Light baryons
The “ordinary” light-flavor baryons are made up of u, d, and s quarks. The three flavors imply
an approximate SU(3) flavor symmetry, which requires that baryons made of these quarks belong
to the multiplets on the right side of3 ⊗ 3 ⊗ 3 = 10S ⊕ 8M ⊕ 8M ⊕ 1A

(15.27)(see the section on “SU(n) Multiplets and Young Diagrams” in this Review). Here the subscripts
indicate symmetric, mixed-symmetry, or antisymmetric states under interchange of any two quarks.
The 1 is a |udsi singlet state (Λ1 ), and the octet contains a similar state (Λ8 ). If these have thesame spin and parity, they can mix. The mechanism is the same as for the mesons (see above). In
the ground state multiplet, the SU(3) flavor singlet Λ1 is forbidden by Fermi statistics. The section
on “SU(3) Isoscalar Factors and Representation Matrices,” shows how relative decay rates in, say,
10 → 8 ⊗ 8 decays may be calculated.
For the light-flavor baryons (no c or b quark), flavor and spin may be combined in an approximate
flavor-spin SU(6), in which the six basic states are d ↑, d ↓, · · · , s ↓ (↑, ↓ = spin up, down). Thenthe baryons belong to the multiplets on the right side of6 ⊗ 6 ⊗ 6 = 56S ⊕ 70M ⊕ 70M ⊕ 20A .

(15.28)

These SU(6) multiplets decompose into flavor SU(3) multiplets as follows:
56 = 4 10 ⊕ 2 8(15.29a)70 = 2 10 ⊕ 4 8 ⊕ 2 8 ⊕ 2 1(15.29b)

31st May, 2024

1415. Quark Model
20 = 2 8 ⊕ 4 1,

(15.29c)where the superscript (2S + 1) gives the net spin S of the quarks for each particle in the SU(3)multiplet. The J P = 1/2+ octet containing the nucleon and the J P = 3/2+ decuplet containing
the ∆(1232) together make up the “ground-state” 56-plet, in which the orbital angular momenta
between the quark pairs are zero (so that the spatial part of the state function is trivially symmetric).
The 70 and 20 require some excitation of the spatial part of the wave function in order to makethe overall wave function symmetric. States with nonzero orbital angular momenta are classified
in SU(6)⊗O(3) supermultiplets.
It is useful to classify the baryons into bands that have the same number N of quanta of
excitation. Each band consists of a number of supermultiplets, specified by (D, LPN ), where D isthe dimensionality of the SU(6) representation, L is the total quark orbital angular momentum,
and P is the total parity. Supermultiplets contained in bands up to N = 12 are given in [58].
The N = 0 band (with positive parity), which contains the nucleon and ∆(1232), consists only of−the (56,0+0 ) supermultiplet. The N = 1 band (with negative parity) consists only of the (70,11 )multiplet and contains the negative-parity baryons with masses below about 1.9 GeV. The N = 2+++
band (with positive parity) contains five supermultiplets: (56,0+2 ), (70,02 ), (56,22 ), (70,22 ), and+(20,1+
2 ), where the (56,00 ) represents the multiplet that contains the first radial excitations having
the same quantum numbers as the ground state.
The wave functions of the non-strange baryons in the harmonic oscillator basis are often labeled
by |X 2S+1 Lπ J P i, where S, L, J, P are as above, X = N or ∆, and π = S, M or A denotes thesymmetry of the spatial wave function. The possible model states for the bands with N=0,1,2
are given in Table 15.7. The assignment of experimentally observed states is only complete andwell established up to the N=1 band. Some more tentative assignments for higher multiplets aresuggested in [59].
In Table 15.6, quark-model assignments are given for many of the established baryons whose
SU(6)⊗O(3) compositions are relatively unmixed. Apart from the mixing of the Λ singlet and octet
states, one must, however, keep in mind that states with same J P but different L, S combinations
can also mix. In the quark model with one-gluon exchange motivated interactions, the size of themixing is determined by the relative strength of the tensor term with respect to the contact term(see below). The mixing is more important for the decay patterns of the states than for theirP−−
positions. An example are the lowest lying (70, 1−
1 ) states with J =1/2 and 3/2 . The physicalstates are| N (1535)1/2− i = cos(ΘS )|N 2 PM 1/2− i − sin(ΘS )|N 4 PM 1/2− i(15.30)| N (1520)3/2− i = cos(ΘD )|N 2 PM 3/2− i − sin(Θ)D |N 4 PM 3/2− i(15.31)and the orthogonal combinations for N (1650)1/2− and N (1700)3/2− . The mixing is large for theJ P =1/2− states (ΘS ≈ -32o ), but small for the J P =3/2− states (ΘD ≈ +6o ) [61–63].
All baryons of the ground state multiplets are known. Many of their properties, in particular
their masses, are in good agreement with even the most basic versions of the quark model, including
harmonic (or linear) confinement and a spin-spin interaction, which is responsible for the octet decuplet mass shifts. A consistent description of the ground-state electroweak properties, however,requires refined relativistic constituent quark models.
The situation for the excited states is much less clear. The assignment of some experimentallyobserved states with strange quarks to model configurations is only tentative and in many cases
candidates are completely missing. Melde, Plessas and Sengl [60] have calculated baryon properties
in relativistic constituent quark models, using one-gluon exchange and Goldstone-boson exchange
for the modeling of the hyperfine interactions (see Sec. 15.7 on Dynamics). Both types of models
31st May, 2024

15. Quark Model

15

Table 15.6: Quark-model assignments for some of the known baryons interms of a flavor-spin SU(6) basis. Only the dominant representation islisted. Assignments for several states, especially for the Λ(1810), Λ(2350),
Ξ(1620), Ξ(1690), Ξ(1820), and Ξ(2030), are merely educated guesses.
† suggestions for assignments and re-assignments from Ref. [60].
JP
1/2+
1/2+
1/2−
3/2−
1/2−
3/2−
5/2−
1/2+
3/2+
5/2+
7/2−
9/2−
9/2+

(D, LPN ) S
Octet members
Singlets(56,0+
)
1/2
N
(939)
Λ(1116)
Σ(1193)
Ξ(1318)0(56,0+
Σ(1660)2 ) 1/2 N (1440) Λ(1600)(70,1−
Σ(1620) Ξ(1620) Λ(1405)
1 ) 1/2 N (1535) Λ(1670)
Σ(1560)†−(70,11 ) 1/2 N (1520) Λ(1690) Σ(1670) Ξ(1820) Λ(1520)(70,1−
Σ(1750) Ξ(1690)
1 ) 3/2 N (1650) Λ(1800)
Σ(1620)†(70,1−
Σ(1940)† Ξ(?)
1 ) 3/2 N (1700) Λ(?)−(70,11 ) 3/2 N (1675) Λ(1830) Σ(1775) Ξ(1950)†(70,0+
Σ(1880) Ξ(?)Λ(1810)†2 ) 1/2 N (1710) Λ(1810)(56,2+)
1/2
N(1720)
Λ(1890)Σ(?)Ξ(?)
2(56,2+
Σ(1915) Ξ(2030)
2 ) 1/2 N (1680) Λ(1820)(70,3−
Σ(?)Ξ(?)Λ(2100)3 ) 1/2 N (2190) Λ(?)(70,3−)
3/2
N(2250)
Λ(?)
Σ(?)
Ξ(?)
3
(56,4+
Σ(?)Ξ(?)
4 ) 1/2 N (2220) Λ(2350)

3/2+ (56,0+0)
3/2+ (56,0+2)
1/2− (70,1−
1)
3/2− (70,1−
1)
5/2+ (56,2+
2)
7/2+ (56,2+
2)
11/2+ (56,4+
4)Decuplet members
3/2 ∆(1232) Σ(1385) Ξ(1530) Ω(1672)
3/2 ∆(1600) Σ(1690)† Ξ(?)Ω(?)
1/2 ∆(1620) Σ(1750)† Ξ(?)Ω(?)
1/2 ∆(1700) Σ(?)Ξ(?)Ω(2012)
3/2 ∆(1905) Σ(?)
Ξ(?)
Ω(?)
3/2 ∆(1950) Σ(2030) Ξ(?)Ω(?)
3/2 ∆(2420) Σ(?)
Ξ(?)
Ω(?)

give qualitatively comparable results and in geneeral, underestimate experimentally observed decay
widths. Nevertheless, in particular on the basis of the observed decay patterns, the authors haveassigned some additional states with strangeness to the SU(3) multiplets and suggest re-assignments
for a few others. Among the new assignments are states with weak experimental evidence (two or
three star ratings) and partly without firm spin/parity assignments, so that further experimental
efforts are necessary before final conclusions can be drawn. We have added their suggestions inTable 15.6.
In the non-strange sector there are two main problems which are illustrated in Fig. 15.4, where
the experimentally observed excitation spectrum of the nucleon (N and ∆ resonances) is comparedto the results of a typical quark model calculation [64]. The lowest states from the N=2 band, theN (1440)1/2+ , and the ∆(1600)3/2+ , appear lower than the negative parity states from the N=1
band (see Table 15.7) and much lower than predicted by most models. Also negative parity ∆states from the N=3 band (∆(1900)1/2− , ∆(1940)3/2− , and ∆(1930)5/2−) are too low in energy.
The low lying states show a clustering in groups of levels around 1700 MeV and 1900 MeV for N ?
states and around 1900 MeV for ∆ states which is not reflected in models.
Furthermore, many more states are predicted than observed. This has been known for a longtime as the ‘missing resonance’ problem [61]. Up to an excitation energy of 2.4 GeV, about 45
N states are predicted, but only 20 are established (four- or three-star; see Note on N and ∆Resonances for the rating of the status of resonances) and 5 are tentative (two- or one-star).
31st May, 2024

15. Quark Model

16

Table 15.7: N and ∆ states in the N=0,1,2 harmonic oscillator bands.
LP denotes angular momentum and parity, S the three-quark spin and‘sym’=A,S,M the symmetry of the spatial wave function. Listed are all
possible spin/parity combinations and assignments of experimentally observed states. Only dominant components are indicated. Assignments in
the N=2 band are partly tentative.
N sym LP
2 A
1+
2 M 2+
2 M 2+
2 M 0+
2 M 0+
2 S
2+
2 S
2+
2 S
0+
2 S
0+
1 M 1−
1 M 1−
0 S
0+
0 S
0+
N sym LP
2 A
1+
2 M 2+
2 M 2+
2 M 0+
2 M 0+
2 S
2+
2 S
2+
2 S
0+
2 S
0+
1 M 1−
1 M 1−
0 S
0+
0 S
0+

S
1/2
3/2
1/2
3/2
1/2
3/2
1/2
3/2
1/2
3/2
1/2
3/2
1/2
S
1/2
3/2
1/2
3/2
1/2
3/2
1/2
3/2
1/2
3/2
1/2
3/2
1/2

N (I = 1/2)
1/2+
3/2+
1/2+
3/2+
5/2+
7/2+
+
+
3/2
5/2
3/2+
1/2+ N (1710) 3/2+ N (1720) 5/2+ N (1680) 1/2+ N (1440) 1/2− N (1650) 3/2− N (1700) 5/2− N (1675) 1/2− N (1535) 3/2− N (1520) 1/2+ N (938) ∆(I = 3/2)
3/2+
5/2+
1/2+ ∆(1750) 1/2+ ∆(1910) 3/2+ ∆(1920) 5/2+ ∆(1905) 7/2+ ∆(1950)
3/2+ ∆(1600) 1/2− ∆(1620) 3/2− ∆(1700) 3/2+ ∆(1232) -However, there is some recent progress. For the N=2 band, candidates have been identified foralmost all predicted states with the exception of the (20,1+
2 ). The total number of states has
perhaps not significantly changed but the number of states with four- or three-star rating has
increased from 14 to 20 compared to the 2018 PDG particle listings. Most of this progress isdue to the programs concentrating on the study of meson photoproduction reactions, while themost recent partial wave analysis of elastic pion scattering and charge exchange data by Arndt
and collaborators [66] found no evidence for almost half of the states listed in this review (andincluded in Fig. 15.4). Such analyses are of course biased against resonances which couple only
weakly to the N π channel. Quark model predictions for the couplings to other hadronic channels
and to photons are given in Ref. [64]. The large experimental effort ongoing at several electron
accelerators to study the baryon resonance spectrum with real and virtual photon-induced meson
production reactions includes the search for as-yet-unobserved states, as well as detailed studies ofthe properties of the low lying states (decay patterns, electromagnetic couplings, magnetic moments,etc.) (see Ref. [67] for reviews). There are two major new aspects of this program. The investigationof single and double polarization observables allows, via the study of interference terms, access to
31st May, 2024

15. Quark Model

17

Mass/(MeV/c2)

N(I=1/2)
exp+

N(2300)1/2+
N(2250)9/2+
N(2220)9/2
N(2190)7/2
N(2120)3/2+
N(2100)1/2
N(2060)5/2+
N(2040)3/2+
N(2000)5/2+
N(1990)7/2+
N(1900)3/2
N(1895)1/2+
N(1880)1/2
N(1875)3/2+
N(1860)5/2

QM

QM
2400

2200

2000

1800

+

N(1720)3/2+
N(1710)1/2
N(1700)5/2+
N(1680)5/2
N(1675)5/2
N(1650)1/2-∆(I=3/2)
exp+∆(2420)11/2∆(2400)9/2+∆(2390)7/2∆(2350)5/2+∆(2300)9/2-∆(2200)7/2∆(2150)1/2+∆(2000)5/2+∆(1950)7/2∆(1940)3/2+∆(1930)5/2+∆(1920)3/2+∆(1910)1/2+∆(1905)5/2∆(1900)1/2∆(1750)1/2∆(1700)3/2-

1600∆(1620)1/2+∆(1600)3/2

1400

N(1535)1/2
N(1520)3/2+

N(1440)1/2

+

1200

∆(1232)3/2

1000+

N(939)1/2

Figure 15.4: Excitation spectrum of the nucleon. Compared are the positions of the excited states
identified in experiment, to those predicted by a relativized quark model calculation. Left handside: isospin I = 1/2 N -states, right hand side: isospin I = 3/2 ∆-states. Experimental: (columns
labeled ’exp’), three- and four-star states are indicated by full lines (two-star dashed lines, one-stardotted lines). At the very left and right of the figure, the spectroscopic notation of these statesis given. Quark model [64, 65]: (columns labeled ’QM’), all states for the N=1,2 bands, low-lying
states for the N=3,4,5 bands. Full lines: at least tentative assignment to observed states, dashedlines: so far no observed counterparts. Many of the assignments between predicted and observed
states are highly tentative.

small partial waves that do not leave a footprint in unpolarized cross sections. An example forthe impact of such data is given by a comparison of results from different multipole analyses ofpion photoproduction [68]. It shows clearly that with the inclusion of polarization observables thereaction model results start to converge. This will in the near future much improve the databasefor excited baryons in the light quark sector.
The other aspect is the study of final states with meson pairs, in particular ππ and πη pairs,
which has made much progress during the last few years. This is important for higher lying states,
which in the quark model may have both possible oscillations excited. Such states can be expected
to decay in sequential processes de-exciting the two oscillations step-by-step so that they couple
strongly to multiple-meson final states but not to single-meson production. Detailed analyses ofsuch data are for example given in [69, 70] and had already significant impact on partial wave
analyses.
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15. Quark Model

The excitation spectrum of hyperons containing s-quarks is even less well explored. Manyexperimental results date back to before 2000, although parameters listed in the review are inmany cases updated by more modern partial wave analyses. A recent review on the quark modelinterpretation of the Λ and Σ states is given in Ref. [71], a new Ω − state was reported in theΞK mass spectrum with high significance from the Belle experiment [72] with a likely J P =3−classification, and Belle also reported on the most significant evidence so far for the Ξ(1620)
2
in the Ξπ mass spectrum [73], which has now warranted an upgrade of this resonance from one
star to two stars. In the near future, significant progress on hyperon spectroscopy is expected fromJLab using the KL facility in conjunction with the GlueX exprimental setup, J-PARC using theK − beam at the extended hadron facility, and the PANDA experiment at FAIR [74, 75].
In quark models, the number of excited states is determined by the effective degrees of freedom,
while their ordering and decay properties are related to the residual quark - quark interaction. An
overview of quark models for baryons is given in [76], recent discussions of baryon spectroscopy
are given in [59, 77]. The effective degrees of freedom in the standard nonrelativistic quark model
are three equivalent valence quarks with one-gluon exchange-motivated, flavor-independent colormagnetic interactions. The QCD aspect of gluon-gluon interactions is emphasized by the hypercentral quark model [78,79], which includes in a natural way three-body forces between the quarks.
A different class of models uses interactions which give rise to a quark - diquark clustering of thebaryons: for a review see [80]. If there is a tightly bound diquark, only two degrees of freedom
are available at low energies, and thus fewer states are predicted. Furthermore, selection rules inthe decay pattern may arise from the quantum numbers of the diquark. However, we note thatthe recent identification of new N ∗ resonances seriously questions the older static diquark-quark
model since these resonances are likely members of the (70,2+2 ), which requires both oscillators inthe baryon to be excited. More states are predicted by collective models of the baryon like thealgebraic approach in [81]. In this approach, the quantum numbers of the valence quarks are distributed over a Y-shaped string-like configuration, and additional states arise e.g., from vibrationsof the strings. More states are also predicted in the framework of flux-tube models, see [82], which
are motivated by lattice QCD. In addition to the quark degrees of freedom, flux-tubes responsible
for the confinement of the quarks are considered as degrees of freedom. These models include hybrid baryons containing explicit excitations of the gluon fields. However, since all half integral J P
quantum numbers are possible for ordinary baryons, such ‘exotics’ will be very hard to identify, andprobably always mix with ordinary states. So far, the experimentally observed number of states isstill far lower even than predicted by the quark–diquark models.
The influence of chiral symmetry on the excitation spectrum of the nucleon has been debatedfrom a somewhat different perspective. Chiral symmetry, the fundamental symmetry of QCD, isstrongly broken for the low lying states, resulting in large mass differences of parity partners likethe J P =1/2+ N (938)1/2+ ground state and the J P =1/2− N (1535)1/2− excitation. However, at
higher excitation energies there is some evidence for parity doublets and even some very tentative
suggestions for full chiral multiplets of N ? and ∆ resonances. An effective restoration of chiralsymmetry at high excitation energies due to a decoupling from the quark condensate of the vacuum
has been discussed (see [83] for recent reviews) as a possible cause. In this case, the mass generating
mechanisms for low and high lying states would be essentially different. As a further consequence,the parity doublets would decouple from pions, so that experimental bias would be worse. However,parity doublets might also arise from the spin-orbital dynamics of the 3-quark system. Presently,
the status of the data does not allow final conclusions.
The most recent developments on the theory side include progress in understanding the physics
of baryons [84, 85] by using the Dyson-Schwinger equations of QCD and Bethe-Salpeter equa31st May, 2024

19

15. Quark Modeltions [86, 87]. In this approach, baryons are relativistic bound states of three quarks, and thetreatment of their interactions arising from QCD is non-perturbative, incorporating aspects ofconfinement and dynamical symmetry breaking. Two paths to solving the three-body problem
are taken: direct solution of the three-body Faddeev equation, and decomposition of baryons intoquark-diquark systems, with all quark pairs able to constitute the diquark. Therefore, these descriptions are different from the older static diquark-quark pictures of the baryon. Moreover, thefirst unquenched lattice-QCD calculations for the excitation spectrum have been presented and are
discussed in Sec. 15.8. The results are basically consistent with the level counting of SU(6)⊗O(3) inthe standard non-relativistic quark model and show no indication for quark-diquark structures orparity doubling. Consequently, there is as yet no indication from lattice that the mis-match between
the excitation spectrum predicted by the standard quark model and experimental observations isdue to inappropriate degrees of freedom in the quark model.
15.5.2 Charmed and bottom baryons
The naming scheme for baryons with c or b quarks follows that of the light baryons: the Λ is an
isosinglet and the Σ an isotriplet with one heavy (s, c or b) quark. The Ξ is an isodoublet whichcontains two heavy quarks, and the Ω an isosinglet with three heavy quarks. The number of c orb quarks is indicated by the subscripts c or b. Hyperons are baryons with at least one s quark.
For charmed baryons the addition of the c quark to the light quarks extends the flavor symmetry
to SU(4)f . Due to the large mass of the c quark, this symmetry is much more strongly broken
than the SU(3)f of the three light quarks. Nevertheless, the SU(4)f representation is still usefulfor bookkeeping purposes. With the additive charm quantum number C the baryons are classified
in a 3-dimensional representation with the three coordinates ( I z , Y , C ). Figure 15.5 shows theSU(4)f weight diagrams.

Figure 15.5: SU(4)f multiplets of ground state baryons made of u, d, s, and c quarks. (a) The
spin 12 20-plet extends the charmless SU(3)f octet to C = 1,2; (b) the spin 32 20-plet extends theSU(3)f decuplet to C = 1, 2, 3.
With four quarks the 64 possible configurations decompose into
4 ⊗ 4 ⊗ 4 = 4̄A ⊕ 20S ⊕ 20M S ⊕ 20M A ,(15.32)(for a review on SU (N ) symmetries see e.g. [88]). The subscripts S and A refer to the symmetry
and antisymmetry properties of the flavor wave functions. The flavor symmetric 20S multiplet,
31st May, 2024

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15. Quark Model

associated with spin- 32 baryons, contains the charmless SU(3)f decuplet at the bottom level. The
20M S and 20M A multiplets correspond to the mixed symmetric and mixed antisymmetric flavour
wave functions of the spin- 21 baryons, with the charmless octet baryons at the bottom level. There
are two dsc and two usc spin- 21 states, labeled Ξc0 , Ξc0 0 and Ξc+ , Ξc0+ . This is because one of theqq pairs can have spin 1 (symmetric) or spin 0 (antisymmetric), giving both the total spin j = 12
with the third quark (see also Fig. 15.6 below).
For C = 1 baryons the flavor decomposition of the diquark, made of u, d, or s quarks, is3 ⊗ 3 = 3̄A ⊕ 6S .

(15.33)For ground-state baryons, the overall antisymmetry of baryon wave function (including color) requires the light diquark to be symmetric under the exchange of spin and flavor, hence both symmetric or both antisymmetric, that is spin 1 for the 6S and spin 0 for the 3̄A . The 3̄ then combines withthe c quark to form the J P = 1/2+ states, while the 6 combines to form J P = 1/2+ or J P = 3/2+ .
The weight diagrams of the 3̄ and 6 ground-state representations are shown in Fig. 15.6. Withineach multiplet the C = 1 baryons obey isospin and SU(3)f mass relations at the expected orders.

Figure 15.6: The SU(3)f 3̄ (a) and 6 (b) ground state J P = 1/2+ representations. The structureof the 6 ground state with J P = 3/2+ is identical to the one in (b).
The antisymmetric quadruplet in (15.32) does not exist in the ground state but is realized forthe first orbital excitations (L = 1). Figure 15.7 shows the weight diagram with the experimentallyobserved states. In the quark model the quadruplet consists of four baryons with spins coupling−−
to 21 , leading to four L = 1 excitations with J P = 12 and four excitations with 23 . The charmed
ones are the partners of the Λ(1405) and Λ(1520) singlets of SU(3)f .
For a detailed review on charmed baryons see Ref. [77]. Quark model predictions for baryons
with two heavy quarks are given in Ref. [89] and lattice results for doubly and triply charmed states
are discussed in Sec. 15.8 of this Review.
The C = 1 ground state baryons have all been observed. Due to their relatively narrow widths
the states are much easier to isolate than the light quark baryon resonances which require intricate
partial wave analyses. The production cross sections are small, but the recent measurements at
the e+ e− B-factories, at the pp̄ Tevatron collider, and at LHCb have boosted the field. The LHCb
collaboration has published evidence for five new narrow Ωc0 states (css) [90] in proton-proton
31st May, 2024

15. Quark Model21◆1⇤c (2595)2✓ ◆3⇤c (2625)+
2+✓<latexit sha1_base64="MUYiNHOv+6noKtGi5rxNzZMeTuo=">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</latexit>◆1⌅c (2790)2✓ ◆3⌅c (2815)0
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0

✓

◆
1
⌅c (2790)2✓ ◆3⌅c (2815)+
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I
✓z ◆
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⇤(1405)2✓ ◆3⇤(1520)2

Figure 15.7: Weight diagram of the 4̄ SU(4)f multiplet with the experimentally observed negative
parity baryons.
collisions. Four of these states have been confirmed by the Belle experiment in e+ e− collisions [91].
Their quantum numbers are still unknown, but they could correspond to the L = 1 ssc orbital
excitations. Constituent quark models, lattice QCD, quark-diquark models, molecular models, andpentaquark states, have been discussed to describe their structure (see e.g. [92] for references). In
the meantime, also four Ωb− states of the bss type have been reported from LHCb [92]. They lie ina narrow range between 6316 - 6350 MeV, but since their intrinsic widths are on the order of justa few MeV, they are well separated in the invariant mass spectra. Spectroscopy and theory in thisfield is rapidly evolving.
++ (ccu) baryon [93]. The quantum numbers
LHCb has also reported a doubly charmed Ξccof this state are undetermined, but its significance is already rated three stars in the current
compilation. Doubly charmed baryons have a very different structure from light baryons, more
resembling heavy ‘double-star’ systems with attached light ‘planets’, which opens a new window+ (ccd) had been reported
for QCD properties. The first candidate for a doubly charmed baryon Ξcc− +earlier by the SELEX experiment [94, 95]. A significance of 6.3σ was claimed for the Λ+c K π
decay and 4.8σ for pD+ K− but searches by other experiments have not confimed it so far (seee.g. [96] and Refs. therein) and it is not included in the particle listings. The SELEX and LHCb++ state lies
masses lie in the predicted 3500 – 3700 MeV mass range (see e.g. [89]). The LHCb Ξcc
about 100 MeV above the SELEX one, with a mass splitting far too large for ucc and dcc isospin
partners. However, it has also been discussed that due to the different production mechanisms thisis not necessarily a contradiction [97].
Figure 15.8 shows the spectrum of the established singly-charmed baryons (3* and 4* baryons
in the Listings with known quantum numbers). The parity of the Λ+
c is that of the c quark, defined
as positive. Spin and parity have not been determined experimentally for most of the states. They
follow the ordering and expectation from the quark model. Candidates for L = 2 orbital excitations++−of the Λc (with J P = 23 and J P = 25 ) have already been observed, as well as a 32 state at 2940
31st May, 2024

15. Quark Model

22

MeV, possibly a radial L = 1 excitation [98].
Charmed baryons

Bottom baryons

3.1

0.8

3
2

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1
[4̄]
2

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3
[4̄]
2

3+
[6]
2

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3+
[6]
2

Mass
difference
[GeV]

1+
[6]
2

0.4

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6.3

5+
2
3+
2
<latexit sha1_base64="5jHgGAy5NZeDImguIFp5F8CuFxE=">AAAB/HicbVDLSsNAFL2pr1pf0S7dDBahIJSkSHVZcOOygn1AGstkOmmHTh7MTIQQ4q+4caGIWz/EnX/jtM1CWw9cOJxzL/fe48WcSWVZ30ZpY3Nre6e8W9nbPzg8Mo9PejJKBKFdEvFIDDwsKWch7SqmOB3EguLA47TvzW7mfv+RCsmi8F6lMXUDPAmZzwhWWhqZ1aEvMMnsPGvmDxcoc1puPjJrVsNaAK0TuyA1KNAZmV/DcUSSgIaKcCylY1uxcjMsFCOc5pVhImmMyQxPqKNpiAMq3WxxfI7OtTJGfiR0hQot1N8TGQ6kTANPdwZYTeWqNxf/85xE+dduxsI4UTQky0V+wpGK0DwJNGaCEsVTTTARTN+KyBTrNJTOq6JDsFdfXie9ZsNuNZp3l7V2vYijDKdwBnWw4QracAsd6AKBFJ7hFd6MJ+PFeDc+lq0lo5ipwh8Ynz/fOpQw</latexit>

3
[4̄]
2
<latexit sha1_base64="5jHgGAy5NZeDImguIFp5F8CuFxE=">AAAB/HicbVDLSsNAFL2pr1pf0S7dDBahIJSkSHVZcOOygn1AGstkOmmHTh7MTIQQ4q+4caGIWz/EnX/jtM1CWw9cOJxzL/fe48WcSWVZ30ZpY3Nre6e8W9nbPzg8Mo9PejJKBKFdEvFIDDwsKWch7SqmOB3EguLA47TvzW7mfv+RCsmi8F6lMXUDPAmZzwhWWhqZ1aEvMMnsPGvmDxcoc1puPjJrVsNaAK0TuyA1KNAZmV/DcUSSgIaKcCylY1uxcjMsFCOc5pVhImmMyQxPqKNpiAMq3WxxfI7OtTJGfiR0hQot1N8TGQ6kTANPdwZYTeWqNxf/85xE+dduxsI4UTQky0V+wpGK0DwJNGaCEsVTTTARTN+KyBTrNJTOq6JDsFdfXie9ZsNuNZp3l7V2vYijDKdwBnWw4QracAsd6AKBFJ7hFd6MJ+PFeDc+lq0lo5ipwh8Ynz/fOpQw</latexit>

3+
[6]
2

<latexit sha1_base64="EjC2wpp0mcNoigCMjvlwbZC6WIU=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSIUhJJUqS4LblxWsA9IY5lMJ+3QyYOZiRBC/BU3LhRx64e482+ctllo64ELh3Pu5d57vJgzqSzr21hb39jc2i7tlHf39g8OzaPjrowSQWiHRDwSfQ9LyllIO4opTvuxoDjwOO1505uZ33ukQrIovFdpTN0Aj0PmM4KVloZmZeALTLKLPGvkD+coc5puPjSrVt2aA60SuyBVKNAeml+DUUSSgIaKcCylY1uxcjMsFCOc5uVBImmMyRSPqaNpiAMq3Wx+fI7OtDJCfiR0hQrN1d8TGQ6kTANPdwZYTeSyNxP/85xE+dduxsI4UTQki0V+wpGK0CwJNGKCEsVTTTARTN+KyATrNJTOq6xDsJdfXiXdRt1u1ht3l9VWrYijBCdwCjWw4QpacAtt6ACBFJ7hFd6MJ+PFeDc+Fq1rRjFTgT8wPn8A4lqUMg==</latexit>

1
[4̄]
2

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1
[6]
2

2.5

1
[4̄]
2

0.2
<latexit sha1_base64="ul7aQCVh/4j/NT29bMLQg5eqDZc=">AAACAnicbVDLSsNAFJ34rPUVdSVuBotQEEpSRV0W3LisYB+QxDKZTtqhk0mYmQhlCG78FTcuFHHrV7jzb5y2WWjrgQuHc+7l3nvClFGpHOfbWlpeWV1bL22UN7e2d3btvf22TDKBSQsnLBHdEEnCKCctRRUj3VQQFIeMdMLR9cTvPBAhacLv1DglQYwGnEYUI2Wknn3oRwJh7ea6nt+fQu35IRL6LA/ynl1xas4UcJG4BamAAs2e/eX3E5zFhCvMkJSe66Qq0EgoihnJy34mSYrwCA2IZyhHMZGBnr6QwxOj9GGUCFNcwan6e0KjWMpxHJrOGKmhnPcm4n+el6noKtCUp5kiHM8WRRmDKoGTPGCfCoIVGxuCsKDmVoiHyGSiTGplE4I7//Iiaddr7kWtfnteaVSLOErgCByDKnDBJWiAG9AELYDBI3gGr+DNerJerHfrY9a6ZBUzB+APrM8fwYKW8g==</latexit><latexit sha1_base64="5jHgGAy5NZeDImguIFp5F8CuFxE=">AAAB/HicbVDLSsNAFL2pr1pf0S7dDBahIJSkSHVZcOOygn1AGstkOmmHTh7MTIQQ4q+4caGIWz/EnX/jtM1CWw9cOJxzL/fe48WcSWVZ30ZpY3Nre6e8W9nbPzg8Mo9PejJKBKFdEvFIDDwsKWch7SqmOB3EguLA47TvzW7mfv+RCsmi8F6lMXUDPAmZzwhWWhqZ1aEvMMnsPGvmDxcoc1puPjJrVsNaAK0TuyA1KNAZmV/DcUSSgIaKcCylY1uxcjMsFCOc5pVhImmMyQxPqKNpiAMq3WxxfI7OtTJGfiR0hQot1N8TGQ6kTANPdwZYTeWqNxf/85xE+dduxsI4UTQky0V+wpGK0DwJNGaCEsVTTTARTN+KyBTrNJTOq6JDsFdfXie9ZsNuNZp3l7V2vYijDKdwBnWw4QracAsd6AKBFJ7hFd6MJ+PFeDc+lq0lo5ipwh8Ynz/fOpQw</latexit><latexit sha1_base64="ul7aQCVh/4j/NT29bMLQg5eqDZc=">AAACAnicbVDLSsNAFJ34rPUVdSVuBotQEEpSRV0W3LisYB+QxDKZTtqhk0mYmQhlCG78FTcuFHHrV7jzb5y2WWjrgQuHc+7l3nvClFGpHOfbWlpeWV1bL22UN7e2d3btvf22TDKBSQsnLBHdEEnCKCctRRUj3VQQFIeMdMLR9cTvPBAhacLv1DglQYwGnEYUI2Wknn3oRwJh7ea6nt+fQu35IRL6LA/ynl1xas4UcJG4BamAAs2e/eX3E5zFhCvMkJSe66Qq0EgoihnJy34mSYrwCA2IZyhHMZGBnr6QwxOj9GGUCFNcwan6e0KjWMpxHJrOGKmhnPcm4n+el6noKtCUp5kiHM8WRRmDKoGTPGCfCoIVGxuCsKDmVoiHyGSiTGplE4I7//Iiaddr7kWtfnteaVSLOErgCByDKnDBJWiAG9AELYDBI3gGr+DNerJerHfrY9a6ZBUzB+APrM8fwYKW8g==</latexit>

2.3<latexit sha1_base64="vSQ8PAJOQZROtqe7W35FDD2YU1g=">AAACAnicbVDLSsNAFJ34rPUVdSVuBovQjSUpRV0W3LisYB+QxDKZTtqhk0mYmQhlCG78FTcuFHHrV7jzb5y2WWjrgQuHc+7l3nvClFGpHOfbWlldW9/YLG2Vt3d29/btg8OOTDKBSRsnLBG9EEnCKCdtRRUjvVQQFIeMdMPx9dTvPhAhacLv1CQlQYyGnEYUI2Wkvn3sRwJh7ea6nt+fQ+35IRK6kQd53644NWcGuEzcglRAgVbf/vIHCc5iwhVmSErPdVIVaCQUxYzkZT+TJEV4jIbEM5SjmMhAz17I4ZlRBjBKhCmu4Ez9PaFRLOUkDk1njNRILnpT8T/Py1R0FWjK00wRjueLooxBlcBpHnBABcGKTQxBWFBzK8QjZDJRJrWyCcFdfHmZdOo196JWv21UmtUijhI4AaegClxwCZrgBrRAG2DwCJ7BK3iznqwX6936mLeuWMXMEfgD6/MHximW9Q==</latexit>

1+
[3̄]
2

1+
[6]
2

<latexit sha1_base64="5jHgGAy5NZeDImguIFp5F8CuFxE=">AAAB/HicbVDLSsNAFL2pr1pf0S7dDBahIJSkSHVZcOOygn1AGstkOmmHTh7MTIQQ4q+4caGIWz/EnX/jtM1CWw9cOJxzL/fe48WcSWVZ30ZpY3Nre6e8W9nbPzg8Mo9PejJKBKFdEvFIDDwsKWch7SqmOB3EguLA47TvzW7mfv+RCsmi8F6lMXUDPAmZzwhWWhqZ1aEvMMnsPGvmDxcoc1puPjJrVsNaAK0TuyA1KNAZmV/DcUSSgIaKcCylY1uxcjMsFCOc5pVhImmMyQxPqKNpiAMq3WxxfI7OtTJGfiR0hQot1N8TGQ6kTANPdwZYTeWqNxf/85xE+dduxsI4UTQky0V+wpGK0DwJNGaCEsVTTTARTN+KyBTrNJTOq6JDsFdfXie9ZsNuNZp3l7V2vYijDKdwBnWw4QracAsd6AKBFJ7hFd6MJ+PFeDc+lq0lo5ipwh8Ynz/fOpQw</latexit>

1+
[3̄]
2
<latexit sha1_base64="ul7aQCVh/4j/NT29bMLQg5eqDZc=">AAACAnicbVDLSsNAFJ34rPUVdSVuBotQEEpSRV0W3LisYB+QxDKZTtqhk0mYmQhlCG78FTcuFHHrV7jzb5y2WWjrgQuHc+7l3nvClFGpHOfbWlpeWV1bL22UN7e2d3btvf22TDKBSQsnLBHdEEnCKCctRRUj3VQQFIeMdMLR9cTvPBAhacLv1DglQYwGnEYUI2Wknn3oRwJh7ea6nt+fQu35IRL6LA/ynl1xas4UcJG4BamAAs2e/eX3E5zFhCvMkJSe66Qq0EgoihnJy34mSYrwCA2IZyhHMZGBnr6QwxOj9GGUCFNcwan6e0KjWMpxHJrOGKmhnPcm4n+el6noKtCUp5kiHM8WRRmDKoGTPGCfCoIVGxuCsKDmVoiHyGSiTGplE4I7//Iiaddr7kWtfnteaVSLOErgCByDKnDBJWiAG9AELYDBI3gGr+DNerJerHfrY9a6ZBUzB+APrM8fwYKW8g==</latexit>

0.0

3+
[6]
2

<latexit sha1_base64="EjC2wpp0mcNoigCMjvlwbZC6WIU=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSIUhJJUqS4LblxWsA9IY5lMJ+3QyYOZiRBC/BU3LhRx64e482+ctllo64ELh3Pu5d57vJgzqSzr21hb39jc2i7tlHf39g8OzaPjrowSQWiHRDwSfQ9LyllIO4opTvuxoDjwOO1505uZ33ukQrIovFdpTN0Aj0PmM4KVloZmZeALTLKLPGvkD+coc5puPjSrVt2aA60SuyBVKNAeml+DUUSSgIaKcCylY1uxcjMsFCOc5uVBImmMyRSPqaNpiAMq3Wx+fI7OtDJCfiR0hQrN1d8TGQ6kTANPdwZYTeSyNxP/85xE+dduxsI4UTQki0V+wpGK0CwJNGKCEsVTTTARTN+KyATrNJTOq6xDsJdfXiXdRt1u1ht3l9VWrYijBCdwCjWw4QpacAtt6ACBFJ7hFd6MJ+PFeDc+Fq1rRjFTgT8wPn8A4lqUMg==</latexit>

1+
[6]
2

6.1

3+
[6]
2

<latexit sha1_base64="EjC2wpp0mcNoigCMjvlwbZC6WIU=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSIUhJJUqS4LblxWsA9IY5lMJ+3QyYOZiRBC/BU3LhRx64e482+ctllo64ELh3Pu5d57vJgzqSzr21hb39jc2i7tlHf39g8OzaPjrowSQWiHRDwSfQ9LyllIO4opTvuxoDjwOO1505uZ33ukQrIovFdpTN0Aj0PmM4KVloZmZeALTLKLPGvkD+coc5puPjSrVt2aA60SuyBVKNAeml+DUUSSgIaKcCylY1uxcjMsFCOc5uVBImmMyRSPqaNpiAMq3Wx+fI7OtDJCfiR0hQrN1d8TGQ6kTANPdwZYTeSyNxP/85xE+dduxsI4UTQki0V+wpGK0CwJNGKCEsVTTTARTN+KyATrNJTOq6xDsJdfXiXdRt1u1ht3l9VWrYijBCdwCjWw4QpacAtt6ACBFJ7hFd6MJ+PFeDc+Fq1rRjFTgT8wPn8A4lqUMg==</latexit><latexit sha1_base64="shKmQCOeWxepyFiFmt71L9BqM8I=">AAACAnicbVDLSsNAFJ3UV62vqCtxM1iEbixJLeqy4MZlBfuAJpbJdNIOnUzCzEQoQ3Djr7hxoYhbv8Kdf+O0zUJbD1w4nHMv994TJIxK5TjfVmFldW19o7hZ2tre2d2z9w/aMk4FJi0cs1h0AyQJo5y0FFWMdBNBUBQw0gnG11O/80CEpDG/U5OE+BEachpSjJSR+vaRFwqE9Xmma9n9GdQ9L0BC1zM/69tlp+rMAJeJm5MyyNHs21/eIMZpRLjCDEnZc51E+RoJRTEjWclLJUkQHqMh6RnKUUSkr2cvZPDUKAMYxsIUV3Cm/p7QKJJyEgWmM0JqJBe9qfif10tVeOVrypNUEY7ni8KUQRXDaR5wQAXBik0MQVhQcyvEI2QyUSa1kgnBXXx5mbRrVfeiWrutlxuVPI4iOAYnoAJccAka4AY0QQtg8AiewSt4s56sF+vd+pi3Fqx85hD8gfX5A8lVlvc=</latexit>

1+
[6]
2

<latexit sha1_base64="ul7aQCVh/4j/NT29bMLQg5eqDZc=">AAACAnicbVDLSsNAFJ34rPUVdSVuBotQEEpSRV0W3LisYB+QxDKZTtqhk0mYmQhlCG78FTcuFHHrV7jzb5y2WWjrgQuHc+7l3nvClFGpHOfbWlpeWV1bL22UN7e2d3btvf22TDKBSQsnLBHdEEnCKCctRRUj3VQQFIeMdMLR9cTvPBAhacLv1DglQYwGnEYUI2Wknn3oRwJh7ea6nt+fQu35IRL6LA/ynl1xas4UcJG4BamAAs2e/eX3E5zFhCvMkJSe66Qq0EgoihnJy34mSYrwCA2IZyhHMZGBnr6QwxOj9GGUCFNcwan6e0KjWMpxHJrOGKmhnPcm4n+el6noKtCUp5kiHM8WRRmDKoGTPGCfCoIVGxuCsKDmVoiHyGSiTGplE4I7//Iiaddr7kWtfnteaVSLOErgCByDKnDBJWiAG9AELYDBI3gGr+DNerJerHfrY9a6ZBUzB+APrM8fwYKW8g==</latexit>

1+
[6]
2

Mass [GeV]<latexit sha1_base64="CpO/TtxFmqv9GOm0zSX9+EDfztI=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBahF0tSRT0WvHisYD+gjWWz3bRLN5u4uymUkN/hxYMiXv0x3vw3btMctPXBwOO9GWbmeRFnStv2t1VYW9/Y3Cpul3Z29/YPyodHbRXGktAWCXkoux5WlDNBW5ppTruRpDjwOO14k9u535lSqVgoHvQsom6AR4L5jGBtJLfvS0ySizSpp4/ng3LFrtkZ0CpxclKBHM1B+as/DEkcUKEJx0r1HDvSboKlZoTTtNSPFY0wmeAR7RkqcECVm2RHp+jMKEPkh9KU0ChTf08kOFBqFnimM8B6rJa9ufif14u1f+MmTESxpoIsFvkxRzpE8wTQkElKNJ8Zgolk5lZExtjkoE1OJROCs/zyKmnXa85VrX5/WWlU8ziKcAKnUAUHrqEBd9CEFhB4gmd4hTdrar1Y79bHorVg5TPH8AfW5w9cbZHB</latexit>

2.7

6.5

5.9

1+
[3̄]
2

1+
[3̄]
2

5.7

Figure 15.8: Mass spectrum of the established C = 1 charmed and B = –1 bottom baryons withknown J P . The flavor symmetry assignments are given by the square brackets. According to theirisospins the Σc,b (Ξc,b ) consist of three (two) charged or neutral states that are nearly degenerate.
The same SU(4)f multiplets can be constructed for the bottom baryons by replacing the c quark
by a b quark.The established 3* and 4* bottom baryons are shown in Fig.15.8. The quadruplet 4̄
contains the two negative parity candidates Λb (5912)0 and Λb (5920)0 . It appears that the confining
potential is only weakly flavour dependent. For example, the mass difference between the Ξb andthe Λb is roughly the same as that between the Ξc and the Λc , the mass splitting between the
spin- 21 Ωb and the Λb close to that between the spin- 21 Ωc and the Λc . The spin- 32 states are alsoheavier than the spin- 12 ones, in agreement with expectations from the spin-spin force.
The bottom hadrons can also be embedded in a larger SU(5)f group that accounts for all
baryons constructed from the five quark flavors. (The existence of baryons with t-quarks is very
unlikely due to the short lifetime of the t-quark.) One predicts the decomposition
5 ⊗ 5 ⊗ 5 = 10A ⊕ 40M S ⊕ 40M A ⊕ 35S .

(15.34)The decuplet is not realized in the ground state. The two 40-plets have mixed symmetry ( 12 -spin)
and the 35-plet is symmetric ( 32 -spin). The SU(4)f spin- 12 multiplet in Fig. 15.5 contains 20 spin- 12
baryons and the corresponding one with bottom quarks an additional 12, giving together 32 states.
Similarly there are 30 spin- 23 baryons. One expects 75 ground state mesons from (15.34). Thus
31st May, 2024

15. Quark Model23

13 ground state baryons containing both b and c quarks are predicted (8 spin- 12 [4Ω + 4Ξ] and 5
spin- 23 [3Ω + 2Ξ]).

15.6 Magnetic moments
The magnetic dipole moment of a baryon is conventionally written relative to the mass of the
proton:µ~ B = gB µN ~s,(15.35)eis the nuclear magneton (in natural units). The factor gB is calculated by adding
where µN = 2mp
the quark contributions. The magnetic moment of quark i with charge qi e and constituent massqi e
mi is µ~ i = 2( 2m)~s, hence for the three light quarks,
i

2
1
1
e
.
µu = κu , µd = − κd , µs = − κs , with κi ≡
3
3
3
2mi

(15.36)The magnetic dipole moment of a baryon is then given by
µB =

3
X

hB ↑ |µi σzi |B ↑i,(15.37)i=1

where |B ↑i is the wavefunction for a baryon with its spin along the z-axis and ~σ = 2~s. Since
the quark model uses the totally symmetric SU(6) wavefunctions (flavor SU(3) × spin SU(2)) one
predicts for the proton:
41eµp = µ u − µd =≡ κ,(15.38)3
32m
assuming mu = md ' m (for the detailed derivation see e.g. Ref. [6]). For the magnetic momentof the neutron one obtains likewise142µn = − µu + µd = − κ,
3
3
3

(15.39)which leads to the simple prediction µn /µp = gn /gp = −2/3.
The famous Frisch and Stern experiment showed for the first time that gp ∼ 5.58  2. Modern
measurements are performed in electromagnetic traps, see the Listings, while the magnetic momentof the neutron is measured with cold neutron beams. The experimental ratio µn /µp = –0.6849793± 0.0000003 is impressively close to the prediction of the quark model (note that using instead theantisymmetric SU(6) wavefunctions would lead to µn /µp = −2 [6]). The constituent mass m of theu and d quarks can be estimated from (15.35) and (15.38): m = 2mp /gp ' 336 MeV.
The magnetic moment of the Λ is easy to predict: the diquark ud has isospin i = 0, since i(s) =0 and i(Λ) = 0, and is therefore an antisymmetric isospin state. The symmetric quark model thenalso requires the spin to be zero, so that the magnetic moment of the Λ stems from the s quark:1µΛ = µs = − κs .
3

(15.40)The magnetic moment of the Λ is obtained from the Larmor precession frequency of the polarization
vector in an homogeneous magnetic field. The latter is derived from the asymmetric distributionof the proton in Λ → π − p [99]. The experimental result is listed in Table 15.8. The constituent
mass of the s quark is then ms = −mµp /3µΛ = 509 MeV.
The magnetic moments of the other baryons (Table 15.8) are predicted from κ and κs . The
magnetic moment of the Σ + is obtained by replacing the d quark in (15.38) by an s quark, andfor the Σ − the u quark by an s quark in (15.39). The magnetic moment of the Σ + has also
31st May, 2024

15. Quark Model

24

Table 15.8: Quark model predictions and measured magnetic dipole moements of the ground state baryons in units of µN ; κ ≡ 2m
= 2.793 µN ande†0κs ≡ 2ms = –3µΛ = 1.84 µN . Σ → Λ transition magnetic moment.
BaryonQuark model
Experimental valuepκ
input 2.793
n− 23 κ
= –1.86 –1.913
1input –0.6138 ± 0.0047
Λ− 3 κs81+Σ= 2.68 2.458 ± 0.010
9 κ + 9 κs1
2
0
= 0.82Σ
9 κ + 9 κs10†Σ
− √3 κ
= –1.61 –1.61 ± 0.08
Σ−
Ξ0
Ξ−
Ω−
∆++
∆+

− 94 κ + 19 κs = –1.04 –1.160 ± 0.025− 29 κ − 49 κs = –1.44 –1.250 ± 0.014
14= –0.51 –0.6507 ± 0.0025
9 κ − 9 κs−κs
= –1.84 –2.024 ± 0.056
2κ
= 5.58 4.52 ± 0.67
κ
= 2.79
2.3
4.5

been measured with the precession method using its decay into pπ 0 [100], that of the Σ − → nπ −likewise, or by detecting the X-rays emitted by cascading Σ − captured in the Coulomb shells oftarget atoms. The magnetic moment is then derived from the fine-structure splitting [101].
The magnetic moments of the Ξ 0 , Ξ − and Ω − are obtained from the polarization of the Λ inthe decays Λπ 0 , Λπ − and ΛK − , respectively. The SU(6) wavefunction of the (spin 3/2) Ω − is the
product of the two symmetric SU(3) flavor and SU(2) spin wavefunctions. Summing over the threes quarks one getsµΩ − =3XhΩ − ↑ |µi σzi |Ω − ↑i = 3µs = 3µΛ .

(15.41)i=1The magnetic moment of the Ω − is hard to measure, because the Ω − is unpolarized at highenergies, in contrast to the other hyperons. Polarized Ω − hyperons have been obtained frompolarized Λ and Ξ 0 impinging on a nuclear target [102].
In quark models with full isospin symmetry the magnetic moments of the ∆-resonances are
simply related to the magnetic moment of the proton by µ∆ = Q∆ · µN , (Q∆ = charge) i.e. themoment of the ∆+ should equal the proton moment and the moment of the ∆++ should be twice aslarge (sum of the three u-quarks). Magnetic moments of the decuplet baryons (with the exceptionof the Ω − ) are very difficult to measure because their lifetimes are so short that spin precession
techniques cannot be used. For the ∆ one can profit from an electromagnetic spin-reorientationtransition inside the large width of the state. This leads to the emission of a magnetic dipole
photon in reactions like π + p → π + pγ or γp → π 0 pγ 0 which is related to the magnetic momentof the resonance. The first reaction has been used to study the magnetic moment of the ∆++ bymeasuring the left-right asymmetry with polarized protons [103]. The second one was used for the∆+ state [104, 105]. Taking all experimental and systematic uncertainties into account the resultspans the range from (2.3 - 4.5)µN , covering the quark model prediction of ≈2.8µN , but is not very
precise.
Table 15.8 lists the current experimental values for the magnetic moments of the ground state
31st May, 2024

15. Quark Model25

baryons, together with the predictions from the quark model. There are significant discrepancies,
but given its crudeness, the quark model performs surprisingly well.

15.7 Dynamics
Quantum chromodynamics (QCD) is well-established as the theory for the strong interactions.
As such, one of the goals of QCD is to predict the spectrum of strongly-interacting particles. To
date, the only first-principles calculations of spectroscopy from QCD use lattice methods. These
are the subject of Sec. 15.8. These calculations are difficult and unwieldy, and many interestingquestions do not have a good lattice-based method of solution. Therefore, it is natural to build
models, whose ingredients are abstracted from QCD, or from the low-energy limit of QCD (suchas chiral Lagrangians) or from the data itself.
Phenomenological models for light quark systems typically include
1. A confining interaction, which is generally spin-independent (e.g., harmonic oscillator or linear
confinement);
2. A strange quark mass somewhat larger than the up and down quark masses, in order to split
the SU(3) multiplets;
3. Different types of spin-dependent interactions:a) commonly used is a color-magnetic flavor-independent interaction modeled after the effects
of gluon exchange in QCD (see e.g., Ref. [106]). For example, in the S-wave states, there isa spin-spin hyperfine interaction of the form
HHF = −αS M

X(~σ λa )i (~σ λa )j ,(15.42)i>jwhere M is a constant with units of energy, λa (a = 1, · · · , 8, ) is the set of SU(3) unitaryspin matrices, defined in the review “SU(3) Isoscalar Factors and Representation Matrices,”and the sum runs over constituent quarks or antiquarks. Spin-orbit interactions, althoughallowed, seem to be small in general, but a tensor term is responsible for the mixing of stateswith the same J P but different L, S combinations.
b) other approaches include flavor-dependent short-range quark forces from instanton effects(see e.g., [107,108]). This interaction acts only on scalar, isoscalar pairs of quarks in a relative
S-wave state:hq 2 ; S, L, T |W |q 2 ; S, L, T i = −4gδS,0 δL,0 δI,0 W(15.43)where W is the radial matrix element of the contact interaction.
c) a rather different and somewhat controversial approach is based on flavor-dependent spinspin forces arising from one-boson exchange. The interaction term is of the form:
HHF ∝

X

V (~rij )λFi · λFj ~σi · ~σj(15.44)i<jwhere the λFi are in flavor space (see e.g., Ref. [109]).
4. In the case of spin-spin interactions, a flavor-symmetric interaction for mixing q q̄ configurations of different flavors (e.g., uū ↔ dd¯ ↔ ss̄), in isoscalar channels, so as to reproduce e.g.,the η - η 0 and ω - φ mesons.
Systems with heavy quarks have their own approaches. Besides potential models to deal with
QQ̄ spectroscopy (which are similar to what we have just described for light quarks) there are a
31st May, 2024

26

15. Quark Modelvariety of effective field theories built on QCD. Nonrelativistic QCD or NRQCD is a nonrelativistic
reduction of the QCD Lagrangian, written as an expansion in powers of the heavy quarks’ velocities.
Its expressions are often used in lattice calculations. Terms in the Lagrangian have obvious quark
model analogs, but are derived directly from QCD. For example, the heavy quark potential is a
derived quantity, extracted from simulations.
There are also many versions of heavy quark effective theory (HQET), an expansion in the
inverse mass of the heavy constituent. The physics input is one of decoupling of scales: theinteractions of the heavy quarks take place at short distance while the light degrees of freedom
involve long distance dynamics, so to some degree they decouple. The form of terms in the expansion
is fixed (often by symmetry) but the coefficients are not. These approaches allow one to interpolate
from the charm sector to the bottom one, or from masses of mesons to baryons. For example, aHamiltonian for one or more heavy quarks bound to light degrees of freedom could be written as
HlQ = mQ + el +κlsl ~ ~+
S · jl + . . .
2mQ 2mQ(15.45)where mQ is the mass of the heavy quark (or heavy quarks where the heavy quarks are treated as a
~ is the angular momentum carried by the heavy degrees of freedom.
single dynamical object) and S~
el , κl and jl involve the light degrees of freedom. We will see an example of the use of this formulain the next section.

15.8 Lattice Calculations of Hadronic Spectroscopy
Lattice calculations are a major source of information about QCD masses and matrix elements.
The necessary theoretical background is given in Sec. 17 of this Review. Here we confine ourselves
to some general comments and illustrations of lattice calculations for spectroscopy.
It might seem a bit out of place to have a section about lattice calculations in a review of
the quark model, since to many readers, the quark model is just a model while QCD could bethought of as a construct which is something deeper than a model. But this review is, despite its
title, actually an introduction to the spectroscopy and related quantities of the strong interactions.
From that perspective, a presentation of lattice results is entirely appropriate.
The input to lattice calculations is a discretized version of the QCD Lagrangian. Lattice calculations measure correlation functions via Monte Carlo simulation and physical observables are
determined through fits of the lattice data to theoretical expectations. It is fair to say that all
lattice calculations make extensive use of quark model ideas – for example, the operators used tocreate and annihilate hadrons are almost always based on the quark model.
There is only a sporadic literature connecting lattice results to the quark model (one recent
example is [110]) but of course the qualitative understanding of lattice results depends as heavily
on quark model ideas as does the qualitative understanding of experimental data.
In general, the cleanest lattice results come from computations of processes in which there is
only one particle in the simulation volume. These quantities include masses of hadrons, simple
decay constants, like pseudoscalar meson decay constants, and semileptonic form factors (such as
the ones appropriate to B → Dlν, Klν, πlν). The cleanest predictions for masses are for states
which are far below any thresholds to open channels, since the effects of final state interactions are
not yet under complete control on the lattice. As a simple corollary, the lightest state in a channel
is easier to study than the heavier ones.
Good-quality modern lattice calculations will present multi-part error budgets with their predictions. Part of the uncertainty is statistical, from sample size. Typically, the quoted statisticaluncertainty includes uncertainty from a fit: it is rare that a simulation computes one global quantity which is the desired observable. Simulations which include virtual quark-antiquark pairs (also
31st May, 2024

27

15. Quark Model

known as “dynamical quarks” or “sea quarks”) are often done at up and down quark mass values
heavier than the experimental ones, and it is then necessary to extrapolate in these quark masses.
Simulations can be carried out at the physical values of the heavier quarks’ masses. They are always
done at nonzero lattice spacing, and so it is necessary to extrapolate to zero lattice spacing. Some
theoretical input is needed to do this. Much of the uncertainty in these extrapolations is systematic, from the choice of fitting function. Other systematics include the effect of finite simulation
volume, the number of flavors of dynamical quarks actually simulated, and technical issues withhow these dynamical quarks are included. The particular choice of a fiducial mass (to normalize
other predictions) is not standardized; there are many possible choices, each with its own set ofstrengths and weaknesses, and determining it usually requires a second lattice simulation from thatused to calculate the quantity under consideration.
A systematic error of major historical interest is the “quenched approximation,” in which dynamical quarks are simply left out of the simulation. This was done because the addition of thesevirtual pairs presented an expensive computational problem. No generally-accepted methodology
has ever allowed one to correct for quenching effects, short of redoing all calculations with dynamical quarks. (All light degrees of freedom must be included in any realistic simulation of a quantum
field theory.) Advances in algorithms and computer hardware have rendered it obsolete.
With these brief remarks, we turn to examples. The field of lattice QCD simulations is vast, andso it is not possible to give a comprehensive review of them in a small space. The history of lattice
QCD simulations is a story of thirty years of incremental improvements in physical understanding,algorithm development, and ever faster computers, which have combined to bring the field to apresent state where it is possible to carry out very high quality calculations. We present a few
representative illustrations, to show the current state of the art.
15.8.1 Spectroscopy of low-lying states
One of the first goals of large scale lattice simulations was to compute masses of the lightest
states in all of the “conventional” q q̄ and qqq channels. We illustrate results from many groups in
Fig. 15.9, a comprehensive summary from 2012 provided by A. Kronfeld (private communication;
see also [124]).
Mesons with a valence structure of identical quark - antiquark pairs such as the eta or eta-prime,¯ + β| s̄si, present more of a challenge, because one must include the effects of| ηi ∼ α| ūu + ddi“annihilation graphs” for the valence q and q̄. Many groups, among them Refs. [21, 118, 125, 126],
have reported calculations of the η and η 0 mesons. The most recent calculation is that of Ref. [126],
which finds masses of 554.7 ± 9.2 and 930 ± 21 MeV for the η and η 0 .
The spectroscopy of mesons containing heavy quarks is truly a high-precision endeavor. Thesesimulations typically use NRQCD for QQ̄ bound states. Figs. 15.10 and 15.11 show the low lying
mass spectrum for charmonium and bottomonium states from several different groups [122,127–130].
Most, but not all, of the results are for the lightest state with a given value of quantum numbers.
We return to this point below.
Fig. 15.12 shows a compilation of lattice results for doubly and triply charmed baryons, provided+ [93] is also shown. Note that the lattice
by S. Meinel [138]. The position of the observed Ξcc
calculations for the mass of this state were predictions, not postdictions.
15.8.2 Excited state spectroscopy
A close look at Fig. 15.9 reveals an issue: while many of the states shown in it are stable under
the strong interactions, some of them (the vector mesons, the Delta) are not. They are resonances.
And so are all the excited states of hadrons listed in the PDG be they “ordinary’,” or “exotic.”
How do lattice calculations deal with such states? This is actually a complicated business, which
is an active area of research.
31st May, 2024

15. Quark Model

28

Figure 15.9: Hadron spectrum from lattice QCD. Comprehensive results for mesons and baryons
are from MILC [111,112], PACS-CS [113], BMW [114], QCDSF [115], and ETM [116]. Results for η
and η 0 are from RBC & UKQCD [21], Hadron Spectrum [117] (also the only ω mass), UKQCD [118],
and Michael, Ottnad, and Urbach [119]. Results for heavy-light hadrons from Fermilab-MILC [120],
HPQCD [121,122], and Mohler and Woloshyn [123]. Circles, squares, diamonds, and triangles standfor staggered, Wilson, twisted-mass Wilson, and chiral sea quarks, respectively. Asterisks represent
anisotropic lattices. Open symbols denote the masses used to fix parameters. Filled symbols (andasterisks) denote results. Red, orange, yellow, green, and blue stand for increasing numbers ofensembles (i.e., lattice spacing and sea quark mass) Black symbols stand for results with 2+1+1
flavors of sea quarks. Horizontal bars (gray boxes) denote experimentally measured masses (widths).
b-flavored meson masses are offset by −4000 MeV.
There are two issues involving excited state spectroscopy, which are entangled in any latticecalculation:
1. There are many states with the same quantum numbers, all of which contribute to latticeobservables
2. In finite volume, states do not decay, they only mix.
To explain what is going on, we have to describe how a lattice calculation determines a mass.
This is done by measuring an observable which involves the creation of a set of quark and gluon
fields at some point on the lattice and their annihilation at other points on the lattice. To find theenergies of zero momentum states, the operators are averaged over position and the observable is
Cij (t) =XxhOi (x, t)Oj (0, 0)i ∝X

h0|Oi |ni hn|Oj |0i exp(−En t).

n31st May, 2024

(15.46)

29

15. Quark Model

Figure 15.10: Selected spectroscopy of the c̄c spectrum, compared to lattice data from Ref. [127]in squares, from Refs. [128, 129] in diamonds, and from Ref. [130] as a fancy diamond. Particles
whose masses are used to fix lattice parameters by Ref. [127] are shown with crosses; octagons label
experimental values. The upper χc1 state is also known as X(3872). The dotted line shows thethreshold value for two charmed mesons.

En is the energy of the nth hadronic state with the quantum numbers of the source or sink. The
reader can see that by taking t large, the correlator is dominated by the contribution from thelightest state, decaying with t as exp(−Emin t). This is a positive thing if one is interested in the
ground state, but the contribution of excited states is exponentially suppressed at large t. The sizeof a given state’s contribution, h0|Oi |ni, can provide information about the content of the state.
As we move away from hadrons which can be created by the simplest quark model operators(appropriate to the lightest meson and baryon multiplets) we encounter a host of new problems:either no good interpolating fields, or too many possible interpolating fields, and many states withthe same quantum numbers. Techniques for dealing with these interrelated problems vary fromcollaboration to collaboration, but all share common features: typically, correlation functions frommany different interpolating fields are used, and the signal is extracted in what amounts to avariational calculation using the chosen operator basis. In addition to mass spectra, wave function
information can be garnered from the form of the best variational wave function.
These calculations give towers of excited states, but the states are not resonances. Latticecalculations do not see decays directly because they are done in boxes of some finite size. Generally,
the energies of the single particle states do not depend strongly on the box size. Multiple particle
states, such as two particle states with equal and opposite momenta, are different, though. The
finite box size quantizes the allowed values of momentum and the spectrum of states, which would
be a continuum in infinite volume, is just a discrete tower. As the box size is varied, some of thesestates can approach the energies of single particle states, and the nearby states will mix and split,
participating in a quantum mechanical avoided level crossing. The mixing and splitting of single
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15. Quark Model

Figure 15.11: Spectroscopy of the b̄b system, adapted from Ref. [122]. Particles whose masses
are used to fix lattice parameters are shown with crosses; octagons label experimental values, andlattice results are shown as squares. The dotted line shows the threshold value for two bottom
mesons.

and multiparticle states is the finite volume data which is used to infer how states in the continuum
interact.
As a simple example, two positive energy but interacting particles in a one dimensional box
have a spatial wave function which looks like cos(k|z| + δ) where δ is the phase shift due to theirinteraction. Periodic boundary conditions in a finite box of length L give a quantization condition
to the allowed momentum of the state (kL + 2δ(E) = 0 mod 2π, for example) which involves thephase shift. Simulations with many values of L and operators for states carrying many values ofmomentum can be combined into information about phase shifts, and then to statements about
the location(s) of singularities in the complex energy plane.
The techniques for studying excited states of mesons and baryons, glueballs, and exotics are
similar. They all use a large basis of trial states: the progression with time started with minimum
quark content states, then added more complicated states (pairs of mesons or tetraquark states),then carried out simulations with multiple volumes with the goal of extracting resonance properties.
The older single volume studies typically presented results for many quantum numbers, which givea broad (though incomplete) view of the QCD spectrum.
An example of meson spectroscopy where this is done, by [48], is shown in Fig. 15.13. The
quark masses are still heavier than their physical values, so the pion is at 392 MeV. The authors
can assign a relative composition of nonstrange and strange quark content to their states, observing,for example, a nonstrange ω and a strange φ. Some states also have a substantial component ofgluonic excitation. Note especially the three exotic channels J P C = 1−+ , 0+− , and 2+− , with statesaround 2 GeV. These calculations will become more realistic as the quark masses are carried lower
and resonance effects are included.
The interesting physics questions of excited baryon spectroscopy to be addressed are precisely
31st May, 2024

15. Quark Model

31

Liu et al., 2010 (a ≈ 0.12 fm)
Briceño et al., 2012 (a = 0)Dürr et al., 2012 (a ≈ 0.07 fm; stat. only)
Namekawa et al., 2013 (a ≈ 0.09 fm; stat. only)
Alexandrou et al., 2014 (a = 0)

Padmanath et al., 2013 (as ≈ 0.12 fm; stat. only)
Brown et al., 2014 (a = 0)Pérez-Rubio et al., 2015 (a ≈ 0.075 fm)
Alexandrou and Kallidonis, 2017 (a ≈ 0.09 fm)
LHCb, 2017

4850

4000
3900

4800

M (MeV)M (MeV)

3800
3700
3600

4750

4700
3500
4650

3400

Ξcc

Ξ∗cc

Ωcc

Ω∗cc

Ωccc

Figure 15.12: Comparison of lattice QCD results for the doubly and triply charmed baryon
masses. Labels are Liu, et al., [131]; Briceno, et al., [132]; Namekawa, et al., [133]; Padmanath, etal., [134]; Alexandrou, et al., [116]; Brown, et al., [135]; Perez-Rubio et al., [136]; Alexandrou andKallidonis 2017, [137]. Only calculations with dynamical light quarks are included; for the doubly
charmed baryons, only calculations were performed at or extrapolated to the physical pion mass areshown. Results without estimates of systematic uncertainties are labeled “stat. only”. The latticespacing values used in the calculations are also given; a = 0 indicates that the results have been
extrapolated to the continuum limit. In the plot of the doubly charmed baryons, the position of+ state [93] is shown with a horizontal line.
the experimentally observed Ξcc

those enumerated in the last section. An example of a calculation involving only single particle
states in a single volume, due to Ref. [139], is shown in Fig. 15.14. Notice that the pion is not yet
at its physical value. The lightest positive parity state is the nucleon, and the Roper resonance has
not yet appeared as a light state.
Glueballs present similar issues. In Fig. 15.3 we showed a figure from [34] presenting a lattice
prediction for the glueball mass spectrum in quenched approximation. A true QCD prediction of
the glueball spectrum requires dynamical light quarks, some way to deal with the mixing of gluestates and quark-antiquark (and beyond) states and (because glueball operators are intrinsicallynoisy) high statistics. Early studies which include the 0++ channel are Refs. [140, 141]. Fig. 15.15
shows results from [140], done with dynamical u, d and s quarks at two lattice spacings, 0.123
and 0.092 fm, with pion masses at 280 and 360 MeV respectively, along with comparisons to thequenched lattice calculation of [33] and to experimental isosinglet mesons. This study shows thatthe effects of quenching seem to be small at its relatively heavy pion masses.
Calculations of resonance properties are more recent. Two body decays with a single open
channel, such as the rho coupled to two pions, have a large literature. A review [142] summarizes
the situation, and example of a calculation of the rho meson decay width is [143]. (See also [144].)
The mass and decay width of the f0 (500) have recently been computed in [145] and [146] and thoseof other mesons in [147]. Ref. [148] studies the decay width of the ∆(1238).
31st May, 2024

15. Quark Model

32
3000

2500

2000

1500

1000

500

Figure 15.13: Isoscalar (green and black) and isovector (blue) spectrum from Ref. [48]. States arelabeled J P C . The quark mass is heavier than its physical value; mπ = 392 MeV. The vertical heightof each box indicates the statistical uncertainty in the mass. Black and green indicate relative
nonstrange and strange composition. Orange outlines show states with a large chromomagnetic
component to their wave function, which the authors argue are hybrid states. Note the exotic
states in the three rightmost columns.
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6

Figure 15.14: Spin-identified spectrum of nucleons and deltas, from lattices where mπ = 396
MeV, in units of the calculated Ω mass, from Ref. [139]. The colors just correspond to the different
J assignments: grey for J = 1/2, red for J = 3/2, green for 5/2, blue for J = 7/2.
Studies which observe a large basis of states across many volumes can probe excited state
spectroscopy as resonances. These calculations are just beginning and the ones we know about are
still performed at unphysically heavy quark masses. Examples include the decays of an exotic J P C =31st May, 2024

33

15. Quark Model

Figure 15.15: Lattice QCD predictions for glueball masses. The open and closed circles are thelarger and smaller lattice spacing data of the full QCD calculation of glueball masses of Ref. [140], atpion masses of 280 and 360 MeV. Squares are the quenched data for glueball masses of Ref. [33]. The
bursts labeled by particle names are experimental states with the appropriate quantum numbers.
1−+ resonance [149] and the spectroscopy of J −− resonances [150]. The results of Refs. [128, 129]
in Fig. 15.10 come from similar analyses. We expect to see considerable progress in this area overthe next few years.
We conclude this section with some remarks about unusual states containing heavy quarks.
Lattice calculations relevant to the extra states observed in the charmonium and bottomonium
spectrum (Sec. 15.4) are difficult, because the states sit high in the spectrum of most channels,
because of the number of nearby multiparticle states, and because it is necessary to have thetechnology to study heavy and light quarks together.
Some lattice calculations are used as input for phenomenological studies of hybrids. The theoretical motivation is that one can imagine states made of a heavy QQ̄ pair along with excitationsof light degrees of freedom, viewed as an excited states of the string connecting them. The analogy is to an excited state of a diatomic molecule. The heavy quark potential, the potential fromorbitally excited gluon configurations, and the “string breaking” potential (from the interaction
between static sources and static-light mesons) are used as inputs in Born - Oppenheimer andcoupled channel calculations for systems whose constituents include heavy quarks and antiquarks.
Examples of such calculations include Refs. [151–153].
The system with the largest lattice literature is a bound state of two heavy quarks and two light¯ The story begins with the spectroscopy of doubly and triply charmed
antiquarks, for example bbūd.
baryons in Fig. 15.12; the lattice calculations in that figure were performed in 2010 - 2017. In 2017
two groups of authors [154,155] realized that one could extrapolate from the masses of Qq̄, Qqq, andQQq states (Q for a heavy quark, q̄ for a light antiquark) to predict the mass of a QQq̄ q̄ tetraquark.
The QQq̄ q̄ state would be stable if the mass of the Q were heavy enough. (This basically arises
because the QQ diquark is in an attractive color-triplet state and the effective binding energy
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15. Quark Model

scales as αs2 mQ .) The J P = 1+ bbūd¯ state was expected to be stable. Simultaneously, a latticecalculation [156] of the mass of a bbūd¯ state showed that it was below threshold to fall apart into apair of B mesons. The I(J P ) = 0(1+ ) state is predicted to be stable by about 128(26) MeV [157] or
189(10) MeV [156]. (These authors put the bbs̄d¯ state at 98 MeV below threshold.) These numbers
are in reasonable agreement with phenomenological estimates from Refs. [154, 155].
These states are easier for the lattice theorist to study than other exotics because they are
the lightest single particle states in their channels. They are difficult because they are just barelybound, and the usual approximation for a lattice calculation of unphysically heavy light quarks
tends to raise their mass. The calculations have to include multiple trial states: the tetraquark
itself plus pairs of mesons.
The situation with regard to the recently discovered Tcc (3875)+ [158] is still unsettled. Bothlattice calculations and HQET estimates put a ccūd¯ state right at threshold, which is a difficult
place to explore on the lattice (or by any other method, for that matter). The Tcc (3875)+ is seen inD0 D0 π + and researchers studying three body decays on the lattice (an open problem, at present)
mark it as an obvious research target.
15.8.3 Electromagnetic effects
As a final part of spectroscopy we mention electromagnetic mass splittings (such as the neutron- proton mass difference). They are interesting but difficult. These calculations are important for
determining the values of the quark masses (for a discussion see Sec. 60 in this Review). Knowing that the neutron is heavier than the proton tells us that these splittings have a complicatedorigin. One part of the shift is because the up and down quarks have slightly different masses.
The second is that the quarks have (different) charges. Phenomenologists (compare Ref. [159])
combine Coulomb forces and spin-dependent electromagnetic hyperfine interactions to model theircharge effects. In order to compute hadronic mass differences on the lattice, electromagnetic interactions must be included in the simulations. This creates a host of technical issues. An important
one is that electromagnetic interactions are long range, but lattice simulations are done in finite
volumes. The theoretical situation is summarized in the recent Flavour Lattice Averaging Group(FLAG) review [160]. A 2014 calculation, Ref. [161], presented the first results for electromagnetic
mass splittings in the baryon octet, with good agreement with observation. Refs. [162–164] have
performed calculations for meson splittings.
There is a small lattice literature associated with magnetic moments of baryons. The calculations are done by including a static magnetic field in the simulation and computing the energy
difference between baryons in different spin states. We are aware of no calculations at physicalquark masses. A recent calculation [165] used 800 and 450 MeV pions. When expressed in unitsof the nucleon magneton at the simulation point (e/(2MN (mπ )) the moments are in reasonableagreement with experiment and with quark model expectations.
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