An introduction to the quark model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
An introduction
to the quark model
Niccolò Cabeo School
available at http://www.ipnl.in2p3.fr/perso/richard/SemConf/Talks.html
Jean-Marc Richard
Institut de Physique Nucléaire de Lyon
Université Claude Bernard (Lyon 1)–IN2P3-CNRS
Villeurbanne, France
Ferrara, Italy,May 2012
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Table of contents
Content
1 Prelude: Few-charge systems
2 Mesons as (q¯q)
3 Baryons as (qqq)
4 Multiquarks
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
More detailed table of contents
1 Few-charge systems
Binary atoms: central potential
Spin forces in atoms
Three-body ions
Four-body molecules (+, +, −, −)
2 History of the quark model
Early hadrons
Generalised isospin, SU(3)
Heavy quarks
3 The quark–antiquark model of mesons
Quantum numbers
Spin averaged spectrum
Improvements
4 Baryons
5 Multiquarks and other exotics
Glueballs, hybrids, molecules
Baryonium
6 Outlook
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Few-Charge systems
Why few-charge systems in lectures about quarks?
a if it is allowed to compare small things with great
Content
1 Binary atoms
2 Spin-dependent forces
3 Three-body ions
4 (+, +, −, −) molecules
Si parva licet componere magnis a
JMR Quark Model
Virgil

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Binary atoms
p 2 1
2 m1
+ p2 1 −
2 m1
e2
,
r12
The centre of mass motion can be removed,
The intrinsic Hamiltonian
can be rescaled to
H = p2 e2
− ,
2 µ r
h = −∆ − r −1 ,
with 2 µ e 4 for E and (2 µ e 2 ) −1 for r.
Similarly, any oscillator can be reduced to −d/dx 2 + x 2 .
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Binary atoms-2
Very characteristic spectrum
1
E = − ,
4 (n + ℓ) 2
where n = 1, 2, . . . is the radial number, ℓ = 0, 1, . . . the orbital
momentum, and n + ℓ the principal quantum number.
Degeneracy of orbital vs. radial excitations,
Infinite number of bound states, even for very small coupling,
In contrast with short-range interactions in nuclear physics,
Many probes: hydrogen-like atoms, muonic atoms, kaonic atoms,
positronium
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Spin forces in atoms
Deduced from the vector character of the exchanged photon,
Automatically included in fully relativistic treatments,
Pauli, Fermi, Breit, etc., derived corrections to be added to NR
Hamiltonians, and treated as perturbation,
In particular, the famous hyperfine correction
Vss = e2
m1 m2
2 π
3 δ(3) (r) σ1.σ2 ,
splits ortho- and para-hydrogen (important transition in
astrophysics; analogue to be measured in antihydrogen),
Note the short-range character,
Note the very specific mass dependence
For (e + , e − ), ∃ other contribution
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Three-body ions
He (α, e − , e − ) is obvious, as any (+q, −, −) with q > 1
q = 1, i.e., H − and similar, less obvious, see Hylleraas,
Chandrasekhar, Bethe, Heisenberg, etc.,
Take mp = ∞ (for simplicity), H − stability resists any f (r1) f (r2),
i.e., Hartree method fails.
Stability demonstrated with better wave functions,
Map of stability for (m ±
1 , m∓
2 , m∓
3 )?
Very stable for H2 + = (e − , p, p)
Marginally stable for H − = (p, e − , e − ) or Ps − = (e + , e − , e − )
Unstable for (p, ¯p, e − )
Stability rather sensitive to the masses, e.g., (p∞, e − , e ′− )
unstable if m ′ differs from m by more than about 10%
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Three-body ions – excitations
H2 + = (e − , p, p) has several excited states,
H − has no stable excited state?
Both true and false
True if you define stability as E < (p, e − )1S + e − ,
False if spontaneous dissociation only into (p, e − )1S + e −
This is the unnatural-parity state of H −
Very sensitive to mp < ∞ and m = m ′ .
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Four-body molecules
Hydrogen molecule and variants best known, (p, p, e + , e − )
In the Born–Oppenheimer–Heitler–London approach, effective
pp potential, which gives the ground-state and the first
excitations. This is a very good approximation,
This corresponds to the two electrons in the lowest state for
given pp separation,
Excited electrons → second set of levels,
Positronium molecule proposed by Wheeler in 1945,
In 1946, Ore publish it does not believe this is the case,
In 1947, Hylleraas and (the very same) Ore have an elegant
proof of the stability
In 2007, indirect experimental evidence
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Systematics of (m +
1
, m+
2 , m−
3 , m−
4 )
Any state with m3 = m4 is stable. Why? Two degenerate
thresholds.
In particular, (m + , m + , m − , m − ) is stable (positronium molecule
and variants)
What about two masses?
(M + , M + , m − , m − ) improves stability.
(M + , m + , M − , m − ) spoils stability. It becomes unstable for
M/m 2.2 (or, of course, M/m 1/2.2)
So, starting from the doubly-symmetric (m + , m + , m − , m − ), and
breaking
Charge conjugation,
or Particle identity
does not produce the same result. Why?
Both ways of breaking symmetry lower the ground state. But in
the second case, the threshold benefits more of symmetry
breaking. Hence, one gets less stability. See the section on
multiquarks.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Some lessons from charge systems
Universality, same potential for p and e +
Scaling,
Level order for atoms, very specific
3- or 4-body systems stable or unstable, depending on the
masses,
Be patient. One could wait up to 60 years to see an exotic state
that is predicted.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
History
History of the quark model
History is Philosophy teaching by examples a
a According to Michel Casevitz, the sentence is not by Thucydid, but a British
commentator
Content
Early hadrons
SU(3)
Quarks and Aces
Heavy quarks
JMR Quark Model
Thucydid

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Very first hadrons
Discovery of the neutron (Chadwick, see, also Joliot-Curie)
Need for a strong interaction between nucleons
Search for underlying symmetry!
The theory of nuclear forces led to important tools!
Pion predicted by Yukawa,
Spin effects according to the quantum number of the pion,
Range ↔ mass of the pion,
Range anticipated from the size of nuclei, and from the ratio of
2-body to 3-body energies (Thomas),
Pion discovered in 1947 at Bristol, with three charge states, π + ,
π 0 and π − , not so easily as their decay are not the same,
Thus in 1947, 7 hadrons seen or expected, p, n, π + , π 0 and π − ,
and also ¯p and ¯n predicted.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
First hadrons: antinucleons
Bevatron built at Berkeley for antinucleons
¯p seen in 1955 by Ypsilantis, Segrè and Chamberlain,
¯n shortly after ( ¯ d a little controversial),
Also the cross-sections of ¯p,
With the unexpected
σann > σel
Because nucleons and antinucleons are not pointlike!
Because ¯ NN annihilation differs from e + e annihilation in QED!
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
First complications: resonances
New baryons: ∆, N ∗ , etc., in πN scattering,
New mesons: ρ, ω, σ produced, and also required, and even
anticipated for describing nuclear forces,
Bootstrap, or Nuclear Democracy, ∆ = πN + · · · ,
and similarly N = π∆ + πN + ρ∆ + · · · , etc.
Everything made of everything,
Partial success, but intricate coupled equations
Difficulty to accommodate mesons as baryon+antibaryon + . . . ,
(Ball, Scotti and Wong), in particular exchange degeneracy
(m(I = 1) m(I = 0)
Next baryon predicted to be J = 5/2 and I = 5/2
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Another complication: strangeness
New particles produced by pairs with strict rules, e.g., Λ with K +
but not with K −
and decaying with similar rules or weakly,
A new quantum number was empirically invented, strangeness,
which is
conserved by strong interactions (production, strong decay)
violated by weak interactions
weak decay linked to ordinary β decay (Gell-Mann, Lévy,
Cabbibo)
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
From SU(2) to SU(3)
SU(2) is a good symmetry for nuclear physics and pion
scattering,
In most cases, strange particles close to the non-strange ones,
For instance, Λ(1.12) close to N, K ∗ (0.89) close to ρ and ω,
Of course this is more complicated for scalar and pseudoscalar
mesons,
This led to extend SU(2) to SU(3)
and imagine that breaking can be described as linear or at most
quadratic in strangeness,
Today, this symmetry, renamed SU(3)F , remains a very useful
concept,
But first, one needs to assign the hadrons into representations
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The Sakata model
(p, n, Λ) in the fundamental representation, 3,
(¯p, ¯n, ¯ Λ) in ¯ 3,
Mesons from 3 × ¯ 3
Higher baryons from 3 × ¯ 3 × 3, etc.
But, as seen for the realisation of bootstrap, it faces serious
difficulties (exchange degeneracy, why Σ is almost as light as Λ?,
etc.)
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The Eightfold way
This led Gell-Mann and Ne’emann to propose to put the 8 lowest
baryons with spin 1/2 in an octet,
•
Σ −
n
•
•
Ξ −
Y
• •Λ Σ 0
• p
• Ξ 0
• Σ +
I3
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The Eightfold way
In most cases, smooth breaking, e.g.,
M = M0 + a Y + b(I(I + 1) − Y 2 /4) ,
lead to the Gell-Mann–Okubo formula
and many similar ones
2(N + Ξ) = 3 Λ + Σ ,
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The prediction of the Ω −
9 baryons with J = (3/2) + were known below 2 GeV. At The
Rochester Conference of 1962 in Geneva, Gell-Mann predicted a
new one with strangeness −3
∆• −
•
Σ ∗−
ƥ 0
Y
•
Σ ∗0
• ∆+
Ξ• ∗− •Ξ∗0 •Ω −
• Σ ∗+
•∆ ++
Discovered by Samios et al. at Brookhaven at the end of 1963
and published in 64.
JMR Quark Model
I3

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Mesons in SU(3)
Octet and singlet, with mixing
One uses to talk about nonet
Y
K• 0
•K+ •
π −
• K− • 0 π η
•η ′
• ¯K 0
• π +
I3
ρ −
•
K• ∗0
• K∗− Y
• 0 ρ ω
•φ
JMR Quark Model
•K ∗+
• ¯K ∗0
• ρ +
I3

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The fundamental representation: quarks
Y
d• •u
• s
I3
ū•
Y
• ¯s
• ¯ d
q b I I3 Y S Q
1 u 3
1 d 3
1
2
1
2
1
2
1 − 2
1
2
3 0 3
1
3 0 − 1
3
1 s 3 0 0 − 2
3 −1 2
3
JMR Quark Model
I3

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The aces
φ
Zweig was puzzled by φ(1020) decay
Mostly into K ¯ K in spite of very little phase-space
π
ρ
φ
He interpreted as due to the content of the Φ and of final-state
mesons, he named “aces”,
But eventually the name “quark” prevailed, and here the notation
(u, d, s), rather than (p, n, λ).
π
JMR Quark Model
ρ
φ
K
K

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The Zweig rule (OZI, A-Z)
the rule explaining the narrowness of φ generalised, with
variants,
e.g., ¯ NN annihilation
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
First quark models
For baryons, mostly
Greenberg, and Dalitz, in particular Les Houches Lectures 1965,
Using the shell-model techniques of nuclear physics,
Both facing the problem of statistics,
And anticipating what will become colour
Indeed, their ∆ − (ddd) has s = 3/2, L = 0, thus everything is
symmetric for three fermions!
see chapter on baryons
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Heavy quarks
Kaon physics always rich,
θ − τ puzzle, P violation, C violation
CP violation in 1964,
Suppression of flavour-changing neutral currents led GIM (1970)
to propose another Q = 2/3 quark, named “charmed” (c)
Not too heavy to get the GIM mechanism working,
Properties of charmed particles anticipated, in particular
Gaillard, Lee and Rosner,
Including (c¯q), (cqq), (ccq), . . . and (c¯c)
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
October 1974 revolution
In November 1974, the J/ψ was discovered simultaneously at
BNL and SLAC, in quite different experiments
Lepton-pair production in hadronic collisions (Ting)
e + e − collisions (Richter)
Not recognised immediately, since extremely narrow,
Eventually identified as (c¯c)
Several other states (ψ ′ , χ, . . . ) seen
Charmed mesons seen also (G. Goldhaber)
As well as charmed baryons
Note: double-charm baryons not yet confirmed!
Beautiful confirmation of the charm prediction
And asymptotic freedom, which make the Zweig rule more
effective for J/ψ than for φ.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Charmonium
Simple models proposed for (c¯c) that work!
A revolution in strong interaction. Predictions with simple tools!
For instance V (r) = −a/r + b r + c and mass mc in the
Schrödinger equation reproduce the experimental spectrum
And properties such as leptonic widths and gamma transitions
Many colleagues said: “Now I believe in quarks”
In short, a real boost for strong interaction physics,
Based on empirical models, which later got support from QCD
See section on mesons
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Top and Bottom
When charm was discovered, the ideas were already rather
advanced on grand unification,
At least quark–lepton symmetry
Note: leptons always ahead,
When the µ was discovered, Rabbi said: “Who ordered the
muon?”
When the τ was discovered (M. Perl), it was said: “ Where are
the associated quarks?”
{τ, ντ } ↔ {b, t}
“Bottom, Top”
Also “Beauty, Truth”
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Upsilon discovery
In 1977, Lederman repeated Ting’s experiment with a more
powerful beam and another detector,
And discovered Υ and Υ ′
Immediately interpreted as (b ¯ b)
See chapter on mesons,
Already Lederman noticed Υ ′ − Υ ψ ′ − J.ψ,
And asked local theorists about a potential such that all ∆E are
independent of the reduced mass,
Answer: V (r) ∝ ln r
B mesons and B baryons also discovered,
As well as Bc = (b¯c)
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Exotic hadrons
Already before the quark model,
For instance, speculations about a Z baryon with S = +1
Within the quark model, exotic = state that cannot be
accommodated as (q¯q ′ ), or (qq ′ q ′′ ).
For instance meson with charm = +2, or baryons with S = +1
Besides flavour?
for mesons, ∃ exotic J PC
not for baryons
question of best beam and target:
e + e − clean but with some restriction
¯p annihilation
formation or production,
low energy or high energy
past or recent excitations: baryonium, glueballs, hybrid hadrons,
molecules
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Mesons: Content
The quark–antiquark model of mesons
Content
Quantum numbers
Spin-averaged spectrum
Improvements
Summary for heavy quarkonia
Light mesons
Heavy-light mesons
Strong decay
Some mathematical developments
I married them
Friar Laurence, Romeo and Juliet
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Quantum numbers
Consider symmetric quarkonia Q ¯ Q
Spin of quarks S, orbital momentum ℓ, spin of the meson J,
parity P and charge conjugation C
Lowest quarkonium states
2 s+1ℓJ 1S0 3S1 1P1 3P0 3P1 3P2 1D2 3D1 3D2 3D3 JPC 0−+ 1−− 1 +− 0 ++ 1 ++ 2 ++ 2−+ 1−− 2−− 3−− Remarks
Some quantum numbers are absent, e.g., J PC = 1 −+
Some J PC occur twice. For instance 1 −− may be a combination
of 3 S1 and 3 D1
In addition, radial number. Here, notation n = 1, 2, . . .. For
instance,
ηc is ηc(1S) or 1 1 S0
η ′ c is ηc(2S) or 2 1 S0
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Radial equation
Assume a simple V (r) without spin dependence, and let
Ψ = uℓ(r)
r
ℓz
Yℓ (ˆr) × spin × colour ,
Thus for u = uℓ(r) (no dependence upon ℓz)
−u ′′ (r) +
ℓ(ℓ + 1)
r 2
u(r) + m V (r) u(r) = m E u(r) ,
with boundary conditions u(0) = u(∞) = 0
Exactly solvable in a few cases
Coulomb, see section on atoms
HO, reduces to −u ′′ (r) + ℓ(ℓ + 1) u(r)/r 2 + r 2 u(r) = ɛ u(r) , with
ɛ = 3 + 4 (n − 1) + 2 ℓ = 3 + 2 N
Linear for ℓ = 0
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Linear potential for ℓ = 0
very similar to the Airy equation
−u ′′ (r) + r u(r) = ɛ u(r) ,
−y ′′ (x) + x y(x) = 0
Aix
0.4
0.2
6 4 2 2 4
0.2
0.4
ɛ = −zero of the Airy function = −an
vn(r) = Ai(r + an)/ Ai ′ (an) .
JMR Quark Model
x

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Scaling
Note the simple scaling properties
−u ′′ (r) +
−u ′′ (r) +
ℓ(ℓ + 1)
r 2
ℓ(ℓ + 1)
r 2
u(r) ± m g r α u(r) = m E u(r) ,
u(r) ± r α u(r) = ɛ u(r) ,
with the scaling in (m g) 1/(2+α) for the distances, and
m −α/(2+α) g 2/(2+α) for the energies.
For a logarithmic potential, m → m ′ gives En,ℓ → E ′ n,ℓ = En,ℓ + C st
The Coulomb-plus-linear −a/r + b r + c can be reduced to
−∆ − λ/r + r − ɛψ(r) = 0 , with only one parameter.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Solving the radial equation
−u ′′ ℓ(ℓ + 1)
(r) +
r 2 u(r) + m V (r) u(r) = m E u(r) ,
See, e.g., Hartree, where V (r) was the effective one-body
potential,
Integrate inwards, and outwards, and fix E by imposing continuity
of both u and u ′ at the matching point,
Or discretise, and solve an approximatively equivalent matrix
eignevalue equation,
Or use a variational method, e.g.,
u(r) =
i
Ci r ℓ+1 exp(−ai r 2 /2) ,
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Simplest quarkonium model
“Funnel” potential V (r) = −a/r + b r + c
Constituent mass mc
Minimal adjustment mc ∼ 1.5, a ∼ 0.4, b ∼ 0.2, and c ∼ −0.35
(all units in powers of GeV)
ur ,V r
2
1
1
2
3
1 2 3 4 5 6 7
Reproduces also (b ¯ b) with mb ∼ 4.5 GeV
JMR Quark Model
r

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Simplest quarkonium model
c¯c b ¯ b
1S 2S 1P 1D 1S 2S 1P 1D 2P
Model 3.07 3.68 3.48 3.78 9.47 9.99 9.87 10.11 10.23
exp. 3.07 3.67 3.52 3.77 9.44 10.01 9.89 10.16 10.26
In particular, the hierarchy of excitations (radial vs. orbital)
corresponds to the observation.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Simplest quarkonium model (c¯c) = dotted lines
M (GeV)
10.2
9.8
9.4
-
-
-
3S
2S
1S
2P
1P
1D
M (GeV)
10.2
9.8
9.4
-
-
-
3S
2S
1S
2P
1P
JMR Quark Model
1D
- 3.80
- 3.40
- 3.00

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Improvements: central potentials
Better potentials
Ng+Tye, Buchmüller, Richardson, etc., include asymptotic
freedom
− a
r
→ −a(r)
r
Schnitzer, . . . , Gonzalez et al., . . . use a softer confinement, due
to pair-creation effects,
etc.
Better simultaneous fit of (b ¯ b) and (c¯c)
Simpler potentials
V (r) = g ln r + C (Quigg + Rosner, etc. )
V (r) = A r α + B (Martin)
as α → 0 becomes logarithmic
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Improvements: relativistic corrections
p 2
2 m → m 2 + p 2 − m ,
Better than a simple renormalisation of the parameters.
However, often used with an instantaneous interaction,
Much better: Bethe–Salpeter equation (Bonn group, etc. )
But much more difficult,
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Improvements: loop corrections
¯D (∗)
(c¯c) (c¯c)
D (∗)
Provide the state above threshold a width,
Can be calculated using the 3 P0 model, and an overall of initial
and final wave functions
Give a mass-shift (dispersive part)
One expects many cancellations. For instance, if D = D ∗ , all 3 PJ
states receive the same shift. But if D ∗ > D, then differential shift
in addition, thus mimicking spin-orbit and tensor forces.
Should be more pronounced near a threshold,
see below ψ ′ − η ′ c
If too large an effect, back to the bootstrap?
¯c
c
JMR Quark Model
q
¯q
¯c
c

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Improvements: fine structure
ψ ′ → χJ + γ , χJ → J/ψ + γ ,
Masses accurately measured with antiprotons
analysed with
to first order
M( 3 P0) = Mt − 2 〈Vls〉 − 4 〈Vt〉 ,
M( 3 P1) = Mt − 〈Vls〉 + 2 〈Vt〉 ,
M( 3 P2) = Mt + 〈Vls〉 − 2
〈Vt〉 ,
5
V (r) + Vss(r) σ1.σ2 + Vls(r) ℓ.s + Vt(r) S12 ,
Mt = 1
M(
9
3 P0) + 3 M( 3 P1) + 5 M( 3
P2) ,
〈Vls〉 = 1
−2 M(
12
3 P0) − 3 M( 3 P1) + 5 M( 3
P1) ,
〈Vt〉 = 5
2 M(
72
3 P0) − 3 M( 3 P1) + M( 3
P1) .
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Improvements: fine structure
1P of (c¯c)
Mt = 3.525 , 〈Vls〉 = 0.035 , and 〈Vt〉 = 0.010 GeV .
1P level of (b ¯ b),
2P
Mt = 9.900 , 〈Vls〉 = 0.014 , and 〈Vt〉 = 0.003 GeV ,
∗Mt = 10.260 , 〈Vls〉 = 0.009 , and 〈Vt〉 = 0.002 GeV . (1)
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Improvements: hyperfine structure
Singlet governed by Vc − 3 Vss
Triplet by Vc+, Vss
So, in perturbation, 3 S1 − 1 S0 = 4 〈Vss〉S
And for P states, 3 Pm − 1 P1 = 4 〈Vss〉P
Where 3 Pm is an average spin-triplet, or say, a fictitious
spin-triplet free of spin-orbit and tensor.
To first order
3 Pm = [M( 3 P0) + 3 M( 3 P1) + 5 M( 3 P2)]/9 ,
but if spin-forces are treated non perturbatively,
3 Pm ≥ [M( 3 P0) + 3 M( 3 P1) + 5 M( 3 P2)]/9 ,
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The shaky history of spin singlets
In 1977, a candidate for ηc claimed in Germany 300 MeV below
the J/ψ, → a lot of excitation
tentatively explained by relativistic dynamics,
difficult to digest in most current models,
not confirmed at SLAC,
and eventually found about 120 MeV below J/ψ
Then η ′ c = ηc(2S) predicted about 70–80 MeV below ψ ′
However, it was pointed out that loop effects tend to decrease
this splitting substantially
Intense search with antiprotons at Fermilbab, but in a range of
too low masses,
Eventually found at Belle, Cleo, etc. about 50 MeV below ψ ′
Interesting process of double-charm production
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Spin singlets: P wave
Even harder for 1 P1, as anticipated (Renard in the late 70s)
First indication in ISR: cooled ¯p on jet target
Resisted for a while formation in a better ¯p beam at Fermilab
Then seen in several experiments
hc = 1 P1 almost coincides with the naive centre of gravity of
triplets,
Probably due to the cancellation of several small effects.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Spin singlets: (b ¯ b)
Some results are very recent
Two remarks
δ(1S) = 0.067 , hb(1P) = 9.898 ,
δ(2S) = 0.049 , hb(2P) = 10.260 GeV .
1 δ(1S) b ¯ b predicted to be 70 ± 9 MeV from Lattice If you cannot
afford Lattice QCD, use a logarithmic potential,
δ(1S) b ¯ b = δ(1S)c¯c
−1/2 mb
mc
∼ 65 MeV .
2 The ratio δ(2S)/δ(1S) is about 1.4 in (b ¯ b) and about 2.3 in (c¯c).
This illustrates how anomalously high is ηc(2S) — or
anomalously low is ψ ′ — due to the neighbouring threshold.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Orbital mixing
Except 3 P0, the states with unnatural parity contain two partial
waves,
For instance 3 S1 and 3 D1 for the states formed in e + e −
ψ = u(r)
r
| 3 S1〉 + w(r)
|
r
3 D1〉 ,
− w ′′ (r)
m +
6
m r 2 + Vc(r) − 3 Vls(r) − 2 Vt(r)
See nuclear-physics textbooks
For instance
− u′′ (r)
m + Vc(r) u(r) + √ 8 Vt(r) w(r) = E u(r) ,
w(r) + √ 8 Vt(r) u(r) = E u(r)
ψ(3770) = a| 3 D1, n = 1〉 + b1| 3 S1, n = 1〉 + b2| 3 S1, n = 2〉 + · · ·
|b2| ≫ |b1|? Not sure!
Contributions of J/ψ ↔ D (∗) ¯ D (∗) ↔ ψ ′′
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The origin of spin dependent forces
Much discussed at the beginning of charmonium
Then discussed by Lattice QCD (Rebbi et al., etc.)
And other non perturbative methods (Brambilla et al.)
Early approaches inspired by
QED
One-boson-exchange picture of nuclear forces
Scalar exchange → central, and spin-orbit,
Vector exchange → central, spin-spin, tensor and spin-orbit
Early models: vector linked to 1/r and scalar linked to
confinement,
One should be careful: some terms come from the reduction of
Dirac operators in terms of Pauli spinors, other come from the
non-relativistic reduction (Thomas precession)
It took some time to get a consistent picture compatible with
Lorentz invariance (Gromes, Eichten-Sucher, etc.)
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Summary for quarkonium
η c (2S)
hadrons
η c (1S)
γ
γ
γ
ψ(2S)
η,π
ππ
0
hadrons
J/ ψ (1S)
hadrons hadrons γ∗ radiative
γ∗ γ
χ
h
c1
(1P)
c (1P)
γ
hadrons
γ
χ c0 (1P)
hadrons π0 χ c2 (1P)
hadrons
J = PC 0−+ 1−− 0 ++ 1 ++ 1 +− 2 ++
γ
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Summary for quarkonium
η b (3S)
η b (2S)
η b (1S)
hadrons
(11020)
(10860)
(4S)
(3S)
hadrons
(2S)
hadrons
(1S)
γ
γ
γ
γ
h b (2P)
h b (1P)
BB threshold
χ b0 (2P)
χ b0 (1P)
χ b1 (2P)
χ b1 (1P)
χ b2 (2P)
χ b2 (1P)
J = PC 0−+ 1−− 1 +− 0 ++ 1 ++ 2 ++
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Summary for quarkonium
η c (2S)
hadrons
η c (1S)
γ
γ
γ
ψ(2S)
η,π
ππ
0
hadrons
J/ ψ (1S)
hadrons hadrons γ∗ radiative
γ∗ γ
χ
h
c1
(1P)
c
(1P)
γ
hadrons
γ
χ c0 (1P)
hadrons π0 χ c2 (1P)
hadrons
J = PC 0−+ 1−− 0 ++ 1 ++ 1 +− 2 ++
γ
η b (3S)
η b (2S)
η b (1S)
hadrons
(11020)
(10860)
(4S)
(3S)
hadrons
(2S)
hadrons
(1S)
γ
γ
γ
γ
h b (2P)
h b (1P)
BB threshold
χ b0 (2P)
χ b0 (1P)
χ b1 (2P)
χ b1 (1P)
χ b2 (2P)
χ b2 (1P)
J = PC 0−+ 1−− 1 +− 0 ++ 1 ++ 2 ++
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Summary for quarkonium
4.0
3.5
3.0
D ∗ s ¯ D ∗ s
Ds ¯ D ∗ s
D ∗ ¯ D ∗
Ds ¯ Ds
D ¯ D ∗
D ¯ D
η ′ c
ηc
☛
✢ ✾
γ
hc
γ
γ
❂
π 0
4320 ÷ 4360
Y (4260)
ψ(4170)
ψ(4040)
❅❘
ππψ ′
❅❘
ππJ/ψ
Z ± (4430)
❄
π ± ψ ′
Y (3940) Z(3930)X(3940)
❄
ωJ/ψ ❄
DD¯ ❄
DD¯ ∗
ψ ′
ψ(3770)
X(3872)
✡
✡ π
✡
✡✢
+ π−J/ψ π + π−π0J/ψ ππJ/ψ
❄ηJ/ψ
J/ψ ❄ ✌ ✢✢
❅ γ
❅❅❅❅❅❅❘
γ
γ χc2
χc1
◆ χc0
J P C : 0−+ 1 +− 1−− 0 ++ 1 ++ 2 ++ ?
M GeV
ππ
η
π 0
γ
γ
JMR Quark Model
γ

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Light mesons
In principle, the quark model not applicable
Though this was attempted!
Two examples:
First positive-parity excitations of (q¯q) with I = 1 are a0(980),
b1(1235), a1(1260) and a2(1320).
In the quark model, they correspond to the partial wave 3 P0, 1 P1,
3 P1 and 3 P2.
Same pattern as for charmonium 1P.
Regge trajectories M 2 vs. J
Linear behaviour reproduced with relativistic kinematics and
V ∝ r
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Heavy-light mesons
Most dangerous sector
Remember: electron more relativistic in H than in Ps
Nevertheless, some properties of the naive quark model applied
to (Q¯q) survive.
Reduced mass
1
µ = 1 1
+
m M
dominated by the light quark.
1
m ,
Thus universal excitation energies and wave functions
One aspect of Heavy quark symmetry
Spin effects ∝ 1/M
D ∗ − D = 2010 − 1870 = 140 MeV
B ∗ − B = 5325 − 5280 = 45 MeV
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Mathematical aspects
Quarkonium physics stimulated studies on the properties of
Schrödinger operators
See Quigg & Rosner, Martin,Bertlmann, Stubbe, Grosse, etc.
Level order
Wave function at the origin
Consequences of flavour independence
With new applications in atomic physics!
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Level order
M (GeV)
10.2
9.8
9.4
-
-
-
3S
2S
1S
2P
1P
1D
Breaking of Coulomb
degeneracy guided by the
sign of ∆V
See alkalin atoms vs. muonic
atoms
Breaking of harmonic
oscillator degeneracy
according to the sign of
JMR Quark Model
d 2 V [r 2 ]
d (r 2 ) 2

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Wave function at the origin
|Φ(0]| 2 governs most decays, in particular leptonic width of
charmonium
Schwinger
pn = |Φn(0)| 2 = 1
4 π u′ n(0) 2 .
u ′ (0) 2 ∞
= 2 µ
0
dV
dr u2 (r) dr .
pn is independent of n for a linear potential
If V ′′ (r) has a given sign, pn ↗ or ↘
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Convexity properties of the spectrum in flavour space
Frequent use of a property of H = A + λ B
The ground-state (or the sum of n first levels) is concave in λ
With λ the inverse reduced mass
(Q ¯ Q) + (q¯q) ≤ 2 (Q¯q) ,
Martin-Bertlmann, Nussinov, Witten, . . .
Consider Vc + σ1.σ2 Vss then
Consider
then
M(Vc) ≥ 1
4 [3 M(Vc + Vss) + M(Vc − 3 Vss]
Vct + Vls(r) ℓ.s + Vt(r) S12 ,
E[Vct] ≥ [EJ=0 + 3 EJ=1 + 5 EJ=2] /9 ,
so we know the sign of the error when treating spin-orbit and
tensor to first order to define a “centre of gravity” of spin-triplet
states.
Important for the interpretation of hc and hb masses.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryons in the quark model
Three quarks in a baryon
a Three makes a company
Content
History
Jacobi coordinates, permutations
The three-body problem
Light baryons, the diquark alternative
Heavy baryons
Spin splittings
Convexity properties
Link between mesons and baryons,
String potential
JMR Quark Model
Tres faciunt collegium a
Latin sentence

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryons: history
Started by Dalitz et al. in the 60s
Many groups, Hey, Kelly, Cutkosky, Stancu, Gromes, Taxil + R.,
Schöberl et al, Guimares, etc., etc.
Best known are Isgur, Karl, Capstick,
The most widely used tool is the harmonic oscillator (HO)
Sometimes difficult to distinguish between nice properties or
difficulties
specific to HO
shared by constituent models
For instance, the location of the Roper resonance! (same
quantum number as the ground state, this generalises the radial
excitation for mesons)
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryons: Jacobi coordinates
q1
x
q2
√ 3y/2
q3
For (qqq)
R = r 1 + r 2 + r 3
,
3
x = r 2 − r 1 ,
y = 2 r 3 − r 1 − r 2
√ 3
For (qqQ), same x and y, R
modified
For (q1q2q3), one should modify
y ∝ (m1 + m2)r 3 − m1 r 1 − m2 r 2
Note: Jacobi coordinates are
convenient, but not compulsory
JMR Quark Model
,

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryons: Jacobi coordinates
From
H =
i
p 2 i
2 mi
+ V
one can remove the c.d.m. free motion and work with the intrinsic
Hamiltonian
h = p2 x
µx
with µx = µy = m for (qqq)
for (qqQ)
+ p2 y
µy
+ V (x, y) ,
µx = m , µ −1
y = (m −1 + 2 M −1 )/3 .
More involved but straightforward for (q1q2q3)
Again not necessary if you use variational methods 〈Ψ|H|Ψ〉 with
Ψ translation invariant.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Permutation symmetry for (qqQ)
For instance Ξ − = (ssd) or Λ = (uds) in the limit where SU(2) is
exact
Ψ = ψ(x, y) ψs ψi ψc ,
should be antisymmetric (A), given that ψc is A,
For instance Λ ground state has I = 0 (A), and Sqq = 0 (A), while
ψ(x, y) is symmetric (S) in x,
For instance, ψ(x, y) ∝ exp[−a x 2 − b y 2 ] in HO.
First orbital excitation of Λ? Keep I = 0. If ψ(x, y) is excited in y,
i.e., ℓy = 1, then keep Sqq = 0, thus Sqqs = 1/2 and two
possibilities
J = 1/2
J = 3/2
with the possibility of spin-orbit splitting among them
Other orbital excitation of Λ? Yes, with ψ(x, y) now odd in x, and
thus Sqq = 1, and various recoupling for Sqqs and J.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Permutation symmetry for (qqQ)
For Ξ − = (ssd) or Σ 0 = (uds) with I = 1, this is inverted, the
ground state has S12 = 1, with two possibilities, J = 1/2 or
J = 3/2, and the possibility of hyperfine splitting.
In the early days of SU(3), the mass difference between Σ 0 and
Λ was a difficulty
In the explicit quark model, it is understood by
spin 1 for (ud) in Σ 0
spin 0 for (ud) in Λ
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Permutation symmetry for (qqq)
Again
Ψ = ψ(x, y) ψs ψi ψc ,
For the ground state of ∆ ++ = (uuu) or Ω − = (sss), this is easy,
each factor is either S or A, where S now means “fully
symmetric” and A “fully antisymmetric”
For the nucleon, one has to introduce the concept of “mixed
symmetry”
The prototype is given by the Jacobi coordinates
x = r 2 − r 1 , y = 2 r 3 − r 1 − r 2
√ 3
Odd or even under P12, but ( j = exp(2 i π/3))
P→[y + i x] = j [y + i x] ,
JMR Quark Model
,

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Permutation symmetry for (qqq)
The Clebsch–Gordan rules for two mixed-symmetry doublets
z = v + i u and Z = V + i U are
ℜe[z Z ∗ ] = u U + v V ,
ℑm[z Z ∗ ] = v U − u V ,
[z Z ] ∗ = (u U − v V ) − i (u V + v U) .
So SM × SM → S, A, or SM.
In particular, the coupling of three spins 1/2 to spin 1/2, with, say
S3 = +1/2 is
Sx = 1
√ 2 [↑↓↑ − ↓↑↑] , Sy = 1
√ 6 [2 ↑↑↓ − ↑↓↑ − ↓↑↑] ,
is completely analogous to (x, y) and form a SM doublet,
So do the isospin wave function for
(1/2) × (1/2) × (1/2) → (1/2)
In the nucleon, the spin–isospin wave function is symmetric
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Permutation symmetry for (qqq)
Most remarkable is the possibility of antisymmetric spin–isospin
wave function
ψx Sx + ψy Sy
√
2
,
Which requires an antisymmetric orbital wave function,
For instance, in the HO
x × y exp[−a(x 2 + y 2 )]
, with ℓ P = 1 + . This state is excited in both coordinates. It has
not yet been seen.
See the discussion on the diquark alternative
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The three-body problem
Several methods, developed earlier in atomic or nuclear physics
HO expansion, Gaussian expansion and other variational
methods
Integro-differential equations: Faddeev, AGS, etc.,
Hyperspherical expansion,
etc.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The harmonic oscillator (HO)
V =
2 K
3
H(m, m, m) = p2 x
m + p2 y
2
r12 + r 2 23 + r 2
31
m + K (x 2 + y 2 ) ,
K
E = (6 + 4 nx + 2ℓx + 4 ny + 2 ℓy)
m
N = 2 nx + ℓx + 2 ny + ℓy)
Levels named after the multiplicity and ℓ P ,
For instance [56, 0 + ] for the ground state with 8 spin 1/2 and 10
spin 3/2, i.e., 2 × 8 + 4 × 10 = 56 states. [56, 0 + ] ′ for Roper.
The first orbital excitation is [70, 1 − ],
the first state with a full antisymmetric orbital wave function is
[20, 1 + ].
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
The harmonic oscillator (HO): (qqQ), (q1q2q3)
H(m, m, M) = p2 x
m + p2 y
µ + K (x 2 + y 2 ) ,
with µ given earlier. Still an exact decoupling,
K
E(m, m, M) =
m (3 + 4 nx
K
+ 2 ℓx) +
µ (3 + 4 ny + 2 ℓy) .
For (q1q2q3), use Jacobi coordinates, and rescale
.
x → x/ √ µx y → y/ √ µy
H(m1, m2, m3) = p 2 x + p2 y + A x 2 + B y 2 + 2 C x y ,
One is left with a 2 × 2 diagonalisation.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Perturbation around HO
0.2∆
0.2∆
0.1∆
0.5∆
V = K (x 2 + y 2 ) + δV ,
One often violates the rules of perturbation theory (δE small as
compared to initial spacings)
Nevertheless, interesting phenomenology,
For instance, hierarchy of N = 2 states
[20, 1 + ]
[70, 1 + ]
[70, 0 + [56, 2
]
+ ]
[56, 0 + ] ′
Except that one would like to push
the lowest state below N = 1!
[56, 0 + ] ′ becomes decoupled with
three body forces (Gromes et al.)
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Converged variational methods
Problems with
More fashionable
Ψ(x, y) =
cn φn(x, y) ,
n
Ψ(x, y) =
γi exp[−(a.i x 2 + biy 2 + 2 ci x.y)] .
i
with restoration of permutation symmetry.
Detailed search of parameter delicate. see Varga et al. or Hiyama et
al.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Hyperspherical expansion
Consider {x, y} as a single 6-d vector
Solve the 6-d Schrödinger equation with a potential which is not
6-d central
Except HO, which is 6-d central
−u ′′ + 3/2)(L + 5/2)
[L] (ϱ)+(L
ϱ2 u [L]+
V [L],[L] ′(ϱ) u [L] ′(ϱ) = E u [L](ϱ) ,
[L] ′
Very good convergence, very systematic
Hypercentral approximation, see Genoa group
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Light-quark baryons
An impressive description of many data with a simple tool,
But some persisting problems,
Let us concentrate on two of them
The Roper resonance = radial excitation of N or ∆
In most models, predicted above the orbital excitations with
P = −1
Similar to ψ ′ > χJ
This is unavoidable with models with ∆V (r) > 0
One suggestion: Yukawa type of interaction among quarks
(Glozmann)
Missing states, e.g., [20, 1 + ]
Absent or not seen since weakly coupled to usual entrance
channels?
A quark-diquark model has been proposed (Lichtenberg, Torino
group)
And is often revisited (Jaffe-Wilczek, Maiani et al., etc.)
Warning: if D = (qq) taken seriously, do you predict ( ¯ DD) or
(DDD)? New spectroscopy!
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryons with a single heavy quarks
M(cqq)
The domain with most recent discoveries in baryon spectroscopy
(MeV)
2900 -
5/2 +
2700 -
2500 -
1/2− 3/2− 2300 - 1/2 +
Λc
3/2 +
1/2 +
Σc
1/2− 3/2− 3/2 +
1/2 +
1/2 +
Ξc
3/2 +
1/2 +
Ωc
1/2− 3/2− 1/2 +
Λb
1/2 +
3/2 +
Σb
3/2 +
1/2 +
Ξb
1/2 +
Ωb
JMR Quark Model
M(bqq)
(MeV)
- 6200
- 6000
- 5800
- 5600

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryons with a single heavy quarks
(Qqq) central forces governed by reduced masses dominated by
q
Excitation spectrum nearly independent upon Q
See the debate about Ωb = (bss) of D0 vs. CDF and LHCb
Flavour independence is important!
Spin splittings in the light quark sector almost independent of Q
Spin splittings involving Q decreases as 1/M
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryons with two heavy quarks
Perhaps the most interesting of ordinary hadrons
For the price of one, get the two extremes
Heavy–heavy motion like in charmonium
Relativistic motion of a light quark, as in D or B mesons
Often described as [(QQ) − q] in a diquark–quark model or
approximation
But the first excitations are in (QQ)!
Then a Born–Oppenheimer picture looks more suited (Fleck et
al.), as for H2 + in atomic physics
Recently the hierarchy of Q − Q vs. q excitations addressed by
Cohen et al., Roberts et al.,
If (QQ) is frozen, then a new heavy-quark symmetry, linking
double-charm baryons to singly-charmed mesons
Experiment: Positive results at SELEX, negative at FOCUS and
BABAR
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryons with three heavy quarks
The ultimate deal of baryon spectroscopy (Bjorken)
The true baryon analogue of charmonium
For instance,look at the hierarchy of levels and compare to the
prediction of static potential computed on the lattice
Let us dream for the future physicists.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Spin splittings in baryons
Very advertised by DGG, Lipkin et al., Isgur and Karl,
As a possible evidence for QCD within the hostile environment of
confinement
In particular
Vss =
i

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Spin splittings in baryons
Long standing problem of spin-orbit forces
IK declared the abolition of spin-orbit forces in baryons, see also
Reinders,
OK, with noticeable exceptions, in particular, the famous
Λ(1405) − Λ(1520) splitting,
Most widely accepted explanation: nearby ¯ K N threshold.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Convexity properties
(MM) + (mm) ≤ 2(Mm) in two-body problems with a given
potential (flavour independence)
Makes it tempting to conjecture about
(m, m, m ′ ) + (M, M, m ′ ) ≤ 2 (m, M, m ′ ) ,
Generally true, so heavy quarks tend to cluster together
But ∃ conterexamples with sharp (unphysical) potentials and very
large mass ratios
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
From mesons to baryons
Historically disconnected, nuclear physicists working on light
baryons, particle physicists working on heavy quarkonia
If colour-octet exchange
V = 1
2 [v(r12 + · · · ] ,
So-called 1/2 rule
Works reasonably well for a combined phenomenology of
mesons and baryons
Challenged by a string picture
B
C
J
A
V (r 1, r 2, r 3) = b min
J (r1J + r2J + r3J) ,
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
String potential for baryons
Link to past work by Fermat, Torricelli and Napoleon
B ′
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B
A
J
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120◦ A ′
C
C ′
C1
B
JMR Quark Model
A
A ′
A1
B1
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Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Mass inequalities for mesons and baryons
If p is the perimeter and Y the minimal Toricelli path
p
2 ≤ Y ≤ p √ 3 ,
The lower bound is saturated for a flat triangle, the upper one for
an equilateral triangle, thus
For the Hamiltonians
H3 = p2 1
1
+ · · · + V ≥
2 m 2
From the variational principle
V ≥ 1
2 [v(r12) + v(r23) + v(r31)] .
2 p1 2 m + p2
2 + v(r12) + · · · .
2 m
2 M(qqq) ≥ 3 M(q¯q) .
Which becomes inverted with different masses, if M/m large
( ¯ Q ¯ Q ¯ Q) + (qqq) ≤ 3 ( ¯ Qq) ,
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Multiquarks and other exotics
Exotic hadrons
Le plus grand dérèglement de l’esprit,
c’est de croire les choses parce qu’on veut qu’elles soient,
et non parce qu’on a vu qu’elles sont en effet. a
Bossuet
a The biggest disorder of the spirit, it is to believe things because we want that they
are, and not because we saw that they are indeed.
Content
Glueballs, hybrids, molecules
Baryonium
Chromomagnetic binding
Chromoelectric binding
Generalised Steiner-tree potential
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Glueballs and hybrids
Glueballs very fashionable in the 80s,
Constituent models, bag models, and later lattice QCD and QCD
SR
Often non exotic, so can be confused with ordinary mesons
Or mix with ordinary mesons
Present status not very clear
Hybrids sometimes seen as (Q ¯ Qg)
or in the Born–Oppenheimer approach as the second potential
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Charmonium hybrids in the early 80s
Using a variant of the bag model
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Light and heavy hybrids
A candidate with J PC = 1 −+ at BNL (Chung)
Perhaps one of X, Y , Z ?
Many theoretical developments (Close, Barnes, Kuti et al.)
Flux tube model (string vibration)
Predicts decay to excited mesons in a first step
One argument in the 80s: (c¯c) is clean, so any extra state should
be clearly visible
We realise now that the situation is also complicated in this
sector.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Molecules
It is regularly rediscovered that the Yukawa mechanism is not
restricted to nucleons,
Any hadron containing light quark(s) can enter a nuclear-type of
interaction,
Even without! Remember ηc–nucleus attraction sometimes
predicted.
The charm sector is no exception
Törnqvist, Manohar & Wise, Ericson & Karl, Swanson, Close and
Thomas, etc., etc., have noticed a possible long-range attraction
between DD ∗ , D ∗ D ∗ or D ¯ D ∗ or D ∗ ¯ D ∗
Weaker than the proton–neutron potential,
But in
− ∆
1
+ g V (r) = [−∆ + m g V (r)]
m m
what matters is m g for the existence of a discrete spectrum.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Molecules
When the X(3872) was discovered, it was considered as a
success for this approach,
Just at the D ¯ D ∗ threshold!
But some more recent measurements better call for a 2P state of
charmonium, in particular
X(3872) → ψ ′ + γ
X(3872) → J/ψ + γ
> 1 ,
Probably a mixture of (c¯c) 2P and molecule,
But do we have two states, or a single (c¯c) with more higher
Fock components than usual?
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Molecules
We learned to be careful in this sector
In 1975, Iwazaki suggested ψ ′ = (c¯cq¯q)
In 1976–77, Voloshin & Okun, and De Rujùla, Georgi & Glashow
molecular structures out of D (∗) and ¯ D (∗)
In particular, DGG were puzzled by ψ(4.04) decaying too often in
D ∗ ¯ D ∗ relative to D ¯ D and D ¯ D ∗ + c.c., as compared to spin
counting and phase-space.
But Le Yaouanc et al., and Eichten et al. have shown this was
due to the node structure of this state.
We were accustomed to orbital excitations (Regge trajectories),
less to radial excitations
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Molecules
Baryon–baryon states (Julia-Diaz & Riska) with charm ≥ 2?
Perhaps a new periodic table, based on charmed baryons
Meson–baryon
Beauty baryons, etc.
the Pandora-box syndrome strikes again!
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryonium
Tentative peaks in antiproton cross-sections in the 70s
Bumps in the inclusive γ spectrum ¯p + p → γ + X
The name “baryonium” was invented for mesons preferentially
coupled to baryon–antibaryon
Two main approaches
Quasi-nuclear baryonium (Shapiro et al., Dover et al.). Today,
would be named “molecular”
With meson-exchange between N and ¯ N, deduced from NN
interaction by the Fermi–Yang rule (G-parity rule)
Annihilation underestimated in this approach,
[(qq)¯ 3 − (¯q¯q)3] structure, with an orbital-momentum barrier
preventing from rearrangement into mesons (Rossi & Veneziano,
Jaffe, etc.), named T-baryonium by Chan H.M. et al.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Baryonium
Chan et al. invented colour chemistry,
In particular, speculated about [(qq)¯ 6 − (¯q¯q)6], named
M-baryonium, narrow for both decay into mesons and decay into
baryon–antibaryon
Note that the clustering into diquarks with such colour structure
was just assumed, not demonstrated from a dynamical
calculation,
Many followers: exotic baryons with (q 4 ¯q) and similar cluster
structure, dibaryons, etc. (de Swart et al., Sorba et al., Nicolescu
et al., etc.)
New experiments with an intense, cooled antiproton beam at
LEAR (CERN). No baryonium confirmed.
Still some enhancements in Jψ → baryon + antibaryon + · · · at
BES, indicatging a strong final-state interaction
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromomagnetic binding
A by now famous paper by DGG suggested the spin-dependent
part of one-gluon-exchange as responsible for spin–spin
splittings in mesons and baryons
It reads
Vss =
i

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromomagnetic binding
Namely attractive and
〈O〉H = 3 〈O〉Λ ,
Thus with Λ = N = ΣΞ all receiving 150 MeV of attraction from
spin-spin
He deduced that H is bound by about 150 MeV below the
degenerate threshold ΛΛ = NΞ = ΣΣ,
More than 20 experiments looked at the H
No positive signal, in particular from S = −2 hypernuclei
Chromomagnetism is remarkable, as it induces a net excess of
attraction in the Hamiltonian, before considering any induced
polarisation in sub-clusters,
The usual situation is: no excess of attraction
For instance Ps2 vs. 2 (e + e − ), both governed by gij/rij, both
have the same gij = −2
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromomagnetic binding
The H was revisited by several theorists (Yazaki et al. Karl et al.,
Rosner, Gignoux et al., in particular for
SU(3)F breaking
Self-consistent calculation of ¯vss = 〈vss(rij)〉
Inclusion of central forces and spin–spin forces in a consistent
6-body calculation
Each effect reduces the binding
And eventually the H is unbound!
The main effect is that ms ↗ splits ΛΛ from other thresholds, ΛΛ
chromomagnetic energy being not penalised, hence the
coherence ΛΛ + NΞ + · · · → H is lost!
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromomagnetic binding
In 1987, Lipkin, and Gignoux et al. realised that
P = ( ¯ Qqqqq)
with (qqqq) = (uuds) or (udds) or (udss)
has the same 150 MeV binding below the [( ¯ Qq) + (qqq)]
threshold in the limit where mQ → ∞ and same assumptions
than Jaffe for the light quark
This was named “pentaquark” (now, one should say: “the
chromomagnetic pentaquark”)
It was searched for in 1 experiment at Fermilab (Ashery et al.),
not conclusive
A re-analysis, including mQ < ∞, indicate that relaxing the the P
likely becomes unbound
Other configurations analysed, see Leandri et al.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromoelectric binding
If chromomagnetism does not work, why not chromo-electricity?
It works! at least in a certain limit
Miracle: all theorists agree! instead of fighting.
Consider first a simple additive model with colour factors, and
next a better modelling of confinement.
The additive model with colour factors reads
V = − 16
3
˜λi. ˜ λj v(rij) ,
i

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromoelectric binding: equal masses
Both Ps2 and (qq¯q¯q), and their thresholds are governed by
H4 =
i
p2 i
2 m − [ pi ] 2
8 m
+ gij v(rij) ,
gij = 2 ,
For this family of Hamiltonians, the highest ground state obtained
for gij = ¯g = 2/15.
Then, the more one departs from this symmetric case, the lower
the binding
Can be measured by the variance of the {gij} set of coefficients
State Pair 12 34 13 24 14 23 ¯g ∆g
Threshold 0 0 1 1 0 0 1/3 0.22
Ps2 −1 −1 1 1 −1 −1 1/3 0.89
T 1/2 1/2 1/4 1/4 1/4 1/4 1/3 0.01
M −1/4 −1/4 5/8 5/8 5/8 5/8 1/3 0.17
i

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromoelectric binding: equal masses
State Pair 12 34 13 24 14 23 ¯g ∆g
Threshold 0 0 1 1 0 0 1/3 0.22
Ps2 −1 −1 1 1 −1 −1 1/3 0.89
T 1/2 1/2 1/4 1/4 1/4 1/4 1/3 0.01
M −1/4 −1/4 5/8 5/8 5/8 5/8 1/3 0.17
Ps2 is more asymmetric than its threshold: it is stable
Both T-type and M-type of tetraquarks are less symmetric than
their threshold, they are unstable
Hence tetraquark with equal masses is penalised by the
non-Abelian character of the theory
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromoelectric binding: unequal masses
Can another asymmetry overcome this problem?
Yes! Remember (M + , M + , m − , m − ) becomes more stable as
M/m departs from 1
The same mechanism makes (QQ¯q¯q) evolving from unbound to
stable
See Ader et al. (1982), Heller & Tjon, Brink & Stancu, Rosina &
Janc, Barnea, Vijande & Valcarce, etc.
The problem is the critical value of M/m required,
(cc¯q¯q) bound or do we need (bb¯q¯q)
Two b or not two b, that is the question!
And what about a better potential
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromoelectric binding: string potential
The additive model
V = − 16
3
˜λi. ˜ λj v(rij) ,
i

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Chromoelectric binding: string potential
The good surprise is that this potential gives better binding than
the simple additive model,
Hence, (cc¯q¯q), marginally bound in the additive model, should
be stable with this improved quark dynamics
Hence, beyond double-charm baryons, one could look at double
charm mesons, a genuine exotic,
For instance, since e + e − → J/ψ + ηc is observed (double charm
production), TQQ + ¯ D + ¯ D + · · · could be observed.
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
String potential: some rigorous results
w12
c12
v1
p
v2
s1
t12
t34
s2
v3
h k
q
v4
c34
JMR Quark Model
w34

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
String potential: some rigorous results
v2
w12
C12
v1
s1
s2
v4
w34
C34
v3
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
String potential: some rigorous results
√
3
Vs ≤ a (x + y)
2 + z√
2 ,
H4 ≤ p2x m + p2y m + p2z + a
m
(x + y)
√
3
2 + z√
2 ,
Stability demonstrated analytically for M/m 6402! (Ay et al.)
q
q
x y
√ 2z
¯q
¯q
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
String potential: pentaquark
Found stable if antisymmetrisation is disregarded
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
String potential: hexaquark
Found stable if antisymmetrisation is disregarded
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
String potential: hexaquark
Found stable if antisymmetrisation is disregarded
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Light pentaquark
Speculation by Diakonov et al.
Indication by Nakano et al.
Confirmed by other groups having data on tapes, never analysed
for such search,
Eventually not confirmed by high-statistics experiments
Also doubts among theorists, in particular about the small width
in this model,
Situation somewhat confusing in constituent models, QCD SR,
and lattice QCD calculations trying to reproduce the light
pentaquark
JMR Quark Model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook
Outlook
Long way from strangeness, SU(3) symmetry, intriguing decay
pattern of the φ(1020) to the present state of art in QCD
Hadron spectroscopy boosted by heavy quarks
The question of exotics remain puzzling,
But some configurations have not yet been investigated
experimentally
JMR Quark Model