an introduction to the quark model

Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook

An introductionto 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 LyonUniversité Claude Bernard (Lyon 1)–IN2P3-CNRS

Villeurbanne, France

Ferrara, Italy,May 2012

JMR Quark Model


Table of contents

Content1 Prelude: Few-charge systems2 Mesons as (qq)

3 Baryons as (qqq)

4 Multiquarks

JMR Quark Model


More detailed table of contents1 Few-charge systems

Binary atoms: central potentialSpin forces in atomsThree-body ionsFour-body molecules (+,+,−,−)

2 History of the quark modelEarly hadronsGeneralised isospin, SU(3)Heavy quarks

3 The quark–antiquark model of mesonsQuantum numbersSpin averaged spectrumImprovements

4 Baryons5 Multiquarks and other exotics

Glueballs, hybrids, moleculesBaryonium

6 Outlook

JMR Quark Model


Few-Charge systems

Why few-charge systems in lectures about quarks?

Si parva licet componere magnis a

Virgil

aif it is allowed to compare small things with great

Content1 Binary atoms2 Spin-dependent forces3 Three-body ions4 (+,+,−,−) molecules

JMR Quark Model


Binary atoms

p21

2 m1+

p21

2 m1− e2

r12,

The centre of mass motion can be removed,The intrinsic Hamiltonian

H =p2

2µ− e2

r,

can be rescaled toh = −∆− r−1 ,

with 2µe4 for E and (2µe2)−1 for r .Similarly, any oscillator can be reduced to −d/dx2 + x2.

JMR Quark Model


Binary atoms-2

Very characteristic spectrum

E = − 14 (n + `)2 ,

where n = 1, 2, . . . is the radial number, ` = 0, 1, . . . the orbitalmomentum, and n + ` 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


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 NRHamiltonians, and treated as perturbation,In particular, the famous hyperfine correction

Vss =e2

m1 m2

2π3δ(3)(r)σ1.σ2 ,

splits ortho- and para-hydrogen (important transition inastrophysics; analogue to be measured in antihydrogen),Note the short-range character,Note the very specific mass dependenceFor (e+,e−), ∃ other contribution

JMR Quark Model


Three-body ions

He (α,e−,e−) is obvious, as any (+q,−,−) with q > 1q = 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


Three-body ions – excitations

H2+ = (e−,p,p) has several excited states,

H− has no stable excited state?Both true and falseTrue 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 6= m′.

JMR Quark Model


Four-body molecules

Hydrogen molecule and variants best known, (p,p,e+,e−)

In the Born–Oppenheimer–Heitler–London approach, effectivepp potential, which gives the ground-state and the firstexcitations. This is a very good approximation,This corresponds to the two electrons in the lowest state forgiven 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 elegantproof of the stabilityIn 2007, indirect experimental evidence

JMR Quark Model


Systematics of (m+1 ,m

+2 ,m

−3 ,m

−4 )

Any state with m3 = m4 is stable. Why? Two degeneratethresholds.In particular, (m+,m+,m−,m−) is stable (positronium moleculeand variants)What about two masses?(M+,M+,m−,m−) improves stability.(M+,m+,M−,m−) spoils stability. It becomes unstable forM/m & 2.2 (or, of course, M/m . 1/2.2)So, starting from the doubly-symmetric (m+,m+,m−,m−), andbreaking

Charge conjugation,or Particle identity

does not produce the same result. Why?Both ways of breaking symmetry lower the ground state. But inthe second case, the threshold benefits more of symmetrybreaking. Hence, one gets less stability. See the section onmultiquarks.

JMR Quark Model


Some lessons from charge systems

Universality, same potential for p and e+

Scaling,Level order for atoms, very specific3- or 4-body systems stable or unstable, depending on themasses,Be patient. One could wait up to 60 years to see an exotic statethat is predicted.

JMR Quark Model


History

History of the quark model

History is Philosophy teaching by examplesa

Thucydid

aAccording to Michel Casevitz, the sentence is not by Thucydid, but a Britishcommentator

ContentEarly hadronsSU(3)Quarks and AcesHeavy quarks

JMR Quark Model


Very first hadrons

Discovery of the neutron (Chadwick, see, also Joliot-Curie)Need for a strong interaction between nucleonsSearch 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 of2-body to 3-body energies (Thomas),Pion discovered in 1947 at Bristol, with three charge states, π+,π0 and π−, not so easily as their decay are not the same,Thus in 1947, 7 hadrons seen or expected, p, n, π+, π0 and π−,and also p and n predicted.

JMR Quark Model


First hadrons: antinucleons

Bevatron built at Berkeley for antinucleonsp 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


First complications: resonances

New baryons: ∆, N∗, etc., in πN scattering,New mesons: ρ, ω, σ produced, and also required, and evenanticipated 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 equationsDifficulty 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


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


From SU(2) to SU(3)

SU(2) is a good symmetry for nuclear physics and pionscattering,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 pseudoscalarmesons,This led to extend SU(2) to SU(3)and imagine that breaking can be described as linear or at mostquadratic in strangeness,Today, this symmetry, renamed SU(3)F , remains a very usefulconcept,But first, one needs to assign the hadrons into representations

JMR Quark Model


The Sakata model

(p,n,Λ) in the fundamental representation, 3,(p, n, Λ) in 3,Mesons from 3× 3Higher baryons from 3× 3× 3, etc.But, as seen for the realisation of bootstrap, it faces seriousdifficulties (exchange degeneracy, why Σ is almost as light as Λ?,etc.)

JMR Quark Model


The Eightfold way

This led Gell-Mann and Ne’emann to propose to put the 8 lowestbaryons with spin 1/2 in an octet,

I3

Y

•Σ− •Σ0 •

Σ+•Λ

•Ξ−

•Ξ0

•n •p

JMR Quark Model


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

2(N + Ξ) = 3 Λ + Σ ,

and many similar ones

JMR Quark Model


The prediction of the Ω−

9 baryons with J = (3/2)+ were known below 2 GeV. At TheRochester Conference of 1962 in Geneva, Gell-Mann predicted anew one with strangeness −3

I3

Y

•Σ∗−

•Σ∗0 •

Σ∗+

•∆− •∆0 •∆++•∆+

•Ξ∗− •Ξ∗0

•Ω−

Discovered by Samios et al. at Brookhaven at the end of 1963and published in 64.

JMR Quark Model


Mesons in SU(3)

Octet and singlet, with mixingOne uses to talk about nonet

I3

Y

•π−

•π0 •

π+•η•η′

•K−

•K0

•K0 •K+

I3

Y

•ρ−

•ρ0

•ρ+

•ω•φ

•K∗−

•K∗0

•K∗0 •K∗+

JMR Quark Model


The fundamental representation: quarks

I3

Y

•d • u

•s

I3

Y

• d•u

•s

q b I I3 Y S Q

u 13

12

12

13 0 2

3

d 13

12 − 1

213 0 − 1

3

s 13 0 0 − 2

3 −1 23

JMR Quark Model


The aces

Zweig was puzzled by φ(1020) decayMostly into K K in spite of very little phase-space

φπ

ρ

φπ

ρ

φ

K

K

He interpreted as due to the content of the Φ and of final-statemesons, he named “aces”,But eventually the name “quark” prevailed, and here the notation(u,d , s), rather than (p,n, λ).

JMR Quark Model


The Zweig rule (OZI, A-Z)

the rule explaining the narrowness of φ generalised, withvariants,e.g., NN annihilation

JMR Quark Model


First quark models

For baryons, mostlyGreenberg, 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 colourIndeed, their ∆−(ddd) has s = 3/2, L = 0, thus everything issymmetric for three fermions!see chapter on baryons

JMR Quark Model


Heavy quarks

Kaon physics always rich,θ − τ puzzle, P violation, C violationCP 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 particularGaillard, Lee and Rosner,Including (cq), (cqq), (ccq), . . . and (cc)

JMR Quark Model


October 1974 revolution

In November 1974, the J/ψ was discovered simultaneously atBNL and SLAC, in quite different experimentsLepton-pair production in hadronic collisions (Ting)e+e− collisions (Richter)Not recognised immediately, since extremely narrow,Eventually identified as (cc)

Several other states (ψ′, χ, . . . ) seenCharmed mesons seen also (G. Goldhaber)As well as charmed baryonsNote: double-charm baryons not yet confirmed!Beautiful confirmation of the charm predictionAnd asymptotic freedom, which make the Zweig rule moreeffective for J/ψ than for φ.

JMR Quark Model


Charmonium

Simple models proposed for (cc) that work!A revolution in strong interaction. Predictions with simple tools!For instance V (r) = −a/r + b r + c and mass mc in theSchrödinger equation reproduce the experimental spectrumAnd properties such as leptonic widths and gamma transitionsMany 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 QCDSee section on mesons

JMR Quark Model


Top and Bottom

When charm was discovered, the ideas were already ratheradvanced on grand unification,At least quark–lepton symmetryNote: leptons always ahead,When the µ was discovered, Rabbi said: “Who ordered themuon?”When the τ was discovered (M. Perl), it was said: “ Where arethe associated quarks?”τ, ντ ↔ b, t“Bottom, Top”Also “Beauty, Truth”

JMR Quark Model


Upsilon discovery

In 1977, Lederman repeated Ting’s experiment with a morepowerful beam and another detector,And discovered Υ and Υ′

Immediately interpreted as (bb)

See chapter on mesons,Already Lederman noticed Υ′ −Υ ' ψ′ − J.ψ,And asked local theorists about a potential such that all ∆E areindependent of the reduced mass,Answer: V (r) ∝ ln rB mesons and B baryons also discovered,As well as Bc = (bc)

JMR Quark Model


Exotic hadrons

Already before the quark model,For instance, speculations about a Z baryon with S = +1Within the quark model, exotic = state that cannot beaccommodated as (qq′), or (qq′q′′).For instance meson with charm = +2, or baryons with S = +1Besides flavour?

for mesons, ∃ exotic JPC

not for baryons

question of best beam and target:e+e− clean but with some restrictionp annihilationformation or production,low energy or high energy

past or recent excitations: baryonium, glueballs, hybrid hadrons,molecules

JMR Quark Model


Mesons: Content

The quark–antiquark model of mesons

I married themFriar Laurence, Romeo and Juliet

ContentQuantum numbersSpin-averaged spectrumImprovementsSummary for heavy quarkoniaLight mesonsHeavy-light mesonsStrong decaySome mathematical developments

JMR Quark Model


Quantum numbers

Consider symmetric quarkonia QQSpin of quarks S, orbital momentum `, spin of the meson J,parity P and charge conjugation C

Lowest quarkonium states2 s+1`J

1S03S1

1P13P0

3P13P2

1D23D1

3D23D3

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

Remarks

Some quantum numbers are absent, e.g., JPC = 1−+

Some JPC occur twice. For instance 1−− may be a combinationof 3S1 and 3D1

In addition, radial number. Here, notation n = 1, 2, . . .. Forinstance,

ηc is ηc(1S) or 1 1S0

η′c is ηc(2S) or 2 1S0

JMR Quark Model


Radial equation

Assume a simple V (r) without spin dependence, and let

Ψ =u`(r)

rY `z` (r)× spin× colour ,

Thus for u = u`(r) (no dependence upon `z)

−u′′(r) +`(`+ 1)

r2 u(r) + m V (r) u(r) = m E u(r) ,

with boundary conditions u(0) = u(∞) = 0Exactly solvable in a few cases

Coulomb, see section on atomsHO, reduces to −u′′(r) + `(`+ 1) u(r)/r 2 + r 2 u(r) = ε u(r) , withε = 3 + 4 (n − 1) + 2 ` = 3 + 2 NLinear for ` = 0

JMR Quark Model


Linear potential for ` = 0

−u′′(r) + r u(r) = εu(r) ,

very similar to the Airy equation

−y ′′(x) + x y(x) = 0

- 6 - 4 - 2 2 4x

- 0.4

- 0.2

0.2

0.4

AiHx L

ε = −zero of the Airy function = −an

vn(r) = Ai(r + an)/Ai′(an) .

JMR Quark Model


Scaling

Note the simple scaling properties

−u′′(r) +`(`+ 1)

r2 u(r)±m g rα u(r) = m E u(r) ,

−u′′(r) +`(`+ 1)

r2 u(r)± rα u(r) = εu(r) ,

with the scaling in (m g)1/(2+α) for the distances, andm−α/(2+α) g2/(2+α) for the energies.For a logarithmic potential, m→ m′ gives En,` → E ′n,` = En,` + Cst

The Coulomb-plus-linear −a/r + b r + c can be reduced to−∆− λ/r + r − εψ(r) = 0 , with only one parameter.

JMR Quark Model


Solving the radial equation

−u′′(r) +`(`+ 1)

r2 u(r) + m V (r) u(r) = m E u(r) ,

See, e.g., Hartree, where V (r) was the effective one-bodypotential,Integrate inwards, and outwards, and fix E by imposing continuityof both u and u′ at the matching point,Or discretise, and solve an approximatively equivalent matrixeignevalue equation,Or use a variational method, e.g.,

u(r) =∑

i

Ci r `+1 exp(−ai r2/2) ,

JMR Quark Model


Simplest quarkonium model

“Funnel” potential V (r) = −a/r + b r + cConstituent mass mc

Minimal adjustment mc ∼ 1.5, a ∼ 0.4, b ∼ 0.2, and c ∼ −0.35(all units in powers of GeV)

1 2 3 4 5 6 7r

- 3

- 2

- 1

1

2

uHr L,V Hr L

Reproduces also (bb) with mb ∼ 4.5 GeV

JMR Quark Model


Simplest quarkonium model

cc bb1S 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 10.23exp. 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


Simplest quarkonium model (cc) = dotted lines

1S

2S

3S

1P

2P

1D

M (GeV)

-9.4

-9.8

-10.2

M (GeV)

-

-

-

9.4

9.8

10.2

1S

2S

3S

1P

2P

1D

-

-

-

3.00

3.40

3.80

JMR Quark Model


Improvements: central potentials

Better potentials

Ng+Tye, Buchmüller, Richardson, etc., include asymptoticfreedom

−ar→ −a(r)

rSchnitzer, . . . , Gonzalez et al., . . . use a softer confinement, dueto pair-creation effects,etc.Better simultaneous fit of (bb) and (cc)

Simpler potentials

V (r) = g ln r + C (Quigg + Rosner, etc. )V (r) = A rα + B (Martin)as α→ 0 becomes logarithmic

JMR Quark Model


Improvements: relativistic corrections

p2

2 m→√

m2 + p2 −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


Improvements: loop corrections

(cc) (cc)

D(∗)

D(∗)

c

c

c

cq

q

Provide the state above threshold a width,Can be calculated using the 3P0 model, and an overall of initialand final wave functionsGive a mass-shift (dispersive part)One expects many cancellations. For instance, if D = D∗, all 3PJstates receive the same shift. But if D∗ > D, then differential shiftin addition, thus mimicking spin-orbit and tensor forces.Should be more pronounced near a threshold,see below ψ′ − η′cIf too large an effect, back to the bootstrap?

JMR Quark Model


Improvements: fine structure

ψ′ → χJ + γ , χJ → J/ψ + γ ,

Masses accurately measured with antiprotonsanalysed with

V (r) + Vss(r)σ1.σ2 + Vls(r) `.s + Vt (r) S12 ,

to first order

M(3P0) = Mt − 2 〈Vls〉 − 4 〈Vt〉 ,

M(3P1) = Mt − 〈Vls〉+ 2 〈Vt〉 ,

M(3P2) = Mt + 〈Vls〉 −25〈Vt〉 ,

Mt =19

[M(3P0) + 3 M(3P1) + 5 M(3P2)

],

〈Vls〉 =1

12

[−2 M(3P0)− 3 M(3P1) + 5 M(3P1)

],

〈Vt〉 =5

72

[2 M(3P0)− 3 M(3P1) + M(3P1)

].

JMR Quark Model


Improvements: fine structure

1P of (cc)

Mt = 3.525 , 〈Vls〉 = 0.035 , and 〈Vt〉 = 0.010 GeV .

1P level of (bb),

Mt = 9.900 , 〈Vls〉 = 0.014 , and 〈Vt〉 = 0.003 GeV ,

2P

∗Mt = 10.260 , 〈Vls〉 = 0.009 , and 〈Vt〉 = 0.002 GeV . (1)

JMR Quark Model


Improvements: hyperfine structure

Singlet governed by Vc − 3 Vss

Triplet by Vc+,Vss

So, in perturbation, 3S1 − 1S0 = 4 〈Vss〉SAnd for P states, 3Pm − 1P1 = 4 〈Vss〉PWhere 3Pm is an average spin-triplet, or say, a fictitiousspin-triplet free of spin-orbit and tensor.To first order

3Pm = [M(3P0) + 3 M(3P1) + 5 M(3P2)]/9 ,

but if spin-forces are treated non perturbatively,

3Pm≥ [M(3P0) + 3 M(3P1) + 5 M(3P2)]/9 ,

JMR Quark Model


The shaky history of spin singlets

In 1977, a candidate for ηc claimed in Germany 300 MeV belowthe J/ψ,→ a lot of excitationtentatively explained by relativistic dynamics,difficult to digest in most current models,not confirmed at SLAC,and eventually found about 120 MeV below J/ψThen η′c = ηc(2S) predicted about 70–80 MeV below ψ′

However, it was pointed out that loop effects tend to decreasethis splitting substantiallyIntense search with antiprotons at Fermilbab, but in a range oftoo low masses,Eventually found at Belle, Cleo, etc. about 50 MeV below ψ′

Interesting process of double-charm production

JMR Quark Model


Spin singlets: P wave

Even harder for 1P1, as anticipated (Renard in the late 70s)First indication in ISR: cooled p on jet targetResisted for a while formation in a better p beam at FermilabThen seen in several experimentshc = 1P1 almost coincides with the naive centre of gravity oftriplets,Probably due to the cancellation of several small effects.

JMR Quark Model


Spin singlets: (bb)

Some results are very recent

δ(1S) = 0.067 , hb(1P) = 9.898 ,δ(2S) = 0.049 , hb(2P) = 10.260 GeV .

Two remarks1 δ(1S)bb predicted to be 70± 9 MeV from Lattice If you cannot

afford Lattice QCD, use a logarithmic potential,

δ(1S)bb = δ(1S)cc

(mb

mc

)−1/2

∼ 65 MeV .

2 The ratio δ(2S)/δ(1S) is about 1.4 in (bb) and about 2.3 in (cc).This illustrates how anomalously high is ηc(2S) — oranomalously low is ψ′ — due to the neighbouring threshold.

JMR Quark Model


Orbital mixing

Except 3P0, the states with unnatural parity contain two partialwaves,For instance 3S1 and 3D1 for the states formed in e+e−

ψ =u(r)

r|3S1〉+

w(r)

r|3D1〉 ,

−u′′(r)

m+ Vc(r) u(r) +

√8 Vt (r) w(r) = E u(r) ,

−w ′′(r)

m+

[6

m r 2 + Vc(r)− 3 Vls(r)− 2 Vt (r)

]w(r) +

√8 Vt (r) u(r) = E u(r)

See nuclear-physics textbooksFor instance

ψ(3770) = a|3D1,n = 1〉+ b1|3S1,n = 1〉+ b2|3S1,n = 2〉+ · · ·

|b2| |b1|? Not sure!Contributions of J/ψ ↔ D(∗)D(∗) ↔ ψ′′

JMR Quark Model


The origin of spin dependent forces

Much discussed at the beginning of charmoniumThen discussed by Lattice QCD (Rebbi et al., etc.)And other non perturbative methods (Brambilla et al.)Early approaches inspired by

QEDOne-boson-exchange picture of nuclear forces

Scalar exchange→ central, and spin-orbit,Vector exchange→ central, spin-spin, tensor and spin-orbitEarly models: vector linked to 1/r and scalar linked toconfinement,One should be careful: some terms come from the reduction ofDirac operators in terms of Pauli spinors, other come from thenon-relativistic reduction (Thomas precession)It took some time to get a consistent picture compatible withLorentz invariance (Gromes, Eichten-Sucher, etc.)

JMR Quark Model


Summary for quarkonium

(2S)ψ

γ∗

ηc(2S)

ηc(1S)

hadrons

hadrons hadrons

hadrons

radiative

hadronshadrons

χc2(1P)

χc0(1P)

(1S)ψJ/

=JPC 0−+ 1−− 0++ 1++ 1+− 2++

χc1(1P)

π0

γ

γ

γ

γγ

γ

γγ∗ hc(1P)

ππη,π0

hadrons

JMR Quark Model



=

BB threshold

(4S)

(3S)

(2S)

(1S)

(10860)

(11020)

hadrons

hadrons

hadrons

γ

γ

γ

γ

ηb(3S)

ηb(2S)

χb1

(1P)χb2

(1P)

χb2

(2P)

hb(2P)

ηb(1S)

JPC 0−+ 1−− 1+− 0++ 1++ 2++

χb0

(2P)χb1

(2P)

χb0

(1P)hb

(1P)

JMR Quark Model



(2S)ψ

γ∗

ηc(2S)

ηc(1S)

hadrons

hadrons hadrons

hadrons

radiative

hadronshadrons

χc2(1P)

χc0(1P)

(1S)ψJ/

=JPC 0−+ 1−− 0++ 1++ 1+− 2++

χc1(1P)

π0

γ

γ

γ

γγ

γ

γγ∗ hc(1P)

ππη,π0

hadrons

=

BB threshold

(4S)

(3S)

(2S)

(1S)

(10860)

(11020)

hadrons

hadrons

hadrons

γ

γ

γ

γ

ηb(3S)

ηb(2S)

χb1

(1P)χb2

(1P)

χb2

(2P)

hb(2P)

ηb(1S)

JPC 0−+ 1−− 1+− 0++ 1++ 2++

χb0

(2P)χb1

(2P)

χb0

(1P)hb

(1P)

JMR Quark Model



Y (3940) Z(3930)X(3940)

ωJ/ψDD

DD∗

??

?

@@RππJ/ψ

@@Rππψ′

?π±ψ′

Z±(4430)4320÷ 4360

?

q

@@@

@@@@R

N

9

=

JPC : 0−+ 1+− 1−− 0++ 1++ 2++ ?

ηc

η′c

hc

J/ψ

ψ′

χc0

χc1

χc2

ψ(3770)

X(3872)

ψ(4040)

ψ(4170)

Y (4260)

γγ

γ

γγγγ

γ

ππη

π0

π0

γ

π+π−J/ψπ+π−π0J/ψ

DD

DD∗DsDs

D∗D∗

DsD∗s

D∗sD

∗s

3.0

3.5

4.0

M GeV

?ππJ/ψηJ/ψ

JMR Quark Model


Light mesons

In principle, the quark model not applicableThough this was attempted!Two examples:First positive-parity excitations of (qq) with I = 1 are a0(980),b1(1235), a1(1260) and a2(1320).In the quark model, they correspond to the partial wave 3P0, 1P1,3P1 and 3P2.Same pattern as for charmonium 1P.Regge trajectories M2 vs. JLinear behaviour reproduced with relativistic kinematics andV ∝ r

JMR Quark Model


Heavy-light mesons

Most dangerous sectorRemember: electron more relativistic in H than in PsNevertheless, some properties of the naive quark model appliedto (Qq) survive.Reduced mass

1µ

=1m

+1M' 1

m,

dominated by the light quark.Thus universal excitation energies and wave functionsOne aspect of Heavy quark symmetrySpin effects ∝ 1/MD∗ − D = 2010− 1870 = 140 MeVB∗ − B = 5325− 5280 = 45 MeV

JMR Quark Model


Mathematical aspects

Quarkonium physics stimulated studies on the properties ofSchrödinger operatorsSee Quigg & Rosner, Martin,Bertlmann, Stubbe, Grosse, etc.Level orderWave function at the originConsequences of flavour independenceWith new applications in atomic physics!

JMR Quark Model


Level order

1S

2S

3S

1P

2P

1D

M (GeV)

-9.4

-9.8

-10.2

Breaking of Coulombdegeneracy guided by thesign of ∆VSee alkalin atoms vs. muonicatomsBreaking of harmonicoscillator degeneracyaccording to the sign of

d2V [r2]

d (r2)2

JMR Quark Model


Wave function at the origin

|Φ(0]|2 governs most decays, in particular leptonic width ofcharmonium

pn = |Φn(0)|2 =1

4πu′n(0)2 .

Schwinger

u′(0)2 = 2µ∫ ∞

0

dVdr

u2(r) dr .

pn is independent of n for a linear potentialIf V ′′(r) has a given sign, pn or

JMR Quark Model


Convexity properties of the spectrum in flavour space

Frequent use of a property of H = A + λBThe ground-state (or the sum of n first levels) is concave in λWith λ the inverse reduced mass

(QQ) + (qq) ≤ 2 (Qq) ,

Martin-Bertlmann, Nussinov, Witten, . . .Consider Vc + σ1.σ2 Vss then

M(Vc) ≥ 14

[3 M(Vc + Vss) + M(Vc − 3 Vss]

ConsiderVct + Vls(r) `.s + Vt (r) S12 ,

thenE [Vct ] ≥ [EJ=0 + 3 EJ=1 + 5 EJ=2] /9 ,

so we know the sign of the error when treating spin-orbit andtensor to first order to define a “centre of gravity” of spin-tripletstates.Important for the interpretation of hc and hb masses.

JMR Quark Model


Baryons in the quark model

Three quarks in a baryon

Tres faciunt collegiuma

Latin sentenceaThree makes a company

ContentHistoryJacobi coordinates, permutationsThe three-body problemLight baryons, the diquark alternativeHeavy baryonsSpin splittingsConvexity propertiesLink between mesons and baryons,String potential

JMR Quark Model


Baryons: history

Started by Dalitz et al. in the 60sMany 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 ordifficulties

specific to HOshared by constituent models

For instance, the location of the Roper resonance! (samequantum number as the ground state, this generalises the radialexcitation for mesons)

JMR Quark Model


Baryons: Jacobi coordinates

b

q1

b

q2

b q3

b

x√3y/2

For (qqq)

R =r1 + r2 + r3

3,

x = r2 − r1 ,

y =2 r3 − r1 − r2√

3,

For (qqQ), same x and y , RmodifiedFor (q1q2q3), one should modifyy ∝ (m1 + m2)r3 −m1 r1 −m2 r2

Note: Jacobi coordinates areconvenient, but not compulsory

JMR Quark Model


Baryons: Jacobi coordinates

From

H =∑

i

p2i

2 mi+ V

one can remove the c.d.m. free motion and work with the intrinsicHamiltonian

h =p2

x

µx+

p2y

µy+ V (x ,y) ,

with µx = µy = m for (qqq)

for (qqQ)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〉 withΨ translation invariant.

JMR Quark Model


Permutation symmetry for (qqQ)

For instance Ξ− = (ssd) or Λ = (uds) in the limit where SU(2) isexact

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), whileψ(x ,y) is symmetric (S) in x ,For instance, ψ(x ,y) ∝ exp[−a x2 − b y2] in HO.First orbital excitation of Λ? Keep I = 0. If ψ(x ,y) is excited in y ,i.e., `y = 1, then keep Sqq = 0, thus Sqqs = 1/2 and twopossibilities

J = 1/2J = 3/2

with the possibility of spin-orbit splitting among themOther orbital excitation of Λ? Yes, with ψ(x ,y) now odd in x , andthus Sqq = 1, and various recoupling for Sqqs and J.

JMR Quark Model


Permutation symmetry for (qqQ)

For Ξ− = (ssd) or Σ0 = (uds) with I = 1, this is inverted, theground state has S12 = 1, with two possibilities, J = 1/2 orJ = 3/2, and the possibility of hyperfine splitting.In the early days of SU(3), the mass difference between Σ0 andΛ was a difficultyIn the explicit quark model, it is understood byspin 1 for (ud) in Σ0

spin 0 for (ud) in Λ

JMR Quark Model


Permutation symmetry for (qqq)

AgainΨ = 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 “fullysymmetric” and A “fully antisymmetric”For the nucleon, one has to introduce the concept of “mixedsymmetry”The prototype is given by the Jacobi coordinates

x = r2 − r1 , y =2 r3 − r1 − r2√

3,

Odd or even under P12, but ( j = exp(2 i π/3))

P→[y + i x ] = j [y + i x ] ,

JMR Quark Model



The Clebsch–Gordan rules for two mixed-symmetry doubletsz = v + i u and Z = V + i U are

<e[z Z ∗] = u U + v V ,

=m[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, sayS3 = +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



Most remarkable is the possibility of antisymmetric spin–isospinwave function

ψx Sx + ψy Sy√2

,

Which requires an antisymmetric orbital wave function,For instance, in the HO

x × y exp[−a(x2 + y2)]

, with `P = 1+. This state is excited in both coordinates. It hasnot yet been seen.See the discussion on the diquark alternative

JMR Quark Model


The three-body problem

Several methods, developed earlier in atomic or nuclear physicsHO expansion, Gaussian expansion and other variationalmethodsIntegro-differential equations: Faddeev, AGS, etc.,Hyperspherical expansion,etc.

JMR Quark Model


The harmonic oscillator (HO)

V =2 K3(r212 + r2

23 + r231)

H(m,m,m) =p2

x

m+

p2y

m+ K (x2 + y2) ,

E = (6 + 4 nx + 2`x + 4 ny + 2 `y )

√Km

N = 2 nx + `x + 2 ny + `y )

Levels named after the multiplicity and `P ,For instance [56,0+] for the ground state with 8 spin 1/2 and 10spin 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


The harmonic oscillator (HO): (qqQ), (q1q2q3)

H(m,m,M) =p2

x

m+

p2y

µ+ K (x2 + y2) ,

with µ given earlier. Still an exact decoupling,

E(m,m,M) =

√Km

(3 + 4 nx + 2 `x ) +

√Kµ

(3 + 4 ny + 2 `y ) .

For (q1q2q3), use Jacobi coordinates, and rescale

x → x/√µx y → y/√µy

.H(m1,m2,m3) = p2

x + p2y + A x2 + B y2 + 2 C x y ,

One is left with a 2× 2 diagonalisation.

JMR Quark Model


Perturbation around HO

V = K (x2 + y2) + δV ,

One often violates the rules of perturbation theory (δE small ascompared to initial spacings)Nevertheless, interesting phenomenology,For instance, hierarchy of N = 2 states

0.5∆

0.1∆

0.2∆

0.2∆

[56,0+]′

[70,0+][56,2+]

[70,1+]

[20,1+]Except that one would like to pushthe lowest state below N = 1![56,0+]′ becomes decoupled withthree body forces (Gromes et al.)

JMR Quark Model


Converged variational methods

Problems withΨ(x ,y) =

∑n

cn φn(x ,y) ,

More fashionable

Ψ(x ,y) =∑

i

γi exp[−(a.i x2 + biy2 + 2 ci x .y)] .

with restoration of permutation symmetry.Detailed search of parameter delicate. see Varga et al. or Hiyama etal.

JMR Quark Model


Hyperspherical expansion

Consider x ,y as a single 6-d vectorSolve the 6-d Schrödinger equation with a potential which is not6-d centralExcept HO, which is 6-d central

−u′′[L](%)+(L + 3/2)(L + 5/2)

%2 u[L]+∑[L]′

V[L],[L]′(%) u[L]′(%) = E u[L](%) ,

Very good convergence, very systematicHypercentral approximation, see Genoa group

JMR Quark Model


Light-quark baryons

An impressive description of many data with a simple tool,But some persisting problems,Let us concentrate on two of themThe Roper resonance = radial excitation of N or ∆In most models, predicted above the orbital excitations withP = −1Similar to ψ′ > χJThis is unavoidable with models with ∆V (r) > 0One suggestion: Yukawa type of interaction among quarks(Glozmann)Missing states, e.g., [20,1+]Absent or not seen since weakly coupled to usual entrancechannels?A quark-diquark model has been proposed (Lichtenberg, Torinogroup)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


Baryons with a single heavy quarksThe domain with most recent discoveries in baryon spectroscopy

-2300

-2500

-2700

-2900

M(cqq)

(MeV)

Λc

1/2+

1/2−3/2−

5/2+

Σc

1/2+

3/2+

Ξc

1/2+

1/2+

3/2+

1/2−3/2−

Ωc

1/2+

3/2+

Λb

1/2+

1/2−3/2−

Σb

1/2+3/2+

Ξb

1/2+

3/2+

Ωb

1/2+

M(bqq)

(MeV)

- 5600

- 5800

- 6000

- 6200

JMR Quark Model


Baryons with a single heavy quarks

(Qqq) central forces governed by reduced masses dominated byqExcitation spectrum nearly independent upon QSee the debate about Ωb = (bss) of D0 vs. CDF and LHCbFlavour independence is important!Spin splittings in the light quark sector almost independent of QSpin splittings involving Q decreases as 1/M

JMR Quark Model


Baryons with two heavy quarks

Perhaps the most interesting of ordinary hadronsFor the price of one, get the two extremes

Heavy–heavy motion like in charmoniumRelativistic motion of a light quark, as in D or B mesons

Often described as [(QQ)− q] in a diquark–quark model orapproximationBut the first excitations are in (QQ)!Then a Born–Oppenheimer picture looks more suited (Fleck etal.), as for H2

+ in atomic physicsRecently the hierarchy of Q −Q vs. q excitations addressed byCohen et al., Roberts et al.,If (QQ) is frozen, then a new heavy-quark symmetry, linkingdouble-charm baryons to singly-charmed mesonsExperiment: Positive results at SELEX, negative at FOCUS andBABAR

JMR Quark Model


Baryons with three heavy quarks

The ultimate deal of baryon spectroscopy (Bjorken)The true baryon analogue of charmoniumFor instance,look at the hierarchy of levels and compare to theprediction of static potential computed on the latticeLet us dream for the future physicists.

JMR Quark Model


Spin splittings in baryons

Very advertised by DGG, Lipkin et al., Isgur and Karl,As a possible evidence for QCD within the hostile environment ofconfinementIn particular

Vss =∑i<j

23

αs

mi mj

2π3δ(3)(r ij )σi .σj ,

Challenged however byInstantons (Bonn group)Loop effects, e.g., N ↔ ∆ + π ↔ N (Törnqvist, Cottingham et al.,etc.)Yukawa-type of forces (Glozmann–Riska, Graz group)Or combinations

JMR Quark Model


Spin splittings in baryons

Long standing problem of spin-orbit forcesIK declared the abolition of spin-orbit forces in baryons, see alsoReinders,OK, with noticeable exceptions, in particular, the famousΛ(1405)− Λ(1520) splitting,Most widely accepted explanation: nearby K N threshold.

JMR Quark Model


Convexity properties

(MM) + (mm) ≤ 2(Mm) in two-body problems with a givenpotential (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 togetherBut ∃ conterexamples with sharp (unphysical) potentials and verylarge mass ratios

JMR Quark Model


From mesons to baryons

Historically disconnected, nuclear physicists working on lightbaryons, particle physicists working on heavy quarkoniaIf colour-octet exchange

V =12

[v(r12 + · · · ] ,

So-called 1/2 ruleWorks reasonably well for a combined phenomenology ofmesons and baryonsChallenged by a string picture

b A

bB

b

C

bJ

V (r1, r2, r3) = b minJ

(r1J + r2J + r3J) ,

JMR Quark Model


String potential for baryons

Link to past work by Fermat, Torricelli and Napoleon

bB

bA

bC

b

A′

b

B′

bC ′

*J

120b

B

bA

bC

b

A′

b B′

bC ′

bC1

b B1

b A1

JMR Quark Model


Mass inequalities for mesons and baryons

If p is the perimeter and Y the minimal Toricelli pathp2≤ Y ≤ p√

3,

The lower bound is saturated for a flat triangle, the upper one foran equilateral triangle, thus

V ≥ 12

[v(r12) + v(r23) + v(r31)] .

For the Hamiltonians

H3 =p2

12 m

+ · · ·+ V ≥ 12

[p2

12 m

+p2

22 m

+ v(r12)

]+ · · ·

.

From the variational principle

2 M(qqq) ≥ 3 M(qq) .

Which becomes inverted with different masses, if M/m large

(QQQ) + (qqq) ≤ 3 (Qq) ,

JMR Quark Model


Multiquarks and other exotics

Exotic hadronsLe 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

BossuetaThe 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.

ContentGlueballs, hybrids, moleculesBaryoniumChromomagnetic bindingChromoelectric bindingGeneralised Steiner-tree potential

JMR Quark Model


Glueballs and hybrids

Glueballs very fashionable in the 80s,Constituent models, bag models, and later lattice QCD and QCDSROften non exotic, so can be confused with ordinary mesonsOr mix with ordinary mesonsPresent status not very clearHybrids sometimes seen as (QQg)

or in the Born–Oppenheimer approach as the second potential

JMR Quark Model


Charmonium hybrids in the early 80sUsing a variant of the bag model

JMR Quark Model


Light and heavy hybrids

A candidate with JPC = 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 stepOne argument in the 80s: (cc) is clean, so any extra state shouldbe clearly visibleWe realise now that the situation is also complicated in thissector.

JMR Quark Model


Molecules

It is regularly rediscovered that the Yukawa mechanism is notrestricted to nucleons,Any hadron containing light quark(s) can enter a nuclear-type ofinteraction,Even without! Remember ηc–nucleus attraction sometimespredicted.The charm sector is no exceptionTörnqvist, Manohar & Wise, Ericson & Karl, Swanson, Close andThomas, etc., etc., have noticed a possible long-range attractionbetween DD∗, D∗D∗ or DD∗ or D∗D∗

Weaker than the proton–neutron potential,But in

−∆

m+ g V (r) =

1m

[−∆ + m g V (r)]

what matters is m g for the existence of a discrete spectrum.

JMR Quark Model


Molecules

When the X (3872) was discovered, it was considered as asuccess for this approach,Just at the DD∗ threshold!But some more recent measurements better call for a 2P state ofcharmonium, in particular

X (3872)→ ψ′ + γ

X (3872)→ J/ψ + γ> 1 ,

Probably a mixture of (cc) 2P and molecule,But do we have two states, or a single (cc) with more higherFock components than usual?

JMR Quark Model


Molecules

We learned to be careful in this sectorIn 1975, Iwazaki suggested ψ′ = (ccqq)

In 1976–77, Voloshin & Okun, and De Rujùla, Georgi & Glashowmolecular structures out of D(∗) and D(∗)

In particular, DGG were puzzled by ψ(4.04) decaying too often inD∗D∗ relative to DD and DD∗ + c.c., as compared to spincounting and phase-space.But Le Yaouanc et al., and Eichten et al. have shown this wasdue to the node structure of this state.We were accustomed to orbital excitations (Regge trajectories),less to radial excitations

JMR Quark Model


Molecules

Baryon–baryon states (Julia-Diaz & Riska) with charm ≥ 2?Perhaps a new periodic table, based on charmed baryonsMeson–baryonBeauty baryons, etc.the Pandora-box syndrome strikes again!

JMR Quark Model


Baryonium

Tentative peaks in antiproton cross-sections in the 70sBumps in the inclusive γ spectrum p + p → γ + XThe name “baryonium” was invented for mesons preferentiallycoupled to baryon–antibaryonTwo main approachesQuasi-nuclear baryonium (Shapiro et al., Dover et al.). Today,would be named “molecular”With meson-exchange between N and N, deduced from NNinteraction by the Fermi–Yang rule (G-parity rule)Annihilation underestimated in this approach,[(qq)3 − (qq)3] structure, with an orbital-momentum barrierpreventing from rearrangement into mesons (Rossi & Veneziano,Jaffe, etc.), named T-baryonium by Chan H.M. et al.

JMR Quark Model


Baryonium

Chan et al. invented colour chemistry,In particular, speculated about [(qq)6 − (qq)6], namedM-baryonium, narrow for both decay into mesons and decay intobaryon–antibaryonNote that the clustering into diquarks with such colour structurewas just assumed, not demonstrated from a dynamicalcalculation,Many followers: exotic baryons with (q4q) and similar clusterstructure, dibaryons, etc. (de Swart et al., Sorba et al., Nicolescuet al., etc.)New experiments with an intense, cooled antiproton beam atLEAR (CERN). No baryonium confirmed.Still some enhancements in Jψ → baryon + antibaryon + · · · atBES, indicatging a strong final-state interaction

JMR Quark Model


Chromomagnetic binding

A by now famous paper by DGG suggested the spin-dependentpart of one-gluon-exchange as responsible for spin–spinsplittings in mesons and baryonsIt reads

Vss =∑i<j

Kmi mj

σi .σj λi .λj vss(rij ) ,

or analogous in the bag modelvss short-rangedIn 1977, Jaffe considered H = (uuddss) in the SU(3)F limit, withJ = 0, assuming

Before spin correction, (uuddss) and (uds) + (uds) are degenerateSpin–spin effect in perturbationvss = 〈vss(rij )〉 the same for all pairs, and the same for baryons andfor the 6-quark system

Thus concentrated on the role of O =∑

i<j σi .σj λi .λj

And discovered a remarkable coherenceJMR Quark Model



Namely attractive and

〈O〉H = 3 〈O〉Λ ,

Thus with Λ = N = ΣΞ all receiving 150 MeV of attraction fromspin-spinHe deduced that H is bound by about 150 MeV below thedegenerate threshold ΛΛ = NΞ = ΣΣ,More than 20 experiments looked at the HNo positive signal, in particular from S = −2 hypernucleiChromomagnetism is remarkable, as it induces a net excess ofattraction in the Hamiltonian, before considering any inducedpolarisation in sub-clusters,The usual situation is: no excess of attractionFor instance Ps2 vs. 2 (e+ e−), both governed by

∑gij/rij , both

have the same∑

gij = −2

JMR Quark Model



The H was revisited by several theorists (Yazaki et al. Karl et al.,Rosner, Gignoux et al., in particular for

SU(3)F breakingSelf-consistent calculation of vss = 〈vss(rij )〉Inclusion of central forces and spin–spin forces in a consistent6-body calculation

Each effect reduces the bindingAnd eventually the H is unbound!The main effect is that ms splits ΛΛ from other thresholds, ΛΛchromomagnetic energy being not penalised, hence thecoherence ΛΛ + NΞ + · · · → H is lost!

JMR Quark Model



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 assumptionsthan Jaffe for the light quarkThis was named “pentaquark” (now, one should say: “thechromomagnetic pentaquark”)It was searched for in 1 experiment at Fermilab (Ashery et al.),not conclusiveA re-analysis, including mQ <∞, indicate that relaxing the the Plikely becomes unboundOther configurations analysed, see Leandri et al.

JMR Quark Model


Chromoelectric binding

If chromomagnetism does not work, why not chromo-electricity?It works! at least in a certain limitMiracle: all theorists agree! instead of fighting.Consider first a simple additive model with colour factors, andnext a better modelling of confinement.The additive model with colour factors reads

V = −163

∑i<j

λi .λj v(rij ) ,

where v(r) is the qq potential of mesons.It generalises the “1/2” rule for baryons.If (qqqq) is solved with this model, without spin corrections nostable tetraquark is foundThis contrasts with the positronium molecule in QEDWhy?

JMR Quark Model


Chromoelectric binding: equal masses

Both Ps2 and (qqqq), and their thresholds are governed by

H4 =∑

i

p2i

2 m− [∑

pi ]2

8 m+∑i<j

gij v(rij ) ,∑i<j

gij = 2 ,

For this family of Hamiltonians, the highest ground state obtainedfor gij = g = 2/15.Then, the more one departs from this symmetric case, the lowerthe bindingCan be measured by the variance of the gij set of coefficients

PPPPPPPState Pair 12 34 13 24 14 23 g ∆g

Threshold 0 0 1 1 0 0 1/3 0.22Ps2 −1 −1 1 1 −1 −1 1/3 0.89T 1/2 1/2 1/4 1/4 1/4 1/4 1/3 0.01M −1/4 −1/4 5/8 5/8 5/8 5/8 1/3 0.17

JMR Quark Model


Chromoelectric binding: equal masses

PPPPPPPPState Pair 12 34 13 24 14 23 g ∆g

Threshold 0 0 1 1 0 0 1/3 0.22Ps2 −1 −1 1 1 −1 −1 1/3 0.89T 1/2 1/2 1/4 1/4 1/4 1/4 1/3 0.01M −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 stableBoth T-type and M-type of tetraquarks are less symmetric thantheir threshold, they are unstableHence tetraquark with equal masses is penalised by thenon-Abelian character of the theory

JMR Quark Model


Chromoelectric binding: unequal masses

Can another asymmetry overcome this problem?Yes! Remember (M+,M+,m−,m−) becomes more stable asM/m departs from 1The same mechanism makes (QQqq) evolving from unbound tostableSee Ader et al. (1982), Heller & Tjon, Brink & Stancu, Rosina &Janc, Barnea, Vijande & Valcarce, etc.The problem is the critical value of M/m required,(ccqq) bound or do we need (bbqq)

Two b or not two b, that is the question!And what about a better potential

JMR Quark Model


Chromoelectric binding: string potential

The additive model

V = −163

∑i<j

λi .λj v(rij ) ,

is already questionable for baryons.The Y -shape interaction seems more suited, and is suggestedby Lattice QCDThe generalisation for tetraquarks has been deviced by Lenz etal., Carlson et al., Vijande et al., etc. It reads

V4 = min(Vf ,Vs) , Vs = min[v(r13) + v(r24), v(r14) + v(r23]

,Vs = connected Steiner-tree,

b

b b

b b

b b

b

b b

JMR Quark Model


Chromoelectric binding: string potential

The good surprise is that this potential gives better binding thanthe simple additive model,Hence, (ccqq), marginally bound in the additive model, shouldbe stable with this improved quark dynamicsHence, beyond double-charm baryons, one could look at doublecharm mesons, a genuine exotic,For instance, since e+e− → J/ψ + ηc is observed (double charmproduction), TQQ + D + D + · · · could be observed.

JMR Quark Model


String potential: some rigorous results

b

v1

b

v2

bw12

bt12

b

c12

b

v3

b

v4

b w34

b t34

b c34

b

s1

b

s2

b

p

b

q

b h b k

JMR Quark Model



v1v2

C12

v3v4

C34

s2

s1

w12b

w34b

bb

rr

b

b

JMR Quark Model



Vs ≤ a

[(x + y)

√3

2+ z√

2

],

H4 ≤p2

x

m+

p2y

m+

p2z

m+ a

[(x + y)

√3

2+ z√

2

],

Stability demonstrated analytically for M/m & 6402! (Ay et al.)

b

q

b

q

b

q

b

q

bb

x y√2 z

JMR Quark Model


String potential: pentaquark

Found stable if antisymmetrisation is disregarded

JMR Quark Model


String potential: hexaquark

Found stable if antisymmetrisation is disregarded

JMR Quark Model


Light pentaquark

Speculation by Diakonov et al.Indication by Nakano et al.Confirmed by other groups having data on tapes, never analysedfor such search,Eventually not confirmed by high-statistics experimentsAlso doubts among theorists, in particular about the small widthin this model,Situation somewhat confusing in constituent models, QCD SR,and lattice QCD calculations trying to reproduce the lightpentaquark

JMR Quark Model


Outlook

Long way from strangeness, SU(3) symmetry, intriguing decaypattern of the φ(1020) to the present state of art in QCDHadron spectroscopy boosted by heavy quarksThe question of exotics remain puzzling,But some configurations have not yet been investigatedexperimentally

JMR Quark Model

an introduction to the quark model

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