Hadron Spectroscopy

Classification and phenomenology

Mikhail Mikhasenko · Lecture 1
AEPSHEP 2026 · Taipei
25 October 2026, 09:00–10:30

Lecture 1 roadmap

A. QCD basics

B. Excited states

C. Observation of hadrons

D. Decays and the hadronic continuum

QCD basics

A · Fields, vacuum and confinement

The QCD Lagrangian

PDG: QCD (2025)

PDG: Quark model (2025)

$$\mathcal L_{\mathrm{QCD}}=\sum_f\bar q_f(i\gamma^\mu D_\mu-m_f)q_f-\frac14G^a_{\mu\nu}G^{a\mu\nu}$$

Quarks carry colour

$$D_\mu=\partial_\mu-i g_sT^aA^a_\mu$$

$q_f$: quark of flavour $f$; $T^a$: colour generators.

Gluons interact with gluons

$$G^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+g_sf^{abc}A^b_\mu A^c_\nu$$

Eight gluon fields, $a=1,\ldots,8$.

The non-Abelian term generates the three- and four-gluon vertices. Gauge-fixing and ghost terms are omitted here.

Conventions: D=partial-igA, [T^a,Tb]=if^{abc}Tc; metric +—; hbar=c=1. The plus sign of the nonlinear field-strength term follows this convention. Draw attention to the two sectors before introducing indices. TikZ sources are in computations/lecture_1/tikz.

The QCD vacuum

Derek Leinweber: original animations

PDG: QCD (2025)

The QCD Lava Lamp · Derek Leinweber, Adelaide

Empty space is not empty of fields.

Even the lowest-energy state contains fluctuating chromo-electric and chromo-magnetic fields.

What are we looking at?

• A lattice-QCD field in a box of $2.4\times2.4\times3.6\;\mathrm{fm}^3$ — a few proton sizes.
• 50 smoothing sweeps reveal the larger-scale lumps.
• Colour shows topological charge density: the local sign and strength of the field’s winding.

Run on your computer: Akio Tomiya’s VisualizingLQCD.jl

A view of a smoothed quantum field, not individual gluons flying through empty space. The animation is not real-time motion.

Exact requested ChargeAPE5LQanimXs30small.gif, retrieved from Derek Leinweber’s original Adelaide VisualQCD Nobel page. The source states a 2.4 by 2.4 by 3.6 fm box and 50 sweeps of APE smearing. This charge animation displays topological charge density, not action density. The companion action animation on the source page is energy-density-like. Smoothing suppresses short-distance fluctuations; visible lumps are processed field structure, not literal particles. Lattice samples have a Euclidean fourth direction, not a real-time movie. The linked Julia package provides related lattice-configuration visualisations; it is not asserted to reproduce this exact Adelaide configuration. fm = 10^-15 m. Source page is saved in research/qcd_intro/sources/adelaide.html.

Confinement and flux tubes

Leinweber: original animations

Bissey et al., PRD 76 (2007) 114512

Biddle et al.: vortices and the potential

Quark–antiquark: $q\bar q$

Three quarks: $qqq$

Pull the quarks apart

The field forms a longer tube of almost constant width. More length costs more energy.

Surfaces: reduction of vacuum action density around static sources.

$$V(r)=\underbrace{-\frac{4\alpha_s}{3r}}_{\text{short-range attraction}}+\underbrace{\sigma r}_{\text{energy of the tube}}+C$$

$C$ sets the energy zero. With light quarks, a long string can break into two hadrons.

What field structures support confinement? Lattice studies link the linear potential to centre vortices.

Exact requested FluxTubeAnim2small.gif and VacuumRespAction16t32_Yshape8small.gif. Meson sources separate from 0.125 to 2.25 fm. Surface heights show suppression of the vacuum action density in the source plane; arrows show its gradient, not electric-field vectors. The Y animation illustrates static three-quark sources and is not a movie of quarks orbiting in a proton. The potential shown is an illustrative colour-singlet heavy quark–antiquark Cornell model, alpha_s=0.30, sigma=0.90 GeV/fm, C=0; hbar c=0.1973269804 GeV fm restores units. It is not fitted to these animations or to lattice data, and the linear model applies before string breaking. Script: research/qcd_intro/scripts/potential.py. The flux-tube surfaces are not a direct image of centre vortices. Biddle et al. 2206.00844 find vortex-only string tensions reproduce the untouched dynamical ensembles studied; this is evidence for the centre-vortex mechanism, not a general analytic proof of QCD confinement.

Centre vortices

Biddle, Kamleh & Leinweber, PRD 102 (2020) 034504

Biddle et al.: static potential (2022)

A 3D slice through the vortex network · Biddle, Kamleh & Leinweber

A familiar vortex — an analogy, not a QCD simulation.

Read the map

Lines mark centre flux; red and blue distinguish its orientation.

A 3D slice cuts the four-dimensional vortex surfaces into lines.

Remove the identified vortices in a lattice calculation → the confining linear rise disappears. Keep them → the confining behaviour survives.

Map is the exact image15.png from HadronPhysics1.pptx slide 6. It represents centre-projected lattice flux, not quark trajectories. Nontrivial Z3 phases are exp(±2pi i/3); the omitted trivial element is 1. Projection is gauge dependent and vortex removal constructs modified ensembles. Quantitative vortex-only reproduction depends on ensemble and procedure; the cited 2022 study recovers full string tension on its dynamical ensembles and approximately two thirds in pure gauge theory. The supplied media/vortex.gif depicts a cyclone labelled CHAMPI near Taiwan: use only as a visual winding analogy, not as physical equivalence with QCD. Its original external provenance is not supplied. Full field/topological density, flux-tube response and centre projection are different observables.

Effective degrees of freedom

Gluons: self-interactions and confinement

Light quarks: $u,d,s$

• Light: $u,d$ nearly massless on hadronic scales; $s$ is heavier.
• Cheap: light quark–antiquark pairs are easier to create.
• Mixed: neutral light mesons can mix flavour components.

Heavy quarks: $c,b$ ($t$)

• Massive: charm and bottom provide a large mass scale.
• Expensive: creating a heavy pair needs much more energy.
• Persistent: a heavy quark usually remains a useful label for its hadron.

Strong interactions conserve net flavour for all quarks.
The top quark decays before forming ordinary hadrons.

IIIIIIuctdsb

123456789101112131415161718192021222324252627282930313233343536

M. Veltman, Diagrammatica

Recreates the attached slide’s composition: bold light/heavy comparison at left and the dense Standard Model Lagrangian at right. The formula is the original PowerPoint slide 7 image17.png, attributed there to Veltman. Corrects the shorthand “massless” and “conserved”: u,d have nonzero current masses; strange is not accurately massless for all purposes; strong interactions conserve net flavour for light quarks as well. Pair creation conserves net flavour. “Mixed” means allowed neutral-hadron flavour superpositions/mixing, not gluon vertices changing a quark’s flavour. Constituent-quark masses include interaction dressing and differ from renormalised Lagrangian masses. Top decays before ordinary hadronisation. Quark tiles omit historical numerical masses and do not equate current and constituent masses.

Quarks and hadron families

PDG: Quark model (2025)

PDG: Light mesons (2025)

LHCb: doubly-charmed baryons (2026)

Start with mesons: a colour-singlet quark and antiquark. Then baryons: three quarks in a colour singlet (antibaryons contain three antiquarks). The cartoons show valence content, not the full wavefunction; gluons and sea pairs are implicit. Coloured dots label constituents schematically and are not a fixed assignment of colour. Meson view counts 10 teaching categories by flavour and isospin, explicitly not 10 particles; light isoscalars can mix nonstrange and strange components. Baryon view shows 6 familiar light-flavour families, then 4 singly-charmed or 4 singly-bottom families. The widget also includes doubly/triply-heavy baryon sectors and exotic configurations, with observation status distinguished. Lambda and Sigma illustrate different isospin at the same uds valence content. Xi_Q and Xi’_Q have different light-quark spin couplings within the same flavour family. Spin and orbital excitations explain why each family contains many states, motivating the next section.

Excited states

B · Spin, orbital motion and excitation multiplets

Radial and orbital multiplets

Godfrey–Isgur, PRD 32 (1985) 189

GIModel.jl: wavefunctions

Wavefunctions and nodes

$$H\psi=E\psi$$

Excitation energy ↑

S · L = 0

P · L = 1

D · L = 2

3S

2P

1D

2S

1P

1S

GIModel charmonium clouds · schematic energy spacings

Radial excitation

• Same orbital angular momentum $L$.
• More nodes in the radial wavefunction.
• $1S\to2S\to3S$: 0 → 1 → 2 radial nodes.

Orbital excitation

• Increase $L$: S → P → D.
• Change the angular shape of the wavefunction.
• The displayed P waves have an angular nodal plane.

Connection to hydrogen

$N=n_r+L+1$, while our radial label is $n=n_r+1$.

Confinement lifts the Coulomb degeneracy: $1P$ and $2S$ need not coincide.

This slide merges Radial and orbital multiplets with Wavefunctions and nodes and the Schrödinger-equation reminder. The upper levels are 3S, 2P and 1D. In nonrelativistic Coulomb hydrogen, N=n_r+L+1 sets the energy; for a general central potential such as confinement, equal N does not imply degeneracy. Our spectroscopic radial label n=n_r+1 differs from hydrogen’s principal N. The staggered positions are a schematic confining spectrum, not a GI mass prediction: 1P lies below 2S, and 1D below 2P and 3S. Each nL central level subsequently generates the spin multiplets built on the following slides. 1S/2S/3S have 0/1/2 radial nodes; 1P/2P have 0/1 radial nodes and the displayed m=0 P waves have an angular nodal plane. The D level is shown without an animation; no D-wave density was computed in the existing asset set.

Reused GIModel charmonium spin-singlet probability clouds, common physical scale, m=0. Only the camera rotates: these are stationary states, not time-dependent densities. Colours indicate depth, not wavefunction sign. The GI Hamiltonian is relativized; the hydrogen connection concerns quantum-number organization rather than identical dynamics. Source computations: research/quark_model. Print/export uses fixed PNGs. Detailed S- and P-wave radial plots are in the appendix.

Spin and spin projection

PDG: CG coefficients (2025)

PDG: Quark model (2025)

$$\mathbf j^2|jm\rangle=\hbar^2j(j+1)|jm\rangle,\qquad j_z|jm\rangle=\hbar m|jm\rangle.$$

Fixed spin $j$; $2j+1$ possible projections: $m=-j,-j+1,\ldots,j$.

Widget adapted from 2027-RUB-KTI Lecture_1/widgets/spin-rotation.html; the expansion is now written out live, with each probability bar under its term. Begin at j=1/2 and theta=0, rotate to 90 degrees, then choose j=1 to show three possible measurement outcomes. The blue arrow is the expectation value, not a classical vector with simultaneous definite components. Its length j hbar differs from sqrt[j(j+1)] hbar, the eigenvalue-derived magnitude. Rotation changes the superposition and probabilities, not j. The mean projection varies continuously; an individual measurement returns one of the allowed discrete m values. The widget illustrates rotated highest-weight states, not every possible spin state for j>1/2.

Adding two spins

PDG: CG coefficients (2025)

PDG: Quark model (2025)

1 · Add angular momenta

$$\mathbf J=\mathbf j_1+\mathbf j_2.$$

Vector addition — with quantized total angular momentum.

2 · Find the allowed total spins

$$j_1\otimes j_2=|j_1-j_2|\oplus\cdots\oplus(j_1+j_2).$$

$J$ increases in steps of one. Each value labels a separate multiplet.

3 · Then resolve the projections

For each $J$, there are $2J+1$ states:

$$M=-J,\ldots,J,\qquad M=m_1+m_2.$$

$$\underbrace{(2j_1+1)(2j_2+1)}_{\text{product states}}=\underbrace{\sum_J(2J+1)}_{\text{coupled states}}.$$

The arrows are a schematic head-to-tail illustration of the angular-momentum operator identity; they are not definite classical spin vectors or a scale drawing of quantum eigenvalues. Determine allowed J first, then its M values. Coupled states are coherent combinations of product states with m1+m2=M, related by Clebsch–Gordan coefficients. This preserves the existing product/coupled-space counting while moving the full basis-transformation formula into the notes: |JM> = sum C(j1 m1,j2 m2;JM)|j1 m1>|j2 m2>.

Example: singlet and triplet

PDG: CG coefficients (2025)

PDG: Quark model (2025)

Triplet $S=1$: symmetric under exchange.

Three projections: $M=1,0,-1$.

Singlet $S=0$: antisymmetric under exchange.

One projection: $M=0$.

Condon–Shortley convention: singlet=(up down - down up)/sqrt2. The opposite overall sign for the singlet is physically equivalent. Explain why the m=0 triplet is symmetric despite opposite spin projections. This supports the later Fermi-statistics quiz.

Spin and orbital motion

PDG: CG coefficients (2025)

PDG: Quark model (2025)

$$j_1^{P_1}\otimes j_2^{P_2}\;\longrightarrow\;S^{P_1P_2},\ S'^{P_1P_2},\ldots\qquad (L=0).$$

³⁄₂⁺ ⊗ ½⁻

S = 1 ladder

S = 2 ladder

Relative motion

1⁻

2⁻

S wave L = 0

0⁺   1⁺   2⁺

1⁺   2⁺   3⁺

P wave L = 1

1⁻   2⁻   3⁻

0⁻   1⁻   2⁻   3⁻   4⁻

D wave L = 2

2⁺   3⁺   4⁺

1⁺   2⁺   3⁺   4⁺   5⁺

F wave L = 3

$$J=|L-S|,\ldots,L+S,\qquad P=P_1P_2(-1)^L.$$

Each initial $S$ generates its own orbital ladder. Repeated $J^P$ values in different columns are distinct coupled configurations.

Editable HTML version of the lecturer’s favourite table, following HadronPhysics1 slides 13–14. First decompose 3/2+ times 1/2- into S=1 and S=2 with product parity negative, before adding any relative orbital motion. Read down each coloured column to build its ladder, then across a row to count the possible J multiplets at fixed L. Each entry still has 2J+1 spin projections. Checks: L=0 gives 3+5=8 states; L=1 gives (1+3+5)+(3+5+7)=24; L=2 gives (3+5+7)+(1+3+5+7+9)=40. Channels with the same J^P can mix if the dynamics permits. The table is an angular-momentum decomposition, not an assertion of mass ordering or guaranteed observable resonances. The constituent parity product is assigned to the L=0 channel; spin alone has no independent intrinsic parity.

Meson quantum numbers

PDG: Quark model (2025)

PDG: CG coefficients (2025)

For a neutral, self-conjugate $q\bar q$ state, charge conjugation exchanges quark and antiquark.

Parity: P = (−1)^L+1Charge conjugation: C = (−1)^L+S

½⁺ ⊗ ½⁻

S = 0 ladder

S = 1 ladder

Relative motion

0⁻⁺

1⁻⁻

S wave L = 0

1⁺⁻

0⁺⁺   1⁺⁺   2⁺⁺

P wave L = 1

2⁻⁺

1⁻⁻   2⁻⁻   3⁻⁻

D wave L = 2

3⁺⁻

2⁺⁺   3⁺⁺   4⁺⁺

F wave L = 3

Forbidden for conventional q q̄: 0⁻⁻, 0⁺⁻, 1⁻⁺, 2⁺⁻, 3⁻⁺, …

Use the neutral self-conjugate member of a flavour multiplet, e.g. rho0, to assign C; charged partners do not have a C eigenvalue. Electrical neutrality alone is insufficient: K0 and D0 are not their own antiparticles. Derivation: swapping q and antiquark gives orbital exchange (-1)^L, spin exchange (-1)^(S+1), and a fermion reordering minus sign; their product is (-1)^(L+S). Intrinsic quark and antiquark parities have opposite sign, giving P=(-1)^(L+1). Each red sign is C, the second sign in J^PC; the first sign remains P. S=0 implies J=L; S=1 gives J=|L-1|,…,L+1 (only J=1 for L=0). Thus the displayed exotic quantum numbers cannot be made by any conventional q-antiquark orbital or radial excitation. They can occur with additional degrees of freedom, e.g. hybrids or multiquark configurations. This is a selection rule, not an assertion that every forbidden combination has been observed. Radial excitation leaves these quantum numbers unchanged.

Isospin

PDG: Quark model (2025)

PDG: CG coefficients (2025)

u ↔︎ dNearly equal masses on the strong-interaction scale.
QCD treats them almost identically → approximate isospin symmetry.

The same addition rules as spin

One light quark $(u,d)$ forms a doublet:

$$I=\tfrac12,\qquad I_3(u)=+\tfrac12,\quad I_3(d)=-\tfrac12.$$

Three light quarks:

$$\tfrac12\otimes\tfrac12\otimes\tfrac12=\tfrac12\oplus\tfrac12\oplus\tfrac32.$$

Nucleons: $I=\tfrac12$ → $p,n$
Delta baryons: $I=\tfrac32$ → $\Delta^{++},\Delta^+,\Delta^0,\Delta^-$

Here “light” means u or d. The $s,c,b$ quarks are isospin singlets: $I=0$.

Neutral mesons: equal-weight mixtures

The $u\bar u$ and $d\bar d$ flavour configurations mix into states of definite isospin:

Isoscalar · $I=0$

$$|0,0\rangle=\frac{|u\bar u\rangle+|d\bar d\rangle}{\sqrt2}.$$

Isovector · $I=1$

$$|1,0\rangle=\frac{|u\bar u\rangle-|d\bar d\rangle}{\sqrt2}.$$

Maximal mixing: equal magnitudes, with a plus or minus relative sign.

Nearly degenerate members of each multiplet — not necessarily equal masses for the $I=0$ and $I=1$ multiplets.

Use the near-degeneracy of u and d masses relative to the hadronic scale to motivate SU(2) flavour symmetry. Isospin is an internal flavour quantum number, not spatial spin. Exact isospin neglects electromagnetism and sets mu=md; real-world breaking splits multiplet members and can mix I assignments. Equal diagonal entries for the uu-bar/dd-bar subspace with isospin-symmetric flavour-changing interactions give symmetric and antisymmetric eigenvectors; degeneracy alone, without such coupling, does not force mixing. “Maximal mixing” means 45-degree, equal-weight components (each has weight one-half in this normalized two-component flavour basis), not a 100 percent conversion probability. A global sign convention does not change the physics.

The isovector expression shown is its neutral I3=0 member; the charged members complete the triplet. The isoscalar formula denotes the nonstrange basis state: physical isoscalars can additionally mix with ss-bar, gluonic and other configurations, so it is not a universal physical-isoscalar wavefunction. Singlet and triplet masses need not be equal, even at exact isospin. For three u/d constituents, the I=1/2 representation occurs twice; total colour, spin, spatial and flavour symmetries must still satisfy Fermi statistics. This prepares the following good/bad diquark quiz.

Baryons: good and bad diquarks

PDG: Quark model (2025)

Jaffe: Exotica, hep-ph/0409065

Heavy quark: $s_Q=\tfrac12$
Light pair: internal S wave, colour $\bar{\mathbf3}$

Good diquark

$j_d^P=0^+$

Antisymmetric flavour

Bad diquark

$j_d^P=1^+$

Symmetric flavour

Relative $L=0$: $\mathbf J=\mathbf s_Q+\mathbf j_d$, $P=+$.

Λ_c · good

Σ_c · bad (isospin)

Ξ_c, Ξ′_c · good + bad

Ω_c · bad (identity)

Cartoons redrawn from the Bormio 2023 lecture (charm-baryon slide): red arrows mark the light-pair spin, 0 (good, antiparallel) or 1 (bad, parallel). Xi_c has both couplings (Xi_c and Xi’_c); Omega_c’s identical ss pair can only be bad in an internal S wave. Source: HadronPhysics1.pptx, slide 17. This is a quark–diquark counting exercise, not a claim that the diquark is an isolated observable particle. The internal diquark orbital momentum and relative Q–diquark L are distinct. For ud, the good configuration has I=0 and the bad one I=1. “Good” and “bad” refer to the relative preference of correlations in the model; both enter baryon spectroscopy. Menti: unscored knowledge check. Present: https://www.mentimeter.com/app/presentation/aliro5rgaxzxriepqgagwg36boz87ekj/present Edit: https://www.mentimeter.com/app/presentation/aliro5rgaxzxriepqgagwg36boz87ekj/edit

A heavy quark and a good diquark

PDG: Quark model (2025)

Jaffe: Exotica, hep-ph/0409065

For $Q[ud]$ with $j_d^P=0^+$ and relative $L=0$, which $J^P$ levels are allowed?

A. $\tfrac12^+$ only

B. $\tfrac32^+$ only

C. $\tfrac12^+,\ \tfrac32^+$

D. $\tfrac12^-$ only

Menti question. Correct answer: A. $\tfrac12\otimes0=\tfrac12$. All constituents have positive parity and L=0. Reveal and discuss the answer after voting; retain the explanation in speaker notes.

A heavy quark and a bad diquark

PDG: Quark model (2025)

Jaffe: Exotica, hep-ph/0409065

For $Q\{ud\}$ with $j_d^P=1^+$ and relative $L=0$, which $J^P$ levels are allowed?

A. $\tfrac12^+$ only

B. $\tfrac32^+$ only

C. $\tfrac12^+,\ \tfrac32^+$

D. $\tfrac12^-,\ \tfrac32^-$

Menti question. Correct answer: C. $\tfrac12\otimes1=\tfrac12\oplus\tfrac32$. These are two J levels with 2+4 magnetic substates. Reveal and discuss the answer after voting; retain the explanation in speaker notes.

Can the good diquark be uu?

PDG: Quark model (2025)

Jaffe: Exotica, hep-ph/0409065

For an internally S-wave, colour-$\bar{\mathbf3}$ $uu$ pair, which spin is compatible with Fermi statistics?

A. $j_d=0$ only

B. $j_d=1$ only

C. Both $j_d=0$ and $j_d=1$

D. Neither

Menti question. Correct answer: B. The spatial and identical-flavour factors are symmetric, while colour is antisymmetric. Spin must therefore be symmetric, j_d=1. The same applies to ss under these internal S-wave and colour-antitriplet assumptions. Reveal and discuss the answer after voting; retain the explanation in speaker notes.

Hyperfine splitting

Godfrey–Isgur, PRD 32 (1985) 189

PDG: Quark model (2025)

$$\mathbf s_1\cdot\mathbf s_2=\frac12[S(S+1)-s_1(s_1+1)-s_2(s_2+1)]$$

For two spin-$\tfrac12$ constituents:

$$\langle\mathbf s_1\cdot\mathbf s_2\rangle=\begin{cases}-3/4&S=0,\\+1/4&S=1.\end{cases}$$

With a fixed spatial wavefunction,

$$\Delta E_S=A\langle\mathbf s_1\cdot\mathbf s_2\rangle.$$

Vary $A$: the singlet and triplet separate while the spin-weighted centroid stays fixed.

Exact spin algebra; illustrative first-order model. Full GIModel spectra follow.

Illustrative first-order spin-spin Hamiltonian with a fixed spatial wavefunction, hbar=1. A includes the spatial matrix element and is varied independently for teaching. The exact eigenvalues are generated with Julia; this is not yet a GIModel scan. At A=80 MeV, singlet shift=-60 MeV and triplet shift=+20 MeV. Spin-weighted centroid is unchanged. Tensor and spin-orbit terms are omitted.

Quark Model calculation

Godfrey–Isgur, PRD 32 (1985) 189

GIModel.jl documentation

PDG: Quark model (2025)

One relativized $q\bar q$ Hamiltonian

18 shared parameters → light, heavy–light and heavy-quarkonium families.

Cornell potential: attraction + confinement

$$V_{\mathrm{Cornell}}(r)=-\frac{4\alpha_s}{3r}+br+c$$

Relativistic motion and spin interactions

$$H=H_{\mathrm{central}}+H_{\mathrm{SS}}+H_{\mathrm{LS}}+H_{\mathrm{tensor}}$$

$$H_{\mathrm{central}}=\sqrt{\mathbf{p}^2+m_1^2}+\sqrt{\mathbf{p}^2+m_2^2}+\widetilde{V}$$

GI adds running coupling, smearing and relativistic factors.

Radial: $1S\to2S\to3S\to\cdots$
Orbital: $1S\to1P\to1D\to1F\to\cdots$

GIModel.jl · masses, wavefunctions and transitions

18 inputs for the ordinary spectrum; isoscalar annihilation mixing requires additional parameters.

The Cornell form is the pedagogical starting point, not the complete GI potential. The relativized Hamiltonian has square-root kinetic energy, running effective coupling, smeared interactions and momentum-dependent relativistic factors. The tilde denotes these modifications schematically; the central interaction need not be purely local. SS is smeared contact hyperfine; LS includes vector and Thomas terms and unequal-mass mixing; tensor includes diagonal and mixing terms.

The author’s docs/model_inputs.md counts 18 numerical physics inputs: 4 constituent masses (u=d), 2 confinement constants, 2 smearing inputs, 4 relativistic exponents, 3 Gaussian coupling weights and 3 Gaussian coupling scales. The 6 Gaussian values are fixed by the published approximation; the other 12 are configurable. Thus 18 shared parameters does not mean 18 independently fitted parameters. Optional isoscalar annihilation mixing introduces 6 more phenomenological inputs. Decay operators can require further inputs. The common Hamiltonian organizes conventional qqbar families and radial/orbital ladders; it does not describe every hadron or continuum dynamics.

Screenshot: live GIModel documentation homepage, 22 October 2026, 1280×960 CSS pixels, device scale 2. No page content altered. Provenance: research/meson_families/documentation.

Building a meson family

Godfrey–Isgur, PRD 32 (1985) 189

GIModel.jl: numerical implementation

PDG: Quark model (2025)

Start with Central only. Seven radial–orbital levels are displayed: 1S, 2S, 3S, 1P, 2P, 1D, 1F. Counting S=0 and S=1 gives 3×2 + 4×4 = 22 meson species for one flavour pair, without counting magnetic substates or charge partners. Spin interactions lift degeneracies; they do not create additional Hilbert-space states. Each column groups one nL multiplet. Labels give JP; C is unavailable for general open flavour. Hover for the model mass and assigned spectroscopic label. Mixed states retain GIModel’s assigned basis label, not a claim of pure spin composition.

Tune the constituent mass mQ from 0.220 to 4.977 GeV, holding the antiquark at 0.220 GeV. Slider positions are separately computed numerical spectra, with no interpolation. The y axis subtracts the spin-independent 1S eigenvalue at the current mass, retaining visible excitation spacings as the heavy rest mass changes. Spin–spin switches the smeared contact operator. Spin–orbit switches vector and Thomas diagonal terms plus antisymmetric same-J mixing. Tensor switches diagonal and off-diagonal tensor terms. This separation is a local adapter around GIModel’s native operator matrices; upstream files are unmodified.

Finite-difference relativistic radial solves, rmax=24 GeV^-1, 450 points; refinement to 650 at light/charm/bottom. The finite spectroscopic mixing basis contains n<=2 S/P/D/F plus 3S; higher partial waves and radial levels are truncated. No continuum channels, annihilation mixing, or experimental state assignments are included. These are quark-model predictions, not the measured spectra on the following slides. The research folder records the convergence comparison and reproducible code. Printed view shows all interactions at the charm mass.

Quarkonium spectra

Godfrey–Isgur, PRD 32 (1985) 189

GIModel.jl

Exact user-supplied comparison image, supplied 23 October 2026. Coloured horizontal segments show model multiplets grouped by radial/orbital excitation; black dots show the labelled experimental states. The two mass axes differ. The graphic retains its original assignments and numbers; these have not been recomputed in this slide. In particular, plotting X(3872) near a 2P level is not an assertion that it is a pure conventional charmonium state. The detailed sibling, bottomonium, charmonium and light-isovector slides remain in Backup. Image provenance: codex-clipboard-37cd3dfa-63b9-4220-be72-7d62320d9ed4.png.

Observation of hadrons

C · Production, discoveries and the experimental spectrum

How we search for hadrons

TikZ Hadron Gallery · collision diagrams

LHCb: excited $\Omega_b$, 2001.00851

LHCb: $B\to DD K$, 2009.00026

Direct (prompt) production

$$pp\to\Omega_b^{**-}+X,\qquad\Omega_b^{**-}\to\Xi_b^0K^-.$$

Reconstruct the long-lived $\Xi_b^0$ and combine it with a kaon from the collision vertex.

Search for peaks in $m(\Xi_b^0K^-)$.

The resonance is prompt; its weakly decaying daughters travel before decaying.

Production in decays

$$B^+\to D^+D^-K^+,\qquad T_{cs}\to D^-K^+.$$

Reconstruct a displaced $B$ decay and inspect the invariant masses of its daughter pairs.

The $D^-K^+$ channel reveals the $T_{cs}(2900)$ structures.

Weak $c$-hadron decays similarly probe lighter hadron resonances.

Diagrams from M. Mikhasenko’s TikzHadronGallery, local source data/figures/collisions. collision_B2DDh is a B-decay topology, not prompt production of the resonance in the D K subsystem. Its generic h is K+ in the displayed example. Charge-conjugate processes are implied; the neutral 2900 structures in D- K+ were originally called X0 and X1. The prompt diagram is adapted from collision_LbK_3pi_prompt: its original Omega_b label was inconsistent with Lambda_b K strong-decay flavour. Daughter labels are corrected to Xi_b0 and Xi_c+, and the Xi_b decay is Xi_c+ pi- as in the Omega_b discovery analysis (Xi_c+ to p K- pi+). The other two pion tracks were removed. Original source is retained unchanged alongside the adapted TeX. Prompt describes the excited parent, even though its daughters have displaced weak-decay vertices. The e+e- scan method is introduced separately before the vector-state examples.

Observed hadrons: the experimental inventory

PDG: Review of Particle Physics

LHCb-FIGURE-2026-010

The particle physicist’s reference book
Photo: PDG · 2024 edition

From a spectrum to a catalogue

Masses, widths, quantum numbers and decay modes bring experimental results together.

756 hadron entries

Including “Further States”: some entries are less established or controversial.

Not all entries have the same level of experimental support.

The lecture uses the inclusive total of 756, matching the sum of the discovery-attribution chart. At the lecturer’s request, this includes the less-established or controversial “Further States”; no subtraction is applied. The book photograph is the official PDG product photograph of the 2024 edition, not a 2026 cover.

A century of hadron discoveries

LHCb-FIGURE-2026-010 · September 2026

New beams and detectors keep revealing new states.

The official yearly-plot caption excludes the separately attributed partial-wave-analysis category. This does not mean that the experiment-attributed discoveries were made without amplitude analyses.

Where the discoveries happened

LHCb-FIGURE-2026-010

Bars span each machine’s first to last discovery—not its operating lifetime.

Official caption: dark bars span each machine’s first-to-last discovery, inside the laboratory’s span in pale colour; they show neither running years nor discovery rates. User-supplied excerpt of the facility-history graphic. The excerpt has no numerical time axis; use this slide to discuss successive experimental settings, not to read precise start or end dates from the bars.

Discoveries also come from amplitude analyses

LHCb-FIGURE-2026-010 · September 2026

LHCb: J/ψϕ amplitude analysis (2016)

LHCb: pentaquark amplitude analysis (2015)

Not every state is an isolated peak

Overlapping resonances can require a partial-wave analysis to separate their contributions.

LHCb discoveries also use amplitude analyses—for example, the J/ψϕ states and hidden-charm pentaquarks.

The bars show discovery attribution, not the analysis method. “Partial-wave fits” is not the total number of states found using such fits.

Official caption: numbers of hadrons discovered by each experiment with count at least ten, by other experiments, and by three categories that are not experiments or collaborations in the present-day sense. Thus the experiment-labelled bins must not be interpreted as excluding partial-wave analyses. LHCb examples are documented in arXiv:1606.07895 and arXiv:1507.03414.

The displayed bins sum to 756, the inclusive total used throughout this lecture. This includes less-established or controversial “Further States”; the count is not a claim that every entry has equal experimental support.

Heavy-flavour baryons: organise the families

PDG: Quark model

PDG: Baryon summary tables

A heavy quark plus two light quarks: Qqq, Q = c or b

Light-quark pair	Charm families	Bottom families
ud · isospin 0	Λc	Λb
uu, ud, dd · isospin 1	Σc, Σc*	Σb, Σb*
us or ds	Ξc, Ξ′c, Ξc*	Ξb, Ξ′b, Ξb*
ss	Ωc, Ωc*	Ωb, Ωb*

Each flavour family can support a ladder of orbital and radial excitations.

Family labels, not a checklist of observed states. Here * denotes the spin-3/2 ground-state partner.

Resolving excited Ω baryons

LHCb: Ωc, PRL 118 (2017) 182001

LHCb: Ωb, PRL 124 (2020) 222001

Charm · css · 2017

Reconstruct Ξc⁺ K⁻: five narrow peaks.

Ξc⁺ → p K⁻ π⁺

Bottom · bss · 2020

Reconstruct Ξb⁰ K⁻: four narrow structures.

Ξb⁰ → Ξc⁺ π⁻, followed by Ξc⁺ → p K⁻ π⁺

Start with the reconstructed final state and its mass distribution. The 2020 paper reports local significances between 3.6 and 7.2 standard deviations for the four structures, so do not call each an independent five-sigma discovery. These plots establish spectral structures; their individual spin-parity assignments are a separate experimental question. Images are original LHCb outreach figures.

Two heavy quarks: the Ξcc⁺⁺ discovery

LHCb, PRL 119 (2017) 112001 · Fig. 2

CERN · 6 July 2017

Reconstruct a weak-decay chain

$$\Xi_{cc}^{++}\to\Lambda_c^+K^-\pi^+\pi^+,\qquad \Lambda_c^+\to pK^-\pi^+$$

Quark content: ccu.
A narrow signal near 3621 MeV.

Check alternative charge combinations and Λc sidebands: the peak is absent.

A historical milestone in 2017: a baryon containing two charm quarks.

New hadrons at the LHC

P. Koppenburg · 24 September 2026

Exotic hadrons at the LHC

P. Koppenburg · 24 September 2026

The exotic-hadron landscape

Hüsken, Spadaro Norella & Polyakov · A Brief Guide to Exotic Hadrons
Illustration: I. Polyakov · Fig. 2

Illustration: I. Polyakov. Figure 2 of Hüsken, Spadaro Norella and Polyakov, A Brief Guide to Exotic Hadrons, https://arxiv.org/abs/2410.06923. The artist’s credit is retained within the original artwork.

Break

Decays and the hadronic continuum

D · Resonances, thresholds and exotic hadrons

QCD states as resonances

PDG: Resonances

Most excited hadrons can decay through the strong interaction:

$$P_0\longrightarrow P_1+P_2.$$

The same quantum numbers belong to a two-hadron continuum: the pair can carry any relative momentum.

A short-lived excitation appears in the interaction of these hadrons — for example, $\pi\pi\to\rho\to\pi\pi$.

For an isolated, narrow resonance,

$$\tau\simeq\frac{\hbar}{\Gamma}.$$

A quark-model level becomes a resonance when coupled to open decay channels.

Adapted from HadronPhysics1.pptx slide 26; original embedded image58.png. The continuum consists of scattering states, not a sharply defined unstable-particle energy eigenstate. Strongly stable hadrons remain possible. The width–lifetime formula is the narrow, approximately exponential-decay limit. In finite volume, lattice QCD has discrete energies; Lüscher’s formalism relates them to infinite-volume scattering. This is not a direct identification of a finite-volume level with a resonance mass.

Resonances

PDG: Resonances

Electrical resonance

Hadronic resonance

An oscillating circuit and a hadronic system share the same language:

Enhanced response near a characteristic energy.

Phase motion as the energy passes through it.

A pole of the complex response function.

Interference can turn a resonance peak into a dip.

Original slide 27, image59.png and image60.png, preserved without redrawing. Curves are illustrative response functions, not experimental data. The hadronic example uses a constant-width Breit–Wigner amplitude. Peak, phase motion and pole are related, but a visible peak by itself is not sufficient evidence for a resonance.

Hadronic amplitude

PDG: Resonances, §50.3

The experiment measures an intensity

$$\frac{dN}{d\Phi}\propto\epsilon(\Phi)\sum_{\text{unobserved spins}}|\mathcal A(\Phi)|^2.$$

$\Phi$: final-state kinematics; $\epsilon$: detection efficiency.

Amplitudes for the same final state interfere before squaring.

A simple resonance model

$$\mathcal A(s)=\frac{N(s)}{m_0^2-s-i\sum_i g_i^2\rho_i(s)},\qquad s=E^2.$$

The production process changes $N(s)$; the resonance pole is shared between reactions.

What goes into the width?

Far from relevant thresholds: a constant $m_0\Gamma$ can approximate the imaginary term.

One important channel: retain $g^2\rho(s)$, with two-body phase space $\rho(s)=2k/\sqrt{s}$.

Several channels: sum their contributions, $\sum_i g_i^2\rho_i(s)$.

Near a threshold, phase space varies rapidly. A constant-width peak can be misleading.

Editable transcription and clarification of original slide 28 (image61.png, image62.png). This is a schematic Flatté-like denominator: dispersive real parts, angular-momentum barriers and coupled-channel matrix structure may be needed. The same isolated pole occurs in different channels, not necessarily an identical numerator or complete background. Spin sums are incoherent only over unobserved orthogonal states; average over initial spins as appropriate. Detector resolution and background must also enter a measured-spectrum fit.

The amplitude in the complex plane

Hanhart & MM., Scattering Theory, §6

PDG: Resonances

What is A?

A transition amplitude (matrix element) for chosen initial and final channels, projected onto a fixed partial wave.

$$A(s)\equiv A^J_{ba}(s).$$

The angular dependence has been projected out; channel and spin labels are suppressed.

What is s?

$$s=(p_1+p_2)^2=E_{\mathrm{cm}}^2.$$

The invariant mass squared of the system whose resonance we study.

Analytic continuation: extend the same analytic amplitude away from real $s$, respecting its singularities and branch choices.

$$s\in\mathbb R\quad\longrightarrow\quad s=x+iy\in\mathbb C.$$

$A(s)$ can already be complex for real $s$.

Bridge between Hadronic amplitude and Riemann sheets. Here A is a schematic fixed-J two-body transition partial wave; the preceding slide also used a production amplitude. Once a channel and partial wave have been specified, angular functions are factored out or projected out. For particles with spin, helicity or LS indices may remain but are suppressed here. s is energy squared, not energy, and may refer to the invariant mass squared of a chosen subsystem in a production reaction. Physical data constrain boundary values A(s+i0), not direct measurements at complex energies. Analytic continuation is not an arbitrary two-dimensional extrapolation or necessarily a Taylor series: it uses an analytic amplitude consistent with those data and physical constraints. The widget uses the lecturer’s original Mathematica figures sheet_0.pdf and sheet_2.pdf (2019 PhD presentation), converted to PNG without redrawing. Red highlights the real-axis amplitude; surface height is Im A. The second frame reveals the continued surface and pole; it is a precomputed illustration, not a live evaluation of the amplitude. Both frames appear in print. The next slide explains why specifying the sheet matters.

Guiding principles for amplitudes

Hanhart & MM., Scattering Theory

PDG: Resonances

I

Analyticity ← Causality

Causal dynamics constrains how the amplitude extends into complex energy.

II

Unitarity ← Probability conservation

Probability is conserved when all accessible final channels are included.

III

Crossing symmetry

Related reactions are described by continuations of the same underlying amplitude.

Consequences: analytic structure

• Hadronic resonances → poles on unphysical sheets.
• Decay-channel thresholds → branch points, joining a non-trivial set of Riemann sheets.

These are guiding principles for constructing and checking hadronic amplitudes, not a replacement for dynamics. Analyticity follows from causality/locality together with the usual field-theory assumptions; it is not a claim that amplitudes are entire functions. Unitarity is S-dagger S=1 and relates discontinuities to physical intermediate states. Crossing relates scattering and annihilation processes by moving a particle between initial and final states as its antiparticle with analytically continued momentum. Crossing is most simply stated for the full invariant amplitude: in a partial-wave representation it generally relates different channels and partial waves, not an isolated A^J(s) to itself. Stable two-body thresholds generate square-root branch points, while multiparticle channels can have more complicated branch structure. Resonance poles are distinct from a visible peak, and crossed-channel dynamics also generates left-hand cuts. The following slide illustrates the simplest one-channel sheet topology.

First and second Riemann sheets

Hanhart & MM., Scattering Theory, §6

PDG: Resonances

Physical sheet

The red curve follows the physical real-energy axis.

Second sheet

Resonance poles influence the amplitude on the physical axis.

Analytic continuation

The upper physical sheet joins the lower second sheet across the cut.

$$k(s)\propto\sqrt{s-s_{\mathrm{th}}}\quad\text{near threshold},\qquad k_{\mathrm{II}}=-k_{\mathrm I}.$$

A threshold is a branch point: the two signs of the square root lead to two sheets.

Original Figure 6 from Hanhart and Mikhasenko, Scattering Theory, arXiv:2606.10634v2 (24 August 2026), PDF page 9. Extracted directly from the PDF without redrawing or changing colours; the three panels and their labels are preserved. This schematic visualizes the amplitude’s sheet structure, not a measured energy spectrum. The chapter’s caption identifies a conjugate pair of resonance poles on sheet II and the physical axis in red. Panel (c) combines sheet I at Im s>0 with sheet II at Im s<0; it is not the entire two-sheet Riemann surface displayed simultaneously. The near-threshold square-root expression is schematic for one stable two-body channel. Additional channels bring additional sheets. In a selected analytic sheet the pole location is complex; conventionally sqrt(s_p)=M-i Gamma/2. Bound states lie on the physical sheet below threshold; virtual states on an unphysical sheet below threshold. Do not confuse amplitude values with the complex s coordinates.

Let’s play the game

Open the app → mmikhasenko.github.io/riemann-sheet-rpg

Interactive break immediately after First and second Riemann sheets. Screenshot supplied by the lecturer, showing Expedition 06. Click the full-slide image to open the live game: https://mmikhasenko.github.io/riemann-sheet-rpg/. The screenshot remains available offline and in print; playing requires access to the website. Ask students to watch the sheet label as they cross a branch cut. Image source: codex-clipboard-7b0a4793-824d-48b5-9e3b-0adb0758f0d2.png, supplied 24 October 2026.

Quark-model states and thresholds

Guo et al., Hadronic molecules

Charmonium and the lowest S-wave thresholds · historical overview

A quark-model level is not isolated from hadron pairs with the same quantum numbers.

$$c\bar c\ \longleftrightarrow\ D^{(*)}\bar D^{(*)}.$$

⟷

The coupling shifts levels and, above an allowed threshold, opens decays.

For a bound state, schematically,

$$|\Psi\rangle=a|Q\bar Q\rangle+\sum_i b_i|H_{1i}H_{2i}\rangle.$$

The hadron-pair part is the molecular component.

Copied figure from slide 30, image66.png, with the original attribution to A. Nefediev’s MIAPbP2023 talk. Retain as a historical conceptual illustration, not an updated mass/assignment compilation. The displayed decomposition is schematic: continuum terms include momentum integrals and depend on the definition of the basis. For unstable resonances the coefficients cannot in general be interpreted as ordinary probabilities. Coupling to the continuum is compatible with a useful quark-model description.

Why a molecule can be a good model

Guo et al., Hadronic molecules

Bound → virtual → resonance: a pole can move continuously as the interaction changes.

These are different pole locations, not three unrelated kinds of constituent.

A shallow S-wave bound state has a long-distance tail:

$$\psi(r)\propto\frac{e^{-\kappa r}}r,\qquad \kappa=\sqrt{2\mu B},\quad B=E_{\mathrm{th}}-E_{\mathrm{bound}}.$$

When $1/\kappa$ exceeds the interaction range, a hadron-pair description becomes especially useful.

Original slide 31, image67.gif; animation credited there to GitHub/mmikhasenko. This is a model demonstration, not measured pole motion. Natural units. Weak binding in an S wave, interaction range, other nearby channels and possible short-distance fine tuning matter. A threshold alone does not guarantee molecular dominance. Bound, virtual and resonant states differ in sheet and pole position; a bound-to-virtual trajectory does not inevitably become a resonance for every potential. The print strip samples the original animation in the three labelled regimes.

Summary and discussion

Hadron families

10 + 24 + X

Conventional hadrons probe QCD in different regimes—from light-quark dynamics to heavy-quark systems.

Exotic hadrons expand the possibilities for how QCD binds matter.

Experimental discoveries

A golden age of hadron spectroscopy: the past 15 years.

Belle / Belle II + BESIII + LHC: new forms of hadronic matter, a richer understanding of what QCD allows—and many puzzles still open.

Hadrons as resonances

S-matrix-driven physics: understand hadrons through their scattering and decay amplitudes.

Intricate mathematics of cuts, poles and Riemann sheets—with ideas reaching far beyond hadron physics.

Hadronic S-matrix → Veneziano amplitude → string theory

10 + 24 uses the lecture’s flavour/isospin grouping: ten meson categories and 24 baryon families (6 light, 8 singly heavy, 6 doubly heavy, 4 triply heavy). Charge partners and antiparticles are grouped; predicted heavy families are included. X denotes the open-ended exotic sector, not a numerical count or the particular X(3872). Red consistently identifies the exotic sector on this slide.

The past fifteen years is an outlook on the field’s exceptionally productive recent period, not a quantitative assertion that every year exceeds historical discovery rates. Belle and its successor Belle II are named together to acknowledge continuity; Belle II did not operate throughout this period. New observations reveal allowed configurations but do not by themselves settle compact versus molecular interpretations.

The S-matrix viewpoint connects observed transitions with the analytic structure of amplitudes. Stable hadrons are asymptotic states; unstable hadrons appear as resonance poles. The historical arrow refers to the hadronic S-matrix and dual-resonance programme that led to Veneziano’s 1968 amplitude and its subsequent string interpretation. It is not a claim that string theory is an established quantitative description of the observed hadron spectrum. Historical source: G. Veneziano, The roots and fruits of string theory, CERN Courier (2018); original paper, Nuovo Cimento A 57 (1968) 190–197, https://cds.cern.ch/record/390478. Amplitude background: Hanhart & Mikhasenko, https://arxiv.org/abs/2606.10634.