Hadron Spectroscopy

Amplitude analysis: how we find hadrons in decays

Mikhail Mikhasenko · Lecture 2
AEPSHEP 2026 · Taipei

Lecture 2 roadmap

A

The puzzle — a peak in J/ψ p

B

Kinematics — phase space, the Dalitz plot, bands and reflections

C

Spin — rotations, decay chains, the full amplitude

D

Mass dependence — line shapes and form factors

E

The full model — Λb → J/ψ p K⁻ and the pentaquarks

F

Near threshold — Tcc⁺

Λb → J/ψ p K⁻, the 2015 pentaquark discovery, frames the lecture: the puzzle at the start, the full model at the end. B⁺ → D⁺D⁻K⁺ is the spinless example in the kinematics section; Tcc⁺ closes as a near-threshold case.

The puzzle

A peak is not yet a particle

A narrow peak in J/ψ p

LHCb · PRL 115, 072001 (2015)

LHCb 2015: about 26 000 $\Lambda_b^0\to J/\psi\,p\,K^-$ decays, almost no background.

The J/ψ p mass shows a narrow peak near 4.45 GeV. Phase space (red) has none.

A $J/\psi\,p$ resonance would contain $c\bar c\,uud$: at least five quarks.

Is it a particle? How would we know?

Original LHCb figure from the 2015 paper (Bormio PcOld: mjpsip-phsp). The red histogram is phase-space simulation normalised to the data. This is the question of the lecture.

Reconstructing Λb → J/ψ p K⁻

LHCb · PRL 115, 072001 (2015)

MM · collision sketches

A Λb is produced in a pp collision and flies a few millimetres before it decays weakly.

Four charged tracks — $p$, $K^-$, $\mu^+$, $\mu^-$ — come from one displaced vertex. The muon pair has the J/ψ mass.

Exclusive: every final-state particle is measured. Each event is a full set of four-momenta.

Schematic, not to scale. The J/ψ decays at the Λb vertex on this scale. Exclusivity is what makes the full kinematics, and therefore an amplitude analysis, possible.

The Dalitz plot of the data

LHCb · PRL 115, 072001 (2015)

Each event is one point: two pair masses squared.

Vertical bands: $\Lambda^*\to pK^-$ resonances, strongest the Λ(1520).

Horizontal band at $m^2(J/\psi p)\approx 19.8$ GeV²: the new structure.

The density is not uniform. Part of it is detector efficiency, the rest is physics.

Original LHCb 2015 Dalitz plot (dlz.pdf). 19.8 GeV² corresponds to 4.45 GeV. Section B explains why phase space alone would fill this plot uniformly.

Kinematics

What can vary in a three-body decay

From events to a probability

Hanhart & MM · arXiv:2606.10634

PDG · Kinematics (2025)

One event tells us little. Many events sample a density over the allowed momenta $x$.

$$dN\;\propto\;\epsilon(x)\;I(x)\;d\Phi(x)$$

Lecture 1: amplitudes for the same final state interfere before squaring.

Today: build A(x) for Λb → J/ψ p K⁻, then fit it to the events.

Three factors, three questions

$d\Phi(x)$ — phase space: what kinematics allows.

$I(x)=\sum_{\text{spins}}|\mathcal A(x)|^2$ — dynamics: what we want to learn.

$\epsilon(x)$ — detector: efficiency, known from simulation.

A structure in the data is physics only if it is in $I(x)$.

Natural units. I(x) is averaged over the initial spin states and summed over unobserved final spins (section C). Resolution and background enter later, in the likelihood.

Phase space

PDG · Kinematics (2025)

Hanhart & MM · arXiv:2606.10634

$$d\Phi_n\;=$$

$$\prod_{i=1}^{n}\frac{d^3\mathbf p_i}{(2\pi)^3\,2E_i}$$

$$(2\pi)^4\,\delta^{(4)}\Big(P-\sum_{i=1}^{n}p_i\Big)$$

Each particle

Three momentum components $\mathbf p_i$. Dividing by $2E_i$ counts states the same way in every frame.

On the mass shell

$E_i=\sqrt{m_i^2+|\mathbf p_i|^2}$: the energy is fixed by the mass, not a new number.

Conservation

Total energy and momentum equal the parent’s: four constraints remove 4 numbers.

Convention: (2π)⁴ inside dΦ; dΓ = |A|² dΦ / (2M). The factors of 2π make dΦ the number of states per unit phase-space volume. The next two slides integrate the δ-function for two and three bodies.

Two-body decay: one fixed momentum

PDG · Kinematics (2025)

In the parent rest frame the daughters fly back to back.

Energy conservation fixes the momentum:

$$q=\frac{\sqrt{\lambda(M^2,m_1^2,m_2^2)}}{2M}$$

Only the direction is free:

$$d\Phi_2=\frac{1}{8\pi}\;\frac{2q}{M}\;\frac{d\Omega}{4\pi}$$

$2q/M\to1$ for massless daughters; $d\Omega/4\pi$: all directions equally likely.

λ(a,b,c) = a² + b² + c² − 2ab − 2ac − 2bc. Derivation in backup.

Convention: (2π)⁴ inside dΦ. q → 0 at threshold: remember this for the barrier factors in section D. Integrated two-body phase space is q/(4πM).

Recursive phase space formulation

Byckling & Kajantie · Particle Kinematics (1973)

H. Jonas, DEMOS, Decay-Web-Page

Group p and K⁻: first $\Lambda_b\to(pK^-)\,J/\psi$, then $(pK^-)\to p\,K^-$.

$$d\Phi_3=\frac{dm^2_{pK}}{2\pi}\;d\Phi_2\;d\Phi_2$$

An exact identity for every event: no resonance is assumed. Resonances will live in the same two steps.

Integrate the three orientation angles:

$$d\Phi_3\;\propto\;dm^2_{pK}\;dm^2_{J/\psi p}$$

H. Jonas, DEMOS, Decay-Web-Page: the chain 0 → (12) + 3, then (12) → 1 + 2.

The full measure: dΦ₃ = dm²₁₂ dm²₂₃ dR / (1024 π⁵ M²) with ∫dR = 8π² (derivation in backup). The constant density in the two pair masses is the Dalitz plot’s defining property.

Five numbers: shape and orientation

R. H. Dalitz · Phil. Mag. 44, 1068 (1953)

JPAC · PRD 101, 034033 (2020)

Three momenta: $3\times3=9$ numbers.

Energy–momentum conservation removes 4.

$$\dim\Phi_n=3n-4=5$$

Two pair masses fix the shape of the momentum triangle: they span the Dalitz plot.

Three Euler angles orient it in space.

Spin projections are labels, not extra coordinates.

Quiz answer: B. Move the rotation slider: the arrows turn, the Dalitz point does not move. Move m(pK) or cos θ: the triangle changes shape and the point moves. The widget shows one rotation angle; a general orientation has three.

The phase space is flat

PDG · Kinematics (2025)

Phase-space events fill the plot uniformly inside the boundary.

A projection counts the height of the plot at each mass: its curved shape is pure kinematics.

In $m$ rather than $m^2$: $\;dN/dm=2m\,dN/dm^2$.

Any non-uniformity in the data is dynamics or detector.

Physical Λb, p, K, J/ψ masses; no dynamics, no detector. Increase the event count to see the histogram approach the red curve. Tick the box to project onto m(pK) instead of m²(pK).

One mass and one angle

JPAC · PRD 101, 034033 (2020)

Fix $m(pK)$: the event moves along a vertical slice.

The position on the slice is an angle: the proton direction in the pK rest frame.

$$m^2_{J/\psi p}=a+b\cos\theta$$

The Dalitz plot is a mass and an angle. Angular distributions sit along every band.

θ is the helicity angle: between the proton and the direction opposite the J/ψ, in the pK rest frame. a and b depend on m(pK) only; cos θ = ±1 are the boundary. This is why angular momentum shows up as patterns along resonance bands.

Bands and reflections

LHCb · PRD 102, 112003 (2020)

Simplest case first: $B^+\to D^+D^-K^+$, all spin 0.

A resonance in $D^-K^+$ makes a band at fixed $m(D^-K^+)$.

Projected onto $m(D^+D^-)$, the same band gives a broad structure: a reflection. No $D^+D^-$ resonance is needed.

Narrow peaks are hard to fake. Broad structures are easy.

Physical B, D and K masses, one Breit–Wigner times P_J(cos θ): exact for spinless particles. Move the D⁻K⁺ mass: its reflection moves across m(D⁺D⁻). The same happens in Λb → J/ψ p K⁻, where Λ* bands reflect into m(J/ψ p). Section C adds spin.

Spin leaves a pattern along a band

JPAC · PRD 101, 034033 (2020)

Spin-$J$ resonance → two spinless particles:

$$\mathcal A\propto d^J_{00}(\theta)=P_J(\cos\theta)$$

$J$ zeros along the band.

Spin also shapes the reflection: bumps in $m(D^+D^-)$ without any $D^+D^-$ state.

B⁺ → D⁺D⁻K⁺ with physical masses; exact for spin-0 particles. Switch J between 0 and 3 and count the zeros. For an isolated band the intensity is |P_J|²; the amplitude’s sign matters only once a second contribution interferes.

B⁺ → D⁺D⁻K⁺: the data

LHCb · PRD 102, 112003 (2020)

LHCb · PRL 125, 242001 (2020)

Horizontal bands: charmonia in D⁺D⁻

m(D⁻K⁺) with the full amplitude fit

The pattern along the D⁻K⁺ band, as in the widget, fixes the spin.

Charmonium bands reflect into m(D⁻K⁺), but cannot explain the peak at 2.9 GeV.

The fit needs D⁻K⁺ states with spin 0 and 1. Quark content $\bar c\bar s ud$: manifestly exotic.

Now named T*_{c̄s̄0}(2870)⁰ and T*_{c̄s̄1}(2900)⁰, originally X0(2900) and X1(2900). The spin assignment comes from the pattern along the D⁻K⁺ band, exactly as in the previous widget. The model-independent PRL showed that D⁺D⁻ resonances alone cannot describe the data.

Spin

Angular patterns, decay chains and frames

Spin rotation, revisited

JPAC · PRD 101, 034033 (2020)

$$d^{\,j}_{m'm}(\theta)$$

$d^{\,j}_{m'm}(\theta)=\langle j,m'|\,e^{-i\theta J_y}\,|j,m\rangle$: the Wigner d-functions are the coefficients above. Amplitudes, with signs.

With three Euler angles, $\;D^{j}_{m'm}(\alpha,\beta,\gamma)=e^{-im'\alpha}\,d^{\,j}_{m'm}(\beta)\,e^{-im\gamma}$.

Lecture 1 widget. It shows the column m = j only: rotating |j, j⟩ spreads it over all projections m. Every angular factor of a decay amplitude is built from these functions.

One decay vertex

JPAC · PRD 101, 034033 (2020)

H. Jonas, DEMOS, Decay-Web-Page

For $A(J,m)\to 1+2$ with helicities $\lambda_1,\lambda_2$:

$$\mathcal A_{m;\lambda_1\lambda_2}=\underbrace{D^{J*}_{m,\lambda_1-\lambda_2}(\phi,\theta,0)}_{\text{rotations: fixed}}\;\underbrace{H_{\lambda_1\lambda_2}}_{\text{fitted}}$$

Helicity: spin along the particle’s momentum.

The angles are those of particle 1 in the rest frame of A.

How many couplings H?

One per allowed $(L,S)$ combination.

Strong $\Lambda(1520)\,[\tfrac32^-]\to p\,K^-$: parity allows only $L=2$ → 1 coupling.

Weak $\Lambda_b\to\Lambda^*J/\psi$: no parity constraint → 4 couplings ($J=\tfrac12$) or 6 ($J\ge\tfrac32$).

Each is a complex number to fit.

D^{J*} carries the angular dependence fixed by rotation symmetry; H holds the dynamics. Convention-dependent phases can be absorbed into H, but must be used consistently across chains. The counting 4 and 6 is the one used in the 2015 LHCb analysis.

A decay chain

JPAC · PRD 101, 034033 (2020)

MM · cascade sketches

$\Lambda_b\to\Lambda^*J/\psi$, then $\Lambda^*\to pK^-$:

$$\mathcal A^{\Lambda^*}=\sum_{\nu}\;\mathcal A^{\Lambda_b\to\Lambda^*\psi}_{\nu}\;\,\mathrm{BW}(m^2_{pK})\;\,\mathcal A^{\Lambda^*\to pK}_{\nu}$$

Two vertices, each in its own rest frame, joined by a propagator.

The Λ* helicity $\nu$ is not observed and is internal: sum the amplitudes.

Labels Λ, μ, λ beside the external lines are spin projections; the internal Λ is the Λ*.

External spin labels (λ_Λb, λ_p, λ_ψ) remain open indices. The J/ψ → μ⁺μ⁻ decay adds one more vertex and two angles. Live: Decay-Web-Page “Amplitude construction” builds this product step by step.

Two chains, one final state

LHCb · PRL 115, 072001 (2015)

Λ* chain: $b\to c\bar c s$; the $s$ quark forms an excited Λ* with $ud$.

Pc chain: the $c\bar c$ pair stays with $uud$ and recoils against $K^-$.

Same particles, same momenta: the amplitudes add, then we square.

Original LHCb diagrams. The total amplitude is the sum over all Λ* and all Pc contributions, each with its own couplings. Both diagrams are Cabibbo favoured.

The full amplitude

JPAC · PRD 101, 034033 (2020)

LHCb · PRL 115, 072001 (2015)

One decay chain: two vertices and a propagator.

$$D^{J_0*}(\Omega_0)$$

angles

$$FF(m_0;\,m_R,\,m_k)$$

form factor

vertex Λb → R k

$$R(m_R^2)$$

line shape

R propagates

$$D^{J_R*}(\Omega_R)$$

angles

$$FF(m_R;\,m_i,\,m_j)$$

form factor

vertex R → i j

Σ over Λ* states

Σ over Pc states

$$\mathcal A_\lambda=\sum_{\text{chains }k}\,c_k\,A^{(k)}_\lambda\qquad\qquad I=\sum_{\lambda}\big|\mathcal A_\lambda\big|^2$$

Rotations: fixed by spin (this section)

Mass dependence: model choices (next section)

$c_k$: complex couplings, fitted

Coherent over chains, incoherent over unobserved spins. Pc chains also need the spin alignment d(ζ) (backup).

A^{(k)} is the product of the five factors with the helicity couplings of both vertices, summed over the isobar helicity. The 2015 analysis: 14 Λ* states and two Pc states, each with several helicity couplings; data were m(pK) and five decay angles, including J/ψ → μ⁺μ⁻. For an unpolarised Λb, averaging over λ_Λb is a sum with weight ½.

Frames, live

H. Jonas, DEMOS, Decay-Web-Page

Start in the laboratory: the Λb flies.

Rotate, then boost, into the Λb rest frame.

Repeat for the pK⁻ pair: its rest frame, with the z axis along its flight direction.

Every angle in the amplitude is measured in such a frame.

Open the Dynamic view

Switch to the website. Press “Frame »” twice: lab → mother rest frame → (12) rest frame. Each step is Rz(−φ), Ry(−θ), Bz(−η). The screenshot is the print fallback.

Mass dependence

Line shapes and vertex form factors

The resonance line shape

PDG · Resonances (2025)

Hanhart & MM · arXiv:2606.10634

$R(m_R^2)$: how the resonance propagates. The typical construction is a Breit–Wigner with a running width:

$$R(\sigma)=\frac{1}{m_R^2-\sigma-i\,m_R\,\Gamma(\sigma)}$$

$$\Gamma(\sigma)=\Gamma_R\,\frac{q}{q_R}\,\frac{m_R}{\sqrt\sigma}\,FF_L^2(\sigma)$$

A peak in $|R|^2$, and a phase that rises through 90° (Lecture 1). The width carries the decay form factor (next slide).

Default: Λ(1520), J^P = 3/2⁻, decaying to pK⁻ in D wave (L = 2). Raise L or move m₀ towards threshold to see the running width matter. Alternatives for broad states or nearby thresholds: Flatté and K-matrix (coupled channels), as in Lecture 1 and the Tcc section.

Form factors

PDG · Resonances (2025)

von Hippel & Quigg · PRD 5, 624 (1972)

$FF(m_0;\,m_1,m_2)$: the vertex where $m_0\to m_1+m_2$, with breakup momentum $q$.

$$FF_L=\Big(\frac{q}{q_0}\Big)^{L}\,\frac{B_L(qr)}{B_L(q_0r)}$$

$q^L$: the angular-momentum barrier. Partial wave $L$ is suppressed near threshold.

Blatt–Weisskopf $B_L$: stops the growth for $q\gtrsim1/r$, with $r\approx$ 1.5–5 GeV⁻¹ (0.3–1 fm).

B₁ = 1/√(1+z²), B₂ = 1/√(9+3z²+z⁴), z = qr. Two vertices, two form factors: production uses p, decay uses q.

Normalised to 1 at the resonance mass m₀, so that the coupling keeps its meaning. Constants differ between conventions; insert the threshold power only once (some codes include q^L inside the Blatt–Weisskopf factor). r is a model parameter, not a measured size; fits often vary it as a systematic. Move r: small r gives the bare q^L, large r flattens the curves above q ≈ 1/r.

The full model: Λb → J/ψ p K⁻

Spin, interference and the pentaquarks

With a proton, the zeros fill in

JPAC · PRD 101, 034033 (2020)

ThreeBodyDecays.jl

Back to $\Lambda^*\to pK^-$: the proton has two helicities, $\pm\tfrac12$.

Its spin is not measured: add $|A|^2$ for both.

Each term has zeros; the sum does not. For $J=\tfrac32$:

$$|d^{3/2}_{\frac12\frac12}|^2+|d^{3/2}_{\frac12,-\frac12}|^2\propto 1+3\cos^2\theta$$

Spin is still visible, but less sharply. LHCb also used the J/ψ → μ⁺μ⁻ angles.

Full helicity amplitudes for Λb → Λ* J/ψ, Λ* → pK⁻ (Dalitz-plot decomposition, as in ThreeBodyDecays.jl; the JS port agrees with the package to 1e−15). Unpolarised Λb, all spins summed; one illustrative helicity coupling at the weak vertex. Switch the Λ* spin: ½ is flat, higher spins change the shape but never reach zero.

Unpolarised and polarised parents

JPAC · PRD 101, 034033 (2020)

LHCb · PLB 724, 27 (2013)

LHCb · Λc⁺ polarimetry, JHEP 07, 228 (2023)

ComPWA · polarimetry

The parent’s spin states form an ensemble, described by a density matrix:

$$I(x)=\sum_{\text{final spins}}\;\sum_{m,m'}\rho_{mm'}\,\mathcal A_m\mathcal A^*_{m'}$$

Unpolarised ($\rho\propto 1$): no preferred direction. Integrate the orientation; the Dalitz variables remain.

Polarised: the production angles — the orientation of the decay relative to the polarisation axis — become anisotropic: $1+\alpha P\cos\theta$.

Λb at LHCb: P compatible with zero. Λc⁺ → pK⁻π⁺ as a polarimeter: paper, website.

LHCb measured the Λb polarisation at 7 TeV: P = 0.06 ± 0.07 ± 0.02. Internal angular correlations (along the bands) remain even for an unpolarised parent. The widget is an illustrative spin-½ → ½ + 0 decay; parity-conserving strong decays have α = 0.

Interference reshapes the plot

JPAC · PRD 101, 034033 (2020)

ThreeBodyDecays.jl

$$|A_1+A_2|^2=|A_1|^2+|A_2|^2+2\,\mathrm{Re}\,A_1A_2^*$$

The cross term follows the relative phase: the Dalitz pattern and the Pc line shape change.

Summing over spins dilutes it, but does not remove it.

The projection alone cannot tell. The fit measures the phase.

Full helicity amplitudes: Λ*(3/2⁻) at 1.75 GeV and Pc(3/2⁻) at 4.45 GeV, one illustrative weak coupling each, alignment included (dpd.js, checked against ThreeBodyDecays.jl). Move φ and watch the region where the bands cross. Untick “coherent” to remove the cross term. Fit fractions need not add to one because of these terms.

Λ* states alone fail

LHCb · PRL 115, 072001 (2015)

m(Kp): the Λ* states describe their own channel

m(J/ψ p): the peak is missed

Fit with all 14 Λ* states and every allowed coupling.

The reflections are included, with all interference.

The narrow structure near 4.45 GeV is still not described.

“Extended” Λ* model of the 2015 paper, no Pc. Red points: the fit. This is the hypothesis test that the reflection widget prepared.

Adding Pc states

LHCb · PRL 115, 072001 (2015)

Add two J/ψ p states to the Λ* model: now the data are described.

A narrow Pc(4450)⁺ (Γ ≈ 40 MeV) and a broad Pc(4380)⁺ (Γ ≈ 200 MeV). Stat. significance: 12σ and 9σ.

Opposite parities; best spins $\tfrac32$ and $\tfrac52$, from the decay angles.

There was more to discover…

Extended Λ* model plus two Pc states. A single Pc improved the fit but did not describe the data. Best fit J^P = (3/2⁻, 5/2⁺); (3/2⁺, 5/2⁻) and (5/2⁺, 3/2⁻) were also acceptable. The spins come from the angular distributions, not from the mass projections shown.

2019: more data, a new picture

LHCb · PRL 122, 222001 (2019)

Nine times more Λb decays

Three narrow peaks in m(J/ψ p)

The 4450 structure splits into Pc(4440)⁺ and Pc(4457)⁺. A new Pc(4312)⁺ appears.

A fit of the mass projection: spins are not determined; the broad Pc(4380)⁺ neither confirmed nor excluded.

The 2019 analysis fitted m(J/ψ p) with a weighting that suppresses Λ* reflections, not a full amplitude analysis. A full amplitude analysis of the larger sample is the next step.

Pentaquarks: measured and interpreted

LHCb · PRL 122, 222001 (2019)

Hanhart & MM · arXiv:2606.10634

Measured

Three narrow peaks: Pc(4312)⁺, Pc(4440)⁺, Pc(4457)⁺, widths 6–20 MeV.

Each sits 2–20 MeV below a $\Sigma_c^+\bar D^{0}$ or $\Sigma_c^+\bar D^{*0}$ threshold. Spins not yet determined.

Interpreted

$\Sigma_c\bar D^{(*)}$ molecules give this pattern naturally (Lecture 1). Compact pentaquarks and threshold effects are also discussed.

Deciding needs spins, partner states and poles — from amplitude analyses.

2019 values: Pc(4312)⁺ M = 4311.9 ± 0.7, Γ = 9.8 ± 2.7 MeV; Pc(4440)⁺ 4440.3 ± 1.3, Γ = 20.6 ± 4.9 MeV; Pc(4457)⁺ 4457.3 ± 0.6, Γ = 6.4 ± 2.0 MeV (statistical uncertainties). Σc⁺D̄⁰ threshold ≈ 4317.5 MeV, Σc⁺D̄⁰ ≈ 4459.5 MeV. In the molecular picture: 4312 as Σc D̄ (½⁻), 4440/4457 as the Σc D̄ pair (½⁻ and 3/2⁻, order not fixed). A Breit–Wigner peak is a parametrisation; a pole and its nature need amplitudes with the right analytic structure (Lecture 1).

2027: the full amplitude analysis

LHCb · PRL 122, 222001 (2019)

The full amplitude analysis of the Run 1 + 2 sample is about to be finished.

It is very complicated.

Why it takes so long

High statistics: small effects become significant; the model must describe them all.

Coupled-channel dynamics: narrow states at the Σc D̄(*) thresholds need more than Breit–Wigners.

Fixes in the formalism: interference between chains needs consistent spin alignment.

The 2015 analysis used Breit–Wigner line shapes and helicity amplitudes; the Dalitz-plot decomposition (JPAC 2020) settled the alignment of the proton and J/ψ spins between chains. The 2019 results came from a fit of the mass projection only.

Tools for amplitude models

K. Habermann · DecaySelector

H. Jonas, DEMOS, Decay-Web-Page

DecaySelector: declare the particles and their $J^P$; it lists the three decay topologies, collects the resonances in each pair, and counts the complete decay chains.

Decay-Web-Page: frames, rotations and the amplitude of each chain.

ThreeBodyDecays.jl: the amplitudes used in today’s widgets.

Open DecaySelector

Switch to the website if time allows: enter Λb → p K⁻ J/ψ with J^P values, add a Λ* in pK⁻ and a Pc in J/ψ p, and watch the chain count. The screenshot shows the empty starting state.

Near threshold: Tcc⁺

When a Breit–Wigner is not enough

A bound state below threshold

Guo et al. · Hadronic molecules, RMP 90, 015004

Bound state: a pole below the continuum

Weaker attraction: the pole moves (Lecture 1)

Below threshold, a bound state is a pole of the amplitude. Above it, free $D^*D$ pairs form the continuum: a cut.

The weaker the binding, the closer the pole to the threshold; then it becomes virtual.

A shallow bound state is large: $\psi\propto e^{-\kappa r}/r$, $\kappa=\sqrt{2\mu B}$. For $T_{cc}^+$, $B\approx0.36$ MeV gives $1/\kappa\approx7.5$ fm.

Bormio talk, frame “Classical bound-state picture (hadronic molecule)”, figure pole_cut. Animation from Lecture 1 (“Why a molecule can be a good model”). μ is the D*⁺D⁰ reduced mass, about 967 MeV. A large size suggests, but does not prove, a molecular structure; the backup shows the compact alternative.

The Tcc⁺ decay amplitude

LHCb · Nature Phys. 18, 751 (2022)

LHCb · Nature Commun. 13, 3351 (2022)

MM et al. (JPAC) · JHEP 08, 080 (2019)

Two D*D channels, three final states

The unitarised model fit to D⁰D⁰π⁺

$J^P=1^+$: S-wave decay to $D^*D$. An isoscalar: equal couplings to $D^{*+}D^0$ and $D^{*0}D^+$.

The model is unitary: the $D^*$ width and all three-body channels enter the line shape. Two parameters: peak position and coupling.

Pole: $\delta m=-360\pm40$ keV, $\Gamma\approx48$ keV, below the $D^{*+}D^0$ threshold. A constant-width Breit–Wigner misses the threshold.

Bormio talk, frame “Tcc⁺ decay amplitude”, with the fit of the unitarised model. Isoscalar wave function |T⟩ = (|D⁰D⁺⟩ − |D⁺D⁰⟩)/√2; no isospin violation in the couplings. The two D⁰ in D⁰D⁰π⁺ are identical: the amplitude is symmetrised. With a plain Breit–Wigner, δm = −273 keV and Γ = 410 keV; the unitarised pole gives Γ ≈ 48 keV. |g| > 7.7 (6.2) GeV at 90 (95)% CL.

Tcc and Tbb: experiment and lattice

LHCb · Nature Phys. 18, 751 (2022)

Colquhoun et al. · PRD 110, 094503

Experiment: T_cc⁺, barely bound

r_h = m_b / m_h

Lattice QCD: binding versus heavy-quark mass

Lattice QCD at $m_\pi\approx 300$ MeV: binding energy versus heavy-quark mass; bottom at $r_h=1$, charm at $r_h\approx3$.

Heavier quarks bind deeper: $T_{bb}$ predicted far below threshold, stable against strong decay.

At the charm mass the lattice point is near threshold; the measured $T_{cc}$ is 0.36 MeV below. $T_{bc}$ has a good chance to be discovered at the LHC.

Lattice: Colquhoun et al., PRD 110, 094503 (2024), as in Figure 5 of the Corfu proceedings. r_h = m_b/m_h: r_h = 1 is the bottom quark, r_h ≈ 3.1 the charm quark. The original figure’s axis label said m_h/m_b and has been replaced. The lattice charm-mass extrapolation is not a precision prediction of the Tcc binding.

What we have discussed

Kinematics

Five numbers per three-body event: two for the shape (the Dalitz plot), three for the orientation.

Phase space is flat; bands, reflections and spin zeros are dynamics.

Spin

Each vertex: rotation × coupling. Chains with the same final state add as amplitudes.

Mass dependence

Line shapes and form factors are model choices, tested by the fit.

Two examples: narrow tetraquarks and pentaquarks

A peak, then a model of the whole decay. Λ* states alone fail; Pc states are needed. Their nature is still open.

T_cc⁺: a narrow peak just below the D*D threshold, where a Breit–Wigner is not enough.

Questions to discuss: which parts of the amplitude are fixed by symmetry, which are model choices, and what does the experiment constrain directly?

The exotic-hadron landscape

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

The closing image from Lecture 1. Illustration: I. Polyakov; Figure 2 of Hüsken, Spadaro Norella and Polyakov, A Brief Guide to Exotic Hadrons, https://arxiv.org/abs/2410.06923.

Backup

Tcc⁺ structure

Backup: Tcc⁺, two extreme spatial configurations

LHCb · Nature Commun. 13, 3351 (2022)

Molecule: D⁰ and D*⁺ held by π, ρ, … exchange

Compact: a (cc) core with ū d̄

Molecular configuration

Two mesons, well separated, bound like van der Waals forces. Entirely coupled to D*⁺D⁰; the lifetime is limited by the D*⁺. Spatially extended.

Compact configuration

A genuine QCD state with a compact (cc) core, about 1 fm. The lifetime depends on how strongly it couples to the continuum.

Bormio talk, frame “Tcc⁺: two extreme spatial configurations”; illustrations from the LHCb Nature Communications paper. These are limiting pictures, not exclusive alternatives.