Quarkonium & bound states

A bound state is simplest when its constituents crawl — when they move so far below $c$ that nonrelativistic quantum mechanics, Schrödinger's equation with a potential $V(r)$ , describes them faithfully. The test is a comparison of energies: if the binding energy is tiny beside the rest energies of the pieces, then by the virial theorem the kinetic energy is tiny too, and the system is nonrelativistic. Hydrogen, with $13.6$ eV of binding against a $511\,000$ eV electron, passes emphatically. Light-quark hadrons fail: a pion's binding is comparable to the effective mass of its $u$ and $d$ quarks, so they are ultrarelativistic and no potential picture survives. Between these extremes sits a gift. The confining scale of the strong force is a few hundred MeV — huge next to a light quark, but small next to the $1.5$ GeV charm quark and $4.7$ GeV bottom quark. So a heavy quark and its own antiquark, $c\bar c$ (charmonium) or $b\bar b$ (bottomonium), form a slow, deeply nonrelativistic bound state: the hydrogen atom of QCD. This article pushes that analogy until it breaks — building the radial Schrödinger equation, replacing the Coulomb law with the QCD Cornell potential (a short-range gluon Coulomb term welded to a long-range linear confining term), and reading off a spectrum of narrow states whose masses, spin splittings, and near-identical level spacings make it one of the sharpest windows on confinement.

Key takeaways

Quarkonium is the nonrelativistic $c\bar c$ and $b\bar b$ bound-state family; because binding $\ll$ heavy-quark rest energy, the Schrödinger equation $H\psi=E\psi$ applies just as for hydrogen and positronium.
The interaction is the Cornell potential $V(r)=-\tfrac{4}{3}\alpha_s/r+\sigma r$ : a $-4/3$ colour-Coulomb term from one-gluon exchange plus a linear confining term with $\sigma\approx0.18\ \mathrm{GeV}^2\approx0.9\ \mathrm{GeV/fm}$ .
The spectrum is hydrogen-like in structure ( $n,\,l,\,J^{PC}$ ) but not in scaling: the $2S{-}1S$ gap is $589$ MeV in charmonium and $563$ MeV in bottomonium — nearly equal despite $m_b\approx3\,m_c$ .
That near-equality kills pure-Coulomb binding (which predicts a factor- $3$ change) and fingerprints an effectively logarithmic potential — the Coulomb-to-linear crossover of the Cornell form.
The states are narrow ( $\Gamma_{J/\psi}\approx93$ keV, a thousand times below a typical hadron) because they sit below open-charm threshold and decay only through OZI-suppressed gluon annihilation.
Key idea: heavy-quark quarkonium is the hydrogen atom of QCD — slow enough for a Schrödinger potential to work — and its spectrum reads the QCD force directly, a Coulomb gluon at short range welded to an unbreakable linear string at long range.

The core insight

Because a heavy quark and antiquark move slowly compared with $c$ , their bound state obeys the ordinary Schrödinger equation with the QCD Cornell potential $V(r)=-\tfrac43\alpha_s/r+\sigma r$ ; the resulting charmonium and bottomonium spectra map the strong force from its short-range Coulomb gluon to its long-range linear confinement.

Where this lives today

Quarkonium is a living precision tool. Lattice QCD now computes the charmonium and bottomonium spectra directly from the QCD path integral — a headline validation of the theory in its strongly coupled regime, and the source of the $\alpha_s$ and $\sigma$ used everywhere else. At the frontier, BESIII in Beijing and Belle II in Japan map these regions for states the simple model does not predict — the " $XYZ$ " exotics such as the $X(3872)$ , tetraquark and molecular candidates sitting suspiciously near open-flavour thresholds where the $q\bar q$ picture breaks down. In heavy-ion collisions at the LHC and RHIC, quarkonium becomes a thermometer: a hot quark–gluon plasma Debye-screens the colour force and "melts" the loosest states first, so the sequential suppression of $\psi'$ , then $\chi_c$ , then $J/\psi$ gauges the plasma temperature and signals deconfinement. On the theory side, the effective field theories NRQCD and pNRQCD turn the nonrelativistic expansion this article sketches into a controlled approximation with computable corrections.

The physical picture

Imagine holding a charm quark still and pulling its antiquark away. At first the two feel a Coulomb-like attraction — a single gluon exchanged between the colour charges, weakened to $\alpha_s$ but otherwise the familiar $1/r$ . Keep pulling and something no atom does happens: the force stops falling off. The gluon field, self-attracting because gluons carry colour, collapses into a narrow flux tube of fixed tension, and the energy climbs linearly with separation. The bound state is a compromise between these regimes, drawn in the first figure with a few levels on its rising wall.

Figure

The Cornell potential V(r) = -(4/3) αs/r + σ r with αs ≈ 0.4 and string tension σ = 0.91 GeV/fm ≈ 15 tonnes of constant force: the attractive one-gluon Coulomb term dominates below about 0.34 fm and the linear confining term dominates beyond it, so the levels 1S, 1P, 2S climb a rising well

Solving the Schrödinger equation in that potential yields a ladder of levels labelled like atomic states — $1S$ , $1P$ , $2S$ , $1D$ — but the rungs barely move when the quark mass triples from charm to bottom. That single fact is the punchline: neither what a Coulomb potential does nor quite what a linear one does. The second figure lays the two spectra on one MeV-per-pixel scale so the near-identical $2S{-}1S$ gaps read off directly from the measured masses.

Figure

Charmonium and bottomonium drawn on one shared MeV-per-pixel scale: the 2S minus 1S gap is 589 MeV for cc̄ and 563 MeV for bb̄ — nearly identical although the bottom quark is about three times heavier, the fingerprint of a logarithmic effective potential rather than a Coulomb one

The core equations

The whole treatment rests on one move: replace the two-body problem by a one-body problem in the relative coordinate, with a reduced mass and a central potential. The angular part factorises into spherical harmonics, and writing $u(r)=rR(r)$ exposes a crucial extra piece.

-\frac{\hbar^2}{2\mu}\frac{d^2u}{dr^2}+\left[V(r)+\frac{\hbar^2\,l(l+1)}{2\mu r^2}\right]u=E\,u,\qquad \mu=\frac{m_q\,m_{\bar q}}{m_q+m_{\bar q}}=\frac{m_q}{2} \qquad (1)

where: $\mu$ — reduced mass; $u(r)=rR(r)$ — reduced radial wavefunction; $l$ — orbital angular-momentum number; $\hbar^2l(l+1)/2\mu r^2$ is the centrifugal barrier. Reading: only special $E$ admit a $u$ vanishing at $r=0$ and normalisable at infinity — the origin of the discrete spectrum. The potential is the physics. QCD supplies a short-distance Coulomb term (one-gluon exchange with a colour factor) and a long-distance linear term (the confining flux tube); the Cornell potential is their sum, and nothing else is needed for the gross spectrum.

V(r)=-\frac{4}{3}\,\frac{\alpha_s}{r}+\sigma\,r \qquad (2)

where: $\alpha_s\approx0.3{-}0.4$ for charmonium; $4/3=C_F=(N_c^2-1)/2N_c$ for $N_c=3$ ; $\sigma\approx0.18\ \mathrm{GeV}^2\approx0.9\ \mathrm{GeV/fm}$ — the string tension. Reading: the first term is attractive and dominant inside $\sim0.34$ fm; the second grows without bound, so there is no ionisation — a quark cannot escape. How the spectrum scales with quark mass is the sharpest diagnostic of $V(r)$ , and it follows from the virial theorem plus a scaling argument. For a pure power law $V\propto r^\nu$ the level spacings obey a clean rule.

\Delta E_{n}\;\propto\;\mu^{-\nu/(\nu+2)} \qquad (3)

where: $\nu$ — power of the potential ( $\nu=-1$ Coulomb, $\nu=+1$ linear, $\nu\to0$ log); $\mu$ — reduced mass. Reading: Coulomb gives $\Delta E\propto\mu$ (spacings grow, as in atoms), linear gives $\Delta E\propto\mu^{-1/3}$ (shrink), and a logarithmic potential gives $\Delta E$ independent of $\mu$ — the observed near-equality. Finally, the vector states ( $J/\psi$ , $\psi'$ , the $\Upsilon$ s) carry the photon's quantum numbers $J^{PC}=1^{--}$ , so they couple to $e^+e^-$ with a width fixed by the wavefunction at the origin — the chance the quark and antiquark sit on top of each other and annihilate.

\Gamma(V\to e^+e^-)=\frac{16\pi\,\alpha^2\,e_q^2}{M_V^2}\,\big|\psi(0)\big|^2 \qquad (4)

where: $e_q$ — quark charge in units of $e$ ( $e_c=\tfrac23$ , $e_b=-\tfrac13$ ); $M_V$ — vector-meson mass; $|\psi(0)|^2$ — squared $S$ -wave wavefunction at zero separation. Reading: this Van Royen–Weisskopf formula turns a leptonic width into a direct probe of the wavefunction, and vanishes for $l\ne0$ since $\psi(0)=0$ .

Derivation AFrom the two-body problem to the radial equation and the centrifugal barrier

The goal is to reduce the six-dimensional Schrödinger problem of a quark and antiquark interacting through a central potential to a single radial equation, exhibiting the centrifugal barrier that pushes higher- $l$ states outward and forbids them from reaching the origin. We separate the centre-of-mass motion, factor the angular dependence into spherical harmonics, and substitute $u=rR$ to reach the clean form that quantises the energy.

Step 1. Split the two-body Hamiltonian into centre-of-mass and relative parts: with $\mu=m_qm_{\bar q}/(m_q+m_{\bar q})$ , the relative motion obeys $[-\hbar^2\nabla^2/2\mu+V(r)]\psi=E\psi$ , and for equal masses $\mu=m_q/2$ . Why: a central potential depends only on $r$ , so the centre of mass drifts freely and all the physics lives in the relative coordinate — exactly as for hydrogen's electron and proton.

Step 2. In spherical coordinates the Laplacian separates, and because $V$ depends only on $r$ the angular solutions are the spherical harmonics: $\psi=R(r)\,Y_l^m(\theta,\phi)$ , with the angular part contributing $l(l+1)$ . Why: rotational symmetry conserves angular momentum, so its eigenfunctions $Y_l^m$ diagonalise the angular part once and for all, leaving only $R(r)$ to find.

Step 3. Substitute $u(r)=rR(r)$ to cancel the first-derivative term, giving $-\tfrac{\hbar^2}{2\mu}u''+[V+\tfrac{\hbar^2 l(l+1)}{2\mu r^2}]u=Eu$ — a 1D problem with an effective potential. Why: the extra $\hbar^2l(l+1)/2\mu r^2$ is the centrifugal barrier — repulsive, growing with $l$ — which keeps orbiting states away from $r=0$ , so only $S$ -waves have $\psi(0)\ne0$ .

Step 4. Impose $u(0)=0$ and $u(r)\to0$ as $r\to\infty$ ; for a confining $V$ every level is bound, and only a discrete set of $E$ satisfies both. Why: quantisation is not put in by hand — it is forced by demanding an acceptable wavefunction at both ends, the same logic behind hydrogen's $E_n=-13.6\ \mathrm{eV}/n^2$ . *Verified: equation (1) reproduces the hydrogenic spectrum when $V=-e^2/4\pi\varepsilon_0 r$ , and the centrifugal term enforces $\psi(0)=0$ for all $l\ge1$ , consistent with the $l=0$ -only leptonic widths of equation (4).*

Go deeper · Derivation B · 1 minThe Cornell potential — colour Coulomb and the confining string

Derivation BThe Cornell potential — colour Coulomb and the confining string

The aim is to justify each half of $V(r)=-\tfrac43\alpha_s/r+\sigma r$ from QCD: that one-gluon exchange between a colour-singlet $q\bar q$ gives an attractive $1/r$ potential with colour factor $C_F=4/3$ , and that the non-abelian gluon field at large separation collapses into a flux tube of constant tension, a term linear in $r$ . Together they interpolate between the perturbative short-distance and confining long-distance regimes, crossing over near $0.34$ fm.

Step 1. At short distance $\alpha_s$ is small (asymptotic freedom), so one-gluon exchange dominates and mimics QED's photon exchange: a $1/r$ potential. Averaging the $SU(3)$ generators over the singlet $q\bar q$ state gives the Casimir $C_F=(N_c^2-1)/2N_c=4/3$ . Why: the $4/3$ is not adjustable — it is fixed by the colour algebra the moment you demand a colour-neutral meson, so $-\tfrac43\alpha_s/r$ is a genuine prediction, not a fit.

Step 2. At large distance $\alpha_s$ grows and perturbation theory fails; gluons, being coloured, attract one another, so the colour field squeezes into a tube of roughly fixed cross-section whose energy is proportional to its length, $\sigma r$ . Why: the linear rise is confinement made quantitative — separation costs energy at a constant rate $\sigma$ , the flux never thins out, and no isolated colour charge can escape.

Step 3. Fix $\sigma$ from data: Regge trajectories and lattice QCD give $\sigma\approx0.18\ \mathrm{GeV}^2$ , which with $\hbar c=0.197\ \mathrm{GeV\,fm}$ is $\sigma\approx0.91\ \mathrm{GeV/fm}$ , a constant force of about $15$ tonnes. Why: the same $\sigma$ that fits light-meson Regge slopes reproduces the charmonium and bottomonium spacings — a cross-check that the linear term is universal, not tuned per system.

Step 4. Locate the crossover by setting the terms equal: $r_\times=\sqrt{\tfrac43\alpha_s\hbar c/\sigma}\approx0.34$ fm for $\alpha_s\approx0.4$ . Charmonium sits on this scale ( $r\sim0.4$ fm), bottomonium tighter ( $r\sim0.2$ fm). Why: because the two systems weight the Coulomb and linear parts differently, comparing their spectra separates the two terms — exactly what the near-equal spacings exploit. *Verified: $C_F=4/3$ from the $SU(3)$ Casimir, $\sigma=0.18\ \mathrm{GeV}^2=0.91\ \mathrm{GeV/fm}$ , and crossover $r_\times=0.34$ fm reproduced numerically for $\alpha_s=0.4$ .*

Go deeper · Derivation C · 1 minWhy charmonium and bottomonium have almost the same level spacings

Derivation CWhy charmonium and bottomonium have almost the same level spacings

The goal is to derive the scaling law $\Delta E\propto\mu^{-\nu/(\nu+2)}$ for a power-law potential $V\propto r^\nu$ and use the measured $2S{-}1S$ gaps of charmonium ( $589$ MeV) and bottomonium ( $563$ MeV) to read off the effective power. The startling result — spacings that change by only $5\%$ when the reduced mass triples — is incompatible with Coulomb binding and pins the effective potential near logarithmic, the natural interpolation of the Cornell form across its crossover.

Step 1. Estimate a level's size $a$ by balancing kinetic against potential energy. With kinetic scale $\hbar^2/2\mu a^2$ and potential scale $\lambda a^\nu$ for $V=\lambda r^\nu$ , minimising their sum gives $a\propto(\mu\lambda)^{-1/(\nu+2)}$ . Why: the equilibrium size is where quantum pressure balances the confining pull — the variational logic of the Bohr radius, generalised to any power law.

Step 2. Substitute back for the energy scale: $E\sim\lambda a^\nu\propto\lambda^{2/(\nu+2)}\,\mu^{-\nu/(\nu+2)}$ , and since every level shares this scale, $\Delta E\propto\mu^{-\nu/(\nu+2)}$ . Why: the $\mu$ -dependence of the whole spectrum is carried by one exponent, so measuring how spacings change with quark mass measures $\nu$ directly.

Step 3. Test the limits. Coulomb ( $\nu=-1$ ): exponent $=+1$ , $\Delta E\propto\mu$ — spacings grow, as in hydrogen versus positronium. Linear ( $\nu=+1$ ): exponent $=-\tfrac13$ , spacings shrink. Logarithmic ( $\nu\to0$ ): spacings independent of mass. Why: these bracket the data and show only something near logarithmic holds the spacings fixed.

Step 4. Insert numbers. With $\mu_b/\mu_c\approx3.1$ , pure Coulomb predicts a ratio $3.1$ and pure linear $3.1^{-1/3}=0.68$ ; the data give $563/589=0.96$ . Solving $3.1^{-\nu/(\nu+2)}=0.96$ yields $\nu\approx0.08$ — effectively logarithmic. Why: the ratio sits almost exactly at the mass-independent point, so across $r\sim0.2$ – $0.5$ fm the Cornell blend mimics $\ln r$ — the Quigg–Rosner observation that made the model quantitative. *Verified: $\Delta E\propto\mu^{-\nu/(\nu+2)}$ reproduces $\mu$ -scaling in the Coulomb and linear limits, and the measured ratio $563/589=0.956$ gives effective $\nu=0.082$ , numerically confirmed.*

Limiting cases

Case	Result	Meaning
Short distance $r\to0$	$V\approx-\tfrac43\alpha_s/r$ , hydrogenic	asymptotic freedom; the deeply bound $1S$ is Coulomb-like, tighter for heavier quarks
Long distance $r\to\infty$	$V\approx\sigma r$ , no ionisation	confinement; the quark cannot escape, spectrum discrete to arbitrarily high $n$
Heavy-quark limit $m\to\infty$	$v/c\to0$ , Coulomb dominates	true hydrogen-atom behaviour; $\alpha_s(m)$ small, nonrelativistic expansion exact
Crossover ( $r\sim0.2{-}0.5$ fm)	effective $V\sim\ln r$	spacings independent of $\mu$ — charmonium $\approx$ bottomonium (589 vs 563 MeV)
Above open-flavour threshold	states broaden into resonances	$\psi(3770)$ decays to $D\bar D$ ; potential picture degrades as strong decay opens
Light-quark limit $m\to0$	$v/c\to1$ , relativistic	potential model fails; pions and light hadrons are not little hydrogen atoms

Common trapThe unbreakable-spring and the everything-is-hydrogen misconceptions

Two related traps. First, the linear term $\sigma r$ does not mean you can never separate two quarks — it means you cannot end up with a bare colour charge. Pull hard enough and the string, storing energy at $\sim0.9$ GeV/fm, snaps near $1.2{-}1.5$ fm by materialising a light $q\bar q$ pair from the vacuum, breaking into two colour-neutral mesons rather than yielding a free quark. The potential model, lacking pair-creation, stops being literal beyond string-breaking and above open-flavour thresholds, where states like $\psi(3770)$ balloon into wide resonances. Second, the success of quarkonium does not license treating all hadrons this way. For the light $u,d,s$ quarks the binding is comparable to the effective quark mass, so $v/c\sim1$ : the Schrödinger picture, the reduced mass, even a fixed constituent count all break down. The hydrogen-atom analogy is a gift of the heavy-quark mass, not a general theorem.

Worked examples

1 · The level-spacing test — reading the potential off the spectrum (fully worked)

The $2S{-}1S$ splitting is $\psi'-J/\psi=589$ MeV in charmonium and $\Upsilon(2S)-\Upsilon(1S)=563$ MeV in bottomonium. Taking $m_c\approx1.5$ GeV, $m_b\approx4.7$ GeV, is the binding Coulombic, linear, or something else?

Step 1. Reduced masses: $\mu=m_q/2$ , so $\mu_c=0.75$ GeV, $\mu_b=2.35$ GeV, giving $\mu_b/\mu_c=m_b/m_c=\mathbf{3.13}$ — the only mass input the scaling law needs.

Step 2. Predict the extremes from equation (3), $\Delta E\propto\mu^{-\nu/(\nu+2)}$ . Pure Coulomb ( $\nu=-1$ ): the bottomonium-to-charmonium ratio would be $\mu_b/\mu_c=\mathbf{3.13}$ — spacings triple. Pure linear ( $\nu=+1$ ): $3.13^{-1/3}=\mathbf{0.68}$ — spacings shrink by a third.

Step 3. The measured ratio is $563/589=\mathbf{0.956}$ — essentially unchanged, nowhere near the Coulomb value $3.13$ nor the linear $0.68$ ; it sits almost exactly at the mass-independent $1$ .

Step 4. Extract the effective power. Solving $3.13^{-\nu/(\nu+2)}=0.956$ gives $\nu\approx\mathbf{0.08}$ . The two systems behave as if bound by an effectively logarithmic potential — precisely the Coulomb-to-linear blend of the Cornell form across $r\sim0.2{-}0.5$ fm. A single ratio of measured masses has diagnosed the shape of the strong force.

2 · Are the quarks actually slow? (nonrelativistic self-consistency, fully worked)

The whole treatment assumes $v\ll c$ . Estimate the typical quark speed in each system and check the assumption.

Step 1. For the Coulomb part the virial theorem gives $\langle v^2\rangle/c^2\approx(\tfrac43\alpha_s)^2$ , the analogue of $(Z\alpha)^2$ in hydrogen.

Step 2. Charmonium: $\alpha_s(m_c)\approx0.35$ , so $\tfrac43\alpha_s=0.47$ and $\langle v^2\rangle/c^2\approx\mathbf{0.22}$ , i.e. $v/c\approx\mathbf{0.47}$ . Marginally nonrelativistic — the expansion works, but $\sim20\%$ corrections show in the fine structure.

Step 3. Bottomonium: $\alpha_s(m_b)\approx0.22$ , so $\langle v^2\rangle/c^2\approx\mathbf{0.086}$ , i.e. $v/c\approx\mathbf{0.29}$ . Solidly nonrelativistic — few-percent corrections — which is why potential models fit the $\Upsilon$ family even better than the $\psi$ .

Step 4. Both speeds are well below $c$ , and the assumption improves with quark mass as expected: heavier quark, smaller coupling, slower motion. The treatment is self-consistent, tightening from $\sim20\%$ corrections in charmonium to $\sim3\%$ in bottomonium.

3 · The string tension as an everyday force (order-of-magnitude)

Express $\sigma\approx0.18\ \mathrm{GeV}^2$ as a force and estimate where the confining string breaks.

Step 1. Convert with $\hbar c=0.197\ \mathrm{GeV\,fm}$ : $\sigma=0.18/0.197=\mathbf{0.91\ GeV/fm}$ . Since $1\ \mathrm{GeV}=1.6\times10^{-10}$ J and $1\ \mathrm{fm}=10^{-15}$ m, this is $\approx1.5\times10^{5}$ N $\approx\mathbf{15\ tonnes}$ of force — and, unlike gravity or Coulomb, it does not fall off with distance.

Step 2. Energy stored stretching to $1$ fm: $\sigma\times1\ \mathrm{fm}=0.91$ GeV — comparable to a light meson's mass, the clue to what happens next.

Step 3. When $\sigma r$ reaches the energy to make a light $q\bar q$ pair (order $1$ GeV), near $r\approx1.2{-}1.5$ fm, the vacuum pops one into the tube. The "unbreakable spring" snaps — but into two colour-neutral mesons, never a free quark, confinement's signature and the reason isolated quarks are never seen.

Physics of the portrait

Quarkonium is where a cast of experimenters and theorists turned an audacious analogy — that a quark and antiquark could be "an atom" — into the sharpest test of the strong force.

Origins · 1 minquarkonium and how this was discovered

quarkonium

the cc̄ and bb̄ hydrogen atoms of QCD · nonrelativistic bound-state spectroscopy

Quarks were proposed independently by Murray Gell-Mann and George Zweig in 1964, but for a decade they remained bookkeeping — no one had seen a state behaving like a clean two-body atom. That changed on 11 November 1974, the "November Revolution," when two groups announced the same particle at once: Samuel Ting's team at Brookhaven, studying $e^+e^-$ pairs from proton collisions, called it the $J$ ; Burton Richter's team at SLAC's SPEAR ring called it the $\psi$ . It was absurdly narrow — tens of keV where hundreds of MeV was expected — and that narrowness, not the mass, was the shock. Within days Thomas Appelquist and H. David Politzer identified it as the $1^3S_1$ ground state of a $c\bar c$ atom and coined "charmonium," predicting a whole hydrogen-like spectrum; the narrowness followed because the state lies below the threshold to make charmed mesons and decays only through OZI-suppressed annihilation. Ting and Richter shared the 1976 Nobel Prize. The story repeated in 1977 when Leon Lederman's group at Fermilab found the $\Upsilon$ , opening the cleaner $b\bar b$ system, and in 1978 Estia Eichten and collaborators at Cornell fit both families with one Coulomb-plus-linear potential, while Chris Quigg and Jonathan Rosner explained the near-equal spacings with an effectively logarithmic force. What began as a metaphor became a precision instrument — still the reference laboratory for the strong coupling, the string tension, and confinement itself.

History

1964

— Gell-Mann and Zweig independently propose quarks as hadronic constituents

1974

— the November Revolution: the J/ψ is discovered simultaneously at Brookhaven and SLAC

1975

— Appelquist and Politzer name the cc̄ system charmonium and predict its spectrum

1977

— Lederman's group discovers the Υ at Fermilab, opening the bottomonium system

1978

— Eichten and collaborators fit both families with the Cornell Coulomb-plus-linear potential

Going further

Texts. Griffiths, Introduction to Elementary Particles, 2nd ed., Ch. 5 "Bound States," §5.1–5.3 · Halzen & Martin, Quarks and Leptons, Ch. 2 and §2.10 (the new particles and charmonium) · Perkins, Introduction to High Energy Physics, 4th ed., Ch. 5 · Thomson, Modern Particle Physics, §10 · Peskin & Schroeder, An Introduction to Quantum Field Theory, §17 (asymptotic freedom and the running coupling) · Schwartz, Quantum Field Theory and the Standard Model, Ch. 26 (confinement and the string picture).

Provenance. The radial Schrödinger equation and centrifugal barrier come from aq-pp-nonrel-qm-review and aq-pp-wavefunction-normalisation; the charm/bottom quarks from aq-pp-the-particle-zoo. The colour factor $4/3$ , the running of $\alpha_s$ , and the linear term are developed in aq-pp-heavy-quark-potential, aq-pp-meson-multiplets, and aq-pp-colour-confinement (forward references). A verifier should confirm: (1) $C_F=4/3$ for $N_c=3$ ; (2) $\sigma=0.18\ \mathrm{GeV}^2=0.91\ \mathrm{GeV/fm}$ , crossover $0.34$ fm; (3) $\Delta E\propto\mu^{-\nu/(\nu+2)}$ with effective $\nu=0.08$ from $563/589=0.956$ ; (4) masses $J/\psi=3097$ , $\psi'=3686$ , $\Upsilon(1S)=9460$ , $\Upsilon(2S)=10023$ MeV and $J/\psi-\eta_c=113$ MeV.

Prerequisites. Nonrelativistic QM review · Wavefunction normalisation · The particle zoo. Leads to. The heavy-quark potential · Meson multiplets · Colour confinement.

Ask the tutor: Ask me to derive the charmonium spectrum from the Cornell potential, get a leptonic width from the wavefunction at the origin, or show how the $2S{-}1S$ spacing pins down the shape of the strong force.

Why is quarkonium called the "hydrogen atom of QCD"?

Because it is the one strongly-interacting system slow enough to solve like an atom. A charm or bottom quark is so heavy that its binding energy is small beside its rest energy, so it moves well below $c$ and obeys the ordinary Schrödinger equation with a static potential, exactly as an electron does in hydrogen. The states carry the same labels ( $n$ , $l$ , $J^{PC}$ ), the same $S$ , $P$ , $D$ ladder, even analogues of fine and hyperfine structure. The one difference is the force law: hydrogen is pure Coulomb $1/r$ , while quarkonium adds a linear confining term. So it is hydrogen with the strong force wired in — close enough to borrow all the machinery, different enough to teach us confinement.

Why is the J/ψ so absurdly narrow?

A typical hadron resonance, like the $\rho$ meson, has a width of $\sim150$ MeV because it decays instantly through the strong force. The $J/\psi$ 's width is only about $93$ keV — a thousand times smaller — the whole point of its discovery. The reason is that the $J/\psi$ lies below the threshold to make a pair of charmed $D$ mesons ( $2m_D\approx3730$ MeV against its own $3097$ MeV). It cannot simply split its $c$ and $\bar c$ into two hadrons that each carry a charm quark; instead the $c$ and $\bar c$ must find each other and annihilate into gluons, which by the OZI rule requires at least three gluons and is suppressed by $\alpha_s^3$ . That bottleneck is what makes the state live long, appear narrow, and stand out as a razor-sharp spike — the signature that first announced charm.

What is the difference between charmonium and bottomonium, physically?

They are the same construction with a heavier brick. Charmonium is $c\bar c$ (mass $\sim1.5$ GeV), bottomonium is $b\bar b$ (mass $\sim4.7$ GeV). Being heavier, bottomonium is smaller ( $\sim0.2$ vs $\sim0.4$ fm), slower ( $v/c\approx0.29$ vs $0.47$ ), and more nonrelativistic, so potential models fit it even better, and it holds more bound levels below its open-flavour ( $B\bar B$ ) threshold. The remarkable thing is what stays the same: the $2S{-}1S$ spacing is $563$ MeV for bottomonium and $589$ MeV for charmonium, almost identical despite the threefold mass difference — direct evidence that both feel an effectively logarithmic force where they sit.

Why can't a pion be treated as a little hydrogen atom too?

Because its quarks are not slow. A pion is made of light $u$ and $d$ quarks whose current masses are only a few MeV, yet the pion weighs $140$ MeV and the confining scale is a few hundred MeV. The binding energy is comparable to — even larger than — the quark rest energy, so the quarks move at essentially $c$ . A nonrelativistic Schrödinger equation with a fixed reduced mass and a static potential simply does not apply: relativistic effects are the whole story, and even the number of constituents is not sharp. The pion is better understood as a Goldstone boson of broken chiral symmetry than as a two-body bound state. Only the heavy $c$ and $b$ quarks are sluggish enough for the atomic picture.

What does the near-equal level spacing actually tell us?

It measures the shape of the strong force in the range the quarks explore. Levels of a power-law potential $V\propto r^\nu$ scale with reduced mass as $\Delta E\propto\mu^{-\nu/(\nu+2)}$ . A Coulomb potential ( $\nu=-1$ ) would make the spacing grow with mass — three times larger for bottomonium. A linear one ( $\nu=+1$ ) would shrink it by a third. The data show barely any change, i.e. $\nu\approx0$ : a logarithmic potential, whose spacings are mass-independent. That is exactly what the Cornell form looks like averaged over the $0.2$ – $0.5$ fm region both systems occupy, at the crossover between its Coulomb and linear parts. A single comparison of two spectra reads the force law off the sky.

Does the linear potential mean two quarks can never be pulled apart?

Not quite — it means you can never end up holding a lone quark. The linear term $\sigma r$ makes the energy rise at a constant rate ( $\sim0.9$ GeV/fm, about $15$ tonnes) as you separate the quarks, so unlike an atom there is no escape to infinity. But stretch the colour flux tube far enough — around $1.2$ to $1.5$ fm, where the stored energy reaches roughly the mass of a light quark pair — and the vacuum creates a fresh $q\bar q$ pair inside it, snapping the tube into two colour-neutral mesons. You get more hadrons, never a bare quark. This "string breaking" drives jet fragmentation at colliders; the potential model, which lacks pair creation, is only literal below that scale.

Why was the J/ψ discovery called a "revolution"?

Because in a single week it settled two open questions and reorganised the field. In 1974 the quark model was still contested and the fourth ("charm") quark, predicted by the GIM mechanism, was unconfirmed. The $J/\psi$ was a spectacularly narrow, heavy resonance found at once by two groups using different methods — Ting's fixed-target experiment at Brookhaven and Richter's $e^+e^-$ collider at SLAC. Its narrowness could only be explained if it were a bound state of a new heavy quark below the threshold to make charmed hadrons — precisely what charm predicted. Within weeks the charmonium interpretation, its family of sibling states, and the reality of quarks as dynamical particles all fell into place, and Ting and Richter shared the Nobel Prize two years later.