The heavy-quark potential

The heavy-quark potential is now computed, not merely fitted. Lattice QCD evaluates the Wilson loop numerically and extracts $V(r)$ point by point from first principles, reproducing the Coulomb-plus-linear shape and pinning $\kappa\approx0.18\text{ GeV}^2$ with no input beyond the QCD action — and it shows where the simple picture ends: at about $1.2\text{ fm}$ the flux tube "breaks", because it becomes energetically cheaper to pop a light $q\bar q$ pair from the vacuum and snap the string into two mesons than to keep stretching it, so the potential flattens rather than rising forever. The systematic version of the model is the effective field theory pNRQCD, which derives $V(r)$ and its spin corrections as a controlled expansion in the heavy-quark velocity. Quarkonium is also a thermometer for the quark-gluon plasma: Matsui and Satz argued in 1986 that in a deconfined medium the colour-Coulomb attraction is Debye-screened, so states melt in sequence as the temperature rises, and CMS and ALICE now observe exactly this sequential suppression of the $\Upsilon(1S)$, $\Upsilon(2S)$ and $\Upsilon(3S)$. At the frontier, states such as the $X(3872)$ sit where no level of the simple Cornell potential belongs, signalling tetraquarks or molecules — the two-body potential is the leading picture of quarkonium, not the whole of it. And the $\Upsilon(4S)$, first bottomonium state above the $B\bar B$ threshold, is the workhorse resonance of the Belle II and BaBar B-factories, produced in $e^+e^-$ precisely because it is $1^{--}$.

Two facts about gluons, and nothing else, determine the shape of the potential. First, at short distances the interaction is one-gluon exchange, and the QCD vertex is the QED vertex with the electric charge promoted to a colour matrix — so the potential is Coulomb-like, $-1/r$, scaled by a colour number. Second, at long distances the gluon field does what the photon field never can: it interacts with itself, and the lines of chromoelectric flux, instead of spreading out like a Coulomb field, are pulled together into a taut tube of roughly constant cross-section. A tube of fixed energy per unit length costs energy $\kappa r$ to stretch, so the potential rises without bound. The first figure is the sum of these two behaviours — a deep well that crosses zero and then climbs a straight line.

The reason this potential earns its keep is that its eigenvalues are a measured spectrum. Feed $V(r)$ into the radial Schrödinger equation for a $c\bar c$ pair and out come discrete levels — a ground state $1S$, a $1P$, a radially excited $2S$, and so on — whose spacings and orderings match the charmonium resonances seen in $e^+e^-$ machines and their decays. The pattern is diagnostic: a pure Coulomb potential would make the $2S$ and $1P$ levels degenerate, while a pure linear one orders them very differently; the observed intermediate ordering is the fingerprint of a potential that is Coulombic inside and linear outside. The second figure is that spectrum, with the states the model must reproduce and the $113\text{ MeV}$ $J/\psi$–$\eta_c$ hyperfine splitting that spin-dependent corrections have to supply.

The whole article is the study of one function and the eigenvalue problem it defines. The potential combines a short-range colour-Coulomb term with a long-range linear one, written here for the colour-singlet $q\bar q$ state that a physical meson occupies.

where: $r$ — quark-antiquark separation; $\alpha_S=g_s^2/4\pi$ — the strong coupling ($\approx0.2$–$0.3$ at these scales); $\tfrac43=C_F$ — the colour factor of the singlet; $\kappa$ — the string tension, $\approx1\text{ GeV/fm}\approx0.18\text{ GeV}^2$. Reading: the first term is the QCD analogue of the Coulomb potential (attractive, $\propto1/r$); the second has no QED counterpart (it grows without bound, and it is what confines). The Coulomb coefficient is not put in by hand — it is the expectation value of the colour operator $\mathbf{T}_1\!\cdot\mathbf{T}_2$ carried by one-gluon exchange, evaluated in the state's colour wavefunction. The spacetime part is identical to QED; only this colour number changes.

where: $\mathbf{T}_i=\tfrac12\boldsymbol\lambda_i$ — the colour generators acting on quark $i$; the singlet value $-\tfrac43$ is attractive, the octet value $+\tfrac16$ is repulsive. Reading: the sign of the short-range force is a colour choice, not a law of nature — nature binds the singlet and rejects the octet, which is why physical hadrons are colour-neutral. The potential earns its physical content only when it is fed into the Schrödinger equation. Because both quarks are heavy and slow, a non-relativistic radial equation with reduced mass $\mu=m_q/2$ governs the bound states.

where: $\mu=m_q/2$ — reduced mass of the equal-mass pair; $\ell$ — orbital angular momentum; $E_{n\ell}$ — binding energy of the $(n,\ell)$ level; the meson mass is $M=2m_q+E_{n\ell}$ plus spin corrections. Reading: this is the hydrogen radial equation with the hydrogen potential replaced by (1); its eigenvalues are the quarkonium spectrum. The levels carry definite parity and charge conjugation, and these quantum numbers — not the mass — decide which states a machine can make. For a fermion-antifermion pair the assignments follow from the intrinsic $q\bar q$ parity and the exchange properties of the wavefunction.

where: $\ell$ — orbital angular momentum; $s$ — total quark spin (0 or 1); $J$ — the meson spin. Reading: the $^3S_1$ ground state ($\ell=0,s=1$) is $1^{--}$, sharing the photon's quantum numbers and so produced directly in $e^+e^-$; the $^1S_0$ state ($s=0$) is $0^{-+}$ and is not. The string tension is not a free number either: the same $\kappa$ that sets the linear potential fixes the slope of the meson Regge trajectories, tying a spectroscopic constant to a scattering one.

where: $\alpha'\approx0.9\text{ GeV}^{-2}$ — the universal Regge slope of light mesons; $r_0$ — the separation at which $V=0$, where the Coulomb well hands over to the linear rise. Reading: $\alpha'\approx0.9\text{ GeV}^{-2}$ gives $\kappa\approx0.18\text{ GeV}^2\approx1\text{ GeV/fm}$, the same tension that the potential model and the lattice both require.

1 · Where the potential turns from attractive to confining (fully worked)

Using $\alpha_S=0.2$ and $\kappa=1\text{ GeV/fm}$, find the separation $r_0$ at which $V_{q\bar q}(r)=0$, and state how strong the confining force is in everyday units.

Step 1. Set the two terms equal: $\tfrac43\,\dfrac{\alpha_S\hbar c}{r_0}=\kappa r_0$, so $r_0=\sqrt{\dfrac{4\alpha_S\hbar c}{3\kappa}}$. The factor $\hbar c=0.1973\text{GeV}\,\cdot\,\text{fm}$ converts the dimensionless $\alpha_S$ into an energy times a length.

Step 2. Evaluate the Coulomb coefficient: $\tfrac43\,\alpha_S\hbar c=\tfrac43\times0.2\times0.1973=0.0526\text{GeV}\,\cdot\,\text{fm}$. Dividing by $\kappa=1\text{ GeV/fm}$ gives $r_0^2=0.0526\text{ fm}^2$, so $r_0=\mathbf{0.23\text{ fm}}$ — just under a quarter of a femtometre, roughly a fifth of the proton radius.

Step 3. The confining force is the slope of the linear term, $\kappa=1\text{ GeV/fm}$. Converting, $1\text{ GeV/fm}=1.6\times10^{-10}\text{ J}/10^{-15}\text{ m}=1.6\times10^{5}\text{ N}$. That is about 16 tonnes of weight — a constant pull that does not weaken with distance, which is why the quarks can never be torn free.

Step 4. Inside $r_0$ the well is Coulombic and the levels look hydrogen-like; outside it the wall goes up linearly. A charmonium $1S$ state has a mean radius of order $0.4$–$0.5\text{ fm}$, straddling $r_0$, which is exactly why charmonium is the delicate case where both terms matter — and why bottomonium, smaller and more Coulombic, is modelled best.

2 · The quantum numbers of quarkonium, and why only $1^{--}$ appears in $e^+e^-$

Assign $J^{PC}$ to the $c\bar c$ ground-state multiplet ($\ell=0$) and explain why the $J/\psi$ but not the $\eta_c$ is produced directly in electron-positron annihilation.

Step 1. For $\ell=0$ the two spin couplings are $s=0$ (singlet, $^1S_0$) and $s=1$ (triplet, $^3S_1$). Apply the parity rule $P=(-1)^{\ell+1}=(-1)^{1}=-1$: both $S$-states have $P=-1$.

Step 2. Apply $C=(-1)^{\ell+s}$. For $^1S_0$: $C=(-1)^{0+0}=+1$, giving $J^{PC}=0^{-+}$ — this is the $\eta_c$ (2984 MeV). For $^3S_1$: $C=(-1)^{0+1}=-1$, giving $J^{PC}=1^{--}$ — this is the $J/\psi$ (3097 MeV).

Step 3. In $e^+e^-$ annihilation the pair annihilates into a single virtual photon, which carries the photon's quantum numbers $J^{PC}=1^{--}$. A resonance can be produced this way only if it matches, so only the $1^{--}$ state, the $J/\psi$, is made directly; the $0^{-+}$ $\eta_c$ must instead be reached through radiative or hadronic decays. The $113\text{ MeV}$ mass gap between them is the hyperfine (spin-spin) splitting, the chromomagnetic analogue of the hydrogen 21 cm splitting.

3 · The near-Coulombic size of bottomonium (estimate)

Treat the bottomonium ground state using only the Coulomb part of the potential, with $m_b=4.7\text{ GeV}$ and an effective $\alpha_S=0.22$. Estimate the Bohr radius and check that the state sits inside the confinement scale.

Step 1. The Coulomb term is $-C_F\alpha_S/r$ with $C_F\alpha_S=\tfrac43\times0.22=0.293$ — a "hydrogen atom" with fine-structure constant $0.293$ instead of $1/137$. The reduced mass is $\mu=m_b/2=2.35\text{ GeV}$.

Step 2. The Bohr radius is $a=1/(\mu\,C_F\alpha_S)=1/(2.35\times0.293)=1.45\text{ GeV}^{-1}$. Converting, $a=1.45\times0.1973\text{ fm}=\mathbf{0.29\text{ fm}}$.

Step 3. This is comfortably inside $r_0\approx0.23$–$0.3\text{ fm}$ and well short of the $\sim1\text{ fm}$ scale where the linear wall dominates, so the $\Upsilon(1S)$ really does live mostly in the Coulombic part of the well — the potential model works best precisely because $b\bar b$ is small and slow. The rough Coulomb binding $E_1=-\tfrac12\mu(C_F\alpha_S)^2\approx-0.10\text{ GeV}$ is only an estimate; the linear term and spin corrections shift the true levels, but the picture — a compact, hydrogen-like heavy atom — is correct.

The confining half of the potential — the linear $\kappa r$ that no perturbative diagram can produce — was made rigorous by one person's reformulation of gauge theory on a lattice, where confinement becomes a countable, calculable statement rather than a hope.

Texts. Thomson, Modern Particle Physics, §10.7.2–10.8 (colour sums and the QCD colour potential, Eqs. 10.36–10.40, Figs. 10.25–10.26) · Halzen & Martin, Quarks and Leptons, Ch. 2 (the quarkonium potential and spectra) · Griffiths, Introduction to Elementary Particles, 2nd ed., §5.3 and §8.4 (charmonium, the quark potential) · Peskin & Schroeder, An Introduction to Quantum Field Theory, §17.3 (the static quark potential and the Wilson loop) · Schwartz, Quantum Field Theory and the Standard Model, Ch. 25 (non-Abelian gauge theories and confinement). The key results to keep in mind: (1) $V=-C_F\alpha_S/r$ follows from one-gluon exchange with the QED spacetime structure and colour factor $-\tfrac43$; (2) the singlet and octet colour factors $-\tfrac43$ and $+\tfrac16$ follow from $\mathbf{T}_1\!\cdot\mathbf{T}_2=\tfrac12[C_{\text{tot}}-2C_F]$; (3) the turning point is $r_0=\sqrt{4\alpha_S\hbar c/3\kappa}=0.23\text{ fm}$ for $\alpha_S=0.2$, $\kappa=1\text{ GeV/fm}$; (4) $\kappa=1/(2\pi\alpha')\approx0.18\text{ GeV}^2$ from $\alpha'\approx0.9\text{ GeV}^{-2}$; (5) the $J^{PC}$ assignments $^3S_1\to1^{--}$, $^1S_0\to0^{-+}$ from $P=(-1)^{\ell+1}$, $C=(-1)^{\ell+s}$.

Prerequisites. Colour & QCD · Colour Factors · Non-relativistic QM Review · The Yukawa Potential. Leads to. Colour Confinement · Quarkonium Bound States · The Running of α_S.

Ask the tutor: Want to see how the eigenvalues of this potential fall on the measured charmonium ladder, or how Wilson's area law turns the flux tube into the exact linear term $\kappa r$?