---
title: Color, Confinement, and Exotic Hadrons
module: The Quark Model
moduleNumber: 4
lessonNumber: 4
order: 404
summary: >
  Color as the gauged SU(3) charge, and the requirement that every physical hadron
  be a color singlet — which selects q-qbar mesons and qqq baryons as the simplest
  states. The R-ratio of e⁺e⁻ annihilation measures three colors directly. Beyond
  the simplest singlets lie glueballs, tetraquarks, and pentaquarks, and the recent
  XYZ states, read as either compact multiquarks or loose hadronic molecules.
topics: [The Quark Model]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 5 §5.9 Color; Ch. 9 §9.1"
  - book: Thomson
    ref: "Ch. 10 — Quantum Chromodynamics, §10.1"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §3.2–3.3 Confinement and color singlets"
---

Color was introduced in the [previous lesson](/particle-physics/quark-model/baryon-spectroscopy)
to rescue Fermi statistics for the $\Delta^{++}$: each quark carries one of three
color charges, and a baryon is the antisymmetric color singlet. Color turns out to
be far more than a bookkeeping index. It is the charge of the strong force — the
gauged $SU(3)$ of quantum chromodynamics — and the demand that physical states carry
no net color is what restricts the hadron spectrum to the combinations actually
observed. This lesson states the color-singlet rule, shows how the $R$-ratio of
electron–positron annihilation counts the colors, and surveys the exotic hadrons
that the rule permits beyond $q\bar q$ and $qqq$.[^grif-color]

## The color-singlet rule

Each quark transforms in the fundamental $\mathbf 3$ of color $SU(3)$, each antiquark
in the $\bar{\mathbf 3}$. The gauge symmetry is exact, and confinement — derived in
the [QCD module](/particle-physics/qcd/asymptotic-freedom-confinement) — states that
the only finite-energy, isolated states are those invariant under color $SU(3)$: the
**color singlets**. A combination of quarks is physical only if its color content
contains the singlet $\mathbf 1$.

This single rule reproduces the quark model's allowed combinations. Decompose the
small products:

- $\mathbf 3 \otimes \bar{\mathbf 3} = \mathbf 8 \oplus \mathbf 1$ — a
  quark–antiquark pair contains a singlet. **Mesons** exist.
- $\mathbf 3 \otimes \mathbf 3 \otimes \mathbf 3 = \mathbf 1 \oplus \mathbf 8
  \oplus \mathbf 8 \oplus \mathbf{10}$ — three quarks contain a singlet.
  **Baryons** exist.
- $\mathbf 3 \otimes \mathbf 3 = \bar{\mathbf 3} \oplus \mathbf 6$ — no singlet. A
  two-quark state $qq$ cannot be color-neutral and does not exist as a free
  particle.
- $\mathbf 3$ alone is not a singlet. A **free quark** cannot exist.

The rule states that a hadron must be "white": the three colors in
a baryon combine to a color-neutral state, as do a color and its anticolor in a
meson. It is the origin of confinement's most visible consequence — the absence of
fractionally charged free particles despite the fractional charges of the quarks.

$$
% caption: The color-singlet combinations. A meson pairs a color with its anticolor
% (red with anti-red, and so on) to a colorless state; a baryon combines one quark of
% each color, red plus green plus blue, to white. A lone quark or a two-quark pair
% carries net color and cannot exist in isolation.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % meson: q qbar
  \begin{scope}[shift={(-3.3,0)}]
    \draw[thick] (-0.6,0) circle (0.4) node {R};
    \draw[thick, fill=white] (0.6,0) circle (0.4) node {$\overline{R}$};
    \draw[black] (-0.2,0) -- (0.2,0);
    \node[black, font=\scriptsize] at (0,-1.0) {meson: color + anticolor};
    \node[black, font=\scriptsize] at (0,-1.45) {singlet};
  \end{scope}
  % baryon: qqq
  \begin{scope}[shift={(3.0,0)}]
    \draw[thick] (0,0.55) circle (0.4) node {R};
    \draw[thick] (-0.6,-0.5) circle (0.4) node {G};
    \draw[thick] (0.6,-0.5) circle (0.4) node {B};
    \node[black, font=\scriptsize] at (0,-1.4) {baryon: R + G + B};
    \node[black, font=\scriptsize] at (0,-1.85) {singlet};
  \end{scope}
\end{tikzpicture}
$$

## Counting colors with the R-ratio

Color is not directly visible — every observed hadron is colorless — but its
multiplicity leaves a quantitative trace in electron–positron annihilation. The
process $e^+ e^- \to \gamma^\ast \to q\bar q$ produces a quark–antiquark pair that
hadronizes; comparing its rate to the point-like $e^+ e^- \to \mu^+\mu^-$ removes
the electromagnetic and phase-space factors and isolates the quark charges and
color count. The ratio is

$$
R = \frac{\sigma(e^+ e^- \to \text{hadrons})}{\sigma(e^+ e^- \to \mu^+\mu^-)}
  = N_c \sum_q e_q^2,
$$

summed over the quark flavors light enough to be pair-produced at the given energy,
with $e_q$ the quark charge in units of the proton charge and $N_c$ the number of
colors.[^grif-Rratio] The color factor $N_c$ appears because each flavor comes in
$N_c$ distinct colors, all produced. Below the charm threshold, only $u$, $d$, $s$
contribute:

$$
R = N_c\left(e_u^2 + e_d^2 + e_s^2\right)
  = 3\left(\tfrac49 + \tfrac19 + \tfrac19\right) = 2.
$$

Above the charm threshold, add $e_c^2 = \tfrac49$; above bottom, add $e_b^2 =
\tfrac19$:

$$
R = 3\left(\tfrac49+\tfrac19+\tfrac19+\tfrac49\right) = \tfrac{10}{3},
\qquad
R = 3\left(\tfrac{10}{9}+\tfrac19\right) = \tfrac{11}{3}.
$$

Without color the same sums give $R = \tfrac23, \tfrac{10}{9}, \tfrac{11}{9}$ — a
factor of three too small. The data sit unambiguously on the $N_c = 3$ predictions:
$R \approx 2$ below charm, stepping up as each new flavor threshold opens. The
$R$-ratio is the cleanest single measurement of the number of colors.

$$
% caption: The R-ratio of electron-positron annihilation against center-of-mass
% energy. Between flavor thresholds R sits on a plateau fixed by the quark charges
% times three colors: about 2 below charm, 10/3 between charm and bottom, 11/3 above.
% The sharp spikes are the vector-meson resonances (rho, omega, phi at low energy;
% J/psi and psi-prime at charm; upsilon at bottom). The dashed lines mark the
% colorless prediction, a factor of three lower.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (10.2,0) node[right, font=\scriptsize] {energy};
  \draw[->, black] (0,0) -- (0,4.3) node[above, font=\scriptsize] {R};
  \foreach \y in {1,2,3,4}
    \draw[black] (-0.1,\y) -- (0.1,\y) node[left=2pt, black, font=\scriptsize] {\y};
  % plateau R=2 (u,d,s)
  \draw[very thick] (0.6,2.0) -- (3.4,2.0);
  % plateau R=10/3 (add charm) ~ 3.33
  \draw[very thick] (4.0,3.33) -- (7.4,3.33);
  % plateau R=11/3 (add bottom) ~ 3.67
  \draw[very thick] (8.0,3.67) -- (9.8,3.67);
  % resonance spikes
  \foreach \x in {1.0,1.4,1.8} \draw[thin] (\x,2.0) -- (\x,3.2);
  \draw[thin] (3.9,3.33) -- (3.9,4.0);
  \draw[thin] (4.2,3.33) -- (4.2,3.85);
  \draw[thin] (7.9,3.67) -- (7.9,4.1);
  % colorless prediction (dashed, factor 3 lower)
  \draw[black, dashed] (0.6,0.67) -- (3.4,0.67);
  \draw[black, dashed] (4.0,1.11) -- (7.4,1.11);
  \node[black, font=\scriptsize, align=left] at (5.6,0.75) {colorless prediction};
  % threshold labels
  \node[black, font=\scriptsize] at (3.7,-0.4) {charm};
  \node[black, font=\scriptsize] at (7.7,-0.4) {bottom};
\end{tikzpicture}
$$

## Exotic color singlets

The color rule permits more than $q\bar q$ and $qqq$. Any combination whose color
content contains a singlet is allowed; the simplest states dominate only because
adding constituents costs energy. The permitted exotics are:

- **Glueballs.** Gluons carry color (the adjoint $\mathbf 8$), so a bound state of
  gluons alone can be a color singlet, e.g. from $\mathbf 8 \otimes \mathbf 8
  \supset \mathbf 1$. A glueball is a hadron with no valence quarks. Candidates
  exist but mix with ordinary $q\bar q$ mesons of the same quantum numbers, which
  makes identification hard.
- **Tetraquarks.** Two quarks and two antiquarks, $qq\bar q\bar q$, can form a
  singlet. The charged states discovered at the $B$-factories and LHCb — such as
  $Z_c(3900)$ and the doubly-charmed $T_{cc}^+$ — cannot be $q\bar q$ because their
  charge and flavor require at least four quarks.
- **Pentaquarks.** Four quarks and one antiquark, $qqqq\bar q$, form a singlet with
  baryon number one. LHCb reported the pentaquark candidates $P_c(4380)^+$ and
  $P_c(4450)^+$ in 2015, seen as peaks in the $J/\psi\,p$ system from
  $\Lambda_b$ decay.[^lhcb]

$$
% caption: Quark content of the exotic color singlets. Ordinary hadrons (top) are a
% quark-antiquark meson and a three-quark baryon; the exotics (bottom) add
% constituents while staying colorless — a four-quark tetraquark and a five-quark
% pentaquark. Antiquarks carry an overbar.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % meson
  \begin{scope}[shift={(-4.6,1.3)}]
    \fill (-0.35,0) circle (2.4pt) node[below, font=\scriptsize] {q};
    \draw[fill=white] (0.35,0) circle (2.4pt) node[below, font=\scriptsize] {$\overline{q}$};
    \draw (-0.35,0) -- (0.35,0);
    \node[black, font=\scriptsize] at (0,0.6) {meson};
  \end{scope}
  % baryon
  \begin{scope}[shift={(-1.4,1.3)}]
    \foreach \x in {-0.4,0,0.4} \fill (\x,0) circle (2.4pt);
    \node[black, font=\scriptsize] at (0,0.6) {baryon};
  \end{scope}
  % tetraquark
  \begin{scope}[shift={(-4.6,-1.0)}]
    \fill (-0.4,0.15) circle (2.4pt);
    \fill (0,0.15) circle (2.4pt);
    \draw[fill=white] (-0.2,-0.35) circle (2.4pt) node[below, font=\scriptsize] {$\overline{q}$};
    \draw[fill=white] (0.4,-0.35) circle (2.4pt) node[below, font=\scriptsize] {$\overline{q}$};
    \node[black, font=\scriptsize] at (0,0.75) {tetraquark};
  \end{scope}
  % pentaquark
  \begin{scope}[shift={(-1.2,-1.0)}]
    \foreach \x in {-0.5,-0.1,0.3} \fill (\x,0.15) circle (2.4pt);
    \fill (-0.1,-0.3) circle (2.4pt);
    \draw[fill=white] (0.5,-0.3) circle (2.4pt) node[below, font=\scriptsize] {$\overline{q}$};
    \node[black, font=\scriptsize] at (-0.1,0.75) {pentaquark};
  \end{scope}
\end{tikzpicture}
$$

## Compact states or hadronic molecules

Knowing that a state has four or five quarks does not fix how they are arranged, and
this is the open question for the exotic spectrum. Two pictures compete.

- **Compact multiquark.** The constituents sit together in one bag, often organized
  as a tightly bound **diquark** (a $qq$ pair in the $\bar{\mathbf 3}$ of color)
  bound to an antidiquark. The state is a single object of hadronic size
  $\sim 1\ \text{fm}$.
- **Hadronic molecule.** The state is two ordinary color-singlet hadrons bound
  loosely at long range, like a nucleus of two mesons. The clearest hint is
  proximity to a threshold: the $X(3872)$ sits within an MeV of the $D^0
  \bar D^{\ast 0}$ threshold, exactly where a weakly bound molecule of those two
  mesons would appear.[^xyz]

$$
% caption: Two interpretations of a four-quark exotic. On the left, a compact
% tetraquark: a diquark bound to an antidiquark within a single hadron-sized region.
% On the right, a hadronic molecule: two ordinary color-singlet mesons bound loosely
% at long range, near their production threshold. Present data do not decisively
% distinguish the two.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % compact
  \begin{scope}[shift={(-3.0,0)}]
    \draw[thick] (0,0) circle (0.95);
    \fill (-0.35,0.3) circle (2.4pt);
    \fill (0.05,0.4) circle (2.4pt);
    \draw[fill=white] (-0.1,-0.35) circle (2.4pt);
    \draw[fill=white] (0.35,-0.15) circle (2.4pt);
    \node[black, font=\scriptsize] at (0,-1.5) {compact tetraquark};
    \node[black, font=\scriptsize] at (0,-1.95) {one hadron-sized bag};
  \end{scope}
  % molecule
  \begin{scope}[shift={(3.0,0)}]
    \draw[thick] (-0.7,0) circle (0.55);
    \draw[thick] (0.7,0) circle (0.55);
    \fill (-0.85,0.1) circle (2.4pt);
    \draw[fill=white] (-0.55,-0.1) circle (2.4pt);
    \fill (0.55,0.1) circle (2.4pt);
    \draw[fill=white] (0.85,-0.1) circle (2.4pt);
    \draw[black, densely dotted] (-0.15,0) -- (0.15,0);
    \node[black, font=\scriptsize] at (0,-1.5) {hadronic molecule};
    \node[black, font=\scriptsize] at (0,-1.95) {two mesons, loosely bound};
  \end{scope}
\end{tikzpicture}
$$

The exotic sector, now a growing catalogue of XYZ states, tests QCD in a regime
where the simple constituent-quark counting of the earlier lessons is no longer
enough. What all these states share with the ordinary hadrons is the color-singlet
rule: every one of them is colorless, whatever its internal arrangement. The
dynamics that binds them — the gauged color $SU(3)$, its running coupling, and
confinement — is the subject of the [QCD module](/particle-physics/qcd/color-su3-gluons),
where color is promoted from the static label of the quark model to the source of
the strong force.

[^grif-color]: Griffiths, _Introduction to Elementary Particles_, §5.9 and §9.1, introduces color as the exact gauge charge and the color-singlet requirement for physical hadrons; Tong, _The Standard Model_ (Cambridge Part III), §3.3, derives the meson and baryon singlets from confinement.

[^grif-Rratio]: Griffiths, §9.1, derives $R = N_c\sum_q e_q^2$ and compares it to the annihilation data as a measurement of $N_c = 3$; the plateau values and thresholds follow the Particle Data Group compilation, [pdg.lbl.gov](https://pdg.lbl.gov).

[^lhcb]: The pentaquark candidates $P_c(4380)^+$ and $P_c(4450)^+$ were reported by the LHCb collaboration in 2015 (R. Aaij et al., _Phys. Rev. Lett._ 115, 072001); the doubly-charmed tetraquark $T_{cc}^+$ by LHCb in 2021. Masses and states from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).

[^xyz]: The $X(3872)$ was discovered by Belle in 2003 (S.-K. Choi et al., _Phys. Rev. Lett._ 91, 262001); its mass coincides with the $D^0\bar D^{\ast 0}$ threshold, motivating the molecular interpretation. Property listings from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).