In a three-body decay \(a \to b\, c\, d\), the kinematics are completely described by two independent Lorentz invariants. A natural choice is the pair of Mandelstam-like variables:

\[s = (p_c + p_d)^2, \qquad t = (p_b + p_d)^2\]

The Dalitz plot — introduced by R.H. Dalitz in 1953 — is a scatter plot of event coordinates \((s, t)\). Because the two-body phase space integration over the third invariant is trivial, the doubly differential decay rate takes the remarkably clean form:

\[\tau_a \frac{d\Gamma}{ds\, dt}(a \to b\, c\, d) = \frac{1}{(2\pi)^3 \, 8 m_a^2} \, |\mathcal{M}|^2(s,t)\]

The crucial consequence: the density of events in the Dalitz plot is directly proportional to \(|\mathcal{M}|^2\). Flat density → constant matrix element. A gradient → momentum-dependent coupling. A sharp band → a resonance propagator.

This makes the Dalitz plot one of the most powerful diagnostic tools in particle physics: you can read off the structure of the amplitude by eye, before writing a single Feynman diagram.

The Kinematic Boundary

Not all \((s,t)\) pairs are kinematically accessible. The physical region — the interior of the Dalitz plot — is bounded by the condition that all three final-state momenta are real and positive. For given masses \(m_a, m_b, m_c, m_d\), this boundary is a closed curve in the \((s,t)\) plane defined by:

\[\lambda(s,\, m_a^2,\, t_{\min/\max}^{}) = 0\]

where \(\lambda(x,y,z) = x^2 + y^2 + z^2 - 2xy - 2yz - 2xz\) is the Källén function. The shape and size of this allowed region itself encodes the masses — which is why the third example below shows a dramatically shrunken plot when \(m_b\) is increased.

Three Examples

All three examples share \(m_a = 500\,\text{MeV}\), \(m_c = m_d = 140\,\text{MeV}\), and differ only in the matrix element and in \(m_b\). The interactive plots below are rendered in real time — try hovering over the boundary to see how the allowed region is determined.

\(m_a=500\,\text{MeV},\; m_b=0,\; m_c=m_d=140\,\text{MeV}\)
\(|\mathcal{M}|^2 = \tfrac{1}{\kappa^4}(m_a^2 - s)^2,\quad [\kappa]=E\)

The density increases toward small \(s\) and is independent of \(t\): the colour bands run horizontally, confirming that \(|\mathcal{M}|^2\) has no \(t\) dependence. The single variable \(s = (p_c+p_d)^2\) controls the amplitude — as \(s \to m_a^2\) the matrix element vanishes, so the plot is darkest in the lower-left corner where \(s\) is minimal.

\(m_a=500\,\text{MeV},\; m_b=0,\; m_c=m_d=140\,\text{MeV}\)
\(|\mathcal{M}|^2 = \tfrac{1}{\kappa^4}(s + 2t - m_a^2 - 2m_c^2)^2,\quad [\kappa]=E\)

Now the colour bands run diagonally: the amplitude depends on the linear combination \(s+2t\). The dark corner in the upper-left and the near-white in the lower-right reflect where \(s+2t - m_a^2 - 2m_c^2\) is largest and smallest, respectively. This pattern is the hallmark of an amplitude that couples to both invariant masses simultaneously.

\(m_a=500\,\text{MeV},\; m_b=135\,\text{MeV},\; m_c=m_d=140\,\text{MeV}\)
\(|\mathcal{M}|^2 = \tfrac{1}{\omega^4}(s + 2t - m_a^2 - 2m_b^2 - m_d^2)^2,\quad [\omega]=E\)

Increasing \(m_b\) from 0 to 135 MeV dramatically reduces the available phase space — the Dalitz plot shrinks to a small, nearly elliptical region in the lower-left corner. The mass of the third particle shifts the kinematic threshold and squeezes the boundary. The gradient structure of \(|\mathcal{M}|^2\) is the same as Example #2, but the allowed region is now so constrained that only a sliver of the full \((s,t)\) range is populated.

What the Plots Are Telling Us

Example #1 — The matrix element \(|\mathcal{M}|^2 \propto (m_a^2 - s)^2\) depends only on \(s\). Accordingly, the density contours are horizontal lines: for fixed \(s\), the rate is the same regardless of \(t\). The distribution peaks at the smallest accessible \(s = (m_c+m_d)^2\), the threshold for the \(c\)–\(d\) pair.

Example #2 — Here \(|\mathcal{M}|^2 \propto (s + 2t - m_a^2 - 2m_c^2)^2\). The argument is a linear combination of both invariants, so the zero of \(\mathcal{M}\) forms a straight line in the \((s,t)\) plane — a nodal line — and the density contours are parallel to it. Events pile up in the corners farthest from the node.

Example #3 — Same gradient structure as Example #2, but with \(m_b = 135\,\text{MeV} \approx m_\pi\). The non-zero mass of the third particle shifts the threshold and shrinks the phase space region dramatically, leaving only a compact island in the lower-left of the \((s,t)\) plane. The lesson: phase space geometry encodes the masses, while density variations within the boundary encode \(|\mathcal{M}|^2\).

A Practical Reading Guide

When confronted with an unknown Dalitz plot, the following questions guide the analysis:

  1. Is the density uniform? If yes, \(|\mathcal{M}|^2 = \text{const}\) and the decay is governed by pure phase space.
  2. Are the contours parallel to one axis? Then \(|\mathcal{M}|^2\) depends only on the other invariant.
  3. Are there bands of high density crossing the plot? These are resonances: an intermediate state \(a \to R\, b \to c\, d\, b\) with propagator \(\sim 1/(s - m_R^2)\) produces a horizontal band at \(s = m_R^2\).
  4. Are there empty regions or nodes? These indicate zeros of the amplitude, often from angular momentum selection rules or destructive interference.
  5. What is the shape of the boundary? This directly constrains the final-state masses through the Källén function.