This Julia package is specifically crafted for creating Dalitz plots, providing users with the capability to visualize the three-body decays of particles by specifying amplitudes. With this package, users enjoy a high degree of flexibility in adjusting amplitude settings, allowing them to tailor the plots to meet their specific needs.

Please note that this package is designed for theoretical calculations of Dalitz plots and is not intended for the analysis of experimental data.

The cross section, denoted by $d\sigma$, can be expressed in terms of amplitudes, ${\cal M}$, as follows:

$d\sigma=F|{\cal M}|^2\frac{1}{S}d\Phi=(2\pi)^{4-3n}F|{\cal M}|^2\frac{1}{S}dR$

Here, $dR = (2\pi)^{3n-4} d\Phi = \prod_{i}\frac{d^3k_i}{2E_i}\delta^4(\sum_{i}k_i-P)$ represents the Lorentz-invariant phase space for $n$ particles, and it is generated using the Monte-Carlo method described in Ref. [F. James, CERN 68-12].

The flux factor $F$ for the cross section is given by: $F=\frac{1}{2E2E'v_{12}}=\frac{1}{4[(p_1\cdot p_2)^2-m_1^2m_2^2]^{1/2}} \frac{|p_1\cdot p_2|}{p_1^0p_2^0}$. In the laboratory or center of mass frame, the relation $\vec{p}_1^2 \vec{p}_2^2 = (\vec{p}_1 \cdot \vec{p}_2)^2$ is utilized. In the laboratory frame, the term $\frac{|p_1\cdot p_2|}{p_1^0 p_2^0}$ simplifies to 1. Additionally, if a boson or zero-mass spinor particle is replaced with a non-zero mass spinor particle, the factor $1/2$ is replaced with the mass of the particle, $m$. The total symmetry factor $S$ is given by $\prod_i m_i!$ if there are $m_i$ identical particles.

For decay width, the flux factor is modified to $F = \frac{1}{2E}$.


To install the "DalitzPlot" package, you can follow the standard Julia package manager procedure. Open Julia and use the following commands:

Using the Julia REPL

using DalitzPlot

Alternatively, if you want to install it directly from the GitHub repository:

Using the Julia REPL:

import Pkg
using DalitzPlot

Or, in the Julia package manager (Pkg) prompt:

pkg> add "https://github.com/junhe1979/DalitzPlot.jl"
using DalitzPlot

These commands will install the "DalitzPlot" package and allow you to use it in your Julia environment.

Define amplitudes with factors for the calculation

Users are required to supply amplitudes with factors $(2\pi)^{4-3n}F|{\cal M}|^2\frac{1}{S}$ within the function, named amp here.

We can take it as 1。

amp(tecm, kf, ch, para)=1.

Define more intricate amplitudes for a 2->3 process.

This function, named amp, calculates amplitudes with factors for a 2->3 process. The input parameters are:

  • tecm: Total energy in the center-of-mass frame.
  • kf: Final momenta generated.
  • ch: Information about the process (to be defined below).
  • para: Additional parameters.

Users are expected to customize the amplitudes within this function according to their specific requirements.

function amp(tecm, kf, ch, para)
    # get kf as momenta in the center-of-mass ,
    #get kf as momenta in laboratory frame
    k1, k2, k3 = getkf(para.p, kf, ch)

    # Incoming particle momentum
    # Center-of-mass frame: p1 = [p 0.0 0.0 E1]
    #p1, p2 = pcm(tecm, ch.mi)
    # Laboratory frame
    p1, p2 = plab(para.p, ch.mi)

    #flux factor for cross section in Laboratory frame
    fac = 1 / (4 * para.p * ch.mi[2] * (2 * pi)^5)

    k12 = k1 + k2
    s12 = cdot(k12, k12)
    m = 3.2
    A = 1e9 / (s12 - m^2 + im * m * 0.1)

    total = abs2(A) * fac* 0.389379e-3

    return total

Define the masses of initial and final particles

The mass of initial and final particles is specified in a NamedTuple (named ch here) with fields mi and mf. Particle names can also be provided for PlotD as namei and namef.

The function for amplitudes with factors is saved as amp.

Example usage:

ch = (mi=[mass_i_1, mass_i_2], mf=[mass_f_1, mass_f_2, mass_f_3], namei=["p^i_{1}", "p^i_{2}"], namef=["p^f_{1}", "p^f_{2}", "p^f_{3}"], amp=amp)

Make sure to replace mass_i_1, mass_i_2, mass_f_1, mass_f_2, and mass_f_3 with the actual masses of the particles (1.0, 1.0, 1.0, 2.0, 3.0 here).

Define the momentum or total energy

Momentum in the Laboratory frame and transfer it to the total energy in the center-of-mass frame.

Example usage:

p_lab = 20.0
tecm = pcm(p_lab, ch.mi)


Calculate the cross section and related spectra using the GENEV function.

The function Xsection takes the momentum of the incoming particle in the Laboratory frame (p_lab), the information about the particles (ch), the axes representing invariant masses (axes), the total number of events (nevtot), the number of bins (Nbin), and additional parameters (para). The function uses the plab2pcm function to transform the momentum from the Laboratory frame to the center-of-mass frame.

Example usage:

res = Xsection(plab2pcm(p_lab, ch.mi), ch, axes=[23, 21], nevtot=Int64(1e7), Nbin=500, para=(p=p_lab, l=1.0), ProgressBars=true)

The results are stored in the variable res as a NamedTuple. Specifically, res.cs0 corresponds to the total cross section, res.cs1 represents the invariant mass spectrum, and res.cs2 captures the data for the Dalitz plot.

Plot Dalitz Plot