Lineshape Parametrizations

This page provides a description of the lineshape parametrizations used in the HadronicLineshapes package. The implementation is consistent with the Review on Resonances by Particle Data Group up to an overall normalization factor.

Breit-Wigner Function

The BreitWigner function is a simplest lineshape used to describe resonances in particle physics. It is defined as:

\[BW(\sigma, m, \Gamma) = \frac{1}{m^2 - \sigma - i m \Gamma}\]

where:

Variable $m$ is the resonance mass,
Variable $\Gamma$ is the total decay width,
Variable $\sigma$ is the Mandelstam variable denoting squared mass of the system.

This parametrization assumes a single channel decay with no form factors or orbital angular momentum contributions. For an elastic resonance with l-wave a energy-dependent expression is used for the width,

\[\Gamma(\sigma) = \Gamma_0 \frac{q(\sigma)}{q_0} \frac{m}{\sqrt{\sigma}} \frac{F_l^2(q(\sigma) d)}{F_l^2(q_0 d)}\]

where $q = q(\sqrt{\sigma}, m_1, m_2)$ is the momentum of the decay products in the center-of-mass frame, and $q_0$ is the momentum evaluated for the nominal mass of the resonance. By default, the Blatt-Weisskopf form factor are used for $F_l(q d)$. The factors include $q^l$ dependence contributing to $q^{2l+1}$ dependence of the width $\Gamma(\sigma)$ on the break up momentum $q$.

Using the BreitWigner Function

To create a BreitWigner instance in the HadronicLineshapes package, you can use the following constructor:

BreitWigner(m::Float64, Γ::Float64)
BreitWigner(m::Float64, Γ::Float64, ma::Float64, mb::Float64, l::Int, d::Float64)

For the first call, the default values for ma, mb, l, and d are set to 0.0, 0.0, 0, and 1.0, respectively, yielding the constant-width Breit-Wigner function.

Multichannel Breit-Wigner Function

The MultichannelBreitWigner function extends the BreitWigner to multiple decay channels. Each channel can have its own mass thresholds, coupling constants, and orbital angular momentum contributions. The function is defined as:

\[\text{MultichannelBW}(\sigma, m, \Gamma_i) = \frac{1}{m^2 - \sigma - i \sum_{k} g_k^2 \cdot \rho_k(\sigma)}\]

where:

Variable $g_k^2$ are the coupling constants for each channel,
Functions $\rho_k(\sigma) = 2q_k/\sqrt{\sigma}$ are the phase space factors for the respective channels.

The total decay width $\Gamma(\sigma)$ is the sum of partial widths across all channels.

Constructing a Multichannel Breit-Wigner

To create a MultichannelBreitWigner, you can specify the parameters for each channel as a NamedTuple within a vector. Here's an example:

MultichannelBreitWigner(
    m::Real,
    channels::Vector{<:NamedTuple{(:gsq, :ma, :mb, :l, :d)}})

Use the BlattWeisskopf form factor function as needed, and remember to ensure all channel parameters are consistent with the physics of your problem.

Flatté Function

The Flatté function is a special case of the MultichannelBreitWigner designed to describe resonances with two S-wave channels. The function is expressed as:

\[\text{Flatté}(\sigma) = \frac{1}{m^2 - \sigma - i \left( g_1^2 \cdot \rho_1(\sigma) + g_2^2 \cdot \rho_2(\sigma) \right)}\]

where:

Variables $g_1$ and $g_2$ are the coupling constants for the two channels,
Functions $\rho_1(\sigma)$ and $\rho_2(\sigma)$ are the corresponding phase space factors.

Using Flatté function

When implementing the Flatté function, use the MultichannelBreitWigner as a base and define the parameters for each of the two channels. Example instantiation:

Flatte(m, gsq1, ma1, mb1, gsq2, ma2, mb2) # with Float64 parameters

This function is widely used for resonances like the $f_0(980)$ with $\pi\pi$ and $K\bar{K}$ channels.