FEAT: implement BreitWigner expression classes

[redeboer]

Closes #438

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[redeboer]

@mmikhasenko have a look at this preview.

I followed your Breit–Wigner implementation, but don't get why the numerator is 1, i.e.

$$\mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{0}\right) = \frac{1}{- i \Gamma_{0} m_{0} + m_{0}^{2} - s}\,.$$

It seems better to me to make the lineshape unitless. With the current definition, I have to redefine AmpForm's relativistic_breit_wigner() as:

$$\Gamma_{0} m_{0} \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{0}\right)\,.$$

Am I missing something here? Seeing as this is considered the default implementation for BreitWigner in amplitude-serialization, it is important to have clear a consensus on this.

[redeboer]

I'm also not sure about the definition of the energy-dependent width of the channels in a MultichannelBreitWigner.

$$\mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(s; \Gamma_{1}, \Gamma_{2}\right) = \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \frac{\Gamma_{1} m_{0} \mathcal{F}_{L_{1}}\left(s, m_{a,1}, m_{b,1}\right)^{2}}{\sqrt{s}} + \frac{\Gamma_{2} m_{0} \mathcal{F}_{L_{2}}\left(s, m_{a,2}, m_{b,2}\right)^{2}}{\sqrt{s}}\right)$$

It results in a weird 'Flatté' lineshape when simplifying:

import sympy as sp
from ampform.dynamics import ChannelArguments, MultichannelBreitWigner

channels = [ChannelArguments(sp.Symbol(f"Gamma{i}")) for i in [1, 2]]
MultichannelBreitWigner(s, m0, channels).doit().simplify()

$$\mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(s; \Gamma_{1}, \Gamma_{2}\right) = - \frac{\sqrt{s}}{i m_{0}^{2} \left(\Gamma_{1} + \Gamma_{2}\right) - \sqrt{s} \left(m_{0}^{2} - s\right)}$$

[redeboer]

Probably best to 'standardize' the symbolic Breit–Wigner as BreitWigner(s, mass, coupling):

$$\mathcal{R}^\mathrm{BW}\left(s; m_0, g\right) = \frac{g^2}{m_0^2 - s - i g^2}$$

One can then customize the parametrization of $g$ with symbolic substitution.

There is a problem though when numerator is decoupled from denominator (production). There, you may want to absorb $g^2$ from the numerator into a weight, while setting $g^2=\Gamma_0 m_0$ in the denominator.

[redeboer]

As for #423 (comment), see RUB-EP1/amplitude-serialization#31. Using EnergyDependentWidth has an implicit normalization of the form factor.

This does not work in a multi-channel Breit–Wigner because the normalization factor $F_L^2(m_0^2)$ is not well defined if $m_0$ lies between the thresholds.

[mmikhasenko]

I love your pages, Remco.

The constant terms in BW is denoted G, not gsq. It is exactly how I defined it for HS3 document, but later I thought that gsq is still better because users assume that the constant 'width' corresponds to the observed width. It's not the case, in the current approach. That is why I fell back to 'gsq'