What are resonances in quantum field theory?

I am currently learning about the coalescence model and femtoscopy, and am very confused about what resonances are. I read here (https://www.quantamagazine.org/how-...pes-reality-20220126/?utm_source=pocket_saves) that they are, as all particles, an excitation of the field in the quantum field theory; but then why do they decay, how is the decaying process exactly; and what makes resonances not a particle, but a resonancy, if both are an excitation of the field?

You can see them as extremely short-living particles. There is no sharp dividing line between these two. Their large decay width makes them more complicated to study and approaches we use for more long-living particles (J/ψ is longliving in this context, for example) don't work well for them.

Strictly speaking resonances are not particles in the modern sense of relativistic QFT. Particles are the asymptotic free one-quantum Fock states of the corresponding fields.

A resonance is defined by a scattering process, where there is a pole of the corresponding propagator in the complex plane with a not too large imaginary part (defining the width/inverse lifetime) in the corresponding Green's function.

Except that these days it has become a synonym for "particle". If a postdoc at a seminar talks about discovering a resonance and you ask him about phase shifts and Argand plots odds are he won't know what you are talking about.

You won't believe, how difficult it can get, if you don't keep in mind that a resonance must be carefully defined by the in and the outstate. E.g., the rho-meson as it occurs in the particle data booklet is defined as the corresponding resonance in the cross section ##\mathrm{e}^+ + \mathrm{e}^- \rightarrow \text{hadrons}##.

Now, from the point of view of effective hadronic field theories, the working horse in my field of relativistic heavy-ion collisions, of course a "##\rho## meson" is produced rather in hadronic interactions. At first glance it's a two-pion excitation, but you also have the creation via baryonic resonances of all kinds, and already "in the vacuum" (i.e., in pp collisions) you get an entirely different "line shape" of the ##\rho## meson than in the PDB definition of the ##\rho##-meson.

Have a look at Fig. 5 in

https://arxiv.org/abs/1203.3557
https://doi.org/10.1140/epja/i2012-12111-9

vanhees71 said:

A resonance is defined by a scattering process, where there is a pole of the corresponding propagator in the complex plane with a not too large imaginary part

Is there really a formal requirement that the width not be too large? I.e. is something different in the maths in that case? Are there examples of poles in propagators that are not considered resonances because their imaginary part is too large?

If the width is "too large", you don't see a clear peak in the corresponding cross section, i.e., it's more like a continuum. One of the paradigmatic measurements is ##\text{e}^+ + \text{e}^- \rightarrow \text{hadrons}##, which is a measurement of the electromagnetic (or electroweak) current-current correlation function. There you also see prominent peaks, the light vector mesons ##\rho##, ##\omega##, and ##\phi##. Then a continuum and then the charmonia (##\text{J}/\psi## et al) and bottomonia (##\Upsilon## et al).