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At the dawn of the modern particle physics era more and more subatomic particles have been discovered. In the world governed by the non-intuitive world of quantum mechanics, the nature and even existence of such states became a heavily debated topic~\cite{HeisenbergBOOK}. It became quickly clear, that a solid mathematical formalism is needed to mitigate the lack of intuition we have dealing with systems at such small scales. A particularly useful approach introduces an operator ($S$-matrix), mapping an incoming (remote past) to an outgoing (remote future) state as
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\begin{align}
  \langle\bm{p}'_1,...,\bm{p}'_n|S|\bm{p}_1,...,\bm{p}_m\rangle
  =:
  \langle\bm{p}'_1,...,\bm{p}'_n|(\mathds{1}+\mathrm{i}T)|\bm{p}_1,...,\bm{p}_m\rangle\,,
\end{align}
%%%%%%%%%%%%%%%%
relating the interaction (encoded in the $T$-matrix) to observable states.

Fundamental principles (causality, probability conservation, particle - antiparticle relations) of nature can then be translated into mathematical constraints on the $T$-matrix elements---analyticity, unitarity and crossing symmetry, see recent review~\cite{Mai:2022eur} and pedagogical introductions~\cite{Willenbrock:2022smq,Mai:2025wjb}. After promoting the kinematic variables (e.g. total energy squared $s$) to their their complex values, unitarity translates the on-shell-ness of (intermediate) multi-particle states into discontinuity constraints of the $T$-matrix elements. Thus, providing a path to define the full Riemann-surface consisting of multiple Riemann sheets. In the case of $N$ two-body coupled-channel system, there are $2^N$ such sheets which altogether can be visualized as (adapted from~\cite{Mai:2022eur})
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\begin{figure}[H]
  \centering
  \includegraphics[width=\textwidth]{RiemannSheets-1.pdf}
\end{figure}
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\noindent
a complex manifold in suitably mapped generalized kinematic variables. For example of such mapping see, e.g.,  Refs.~\cite{Kato:1965iee,Yamada:2022xam} and \cref{fig:example-b}, and [S. Dawids part] for a Riemann surface in the context of Efimov physics. Analyticity implies that besides bound states (poles at real energies below production threshold), the physical sheet [+\ldots+] is pole-free, constrained through experiments, or results of numerical lattice QCD calculations for real kinematic variables. For discussion of implications of crossing-symmetry not further tackled here see, e.g., Refs.~\cite{gribov_2008,Mandelstam:1958xc}.



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\begin{figure}[t]
  \centering
  \includegraphics[width=\textwidth]{TFF-1.pdf}
  \caption{
  Connection of photon-/meson-induced transitions preserving final-state unitarity and resulting pole position of the Roper-resonance including the associated transverse charge density ($\rho_T^{pN^*}$) of the $p\to N(1440)$ transition from the residuum of the resonance pole. For details see Refs.~\cite{Ronchen:2018ury,Wang:2024byt,Hanhart:2026hdi}.
  }
  \label{fig:example-a}
\end{figure}
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Bound states represent only a small part of currently known states of strongly interacting matter, most of which are unstable~\cite{ParticleDataGroup:2024cfk} see also reviews~\cite{Thiel:2022xtb,Doring:2025wms}. Within the $S$-matrix approach, such states are associated with poles in the complex plane on unphysical Riemann sheets. There, they are analytically connected to the physical (real) axis leading in most simple cases (single channel, narrow resonances) to excitation in observable quantities (cross sections etc.). There are several \emph{universal} aspects of this:
\begin{itemize}
  \item[(a)] Pole positions are reaction independent as long as all relevant constraints are taken into account. Couplings to particular decay channels are then encoded in the complex-valued residua of these poles. Phenomenologically, this is quite relevant, uniting input from different experiments, for example through modern dynamical coupled-channel models~\cite{Doring:2025sgb}. There, among other baryon resonances, the pole of the famous Roper-resonance $N(1440)$~\cite{Roper:1964zza} (see also \cite{Burkert:2025aze,Doring:2025wms}) has the same position in either $(e)\gamma^*p\to MB(e)$, $\gamma p\to MB$, or $MB\to MB$ transitions ($MB$ refers to a meson-baryon pair). Evaluating residua of the poles in individual channels allows one to study photo-excitation of the resonance itself and since the photon couples differently to $u$- and $d$-quarks also the associated charge density. Indeed, this allows to peek into the resonance formation as depicted exemplarily for the $N(1440)$ in \cref{fig:example-a}.
  \item[(b)] Bound states and resonances represent the same phenomena with the only difference being the Riemann sheets on which these are found. Among others this allows one to explore the hadron spectrum in hypothetical scenarios, for example increasing the quark masses above physical values. Two examples of the so-called chiral trajectory ($s_{\rm pole}(m_q)$) of the famous $f_0(500)$, $\Lambda(1405)$ and its partners is shown in \cref{fig:example-b}.
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  \begin{figure}[t]
    \centering
    \includegraphics[width=0.49\textwidth]{allpoles.pdf}\hfill
    \includegraphics[width=0.49\textwidth]{RES__POLEMOVE.pdf}
    \caption{
    Examples of resonance pole movement through a variation of quark masses from the physical value to unphysical values.
    Left: $f_0(500)$~\cite{Guo:2018zss} pole positions with the Riemann surface mapped to a 2D plane of complex-valued modulus of the three-momentum in
the center-of-mass system ($p$).
    Right: Excited $\Lambda$ $(S=-1,I=0,J^{P}=1/2^{-})$ baryons~\cite{Guo:2023wes} on a quark mass trajectory restoring SU(3) symmetry. Black lines show the corresponding movement of the two-body thresholds in pertinent steps (empty dots). On that trajectory, resonance become virtual bound states turning then to bound states visualized here through small negative and positive displacements from the real axis, respectively.
    }
    \label{fig:example-b}
  \end{figure}
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  There, through a specific model derived from a unitary implementation of Chiral Perturbation Theory (see, e.g., \cite{Oller:2000fj,Borasoy:2005ie,Garcia-Recio:2003ejq,Mai:2014xna} and recent review~\cite{Mai:2020ltx}), the resonance poles can be tracked ``moving'' to the real energy axis on the second Riemann sheet, thus, becoming virtual bound states. At higher quark masses closer to the SU(3) limit these move to the first (physical) Riemann sheet below the production threshold becoming bound states. Currently, lattice QCD calculations in this sector become available~\cite{BaryonScatteringBaSc:2023ori} allowing to test the pole trajectories explicitly~\cite{BaryonScatteringBaSc:2023zvt,Pittler:2025upn}.
\end{itemize}



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\begin{figure}[t]
  \centering
  \includegraphics[width=0.49\textwidth]{ellipse_omega_scatter.pdf}\hfill
  \includegraphics[width=0.49\textwidth]{glue_pics-1.pdf}
  \caption{
  Three-body resonances $\omega(782)$~\cite{Yan:2024gwp} and $\pi(1300)$~\cite{Yan:2025mdm} from lattice QCD at unphysical quark masses ($M_\pi=208,\,305\,{\rm MeV}$) and their extrapolations to the physical point ($M_\pi^{phys}$).
  }
  \label{fig:example-c}
\end{figure}
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Studying $S$-matrix constraints, led also recently to other fruitful insights in hadron physics and beyond. For example, unitarity condition mentioned before, is directly related to the on-shell configurations of intermediate states. Within the context of lattice QCD computations performed in a finite volume (e.g., cubic box of the size $L^3$), this has an interesting effect. Specifically, it turns out that on-shell configurations provide the only polynomial (in box-size $L$) effects, while the off-shell states lead to exponentially suppressed effects. A heuristic picture of this is that on-shell states can propagate arbitrary far from a "classical" trajectory thus feeling the size of the box, while off-shell states decay exponentially close to that trajectory. For not too small boxes (in typical modern setups $M_\pi L\approx 4\mapsto e^{-M_\pi L}\approx 1\%$)) exponentially suppressed terms can be neglected, such that this separation can be utilized deriving of two- \cite{Doring:2011vk,Dawid:2024oey} or three-particle quantization conditions~\cite{Mai:2017bge}, used recently to study $a_1(1260)$~\cite{Mai:2021nul} and rediscover $\omega(782)$~\cite{Yan:2024gwp} and $\pi(1300)$~\cite{Yan:2025mdm} from lattice QCD, see \cref{fig:example-c}. Unitarity serves further as an important anchor point comparing various quantization conditions~\cite{Jackura:2019bmu}, see also reviews~\cite{Hansen:2019nir,Mai:2021lwb} and [Ref. A. Jackura part]. Methods related to this have also been recently applied in an interesting study of Efimov effect~\cite{Dawid:2023kxu} [S. Dawid and Deltuva  part]. Understanding the on-shell configurations in the crossed channels is of large relevance for states with large mass separation in e.g. systems including pions in addition to nucleons or heavy mesons. The issue is that in the crossed channels light particles can appear as intermediate states, which leads to the appearance of the sub-threshold (left-hand) cuts (in the Mandelstam $s$ of the studied process) not too far below the reaction threshold. This is the case for the $\pi N$ scattering~\cite{Doring:2009yv} or the exotic $T_{cc}^+(3875)$ studied in finite (lattice QCD) and infinite volume, see, e.g., Refs.\cite{Meng:2023bmz,Du:2021zzh,Collins:2024sfi,Dawid:2025wsn}.

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