Resonance on Unphysical Riemann Sheets

• Resonance properties on unphysical Riemann sheets are defined by the analytic continuation of scattering amplitudes, revealing unstable states as pole singularities beyond the physical sheet.
• The framework uses branch cuts and multi-sheeted Riemann surfaces to systematically extract resonance masses, widths, and coupling constants from experimental and model data.
• This methodology is pivotal in analyzing coupled-channel scattering, quantum field theories, and non-Hermitian systems, thereby connecting complex analytic structures with observable phenomena.

Resonance properties on unphysical Riemann sheets comprise a central aspect of analytic S-matrix theory and the modern theory of scattering amplitudes, parameterizing the unstable states that manifest as poles of analytically continued amplitudes away from the physical sheet. The phenomenon is inherently tied to the multivaluedness induced by branch cuts associated with thresholds and inelastic channels, requiring an appropriate multi-sheeted Riemann surface for a single-valued analytic description. Resonance parameters and the relevant analytic continuation prescriptions depend intricately on the structure of these Riemann sheets, and their identification is crucial for extracting physical masses, widths, and coupling constants of unstable hadronic states, as well as for understanding subtle spectral features in non-Hermitian, PT-symmetric, or open quantum systems.

1. Analytic Structure, Branch Points, and Riemann Sheets

The analytic properties of scattering amplitudes arise from fundamental quantum principles—primarily causality and unitarity—which enforce analyticity in specific complex domains. When inelastic thresholds or multiparticle channels open, the S-matrix develops branch points and associated branch cuts at the corresponding energies (e.g., $z = m_\pi + M_N$, $z = m_\pi + M_\Delta$ in $\pi N$ scattering (Yang, 2011); see also the Lee and $S\varphi\varphi$ models (Wolkanowski, 2013)). Each square-root branch cut (e.g., $\sqrt{z-z_b}$) doubles the necessary sheets to maintain single-valuedness. For $n$ additive square-root branch points, a $2^n$-sheeted Riemann surface emerges, where analytic continuation of the amplitude to other sheets is made possible by traversing the cut(s).

The analytic continuation through a cut—across which the discontinuity is typically determined via the Sokhotski–Plemelj formula—leads to the essential distinction between the physical (first) Riemann sheet and its unphysical counterparts. Singularities not on the physical sheet (i.e., not directly reachable from the physical real axis without traversing a cut) encode the existence and properties of unstable resonances.

2. Multiple Poles and Consequences for Resonances

A direct consequence of the multi-sheeted structure is the unavoidable emergence of multiple image poles for a single resonance, a phenomenon particularly emphasized by model analytic functions such as

$$f_1(z) = \sqrt{z-z_a} + \sqrt{z-z_b} + \frac{g(z)}{z-z_0},$$

where, after full construction of a four-sheeted Riemann surface, the simple pole at $z_0$ appears in all sheets with identical residue if $g(z)$ is analytic (Yang, 2011). In more intricate structures (e.g., with poles of the form $h(z)/(\sqrt{z}-z_0)$), the multiplicity and residue structure of poles is sheet-dependent; residues may differ and poles may only manifest in certain sheets.

Practically, this is observed in the $\pi N$ system where, as the mass of the pion decreases and the $\Delta$ becomes unstable with its branch points moving into the complex plane, the Roper resonance exhibits a two-pole structure: one sheet connected to the real axis and a second “behind” the complex $\pi\Delta$ threshold, both associated with nearly degenerate pole positions. The mathematical inevitability of such multiple poles is underscored: physically, the resonance is unique, but the analytic representation on the full Riemann surface maps it to multiple poles, each contributing to physical observables through their analytic “echoes” (Yang, 2011).

3. Analytic Continuation, Unitarity, and Extraction of Resonance Parameters

Resonances are identified as poles of the analytically continued amplitude on the relevant unphysical sheet. For example, in effective quantum field theories coupling a “seed” scalar to two-particle states, the dressed propagator after Dyson resummation takes the form

$$\Delta_S(p^2) = \frac{1}{p^2 - M_0^2 + g^2\Sigma(p^2)},$$

where $\Sigma(p^2)$ contains a (multi-valued) logarithmic structure with branch cuts above threshold (Wolkanowski, 2013). Analytic continuation is implemented by shifting the branch cut (e.g., via $\Sigma_{II}(z) = \Sigma(z) + i/2$), yielding the propagator on the second sheet. The resonance pole is then obtained by solving $p^2 - M_0^2 + g^2 \Sigma_{II}(p^2) = 0$, from which the pole position encodes the resonance mass and width (see also the practical recipes for numerical continuation and discontinuity computations in scalar models (Wolkanowski, 2013)).

For coupled-channel and three-body systems, such as in Roy equation analyses for $\pi\pi$ scattering at unphysical pion masses (Cao et al., 2023), or relativistic three-body $D^0D^{*+}-D^+D^{*0}$ systems (Zhang, 3 Feb 2024), the analytic structure is more complex: each open channel introduces additional branching, and unitarity demands that the analytic continuation respects the full multichannel S-matrix structure, including both right-hand and left-hand cuts. Numerically, the continuation is achieved via contour deformations in the energy or momentum variables, moving around branch points or threshold cuts to reach the desired sheet where the resonance pole lies (Dawid, 2023, Zhang, 3 Feb 2024).

4. Example Case Studies and Physical Systems

• $\pi N$ Roper resonance: The two-pole structure observed near the $\pi\Delta$ threshold is a robust consequence of the analytic continuation through the complex $\pi\Delta$ branch point. Both poles are necessary to reproduce the partial wave analysis data in the $P_{11}$ channel; neglecting the full multi-sheeted Riemann surface would yield an incomplete characterization of the resonance (Yang, 2011).
• Scalar resonance with hadronic loops: In quantum-field-theoretic models, as the coupling increases, a single resonance can “descend” in the complex plane (widening the width) and, at strong coupling, an additional “gap pole” (stable bound state) may emerge on the first Riemann sheet below threshold (Wolkanowski, 2013).
• Dyson-Schwinger/Bethe-Salpeter equations: In continuum QCD approaches, resonance poles on the second sheet are calculated via analytic continuation of the 4-point amplitude, with the physical mass and width provided by the location in complex $s$ ($P^2$ in total momentum) (Eichmann, 2019).
• Three-body amplitudes: Analytic continuation in the presence of three-body unitarity and unstable particles often requires deformation of multiple complex contours. Poles corresponding to e.g. the $T_{cc}^+$ tetraquark are found on the unphysical sheets reached by rotating both the external and internal (self-energy) momentum loop variables (Zhang, 3 Feb 2024).
• Resonances in pseudo-Riemannian hyperbolic spaces: The meromorphic continuation of the Laplace-Beltrami resolvent introduces poles (resonances) whose locations and multiplicities (including possible order-two poles and infinite-dimensional residue representations) depend on the rank, signature, and field structure of the underlying symmetric space (Frahm et al., 2023).

5. Impact of Sheet Structure: Physical Observables, Multiplicities, and Restrictions

The presence and multiplicity of resonance poles on different sheets affect observable scattering quantities. For instance, in even-dimensional quantum scattering (Christiansen et al., 2013), the meromorphic continuation occurs on the Riemann surface of the logarithm, $\Lambda$, with resonance multiplicities on different sheets concretely linked to the zeros of scalar functions defined on the physical sheet, and subject to identities generalizing those in odd dimensions. Notably, certain resonance phenomena (e.g., the absence of “pure imaginary” resonances for $V \geq 0$ in even dimensions) are intimately connected to the sheet structure and correct analytic symmetries.

Specific features such as redundant poles and zeros (arising, for example, from analytic continuation of Bessel function parameters in exponential potentials (Moroz et al., 2019)) play a structural role in completeness relations, even if not directly tied to physical bound states.

The modeling of open quantum systems, non-Hermitian band structures as Riemann surfaces (Wang et al., 22 Jan 2024), or resonance behaviors in PT-symmetric systems similarly relies critically on the explicit identification of sheet structures and their associated analytic topology.

6. Methodological Frameworks and Generalizations

The analytic extraction of resonance properties on unphysical Riemann sheets is operationalized by:

• Employing parametric continuation (e.g., in coupling strength, mass, or momentum) to track pole trajectories through the sheets (Wolkanowski, 2013, Zhang, 3 Feb 2024).
• Constructing spectral/Hilbert space representations (e.g., in Fredholm determinants for high-order differential operators (Korotyaev, 2016) or in spectral decompositions for Laplacian resolvents (Frahm et al., 2023)).
• Utilizing dispersion relations rooted in causality to relate real and imaginary parts of amplitudes, enabling analytic continuation and precise pole identification (e.g., via the Cauchy formula, subtracted dispersion relations, and their implementation in partial wave amplitudes (Kubis, 2 Oct 2025)).
• Leveraging advanced mathematical frameworks (e.g., Roy equations (Cao et al., 2023), analytic Fredholm theory (Motovilov, 2018), boundary matrices and Cartwright class theory (Hoop et al., 2022)).

The topology of the Riemann surface, the location and number of branch cuts, and the algebraic structure of the multivalued functions involved are intrinsically tied to the physics: the mapping between analytic singularities (poles, virtual states, threshold effects) and the physical resonant content is determined by these analytic features.

7. Broader Implications and Applications

The quantification and analytic continuation of resonances on unphysical Riemann sheets underlie the standard methodology for extracting resonance parameters from experimental data, including pole masses and widths of hadrons, multi-pion or exotic resonances, leaky modes in elastic systems, and characterizing edge states in non-Hermitian lattices.

The identification of multiple poles, “companion” or “shadow” poles, or even unanticipated bound states dynamically generated below threshold, all become natural within the full Riemann surface framework. This theoretical paradigm is essential not only in relativistic quantum field theory and hadronic physics, but extends to quantum chaos, condensed matter, mathematical scattering theory, and PT-symmetric or nonlocal field theory, where the proximity and nature of sheet structure determines physical, spectral, or even topological features of the system.

Understanding resonance properties on unphysical Riemann sheets thus provides the analytic machinery necessary to connect underlying quantum dynamics—through multichannel effects, thresholds, and crossings—to observable characteristics of unstable states, including their dynamical roles in spectral, transport, or decay phenomena across a variety of physical contexts.