Unphysical Riemann Sheets: Analytic and Physical Insights

• Unphysical Riemann Sheets are auxiliary branches of Riemann surfaces obtained by analytic continuation across branch cuts, essential for unveiling resonance structures in scattering amplitudes, nonlinear waves, and non-Hermitian systems.
• Their labeling is determined by the number and type of branch cut crossings and winding numbers, enabling systematic tracking of analytic continuations and singularity classification in complex energy planes.
• Resonance poles appearing on unphysical sheets govern physical observables by revealing unstable states and influencing scattering processes, as demonstrated in models such as Friedrichs-Faddeev and non-Hermitian band theories.

A Riemann surface associated with a multivalued analytic function is partitioned into distinct “sheets” by its set of branch points and branch cuts. The “physical” (principal) sheet is defined by specific boundary or asymptotic conditions conforming to the physical problem, while all other sheets reached by analytic continuation across branch cuts are termed unphysical Riemann sheets. Such sheets underlie a broad class of phenomena in mathematical physics, including resonances in scattering amplitudes, the singularity structure of nonlinear waves, properties of analytic continuations in complex RG flows, and the topology of non-Hermitian band structures. The analytic properties, labeling, and physical interpretation of unphysical sheets are model-dependent but share common foundational features rooted in the theory of analytic continuation and monodromy.

1. Sheet Structure: Analytic Continuation and Branch Cuts

Unphysical Riemann sheets arise when a multi-valued complex function defined by algebraic, transcendental, or integral equations is analytically continued across branch cuts emanating from its branch points. The canonical example is the square-root or logarithmic branch in a complex variable, but in physical models, much richer structures occur.

In two-body and multi-body scattering theory, the analytic continuation of amplitude functions (e.g., the S-matrix or T-matrix) through branch cuts associated with reaction thresholds (two-body/three-body thresholds, unitarity cuts) leads to a multi-sheeted Riemann surface. Each cut is associated with a particular channel or kinematic configuration; winding around the cut or encircling the branch point moves the amplitude onto a corresponding unphysical sheet. In models like the Friedrichs-Faddeev system, traversing the branch cut along the continuous spectrum $[a,b]$ from the upper to the lower half-plane transfers functions from the physical sheet to an adjacent unphysical one. The number and structure of sheets can be finite (as in models with discrete thresholds) or infinite (as in systems with logarithmic cuts, multiple thresholds, or in the limit of infinite volume or continuum models) (Correia et al., 15 Dec 2025, Motovilov, 2018, Dawid, 2023).

2. Physical Versus Unphysical Sheets: Labeling and Operational Criteria

The distinction between physical and unphysical sheets is defined operationally via analyticity, boundary conditions, and monodromy properties. The physical (principal) sheet is typically the one containing the real axis or positive energy region where the original boundary conditions or asymptotics are specified—e.g., $M^2\to\infty$ as $b\to 0$ in the large-N $O(N)$ sigma model (Meurice et al., 2011), or the upper half plane for energy variable $z$ in Friedrichs-Faddeev models (Motovilov, 2018).

Labeling of sheets is governed by the number and type of cuts crossed. For a collection of $N$ independent square-root cuts (e.g., $N$ distinct two-body thresholds), sheets are labeled by a multi-index $(n_1,\ldots,n_N) \in \mathbb{Z}_2^N$, with $n_i=0$ (not wound) or $n_i=1$ (wound once). For logarithmic cuts or in the presence of infinite thresholds, labels may be drawn from $\mathbb{Z}^N$ (arbitrary winding). In multivariable situations (e.g., CFT correlators in Lorentzian signature or non-Hermitian bands), sheets are labeled by tuples specifying the winding order and direction around each branch point (Correia et al., 15 Dec 2025, Kundu et al., 2 May 2025, Wang et al., 2024).

3. Resonances and Poles on Unphysical Sheets

The physical significance of unphysical sheets is exemplified by the appearance of resonance poles—zeros of $\det S(s)$ or $\det[1 - VG]$—located away from the physical region but which dominate analytic continuations and physical cross sections. In two-body scattering, analytic continuation reveals resonance poles on the second or higher unphysical sheets of the S-matrix (or T-matrix), corresponding to unstable particle states. Such poles are not present on the physical sheet but influence physical amplitudes via analytic continuation and their proximity to the real axis (Correia et al., 15 Dec 2025, Motovilov, 2018, Dawid, 2023, Zhang, 2024). The positions, residues, and couplings of these poles are universal and factorize across channels due to analyticity and unitarity. This factorization property ensures that, upon continuation to an unphysical sheet, the pole structure remains encoded in the analytic data of the physical sheet (Correia et al., 15 Dec 2025, Motovilov, 2018).

In three-body and coupled-channel settings, the Riemann surface is stratified by multiple cuts: leads to a rapid proliferation of unphysical sheets. For example, in the $D^0 D^{*+}$-$D^+ D^{*0}$ hadronic system, each two-body self-energy and three-body breakup threshold yields a new type of discontinuity, and the physically relevant resonance (e.g., $T_{cc}^+$) appears as a pole only after analytic continuation across the appropriate sequence of two- and three-body cuts (Zhang, 2024). Contour deformation in the space of internal or spectator momenta ensures that analytic continuations land on the correct unphysical sheet to reveal the resonance pole (Dawid, 2023, Zhang, 2024).

4. Examples in Wave, RG, and Band-Structure Problems

Beyond scattering theory, unphysical sheets structure the analysis of nonlinear water waves, renormalization group flows, and band theory:

• Stokes waves: The analytic continuation of the conformal map describing a finite-amplitude periodic gravity wave generates an infinite ladder of unphysical sheets as square-root branch cuts are crossed. Each higher sheet features new coupled singularities, and in the limit $v_c\to 0$, the coalescence of an infinite set of such singularities reproduces the $2/3$-power law singularity of the limiting Stokes wave crest (Lushnikov, 2015).
• Complex RG flows: In two-dimensional O(N) sigma models, the map from coupling to mass gap is $(q+1)$-valued—i.e., it lives on a Riemann surface with $(q+1)$ physical and unphysical sheets. Fisher’s zeros of the partition function appear as strings terminating near unphysical-sheet singularities, thus controlling the convergence and large-order behavior of the perturbative expansion (Meurice et al., 2011).
• Non-Hermitian band structures: In one-dimensional non-Hermitian lattice models, the band structure as a function of complex momentum forms a multi-sheeted Riemann surface, with exceptional points as branch points. The physical sheet corresponds to Bloch states with real $k$, while open-boundary spectra and the skin effect manifest as phenomena on unphysical sheets behind branch cuts (Wang et al., 2024).

5. Monodromy, Topology, and Sheet Classification

The full sheet structure is governed by the monodromy representation—how analytic continuation along paths in the space of variables (energy, momentum, cross-ratios, etc.) permutes the function’s multiple branches. In two-band non-Hermitian Hamiltonians, each branch point (exceptional point) corresponds to a simple transposition. A “braid word” records the effect of traversing the Brillouin zone and crossing the sequence of cuts, and the winding structure constrains the possible topological features of the open-boundary spectrum (Wang et al., 2024).

In conformal field theory four-point correlators on the Lorentzian cylinder, Riemann sheet labels enumerate the winding numbers around all singularities in both holomorphic and anti-holomorphic sectors. Only a small subset of these sheets are realized by causal orderings (the “physical” sheets), while the vast bulk remain unphysical and currently lack direct physical interpretation (Kundu et al., 2 May 2025).

6. Analytic Properties and Resonance Extraction Techniques

The extraction of resonance information from lattice QCD, EFT, or analytic model calculations requires careful analytic continuation onto unphysical sheets. This is typically achieved via:

• Contour deformation in integration variables to circumvent singularities or traverse branch cuts;
• Monodromy tracking to account for the jumps in amplitude as cuts are crossed, using explicit mapping between physical and unphysical amplitudes (as in the Motovilov representation for the $T$-matrix (Motovilov, 2018));
• Fredholm determinant condition applied on the desired unphysical sheet, whose zeros correspond to resonance energies;
• Residue analysis at the pole, which yields effective couplings to all relevant channels, universally and with channel factorization (Correia et al., 15 Dec 2025, Dawid, 2023, Zhang, 2024).

A resonance is only physically meaningful if it appears on the correct unphysical sheet—i.e., the sheet determined by the analytic structure that corresponds to crossing the relevant sequence of physical cuts.

7. Physical Interpretation and Limitations

Unphysical sheets are indispensable for encoding the global analytic structure of physical quantities. While only the physical sheet is directly realized in experiment or classical solution, unphysical sheets control the singularity structure, resonance positions, large-order behavior, and topological features of the underlying model. In finite systems or mathematical models, they delimit the radius of convergence of expansions, track the sensitivity to parameter variations, and underpin the analytic continuation procedures essential for extracting resonance properties.

Although the vast majority of unphysical sheets in models with infinite branching structure (e.g., CFT on the Lorentzian cylinder, non-Hermitian bands) may never correspond to experimentally reachable configurations, their presence is a mathematical necessity, and in certain circumstances (e.g., string termination points for Fisher zeros or resonance poles just below the real axis) they have direct physical consequences (Kundu et al., 2 May 2025, Meurice et al., 2011, Dawid, 2023).

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References

• "Branch cuts of Stokes wave on deep water. Part II: Structure and location of branch points in infinite set of sheets of Riemann surface" (Lushnikov, 2015)
• "Resonances: Universality and Factorization on Higher Sheets" (Correia et al., 15 Dec 2025)
• "Complex RG flows for 2D nonlinear O(N) sigma models" (Meurice et al., 2011)
• "Unphysical energy sheets and resonances in the Friedrichs-Faddeev model" (Motovilov, 2018)
• "One-dimensional non-Hermitian band structures as Riemann surfaces" (Wang et al., 2024)
• "Monodromies of CFT correlates on the Lorentzian Cylinder" (Kundu et al., 2 May 2025)
• "Analytic continuation of the finite-volume three-particle amplitudes" (Dawid, 2023)
• "The relativistic three-body scattering and the $D^0D^{*+}-D^+D^{*0}$ system" (Zhang, 2024)