Riemann Sheets Research Articles

Pole structure of Pψ N(4312)+ via machine learning and uniformized S-matrix

Abstract We probed the pole structure of the Pψ N(4312)+ using a trained deep neural network. The training dataset was generated using uniformized independent S-matrix poles to ensure that the resulting interpretation is as model-independent as possible. To prevent possible ambiguity in the interpretation of the pole structure, we included the contribution from the off-diagonal elements of the S-matrix. Our trained model favors a possible three-pole structure of Pψ N(4312)+ signal, with one pole on each of the unphysical sheets. The two poles on two different Riemann sheets can be associated with a pole-shadow pair which is a characteristic of a true resonance. On the other hand, the last isolated pole is most likely associated with the coupled-channel effect in the meson-baryon system. The combined effect of these poles produced a peak below the ΣcD threshold which mimics the line shape of a hadronic molecule.

Tracking Rayleigh–Bloch waves swapping between Riemann sheets

Rayleigh–Bloch waves are modes localized to periodic arrays of scatterers with unbounded unit cells. Here, Rayleigh–Bloch waves are studied for line arrays of sound-hard circular scatterers embedded in a two-dimensional acoustic medium, for which it has recently been shown that Rayleigh–Bloch waves exist for higher frequencies than previously thought. Moreover, it was shown that Rayleigh–Bloch waves can cut off (disappear) and cut on (reappear), and additional Rayleigh–Bloch waves can cut on and interact with the existing ones. These complicated behaviours are reconsidered using a family of quasi-periodic Green’s functions that allow particular plane-wave components to become unbounded away from the array. The Green’s function formulation is combined with the block Sakurai–Sugiura method to trace the trajectories of the Rayleigh–Bloch wavenumbers as they swap between Riemann sheets that are categorized according to the unbounded plane wave(s). A detailed analysis is presented for three different scatterer radius values, and contrasting qualitative behaviours are identified. The findings are consistent with those published previously, extend to higher frequencies than allowed by the previous approach, and provide new understanding of Rayleigh–Bloch waves around the critical frequency intervals where they cut on/cut off/interact.

Nonreciprocal conversion based on line trajectory near exceptional points

Spatio-temporal permittivity modulation can simultaneously impart wave vector and frequency shifts during an indirect photonic transition process where nonreciprocal responses are accomplished. Here, we combine the nonreciprocity with exceptional points (EPs) by employing line parameter trajectory to a spatio-temporally modulated waveguide. The instantaneous state evolutions on direction-dependent Riemann sheets are thus different in forward and backward directions. Light propagates through the structure is nonreciprocal. Forwardly, an arbitrary incident wave will convert to a combination of the two waveguide modes with same intensity. Backwardly, the waves almost entirely convert to one specific mode. The conversion efficiency is more than 80% and robust to the line design. The operating wavelength is in the telecommunications band, conducive to integration with other chips. Compared to the widely used loop path, the line path has the merit of easy preparation in experiments as only one parameter is changed. The research sheds light on the optical devices such as isolator and amplifier.

Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation

Abstract Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on tracking the dynamics of the singularities in the complexified space domain all the way from the initial time until the blow-up time, which occurs when the singularities reach the real axis. This widely applicable approach gives forewarning of the possibility of blow up and an understanding of the influence of singularities on the solution behaviour on the real axis, aiding the (perhaps surprisingly involved) asymptotic analysis of the real-line behaviour. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales over which blow up is approached. The solution to the nonlinear heat equation in the complex spatial plane is shown to be related asymptotically to a nonlinear ordinary differential equation. This latter equation is studied in detail, including its computation on multiple Riemann sheets, providing further insight into the singularities of blow-up solutions of the nonlinear heat equation when viewed as multivalued functions in the complex space domain and illustrating the potential intricacy of singularity dynamics in such (non-integrable) nonlinear contexts.

Relativistic three-body scattering and the D0D*+−D+D*0 system

Scattering amplitudes involving three-particle scattering processes are investigated within the isobar approximation which respects constraints from two- and three-body unitarity. The particular system considered is the D0D*+−D+D*0, where the D*+ (D*0) enters as a p-wave D+π0 or D0π+ (D0π0 or D+π−) resonance. The interaction potentials in the coupled-channel D0D*+−D+D*0 system contain the σ, ρ, ω, and π-exchange. The analytic continuation of the amplitudes across the three-body unitary cuts is investigated to search for poles on the unphysical Riemann sheets. Associated with an unstable particle D*+ (D*0) is a complex two-body unitarity cut, through which one can further analytically continue into another unphysical Riemann sheet. Dynamical singularities emerged from the π-exchange potential are stressed. The pole generated from the D0D*+−D+D*0 interaction and its line shape in D0D0π+ break-up production are in agreement with double-charmed tetraquark Tcc+ observed by the LHCb Collaboration. Published by the American Physical Society 2024

Coupled-Channel Analysis of the χ_{c1}(3872) Line Shape with BESIII Data.

We perform a study of the χ_{c1}(3872) line shape using the data samples of e^{+}e^{-}→γχ_{c1}(3872), χ_{c1}(3872)→D^{0}D[over ¯]^{0}π^{0}, and π^{+}π^{-}J/ψ collected with the BESIII detector. The effects of the coupled channels and the off-shell D^{*0} are included in the parametrization of the line shape. The line shape mass parameter is obtained to be M_{X}=(3871.63±0.13_{-0.05}^{+0.06}) MeV. Two poles are found on the first and second Riemann sheets corresponding to the D^{*0}D[over ¯]^{0} branch cut. The pole location on the first sheet is much closer to the D^{*0}D[over ¯]^{0} threshold than the other, and is determined to be 7.04±0.15_{-0.08}^{+0.07} MeV above the D^{0}D[over ¯]^{0}π^{0} threshold with an imaginary part -0.19±0.08_{-0.19}^{+0.14} MeV.

Virtual states in the coupled-channel problems with an improved complex scaling method

We improve the complex scaling method (CSM) to obtain virtual states, which is challenging in the conventional CSM. Our approach solves the Schrödinger equation in momentum space as an eigenvalue problem by choosing flexible contours. It proves to be highly effective in identifying the poles across different Riemann sheets in multichannel scatterings. This approach is more straightforward and efficient than employing the Lippmann-Schwinger equation and using a root-finding algorithm to search for the zeros of the Fredholm determinant. This advancement significantly extends the capabilities of the CSM in accurately characterizing the virtual states in quantum systems. Published by the American Physical Society 2024

Three-body unitary coupled-channel approach to radiative J/ψ decays and η(1405/1475)

Recent BESIII data on radiative J/ψ decays from ∼1010 J/ψ samples should significantly advance our understanding of the controversial nature of η(1405/1475). This motivates us to develop a three-body unitary coupled-channel model for radiative J/ψ decays to three-meson final states of any partial wave (JPC). Basic building blocks of the model are bare resonance states such as η(1405/1475) and f1(1420), and πK, KK¯, and πη two-body interactions that generate resonances such as K*(892), K0*(700), and a0(980). This model reasonably fits KSKSπ0 Dalitz plot pseudodata generated from the BESIII’s JPC=0−+ amplitude for J/ψ→γKSKSπ0. The experimental branching ratios of η(1405/1475)→ηππ and η(1405/1475)→γρ relative to that of η(1405/1475)→KK¯π are simultaneously fitted. Our 0−+ amplitude is analytically continued to find three poles, two of which correspond to η(1405) on different Riemann sheets of the K*K¯ channel, and the third one for η(1475). This is the first pole determination of η(1405/1475) and, furthermore, the first-ever pole determination from analyzing experimental Dalitz plot distributions with a manifestly three-body unitary coupled-channel framework. Process-dependent ηππ, γπ+π−, and πππ lineshapes of J/ψ→γ(0−+)→γ(ηππ), γ(γρ), and γ(πππ) are predicted, and are in reasonable agreement with data. A triangle singularity is shown to play a crucial role to cause the large isospin violation of J/ψ→γ(πππ). Published by the American Physical Society 2024

A method for transforming aliased modes to true modes based on density clustering and Riemann sheets selection in acoustic logging dispersion inversion

Dispersion curves obtained from well logs provide crucial formation information around the borehole. An essential step in the inversion of such curves is to eliminate aliased modes, which can potentially interfere with true modes at higher frequencies. The traditional approach involves manually matching the inversion slowness values with the corresponding wave modes at each frequency, which is time consuming and labor intensive. In addition, the traditional method is applicable for the elimination of the aliased modes that are far from true modes, but it is ineffective in resolving aliased modes that approach or even intersect with the true modes. To address these limitations, a novel method is developed to transform aliased modes into true modes. The proposed method involves two key steps. First, the slownesses of individual modes are automatically picked by the density clustering algorithm. Second, based on the Riemann sheets selection of slowness, aliased modes are transformed into true modes by using the single valuedness of the wave-mode spectra and the correspondence between the amplitude spectrum and the slowness. In addition, a theoretical background is provided by introducing the concept of Riemann sheets to explain the origin of aliased modes in the computation. The proposed method can obtain comprehensive and precise dispersion curves, even in cases for which the true modes are interfered with or crossed by aliased modes. The accuracy of the method is validated by comparing the inversion results with the forward model’s dispersion curves. Furthermore, the robustness of the approach is determined by applying it to synthetic waveforms with noise and field waveforms.

Three-body unitary coupled-channel analysis on η(1405/1475)

The recent BESIII data on $J/\psi\to\gamma(K_SK_S\pi^0)$, which is significantly more precise than earlier $\eta(1405/1475)$-related data, enables quantitative discussions on $\eta(1405/1475)$ at the previously unreachable level. We conduct a three-body unitary coupled-channel analysis of experimental Monte-Carlo outputs for radiative $J/\psi$ decays via $\eta(1405/1475)$: $K_SK_S\pi^0$ Dalitz plot distributions from the BESIII, and branching ratios of $\gamma(\eta\pi^+\pi^-)$ and $\gamma(\gamma\pi^+\pi^-)$ final states relative to that of $\gamma(K\bar{K}\pi)$. Our model systematically considers (multi-)loop diagrams and an associated triangle singularity, which is critical in making excellent predictions on $\eta(1405/1475)\to \pi\pi\pi$ lineshapes and branching ratios. The $\eta(1405/1475)$ pole locations are revealed for the first time. Two poles for $\eta(1405)$ are found on different Riemann sheets of the $K^*\bar{K}$ channel, while one pole for $\eta(1475)$. The $\eta(1405/1475)$ states are described with two bare states dressed by continuum states. The lower bare state would be an excited $\eta^\prime$, while the higher one could be an excited $\eta^{(\prime)}$, hybrid, glueball, or their mixture. This work presents the first-ever pole determination based on a manifestly three-body unitary coupled-channel framework applied to experimental three-body final state distributions (Dalitz plots).

Component wave calculation and analysis of the acoustic field in a borehole within a three-phase porous medium

The knowledge of acoustic wave propagation in a borehole embedded in gas hydrate-bearing sediments is of great significance for the exploitation of gas hydrate. A gas hydrate-bearing sediment is a typical three-phase porous medium containing two solids and one fluid. However, until now, the borehole acoustic wavefield and its component waves within such a porous medium have never been calculated. In this work, a real-axis integration method is proposed to calculate the borehole acoustic field embedded in a three-phase porous medium based on the Biot-type three-phase theory. Meanwhile, a component wave approach, combined with the branch-cut integral method and the residue theorem, including residues at leaky poles is proposed to study the borehole wave propagation of a three-phase porous medium. The branch points and poles of the potential acoustic wave function are obtained, which correspond to the normal and leaky modes on various Riemann sheets. On this basis, the excitation intensity and waveforms of each component are obtained. The result shows that the waveform summed up from all individual waves agrees well with the full waveform calculated by real-axis integration, which provides a theoretical basis for the subsequent inversion of reservoir parameters by using the information of various mode waves.

Analytic Map of Three-Channel S Matrix: Generalized Uniformization and Mittag-Leffler Expansion.

We explore the analytic structure of the three-channel S matrix by generalizing uniformization and making a single-valued map for the three-channel S matrix. First, by means of the inverse Jacobi's elliptic function we construct a transformation from eight Riemann sheets of the center-of-mass energy complex plane onto a torus, on which the three-channel S matrix is represented single-valued. Second, we show that the Mittag-Leffler expansion, a pole expansion, of the three-channel scattering amplitude includes not only topologically trivial but also nontrivial contributions and is given by the Weierstrass zeta function. Finally, taking a simple nonrelativistic effective field theory with contact interaction for the S=-2, I=0, J^{P}=0^{+}, ΛΛ-NΞ-ΣΣ coupled-channel scattering, we demonstrate that as a function of the uniformization variable the scattering amplitude is, in fact, given by the Mittag-Leffler expansion and is dominated by contributions from neighboring poles.

Parity-dependent unidirectional and chiral photon transfer in reversed-dissipation cavity optomechanics

Nonreciprocal elements, such as isolators and circulators, play an important role in classical and quantum information processing. Recently, strong nonreciprocal effects have been experimentally demonstrated in cavity optomechanical systems. In these approaches, the bandwidth of the nonreciprocal photon transmission is limited by the mechanical resonator linewidth, which is arguably much smaller than the linewidths of the cavity modes in most electromechanical or optomechanical devices. In this work, we demonstrate broadband nonreciprocal photon transmission in the reversed-dissipation regime, where the mechanical mode with a large decay rate can be adiabatically eliminated while mediating anti-PT-symmetric dissipative coupling with two kinds of phase factors. Adjusting the relative phases allows the observation of periodic Riemann-sheet structures with distributed exceptional points (Eps). At the Eps, destructive quantum interference breaks both the T- and P-inversion symmetry, resulting in unidirectional and chiral photon transmissions. In the reversed-dissipation regime, the nonreciprocal bandwidth is no longer limited by the mechanical mode linewidth but is improved to the linewidth of the cavity resonance. Furthermore, we find that the direction of the unidirectional and chiral energy transfer could be reversed by changing the parity of the Eps. Extending non-Hermitian couplings to a three-cavity model, the broken anti-PT-symmetry allows us to observe high-order Eps, at which a parity-dependent chiral circulator is demonstrated. The driving-phase controlled periodical Riemann sheets allow observation of the parity-dependent unidirectional and chiral energy transfer and thus provide a useful cell for building up nonreciprocal array and realizing topological, e.g., isolators, circulators, or amplifiers.

Uniformizing Lee-Yang singularities

Motivated by the search for the QCD critical point, we discuss how to obtain the singular behavior of a thermodynamic system near a critical point, namely the Lee-Yang singularities, from a limited amount of local data generated in a different region of the phase diagram. We show that by using a limited number of Taylor series coefficients, it is possible to reconstruct the equation of state past the radius of convergence, in particular in the critical region. Furthermore we also show that it is possible to extend this reconstruction to go from a crossover region to the first-order transition region in the phase diagram, using a uniformizing map to pass between Riemann sheets. We illustrate these ideas via the Chiral Random Matrix Model and the Ising Model.

Pole position of the a1(1260) resonance in a three-body unitary framework

Masses, widths, and branching ratios of hadronic resonances are quantified by their pole positions and residues with respect to transition amplitudes on the Riemann sheets of the complex energy-plane. In this study we discuss the analytic structure in the physical energy region of three-body scattering amplitudes on such manifolds. As an application, we determine the pole position of the $a_1(1260)$ meson from the ALEPH experiment by allowing for $\pi\rho$ coupled channels in S- and D-wave. We find it to be $\sqrt{s_0}=(1232^{+15+9}_{-0-11}-i266^{+0+15}_{-22-27})~\text{MeV}$.

Near-threshold spectrum from a uniformized Mittag-Leffler expansion: Pole structure of the Z(3900)

We demonstrate how S-matrix poles manifest themselves as the physical spectrum near the upper threshold in the context of the two-channel uniformized Mittag-Leffler expansion, an expression written as a sum of pole terms under an appropriate variable where the S-matrix is made single-valued (uniformization). We show that the transition of the spectrum is continuous as a S-matrix pole moves across the boundaries of the complex energy Riemann sheets and that the physical spectrum peaks at or near the upper threshold when the S-matrix pole is positioned sufficiently close to it on the uniformized plane. There is no essential difference on which sheet the pole is positioned. What is important is the existence of a pole near the upper threshold and the distance between the pole and the physical region, not on which complex energy sheet the pole is positioned. We also point out that when the pole is close to the upper threshold, the complex pole does not have the usual meaning of the resonance. Neither the real part represents the peak energy, nor the imaginary part represents the half width. Subsequently, we try to understand the current status of $Z(3900)$ from the viewpoint of the uniformized Mittag-Leffler expansion reflecting in particular, Phys.Rev.Lett.117, 242001 (2016) in which they concluded that $Z(3900)$ is not a conventional resonance but a threshold cusp. We point out that their results turn out to indicate the existence of S-matrix poles near the $\bar D D^*$ threshold, which is most likely the origin of the peak found in their calculation of the near-threshold spectrum. In order to support our argument, we set up a separable potential model which shares common behavior of poles near the $\bar D D^*$ threshold to the above-mentioned reference and show in our model that the structures near the $\bar D D^*$ threshold are indeed caused by these near-threshold poles.

Self-consistent O(4) model spectral functions from analytically continued functional renormalization group flows

In this paper we explore practicable ways for self-consistent calculations of spectral functions from analytically continued functional renormalization group (aFRG) flow equations. As a particularly straightforward one we propose to include parametrizations of self-energies based on explicit analytic one-loop expressions. To exemplify this scheme we calculate the spectral functions of pion and sigma meson of the $O(4)$ model at vanishing temperature in the broken phase. Comparing the results with those from previous aFRG calculations, we explicitly demonstrate how self-consistency at all momenta fixes the tight relation between particle masses and decay thresholds. In addition, the two-point functions from our new semianalytic FRG scheme have the desired domain of holomorphy built in and can readily be studied in the entire cut-complex frequency plane, on physical as well as other Riemann sheets. This is very illustrative and allows, for example, to trace the flow of the resonance pole of the sigma meson across an unphysical sheet. In order to assess the limitations due to the underlying one-loop structure, we also introduce a fully self-consistent numerical scheme based on spectral representations with scale-dependent spectral functions. The most notable improvement of this numerically involved calculation is that it describes the three-particle resonance decay of an off-shell pion, ${\ensuremath{\pi}}^{*}\ensuremath{\rightarrow}\ensuremath{\sigma}\ensuremath{\pi}\ensuremath{\rightarrow}3\ensuremath{\pi}$. Apart from this further conceptual improvement, overall agreement with the results from the considerably simpler semianalytic one-loop scheme is very encouraging, however. The latter can therefore provide a sound and practicable basis for self-consistent calculations of spectral functions in more realistic effective theories for warm and dense matter.

Thermal study of a coupled-channel system: a brief review

We demonstrate the construction of a density of states from the S-matrix describing a coupled-channel (S-wave pi pi , K {bar{K}}) system, and examine the influences from various structures of particle dynamics: poles, roots, and Riemann sheets.

Unstable States in a Model of Nonrelativistic Quantum Electrodynamics: Corrections to the Lorentzian Distribution

We revisit the Lee-Friedrichs model as a model of atomic resonances in the hydrogen atom, using the dipole-moment matrix-element functions which have been exactly computed by Nussenzveig. The Hamiltonian H of the model is positive and has absolutely continuous spectrum. Although the return probability amplitude \(R_{\Psi }(t) = (\Psi , \exp (-iHt) \Psi )\) of the initial state \(\Psi \), taken as the so-called Weisskopf–Wigner (W.W.) state, cannot be computed exactly, we show that it equals the sum of an exponentially decaying term and a universal correction \(O(\beta ^{2}\frac{1}{t})\), for large positive times t and small coupling constants \(\beta \), improving on some results of King (Lett Math Phys 23:215–222, 1991). The remaining, non-universal, part of the correction is also shown to be of the same qualitative type. The method consists in approximating the matrix element of the resolvent operator operator in the W.W. state by a Lorentzian distribution. No use is made of complex energies associated to analytic continuations of the resolvent operator to ”physical” Riemann sheets. Other new results are presented, in particular a physical interpretation of the corrections, and the characterization of the so-called sojourn time \(\tau _{H}(\Psi )= \int _{0}^{\infty } |R_{\Psi }(t)|^{2} dt\) as the average lifetime of the decaying state, a standard quantity in (quantum) probability.

Density of states of a coupled-channel system

We demonstrate how an effective density of states can be derived from the S-matrix describing a coupled-channel system. Besides the locations of poles, the phase of the determinant of the S-matrix encodes essential details in characterizing the dynamics of resonant and non-resonant interactions. The density of states is computed for the two channel scattering problem ($\pi\pi, K \bar{K}$, S-wave), and the influences from the various dynamical structures: poles, roots, branch cuts, and Riemann sheets, are examined.