Complex Energy Plane Research Articles

Dynamical generation of resonances in the P33 partial wave

We investigate the formation of resonances in the P33 partial wave with the emphasis on possible emergence of dynamically generated quasi-bound states as a consequence of a strong $p$-wave pion attractive interaction in this partial wave, as well as their possible interaction with the genuine quark excited states. By using the Laurent-Pietarinen expansion we follow the evolution of the $S$-matrix poles in the complex energy plane as a function of the interaction strength. Already without introducing a genuine quark resonant state, two physically interesting resonances emerge with pole masses around 1200 MeV and 1400 MeV, with the dominant $\pi N$ and $\pi\Delta$ component, respectively. The added genuine resonant state in the $(1s)^3$ quark configuration mixes with the lower dynamically generated resonance forming the physical $\Delta(1232)$ resonance, and pushes the second dynamical resonance to around 1500 MeV, which allows it to be identified with the $\Delta(1600)$ resonance. Adding a second resonant state with one quark promoted to the $2s$ orbit generates another pole whose evolution remains well separated from the lower two poles. We calculate the helicity amplitudes at the pole and suggest that their $Q^2$ dependence could be a decisive test to discriminate between different models of the $\Delta(1600)$ resonance.

Reflection Time Difference as a Probe of S-Matrix Zeroes in Chaotic Resonance Scattering

Motivated by recent interest in the zeroes of S-matrix entries in the complex energy plane, I use the Heidelberg model of resonance scattering to introduce the notion of Reflection Time Difference which is shown to play the same role for the {\it zeroes} as the Wigner time delay plays for the S- matrix {\it poles}.

Quark-Mass Dependence of Elastic πK Scattering from QCD.

We present a determination of the isospin-1/2 elastic πK scattering amplitudes in S and P partial waves using lattice quantum chromodynamics. The amplitudes, constrained for a large number of real-valued energy points, are obtained as a function of light-quark mass, corresponding to four pion masses between 200 and 400MeV, at a single lattice spacing. Below the first inelastic threshold, the P-wave scattering amplitude is dominated by a single pole singularity that evolves from being a stable bound state at the highest quark mass into a narrow resonance that broadens as the pion and kaon masses are reduced. As in experiment, the S-wave amplitude does not exhibit an obviously resonant behavior, but instead shows a slow rise from threshold, which is not inconsistent with the presence of a κ/K_{0}^{⋆}(700)-like resonance at the considered quark masses. As has been found in analyses of experimental scattering data, simple analytic continuations into the complex energy plane of precisely determined lattice QCD amplitudes on the real energy axis are not sufficient to model-independently determine the existence and properties of this state. The spectra and amplitudes we present will serve as an input for increasingly elaborate amplitude analysis techniques that implement more of the analytic structure expected at complex energies.

Can a $$\varLambda nn$$ Resonance Constrain the $$\varLambda n$$ Amplitude?

We address the question of constraints on the $$\varLambda n$$ amplitude as a result of observing a $$\varLambda nn$$ resonance above the $$\varLambda nn$$ three-body threshold. To examine this question, we will first demonstrate that by starting with a $$\varLambda n$$ interaction based upon the experimental $$\varLambda p$$ data and then scaling the $$\varLambda n$$ potential by $$\approx 8\%$$ one can generate a $$\varLambda nn$$ resonance, and with a sufficiently large scaling one can generate a $$\varLambda nn$$ bound state. This is achieved using the three-body Faddeev equations with rank one separable potentials for all pairwise interactions. The use of separable potentials is motivated by two factors: (i) the Faddeev equations reduce to a set of one dimensional integral equations that can be analytically continued into the complex energy plane where resonances reside. (ii) The $$\varLambda n$$ amplitude can be defined in terms of the effective range parameters. As a result we can explore the question: What constraint can be placed on the $$\varLambda n$$ effective range parameters by the $$\varLambda nn$$ resonance parameters?

S -matrix pole symmetries for non-Hermitian scattering Hamiltonians

The complex eigenvalues of some non-Hermitian Hamiltonians, e.g. parity-time symmetric Hamiltonians, come in complex-conjugate pairs. We show that for non-Hermitian scattering Hamiltonians (of a structureless particle in one dimension) possesing one of four certain symmetries, the poles of the $S$-matrix eigenvalues in the complex momentum plane are symmetric about the imaginary axis, i.e. they are complex-conjugate pairs in complex-energy plane. This applies even to states which are not bounded eigenstates of the system, i.e. antibound or virtual states, resonances, and antiresonances. The four Hamiltonian symmetries are formulated as the commutation of the Hamiltonian with specific antilinear operators. Example potentials with such symmetries are constructed and their pole structures and scattering properties are calculated.

Pseudoscalar meson and baryon octet interaction with strangeness zero in the unitary coupled-channel approximation

The interaction of the pseudoscalar meson and the baryon octet is investigated by solving the Bethe-Salpeter equation in the unitary coupled-channel approximation. In addition to the Weinberg-Tomozawa term, the contribution of the and channel potentials in the S-wave approximation are taken into account. In the sector of isospin and strangeness , a pole is detected in a reasonable region of the complex energy plane of in the center-of-mass frame by analyzing the behavior of the scattering amplitude, which is higher than the threshold and lies on the third Riemann sheet. Thus, it can be regarded as a resonance state and might correspond to the particle of the Particle Data Group (PDG) review. The coupling constants of this resonance state to the , , and channels are calculated, and it is found that this resonance state couples strongly to the hidden strange channels. Apparently, the hidden strange channels play an important role in the generation of resonance states with strangeness zero. The interaction of the pseudoscalar meson and the baryon octet is repulsive in the sector of isospin and strangeness , so that no resonance state can be generated dynamically.

On chiral excitations with exotic quantum numbers

We consider the flavour sextet of charmed meson resonances with JP=1+ quantum numbers that is predicted by the leading order chiral Lagrangian with up, down and strange quarks. The effect of chiral correction terms as determined previously from QCD lattice data is worked out. Pole masses in the complex energy plane are derived. The most promising signal from such states accessible in experiments like Belle, LHCb and PANDA is foreseen in the s-wave πD⁎ phase shift and the ηD⁎ invariant mass distribution. For physical quark masses a rapid variation of the phase shift in-between the ηD⁎ and the K¯Ds⁎ thresholds is predicted.

Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics

Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequence of this symmetry unification is the emergence of unique non-equilibrium topological phases that have no counterparts in Hermitian systems. We illustrate this by presenting a non-Hermitian counterpart of the Majorana chain in an insulator with time-reversal symmetry and that of the quantum spin Hall insulator in a superconductor with particle-hole symmetry. Our work establishes a fundamental symmetry principle in non-Hermitian physics and paves the way towards a unified framework for non-equilibrium topological phases.

Circular transmission resonances and magnetic field effects in a ring of quantum dots connected to external leads in the meta-configuration

The transmission and the circular transmission are investigated for a ring of quantum dots (in a benzene-type configuration) connected to external leads in the meta-configuration. A computational method utilizing the tight-binding approximation to the Schrodinger equation is used to solve for the transmission probabilities as a function of the electron energy and external magnetic flux. The flux dependence is incorporated into the model using a standard procedure involving the Aharonov–Bohm effect. The positions of the transmission zeros and poles in the complex energy plane, and their possible interference with or even complete cancellation of each other, are shown to correlate with the amplitude and structure of the circular transmission resonances. Large-amplitude resonances of the circular transmission are found to occur when two poles of the transmission are separated along the imaginary axis. These resonances demonstrate a high degree of flux sensitivity at specific energy values and flux ranges. A small change in flux causes the orientation of the resonance poles in the complex energy plane to rotate parallel to the real energy axis, resulting in a concurrent decrease in the circular transmission amplitude. The flux-dependent interference between the transmission poles and zeros in the complex energy plane leads to a decrease of the circular transmission resonance amplitudes. The circular transmission and its corresponding current–voltage characteristic provide more information related to the external flux than can be obtained from the normal transmission alone.

KN scattering amplitude revisited in a chiral unitary approach and a possible broad resonance in S = +1 channel

Abstract We revisit the $KN$ scattering amplitude in order to investigate the possibility of the existence of a broad resonance in the $I=0$$KN$ channel around the energy of 1617 MeV with 305 MeV width. We use the chiral unitary model to describe the $KN$ scattering amplitudes and determine the model parameters so as to reproduce the differential cross sections of the $K^{+}N$ scatterings and the $I=0$ and 1 total cross sections up to $p_{\rm lab} = 800$ MeV/c, from which inelastic contributions start to be significant. Performing analytic continuation of the determined amplitude to the complex energy plane, we find a pole for a broad resonance state. We point out that the rapid increase appearing in the $I=0$ total cross section around $p_{\rm lab}=500$ MeV/c is a hint of the possible broad resonance of strangeness $S=+1$.

The Connection between Resonances and Bound States in the Presence of a Coulomb Potential.

The connection between resonant metastable states and bound states with changing potential strength in the presence of a Coulomb potential is fundamentally different from the case of short-range potentials. This phenomenon is central to the physics of dissociative recombination of electrons with molecular cations. Here, it is verified computationally that there is no direct connection between the resonance pole of the S-matrix and any pole in the bound state spectrum. A detailed analysis is presented of the analytic structure of the scattering matrix, in which the resonance pole remains distinct in the complex k-plane while a new state appears in the bound state spectrum. A formulation of quantum-defect theory is developed based on the scattering matrix, which nonetheless exposes a close analytic relation between the resonant and bound state poles and thereby reveals the connection between quantum-defect theory and analytic S-matrix theory in the complex energy and momentum planes. One-channel and multichannel versions of the expressions with numerical examples for simple models are given, and the formalism is applied to give a unified picture of ab initio electronic structure and scattering calculations for e-O2+ and e-H2+ scattering.

Pole position of the a1(1260) from τ -decay

We perform an analysis of the three-pion system with quantum numbers $J^{PC}=1^{++}$ produced in the weak decay of $\tau$ leptons. The interaction is known to be dominated by the axial meson $a_1(1260)$. We build a model based on approximate three-body unitarity and fix the free parameters by fitting it to the ALEPH data on $\tau^-\to \pi^-\pi^+\pi^-\,\nu_\tau$ decay. We then perform the analytic continuation of the amplitude to the complex energy plane. The singularity structures related to the $\pi\pi$ subchannel resonances are carefully addressed. Finally, we extract the $a_1(1260)$ pole position $m_p^{(a_1(1260))}-i\Gamma_p^{(a_1(1260))}/2$ with $m_p^{(a_1(1260))} = (1209 \pm 4^{+12}_{-9})\text{MeV}$, $\Gamma_p^{(a_1(1260))} = (576 \pm 11 ^{+89}_{-20})\text{MeV}$}.

Ionization by an Oscillating Field: Resonances and Photons

We describe new exact results for a model of ionization of a bound state in a 1d delta function potential, induced by periodic oscillations of the potential of period $$2\pi /\omega $$ . In particular we have obtained exact expressions, in the form of Borel summed transseries for the energy distribution of the emitted particle as a function of time, $$\omega $$ and strength $$\alpha $$ of the oscillation of the potential. These show peaks in the energy distribution, separated by $$\hbar \omega $$ , which look like single or multi-photon absorption. The peaks are very sharp when the time is large and the strength of the oscillating potential is small but are still clearly visible for large fields, and even for time-periods of a few oscillations. These features are similar to those observed in laser induced electron emission from solids or atoms (Phys Rev Lett 105:257601, 2010). For large $$\alpha $$ the model exhibits peak-suppression. The ionization probability is not monotone in the strength of the oscillating potential: there are windows of much slower ionization at special pairs $$(\alpha ,\omega )$$ . This shows that ionization processes by time-periodic fields exhibit universal features whose mathematical origin are resonances which pump energy into the system represented by singularities in the complex energy plane. All these features are proven in our simple model system without the use of any approximations.

Four-parameter $1/r^2$ singular short-range potential with rich bound states and a resonance spectrum

Метод трехдиагонального представления применяется с целью расширить класс точно решаемых квантовых систем, для чего используется квадратично интегрируемый базис, в котором матрица волнового оператора трехдиагональна. При этом волновое уравнение принимает вид рекуррентного соотношения для коэффициентов разложения волновой функции трех последовательных порядков, решение которого в терминах ортогональных полиномов эквивалентно решению исходной задачи. Получены S-волновые связанные состояния для нового четырехпараметрического потенциала с особенностью типа $1/r^2$, но короткодействующего, который обладает сложной конфигурационной структурой и богатыми спектральными свойствами. Частица при рассеянии на таком потенциале должна преодолевать потенциальный барьер, а затем может быть захвачена в ловушку в потенциальной яме в резонансном или связанном состоянии. С использованием комплексного вращения продемонстрированы богатые спектральные свойства потенциала в случае ненулевого углового момента и показано, как эта структура изменяется с изменением параметров потенциала.

Effective-range function methods for charged particle collisions

Different versions of the effective-range function method for charged particle collisions are studied and compared. In addition, a novel derivation of the standard effective-range function is presented from the analysis of Coulomb wave functions in the complex plane of the energy. The recently proposed effective-range function denoted as $\Delta_\ell$ [Phys. Rev. C 96, 034601 (2017)] and an earlier variant [Hamilton et al., Nucl. Phys. B 60, 443 (1973)] are related to the standard function. The potential interest of $\Delta_\ell$ for the study of low-energy cross sections and weakly bound states is discussed in the framework of the proton-proton ${}^1S_0$ collision. The resonant state of the proton-proton collision is successfully computed from the extrapolation of $\Delta_\ell$ instead of the standard function. It is shown that interpolating $\Delta_\ell$ can lead to useful extrapolation to negative energies, provided scattering data are known below one nuclear Rydberg energy (12.5 keV for the proton-proton system). This property is due to the connection between $\Delta_\ell$ and the effective-range function by Hamilton et al. that is discussed in detail. Nevertheless, such extrapolations to negative energies should be used with caution because $\Delta_\ell$ is not analytic at zero energy. The expected analytic properties of the main functions are verified in the complex energy plane by graphical color-based representations.

ρ resonance from the I = 1 ππ potential in lattice QCD

We calculate the phase shift for the I = 1 ππ scattering in 2+1 flavor lattice QCD at mπ = 410 MeV, using all-to-all propagators with the LapH smearing. We first investigate the sink operator independence of the I = 2 ππ scattering phase shift to estimate the systematics in the LapH smearing scheme in the HAL QCD method at mπ = 870 MeV. The difference in the scattering phase shift in this channel between the conventional point sink scheme and the smeared sink scheme is reasonably small as long as the next-toleading analysis is employed in the smeared sink scheme with larger smearing levels. We then extract the I = 1 ππ potential with the smeared sink operator, whose scattering phase shift shows a resonant behavior (ρ resonance). We also examine the pole of the S-matrix corresponding to the ρ resonance in the complex energy plane.

N* resonances from K $\Lambda$ Λ amplitudes in sliced bins in energy

The two reactions $\gamma p\to K^+\Lambda$ and $\pi^-p\to K^0\Lambda$ are analyzed to determine the leading photoproduction multipoles and the pion-induced partial wave amplitudes in slices of the invariant mass. The multipoles and the partial-wave amplitudes are simultaneously fitted in a multichannel Laurent+Pietarinen model (L+P model), which determines the poles in the complex energy plane on the second Riemann sheet close to the physical axes. The results from the L+P fit are compared with the results of an energy-dependent fit based on the Bonn-Gatchina (BnGa) approach. The study confirms the existence of several poles due to nucleon resonances in the region at about 1.9\,GeV with quantum numbers $J^P = 1/2^+$, $3/2^+, 1/2^-, 3/2^-, 5/2^-$.

Ground state atoms confined in a real Rydberg and complex Rydberg-Scarf II potential

In this work, a system of two ground state atoms confined in a one-dimensional real Rydberg potential was modeled. The atom-atom interaction was considered as a nonlocal separable potential (NLSP) of rank one. This potential was assumed because it leads to an analytical solution of the Lippmann-Schwinger equation. The NLSPs are useful in the few body problems that the many-body potential at each point is replaced by a projective two-body nonlocal potential operator. Analytical expressions for the confined particle resolvent were calculated as a key function in this study. The contributions of the bound and virtual states in the complex energy plane were obtained via the derived transition matrix. Since the low energy quantum scattering problems scattering length is an important quantity, the behavior of this parameter was described versus the reduced energy considering various values of potential parameters. In a one-dimensional model, the total cross section in units of the area is not a meaningful property; however, the reflectance coefficient has a similar role. Therefore the reflectance probability and its behavior were investigated. Then a new confined potential via combining the complex absorbing Scarf II potential with the real Rydberg potential, called the Rydberg-Scarf II potential, was introduced to construct a non-Hermitian Hamiltonian. In order to investigate the effect of the complex potential, the scattering length and reflectance coefficient were calculated. It was concluded that in addition to the competition between the repulsive and attractive parts of both potentials, the imaginary part of the complex potential has an important effect on the properties of the system. The complex potential also reduces the reflectance probability via increasing the absorption probability. For all numerical computations, the parameters of a system including argon gas confined in graphite were considered.

Threshold energies and poles for hadron physical problems by a model-independent universal algorithm

In this work we show how by using a Padé type analytical continuation scheme, based on the Schlessinger point method, it is possible to find higher production thresholds in hadron physical problems. Recently, an extension of this numerical approach to the complex energy plane enabled the calculations of auto-ionization decay resonance poles in atomic and molecular systems. Here we use this so-called Resonances via Padé (RVP) method, to show its convergence beyond the singular point in hadron physical problems. In order to demonstrate the capabilities of the RVP method, two illustrations for the ability to identify singularities and branch points are given. In addition, two applications for hadron physical problems are given. In the first one, we identify the decay thresholds from a numerically calculated spectral function. In the second one, we use experimental data. First, we calculate the resonance pole of the f0(500) or σ meson using the S0 partial wave amplitude for ππ scattering in very good agreement with the literature. Second, we use data on the cross section ratio R(s) for e+e− collisions and discuss the prediction of decay thresholds which proves to be difficult if the data is noisy.

Reconfiguration of quantum states in {mathscr{P}}{mathscr{T}}-symmetric quasi-one-dimensional lattices

We demonstrate mesoscopic transport through quantum states in quasi-1D lattices maintaining the combination of parity and time-reversal symmetries by controlling energy gain and loss. We investigate the phase diagram of the non-Hermitian system where transitions take place between unbroken and broken {mathscr{P}}{mathscr{T}}-symmetric phases via exceptional points. Quantum transport in the lattice is measured only in the unbroken phases in the energy band–but not in the broken phases. The broken phase allows for spontaneous symmetry-broken states where the cross-stitch lattice is separated into two identical single lattices corresponding to conditionally degenerate eigenstates. These degeneracies show a lift-up in the complex energy plane, caused by the non-Hermiticity with {mathscr{P}}{mathscr{T}}-symmetry.