Bound-state Poles Research Articles

Nonperturbative anomalous thresholds

Feynman diagrams (notably the triangle diagram) involving heavy enough particles contain branch cuts on the physical sheet—anomalous thresholds—which, unlike normal thresholds and bound-state poles, do not correspond to any asymptotic n-particle state. “Who ordered that?” We show that anomalous thresholds arise as a consequence of established S-matrix principles and two reasonable assumptions: unitarity below the physical region and analyticity in the mass. We find explicit nonperturbative formulas for the anomalous threshold singularity and test them against the Coleman-Thun poles of the E8 integrable model. Published by the American Physical Society 2024

Revisiting O(N) σ model at unphysical pion masses and high temperatures

Roy-equation analysis on lattice data of ππ scattering phase shifts at mπ=391 MeV reveals that the lowest f0 meson becomes a bound state under this condition. In addition, there is a pair of complex poles below threshold generated by crossing symmetry [X.-H. Cao , ]. We use the N/D method to partially recover crossing symmetry of the O(N) σ model amplitude at leading order of 1/N expansion, and qualitatively reproduce the pole structure and pole trajectories with varying pion masses as revealed by Roy-equation analyses. The σ pole trajectory with varying temperature is also discussed and found to be similar to its properties when varying mπ. As the temperature increases, the complex σ poles firstly move from the second Riemann sheet to the real axis becoming two virtual state poles, and then one virtual state pole moves to the first sheet turning into a bound state pole and finally tends to the pion pole position at high temperature which is as expected from the chiral symmetry restoration. Our results provide further evidence that the lowest f0 state extracted from experiments and lattice data plays the role of the σ meson in the spontaneous breaking of chiral symmetry. Finally, we also briefly discuss the problems of the effective potential in the situation when mπ and the temperature become large. Published by the American Physical Society 2024

Unitary model analysis of f0(500) pole positions by continuously varying mπ : Comparison with discrete lattice predictions

Resonance, bound-state, and virtual-state pole positions of the f0(500) scalar meson are computed as a continuous function of pion mass in the framework of a unitarized and analytic coupled-channel model for scalar mesons, described as dynamical quark-antiquark states. The f0(500) is modeled with both light and strange qq¯ seeds, mixing with each other through the common S-wave ππ, KK¯, and ηη meson-meson decay channels. The few model parameters are fitted to experimental S-wave ππ phase shifts up to 1 GeV. In the case of the physical π± mass of 139.57 MeV, resonance poles at (460−i222) MeV and (978−i37.2) MeV are found for the f0(500) and f0(980), respectively. Resonance, bound-state, and virtual-state pole trajectories are computed and plotted as a function of pion masses up to 500 MeV, both in the complex-energy and complex-momentum planes. The results are discussed and compared to the most advanced lattice QCD computations employing interpolators that correspond to the qq¯ and meson-meson channels in the present model, that is, for a few discrete values of the unphysical pion mass in those lattice calculations. Published by the American Physical Society 2024

Density-Induced Hadron–Quark Crossover via the Formation of Cooper Triples

We discuss the hadron–quark crossover accompanied by the formation of Cooper triples (three-body counterpart of Cooper pairs) by analogy with the Bose–Einstein condensate to Bardeen–Cooper–Schrieffer crossover in two-component fermionic systems. Such a crossover is different from a phase transition, which often involves symmetry breaking. We calculate the in-medium three-body energy from the three-body T-matrix with a phenomenological three-body force characterizing a bound hadronic state in vacuum. With increasing density, the hadronic bound-state pole smoothly undergoes a crossover toward the Cooper triple phase where the in-medium three-body clusters coexist with the quark Fermi sea. The relation to the quarkyonic matter model can also be found in a natural manner.

Signature of a Doubly Charm Tetraquark Pole in DD^{*} Scattering on the Lattice.

The doubly charm tetraquark with flavor ccu[over ¯]d[over ¯] and isospin I=0 is investigated by calculating the DD^{*} scattering amplitude with lattice QCD. The simulation is done on CLS ensembles with dynamical u/d, s quarks and m_{π}≃280 MeV for two charm quark masses, one slightly larger and one slightly lower than the physical value. The scattering amplitudes for partial waves l=0, 1 are extracted near threshold via the Lüscher method by considering systems with total momenta PL/(2π)=0,1,sqrt[2],2 on two spatial volumes. A virtual bound state pole in the DD^{*} scattering amplitude with l=0 is found 9.9_{-7.1}^{+3.6} MeV below the DD^{*} threshold for the charm quark mass closer to the physical value. This pole is likely related to the doubly charm tetraquark discovered by LHCb less than 1MeV below the D^{0}D^{*+} threshold. Future lattice simulations closer to the continuum limit and physical quark masses would be valuable to establish this connection systematically.

Scrutinizing ππ scattering in light of recent lattice phase shifts

In this paper, the $IJ=00, 11, 20$ partial wave $\pi\pi$ scattering phase shifts determined by the lattice QCD approach are analyzed by using a novel dispersive solution of the S-matrix, i.e. the PKU representation, in which the unitarity and analyticity of scattering amplitudes are automatically satisfied and the phase shifts are conveniently decomposed into the contributions of the cuts and various poles, including bound states, virtual states and resonances. The contribution of the left-hand cut is estimated by the $SU(2)$ chiral perturbation theory to $\mathcal{O}(p^4)$. The Balanchandran-Nuyts-Roskies relations are considered as constraints to meet the requirements of the crossing symmetry. It is found that the $IJ=00$ $\pi\pi$ scattering phase shifts obtained at $m_\pi=391$ MeV by Hadron Spectrum Collaboration (HSC) reveal the presence of both a bound state pole and a virtual state pole below the $\pi\pi$ threshold rather than only one bound state pole for the $\sigma$. To reproduce the lattice phase shifts at $m_\pi=391$ MeV, a virtual-state pole in the $IJ=20$ channel is found to be necessary in order to balance the left-hand cut effects from the chiral amplitudes. Similar discussions are also carried out for the lattice results with $m_\pi=236$ MeV from HSC. The observed behaviors of the pole positions with respect to the variation of the pion masses can provide deep insights into our understanding of the dynamical origin of $\sigma$ resonance.

Exploring smoking-gun signals of the Schwinger mechanism in QCD

In Quantum Chromodynamics, the Schwinger mechanism endows the gluons with an effective mass through the dynamical formation of massless bound-state poles that are longitudinally coupled. The presence of these poles affects profoundly the infrared properties of the interaction vertices, inducing crucial modifications to their fundamental Ward identities. Within this general framework, we present a detailed derivation of the non-Abelian Ward identity obeyed by the pole-free part of the three-gluon vertex in the soft-gluon limit, and determine the smoking-gun displacement that the onset of the Schwinger mechanism produces to the standard result. Quite importantly, the quantity that describes this distinctive feature coincides formally with the bound-state wave function that controls the massless pole formation. Consequently, this signal may be computed in two independent ways: by solving an approximate version of the pertinent Bethe-Salpeter integral equation, or by appropriately combining the elements that enter in the aforementioned Ward identity. For the implementation of both methods we employ two- and three-point correlation functions obtained from recent lattice simulations, and a partial derivative of the ghost-gluon kernel, which is computed from the corresponding Schwinger-Dyson equation. Our analysis reveals an excellent coincidence between the results obtained through either method, providing a highly nontrivial self-consistency check for the entire approach. When compared to the null hypothesis, where the Schwinger mechanism is assumed to be inactive, the statistical significance of the resulting signal is estimated to be 3 standard deviations.

QFT, EFT and GFT

We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist quite generally for the fully crossing symmetric situation. Numerical implementation of the GFT constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows a systematic exploration. A comparison of our findings obtained using GFT techniques and other results in the literature is made. We study both the three-channel as well as the two-channel crossing-symmetric cases, the latter having some crucial differences. We also consider bound state poles as well as massless poles in EFTs. Finally, we consider nonlinear constraints arising from the positivity of certain Toeplitz determinants, which occur in the trigonometric moment problem.

Nonperturbative decay dynamics in metamaterial waveguides

In this work, we investigate the nonperturbative decay dynamics of a quantum emitter coupled to a composite right-/left-handed transmission line. Our theory captures the contributions from the different spectral features of the waveguide, providing an accurate prediction beyond the weak coupling regime and illustrating the multiple possibilities offered by the nontrivial dispersion of metamaterial waveguides. We show that the waveguide is characterized by a bandgap with two asymmetric edges: (i) a mu-near-zero band edge, where spontaneous emission is inhibited and an unstable pole is smoothly transformed into a bound state, and (ii) an epsilon-near-zero band edge, where the decay rate diverges and unstable and real (bound state) poles coexist. In both cases, branch cut singularities contribute with fractional decay dynamics whose nature depends on the properties of the band edges.

A new Δ method for bound state asymptotic normalization coefficients with a finite limit of the nuclear part of the effective-range function at zero energy

A new bound state Δ method for finding an asymptotic normalization coefficient (ANC) is proposed. This is valid, while the standard effective-range function (ERF) Kl(k2) method does not work when the product of colliding particles charges increases. The denominator of the strict expression of the re-normalized scattering amplitude f˜l includes the factor dl(E)=Δl(E)+hr(E)−h(η), where η=1/aBk, aB is the Bohr radius. In the physical region, the Coulomb term hr=Reh(η). So an analytical continuation of hr(E) from the physical region to E≤0 can be found by fitting hr(E) for E≥0. The related analytical continuation of h(η) consists in a simple sign change, E→−E, using an explicit dependence of h(η) on E. This is important that for E≤0, abs(Δh)=abs(h−hr)=0 at E=0, and it increases when abs(E) does. Thus, a new real equation, dl(−ε)=0, is obtained for a binding energy ε. It is applied to find a residue W of f˜l at the bound state pole E=−ε, the nuclear vertex constant (NVC) and (ANC). The Coulomb-nuclear phase shift δlcs, cotδlcs and a finite limit of the nuclear part Δl(k2) of Kl(k2) are also derived for an arbitrary orbital momentum l when E→0. It is shown that cotδlcs has an essential singularity at zero energy, but Δl(k2) does not. The explicit finite limit of Δl(k2) when E→0 is found using the expression for Kl(k2). These results are in agreement with those for the S-wave scattering, which are widely accepted. The ANCs for the ground and first excited bound states for the vertex Be7↔3He4He are calculated using the proposed new method, and are compared with those for the approximate method when dl(E)=Δl(E) proposed by Ramírez Suárez and Sparenberg (2017). The fit of Δl(E) is found from the experimental phase-shift input data and the additional equation dl(−ε)=0.

Redundant poles of the S-matrix for the one-dimensional Morse potential

We analyze the structure of the scattering matrix, S(k), for the one-dimensional Morse potential. We show that, in addition to a finite number of bound state poles and an infinite number of antibound poles, there exist infinite redundant poles, on the positive imaginary axis, which do not correspond to either of the other types. We explain in detail the role of these redundant poles, in particular when they coincide with the bound poles. This can be solved analytically and exactly. In addition, we obtain wave functions for all these poles and ladder operators connecting them.

Possible molecular states in B(*)B(*) scatterings

We present that, if unitarizing the $B^{(*)}B^{(*)}$ scattering amplitudes in the constituent interchange model, one can find two bound state poles for $(I_{tot},S_{tot})=(0,1)$ $BB^*$ and $B^*B^*$ system, which corresponds to two $I(J^P)=0(1^+)$ doubly bottomed molecular states. Furthermore, it is noticed that the virtual states in $(1,0)$ $BB$, $(1,1)$ $BB^*$, $(1,0)$ $B^*B^*$, and $(1,2)$ $B^*B^*$ systems could produce enhancements of the module squares of the scattering $T$-matrix just above the related thresholds, which might correspond to $I(J^P)=1(0^+)$, $1(1^+)$, and $1(2^+)$ doubly bottomed molecular states, respectively. The calculation may be helpful for searching for the doubly bottomed molecular state in future experiments.

Effective Quantum Field Theory for the Thermodynamical Bethe Ansatz

We construct an effective Quantum Field Theory for the wrapping effects in 1+1 dimensional models of factorised scattering. The recently developed graph-theoretical approach to TBA gives the perturbative desctiption of this QFT. For the sake of simplicity we limit ourselves to scattering matrices for a single neutral particle and no bound state poles, such as the sinh-Gordon one. On the other hand, in view of applications to AdS/CFT, we do not assume that the scattering matrix is of difference type. The effective QFT involves both bosonic and fermionic fields and possesses a symmetry which makes it one-loop exact. The corresponding path integral localises to a critical point determined by the TBA equation.

On the existence of N*(890) resonance in S11 channel of πN scatterings

Low-energy partial-wave πN scattering data is reexamined with the help of the production representation of partial-wave S matrix, where branch cuts and poles are thoroughly under consideration. The left-hand cut contribution to the phase shift is determined, with controlled systematic error estimates, by using the results of O(p3) chiral perturbative amplitudes obtained in the extended-onmass- shell scheme. In S11 and P11 channels, severe discrepancies are observed between the phase shift data and the sum of all known contributions. Statistically satisfactory fits to the data can only be achieved by adding extra poles in the two channels. We find that a S11 resonance pole locates at $$\sqrt {{z_r}} $$ = (0:895–0:081)–(0:164–0:023)i GeV, on the complex s-plane. On the other hand, a P11 virtual pole, as an accompanying partner of the nucleon bound-state pole, locates at $$\sqrt {{z_v}} $$ = (0:966–0:018) GeV, slightly above the nucleon pole on the real axis below threshold. Physical origin of the two newly established poles is explored to the best of our knowledge. It is emphasized that the O(p3) calculation greatly improves the fit quality comparing with the previous O(p2) one.

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.

Born Level Bound States

Bound state poles in the $S$-matrix of perturbative QED are generated by the {\em divergence} of the expansion in $\alpha$. The perturbative corrections are necessarily singular when expanding around free, \order{\alpha^0} $in$ and $out$ states that have no overlap with finite-sized atomic wave functions. Nevertheless, measurables such as binding energies do have well-behaved expansions in powers of $\alpha$ (and $\log\alpha$). It is desirable to formulate the concept of "lowest order" for gauge theory bound states such that higher order corrections vanish in the $\alpha \to 0$ limit. This may allow to determine a lowest order term for QCD hadrons which incorporates essential features such as confinement and chiral symmetry breaking, and thus can serve as the starting point of a useful perturbative expansion. I discuss a "Born" (no loop, lowest order in $\hbar$) approximation. Born level states are bound by gauge fields which satisfy the classical field equations. Gauss' law determines a distinct field $A^0(\xv)$ for each instantaneous position of the charges. A Poincar\'e covariant boundary condition for the gluon field leads to a confining potential for $q\bar q$ and $qqq$ states. In frames where the bound state is in motion the classical gauge field is obtained by a Lorentz boost of the rest frame field.

Two-body wave functions and compositeness from scattering amplitudes: General properties with schematic models

For a general two-body bound state in quantum mechanics, both in the stable and decaying cases, we establish a way to extract its two-body wave function in momentum space from the scattering amplitude of the constituent two particles. For this purpose, we first show that the two-body wave function of the bound state corresponds to the residue of the off-shell scattering amplitude at the bound state pole. Then, we examine our scheme to extract the two-body wave function from the scattering amplitude in several schematic models. As a result, the two-body wave functions from the Lippmann--Schwinger equation coincides with that from the Schr\"{o}dinger equation for an energy-independent interaction. Of special interest is that the two-body wave function from the scattering amplitude is automatically scaled; the norm of the two-body wave function, to which we refer as the compositeness, is unity for an energy-independent interaction, while the compositeness deviates from unity for an energy-dependent interaction, which can be interpreted to implement missing channel contributions. We also discuss general properties of the two-body wave function and compositeness for bound states in the schematic models.

Extraction of parameters from final state interactions

It is argued that final state enhancements in production reactions at large momentum transfers, such as pp -> K^+ Lambda p, are primarily sensitive to the position of a virtual bound state pole in the Lambda p system rather than the Lambda p scattering length and effective range. These arguments are supported by a study of the dispersion relation derived to describe such processes as a function of the cut-off energy. This shows that the position of the virtual bound state is independent of the cut-off energy.

Virtual states and the generalized completeness relation in the Friedrichs model

We study the well-known Friedrichs model, in which a discrete state is coupled to a continuum state. By examining the pole behaviors of the Friedrichs model in a specific form factor thoroughly, we find that, in general, when the bare discrete state is below the threshold of the continuum state, there should also be a virtual-state pole accompanying the bound-state pole originating from the bare discrete state as the coupling is turned on. There are also other second-sheet poles originating from the singularities of the form factor. We give a general argument for the existence of these two kinds of states. As the coupling is increased to a certain value, the second-sheet poles may merge and become higher-order poles. We then discuss the completeness relations incorporating bound states, virtual states, and resonant states corresponding to higher-order poles.

Threshold singularities, dispersion relations and fixed-order perturbative calculations

We show how to correctly treat threshold singularities in fixed-order perturbative calculations of the electron anomalous magnetic moment and hadronic pair production processes such as top pair production. With respect to the former, we demonstrate the equivalence of the "non-perturbative", resummed treatment of the vacuum polarization contribution, whose spectral function exhibits bound state poles, with the fixed-order calculation by identifying a threshold localized term in the four-loop spectral function. In general, we find that a modification of the dispersion relation by threshold subtractions is required to make fixed-order calculations well-defined and provide the subtraction term. We then solve the apparent problem of a divergent convolution of the partonic cross section with the parton luminosity in the computation of the top pair production cross section starting from the fourth-order correction. We find that when the computation is performed in the usual way as an integral of real and virtual corrections over phase space at a given order in the expansion in the strong coupling, an additional contribution has to be added at N3LO.