Effective Range Expansion Research Articles

Renormalization of three-body interaction in DDK system

In systems comprising three identical particles, the Efimov effect emerges in the unitary limit. This article, utilizing the method of effective field theory along with the effective range expansion (ERE) and low-momentum expansion, reproduces the Efimov effect beyond unitary limit and identical case and applies to DDK system. It also links the high-energy scale with the ultraviolet cutoff in the three-body system. We discuss a method for obtaining exact solutions for the three-body wave function below the threshold of dimer production, that is D and Ds0⁎(2317). It turns out that introducing this scale is a necessary condition for the emergence of DDK three-body bound state. Since without this scale, the wave function would be solved below threshold with arbitrary binding energy, which violates the conditions of unitarity and energy quantization. The calculation involving this scale results in a discrete spectrum below the threshold and restores unitarity, making the effects induced by this scale significant for the formation of DDK bound state. In order to establish this in a complete formalism, the paper further gives an analytical expression for the three-body coupling constant, and compares it to the numerical calculation in DDK system.

Structure of near-threshold resonances with new interpretation scheme of complex compositeness

The nature of near-threshold resonances is quantitatively studied with a new interpretation scheme using the complex compositeness. A difficulty was known in the understanding of the internal structure of unstable resonances because their complex compositeness is not an interpretable measure. To overcome this problem, we develop a new interpretation scheme respecting the ambiguous aspects of the identification of the internal structure of resonances. We then apply the interpretation scheme to the near-threshold resonances slightly above the threshold, described by the effective range expansion. With the new interpretation scheme, we show that near-threshold resonances are dominated by the non-molecular component. Namely, even in the near-threshold region, the nature of resonances is sharply contrasted with bound states whose internal structure is usually molecular dominant.

Scattering of dark pions in Sp(4) gauge theory

Analyses of astrophysical data provide first hints on the self-interactions of dark matter at low energies. Lattice calculations of dark matter theories can be used to investigate them, especially in the case of strongly interacting dark matter. We consider Sp(4) gauge theory with two fundamental fermions as a candidate theory. We compute the scattering phase shift for the scattering of two identical dark pions and determine the parameters of the effective range expansion. Our exploratory results in the supposedly most common interaction channel provide a lower limit for the dark matter mass when compared to astrophysical data. We also provide first benchmarks of velocity-weighted cross sections in the relevant nonrelativistic domain. Published by the American Physical Society 2024

Confronting the Lippmann–Schwinger equation and the [formula omitted] method for coupled-wave separable potentials

We study a family of separable potentials with and without added contact interactions by solving the associated Lippmann–Schwinger equation with two coupled partial waves. The matching of the resulting amplitude matrix with the effective-range expansion is studied in detail. When a counterterm is included in the potential we also carefully discuss its renormalization. Next, we use the matrix N/D method and study whether the amplitude matrices from the potentials considered admit an N/D representation in matrix form. As a novel result we show that it is typically not possible to find such matrix representation for the coupled partial-wave case. However, a separate N/D representation for each coupled partial wave — a valid option known in the literature — is explicitly implemented and numerically solved in cases where the matrix N/D method is unavailable.

Toward the quark mass dependence of Tcc+ from lattice QCD

The DD* scattering phase shifts in the Tcc+=ccu¯d¯ channel are extracted from lattice QCD for five different charm quark masses and a fixed light-quark mass corresponding to mπ≃280 MeV. The phase shifts are analyzed employing two approaches: effective range expansion and Lippmann-Schwinger equation derived in the effective field theory. In the latter case, the results imply an attraction at short range parametrized by contact terms and a slight repulsion at long range mediated by one-pion exchange with mπ>mD*−mD. The poles in the amplitude across the complex energy plane are extracted and their trajectories are discussed as the charm quark mass is varied. Two complex conjugate poles corresponding to a resonance below threshold are found for mc close to the physical value. They turn into a pair of virtual states at the largest mc studied. With further increasing mc, one virtual pole representing Tcc+ is expected to move towards the two-body threshold and turn into a bound state. The light-quark mass dependence of the Tcc+ pole is briefly discussed using the data on DD* scattering from other lattice collaborations. Published by the American Physical Society 2024

Lüscher equation with long-range forces

We derive the modified Lüscher equation in the presence of the long-range force caused by the exchange of a light particle. It is shown that the use of this equation enables one to circumvent the problems related to the strong partial-wave mixing and the t-channel sub-threshold singularities. It is also demonstrated that the present method is intrinsically linked to the so-called modified effective-range expansion (MERE) in the infinite volume. A detailed comparison with the two recently proposed alternative approaches is provided.

Shallow Bound States and Hints for Broad Resonances with Quark Content b[over ¯]c[over ¯]ud in B-D[over ¯] and B^{*}-D[over ¯] Scattering from Lattice QCD.

We present the first determination of the energy dependence of the B-D[over ¯] and B^{*}-D[over ¯] isospin-0, S-wave scattering amplitudes both below and above the thresholds using lattice QCD, which allows us to investigate rigorously whether mixed bottom-charm b[over ¯]c[over ¯]ud tetraquarks exist as bound states or resonances. The scattering phase shifts are obtained using Lüscher's method from the energy spectra in two different volumes. To ensure that no relevant energy level is missed, we use large, symmetric 7×7 and 8×8 correlation matrices that include, at both source and sink, B^{(*)}-D[over ¯] scattering operators with the lowest three or four possible back-to-back momenta in addition to local b[over ¯]c[over ¯]ud operators. We fit the energy dependence of the extracted scattering phase shifts using effective-range expansions. We observe sharp peaks in the B^{(*)}-D[over ¯] scattering rates close to the thresholds, which are associated with shallow bound states, either genuine or virtual, a few MeV or less below the B^{(*)}-D[over ¯] thresholds. In addition, we find hints for resonances with masses of order 100MeV above the thresholds and decay widths of order 200MeV.

Contact interactions, self-adjoint extensions, and low-energy scattering

Low-energy scattering is well described by the effective-range expansion. In quantum mechanics, a tower of contact interactions can generate terms in this expansion after renormalization. Scattering parameters are also encoded in the self-adjoint extension of the Hamiltonian. We briefly review this well-known result for two particles with s-wave interactions using impenetrable self-adjoint extensions, including the case of harmonically trapped two-particle states. By contrast, the one-dimensional scattering problem is surprisingly intricate. We show that the families of self-adjoint extensions correspond to a coupled system of symmetric and antisymmetric outgoing waves, which is diagonalized by an SU(2) transformation that accounts for mixing and a relative phase. This is corroborated by an effective theory computation that includes all four energy-independent contact interactions. The equivalence of various one-dimensional contact interactions is discussed and scrutinized from the perspective of renormalization. As an application, the spectrum of a general point interaction with a harmonic trap is solved in one dimension.

Analysis of the Ξ(1620) resonance and K¯Λ scattering length with a chiral unitary approach

We study the Ξ(1620) resonance near the K¯Λ threshold in the light of recent experimental constraints. The Belle Collaboration have found a resonance peak of Ξ(1620) slightly below the K¯0Λ threshold in the π+Ξ− invariant mass spectrum, and the ALICE Collaboration have determined the K−Λ scattering length from the measurement of the momentum correlation functions in heavy ion collisions. Using the effective range expansion, we classify the nature of the pole of the near-threshold eigenstate in terms of the scattering length, in the presence of the decay channel. It is shown that the quasibound state below the threshold can be described by only the scattering length, while the description of the resonance above the threshold requires the contribution from the effective range. Based on the chiral unitary approach, we construct a theoretical model which generates the pole of Ξ(1620) below the K¯Λ threshold with relatively narrow width, as reported by the Belle Collaboration. It is quantitatively demonstrated that the spectrum of the Ξ(1620) quasibound state is distorted by the effect of the nearby K¯Λ threshold. We then construct another model which reproduces the K−Λ scattering length by the ALICE Collaboration. In this case, the eigenstate pole does not appear in the physically relevant Riemann sheet, and the spectrum shows a cusp structure at the K¯Λ threshold. We finally examine the compatibility of the value of the K¯Λ scattering length and the subthreshold pole of Ξ(1620) including the experimental uncertainties. Published by the American Physical Society 2024

An Alternative Scheme for Pionless EFT: Neutron-Deuteron Scattering in the Doublet S-Wave

Using the effective-range expansion for the two-body amplitudes may generate spurious sub-threshold poles outside of the convergence range of the expansion. In the infinite volume, the emergence of such poles leads to the breakdown of unitarity in the three-body amplitude. We discuss the extension of our alternative subtraction scheme for including effective range corrections in pionless effective field theory for spinless bosons to nucleons. In particular, we consider the neutron-deuteron system in the doublet S-wave channel explicitly.

Interactions of omega mesons in nuclear matter and with nuclei

In-medium interactions of omega -mesons in infinite nuclear matter and finite nuclei are investigated in a microscopic approach, focused on the particle-hole excitations of the medium involving nucleonic NN^{-1} and N^*N^{-1} modes, where N^* denotes a nucleon resonance. The nuclear polarization tensors include relativistic mean-field dynamics by self–consistent scalar and vector fields. The resulting self-energies are transmitted to finite nuclei in local density approximation. Real and imaginary parts of longitudinal and transversal self-energies are discussed. The relation of the present approach to meson cloud models is addressed and an ambiguity is pointed out. Applications to recent data on the in–medium width of omega mesons scattered on a Niobium target serve to determine unknown N^*Nomega in-medium coupling constants. The data are well described by N^*N^{-1} self–energies containing S–wave and P-wave N^* resonances. Exploratory investigations, however, show that the spectroscopic composition of self-energies depends crucially on the near-threshold properties of the width which at present is known only within large error bars. The calculations predict the prevalence of transversal self–energies, implying that vector current conservation is still maintained in the nuclear medium by slightly more than 90%. Schrödinger-equivalent potentials are derived and scattering lengths and effective range parameters are extracted for the longitudinal and transversal channel. Longitudinal and transversal spectral distributions are discussed and the dependencies on momentum and nuclear density are investigated. Schrödinger-type omega +{}^{93}Nb optical potentials are constructed. Low-energy parameters are determined, are used to study the pole structure of the S-matrix at threshold. The effective range expansion of the omega-nucleus K-matrix led to a omega +{}^{93}Nb bound states with binding energy Re(varepsilon _B)=-448 keV but of width Gamma _B=4445 keV.

Role of Left-Hand Cut Contributions on Pole Extractions from Lattice Data: Case Study for T_{cc}(3875)^{+}.

We discuss recent lattice data for the T_{cc}(3875)^{+} state to stress, for the first time, a potentially strong impact of left-hand cuts from the one-pion exchange on the pole extraction for near-threshold exotic states. In particular, if the left-hand cut is located close to the two-particle threshold, which happens naturally in the DD^{*} system for the pion mass exceeding its physical value, the effective-range expansion is valid only in a very limited energy range up to the cut and as such is of little use to reliably extract the poles. Then, an accurate extraction of the pole locations requires the one-pion exchange to be implemented explicitly into the scattering amplitudes. Our findings are general and potentially relevant for a wide class of hadronic near-threshold states.

Low-energy deuteron–alpha elastic scattering in cluster effective field theory

In this paper, we study the low-energy $$d{-}\alpha $$ elastic scattering within the two-body cluster effective field theory (EFT) framework. The importance of the $$d(\alpha ,\alpha ) d$$ scattering in the $$^6 \text {Li} $$ production reaction leads us to study this system in an effective way. In the beginning, the scattering amplitudes of each channel are written in a cluster EFT with two-body formalism. Using the effective range expansion analysis for the elastic scattering phase shift of S, P and D partial waves, the unknown EFT low-energy coupling constants are determined and the leading and next-to-leading orders EFT results for the phase shift in each channel are presented. To verify the accuracy of the results, we compare experimental phase shift and differential cross section data with obtained results. The accuracy of the EFT results and consistency with the experimental data indicate that the EFT is an effective approach for describing low-energy systems.

Elastic nucleon-pion scattering at mπ = 200 MeV from lattice QCD

Elastic nucleon-pion scattering amplitudes are computed using lattice QCD on a single ensemble of gauge field configurations with Nf=2+1 dynamical quark flavors and mπ=200MeV. The s-wave scattering lengths with both total isospins I=1/2 and I=3/2 are inferred from the finite-volume spectrum below the inelastic threshold together with the I=3/2p-wave containing the Δ(1232) resonance. The amplitudes are well-described by the effective range expansion with parameters constrained by fits to the finite-volume energy levels, enabling a determination of the I=3/2 scattering length with statistical errors below 5%, while the I=1/2 scattering length is somewhat less precisely evaluated. Systematic errors due to excited states and the influence of higher partial waves are controlled, providing a step toward future computations down to physical light quark masses with multiple lattice spacings and volumes.

Effective range expansion for narrow near-threshold resonances

We discuss some general features of the effective range expansion, the content of its parameters with respect to the nature of the pertinent near-threshold states and the necessary modifications in the presence of coupled channels, isospin violations and unstable constituents. As illustrative examples, we analyse the properties of the χc1(3872) and Tcc+ states supporting the claim that these exotic states have a predominantly molecular nature.

Spurious poles in a finite volume

Using effective-range expansion for the two-body amplitudes may generate spurious sub-threshold poles outside of the convergence range of the expansion. In the infinite volume, the emergence of such poles leads to the inconsistencies in the three-body equations, e.g., to the breakdown of unitarity. We investigate the effect of the spurious poles on the three-body quantization condition in a finite volume and show that it leads to a peculiar dependence of the energy levels on the box size L. Furthermore, within a simple model, it is demonstrated that the procedure for the removal of these poles, which was recently proposed in ref. [1] in the infinite volume, can be adapted to the finite-volume calculations. The structure of the exact energy levels is reproduced with an accuracy that systematically improves order by order in the EFT expansion.

Generalization of Weinberg’s compositeness relations

We generalize the time-honored Weinberg's compositeness relations by including the range corrections through considering a general form factor. In Weinberg's derivation, he considered the effective range expansion up to $\mathcal{O}({p}^{2})$ and made two additional approximations: neglecting the nonpole term in the Low equation and approximating the form factor by a constant. We lift the second approximation and work out an analytic expression for the form factor. For a positive effective range, the form factor is of a single-pole form. An integral representation of the compositeness is obtained and is expected to have a smaller uncertainty than that derived from Weinberg's relations. We also establish an exact relation between the wave function of a bound state and the phase of the scattering amplitude neglecting the nonpole term. The deuteron is analyzed as an example, and the formalism can be applied to other cases where range corrections are important.

Composite nature of Z_b states from data analysis

We use a near-threshold parameterization with explicit inclusion of the Castillejo–Dalitz–Dyson poles, which is more general than the effective range expansion, to study the bottomonium-like states Z_b(10610) and Z_b(10650). In terms of the partial-wave amplitude, we fit the event number distribution of B^{(*)}{bar{B}}^* system to the experimental data for these resonances from Belle Collaboration. The data could be described very well in our method, which supports the molecular interpretation. Then the relevant physical quantities are obtained, including the B^{(*)}bar{B}^* scattering length (a), effective range (r), and residue squared (gamma _s^2) of the pole in the complex plane. In particular, we find the compositeness can range from about 0.4 up to 1 for the B{bar{B}}^* (B^*{bar{B}}^*) component in the resonance Z_b(10610) (Z_b(10650)).

Femtoscopy of the J/ψ-nucleon interaction

I discuss the prospects of using femtoscopy in high-energy protonproton and heavy-ion collisions to learn about the low-energy J/ψ-nucleon interaction. Femtoscopy is a technique that makes it possible to obtain spatiotemporal information on particle production sources at the femtometer scale through measurements of two-hadron momentum correlation functions. These correlation functions also provide information on low-energy hadron-hadron forces as final-state effects. In particular, such correlation functions give access to the forward scattering amplitude. One can express the forward amplitude as the product of the J/ψ chromopolarizability and the nucleon’s average chromoelectric gluon distribution, the latter being relevant to the problem of the origin of the nucleon mass. I will present the results of a recent study using the information on the J/ψ-nucleon interaction from lattice QCD simulations to compute J/ψ-nucleon correlation functions. The calculated correlation functions show clear sensitivity to the final-state interaction. I conclude discussing open issues regarding the use of the effective range expansion formula to fit experimental data for small scattering lengths and large effective range parameters.

Compositeness and several applications to exotic hadronic states with heavy quarks

Several methods for studying the nature of a resonance are applied to resonances recently discovered in the bottonomium and charmonium sectors. We employ the effective-range expansion, the saturation of the width and compositeness of a resonance, as well as direct fits to data. The latter stem from generic S -matrix parameterization that account for relevant dynamical features associated to channels that couple strongly in an energy region around the resonance masses, in which their thresholds also lie. We report on results obtained with these methods for the resonances Zb(10610), Zb(10650), Zcs(3985), Zc(3900), X(4020), X(6900), X(6825), and Pcs(4459).