Three-body Unitarity Research Articles

Effects of final state interactions on Landau singularities

In certain kinematic and particle mass configurations, triangle singularities may lead to line-shapes which mimic the effects of resonances. This well-known effect is scrutinized here in the presence of final-state rescattering. The goal is achieved first by utilizing general arguments provided by Landau equations, and second by applying a modern scattering formalism with explicit two- and three-body unitarity.

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 approach to Tcc+ including three-body effects

A coupled-channel approach is applied to the charged tetraquark state ${T}_{cc}^{+}$ recently discovered by the LHCb Collaboration. The parameters of the interaction are fixed by a fit to the observed line shape in the three-body ${D}^{0}{D}^{0}{\ensuremath{\pi}}^{+}$ channel. Special attention is paid to the three-body dynamics in the ${T}_{cc}^{+}$ due to the finite life time of the ${D}^{*}$. An approach to the ${T}_{cc}^{+}$ is argued to be self-consistent only if both manifestations of the three-body dynamics, the pion exchange between the $D$ and ${D}^{*}$ mesons and the finite ${D}^{*}$ width, are taken into account simultaneously to ensure that three-body unitarity is preserved. This is especially important to precisely extract the pole position in the complex energy plane whose imaginary part is very sensitive to the details of the coupled-channel scheme employed. The ${D}^{0}{D}^{0}$ and ${D}^{0}{D}^{+}$ invariant mass distributions, predicted based on this analysis, are in good agreement with the LHCb data. The low-energy expansion of the ${D}^{*}D$ scattering amplitude is performed and the low-energy constants (the scattering length and effective range) are extracted. The compositeness parameter of the ${T}_{cc}^{+}$ is found to be close to unity, which implies that the ${T}_{cc}^{+}$ is a hadronic molecule generated by the interactions in the ${D}^{*+}{D}^{0}$ and ${D}^{*0}{D}^{+}$ channels. Employing heavy-quark spin symmetry, an isoscalar ${D}^{*}{D}^{*}$ molecular partner of the ${T}_{cc}^{+}$ with ${J}^{P}={1}^{+}$ is predicted under the assumption that the $D{D}^{*}\text{\ensuremath{-}}{D}^{*}{D}^{*}$ coupled-channel effects can be neglected.

Three-body unitarity versus finite-volume π+π+π+ spectrum from lattice QCD

Strong three-body interactions above threshold govern the dynamics of many exotics and conventional excited mesons and baryons. Three-body finite-volume energies calculated from lattice QCD promise an ab-initio understanding of these systems. We calculate the three-$\pi^+$ spectrum unraveling the three-body dynamics that is tightly intertwined with the $S$-matrix principle of three-body unitarity and compare it with recent lattice QCD results. For this purpose, we develop a formalism for three-body systems in moving frames and apply it numerically.

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 $m_p^{(a_1(1260))} = (1209 \pm 4^{+12}_{-9})\text{MeV}$, $\Gamma_p^{(a_1(1260))} = (576 \pm 11 ^{+89}_{-20})\text{MeV}$}.

Developments of the Jülich–Bonn Dynamical Coupled-Channel Analysis

A new analysis of $$K\varLambda $$ photoproduction is presented, including the latest data measured at Jefferson Lab and other facilities. Evidence for two states has been found, the $$N(1900)3/2^+$$ and $$N(2060)5/2^-$$ resonances that have been seen in other analysis. Future extensions are discussed like the use of model selection techniques to achieve a minimal resonance content of the amplitude. Also, conceptual aspects like three-body unitarity are highlighted that can help to provide a connection between phenomenology and ab-initio lattice QCD calculations.

Three-body unitarity in the finite volume

The physical interpretation of lattice QCD simulations, performed in a small volume, requires an extrapolation to the infinite volume. A method is proposed to perform such an extrapolation for three interacting particles at energies above threshold. For this, a recently formulated relativistic $3\to 3$ amplitude based on the isobar formulation is adapted to the finite volume. The guiding principle is two- and three-body unitarity that imposes the imaginary parts of the amplitude in the infinite volume. In turn, these imaginary parts dictate the leading power-law finite-volume effects. It is demonstrated that finite-volume poles arising from the singular interaction, from the external two-body sub-amplitudes, and from the disconnected topology cancel exactly leaving only the genuine three-body eigenvalues. The corresponding quantization condition is derived for the case of three identical scalar-isoscalar particles and its numerical implementation is demonstrated.

Three-body unitarity with isobars revisited

The particle exchange model of hadron interactions can be used to describe three-body scattering under the isobar assumption. In this study we start from the 3->3 scattering amplitude for spinless particles, which contains an isobar-spectator scattering amplitude. Using a Bethe-Salpeter Ansatz for the latter, we derive a relativistic three-dimensional scattering equation that manifestly fulfills three-body unitarity and two-body unitarity for the sub-amplitudes. This property holds for energies above breakup and also in the presence of resonances in the sub-amplitudes.

Coupled channels approach to photo-meson production on the nucleon

The coupled channels Lagrangian approach of the Giessen model (GiM) for meson production on the nucleon is discussed and applied to a selected set of meson production channels on the nucleon, ranging from πN → πN and γ N → π N , eta-production and associated strangeness production to 2π N channels in the resonance energy region. We present an updated coupled-channel analysis of eta-meson production including all recent photoproduction data on the proton. The dip structure observed in the differential cross sections at c.m. energies W=1.68 GeV is explained by destructive interference between the S 11 (1535) and S 11 (1560) states, not confirming the postulated sharp state. Kaon production on the nucleon is investigated in K Λ and K Σ exit channels. The approach to 2π N production has been significantly improved by using the isobar approximation with σN and π Δ1232 intermediate states. Three-body unitarity is maintained up to interference between the isobar subchannels. We obtain R σ N (1440)=27+4 -9 % and R π Δ (1440)=12+5 -3 % for the for the σ N and π Δ1232 decay branching ratios of N *(1440) respectively. The extracted π N inelasticities and reaction amplitudes are consistent with the results of other groups.

2πproduction in the Giessen coupled-channels model

We present a coupled-channel Lagrangian approach (GiM) to describe the $\pi N \to \pi N$, $2\pi N$ scattering in the resonance energy region. The $2\pi N$ production has been significantly improved by using the isobar approximation with $\sigma N$ and $\pi \Delta(1232)$ in the intermediate state. The three-body unitarity is maintained up to interference pattern between the isobar subchannels. The scattering amplitudes are obtained as a solution of the Bethe-Salpeter equation in the $K$ matrix approximation. As a first application we perform a partial wave analysis of the $\pi N \to \pi N$, $\pi^0\pi^0 N$ reactions in the Roper resonance region. We obtain $R_{\sigma N}(1440)=27^{+4}_{-9}$\,\% and $R_{\sigma N}(1440)=12^{+5}_{-3}$\,\% for the $\sigma N$ and $\pi \Delta$ decay branching ratios of $N^*(1440)$ respectively. The extracted $\pi N$ inelasticities and reaction amplitudes are consistent with the results from other groups.

Coupled-channel analysis ofD+→K−π+π+decay

We perform a coupled-channel analysis of pseudodata for the D+-> K-pi+pi+ Dalitz plot. The pseudodata are generated from the isobar model of the E791 Collaboration, and are reasonably realistic. We demonstrate that it is feasible to analyze the high-quality data within a coupled-channel framework that describes the final state interaction of D+-> K-pi+pi+ as multiple rescatterings of three pseudoscalar mesons through two-pseudoscalar-meson interactions in accordance with the two-body and three-body unitarity. The two-pseudoscalar-meson interactions are designed to reproduce empirical pi pi and pi bar K scattering amplitudes. Furthermore, we also include mechanisms that are beyond simple iterations of the two-body interactions, i.e., a three-meson force, derived from the hidden local symmetry model. A picture of hadronic dynamics in D+-> K-pi+pi+ described by our coupled-channel model is found to be quite different from those of the previous isobar-type analyses. For example, we find that the D+-> K-pi+pi+ decay width can get almost triplicated when the rescattering mechanisms are turned on. Among the rescattering mechanisms, those associated with the rho(770) bar K0 channel, which contribute to D+-> K-pi+pi+ only through a channel coupling, give a large contribution, and significantly improve the quality of the fits. The K-pi+ s-wave amplitude from our analysis is reasonably consistent with those extracted from the E791 model independent partial-wave analysis; the hadronic rescattering and the coupling to the rho(770) bar K0 channel play a major role here. We also find that the dressed D+ decay vertices have phases, induced by the strong rescatterings, that strongly depend on the momenta of the final pseudoscalar mesons. Although the conventional isobar-type analyses have assumed the phases to be constant, this common assumption is not supported by our more microscopic viewpoint.

RESONANCE DYNAMICS IN COUPLED CHANNELS

General properties of the S-matrix, such as constraints from two- and three-body unitarity as well as gauge invariance, are discussed and illustrated for the example of a dynamical coupled channel approach. The Jülich model has been updated to analyze πN, ηN, and KY production as well as pion photoproduction. Partial wave amplitudes and resonance properties are determined.

Extraction of meson resonances from three-pions photo-production reactions

We investigate the model dependence of meson resonance properties extracted from the Dalitz-plot analysis of the three-pions photo-production reactions on the nucleon. Within a unitary model developed in our earlier work, we generate Dalitz-plot distributions as data to perform an isobar model fit that is similar to most of the previous analyses of three-pion production reactions. It is found that the resonance positions from the two models agree well when both fit the data accurately, except for the resonance poles near branch points. The residues of the resonant amplitudes extracted from the two models and by the usual Breit-Wigner procedure agree well only for the isolated resonances with narrow widths. For overlapping resonances, most of the extracted residues could be drastically different. Our results suggest that even with high precision data, the resonance extraction should be based on models within which the amplitude parametrization is constrained by a three-body unitarity condition.

Precise calculation of the two-step process forK−d→πΣnin theΛ(1405) resonance region

The reaction K- d -> pi Sigma N is investigated taking into account single scattering and the two-step process due to Kbar N -> pi Sigma rescattering. The influence of some common approximations are examined. It is found that the treatment of the kinematics in the Green's function that appears in the loop integral of the rescattering process has a rather strong impact on the resulting lineshape of the pi Sigma invariant mass spectrum. Specifically, a calculation with correct kinematics where the three-body unitarity cut due to the nK- p threshold occurs at the physical value yields a pronounced peak in the invariant mass spectrum at this threshold and, at the same time, suppresses the signal in the region of the Lambda(1405) resonance. On the other hand, an approximation applied in past calculations shifts that threshold down and, consequently, leads to an accidental and therefore erroneous enhancement of the signal of the Lambda(1405) in the pi Sigma invariant mass spectrum.

Signature of Strange Dibaryon in Kaon-Induced Reaction

We examine how the signature of the strange-dibaryon resonances in the barKNN-piSigmaN system shows up in the scattering amplitude on the physical real energy axis within the framework of Alt-Grassberger-Sandhas (AGS) equations. The so-called point method is applied to handle the three-body unitarity cut in the amplitudes. We also discuss the possibility that the strange-dibaryon production reactions can be used for discriminating between existing models of the two-body barKN-piSigma system with Lambda(1405).

Unitary coupled-channels model for three-mesons decays of heavy mesons

A unitary coupled-channels model is presented for investigating the decays of heavy mesons and excited meson states into three light pseudoscalar mesons. The model accounts for the three-mesons final state interactions in the decay processes, as required by both the three-body and two-body unitarity conditions. In the absence of the Z-diagram mechanisms that are necessary consequences of the three-body unitarity, our decay amplitudes are reduced to a form similar to those used in the so-called isobar-model analysis. We apply our coupled-channels model to the three-pions decays of a1(1260), pi2(1670), pi2(2100), and D0 mesons, and show that the Z-diagram mechanisms can contribute to the calculated Dalitz plot distributions by as much as 30% in magnitudes in the regions where f0(600), rho(770), and f2(1270) dominate the distributions. Also, by fitting to the same Dalitz plot distributions, we demonstrate that the decay amplitudes obtained with the unitary model and the isobar model can be rather different, particularly in the phase that plays a crucial role in extracting the CKM CP-violating phase from the data of B meson decays. Our results indicate that the commonly used isobar model analysis must be extended to account for the final state interactions required by the three-body unitarity to reanalyze the three-mesons decays of heavy mesons, thereby exploring hybrid or exotic mesons, and signatures of physics beyond the standard model.

Two- and three-charged-particle nuclear scattering in momentum space: A two-potential theory and a boundary condition model

The two- and three-charged-particle nuclear scattering problems are investigated in momentum space. The three-body equations with a short-range nuclear potential, a three-body force potential, and the long-range Coulomb potential are presented in a mathematically rigorous way within a generalized two-potential theory. To remove the serious singularity in the two- and three-body Coulomb problems in momentum space, we have proposed a novel boundary condition to the phase shift that arises from a potential difference, the so-called auxiliary potential (AP), between the Coulomb potential and the screened Coulomb one: ${V}^{\ensuremath{\phi}}={V}^{C}\ensuremath{-}{V}^{R}$. Furthermore, we point out the importance of the off-shell amplitude for the AP, by which one can uniquely obtain the two-body on-shell and off-shell Coulomb amplitude in momentum space. Therefore, this formulation is also useful for atomic systems, which is another benefit. It is recalled that the traditional phase-shift renormalization theory, in which the screened Coulomb amplitude is sandwiched by renormalized phase factors ${e}^{i\ensuremath{\phi}}$, is not consistent with two-potential theory. Some ambiguities or misunderstandings of the traditional methods for handling the Coulomb problem are clarified. Finally, the three-body unitarity relation is proved for the amplitude generated from these three kinds of potential. Moreover, a generalized two-potential theory is presented. It is pointed out that the asymptotic three-body nuclear wave function with the Coulomb potential could not be written by a product of two individual wave functions ${\stackrel{~}{\ensuremath{\psi}}}_{x}(x){\stackrel{~}{\ensuremath{\psi}}}_{y}(y)$ defined in terms of Jacobi coordinates $x$ and $y$, respectively.

Unitarity and the Bethe-Salpeter equation

We investigate the relation between different three-dimensional reductions of the Bethe-Salpeter equation and the analytic structure of the resultant amplitudes in the energy plane. This correlation is studied for both the $\phi^2\sigma$ interaction Lagrangian and the $\pi N$ system with $s$-, $u$-, and $t$-channel pole diagrams as driving terms. We observe that the equal-time equation, which includes some of the three-body unitarity cuts, gives the best agreement with the Bethe-Salpeter result. This is followed by other 3-D approximations that have less of the analytic structure.

The role of meson retardation in the [formula omitted] interaction above pion threshold

A model is developed for the hadronic interaction in the two-nucleon system above pion threshold which is based on meson, nucleon and Δ degrees of freedom and which includes full meson retardation in the exchange operators. For technical reasons, the model allows maximal one meson to be present explicitly. Thus the Hilbert space contains besides NN and NΔ also configurations consisting of two nucleons and one meson. For this reason, only two- and three-body unitarity is obeyed, and the model is suited for reactions in the two-nucleon sector, where only one pion is produced or absorbed. Starting from a realistic pure nucleonic retarded potential, which had to be renormalized because of the additional π and Δ degrees of freedom, a reasonable fit to experimental NN-scattering data could be achieved.

Protons in collision with hydrogen atoms: Influence of unitarity and multiple scattering

Three-body integral equations when applied to collisions of protons with hydrogen atoms yield the amplitudes for direct scattering and electron exchange which automatically satisfy two-body and (if solved exactly also) three-body unitarity. As a consequence, differential cross sections (DCS) for both reactions are calculated on equal footing, and the total exchange cross section (TCS) results from the corresponding DCS. This is to be contrasted with standard models used in atomic physics which either are of the (Distorted-Wave) Born type (to be applied at higher energies), or solve the Schrodinger equation by expanding the wave function in a basis (close-coupling method which has problems especially for rearrangement processes). Hence, at higher energies in general separate models have to be developed to describe the DCS for either direct scattering or electron exchange, and frequently also for the TCS. For instance, the most sophisticated traditional models, which provide a very good reproduction of the TCS data, are the continuum distorted wave [1] and the boundary corrected first Born model [2]. Both, however, fail to describe differential cross sections which represent a much more stringent test. On the other hand, three-body integral equations suffer from the principal defect that their kernels are not compact when particles with charges of different sign are involved, as it happens in applications to atomic reactions (references can be traced from [3]). The consequence is that naive application of standard solution methods of integral equations theory would not be justified. Additional, practical difficulties arise from the complicated singularity structure of the off-shell twoparticle Coulomb T-matrix which is the basic dynamical ingredient. As is well known, the latter develops nasty singularities in the on-shell limit and, in case of attraction, has in addition an infinite number of poles.