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Register a new userBound states in the B-matrix formalism for the three-body scattering
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We consider a model of relativistic three-body scattering with a bound state in the two-body sub-channel. We show that the naive K-matrix type parametrization, here referred to as the B-matrix, has nonphysical singularities near the physical region. We show how to eliminate such singularities by using dispersion relations and also show how to reproduce unitarity relations by taking into account all relevant open channels.
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We propose a three-potential formalism for the three-body Coulomb scattering problem. The corresponding integral equations are mathematically well-behaved and can succesfully be solved by the Coulomb-Sturmian separable expansion method. The results show perfect agreements with existing low-energy $n-d$ and $p-d$ scattering calculations.
The Bethe-Salpeter equation for three bosons with zero-range interaction is solved for the first time. For comparison the light-front equation is also solved. The input is the two-body scattering length and the outputs are the three-body binding energies, Bethe-Salpeter amplitudes and light-front wave functions. Three different regimes are analyzed: ({it i}) For weak enough two-body interaction the three-body system is unbound. ({it ii}) For stronger two-body interaction a three-body bound state appears. It provides an interesting example of a deeply bound Borromean system. ({it iii}) For even stronger two-body interaction this state becomes unphysical with a negative mass squared. However, another physical (excited) state appears, found previously in light-front calculations. The Bethe-Salpeter approach implicitly incorporates three-body forces of relativistic origin, which are attractive and increase the binding energy.
Recently, a framework has been developed to study form factors of two-hadron states probed by an external current. The method is based on relating finite-volume matrix elements, computed using numerical lattice QCD, to the corresponding infinite-volume observables. As the formalism is complicated, it is important to provide non-trivial checks on the final results and also to explore limiting cases in which more straightforward predications may be extracted. In this work we provide examples on both fronts. First, we show that, in the case of a conserved vector current, the formalism ensures that the finite-volume matrix element of the conserved charge is volume-independent and equal to the total charge of the two-particle state. Second, we study the implications for a two-particle bound state. We demonstrate that the infinite-volume limit reproduces the expected matrix element and derive the leading finite-volume corrections to this result for a scalar current. Finally, we provide numerical estimates for the expected size of volume effects in future lattice QCD calculations of the deuterons scalar charge. We find that these effects completely dominate the infinite-volume result for realistic lattice volumes and that applying the present formalism, to analytically remove an infinite-series of leading volume corrections, is crucial to reliably extract the infinite-volume charge of the state.
For solving the $2to 2,3$ three-body Coulomb scattering problem the Faddeev-Merkuriev integral equations in discrete Hilbert-space basis representation are considered. It is shown that as far as scattering amplitudes are considered the error caused by truncating the basis can be made arbitrarily small. By this truncation also the Coulomb Greens operator is confined onto the two-body sector of the three-body configuration space and in leading order can be constructed with the help of convolution integrals of two-body Greens operators. For performing the convolution integral an integration contour is proposed that is valid for all energies, including bound-state as well as scattering energies below and above the three-body breakup threshold.
Self-consistent Greens function theory has recently been extended to the basic formalism needed to account for three-body interactions [A. Carbone, A. Cipollone, C. Barbieri, A. Rios, and A. Polls, (Phys. Rev. C 88, 054326 (2013))]. The contribution of three-nucleon forces has so far been included in ab initio calculations on nuclear matter and finite nuclei only as averaged two-nucleon forces. We derive the working equations for all possible two- and three-nucleon terms that enter the expansion of the self-energy up to the third order, thus including the interaction-irreducible (i.e., not averaged) diagrams with three-nucleon forces that have been previously neglected. We employ the algebraic diagrammatic construction up to the third order as an organization scheme for generating a non perturbative self-energy, in which ring (particle-hole) and ladder (particle-particle) diagrams are resummed to all orders. We derive expressions of the static and dynamic self-energy up to the third order, by taking into account the set of diagrams required when either the skeleton or nonskeleton expansions of the single-particle propagator are assumed. A hierarchy of importance among different diagrams is revealed, and a particular emphasis is given to a third-order diagram (see Fig. 2c) that is expected to play a significant role among those featuring an interaction-irreducible three-nucleon force. A consistent formalism to resum at infinite order correlations induced by three-nucleon forces in the self-consistent Greens function theory is now available and ready to be implemented in the many-body solvers.
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