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Register a new userTwo-Fermion Bound States within the Bethe-Salpeter Approach
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Authors
S. M. Dorkin
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To solve the spinor-spinor Bethe-Salpeter equation in Euclidean space we propose a novel method related to the use of hyperspherical harmonics. We suggest an appropriate extension to form a new basis of spin-angular harmonics that is suitable for a representation of the vertex functions. We present a numerical algorithm to solve the Bethe-Salpeter equation and investigate in detail the properties of the solution for the scalar, pseudoscalar and vector meson exchange kernels including the stability of bound states. We also compare our results to the non relativistic ones and to the results given by light front dynamics.
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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.
The relativistic properties of the three-nucleon system are investigated using the Faddeev equations within the Bethe-Salpeter approach. The nucleon-nucleon interaction is chosen in a separable form. The Gauss quadrature method is used to calculate the integrals. The system of the integral equations are solving by iterations method. The binding energy and the partial-wave amplitudes (1S 0 and 3S 1) of the triton are found.
In theoretical hadron physics mesons are a center of attention. Constructed in a simpler way than baryons in the quark model, they still present a considerable challenge if one aims at an understanding of all their aspects in terms of quarks and gluons in the context of Quantum Chromodynamics, the quantum field theory of the strong interaction. Complementary to (constituent-) quark models, reductions of the Bethe-Salpeter equation, lattice QCD, and effective field theories, the Dyson-Schwinger-equation approach has emerged as a well-suited formalism for the covariant study of hadron properties. In particular, radially excited mesons exhibit a sensitivity to long-range strong-interaction physics. This sensitivity has recently been studied with the help of the Bethe-Salpeter equation. Here these studies are reviewed and continued together with an account of possible future developments.
The mass spectrum of heavy pseudoscalar mesons, described as quark-antiquark bound systems, is considered within the Bethe-Salpeter formalism with momentum dependent masses of the constituents. This dependence is found by solving the Schwinger-Dyson equation for quark propagators in rainbow-ladder approximation. Such an approximation is known to provide both a fast convergence of numerical methods and accurate results for lightest mesons. However, as the meson mass increases, the method becomes less stable and special attention must be devoted to details of numerical means of solving the corresponding equations. We focus on the pseudoscalar sector and show that our numerical scheme describes fairly accurately the $pi$, $K$, $D$, $D_s$ and $eta_c$ ground states. Excited states are considered as well. Our calculations are directly related to the future physics programme at FAIR.
The mass spectrum of heavy pseudoscalar mesons, described as quark-antiquark bound systems, is considered within the Bethe-Salpeter formalism with momentum-dependent masses of the constituents. This dependence is found by solving the Schwinger-Dyson equation for quark propagators in rainbow-ladder approximation. Such an approximation is known to provide both a fast convergence of numerical methods and accurate results for lightest mesons. However, as the meson mass increases, the method becomes less stable and special attention must be devoted to details of numerical means of solving the corresponding equations. We focus on the pseudoscalar sector and show that our numerical scheme describes fairly accurately the $pi$, $K$, $D$, $D_s$ and $eta_c$ ground states. Excited states are considered as well. Our calculations are directly related to the future physics programme at FAIR.
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