We consider an exotic baryon (pentaquark) as a bound state of two-body systems composed of a baryon (nucleon) and a meson. We used a baryon-meson picture to reduce a complicated five-body problem to simple two-body problems. The homogeneous Lippmann-Schwinger integral equation is solved in configuration space by using one-pion exchange potential. We calculate the masses of pentaquarksθc(uuddc¯)andθb(uuddb¯).

In this study, we investigate the relativistic Klein-Gordon equation analytically for the Deng-Fan potential and Hulthen plus Eckart potential under the equal vector and scalar potential conditions. Accordingly, we obtain the energy eigenvalues of the molecular systems in different states as well as the normalized wave function in terms of the generalized Laguerre polynomials function through the NU method, which is an effective method for the exact solution of second-order linear differential equations.

In this work, we calculated the masses of the ground and radial excited states of heavy tetraquarks in the framework of the nonrelativistic quark model for two wave states nS and nP. The potential used is the parabolic potential plus the spin term, and the method we used to solve the Schrödinger equation is the NU method. The Regge trajectories in the $$(n_{r}, M^{2})$$ ( n r , M 2 ) plane are linear and we obtained the parameters $$\beta $$ β and $$\beta _{0}$$ β 0 for each state. The results are in good agreement with the available experimental data and we predicted the structures of tetraquarks X(4.500), X(4.700), X(4.740), X(6.900), X(7.200), Y(4.230), Y(4.660), and $$Z_b(10.650)$$ Z b ( 10.650 ) .

Since in recent years most of the heavy tetraquarks are discovered by Belle, LHCb, BESIII, etc., it motivated us to study these exotic hadrons. To the extent of our knowledge about the mass characteristics of heavy tetraquarks, we utilize the spinless relativistic Bethe–Salpeter equation by applying the Cornell potential in order to calculate the eigenvalue and mass of heavy tetraquarks. For this purpose, we present an ansatz solution to obtained Schrödinger-like equation to calculate the ground-state energy of diquarks and tetraquarks. Eventually, we compare our obtained predictions for heavy tetraquark masses with available experimental and theoretical data.

In this paper, we study four-body systems and consider them as diquark–antidiquark. We use the Bethe–Salpeter equation (Schrödinger equation with relativistic kinematics) to calculate the binding energies and the mass of heavy tetraquarks with hidden charm and bottom structures. To solve the Bethe–Salpeter equation, we use the Hellmann potential and the Ansatz method and calculate the mass of tetraquarks X(3.872), Y(3.940), $$Z_{cs}(3.985),$$ Z cs ( 3.985 ) , X(4.140), $$Z_{b}(10.610),$$ Z b ( 10.610 ) , and $$Z_{b}(10.650).$$ Z b ( 10.650 ) . Our results are in good agreement with theoretical and experimental data.

In the present study, we consider baryons as three-body bound systems according to the hypercentral constituent quark model in configuration space and solve the three-body Klein-Gordon equation. Then, we analyze perturbative spin-dependent and isospin-dependent interaction effects. To find the analytical solution, we use screened potential and calculate the eigenfunctions and eigenvalues of some baryons. We consider exclusive semileptonic decays of bottom and charm baryons and apply the differential decay width with the Isgur-Wise function and arrive at the rates for some semileptonic baryon decays. The results prove more enhanced compared to recent works and comply well with the experimental data.