Relativistic five-quark equations and the nature of $\theta$ pentaquarks

The relativistic five-quark equations are found in the framework of the dispersion relation technique. The solutions of these equations using the method based on the extraction of the leading singularities of the amplitudes are obtained. The five-quark amplitudes for the low-lying pentaquarks including the $u$, $d$, $s$ quarks are calculated. The poles of these amplitudes determine the masses of $\theta$ pentaquarks. The mass spectra of the isotensor $\theta$ pentaquarks with $J^P={\frac{1}{2}}^{\pm },{\frac{3}{2}}^{\pm }$ are calculated.

Figure 1

Graphic representation of the equations for the five-quark subamplitudes $A_k$ ($k=1–4$) in the case of ${\theta }^{+++}({\theta }^{-})$ pentaquarks.

Figure 2

Graphic representation of the equations for the five-quark subamplitudes $A_k$ ($k=1–6$) corresponding to the ${\theta }^{++}({\theta }^0)$ pentaquarks.

Figure 3

Graphic representation of the equations for the five-quark subamplitudes $A_k$ ($k=1–7$) corresponding to the ${\theta }^{+}$ pentaquark.

Figure 4

Second term for the first subamplitude (Fig. 1) as six diagrams.

Figure 5

Contours of integration 1,2 in the complex plane $s_{13}$ for the functions $J_1,J_3$.

Figure 6

Contours of integration 1,2,3 in the complex plane (a) $s_{134}$ and (b) $s_{13}$ for the function $J_2$.

Figure 7

Contours of integration 1,2 in the complex plane $s_{24}$ for the function $J_3$.