Bound states in the $B$-matrix formalism for the three-body scattering

We consider a model of relativistic three-body scattering with a bound state in the two-body subchannel. We show that the naiïve $K$-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.

Figure 1

Diagrammatic representation of (a) ${\mathcal{A}}_{33}$, as given by Eq. (1), and (b) the $B$-matrix kernel of Eq. (2). As explained in Appendix pp1, a single external line represents a spectator, while a double external line represents an isobar. A solid circle with both external isobars and spectators is the three-body connected amplitude ${\mathcal{A}}_{33,{\mathit{p}}^{\prime }\mathit{p}}$, and a solid circle only with external isobars is the two-body amplitude ${\mathcal{F}}_{\mathit{p}}$.

Figure 2

Diagrammatic representation of the ladder Eq. (2). Here, the black box represents ${\mathcal{D}}_{{\mathit{p}}^{\prime }\mathit{p}}$.

Figure 3

Diagrammatic representation of the multichannel $B$-matrix framework, Eqs. (18)–(21). Amplitudes ${\mathcal{A}}_{mn}$ are represented by solid circles and can be differentiated from the different types of external legs. The dashed line represents the two-body bound state of mass $M$. The ${\mathcal{B}}_{33}$ kernel is decomposed as in Fig. 1, while other, real kernels describe just short-range interactions.

Figure 4

The kernel $\mathcal{I}(s)$ for two different choices of ${\sigma }_{\min }$. The bound state has a mass $M^2=3m^2$, which corresponds to the bound state–spectator threshold energy $s_{\mathrm{th},2}\approx 7.456m^2$. For $s=0$, the singularity from the definition of the three-body phase space factor $\tau (\sigma ,s)$ occurs. The three-body threshold occurs at $s_{\mathrm{th},3}=(3m{)}^2$. The points of nonanalyticity are highlighted by the dashed red lines. (a) EFT-like model with cutoff ${\sigma }_{\min }/m^2=0$ and (b) B-matrix model with the physical limit of integration ${\sigma }_{\min }/m^2=4$.

Figure 5

The comparison between the $\mathbf{3}\to \mathbf{3}$ amplitudes in the model with undispersed (a) and dispersed (b) integral kernel $\mathcal{I}(s)$. The values of the couplings are $g_{22}=1$, $G=0$, and $g_{33}=100$ or $g_{33}=30$. The two-body bound-state pole is at $M^2=3m^2$, which corresponds to the bound state–spectator threshold energy $s_{\mathrm{th},2}\approx 7.456m^2$. The three-body threshold is at $s=9m^2$. The points of nonanalyticity are highlighted by the dashed, vertical, red lines. As can be seen, the analytic structure of original model is insensitive to the changes of $g_{33}$ and has a nonphysical branch cut below $s_{\mathrm{th},2}$. The dispersed model ${\tilde{a}}_{33,d}(s)$ has no singularities below the three-body threshold other than the three-body bound-state pole at $s_{*}/m^2\approx 3.618$ for large enough $g_{33}$. For coupling which is too small, the pole does not appear on the real axis.

Figure 6

The dispersed kernel ${\mathcal{I}}_d(s)$. The two-body bound state pole is at $M^2=3m^2$, which corresponds to the energy $s_{\mathrm{th},2}\approx 7.456m^2$. Three-body threshold $s_{\mathrm{th},3}$ is highlighted by the dashed red line. Above $s_{\mathrm{th},3}$, the dispersed integral ${\mathcal{I}}_d(s)$ is identical to the original model $\mathcal{I}(s)$.

Figure 7

Motivation for the $G=0$ condition. In the microscopic model, the contact interaction amplitudes $g_{mn}$ (small black circles) are explained as a heavy particle exchange (“zigzag” line), which couples with a strength $\lambda$ to three particles and $\eta$ to a bound state and a particle. This gives $g_{22}\sim {\eta }^2$, $g_{33}\sim {\lambda }^2$, and $g_{32}\sim \eta \lambda$, which satisfies $g_{32}^2=g_{22}{g}_{33}$.

Figure 8

Panel (a): dispersed kernel $\mathrm{Im}{\mathcal{I}}_d(s_{\mathrm{th},2})$ as a function of $a_0$ on the logarithmic plot. Color lines correspond to the inverted couplings of the Fig. 8, while color points are solutions of the ${\mathcal{I}}_d(s_{\mathrm{th},2})=1/g_{33}$ three-body bound-state equation. Panel (b): dependence of the bound state–spectator scattering length $b_0$, defined in Eq. (62) for $G=0$, on the two-body dimensionless scattering length $a_0$. Results for various values of $g_{33}$ are shown, while the coupling $g_{22}=1$. It can be seen that for large enough $g_{33}$ and particular values of $a_0$ a shallow three-body bound state is developed, with a mass equal to $M+m$.

Figure 9

A three-particle state in the (a) CMF and (b) HF. Lorentz boost with $\mathbf{\beta }=-(\mathit{P}-\mathit{p})/(E-{\omega }_{\mathit{p}})$ transforms between the frames. The angular momentum of the pair $(\ell ,m_{\ell })$ is defined in the HF with respect to the spherical angles of ${\mathit{q}}_{\mathit{p}}^{\star }$.

Figure 10

The trajectories of the movable singularities ${\sigma }_3$ and ${\sigma }_4$ as a functions of $s$. Here, $s$ is decreasing from $s/m^2=9$ to $-3$. The ${\sigma }_3$ point travels on the real axis until it reaches $s/m^2=1$, where it becomes complex, with a positive imaginary part. The ${\sigma }_4$ moves on the real axis to the left until it reaches the value ${\sigma }_4=0$ for $s/m^2=1$, and then it moves to the right and becomes complex for $s=0$.

Figure 11

The integrand $\mathcal{J}(\sigma ,s)$ of Eq. (48) as a function of $\sigma$ for fixed, real values of $s$, decreasing from $s/m^2=10$ to $s/m^2=0$. The bound-state pole occurs at $M^2=3m^2$. The $s$-dependent branch points ${\sigma }_{3,4}$ are specified by arrows, while fixed points ${\sigma }_0/m^2=0$, ${\sigma }_b/m^2=3$, and ${\sigma }_2/m^2=4$ are marked with dashed vertical lines. The plot was obtained for $\epsilon /m^2={10}^{-4}$.