Two-body wave functions and compositeness from scattering amplitudes: General properties with schematic models

For a general two-body bound state in quantum mechanics, both in the stable and decaying cases, I establish a way to extract its two-body wave function in momentum space from the scattering amplitude of the constituent two particles. For this purpose, I first show that the two-body wave function of the bound state corresponds to the residue of the off-shell scattering amplitude at the bound state pole. Then, I examine my scheme to extract the two-body wave function from the scattering amplitude in several schematic models. As a result, the two-body wave function from the Lippmann-Schwinger equation coincides with that from the Schrödinger equation for an energy-independent interaction. Of special interest is that the two-body wave function from the scattering amplitude is automatically scaled; the norm of the two-body wave function, to which I refer as the compositeness, is unity for an energy-independent interaction, while the compositeness deviates from unity for an energy-dependent interaction, which can be interpreted to implement missing-channel contributions. I also discuss general properties of the two-body wave function and compositeness for bound states with the schematic models.

Figure 1

Interaction (74) as a function of the radial coordinate $r$ with its strength $v_0=-35\text{MeV}$ and $v_1=0$. I also show the binding energies of the bound states $B_{\mathrm{E}}\equiv m+M-E_{\mathrm{pole}}:B_{\mathrm{E}}\left(0s\right)=23.2\text{MeV}$, $B_{\mathrm{E}}\left(0p\right)=13.1\text{MeV}$, $B_{\mathrm{E}}\left(0d\right)=2.6\text{MeV}$, and $B_{\mathrm{E}}\left(1s\right)=2.1\text{MeV}$.

Figure 2

Density distributions $\text{P}\left(q\right)$ obtained from the Schrödinger equation (Schr) and from the Lippmann-Schwinger equation (LS) with the interaction strength $v_0=-35\text{MeV}$ and $v_1=0$. The solutions of the Schrödinger equation are normalized by hand, while those of the Lippmann-Schwinger equation are automatically scaled.

Figure 3

Density distributions $\text{P}\left(q\right)$ for the $0s$ state obtained from the Lippmann-Schwinger equation (LS) with several values of $v_1$.

Figure 4

Compositeness $X$ as a function of $v_1$. Lines and points are obtained from the formulas in Eqs. (52) and (79), respectively, with the wave function from the scattering amplitude.

Figure 5

Interaction $V$ (80) as a function of the radial coordinate $r$ with its parameters $v_0=-50\text{MeV}$, $v_1=0$, $b_1=2.5\text{fm}$, and $b_2=5.0\text{fm}$. I also show the eigenenergy of the discrete resonance state measured from the threshold, $m+M-\text{Re}{E}_{\mathrm{pole}}$.

Figure 6

(a) Real and (b) imaginary parts of the density distributions $\text{P}\left(q\right)$ obtained from the Schrödinger equation (Schr) and Lippmann-Schwinger equation (LS). The scaling angle is $\theta ={10}^{\circ }$, ${20}^{\circ }$, and ${30}^{\circ }$.

Figure 7

Compositeness $X$ as a function of $v_1$. Lines and points are obtained from the formulas in Eqs. (73) and (81), respectively, with the wave function from the scattering amplitude.

Figure 8

Interaction $V_{11}$ (82) as a function of the radial coordinate $r$ with its strength $v_0=-650\text{MeV}$ and $v_1=0$. I also show the binding energies of the discrete bound states $B_{\mathrm{E}}\equiv m_1+M_1-\text{Re}{E}_{\mathrm{pole}}$ with $x=0$ and $x=0.5$. The shaded band for $x=0.5$ indicates the range of $B_{\mathrm{E}}\pm \text{Im}{E}_{\mathrm{pole}}$.

Figure 9

Density distributions $\text{P}\left(q\right)$ obtained from the Schrödinger equation (Schr) and Lippmann-Schwinger equation (LS) with the interaction parameters $v_0=-650\text{MeV}$, $v_1=0$, and $x=0.5$. The scaling angle is $\theta ={20}^{\circ }$.

Figure 10

Compositeness $X$ as a function of $v_1$. Lines and points are obtained from the formulas in Eqs. (73) and (84), respectively, with the wave function from the scattering amplitude.

Figure 11

Diagram for the description of the physical unstable particle $A$. Double and dashed lines represent the bare particle and decay channel for $A$, respectively.

Figure 12

Diagram for the self-energy of an unstable particle $A$ in the $AB$ propagator. Dashed lines represent the decay channel for $A$.

Figure 13

Density distribution $\text{P}\left(q\right)$ obtained from the Schrödinger equation (Schr) and Lippmann-Schwinger equation (LS) for the bound state of unstable and stable particles. The scaling angle is $\theta ={20}^{\circ }$.