The \(\pi^+\pi^-\) atom
Goal: Study the properties of the \(\pi^+\pi^-\) atom taking into account the Coulomb and the strong interactions. Compute the binding energy and its correction due to the strong interaction, and compare them with experimental data.
Coulomb binding energy: Compute the energy of the \(1s\) state with only the electromagnetic interaction taken into account. Estimate the Bohr radius.
Strong-interaction correction: Compute the correction \(\Delta E\) to the binding energy of the \(1s\) state from \[\Delta E = \frac{\int\limits_0^\infty |R_{1s}|^2V_{\text{strong}}(r) r^2 dr}{\int\limits_0^\infty |R_{1s}|^2r^2 dr},\] using three different potentials for the strong interaction:
A contact potential (delta function): \[V_{\text{strong}}(r) = -\frac{4\pi a_0}{\mu} \delta^3(r),\] where \(\mu = m_\pi\) and \(a_0 \approx 0.22\,m_\pi^{-1} \approx 0.31\,\mathrm{fm}\) is the \(\pi\pi\) scattering length in the isospin \(I=0\) state.
A finite-range potential (square well): \[V_{\text{strong}}(r) = \begin{cases} -V_0, & r \leq R_{\text{strong}}, \\ 0, & r > R_{\text{strong}}, \end{cases}\] where \(V_0 \sim 100\,\mathrm{MeV}, \quad R_{\text{strong}} \sim 1\,\mathrm{fm}\).
A Yukawa potential: \[V_{\text{strong}}(r) = -g^2 \frac{e^{-\mu_s r}}{r},\] where \(g^2\) is the strong coupling constant and \(\mu_s \sim 1\,\mathrm{fm}^{-1}\).
Compare the values of \(\Delta E\) you obtain with experimental data.
Plot \(\Delta E\) as a function of the parameters \(a_0\), \(V_0\), \(g^2\) for the three potential models and compare with experiment.
Analyze the contribution of the strong interaction to the total binding energy.