  We show the system of a heavy charged particle and a neutral atom can be
described by a low-energy effective field theory where the attractive $1/r^4$
induced dipole potential determines the long-distance/low-energy wave
functions. The $1/r^4$ interaction is renormalized by a contact interaction at
leading order. Derivative corrections to that contact interaction give rise to
higher-order terms. We show that this ``Induced-dipole EFT'' (ID-EFT)
reproduces the $\pi^+$-hydrogen phase shifts of a more microscopic potential,
the Temkin-Lamkin potential, over a wide range of energies. Already at leading
order it also describes the highest-lying excited bound states of the
pionic-hydrogen ion. Lower-lying bound states receive substantial corrections
at next-to-leading order, with the size of the correction proportional to their
distance from the scattering threshold. Our next-to-leading order calculation
shows that the three highest-lying bound states of the Temkin-Lamkin potential
are well-described in ID-EFT.
