  We perform theoretical calculations to investigate the naturally occurring
high-frequency cutoff in a circuit comprising a flux qubit coupled inductively
to a transmission line resonator (TLR). Specifically, a decoupling occurs
between the qubit and the high-frequency modes. The coupling strength between
the qubit and resonator modes increases with mode frequency $\omega$ as
$\sqrt{\omega}$ at low frequencies and decreases as $1/\sqrt{\omega}$ at high
frequencies. This result is similar to those of past studies that considered
somewhat similar circuit designs. By avoiding the approximation of ignoring the
qubit-TLR coupling in certain steps in the analysis, we obtain effects not
captured in previous studies. In particular, we obtain a resonance effect that
shifts the TLR mode frequencies close to qubit oscillation frequencies. We
derive expressions for the TLR mode frequencies, qubit-TLR coupling strengths
and qubit Lamb shift. We identify features in the spectrum of the system that
can be used in future experiments to test and validate the theoretical model.
