= Solution
After $\lambda_a$ and $\lambda_b$ have been matched to the same measured coupling, the nonanalytic low-energy dependence from the light scalar loops is identical. For $|P|\ll M$, a heavy loop has the local expansion
$$
I(P;M,\Lambda)=I(0;M,\Lambda)+O(P^2/M^2).
$$
Its constant term is already absorbed into the matched quartic coupling, while the remaining terms are higher-dimensional local interactions suppressed by powers of $M$. Dependence on the finite ultraviolet cutoff is similarly suppressed by powers of $\Lambda$. This is <heavy-field decoupling>: low-energy scattering agrees up to $O(p^2/M^2,p^2/\Lambda^2)$ corrections after all relevant parameters are matched. At energies comparable to $M$, the two theories differ sharply because the second theory contains wrong-statistics heavy states.
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