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-label=”fig:double-loop”}](sensitivity_results.pdf){width=”0.99columnwidth”}
The fact that the sensitivity of the process is suppressed by the factor $M^2_N/m^2_sigma$ is a generic feature of the case where the loop on the DM is replaced by loops on SM particles. The reason behind this suppression is explained in Fig. [fig:double-loop], where we compare the behavior of the minimum and maximum sensitivity in the $M_sigma-M_N$ plane, considering two loops on SM particles and two loops on DM particles as in Eq. ([eq:upper-bound]). The top panel shows the case with no interplay between them; while the bottom panel shows the case where they are kept at their minimum point. For the top case, one can read off that the maximum is suppressed by the factor $M^2_N/m^2_sigma$ even if there is no interplay between the two loops, and the minimum is also suppressed by the same factor. On the other hand, for the bottom case, one can read off that the maximum is not suppressed by the factor $M^2_N/m^2_sigma$, and the minimum is enhanced by the same factor. The reason behind this is the following: the minimum of the loop on the DM is below the maximum of the loop on the SM. Since the loop on the DM is precisely the contribution to the Weinberg operator in Eq. ([eq:Weinberg-operator]), one can expect it to be suppressed by the factor $M^2_N/m^2_sigma$ compared to the loop on the SM. However, this suppression is canceled out by the fact that the loop on the SM is enhanced by the factor $m_sigma^2/m^2_chi$; therefore, the minimum of the process is enhanced by the same factor. The relative enhancement of the loop on the SM as compared to that of the DM is the origin of the suppression of the sensitivity.
![Contour plot for the process in Eq. ([eq:process]). Green lines represent the regions of parameter space leading to resonant single-loop signals. Yellow lines represent the region of parameter space leading to resonant double-loop signals. The yellow dot-d
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