Recently, symmetry arguments within certain classes of theories have been put forth that question Drell-Yan monopole production [31]. In this work, we remain agnostic about this traditional monopole channel and include it for direct comparison with existing searches.

Ultrahigh energy neutrino collisions have been also previously considered, e.g., [33, 34, 35, 36].

We note that the resultant cross section is $\sim 10$ times larger than that of spin-zero model but $\sim 10$ times smaller than that of spin-one model [23], $g_D=1$, and velocity ($\beta$)-independent monopole model.

In order to discuss a wider range of theoretical models, some studies have also advocated for considering a $\beta$-dependent coupling $g\beta =g\sqrt{1-4M^2/s^2}$ (e.g., [43]). The $\beta$ dependence of the model can suppress the cross section by a factor of 2.

We note that at masses that are larger than the kinematic reach of the LHC, the measurements for $pp\to X$ have increased uncertainty, stemming from cosmic ray data that has uncertainties on the order of $\sim 30\%$ (see, e.g., [25]). We note that $\sigma (\sqrt{s})$ varies between $\sim 40\mathrm{mb}$ and $\sim 100\mathrm{mb}$ as $\sqrt{s}$ ranges from 10 to 100 TeV.

The background of cosmic ray muons depends on zenith angle and so experimental cuts are necessarily more severe for down-going monopoles. Because the monopole flux is attenuated at large zenith angles, there is a competition between these two effects such that an optimum zenith angle will be achieved at intermediate values ${\theta }_z\sim \pi /6-\pi /3$.

We do not consider here searches focusing on GUT monopoles, e.g., Super-Kamiokande [8].