Much theoretical effort and automatization are required to confront new physics models with experimental data for many types of particle reactions at future colliders. In this context, we extend GRACE, an automatic calculation system for invariant amplitudes, to incorporate particles and interactions in the Georgi-Machacek model. With the extended GRACE system, we study fermiophobic Higgs boson production processes at $e^+ e^-$ and $e^- e^-$ colliders in the model. The results show some advantages of $e^- e^-$ colliders over $e^+ e^-$ colliders for new physics search and thus its complementary role.
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