In theories with extra spatial dimensions, the behavior of particles and their interactions can be influenced by the geometry and properties of those additional dimensions. However, the relationship between the masses of particles arising from the Higgs mechanism and the number of spatial dimensions is not straightforward.
In the Standard Model of particle physics, the Higgs mechanism is responsible for giving mass to elementary particles such as quarks and leptons. The Higgs field permeates all of space, and particles interact with it, acquiring mass through this interaction. The strength of the interaction between particles and the Higgs field is determined by the coupling constant associated with the Higgs boson.
In theories with extra dimensions, the specific details of the interaction between particles and the Higgs field can be modified. The behavior of the Higgs field and its associated particles may depend on the geometry and size of the extra dimensions. However, it is not necessarily the case that the masses of particles would be greater in higher-dimensional scenarios.
The relationship between the number of spatial dimensions and particle masses is a complex and model-dependent issue. The masses of particles are determined by a combination of factors, including the Higgs field vacuum expectation value, the strength of the Higgs coupling, and the interactions with other fields in the theory. The precise impact of additional dimensions on these factors depends on the specific framework and assumptions of the theory under consideration.
It's worth noting that the existence of extra dimensions and their impact on particle physics is still an area of active research, and various models and theories propose different scenarios. Exploring the consequences of higher-dimensional theories and their connections to experimental observations remains a challenging and ongoing endeavor in theoretical physics.