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This thesis explores a fundamental question in field theory: If we modify a theory by introducing a small parameter, can we recover the original theory by setting that parameter back to zero? While seemingly simple, this question presents a major challenge in gauge theories where mass is added manually. In fact, some have even argued that such theories should be completely discarded as viable descriptions of nature.

Focusing on massive gauge theories, particularly those involving vector fields, this work offers a fresh perspective that diverges from conventional field theory textbooks. Beginning with the basic field theory, it introduces key concepts in a clear and accessible manner, making it suitable even for advanced undergraduate students. By integrating ideas from cosmological perturbation theory and massive gravity, the book provides new insights into the behavior of field theories-especially in extreme limits where standard particle physics tools fail.