Recall in a global transformation of the field \(\phi\), \(\phi(x)\rightarrow \textbf{U}\phi(x)\), where the linear transformation \(U\) does not depend on spatial location. In a local transformation, \(\phi(x)\rightarrow \textbf{U}(x)\phi(x)\), which naturally leads to a new definition of derivative

$$ n^\mu D_\mu\phi(x)\equiv\lim\limits_{\epsilon\rightarrow 0}\frac{\phi(x+\epsilon n)-W(x+\epsilon n,x)\phi(x)}{\epsilon},$$

where \(W(x+\epsilon n,x)\) is the Wilson line connecting the beginning and end of an ordinary derivative. Since \(W(x,x)=1\),

$$ W(x+\epsilon n,x)=1+\epsilon n^\mu \cdot(\text{something}_\mu)+\mathcal{O}(\epsilon^2).$$

By convention something\(=ieA\), therefore

$$D_\mu=\partial_\mu-ieA_\mu.$$

Inevitably, the dreaded gauge vector field \(A_\mu\) invades our life. The gauge fields encode the way the Wilson line transforms. By convention, we define the behavior of these gauge fields by requiring covariant derivative \(D_\mu \phi \) to transform like a vector under the local transformation \(U(x)\) in question. For example, for U(1) transformation \(\phi(x)\rightarrow e^{i\alpha(x)}\phi(x)\), to maintain

$$ D_\mu \phi(x) \rightarrow e^{i\alpha(x)}D_\mu\phi(x),$$

we need \(A_\mu\rightarrow A_\mu+\frac{1}{e}\partial_\mu\alpha(x)\).