Could the Higgs field be noted $H^{\mu}$, in the same spirit as $W^{\mu}$ and $Z^{\mu}$? The $W$ and $Z$ particles are noted in Lagrangian in the form of $W^{\mu}$ and $Z^{\mu}$, in order to construct quantities such as $W_{\mu}W^{\mu}$ and $Z_{\mu}Z^{\mu}$.
Could the Higgs (that appears in $\frac{v+H}{\sqrt{2}}$) also be noted $H^{\mu}$?
If so, what about quantities such as:
$W_{\mu}W^{\mu}H$?
Should a $\mu$ be put on the $H$?
If not, why?
 A: After reading your comment below your question: a field is a function of space-time, no matter its spin: roughly speaking, a field with spin 0

*

*has scalar values, because its spin is 0

*depends on space and time variables, because it's a field

while a spin 1 field

*

*has vector values, because its spin is 1

*depends on space and time variables, because it's a field

So the Higgs field doesn't have any Lorentz index ($H$), while the $W$ field has a Lorentz index ($W^\mu$).
A term like $W^\mu W_\mu H$ is an acceptable interaction term because it's a Lorentz scalar (no Lorentz index left). It describes any interaction between a Higgs and two $W$, for example, two $W$ interacting to produce a Higgs.
Edit:
A vector field is a function of space and time that takes vector values.

*

*In classical physics, they're traditionally written $\vec{F}(x,y,z,t)$, the arrow meaning that the field has vector values in $\mathbb{R}^3$ or $\mathbb{C}^3$.

*In (non quantum) relativity, they're traditionally written $F^\mu(x,y,z,t)$. It's exactly the same thing, except that the vector values are in $\mathbb{R}^4$ or $\mathbb{C}^4$.

It's a bit more complicated than that in quantum field theory because the fields don't have numerical values, but this idea is the same (they're representations of the Lorentz group in 4 dimensions).
