What precisely and mathematically does it mean to say gauge bosons as elementary particles? In standard particle physics textbook, we say that photons, gluons and $W$ and $Z$ bosons are gauge bosons as elementary particles.
However the gauge bosons are vector bosons and they carry the form of one form gauge field, and have gauge invariant form such as a closed line as
$$
tr_R \left[\exp(i \oint  A)\right],
$$
for some representation $R$.
So gauge field is better defined precisely and mathematically as 1 dimensional loop, instead of 0 dimensional particle.
My Question
So why can we regard photons, gluons and $W$ and $Z$ bosons are gauge bosons as 0 dimensional elementary particles? In fact, it makes better
reasons to consider
$$
tr_R \left[\exp(i \oint  W^{\pm})\right],
$$
$$
tr_R \left[\exp(i \oint  Z^{0})\right],
$$
And how do we measure them as particles in experiments? (the lifetime width of the excitations) instead of regarding gauge bosons as 1 dimensional loop?
 A: This is my understanding of the Quantum Field Theory (QFT) as used in particle physics models, after years working in experiments in particle physics to verify  or find discrepancies in the Standard Model.
To the point particles in the table of the standard model, the  QFT mathematical model assigns a field, an electron field, an electron neutrino field....., which mathematically is the plane wave solution of the corresponding quantum mechanical equation. The fields cover all of space time. On these fields creation and annihilation operators work in creating and annihilating the particles in the table, so an interaction of real particles can be modeled finally with Feynman diagrams.
For the theory to be correct, there should exist the corresponding particle  as a real particle too, in order  to be created and annihilated, on par with the existence of electrons, muons, etc.in the table.  They are called gauge bosons because of the way they are used in the theory of the SM, as the carriers of the weak interaction.
In my time the  W and the Z bosons were found  with the masses in the table, confirming the predictions of the standard model.
That is why we were desperately looking for a particle Higgs, because the Higgs field of the SM had to have the corresponding particle, which finally the LHC managed to find in the data of the CMS and Atlas experiments.
Here is the measurement of the Higgs boson by CMS, verifying the standard model hypotheses, in the invariant mass of the channel of two gammas.

In comments you ask:

compare the W/Z vs Higgs.

The gauge bosons have vector spin 1, the Higgs spin 0.  (intrinsic spin is introduced for elementary particles in the table in order for angular momentum to be conserved in interactions and decays). They have different functions within the model. The gauge bosons, this includes photons, are the virtual particles exchanged in the lowest order  Feynman diagram for the given force (electromagnetic, strong, weak), and allowing for change in quantum numbers at the vertices in all orders. The Higgs mechanism is introduced in the model in order to describe mathematically electroweak symmetry breaking. ( this has the history, of how experimentally observed partial symmetries led to the model)The structure of a QFT model is such that fields cannot exist without the corresponding real (with an on mass four vector) particle. That is why it was necessary to find a Higgs particle, to confirm the standard model. There are other models than the SM, where the Higgs is a composite particle, technicolor for example.
One must point out that the photon, due to its zero mass can be easily seen to manifest as a particle, for example see this experiment.
A: Fundamental particles in quantum mechanics are defined to be the quanta of fundamental fields. For example, when we apply the laws of quantum mechanics to the electromagentic field, we find out the the energy and momentum of the field can only come in discrete bundles (quanta) which we call photons. For electrons, we say that there is an electron field. When we apply the laws of quantum mechanics to this field, we again get discrete bundles that correspond to electrons and positrons. For quarks, we say that there is a gauge field, and when we quantize the gauge field, we again get discrete bundles.
In every case, we start by simply declaring that a field exists. In this sense, the field is fundamental.  It's existence is inferred from experiment and not derived from some other theory. Then, the quanta of these fields are referred to as fundamental particles. They are fundamental because they are the quanta of fundamental fields.
