I understand from the internet that the Higgs particle is highly unstable! It decays as soon as it is created. If it is so unstable, how one can say that the Higgs field exist? Just like, if photons are unstable, there won't be electromagnetic field. I cannot imagine a field whose particle is highly unstable. Also, it is said that the mass depends up on how the matter interact with this higgs field. If the higgs bosons are highly unstable, how matter can interact with those unstable higgs bosons?
A field and a particle are two different concepts and it is well that one should separate them.
A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or a tensor, respectively. For example, the Newtonian gravitational field is a vector field: specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point. Moreover, within each category (scalar, vector, tensor), a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators respectively.
We are in quantum mechanics when we are talking of the Higgs field.
The Higgs boson or Higgs particle is an elementary particle in the Standard Model of Particle physics. Its main relevance is that it is the smallest possible excitation of the Higgs field
The field in quantum mechanics is defined via operators.
So it is two separate concepts. The Higgs field exists from low to high energies, but it is only at high energies that the Higgs boson can be on mass shell and subsequently decay. The Higgs boson is not a gauge boson, it does not define the interactions possible within the standard model between particles, as the photon, W and Z or the gluons do. It is a manifestation of the underlying field everywhere, when the energy available allows it, and the vacuum expectation value of this field is 246 GeV..
Particles acquire mass through the symmetry breaking mechanism of the Higgs field, not interacting with the Higgs boson/particle which is just a manifestation that had to exist if the symmetry breaking model is correct.
The Higgs Field is believed to permeate the universe, and the Higgs Boson is just an excitation of one of the four components the Higgs Field! The Higgs field needs high amounts of energy to be excited, so when Higgs Boson is "created", its energy level is usually many orders of magnitude higher than the ground energy level of its surroundings, and hence the Particle decays. Much like how water kept on a higher level would flow down to a lower level when possible. In simple terms particles gain their masses when they interact the Higgs field, and not with the actual particle itself. Similarly, the photon is just an excitation of the electromagnetic field, and it is not necessary photons to exist wherever the electromagnetic field exists. Hope this has been clear enough. In the universe objects always seek to occupy the lowest energy state, so heavy particles like the Higgs, or even Neutrons (free ones) always decay into particles of lower mass/energy. The Higgs Boson being unstable doesn't imply the higgs field does not exist.
Particles can only decay into other pairs of particles whose total mass is lesser than the particle which is undergoing decay, and that is why Photons are "stable" (since they are massless.)