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The Higgs field is known as a physical field that covers the entire universe, giving particles their mass. However, that got me thinking if the Higgs field not only gives mass to other particles, but also makes up particles, with particles and their various characteristics being excitations of the underlying Higgs field. Is that what the Higss field really is, a field where particles arise, or does it only give particles their mass alone to allow gravity to take effect?

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A look at the wikipedia article will clear up confusions in your question.

In the standard model, at temperatures high enough that electroweak symmetry is unbroken, all elementary particles are massless. At a critical temperature the Higgs field becomes tachyonic, the symmetry is spontaneously broken by condensation, and the W and Z bosons acquire masses. (EWSB, ElectroWeak Symmetry Breaking, is an abbreviation used for this.)

So there exists a critical temperature ( average kinetic energy per particle in the primordial soup) where there is a transition from massless to massive gauge bosons and particles, not a creation of particles.

There exists group algebra of the standard model on how the various Higgs field components mathematically give mass to the gauge bosons

after symmetry breaking, these three of the four degrees of freedom in the Higgs field mix with the three W and Z bosons (W+, W− and Z), and are only observable as spin components of these weak bosons, which are now massive; while the one remaining degree of freedom becomes the Higgs boson—a new scalar particle.

So the only "new" particle is the Higgs boson, one and only in the standard model, maybe some more in theories beyond the standard model. The other particles in the table acquire mass by this symmetry breaking, they already existed with zero mass before.

Have a look at this article on the Higgs field. This article gives a view of how the electron acquires mass through the Higgs field.

To address this part of the question:

or does it only give particles their mass alone to allow gravity to take effect?

The Higgs gives mass to the elementary particles in the table in the standard model link above. Composite particles as protons ,neutrons and then atoms and molecules , have the largest contribution to their mass from the consequences of special relativity, where mass and energy are connected. The internal interactions of the quark and gluon components of the proton and neutron generate an invariant mass much larger than the summed mass of the three quarks composing them ( gluons have zero mass).

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Perhaps you could develop a mathematical therory following your lines of thouhght, but for current theories, the higgs is another field, different that the field of the particles to wich it gives them mass. The do not actually give them mass, but the interactions with them "looks like" mass for any macroscopic practical uses.

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