I Giving mass to other particles?

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They say at the Big Bang all the particles does not have mass. It’s the Higgs Bosons give them mass...
Higgs bosons are very heavy particles (probably 1000 times heavier than a protons) and very unstable. Now we can create them in particle accelerators like LHC, like countless of other particles.
But wait. This one can give mass to particles without mass, does this violate the conservation of mass or energy? Or the laws of physics is different at the beginning of time? Also I heard a Boson particle is the one usually carries a force (like gravity, electromagnetic forces and nuclear force). If Higgs boson was a boson, what kind of force does it carry? And what makes it capable of bringing mass? Does this process still work in the universe now (like giving mass to photons)?
 
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Summary: They say at the Big Bang all the particles does not have mass. It’s the Higgs Bosons give them mass...

This one can give mass to particles without mass, does this violate the conservation of mass or energy? Or the laws of physics is different at the beginning of time?
The laws of physics were the same back then as they are now. Strictly speaking none of the particles in the SM are massive in the traditional sense (dirac mass, quadratic scalar mass), it's only when you take the weak field limit in the Higgs doublet that you get terms that look like mass terms.

Summary: They say at the Big Bang all the particles does not have mass. It’s the Higgs Bosons give them mass...

If Higgs boson was a boson, what kind of force does it carry? And what makes it capable of bringing mass? Does this process still work in the universe now (like giving mass to photons)?
The "force" bosons originate from lorentz vector fields while the higgs comes from a lorentz scalar (technically a lorentz doublet under ##SU(2)_L##), this makes a world of difference as the higgs field doesn't implement a local gauge symmetry like the vector bosons do. In a sense I guess you could consider the higgs to be a "force" in the sense that it can mediate interactions.
 
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It’s the Higgs Bosons give them mass...
Higgs field, not the boson itself.
 
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Summary: They say at the Big Bang all the particles does not have mass. It’s the Higgs Bosons give them mass...
No, that's not what "they" say.

Particles gain mass in the Standard Model via spontaneous symmetry breaking, which involves the Higgs field. The Higgs boson that was observed in the LHC is what is left over from the Higgs field after the spontaneous symmetry breaking and the gaining of mass by other particles. The observed mass of the Higgs boson is therefore separate from and not connected to the masses of the other particles.
 

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