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These two statements seem to conflict, but since we're here they can't:
However, the Higgs field theory states that the field was only created AFTER the Big Bang. This implies that all particles from which hydrogen could be formed were massless. Hydrogen consists of 2 charged particles.
If you could provide an answer without complex math I would really appreciate it.
Hydrogen was not "created at the moment of the big bang".
Particles (leptons and quarks) can attain mass via the Higgs field after the epoch of electroweak symmetry breaking, that occurred about a picosecond $(10^{-12}$s) after the big bang. Only after this was it possible to form the building blocks of hydrogen.
In fact the formation of stable protons comes "much" later at about a millionth of a second. Most of the mass of a proton does not come about from the Higgs field, but from the binding energy of its constituent quarks.
Hydrogen (the atom, rather than the proton nucleus) forms much, much later; about 400,000 years after the big bang.
After the Big Bang, matter has formed in many steps. These are calles epoches.
Their length grows typically exponentially. A more detailed description can you read here.
The Higgs field always existed, but its value were different. In the very early times, around between $10^{-43}$ and $10^{-36}$ seconds (see Grand unification epoch), its value could change, just like like electromagnetic field can change. Since then, its value is its current one.
The temperature of the Universe was higher than $10^{27}$ K at the time.
For atomic hydrogen, we need
It happened much-much later, around 300000 year after the Big Bang. Before that, atomic hydrogen couldn't exist, it was too warm for that.
A remark: the visible hydrogen of the stars is still not atomic hydrogen, because the stars are still too hot for it to exist. It is in plasma state, a mixture of protons of electrons, without atomic hydrogen.
Your basic assumption that Hydrogen was created at the Big Bang is wrong.
The first protons did not appear until the Hadron Epoch when the temperature of the universe was "low" enough to allow quarks to bind together into, among other things, protons.
Hydrogen is an atom formed from an electron and a proton and optional neutrons (for Deuterium and Tritium isotopes of Hydrogen). These did not form until the Recombination Epoch when temperatures had cooled enough to allow electrons to bind with protons to form Hydrogen atoms.
Note that the binding energies are also the energy required to break up particles that are bound, so at higher energies (= higher average temperatures) the atoms and hadrons would not remain in their bound states. The binding energy of a proton to an electron is orders of magnitude lower than the binding energy of quarks to quarks, hence the Hadron epoch happens at a much early phase (a much higher temperature).
Electrically charged particles cannot be massless.
That's an interesting point and there is an interesting question on Physics SE about this. There are arguments for saying it is not impossible on theoretical grounds, but also arguments for saying that we haven't seen them so they don't exist. Suffice to say that the statement you make is subject to ongoing debate and probably a good one to start an argument between particle physicists at parties (assuming you ever see a particle physicist in the wild, let alone at a party :-)).
