It's not so much that there aren't 3 different forces, but rather that they are linked to each other in a way that cause them to "mix".
In particular, electromagnetism (EM) and the weak force (WF) are represented mostly by the U(1) group and SU(2) groups, respectively.
In pure electromagnetism, U(1)_em is what we're observing. This group is linked to a field caused by electric charge.
But if we drill into it, there is also and underlying U(1)_y, that is linked to a hypercharge that is a combination of of electric charge and the WF interaction strength.
The physics of the combined electroweak force is defined by the combined gauge group:
SU(2)_L x U(1)_y
From this fundamental physics we get (as energies get low enough) the Higgs mechanism through "spontaneous symmetry breaking".
This causes two new independent fields from a linear combination of the fields associated with the two above gauge symmetries. One of this gives rise to observable photons, and one gives rise to observable (kind of) W, Z+ and Z-.
And also quite significantly, this mechanism also gives rise to the Higgs field itself, which in turn provides mass (inertia). Without the Higgs mechanism, the particles arising from "pure" U(1) and SU(2) fields would be massless.
In particular, electromagnetism (EM) and the weak force (WF) are represented mostly by the U(1) group and SU(2) groups, respectively.
In pure electromagnetism, U(1)_em is what we're observing. This group is linked to a field caused by electric charge.
But if we drill into it, there is also and underlying U(1)_y, that is linked to a hypercharge that is a combination of of electric charge and the WF interaction strength.
The physics of the combined electroweak force is defined by the combined gauge group:
SU(2)_L x U(1)_y
From this fundamental physics we get (as energies get low enough) the Higgs mechanism through "spontaneous symmetry breaking".
This causes two new independent fields from a linear combination of the fields associated with the two above gauge symmetries. One of this gives rise to observable photons, and one gives rise to observable (kind of) W, Z+ and Z-.
And also quite significantly, this mechanism also gives rise to the Higgs field itself, which in turn provides mass (inertia). Without the Higgs mechanism, the particles arising from "pure" U(1) and SU(2) fields would be massless.