Well, I know if this question was asked about another particle the answer would be that the charge of a particle in an intrinsic property and that’s that(as far as we currently know), and that might also be the case for the Higgs, but I am asking the question because the Higgs field interacts with particles to give them mass and does not affect the charge, so I was wondering if the Higgs field has a property that makes the Higgs boson chargeless, I am not sure if the question makes sense, but I would appreciate an explanation.
because the Higgs field interacts with particles to give them mass and does not affect the charge
The Higgs field does not interact with particles in the definition of interaction for elementary particles. Particles are the effect of creation and annihilation operators on the fields defined over all space for the elementary particles seen in the table of the standard model.
See also my answer here, on a similar question, where I try to make clear the difference between interaction and the Higgs mechanism.
One of the requirements for the standard model is the KISS tool,( Keep It Simple ..), form the model to fit the data available and stop there. So the answer is the same as with the other particles and their fields "the charge of a particle in an intrinsic property and that’s that(as far as we currently know)" .
That the Higgs boson in the standard model is neutral is because with this hypothesis the standard model fits in a satisfactory way the great majority of data. In addition, the Higgs mechanism was postulated in the model to explain the masses of the electroweak gauge bosons. The discovery of the Higgs boson, which has to exist if there is a Higgs field, is an extra verification of the model presently.
There are models that have charged Higgs bosons. If candidates for charged Higgs bosons appear in experiments, then the standard model would have to be extended or embedded in a new model.
At electroweak symmetrybreaking, three components of the Higgs field are absorbed by the SU(2) and U(1) gauge bosons (Higgs mechanism), and become the longitudial component of the (massive) W and Z bosons. The Z boson is EM neutral. The remaining EM neutral component of the Higgs field can manifest as the Higgs boson which is EM neutral too.
In the standard model, the Higgs field is an SU(2) doublet (i.e. the standard representation with two complex components called isospin), which is a scalar under Lorentz transformations. Its electric charge is zero; its weak isospin is 1⁄2; its weak hypercharge (the charge for the U(1) gauge group) is 1 . Under U(1) rotations, it is multiplied by a phase, which thus mixes the real and imaginary parts of the complex spinor into each other, combining to the standard two-component complex representation of the group U(2).
The field itself is defined as having no EM charge.