This is the table of elementary particles in the standard model of particle physics:

The model is called standard because it describes the grand majority of existing data and has been predictive of new phenomena, as the discovery of the Higgs boson.

The standard model lagrangian develops a field theory of interactions for these particles, the measurable quantities defining integrals represented by Feynman diagrams.

For each particle in the table a field over all (x,y,z,t) is postulated on which creation and annihilation operators define the particles, so there exists in all of space a photon field , an electron field,... and thus a Higgs field.

If an extension of the standard model in the future has more elementary particles, including more Higgs ones, these also will be defined over all space on which the corresponding creation and annihilation operators will operate.

Would I say "Interacting with the Higgs field" or "Interacting with a Higgs field"?

From the introduction above it should be evident that one "operates" on the quantum mechanical fields with the appropriate operators that will give the measurable interactions and described by the appropriate feynman diagrams .

Is the higgs field composed of scalar or vector units?

The Higgs field is the corresponding ground state wave function describing a particle in the appropriate quantum mechanical equation, fermions will be with solutions of the Dirac equation bosons (as the Higgs) with the Klein Gordon. They are operator fields, creation and annihilation operators operating and giving number of particles. See this answer for the type of fields .

This is the table of elementary particles in the standard model of particle physics:

The model is called standard because it describes the grand majority of existing data and has been predictive of new phenomena, as the discovery of the Higgs boson.

The standard model lagrangian develops a quantum field theory of interactions for these particles, the measurable quantities defining integrals represented by Feynman diagrams.

For each particle in the table a field over all (x,y,z,t) is postulated on which creation and annihilation operators define the particles, so there exists in all of space a photon field , an electron field,... and thus a Higgs field.

If an extension of the standard model in the future has more elementary particles, including more Higgs ones, these also will be defined over all space on which the corresponding creation and annihilation operators will operate.

Would I say "Interacting with the Higgs field" or "Interacting with a Higgs field"?

From the introduction above it should be evident that one "operates" on the quantum mechanical fields with the appropriate operators that will give the measurable interactions and described by the appropriate feynman diagrams . At the moment there is only one higgs field verified, the one in the standard model

Is the higgs field composed of scalar or vector units?

The Higgs field is the corresponding ground state wave function describing a particle in the appropriate quantum mechanical equation, fermions will be with solutions of the Dirac equation bosons (as the Higgs) with the Klein Gordon. They are operator fields, creation and annihilation operators operating and giving number of particles. See this answer for the type of fields .

This is the table of elementary particles in the standard model of particle physics:

The model is called standard because it describes the grand majority of existing data and has been predictive of new phenomena, as the discovery of the Higgs boson.

The standard model lagrangian develops a field theory of interactions for these particles, the measurable quantities defining integrals represented by Feynman diagrams.

For each particle in the table a field over all (x,y,z,t) is postulated on which creation and annihilation operators define the particles, so there exists in all of space a photon field , an electron field,... and thus a Higgs field.

If an extension of the standard model in the future has more elementary particles, including more Higgs ones, these also will be defined over all space on which the corresponding creation and annihilation operators will operate.

This is the table of elementary particles in the standard model of particle physics:

The model is called standard because it describes the grand majority of existing data and has been predictive of new phenomena, as the discovery of the Higgs boson.

The standard model lagrangian develops a field theory of interactions for these particles, the measurable quantities defining integrals represented by Feynman diagrams.

For each particle in the table a field over all (x,y,z,t) is postulated on which creation and annihilation operators define the particles, so there exists in all of space a photon field , an electron field,... and thus a Higgs field.

If an extension of the standard model in the future has more elementary particles, including more Higgs ones, these also will be defined over all space on which the corresponding creation and annihilation operators will operate.

Would I say "Interacting with the Higgs field" or "Interacting with a Higgs field"?

From the introduction above it should be evident that one "operates" on the quantum mechanical fields with the appropriate operators that will give the measurable interactions and described by the appropriate feynman diagrams .

Is the higgs field composed of scalar or vector units?

The Higgs field is the corresponding ground state wave function describing a particle in the appropriate quantum mechanical equation, fermions will be with solutions of the Dirac equation bosons (as the Higgs) with the Klein Gordon. They are operator fields, creation and annihilation operators operating and giving number of particles. See this answer for the type of fields .