mass - How does the Higgs field enable masses to attract each other?

Please refrain from quoting general relativity. I'd like to know whether the exchange of virtual Higgs bosons, which are responsible for giving particles their masses, can produce an attractive force between particles.

You may mention other virtual particles such as the graviton if it is helpful. This is really confusing me because I kept wondering how the momentum is transferred by the virtual particle presuming a virtual Higgs boson or a virtual graviton (theoretically).

Answer

The Higgs field mediates so-called Yukawa interactions between fermions. These interactions are of the form $$ y H \bar Q d_R + \text{h.c} $$ and result in masses by the Higgs mechanism. The Higgs field $H$ acquires a vacuum expectation value (VEV) because of the Mexican hat shape of the Higgs potential, $H \to \frac1{\sqrt 2}(0, h + v)$. This results in masses $$ \frac1{\sqrt 2} y v \bar{q} q = m \bar{q} q $$ and interactions of the form $$ \frac1{\sqrt 2} y h \bar{q} q = \frac{m}{\sqrt 2 v} h \bar{q} q $$ The Yukawa interactions generate an attractive potential between fermions exchanging Higgs bosons. Thus, the Higgs field enables fermions to attract each other via the same Yukawa interaction that generates mass, and the strength of the attraction is proportional to mass.

Remember, though, that the Higgs field and Yukawa interactions have nothing to do with gravity.