When I first learned about the four fundamental forces of nature, I assumed that they were just the only four kind of interactions there were. But after learning a little field theory, there are many other kinds of couplings, even in the standard model. So why isn't the Yukawa Higgs coupling considered a fifth fundamental force? To be a fundamental force, does there need to be a gauge boson mediating it?
The Higgs exchange between matter particles can certainly be called a force. Whether it can be viewed a fundamental force, is a matter of taste. But there is one important distinction between the force due to the Higgs exchange and the usual fundamental interactions. The strong and electroweak interactions are described as gauge interactions. It means that they are not put by hand but arise automatically when you require that the matter fields be invariant under local certain internal transformations (phase rotations, color transformations, etc.). In contrast to that, the "Higgs force" is put into the model by hand, as its presence is not driven by any symmetry consideration. You can equally nicely consider a model with zero Yukawa coupling.
Forces and interactions are similar, so maybe for the purposes of this discussion we should define force as "interaction with a massless particle," as this gives the possibility of macroscopic-range "forces" (barring impeding effects such as confinement or symmetry breaking). After all, gravity is a massless but non-Yang-Mills force (mediated by spin-2, not spin-1, particle). There are some constraints on which spins (representations) of massless particles are possible in quantum field theory, but I'm not familiar enough to try to cite them off-hand, sorry.
Scalar fields do transfer momentum in classical physics. Just take a look at acoustic signals in a gas. A strong sound can cause your windows to rattle. A well known example of energy transference by means of sound (pressure waves) is demonstrated with tuning forks. Quantum theory speaks of sound as particles (phonons), the discrete quanta of quantized mechanical vibrations of a christal lattice, whose longitudinal mode corresponds to our macroscopic perception of sound. In quantum field theory all fields need to be quantized in the end. If I've understood it correctly, the Higgs particle represents the discrete quanta of the quantized Higgs field. Since the Higgs field is a scalar field, its quanta are bosons. They carry kinetic energy, and this energy can be transmitted to all quantum fields that the Higgs field interacts with. You just need to take a look at the Lagrangian to see all the fields that the Higgs interacts with. Alternatively, look for the various Feynman diagrams that include Higgs lines. You can then easily see all the particles it interacts with.
The theory of particle physics is mathematically organized as a gauge theory with the group structure of SU(3)xSU(2)xU(1). These have exchange bosons, which before weak symmetry breaking have a zero mass, and after symmetry breaking have been associated with the three fundamental forces , electromagnetism, weak, strong. This association comes from the mathematical continuity that is necessary going from the microscopic frame of quantum mechanics to the macroscopic of classical electrodynamics, extended to the strong and weak forces.
A force in the micro world of Feynman diagrams is any dp/dt, momentum transfer, in an interaction exchanging particles, for example in this Compton scattering diagrams which is for calculating the crossection of a photon scattering off an electron:
There is a dp/dt, and the virtual exchange particle is the electron. It is an electromagnetic interaction because the incoming vertex is electromagnetic and has the coupling constant of electromagnetism. The electron exchange between vertices is not a fundamental force.
Fundamental are the exchanges associated with the gauge bosons, as the lower order diagrams which give most of the probability for the interaction are the simple exchange of a gauge boson, when it is allowed by quantum numbers. It is the simple exchange of virtual photons that will build up the classical electric and magnetic potentials of the classical electromagnetic interaction.
The Higgs field which is associated with the existence of the mass of the elementary particles does not transfer momentum, and thus is not a force; the exchanges of the Higgs boson are on the same level as for example the exchanges of electrons, (as seen in the diagram above,) in the appropriate diagrams, i.e. a simple transfer of dp/dt. Thus the Higgs mechanism is not related to a new fundamental force.
At the quantum level it is the coupling constants that define the type of interaction and are fundamental. The Higgs boson is in the electroweak sector and, as a neutral elementary particle, interacts only with the weak coupling constant at the vertices.
The Higgs field is not a vector field like, say, the vector potential of EM is. It arises from the observed coupling of massive particles to the weak field. So the Higgs boson is not a force-exchanging gauge boson in the same way as the other bosons of the Standard Model.
If gravity is a fundamental force, then the Higgs mechanism is also. This is is true whether they are related or not. The Higgs mechanism is certainly the source of the inertial mass that inspired Newton to quantify what a force is, and how it behaves.
Gravity, like the Higgs mechanism, can add mass/energy to matter in bulk (like constant acceleration), bend or even confine other force carrying bosons (photons) in space. This happens in the event horizon of black holes.
Not only is the Higgs mechanism a fundamental force, but the way it interacts is unquestionably unique among the fundamental forces. It slows down and limits the range of electroweak force carriers (W,Z) by breaking local symmetry, giving them mass without violating conservation of mass/energy.