Yukawa interaction


In particle physics, Yukawa's interaction or Yukawa coupling, named after Hideki Yukawa, is an interaction between a scalar field ϕ and a Dirac field ψ of the type
A Yukawa interaction can be used to describe the nuclear force between nucleons, mediated by pions. A Yukawa interaction is also used in the Standard Model to describe the coupling between the Higgs field and massless quark and lepton fields. Through spontaneous symmetry breaking, these fermions acquire a mass proportional to the vacuum expectation value of the Higgs field.

The action

The action for a meson field interacting with a Dirac baryon field is
where the integration is performed over dimensions.
The meson Lagrangian is given by
Here, is a self-interaction term. For a free-field massive meson, one would have where is the mass for the meson. For a self-interacting field, one will have where λ is a coupling constant. This potential is explored in detail in the article on the quartic interaction.
The free-field Dirac Lagrangian is given by
where is the real-valued, positive mass of the fermion.
The Yukawa interaction term is
where g is the coupling constant for scalar mesons and
for pseudoscalar mesons. Putting it all together one can write the above more explicitly as

Classical potential

If two fermions interact through a Yukawa interaction with Yukawa particle mass, the potential between the two particles, known as the Yukawa potential, will be:
which is the same as a Coulomb potential except for the sign and the exponential factor. The sign will make the interaction attractive between all particles. This is explained by the fact that the Yukawa particle has spin zero and even spin always results in an attractive potential. or the graviton results in forces always attractive, while odd-spin bosons like the gluons, the photon or the rho meson. The negative sign of the exponential gives the interaction an effectively finite range, so that particles at great distances will hardly interact any longer.
As for other forces, the form of the Yukawa potential has a geometrical interpretation in term of the field line picture introduced by Faraday: The part results from the dilution of the field line flux in space. The force is proportional to the number of field lines crossing an elementary surface. Since the field lines are emitted isotropically from the force source and since the distance between the elementary surface and the source varies the apparent size of the surface as, the force also follows the -dependence. This is equivalent to the part of the potential. In addition, the exchanged mesons are instables and have a finite lifetime. The disappearance of the mesons causes a reduction of the flux through the surface that results in the additional exponential factor of the Yukawa potential. Massless particles such as photons are stables and thus yield only potentials. or for very short distances for the strong interaction

Spontaneous symmetry breaking

Now suppose that the potential has a minimum not at but at some non-zero value. This can happen, for example, with a potential form such as with set to an imaginary value. In this case, the Lagrangian exhibits spontaneous symmetry breaking. This is because the non-zero value of the field, when operated on by the vacuum, has a non-zero expectation called the vacuum expectation value of. In the Standard Model, this non-zero expectation is responsible for the fermion masses, as shown below.
To exhibit the mass term, the action can be re-expressed in terms of the derived field, where is constructed to be a constant independent of position. This means that the Yukawa term has a component
and since both g and are constants, this term resembles a mass term for a fermion with mass. This mechanism, the Higgs mechanism, is the means by which spontaneous symmetry breaking gives mass to fermions. The field is known as the Higgs field. The Yukawa coupling for any given fermion in the Standard Model is an input to the theory. The ultimate source of these couplings is unknown, and would be something that a deeper theory should explain.

Majorana form

It is also possible to have a Yukawa interaction between a scalar and a Majorana field. In fact, the Yukawa interaction involving a scalar and a Dirac spinor can be thought of as a Yukawa interaction involving a scalar with two Majorana spinors of the same mass. Broken out in terms of the two chiral Majorana spinors, one has
where is a complex coupling constant, is a complex number, and is the number of dimensions, as above.

Feynman rules

The article Yukawa potential provides a simple example of the Feynman rules and a calculation of a scattering amplitude from a Feynman diagram involving a Yukawa interaction.