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
Contents
The Yukawa interaction can be used to describe the nuclear force between nucleons (which are fermions), mediated by pions (which are pseudoscalar mesons). The Yukawa interaction is also used in the Standard Model to describe the coupling between the Higgs field and massless quark and lepton fields (i.e., the fundamental fermion particles). 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 φ interacting with a Dirac baryon field ψ is
where the integration is performed over d dimensions (typically 4 for four-dimensional spacetime). The meson Lagrangian is given by
Here,
The free-field Dirac Lagrangian is given by
where m is the positive, real mass of the fermion.
The Yukawa interaction term is
where g is the (real) 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
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 (the electromagnetic interaction is repulsive for same electrical charge sign particles). This is explained by the fact that the Yukawa particle has spin zero and even spin always results in an attractive potential. The negative sign of the exponential gives the interaction an effectively finite range, so that particles at great distances will hardly interact any longer (interaction forces drop off exponentially with increasing separation).
Spontaneous symmetry breaking
Now suppose that the potential
To exhibit the mass term, the action can be re-expressed in terms of the derived field
and since both g and
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 g is a complex coupling constant and m is a complex number.
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 the Yukawa interaction.