- (scalar) or (pseudoscalar).
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.
where the integration is performed over d dimensions (typically 4 for four-dimensional spacetime). 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 (renormalizable, polynomial) 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 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
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 (the electromagnetic interaction is repulsive for identical particles). This is explained by the fact that the Yukawa particle has spin zero and even spin always results in an attractive potential[clarification needed]. The exponential will give the interaction a finite range, so that particles at great distances will hardly interact any longer.
Spontaneous symmetry breaking
Now suppose that the potential has a minimum not at but at some non-zero value . This can happen if one writes (for example) and then sets to an imaginary value. In this case, one says that the Lagrangian exhibits spontaneous symmetry breaking. The non-zero value of is called the vacuum expectation value of . In the Standard Model, this non-zero value is responsible for the fermion masses, as shown below.
To exhibit the mass term, one re-expresses the action in terms of the field , where is now understood to be a constant independent of position. We now see that the Yukawa term has a component
and since both g and are constants, this term looks exactly like a mass term for a fermion with mass . This is the mechanism by which spontaneous symmetry breaking gives mass to fermions. The field is known as the Higgs field.
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
- Itzykson, Claude; Zuber, Jean-Bernard (1980). Quantum Field Theory. New York: McGraw-Hill. ISBN 0-07-032071-3.
- Bjorken, James D.; Drell, Sidney D. (1964). Relativistic Quantum Mechanics. New York: McGraw-Hill. ISBN 0-07-232002-8.
- Peskin, Michael E.; Schroeder, Daniel V. (1995). An Introduction to Quantum Field Theory. Addison-Wesley. ISBN 0-201-50397-2.