- (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 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[clarification needed]. 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 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.
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.