Yukawa interaction

In particle physics, Yukawa's interaction, named after Hideki Yukawa, is an interaction between a scalar field ϕ and a Dirac field ψ of the type

$V\approx g{\bar {\psi }}\phi \psi$ (scalar) or $g{\bar {\psi }}i\gamma ^{5}\phi \psi$ (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.

The action

The action for a meson field φ interacting with a Dirac baryon field ψ is

$S[\phi ,\psi ]=\int d^{d}x\;\left[{\mathcal {L}}_{\mathrm {meson} }(\phi )+{\mathcal {L}}_{\mathrm {Dirac} }(\psi )+{\mathcal {L}}_{\mathrm {Yukawa} }(\phi ,\psi )\right]$

where the integration is performed over d dimensions (typically 4 for four-dimensional spacetime). The meson Lagrangian is given by

${\mathcal {L}}_{\mathrm {meson} }(\phi )={\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -V(\phi ).$

Here, $V(\phi )$ is a self-interaction term. For a free-field massive meson, one would have $V(\phi )={\frac {1}{2}}\mu ^{2}\phi ^{2}$ where $\mu$ is the mass for the meson. For a (renormalizable, polynomial) self-interacting field, one will have $V(\phi )={\frac {1}{2}}\mu ^{2}\phi ^{2}+\lambda \phi ^{4}$ 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

${\mathcal {L}}_{\mathrm {Dirac} }(\psi )={\bar {\psi }}(i\partial \!\!\!/-m)\psi$

where m is the positive, real mass of the fermion.

The Yukawa interaction term is

${\mathcal {L}}_{\mathrm {Yukawa} }(\phi ,\psi )=-g{\bar {\psi }}\phi \psi$

where g is the (real) coupling constant for scalar mesons and

${\mathcal {L}}_{\mathrm {Yukawa} }(\phi ,\psi )=-g{\bar {\psi }}i\gamma ^{5}\phi \psi$

for pseudoscalar mesons. Putting it all together one can write the above more explicitly as

$S[\phi ,\psi ]=\int d^{d}x\left[{\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -V(\phi )+{\bar {\psi }}(i\partial \!\!\!/-m)\psi -g{\bar {\psi }}\phi \psi \right].$

Classical potential

If two fermions interact through a Yukawa interaction with Yukawa particle mass $\mu$, the potential between the two particles, known as the Yukawa potential, will be:

$V(r)=-{\frac {g^{2}}{4\pi }}{\frac {1}{r}}e^{-\mu r}$

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. 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 $V(\phi )$ has a minimum not at $\phi =0$ but at some non-zero value $\phi _{0}$. This can happen, for example, with a potential form such as $V(\phi )=\mu ^{2}\phi ^{2}+\lambda \phi ^{4}$ with $\mu$ set to an imaginary value. In this case, the Lagrangian exhibits spontaneous symmetry breaking. This is because the non-zero value of the $\phi$ field, when operated on by the vacuum, has a a non-zero expectation called the vacuum expectation value of $\phi$. 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 ${\tilde {\phi }}=\phi -\phi _{0}$, where $\phi _{0}$ is constructed to be a constant independent of position. This means that the Yukawa term has a component

$g\phi _{0}{\bar {\psi }}\psi$

and since both g and $\phi _{0}$ are constants, this term resembles a mass term for a fermion with mass $g\phi _{0}$. This mechanism, the Higgs mechanism, is the means by which spontaneous symmetry breaking gives mass to fermions. The field ${\tilde {\phi }}$ is known as the Higgs field.

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

$S[\phi ,\chi ]=\int d^{d}x\left[{\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -V(\phi )+\chi ^{\dagger }i{\bar {\sigma }}\cdot \partial \chi +{\frac {i}{2}}(m+g\phi )\chi ^{T}\sigma ^{2}\chi -{\frac {i}{2}}(m+g\phi )^{*}\chi ^{\dagger }\sigma ^{2}\chi ^{*}\right]$

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.

References

• 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.