In particle physics, Yukawa interaction, named after Hideki Yukawa, is an interaction between a scalar field $\backslash phi$ and a Dirac field $\backslash Psi$ of the type

$V\; \backslash approx\; g\backslash bar\backslash Psi\; \backslash phi\; \backslash Psi$ (scalar) or $g\; \backslash bar\; \backslash Psi\; \backslash gamma^5\; \backslash phi\; \backslash Psi$ (pseudoscalar).

The Yukawa interaction can be used to describe the strong 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 electron fields. Through spontaneous symmetry breaking, the fermions acquire a mass proportional to the vacuum expectation value of the Higgs field.

The action

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

$S*=\backslash int\; d^dx\; \backslash ;\backslash left[$

\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

$\backslash mathcal\{L\}\_\backslash mathrm\{meson\}(\backslash phi)\; =$

\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -V(\phi).

Here, $V(\backslash phi)$ is a self-interaction term. For a free-field massive meson, one would have $V(\backslash phi)=\backslash mu^2\backslash phi^2$ where $\backslash mu$ is the mass for the meson. For a (renormalizable) self-interacting field, one will have $V(\backslash phi)=\backslash mu^2\backslash phi^2\; +\; \backslash lambda\backslash phi^4$ where λ is a coupling constant. This potential is explored in detail in the article phi to the fourth.

The free-field Dirac Lagrangian is given by

$\backslash mathcal\{L\}\_\backslash mathrm\{Dirac\}(\backslash psi)\; =$

\bar{\psi}(i\partial\!\!\!/-m)\psi

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

The Yukawa interaction term is

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

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

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

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

$S*=\backslash int\; d^dx$

\left[\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -V(\phi) + \bar{\psi}(i\partial\!\!\!/-m)\psi -g \bar{\psi}\phi\psi \right]

Spontaneous symmetry breaking

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Now suppose that the potential $V(\backslash phi)$ has a minimum not at $\backslash phi=0$ but at some non-zero value $\backslash phi\_0$. This can happen if one writes (for example) $V(\backslash phi)=\backslash mu^2\backslash phi^2\; +\; \backslash lambda\backslash phi^4$ and then sets μ to an imaginary value. In this case, one says that the Lagrangian exhibits spontaneous symmetry breaking. The non-zero value of φ is called the vacuum expectation value of φ. 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 $\backslash tilde\; \backslash phi\; =\; \backslash phi-\backslash phi\_0$, where $\backslash phi\_0$ is now understood to be a constant independent of position. We now see that the Yukawa term has a component

$g\backslash phi\_0\; \backslash bar\backslash psi\backslash psi$

and since both g and $\backslash phi\_0$ are constants, this term looks exactly like a mass term for a fermion with mass $g\backslash phi\_0$. This is the mechanism by which spontaneous symmetry breaking gives mass to fermions. The field $\backslash tilde\backslash phi$ is known as the Higgs field.

Majorana form

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It's 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

$Sd^dx\; \backslash left[\backslash frac\{1\}\{2\}\backslash partial^\backslash mu\backslash phi\; \backslash partial\_\backslash mu\; \backslash phi\; -V(\backslash phi)+\backslash chi^\backslash dagger\; i\backslash bar\{\backslash sigma\}\backslash cdot\backslash partial\backslash chi+\backslash frac\{i\}\{2\}(m+g\; \backslash phi)\backslash chi^T\; \backslash sigma^2\; \backslash chi-\backslash frac\{i\}\{2\}(m+g\; \backslash phi)^*\; \backslash chi^\backslash dagger\; \backslash sigma^2\; \backslash chi^*\backslash right$

where g is a complex coupling constant and m is a complex number.

Feynman rules

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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.

See also

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• Standard Model

References

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• Claude Itzykson and Jean-Bernard Zuber, Quantum Field Theory, (1980) McGraw-Hill Book Co. New York ISBN 0-07-032071-3
• James D. Bjorken and Sidney D. Drell, Relativistic Quantum Mechanics (1964) McGraw-Hill Book Co. New York ISBN 07-005493-2

Quantum field theory | Standard Model | Electroweak theory | Nuclear physics

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