Yukawa interaction

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In particle physics, Yukawa 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 \gamma^5 \phi \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.

Contents

[edit] The action

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

S[\phi,\psi]=\int d^dx \;\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(φ) is a self-interaction term. For a free-field massive meson, one would have V(φ) = μ2φ2 where μ is the mass for the meson. For a (renormalizable) self-interacting field, one will have V(φ) = μ2φ2 + λφ4 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

\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 \gamma^5 \phi \psi

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

S[\phi,\psi]=\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]

[edit] Classical potential

If two scalar mesons interact through a Yukawa interaction, the potential between the two particles will be:

V(r) = -\frac{g^2}{4\pi} \frac{1}{r} e^{-m_\psi 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 electromagnetisch 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 exponential will give the interaction a finite range, so that particles at great distances will hardly interact any longer.

[edit] Spontaneous symmetry breaking

Now suppose that the potential V(φ) has a minimum not at φ = 0 but at some non-zero value φ0. This can happen if one writes (for example) V(φ) = μ2φ2 + λφ4 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 \tilde \phi = \phi-\phi_0, where φ0 is now understood to be a constant independent of position. We now see that the Yukawa term has a component

g\phi_0 \bar\psi\psi

and since both g and φ0 are constants, this term looks exactly like a mass term for a fermion with mass gφ0. This is the mechanism by which spontaneous symmetry breaking gives mass to fermions. The field \tilde\phi is known as the Higgs field.

[edit] Majorana form

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

S[\phi,\chi]=\int d^dx \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.

[edit] 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.

[edit] See also

[edit] References

  • 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 0-07-232002-8
  • Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (1995), Addison-Wesley Publishing Company, ISBN 0-201-50397-2
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