- Yukawa interaction
In
particle physics , Yukawa's interaction, named afterHideki Yukawa , is an interaction between a scalar field phi and aDirac field Psi of the type:V approx garPsi phi Psi (scalar) or g ar Psi gamma^5 phi Psi (pseudoscalar ).The Yukawa interaction can be used to describe the
strong nuclear force betweennucleon s (which arefermion s), mediated bypion s (which are pseudoscalarmeson s). The Yukawa interaction is also used in theStandard Model to describe the coupling between theHiggs field and masslessquark andelectron fields. Throughspontaneous symmetry breaking , the fermions acquire a mass proportional to thevacuum expectation value of the Higgs field.The action
The action for a
meson field φ interacting with aDirac 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) ight]
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)=mu^2phi^2 where mu is the mass for the meson. For a (
renormalizable ) self-interacting field, one will have V(phi)=mu^2phi^2 + lambdaphi^4 where λ is a coupling constant. This potential is explored in detail in the articlephi to the fourth .The free-field Dirac Lagrangian is given by
:mathcal{L}_mathrm{Dirac}(psi) = ar{psi}(ipartial!!!/-m)psi
where "m" is the positive, real mass of the fermion.
The Yukawa interaction term is :mathcal{L}_mathrm{Yukawa}(phi,psi) = -garpsi phi psi
where "g" is the (real)
coupling constant for scalar mesons and:mathcal{L}_mathrm{Yukawa}(phi,psi) = -garpsi 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) +ar{psi}(ipartial!!!/-m)psi -g ar{psi}phipsi ight]
Classical potential
If two scalar mesons interact through a Yukawa interaction, the potential between the two particles, known as the
Yukawa potential , will be::V(r) = frac{g^2}{4pi} 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 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 exponential will give the interaction a finite range, so that particles at great distances will hardly interact any longer.pontaneous 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 if one writes (for example) V(phi)=mu^2phi^2 + lambdaphi^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 thevacuum expectation value of φ. In theStandard 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 ilde phi = phi-phi_0, where phi_0 is now understood to be a constant independent of position. We now see that the Yukawa term has a component
:gphi_0 arpsipsi
and since both "g" and phi_0 are constants, this term looks exactly like a mass term for a fermion with mass gphi_0. This is the mechanism by which spontaneous symmetry breaking gives mass to fermions. The field ildephi is known as the
Higgs field .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^muphi partial_mu phi -V(phi)+chi^dagger iar{sigma}cdotpartialchi+frac{i}{2}(m+g phi)chi^T sigma^2 chi-frac{i}{2}(m+g phi)^* chi^dagger sigma^2 chi^* ight]
where "g" is a complex
coupling constant and m is acomplex number .Feynman rules
The article
Yukawa potential provides a simple example of the Feynman rules and a calculation of ascattering amplitude from aFeynman diagram involving the Yukawa interaction.ee also
*
Standard Model 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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