Yukawa interaction

In

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

The Yukawa interaction was developed to model the

If two fermions interact through a Yukawa interaction mediated by a Yukawa particle of mass ${\displaystyle \mu }$, the potential between the two particles, known as the Yukawa potential, will be:

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

which is the same as a

) of the mesons causes a reduction of the flux through the surface that results in the additional exponential factor ${\displaystyle ~e^{-\mu r}~}$ of the Yukawa potential. Massless particles such as

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

The action for a meson field ${\displaystyle \phi }$ interacting with a

field ${\displaystyle \psi }$ is

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

where the integration is performed over n dimensions; for typical four-dimensional spacetime n = 4, and ${\displaystyle \mathrm {d} ^{4}x\equiv \mathrm {d} x_{1}\,\mathrm {d} x_{2}\,\mathrm {d} x_{3}\,\mathrm {d} x_{4}~.}$

The meson Lagrangian is given by

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

Here, ${\displaystyle ~V(\phi )~}$ is a self-interaction term. For a free-field massive meson, one would have ${\textstyle ~V(\phi )={\frac {1}{2}}\,\mu ^{2}\,\phi ^{2}~}$ where ${\displaystyle \mu }$ is the mass for the meson. For a (

${\textstyle V(\phi )={\frac {1}{2}}\,\mu ^{2}\,\phi ^{2}+\lambda \,\phi ^{4}}$

The free-field Dirac Lagrangian is given by

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

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

The Yukawa interaction term is

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

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

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

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

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

Yukawa coupling to the Higgs in the Standard Model

A Yukawa coupling term to the

Suppose that the potential ${\displaystyle ~V(\phi )~}$ has its minimum, not at ${\displaystyle ~\phi =0~,}$ but at some non-zero value ${\displaystyle ~\phi _{0}~.}$ This can happen, for example, with a potential form such as ${\displaystyle ~V(\phi )=\lambda \,\phi ^{4}~-\mu ^{2}\,\phi ^{2}}$. In this case, the Lagrangian exhibits spontaneous symmetry breaking. This is because the non-zero value of the ${\displaystyle ~\phi ~}$ field, when operating on the vacuum, has a non-zero vacuum expectation value of ${\displaystyle ~\phi ~.}$

In the Standard Model, this non-zero expectation is responsible for the fermion masses despite the chiral symmetry of the model apparently excluding them. To exhibit the mass term, the action can be re-expressed in terms of the derived field ${\displaystyle \phi '=\phi -\phi _{0}~,}$ where ${\displaystyle ~\phi _{0}~}$ is constructed to be independent of position (a constant). This means that the Yukawa term includes a component

$${\displaystyle ~g\,\phi _{0}\,{\bar {\psi }}\,\psi ~}$$

${\displaystyle \phi _{0}}$

${\displaystyle ~g\,\phi _{0}~.}$

${\displaystyle \phi '~}$

The Yukawa coupling for any given fermion in the Standard Model is an input to the theory. The ultimate reason for these couplings is not known: it would be something that a better, deeper theory should explain.

Majorana form

It is also possible to have a Yukawa interaction between a scalar and a

Majorana spinors, one has

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

where g is a complex coupling constant, m is a complex number, and n is the number of dimensions, as above.

See also

• The article Yukawa potential provides a simple example of the Feynman rules and a calculation of a scattering amplitude from a Feynman diagram involving a Yukawa interaction.

References