Yukawa Interaction

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. 

The Yukawa interaction can be used to describe the strong nuclear force between nucleons (which are fermions), mediated by pions (which are scalar 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

The action for a meson field interaction with a Dirac fermion field is:

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

Here, V(φ(x)) 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

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

The Yukawa interaction term is

where g is the (real) coupling constant. Putting it all together, and dropping the explicit dependence on position x, one can write the above far more compactly as

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 , where φ0 is now understood to be a constant independent of position. We now see that the Yukawa term has a component

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 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

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

Yukawa potential

A Yukawa potential (also called a screened Coulomb potential) is a potential of the form

Hideki Yukawa showed in the 1930s that such a potential arises from the exchange of a massive scalar field such as the field of the pion whose mass is m. Since the field mediator is massive the corresponding force has a certain range due to its decay.

In the above equation, the potential is negative, denoting that the force is attractive. The constant g is a real number; it is equal to the coupling constant between the meson field and the fermion/fermion field with which it interacts. In the case of nuclear physics, the fermions would be the proton and the neutron.

Fourier transform

The easiest way to understand that the Yukawa potential is associated with a massive field is by examining its Fourier transform. One has

where the integral is performed over all possible values of the 3-vector momentum k. In this form, the fraction 4π / (k2 + m2) is seen to be the propagator or Green's function of the Klein-Gordon equation.

Feynman amplitude

The Yukawa potential can be derived as the lowest order amplitude of the interaction of a pair of fermions. The Yukawa interaction couples the fermion field ψ(x) to the meson field φ(x) with the coupling term

The scattering amplitude for two fermions, one with initial momentum p1 and the other with momentum p2, exchanging a meson with momentum k, is given by the Feynman diagram below. 

 

 

The Feynman rules for each vertex associate a factor of g with the amplitude; since this diagram has two vertices, the total amplitude will have a factor of g2. The line in the middle, connecting the two fermion lines, represents the exchange of a meson. The Feynman rule for a particle exchange is to use the propagator; the propagator for a massive meson is - 4π / (k2 + m2). Thus, we see that the Feynman amplitude for this graph is nothing more than

 

 

Klein-Gordon equation

The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is a relativistic version (describing scalar (or pseudoscalar) spinless particles) of the Schrödinger equation.

The Schrödinger equation for a free particle is

where is the momentum operator, using natural units where .

The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's special theory of relativity.

It is natural to try to use the identity from special relativity

for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation

This, however, is a cumbersome expression to work with because of the square root. Cumbersomeness, however, doesn't really count as an objection. But this equation, as it stands, is non-local.

Klein and Gordon instead worked with the square of this equation (the Klein-Gordon equation for a free particle), which in covariant notation reads:  (∂2 + m2)ψ = 0,  where 2 is the d'Alembert operator.

The Klein-Gordon equation was actually first found by Schrödinger, before he made the discovery of the equation that now bears his name. He rejected it because he couldn't make it fit data (the equation doesn't take into account the spin of the electron); the way he found his equation was by making simplifications in the Klein-Gordon equation.

The Klein-Gordon equation may also be derived out of purely information-theoretic considerations. See extreme physical information.

In 1926, soon after the Schrödinger equation was introduced, Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the gauge theory for the wave equation. The Klein-Gordon equation for a free particle has a simple plane wave solution.

Relativistic free particle solution

If the particle is charge-neutral and spinless, and relativistic effects cannot be ignored, we may use the Klein-Gordon equation to describe the wave function. The Klein-Gordon equation for a free particle is written

with the same solution as in the non-relativistic case:

except with the constraint

Just as with the non-relativistic particle, we have for energy and momentum:

Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles:

For massless particles, we may set m=0 in the above equations. We then recover the relationship between energy and momentum for massless particles: