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In particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
, Yukawa's interaction, named after Hideki Yukawa
Hideki Yukawa

n? , was a Japanese theoretical physicist and the first Japanese Nobel prize....
, is an interaction between a scalar field and a Dirac field of the type (scalar) or (pseudoscalar
Pseudoscalar

In physics, a pseudoscalar is a quantity that behaves like a scalar , except that it changes sign under a Parity such as improper rotations while a true scalar does not....
).

The Yukawa interaction can be used to describe the strong nuclear force between nucleon
Nucleon

In physics, a nucleon is a collective name for two baryons: the neutron and the proton. They are constituents of the atomic nucleus and until the 1960s were thought to be elementary particles....
s (which are fermion
Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
s), mediated by pion
Pion

In particle physics, a pion is any of three subatomic particles: , and . Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force....
s (which are pseudoscalar meson
Meson

In particle physics, mesons are subatomic particles composed of one quark and one antiquark. They are part of the hadron particle family ? particles made of quarks....
s). The Yukawa interaction is also used in the Standard Model
Standard Model

The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....
 to describe the coupling between the Higgs field and massless quark
Quark

Quarks are a type of elementary particle and major constituents of matter. They are the only particles in the Standard Model to experience all four fundamental interaction, which are also known as fundamental interactions....
 and electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
 fields.






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In particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
, Yukawa's interaction, named after Hideki Yukawa
Hideki Yukawa

n? , was a Japanese theoretical physicist and the first Japanese Nobel prize....
, is an interaction between a scalar field and a Dirac field of the type (scalar) or (pseudoscalar
Pseudoscalar

In physics, a pseudoscalar is a quantity that behaves like a scalar , except that it changes sign under a Parity such as improper rotations while a true scalar does not....
).

The Yukawa interaction can be used to describe the strong nuclear force between nucleon
Nucleon

In physics, a nucleon is a collective name for two baryons: the neutron and the proton. They are constituents of the atomic nucleus and until the 1960s were thought to be elementary particles....
s (which are fermion
Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
s), mediated by pion
Pion

In particle physics, a pion is any of three subatomic particles: , and . Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force....
s (which are pseudoscalar meson
Meson

In particle physics, mesons are subatomic particles composed of one quark and one antiquark. They are part of the hadron particle family ? particles made of quarks....
s). The Yukawa interaction is also used in the Standard Model
Standard Model

The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....
 to describe the coupling between the Higgs field and massless quark
Quark

Quarks are a type of elementary particle and major constituents of matter. They are the only particles in the Standard Model to experience all four fundamental interaction, which are also known as fundamental interactions....
 and electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
 fields. Through spontaneous symmetry breaking
Spontaneous symmetry breaking

In physics, spontaneous symmetry breaking occurs when a system that is symmetry in physics with respect to some symmetry group goes into a vacuum state that is not symmetric....
, the fermions acquire a mass proportional to the vacuum expectation value
Vacuum expectation value

In quantum field theory the vacuum expectation value of an Operator is its average, expected value in the Vacuum#The quantum-mechanical vacuum....
 of the Higgs field.

The action

The action
Action (physics)

In modern physics, action is an attribute of the development of a physical system over a period of time, namely amount by which the Phase of the wave function has changed....
 for a meson
Meson

In particle physics, mesons are subatomic particles composed of one quark and one antiquark. They are part of the hadron particle family ? particles made of quarks....
 field f interacting with a Dirac fermion
Dirac fermion

In particle physics, a Dirac fermion is a fermion which is not its own anti-particle. It is named for Paul Dirac. All fermions in the Standard Model, except possibly neutrinos, are Dirac fermions....
 field ? is

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

The Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics....
 is given by

.

Here, is a self-interaction term. For a free-field massive meson, one would have where is the mass for the meson. For a (renormalizable) self-interacting field, one will have where ? 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 positive, real mass of the fermion.

The Yukawa interaction term is

where g is the (real) coupling constant
Coupling constant

In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian mechanics of a system can be separated into a kinetic part and an interaction part....
 for scalar mesons and

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

Classical potential

If two scalar mesons interact through a Yukawa interaction, the potential between the two particles, known as the Yukawa potential
Yukawa potential

A Yukawa potential is a potential of the formHideki 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 ....
, will be:

V(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.

Spontaneous symmetry breaking

Now suppose that the potential has a minimum not at but at some non-zero value . This can happen if one writes (for example) and then sets µ to an imaginary value. In this case, one says that the Lagrangian exhibits spontaneous symmetry breaking
Spontaneous symmetry breaking

In physics, spontaneous symmetry breaking occurs when a system that is symmetry in physics with respect to some symmetry group goes into a vacuum state that is not symmetric....
. The non-zero value of f is called the vacuum expectation value
Vacuum expectation value

In quantum field theory the vacuum expectation value of an Operator is its average, expected value in the Vacuum#The quantum-mechanical vacuum....
 of f. In the Standard Model
Standard Model

The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....
, 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 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 are constants, this term looks exactly like a mass term for a fermion with mass . 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
Chirality (physics)

A phenomenon is said to be chiral if it is not identical to its mirror image . The Spin of a particle may be used to define a handedness for that particle....
 Majorana spinors, one has

where g is a complex coupling constant
Coupling constant

In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian mechanics of a system can be separated into a kinetic part and an interaction part....
 and m is a complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
.

Feynman rules

The article Yukawa potential
Yukawa potential

A Yukawa potential is a potential of the formHideki 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 ....
 provides a simple example of the Feynman rules and a calculation of a scattering amplitude
Scattering amplitude

The scattering amplitude describes the amplitude of an outgoing, elementary, spherical wave relative to a plane, incoming wave scattered on a point size particle....
 from a Feynman diagram
Feynman diagram

In quantum field theory a Feynman diagram is an intuitive graphical representation of a contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory....
 involving the Yukawa interaction.

See also

  • Standard Model
    Standard Model

    The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....