NationMaster - Encyclopedia: 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 Particles erupt from the collision point of two relativistic (100GeV) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Hideki Yukawa Hideki Yukawa (æ¹¯å· ç§€æ¨¹, January 23, 1907 - September 8, 1981) was a Japanese theoretical physicist and the first Japanese person to win the Nobel prize. ... The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... In physics, a field is an assignment of a quantity to every point in space (or more generally, spacetime). ...

$V approx gbarPsi phi Psi$ .

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 strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ... In physics a nucleon is a collective name for the two baryons: the neutron and the proton. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... In particle physics, pion (short for the Greek pi meson = P middle) is the collective name for three subatomic particles discovered in 1947: Ï€0, Ï€+ and Ï€âˆ’. Pions are the lightest mesons. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ... Higgs bosons are hypothetical elementary particles predicted to exist by the Standard Model of particle physics. ... Quarks are one of the two basic constituents of matter in the Standard Model of particle physics. ... Properties The electron is a fundamental subatomic particle that carries a negative electric charge. ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... In quantum field theory the vacuum expectation value (also called condensate) of an operator is its average, expected value in the vacuum. ...

1 The action
2 Spontaneous symmetry breaking
3 Majorana form
4 Feynman rules
5 See also
6 References

The action

The action for a meson field interaction with a Dirac fermion field is In physics, the action principle is an assertion about the nature of motion, from which the trajectory of an object subject to forces can be determined. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ...

$S[phi,psi]=int d^dx ;left[ mathcal{L}_mathrm{meson}(x) + mathcal{L}_mathrm{Dirac}(x) + mathcal{L}_mathrm{Yukawa}(x) right]$

where the integration is performed over d dimensions (typically 4 for four-dimensional spacetime). The meson Lagrangian is given by A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

$mathcal{L}_mathrm{meson}(x) = frac{1}{2}partial^mu phi(x) partial_mu phi(x) -V(phi(x))$ .

Here, $V (φ(x))$ is a self-interaction term. For a free-field massive meson, one would have $V (φ) = μ 2 φ 2$ where $μ$ is the mass for the meson. For a (renormalizable) self-interacting field, one will have $V (φ) = μ 2 φ 2 + λφ 4$ where λ is a coupling constant. This potential is explored in detail in the article phi to the fourth. In physics, the adjective renormalizable refers to a theory (usually a quantum field theory) in which all ultraviolet divergences, infinities and other seemingly meaningless results can be cured by the process of renormalization. ... This article is in need of attention from an expert on the subject. ...

The free-field Dirac Lagrangian is given by

$mathcal{L}_mathrm{Dirac}(x) = bar{psi}(x)(ipartial!!!/-m)psi(x)$

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

The Yukawa interaction term is

$mathcal{L}_mathrm{Yukawa}(x) = gbarpsi(x) phi(x) psi(x)$

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 In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ...

$S[phi,psi]=int d^dx left[frac{1}{2}partial^mu phi partial_mu phi -V(phi) + bar{psi}(ipartial!!!/-m)psi -g bar{psi}phipsi right]$

Spontaneous symmetry breaking

Now suppose that the potential $V (φ)$ has a minimum not at $φ = 0$ but at some non-zero value $φ 0$ . This can happen if one writes (for example) $V (φ) = μ 2 φ 2 + λφ 4$ 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 the vacuum expectation value of φ. In the Standard Model, this non-zero value is responsible for the fermion masses, as shown below. Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... In quantum field theory the vacuum expectation value (also called condensate) of an operator is its average, expected value in the vacuum. ... The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ...

To exhibit the mass term, one re-expresses the action in terms of the field $tilde phi = phi-phi_0$ , where $φ 0$ is now understood to be a constant independent of position. We now see that the Yukawa term has a component

$gphi_0 barpsipsi$

and since both g and $φ 0$ are constants, this term looks exactly like a mass term for a fermion with mass $g φ 0$ . This is the mechanism by which spontaneous symmetry breaking gives mass to fermions. The field $tildephi$ is known as the Higgs field. Higgs bosons are hypothetical elementary particles predicted to exist by the Standard Model of particle physics. ...

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 ... Chirality refers to several phenomena, all having to do with objects that differ from their mirror image. ...

$S[phi,chi]=int d^dx left[frac{1}{2}partial^muphi partial_mu phi -V(phi)+chi^dagger ibar{sigma}cdotpartialchi+frac{i}{2}(m+g phi)chi^T sigma^2 chi-frac{i}{2}(m+g phi)^* chi^dagger sigma^2 chi^*right]$

where g is a complex coupling constant and m is a complex number. In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i represents the imaginary number, i2 = âˆ’1. ...

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. 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 . ... The S-matrix is the matrix in quantum mechanics or quantum field theory that relates the final state in the infinite future and the initial state in the infinite past. ... In this Feynman diagram, electrons annihilate and become a quark-antiquark pair. ...

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 07-005493-2

Categories: Quantum field theory | Standard Model | Electroweak theory | Nuclear physics

Results from FactBites:

Hadrons, baryons, mesons (1130 words)
	Yukawa worked out a potential for the force and predicted its mass based on the uncertainty principle from measurements of the apparent range of the strong force in nuclei.
	Particles that interact by the strong interaction are called hadrons.
	We now know that the pion is a meson, a composite particle, and the current view is that the strong interaction is an interaction between quarks, but the Yukawa theory stimulated a major advance in the understanding of the strong interaction.

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Contents

The action GA_googleFillSlot("encyclopedia_square");

Spontaneous symmetry breaking

Majorana form

Feynman rules

See also

References

The action