Editing Yukawa interaction

{{Short description|Type of interaction between scalars and fermions}}
In [[particle physics]], '''Yukawa's interaction''' or '''Yukawa coupling''', named after [[Hideki Yukawa]], is an interaction between particles according to the [[Yukawa potential]]. Specifically, it is between a [[scalar field (quantum field theory)|scalar field]] (or [[pseudoscalar]] field) <math>\ \phi\ </math> and a [[Dirac field]] <math>\ \psi\ </math> of the type
{{block indent | em = 1.5 | text = <math>~ V \approx g \, \bar\psi \, \phi \, \psi </math>{{pad|1em}} (scalar) {{pad|2em}} or {{pad|2em}}<math> g \, \bar\psi \, i \,\gamma^5 \, \phi \, \psi </math>{{pad|1em}} (pseudoscalar).}}

The Yukawa interaction was developed to model the [[strong interaction|strong force]] between [[hadrons]]. A Yukawa interaction is thus used to describe the [[nuclear force]] between [[nucleon]]s mediated by [[pion]]s (which are pseudoscalar [[meson]]s).

A Yukawa interaction is also used in the [[Standard Model]] to describe the coupling between the [[Higgs field]] and massless [[quark]] and [[lepton]] fields (i.e., the fundamental [[fermion]] particles). Through [[spontaneous symmetry breaking]], these fermions acquire a mass proportional to the [[vacuum expectation value]] of the Higgs field. This Higgs-fermion coupling was first described by [[Steven Weinberg]] in 1967 to model lepton masses.<ref>{{cite journal |last=Weinberg |first=Steven |date=1967-11-20 |title=A Model of Leptons |journal=Physical Review Letters |volume=19 |issue=21 |pages=1264–1266 |doi=10.1103/PhysRevLett.19.1264 | bibcode=1967PhRvL..19.1264W |doi-access=free }}</ref>

==Classical potential==
{{main|Yukawa potential}}

If two [[fermion]]s interact through a Yukawa interaction mediated by a '''Yukawa particle''' of mass <math>\mu</math>, the potential between the two particles, known as the ''Yukawa potential'', will be:

<math display="block">V(r) = -\frac{g^2}{\,4\pi\,} \, \frac{1}{\,r\,} \, e^{-\mu r}</math>

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 same electrical charge sign particles). This is explained by the fact that the Yukawa particle has spin zero and even spin always results in an attractive potential. (It is a non-trivial result of [[quantum field theory]]<ref>{{cite book |author=A. Zee |title=Quantum Field Theory in a Nutshell |edition=2nd |publisher=World Scientific |year=2010 |chapter=I.5 |isbn=978-0691140346}}</ref> that the exchange of even-spin [[bosons]] like the [[pion]] (spin 0, Yukawa force) or the [[graviton]] (spin 2, [[gravity]]) results in forces always attractive, while odd-spin bosons like the [[gluons]] (spin 1, [[strong interaction]]), the [[photon]] (spin 1, [[electromagnetic force]]) or the [[rho meson]] (spin 1, Yukawa-like interaction) yields a force that is attractive between opposite charge and repulsive between like-charge.) The negative sign in the exponential gives the interaction a finite effective range, so that particles at great distances will hardly interact any longer (interaction forces fall off exponentially with increasing separation).

As for other forces, the form of the Yukawa potential has a geometrical interpretation in term of the [[field line]] picture introduced by [[Faraday]]: The {{Math|{{sfrac|1|''r''}}}} part results from the dilution of the field line flux in space. The force is proportional to the number of field lines crossing an elementary surface. Since the field lines are emitted isotropically from the force source and since the distance {{Mvar|r}} between the elementary surface and the source varies the apparent size of the surface (the [[solid angle]]) as {{Math|{{sfrac|1|''r''<sup>2</sup>}}}} the force also follows the {{Math|{{sfrac|1|''r''<sup>2</sup>}}}}&nbsp;dependence. This is equivalent to the {{Math|{{sfrac|1|''r''}}}} part of the potential. In addition, the exchanged mesons are unstable and have a finite lifetime. The disappearance ([[radioactive decay]]) of the mesons causes a reduction of the flux through the surface that results in the additional exponential factor <math>~e^{-\mu r}~</math> of the Yukawa potential. Massless particles such as [[photons]] are stable and thus yield only {{Math|{{sfrac|1|''r''}}}} potentials. (Note however that other massless particles such as [[gluons]] or [[gravitons]] do not generally yield {{Math|{{sfrac|1|''r''}}}} potentials because they interact with each other, distorting their field pattern. When this self-interaction is negligible, such as in weak-field gravity ([[Newtonian gravitation]]) or for very short distances for the [[strong interaction]] ([[asymptotic freedom]]), the {{Math|{{sfrac|1|''r''}}}} potential is restored.)

==The action==
The Yukawa interaction is an interaction between a [[scalar field (quantum field theory)|scalar field]] (or [[pseudoscalar]] field) {{mvar|ϕ}} and a [[Dirac field]] {{mvar|ψ}} of the type
{{block indent | em = 1.5 | text = <math> V \approx g\,\bar\psi \,\phi \,\psi </math>{{pad|1em}} (scalar) {{pad|2em}} or {{pad|2em}}<math> g \,\bar\psi \,i\,\gamma^5 \,\phi \,\psi </math>{{pad|1em}} ([[pseudoscalar]]).}}

The [[action (physics)|action]] for a [[meson]] field <math>\phi</math> interacting with a [[Dirac field|Dirac]] [[baryon]] field <math>\psi</math> is

<math display="block">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
</math>

where the integration is performed over {{mvar|n}} dimensions; for typical four-dimensional spacetime {{math|1=''n'' = 4}}, and <math>\mathrm{d}^{4}x \equiv \mathrm{d}x_1 \, \mathrm{d}x_2 \, \mathrm{d}x_3 \, \mathrm{d}x_4 ~.</math>

The meson [[Lagrangian (field theory)|Lagrangian]] is given by
<math display="block">\mathcal{L}_\mathrm{meson}(\phi) = 
\frac{1}{2}\partial^\mu \phi \; \partial_\mu \phi - V(\phi)~.</math>

Here, <math>~V(\phi)~</math> is a self-interaction term. For a free-field massive meson, one would have <math display="inline">~V(\phi)=\frac{1}{2}\,\mu^2\,\phi^2~</math> where <math>\mu</math> is the mass for the meson. For a ([[renormalizable]], polynomial) self-interacting field, one will have <math display="inline">V(\phi) = \frac{1}{2}\,\mu^2\,\phi^2 + \lambda\,\phi^4</math> where {{mvar|λ}} is a coupling constant. This potential is explored in detail in the article on the [[quartic interaction]].

The free-field Dirac Lagrangian is given by
<math display="block">\mathcal{L}_\mathrm{Dirac}(\psi) = 
\bar{\psi}\,\left( i\,\partial\!\!\!/ - m \right)\,\psi </math>

where {{mvar|m}} is the real-valued, positive mass of the fermion.

The Yukawa interaction term is 
<math display="block">\mathcal{L}_\mathrm{Yukawa}(\phi,\psi) = -g\,\bar\psi \,\phi \,\psi</math>

where {{mvar|g}} is the (real) [[coupling constant]] for scalar mesons and

<math display="block">\mathcal{L}_\mathrm{Yukawa}(\phi,\psi) = -g\,\bar\psi \,i \,\gamma^5 \,\phi \,\psi</math>

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

<math display="block">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 ~.</math>

==Yukawa coupling to the Higgs in the Standard Model==
A Yukawa coupling term to the [[Higgs field]] affecting [[spontaneous symmetry breaking]] in the Standard Model is responsible for fermion masses in a symmetric manner.

Suppose that the potential <math>~V(\phi)~</math> has its minimum, not at <math>~\phi = 0~,</math> but at some non-zero value <math>~\phi_0~.</math> This can happen, for example, with a potential form such as <math>~V(\phi) = \lambda\,\phi^4~ - \mu^2\,\phi^2 </math>. In this case, the Lagrangian exhibits [[spontaneous symmetry breaking]]. This is because the non-zero value of the <math>~\phi~</math> field, when operating on the vacuum, has a non-zero [[vacuum expectation value]] of <math>~\phi~.</math>

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 <math> \phi' = \phi - \phi_0~,</math> where <math>~\phi_0~</math> is constructed to be independent of position (a constant). This means that the Yukawa term includes a component
<math display="block">~g \, \phi_0 \, \bar\psi \, \psi~</math>
and, since both {{mvar|g}} and <math>\phi_0</math> are constants, the term presents as a mass term for the fermion with equivalent mass <math>~g\,\phi_0~.</math> This mechanism is the means by which spontaneous symmetry breaking gives mass to fermions. The scalar field <math>\phi'~</math> is known as the [[Higgs field]].

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 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 [[Chirality (physics)|chiral]] Majorana spinors, one has
<math display="block">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^\top \,\sigma^2 \,\chi - \frac{i}{2}\,(m + g \,\phi)^* \, \chi^\dagger \,\sigma^2 \, \chi^*\,\right] \mathrm{d}^{n}x</math>

where {{mvar|g}} is a complex [[coupling constant]], {{mvar|m}} is a [[complex number]], and {{mvar|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==
{{reflist|25em}}

{{refbegin|25em|small=yes}}
*{{cite book |authorlink=Claude Itzykson |first1=Claude |last1=Itzykson |first2=Jean-Bernard |last2=Zuber |title=Quantum Field Theory |url=https://archive.org/details/quantumfieldtheo0000itzy |url-access=registration |year=1980 |publisher=McGraw-Hill |location=New York |isbn=0-07-032071-3 }}
*{{cite book |authorlink=James D. Bjorken |first1=James D. |last1=Bjorken |authorlink2=Sidney Drell |first2=Sidney D. |last2=Drell |title=Relativistic Quantum Mechanics |url=https://archive.org/details/relativisticquan0000bjor |url-access=registration |year=1964 |publisher=McGraw-Hill |location=New York |isbn=0-07-232002-8 }}
*{{cite book |authorlink=Michael Peskin |first1=Michael E. |last1=Peskin |first2=Daniel V. |last2=Schroeder |title=An Introduction to Quantum Field Theory |url=https://archive.org/details/introductiontoqu0000pesk |url-access=registration |year=1995 |publisher=Addison-Wesley |isbn=0-201-50397-2 }}
{{refend}}

{{Quantum field theories}}

[[Category:Quantum field theory]]
[[Category:Standard Model]]
[[Category:Electroweak theory]]