Yukawa interaction

In particle physics, the Yukawa interaction is an interaction between a scalar or pseudoscalar field with spin 0 and a fermionic field with spin ½. It was first proposed by Hideki Yukawa in 1935 to explain the bond between protons and neutrons , which are both fermions, in the nucleus . The scalar particle postulated by Yukawa was detected in the form of the pion in cosmic rays in 1947 . For his theory, Yukawa received the Nobel Prize in Physics in 1949. ${\displaystyle \phi }$ ${\displaystyle \psi }$

At the level of elementary particles, the cohesion of the atomic nucleus is now explained by quantum chromodynamics , the pseudo Goldstone boson of which is the pion as a composite particle. Instead, in the Standard Model of elementary particle physics , the Yukawa interaction finds its place in the interaction between the Higgs field and the elementary fermions, which assigns them their mass within the framework of the Higgs mechanism .

Mathematical description

The interaction between elementary particles is described by the Lagrangian . This contains kinetic terms that describe the movement of the particles and the interaction terms. The interaction term for the Yukawa interaction between a scalar and a fermion is

${\displaystyle {\mathcal {L}}_{\mathrm {SF} }=-g{\bar {\psi }}\phi \psi }$

with a coupling constant . For a pseudoscalar particle the interaction term is ${\displaystyle g}$

${\displaystyle {\mathcal {L}}_{\mathrm {PF} }=-\mathrm {i} g{\bar {\psi }}\gamma ^{5}\phi \psi }$

with the fifth Dirac matrix . ${\displaystyle \gamma ^{5}}$

The full Lagrangian for a theory with a Yukawa interaction is then:

${\displaystyle {\mathcal {L}}={\bar {\psi }}(\mathrm {i} \gamma ^{\mu }\partial _{\mu }-m_{\psi })\psi +{\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {1}{2}}m_{\phi }^{2}\phi ^{2}-g{\bar {\psi }}\phi \psi }$

Are there

• ${\displaystyle \gamma ^{\mu }}$ the Dirac matrices,
• ${\displaystyle m_{\psi }}$ the mass of the fermion and
• ${\displaystyle m_{\phi }}$ the mass of the scalar particle.

Relation to the Yukawa potential

The Yukawa potential is the potential induced by a Yukawa interaction. It reads in the leading order of the disturbance series

${\displaystyle V(r)=-{\frac {g^{2}}{4\pi r}}e^{-m_{\phi }r}}$

and expands the Coulomb potential of the electrostatics by an exponentially decreasing factor that is dependent on the mass of the scalar. The Coulomb potential is reached by the limit as it is fulfilled for the photon. In this leading order there is no difference whether the interacting particle is a scalar or a vector like the photon. ${\displaystyle m\to 0}$

The potential results from the scattering of two particles in a Yukawa interaction. The matrix element of the S matrix is then in a non-relativistic approximation ${\displaystyle {\mathcal {M}}}$

${\displaystyle {\mathcal {M}}={\frac {4m_{\psi }^{2}g^{2}}{{\vec {q}}^{2}+m_{\phi }^{2}}}}$

where denotes the transmitted pulse and the factor is a relic from the normalization. Then the potential in the momentum space reads according to Fermi's Golden Rule ${\displaystyle {\vec {q}}}$ ${\displaystyle 4m_{\psi }^{2}}$

${\displaystyle {\tilde {V}}({\vec {q}})={\frac {-g^{2}}{{\vec {q}}^{2}+m_{\phi }^{2}}}}$

and after a Fourier transformation into the spatial space

${\displaystyle V(r)=\int {\frac {\mathrm {d} ^{3}{\vec {q}}}{(2\pi )^{3}}}{\tilde {V}}({\vec {q}})=-{\frac {g^{2}}{4\pi r}}e^{-m_{\phi }r}}$

literature

• Michael E. Peskin and Daniel V. Schroeder: An Introduction to Quantum Field Theory . Westview Press, Boulder 1995, ISBN 0-201-50397-2 , pp. 116-123 .

Individual evidence

1. ^ Hideki Yukawa: On the Interaction of Elementary Particles . In: Progress of Theoretical Physics . tape 17 , no. 48 , 1935, pp. 1-9 .