Yukawa potential

Yukawa potential
Name	Yukawa potential
Named after	Hideki Yukawa
Field	Theoretical physics, Nuclear physics, Particle physics
First published	1935
Related	Yukawa interaction, Meson theory, Pion

Yukawa potential The Yukawa potential is a central-force potential introduced by Hideki Yukawa to describe the short-range interaction between nucleons mediated by mesons. It provided the first successful quantitative model connecting a massive carrier particle to an exponentially screened force, influencing developments in Nuclear physics, Particle physics, and Quantum field theory. The potential established conceptual links between experimental findings at institutions such as Cavendish Laboratory and theoretical programs at universities like Kyoto University and University of Tokyo.

Introduction

Yukawa proposed a potential to account for the observed range of the nuclear force after puzzles noted in experiments at facilities including Cavendish Laboratory and measurements by researchers associated with Copenhagen interpretation-era laboratories. The form captured the inverse-square behavior at short distances and an exponential decay at distances set by the inverse mass of the carrier particle, motivating searches for the Pion and informing the award of the Nobel Prize in Physics to Yukawa. The concept also influenced later work at institutions like Brookhaven National Laboratory and CERN, where meson exchange models were compared with scattering data.

Mathematical Formulation

The Yukawa potential V(r) for two pointlike sources separated by a radial distance r is conventionally written as V(r) = -g^2 (e^{-μ r}) / r, where g denotes a coupling constant and μ is the inverse range parameter proportional to the mass of the mediator. This expression follows from solving the static Green's function of the Klein–Gordon operator (□ + μ^2) in three spatial dimensions, a procedure with antecedents in methods developed at Princeton University and Harvard University for propagator analysis. The Fourier transform of the potential yields a propagator proportional to 1/(k^2 + μ^2), which mirrors structures used in perturbation theory at Institute for Advanced Study and in textbooks associated with Cornell University.

Boundary conditions impose regularity at r → 0 and exponential decay as r → ∞; these behaviors relate to partial-wave expansions employed in analyses at Imperial College London and Massachusetts Institute of Technology. Dimensional generalizations replace 1/r by appropriate Green's functions in D spatial dimensions, a topic explored in seminars at École Normale Supérieure and workshops at Max Planck Institute for Physics.

Physical Interpretation and Applications

Physically, μ corresponds to the mass of the exchanged boson, so the potential predicts a force range ≈ μ^{-1}. Yukawa originally associated μ with a particle mass near that later measured for the Pion by experiments at Chicago Pile-1-era collaborations and beam lines at CERN. The model provides qualitative understanding of nucleon–nucleon binding energies relevant to studies at Los Alamos National Laboratory and to phenomenological potentials used by groups at Oxford University.

Beyond nucleon interactions, Yukawa-like forms appear in screening phenomena in plasmas examined at Princeton Plasma Physics Laboratory and in condensed-matter analogues investigated at Bell Labs and Stanford University. In cosmology, finite-range interactions with Yukawa kernels inform fifth-force searches pursued by teams at Kavli Institute for the Physics and Mathematics of the Universe and tests of modifications to gravity by collaborations involving European Space Agency missions.

Solutions and Scattering

The Schrödinger equation with a Yukawa potential admits bound states for sufficiently strong coupling g, studied using variational methods developed in courses at University of Cambridge and perturbative expansions akin to work at Yale University. Scattering by a Yukawa potential is analyzed via partial-wave decomposition; phase shifts δ_l(k) are computed numerically in programs influenced by algorithms from Los Alamos National Laboratory and approximated analytically through the Born approximation, originally formulated by Max Born whose colleagues at University of Göttingen contributed to scattering theory.

Resonances and bound-state poles in the complex momentum plane have parallels with S-matrix theory championed by researchers at S-matrix Theory-adjacent centers and with dispersion-relation techniques from École Polytechnique. Empirical nucleon–nucleon scattering data from Brookhaven National Laboratory and TRIUMF have been used to fit Yukawa-inspired potentials augmented by tensor and spin-dependent terms introduced in models from Argonne National Laboratory and Paris Observatory groups.

Relation to Quantum Field Theory

In quantum field theory, the Yukawa potential arises as the nonrelativistic limit of single-boson exchange between fermions described by a Yukawa interaction term ψ̄ψφ, an insight developed in parallel with field-theoretic approaches at CERN and Fermi National Accelerator Laboratory. The momentum-space denominator 1/(k^2 + μ^2) corresponds to the Feynman propagator of a massive scalar field, a cornerstone in perturbative calculations taught at California Institute of Technology and formalized in path-integral treatments by researchers at Institute for Advanced Study.

Renormalization considerations and loop corrections modify the effective potential; such quantum corrections were pursued in seminars at Princeton University and in renormalization group treatments associated with Stanford Linear Accelerator Center. The Yukawa interaction also appears in effective field theories developed by groups at Perimeter Institute and in chiral Lagrangian approaches used by collaborations at European Organization for Nuclear Research.

Extensions and Generalizations

Generalizations include screened Coulomb potentials in plasma physics modeled at Princeton Plasma Physics Laboratory, Yukawa potentials with multiple exchange masses as in multi-meson models used by teams at Oak Ridge National Laboratory, and anisotropic or noncentral extensions incorporating spin–orbit and tensor forces developed by theorists at University of Chicago and University of Illinois Urbana-Champaign. Lattice field theory calculations at Rutherford Appleton Laboratory and Brookhaven National Laboratory probe nonperturbative modifications, while modifications in higher-dimensional theories appear in investigations at Perimeter Institute and proposals linked to String Theory groups.

Applied mathematics communities at Institut Henri Poincaré and numerical analysis groups at Siemens-affiliated research centers have explored efficient evaluation of Yukawa kernels and Green's functions for many-body simulations used in molecular dynamics packages developed by teams at Argonne National Laboratory and Lawrence Berkeley National Laboratory.

Category:Potentials