The Yukawa potential calculator determines the effective potential between two nucleons mediated by meson exchange, via the formula V(r) = -g²/(4π) · (ℏc/r) · exp(-mr/ℏc). Postulated by Hideki Yukawa (Kyoto University 1935) in "On the Interaction of Elementary Particles" Proc. Phys. Math. Soc. Japan 17:48, it was a turning point in modern nuclear physics: introducing the concept of force-mediating particle (gauge boson concept), predicting the existence of mesons with intermediate mass ~100 MeV/c² as quanta exchanged between protons and neutrons. Yukawa won the Nobel Prize in Physics 1949 after Cecil Powell (University of Bristol 1947) observed the pion π in cosmic-ray emulsions on Pic du Midi.

1. Yukawa 1935 formula and parameters

The potential between two color charges at distance r mediated by exchange of a mediator of mass m is V(r) = -g²/(4π) · (ℏc/r) · exp(-r/λ), with λ = ℏc/m Compton range of the mediator (for π: λ = 197.3 MeV·fm / 139.57 MeV ≈ 1.41 fm). The exp(-r/λ) term provides the characteristic exponential cutoff beyond Compton wavelength, distinguishing Yukawa from Coulomb (infinite r⁻¹ power). The coupling constant g is dimensionless: for pion-nucleon g_πN ≈ 13.5, giving α_πN = g²/(4π) ≈ 14.5 (strong). The m → 0 limit recovers Coulomb (massless QCD gluons but confined). The derivative dV/dr gives attractive force F = -dV/dr.

2. Mesons and W/Z bosons as mediators

• Pion π± π⁰: m ≈ 140 MeV, λ ≈ 1.41 fm — residual nuclear force between nucleons.
• Kaon K± K⁰: m ≈ 494 MeV, λ ≈ 0.40 fm — strangeness, Powell 1947.
• Eta η: m ≈ 548 MeV, λ ≈ 0.36 fm — SU(3) singlet meson.
• Rho ρ, omega ω: m ≈ 770-783 MeV, λ ≈ 0.25 fm — vector mesons.
• Phi φ: m ≈ 1019 MeV, λ ≈ 0.19 fm — INFN-LNF DAFNE 1999 phi-factory.
• J/ψ, Υ: charm/bottomonium, m = 3.1 / 9.5 GeV — Brookhaven Ting-Richter Nobel 1976.
• W± Z⁰ bosons: m ≈ 80-91 GeV, λ ≈ 0.002 fm — Carlo Rubbia CERN UA1 Nobel 1984.
• Gluon: m = 0, infinite range but confined within hadrons (QCD asymptotic freedom).

3. Range of validity of Yukawa model

The Yukawa model is a single-meson exchange approximation valid for nucleon-nucleon at distances 1-2 fm. At distances < 1 fm multi-meson contributions (rho, omega, sigma, two-pion) and core repulsion become important — modern models (Argonne v18, CD-Bonn, Nijmegen) use phenomenological potentials with 30-40 parameters. At distances >> 2 fm pion dominates (one-pion exchange OPE Brueckner-Bethe). For quark-quark within nucleons: non-perturbative QCD, lattice QCD (Wilson loop, INFN Pisa Cnaf, Cineca Bologna 1969 Marconi). For weak interactions: W/Z exchange, mass = 80-91 GeV, range λ_W ≈ 0.002 fm — Fermi contact 4-fermion approximation. For electromagnetic QED: massless photon, infinite range, V = -α·ℏc/r Coulomb.

4. Worked numerical examples

Example 1 — Pion between two nucleons at r = 1 fm. m_π = 139.57 MeV, λ = 197.33/139.57 = 1.414 fm. r/λ = 0.707. exp(-0.707) = 0.493. V = -(13.5²/4π) · 197.33 · 0.493 / 1 = -1432 MeV (attractive). Force F ≈ 2.3 × 10⁻⁵ N at short nuclear range.

Example 2 — W± bosons at r = 1 fm. m_W = 80.4 GeV, λ = 0.00245 fm. r/λ = 408. exp(-408) ≈ 10⁻¹⁷⁷ — completely suppressed. The weak force at 1 fm is essentially zero, only exists at contact (subnuclear range).

Example 3 — Yukawa vs Coulomb at r = 0.5 fm (inside nucleon). Pion: V_Y = -3000 MeV. Coulomb (e+e-): V_C = -2.88 MeV. Ratio ~ 1000× — nuclear force is enormously stronger than EM at those scales. At r > 2.5 fm Yukawa < Coulomb (exponential damping).

5. Practical applications of Yukawa potential

• Nuclear models: Brueckner-Hartree-Fock, Argonne AV18, CD-Bonn, Nijmegen NN potentials — theoretical basis for heavy nuclei, INFN-Catania Theory + INFN-Padova.
• Lattice QCD: Cineca Bologna 1969 Marconi100 30 PFlops + Leonardo HPC 250 PFlops + INFN CNAF Bologna compute hadron spectra and Wilson loops.
• DAFNE Frascati experiments: INFN-LNF DAFNE 1999 phi-factory φ → KK, KLOE-2 detector, kaon CP violation studies.
• XENON-nT dark matter: INFN-LNGS Gran Sasso 1987, 8.4 t liquid Xe. Yukawa-like with Z' mediator for hypothetical light DM.
• Cosmic ray physics Pancini-Conversi-Piccioni Rome 1947: muon (mesotron) discovery via Yukawa-like absorption with magnet and Wilson chamber.
• LHC ATLAS-CMS CERN: Higgs search (m=125 GeV, λ_H≈0.0016 fm) and extra mediators (Z', W' BSM); INFN-Rome + Pisa + Padova.
• Borexino+SOX neutrino research: INFN-LNGS, Yukawa exchange in neutrino-electron interactions Z⁰ exchange.
• INAF nuclear astrophysics: INAF Bologna stellar nucleosynthesis BBN/SN, nuclear reaction rates Y(α,n) Y(p,γ).

6. Comparison with other fundamental potentials

Yukawa 1935: residual nuclear force, range 1-2 fm, massive mediator. Coulomb 1785: V = -e²/(4πε₀r), electrostatic, massless photon mediator, infinite range. Newton gravity: V = -GMm/r, hypothetical massless graviton. Fermi 4-fermion: G_F · ψ̄ψψ̄ψ contact, weak old-fashioned. Non-perturbative QCD: linear confinement V ∝ r for quark-quark at r > 1 fm (Wilson loop). Higgs Yukawa: V_H = -(λ_H · ℏc/r) · exp(-m_H r/ℏc), m_H = 125 GeV, λ ≈ 0.0016 fm — scalar mediator. Reid soft-core (Reid 1968): phenomenological nuclear potential with repulsive core. Argonne v18: 18 operators, complete NN scattering parametrization. Chiral Effective Field Theory: Weinberg-Tomozawa, power expansion Q/Λ_χ with low-energy constants LECs.

7. Modern developments and corrections

The QCD corrections to the Yukawa model include: multi-meson exchange (two-pion exchange TPE, Goldstone bosons), chiral perturbation theory (Weinberg 1990, INFN Pisa), lattice QCD for quark-quark within hadrons (Cineca Leonardo 2024). For heavy nuclei: ab initio no-core shell model (NCSM), Green's Function Monte Carlo (GFMC) implemented on INFN-CNAF HPC. For BSM physics: Z' boson, W' boson, Higgs portal Yukawa-like with dark matter (LZ Stanford, XENON-nT INFN-LNGS Gran Sasso). For extra dimensions Randall-Sundrum: modified Yukawa with TeV cutoff. For electron-nucleon scattering: Born approximation with Sachs form factors G_E G_M (JLab Hall A INFN-Rome collaboration). Software: FeynCalc Mathematica (T. Hahn), FORM (Vermaseren), MadGraph (UCLouvain), Pythia (Lund), HERWIG (Manchester) implement Yukawa for Monte Carlo LHC events.

Updated July 2026