Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 4 / b

Restoring units, the decaying Yukawa potential is $V(r)=C{e}^{-r/{\lambda }_m}/r$ with range ${\lambda }_m=\hslash /(mc)$. For $r_{\ast }=10^{-18}\mathrm{m}$,The geometric $1/r$ factor belongs in the potential ratio; the exponential alone is the additional suppression relative to massless exchange. Using $\hslash c=197.32698\mathrm{MeV}\mathrm{fm}$ givesThe following are ratios of potential magnitudes to the magnitude at $r_{\ast }$, with the same coupling held fixed:

For example, the pion's nuclear exponential factor is $e^{-0.68346}=0.505$, whereas the Higgs nuclear exponent is about $-632.83$. The huge atomic suppressions are most reliably reported through logarithms, not through floating-point exponentiation that silently underflows to zero. These are potential energy ratios, not force ratios or absolute comparisons of different interaction strengths.

A pion mass gives a Yukawa potential range of order a femtometre, so meson exchange can describe the short range of the residual nuclear force. The nuclear suppression relative to a massless potential of the same coupling is only about a factor of two at one femtometre. A Higgs-mass mediator instead has a range around $10^{-18}\mathrm{m}$ and a further factor $e^{-633}$ by nuclear separations; direct Higgs exchange cannot supply an appreciable long-range nuclear binding interaction at those separations. The range comparison does not replace the spin, coupling and many-body physics of actual nuclear forces.

At atomic separations both massive exchange potentials are exponentially negligible. Molecular binding needs an interaction that survives at atomic distances; actual chemical bonds is governed principally by electromagnetism and quantum mechanics, with a massless photon rather than a scalar mediator. A hypothetical massless scalar would also have an unscreened $1/r$ potential, but its coupling and spin-dependent phenomenology would be different. The Higgs mechanism affects particle masses; that does not imply an appreciable direct Higgs-exchange force between atoms separated by $10^{-10}\mathrm{m}$.